Poisson Structures and Their Normal Forms (Progress in Mathematics)

Progress in Mathematics

Volume 242

Series Editor

H. BassJ. OesterléA. Weinstein

Jean-Paul DufourNguyen Tien Zung

Poisson Structures and Their Normal Forms

Birkhäuser VerlagBasel Boston Berlin

Authors:

2000 Mathematics Subject Classifi cation 53Dxx, 53Bxx, 85Kxx, 70Hxx

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

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ISBN 3-7643-7334-2 Birkhäuser Verlag, Basel – Boston – Berlin

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© 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, SwitzerlandPart of Springer Science+Business MediaPrinted on acid-free paper produced of chlorine-free pulp. TCF ∞Printed in GermanyISBN-10: 3-7643-7334-2 e-ISBN: 3-7643-7335-0ISBN-13: 978-3-7643-7334-4

9 8 7 6 5 4 3 2 1 www.birkhauser.ch

Jean-Paul DufourDépartement de mathématiqueUniversité de Montpellier 2 place Eugène Bataillon34095 MontpellierFrancee-mail: [email protected]

Nguyen Tien ZungLaboratoire Émile Picard, UMR 5580 CNRSInstitut de MathématiquesUniversité Paul Sabatier118 route de Narbonne31062 ToulouseFrancee-mail: [email protected]

a HeleneJ.-P. D.

a MaiN. T. Z.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Generalities on Poisson Structures1.1 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Poisson tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Poisson morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Local canonical coordinates . . . . . . . . . . . . . . . . . . . . . . 131.5 Singular symplectic foliations . . . . . . . . . . . . . . . . . . . . . 161.6 Transverse Poisson structures . . . . . . . . . . . . . . . . . . . . . 211.7 Group actions and reduction . . . . . . . . . . . . . . . . . . . . . 231.8 The Schouten bracket . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.8.1 Schouten bracket of multi-vector fields . . . . . . . . . . . . 271.8.2 Schouten bracket on Lie algebras . . . . . . . . . . . . . . . 311.8.3 Compatible Poisson structures . . . . . . . . . . . . . . . . 33

1.9 Symplectic realizations . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Poisson Cohomology2.1 Poisson cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1 Definition of Poisson cohomology . . . . . . . . . . . . . . . 392.1.2 Interpretation of Poisson cohomology . . . . . . . . . . . . 402.1.3 Poisson cohomology versus de Rham cohomology . . . . . . 412.1.4 Other versions of Poisson cohomology . . . . . . . . . . . . 422.1.5 Computation of Poisson cohomology . . . . . . . . . . . . . 43

2.2 Normal forms of Poisson structures . . . . . . . . . . . . . . . . . . 442.3 Cohomology of Lie algebras . . . . . . . . . . . . . . . . . . . . . . 49

2.3.1 Chevalley–Eilenberg complexes . . . . . . . . . . . . . . . . 492.3.2 Cohomology of linear Poisson structures . . . . . . . . . . . 512.3.3 Rigid Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 53

2.4 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.4.1 Spectral sequence of a filtered complex . . . . . . . . . . . . 542.4.2 Leray spectral sequence . . . . . . . . . . . . . . . . . . . . 562.4.3 Hochschild–Serre spectral sequence . . . . . . . . . . . . . . 57

viii Contents

2.4.4 Spectral sequence for Poisson cohomology . . . . . . . . . . 592.5 Poisson cohomology in dimension 2 . . . . . . . . . . . . . . . . . . 60

2.5.1 Simple singularities . . . . . . . . . . . . . . . . . . . . . . . 612.5.2 Cohomology of Poisson germs . . . . . . . . . . . . . . . . . 632.5.3 Some examples and remarks . . . . . . . . . . . . . . . . . . 68

2.6 The curl operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.6.1 Definition of the curl operator . . . . . . . . . . . . . . . . . 692.6.2 Schouten bracket via curl operator . . . . . . . . . . . . . . 712.6.3 The modular class . . . . . . . . . . . . . . . . . . . . . . . 722.6.4 The curl operator of an affine connection . . . . . . . . . . 73

2.7 Poisson homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3 Levi Decomposition3.1 Formal Levi decomposition . . . . . . . . . . . . . . . . . . . . . . 783.2 Levi decomposition of Poisson structures . . . . . . . . . . . . . . . 813.3 Construction of Levi decomposition . . . . . . . . . . . . . . . . . . 843.4 Normed vanishing of cohomology . . . . . . . . . . . . . . . . . . . 883.5 Proof of analytic Levi decomposition theorem . . . . . . . . . . . . 923.6 The smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Linearization of Poisson Structures4.1 Nondegenerate Lie algebras . . . . . . . . . . . . . . . . . . . . . . 1054.2 Linearization of low-dimensional Poisson structures . . . . . . . . . 107

4.2.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . 1074.2.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . 1084.2.3 Four-dimensional case . . . . . . . . . . . . . . . . . . . . . 110

4.3 Poisson geometry of real semisimple Lie algebras . . . . . . . . . . 1124.4 Nondegeneracy of aff(n) . . . . . . . . . . . . . . . . . . . . . . . . 1174.5 Some other linearization results . . . . . . . . . . . . . . . . . . . . 122

4.5.1 Equivariant linearization . . . . . . . . . . . . . . . . . . . . 1224.5.2 Linearization of Poisson–Lie tensors . . . . . . . . . . . . . 1224.5.3 Poisson structures with a hyperbolic Rk-action . . . . . . . 1244.5.4 Transverse Poisson structures to coadjoint orbits . . . . . . 1254.5.5 Finite determinacy of Poisson structures . . . . . . . . . . . 126

5 Multiplicative and Quadratic Poisson Structures5.1 Multiplicative tensors . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2 Poisson–Lie groups and r-matrices . . . . . . . . . . . . . . . . . . 1325.3 The dual and the double of a Poisson-Lie group . . . . . . . . . . . 1365.4 Actions of Poisson–Lie groups . . . . . . . . . . . . . . . . . . . . . 139

5.4.1 Poisson actions of Poisson–Lie groups . . . . . . . . . . . . 1395.4.2 Dressing transformations . . . . . . . . . . . . . . . . . . . 1425.4.3 Momentum maps . . . . . . . . . . . . . . . . . . . . . . . . 144

5.5 r-matrices and quadratic Poisson structures . . . . . . . . . . . . . 145

Contents ix

5.6 Linear curl vector fields . . . . . . . . . . . . . . . . . . . . . . . . 1475.7 Quadratization of Poisson structures . . . . . . . . . . . . . . . . . 1505.8 Nonhomogeneous quadratic Poisson structures . . . . . . . . . . . 156

6 Nambu Structures and Singular Foliations

6.1 Nambu brackets and Nambu tensors . . . . . . . . . . . . . . . . . 1596.2 Integrable differential forms . . . . . . . . . . . . . . . . . . . . . . 1656.3 Frobenius with singularities . . . . . . . . . . . . . . . . . . . . . . 1686.4 Linear Nambu structures . . . . . . . . . . . . . . . . . . . . . . . 1716.5 Kupka’s phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.6 Linearization of Nambu structures . . . . . . . . . . . . . . . . . . 182

6.6.1 Decomposability of ω . . . . . . . . . . . . . . . . . . . . . 1846.6.2 Formal linearization of the associated foliation . . . . . . . 1856.6.3 The analytic case . . . . . . . . . . . . . . . . . . . . . . . . 1886.6.4 Formal linearization of Λ . . . . . . . . . . . . . . . . . . . 1886.6.5 The smooth elliptic case . . . . . . . . . . . . . . . . . . . . 190

6.7 Integrable 1-forms with a non-zero linear part . . . . . . . . . . . . 1926.8 Quadratic integrable 1-forms . . . . . . . . . . . . . . . . . . . . . 1976.9 Poisson structures in dimension 3 . . . . . . . . . . . . . . . . . . . 199

7 Lie Groupoids

7.1 Some basic notions on groupoids . . . . . . . . . . . . . . . . . . . 2037.1.1 Definitions and first examples . . . . . . . . . . . . . . . . . 2037.1.2 Lie groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.1.3 Germs and slices of Lie groupoids . . . . . . . . . . . . . . . 2087.1.4 Actions of groupoids . . . . . . . . . . . . . . . . . . . . . . 2087.1.5 Haar systems . . . . . . . . . . . . . . . . . . . . . . . . . . 209

7.2 Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 2107.3 Proper Lie groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . 213

7.3.1 Definition and elementary properties . . . . . . . . . . . . . 2137.3.2 Source-local triviality . . . . . . . . . . . . . . . . . . . . . 2157.3.3 Orbifold groupoids . . . . . . . . . . . . . . . . . . . . . . . 216

7.4 Linearization of Lie groupoids . . . . . . . . . . . . . . . . . . . . . 2177.4.1 Linearization of Lie group actions . . . . . . . . . . . . . . 2177.4.2 Local linearization of Lie groupoids . . . . . . . . . . . . . . 2187.4.3 Slice theorem for Lie groupoids . . . . . . . . . . . . . . . . 222

7.5 Symplectic groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.5.1 Definition and basic properties . . . . . . . . . . . . . . . . 2237.5.2 Proper symplectic groupoids . . . . . . . . . . . . . . . . . 2277.5.3 Hamiltonian actions of symplectic groupoids . . . . . . . . 2327.5.4 Some generalizations . . . . . . . . . . . . . . . . . . . . . . 233

x Contents

8 Lie Algebroids8.1 Some basic definitions and properties . . . . . . . . . . . . . . . . . 235

8.1.1 Definition and some examples . . . . . . . . . . . . . . . . . 2358.1.2 The Lie algebroid of a Lie groupoid . . . . . . . . . . . . . 2378.1.3 Isotropy algebras . . . . . . . . . . . . . . . . . . . . . . . . 2388.1.4 Characteristic foliation of a Lie algebroid . . . . . . . . . . 2398.1.5 Lie pseudoalgebras . . . . . . . . . . . . . . . . . . . . . . . 239

8.2 Fiber-wise linear Poisson structures . . . . . . . . . . . . . . . . . . 2408.3 Lie algebroid morphisms . . . . . . . . . . . . . . . . . . . . . . . . 2428.4 Lie algebroid actions and connections . . . . . . . . . . . . . . . . . 2438.5 Splitting theorem and transverse structures . . . . . . . . . . . . . 2468.6 Cohomology of Lie algebroids . . . . . . . . . . . . . . . . . . . . . 2498.7 Linearization of Lie algebroids . . . . . . . . . . . . . . . . . . . . . 2528.8 Integrability of Lie brackets . . . . . . . . . . . . . . . . . . . . . . 257

8.8.1 Reconstruction of groupoids from their algebroids . . . . . 2578.8.2 Integrability criteria . . . . . . . . . . . . . . . . . . . . . . 2598.8.3 Integrability of Poisson manifolds . . . . . . . . . . . . . . . 262

AppendixA.1 Moser’s path method . . . . . . . . . . . . . . . . . . . . . . . . . . 263A.2 Division theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269A.3 Reeb stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271A.4 Action-angle variables . . . . . . . . . . . . . . . . . . . . . . . . . 273A.5 Normal forms of vector fields . . . . . . . . . . . . . . . . . . . . . 276

A.5.1 Poincare–Dulac normal forms . . . . . . . . . . . . . . . . . 276A.5.2 Birkhoff normal forms . . . . . . . . . . . . . . . . . . . . . 278A.5.3 Toric characterization of normal forms . . . . . . . . . . . . 280A.5.4 Smooth normal forms . . . . . . . . . . . . . . . . . . . . . 282

A.6 Normal forms along a singular curve . . . . . . . . . . . . . . . . . 283A.7 The neighborhood of a symplectic leaf . . . . . . . . . . . . . . . . 286

A.7.1 Geometric data and coupling tensors . . . . . . . . . . . . . 286A.7.2 Linear models . . . . . . . . . . . . . . . . . . . . . . . . . . 290

A.8 Dirac structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292A.9 Deformation quantization . . . . . . . . . . . . . . . . . . . . . . . 294

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Preface

“Il ne semblait pas que cette importante theorie put encore etre perfectionnee,lorsque les deux geometres qui ont le plus contribue a la rendre complete, en ontfait de nouveau le sujet de leurs meditations. . . ”. By these words, Simeon DenisPoisson announced in 1809 [293] that he had found an improvement in the theory ofLagrangian mechanics, which was being developed by Joseph-Louis Lagrange andPierre-Simon Laplace. In that pioneering paper, Poisson introduced (we slightlymodernize his writing) the notation

(a, b) =n∑

i=1

(∂a

∂qi

∂b

∂pi− ∂a

∂pi

∂b

∂qi

), (0.1)

where a and b are two functions of the coordinates qi and the conjugate quantitiespi = ∂R

∂qifor a mechanical system with Lagrangian function R. He proved that, if

a and b are first integrals of the system then (a, b) also is. This (a, b) is nowadaysdenoted by a, b and called the Poisson bracket of a and b. Mathematicians ofthe 19th century already recognized the importance of this bracket. In particular,William Hamilton used it extensively to express his equations in an essay in 1835[168] on what we now call Hamiltonian dynamics. Carl Jacobi in his “Vorlesungenuber Dynamik” around 1842 (see [185]) showed that the Poisson bracket satisfiesthe famous Jacobi identity:

a, b, c+ b, c, a+ c, a, b = 0. (0.2)

This same identity is satisfied by Lie algebras, which are infinitesimal versions ofLie groups, first studied by Sophus Lie and his collaborators in the end of the 19thcentury [213].

In our modern language, a Poisson structure on a manifold M is a 2-vectorfield Π (Poisson tensor) on M , such that the corresponding bracket (Poissonbracket) on the space of functions on M , defined by

f, g := 〈df ∧ dg, Π〉 , (0.3)

satisfies the Jacobi identity. (M, Π) is then called a Poisson manifold. This notionof Poisson manifolds generalizes both symplectic manifolds and Lie algebras. The

xii Preface

Poisson tensor of the original bracket of Poisson is

Π =n∑

i=1

∂

∂pi∧ ∂

∂qi, (0.4)

which is nondegenerate and corresponds to a symplectic 2-form, namely

ω =n∑

i=1

dpi ∧ dqi . (0.5)

On the other hand, each finite-dimensional Lie algebra gives rise to a linear Poissontensor on its dual space and vice versa.

Poisson manifolds play a fundamental role in Hamiltonian dynamics, wherethey serve as phase spaces. They also arise naturally in other mathematical prob-lems as well. In particular, they form a bridge from the “commutative world” tothe “noncommutative world”. For example, Lie groupoids give rise to noncommu-tative operator algebras, while their infinitesimal versions, called Lie algebroids,are nothing but “fiber-wise linear” Poisson structures. Poisson geometry, i.e., thegeometry of Poisson structures, which began as an outgrowth of symplectic geom-etry, has seen rapid growth in the last three decades, and has now become a verylarge theory, with interactions with many other domains of mathematics, includ-ing Hamiltonian dynamics, integrable systems, representation theory, quantumgroups, noncommutative geometry, singularity theory, and so on.

This book arises from its authors’ efforts to study Poisson structures, and inparticular their normal forms. As a result, the book aims to offer a quick intro-duction to Poisson geometry, and to give an extensive account on known resultsabout the theory of normal forms of Poisson structures and related objects. Thistheory is relatively young. Though some earlier results may be traced back to V.I.Arnold, it really took off with a fundamental paper of Alan Weinstein in 1983[346], in which he proved a formal linearization theorem for Poisson structures, alocal symplectic realization theorem, and the following splitting theorem: locallyany Poisson manifold can be written as the direct product of a symplectic mani-fold with another Poisson manifold whose Poisson tensor vanishes at a point. Sincethen, a large number of other results have emerged, many of them very recently.

Here is a brief summary of this book, which only highlights a few importantpoints from each chapter. For a more detailed list of what the book has to offer,the reader may look at the table of contents.

The book consists of eight chapters and some appendices. Chapter 1 is basedon lectures given by the authors in Montpellier and Toulouse for graduate stu-dents, and is a small self-contained introduction to Poisson geometry. Amongother things, we show how Poisson manifolds can be viewed as singular foliationswith symplectic leaves, and also as quotients of symplectic manifolds. The readerwill also find in this chapter a section about the Schouten bracket of multi-vectorfields, which was discovered by Schouten in 1940 [311], and whose importance goesbeyond Poisson geometry.

Preface xiii

Starting from Chapter 2, the book contains many recent results which havenot been previously available in book form. A few results in this book are evenoriginal and not published elsewhere.

Chapter 2 is about Poisson cohomology, a natural and important invari-ant introduced by Andre Lichnerowicz in 1977 [211]. In particular, we show therole played by this cohomology in normal form problems, and its relations withde Rham cohomology of manifolds and Chevalley–Eilenberg cohomology of Liealgebras. Some known methods for computing Poisson cohomology are brieflydiscussed, including standard tools from algebraic topology such as the Mayer–Vietoris sequence and spectral sequences, and also tools from singularity theory.Many authors, including Viktor Ginzburg, Johannes Huebschmann, Mikhail Kara-sev, Jean-Louis Koszul, Izu Vaisman, Ping Xu, etc., contributed to the understand-ing of Poisson cohomology, and we discuss some of their results in this chapter.However, the computation of Poisson cohomology remains very difficult in general.

Chapter 3 is about a kind of normal form for Poisson structures, which arecomparable to Poincare–Birkhoff normal forms for vector fields, and which arecalled Levi decompositions because they are analogous to Levi–Malcev decom-positions for finite-dimensional Lie algebras. The results of this chapter are duemainly to Aissa Wade [342] (the formal case), the second author and Monnier[369, 263] (the analytic and smooth cases). The proof of the formal case is purelyalgebraic and relatively simple. The analytic and smooth cases make use of thefast convergence methods of Kolmogorov and Nash–Moser.

Chapter 4 is about linearization of Poisson structures. The results of Chapter3 are used in this chapter. In particular, Conn’s linearization results for Poissonstructures with a semi-simple linear part [80, 81] may be viewed as special casesof Levi decomposition. Among results discussed at length in this chapter, we willmention here Weinstein’s theorem on the smooth degeneracy of real semisimpleLie algebras of real rank greater than or equal to 2 [348], and our result on theformal and analytic nondegeneracy of the Lie algebra aff(n) [120].

In Chapter 5 we explain the links among quadratic Poisson structures, r-matrices, and the theory of Poisson–Lie groups introduced by Drinfeld [107]. Sofar, all quadratic Poisson structures known to us can be obtained from r-matrices,which have their origins in the theory of integrable systems. Some important con-tributions of Semenov–Tian–Shansky, Lu, Weinstein and other people can be foundin this chapter. We then show how the curl vector field (also known as modular vec-tor field) led the first author and other people to a classification of “nonresonant”quadratic Poisson structures, and quadratization results for Poisson structureswhich begin with a nonresonant quadratic part. Let us mention that Poisson–Liegroups are classical versions of quantum groups, a subject which is beyond thescope of this book.

Chapter 6 is devoted to n-ary generalizations of Poisson structures, whichgo under the name of Nambu structures. Though originally invented by physicistsNambu [275] and Takhtajan [328], these Nambu structures turn out to be dual tointegrable differential forms and play an important role in the theory of singular

xiv Preface

foliations. A linearization theorem for Nambu structures [119] is given in this chap-ter. Its proof at one point makes use of Malgrange’s “Frobenius with singularities”theorem [233, 234]. Malgrange’s theorem is also discussed in this chapter, togetherwith many other results on singular foliations and integrable differential forms.In particular, we present generalizations of Kupka’s stability theorem [204], whichare due to de Medeiros [244, 245], Camacho and Lins Neto [59], and ourselves.

Chapter 7 deals with Lie groupoids. Among other things, it contains a re-cent slice theorem due to Weinstein [354] and the second author [370]. This slicetheorem is a normal form theorem for proper Lie groupoids near an orbit, and gen-eralizes the classical Koszul–Palais slice theorem for proper Lie group actions. Wealso discuss symplectic groupoids, an important object of Poisson geometry intro-duced independently by Karasev [189], Weinstein [349], and Zakrzewski [364] in the1980s. A local normal form theorem for proper symplectic groupoids is also given.

Chapter 8 is about Lie algebroids, introduced by Pradines [294] in 1967 asinfinitesimal versions of Lie groupoids. They correspond to fiber-wise linear Pois-son structures, and many results about general Poisson structures, including thesplitting theorem and the Levi decomposition, apply to them. Our emphasis isagain on their local normal forms, though we also discuss cohomology of Lie alge-broids, and the problem of integrability of Lie algebroids, including a recent strongtheorem of Crainic and Fernandes [86].

Finally, Appendix A is a collection of discussions which help make the bookmore self-contained or which point to closely related subjects. It contains, amongother things, Vorobjev’s description of a neighborhood of a symplectic leaf [340],toric characterization of Poincare–Birkhoff normal forms of vector fields, a briefintroduction to deformation quantization, including a famous theorem of Kontse-vich [195] on the existence of deformation quantization for an arbitrary Poissonstructure, etc.

The book is biased towards what we know best, i.e., local normal forms. Maythe specialists in Poisson geometry forgive us for not giving more discussions onother topics, due to our lack of competence. Familiarity with symplectic manifoldsis not required, though it will be helpful for reading this book. There are manynice books readily available on symplectic geometry. On the other hand, books onPoisson geometry are relatively rare. The only general introductory reference todate is Vaisman [333]. Some other references are Cannas da Silva and Weinstein[60] (a nice book about geometric models for noncommutative algebras, wherePoisson geometry plays a key role), Karasev and Maslov [190] (a book on Poissonmanifolds with an emphasis on quantization), Mackenzie [228] (a general referenceon Lie groupoids and Lie algebroids), Ortega and Ratiu [288] (a comprehensivebook on symmetry and reduction in Poisson geometry), and a book in preparationby Xu [362] (with an emphasis on Poisson groupoids). We hope that our book iscomplementary to the above books, and will be useful for students and researchersinterested in the subject.

Preface xv

Acknowledgements. During the preparation of the book, we benefited from collab-oration and discussions with many colleagues. We would like to thank them all,and in particular Rui Fernandes, Viktor Ginzburg, Kirill Mackenzie, Alberto Med-ina, Philippe Monnier, Michel Nguiffo Boyom, Tudor Ratiu, Pol Vanhaecke, AissaWade, Alan Weinstein, Ping Xu, Misha Zhitomirskii. Special thanks go to Rui L.Fernandes who carefully read the preliminary version of this book and made nu-merous suggestions which greatly helped us to improve the book. N.T.Z. thanksMax-Planck Institut fur Mathematik, Bonn, for hospitality during 07-08/2003,when a part of this book was written.

Chapter 1

Generalities onPoisson Structures

1.1 Poisson brackets

Definition 1.1.1. A C∞-smooth Poisson structure on a C∞-smooth finite-dimen-sional manifold M is an R-bilinear antisymmetric operation

C∞(M)× C∞(M)→ C∞(M), (f, g) −→ f, g (1.1)

on the space C∞(M) of real-valued C∞-smooth functions on M , which verifies theJacobi identity

f, g, h+ g, h, f+ h, f, g = 0 (1.2)

and the Leibniz identity

f, gh = f, gh + gf, h, ∀f, g, h ∈ C∞(M). (1.3)

In other words, C∞(M), equipped with , , is a Lie algebra whose Lie bracketsatisfies the Leibniz identity. This bracket , is called a Poisson bracket . A man-ifold equipped with such a bracket is called a Poisson manifold .

Similarly, one can define real analytic, holomorphic, and formal Poisson man-ifolds, if one replaces C∞(M) by the corresponding sheaf of local analytic (re-spectively, holomorphic, formal) functions. In order to define Ck-smooth Poissonstructures (k ∈ N), we will have to express them in terms of 2-vector fields. Thiswill be done in the next section.

Remark 1.1.2. In this book, when we say that something is smooth without makingprecise its smoothness class, we usually mean that it is C∞-smooth. However, mostof the time, being C1-smooth or C2-smooth will also be good enough, thoughwe don’t want to go into these details. Analytic means either real analytic or

2 Chapter 1. Generalities on Poisson Structures

holomorphic. Though we will consider only finite-dimensional Poisson structures inthis book, let us mention that infinite-dimensional Poisson structures also appearnaturally (especially in problems of mathematical physics), see, e.g., [281, 285] andreferences therein.

Example 1.1.3. One can define a trivial Poisson structure on any manifold byputting f, g = 0 for all functions f and g.

Example 1.1.4. Take M = R2 with coordinates (x, y) and let p : R2 −→ R be anarbitrary smooth function. One can define a smooth Poisson structure on R2 byputting

f, g =(

∂f

∂x

∂g

∂y− ∂f

∂y

∂g

∂x

)p . (1.4)

Exercise 1.1.5. Verify the Jacobi identity and the Leibniz identity for the abovebracket. Show that any smooth Poisson structure of R2 has the above form.

Definition 1.1.6. A symplectic manifold (M, ω) is a manifold M equipped with anondegenerate closed differential 2-form ω, called the symplectic form.

The nondegeneracy of a differential 2-form ω means that the correspondinghomomorphism ω : TM → T ∗M from the tangent space of M to its cotangentspace, which associates to each vector X the covector iXω, is an isomorphism. HereiXω = Xω is the contraction of ω by X and is defined by iXω(Y ) = ω(X, Y ).

If f : M → R is a function on a symplectic manifold (M, ω), then we candefine its Hamiltonian vector field , denoted by Xf , as follows:

iXfω = −df . (1.5)

We can also define on (M, ω) a natural bracket, called the Poisson bracket of ω,as follows:

f, g = ω(Xf , Xg) = −〈df, Xg〉 = −Xg(f) = Xf (g). (1.6)

Proposition 1.1.7. If (M, ω) is a smooth symplectic manifold, then the bracketf, g = ω(Xf , Xg) is a smooth Poisson structure on M .

Proof. The Leibniz identity is obvious. Let us show the Jacobi identity. Recall thefollowing Cartan’s formula for the differential of a k-form η (see, e.g., [41]):

dη(X1, . . . , Xk+1) =k+1∑i=1

(−1)i−1Xi

(η(X1, . . . , Xi, . . . , Xk+1)

)+

∑1≤i<j≤k+1

(−1)i+jη([Xi, Xj], X1, . . . , Xi, . . . , Xj, . . . , Xk+1

), (1.7)

1.1. Poisson brackets 3

where X1, . . . , Xk+1 are vector fields, and the hat means that the correspondingentry is omitted. Applying Cartan’s formula to ω and Xf , Xg, Xh, we get:

0 = dω(Xf , Xg, Xh)= Xf (ω(Xg, Xh)) + Xg(ω(Xh, Xf)) + Xh(ω(Xf , Xg))− ω([Xf , Xg], Xh)− ω([Xg, Xh], Xf)− ω([Xh, Xf ], Xg)

= Xfg, h+ Xgh, f+ Xhf, g+ [Xf , Xg](h) + [Xg, Xh](f) + [Xh, Xf ](g)

= f, g, h+ g, h, f+ h, f, g+ Xf(Xg(h))−Xg(Xf (h))+ Xg(Xh(f))−Xh(Xg(f)) + Xh(Xf (g))−Xf (Xh(g))

= 3(f, g, h+ g, h, f+ h, f, g).

Thus, any symplectic manifold is also a Poisson manifold, though the inverseis not true.

The classical Darboux theorem says that in the neighborhood of every pointof (M, ω) there is a local system of coordinates (p1, q1, . . . , pn, qn), where 2n =dimM , called Darboux coordinates or canonical coordinates , such that

ω =n∑

i=1

dpi ∧ dqi . (1.8)

A proof of Darboux’s theorem will be given in Section 1.4. In such a Darbouxcoordinate system one has the following expressions for the Poisson bracket andthe Hamiltonian vector fields:

f, g =n∑

i=1

(∂f

∂pi

∂g

∂qi− ∂f

∂qi

∂g

∂pi

), (1.9)

Xh =n∑

i=1

∂h

∂pi

∂

∂qi−

n∑i=1

∂h

∂qi

∂

∂pi. (1.10)

The Hamiltonian equation of h (also called the Hamiltonian system of h), i.e., theordinary differential equation for the integral curves of Xh, has the following form,which can be found in most textbooks on analytical mechanics:

qi =∂h

∂pi, pi = − ∂h

∂qi. (1.11)

In fact, to define the Hamiltonian vector field of a function, what one reallyneeds is not a symplectic structure, but a Poisson structure: The Leibniz identitymeans that, for a given function f on a Poisson manifold M , the map g −→ f, g isa derivation. Thus, there is a unique vector field Xf on M , called the Hamiltonianvector field of f , such that for any g ∈ C∞(M) we have

Xf (g) = f, g . (1.12)


Exercise 1.1.8. Show that, in the case of a symplectic manifold, Equation (1.5)and Equation (1.12) give the same vector field.

Example 1.1.9. If N is a manifold, then its cotangent bundle T ∗N has a uniquenatural symplectic structure, hence T ∗N is a Poisson manifold with a naturalPoisson bracket. The symplectic form on T ∗N can be constructed as follows. De-note by π : T ∗N → N the projection which assigns to each covector p ∈ T ∗

q N itsbase point q. Define the so-called Liouville 1-form θ on T ∗N by

〈θ, X〉 = 〈p, π∗X〉 ∀ X ∈ Tp(T ∗N).

In other words, θ(p) = π∗(p), where on the left-hand side p is considered as apoint of T ∗N and on the right-hand side it is considered as a cotangent vectorto N . Then ω = dθ is a symplectic form on N : ω is obviously closed; to seethat it is nondegenerate take a local coordinate system (p1, . . . , pn, q1, . . . , qn) onT ∗N , where (q1, . . . , qn) is a local coordinate system on N and (p1, . . . , pn) are thecoefficients of covectors

∑pidqi(q) in this coordinate system. Then θ =

∑pidqi

and ω = dθ =∑

dpi ∧ dqi, i.e., (p1, . . . , pn, q1, . . . , qn) is a Darboux coordinatesystem for ω. In classical mechanics, one often deals with Hamiltonian equations ona cotangent bundle T ∗N equipped with the natural symplectic structure, where Nis the configuration space, i.e., the space of all possible configurations or positions;T ∗N is called the phase space.

A function g is called a first integral of a vector field X if g is constant withrespect to X : X(g) = 0. Finding first integrals is an important step in the study ofdynamical systems. Equation (1.12) means that a function g is a first integral of aHamiltonian vector field Xf if and only if f, g = 0. In particular, every functionh is a first integral of its own Hamiltonian vector field: Xh(h) = h, h = 0 due tothe anti-symmetricity of the Poisson bracket. This fact is known in physics as theprinciple of conservation of energy (here h is the energy function).

The following classical theorem of Poisson [293] allows one sometimes to findnew first integrals from old ones:

Theorem 1.1.10 (Poisson). If g and h are first integrals of a Hamiltonian vectorfield Xf on a Poisson manifold M , then g, h also is.

Proof. Another way to formulate this theorem is

g, f = 0h, f = 0

⇒ g, h, f = 0. (1.13)

But this is a corollary of the Jacobi identity. Another immediate consequence of the definition of Poisson brackets is the

following lemma:

Lemma 1.1.11. Given a smooth Poisson manifold (M, , ), the map f → Xf

is a homomorphism from the Lie algebra C∞(M) of smooth functions under the

1.2. Poisson tensors 5

Poisson bracket to the Lie algebra of smooth vector fields under the usual Liebracket. In other words, we have the following formula:

[Xf , Xg] = Xf,g. (1.14)

Proof. For any f, g, h ∈ C∞(M) we have [Xf , Xg] h = Xf (Xgh) − Xg (Xfh) =f, g, h − g, f, h = f, g, h = Xf,gh. Since h is arbitrary, it meansthat [Xf , Xg] = Xf,g.

1.2 Poisson tensors

In this section, we will express Poisson structures in terms of 2-vector fields whichsatisfy some special conditions.

Let M be a smooth manifold and q a positive integer. We denote by ΛqTMthe space of tangent q-vectors of M : it is a vector bundle over M , whose fiberover each point x ∈ M is the space ΛqTxM = Λq(TxM), which is the exterior(antisymmetric) product of q copies of the tangent space TxM . In particular,Λ1TM = TM . If (x1, . . . , xn) is a local system of coordinates at x, then ΛqTxM

admits a linear basis consisting of the elements∂

∂xi1

∧ · · · ∧ ∂

∂xiq

(x) with i1 <

i2 < · · · < iq. A smooth q-vector field Π on M is, by definition, a smooth sectionof ΛqTV , i.e., a map Π from V to ΛqTM , which associates to each point x of Ma q-vector Π(x) ∈ ΛqTxM , in a smooth way. In local coordinates, Π will have alocal expression

Π(x) =∑

i1<···<iq

Πi1...iq

∂

∂xi1

∧· · ·∧ ∂

∂xiq

=1q!

∑i1...iq

Πi1...iq

∂

∂xi1

∧· · ·∧ ∂

∂xiq

, (1.15)

where the components Πi1...iq , called the coefficients of Π, are smooth functions.The coefficients Πi1...iq are antisymmetric with respect to the indices, i.e., if wepermute two indices then the coefficient is multiplied by−1. For example, Πi1i2... =−Πi2i1.... If Πi1...iq are Ck-smooth, then we say that Π is Ck-smooth, and so on.

Smooth q-vector fields are dual objects to differential q-forms in a naturalway. If Π is a q-vector field and α is a differential q-form, which in some localsystem of coordinates are written as Π(x) =

∑i1<···<iq

Πi1...iq

∂∂xi1

∧ · · · ∧ ∂∂xiq

and α =∑

i1<···<iqai1...iqdxi1 ∧ · · · ∧ dxiq , then their pairing 〈α, Π〉 is a function

defined by〈α, Π〉 =

∑i1<···<iq

Πi1...iqai1...iq . (1.16)

Exercise 1.2.1. Show that the above definition of 〈α, Π〉 does not depend on thechoice of local coordinates.

In particular, smooth q-vector fields on a smooth manifold M can be consid-ered as C∞(M)-linear operators from the space of smooth differential q-forms onM to C∞(M), and vice versa.


A Ck-smooth q-vector field Π will define an R-multilinear skewsymmetricmap from C∞(M)× · · · × C∞(M) (q times) to C∞(M) by the following formula:

Π(f1, . . . , fq) := 〈Π, df1 ∧ · · · ∧ dfq〉 . (1.17)

Conversely, we have:

Lemma 1.2.2. An R-multilinear map Π : C∞(M)× · · · × C∞(M) → Ck(M) arisesfrom a Ck-smooth q-vector field by Formula (1.17) if and only if Π is skewsym-metric and satisfies the Leibniz rule (or condition):

Π(fg, f2, . . . , fq) = fΠ(g, f2, . . . , fq) + gΠ(f, f2, . . . , fq). (1.18)

A map Π which satisfies the above conditions is called a multi-derivation, andthe above lemma says that multi-derivations can be identified with multi-vectorfields.

Proof (sketch). The “only if” part is straightforward. For the “if” part, we haveto check that the value of Π(f1, . . . , fq) at a point x depends only on the valueof df1, . . . ,dfq at x. Equivalently, we have to check that if df1(x) = 0 thenΠ(f1, . . . , fq)(x) = 0. If df1(x) = 0 then we can write f1 = c +

∑i xigi where

c is a constant and xi and gi are smooth functions which vanish at x. Accord-ing to the Leibniz rule we have Π(1 × 1, f2, . . . , fq) = 1 × Π(1, f2, . . . , fq) + 1 ×Π(1, f2, . . . , fq) = 2Π(1, f2, . . . , fq), hence Π(1, f2, . . . , fq) = 0. Now according tothe linearity and the Leibniz rule we have Π(f1, . . . , fq)(x) = cΠ(1, f2, . . . , fq)(x)+∑

xi(x)Π(gi, f2, . . . , fq)(x) +∑

gi(x)Π(xi, f2, . . . , fq)(x) = 0. In particular, if Π is a Poisson structure, then it is skewsymmetric and satisfies

the Leibniz condition, hence it arises from a 2-vector field, which we will also denoteby Π:

f, g = Π(f, g) = 〈Π, df ∧ dg〉 . (1.19)

A 2-vector field Π, such that the bracket f, g := 〈Π, df ∧ dg〉 is a Poissonbracket (i.e., satisfies the Jacobi identity f, g, h+ g, h, f+ h, f, g= 0for any smooth functions f, g, h), is called a Poisson tensor , or also a Poissonstructure. The corresponding Poisson bracket is often denoted by , Π. If thePoisson tensor Π is a Ck-smooth 2-vector field, then we say that we have a Ck-smooth Poisson structure, and so on.

In a local system of coordinates (x1, . . . , xn) we have

Π =∑i<j

Πij∂

∂xi∧ ∂

∂xj=

12

∑i,j

Πij∂

∂xi∧ ∂

∂xj, (1.20)

where Πij = 〈Π, dxi ∧ dxj〉 = xi, xj, and

f, g = 〈∑i<j

xi, xj∂

∂xi∧ ∂

∂xj,∑i,j

∂f

∂xi

∂g

∂xjdxi∧dxj〉 =

∑i,j

Πij∂f

∂xi

∂g

∂xj. (1.21)

1.2. Poisson tensors 7

Example 1.2.3. The Poisson tensor corresponding to the standard symplecticstructure ω =

∑nj=1 dxj ∧ dyj on R2n is

∑nj=1

∂∂xj

∧ ∂∂yj

.

Notation 1.2.4. In this book, if functions f1, . . . , fp depend on variables x1, . . . , xp,and maybe other variables, then we will denote by

∂(f1, . . . , fp)∂(x1, . . . , xp)

:= det(

∂fi

∂xj

)p

i,j=1

(1.22)

the Jacobian determinant of (f1, . . . , fp) with respect to (x1, . . . , xp). For example,

∂(f, g)∂(xi, xj)

:=∂f

∂xi

∂g

∂xj− ∂f

∂xj

∂g

∂xi. (1.23)

With the above notation, we have the following local expression for Poissonbrackets:

f, g =∑i,j

xi, xj∂f

∂xi

∂g

∂xj=∑i<j

xi, xj∂(f, g)

∂(xi, xj). (1.24)

Due to the Jacobi condition, not every 2-vector field will be a Poisson tensor.

Exercise 1.2.5. Show that the 2-vector field ∂∂x ∧ ( ∂

∂y +x ∂∂z ) in R3 is not a Poisson

tensor.

Exercise 1.2.6. Show that if X1, . . . , Xm are pairwise commuting vector fields andaij are constants, then

∑ij aijXi ∧Xj is a Poisson tensor.

To study the Jacobi identity, we will use the following lemma:

Lemma 1.2.7. For any C1-smooth 2-vector field Π, one can associate to it a3-vector field Λ defined by

Λ(f, g, h) = f, g, h+ g, h, f+ h, f, g (1.25)

where k, l denotes 〈Π, dk ∧ dl〉 (i.e., the bracket of Π).

Proof. It is clear that the right-hand side of Formula (1.25) is R-multilinear andantisymmetric. To show that it corresponds to a 3-vector field, one has to verifythat it satisfies the Leibniz rule with respect to f , i.e.,

f1f2, g, h+ g, h, f1f2+ h, f1f2, g= f1(f2, g, h+ g, h, f2+ h, f2, g)

+ f2(f1, g, h+ g, h, f1+ h, f1, g).

This is a simple direct verification, based on the Leibniz rule ab, c = ab, c +ba, c for the bracket of the 2-vector field Π. It will be left to the reader as anexercise.


Direct calculations in local coordinates show that

Λ(f, g, h) =∑ijk

(∮ijk

∑s

∂Πij

∂xsΠsk

)∂f

∂xi

∂g

∂xj

∂h

∂xk, (1.26)

where∮

ijkaijk means the cyclic sum aijk + ajki + akij . In other words,

Λ =∑

i<j<k

(∮ijk

∑s

∂Πij

∂xsΠsk

)∂

∂xi∧ ∂

∂xj∧ ∂

∂xk. (1.27)

Clearly, the Jacobi identity for Π is equivalent to the condition that Λ = 0.Thus we have:

Proposition 1.2.8. A 2-vector field Π =∑

i<j Πij∂

∂xi∧ ∂

∂xjexpressed in terms of

a given system of coordinates (x1, . . . , xn) is a Poisson tensor if and only if itsatisfies the following system of equations:∮

ijk

∑s

∂Πij

∂xsΠsk = 0 (∀ i, j, k) . (1.28)

An obvious consequence of the above proposition is that the condition fora 2-vector field to be a Poisson structure is a local condition. In particular, therestriction of a Poisson structure to an open subset of the manifold is again aPoisson structure.

Example 1.2.9. Constant Poisson structures on Rn: Choose arbitrary constantsΠij . Then Equation (1.28) is obviously satisfied. The canonical Poisson structureon R2n, associated to the canonical symplectic form ω =

∑dqi ∧ dpi, is of this

type.

Example 1.2.10. Any 2-vector field on a two-dimensional manifold is a Poissontensor. Indeed, the 3-vector field Λ in Lemma 1.2.7 is identically zero becausethere are no nontrivial 3-vectors on a two-dimensional manifold. Thus the Jacobiidentity is nontrivial only starting from dimension 3.

Example 1.2.11. Let V be a finite-dimensional vector space over R (or C). A linearPoisson structure on V is a Poisson structure on V for which the Poisson bracket oftwo linear functions is again a linear function. Equivalently, in linear coordinates,the components of the corresponding Poisson tensor are linear functions. In thiscase, by restriction to linear functions, the operation (f, g) → f, g gives rise toan operation [ , ] : V ∗×V ∗ −→ V ∗, which is a Lie algebra structure on V ∗, whereV ∗ is the dual linear space of V .

Conversely, any Lie algebra structure on V ∗ determines a linear Poissonstructure on V . Indeed, consider a finite-dimensional Lie algebra (g, [ , ]). Foreach linear function f : g∗ −→ R we denote by f the element of g corresponding

1.3. Poisson morphisms 9

to it. If f and g are two linear functions on g∗, then we put f, g(α) = 〈α, [f , g]〉for every α in g∗. If we choose a linear basis e1, . . . , en of g, with [ei, ej ] =

∑ckijek,

then we have xi, xj =∑

ckijxk where xl is the function such that xl = el. By

taking (x1, . . . , xn) as a linear system of coordinates on g∗, it follows from theJacobi identity for [ , ] that the functions Πij = xi, xj verify Equation (1.28).Thus we get a Poisson structure on g∗. This Poisson structure can be definedintrinsically by the following formula:

f, g(α) = 〈α, [df(α), dg(α)]〉 , (1.29)

where df(α) and dg(α) are considered as elements of g via the identification(g∗)∗ = g. Thus, there is a natural bijection between finite-dimensional linearPoisson structures and finite-dimensional Lie algebras. One can even try to studyLie algebras by viewing them as linear Poisson structures (see, e.g., [61]).

Remark 1.2.12. Multi-vector fields are also known as antisymmetric contravarianttensors , because their coefficients change contravariantly under a change of localcoordinates. In particular, the local expression of a Poisson bracket will changecontravariantly under a change of local coordinates: Let x = (x1, . . . , xn) andy = (y1, . . . , yn) be two local coordinate systems on the same open subset of aPoisson manifold (M, , ). Viewing yi as functions of (x1, . . . , xn), we have

yi, yj =∑r<s

∂(yi, yj)∂(xr, xs)

xr, xs . (1.30)

Denote Πrs(x) = xr , xs (x), Π′ij(y) = yi, yj (y). Then the above equation can

be rewritten as

Π′ij(y(x)) =

∑r<s

∂(yi, yj)∂(xr , xs)

(x)Πrs(x). (1.31)

Exercise 1.2.13. Consider the Poisson structure on R2 defined by x, y = ex.Show that in the new coordinates (u, v) = (x, ye−x) the Poisson tensor will havethe standard form ∂

∂u ∧ ∂∂v .

Exercise 1.2.14. Let Π =∑

Πij∂/∂xi ∧ ∂/∂xj be a constant Poisson structure onRn, i.e., the coefficients Πij are constants. Show that there is a number p ≥ 0 anda linear coordinate system (y1, . . . , yn) in which the Poisson bracket has the form

f, g =∂(f, g)

∂ (y1, y2)+

∂(f, g)∂ (y3, y4)

+ · · ·+ ∂(f, g)∂ (y2p−1, y2p)

. (1.32)

1.3 Poisson morphisms

Definition 1.3.1. If (M1, , 1) and (M2, , 2) are two smooth Poisson manifolds,then a smooth map φ from M1 to M2 is called a smooth Poisson morphism orPoisson map if the associated pull-back map φ∗ : C∞(M2) → C∞(M1) is a Liealgebra homomorphism with respect to the corresponding Poisson brackets.


In other words, φ : (M1, , 1) → (M2, , 2) is a Poisson morphism if

φ∗f, φ∗g1 = φ∗f, g2 ∀ f, g ∈ C∞(M2) . (1.33)

Of course, Poisson manifolds together with Poisson morphisms form a cat-egory: the composition of two Poisson morphisms is again a Poisson morphism,and so on. Notice that a Poisson morphism which is a diffeomorphism will auto-matically be a Poisson isomorphism: the inverse map is also a Poisson map.

Similarly, a map φ : (M1, ω1) → (M2, ω2) is called a symplectic morphism ifφ∗ω2 = ω1. Clearly, a symplectic isomorphism is also a Poisson isomorphism. How-ever, a symplectic morphism is not a Poisson morphism in general. For example,if M1 is a point with a trivial symplectic form, and M2 is a symplectic manifoldof positive dimension, then any map φ : M1 → M2 is a symplectic morphism butnot a Poisson morphism.

Example 1.3.2. If φ : h → g is a Lie algebra homomorphism, then the lineardual map φ∗ : g∗ → h∗ is a Poisson map, where g∗ and h∗ are equipped withtheir respective linear Poisson structures. The proof of this fact will be left to thereader as an exercise. In particular, if h is a Lie subalgebra of g, then the canonicalprojection g∗ → h∗ is Poisson.

Example 1.3.3. If φ is a diffeomorphism of a manifold N , then it can be liftednaturally to a diffeomorphism φ∗ : T ∗N → T ∗N covering φ. By definition, φ∗ pre-serves the Liouville 1-form θ (see Example 1.1.9), hence it preserves the symplecticform dθ. Thus, φ∗ is a Poisson isomorphism.

Example 1.3.4. Direct product of Poisson manifolds. Let (M1, , 1) and (M2, , 2)be two Poisson manifolds. Then their direct product M1 ×M2 can be equippedwith the following natural bracket:

f (x1, x2) , g (x1, x2) = fx2 , gx21 (x1) + fx1 , gx12 (x2) (1.34)

where we use the notation hx1(x2) = hx2(x1) = h(x1, x2) for any function h onM1 × M2, x1 ∈ M1 and x2 ∈ M2. Using Equation (1.28), one can verify easilythat this bracket is indeed a Poisson bracket on M1 ×M2. It is called the productPoisson structure. With respect to this product Poisson structure, the projectionmaps M1 ×M2 →M1 and M1 ×M2 →M2 are Poisson maps.

Exercise 1.3.5. Let M1 = M2 = Rn with trivial Poisson structure. Find a nontrivialPoisson structure on M1×M2 = R2n for which the two projections M1×M2 →M1

and M1 ×M2 →M2 are Poisson maps.

Exercise 1.3.6. Show that any Poisson map from a Poisson manifold to a symplec-tic manifold is a submersion.

A vector field X on a Poisson manifold (M, Π), is called a Poisson vectorfield if it is an infinitesimal automorphism of the Poisson structure, i.e., the Liederivative of Π with respect to X vanishes:

LXΠ = 0 . (1.35)

1.3. Poisson morphisms 11

Equivalently, the local flow (ϕtX) of X , i.e., the one-dimensional pseudo-group

of local diffeomorphisms of M generated by X , preserves the Poisson structure:∀t ∈ R, (ϕt

X) is a Poisson morphism wherever it is well defined.By the Leibniz rule we have LX(f, g) = LX(〈Π, df ∧ dg〉) = 〈LXΠ, df ∧

dg〉+ 〈Π, dLXf ∧dg〉+ 〈Π, df ∧dLXg〉 = 〈LXΠ, df ∧dg〉+X(f), g+f, X(g).So another equivalent condition for X to be a Poisson vector field is the following:

Xf, g+ f, Xg = Xf, g . (1.36)

When X = Xh is a Hamiltonian vector field, then Equation (1.36) is nothing butthe Jacobi identity. Thus any Hamiltonian vector field is a Poisson vector field. Theinverse is not true in general, even locally. For example, if the Poisson structure istrivial, then any vector field is a Poisson vector field, while the only Hamiltonianvector field is the trivial one.

Exercise 1.3.7. Show that on R2n with the standard Poisson structure∑ ∂

∂xi∧ ∂

∂yi

any Poisson vector field is also Hamiltonian.

Example 1.3.8. Infinitesimal version of Example 1.3.3. If X is a vector field on amanifold N , then X admits a unique natural lifting to a vector field X on T ∗Nwhich preserves the Liouville 1-form. In a local coordinate system (p1, . . . , pn,q1, . . . , qn) on T ∗N , where (q1, . . . , qn) is a local coordinate system on N andthe Liouville 1-form is θ =

∑i pidqi (see Example 1.1.9), we have the following

expression for X:

If X =∑

i

αi(q)∂

∂qithen X =

∑i

αi(q)∂

∂qi−∑i,j

∂αi(q)∂qj

pi∂

∂pj.

The vector field X is in fact the Hamiltonian vector field of the function

X (p1, . . . , pn, q1, . . . , qn) =∑

i

αi(q)pi

on T ∗N . This function X is nothing else than X itself, considered as a fiber-wiselinear function on T ∗N .

Example 1.3.9. Let G be a connected Lie group, and denote by g the Lie algebraof G. By definition, g is isomorphic to the Lie algebra of left-invariant tangentvector fields of G (i.e., vector fields which are invariant under left translationsLg : h → gh on G). Denote by e the neutral element of G. For each Xe ∈ TeG,there is a unique left-invariant vector field X on G whose value at e is Xe (Xobtained from Xe by left translations), so we may identify TeG with g via thisassociation Xe → X . We will write TeG = g, and T ∗

e G = g∗ by duality. Considerthe left translation map

L : T ∗G→ g∗ = T ∗e G, L(p) = (Lg)∗p = Lg−1p ∀ p ∈ T ∗

g G, (1.37)

where Lg−1p means the push-forward (Lg−1)∗p of p by Lg−1 (we will often omitthe subscript asterisk when writing push-forwards to simplify the notation).


Theorem 1.3.10. The above left translation map L : T ∗G→ g∗ is a Poisson map,where T ∗G is equipped with the standard symplectic structure, and g∗ is equippedwith the standard linear Poisson structure (induced from the Lie algebra structureof g).

Proof (sketch). It is enough to verify that, if x, y are two elements of g, consideredas linear functions on g∗, then we have

L∗x, L∗y = L∗([x, y]).

Notice that L∗x is nothing else than x itself, considered as a left-invariant vectorfield on G and then as a left-invariant fiber-wise linear function on T ∗G. By theformulas given in Example 1.3.8, the Hamiltonian vector field XL∗x of L∗x is thenatural lifting to T ∗G of x, considered as a left-invariant vector field on G. Sincethe process of lifting of vector fields from N to T ∗N preserves the Lie bracket forany manifold N , we have

[XL∗x, XL∗y] = XL∗[x,y].

It follows from the above equation and Lemma 1.1.11 that L∗x, L∗y andL∗([x, y]) have the same Hamiltonian vector field on T ∗G. Hence these two func-tions differ by a function which vanishes on the zero section of T ∗G and whoseHamiltonian vector field is trivial on T ∗G. The only such function is 0, soL∗x, L∗y = L∗([x, y]). Exercise 1.3.11. Show that the right translation map R : T ∗G → g∗ = T ∗

e G,defined by L(p) = (Rg)∗p ∀ p ∈ T ∗

g G, is an anti-Poisson map. A map φ : (M, Π) →(N, Λ) is called an anti-Poisson map if φ : (M, Π)→ (N,−Λ) is a Poisson map.

Given a subspace V ∈ TxM of a tangent space TxM of a symplectic mani-fold (M, ω), we will denote by V ⊥ the symplectic orthogonal to V : V ⊥ = X ∈TxM | ω(X, Y ) = 0 ∀ Y ∈ V . Clearly, V = (V ⊥)⊥. V is called Lagrangian(resp. isotropic, coisotropic, symplectic) if V = V ⊥ (resp. V ⊂ V ⊥, V ⊃ V ⊥,V ∩ V ⊥ = 0). A submanifold of a symplectic manifold is called Lagrangian (resp.isotropic, coisotropic, resp. symplectic) if its tangent spaces are so. Lagrangiansubmanifolds play a central role in symplectic geometry, see, e.g., [345, 243]. Inparticular, we have the following characterization of symplectic isomorphisms interms of Lagrangian submanifolds:

Proposition 1.3.12. A diffeomorphism φ : (M, ω1) → (M2, ω2) is a symplecticisomorphism if and only if its graph ∆ = (x, φ(x)) ⊂ M1 ×M2 is a Lagrangianmanifold of M1 ×M2, where M2 means M2 together with the opposite symplecticform −ω2.

The proof is almost obvious and is left as an exercise. A subspace V ⊂ TxM of a Poisson manifold (M, Π) is called coisotropic if for

any α, β ∈ T ∗xM such that 〈α, X〉 = 〈β, X〉 = 0 ∀ X ∈ V we have 〈Π, α ∧ β〉 = 0.

1.4. Local canonical coordinates 13

In other words, V ⊂ (V )⊥, where V = α ∈ T ∗x M | 〈α, X〉 = 0 ∀ X ∈ V is the

annulator of V and (V )⊥ = β ∈ T ∗xM | 〈Π, α∧β〉 = 0 ∀ α ∈ V is the “Poisson

orthogonal” of V . A submanifold N of a Poisson manifold is called coisotropic ifits tangent spaces are coisotropic.

Proposition 1.3.13. A map φ : (M1, Π1) → (M2, Π2) between two Poisson mani-folds is a Poisson map if and only if its graph Γ(φ) := (x, y) ∈ M1 ×M2; y =φ(x) is a coisotropic submanifold of (M1, Π1)× (M2,−Π2).

Again, the proof will be left as an exercise.

1.4 Local canonical coordinates

In this section, we will prove the splitting theorem of Alan Weinstein [346], whichsays that locally a Poisson manifold is a direct product of a symplectic manifoldwith another Poisson manifold whose Poisson tensor vanishes at a point. Thissplitting theorem, together with the Darboux theorem which will be proved at thesame time, will give us local canonical coordinates for Poisson manifolds.

Given a Poisson structure Π (or more generally, an arbitrary 2-vector field)on a manifold M , we can associate to it a natural homomorphism

= Π : T ∗M −→ TM, (1.38)

which maps each covector α ∈ T ∗xM over a point x to a unique vector (α) ∈ TxM

such that〈α ∧ β, Π〉 = 〈β, (α)〉 (1.39)

for any covector β ∈ T ∗xM . We will call = Π the anchor map of Π.

The same notations (or Π) will be used to denote the operator whichassociates to each differential 1-form α the vector field (α) defined by ((α))(x) =(α(x)). For example, if f is a function, then (df) = Xf is the Hamiltonian vectorfield of f .

The restriction of Π to a cotangent space T ∗xM will be denoted by x or

Π(x). In a local system of coordinates (x1, . . . , xn) we have

( n∑

i=1

aidxi

)=∑ij

xi, xj ai∂

∂xj=∑ij

Πijai∂

∂xj.

Thus x is a linear operator, given by the matrix [Πij(x)] in the linear bases

(dx1, . . . , dxn) and(

∂∂x1

, . . . , ∂∂xn

).

Definition 1.4.1. Let (M, Π) be a Poisson manifold and x a point of M . Thenthe image Cx := Imx of x is called the characteristic space at x of the Pois-son structure Π. The dimension dim Cx of Cx is called the rank of Π at x, andmaxx∈M dim Cx is called the rank of Π. When rankΠx = dimM we say that Π


is nondegenerate at x. If rankΠx is a constant on M , i.e., does not depend on x,then Π is called a regular Poisson structure.

Example 1.4.2. The constant Poisson structure∑s

i=1∂

∂xi∧ ∂

∂xi+son Rm (m ≥ 2s)

is a regular Poisson structure of rank 2s.

Exercise 1.4.3. Show that rankΠx is always an even number, and that Π is non-degenerate everywhere if and only if it is the associated Poisson structure of asymplectic structure.

The characteristic space Cx admits a unique natural antisymmetric nonde-generate bilinear scalar product, called the induced symplectic form: if X and Yare two vectors of Cx, then we put

(X, Y ) := 〈β, X〉 = 〈Π, α ∧ β〉 = −〈Π, β ∧ α〉 = −〈α, Y 〉 = −(Y, X) (1.40)

where α, β ∈ T ∗xM are two covectors such that X = α and Y = β.

Exercise 1.4.4. Verify that the above scalar product is anti-symmetric nondegen-erate and is well defined (i.e., does not depend on the choice of α and β). WhenΠ is nondegenerate then the above formula defines the corresponding symplecticstructure on M .

Theorem 1.4.5 (Splitting theorem [346]). Let x be a point of rank 2s of a Pois-son m-dimensional manifold (M, Π): dim Cx = 2s where Cx is the characteristicspace at x. Let N be an arbitrary (m − 2s)-dimensional submanifold of M whichcontains x and is transversal to Cx at x. Then there is a local system of coor-dinates (p1, . . . , ps, q1, . . . , qs, z1, . . . , zm−2s) in a neighborhood of x, which satisfythe following conditions:

a) pi(Nx) = qi(Nx) = 0 where Nx is a small neighborhood of x in N .b) qi, qj = pi, pj = 0 ∀ i, j; pi, qj = 0 if i = j and pi, qi = 1 ∀ i.c) zi, pj = zi, qj = 0 ∀ i, j.d) zi, zj(x) = 0 ∀ i, j.

A local coordinate system which satisfies the conditions of the above theoremis called a system of local canonical coordinates . In such canonical coordinates wehave

f, g =∑i,j

zi, zj∂f

∂zi

∂g

∂zj+

s∑i=1

∂(f, g)∂(pi, qi)

= f, gN + f, gS , (1.41)

where

f, gS =s∑

i=1

∂(f, g)∂(pi, qi)

(1.42)

defines the nondegenerate Poisson structure∑

∂∂pi∧ ∂

∂qion the local submanifold

S = z1 = · · · = zm−2s = 0, and

f, gN =∑u,v

zi, zj∂f

∂zi

∂g

∂zj(1.43)

1.4. Local canonical coordinates 15

defines a Poisson structure on a neighborhood of x in N . Notice that, sincezi, pj = zi, qj = 0 ∀ i, j, the functions zi, zj do not depend on the vari-ables (p1, . . . , ps, q1, . . . , qs). The equality zi, zj(x) = 0 ∀ i, j means that thePoisson tensor of , N vanishes at x.

Formula (1.41) means that the Poisson manifold (M, Π) is locally isomorphic(in a neighborhood of x) to the direct product of a symplectic manifold (S,

∑s1 dpi∧

dqi) with a Poisson manifold (Nx, , N) whose Poisson tensor vanishes at x. That’swhy Theorem 1.4.5 is called the splitting theorem for Poisson manifolds: locally,we can split a Poisson structure in two parts – a regular part and a singular partwhich vanishes at a point.

Proof of Theorem 1.4.5. If Π(x) = 0 then s = 0 and there is nothing to prove.Suppose that Π(x) = 0. Let p1 be a local function (defined in a small neighborhoodof x in M) which vanishes on N and such that dp1(x) = 0. Since Cx is transversalto N , there is a vector Xg(x) ∈ Cx such that 〈Xg(x), dp1(x)〉 = 0, or equivalently,Xp1(g)(x) = 0, where Xp1 denotes the Hamiltonian vector field of p1 as usual.Therefore Xp1(x) = 0. Since Cx (dp1)(x) = Xp1(x) = 0 and is not tangentto N , there is a local function q1 such that q1(N) = 0 and Xp1(q1) = 1 in aneighborhood of x, or

Xp1q1 = p1, q1 = 1 . (1.44)

Moreover, Xp1 and Xq1 are linearly independent (Xq1 = λXp1 would imply thatp1, q1 = −λXp1(p1) = 0), and we have

[Xp1 , Xq1 ] = Xp1,q1 = 0 . (1.45)

Thus Xp1 and Xq1 are two linearly independent vector fields which commute.Hence they generate a locally free infinitesimal R2-action in a neighborhood of x,which gives rise to a local regular two-dimensional foliation. As a consequence, wecan find a local system of coordinates (y1, . . . , ym) such that

Xq1 =∂

∂y1, Xp1 =

∂

∂y2. (1.46)

With these coordinates we have q1, yi = Xq1 (yi) = 0 and p1, yi = Xp1 (yi) =0, for i = 3, . . . , m. Poisson’s Theorem 1.1.10 then implies that p1, yi, yj =q1, yi, yj = 0 for i, j ≥ 3, whence

yi, yj = ϕij (y3, . . . , yn) ∀ i, j ≥ 3 ,p1, q1 = 1 ,p1, yj = q1, yj = 0 ∀ j ≥ 3 .

(1.47)

We can take (p1, q1, y3, . . . , yn) as a new local system of coordinates. In fact, theJacobian matrix of the map ϕ : (y1, y2, y3, . . . , ym) → (p1, q1, y3, . . . , ym) is of theform ⎛⎝ 0 1

−1 0 ∗0 Id

⎞⎠ (1.48)


(because ∂q1∂y1

= Xq1q1 = 0, ∂q1∂y2

= Xp1q1 = q1, p1 = 1, . . .), which has a non-zerodeterminant (equal to 1). In the coordinates (q1, p1, y3, . . . , ym), we have

Π =∂

∂p1∧ ∂

∂q1+

12

∑i,j≥3

Π′ij(y3, . . . , ym)

∂

∂yi∧ ∂

∂yj. (1.49)

The above formula implies that our Poisson structure is locally the productof a standard symplectic structure on a plane (p1, q1) with a Poisson structureon a (m − 2)-dimensional manifold (y3, . . . , ym). In this product, N is also thedirect product of a point (= the origin) of the plane (p1, q1) with a local subman-ifold in the Poisson manifold (y3, . . . , ym). The splitting theorem now follows byinduction on the rank of Π at x.

Remark 1.4.6. In the above theorem, when m = 2s, we recover Darboux’s theoremwhich gives local canonical coordinates for symplectic manifolds. If (M, Π) is aregular Poisson structure, then the Poisson structure of Nx in the above theoremmust be trivial, and we get the following generalization of Darboux’s theorem:any regular Poisson structure is locally isomorphic to a standard constant Poissonstructure.

Exercise 1.4.7. Prove the following generalization of Theorem 1.4.5. Let N be asubmanifold of a Poisson manifold (M, Π), and x be a point of N such that TxN +Cx = TxM and TxN ∩Cx is a symplectic subspace of Cx, i.e., the restriction of thesymplectic form on the characteristic space Cx to TzN∩Cx is nondegenerate. (Sucha submanifold N is sometimes called cosymplectic.) Then there is a coordinatesystem in a neighborhood of x which satisfies the conditions a), b), c) of Theorem1.4.5, where 2s = dimM − dim N = dimCx − dim(TxN ∩ Cx).

1.5 Singular symplectic foliations

A smooth singular foliation in the sense of Stefan–Sussmann [320, 327] on a smoothmanifold M is by definition a partition F = Fα of M into a disjoint unionof smooth immersed connected submanifolds Fα, called leaves , which satisfiesthe following local foliation property at each point x ∈ M : Denote the leaf thatcontains x by Fx, the dimension of Fx by d and the dimension of M by m. Thenthere is a smooth local chart of M with coordinates y1, . . . , ym in a neighborhoodU of x, U = −ε < y1 < ε, . . . ,−ε < ym < ε, such that the d-dimensionaldisk yd+1 = · · · = ym = 0 coincides with the path-connected component of theintersection of Fx with U which contains x, and each d-dimensional disk yd+1 =cd+1, . . . , ym = cm, where cd+1, . . . , cm are constants, is wholly contained in someleaf Fα of F . If all the leaves Fα of a singular foliation F have the same dimension,then one says that F is a regular foliation.

A singular distribution on a manifold M is the assignment to each pointx of M a vector subspace Dx of the tangent space TxM. The dimension of Dx

1.5. Singular symplectic foliations 17

may depend on x. For example, if F is a singular foliation, then it has a naturalassociated tangent distribution DF : at each point x ∈ V , DF

x is the tangent spaceto the leaf of F which contains x.

A singular distribution D on a smooth manifold is called smooth if for anypoint x of M and any vector X0 ∈ Dx, there is a smooth vector field X defined ina neighborhood Ux of x which is tangent to the distribution, i.e., X(y) ∈ Dy ∀ y ∈Ux, and such that X(x) = X0. If, moreover, dimDx does not depend on x, thenwe say that D is a smooth regular distribution.

It follows directly from the local foliation property that the tangent distri-bution DF of a smooth singular foliation is a smooth singular distribution.

An integral submanifold of smooth singular distribution D on a smooth man-ifold M is, by definition, a connected immersed submanifold W of M such thatfor every y ∈ W the tangent space TyW is a vector subspace of Dy. An integralsubmanifold W is called maximal if it is not contained in any other integral sub-manifold; it is said to be of maximum dimension if its tangent space at every pointy ∈ W is exactly Dy.

We say that a smooth singular distribution D on a smooth manifold M isan integrable distribution if every point of M is contained in a maximal integralmanifold of maximum dimension of D.

Let C be a family of smooth vector fields on M . Then it gives rise to a smoothsingular distribution DC : for each point x ∈ M , DC

x is the vector space spannedby the values at x of the vector fields of C. We say that DC is generated by C.

A distribution D is called invariant with respect to a family of smooth vectorfields C if it is invariant with respect to every element of C: if X ∈ C and (ϕt

X)denotes the local flow of X , then we have (ϕt

X)∗Dx = DϕtX(x) wherever ϕt

X(x) iswell defined.

The following result, due to Stefan [320] and Sussmann [327] (see also Dazord[93]), gives an answer to the following question: what are the conditions for asmooth singular distribution to be the tangent distribution of a singular foliation?

Theorem 1.5.1 (Stefan–Sussmann). Let D be a smooth singular distribution on asmooth manifold M . Then the following three conditions are equivalent:

a) D is integrable.b) D is generated by a family C of smooth vector fields, and is invariant with

respect to C.c) D is the tangent distribution DF of a smooth singular foliation F .

Proof (sketch). a) ⇒ b). Suppose that D is integrable. Let C be the family ofall smooth vector fields which are tangent to D. The smoothness condition of Dimplies that D is generated by C. It remains to show that if X is an arbitrarysmooth vector field tangent to D, then D is invariant with respect to X . Let x bean arbitrary point in M , and denote by F(x) the maximal invariant submanifoldof maximum dimension which contains x. Then by definition (the condition ofmaximum dimension), for every point y ∈ F(x) we have TyF(x) = Dy, which


implies that the vector field X , when restricted to F(x), is tangent to F(x). Inparticular, the local flow ϕt

X can be restricted to F(x), i.e., F(x) is an invariantmanifold for this local flow. Moreover, if ϕτ

X(x) is well defined for some τ > 0,then the point ϕτ

X(x) lies on F(x). This fact follows from the maximality condi-tion on F(x). (Note that, the union of two invariant submanifolds of maximumdimension is again an invariant submanifold of maximum dimension if it is con-nected.) Because X is tangent to F(x), we have (ϕτ

X)∗(TxF(x)) = TϕτX(x)F(x).

But TxF(x) = Dx and TϕτX(x)F(x) = Dϕτ

X(x), hence (ϕτX)∗Dx = Dϕτ

X(x).b) ⇒ c). Suppose that D is generated by a family C of smooth vector fields,

and is invariant with respect to C. Let x be an arbitrary point of M , denoteby d the dimension of Dx, and choose d vector fields X1, . . . , Xd of C such thatX1(x), . . . , Xd(x) span Dx. Denote by φt

1, . . . , φtd the local flow of X1, . . . , Xd re-

spectively. The map(s1, . . . , sd) → φs1

1 · · · φsd

d (x) (1.50)

is a local diffeomorphism from a d-dimensional disk to a d-dimensional submani-fold containing x in M . The invariance of D with respect to C implies that thissubmanifold is an integral submanifold of maximum dimension. Gluing these localintegral submanifolds together (wherever they intersect), we obtain a partitionof M into a disjoint union of connected immersed integral submanifolds of maxi-mum dimension, called leaves. To see that this partition satisfies the local foliationproperty of singular foliations, we can proceed by induction on the dimension ofDx: If dimDx = 0, then the local foliation property at x is empty. If dimDx > 0,then there is a vector field X ∈ C such that X(x) = 0. Then the trajectories ofX lie on the leaves, and we can take the quotient of a small neighborhood of xby the trajectories of X to reduce the dimension of M and of the leaves by 1.The invariance with respect to C and the local foliation property does not changeunder this reduction.

c) ⇒ a): If D = DF is the tangent distribution of a singular foliation F ,then the leaves of F are maximal invariant submanifolds of maximum dimensionfor D. Definition 1.5.2. An involutive distribution is a distribution D such that if X, Yare two arbitrary smooth vector fields which are tangent to D, then their Liebracket [X, Y ] is also tangent to D.

It is clear from Theorem 1.5.1 that if a singular distribution is integrable,then it is involutive. Conversely, for regular distributions we have:

Theorem 1.5.3 (Frobenius). If a smooth regular distribution is involutive then itis integrable, i.e., it is the tangent distribution of a regular foliation.

Proof (sketch). One can use Formula (1.50) to construct local invariant submani-folds of maximum dimension and then glue them together, just like in the proof ofTheorem 1.5.1. One can also see Theorem 1.5.3 as a special case of Theorem 1.5.1,by first showing that a regular involutive distribution is invariant with respect tothe family of all smooth vector fields which are tangent to it.

1.5. Singular symplectic foliations 19

Example 1.5.4. Consider the following singular foliation D on R2 with coordinates(x, y): D(x,y) = T(x,y)R

2 if x > 0, and D(x,y) is spanned by ∂∂x if x ≤ 0. Then D is

smooth involutive but not integrable.

The above example shows that, if in Frobenius’ theorem we omit the wordregular, then it is false. The reason is that, though Formula (1.50) still provides uswith local invariant submanifolds, they are not necessarily of maximum dimension.However, the situation in the finitely generated case is better. A smooth distribu-tion D on a manifold M is called locally finitely generated if for any x ∈ M thereis a neighborhood U of x such that the C∞(U)-module of smooth tangent vectorfields to D in U is finitely generated: there is a finite number of smooth vectorfields X1, . . . , Xn in U which are tangent to D, such that any smooth vector fieldY in U which is tangent to D can be written as Y =

∑ni=1 fiXi with fi ∈ C∞(U).

Theorem 1.5.5 (Hermann [171]). Any locally finitely generated smooth involutivedistribution on a smooth manifold is integrable.

See [366] for a simple proof of Theorem 1.5.5. Consider now a smooth Poisson manifold (M, Π). Denote by C its character-

istic distribution. Recall that

Cx = Imx = Xf(x), f ∈ C∞(M) ∀ x ∈M . (1.51)

Since the Hamiltonian vector fields preserve the Poisson structure, they also pre-serve the characteristic distribution. Thus, according to Stefan–Sussmann’s theo-rem, the characteristic foliation C is completely integrable and corresponds to asingular foliation, which we will denote by F = FΠ. For the reasons which willbecome clear below, this singular foliation is called the symplectic foliation of thePoisson manifold (M, Π).

For each point x ∈ M , denote by F(x) the leaf of F which contains x. Localcharts of F(x) are readily provided by Theorem 1.4.5: If

(p1, . . . , ps, q1, . . . , qs, z1, . . . , zm−2s)

is a local canonical system of coordinates at a point x ∈ M , then the submanifoldz1 = · · · = zm−2s = 0 is an open subset of F(x), and it has a natural symplecticstructure with Darboux coordinates (pi, qi). Notice that this symplectic structuredoes not depend on the choice of coordinates: at each point of z1 = · · · = zm−2s =0, it coincides with the symplectic form on the characteristic space. Thus, oneach leaf F(x) we have a unique natural symplectic structure, which at each pointcoincides with the symplectic form on the corresponding characteristic space. Italso follows from Assertions b), d) of Theorem 1.4.5 that the injection i : F →Mis a Poisson morphism: if f, g are two functions on M and y ∈ F(x), then

f, g(y) = f |F(x), g|F(x)x(y), (1.52)

where , x is the Poisson bracket of the symplectic form on F(x). In other words,we have:


Theorem 1.5.6 ([346]). Every leaf F(x) of the symplectic foliation FΠ of a Pois-son manifold (M, Π) is an immersed symplectic submanifold, the immersion beinga Poisson morphism. The Poisson structure Π is completely determined by thesymplectic structures on the leaves of FΠ.

Example 1.5.7. Symplectic foliation of linear Poisson structures. Let G be a con-nected Lie group, g its Lie algebra, and g∗ the dual of g. Recall that g∗ has a naturallinear Poisson structure, also known as the Lie–Poisson structure, defined by

f, h(α) = 〈α, [df(α), dh(α)]〉. (1.53)

Denote the neutral element of G by e, and identify g with TeG. G acts on g bythe adjoint action Adg(x) = (u → gug−1)∗e(x) and on g∗ by the coadjoint action(the induced dual action) Ad∗

g(α)(x) = α(Adg−1x), α ∈ g∗, x ∈ g, g ∈ G. Thisaction is generated infinitesimally by the coadjoint action of g on g∗ defined byad∗x(α)(y) = 〈α, [y, x]〉 = −〈α, adxy〉.Theorem 1.5.8. The symplectic leaves of the Lie–Poisson structure on the dual ofan arbitrary finite-dimensional Lie algebra coincide with the orbits of the coadjointrepresentation on it.

Proof. Due to the Leibniz rule, the tangent spaces to the symplectic leaves, i.e.,the characteristic spaces, are generated by the Hamiltonian vector fields of linearfunctions. If f, h are two linear functions on g∗, also considered as two elementsof g by duality, and α is a point of g∗, then we have

Xf (h)(α) = 〈α, [f, h]〉 = −〈ad∗f (α), h〉. (1.54)

It implies that the tangent spaces of symplectic leaves are the same as the tangentspaces of coadjoint orbits. It follows that coadjoint orbits are open closed subsetsof symplectic leaves, so they coincide with symplectic leaves because symplecticleaves are connected by definition.

A corollary of the above theorem is that the orbits of the coadjoint represen-tation of a finite-dimensional Lie algebra are of even dimension and equipped witha natural symplectic form. This symplectic form is also known as the Kirillov–Kostant–Souriau form. Let us mention that coadjoint orbits play a very impor-tant role in the theory of unitary representations of Lie groups (the so-called orbitmethod), see, e.g., [193].

Exercise 1.5.9. Describe the symplectic leaves of so∗(3) and sl∗(2).

Remark 1.5.10. A direct way to define the symplectic foliation of a Poisson man-ifold (M, Π) is as follows: two points x, y are said to belong to the same leaf ifthey can be connected by a piecewise-smooth curve consisting of integral curvesof Hamiltonian vector fields. Then it is a direct consequence of the splitting The-orem 1.4.5 that the corresponding partition of M into leaves satisfies the localfoliation property. Thus, in fact, we can use the splitting Theorem 1.4.5 instead

1.6. Transverse Poisson structures 21

of the Stefan–Sussmann Theorem 1.5.1 in order to show that on (M, Π) thereis a natural associated foliation whose tangent distribution is the characteristicdistribution.

1.6 Transverse Poisson structures

Let N be a smooth local (i.e., sufficiently small) disk of dimension m − 2s ofan m-dimensional Poisson manifold (M, Π), which intersects transversally a 2s-dimensional leaf F(x) of the symplectic foliation F of (M, Π) at a point x. Inother words, N contains x and is transversal to the characteristic space Cx. Thenaccording to the splitting Theorem 1.4.5, there are local canonical coordinates ina neighborhood of x, which will define on N a Poisson structure. This Poissonstructure on N is called the transverse Poisson structure at x of the Poissonmanifold (M, Π).

To justify the above definition of transverse Poisson structures, we must showthat the Poisson structure on N given by Theorem 1.4.5 does not depend on thechoice of local canonical coordinates, nor on the choice of N itself, modulo localPoisson diffeomorphisms.

Theorem 1.6.1. With the above notations, we have:a) The local Poisson structure on N given by Theorem 1.4.5 does not depend on

the choice of local canonical coordinates.b) If x0 and x1 are two points on the symplectic leaf F(x), and N0 and N1 are

two smooth local disks of dimension m−2s which intersect F(x) transversallyat x0 and x1 respectively, then there is a smooth local Poisson diffeomorphismfrom (N0, x0) to (N1, x1).

Proof. a) Theorem 1.4.5 implies that the local symplectic leaves near point x aredirect products of the symplectic leaves of a neighborhood of x in N with the localsymplectic manifold (p1, . . . , ps, q1, . . . , qs). In particular, the symplectic leavesof N are connected components of intersections of the symplectic leaves of Mwith N , and the symplectic form on the symplectic leaves of N is the restrictionof the symplectic form of the leaves of M to those intersections. This geometriccharacterization of the symplectic leaves of N and their corresponding symplecticforms shows that they do not depend on the choice of local canonical coordinates.Hence, according to Theorem 1.5.6, the Poisson structure of N does not dependon the choice of local canonical coordinates.

b) Let N0 and N1 be two local disks which intersect a symplectic leaf F(x)transversally at x0 and x1 respectively. Then there is a smooth one-dimensionalfamily of local submanifolds Nt (0 ≤ t ≤ 1), connecting N0 to N1, such that Nt

intersects Fα transversely at a point xt. Point xt depends smoothly on t. Accordingto (a parameterized version of) Theorem 1.4.5, there is a smooth family of localfunctions

(pt1, . . . , p

ts, q

t1, . . . , q

ts, z

t1, . . . , z

tm−2s),


such that for each t ∈ [0, 1], (pt1, . . . , p

ts, q

t1, . . . , q

ts, z

t1, . . . , z

tm−2s) is a local canoni-

cal system of coordinates for (M, Π) in a neighborhood of xt such that pti(Nt) =

qti(Nt) = 0.

For each point y ∈ Nt close enough to xt, define a tangent vector Yt(y) ∈TyM as follows: For each τ near t, the disk Nτ intersects the local submanifoldzt

1 = zt1(y), . . . , zt

m−2s = ztm−2s(y) transversally at a unique point yτ . The map

τ → yτ is smooth. Vector Yt(y) is defined to be the derivation of this map at τ = t.In particular, Yt(xt) is the derivation of the map τ → xτ at τ = t.

There is a unique cotangent vector βt(y) ∈ T ∗y M such that βt(y) annulates

TyNt (the tangent space of Nt at y), and βt(y) = Yt(y). For each t ∈ [0, 1] wecan choose a function ft defined in a neighborhood of Nt, in such a way that ft

depends smoothly on t, and that dft(y) = βt(y) ∀ y ∈ Nt. It implies that we haveXt(y) = Yt(y) ∀ y ∈ Nt, where Xt = Xft denotes the Hamiltonian vector fieldof ft.

Denote by ϕt the local flow of the time-dependent Hamiltonian vector fieldXt (where t is considered as the time variable). Then of course ϕt preserves thePoisson structure of V (wherever ϕt is defined). From the construction of Xt, wealso see that ϕt moves x0 to xt, and it moves a sufficiently small neighborhood of x0

in N0 into Nt. In particular, ϕ1 defines a local diffeomorphism from N0 to N1. Sinceϕ1 preserves the Poisson structure of M , it also preserves the Poisson structureof N0. (As explained in the first part of this theorem, the Poisson structure ofN0 depends only on Π and N0, and does not depend on other things like localcanonical coordinates.) In other words, ϕ1 defines a local Poisson diffeomorphismfrom (N0, x0) to (N1, x1).

In practice, the transverse Poisson structure may be calculated by the follow-ing so-called Dirac’s constrained bracket formula, or Dirac’s formula1 for short.

Proposition 1.6.2 (Dirac’s formula). Let N be a local submanifold of a Poissonmanifold (M, Π) which intersects a symplectic leaf transversely at a point z. Letψ1, . . . , ψ2s, where 2s = rankΠ(z), be functions in a neighborhood U of z such that

N = x ∈ U | ψi(x) = constant. (1.55)

Denote by Pij = ψi, ψj and by (P ij) the inverse matrix of (Pij)2si,j=1. Then the

bracket formula for the transverse Poisson structure on N is given as follows:

f, gN(x) = f , g(x)−2s∑

i,j=1

f , ψi(x)P ij(x)ψj , g(x) ∀ x ∈ N, (1.56)

where f, g are functions on N and f , g are extensions of f and g to U . The aboveformula is independent of the choice of extensions f and g.

1According to Weinstein [347], Dirac’s formula was actually found by T. Courant and R. Mont-gomery, who generalized a constraint procedure of Dirac.

1.7. Group actions and reduction 23

Proof (sketch). If one replaces f by ψk (∀ k = 1, . . . , 2s) in the above formula,then the right-hand side vanishes. If f and f are two extensions of f , then we canwrite f = f +

∑2si=1(ψi − ψi(z))hi. Using the Leibniz rule, one verifies that the

right-hand side in Formula (1.56) does not depend on the choice of f . By anti-symmetricity, the right-hand side does not depend on the choice of g either. Finally,we can choose f and g to be independent of pi, qi in a canonical coordinate system(p1, . . . , ps, q1, . . . , qs, z1, . . . , zm−2s) provided by the splitting Theorem 1.4.5. Forthat particular choice we have f , ψi(x) = 0 and f , g(x) = f, gN(x).

1.7 Group actions and reduction

Recall that, a left action of a Lie group G on manifold M is a (smooth) map : G×M →M such that

(e, z) = z ∀ z ∈ M, (1.57)

where e denotes the neutral element of G, and

(gh, z) = (g, (h, z)) ∀ g, h ∈ G, z ∈ M. (1.58)

Similarly, a right action of a Lie group G on a manifold M is a map ρ :M ×G→M such that ρ(z, e) = z ∀ z ∈ M and

ρ(z, gh) = ρ(ρ(z, g), h) ∀ g, h ∈ G, z ∈ M. (1.59)

If we write the right action ρ(g)(z) of g on z as z.g, then we have z.(gh) = (z.g).h.Sometimes, for convenience, a right action will also be denoted as ρ : G×M →M .

If ρ is a right action, then ρ(g, z) := ρ(z, g−1) is a left action, and vice versa.So left actions and right actions are essentially the same. A (left) action of G onM may also be defined as a homomorphism ρ : G → Diff(M), ρ(g) := ρ(g, .),from G to the group of diffeomorphisms of M .

Recall that, a Lie algebra action of a Lie algebra g on a manifold M is a linearmap ξ : g → V1(M) from g to the space of vector fields on M , which preservesthe Lie bracket, i.e., [ξ(x), ξ(y)] = ξ([x, y]) ∀ x, y ∈ g. In other words, ξ is a Liehomomorphism from g to V1(M).

Example 1.7.1. An action of a Lie algebra g on a vector space V is a map η :g× V → V such that

η([x, y])(v) = η(x)(η(y)(v)) − η(y)(η(x)(v)).

If η is an action of g on a vector space V , then the corresponding action of g onV , considered as a manifold, is given by

−η : (x, v) → −η(x)(v) , (1.60)

where η(x)(v) is considered as a tangent vector at v to TvV . (Notice the minussign there.)


Let ρ : M ×G→M be a smooth right action of a Lie group G on a manifoldM . For each x ∈ g, define ξx to be the generator of the one-dimensional groupaction ρ(exp(tx)) on M . In other words, ρ(exp(tx)) is the time-t flow of the vectorfield ξx: ρ(exp(tx)) = exp(tξx). Then the map ξ : g → V1(M), ξ(x) = ξx, is a Liealgebra action of g on M .

If a left action : G×M → M is said to be generated by a Lie algebra actionξ : g → V1(M) of g, then it means that (exp(x), z) = exp(−ξ(x))(z) ∀ x ∈ g,z ∈M .

Motivated in particular by the fact that Hamiltonian systems and their phasespaces often admit symmetries, we are interested in Lie group and Lie algebraactions on Poisson manifolds M . If a connected Lie group G acts on a Poissonmanifold (M, Π), in such a way that the action preserves the Poisson structure Π,then the space C∞(M)G of G-invariant functions on M is a Poisson algebra in thefollowing natural sense:

Definition 1.7.2. An associative commutative algebra A together with an anti-symmetric bracket , : A × A → is called a Poisson algebra if , satisfies theLeibniz identity f, gh = f, gh + gf, h and the Jacobi identity f, g, h+g, h, f+ h, f, g = 0 ∀ f, g, h ∈ A.

Recall that, an action of a Lie group G on a manifold M is called a properaction if the corresponding map ρ : G ×M → M is a proper map, i.e., for anycompact K ⊂ M its preimage ρ−1(K) is also compact. In particular, if G is acompact Lie group, then any action of G is automatically proper. Assume thatwe have a free and proper action of a Lie group G on a manifold M . Then thequotient M/G of M by the action of G is a manifold, and C∞(M) can be naturallyidentified with C∞(M)G via the pull-back of the projection map M → M/G.Hence M/G has a natural reduced Poisson structure, for which the projectionmap M → M/G is a Poisson map. Note that dim M/G = dim M − dim G in thiscase. If h ∈ C∞(M)G is a G-invariant function on M , and h its projection onM/G, then the Hamiltonian vector field Xh projects to the Hamiltonian vectorfield Xh under the projection M →M/G. The Hamiltonian system of h on M/Gis called the reduced Hamiltonian system of the Hamiltonian system of h on M .

Example 1.7.3. Consider the left action of a connected Lie group G on T ∗G.Then, according to Theorem 1.3.10, the corresponding reduced Poisson manifoldis isomorphic to g∗, with the left translation map L : T ∗G→ g∗ as the projectionmap. For example, consider the Euler top, i.e., the rotational movement of a rigidbody around its fixed center of gravity. The configuration space is SO(3) (the spaceof all possible rotational positions), and the corresponding Hamiltonian system isa system on T ∗SO(3), which can be written in reduced form as a Hamiltoniansystem on so∗(3). A Hamiltonian system on the dual g∗ of a Lie algebra with thecorresponding linear Poisson structure is often called an Euler equation.

A particularly interesting class of actions on Poisson manifolds consists ofthe so-called Hamiltonian actions.

1.7. Group actions and reduction 25

Definition 1.7.4. A Hamiltonian action of a Lie algebra g on a Poisson manifold(M, Π) is a Lie algebra action ξ : g → V1(M) which is induced from a Poissonmorphism µ : (M, Π) → g∗ (where g∗ is equipped with the standard Lie–Poissonstructure) by the following formula:

ξ(x) = Xµ∗(x) ∀ x ∈ g, (1.61)

where on the right-hand side of the above equation x is considered as a linearfunction on g∗, and Xµ∗(x) denotes the Hamiltonian vector field of µ∗(x). ThisPoisson morphism µ : (M, Π) → g∗ is called an equivariant momentum map, ormomentum map2 for short. A left (resp. right) Hamiltonian action of a connectedLie group G on a Poisson manifold (M, Π) is a left (resp. right) action of G on Mwhose corresponding Lie algebra action is a Hamiltonian action.

Remark 1.7.5. If µ : (M, Π) → g∗ is a Poisson morphism, then it automaticallygives rise to a Hamiltonian action of g on M by Formula (1.61). The adjectiveequivariant in the above definition is due to the fact that the momentum mapµ : (M, Π) → g∗ is a Poisson map if and only if it intertwines the (left) action ofG on M with the coadjoint action of G on g∗. The proof of this fact will be leftto the reader as an exercise.Remark 1.7.6. It may happen that a map µ : (M, Π) → g∗ is not Poisson, butstill defines a Lie algebra action ξ : g → V1(M) by Formula (1.61). In that case ξis called a weakly Hamiltonian action, and µ a non-equivariant momentum map.

Example 1.7.7. Consider the right action of a connected Lie group G on itself.Then its associated Lie algebra action of g on G is given by left-invariant vectorfields. The right action of G on itself can be lifted to a right Hamiltonian action ofG on T ∗G, whose momentum map is the natural Poisson morphism L : T ∗G → g∗

given in Theorem 1.3.10. Thus, the map L : T ∗G → g∗ is at the same time theprojection map for the left action of G and the momentum map for the right actionof G on T ∗G.

Example 1.7.8. If a connected Lie group G acts on a manifold N on the right, thenits natural lifting to T ∗N is a Hamiltonian action on T ∗N , whose momentum mapµ : T ∗N → g∗ is defined as follows:

〈µ(p), x〉 = 〈p, ξx(π(p))〉, (1.62)

where p ∈ T ∗N , x ∈ g, π : T ∗N → N is the projection, ξx is the vector field onN which generates the action of the one-dimensional group exp(tx). This examplegeneralizes the previous example.

If the action of G on M is Hamiltonian with a given equivariant momentummap µ : M → g∗, then µ−1(0) is invariant under the action of G, and we can formanother reduced Poisson space, denoted by M//G, as follows:

M//G = µ−1(0)/G. (1.63)2The terminology momentum map is the English translation of the French term application

de moment introduced by Souriau [319]. Some people prefer to call it moment map.


Theorem 1.7.9. With the above notations, if the action of G on µ−1(0) is free andproper, then M//G = µ−1(0)/G has a unique natural Poisson structure comingfrom the Poisson structure of M , called the reduced Poisson structure.

Proof. Since the action of G on µ−1(0) is free, its infinitesimal action is also free,i.e., ξx(z) = 0 ∀ 0 = x ∈ g, z ∈ µ−1(0), where ξ : g → V−1(M) denotes thecorresponding action of g. It implies that dfµ(z) : TzM → Tµ(z)g

∗ is surjectivefor any z ∈ µ−1(0). In particular, 0 is a regular value of the momentum map, andµ−1(0) is a closed submanifold of M . Let f, h be any two functions on µ−1(0)/G,viewed also as G-invariant functions on µ−1(0). Extend them to two G-invariantfunctions f , h on a neighborhood of µ−1(0) in M . Since the Poisson structureon M is G-invariant, the Poisson bracket f , h is also G-invariant. Similarly tothe proof of Dirac’s formula (1.56), it is easy to see that the restriction of f , hto µ−1(0) depends only on f, h but not on the extensions f , h. We can definethe Poisson bracket of f, h on µ−1(0)/G to be the projection of f , h|µ−1(0) toµ−1(0)/G. Exercise 1.7.10. Show that (assuming that the action of G is Hamiltonian, properand free) the inclusion map M//G→M/G is a Poisson morphism. When M is asymplectic manifold, then M//G is a symplectic leaf of M/G.

Remark 1.7.11. The Poisson reduction M//G is also known as the Marsden–Ratiureduction [238]. When M is symplectic, it is known as the Marsden–Weinstein–Meyer reduction [237, 247].

Example 1.7.12 ([348]). If G is a Lie group, B is a principal G-bundle over amanifold X (i.e., G acts on B freely on the right and B/G = X), and M is aPoisson manifold together with a Hamiltonian G-action, then the triple (G, B, M)is called a classical Yang–Mills–Higgs setup. Given a classical Yang–Mills–Higgssetup (G, B, M), the product Hamiltonian action of G on T ∗B × M is freeand proper, so by Theorem 1.7.9, we can form the reduced Poisson manifoldY(G, B, M) = (T ∗B × M)//G. This Poisson manifold Y(G, B, M) is called theYang–Mills–Higgs phase space for a classical particle with configuration space X ,gauge group G and internal phase space M .

When the action of G is proper but not free, the reduced spaces M/G andM//G are no longer manifolds in general, but they are still Hausdorff spaces withthe structure of a stratified manifold. We will not go into the details of this so-called singular reduction theory (see, e.g., [317, 23, 287, 90] and references therein).We refer the reader to the book by Ortega and Ratiu [288] for a comprehensivetreatment of reduction theory.

1.8. The Schouten bracket 27

1.8 The Schouten bracket

1.8.1 Schouten bracket of multi-vector fields

Recall that, if A =∑

i ai∂

∂xiand B =

∑i bi

∂∂xi

are two vector fields written in alocal system of coordinates (x1, . . . , xn), then the Lie bracket of A and B is

[A, B] =∑

i

ai

(∑j

∂bj

∂xi

∂

∂xj

)−∑

i

bi

(∑j

∂aj

∂xi

∂

∂xj

). (1.64)

We will redenote ∂∂xi

by ζi and consider them as formal, or odd variables3

(formal in the sense that they don’t take values in a field, but still form an algebra,and odd in the sense that ζiζj = −ζjζi, i.e., ∂

∂xj∧ ∂

∂xj= − ∂

∂xj∧ ∂

∂xi). We can write

A =∑

i aiζi and B =∑

i biζi and consider them formally as functions of variables(xi, ζi) which are linear in the odd variables (ζi). We can write [A, B] formally as

[A, B] =∑

i

∂A

∂ζi

∂B

∂xi−∑

i

∂B

∂ζi

∂A

∂xi. (1.65)

The above formula makes the Lie bracket of two vector fields look pretty muchlike the Poisson bracket of two functions in a Darboux coordinate system.

Now if Π =∑

i1<···<ipΠi1...ip

∂∂xi1

∧ · · · ∧ ∂∂xip

is a p-vector field, then we willconsider it as a homogeneous polynomial of degree p in the odd variables (ζi):

Π =∑

i1<···<ip

Πi1...ipζi1 . . . ζip . (1.66)

It is important to remember that the variables ζi do not commute. In fact,they anti-commute among themselves, and commute with the variables xi:

ζiζj = −ζjζi; xiζj = ζjxi; xixj = xjxi. (1.67)

Due to the anti-commutativity of (ζi), one must be careful about the signswhen dealing with multiplications and differentiations involving these odd vari-ables. The differentiation rule that we will adopt is as follows:

∂(ζi1 . . . ζip)∂ζip

:= ζi1 . . . ζip−1 . (1.68)

Equivalently,∂(ζi1 ...ζip )

∂ζik

= (−1)p−kζi1 . . . ζik. . . ζip , where the hat means that ζik

is missing in the product (1 ≤ k ≤ p).

3The name odd variable comes from the theory of supermanifolds, though it is not necessaryto know what a supermanifold is in order to understand this section.


If

A =∑

i1<···<ia

Ai1,...,ia

∂

∂xi1

∧ · · · ∧ ∂

∂xia

=∑

i1,...,ia

Ai1,...,iaζi1 . . . ζia (1.69)

is an a-vector field, and

B =∑

i1<···<ib

Bi1,...,ib

∂

∂xi1

∧ · · · ∧ ∂

∂xib

=∑

i1,...,ia

Bi1,...,iaζi1 . . . ζib(1.70)

is a b-vector field, then generalizing Formula (1.65), we can define a bracket of Aand B as follows:

[A, B] =∑

i

∂A

∂ζi

∂B

∂xi− (−1)(a−1)(b−1)

∑i

∂B

∂ζi

∂A

∂xi. (1.71)

Clearly, the bracket [A, B] of A and B as defined above is a homogeneouspolynomial of degree a+ b−1 in the odd variables (ζi), so it is a (a+ b−1)-vectorfield.

Theorem 1.8.1 (Schouten–Nijenhuis). The bracket defined by Formula (1.71) sat-isfies the following properties:

a) Graded anti-commutativity: if A is an a-vector field and B is a b-vector fieldthen

[A, B] = −(−1)(a−1)(b−1)[B, A] . (1.72)

b) Graded Leibniz rule: if A is an a-vector field, B is a b-vector field and C isa c-vector field then

[A, B ∧C] = [A, B] ∧ C + (−1)(a−1)bB ∧ [A, C] , (1.73)

[A ∧B, C] = A ∧ [B, C] + (−1)(c−1)b[A, C] ∧B . (1.74)

c) Graded Jacobi identity:

(−1)(a−1)(c−1)[A, [B, C]] + (−1)(b−1)(a−1)[B, [C, A]]

+ (−1)(c−1)(b−1)[C, [A, B]] = 0 . (1.75)

d) If A = X is a vector field then

[X, B] = LXB , (1.76)

where LX denotes the Lie derivative by X. In particular, if A and B are twovector fields, then the Schouten bracket of A and B coincides with their Liebracket. If A = X is a vector field and B = f is a function (i.e., a 0-vectorfield), then we have

[X, f ] = X(f) = 〈df, X〉 . (1.77)


Proof. Assertion a) follows directly from the definition.b) The differentiation rule (1.68) implies that

∂(B ∧ C)∂ζi

= B∂C

∂ζi+ (−1)c ∂B

∂ζiC.

Hence we have

[A, B ∧C] =∑ ∂A

∂ζi

∂(B ∧ C)∂xi

− (−1)(a−1)(b+c−1)∑ ∂(B ∧C)

∂ζi

∂A

∂xi

=∑ ∂A

∂ζi

∂B

∂xiC +

∑ ∂A

∂ζiB

∂C

∂xi− (−1)(a−1)(b+c−1)

∑B

∂C

∂ζi

∂A

∂xi

−(−1)(a−1)(b+c−1)+c∑ ∂B

∂ζiC

∂A

∂xi

=∑ ∂A

∂ζi

∂B

∂xiC − (−1)(a−1)(b+c−1)+c+ac

∑ ∂B

∂ζi

∂A

∂xiC

+(−1)(a−1)b

(−(−1)(a−1)(c−1)

∑B

∂C

∂ζi

∂A

∂xi+∑

B∂A

∂ζi

∂C

∂xi

)= [A, B] ∧ C + (−1)(a−1)bB ∧ [A, C].

The proof of Formula (1.74) is similar.c) By direct calculations we have

(−1)(a−1)(c−1)[A, [B, C]] = S1 + S2 + S3 + S4 ,

where

S1 = (−1)(a−1)(c−1)∑i,j

∂A

∂ζj

∂2B

∂xj∂ζi

∂C

∂xi− (−1)(a−1)(b−1)

∑i,j

∂B

∂ζi

∂2C

∂xi∂ζj

∂A

∂xj,

S2 = (−1)(a−1)(c−1)∑i,j

∂A

∂ζj

∂B

∂ζi

∂2C

∂xi∂xj− (−1)(c−1)(b−1)

∑i,j

∂C

∂ζi

∂A

∂ζj

∂2B

∂xi∂xj,

S3 = (−1)(b−1)(a−1)∑i,j

∂2B

∂ζj∂xi

∂C

∂ζi

∂A

∂xj− (−1)(c−1)(b−1)

∑i,j

∂2C

∂ζi∂xj

∂A

∂ζj

∂B

∂xi,

S4 = (−1)(b−1)(a+c)+b∑i,j

∂2C

∂ζj∂ζi

∂B

∂xi

∂A

∂xj− (−1)(a−1)(b−1)+c

∑i,j

∂2B

∂ζj∂ζi

∂C

∂xi

∂A

∂xj

= (−1)(b−1)(c−1)+a∑i,j

∂2C

∂ζj∂ζi

∂A

∂xi

∂B

∂xj− (−1)(a−1)(b−1)+c

∑i,j

∂2B

∂ζj∂ζi

∂C

∂xi

∂A

∂xj

(because ∂2C∂ζi∂ζj

= − ∂2C∂ζj∂ζi

).Each of the summands S1, S2, S3, S4 will become zero when adding similar

terms from (−1)(b−1)(a−1)[B, [C, A]] and (−1)(c−1)(b−1)[C, [A, B]].


d) If f is a function and X =∑

i ξi∂

∂xiis a vector field, then ∂f

∂ζi= 0,

and [X, f ] =∑

∂X∂ζi

∂f∂xi

=∑

ξi∂f∂xi

= X(f). When A and B are vector fields,Formula (1.71) clearly coincides with Formula 1.65. When B is a multi-vectorfield, Assertion d) can be proved by induction on the degree of B, using theLeibniz rules given by Assertion b).

A priori, the bracket of an a-vector field A with a b-vector field B, as definedby Formula (1.71), may depend on the choice of local coordinates (x1, . . . , xn).However, the Leibniz rules (1.73) and (1.74) show that the computation of [A, B]can be reduced to the computation of the Lie brackets of vector fields. Since theLie bracket of vector fields does not depend on the choice of local coordinates, itfollows that the bracket [A, B] is in fact a well-defined (a+b−1)-vector field whichdoes not depend on the choice of local coordinates.

Definition 1.8.2. If A is an a-vector field and B is a b-vector field, then the uniquelydefined (a+ b−1)-vector field [A, B], given by Formula (1.71) in each local systemof coordinates, is called the Schouten bracket of A and B.

Remark 1.8.3. Our sign convention in the definition of the Schouten bracket is thesame as Koszul’s [201], but different from Vaisman’s [333] and some other authors.

The Schouten bracket was first discovered by Schouten [311, 312]. Theorem1.8.1 is essentially due to Schouten [311, 312] and Nijenhuis [280]. The gradedJacobi identity (1.75) means that the Schouten bracket is a graded Lie bracket:the space V(M) =

⊕p≥0 Vp(M), where Vp(M) is the space of smooth p-vector

fields on a manifold M , is a graded Lie algebra, also known as Lie super-algebra,under the Schouten bracket, if we define the grade of Vp(M) to be p− 1. In otherwords, we have to shift the natural grading by −1 for V(M) together with theSchouten bracket to become a graded Lie algebra in the usual sense.

Another equivalent definition of the Schouten bracket, due to Lichnerowicz[211], is as follows. If A is an a-vector field, B is a b-vector field, and η is an(a + b− 1)-form then

〈η, [A, B]〉 = (−1)(a−1)(b−1)〈d(iBη), A〉 − 〈d(iAη), B〉+ (−1)a〈dη, A ∧B〉. (1.78)

In this formula, iA : Ω(M) → Ω(M) denotes the inner product of differentialforms with A, i.e., 〈iAβ, C〉 = 〈β, A∧C〉 for any k-form β and (k− a)-vector fieldC. If k < a then iAβ = 0.

More generally, we have the following useful formula, due to Koszul [201]4.

Lemma 1.8.4. For any A ∈ Va(M), B ∈ Vb(M) we have

i[A,B] = (−1)(a−1)(b−1)iA d iB − iB d iA

+ (−1)aiA∧B d + (−1)bd iA∧B. (1.79)

4Koszul [201] wrote (1.79) as i[A,B] = [[iA,d], iB ]. But the brackets on the right-hand sidemust be understood as graded commutators of graded endomorphisms of Ω(M), not usualcommutators.


Proof. By induction, using the Leibniz rule.

Yet another equivalent definition of the Schouten bracket, via the so-calledcurl operator, will be given in Section 2.6.

The Schouten bracket offers a very convenient way to characterize Poissonstructures and Hamiltonian vector fields:

Theorem 1.8.5. A 2-vector field Π is a Poisson tensor if and only if the Schoutenbracket of Π with itself vanishes:

[Π, Π] = 0 . (1.80)

If Π is a Poisson tensor and f is a function, then the corresponding Hamiltonianvector field Xf satisfies the equation

Xf = −[Π, f ] . (1.81)

Proof. It follows directly from Formula (1.71) that Equation (1.80), when ex-pressed in local coordinates, is the same as Equation (1.28). Thus the first partof the above theorem is a consequence of Proposition 1.2.8. The second partalso follows directly from Formula (1.71) and the definition of Xf : −[Π, f ] =−[∑

i<j Πij∂

∂xi∧ ∂

∂xj, f ] = −

∑i,j Πij

∂∂xi

∂f∂xj

=∑

i,j∂f∂xi

Πij∂

∂xj= Xf .

By abuse of language, we will call Equation (1.80) the Jacobi identity, becauseit is equivalent to the usual Jacobi identity (1.2).

Exercise 1.8.6. Let Π be a smooth Poisson tensor on a manifold M . Using Theorem1.8.5, show that the following two statements are equivalent: a) rankΠ ≤ 2; b) fΠis a Poisson tensor for every smooth function f on M .

Exercise 1.8.7. Show that, if Λ is a p-vector field on a Poisson manifold (M, Π),then the Schouten bracket [Π, Λ] can be given, in terms of multi-derivations, asfollows:

[Π, Λ](f1, . . . , fp+1) =p+1∑i=1

(−1)i+1fi, Λ(f1, . . . fi . . . , fp+1)

+∑i<j

(−1)i+jΛ(fi, fj, f1, . . . fi . . . fj . . . , fp+1), (1.82)

where the hat over fi and fj means that these terms are missing in the expression.(Hint: one can use Formula (1.78)).

1.8.2 Schouten bracket on Lie algebras

The Schouten bracket on V(M) extends the Lie bracket on V1(M) by the gradedLeibniz rule. Similarly, by the graded Leibniz rule (1.73,1.74), we can extend


the Lie bracket on any Lie algebra g to a natural graded Lie bracket on ∧g =⊕∞k=0 ∧kg, where ∧kg means g ∧ · · · ∧ g (k times), which will also be called the

Schouten bracket . More precisely, we have:

Lemma 1.8.8. Given a Lie algebra g over K, there is a unique bracket on ∧g =⊕∞

k=0 ∧k g which extends the Lie bracket on g and which satisfies the followingproperties, ∀ A ∈ ∧ag, B ∈ ∧bg, C ∈ ∧cg:

a) Graded anti-commutativity:

[A, B] = −(−1)(a−1)(b−1)[B, A]. (1.83)

b) Graded Leibniz rule:

[A, B ∧ C] = [A, B] ∧ C + (−1)(a−1)bB ∧ [A, C], (1.84)

[A ∧B, C] = A ∧ [B, C] + (−1)(c−1)b[A, C] ∧B. (1.85)

c) Graded Jacobi identity:

(−1)(a−1)(c−1)[A, [B, C]] + (−1)(b−1)(a−1)[B, [C, A]]+(−1)(c−1)(b−1)[C, [A, B]] = 0.

(1.86)

d) The bracket of any element in ∧g with an element in ∧0g = K is zero.

Proof. The proof is straightforward and is left to the reader as an exercise. Remarkthat, another equivalent way to define the Schouten bracket on ∧g is to identify∧g with the space of left-invariant multi-vector fields on G, where G is the simply-connected Lie group whose Lie algebra is g, then restrict the Schouten bracket onV(G) to these left-invariant multi-vector fields.

If ξ : g → V1(M) is an action of a Lie algebra G on a manifold M , then itcan be extended in a unique way by wedge product to a map

∧ξ : ∧g → V(M).

For example, if x, y ∈ g then ∧ξ(x ∧ y) = ξ(x) ∧ ξ(y).

Lemma 1.8.9. If ξ : g→ V1(M) is a Lie algebra homomorphism, then its extension∧ξ : ∧g → V(M) preserves the Schouten bracket, i.e.,

∧ξ([α, β]) = [∧ξ(α),∧ξ(β)] ∀ α, β ∈ ∧g.

Proof. The proof is straightforward, by induction, based on the Leibniz rule. Notation 1.8.10. For an element α ∈ ∧g, we will denote by α+ the left-invariantmulti-vector field on G whose value at the neutral element e of G is α, i.e., α+(g) =Lgα, where Lg means the left translation by g. Similarly, α− denotes the right-invariant multi-vector field α−(g) = Rgα, where Rg means the right translationby g.


As a direct consequence of Lemma 1.8.9, we have:

Theorem 1.8.11. For an element r ∈ g∧g, where g is the Lie algebra of a connectedLie group G, the following three conditions are equivalent:

a) r satisfies the equation [r, r] = 0,b) r+ is a left-invariant Poisson structure on G,c) r− is a right-invariant Poisson structure on G.

Proof. Obvious.

The equation [r, r] = 0 is called the classical Yang–Baxter equation5 [141], orCYBE for short. This equation will be discussed in more detail in Chapter 5.

Example 1.8.12. If x, y ∈ g such that [x, y] = 0 and x ∧ y = 0, then r = x ∧ ysatisfies the classical Yang–Baxter equation, and the corresponding left- and right-invariant Poisson structures on G have rank 2.

1.8.3 Compatible Poisson structures

Definition 1.8.13. Two Poisson tensors Π1 and Π2 are called compatible if theirSchouten bracket vanishes:

[Π1, Π2] = 0. (1.87)

Another equivalent definition is: two Poisson structures Π1 and Π2 are com-patible if Π1 +Π2 is also a Poisson structure. Indeed, we have [Π1 +Π2, Π1 +Π2] =[Π1, Π1] + [Π2, Π2] + 2[Π1, Π2] = 2[Π1, Π2], provided that [Π1, Π1] = [Π2, Π2] = 0.So Equation (1.87) is equivalent to [Π1 + Π2, Π1 + Π2] = 0.

If Π1 and Π2 are two compatible Poisson structures, then we have a wholetwo-dimensional family of compatible Poisson structures (or projective one-dim-ensional family): for any scalars c1 and c2, c1Π1 + c2Π2 is a Poisson structure.Such a family of Poisson structures is often called a pencil of Poisson structures .

Example 1.8.14. The linear Poisson structure x1∂

∂x2∧ ∂

∂x3+x2

∂∂x3∧ ∂

∂x1+x3

∂∂x1∧ ∂

∂x2

on so∗(3) = R3 can be decomposed into the sum of two compatible linear Poissonstructures (x1

∂∂x2

− x2∂

∂x1) ∧ ∂

∂x3and x3

∂∂x1

∧ ∂∂x2

.

Example 1.8.15. If r1, r2 ∈ g ∧ g are solutions of the CYBE [r, r] = 0, then r+1

and r−2 form a pair of compatible Poisson structures on G, where G is a Lie groupwhose Lie algebra is g.

Example 1.8.16 ([253]). On the dual g∗ of a Lie algebra g, besides the standard Lie–Poisson structure f, gLP (x) = 〈[df(x), dg(x)], x〉, consider the following constantPoisson structure:

f, ga(x) = 〈[df(x), dg(x)], a〉, (1.88)

5The Yang–Baxter equation has its origins in integrable models in statistical mechanics, andis one of the main tools in the study of integrable systems (see, e.g., [187]).


where a is a fixed element of g∗. This constant Poisson structure , a and theLie–Poisson structure , LP are compatible. In fact, their sum is the affine (i.e.,nonhomogeneous linear) Poisson structure

f, g(x) = 〈[df(x), dg(x)], x + a〉, (1.89)

which can be obtained from the linear Poisson structure , LP by the pull-backof the translation map x → x + a on g∗.

Exercise 1.8.17. Suppose that Π1 is a nondegenerate Poisson structure, i.e., itcorresponds to a symplectic form ω1. For a Poisson structure Π2, denote by ω2

the differential 2-form defined as follows:

ω2(X, Y ) = 〈Π2, iXω1 ∧ iY ω1〉 ∀ X, Y ∈ V1(M).

Show that [Π1, Π2] = 0 if and only if dω2 = 0.

Exercise 1.8.18 ([37]). Consider a complex pencil of holomorphic Poisson struc-tures λ1Π1 + λ2Π2, γ1, γ2 ∈ C. Let S be the set of points (γ1, γ2) ∈ C2 such thatthe rank of γ1Π1 + γ2Π2 is smaller than the rank of a generic Poisson structurein the pencil. Show that if (γ1, γ2), (δ1, δ2) ∈ C2 \ S are two arbitrary “regular”points of the pencil (which may coincide), f is a Casimir function for γ1Π1 +γ2Π2

and g is a Casimir function for δ1Π1 + δ2Π2, then f, gΠ1 = f, gΠ2 = 0.

Remark 1.8.19. A vector field X on a manifold is called a bi-Hamiltonian system ifit is Hamiltonian with respect to two compatible Poisson structures: X = XΠ1

H1=

XΠ2H2

. Bi-Hamiltonian systems often admit large sets of first integrals, which makethem into integrable Hamiltonian systems. Conversely, a vast majority of knownintegrable systems turn out to be bi-Hamiltonian. The theory of bi-Hamiltoniansystems starts with Magri [232] and Mischenko–Fomenko [253], and there is nowa very large amount of articles on the subject. See, e.g., [2, 19, 20, 38, 102] for anintroduction to the theory of integrable Hamiltonian systems.

1.9 Symplectic realizations

We have seen in Section 1.5 that Poisson manifolds can be viewed as singularfoliations by symplectic manifolds. In this section, we will discuss another way tolook at Poisson manifolds, namely as quotients of symplectic manifolds.

Definition 1.9.1. A symplectic realization of a Poisson manifold (P, Π) is a symplec-tic manifold (M, ω) together with a surjective Poisson submersion Φ : (M, ω) →(P, Π) (i.e., a submersion which is a Poisson map).

For example, Theorem 1.3.10 says that if G is a Lie group, then T ∗G togetherwith the left translation map L : T ∗G→ g∗ is a symplectic realization for g∗.

The existence of symplectic realization for arbitrary Poisson manifolds is animportant result due to Karasev [189] and Weinstein [349]:

1.9. Symplectic realizations 35

Theorem 1.9.2 (Karasev–Weinstein). Any smooth Poisson manifold of dimensionn admits a symplectic realization of dimension 2n.

In fact, the result of Karasev and Weinstein is stronger: any Poisson man-ifold can be realized by a local symplectic groupoid (see Section 8.8). In thissection, we will give a pedestrian proof of Theorem 1.9.2. First let us show a lo-cal version of it, which can be proved by an explicit formula. We will say thatΦ : (M, ω, L) → (P, Π) is a marked symplectic realization of (P, Π), where Lis a Lagrangian submanifold of M , if it is a symplectic realization such thatΦ|L : L → P is a diffeomorphism. Note that in this case we automatically havedimM = 2 dimL = 2 dimP .

Theorem 1.9.3 ([346]). Any point z of a smooth Poisson manifold (P, Π) has anopen neighborhood U such that (U, Π) admits a marked symplectic realization.

Proof. Denote by (x1, . . . , xn) a local system of coordinates at z. We will lookfor functions wi(x, y), i = 1, . . . , n, x = (x1, . . . , xn), viewed as functions in aneighborhood of z which depend smoothly on n parameters y = (y1, . . . , yn), suchthat wi(x, 0) = xi, and if we denote by xi = xi(w, y) (w = (w1, . . . , wn)) theinverse functions, then the map Θ : (w, y) → x(w, y) is a Poisson submersionfrom a symplectic manifold M with coordinates (w, y) and standard symplecticstructure ω =

∑i dwi∧dyi to a neighborhood (U, Π) of z. We may also view (x, y)

as a local coordinate system on M . The condition that Θ be a Poisson map canbe written as:

xi, xjω(x, y) = xi, xjΠ(x) (∀ i, j = 1, . . . , n), (1.90)

orn∑

h=1

( ∂xi

∂wh

∂xj

∂yh− ∂xi

∂yh

∂xj

∂wh

)= xi, xjΠ (∀ i, j = 1, . . . , n). (1.91)

Viewing the above equation as a matrix equation, and multiplying it by (∂wk

∂xi)k,i

on the left and (∂wl

∂xj)j,l on the right of each side, we get

(∂wk

∂xi

)k,i

(n∑

h=1

(∂xi

∂wh

∂xj

∂yh− ∂xi

∂yh

∂xj

∂wh

))ij

(∂wl

∂xj

)j,l

=(

∂wk

∂xi

)k,i

(xi, xjΠ)ij

(∂wl

∂xj

)j,l

, (1.92)

which means∂wl

∂yk− ∂wk

∂yl= wk, wlΠ (∀ k, l = 1, . . . , n). (1.93)

Equation (1.93) with the initial condition wi(x, 0) = xi has the following explicitlocal solution: denote by ϕt

y the local time-t flow of the local Hamiltonian vector


field Xfy of the local function fy =∑

i yixi on (P, Π). Then put (noting that ϕ1y

is well defined in a neighborhood of z when y is small enough)

wi(x, y) =∫ 1

0

xi ϕtydt . (1.94)

A straightforward computation, which will be left as an exercise (see [333, 346]),shows that this is a solution of (1.93). The local Lagrangian submanifold in ques-tion can be given by L = y = 0.

Proposition 1.9.4. If Φ1 : (M1, ω1, L1) → (P, Π) and Φ2 : (M2, ω2, L2) → (P, Π)are two marked symplectic realizations of a Poisson manifold (P, Π), then there isa unique symplectomorphism Ψ : U(L1) → U(L2) from a neighborhood U(L1) ofL1 in (M1, ω1) to a neighborhood U(L2) of L2 in (M2, ω2), which sends L1 to L2

and such that Φ1|U(L1) = Φ2|U(L2) Ψ.

Proof (sketch). Clearly, ψ = (Φ2|L2)−1 Φ1|L1 : L1 → L2 is a diffeomorphism.We want to extend it to a symplectomorphism Ψ from a neighborhood of L1 toa neighborhood of L2 which satisfies the conditions of the theorem. Let f be afunction on P . Then Ψ must send Φ∗

1f to Φ∗2f , hence it sends the Hamiltonian

vector field XΦ∗1f to XΦ∗

2f . If x1 ∈M1 is a point close enough to L1, then there isa point y1 ∈ L1 and a function f on P such that x1 = φ1

Φ∗1f (y1), where φt

g denotesthe time-t flow of the Hamiltonian vector field Xg of the function g, and we musthave

Ψ(x1) = φ1Φ∗

2f (ψ(y1)). (1.95)

This formula shows the uniqueness of Ψ (if it can be defined) in a neighborhood ofL1. To show that this formula also defines Ψ unambiguously, we will find the graphof Ψ in M1 ×M2. Consider the distribution D on M1 ×M2, generated at eachpoint (x1, x2) ∈M1×M2 by the tangent vectors of the type (XΦ∗

1f (x1), XΦ∗2f (x2)).

The fact that Φ1 and Φ2 are Poisson submersions imply that this distribution isregular involutive of dimension n = dimP , so we have an n-dimensional foliation.The graph of Ψ is nothing but the union of the leaves which go through then-dimensional submanifold (y1, ψ(y1)) | y1 ∈ L1.

It remains to show that Φ is symplectic, and sends Φ∗1f to Φ∗

2f for anyfunction f on P . To show that Φ is symplectic, it suffices to show that its graph inM1 ×M2 is Lagrangian (see Proposition 1.3.12). Since the involutive distributionD is generated by Hamiltonian vector fields (XΦ∗

1f , XΦ∗2f ), and the property of

being Lagrangian is invariant under Hamiltonian flow, it is enough to show thatthe tangent spaces to the graph of Ψ at points (y1, ψ(y1)), y1 ∈ L1, are Lagrangian.But this last fact can be verified immediately.

Since Ψ is symplectic and Ψ∗XΦ∗1f = XΦ∗

2f by construction, it means thatΨ∗(Φ∗

1f) is equal to Φ∗2f up to a constant. But this constant is zero, because these

two functions coincide on L2 by the construction of Ψ. Thus Ψ∗(Φ∗1f) = Φ∗

2f forany function f on P , implying that Φ1 = Φ2 Ψ.

1.9. Symplectic realizations 37

Theorem 1.9.2 is now a direct consequence of the local realization Theorem1.9.3 and the uniqueness Proposition 1.9.4: there is a unique way to glue localmarked symplectic realizations together, which glues the marked Lagrangian sub-manifolds together on their overlaps, to get a marked symplectic realization of agiven Poisson manifold. Remark 1.9.5. Of course, (non-marked) symplectic realizations of a Poisson man-ifold (P, Π) of dimension n are not necessarily of dimension 2n. For example, if(M, ω) is a symplectic realization of (P, Π) and (N, σ) is a symplectic manifold,then M × N is also a symplectic realization of P . And if (P, Π) is symplecticthen it is a symplectic realization of itself. Proposition 1.9.4 can be generalizedto an “essential uniqueness” result for non-marked local symplectic realizations(see [346]).

An important notion in symplectic geometry, directly related to symplecticrealizations, is the following:

Definition 1.9.6 ([209]). A foliation F on a symplectic manifold (M, ω) is calleda symplectically complete foliation if the symplectically orthogonal distribution(TF)⊥ to F is integrable.

In other words, F is a symplectically complete foliation if there is anotherfoliation F ′ such that TxF = (TxF ′)⊥ ∀ x ∈ M . In this case, the pair (F ,F ′)is called a dual pair . For example, any Lagrangian foliation is a symplecticallycomplete foliation which is dual to itself.

Theorem 1.9.7 (Libermann [209]). Let Φ : (M, ω)→ P be a surjective submersionfrom a symplectic manifold (M, ω) to a manifold P , such that the level sets of Φare connected. Denote by F the foliation whose leaves are level sets of Φ. Thenthere is a (unique) Poisson structure Π on P such that Φ : (M, ω) → (P, Π) isPoisson if and only if F is symplectically complete.

Proof (sketch). The symplectically orthogonal distribution (TF)⊥ to F is gener-ated by Hamiltonian vector fields of the type XΦ∗f where f is a function on P .The integrability of (TF)⊥ is equivalent to the fact that [XΦ∗f , XΦ∗g] is tangent to(TF)⊥ for any functions f, g on P . In other words, XΦ∗f,Φ∗g is tangent to (TF)⊥,i.e., Φ∗f, Φ∗g is constant on the leaves of F . Since the leaves of F are level setsof Φ, it means that there is a function h on P such that Φ∗f, Φ∗g = Φ∗h. Inother words, f, g := h is a Poisson bracket on P such that Φ is Poisson.

Chapter 2

Poisson Cohomology

2.1 Poisson cohomology

2.1.1 Definition of Poisson cohomology

Poisson cohomology was introduced by Lichnerowicz [211]. Its existence is basedon the following simple lemma.

Lemma 2.1.1. If Π is a Poisson tensor, then for any multi-vector field A we have

[Π, [Π, A]] = 0 . (2.1)

Proof. By the graded Jacobi identity (1.75) for the Schouten bracket, if Π is a2-vector field and A is an a-vector field, then

(−1)a−1[Π, [Π, A]]− [Π, [A, Π]] + (−1)a−1[A, [Π, Π]] = 0 .

Moreover, [A, Π] = −(−1)a−1[Π, A] due to the graded anti-commutativity, hence[Π, [Π, A]] = − 1

2 [A, [Π, Π]]. Now if Π is a Poisson structure, then [Π, Π] = 0, andtherefore [Π, [Π, A]] = 0.

Let (M, Π) be a smooth Poisson manifold. Denote by δ = δΠ : V(M) −→V(M) the R-linear operator on the space of multi-vector fields on M , defined asfollows:

δΠ(A) = [Π, A]. (2.2)

Then Lemma 2.1.1 says that δΠ is a differential operator in the sense thatδΠ δΠ = 0. The corresponding differential complex (V(M), δ), i.e.,

· · · −→ Vp−1(M) δ−→ Vp(M) δ−→ Vp+1(M) −→ · · · , (2.3)

will be called the Lichnerowicz complex . The cohomology of this complex is calledPoisson cohomology.

40 Chapter 2. Poisson Cohomology

By definition, Poisson cohomology groups of (M, Π), i.e., the cohomologygroups of the Lichnerowicz complex (2.3), are the quotient groups

HpΠ(M) =

ker(δ : Vp(M) −→ Vp+1(M))Im(δ : Vp−1(M) −→ Vp(M))

. (2.4)

The above Poisson cohomology groups are also denoted by Hp(M, Π), or alsoHp

LP (M, Π), where LP stands for Lichnerowicz–Poisson.Remark 2.1.2. Poisson cohomology groups can be very big, infinite-dimensional.For example, when Π = 0 then H

Π(M) :=⊕

k HkΠ(M) = V(M). Poisson coho-

mology groups of smooth Poisson manifolds have a natural induced topology fromthe Frechet spaces of multi-vector fields, which make them into not-necessarily-separated locally convex topological vector spaces (see Ginzburg [143, 144]).

2.1.2 Interpretation of Poisson cohomology

The zeroth Poisson cohomology group H0Π(M) is the group of functions f ∈

C∞(M) such that Xf = −[Π, f ] = 0. In other words, H0Π(M) is the space of

Casimir functions of Π, i.e., the space of first integrals of the associated symplec-tic foliation.

The first Poisson cohomology group H1Π(M) is the quotient of the space of

Poisson vector fields (i.e., vector fields X such that [Π, X ] = 0) by the space ofHamiltonian vector fields (i.e., vector fields of the type [Π, f ] = X−f ). Poisson vec-tor fields are infinitesimal automorphisms of the Poisson structures, while Hamil-tonian vector fields may be interpreted as inner infinitesimal automorphisms. ThusH1

Π(M) may be interpreted as the space of outer infinitesimal automorphisms of Π.The second Poisson cohomology group H2

Π(M) is the quotient of the space of2-vector fields Λ which satisfy the equation [Π, Λ] = 0 by the space of 2-vector fieldsof the type Λ = [Π, Y ]. If [Π, Λ] = 0 and ε is a formal (infinitesimal) parameter,then Π + εΛ satisfies the Jacobi identity up to terms of order ε2:

[Π + εΛ, Π + εΛ] = ε2[Λ, Λ] = 0 mod ε2. (2.5)

So one may view Π + εΛ as an infinitesimal deformation of Π in the space ofPoisson tensors. On the other hand, up to terms of order ε2, Π + ε[Π, Y ] is equalto (ϕε

Y )∗Π, where ϕεY denotes the time-ε flow of Y . Therefore Π + ε[Π, Y ] is a

trivial infinitesimal deformation of Π up to an infinitesimal diffeomorphism. Thus,H2

Π(M) is the quotient of the space of all possible infinitesimal deformations of Πby the space of trivial deformations. In other words, H2

Π(M) may be interpretedas the moduli space of formal infinitesimal deformations of Π. For this reason,the second Poisson cohomology group plays a central role in the study of normalforms of Poisson structures.

The third Poisson cohomology group H3Π(M) may be interpreted as the space

of obstructions to formal deformation. Suppose that we have an infinitesimal de-formation Π+εΛ, i.e., [Π, Λ] = 0. Then a priori, Π+εΛ satisfies the Jacobi identity

2.1. Poisson cohomology 41

only modulo ε2. To make it satisfy the Jacobi identity modulo ε3, we have to adda term ε2Λ2 such that

[Π + εΛ + ε2Λ2, Π + εΛ + ε2Λ2] = 0 mod ε3. (2.6)

The equation to solve is 2[Π, Λ2] = −[Λ, Λ]. This equation can be solved if andonly if the cohomology class of [Λ, Λ] in H3

Π(M) is trivial. Similarly, if (2.6) isalready satisfied, to find a term ε3Λ3 such that

[Π + εΛ + ε2Λ2 + ε3Λ3, Π + εΛ + ε2Λ2 + ε3Λ3] = 0 mod ε4, (2.7)

we have to make sure that the cohomology class of [Λ, Λ2] in H3Π(M) vanishes,

and so on.The Poisson tensor Π is itself a cocycle in the Lichnerowicz complex. If the

cohomology class of Π in H2Π(M) vanishes, i.e., there is a vector field Y such that

Π = [Π, Y ], then Π is called an exact Poisson structure.

2.1.3 Poisson cohomology versus de Rham cohomology

Recall that, the Poisson structure Π gives rise to a homomorphism

= Π : T ∗M −→ TM, (2.8)

which associates to each covector α a unique vector (α) such that

〈α ∧ β, Π〉 = 〈β, (α)〉 (2.9)

for any covector β. This homomorphism is an isomorphism if and only if Π isnondegenerate, i.e., is a symplectic structure. By taking exterior powers of theabove map, we can extend it to a homomorphism

: ΛpT ∗M −→ ΛpTM, (2.10)

and hence a C∞(M)-linear homomorphism

: Ωp(M) −→ Vp(M), (2.11)

where Ωp(M) denotes the space of smooth differential forms of degree p on M .Recall that is called the anchor map of Π.

Lemma 2.1.3. For any smooth differential form η on a given smooth Poisson man-ifold (M, Π) we have

(dη) = −[Π, (η)] = −δΠ((η)) . (2.12)

Proof. By induction on the degree of η, using the Leibniz rule. If η is a functionthen (η) = η and (dη) = −[Π, η] = Xη, the Hamiltonian vector field of η. If η =df is an exact 1-form then (dη) = 0 and [Π, (η)] = [Π, Xf ] = 0, hence Equation


(2.12) is satisfied. If Equation (2.12) is satisfied for a differential p-form η and adifferential q-form µ, then its also satisfied for their exterior product η∧µ. Indeed,we have (d(η∧µ)) = (dη∧µ+(−1)pη∧dµ) = (dη)∧ (µ)+(−1)p(η)∧ (dµ) =−[Π, (η)] ∧ (µ)− (−1)p(η) ∧ [Π, (µ)] = −[Π, (η) ∧ (µ)] = −[Π, (η ∧ µ)].

The above lemma means that, up to a sign, the operator intertwines theusual differential operator d of the de Rham complex

· · · −→ Ωp−1(M) d−→ Ωp(M) d−→ Ωp+1(M) −→ · · · (2.13)

with the differential operator δΠ of the Lichnerowicz complex. In particular, itinduces a linear homomorphism of the corresponding cohomologies. In other words,we have:

Theorem 2.1.4 ([211]). For every smooth Poisson manifold (M, Π), there is a nat-ural homomorphism

∗ : HdR(M) =

⊕p

HpdR(M) −→ H

Π(M) =⊕

p

HpΠ(M) (2.14)

from its de Rham cohomology to its Poisson cohomology, induced by the map =Π. If M is a symplectic manifold, then this homomorphism is an isomorphism.

When M is symplectic, is an isomorphism, and that’s why ∗ is also anisomorphism. Remark 2.1.5. The de Rham cohomology has a graded Lie algebra structure, givenby the cap product (induced from the exterior product of differential forms). Sodoes the Poisson cohomology. The Lichnerowicz homomorphism ∗ : H

dR(M) −→H

Π(M) in the above theorem is not only a linear homomorphism, but also analgebra homomorphism.Remark 2.1.6. If (M, Π) is not symplectic then the map ∗ : H

dR(M)→ HΠ(M) is

not an isomorphism in general. In particular, while de Rham cohomology groupsof manifolds of “finite type” (e.g., compact manifolds) are of finite dimensions,Poisson cohomology groups may have infinite dimension in general. An interest-ing and largely open question is: what are the conditions for the Lichnerowiczhomomorphism to be injective or surjective?

2.1.4 Other versions of Poisson cohomology

If, in the Lichnerowicz complex, instead of smooth multi-vector fields, we considerother classes of multi-vector fields, then we arrive at other versions of Poissoncohomology. For example, if Π is an analytic Poisson structure, and one considersanalytic multi-vector fields, then one gets analytic Poisson cohomology.

Recall that, a germ of an object (e.g., a function, a differential form, a Rie-mannian metric, etc.) at a point z is an object defined in a neighborhood of z.Two germs at z are considered to be the same if there is a neighborhood of z in

2.1. Poisson cohomology 43

which they coincide. When considering a germ of smooth (resp. analytic) Poissonstructure Π at a point z, it is natural to talk about germified Poisson cohomology:the space V(M) in the Lichnerowicz complex is replaced by the space of germsof smooth (resp. analytic) multi-vector fields. More generally, given any subset Nof a Poisson manifold (M, Π), one can define germified Poisson cohomology at N .Similarly, one can talk about formal Poisson cohomology. By convention, the germof a formal multi-vector field is itself. Viewed this way, formal Poisson cohomologyis the formal version of germified Poisson cohomology.

If M is not compact, then one may be interested in Poisson cohomology withcompact support, by restricting one’s attention to multi-vector fields with compactsupport. Remark that Theorem 2.1.4 also holds in the case with compact support:if (M, Π) is a symplectic manifold then its de Rham cohomology with compactsupport is isomorphic to its Poisson cohomology with compact support.

If one considers only multi-vector fields which are tangent to the character-istic distribution, then one gets tangential Poisson cohomology. (A multi-vectorfield λ is said to be tangent to a distribution D on a manifold M if at each pointx ∈M one can write Λ(x) =

∑aivi1 ∧ · · · ∧ vis where vij are vectors lying in D.)

It is easy to see that the homomorphism ∗ in Theorem 2.1.4 also makes sense fortangential Poisson cohomology (and tangential de Rham cohomology).

The above versions of Poisson cohomology also have a natural interpretation,similar to the one given for smooth Poisson cohomology.

2.1.5 Computation of Poisson cohomology

If a Poisson structure Π on a manifold M is nondegenerate (i.e., symplectic),then Poisson cohomology of Π is the same as de Rham cohomology of M . Thereare many tools for computing de Rham cohomology groups, and these groupshave probably been computed for most “familiar” manifolds, see, e.g., [41, 138].However, when Π is not symplectic, H

Π(M) is much more difficult to computethan H

dR(M) in general, and at the moment of writing of this book, there arefew Poisson (non-symplectic) manifolds for which Poisson cohomology has beencomputed. For one thing, H

Π(M) can have infinite dimension even when M iscompact, and the problem of determining whether H

Π(M) is finite dimensional ornot is already a difficult open problem for most Poisson structures that we know of.

Nevertheless, various tools from algebraic topology and homological algebracan be adapted to the problem of computation of Poisson cohomology. One of themis the classical Mayer–Vietoris sequence (see, e.g., [41]). The following Poissoncohomology version of Mayer–Vietoris sequence is absolutely analogous to its deRham cohomology version.

Proposition 2.1.7 ([333]). Let U and V be two open subsets of a smooth Poissonmanifold (M, Π). Then

0 −→ V(U ∪ V ) α−→ V(U)⊕ V(V )β−→ V(U ∩ V ) −→ 0, (2.15)


where α(Λ) = (Λ|U , Λ|V ) is the restriction map, and β(Λ1, Λ2) = Λ1|U∩V −Λ2|U∩V

is the difference map, is an exact sequence of smooth Lichnerowicz complexes, andthe corresponding cohomological long exact sequence (called the Mayer–Vietorissequence) has the form

· · · −→ Hk(U ∪ V, Π) α∗−→ Hk(U, Π)⊕Hk(V, Π)β∗−→

Hk(U ∩ V, Π) −→ Hk+1(U ∪ V, Π) α∗−→ · · · . (2.16)

The proof of Proposition 2.1.7 is also absolutely similar to the proof of itsde Rham version. The above Mayer–Vietoris sequence reduces the computation ofPoisson cohomology on a manifold to the computation of Poisson cohomology onsmall open sets (which contain singularities of the Poisson structure). To study(germified) Poisson cohomology of singularities of Poisson structures, one can tryto use the tools from singularity theory. This will be done in Section 2.5 for Poissonstructures in dimension 2.

Another standard tool is the spectral sequence, which will be discussed inSection 2.4.

In the case of linear Poisson structures, Poisson cohomology is intimatelyrelated to Lie algebra cohomology, also known as Chevalley–Eilenberg cohomology,which will be discussed in Section 2.3.

In Chapter 8, we will interpret Poisson cohomology as a particular case ofcohomology of Lie algebroids. This leads to a definition and study of Poissoncohomology from a purely algebraic point of view, as was done by Huebschmann[180].

Some other methods for computing and studying Poisson cohomology in-clude: the use of symplectic groupoids1 to reduce the computation of Poisson co-homology of certain Poisson manifolds to the computation of de Rham cohomologyof other manifolds [359]; the van Est map which relates Lie algebroid cohomologywith differentiable cohomology of Lie groupoids [355, 85]; comparison of Poissoncohomology of Poisson manifolds which are Morita equivalent [147, 146, 145, 85];equivariant Poisson cohomology [144].

2.2 Normal forms of Poisson structures

Consider a Poisson structure Π on a manifold M . In a given system of coordinates(x1, . . . , xm), Π has the expression

Π =∑i<j

Πij∂

∂xi∧ ∂

∂xj=

12

∑i,j

Πij∂

∂xi∧ ∂

∂xj. (2.17)

A priori, the coefficients Πij of Π may be very complicated, non-polynomial func-tions. The idea of normal forms is to simplify these coefficients in the expressionof Π.

1Symplectic groupoids will be introduced in Section 7.5.

2.2. Normal forms of Poisson structures 45

A (local) normal form of Π is a Poisson structure

Π′ =∑i<j

Π′ij

∂

∂x′i

∧ ∂

∂x′j

=12

∑i,j

Π′ij

∂

∂x′i

∧ ∂

∂x′j

(2.18)

which is (locally) isomorphic to Π, i.e., there is a (local) diffeomorphism ϕ : (xi) →(x′

i) called a normalization such that ϕ∗Π = Π′, such that the functions Π′ij are

“simpler” than the functions Πij . The ideal would be that Π′ij were constant

functions. According to Remark 1.4.6, such a local normal form exists when Π isa (locally) regular Poisson structure.

Near a singular point of Π, we can use the splitting Theorem 1.4.5 to writeΠ as the direct sum of a constant symplectic structure with a Poisson structurewhich vanishes at a point. The local normal form problem for Π is then reducedto the problem of local normal forms for a Poisson structure which vanishes at apoint.

Having this in mind, we now assume that Π vanishes at the origin 0 of agiven local coordinate system (x1, . . . , xm). Denote by

Π = Π(k) + Π(k+1) + · · ·+ Π(k+n) + · · · (k ≥ 1) (2.19)

the Taylor expansion of Π in the coordinate system (x1, . . . , xm), where for eachh ∈ N, Π(h) is a 2-vector field whose coefficients Π(h)

ij are homogeneous polynomialfunctions of degree h. Π(k), assumed to be nontrivial, is the term of lowest degreein Π, and is called the homogeneous part , or principal part of Π. If k = 1 then Π(1)

is called the linear part of Π, and so on. This homogeneous part can be definedintrinsically, i.e., it does not depend on the choice of local coordinates.

At the formal level, the Jacobi identity for Π can be written as

0 = [Π, Π] = [Π(k) + Π(k+1) + · · · , Π(k) + Π(k+1) + · · · ]= [Π(k), Π(k)] + 2[Π(k), Π(k+1)] + 2[Π(k), Π(k+2)] + [Π(k+1), Π(k+1)] + · · · ,

which leads to (by considering terms of the same degree):

[Π(k), Π(k)] = 0,2[Π(k), Π(k+1)] = 0,

2[Π(k), Π(k+2)] + [Π(k+1), Π(k+1)] = 0,. . .

(2.20)

In particular, the homogeneous part Π(k) of Π is a Poisson structure, andΠ may be viewed as a deformation of Π(k). A natural homogenization questionarises: is this deformation trivial? In other words, is Π locally (or formally) iso-morphic to its homogeneous part Π(k)? That’s where Poisson cohomology comesin, because, as explained in Subsection 2.1.2, Poisson cohomology governs (formal)deformations of Poisson structures.


When k = 1, one talks about the linearization problem, and when k = 2 onetalks about the quadratization problem, and so on. These problems, for Poissonstructures and related structures like Nambu structures, Lie algebroids and Liegroupoids, will be studied in detail in the subsequent chapters of this book. Here wewill discuss, at the formal level, a more general problem of quasi-homogenization.

Denote by

Z =n∑

i=1

wixi∂

∂xi, wi ∈ N (2.21)

a given diagonal linear vector field with the following special property: its coeffi-cients wi are positive integers. Such a vector field is called a quasi-radial vectorfield . (When wi = 1 ∀i, we get the usual radial vector field .)

A multi-vector field Λ is called quasi-homogeneous of degree d (d ∈ Z) withrespect to Z if

LZΛ = dΛ. (2.22)

For a function f , it means Z(f) = df . For example, a monomial k-vector field

( n∏i=1

xai

i

) ∂

∂xj1

∧ · · · ∧ ∂

∂xjk

, ai ∈ Z≥0, (2.23)

is quasi-homogeneous of degree∑n

i=1 aiwi −∑k

s=1 wjs . In particular, monomialterms of high degree in the usual sense (i.e., with large

∑ai) also have high

quasi-homogeneous degree. As a consequence, quasi-homogeneous (smooth, formalor analytic) multi-vector fields are automatically polynomial in the usual sense.Note that the quasi-homogeneous degree of a monomial multi-vector field can benegative, though it is always greater than or equal to −∑n

i=1 wi.Given a Poisson structure Π with Π(0) = 0, by abuse of notation, we will

now denote byΠ = Π(d1) + Π(d2) + · · · , d1 < d2 < · · · (2.24)

the quasi-homogeneous Taylor expansion of Π with respect to Z, where each termΠ(di) is quasi-homogeneous of degree di. The term Π(d1), assumed to be nontrivial,is called the quasi-homogeneous part of Π. Similarly to the case with usual homo-geneous Taylor expansion, the Jacobi identity for Π implies the Jacobi identityfor Π(d1), which means that Π(d1) is a quasi-homogeneous Poisson structure, andΠ may be viewed as a deformation of Π(d1). The quasi-homogenization problemis the following: is there a transformation of coordinates which sends Π to Π(d1),i.e., which kills all the terms of quasi-homogeneous degree > d1 in the expressionof Π?

In order to treat this quasi-homogenization problem at the formal level, wewill need the quasi-homogeneous graded version of Poisson cohomology.

Let Π(d) be a Poisson structure on an n-dimensional space V = Kn, whichis quasi-homogeneous of degree d with respect to a given quasi-radial vector field

2.2. Normal forms of Poisson structures 47

Z =∑n

i=1 wixi∂

∂xi. For each r ∈ Z, denote by Vk

(r) = Vk(r)(K

n) the space of quasi-homogeneous polynomial k-vector fields on Kn of degree r with respect to Z. Ofcourse, we have

Vk = ⊕rVk(r), (2.25)

where Vk = Vk(Kn) is the space of all polynomial vector fields on Kn. Note that,if Λ ∈ Vk

(r) then

LZ [Π(d), Λ] = [LZΠ(d), Λ] + [Π(d),LZΛ] = (d + r)[Π(d), Λ],

i.e., δΠ(d)Λ = [Π(d), Λ] ∈ Vk+1(r+d). The group

Hk(r)(Π

(d)) =ker(δΠ(d) : Vk

(r) −→ Vk+1(r+d))

Im(δΠ(d) : Vk−1(r−d) −→ Vk

(r))(2.26)

is called the kth quasi-homogeneous of degree r Poisson cohomology group ofΠ(d). Of course, there is a natural injection from Hk

(r)(Π(d)) to the usual (formal,

analytic or smooth) Poisson cohomology group Hk(Π(d)) of Π(d) over Kn. WhileHk(Π(d)) may be of infinite dimension, Hk

(r)(Π(d)) is always of finite dimension

(for each r).Return now to the quasi-homogeneous Taylor series Π = Π(d1) + Π(d2) + · · · .

The Jacobi identity for Π implies that [Π(d1), Π(d2)] = 0, i.e., Π(d2) is a quasi-homogeneous cocycle in the Lichnerowicz complex of Π(d1). If this term Π(d2) is acoboundary, i.e., Π(d2) = [Π(d1), X(d2−d1)] for some quasi-homogeneous vector fieldX(d2−d1) = X

(d2−d1)i ∂/∂xi, then the coordinate transformation x′

i = xi−X(d2−d1)i

will kill the term Π(d2) in the expression of Π. More generally, we have:

Proposition 2.2.1. With the above notations, suppose that Π(dk) = [Π(d1), X ]+Λ(dk)

for some k > 1, where X = Xi∂/∂xi is a quasi-homogeneous vector field of degreedk − d1. Then the diffeomorphism (coordinate transformation) φ : (xi) → (x′

i) =(xi −Xi) transforms Π into

φ∗Π = Π(d1) + · · ·+ Π(dk−1) + Λ(dk) + Π(dk+1) · · · . (2.27)

In other words, this transformation suppresses the term [Π(d1), X ] without chang-ing the terms of degree strictly smaller than dk.

Proof. Denote by Γ = φ∗Π. For the Poisson structure Π we have

x′i, x

′j =

∑uv

∂x′i

∂xu

∂x′j

∂xvxu, xv =

∑uv

∂x′i

∂xu

∂x′j

∂xvΠuv

=∑uv

(δui −

∂Xi

∂xu)(δv

j −∂Xj

∂xv)(Π(d1) + Π(d2) + · · · )uv,


where δui is the Kronecker symbol, and the terms of degree smaller than or equal

to dk in this expression give

(Π(d1) + · · ·+ Π(dk))ij −∑

u

∂Xi

∂xuΠ(d1)

uj −∑

v

∂Xj

∂xvΠ(d1)

iv .

On the other hand, by definition, x′i, x

′j is equal to Γij φ. But the terms of

degree smaller than or equal to dk in the expansion of Γij φ are

(Γ(d1) + · · ·+ Γ(dk))ij −∑

s

Xs

∂Γ(d1)ij

∂xs.

Comparing the terms of degree d1, . . . , dk−1, we get Γ(d1)ij = Π(d1)

ij , . . . ,Γ(dk−1)ij =

Π(dk−1)ij . As for the terms of degree dk, they give

Γ(dk)ij −

∑s

Xs

∂Π(d1)ij

∂xs= Π(dk)

ij −∑

u

∂Xi

∂xuΠ(d1)

uj −∑

v

∂Xj

∂xvΠ(d1)

iv .

As we have

[X, Π(d1)]ij =∑

s

Xs

∂Π(d1)ij

∂xs−∑

u

∂Xi

∂xuΠ(d1)

uj −∑

v

∂Xj

∂xvΠ(d1)

iv ,

it follows that

Γ(dk)ij = Π(dk)

ij + [X, Π(d1)]ij = Π(dk)ij − [Π(d1), X ]ij = Λ(dk)

ij .

The proposition is proved.

Theorem 2.2.2. If the quasi-homogeneous Poisson cohomology groups H2(r)(Π

(d))of a quasi-homogeneous Poisson structure Π(d) of degree d are trivial for all r > d,then any Poisson structure admitting a formal quasi-homogeneous expansion Π =Π(d) + Π(d2) + · · · is formally isomorphic to its quasi-homogeneous part Π(d).

Proof. Use Proposition 2.2.1 to kill the terms of degree strictly greater than d inΠ consecutively.

Example 2.2.3. One can use Theorem 2.2.2 to show that any Poisson structure onK2 of the form Π = f ∂

∂x ∧ ∂∂y , where f = x2 + y3+ higher-order terms, is formally

isomorphic to (x2 + y3) ∂∂x ∧ ∂

∂y . (This is a simple singularity studied by Arnold,see Theorem 2.5.2.) The quasi-radial vector field in this case is Z = 3x ∂

∂x + 2y ∂∂y .

2.3. Cohomology of Lie algebras 49

2.3 Cohomology of Lie algebras

Let Π(1) be a linear Poisson structure on a vector space Kn. Denote by g =((Kn)∗, , Π(1)) the Lie algebra corresponding to Π(1). We will see in this sectionthat Poisson cohomology groups of Π(1) are special cases of Lie algebra cohomologyof g.

2.3.1 Chevalley–Eilenberg complexes

Let W be a g-module, i.e., a vector space together with a Lie algebra homomor-phism ρ : g → End(W ) from g to the Lie algebra of endomorphisms of W . Inother words, ρ is a linear map such that ρ([x, y]) = ρ(x).ρ(y)−ρ(y).ρ(x) ∀x, y ∈ g.The action of an element x ∈ g on a vector v ∈ W is defined by

x.v = ρ(x)(v). (2.28)

One associates to W the following complex, called Chevalley–Eilenberg complex ofg with coefficients in W [77]:

· · · δ−→ Ck−1(g, ρ) δ−→ Ck(g, ρ) δ−→ Ck+1(g, ρ) δ−→ · · · , (2.29)

whereCk(g, ρ) = (∧kg∗)⊗W (2.30)

(k ≥ 0) is the space of k-multilinear antisymmetric maps from g to W : an elementθ ∈ Ck(g, ρ) may be presented as a k-multilinear antisymmetric map from g toW , or a linear map from ∧kg to W :

θ(x1, . . . , xk) = θ(x1 ∧ · · · ∧ xk) ∈W, xi ∈ g. (2.31)

The operator δ = δCE : Ck(g, ρ) → C(k+1)(g, ρ) in the Chevalley–Eilenberg com-plex is defined as follows:

(δθ)(x1, . . . , xk+1) =∑

i

(−1)i+1ρ(xi)(θ(x1, . . . , xi, . . . , xk+1))

+∑i<j

(−1)i+jθ([xi, xj ], x1, . . . , xi . . . xj , . . . , xk+1), (2.32)

the symbol above a variable means that this variable is missing in the list.It is a classical result [77], which follows directly from the Jacobi identity

of g, that δCE δCE = 0. It means that the Chevalley–Eilenberg complex is adifferential complex with differential operator δ = δCE . Its cohomology groups

Hk(g, ρ) = Hk(g, W ) =ker(δ : Ck(g, ρ) −→ Ck+1(g, ρ))Im(δ : Ck−1(g, ρ) −→ Ck(g, ρ))

(2.33)

are called cohomology groups of g with coefficients in W (or with respect to therepresentation ρ).


Remark 2.3.1. Formula (2.32) is absolutely analogous to Cartan’s formula (1.7).This construction of differential operators is sometimes referred to as Cartan–Chevalley–Eilenberg construction. If G is a connected Lie group with Lie algebrag, then the space Ω

L(G) of left-invariant differential forms on G is a subcomplexof the de Rham complex of G which is naturally isomorphic to the Chevalley–Eilenberg C(g, R) for the trivial action of g on R, which implies that their coho-mologies are also isomorphic:

HL(G) ∼= H(g, R). (2.34)

(The isomorphism from ΩL(G) to C(g, R) associates to each left-invariant dif-

ferential form on G its value at the neutral element e of G, after the identifica-tion of g∗ with T ∗

e G.) In particular, when G is compact, the averaging processα →

∫G

L∗gαdg (where α denotes a differential form on G, and Lg denotes the left

translation by g ∈ G) induces an isomorphism from HdR(G) to H

L(G), and wehave H

dR(G) ∼= H(g, R).

Exercise 2.3.2. Show that, given a smooth Poisson manifold (M, Π), its Lichnerow-icz complex can be identified with a subcomplex of the Chevalley–Eilenberg com-plex of the (infinite-dimensional) Lie algebra C∞(M) with coefficients in C∞(M)(with respect to the adjoint action given by the Poisson bracket), which consistsof cochains which are multi-derivations. (Hint: use Formula (1.82)).

In general, the problem of computation of H(g, W ) for a finite-dimensional g-module W of a finite-dimensional Lie algebra g is a problem of linear algebra: onesimply has to deal with finite-dimensional systems of linear equations. However,even for low-dimensional Lie algebras, these systems of linear equations often havehigh dimensions and require thousands or millions of computations, so it is noteasy to do it by hand.

Fortunately, cohomology of semisimple Lie algebras is relatively simple, duein part to the following results, known as Whitehead’s lemmas .

Theorem 2.3.3 (Whitehead). If g is semisimple, and W is a finite-dimensionalg-module, then H1(g, W ) = 0 and H2(g, W ) = 0.

Theorem 2.3.4 (Whitehead). If g is semisimple, and W is a finite-dimensionalg-module such that W g = 0, where W g = w ∈ W | x.w = 0 ∀ x ∈ g denotes theset of elements in W which are invariant under the action of g, then Hk(g, W ) =0 ∀ k ≥ 0.

See, e.g., [186] for the proof of Whitehead’s lemmas. A refined (normed)version of Theorem 2.3.3 will be proved in Chapter 3. Let us also mention that ifg is simple then dimH3(g, K) = 1. Combining the two Whitehead’s lemmas withthe fact that any finite-dimensional module W of a semisimple Lie algebra g iscompletely reducible, one gets the following formula:

H(g, W ) = H(g, K)⊗W g =⊕

k =1,2

Hk(g, K)⊗W g. (2.35)


Remark 2.3.5. If W is a smooth Frechet module of a compact Lie group G and g isthe Lie algebra of G, then the formula H(g, W ) = H(g, R)⊗W g is still valid, seeGinzburg [144]. In particular, if a compact Lie group G acts on a smooth manifoldM , then C∞(M) is a smooth Frechet G-module, and we have

H(g, C∞(M)) = H(g, R)⊗ (C∞(M))g. (2.36)

Remark 2.3.6. Cohomology of Lie algebras is closely related to differentiable (orcontinuous) cohomology of Lie groups, via the so-called Van Est map and Van Estspectral sequence. See, e.g., [40, 157].

2.3.2 Cohomology of linear Poisson structures

Consider now the case W = Sqg, the q-symmetric power of g together with theadjoint action of g:

ρ(x)(xi1 . . . xiq ) =q∑

s=1

xi1 . . . [x, xis ] . . . xiq . (2.37)

Since g = ((Kn)∗, , Π(1)), the space W = Sqg can be naturally identified withthe space of homogeneous polynomials of degree q on Kn, and we can write

ρ(x).f = x, f, (2.38)

where f ∈ Sqg, and x, f denotes the Poisson bracket of x with f with respectto Π(1).

Denote by Vp(q) = Vp

(q)(Kn) the space of homogeneous p-vector fields of degree

q. (It is the same as the space of quasi-homogeneous p-vector fields of quasi-homogeneous degree q− p with respect to the radial vector field

∑xi∂/∂xi.) Vp

(q)

can be identified with Cp(g,Sqg) as follows: For

A =∑

i1<···<ip

Ai1,...,ip

∂

∂xi1

∧ · · · ∧ ∂

∂xip

∈ Vp(q) (2.39)

define θA ∈ Cp(g,Sqg) by

θA(xi1 , . . . , xip) = Ai1,...,ip . (2.40)

Lemma 2.3.7. With the above identification A↔ θA, the Lichnerowicz differentialoperator δLP = [Π(1), .] : Vp

(q) −→ Vp+1(q) is identified with the Chevalley–Eilenberg

differential operator δCE : Cp(g,Sqg) −→ Cp+1(g,Sqg).

Proof. We must show that θ[Π(1),A] = δCEθA for A ∈ Vp(q). Write

Π(1) = 12

∑i,j,k ck

ijxk∂

∂xi∧ ∂

∂xj, and A = 1

p!

∑i1,...,ip

Ai1,...,ip

∂∂xi1

∧ · · · ∧ ∂∂xip

.


Denote the Poisson bracket of Π(1) by , . By the Leibniz rule, we have

[Π(1), A] = E1 + E2

where

E1 =1p!

∑[Π(1), Ai1,...,ip ] ∧ ∂

∂xi1

∧ · · · ∧ ∂

∂xip

=1p!

∑i1,...,ip,i

xi, Ai1,...,ip∂

∂xi∧ ∂

∂xi1

∧ · · · ∧ ∂

∂xip

and

E2 =1p!

∑Ai1,...,ip

[Π(1),

∂

∂xi1

∧ · · · ∧ ∂

∂xip

]=

12p!

∑i1,...,ip,i,j,s

(−1)sAi1,...,ipcis

ij

∂

∂xi∧ ∂

∂xj∧ ∂

∂xi1

∧ · · · ∂

∂xis

· · · ∧ ∂

∂xip

.

It means that

E1(dxi1 , . . . ,dxip+1) =∑

s

(−1)s+1xis , Ai1,...is...,ip+1

and

E2(dxi1 , . . . ,dxip+1) =∑

u<v;k

(−1)u+vAk,i1,...iu...iv ...,ip+1ckiuiv

.

On the other hand, we have

δθA(xi1 , . . . , xip+1)

=∑

u

(−1)u+1ρ(xiu)A(xi1 , . . . xiu . . . , xip+1)

+∑u<v

(−1)u+vA([xiu , xiv ], xi1 , . . . xiu . . . xiv . . . , xip+1)

=∑

u

(−1)u+1xiu , Ai1,...iu...,ip+1

+∑

u<v;k

(−1)u+vckiuiv

A(xk, xi1 , . . . xiu . . . xiv . . . , xip+1).

It remains to compare the above formulas.

An immediate consequence of Lemma 2.3.7 and Theorem 2.2.2 is the following:


Theorem 2.3.8 ([346]). If g is a finite-dimensional Lie algebra such that

H2(g,Skg) = 0 ∀ k ≥ 2,

then any formal Poisson structure Π which vanishes at a point and whose linearpart Π(1) at that point corresponds to g is formally linearizable. In particular, itis the case when g is semisimple.

Remark 2.3.9. In Lemma 2.3.7, the fact that A is homogeneous is not so important.What is important is that the module W in question can be identified with asubspace of the space of functions on Kn, where the action of g is given by thePoisson bracket, i.e., by Formula (2.38). The following smooth (as compared tohomogeneous) version of Lemma 2.3.7 is also true, with a similar proof (see, e.g.,[144, 148, 221, 222]): if U is an Ad∗-invariant open subset of the dual g∗ of theLie algebra g of a connected Lie group G (or more generally, an open subset ofa dual Poisson–Lie group G∗ which is invariant under the dressing action of G –Poisson–Lie groups will be introduced later in the book), then

HΠ(U) ∼= H(g, C∞(U)) , (2.41)

where the action of g on C∞(U) is induced by the coadjoint (or dressing) ac-tion, and a natural isomorphism exists already at the level of cochain complexes.In particular, if G is compact semisimple, then this formula together with theFrechet-module version of Whitehead’s lemmas (Remark 2.3.5) leads to the fol-lowing formula (see [148]):

HΠ(U) = H(g)⊗ (C∞(U))G =

⊕k =1,2

Hk(g)⊗ (C∞(U))G. (2.42)

2.3.3 Rigid Lie algebras

Theorem 2.3.8 means that the second cohomology group

H2(g,S≥2g) =⊕k≥2

H2(g,Skg)

governs nonlinear deformations of the linear Poisson structure of g∗. Meanwhile,the group H2(g, g) governs deformations of g itself (or equivalently, linear defor-mations of the Poisson structure on g∗ associated to g).

An n-dimensional Lie algebra g over K can be determined by its structureconstants ck

ij with respect to a given basis (xi): [xi, xj ]g =∑

ckijxk. The n3-tuple of

coefficients (ckij) is called a Lie algebra structure of dimension n. The set A(n, K) ⊂

Kn3of all Lie algebra structures of dimension n is an algebraic variety (the Jacobi

identity and the anti-commutativity give the system of algebraic equations whichdetermine this set). The full linear group GL(n, K) acts naturally on A(n, K) bychanges of basis, and two Lie algebra structures are isomorphic if and only if


they lie on the same orbit of GL(n, K). An n-dimensional Lie algebra g is calledrigid if the orbit of its structure is an open subset of A(n, K) with respect to theusual topology induced from the Euclidean topology of Kn3

; equivalently, any Liealgebra g′ close enough to g is isomorphic to g.

Theorem 2.3.10 (Nijenhuis–Richardson [279]). If g is a finite-dimensional Lie al-gebra such that H2(g, g) = 0, then g is rigid. In particular, semisimple Lie algebrasare rigid.

Remark 2.3.11. The condition H2(g, g) = 0 is a sufficient but not a necessarycondition for the rigidity of a Lie algebra. For example, Richardson [300] showedthat, for any odd integer n > 5, the semi-direct product ln = sl(2, K) W 2n+1,where W 2n+1 is the (2n + 1)-dimensional irreducible sl(2, K)-module, is rigid buthas H2(ln, ln) = 0. In fact, H2(g, g) = 0 means that there are nontrivial infinitesi-mal deformations, but not every infinitesimal deformation can be made into a truedeformation. See, e.g., [62, 63, 151].

2.4 Spectral sequences

2.4.1 Spectral sequence of a filtered complex

Spectral sequences are one of the main tools for computing cohomology groups.The general idea is as follows.

Let (C = ⊕k∈Z+Ck, δ) be a differential complex. It means that Ck (k ≥ 0)are vector spaces (or more generally, Abelian groups), and δ : Ck → Ck+1 arelinear operators such that δ δ = 0.

Assume that (C, δ) admits a filtration (Ch)h∈N. It means that each Ck isfiltered by subspaces

Ck = Ck0 ⊃ Ck

1 ⊃ Ck2 ⊃ · · · , (2.43)

such that δCkh ⊂ Ck+1

h ∀ k, h. In other words, Ch =⊕

k Ckh is a differential

subcomplex of Ch−1 for h ≥ 1, and C0 = C. Put Ch = C if h < 0 by convention.Using this filtration, one decomposes cohomology groups

Hk(C) =Zk

Bk=

ker(δ : Ck −→ Ck+1)Im(δ : Ck−1 −→ Ck)

(2.44)

into smaller pieces Hkh(C)/Hk

h+1(C), where Hkh(C) consists of the elements of

Hk(C) which can be represented by cocycles lying in Ckh . The group⊕

h≥0

Hkh(C)/Hk

h+1(C) (2.45)

is called the graded version of Hk(C); it is linearly isomorphic to Hk(C) if, say,the filtration is finite, i.e., Cn = 0 for some n ∈ N.

2.4. Spectral sequences 55

A way to compute Hkh(C)/Hk

h+1(C) and Hk(C) is to use the spectral sequence(Ep,q

r )r≥0 of the above filtered complex. By definition,

Ep,qr =

Zp,qr + Cp+q

p+1

Bp,qr + Cp+q

p+1

(2.46)

whereZp,q

r =

y ∈ Cp+qp | δy ∈ Cp+q+1

p+r

(2.47)

andBp,q

r =y ∈ Cp+q

p | y = δz, z ∈ Cp+q−1p−r+1

. (2.48)

Clearly, Bp,qr ⊂ Zp,q

r , Bp,qr ⊂ Bp,q

r+1 and Zp,qr ⊃ Zp,q

r+1. Hence Ep,qr is well de-

fined, and is bigger than Ep,qr+1. (There is a surjection from a subgroup of Ep,q

r toEp,q

r+1.) As r tends to ∞, the group Ep,qr gets smaller and smaller, and it approxi-

mates better and better the group Hp+qp (C)/Hp+q

p+1 (C). In fact, if the filtration isof finite length, i.e., Cn = 0 for some n ∈ N, then

Ep,qr =

Zp+q ∩ Cp+qp + Cp+q

p+1

Bp+q ∩Cp+qp + Cp+q

p+1

∼= Hp+qp (C)/Hp+q

p+1 (C) ∀ r ≥ n, p. (2.49)

In general, one says that (Ep,qr ) converges if its limit

Ep,q∞ = lim

r→∞Ep,qr (2.50)

is isomorphic to Hp+qp (C)/Hp+q

p+1 (C).The terms Ep,q

r of the spectral sequence can be computed inductively on r(that’s why they are useful for computing Hp+q

p (C)/Hp+qp+1 (C)). The zeroth term is:

Ep,q0 = Cp+q

p /Cp+qp+1 . (2.51)

In other words, E0 = ⊕Ep,q0 is just the graded version of the complex C. For

r ≥ 0, the differential operator δ induces an operator on (Ep,qr ), denoted by δr:

δr : Ep,qr → Ep+r,q−r+1

r . (2.52)

(The image of y ∈ Zp,qr mod Bp,q

r + Cp+qp+1 under δr is δy ∈ Zp+r,q−r+1

r modBp+r,q−r+1

r + Cp+qp+r+1. One verifies directly that δr is well defined.)

Since δ δ = 0, we also have δr δr = 0, i.e., δr is a differential operator. Itturns out that Ep,q

r+1 is nothing but the cohomology of Ep,qr with respect to δr:

Ep,qr+1 =

ker(δr : Ep,qr −→ Ep+r,q−r+1

r )Im(δr : Ep−r,q+r−1

r −→ Ep,qr )

. (2.53)

Exercise 2.4.1. Verify Formula (2.53), starting from (2.46), (2.47) and (2.48).

Remark 2.4.2. In the literature, Formula (2.53) is often used in the definition ofspectral sequences, and Formulas (2.46), (2.47) and (2.48) only show up after ornot at all.


2.4.2 Leray spectral sequence

As a first example of spectral sequences, let us consider a locally trivial fibrationπ : M → N of a manifold M over a connected manifold N with fibers diffeomorphicto F . The de Rham complex Ω(M) of differential forms on M admits a naturalfiltration with respect to this fibration: Ωk

h(M) (h ≥ 0) is the subspace of Ωk(M)consisting of k-forms ω which satisfy the following condition:

ωx(X1, . . . , Xk) = 0 ∀x ∈ M, X1, . . . , Xk ∈ TxM s.t. π∗X1 = · · · = π∗Xk−h+1 = 0.(2.54)

The associated spectral sequence of this filtration is known as the Leray spectralsequence. Its zeroth term Ep,q

0 can be written as follows:

Ep,q0∼= Ωp(N, Ωq(F )). (2.55)

More precisely, Ep,q0 = Ωp+q

p (M)/Ωp+qp+1(M) is naturally isomorphic to the space of

vector-valued p-forms on N with values in the vector bundle over N whose fiberover a point y ∈ N is the space of q-forms on the fiber Fy = π−1(y) of the fibrationof M over N . The first and second terms are

Ep,q1 = Ωp(N, Hq(F )) (2.56)

andEp,q

2 = Hp(N, Hq(F )). (2.57)

In the above formulas, Hq(F ) must be understood as a vector bundle over Nwhose fiber over y ∈ N is Hq

dR(Fy), i.e., it is a local system of coefficients. If N issimply connected then this bundle is automatically trivial and we can write

Ep,q2 = Hp

dR(N)⊗HqdR(F ). (2.58)

Example 2.4.3. The de Rham cohomology of the special unitary groups SU(n)can be computed inductively on n with the help of the Leray spectral sequenceassociated to the natural fibration of SU(n) over S2n−1 with fiber SU(n−1) (thisfibration is obtained via the natural action of SU(n) on the unit sphere S2n−1 inCn). When n = 2, SU(2) is diffeomorphic to the three-dimensional sphere S3, sowe will simply write H

dR(SU(2)) = HdR(S3). When n = 3, the second terms of

the Leray spectral sequence of the fibration SU(2)→ SU(3)→ S5 are as follows:

E0,32 = R, E5,3

2 = R,

E0,02 = R, E5,0

2 = R,

(the other second terms are zero). The differential δ2 is automatically trivial,because, for example, it maps the nontrivial term E0,3

2 to the trivial term E2,22 .

Similarly, all the other differentials δr, r ≥ 2, are trivial, because there are onlyfour nontrivial cells E0,3, E5,3, E0,0, E5,0, and no differential δr connects two of


these cells. It means that the Leray spectral sequence degenerates at E2, implyingthat

HdR(SU(3)) ∼= H

dR(SU(2))⊗HdR(S5) ∼= H

dR(S3 × S5). (2.59)

This isomorphism between HdR(SU(3)) and H

dR(S3 × S5) is actually an algebraisomorphism, because the Leray spectral sequence is compatible with the productstructure of the de Rham cohomology in a natural sense. Using this compatibility,one can show inductively on n that the Leray spectral sequence for the fibrationSU(n − 1) → SU(n) → Sn−1 degenerates at the second term E2 for any n ≥ 3,leading to the following algebra isomorphism:

HdR(SU(n)) ∼= H

dR(S3 × S5 × · · · × S2n−1). (2.60)

See, e.g., [41, 138] for details and other applications of Leray spectral sequencesand other spectral sequences in topology.

2.4.3 Hochschild–Serre spectral sequence

Given a Lie algebra l, a Lie subalgebra r ⊂ l, and an l-module W , the Chevalley–Eilenberg complex C(l, W ) has the following natural filtration with respect to r:

Ckh(l, W ) =

f ∈ Ck(l, W ) | f(x1, . . . , xk) = 0 if x1, . . . , xk−h+1 ∈ r

. (2.61)

By convention, Ck0 (l, W ) = Ck(l, W ), and Ck

h(l, W ) = 0 if h > k. One checksdirectly that δCh(l, W ) ⊂ Ch(l, W ), i.e., it is really a filtered complex. The cor-responding spectral sequence is known as the Hochschild–Serre spectral sequence[178].

We will be mainly interested in a special case of this spectral sequence, whenr is an ideal of l, and the quotient Lie algebra g = l/r is semisimple. (This is thecase, for example, when r is the radical , i.e., the maximal solvable ideal of l.) Inthis case, the Hochschild–Serre spectral sequence leads to the following theorem.

Theorem 2.4.4 (Hochschild–Serre [178]). Let l be a finite-dimensional Lie algebraover K (K = R or C), r be an ideal of l such that g = l/r is semisimple, and Wbe a finite-dimensional l-module. Then

Hk(l, W ) ∼=⊕

i+j=k

Hi(g, K)⊗Hj(r, W )g ∀ k ≥ 0. (2.62)

In the above theorem, Hj(r, W ) has a natural structure of g-module whichwill be explained below, and Hj(r, W )g is the subspace of Hj(r, W ) consisting ofelements which are invariant under the action of g.

Proof (sketch). Since g is semisimple, by the classical Levi–Malcev theorem thereis a Lie algebra injection g → l whose composition with the projection map l →l/r = g is identity. (See, e.g., [42, 335], and the beginning of Chapter 3.) In otherwords, we may assume that g is a Lie subalgebra of l. As a vector space, l is the


direct sum of g with r. As a Lie algebra, l can be written a semi-direct productl = g r.

By definition, the zeroth term Ep,q0 of the spectral sequence is

Ep,q0 = Cp+q

p (l, W )/Cp+qp+1 (l, W ). (2.63)

This space can be naturally identified with Cp(g, Cq(r, W )). Indeed, if we de-note by (x1, . . . , xm, y1, . . . , yn−m) a basis of l such that (x1, . . . , xm) span g and(y1, . . . , yn−m) span r, then an element f ∈ Cp+q

p (l, W ) mod Cp+qp+1 (l, W ) is com-

pletely determined by its value on elements of the type

xi1 ∧ · · · ∧ xip ∧ yj1 ∧ · · · ∧ yjq .

The map

θf : xi1 ∧ · · · ∧ xip →(yj1 ∧ · · · ∧ yjq → f(xi1 ∧ · · · ∧ xip ∧ yj1 ∧ · · · ∧ yjq)

)is a linear map from ∧pg to Cq(r, W ), i.e., θf ∈ Cp(g, Cq(r, W )). Note thatCq(r, W ) = ∧qr ⊗ W is a g-module: g acts on W by the restriction of the ac-tion of l; it acts on r by the adjoint action of g in l, and on r∗ by the dual action.It is clear that the correspondence f ↔ θf is one-to-one. Thus, we can write

Ep,q0∼= Cp(g, Cq(r, W )). (2.64)

The next step is to look at the first spectral term Ep,q1 , which is the co-

homology of Ep,q0 with respect to δ0 : Ep,q

0 → Ep,q+10 . Using the identification

Ep,q0∼= Cp(g, Cq(r, W )), we can write δ0 as

δ0 : Cp(g, Cq(r, W )) → Cp(g, Cq+1(r, W )). (2.65)

It follows thatEp,q

1∼= Cp(g, Hq(r, W )). (2.66)

Similarly, we haveEp,q

2∼= Hp(g, Hq(r, W )). (2.67)

Whitehead’s lemma (see Formula (2.35)) implies that

Ep,q2∼= Hp(g, Hq(r, W )g) ∼= Hp(g, K)⊗Hq(r, W )g. (2.68)

If f ∈ ∧pg∗ and g ∈ ∧qr∗ ⊗W are cocycles, and moreover g is invariant under theaction of g on ∧qr∗ ⊗W , then their product

f ⊗ g ∈ ∧pg∗ ⊗ ∧qr∗ ⊗W ⊂ ∧p+ql∗ ⊗W (2.69)

is a cocycle. Equation (2.68) means that elements of Ep,q2 can be written as linear

combinations of such cocycles f ⊗ g. In particular, any element of Ep,q2 can be


represented by a cocycle in Zp+q(l, W ). It implies that δ2 = δ3 = · · · = 0, and theHochschild–Serre spectral sequence degenerates (stabilizes) at E2, i.e.,

Ep,q2 = Ep,q

3 = · · · = Ep,q∞ .

Since the filtration is clearly of finite length, we have

Hp+qp (l, W )/Hp+q

p+1 (l, W ) ∼= Ep,q∞ ∼= Hp(g, K)⊗Hq(r, W )g

and

Hk(l, W ) ∼=⊕

p

Hkp (l, W )

Hkp+1(l, W )

∼=⊕

p+q=k

Hp(g, K)⊗Hq(r, W )g.

2.4.4 Spectral sequence for Poisson cohomology

Given a smooth Poisson manifold (M, Π), there is a natural filtration of the Lich-nerowicz complex, induced by the characteristic distribution as follows. Denote byVq

k(M, Π) the space of smooth q-vector fields Λ on M with the following property:Λ(x) ∈ ∧k(Imx) ∧ ∧q−kTxM ∀ x ∈ M . In other words, Λ(x) is a linear combi-nation of q-vectors of the type Y1 ∧ · · · ∧ Yq with Y1, . . . , Yk ∈ Imx. It is clearthat Π ∈ V2

2 (M, Π), V(M) = V0 (M, Π) ⊃ V

1 (M, Π) ⊃ · · · ⊃ V(rankΠ)(M, Π) ⊃

V(rankΠ)+1(M, Π) = 0, and [Π, Λ] ∈ Vp+1

k (M, Π) if Λ ∈ Vqk(M, Π). So we have a

filtration of finite length. The corresponding spectral sequence was first writtendown explicitly by Vaisman [332, 333], and by Karasev and Vorobjev [338, 339].

Let us mention that the zeroth column Ep,00 of the zeroth term of the above

spectral sequence consists of multi-vector fields which are tangent to the char-acteristic distribution. Consequently, the zeroth column Ep,0

1 of the first term ofthe above spectral sequence consists of tangential Poisson cohomology groups,mentioned in Subsection 2.1.4.

The use of the above spectral sequence in the computation of Poisson co-homology has yielded only limited success so far, mainly in the case when thePoisson structure is regular and the characteristic symplectic foliation is a fibra-tion [338, 332]. For this reason, we will not write down explicitly the above spectralsequence in the general case (the reader may try to do it as an exercise). Instead,we will give here a concrete simple example.

Example 2.4.5. Let M = P×Bn, where P is a closed manifold such that H1dR(P ) =

0, and Bn is an open ball of dimension n = dimH2dR(P ). Let Π be a regular Poisson

structure on M , whose symplectic leaves are P × y, y ∈ Bn, such that the mapBn → H2

dR(P ), y → [ωy], where ωy is the symplectic form of the symplectic leafP × y of Π, is a diffeomorphism from B to its image. Then the second Poissoncohomology of M vanishes: H2

Π(M) = 0. To see this, decompose any 2-vector fieldΛ such that [Λ, Π] = 0 into the sum of three parts, Λ = Λxx+Λxy+Λyy, where Λxx

is tangent to the symplectic leaves, Λxy =∑n

i=1 Xi ∧ ∂/∂yi with Xi being vector


fields tangent to the symplectic leaves and (yi) being a system of coordinates onBn, and Λyy =

∑fij∂/∂yi ∧ ∂/∂yj. The condition [Λ, Π] = 0 is equivalent to the

following system of equations:

[Λxx, Π] = −∑

i

αi ∧ [∂/∂yi, Π], (2.70)∑i

[Xi, Π] ∧ ∂/∂yi =∑i,j

fij [∂/∂yj, Π] ∧ ∂/∂yi, (2.71)

∑i,j

[fij , Π] ∧ ∂/∂yi ∧ ∂/∂yj = 0. (2.72)

The second equation means that [X, Π] =∑

j fij [∂/∂yj, Π] ∀ i. If we fix a symplec-tic leaf y = constant, then [Xi, Π] is exact on that leaf while

∑j fij [∂/∂yj, Π]

is not exact unless fij = 0 because of the hypothesis that y → [ωy] is a dif-feomorphism. Thus the equation [Λ, Π] = 0 implies that fij = 0, i.e., Λyy = 0,and [Xi, Π] = 0 ∀ i. It follows from the hypothesis H1

dR(P ) = 0 that Xi is ex-act on each symplectic leaf, hence we can write Xi = [gi, Π]. The 2-vector fieldΛ′ = Λ +

∑[gi∂/∂yi, Π] is tangent to the symplectic leaves (i.e., Λ′ = Λ′

xx), and[Λ′, Π] = 0. It follows again from the hypothesis that y → [ωy] is a diffeomorphismthat Λ′ is exact, Λ′ = [Z, Π], and so is Λ. Thus, any 2-cocycle is a coboundary, andH2

Π(M) = 0. In terms of spectral sequences, the decomposition Λ = Λxx+Λxy+Λyy

corresponds to the decomposition H2Π(M) ∼= E0,2

∞ ⊕ E1,1∞ ⊕ E2,0

∞ , and we showedthat each of the three summands in this cohomology decomposition is trivial.

Exercise 2.4.6. Write down more explicitly the spectral sequence for the Poissoncohomology of the above example.

Remark 2.4.7. There are some other natural filtrations of the Lichnerowicz com-plex, e.g., the filtration associated to a momentum map, studied by Viktor Ginz-burg [143, 144], and the filtration given by the powers of an ideal (usually themaximal ideal) of functions at a point where the Poisson structure vanishes). Thislast filtration is a general one, appearing in the study of local normal forms of manydifferent objects – we already used it in Section 2.2, without even mentioning thespectral sequence.

2.5 Poisson cohomology in dimension 2

Consider a Poisson structure Π on a two-dimensional surface Σ. For simplicity,assume that Σ is orientable. Then Π can be written as

Π = fΛ (2.73)

where Λ is a given 2-vector field on Σ which does not vanish anywhere.If the zero-level set f−1(0) of f is empty, then Π is a symplectic structure,

and (smooth) Poisson cohomology of Π is the same as the de Rham cohomology

2.5. Poisson cohomology in dimension 2 61

of Σ. The more interesting and difficult case is when f−1(0) = ∅. Using Mayer–Vietoris sequences, the computation of HΠ(Σ) can be reduced to the computationof Poisson cohomology of Π over small neighborhoods of points x ∈ f−1(0). LetU x be a small open disk containing a point x such that f(x) = 0. Under “rea-sonable assumptions” (for example, when x is either a regular point or an isolatedsingular point of f), one can show that HΠ(U) is isomorphic to the germified(smooth) cohomology of Π at x. Thus, the computation of HΠ(Σ) can often bereduced to the computation of germified cohomology.

In this section, we will present the results of Monnier [262] on germifiedPoisson cohomology of a large class of two-dimensional Poisson structures. Thisclass consists of Poisson tensors of the form

Π = f(1 + h)∂

∂x∧ ∂

∂y, (2.74)

where f is a quasi-homogeneous polynomial of degree D with respect to a quasi-radial vector field Z = w1x

∂∂x + w2

∂∂y (w1, w2 ∈ N), i.e., Z(f) = Df , and either

h = 0 or h is a non-constant quasi-homogeneous polynomial of degree D−w1−w2.(If D−w1 −w2 can’t be written as D−w1 −w2 = m1w1 + m2w2 with m1, m2 ∈Z, m1, m2 ≥ 0, m1 + m2 > 0, then h = 0.)

One of the reasons why this class of Poisson structures is interesting is thatit contains all simple singularities of two-dimensional Poisson structures in thesense of Arnold [15]. Before showing the cohomology of these Poisson structures,we will first discuss briefly Arnold’s classification of simple singularities.

2.5.1 Simple singularities

Two germs f1, f2 of (smooth or analytic) functions on K2, where K = R or C, arecalled R-equivalent if there is a germ of diffeomorphism φ : (K2, 0)→ (K2, 0) suchthat f1 = f2 φ. A germ g is called adjacent to f if there is a sequence of germsgn, n ∈ N, such that gn is R-equivalent to g and limn→∞ gn = f . This limit maybe understood in a formal sense, i.e., for any k ∈ N the k-jet of gn at 0 tends tothe k-jet of f when n →∞.

A germ of function f such that f(0) = 0 and df(0) = 0 is called a sim-ple singularity if up to R-equivalence there are only a finite number of functiongerms adjacent to it. Simple singularities of functions of many variables are definedsimilarly.

Theorem 2.5.1 (Arnold [13]). The following is a complete list of simple singulari-ties, up to R-equivalence, of functions of two variables:

A±k (k ≥ 1) D±

k (k ≥ 4) E±6 E7 E8

x2 ± yk+1 x2y ± yk−1 x3 ± y4 x3 + xy3 x3 + y5

If K = C, or if k is even in the real Ak case, then the symbols ± can be omittedor replaced by +.


The above classification also holds in higher dimensions: to get any simplesingularity with n variables, one simply adds a Morse term

∑ni=3±x2

i to a simplesingularity with two variables. The letters A, D, E in the above classification aredue to natural relations between these simple singularities and Dynkin diagramsof the same names. See, e.g., [16], for details.

Two Poisson germs Π1 = f1∂∂x ∧ ∂

∂y and Π2 = f2∂∂x ∧ ∂

∂y are equivalent ifthere exists a germ of diffeomorphism ϕ : (K2, 0)→ (K2, 0) such that ϕ∗Π1 = Π2.This condition means that

f2 ϕ = Jac(ϕ)f1, (2.75)

where Jac(ϕ) denotes the Jacobian of ϕ. Due to the additional term Jac(ϕ),Poisson tensors in dimension 2 do not behave exactly like functions under diffeo-morphisms, but quite similarly.

Theorem 2.5.2 (Arnold [15]). Let f ∈ F(K2, 0) be a simple singularity. Then, thePoisson germ Π = f ∂

∂x ∧ ∂∂y is equivalent, up to a multiplicative constant, to a

Poisson germ of the type g ∂∂x ∧ ∂

∂y , where g is in the following list (where λ is aconstant):

A2p : x2 + y2p+1, p ≥ 1A±

2p−1 : (x2 ± y2p)(1 + λyp−1), p ≥ 1D±

2p : (x2y ± y2p−1)(1 + λyp−1), p ≥ 2D2p+1 : (x2y + y2p)(1 + λx), p ≥ 2E6 : x3 + y4

E7 : (x3 + xy3)(1 + λy2)E8 : x3 + y5

(2.76)

When K = C, the symbols ± can be omitted or replaced by +.

Remark 2.5.3. A simple case excluded from the above theorem is when f(0) = 0but df(0) = 0. It is easy to see that in this case Π is isomorphic to the linearPoisson structure x ∂

∂x ∧ ∂∂y (see Theorem 4.2.1).

The proof of Theorem 2.5.2 is based on Theorem 2.5.1 and the following

Theorem 2.5.4 ([15]). Let Π = fa ∂∂x ∧ ∂

∂y , where f is a quasi-homogeneous poly-nomial of degree D > 0 with respect to a given quasi-radial vector field Z =w1x

∂∂x + w2y

∂∂y (w1, w2 ∈ N), and a(0) = 0. Then up to a multiplicative con-

stant, the germ of Π at 0 is equivalent to a germ of Poisson structure of the typef(1 + h) ∂

∂x ∧ ∂∂y (with the same f), where either h = 0 or h is a non-constant

quasi-homogeneous polynomial of degree D − w1 − w2.

Proof (sketch). In order to preserve f , one uses only coordinate transformationsgenerated by vector fields of the type αZ, where α is a function. In the formalcategory, one shows easily that all the quasi-homogeneous terms in a, except theconstant term and the terms of quasi-homogeneous degree D − w1 − w2, can bekilled consecutively by such coordinate transformations. To prove the result inanalytic and smooth categories, one first kills the terms of degree < D −w1 −w2


as above to arrive at a Poisson germ of the type f(1 + h + R) ∂∂x ∧ ∂

∂y , wheredeg(h) = D − w1 − w2 and the Taylor expansion of R contains only terms ofdegree > D−w1 −w2. One then kills R by Moser’s path method. For details, see[15, 262].

2.5.2 Cohomology of Poisson germs

In this subsection, we will denote by

Π0 = f∂

∂x∧ ∂

∂y(2.77)

a Poisson structure on K2, where f is a quasi-homogeneous polynomial functionof degree D with respect to a given quasi-radial vector field Z = w1x

∂∂x + w2y

∂∂y ,

where w1, w2 ∈ N. Recall that it means that f is polynomial and LZf = Df . Wewill also denote by

Π = f(1 + h)∂

∂x∧ ∂

∂y(2.78)

a deformation of Π, where h is a quasi-homogeneous polynomial function of degreeD − w1 − w2 with respect to Z. (If D − w1 − w2 ≤ 0 then h = 0.)

We will denote by F(K2, 0),V1(K2, 0) and V2(K2, 0) the space of germs offunctions, vector fields and 2-vector fields on K2, respectively. We will view Π0 andΠ as germs of Poisson structures at 0, and denote their germified cohomology byHgerm(Π0) and Hgerm(Π). The spaces of germified k-cocycles and k-coboundariesof the Lichnerowicz complex of Π (resp. Π0) will de denoted by Zk

germ(Π) andBk

germ(Π) (resp. Zkgerm(Π0) and Bk

germ(Π0)).Actually, by a germ, we will mean either a C∞-smooth germ (K = R), a real

analytic germ (K = R), a holomorphic germ (K = C), or a formal series (K = R

or C). So there are several versions of this germified cohomology. However, theresulting cohomology groups (for the Poisson structures (2.77) and (2.78)) will bethe same, so we consider these different versions together.

Recall that f is said to have finite codimension (or equivalently, the germof f at 0 is an isolated singularity, if indeed it is a singularity), if its multiplicityc = dimQf , where Qf = F(K2, 0)/If , is finite. Here F(K2, 0) is the space of germsof functions on K2, and If denotes the ideal generated by ∂f

∂x and ∂f∂y . This ideal

consists of germs of functions of the type Y (f), Y ∈ V1(K2, 0).

Theorem 2.5.5 (Monnier [262]). With the above notations, assume that f has finitecodimension. Then we have:

a) H0germ(Π) ∼= H0

germ(Π0) ∼= K and Hkgerm(Π) = Hk

germ(Π0) = 0 if k ≥ 3.b) H1

germ(Π) ∼= H1germ(Π0) ∼= Kr+1, where r is the dimension of the space of

polynomials which are quasi-homogeneous of degree D−w1−w2 with respectto Z.

c) H2germ(Π) ∼= H2

germ(Π0) ∼= Kr+c, where r is defined as above, and c is themultiplicity of f at 0.


Assertion a) in the above theorem is clear, because the only Casimir functionsare constants, and the dimension of the manifold is 2. The proof of b) and c)consists of several lemmas.

In this subsection, for a function g on K2, we will denote by Hg the Hamil-tonian vector field of g with respect to the standard symplectic form dx ∧ dy:

Hg =∂g

∂x

∂

∂y− ∂g

∂y

∂

∂x. (2.79)

Lemma 2.5.6. Let X be in V1(K2, 0). Then we have:

i) If Div X = 0, where Div X denotes the divergence of X with respect to thevolume form dx ∧ dy, then there exists g ∈ F(K2, 0) such that X = Hg.

ii) If X(f) = 0, then X = αHf for some α ∈ F(K2, 0).

Proof. Write X = A ∂∂x+B ∂

∂y . Consider the 1-form θ = −Bdx+Ady = iX(dx∧dy).i) If Div X = 0 then dω = 0, which implies that θ = −dg for some g ∈ F(K2),

and so X = Hg.ii) If X(f) = 0 then df ∧ θ = 0. Since f has finite codimension, de Rham’s

division theorem [98] implies that θ = −αdf for some α ∈ F(K2, 0).

Lemma 2.5.7. Let X ∈ Z1(Π0). Then there exists α ∈ F(K2) such that

X = αHf + Div XD Z.

Proof. 0 = LX(f ∂∂x ∧ ∂

∂y ) = (X(f) − (Div X)f) ∂∂x ∧ ∂

∂y , implying that X(f) =(Div X)f = Div X

D Z(f). Now apply Lemma 2.5.6.

Lemma 2.5.8. Let X ∈ Z1(Π0) be such that ord(X) > D−w1−w2, where ord(X)denotes the order of X at 0 with respect to Z (if X0 is the quasi-homogeneous partof X with respect to Z, then LZX0 = ord(X)X0). Then X ∈ B1(Π0).

Proof. Suppose for the moment that Div X = 0. Then X(f) = (Div X)f = 0,hence X = γHf by Lemma 2.5.6. Since ord(X) > D−w1−w2 and ord(Hf ) = D−w1−w2, we have ord(γ) > 0, so γ(0) = 0. It is easy to see that, because γ(0) = 0,there exists µ ∈ F(K2, 0) such that Z(µ) = γ. We have ord(µ) = ord(γ) > 0.Hence ord(Hf (µ)) ≥ ord(µ) + ord(Hf ) > ord(Hf ) = D − w1 − w2. On the otherhand, we have

0 = Div X = Hf (γ) = Hf (Z(µ)) = Z(Hf (µ)) + [Hf , Z](µ)= Z(Hf (µ))− (D − w1 − w2)Hf (µ),

i.e., Hf (µ) either vanishes or is quasi-homogeneous of degree D−w1−w2. ThereforeHµ(f) = −Hf (µ) = 0. By Lemma 2.5.6, there exists ν ∈ F(K2, 0) such that∂µ∂x = ν ∂f

∂x and ∂µ∂y = ν ∂f

∂y . Therefore, Z(µ) = ν Z(f), which means that γ = Dνf .We deduce that X = fY with Y ∈ V1(K2, 0). Finally, since X ∈ Z1(Π0), we


have Div Y = 0, hence Y = Hg for some g ∈ F(K2) by Lemma 2.5.6. ThusX = fHg = Xg (the Hamiltonian vector field of g with respect to Π0).

Consider now the case Div X = 0. If we find β ∈ F(K2, 0) such that Div X =Div Xβ, then the 1-cocycle X−Xβ satisfies Div(X −Xβ) = 0, which implies thatX − Xβ = Xε and X = Xβ+ε for some ε ∈ F(K2). Since Div Xβ = Hβ(f) =−Hf(β), we are looking for β such that Hf (β) = −Div X .

By Lemma 2.5.7, we have X = αHf + Div XD Z for some α ∈ F(K2, 0). Taking

the divergence of the two sides of this equation, we get:

Z(Div X)− (D − w1 − w2)Div X = −DHf(α).

Note that, because ord(X) > D − w1 − w2, we have ord(α) > 0, which impliesthat there is β ∈ F(K2) such that Z(β) = Dα (and moreover, ord(β) > 0). Simplecalculations show that Z(Div X + Hf (β)) = (D−w1 −w2)(Div X + Hf (β)), i.e.,Div X+Hf(β) is either 0 or quasi-homogeneous of degree D−w1−w2. On the otherhand, by evaluating directly the order, we get ord(Div X +Hf (β)) > D−w1−w2.Therefore Div X = −Hf (β).

Denote by e1, . . . , er a linear basis of the space of quasi-homogeneous poly-nomials of degree D−w1−w2. For X ∈ Z1

germ(Π0) (resp. Z1germ(Π0)), we will denote

by [X ]Π0

(resp. [X ]Π) its cohomological class in H1

germ(Π0) (resp. H1germ(Π)).

Lemma 2.5.9. The family[Hf ]Π0

, [e1Z]Π0, . . ., [erZ]Π0

is a linear basis of

H1germ(Π0). In particular, dimH1(Π0) = r + 1 < ∞.

Proof. First we prove that H1germ(Π0) is spanned by this family.

Lemma 2.5.8 implies that every X in Z1germ(Π0) is cohomologous to a poly-

nomial vector field of quasi-homogeneous degree ≤ D − w1 − w2. (Decompose Xas X = X1 + X2 where ord(X1) > D − w1 − w2 and X2 is a polynomial vectorfield of quasi-homogeneous degree at most D − w1 − w2. Then both X1 and X2

belong to Z1germ(Π0), and moreover X1 is a coboundary by Lemma 2.5.8.)

On the other hand, if X ∈ Z1germ(Π0) is quasi-homogeneous of degree deg X <

D−w1−w2, then X = 0. Indeed, according to Lemma 2.5.7, we have X = Div XD Z,

and so

Div X =Div X

DDiv Z + Z

(Div X

D

),

which implies that (D − w1 − w2 − deg X)Div X = 0.The above arguments show that H1

germ(Π0) is spanned by quasi-homogeneous1-cocycles of degree D−w1−w2. If X ∈ Z1(Π0) is such a cocycle, then by Lemma2.5.7 we have X = αHf + Div X

D Z, where α ∈ K and Div X is a quasi-homogeneouspolynomial of degree D − w1 − w2. Therefore the family

[Hf ]Π0

, [e1Z]Π0, . . .,

[erZ]Π0

generates H1

germ(Π0).Now let us prove that this family is free. Suppose that

∑i λieiZ + αHf ∈

B1germ(Π0), where α, λ1, . . . , λr are scalars. Then

∑i λieiZ + αHf = 0. Indeed, if

g is a quasi-homogeneous polynomial, then deg Xg = deg Hf + deg g = D − w1 −


w2 + deg g, which is strictly larger than D − w1 − w2 as soon as g = constant.Consequently, Div

(∑i λieiZ + αHf

)= 0, i.e.,

∑i λiei = 0. We deduce that

λ1, . . . , λr are 0, and so α = 0.

Lemma 2.5.10.[(1 + h)Hf ]Π , [(1 + h)e1W ]Π , . . . , [(1 + h)erW ]Π

is a basis of

H1germ(Π). In particular, H1

germ(Π) ∼= H1germ(Π0).

Proof. Notice that X ∈ Z1germ(Π) (resp. X ∈ B1

germ(Π)) if and only if X1+h ∈

Z1germ(Π0) (resp. B1

germ(Π0)).

Lemma 2.5.11. Let g ∈ F(K2, 0).

i) If g ∂∂x ∧ ∂

∂y ∈ B2germ(Π0), then g ∈ If .

ii) If the ∞-jet at 0 of g does not contain components of quasi-homogeneousdegree 2D − w1 − w2, then g ∂

∂x ∧ ∂∂y ∈ B2

germ(Π0) if and only if g ∈ If .

Proof. i) For X ∈ V1(K2, 0), we have

[Π0, X ] =((Div X)f −X(f)

) ∂

∂x∧ ∂

∂y. (2.80)

So g ∂∂x ∧ ∂

∂y ∈ B2germ(Π0) if and only if g can be written as g = X(f)−(Div X)f =

Y (f) ∈ If , where Y = X − Div XD Z.

ii) Assume that g = Y (f) ∈ If , and that the ∞-jet of g does not con-tain a component of degree 2D − w1 − w2. We will find X ∈ V1(K, 0) such thatg = X(f) − (Div X)f . In the analytic (or formal) case, write g =

∑i≥0 g(i) and

Y =∑

i≥D−max(w1,w2)Y (i−D), where g(i) is quasi-homogeneous of degree i and

Y (i−D) is quasi-homogeneous of degree i−D. Note that Y (D−w1−w2) = 0. Directcalculations show that X can be given by

X = Y +∑

i=2D−w1−w2

Div Y (i−D)

2D − w1 − w2 − iZ . (2.81)

In the C∞-smooth case, using the result from the formal case and Borel’s theoremabout existence of smooth functions with an arbitrary given Taylor series, we canwrite g = Y (f) = Y1(f) + X1(f) − (Div X1)f , where X1, Y1 are smooth, and Y1

is flat at 0. The equation

Z(α)− (D − w1 − w2)α = −Div Y1 (2.82)

has a smooth (actually flat) solution α, and we can put X = αZ + X1.

Remark 2.5.12. Lemma 2.5.11 holds true even when f is not of finite codimension,and implies that the map g ∂

∂x ∧ ∂∂y → g induces a surjection from H2

germ(Π0) =V2(K, 0)/B2

germ(Π0) to Qf = F(K, 0)/If . Therefore, if f is of infinite codimension,i.e., dim Qf = ∞, then dimH2

germ(Π0) = ∞.


Denote by u1, . . . , uc a monomial linear basis of Qf = F(K2, 0)/If , c =dimQf . In other words, u1, . . . , uc are monomial functions which span F(K2, 0)mod If . We will need the following result from the singularity theory of functions(see [16]): such a monomial linear basis always exists, and moreover deg(ui) ≤2(D−w1−w2) ∀i, where deg denotes the quasi-homogeneous degree. In particular,any quasi-homogeneous function of degree > 2(D − w1 − w2) belongs to If .

Denote by e1, . . . , er a basis of the space of quasi-homogeneous polynomialsof degree D−w1−w2. For a function g ∈ F(K2), denote by [g]Π0

the cohomologyclass of g ∂

∂x ∧ ∂∂y in H2

germ(Π0).

Lemma 2.5.13. The family[e1f ]Π0

, . . . , [erf ]Π0, [u1]Π0

, . . . , [uc]Π0

is a basis of

H2germ(Π0). In particular, dimH2

germ(Π0) = r + c.

Proof. First let us show that this family generates dim H2germ(Π0). Let g∈F(K2,0).

Write g =∑c

i=1 λiui + ξ where λi ∈ K and ξ ∈ If . Invoking Lemma 2.5.11, wecan write

g =c∑

i=1

λiui + g′ mod B2germ(f),

where g′ is a quasi-homogeneous polynomial of degree 2D − w1 − w2, and

B2germ(f) =

ψ ∈ F(K2, 0) | ψ ∂

∂x∧ ∂

∂y∈ B2

germ(Π0)

. (2.83)

Since deg(g′) = 2D−w1 −w2 > 2(D−w1 −w2), g′ belongs to If , i.e., g′ = X(f)for some quasi-homogeneous vector field X of degree D − w1 − w2. So we canwrite g′ = (Div X)f +

(X(f)− (Div X)f

)≡ (Div X)f mod B2

germ(Π0). Div X isquasi-homogeneous of degree D−w1−w2, so [(Div X)f ]Π0

is in the linear hull of[e1f ]Π0

, . . . , [erf ]Π0.

Now let us show that this family is free. Let g1 =∑r

i=1 λiei and g2 =∑cj=1 µjuj , with λi, µj ∈ K, such that g1f +g2 ∈ B2

germ(f) ⊂ If . Since deg(g1f) =2D − w1 − w2 > 2(D − w1 − w2), we have g1f ∈ If , so g2 is also in If . It ispossible only if g2 = 0, and µj = 0 ∀j. We are left with g1f ∈ B2

germ(f), i.e.,g1f = X(f) − (Div X)f for some quasi-homogeneous vector field X of degreeD − w1 − w2. Denote by Y = g1+Div X

D Z, then (X − Y )(f) = 0. Therefore,by Lemma 2.5.6, X = Y + αHf with α ∈ K. Note that Y ∈ Z1

germ(Π0) andHf ∈ B1

germ(Π0). Hence X ∈ Z1germ(Π0), i.e., g1 = X(f) − (Div X)f = 0, which

implies that λ1 = · · · = λr = 0.

Lemma 2.5.14. The family[e1f ]Π , . . ., [erf ]Π , [u1]Π , . . ., [uc]Π

is a basis of

H2germ(Π). In particular, H2

germ(Π) ∼= H2germ(Π0) ∼= Kr+c.

The proof of Lemma 2.5.14 is similar to the proof of Lemma 2.5.13, thoughsomewhat more complicated. We will leave it as an exercise (see [262]).


2.5.3 Some examples and remarks

Example 2.5.15. The case when f = x is regular: Π = Π0 = x ∂∂x ∧ ∂

∂y . In thiscase, we have w1 = w2 = D = 1, If = F(K2, 0), r = c = 0, H1

germ(Π) ∼= K ∂∂y , and

H2germ(Π) = 0.

Example 2.5.16. Morse singularity (A1): Suppose that Π = (x2 +y2) ∂∂x ∧ ∂

∂y . Herewe have w1 = w2 = 1 and D = 2. The only polynomials of degree D − w1 − w2

are the scalars. In particular, r = 1 Moreover, Qx2+y2 ∼= K.1, implying that c =dimQx2+y2 = 1. Hence dimH1

germ(Π) = r + 1 = 2 and dimH2germ(Π) = r + c = 2.

More precisely, these cohomology groups can be written as follows:

H1germ(Π) ∼= K

(y

∂

∂x− x

∂

∂y

)⊕K

(x

∂

∂x+ y

∂

∂y

),

H2germ(Π) ∼= K

(∂

∂x∧ ∂

∂y

)⊕K

(f

∂

∂x∧ ∂

∂y

).

The origin is the only singular point of f = x2 + y2, and when K = R the abovegermified cohomology groups are naturally isomorphic to the smooth Poisson co-homology groups of (x2+y2) ∂

∂x ∧ ∂∂y on R2. These groups were computed earlier in

[143] and [273]. Vaisman [333] also mentioned them, without actually computingthem. Similarly, the Poisson structure (x2− y2) ∂

∂x ∧ ∂∂y has the same cohomology.

Example 2.5.17. Singularity D2p+1, p ≥ 2: Suppose that

Π = (x2y + y2p)(1 + x)∂

∂x∧ ∂

∂y.

Then w1 = 2p − 1, w2 = 2 and D = 4p. The the family 1, x, y, y2, . . . , y2p−1is a monomial basis of Qx2y+y2p , and space of polynomials of quasi-homogeneousdegree D−w1−w2 is spanned by x. In particular, r = 1, c = 2p+1, dimH1

germ(Π) =2, and dimH2

germ(Π) = 2p + 2.

Example 2.5.18. The algebraic Poisson cohomology of homogeneous Poisson struc-tures on K2 was computed by Roger and Vanhaecke in [302] (algebraic means thatone considers only cochains with algebraic coefficients), by direct algebraic calcula-tions instead of using singularity theory. The result is as follows: Let Π = f ∂

∂x ∧ ∂∂y

be a homogeneous Poisson structure of degree n, where f =∏n

i=1(x−aiy), ai ∈ K.Then dim H2

Π(K2) = ∞ if φ is not reduced, i.e., if ai = aj for some i = j. Ifai = aj ∀i = j then H1

Π(K2) ∼= Kn and H2Π(K2) ∼= Kn(n−1). They are isomorphic

to germified Poisson cohomology groups of Π: r = n−1 and c = (n−1)2 in this case.

In principle, cohomology of Poisson structures on closed two-dimensionalsurfaces can be computed via local results and Mayer–Vietoris sequences. Forexample, consider a Poisson structure of the type Λ = fΛ0 on an orientableclosed surface Σ of genus g, where Π0 is nondegenerate, f is a function such

2.6. The curl operator 69

that the zero level set S = f = 0 of f is regular and consists of n connectedcomponents (n circles on which Λ vanishes). Then Poisson cohomology of such a Λwas computed by Radko [296], based on some earlier computations of Roytenberg[305]. The result is: H0(Σ, Λ) = R1, H1(Σ, Λ) ∼= Rn ⊕ H1

dR(Σ) ∼= Rn+2g, andH2(Σ, Λ) ∼= Rn⊕H2

dR(Σ) ∼= Rn+1 (the Lichnerowicz map is injective in this case).The dimension n + 1 of H2(Σ, Λ) corresponds to n + 1 numerical invariants of Λ,which may be described as follows:• Near each circle on which Λ vanishes (connected component of the set S),

there is a coordinate system (θ, x), where θ ∈ R/Z is a periodic coordinate,in which Λ = x

c∂∂θ ∧ ∂

∂x for some positive constant c. This number c is aninvariant of Λ, and is the period of a curl vector field of Λ on the circle (seeSection 2.6): any curl vector field of Λ will have S as a union of periodic orbitsand gives an orientation of S, and this orientation is also an invariant of λ.Since there are n circles on which Λ vanishes, we have n periods c1, . . . , cn.

• Let U be a neighborhood of S, such that each connected component of U isan annulus Ui with canonical coordinates (θi, xi) such as above (Ui = −εi <xi < εi for some positive εi). Λ is nondegenerate on Σ \U , and the number

V ol(Λ) =∫

Σ\U

ω,

where ω is the symplectic form dual to Λ, does not depend on the choice ofU . This number is called the regularized volume of (Σ, Λ) and is an invariantof Λ.

Theorem 2.5.19 (Radko [296])). With the above notations and assumptions, thePoisson structure Λ = fΛ0 on Σ is uniquely determined, up to Poisson isomor-phisms, by the topological configuration of the oriented singular set S, the periodsc1, . . . , cn, and the regularized volume V ol(Λ).

The proof of the above theorem, based on Moser’s path method (see Ap-pendix A.1), is relatively simple. See [296, 303] for some generalizations of theabove results to the case when Σ is not necessarily closed and S may be a sin-gular level of f , and [240] for a generalization of Theorem 2.5.19 to the case ofmulti-vector fields of top degree on a manifold.

2.6 The curl operator

2.6.1 Definition of the curl operator

Recall that, if A is an a-vector field and ω is a differential p-form with p ≥ a, thenthe inner product of ω by A is a unique (p − a)-form, denoted by iAω or Aω,such that

〈iAω, B〉 = 〈ω, A ∧B〉 (2.84)

for any (p− a)-vector field B. If p < a then we put iAω = 0 by convention.


For example, if X is a vector field then iXω(X1, . . . , Xp−1) = 〈iXω, X1∧· · ·∧Xp−1〉 = 〈ω, X ∧X1 ∧ · · · ∧Xp−1〉 = ω(X, X1, . . . , Xp−1).

Similarly, when a ≥ p, then we can define the inner product of an a-vectorfield A by a p-form η to be a unique (a− p)-vector field, denoted by iηA or ηA,such that

〈β, iηA〉 = 〈β ∧ η, A〉 (2.85)

for any (a− p)-form β.Warning: Due to the noncommutativity of the wedge product, one must be

careful with the signs when dealing with inner products. Also, our sign conventionmay be different from some other authors.

Exercise 2.6.1. If f is a function and A a multi-vector field then

idfA = [A, f ] . (2.86)

In particular, the Hamiltonian vector field of f with respect to a given Poissonstructure Π is Xf = Π(df) = −idfΠ.

Let Ω be a smooth volume form on an m-dimensional manifold M , i.e., anowhere vanishing differential m-form. Then for every p = 0, 1, . . . , m, the map

Ω : Vp(M) −→ Ωm−p(M) (2.87)

defined by Ω(A) = iAΩ, is a C∞(M)-linear isomorphism from the space Vp(M)of smooth p-vector fields to the space Ωm−p(M) of smooth (m − p)-forms. Theinverse map of Ω is denoted by Ω : Ωn−p(M) −→ Vp(M), which can be definedby Ω(η) = iηΩ, where Ω is the dual m-vector field of Ω, i.e., 〈Ω, Ω〉 = 1.

Exercise 2.6.2. Prove the formula (ηΩ)Ω = η.

Denote by DΩ : Vp(M) −→ Vp−1(M) the linear operator defined by DΩ =Ω# d Ω. Then we have the following commutative diagram:

Vp(M) Ω

−−−−−→ Ωm−p(M)

DΩ

$$ d

Vp−1(M) Ω

−−−−−→ Ωm−p+1(M)

(2.88)

Since d d = 0, we also have DΩ DΩ = 0.

Definition 2.6.3. The above operator DΩ is called the curl operator (with respectto the volume form Ω). If A is an a-vector field then DΩA is called the curl of A(with respect to Ω).

Example 2.6.4. The curl DΩX of a vector field X is nothing but the divergence ofX with respect to the volume form Ω: (DΩX)Ω = Ω(DΩX) = diXΩ = LXΩ =(DivΩX)Ω, which implies that DΩX = DivΩX .


In a local system of coordinates (x1, . . . , xn) with Ω = dx1 ∧ · · · ∧ dxn, anddenoting ∂

∂xiby ζi as in Section 1.8, we have the following convenient formal

formula for the curl operator:

DΩA =∑

i

∂2A

∂xi∂ζi. (2.89)

The following proposition shows what happens to the curl when we changethe volume form.

Proposition 2.6.5. If f is a non-vanishing function, then we have

DfΩA = DΩA + [A, ln |f |] . (2.90)

Proof. We have DfΩA−DΩA = Ω#Ω(DfΩA−DΩA) = Ω#( 1f diA(fΩ)− diAΩ)

= Ω#(d ln |f | ∧ iAΩ) = id ln |f |Ω#(iAΩ) = id ln |f |A = [A, ln |f |]. Remark 2.6.6. It follows from the above proposition that, if we multiply the volumeform by a non-zero constant, then the curl operator does not change. In particular,the curl operator DΩ can be defined on non-orientable manifolds as well. Non-orientable manifolds don’t admit global volume forms in the sense of non-vanishingdifferential forms of top degree, but they do admit measure-theoretic volume formswith smooth positive distribution. Such a measure-theoretic volume form is a non-oriented (or absolute) version of differential volume forms, and is also called adensity. Proposition 2.6.5 implies that one can replace a volume form by a densityin the definition of the curl operator.

2.6.2 Schouten bracket via curl operator

Theorem 2.6.7 (Koszul [201]). If A is an a-vector field, B is a b-vector field andΩ is a volume form then

[A, B] = (−1)bDΩ(A ∧B)− (DΩA) ∧B − (−1)bA ∧ (DΩB). (2.91)

Proof. By Formula (2.89) and Formula (1.71) we have:(−1)bDΩ(A ∧B) = (−1)b

∑ ∂2(A∧B)∂xi∂ζi

=∑ ∂

∂xi

(∂A∂ζi

B + (−1)bA ∂B∂ζi

)=∑

∂2A∂xi∂ζi

B + (−1)bA∑

∂2B∂xi∂ζi

+ ∂A∂ζi

∂B∂xi

+ (−1)b∑

∂A∂xi

∂B∂ζi

= (DΩA)B + (−1)bA(DΩB) +(

∂A∂ζi

∂B∂xi

− (−1)(b−1)(a−1)∑

∂B∂ζi

∂A∂xi

)= (DΩA)B + (−1)bA(DΩB) + [A, B].

The curl operator is, up to a sign, a derivation of the Schouten bracket. Moreprecisely, we have the following formula:

DΩ[A, B] = [A, DΩB] + (−1)b−1[DΩA, B]. (2.92)

Exercise 2.6.8. Prove the above formula, either by direct calculations, or by usingTheorem 2.6.7 and the fact that DΩ DΩ = 0.


2.6.3 The modular class

A particularly important application of the curl operator in Poisson geometry isthe curl vector field DΩΠ, also called modular vector field , of a Poisson structureΠ with respect to a volume form Ω. This curl vector field is an infinitesimalautomorphism of the Poisson structure, i.e., it is a Poisson vector field. Moreover,it also preserves the volume form:

Lemma 2.6.9. If Π is a Poisson tensor and Ω a volume form, then

[DΩΠ, Π] = 0 and L(DΩΠ)Ω = 0. (2.93)

Proof. It follows from Formula (2.92) and the fact that [Π, Π] = 0 that we have0 = DΩ[Π, Π] = [Π, DΩΠ]−[DΩΠ, Π] = −2[DΩΠ, Π]. Hence we have [DΩΠ, Π] = 0.

To prove the second equality, we don’t even need the fact that Π is a Poissonstructure. Indeed, we have L(DΩΠ)Ω = i(DΩΠ)dΩ+di(DΩΠ)Ω = d(d(ΠΩ)) = 0. Exercise 2.6.10. Show that the curl vector field of the linear Poisson structureΠ = y ∂

∂x ∧ ∂∂y with respect to the volume form dx ∧ dy is ∂

∂x , and it is not aHamiltonian vector field.

Lemma 2.6.9 means that the curl vector field DΩΠ is a 1-cocycle in theLichnerowicz complex. Proposition 2.6.5 implies that if we change the volume form(or more precisely, the density, see Remark 2.6.6) then this cocycle changes by acoboundary. Thus the cohomology class of the curl vector field DΩΠ in H1(M, Π)does not depend on the choice of the volume form Ω.

Definition 2.6.11. If (M, Π) is a Poisson manifold and Ω a smooth density on M ,then the cohomology class of the curl vector field DΩΠ in H1(M, Π) is called themodular class of (M, Π). If this class is trivial, then (M, Π) is called a unimodularPoisson manifold .

Definition 2.6.12. A density Ω on a Poisson manifold (M, Π) is called an invariantdensity if it is preserved by all Hamiltonian vector fields on M : LXf

Ω = 0 ∀ f ∈C∞(M).

Lemma 2.6.13. If Π is a Poisson structure and Ω is a smooth density, then DΩΠ =0 if and only if Ω is an invariant density. In particular, a Poisson manifold isunimodular if and only if it admits a smooth invariant density.

We will leave the proof of the above lemma as an exercise. Exercise 2.6.14. Show that, if (M2n, ω) is a symplectic manifold of dimension 2n,then it is unimodular as a Poisson manifold. Up to multiplication by a constant,the only invariant volume form on M is the so-called Liouville form

Ω =1n!∧n ω. (2.94)

(Don’t confuse this Liouville volume form with the Liouville 1-form mentioned inExample 1.1.9.)


Exercise 2.6.15. A unimodular Lie algebra is a Lie algebra g such that for anyx ∈ g, the linear operator adx : g → g is traceless. In other words, g is calledunimodular if its adjoint action preserves a standard volume form. Show that alinear Poisson structure is unimodular if and only if its corresponding Lie algebrais unimodular.

For more about the modular class, see, e.g., [1, 127, 130, 146, 181]. For thetheory of (secondary) characteristic classes of Poisson manifolds (and Lie alge-broids), of which the modular class is a particular case, see Fernandes [130] andCrainic [85].

2.6.4 The curl operator of an affine connection

Recall that a linear connection on a vector bundle E over a manifold M is anR-bilinear map

∇ : V1(M)× Γ(E)→ Γ(E), (X, ξ) → ∇Xξ, (2.95)

(where Γ(E) denotes the space of sections of E), which is C∞(M)-linear withrespect to X , i.e., ∇fXξ = f∇Xξ ∀f ∈ C∞(M), and which satisfies the Leibnizrule with respect to ξ, i.e., ∇X(fξ) = f∇Xξ + X(f)ξ. A linear connection is alsocalled a covariant derivation on E.

Let ∇ be an affine connection on a manifold M , i.e., a linear connection onthe tangent bundle TM of M . By the Leibniz rule, one can extend ∇ to a map

∇ : V1(M)× V(M)→ V(M) (2.96)

(and more generally, to a covariant derivation on all kinds of tensor fields on M).For example, ∇X(Y ∧ Z) = (∇XY ) ∧ Z + Y ∧ (∇XZ). The operator

D∇ =∑

k

idxk ∇∂/∂xk

: V(M) → V(M), (2.97)

where (x1, . . . , xm) denotes a system of coordinates on M , is called the curl oper-ator of ∇.

Exercise 2.6.16. Show that the above definition of D∇ does not depend on thechoice of local coordinates.

Recall that, an affine connection ∇ on a manifold M is called torsionless if∇XY − ∇Y X = [X, Y ] for all X, Y ∈ V1(M). We have the following statement,similar to Theorem 2.6.7:

Theorem 2.6.17 (Koszul [201]). If A is an a-vector field, B is a b-vector field and∇ is a torsionless affine connection, then

[A, B] = (−1)bD∇(A ∧B)− (D∇A) ∧B − (−1)bA ∧ (D∇B). (2.98)


Proof. By induction, using the Leibniz identity.

If ∇ is a flat torsionless connection with (x1, . . . , xm) as a trivializing co-ordinate system, i.e., ∇∂/∂xi

∂/∂xj = 0 ∀i, j, then Formula (2.97) coincides withFormula (2.89).

2.7 Poisson homology

Given a Poisson manifold (M, Π), there is another differential complex associatedto it, with the following differential operator:

δKB = [iΠ, d] = iΠ d− d iΠ : Ωk(M)→ Ωk−1(M). (2.99)

Here KB stands for Koszul–Brylinski. This operator was introduced byKoszul [201], with a more explicit expression given by Brylinski [51]:

δKB(f0df1 ∧ · · · ∧ dfk) =∑

i

(−1)i+1f0, fidf1 ∧ · · · dfi · · · ∧ dfk+

+∑i<j

(−1)i+jf0dfi, fj ∧ df1 ∧ · · · dfi · · · dfj · · · ∧ dfk. (2.100)

The fact thatδKB δKB = 0 (2.101)

follows easily from Formula (2.100) and the Jacobi identity for the Poisson bracket.The homology2 groups

Hk(M, Π) =ker(δKB : Ωk(M) −→ Ωk−1(M))Im(δKB : Ωk+1(M) −→ Ωk(M))

(2.102)

of the differential complex

· · · −→ Ωk+1(M) δKB−→ Ωk(M) δKB−→ Ωk−1(M) −→ · · · (2.103)

are called Poisson homology groups.Poisson homology is a natural and important invariant of Poisson manifolds.

However, for the study of normal forms of Poisson structures, it is the Poissoncohomology which is more relevant, so we will only mention here briefly somemain features of Poisson homology.

It follows directly from Formula (2.99) that

δKB d + d δKB = 0. (2.104)

2The word homology is used instead of cohomology because δKB is of degree −1, i.e., it sendsΩk to Ωk−1.

2.7. Poisson homology 75

This equality implies that the space of differential forms on M together with(d, δKB) is a double complex, and there is a spectral sequence associated to thisdouble complex [51]. In analogy with Riemannian geometry, one says that a dif-ferential form β is a Poisson harmonic form if dβ = δKBβ = 0 [51]. On a compactRiemannian manifold, every de Rham cohomology class can be represented bya unique harmonic form (see, e.g., [41]). The analogue of this fact in the Pois-son case does not hold in general, even for symplectic manifolds [133, 241]. Infact, Mathieu [241] proved that, on a compact symplectic manifold (M2n, ω), ev-ery de Rham cohomology class has a symplectic harmonic representative if andonly if (M2n, ω) satisfies the so-called strong Lefschetz theorem about Kahlermanifolds, i.e., the cup product map β → ∧kω ∧ β induces an isomorphism[ω]k : Hn−k(M) → Hn+k(M) for any k ≤ n. Another proof of this fact is givenby Yan [363].

Bhaskara and Viswanath [30, 31] showed that there is a natural pairingbetween Poisson cohomology and Poisson homology, with values in the zerothPoisson homology group. In fact, it follows directly from Formula (1.8.1) that ifΛ ∈ Vk−1(M) and β ∈ Ωk(M) then

〈β, [Π, Λ]〉 − 〈δKBβ, Λ〉 = (−1)kδKB(iΛβ). (2.105)

This formula implies that the usual pairing between k-vector fields and k-formsinduces a pairing

Hk(M, Π)×Hk(M, Π)→ H0(M, Π). (2.106)

When (M2n, Π) is a compact symplectic manifold, there is a natural involu-tion

: Ωk(M)→ Ω2n−k(M) (2.107)

(defined by α = iαΩ, where Ω = (1/n!) ∧n ω is the standard symplectic formand = Π is the anchor map of Π), and δKB can be written as

δ = ± d . (2.108)

It follows that, in this case, Hk(M, Π) is naturally isomorphic to H2n−kdR (M) [51],

and Formula (2.105) gives the usual Poincare duality if we identify Hk(M, Π) withHk

dR(M).Huebschmann [180] gave a purely algebraic definition of Poisson homology in

terms of the Tor functor (and Poisson cohomology in terms of the Ext functor).Xu [360] showed that if (M, Π) is an orientable unimodular Poisson manifold of di-mension n then Hk(M, Π) ∼= Hn−k(M, Π). The duality between Poisson homologyand Poisson cohomology was also studied in [181, 127]. There are natural relationsbetween Poisson homology and Hochschild homology, see, e.g., [51, 129, 27]. Thepaper [129] also contains explicit computations of Poisson homology for somePoisson–Lie groups. Similarly to Poisson cohomology, it is very hard to computePoisson homology in general.

Chapter 3

Levi Decomposition

In this chapter, we will discuss a type of local normal forms for Poisson structureswhich vanish at a point, called Levi normal forms, or Levi decompositions. ALevi normal form is a kind of partial linearization of a Poisson structure, and in“good” cases this leads to a true linearization. The name Levi decomposition comesfrom the analogy with the classical Levi decomposition for finite-dimensional Liealgebras. Let us briefly recall here the classical theory (see, e.g., [42, 335]):

Let l be a finite-dimensional Lie algebra. Denote by r the radical of l, i.e., themaximal solvable ideal of l. Then the quotient Lie algebra g = l/r is semi-simple,and we have the following exact sequence:

0 → r→ l → g→ 0. (3.1)

The classical Levi–Malcev theorem says that the above sequence splits, i.e., thereis an injective Lie algebra homomorphism ı : g→ l such that its composition withthe projection map l→ g is identity. The image ı(g) of g in l is called a Levi factorof l. Up to conjugations in l, the Levi factor of l is unique. We will identify g withı(g). Then g acts on r by the adjoint action in l, and l can be decomposed into asemi-direct product of g with r:

l = g r. (3.2)

The above decomposition is called the Levi decomposition of l.In the study of Poisson structures or other structures involving Lie brackets,

we often have infinite-dimensional Lie algebras. So the idea is to find analogs ofthe Levi–Malcev theorem which hold for these infinite-dimensional Lie algebras.These analogs will give interesting information about Poisson structures.

In Section 3.1 we will give a formal infinite-dimensional analog of the Levi–Malcev theorem, and illustrate its use in the example of singular foliations. Then inthe rest of this chapter, we will discuss Levi decomposition for Poisson structures.

78 Chapter 3. Levi Decomposition

3.1 Formal Levi decomposition

Let L be a Lie algebra of infinite dimension. Suppose that L admits a filtration

L = L0 ⊃ L1 ⊃ L2 ⊃ · · · , (3.3)

such that ∀i, j ≥ 0, [Li,Lj ] ⊂ Li+j and dim(Li/Li+1) < ∞. Then we say that Lis a pro-finite Lie algebra, and call the inverse limit

L = lim∞←i

L/Li (3.4)

the formal completion of L (with respect to a given pro-finite filtration).

Example 3.1.1. Let L be the Lie algebra of smooth vector fields on Rn whichvanish at the origin 0, and Lk be the ideal of L consisting of vector fields withzero k-jet at 0. Then L is pro-finite, and its formal completion is the algebra offormal vector fields at 0.

Given a pro-finite Lie algebra L as above, denote by r the radical of l = L/L1

and by g the semisimple quotient l/r. Denote by R the preimage of r under theprojection L → l = L/L1. Then R is an ideal of L, called the pro-solvable radical ,and we have L/R ∼= l/r = g. Denote by R = lim←R/Li the formal completion ofR. Then we have the following exact sequences:

0 →R→ L → g → 0, (3.5)

0 → R → L → g → 0. (3.6)

The exact sequence (3.5) does not necessarily split, but its formal completion(3.6) always does:

Theorem 3.1.2. With the above notations, there is a Lie algebra injection ı : g→ Lwhose composition with the projection map L → g is the identity map. Up toconjugations in L, such an injection is unique.

Proof. By induction, for each k ∈ N we will construct an injection ık : g → L/Lk,whose composition with the projection map L/Lk → g is identity, and moreoverthe following compatibility condition is satisfied: the diagram

gık+1−−−−−→ L/Lk+1

Id

$$ proj.

gık−−−−−→ L/Lk

(3.7)

is commutative. Then ı = lim← ık will be the required injection. When k = 1, ı1 isgiven by the Levi–Malcev theorem. If we forget about the compatibility condition,

3.1. Formal Levi decomposition 79

then the other ık, k > 1, can also be provided by the Levi–Malcev theorem. Butto achieve the compatibility condition, we will construct ık+1 directly from ık.

Assume that ık has been constructed. Denote by ρ : g→ L/Lk+1 an arbitrarylinear map which lifts the injective Lie algebra homomorphism ık : g → L/Lk. Wewill modify ρ into a Lie algebra injection.

Note that Lk/Lk+1 is a g-module. The action of g on Lk/Lk+1 is definedas follows: for x ∈ g, v ∈ Lk/Lk+1, put x.v = [ρ(x), v] ∈ Lk/Lk+1. If x, y ∈ gthen [ρ(x), ρ(y)] − ρ([x, y]) ∈ Lk/Lk+1 ⊂ L1/Lk+1, and therefore [[ρ(x), ρ(y)] −ρ([x, y]), v] = 0 because [L1/Lk+1,Lk/Lk+1] = 0. The Jacobi identity in L/Lk+1

then implies that x.(y.v) − y.(x.v) = [x, y].v, so Lk/Lk+1 is a g-module.Define the following 2-cochain f : g ∧ g → Lk/Lk+1:

x ∧ y ∈ g ∧ g → f(x, y) = [ρ(x), ρ(y)] − ρ([x, y]) ∈ Lk/Lk+1. (3.8)

One verifies directly that f is a 2-cocycle of the corresponding Chevalley–Eilenbergcomplex: denoting by

∮xyz

the cyclic sum in (x, y, z), we have

δf(x, y, z) =∮

xyz

(x.f(y, z)− f([y, z], x)

)=∮

xyz

([ρ(x), [ρ(y), ρ(z)] − ρ([y, z])

]− [ρ[y, z], ρ(x)] + ρ([[y, z], x])

)=∮

xyz

[ρ(x), [ρ(y), ρ(z)]

]+∮

xyz

ρ([[y, z], x]) = 0 + 0 = 0.

Since g is semisimple, by Whitehead’s lemma every 2-cocycle of g is a2-coboundary. In particular, there is a 1-cochain φ : g → Lk/Lk+1 such thatδφ = f, i.e.,

[ρ(x), φ(y)] − [ρ(y), φ(x)] − φ([x, y]) = [ρ(x), ρ(y)] − ρ([x, y]). (3.9)

It implies that the linear map ık+1 = ρ−φ is a Lie algebra homomorphism from gto L/Lk+1. Since the image of φ lies in Lk/Lk+1, it is clear that ık+1 is a lifting ofık. Thus ık+1 satisfies our requirements. By induction, the existence of ı is proved.

The uniqueness of ı up to conjugations in L is proved similarly. Supposethat ık+1, ı

′k+1 : g → L/Lk+1 are two different injections which lift ık. Then

ı′k+1 − ık+1 is a 1-cocycle, and therefore a 1-coboundary by Whitehead’s lemma.Denote by α an element of Lk/Lk+1 such that δα is this 1-coboundary. Then theinner automorphism of L/Lk+1 given by

v ∈ L/Lk+1 → Adexpαv = v + [α, v] (3.10)

(because the other terms vanish) is a conjugation in L/Lk+1 which intertwinesık+1 and ı′k+1, and which projects to the identity map on L/Lk.

The image ı(g) of g in L, where ı is given by Theorem 3.1.2, is called a formalLevi factor of L.


Remark 3.1.3. The above proof can be modified slightly to yield a proof of theclassical Levi–Malcev theorem, pretty close to the one given in [335]. (Put L1 =the radical of L in the finite-dimensional case.)Remark 3.1.4. Every semisimple subalgebra of a finite-dimensional Lie algebra iscontained in a Levi factor. Similarly, each semisimple subalgebra of a pro-finiteLie algebra is formally contained in a formal Levi factor. These facts can also beproved by a slight modification of the uniqueness part of the proof of Theorem3.1.2.

Relations between Levi decomposition and linearization problems were ob-served, for example, by Flato and Simon [135] in their work on linearization offield equations. Here we will show a simple example of such relations, involvingsingular foliations.

Let F be a singular holomorphic foliation in a neighborhood of 0 in Cn.Holomorphic means that F is generated by holomorphic vector fields. We willassume that the rank of F at 0 is 0, i.e., X(0) = 0 for any tangent vector fieldX tangent to F . Denote by X (F) the Lie algebra of germs at 0 of holomorphicvector fields tangent to F . Denote by X (1)(F) the Lie algebra consisting of linearparts of elements of X (F) at 0. Then X (1)(F) is a Lie algebra of linear vectorfields. Denote by F (1) the singular foliation generated by X (1)(F) and call it thelinear part of F .

Theorem 3.1.5 (Cerveau [71]). With the above notations, if X (1)(F) is semisimpleand dimF = dimF (1), then F is formally linearizable at 0, i.e., it is formallyisomorphic to F (1).

Proof. X (F) is a pro-finite Lie algebra with the standard filtration given by theorder of vanishing of vector fields at 0, hence it admits a formal Levi factor g. WhenX (1)(F) is semisimple, then g is isomorphic to X (1)(F). Since g is semisimple, itsformal action on Cn is formally linearizable by a classical theorem of Hermann(Theorem 3.1.6). Suppose that the action of g has been linearized. It means thatg consists of linear vector fields, hence it coincides with X (1)(F). In other words,after the formal linearization, we have an inclusion X (1)(F) ⊂ X (F), hence F (1) ⊂F . But F and F (1) have the same dimension by assumptions, hence they mustcoincide. Theorem 3.1.6 (Hermann [172]). If g ⊂ V1

formal,0(Kn) is a finite-dimensional

semisimple subalgebra of the Lie algebra V1formal,0(K

n) of formal vector fields onKn which vanish at 0, where K = R or C, then there is a formal coordinatesystem (z1, . . . , zn) of Kn at 0, with respect to which the elements of g have linearcoefficients.

Proof (sketch). The proof follows the usual formal normalization procedure, andis based on Whitehead’s lemma H1(g, W ) = 0. Let X1, . . . , Xd be a basis of g.Suppose that, in a coordinate system (z1, . . . , zn), we have

Xi = X(1)i + X

(s)i + X

(s+1)i + · · · (3.11)

3.2. Levi decomposition of Poisson structures 81

with s ≥ 2, where X(s)i is a vector field whose coefficients are homogeneous of

degree s, and so on. We want to kill the term X(s)i in the expression of Xi by a

coordinate transformation of the type z′i = zi+ terms of degree ≥ s. Due to theJacobi identity, the map Xi → X

(s)i is a 1-cocycle of g with coefficients in the

g-module of homogeneous vector fields of degree s. By Whitehead’s lemma, this1-cocycle is a coboundary, i.e., we can write

X(s)i = [X(1)

i , Y ], (3.12)

where Y =∑

j fj∂/∂zj is homogeneous of degree s. Put z′i = zi − fi. This co-ordinate transformation will kill the term of degree s in the Taylor expansionof Xi.

3.2 Levi decomposition of Poisson structures

Let Π be a Poisson structure in a neighborhood of 0 in Kn, where K = R or C,which vanishes at 0: Π(0) = 0. Denote by Π(1) the linear part of Π at 0, and byl the Lie algebra of linear functions on Kn under the linear Poisson bracket ofΠ(1). Let g ⊂ l be a semisimple subalgebra of l. If Π is formal or analytic, wewill assume that g is a Levi factor of l. If Π is smooth (but not analytic), we willassume that g is a maximal compact semisimple subalgebra of l, and we will callsuch a subalgebra a compact Levi factor . Denote by (x1, . . . , xm, y1, . . . , yn−m) alinear basis of l, such that x1, . . . , xm span g (dim g = m), and y1, . . . , yn−m spana complement r of g with respect to the adjoint action of g on l, i.e., [g, r] ⊂ r. (Inthe formal and analytic cases, r is the radical of l; in the smooth case it is not theradical in general.) Denote by ck

ij and akij the structural constants of g and of the

action of g on r respectively: [xi, xj ] =∑

k ckijxk and [xi, yj ] =

∑k ak

ijyk.

Definition 3.2.1. With the above notations, we will say that Π admits a formal(resp. analytic, resp. smooth) Levi decomposition or Levi normal form at 0, withrespect to the (compact) Levi factor g, if there is a formal (resp. analytic, resp.smooth) coordinate system

(x∞1 , . . . , x∞

m , y∞1 , . . . , y∞

n−m),

with x∞i = xi+ higher-order terms and y∞

i = yi+ higher-order terms, such thatin this system of coordinates we have

Π =∑i<j

ckijx

∞k

∂

∂x∞i

∧ ∂

∂x∞j

+∑

akijy

∞k

∂

∂x∞i

∧ ∂

∂y∞j

+∑i<j

Pij∂

∂y∞i

∧ ∂

∂y∞j

, (3.13)

where Pij are formal (resp. analytic, resp. smooth) functions.

Remark 3.2.2. Another way to express Equation (3.13) is as follows:

x∞i , x∞

j =∑

ckijx

∞k and x∞

i , y∞j =

∑ak

ijy∞k . (3.14)


In other words, the Poisson brackets of x-coordinates with x-coordinates, and ofx-coordinates with y-coordinates, are linear. Yet another way to say it is that theHamiltonian vector fields of x∞

i are linear:

Xx∞i

=∑

ckijx

∞k

∂

∂x∞j

+∑

akijy

∞k

∂

∂y∞j

. (3.15)

In particular, the vector fields Xx∞1

, . . . , Xx∞m

form a Lie algebra isomorphic tog, and we have an infinitesimal linear Hamiltonian action of g on (Kn, Π), whosemomentum map µ : Kn → g∗ is defined by 〈µ(z), xi〉 = xi(z).

Theorem 3.2.3 (Wade [342]). Any formal Poisson structure Π in Kn (K = R orC) which vanishes at 0 admits a formal Levi decomposition.

Proof. Denote by L the algebra of formal functions in Kn which vanish at 0,under the Lie bracket of Π. Then it is a pro-finite Lie algebra, whose completionis itself. The Lie algebra L/L1, where L1 is the ideal of L consisting of functionswhich vanish at 0 together with their first derivatives, is isomorphic to the Liealgebra l of linear functions on Kn whose Lie bracket is given by the linear Poissonstructure Π(1). By Theorem 3.1.2, L admits a Levi factor, which is isomorphicto the Levi factor g of l. Denote by x∞

1 , . . . , x∞m a linear basis of a Levi factor

of L, x∞i , x∞

j =∑

k ckijx

∞k where ck

ij are structural constants of g. Then theHamiltonian vector fields Xx∞

1, . . . , Xx∞

mgive a formal action of g on Kn. By

Hermann’s formal linearization Theorem 3.1.6, this formal action can be linearizedformally, i.e., there is a formal coordinate system (x0

1, . . . , y0n−m) in which we have

Xx∞i

=∑

ckijx

0k

∂

∂x0j

+∑

akijy

0k

∂

∂y0j

. (3.16)

A priori, it may happen that x0i = x∞

i , but in any case we have x0i = x∞

i + higher-order terms, and Xx∞

i(x∞

j ) =∑

k ckijx

∞k , Xx∞

i(y0

j ) =∑

k akijy

0k. Renaming y0

i byy∞

i , we get a formal coordinate system (x∞1 , . . . , y∞

n−m) which puts Π in formalLevi normal form. Remark 3.2.4. A particular case of Theorem 3.2.3 is the following formal lineariza-tion theorem of Weinstein [346] mentioned in Chapter 2: if the linear part of Π at0 is semisimple (i.e., it corresponds to a semisimple Lie algebra l = g), then Π isformally linearizable at 0.Remark 3.2.5. As observed by Chloup [79], Theorem 3.2.3 may also be viewed asa consequence of Hochschild–Serre’s Theorem 2.4.4. Indeed, according to Theorem2.4.4 and Whitehead’s lemma, we have

H2(l,Spl) ∼= H0(g, K)⊗H2(r,Spl)g ∼= H2(r,Spl)g ∀ p. (3.17)

It means that any nonlinear term in the Taylor expansion of Π, which is representedby a 2-cocycle of l with values in Sl = ⊕pSpl, can be “pushed to r”, i.e., pushedto the “y-part” (consisting of terms Pij∂/∂yi ∧ ∂/∂yj) of Π.

3.2. Levi decomposition of Poisson structures 83

In the analytic case, we have:

Theorem 3.2.6 ([369]). Any analytic Poisson structure Π in a neighborhood of 0 inKn, where K = R or C, which vanishes at 0, admits an analytic Levi decomposition.

Remark 3.2.7. If in the above theorem, l = ((Kn)∗, ., .Π(1)) is a semi-simple Liealgebra, i.e., g = l, then we recover the following analytic linearization theorem ofConn [80]: any analytic Poisson structure with a semi-simple linear part is locallyanalytically linearizable. When l = g⊕ K, then a Levi decomposition of Π is stillautomatically a linearization (because y1, y1 = 0), and Theorem 3.2.6 impliesthe following result of Molinier [258] and Conn (unpublished): If the linear partof an analytic Poisson structure Π which vanishes at 0 corresponds to l = g⊕ K,where g is semisimple, then Π is analytically linearizable in a neighborhood of 0.

Remark 3.2.8. The existence of a local analytic Levi decomposition of Π is es-sentially equivalent to the existence of a Levi factor (and not just a formal Levifactor) for the Lie algebra O of germs at 0 of analytic functions under the Poissonbracket of Π. Indeed, if Π is in analytic Levi normal form with respect to a co-ordinate system (x1, . . . , yn−m), then the functions x1, . . . , xm form a linear basisof a Levi factor of O. Conversely, suppose that O admits a Levi factor with alinear basis x1, . . . , xm. Then Xx1 , . . . , Xxm generate a local analytic action of gon Kn. According to the Kushnirenko-Guillemin–Sternberg analytic linearizationtheorem for analytic actions of semisimple Lie algebras [160, 205], we may assumethat

Xxi =∑

ckijx

0k

∂

∂x0j

+∑

akijy

0k

∂

∂y0j

(3.18)

in a local analytic system of coordinates (x01, . . . , y

0n−m), where x0

i = xi+ higher-order terms. Renaming y0

i by yi, we get a local analytic system of coordinates(x1, . . . , yn−m) which puts Π in Levi normal form.

In the smooth case, we have:

Theorem 3.2.9 (Monnier–Zung [263]). For any n ∈ N and p ∈ N ∪ ∞ there isp′ ∈ N ∪ ∞, p′ < ∞ if p < ∞, such that the following statement holds: LetΠ be a Cp′

-smooth Poisson structure in a neighborhood of 0 in Rn. Denote by lthe Lie algebra of linear functions in Rn under the Lie–Poisson bracket Π1 whichis the linear part of Π, and by g a compact Levi factor of l. Then there exists aCp-smooth Levi decomposition of Π with respect to g in a neighborhood of 0 .

Remark 3.2.10. The condition that g be compact in Theorem 3.2.9 is in a sensenecessary, already in the case when l = g. (See Section 4.3.)

Remark 3.2.11. Remark 3.2.7 and Remark 3.2.8 also apply to the smooth case(provided that g is compact). In particular, when l = g, one recovers from Theorem3.2.9 the following smooth linearization theorem of Conn [81]: any smooth Poissonstructure whose linear part is compact semisimple is locally smoothly linearizable.When l = g ⊕ R with g compact semisimple, Theorem 3.2.9 still gives a smooth


linearization. And the existence of a local smooth Levi decomposition is equivalentto the existence of a compact Levi factor.

In the rest of this chapter, we will give a full proof of Theorem 3.2.6, andthen a sketch of the proof of Theorem 3.2.9, which is similar but more technical.These proofs of Theorem 3.2.6 and Theorem 3.2.9 are inspired by and based onConn’s work [80, 81], and use a normed version of Whitehead’s lemma (on vanish-ing cohomology of semisimple Lie algebras) and the fast convergence method (ofKolmogorov in the analytic case and Nash–Moser in the smooth case) in order toshow the convergence of a formal coordinate transformation putting the Poissonstructure in Levi normal form.

3.3 Construction of Levi decomposition

In this section we will construct, by a recurrence process, a formal system ofcoordinates (x∞

1 , . . . , x∞m , y∞

1 , . . . , y∞n−m) which satisfy Relations (3.14) for a given

local analytic Poisson structure Π. We will later use analytic estimates to showthat our construction actually yields a local analytic system of coordinates.

Each step in our recurrence process consists of two substeps: the first substepis to find an almost Levi factor. The second substep consists of almost linearizingthis almost Levi factor.

We begin the first step with the original linear coordinate system

(x01, . . . , x

0m, y0

1, . . . , y0n−m) = (x1, . . . , xm, y1, . . . , yn−m) . (3.19)

For each positive integer l, after Step l we will find a local coordinate system(xl

1, . . . , xlm, yl

1, . . . , yln−m) with the following properties (3.20), (3.21), (3.24):

(xl1, . . . , x

lm, yl

1, . . . , yln−m) = (xl−1

1 , . . . , xl−1m , yl−1

1 , . . . , yl−1n−m) φl , (3.20)

where φl is a local analytic diffeomorphism of (Kn, 0) of the type

φl(z) = z + terms of order ≥ 2l−1 + 1 . (3.21)

The space (Kn, 0) above is fixed (our local Poisson manifold). The functionsxl−1

1 , xl1, etc. are local functions on that fixed space.

Denote byX l

i = Xxli

(i = 1, . . . , m) (3.22)

the Hamiltonian vector field of xli with respect to our Poisson structure Π. Then

we haveX l

i = X li + Y l

i , (3.23)

whereX l

i =∑jk

ckijx

lk

∂

∂xlj

+∑jk

akijy

lk

∂

∂ylj

, Y li ∈ O(|z|2l+1) , (3.24)

3.3. Construction of Levi decomposition 85

i.e., X li is the linear part of X l

i = Xxliin the coordinate system (xl

1, . . . , yln−m), ck

ij

and akij are structural constants as appeared in Theorem 3.2.6, and Y l

i = X li − X l

i

does not contain terms of order ≤ 2l.Condition (3.24) may be rewritten as

xli, x

lj =

∑k

ckijx

lk modulo terms of order ≥ 2l + 1 , (3.25)

xli, y

lj =

∑k

akijy

lk modulo terms of order ≥ 2l + 1 . (3.26)

So we may say that the functions (xl1, . . . , x

lm) form an almost Levi factor ,

and their corresponding Hamiltonian vector fields are almost linearized , up toterms of order 2l + 1.

Of course, when l = 0, then Relation (3.24) is satisfied by the assump-tions of Theorem 3.2.6. Let us show how to construct the coordinate system(xl+1

1 , . . . , yl+1n−m) from the coordinate system (xl

1, . . . , yln−m). Denote

Ol = local analytic functions in (Kn, 0) without terms of order ≤ 2l . (3.27)

Due to Relations (3.20) and (3.21), it doesn’t matter if we use the orig-inal coordinate system (x1, . . . , xm, y1, . . . , yn−m) or the new one (xl

1, . . . , xlm,

yl1, . . . , y

ln−m) in the above definition of Ol. It follows from Relation (3.24) that

f lij := xl

i, xlj −

∑k

ckijx

lk = Y l

i (xlj) ∈ Ol . (3.28)

Denote by (ξ1, . . . , ξm) a fixed basis of the semi-simple algebra g, with

[ξi, ξj ] =∑

k

ckijξk . (3.29)

Then g acts on O via vector fields X l1, . . . , X

lm, and this action induces the

following linear action of g on the finite-dimensional vector space Ol/Ol+1: ifg ∈ Ol, considered modulo Ol+1, then we put

ξi · g := X li(g) =

∑jk

ckijx

lk

∂g

∂xlj

+∑jk

akijy

lk

∂g

∂ylj

mod Ol+1 . (3.30)

Notice that if g ∈ Ol then Y li (g) ∈ Ol+1, and hence we have

ξi · g = X l(g) mod Ol+1 = xli, g mod Ol+1 . (3.31)

The functions f lij in (3.28) form a 2-cochain f l of g with values in the g-

module Ol/Ol+1:

f l : g ∧ g → Ol/Ol+1

f l(ξi ∧ ξj) := f lij mod Ol+1 = xl

i, xlj −

∑k

ckijx

lk mod Ol+1 . (3.32)


In other words, if we denote by g∗ the dual space of g, and by (ξ∗1 , . . . , ξ∗m) thebasis of g∗ dual to (ξ1, . . . , ξm), then we have

f l =∑i<j

ξ∗i ∧ ξ∗j ⊗ (f lij mod Ol+1) ∈ ∧2g∗ ⊗Ol/Ol+1 . (3.33)

It follows from (3.28), and the Jacobi identity for the Poisson bracket of Πand the algebra g, that the above 2-cochain is a 2-cocycle. Because g is semi-simple,we have H2(g,Ol/Ol+1) = 0, i.e., the second cohomology of g with coefficients ing-module Ol/Ol+1 vanishes, and therefore the above 2-cocycle is a coboundary. Inother words, there is a 1-cochain

wl ∈ g∗ ⊗Ol/Ol+1 (3.34)

such thatf l(ξi ∧ ξj) = ξi · wl(ξj)− ξj · wl(ξi)− wl

(∑k

ckijξk

). (3.35)

Denote by wli the element of Ol which is a polynomial of order ≤ 2l+1 in

variables (xl1, . . . , x

lm, yl

1, . . . , yln−m) such that the projection of wl

i in Ol/Ol+1 iswl(ξi).Remark 3.3.1. Remember that wl

i are local functions on our fixed space (Kn, 0).They are not functions of variables (xl

1, . . . , xlm, yl

1, . . . , yln−m) per se, but when

expressed in terms of these variables they become polynomial functions.Define xl+1

i as follows:

xl+1i = xl

i − wli ∀ i = 1, . . . , m . (3.36)

Then it follows from (3.28) and (3.35) that we have

xl+1i , xl+1

j −∑

k

ckijx

l+1k ∈ Ol+1 for i, j ≤ m . (3.37)

This concludes our first substep (the (xl+1i ) form a better “almost Levi factor”

than (xli)). Let us now proceed to the second substep.

Denote by Y l the space of local analytic vector fields of the type u =∑n−mi=1 ui∂/∂yl

i (with respect to the coordinate system (xl1, . . . , y

ln−m)), with ui

being local analytic functions. For each natural number k, denote by Y lk the fol-

lowing subspace of Y l:

Y lk =

u =

n−m∑i=1

ui∂/∂yli

∣∣∣ ui ∈ Ok

. (3.38)

Then Y l, as well as Y ll /Y l

l+1, are g-modules under the following action:

ξi ·∑

j

uj∂/∂ylj := [X l

i , u] =[∑

jk

ckijx

lk

∂

∂xlj

+∑jk

akijy

lk

∂

∂ylj

,∑

j

uj∂/∂ylj

]. (3.39)

3.3. Construction of Levi decomposition 87

The above linear action of g on Yl/Yl+1 can also be written as

ξi ·∑

j

uj∂/∂ylj =

∑j

(xli, uj −

∑k

akijuk)∂/∂yl

j mod Y ll+1 . (3.40)

Define the following 1-cochain of g with values in Y ll /Y l

l+1:

m∑i=1

(ξ∗i ⊗

( n−m∑j=1

(xl+1i , yl

j−∑

k

akijy

lk)∂/∂yl

j mod Y ll+1

))∈ g∗⊗Y l

l /Y ll+1 . (3.41)

Due to Relation (3.37), the above 1-cochain is a 1-cocycle. Since g is semi-simple, we have H1(g,Y l

l /Y ll+1) = 0, and the above 1-cocycle is a 1-coboundary.

In other words, there exists a vector field

n−m∑j=1

vlj∂/∂yl

j ∈ Y ll , (3.42)

with vlj being polynomial functions of degree ≤ 2l+1 in variables (xl

1, . . . , yln−m),

such that for every i = 1, . . . , m we have∑j

(xl+1

i , ylj −

∑ak

ijylk

)∂/∂yl

j =∑

j

(xl

i, vlj −

∑ak

ijvlk

)∂/∂yl

j mod Y ll+1 .

(3.43)We now define the new system of coordinates as

xl+1i = xl

i − wli (i = 1, . . . , m),

yl+1i = yl

i − vli (i = 1, . . . , n−m),

(3.44)

where functions wli, v

li ∈ Ol are chosen as above. In particular, Relations (3.37)

and (3.43) are satisfied, which means that

xl+1i , xl+1

j −∑ckijx

l+1k ∈ Ol+1 ,

xl+1i , yl+1

j −∑

akijy

l+1k ∈ Ol+1 ,

(3.45)

i.e., Relation (3.24) is satisfied with l replaced by l+1. Of course, Relations (3.20)and (3.21) are also satisfied with l replaced by l + 1, and with φl+1 being the mapwhich when written in the variables (xl

1, . . . , yln−m) has the form

φl+1 = Id + ψl+1 , (3.46)

whereψl+1 = −(wl

1, . . . , wlm, vl

1, . . . , vln−m) ∈ (Ol)n . (3.47)


Remark 3.3.2. We stress here the fact that Formula (3.46) is valid with respectto the coordinate system (xl

1, . . . , yln−m) only. In particular, the sum there is

taken with respect to the local linear structure given by the coordinate system(xl

1, . . . , yln−m) and not by the original coordinate system (x1, . . . , yn−m). If we

want to express φl+1 in terms of the original coordinate system then it will bemuch more complicated.

Recall that, by the above construction, wl1, . . . , w

lm, vl

1, . . . , vln−m are polyno-

mial functions of degree ≤ 2l+1 in variables (xl1, . . . , y

ln−m), which do not contain

terms of degree ≤ 2l.Define the limits

(x∞1 , . . . , y∞

n−m) = liml→∞

(xl1, . . . , y

ln−m) ,

Φ∞ = liml→∞

Φl where Φl = φ1 · · · φl .(3.48)

It is clear that the above limits exist in the formal category,

(x∞1 , . . . , y∞

n−m) = (x01, . . . , y

0n−m) Φ∞, (3.49)

and the formal coordinate system (x∞1 , . . . , y∞

n−m) satisfies Relation (3.14).The above construction works not only for local analytic Poisson structures,

but also for formal Poisson structures, so it gives us another proof of Theorem3.2.3. To prove Theorem 3.2.6, it remains to show that, when Π is analytic, wecan choose functions wl

i, vli in such a way that (x∞

1 , . . . , y∞n−m) is in fact a local

analytic system of coordinates.Remark 3.3.3. The above construction of formal Levi decomposition differs fromthe construction of Wade [342] and Weinstein [353]. Their construction is simpler(they don’t almost linearize the almost Levi factor at each step, and they kill onlyone term at each step), and is good enough to show the existence of a formalLevi decomposition. However, in order to prove the existence of an analytic Levidecomposition, using Kolmogorov’s fast convergence method, one needs to kill abunch of terms at each step, and that’s why the second substep (almost linearizingan almost Levi factor) is important.

3.4 Normed vanishing of cohomology

In this section, using normed vanishing of first and second cohomology groups ofg, we will obtain some estimates on wl

i = xli − xl+1

i and vli = yl

i − yl+1i . For some

basic results on semi-simple Lie algebras and their representations which will beused below, one may consult a book on Lie algebras, e.g., [186, 335].

We will denote by gC the algebra g if K = C, and the complexification of gif K = R. So gC is a complex semi-simple Lie algebra of dimension m. Denote byg0 the compact real form of gC, and identify gC with g0⊗R C. Fix an orthonormalbasis (e1, . . . , em) of gC with respect to the Killing form: 〈ei, ej〉 = δij . We may

3.4. Normed vanishing of cohomology 89

assume that e1, . . . , em ∈√−1g0. Denote by Γ =

∑i e2

i the Casimir element ofgC: Γ lies in the center of the universal enveloping algebra U(gC) and does notdepend on the choice of the basis (ei). When K = R then Γ is real, i.e., Γ ∈ U(g).

Let W be a finite-dimensional complex linear space endowed with a Hermitianmetric denoted by 〈, 〉. If v ∈W then its norm is denoted by ‖v‖ =

√〈v, v〉. Assume

that W is a Hermitian g0-module. In other words, the linear action of g0 on W isvia infinitesimal unitary (i.e., skew-adjoint) operators. W is a gC-module via theidentification gC = g0 ⊗R C. We have the decomposition W = W0 + W1, whereW1 = gC ·W (the image of the representation), and gC acts trivially on W0. SinceW1 is a gC-module, it is also a U(gC)-module. The action of Γ on W1 is invertible:Γ ·W1 = W1, and we will denote by Γ−1 the inverse mapping.

Denote by g∗C

the dual of gC, and by (e∗1, . . . , e∗m) the basis of g∗

Cdual to

(e1, . . . , em). If w ∈ g∗C⊗W is a 1-cochain and f : ∧2g∗

C⊗W is a 2-cochain with

values in W , then we will define the norm of f and w as follows:

‖w‖ = maxi‖w(ei)‖ , ‖f‖ = max

i,j‖f(ei ∧ ej)‖ . (3.50)

Since H2(g, K) = 0, there is a (unique) linear map h0 : ∧2g∗ → g∗ such thatif u ∈ ∧2g∗ is a 2-cocycle for the trivial representation of g in K (i.e., u([x, y], z)+u([y, z], x) + u([z, x], y) = 0 for any x, y, z ∈ g), then u = δh0(u), i.e., u(x, y) =h0(u)([x, y]). By complexifying h0 if K = R, and taking its tensor product withthe projection map P0 : W →W0, we get a map

h0 ⊗ P0 : ∧2g∗C ⊗W → g∗C ⊗W0 . (3.51)

Define another map

h1 : ∧2g∗C ⊗W → g∗C ⊗W1 (3.52)

as follows: if f ∈ ∧2g∗C⊗W then we put

h1(f) =∑

i

e∗i ⊗(Γ−1 ·

∑j

(ej · f(ei ∧ ej)))

. (3.53)

Then the maph = h0 ⊗ P0 + h1 : ∧2g∗C ⊗W → g∗C ⊗W (3.54)

is an explicit homotopy operator , in the sense that if f ∈ ∧2g∗C⊗W is a 2-cocycle

(i.e., δf = 0 where δ denotes the differential of the Chevalley–Eilenberg complex· · · → ∧kg∗

C⊗W → ∧k+1g∗

C⊗W → · · · ), then f = δ(h(f)).

Similarly, the map h : g∗C⊗W →W defined by

h(w) = Γ−1 ·(∑

i

ei · w(ei))

(3.55)

is also a homotopy operator, in the sense that if w ∈ g∗C⊗W is a 1-cocycle then

w = δ(h(w)).


When K = R, i.e., when gC is the complexification of g, then the abovehomotopy operators h are real, i.e., they map real cocycles into real cochains.

The above formulas make it possible to control the norm of a primitive ofa 1-cocycle w or a 2-cocycle f in terms of the norm of w or f . More precisely,we have the following lemma about normed vanishing of cohomology, which is anormed version of Whitehead’s lemma which says that H1(g, W ) = H2(g, W ) = 0.

Lemma 3.4.1 (Conn). There is a positive constant D (which depends on g but doesnot depend on W ) such that with the above notations we have

‖h(f)‖ ≤ D‖f‖ and ‖h(w)‖ ≤ D‖w‖ (3.56)

for any 1-cocycle w and any 2-cocycle f of gC with values in W .

Remark 3.4.2. The above lemma is essentially due to Conn (see Proposition 2.1of [80]). Conn stated the result only for some particular modules that he needed,but his proof, which we give below, works without any change for other Hermitianmodules.

Proof (sketch). We can decompose W into an orthogonal sum (with respect tothe Hermitian metric of W ) of irreducible modules of g0. The above homotopyoperators decompose correspondingly, so it is enough to prove the above lemmafor the case when W is a nontrivial irreducible module, which we will now suppose.Let λ = 0 denote the highest weight of the irreducible g0-module W , and by δone-half the sum of positive roots of g0 (with respect to a fixed Cartan subalgebraand Weyl chamber). Then Γ acts on W by multiplication by the scalar 〈λ, λ+2δ〉,which is greater than or equal to ‖λ‖2. Denote by J the weight lattice of g0, andD = m(minγ∈J ‖γ‖)−1. Then D < ∞ does not depend on W , and ‖λ‖2 > m‖λ‖

D ,which implies that the norm of the inverse of the action of Γ on W is smaller thanor equal to D

m‖λ‖ . On the other hand, the norm of the action of ei on W is smallerthan or equal to ‖λ‖ for each i = 1, . . . , m (recall that

√−1ei ∈ g0 and 〈ei, ei〉 = 1),

hence the norm of the operator∑m

i=1 ei · Γ−1 : W → W is smaller than or equalto D. Now apply Formulas (3.53) and (3.55). The lemma is proved.

Let us now apply the above lemma to g-modules Ol/Ol+1 and Y ll /Y l

l+1 intro-duced in the previous section. Recall that g is a Levi factor of l, the space of linearfunctions in Kn, which is a Lie algebra under the linear Poisson bracket Π(1). g actson l by the (restriction of the) adjoint action, and on Kn by the coadjoint action.By complexifying these actions if necessary, we get a natural action of gC on (Cn)∗

(the dual space of Cn) and on Cn. The elements x1, . . . , xm, y1, . . . , yn−m of theoriginal linear coordinate system in Kn may be view as a basis of (Cn)∗. Noticethat the action of gC on (Cn)∗ preserves the subspace spanned by (x1, . . . , xm) andthe subspace spanned by (y1, . . . , yn−m). Fix a basis (z1, . . . , zn) of (Cn)∗, suchthat the Hermitian metric of (Cn)∗ for which this basis is orthonormal is preserved

3.4. Normed vanishing of cohomology 91

by the action of g0, and such that

zi =∑j≤m

Aijxj +∑

j≤n−m

Ai,j+myj , (3.57)

with the constant transformation matrix (Aij) satisfying the following condition:

Aij = 0 if (i ≤ m < j or j ≤ m < i) . (3.58)

Such a basis (z1, . . . , zn) always exists, and we may view (z1, . . . , zn) as a linearcoordinate system on Cn. We will also define local complex analytic coordinatesystems (zl

1, . . . , zln) as follows:

zli =

∑j≤m

Aijxlj +

∑j≤n−m

Ai,j+mylj . (3.59)

Let l be a natural number, ρ a positive number, and f a local complexanalytic function of n variables. Define the following ball Bl,ρ and L2-norm ‖f‖l,ρ,whenever it makes sense:

Bl,ρ =x ∈ Cn |

√∑|zl

i(x)|2 ≤ ρ

, (3.60)

‖f‖l,ρ =

√1Vρ

∫Sl,ρ

|f(x)|2dµl , (3.61)

where dµl is the standard volume form on the boundary Sl,ρ = ∂Bl,ρ of thecomplex ball Bl,ρ with respect to the coordinate system (zl

1, . . . , zln), and Vρ is the

volume of Sl,ρ, i.e., of a (2n− 1)-dimensional sphere of radius ρ.We will say that the ball Bl,ρ is well defined if it is analytically diffeomorphic

to the standard ball of radius ρ via the coordinate system (zl1, . . . , z

ln), and will

use ‖f‖l,ρ only when Bl,ρ is well defined. When Bl,ρ is not well defined we simplyput ‖f‖l,ρ = ∞. We will write Bρ and ‖f‖ρ for B0,ρ and ‖f‖0,ρ respectively. Iff is a real analytic function (the case when K = R), we will complexify it beforetaking the norms.

It is well known (see, e.g., Chapter 1 of [306]) that the L2-norm ‖f‖ρ is givenby a Hermitian metric, in which the monomial functions form an orthogonal basis:if f =

∑α∈Nn aα

∏i zαi

i and g =∑

α∈Nn bα

∏i zαi

i then the scalar product 〈f, g〉ρis given by

〈f, g〉ρ =∑

α∈Nn

α!(n− 1)!(|α|+ n− 1)!

ρ2|α|aαbα , (3.62)

(where α! =∏

i αi!, |a| =∑

αi, and b is the complex conjugate of b), and the norm‖f‖ρ is given by

‖f‖ρ =( ∑

α∈Nn

α!(n− 1)!(|α| + n− 1)!

|aα|2ρ2|α|)1/2

. (3.63)


The above scalar product turns Ol/Ol+1 into a Hermitian space, if we con-sider elements of Ol/Ol+1 as polynomial functions of degree less than or equal to2l+1 and which do not contain terms of order ≤ 2l. Of course, when K = R wewill have to complexify Ol/Ol+1, but will redenote (Ol/Ol+1)C by Ol/Ol+1, forsimplicity.

For the space Y l of local vector fields of the type u =∑n−m

i=1 ui∂/∂zli+m

(due to (3.58) and (3.59), this is the same as the space of vector fields of the type∑n−mi=1 u′

i∂/∂yli defined in the previous section, up to a complexification if K = R),

we define the L2-norms in a similar way:

‖u‖l,ρ =

√√√√ 1Vρ

∫Sl,ρ

n−m∑i=1

|ui(x)|2dµl . (3.64)

These L2-norms are given by Hermitian metrics similar to (3.62), which makeY l

l/Y ll+1 into Hermitian spaces.Remark that if u = (u1, . . . , un−m) then∑

i

‖ui‖l,ρ ≥ ‖u‖l,ρ ≥ maxi‖ui‖l,ρ . (3.65)

It is an important observation that, since the action of g0 on Cn preservesthe Hermitian metric of Cn, its actions on Ol/Ol+1 and Y l

l/Y ll+1, as given in

the previous section, also preserve the Hermitian metrics corresponding to thenorms ‖f‖l,ρ and ‖u‖l,ρ (with the same l). Thus, applying Lemma 3.4.1 to thesegC-modules, we get:

Lemma 3.4.3. There is a positive constant D1 such that for any l ∈ N and anypositive number ρ there exist local analytic functions wl

1, . . . , wlm, vl

1, . . . , vln−m,

which satisfy the relations of the previous section (in particular Relation (3.35)and Relation (3.43)), and which have the following additional property wheneverBl,ρ is well defined:

maxi‖wl

i‖l,ρ ≤ D1. maxi,j

∥∥∥xli, x

lj −

∑k

ckijx

lk

∥∥∥l,ρ

(3.66)

andmax

i‖vl

i‖l,ρ ≤ D1. maxi,j

∥∥∥xli − wl

i, ylj −

∑k

akijy

lk

∥∥∥l,ρ

. (3.67)

3.5 Proof of analytic Levi decomposition theorem

Besides the L2-norms defined in the previous section, we will need the followingL∞-norms: If f is a local function then put

|f |l,ρ = supx∈Bl,ρ

|f(x)| , (3.68)

3.5. Proof of analytic Levi decomposition theorem 93

where the complex ball Bl,ρ is defined by (3.60). Similarly, if g = (g1, . . . , gN ) is avector-valued local map then put |g|l,ρ = supx∈Bl,ρ

√∑i |gi(x)|2. For simplicity,

we will write |f |ρ for |f |0,ρ.For the Poisson structure Π, we will use the following norms:

|Π|l,ρ := maxi,j=1,...,n

∣∣zli, z

lj∣∣l,ρ

. (3.69)

Due to the following lemma, we will be able to use the norms |f |ρ and ‖f‖ρ

interchangeably for our purposes, and control the norms of the derivatives:

Lemma 3.5.1. For any ε > 0 there is a finite number K < ∞ depending on ε suchthat for any integer l > K, positive number ρ, and local analytic function f ∈ Ol

we have|f |(1+ε/l2)ρ ≥ exp(2l/2)|f |(1+ε/2l2)ρ ≥ ρ|df |ρ , (3.70)

and|f |(1−ε/l2)ρ ≤ ‖f‖ρ ≤ |f |ρ . (3.71)

We will postpone the proof of Lemma 3.5.1 a little bit. Now we want to showa key proposition which, together with a simple lemma, will imply Theorem 3.2.6.

Proposition 3.5.2. Under the assumptions of Theorem 3.2.6, there exists a constantC, such that for any positive number ε < 1/4, there is a natural number K = K(ε)and a positive number ρ = ρ(ε), such that for any l ≥ K we can construct a localanalytic coordinate system (xl

1, . . . , yln−m) as in the previous sections, with the

following additional properties (using the previous notations):(i)l (Chains of balls) The ball Bl,exp(1/l)ρ is well defined, and if l > K we have

Bl−1,exp( 1l − 2ε

l2)ρ ⊂ Bl,exp(1/l)ρ ⊂ Bl−1,exp( 1

l + 2εl2

)ρ . (3.72)

(ii)l (Norms of changes) If l > K then we have

|ψl|l−1,exp( 1l−1− ε

(l−1)2)ρ < ρ . (3.73)

(iii)l (Norms of the Poisson structure):

|Π|l,exp(1/l)ρ ≤ C. exp(−1/√

l)ρ . (3.74)

Theorem 3.2.6 follows immediately from the first part of Proposition 3.5.2and the following lemma:

Lemma 3.5.3. If there is a finite number K such that Condition (i)l of Proposition3.5.2 is satisfied for all l ≥ K, then the formal coordinate system

(x∞1 , . . . , x∞

m , y∞1 , . . . , y∞

n−m)

is convergent (i.e., locally analytic).


The main idea behind Lemma 3.5.3 is that, if Condition (i)l is true for any l ≥K, then the infinite intersection

⋂∞l=K Bl,exp(1/l)ρ contains an open neighborhood

of 0, implying a positive radius of convergence.The second and third parts of Proposition 3.5.2 are needed for the proof of

the first part. Proposition 3.5.2 will be proved by recurrence: By taking ρ smallenough, we can obviously achieve Conditions (iii)K and (i)K (Condition (ii)K isvoid). Then provided that K is large enough, when l ≥ K we have that Condition(ii)l implies Conditions (i)l and (iii)l, and Condition (iii)l in turn implies Condition(ii)l+1. In other words, Proposition 3.5.2 is a direct consequence of the followingthree technical lemmas:

Lemma 3.5.4. There exists a finite number K (depending on ε) such that if Con-dition (ii)l+1 is satisfied and l ≥ K, then Condition (i)l+1 is also satisfied.

Lemma 3.5.5. There exists a finite number K (depending on ε) such that if Con-dition (iii)l (of Proposition 3.5.2) is satisfied and l ≥ K, then Condition (ii)l+1 isalso satisfied.

Lemma 3.5.6. There exists a finite number K (depending on ε) such that if Con-ditions (ii)l+1 and (iii)l are satisfied and l ≥ K, then Condition (iii)l+1 is alsosatisfied.

The lemmas of this section will be proved now, one by one. But first let usmention here the main ingredients behind the last three ones: The proof of Lemma3.5.4 and Lemma 3.5.6 is straightforward and uses only the first part of Lemma3.5.1. Lemma 3.5.5 (the most technical one) follows from the estimates on theprimitives of cocycles as provided by Lemma 3.4.3.

Proof of Lemma 3.5.1. Let f be a local analytic function in (Cn, 0). To make anestimate on df , we use the Cauchy integral formula. For z ∈ Bρ, denote by γi

the following circle: γi = v ∈ Cn | vj = zj if j = i , |vi − zi| = ερ/2l2. Thenγi ⊂ B(1+ε/l2)ρ, and we have∣∣∣∣ ∂f

∂zi(z)

∣∣∣∣ =12π

∣∣∣∣∮γi

f(v)dv

(v − z)2

∣∣∣∣ ≤ 2l2

ερ|f |(1+ε/2l2)ρ ,

which implies that exp(2l/2)|f |(1+ε/2l2)ρ ≥ ρ|df | when l is large enough.Now let f ∈ Ol such that |f |(1+ε/l2)ρ < ∞. We want to show that if

x ∈ B(1+ε/2l2)ρ then |f(x)| ≤ exp(2l/2)|f |(1+ε/l2)ρ (provided that l is large enoughcompared to 1/ε). Fix a point x ∈ B(1+ε/2l2)ρ and consider the following holomor-phic function of one variable: g(z) = f( x

|x|z). This function is holomorphic in thecomplex one-dimensional disk B1

(1+ε/l2)ρ of radius (1 + ε/l2)ρ, and is bounded by

|f |(1+ε/l2)ρ in this disk. Because f ∈ Ol, we have that g(z) is divisible by z2l

, thatis g(z)/z2l

is holomorphic in B1(1+ε/l2)ρ. By the maximum principle we have

|f(x)||x|2l =

∣∣∣∣g(|x|)|x|2l

∣∣∣∣ ≤ max|z|=(1+ε/l2)ρ

∣∣∣∣g(z)z2l

∣∣∣∣ ≤ |f |(1+ε/l2)ρ

((1 + ε/l2)ρ)2l ,


which implies that

|f(x)| ≤(

1 + ε/2l2

1 + ε/l2

)2l

|f |(1+ε/l2)ρ ≈ exp(− 2l

2εl2)|f |(1+ε/l2)ρ

≤ exp(−2l/2)|f |(1+ε/l2)ρ

(when l is large enough). Thus we have proved that there is a finite number Kdepending on ε such that

|f |(1+ε/l2)ρ ≥ exp(2l/2)|f |(1+ε/2l2)ρ

for any l > K and any f ∈ Ol.To compare the norms of f , we use Cauchy–Schwartz inequality: for f =∑

α∈Nk cα

∏i zαi

i and |z| = (1 − ε/2l2)ρ we have

|f(z)| ≤∑

α∈Nk

|cα|∏

i

|zi|αi

≤(∑

α

|cα|2α!(n− 1)!

(|α|+ n− 1)!ρ2|α|

)1/2(∑α

(|α|+ n− 1)!α!(n− 1)!

ρ−2|α|∏i

|zi|2α)1/2

= ‖f‖ρ

(1−

∑i

|zi|2ρ2

)−n/2

= ‖f‖ρ

(1− (1− ε/2l2)2

)−n/2 ≤ (2l)n

εn/2‖f‖ρ .

It means that for any local analytic function f we have

|f |(1−ε/2l2)ρ ≤(2l)n

εn/2‖f‖ρ .

Now if f ∈ Ol, we can apply Inequality (3.70) to get

|f |(1−ε/l2)ρ ≤ exp(−2l/2)|f |(1−ε/2l2)ρ ≤(2l)n

εn/2exp(−2l/2)‖f‖ρ ≤ ‖f‖ρ ,

provided that l is large enough compared to 1/ε. Lemma 3.5.1 is proved.

Proof of Lemma 3.5.3. The main point is to show that the limit⋂∞

l=K Bl,ρ con-tains a ball Br of positive radius centered at 0. Then for x ∈ Br, we have x ∈ Bl,ρ,implying ‖(zl

1(x), . . . , zln(x))‖ < ρ is uniformly bounded, which in turn implies that

the formal functions z∞i = liml→∞ zli are analytic functions inside Br (recall that

(zl1, . . . , z

ln) is obtained from (xl

1, . . . , yln−m) by a constant linear transformation

(Aij) which does not depend on l).Recall the following fact of complex analysis, which is a consequence of the

maximum principle: if g is a complex analytic map from a complex ball of radiusρ to some linear Hermitian space such that g(0) = 0 and |g(x)| ≤ C for all |x| < ρ


and some constant C, then we have |g(x)| ≤ C|x|/ρ for all x such that |x| < ρ. Ifl1, l2 ∈ N and r1, r2 > 0, s > 1, then applying this fact we get:

If Bl1,r1 ⊂ Bl2,r2 then Bl1,r1/s ⊂ Bl2,r2/s . (3.75)

(Here r1 plays the role of ρ, r2 plays the role of C, and the coordinate trans-formation from (zl1

1 , . . . , zl1n ) to (zl2

1 , . . . , zl2n ) plays the role of g in the previous

statement.)Using Formula (3.75) and Condition (i)l recursively, we get

Bl,ρ ⊃ Bl−1,exp(−1/l2)ρ ⊃ Bl−2,exp(−1/l2−1/(l−1)2)ρ

⊃ · · · ⊃ BK,exp(−∑lk=K 1/k2)ρ .

Since c = exp(−∑∞

k=K 1/k2) is a positive number, we have⋂∞

l=K Bl,ρ ⊃BK,cρ, which clearly contains an open neighborhood of 0. Lemma 3.5.3 is proved.

Proof of Lemma 3.5.4. Suppose that Condition (ii)l+1 is satisfied. For simplicityof exposition, we will assume that the coordinate system (zl

1, . . . , zln) coincides with

the coordinate system (xl1, . . . , y

ln−m) (The more general case, when (zl

1, . . . , zln)

is obtained from (xl1, . . . , y

ln−m) by a constant linear transformation, is essentially

the same.) Suppose that we have

|ψl+1|l,exp(1/l−ε/l2)ρ < ρ .

Then it follows from Lemma 3.5.1 that, provided that l is large enough:

|dψl+1|l,exp(1/l−2ε/l2)ρ <12n

.

(In order to define |dψl+1|l,exp(1/l−2ε/l2)ρ, consider dψl+1 as an n2-vector-valuedfunction in variables (zl

1, . . . , zln).) Hence the map φl+1 = Id + ψl+1 is injective

in Bl,exp(1/l−2ε/l2)ρ: if x, y ∈ Bl,ρl, x = y, then ‖φl+1(x) − φl+1(y)‖ ≥ ‖x − y‖ −

‖ψl+1(x)−ψl+1(y)‖ ≥ ‖x−y‖−n|dψl+1|exp(1/l−2ε/l2)ρ‖x−y‖ ≥ (1−1/2)‖x−y‖ >

0. (Here (x − y) means the vector (zl1(x) − zl

1(y), . . . , zln(x) − zl

n(y)), i.e., theirdifference is taken with respect to the coordinate system (zl

1, . . . , zln).)

It follows from Lemma 3.5.1 that

|φl+1|l,exp(1/l−2ε/l2)ρ = |Id + ψl+1|l,exp(1/l−2ε/l2)ρ

≤ |Id|l,exp(1/l−2ε/l2)ρ + |ψl+1|l,exp(1/l−2ε/l2)ρ

< exp(1/l− 2ε/l2)ρ +ε

4l2exp(1/l− 2ε/l2)ρ < exp(1/l− ε/l2)ρ .

In other words, we have

φl+1(Bl,exp(1/l−2ε/l2)ρ) ⊂ Bl,exp(1/l−ε/l2)ρ . (3.76)


Applying Formula (3.75) to the above relation, noticing that 1/l − 2ε/l2 >1/(l + 1), and simplifying the obtained formula a little bit, we get

φl+1(Bl,exp(1/(l+1)−2ε/(l+1)2)ρ) ⊂ Bl,exp(1/(l+1))ρ . (3.77)

We will show that φ−1l+1 is well defined in Bl,exp(1/(l+1))ρ, and

φ−1l+1(Bl,exp(1/(l+1))ρ) = Bl+1,exp(1/(l+1))ρ ⊂ Bl,exp(1/(l+1)+2ε/(l+1)2)ρ . (3.78)

Indeed, denote by Sl,exp(1/l−2ε/l2)ρ the boundary of Bl,exp(1/l−2ε/l2)ρ. Then

φl+1(Sl,exp(1/l−2ε/l2)ρ) ⊂ Bl,exp(1/l−ε/l2)ρ

and is homotopic to Sl,exp(1/l−2ε/l2)ρ via a homotopy which does not intersectBl,exp(1/(l+1))ρ. It implies, via the classical Brower’s fixed point theorem, thatφl+1(Bl,exp(1/l−2ε/l2)ρ) must contain Bl,exp(1/(l+1))ρ. Because φl+1 is injective in(Bl,exp(1/l−2ε/l2)ρ), it means that the inverse map is well defined in Bl,exp(1/(l+1))ρ,with

φ−1l+1(Bl,exp(1/(l+1))ρ) ⊂ Bl,exp(1/l−2ε/l2)ρ .

In particular, Bl+1,exp(1/(l+1))ρ = φ−1l+1(Bl,exp(1/(l+1))ρ) is well defined. Lemma

3.5.4 then follows from (3.77) and (3.78). Proof of Lemma 3.5.5. Suppose that Condition (iii)l is satisfied. Then accordingto (3.28) we have:

‖f lij‖l,exp(1/l)ρ ≤ |f l

ij |l,exp(1/l)ρ =∣∣∣xl

i, xlj −

∑k

ckijx

lk

∣∣∣l,exp(1/l)ρ

≤ C1|Π|l,exp(1/l)ρ +∑

k

|ckij‖xl

k|l,ρ ≤ C1.C.ρ + C2. exp(1/l)ρ∑

k

|ckij | < C3ρ ,

(3.79)

where C3 is some positive constant (which does not depend on l).We can apply the above inequality ‖f l

ij‖l,exp(1/l)ρ < C3ρ and Lemma 3.4.3to find a positive constant C4 (which does not depend on l) and a solution wl

i of(3.37), such that ∥∥wl

i

∥∥l,exp(1/l)ρ

< C4ρ . (3.80)

Together with Lemma 3.5.1, the above inequality yields∣∣dwli

∣∣l,exp(1/l−ε/2l2)ρ

< C4, (3.81)

provided that l is large enough. Applying Lemma 3.5.1 and the assumption that|Π|l,exp(1/l)ρ < Cρ to the above inequality, we get∣∣wl

i, ylj∣∣l,exp(1/l−ε/2l2)ρ

< C5ρ


for some constant C5 (which does not depend on l). Using this inequality, andinequalities similar to (3.79), we get that the norm ‖.‖l,exp(1/l−ε/2l2)ρ of the 1-cocycle given in Formula (3.41) is bounded from above by C6ρ, where C6 is someconstant which does not depend on L. Using Lemma 3.4.3, we find a solution vL

i

to Equation 3.43 such that∥∥vli

∥∥l,exp(1/l−ε/2l2)ρ

< C6ρ , (3.82)

where C6 is some constant which does not depend on l. Lemma 3.5.5 (fr l largeenough compared to C6) now follows directly from Inequalities (3.80), (3.82) andLemma 3.5.1. Proof of Lemma 3.5.6. Suppose that Condition (ii)l+1 is satisfied. By Lemma3.5.4, Condition (i)l+1 is also satisfied. In particular,

Bl+1,exp(1/(l+1))ρ ⊂ Bl,exp(1/(l+1)+2ε/(l+1)2)ρ ⊂ Bl,exp(1/l−2ε/l2)ρ

(for ε < 1/4 and l large enough). Thus we have

|zl+1i , zl+1

j |l+1,exp(1/(l+1))ρ ≤ |zl+1i , zl+1

j |l,exp(1/l−2ε/l2)ρ ≤ T 1 + T 2 + T 3 + T 4

where

T 1 = |zli, z

lj|l,exp(1/l−2ε/l2)ρ ,

T 2 = |zl+1i − zl

i, zl+1j |l,exp(1/l−2ε/l2)ρ ,

T 3 = |zl+1i , zl+1

j − zlj|l,exp(1/l−2ε/l2)ρ ,

T 4 = |zl+1i − zl

i, zl+1j − zl

j|l,exp(1/l−2ε/l2)ρ .

For the first term, we have

T 1 ≤ |zli, z

lj|l,exp(1/l)ρ ≤ |Π|l,exp(1/l)ρ ≤ C. exp(−1/

√l)ρ .

Notice that C exp(−1/√

l + 1)ρ−C exp(−1/√

l)ρ > Cl2 ρ (for l large enough). So to

verify Condition (iii)l+1, it suffices to show that T 2 + T 3 + T 4 < Cl2 ρ. But this last

inequality can be achieved easily (provided that l is large enough) by Conditions(ii)l+1, (iii)l and Lemma 3.5.1. Lemma 3.5.6 is proved.

3.6 The smooth case

In this section we will give a sketch of the proof of Theorem 3.2.9, referring thereader to [263] for the details, which are quite long. Or rather, we will show whatmodifications need to be made to the proof of the analytic Levi decompositiontheorem 3.2.6 in order to obtain the proof of Theorem 3.2.9.

In this section, g will be a compact semisimple Lie algebra.

3.6. The smooth case 99

Denote by (ξ1, . . . , ξm) a fixed basis of g, which is orthonormal with respectto a fixed positive definite invariant metric on g. Denote by ck

ij the structuralconstants of g with respect to this basis:

[ξi, ξj ] =∑

k

ckijξk . (3.83)

Since g is compact, we may extend (ξ1, . . . , ξm) to a basis

(ξ1, . . . , ξm, y1, . . . , yn−m)

of l such that the corresponding Euclidean metric is preserved by the adjoint actionof g. The algebra g acts on l∗ = Rn via the coadjoint action of l ζ(z) := ad∗

ζ(z) forζ ∈ g ⊂ l, z ∈ Rn = l∗. The basis (ξ1, . . . , ξm, y1, . . . , yn−m) of l may be viewed asa coordinate system (x1, . . . , xm, y1, . . . , yn−m) on Rn (with xi = ξi).

Denote by G the compact simply-connected Lie group whose Lie algebra isg. Then the above action of g on Rn integrates into an action of G on Rn (thecoadjoint action). The action of G on Rn preserves the Euclidean metric of Rn

given by ‖z‖2 =∑ |xi(z)|2 +

∑ |yj(z)|2.For each positive number r > 0, denote by Br the closed ball of radius r

in Rn centered at 0. The group G (and hence the algebra g) acts linearly on thespace of functions on Br via its action on Br: for each function F and elementg ∈ G we put

g(F )(z) := F (g−1(z)) = F (Ad(g−1)z). (3.84)

In the smooth case, we will use Ck-norms and Sobolev norms. For eachnonnegative integer k ≥ 0 and each pair of real-valued functions F1, F2 on Br, wewill define the Sobolev inner product of F1 with F2 with respect to the SobolevHk-norm as follows:

〈F1, F2〉Hk,r :=∑|α|≤k

∫Br

( |α|!α!

)(∂|α|F1

∂zα(z)

)(∂|α|F2

∂zα(z)

)dµ(z). (3.85)

The Sobolev Hk-norm of a function F on Br is

‖F‖Hk,r :=

√〈F, F 〉Hk,r . (3.86)

We will denote by Cr the subspace of the space C∞(Br) of smooth real-valuedfunctions on Br, which consists of functions vanishing at 0 whose first derivativesalso vanish at 0. Then the action of G on Cr defined by (3.84) preserves the Sobolevinner products (3.85).

Denote by Yr the space of smooth vector fields on Br of the type

u =n−m∑i=1

ui∂/∂yi , (3.87)


such that ui vanish at 0 and their first derivatives also vanish at 0. Then Yr is ag-module under the following action:

ξi ·∑

j

uj∂/∂yj :=[∑

jk

ckijxk

∂

∂xj+∑jk

akijyk

∂

∂yj,∑

j

uj∂/∂yj

], (3.88)

where Xi =∑

jk ckijxk∂/∂xj +

∑jk ak

ijyk∂/∂yj are the linear vector fields whichgenerate the linear orthogonal coadjoint action of g on Rn.

Equip Yr with Sobolev inner products:

〈u, v〉Hk,r :=n−m∑i=1

〈ui, vi〉k,r , (3.89)

and denote by YHk,r the completion of Yr with respect to the corresponding Hk,r-

norm. Then YHk,r is a separable real Hilbert space on which g and G act orthogo-

nally.The Ck-norms can be defined as follows:

‖F‖k,r := sup|α|≤k

supz∈Br

|DαF (z)| (3.90)

for F ∈ Cr, where the sup runs over all partial derivatives of degree |α| at most k.Similarly, for u =

∑n−mi=1 ui∂/∂yi ∈ Yr we put

‖u‖k,r := supi

sup|α|≤k

supz∈Br

|Dαui(z)|. (3.91)

The Ck norms ‖.‖k,r are related to the Sobolev norms ‖.‖Hk,r as follows:

‖F‖k,r ≤ C‖F‖Hk+s,r and ‖F‖H

k,r ≤ C(n + 1)k‖F‖k,r (3.92)

for any F in Cr or Yr and any k ≥ 0, where s = [n2 ]+1 and C is a positive constant

which does not depend on k. In other words, Ck norms and Sobolev norms are“tamely equivalent”. A priori, the constant C depends on r, but later on we willalways assume that 1 ≤ r ≤ 2, and so may assume C to be independent of r. Theabove inequality is a version of the classical Sobolev’s lemma for Sobolev spaces.

Similarly to the analytic case, we will need the following normed version ofWhitehead’s lemma (cf. Proposition 2.1 of [81]):

Lemma 3.6.1 (Conn). For any given positive number r, and W = Cr or Yr withthe above action of g, consider the (truncated ) Chevalley–Eilenberg complex

Wδ0→W ⊗ ∧1g∗ δ1→W ⊗ ∧2g∗ δ2→W ⊗ ∧3g∗.

Then there is a chain of operators

Wh0← W ⊗ ∧1g∗ h1←W ⊗ ∧2g∗ h2←W ⊗ ∧3g∗


such thatδ0 h0 + h1 δ1 = IdW⊗∧1g∗ ,

δ1 h1 + h2 δ2 = IdW⊗∧2g∗ .(3.93)

Moreover, there exists a constant C > 0, which is independent of the radius r ofBr, such that

‖hj(u)‖Hk,r ≤ C‖u‖H

k,r j = 0, 1, 2 (3.94)

for all k ≥ 0 and u ∈ W ⊗ ∧j+1g∗. If u vanishes to an order l ≥ 0 at the origin,then so does hj(u).

Strictly speaking, Conn [81] proved the above lemma only in the case wheng = l and for the module Cr, but his proof is quite general and works perfectly inour situation without any modification. In fact, in order to prove Lemma 3.6.1, it issufficient to show that W is an infinite direct sum of finite-dimensional orthogonalmodules, and then repeat the proof of Lemma 3.4.1.

For simplicity, in the sequel we will denote the homotopy operators hj in theabove lemma simply by h. Homotopy relation (3.93) will be rewritten simply as

Id− δ h = h δ . (3.95)

The meaning of the last equality is as follows: if u is a 1-cocycle or 2-cocycle, thenit is also a coboundary, and h(u) is an explicit primitive of u: δ(h(u)) = u. If u isa “near cocycle” then h(u) is also a “near primitive” for u.

Combining Inequality (3.94) with Sobolev inequalities, we get the followingestimate for the homotopy operators h with respect to Ck norms:

‖h(u)‖k,r ≤ C(n + 1)k+s‖u‖k+s,r ∀ j = 0, 1, 2 (3.96)

for all k ≥ 0 and u ∈ W ⊗ ∧j+1g∗ where W = Cr or Yr. Here s = [n2 ] + 1, C is a

positive constant which does not depend on k (and r provided that 1 ≤ r ≤ 2).It is well known that the space C∞(Br) with Ck norms (3.90) is a tame

Frechet space (see, e.g., [167] for the theory of tame Frechet spaces). Since Cr isa tame direct summand of C∞(Br), it is also a tame Frechet space. Similarly,Yr with norms (3.91) is a tame Frechet space as well. What we will use here isthe fact that tame Frechet spaces admit smoothing operators and interpolationinequalities:

For each t > 1 there is a linear operator S(t) = Sr(t) from Cr to itself, calleda smoothing operator , with the following properties:

‖S(t)F‖p,r ≤ Cp,qt(p−q)‖F‖q,r (3.97)

and‖(Id− S(t))F‖q,r ≤ Cp,qt

(q−p)‖F‖p,r (3.98)

for any F ∈ Cr, where p, q are any nonnegative integers such that p ≥ q, Id denotesthe identity map, and Cp,q denotes a constant which depends on p and q.


The second inequality means that S(t) is close to identity and tends to iden-tity when t → ∞. The first inequality means that F becomes “smoother” whenwe apply S(t) to it. That’s why S(t) is called a smoothing operator. A priori,the constants Cp,q also depend on the radius r. But later on, we will always have1 ≤ r ≤ 2, and so we may choose Cp,q to be independent of r.

There is a similar smoothing operator from Yr to itself, which by abuse oflanguage we will also denote by S(t) or Sr(t). We will assume that inequalities(3.97) and (3.98) are still satisfied when F is replaced by an element of Yr.

For any F in Cr or Yr , and nonnegative integers p1 ≥ p2 ≥ p3, we have thefollowing interpolation inequality:

(‖F‖p2,r)p3−p1 ≤ Cp1,p2,p3(‖F‖p1,r)p3−p2(‖F‖p3,r)p2−p1 , (3.99)

where Cp1,p2,p3 is a positive constant which may depend on p1, p2, p3.Similarly to the analytic case, in order to prove Theorem 3.2.9, we will

construct by recurrence a sequence of local smooth coordinate systems (xd,yd) :=(xd

1,...,xdm,yd

1 ,...,ydn−m), which converges to a local coordinate system (x∞,y∞)=

(x∞1 ,...,x∞

m ,y∞1 ,...,y∞

n−m), in which the Poisson structure Π has the desired form.Here (x0, y0) = (x1, . . . , xm, y1, . . . , yn−m) is the original linear coordinate system.

For simplicity of exposition, we will assume that Π is C∞-smooth. However,in every step of the proof of Theorem 3.2.9, we will only use differentiability ofΠ up to some finite order, and that’s why our proof will also work for finitely(sufficiently highly) differentiable Poisson structures.

We will denote by Θd the local diffeomorphisms of (Rn, 0) such that

(xd, yd)(z) = (x0, y0) Θd(z) , (3.100)

where z denotes a point of (Rn, 0).Denote by Πd the Poisson structure obtained from Π by the action of Θd:

Πd = (Θd)∗Π . (3.101)

Of course, Π0 = Π. Denote by ., .d the Poisson bracket with respect to thePoisson structure Πd. Then we have

F1, F2d(z) = F1 Θd, F2 Θd(Θd−1(z)) . (3.102)

Assume that we have constructed (xd, yd) = (x, y)Θd. Let us now construct(xd+1, yd+1) = (x, y) Θd+1. Similarly to the analytic case, this construction con-sists of two steps: 1) find an “almost Levi factor”, i.e., coordinates xd+1

i such thatthe error terms xd+1

i , xd+1j −∑

k ckijx

d+1k are small, and 2) “almost linearize”

it, i.e., find the remaining coordinates yd+1 such that in the coordinate system(xd+1, yd+1) the Hamiltonian vector fields of the functions xd+1

i are very closeto linear ones. In fact, we will define a local diffeomorphism θd+1 of (Rn, 0) andthen put Θd+1 = θd+1 Θd. In particular, we will have Πd+1 = (θd+1)∗Πd and(xd+1, yd+1) = (xd, yd) (Θd)−1 θd+1 Θd.


Similarly to the analytic case, consider the 2-cochain

fd =∑ij

fdij ⊗ ξ∗i ∧ ξ∗j (3.103)

of the Chevalley–Eilenberg complex associated to the g-module Cr, where now

fdij(x, y) = xi, xjd −

m∑k=1

ckijxk, (3.104)

and r = rd depends on d and can be chosen as follows:

rd = 1 +1

d + 1. (3.105)

In particular, r0 = 2, rd/rd+1 ∼ 1+ 1d2 , and limd→∞ rd = 1 is positive. This choice

of radii rd means in particular that we will be able to arrange it so that the Poissonstructure Πd = (Θd)∗Π is defined in the closed ball of radius rd. (For this to hold,we will have to assume that Π is defined in the closed ball of radius 2, and showby recurrence that Brd

⊂ θd(Brd−1) for all d ∈ N.)Put

ϕd+1 :=∑

i

ϕd+1i ⊗ ξ∗i = S(td)

(h(fd)

), (3.106)

where h is the homotopy operator as given in Lemma 3.6.1, S is the smoothingoperator and the parameter td is chosen as follows: take a real constant t0 > 1(which later on will be assumed to be large enough) and define the sequence (td)d≥0

by td+1 = t3/2d . In other words, we have

td = exp

((32

)d

ln t0

), ln t0 > 0 . (3.107)

The above choice of smoothing parameter td is a standard one in problems in-volving the Nash–Moser method, see, e.g., [166, 167]. The number 3

2 in the aboveformula is just a convenient choice. The main point is that this number is greaterthan 1 (so we have a very fast increasing sequence) and smaller than 2 (where 2corresponds to the fact that we have a fast convergence algorithm which “quadra-tizes” the error term at each step, i.e., goes from an “ε-small” error term to an“ε2-small” error term).

According to Inequality (3.96), in order to control the Ck-norm of h(fd) weneed to control the Ck+s-norm of fd, i.e., we face a “loss of differentiability”.That’s why in the above definition of ϕd+1 we have to use the smoothing operatorS, which will allow us to compensate for this loss of differentiability. This is astandard trick in the Nash–Moser method.


Next, consider the 1-cochain

gd =∑

i

(∑α

xi − h(fd)i, yαd −n−m∑β=1

aβiαyβ

∂

∂yα

)⊗ ξ∗i (3.108)

of the Chevalley–Eilenberg complex associated to the g-module Yr, where r =rd = 1 + 1

d+1 , and put

ψd+1 :=∑α

ψd+1α

∂

∂yα= S(td)

(h(gd)

), (3.109)

where h is the homotopy operator as given in Lemma 3.6.1, and S(td) is thesmoothing operator (with the same td as in the definition of ϕd+1).

Now define θd+1 to be a local diffeomorphism of Rn given by

θd+1 := Id− (ϕd+11 , . . . , ϕd+1

m , ψd+11 , . . . , ψd+1

n−m) . (3.110)

This finishes our construction of Θd+1 = θd+1Θd and (xd+1, yd+1) = (x, y)Θd+1. This construction is very similar to the analytic case, except mainly for theuse of the smoothing operator. Another difference is that, for technical reasons,in the smooth case we use the original coordinate system and the transformedPoisson structures Πd for determining the error terms, while in the analytic casethe original Poisson structure and the transformed coordinate systems are used. Inparticular, the closed balls used here are always balls with respect to the originalcoordinate system – this allows us to easily compare the Sobolev norms of functionson them, i.e., bigger balls correspond to bigger norms.

The technical part of the proof (see [263]) consists of a series of lemmas whichshow that the above construction actually yields a smooth Levi normalization inthe limit, provided that Π is defined on the closed ball of radius 2 and is sufficientlyclose to its linear part there. If Π does not satisfy these conditions, then we mayuse the following homothety trick to make it satisfy: replace Π by Πt = 1

t G(t)∗Πwhere G(t) : z → tz is a homothety, t > 0. The limit limt→∞ Πt is equal to thelinear part of Π. So by choosing t high enough, we may assume that Πt is definedon the closed ball of radius 2 and is sufficiently close to its linear part there. If Θis a local smooth Levi normalization for Πt, then G(1/t) Θ G(t) will be a localsmooth Levi normalization for Π.Remark 3.6.2. In [263] there is also an abstract Nash–Moser normal form theorem,which can be applied to the problem of smooth Levi decomposition of Poissonstructures, and hopefully to other smooth normal form problems as well.

Chapter 4

Linearization ofPoisson Structures

4.1 Nondegenerate Lie algebras

Let Π be a Poisson structure which vanishes at a point z: Π(z) = 0. Denote by

Π = Π(1) + Π(2) + · · · , (4.1)

the Taylor expansion of Π in a local coordinate system centered at z, where Π(k)

is a homogeneous 2-vector field of degree k. Recall that, the terms of degree k ofthe equation [Π, Π] = 0 give

k∑i=1

[Π(i), Π(k+1−i)

]= 0. (4.2)

In particular, [Π(1), Π(1)] = 0, i.e., the linear part Π(1) of Π is a linear Poissonstructure. One says that Π is locally smoothly (resp. analytically, resp. formally)linearizable if there is a local smooth (resp. analytic, resp. formal) diffeomorphismφ (a coordinate transformation) such that φ∗Π = Π(1).

Definition 4.1.1 ([346]). A finite-dimensional Lie algebra g is called formally (resp.analytically, resp. smoothly) nondegenerate if any formal (resp. analytic, resp.smooth) Poisson structure Π which vanishes at a point and whose linear partat that point corresponds to g is formally (resp. analytically, resp. smoothly)linearizable.

In other words, a Lie algebra is nondegenerate if any Poisson structure, whoselinear part corresponds to this algebra, is completely determined by its linear partup to local isomorphisms.

106 Chapter 4. Linearization of Poisson Structures

The above definition begs the question: which Lie algebras are nondegenerateand which are degenerate? This question is the main topic of this chapter. One ofthe main tools for studying it is the Levi decomposition, treated in the previouschapter. The question is still largely open, though we now know of several seriesof nondegenerate Lie algebras, and many degenerate ones (it is much easier ingeneral to find degenerate Lie algebras than to find nondegenerate ones).

As explained in Chapter 2, formal deformations of Poisson structures aregoverned by Poisson cohomology, and for linear Poisson structures Poisson coho-mology is a special case of Chevalley–Eilenberg cohomology of the correspondingLie algebras. In particular, if l is a Lie algebra such that H2(l,Skl) = 0 ∀ k ≥ 2,then it is formally nondegenerate (Theorem 2.3.8). Let us recall here a special caseof this result:

Theorem 4.1.2 ([346]). Any semisimple Lie algebra is formally nondegenerate.

In general, it is much more difficult to study smooth or analytic nondegener-acy of Lie algebras than to study their formal nondegeneracy, because the formerproblem involves not only algebra (cohomology of Lie algebras) but also geometryand analysis (to show the analyticity or smoothness of coordinate transforma-tions).

The first significant results about analytic and smooth nondegenerate Liealgebras are due to Conn [80, 81], and are already mentioned in Chapter 3 asspecial cases of Levi decomposition theorems. Let us recall here Conn’s results.

Theorem 4.1.3 ([80]). Any semisimple Lie algebra is analytically nondegenerate.

Theorem 4.1.4 ([81]). Any compact semisimple Lie algebra is smoothly nondegen-erate.

On the other hand, most non-compact real semisimple Lie algebras aresmoothly degenerate (see Section 4.3).

Related to the notion of (formal) nondegeneracy is the notion of rigidity ofLie algebras, mentioned in Subsection 2.3.3, and also the notion of strong rigidity[39]. Recall that H2(g, g) is the cohomology group which governs infinitesimaldeformations of a Lie algebra g. This group is somehow related to the group⊕k≥2H

2(g, Skg), but they are not the same. Not surprisingly, there are Lie algebraswhich are rigid but degenerate (e.g., saff(2), see Example 4.1.5), Lie algebras whichare non-rigid but nondegenerate (e.g., a three-dimensional solvable Lie algebraK A K2, where A is a nonresonant 2× 2 matrix, see Theorem 4.2.2), Lie algebraswhich are both rigid and nondegenerate (e.g., semisimple Lie algebras), and Liealgebras which are both non-rigid and degenerate (e.g., Abelian Lie algebras).

Example 4.1.5. Denote by saff(2, K) = sl(2, K)K2 the Lie algebra of infinitesimalarea-preserving affine transformations on K2. Then saff(2, K) is rigid but degen-erate. The linear Poisson structure corresponding to saff(2) has the form Π(1) =2e∂h∧∂e−2f∂h∧∂f+h∂e∧∂f+y1∂h∧∂y1−y2∂h∧∂y2+y1∂e∧∂y2+y2∂f∧∂y1 in anatural system of coordinates. Now put Π = Π(1)+Π with Π = (h2+4ef)∂y1∧∂y2.

4.2. Linearization of low-dimensional Poisson structures 107

Then Π is a Poisson structure, vanishing at the origin, with a linear part corre-sponding to saff(2). For Π(1) the set where the rank is less than or equal to 2is a codimension 2 linear subspace (given by the equations y1 = 0 and y2 = 0).For Π the set where the rank is less than or equal to 2 is a two-dimensional cone(the cone given by the equations y1 = 0, y2 = 0 and h2 + 4ef = 0). So these twoPoisson structures are not isomorphic, even formally. The rigidity of saff(2) willbe left to the reader as an exercise (see [300, 63]).

Example 4.1.6. The Lie algebra e(3) = so(3)R3 of rigid motions of the Euclideanspace R3 is degenerate and non-rigid. The linear Poisson structure correspondingto e(3) has the form Π(1) = x1∂x2 ∧ ∂x3 + x2∂x3 ∧ ∂x1 + x3∂x1 ∧ ∂x2 + y1∂x2 ∧∂y3 + y2∂x3 ∧ ∂y1 + y3∂x1 ∧ ∂y2 in a natural system of coordinates. Now putΠ = Π(1) + Π with Π = (x2

1 + x22 + x2

3)(x1∂x2 ∧ ∂x3 + x2∂x3 ∧ ∂x1 +x3∂x1 ∧ ∂x2).For Π(1) the set where the rank is less than or equal to 2 is a dimension 3 subspace(given by the equation y1 = y2 = y3 = 0), while for Π the set where the rank is lessthan or equal to 2 is the origin. A Lie algebra not isomorphic to e(3) but adjacentto e(3) is so(3, 1), the Lie algebra of infinitesimal linear automorphisms of theMinkowski space. Here the adjective adjacent means that, in the variety of all Liealgebra structures of dimension 6 (see Subsection 2.3.3), the GL(6)-orbit whichcorresponds to e(3) lies in the closure (with respect to the Euclidean topology)of the GL(6)-orbit which corresponds to so(3, 1). One also says that e(3) is acontraction of so(3, 1). If a Lie algebra is a contraction of another Lie algebra,then it is not rigid.

A strongly rigid Lie algebra is a Lie algebra g whose universal envelopingalgebra U(g) is rigid as an associative algebra [39]. It is easy to see that if g isstrongly rigid then it is rigid. A sufficient condition for g to be strongly rigidis H2(g, Skg) = 0 ∀ k ≥ 0, and if this condition is satisfied then g is calledinfinitesimally strongly rigid [39]. Obviously, if g is infinitesimally strongly rigid,then it is formally nondegenerate. In fact, we have the following result, due toBordemann, Makhlouf and Petit:

Theorem 4.1.7 ([39]). If g is a strongly rigid Lie algebra, then it is formally non-degenerate.

We refer to [39] for the proof of the above theorem, which is based on Kont-sevich’s theorem [195] on the existence of deformation quantization of Poissonstructures (see Appendix A.9).

4.2 Linearization of low-dimensional Poisson structures

4.2.1 Two-dimensional case

Up to isomorphisms, there are only two Lie algebras of dimension 2: the trivial,i.e., Abelian one, and the solvable Lie algebra K K, which has a basis (e1, e2)


with [e1, e2] = e1. This Lie algebra is isomorphic to the Lie algebra of infinitesimalaffine transformations of the line, so we will denote it by aff(1).

The Abelian Lie algebra of dimension 2 is of course degenerate. For example,the quadratic Poisson structure (x2

1 + x22)

∂∂x1

∧ ∂∂x2

is nontrivial and is not locallyisomorphic to its linear part, which is trivial.

On the other hand, we have:

Theorem 4.2.1 ([14]). The Lie algebra aff(1) is formally, analytically and smoothlynondegenerate.

Proof. We begin with x, y = x + · · · . Putting x′ = x, y, y′ = y, we obtainx′, y′ = ∂x′

∂x x, y = x′a(x′, y′), where a is a function such that a(0) = 1. Wefinish with a second change of coordinates x′′ = x′, y′′ = f(x′, y′), where f is afunction such that ∂f

∂y′ = 1/a.

4.2.2 Three-dimensional case

Every Lie algebra of dimension 3 over R or C is of one of the following three types,where (e1, e2, e3) denote a basis:

• so(3) with brackets [e1, e2] = e3, [e2, e3] = e1, [e3, e1] = e2.

• sl(2) with brackets [e1, e2] = e3, [e1, e3] = e1, [e2, e3] = −e2. (Recall thatso(3, R) sl(2, R), so(3, C) ∼= sl(2, C).)

• semi-direct products K A K2 where K acts linearly on K2 by a matrix A.In other words, we have brackets [e2, e3] = 0, [e1, e2] = ae2 + be3, [e1, e3] =ce2 + de3, and A is the 2× 2-matrix with coefficients a, b, c and d. (Differentmatrices A may correspond to isomorphic Lie algebras.)

The Lie algebras sl(2) and so(3) are simple, so they are formally and ana-lytically nondegenerate, according to Weinstein’s and Conn’s theorems.

The fact that the compact simple Lie algebra so(3, R) is smoothly nonde-generate (it is a special case of Conn’s Theorem 4.1.4) is due to Dazord [94]. InChapter 6, we will extend this result of Dazord to the case of elliptic singularitiesof Nambu structures, using arguments similar to his.

On the other hand, sl(2, R) is known to be smoothly degenerate (see [346]).A simple construction of a smooth nonlinearizable Poisson structure whose linearpart corresponds to sl(2, R) is as follows: In a linear coordinate system (y1, y2, y3),write

Π(1) = y3∂

∂y1∧ ∂

∂y2− y2

∂

∂y1∧ ∂

∂y3− y1

∂

∂y2∧ ∂

∂y3= X ∧ Y,

where X = y2∂

∂y3− y3

∂∂y2

, and Y = ∂∂y1

+ y1y22+y2

3

(y2

∂∂y2

+ y3∂

∂y3

). This linear

Poisson structure corresponds to sl(2, R) and has C = y22 + y2

3 − y21 as a Casimir

function. Denote by Z a vector field on R3 such that Z = 0 when y22 +y2

3−y21 ≥ 0,

and Z = G(C)√y22+y2

3

(y2

∂∂y2

+ y3∂

∂y3

)when y2

2 + y23 − y2

1 > 0, where G is a flat


function such that G(0) = 0 and G(C) > 0 when C > 0. Then Z is a flat vectorfield such that [Z, X ] = [Z, Y ] = 0. Hence Π = (X −Z)∧ Y is a Poisson structurewhose linear part is Π(1) = X ∧ Y . While X is a periodic vector field, the integralcurves of X − Z in the region y2

2 + y23 − y2

1 > 0 are spiraling towards the coney2

2 + y23 − y2

1 = 0. Thus, while almost all the symplectic leaves of Π(1) areclosed, the symplectic leaves of Π in the region y2

2 + y23 − y2

1 > 0 contain thecone y2

2 + y23 − y2

1 = 0 in their closure (also locally in a neighborhood of 0).This implies that the symplectic foliation of Π is not locally homeomorphic to thesymplectic foliation of Π(1). Hence Π can’t be locally smoothly equivalent to Π(1).

For solvable Lie algebras K A K2, we have the following result:

Theorem 4.2.2 ([110]). The Lie algebra R2×A R is smoothly (or formally) nonde-generate if and only if A is nonresonant in the sense that there are no relationsof the type

λi = n1λ1 + n2λ2 (i = 1 or 2), (4.3)

where λ1 and λ2 are the eigenvalues of A, n1 and n2 are two nonnegative integerswith n1 + n2 > 1.

Proof. Let Π be a Poisson structure on a three-dimensional manifold which van-ishes at a point with a linear part corresponding to R2 ×A R with a nonresonantA. In a system of local coordinates (x, y, z), centered at the considered point, wehave

z, x = ax + by + O(2), z, y = cx + dy + O(2), x, y = O(2), (4.4)

where a, b, c, d are the coefficients of A, and O(2) means terms of degree at least2. It follows that the curl vector field DΩΠ (see Section 2.6), with respect to anyvolume form Ω, has the form (a + c)∂/∂z + Y , where Y is a vector field vanishingat the origin. But the nonresonance hypothesis imposes that the trace of A is notzero; so DΩΠ doesn’t vanish in a neighborhood of the origin. We can straighten itand suppose that the coordinates (x, y, z) are chosen such that DΩΠ = ∂z.

Now the three-dimensional hypothesis implies Π ∧Π = 0 and, using formula

[Π, Π] = DΩ(Π ∧Π)±DΩ(Π) ∧Π

(see Section 2.6), we obtainDΩ(Π) ∧Π = 0.

In the above coordinates this gives ∂/∂z ∧Π = 0 and, so,

Π = ∂/∂z ∧X,

where X is a vector field. Now we recall the basic formula

[DΩ(Π), Π] = 0

which have the consequence that we can suppose that X depends only on thecoordinates x and y.


Because of the form of the linear part of Π, X is a vector field which vanishesat the origin but with a nonresonant linear part. Hence, up to a smooth (or formal)change of coordinates x and y, we can linearize X (see Appendix A.5). This givesthe smooth (or formal) nondegeneracy of R2 ×A R.

To prove the “only if” part, we start with a linear Poisson structure

Π(1) = ∂/∂z ∧X(1),

where X(1) is a linear resonant vector field. Every resonance relation permitsthe construction of a polynomial perturbation X of X(1) which is not smoothlyisomorphic to X(1), even up to a product with a function (see Appendix A.5).Then it is not difficult to prove that ∂/∂z ∧ X is a polynomial perturbation ofΠ(1) which is not equivalent to it. Remark 4.2.3. The same proof shows that algebras of type K2 ×A K (where K =R or C) are analytic nondegenerate if we add to the nonresonance condition aDiophantine condition on the eigenvalues of A (see Appendix A.5).

4.2.3 Four-dimensional case

The results on (non)degeneracy of four-dimensional Lie algebras presented in thissubsection are taken from Molinier’s thesis [258].

According to [292], every four-dimensional Lie algebra over K, where K = R

or C, belongs to (at least) one of the following four types:Type 1: direct products K×L3 where L3 is a three-dimensional algebra (see the

preceding paragraph for a classification).Type 2: semi-direct products KA K3, where K3 is the commutative Lie algebra

of dimension 3, and K acts on K3 by a matrix A.Type 3: semi-direct products K A H3, where H3 is the three-dimensional Hei-

senberg Lie algebra: H3 has a basis (x, y, z) such that [x, y] = z, [x, z] =[y, z] = 0.

Type 4: semi-direct products K2 K2.In the first type we have K× sl(2) and K× so(3), which are the same when

K = C, and the cases where L3 is a semi-direct product K K2. The Levi decom-position theorems from Chapter 3 imply that K×sl(2) and K×so(3) are formallyand analytically nondegenerate, and that R× so(3, R) is smoothly nondegenerate.However, R× sl(2, R) is smoothly degenerate: just repeat the proof, given in theprevious subsection, of the fact that sl(2) is smoothly degenerate. The case whereL3 is a semi-direct product is degenerate (formally, analytically and smoothly): ifwe choose coordinates (u, v, x, y) such that the corresponding linear Poisson tensorhas the form

(ax + by)∂

∂u∧ ∂

∂x+ (cx + dy)

∂

∂v∧ ∂

∂y, (4.5)

then we can add a quadratic term v2 ∂∂u ∧ ∂

∂v to get a nonlinearizable Poissontensor.


Every algebra of the second type is degenerate. To prove this we choose co-ordinates (u, x1, x2, x3) such that in these coordinates the corresponding linearPoisson structure has brackets u, xi =

∑j aj

ixj , (the others brackets are triv-ial), where aj

i are the coefficients of the matrix A. Up to isomorphisms, we cansuppose that A is in Jordan form and we can also replace A by λA where λ is anynonvanishing constant. So we have the following list of cases:

u, x1 = 0, u, x2 = x1, u, x3 = x2; (4.6)u, x1 = ax1, u, x2 = x2, u, x3 = x2 + x3; (4.7)u, x1 = x1, u, x2 = 0, u, x3 = x2; (4.8)u, x1 = x1, u, x2 = x1 + x2, u, x3 = x2 + x3; (4.9)u, x1 = x1, u, x2 = ax2, u, x3 = bx3; (4.10)u, x1 = ax1, u, x2 = bx2 − x3, u, x3 = x2 + bx3. (4.11)

Each of the above cases can be perturbed to a nonlinearizable Poisson structure byadding a quadratic term: ∂x3 ∧ (x2

1∂x1 + x1x2∂x2) for (4.6); x22∂x3 ∧ ∂x2 for (4.7)

and (4.8); x21∂x3 ∧ ∂x2 for (4.9); x2x3∂x3 ∧ ∂x2 for (4.10); and (x2

2 +x23)∂x3 ∧ ∂x2

for (4.11).Similarly, every algebra of the third type is also degenerate. To prove this

we choose coordinates (u, x1, x2, x3) such that the corresponding linear Poissonstructure has brackets x3, x2 = x1, u, xi =

∑j aj

ixj , (the other brackets arezero), where aj

i are the coefficients of the matrix A. Up to isomorphisms, we havethe following cases:

u, x1 = 2x1, u, x2 = x2, u, x3 = x2 + x3; (4.12)u, x1 = (1 + b)x1, u, x2 = x2, u, x3 = bx3; (4.13)u, x1 = 2ax1, u, x2 = ax2 − x3, u, x3 = x2 + ax3. (4.14)

Each case can be perturbed to a nonlinearizable Poisson structure by adding aquadratic term: x2

2∂x3∧∂x2 for (4.12); x2x3∂x3∧∂x2 for (4.13); and (x22+x2

3)∂x3∧∂x2 for (4.14).

Finally, if g is a Lie algebra of the last type, which does not belong to theprevious three types, then it admits a basis (u, v, x, y), with either the brackets

[u, x] = x, [v, y] = y; (4.15)

or the brackets

[u, x] = x, [u, y] = y , [v, x] = −y, [v, y] = x. (4.16)

When K = C then the above two cases are the same and are isomorphic toaff(1, C) × aff(1, C). When K = R, these are the two real versions of aff(1, C) ×aff(1, C). We will see in Section 4.4 that (4.15) is formally, analytically andsmoothly nondegenerate, and therefore (4.16) is also formally and analyticallynondegenerate. We don’t know whether (4.16) is smoothly nondegenerate.


4.3 Poisson geometry of real semisimple Lie algebras

We refer to [170, 194, 272] for the theory of real semisimple Lie algebras usedin this section. Recall that, if g is a real semisimple Lie algebra, then there isa (unique up to isomorphisms) Cartan involution θ : g → g, and we can writeg = k + s, where k = x ∈ g, θ(x) = x and s = x ∈ g, θ(x) = −x. Denote bygC the complexification of g, and by g0 = k +

√−1s ⊂ gC. Then g0 is a compact

real semisimple Lie algebra (the compact part of gC; if g is compact then s = 0and g = g0 = k). The decomposition g = k + s is called the Cartan decompositionof g, and k is a subalgebra of g, called the compact part of g. We also have[k, s] ⊂ s, [s, s] ⊂ k. At the group level, K is the maximal compact subgroup of G,and S = G/K is a symmetric space. The real rank of g is the same as the rankof the symmetric space S, and is equal to the dimension of a maximal Abeliansubalgebra of g which is contained in s.

In this section, we will explain the following result of Weinstein [348] aboutthe existence of smooth non-Hamiltonian Poisson vector fields for linear non-compact real semisimple Poisson structures.

Theorem 4.3.1 ([348]). Let g be a real semisimple Lie algebra of real rank r ≥ 1.Then for the corresponding linear Poisson structure Π(1) on the dual space g∗,there are r smooth Poisson vector fields X1, . . . , Xr which are flat at the originand which commute pair-wise, and such that, in an open subset of g∗ whose closurecontains the origin, these vector fields span an r-dimensional distribution whoseintersection with the characteristic distribution (i.e., tangent spaces to coadjointorbits) is trivial.

A direct consequence of Theorem 4.3.1 is the following:

Theorem 4.3.2 (Weinstein [348]). Any non-compact real semisimple Lie algebra gof real rank at least 2 is smoothly degenerate.

Proof of Theorem 4.3.2. Let X1, . . . , X2 be smooth Poisson vector fields on g∗

given by Theorem 4.3.1, where r ≥ 2 is the real rank of g. Then since [X1, X2] = 0,[X1, Π(1)] = [X2, Π(1)] = 0 and [Π(1), Π(1)] = 0, where Π(1) denotes the linearPoisson structure, by the Leibniz rule we obtain that [Π(1) +X1 ∧X2, Π(1) +X1 ∧X2] = 0. In other words, Π = Π(1) + X1 ∧ X2 is a smooth Poisson structure.The linear part of Π is Π(1) because X1 and X2 are flat. In an open region whoseclosure contains the origin, where X1 and X2 are linearly independent and aretransversal to the symplectic leaves of Π(1), the characteristic distribution of Π isspanned by X1, X2 and the characteristic distribution of Π(1). Hence the rank of Π(in any neighborhood of the origin) is equal to the rank of Π(1) plus 2. Since Π andΠ(1) don’t have the same rank (locally near 0), they can’t be locally isomorphic.In other words, Π is not locally smoothly linearizable.

Proof of Theorem 4.3.1. Denote by G = KAN the Iwasawa decomposition of G,where G is the connected simply-connected Lie group whose Lie algebra is g. K

4.3. Poisson geometry of real semisimple Lie algebras 113

is the compact part of G, A is Abelian, N is nilpotent. Denote by M ⊂ K thecentralizer of A in K, and by P = MAN the minimal parabolic subgroup. Denoteby g = k+a+n the corresponding Iwasawa decomposition of g at the algebra level,by m the Lie algebra of M , m ⊂ k, and by p = m + a + n the parabolic subalgebra.The real rank of g is r = dim a ≥ 1 by assumptions. The adjoint action of a on gis diagonalizable with real weights, and we have a linear decomposition of g into asum of eigenspaces of this adjoint action of a, called the real root decomposition.n is spanned by the eigenspaces of positive roots, while m + a is the eigenspace ofweight 0. Denote by a+ the open (without boundary) positive Weyl chamber of awith respect to a root system of the real root decomposition of g. Denote

g+ = AdK(m + a+ + n) = AdKp+, p+ = m + a+ + n . (4.17)

We will need the following lemma of Weinstein:

Lemma 4.3.3 ([348]). If two elements of p+ are conjugate by an element g of G,then g belongs to P .

See ([348], Lemma (2.3)) for the proof of the above lemma, which uses stan-dard results from the theory of real semisimple Lie algebras.

For each a ∈ a, denote

Pa = AdK(m + a + n) . (4.18)

Consider the following map Eab : Pa → Pb, for each pair a, b ∈ a with a = 0:

Eab(Adh(m + a + n)) = Adh(m + b + n) ∀ h ∈ K, m ∈ m, n ∈ n . (4.19)

Lemma 4.3.4. With the above notations, we have:a) g+ =

⋃a∈a+

Pa is an open cone in g.b) Each Pa is saturated by adjoint orbits of G: if x ∈ Pa then AdG(x) ⊂ Pa.c) If a, b ∈ a+, a = b, then Pa is disjoint from Pb: Pa ∩ Pb = ∅.d) dimPa = dim g− r ∀ a ∈ a, where r = dim a ≥ 1 is the real rank. If a ∈ a+

then Pa is a smooth submanifold of g.e) The map Eab : Pa → Pb is well defined for all a ∈ a+ and b ∈ a. If a, b ∈ a+

then Eab is a diffeomorphism. When b = 0, Ea0 is a local diffeomorphismalmost everywhere.

The proof of the above lemma follows directly from the root decomposition ofg with respect to a and Lemma 4.3.3. It will be left to the reader as an exercise.

Identify g with g∗ via the Killing form. Then g is a linear Poisson manifoldwhose symplectic leaves are the adjoint orbits. Each Pa is saturated by symplecticleaves of g and has a natural induced Poisson structure.

Proposition 4.3.5. Eab : Pa → Pb is a Poisson isomorphism for any a, b ∈ a+.


Proof of Proposition 4.3.5. Put c = a− b, and denote Eab simply by E . Since E isequivariant under the adjoint action of K, it is enough to show that E preservesthe Poisson tensor at points of the type x = m + a + n ∈ m + a + n ⊂ Pa.

For an element g ∈ g, we will write g = gk +ga +gn, where gk ∈ k, ga ∈ a, gn ∈n. The map E maps x to x0 = m+ b+n. By direct computations, one verifies that

E∗(x)([g, x]) = [g, x0] + [gn, c] = [g, x]− [gk, c] . (4.20)

In other words, the element [g, x] ∈ g, considered as a tangent vector at x toAdG(x) ⊂ Pa, will be mapped under E to [g, x0] + [gn, c], considered as a tangentvector to Pb at x0.

If y∗ ∈ g∗ is an element of g∗, considered as a covector at x0, then its pull-back by E is an element y∗ ∈ g∗, considered as a covector at x and determinedonly up to elements which vanish on Tx(Pa), such that

〈ad∗xy∗, g〉 = 〈y∗, [g, x]〉 = 〈y∗, [g, x]− [gk, c]〉 = 〈ad∗

xy∗ − (ad∗cy

∗)k, g〉, (4.21)

where (ad∗cy

∗)k denotes the k∗-component of ad∗ay∗ with respect to the decompo-

sition g∗ = k∗ + a∗ + n∗ which is dual to the decomposition g = k + a + n.Denote by y, y the elements of g which correspond to y∗, y∗ under the identifi-

cation of g with g∗ via the Killing form, then we can rewrite the above equation as:

adxy = adxy − (adcy)k , (4.22)

where (adcy)k is the image of (ad∗cy

∗)k in g under the identification of g∗ with g,and not the k-component of adcy (under this identification via the Killing form,k∗ is identified with m + n rather than k).

The Hamiltonian vector (i.e., the image under the map corresponding tothe linear Poisson structure on g) of y∗ at x is adxy and the Hamiltonian vector ofy∗ at x0 is adx0y. To show that E preserves the Poisson structure, we must showthat

E∗(x)(adxy) = adx0y. (4.23)

According to Formula (4.22), we have

E∗(x)(adxy) = E∗(x)(adxy)− E∗(x)((adcy)k) .

Notice that (adcy)k = [c, yn] ∈ n (because 〈n, a〉 = 〈k, a〉 = 0). Hence (adcy)k

is invariant under E∗(x), because E∗(x) is identity on m + n. In other words, wehave E∗(x)((adcy)k) = (adcy)k = [c, yn]. Now applying Formula (4.20), we getE∗(x)(adxy) = E∗(x)(adxy)− [c, yn] = [x0, y]. Proposition 4.3.5 is proved.

An immediate consequence of Proposition 4.3.5 is that the open cone g+ =⋃a∈a+

Pa, as a Poisson manifold, is isomorphic to the direct product Pa× a+ (forany a ∈ a+), where the Poisson structure on a+ is trivial.

Since dim a+ = r, we can construct r smooth vector fields Y1, . . . , Yr on a+

with the following properties: they commute pairwise and are linearly independent

4.3. Poisson geometry of real semisimple Lie algebras 115

also everywhere on a+, and they extend to smooth vector fields on a which vanishon the complement of a+. (We will leave the actual construction of these vectorfields to the reader as an exercise.) Then lifting Y1, . . . , Yr to g+ via the Poissonisomorphism g+

∼= Pa × a+, we get r smooth pairwise commuting Poisson vectorfields on g+, which can be extended to g by making them vanish on the complementof g+. These are the Poisson vector fields that we are looking for. Theorem 4.3.1is proved. Remark 4.3.6. The above proof of Theorem 4.3.1 is a simplified version of Wein-stein’s original proof in [348], where he interpreted g+ as a classical Yang–Mills–Higgs phase space (see Example 1.7.12): g+ is isomorphic to Y(P, G, m∗×a∗+), theclassical Yang–Mills–Higgs phase space of configuration space G/P, gauge group Pand internal phase space m∗, where P = MAN (the real Borel subgroup of G), Gis viewed as a principal P bundle over G/P , and the action of P on m∗× a∗+ ⊂ p∗

is the coadjoint action. Weinstein [348] showed that classical Yang–Mills–Higgssetups form a category with natural morphisms, and deduced from this fact thatg+∼= Y(P, G, m∗ × a∗+) is isomorphic to Y(M, K, m∗ × a∗) where K is viewed as a

principal bundle over K/M (basically because K/M = G/P ). On the other hand,Y(M, K, m∗×a∗+) is clearly isomorphic to Y(M, K, m∗)×a∗+, with the trivial Pois-son structure on a∗+. We refer to [348] for more details and interesting connectionswith representation theory.

When g is a non-compact real semisimple Lie algebra with a Cartan decom-position g = k + s, then the semi-direct product

g1 = k s, (4.24)

where s is considered as a vector space with a trivial bracket on which k acts, iscalled the Cartan motion algebra of g. Let us mention here another interestingresult from [348]:

Theorem 4.3.7 (Weinstein [348]). With the above notations, there is a K-equivari-ant Poisson isomorphism from g+ to an open cone in the dual of the Cartanmotion algebra g1.

As an exercise, the reader may try to find for himself a proof of Theorem4.3.7 similar to the above proof of Proposition 4.3.5.

Weinstein’s Theorem 4.3.2 leaves out the case of real semisimple Lie algebrasof real rank 1. The rest of this section is devoted to the discussion of this case.

From the classification of real simple Lie algebras (see, e.g., [170, 194, 272]),we have the following complete list of real simple Lie algebras of real rank 1:so(n, 1), su(n, 1), sp(n, 1), and a real form of F2. (There is an isomorphismso(2, 1) ∼= su(1, 1) ∼= sl(3, R); the other algebras in the list are distinct.) A generalreal semisimple Lie algebra of real rank 1 is a direct sum of one of the abovealgebras with a compact semisimple Lie algebra. The rank-1 symmetric spacescorresponding to so(n, 1), su(n, 1) and sp(n, 1) are the hyperbolic space Hn, thecomplex hyperbolic space, and the quaternionic hyperbolic space, respectively.


Conjecture 4.3.8. A real semisimple Lie algebra g of real rank 1 is smoothly non-degenerate if and only if its compact part k is semisimple.

Among the real simple Lie algebras of real rank 1, only the algebras su(n, 1)have a non-semisimple compact part. (The compact part of su(n, 1) is isomorphicto su(n)⊕R; at the group level, the compact part of SU(n, 1) is S(U(n)×U(1)) ∼=SU(n) × T1.) Hence, the “only if” part of Conjecture 4.3.8 is provided by thefollowing theorem.

Theorem 4.3.9 (Monnier–Zung). The Lie algebra su(n, 1) is smoothly degeneratefor any n ∈ N.

Proof. Denote the linear Poisson structure on su∗(n, 1) by Π(1). Denote by K themaximal compact subgroup in SU(n, 1), and k ∼= su(n)⊕R its Lie algebra. Denoteby Y the non-Hamiltonian Poisson vector field on su∗(n, 1) given by Theorem4.3.1. By construction, Y is invariant under the coadjoint action of K. Denote byX = Xc the Hamiltonian vector field generated by a nontrivial element c in thecenter of k (i.e., c lies in R in the decomposition k ∼= su(n) ⊕ R). Note that X isa periodic vector field. We will choose c so that the period of the flow of X onsu∗(n, 1) is exactly 1, and we have an analytic Hamiltonian T1-action on su∗(n, 1)generated by X .

Choose a (singular) closed 1-form α such that α is K-invariant, Y -invariant, and

〈α, X〉 = 1.

Denote by Z = iαΠ(1) the singular locally Hamiltonian vector field generated by α.The 1-form α is singular at points x ∈ su∗(n, 1) such that X(x) = 0. A simple

direct verification shows that the set S = x ∈ su∗(n, 1), X(x) = 0 is a linearsubspace of su∗(n, 1) which does not intersect the “positive” part su∗(n, 1)+ ofsu∗(n, 1) (using the notations of the proof of Theorem 4.3.1 and identifying su(n, 1)with su∗(n, 1)), so we can choose α smooth on su∗(n, 1)+. Since, by construction,the vector field Y is trivial outside of su∗(n, 1)+, we can arrange it so that Z ∧ Yis smooth. Since α is Y -invariant, Z is also Y -invariant, i.e., [Z, Y ] = 0. Now put

Π = Π(1) + Z ∧ Y.

Then Π is a Poisson structure (of the same rank as Π(1)). Look at the Hamiltonianvector field of c with respect to Π: by construction, this vector field is equal toX + Y instead of X . So instead of being periodic, its flow is now spiraling. As aconsequence, there are symplectic leaves of Π in su∗(n, 1)+ which are not closed(in an arbitrary small neighborhood of the origin), so Π can’t be locally isomorphicto Π(1) for topological reasons.

Let us give here some arguments in support of the “if” part of Conjecture4.3.8. Let Π be a C∞-smooth Poisson structure which vanishes at a point, andwhose linear part corresponds to a semisimple Lie algebra g = k+ s of real rank 1,

4.4. Nondegeneracy of aff(n) 117

such that its compact part k is semisimple. Then one can apply the smooth Levidecomposition Theorem 3.2.9 to k. After that, it remains to linearize the “s-part”of Π. But this part has a “hyperbolic behavior” because g is of real rank 1, so onecan probably use some recent results and techniques on local linearization of hyper-bolic dynamical systems due to Chaperon [74, 75] and other people. For example,consider the simplest case g = so(3, 1) = so(3)+ s, dim g = 6, dim s = 3. The rankof the corresponding linear Poisson structure is 4. By geometric arguments, onecan probably show that the rank of a smooth nonlinear Poisson structure Π withthis linear part is also 4 (it is at least 4, and can’t be 6). The smooth Levi normalform gives a linear Hamiltonian Spin(3)-action. This action is locally free almosteverywhere, and its orbits lie on the symplectic leaves of Π. Taking the quotientof R6 by this action, one gets a three-dimensional cone, on which the quotientof the symplectic leaves are curves, i.e., one gets a one-dimensional foliation in athree-dimensional cone. Due to the hyperbolic behavior, one can probably linearizethis foliation. Lifting this linearization to R6, one gets a smooth linearization ofthe characteristic foliation of Π. The last step is to use Moser’s path method tolinearize Π.

4.4 Nondegeneracy of aff(n)

Theorem 4.4.1 ([120]). For any natural number n, the Lie algebra aff(n, K) =gl(n, K) Kn of affine transformations of Kn, where K = R or C, is formally andanalytically nondegenerate.

Proof. We will prove the above theorem in the analytic case. The formal caseis absolutely similar, if not simpler. Denote by l = g r a Levi decompositionfor a (real or complex) Lie algebra l, where s is semisimple and r is the solvableradical of l. Let Π be an analytic Poisson structure vanishing at a point 0 in amanifold whose linear part at 0 corresponds to l. According to the analytic Levidecomposition Theorem 3.2.6, there exists a local analytic system of coordinates(x1, . . . , xm, y1, . . . , yd) in a neighborhood of 0, where m = dim g and d = dim r,such that in these coordinates we have

xi, xj =∑

ckijxk , xi, yr =

∑as

irys , (4.25)

where ckij are structural constants of s and as

ir are constants. This gives what wecall a semilinearization for Π. Note that the remaining Poisson brackets yr, ysare nonlinear in general.

We now restrict our attention to the case where l = aff(n), m = n2 − 1,d = n + 1, g = sl(n), r = R(Id) Kn where Id acts on Kn by the identity map.The following lemma says that we may have a semilinearization associated tothe decomposition aff(n) = gl(n) Kn (which is slightly better than the Levidecomposition).


Lemma 4.4.2. There is a local analytic coordinate system

(x1, . . . , xn2−1, y0, y1, . . . , yn)

which satisfy Relations (4.25), with the following extra properties: y0, yr = yr

for r = 1, . . . , n; xi, y0 = 0 ∀i.Proof. We can assume that the coordinates yr are chosen so that Relations (4.25)are already satisfied, and y0 corresponds to Id in R(Id)Kn. Then the Hamiltonianvector fields Xxi are linear and form a linear action of sl(n). Because of (4.25),we have that xi, y0 = 0, which implies that [Xxi , Xy0 ] = 0, i.e., Xy0 is invariantunder the sl(n) action. Moreover we have Xy0(xi) = 0 (i.e., Xy0 does not containcomponents ∂/∂xi), and Xy0 =

∑n1 yi∂/∂yi+ nonlinear terms. Hence we can

use (the parametrized equivariant version of) Poincare linearization theorem tolinearize Xy0 in a sl(n)-invariant way. After this linearization, we have that Xy0 =∑n

1 yi∂/∂yi. In other words, Relations (4.25) are still satisfied, and moreover wehave y0, yi = Xy0(yi) = yi. Remark 4.4.3. Lemma 4.4.2 still holds if we replace aff(n) by any Lie algebra ofthe type (g⊕Ke0) n where g is semisimple and e0 acts on n by the identity map(or any matrix whose corresponding linear vector field is nonresonant and satisfiesa Diophantine condition).

We will redenote y0 in Lemma 4.4.2 by xn2 . Then Relations (4.25) are stillsatisfied. We will work in a coordinate system (x1, . . . , xn2 , y1, . . . , yn) providedby this lemma . We will fix the variables x1, . . . , xn2 , and consider them as linearfunctions on gl∗(n) (they give a Poisson projection from our (n2 + n)-dimensionalspace to gl∗(n)). Denote by F1, . . . , Fn the n basic Casimir functions for gl∗(n).(If we identify gl(n) with its dual via the Killing form, then F1, . . . , Fn are ba-sic symmetric functions of the eigenvalues of n × n matrices.) We will considerF1(x), . . . , Fn(x) as functions in our (n2 + n)-dimensional space, which do notdepend on variables yi. Denote by X1, . . . , Xn the Hamiltonian vector fields ofF1, . . . , Fn.

Lemma 4.4.4. The vector fields X1, . . . , Xn do not contain components ∂/∂xi.They form a system of n linear commuting vector fields on Kn (the space of y =(y1, . . . , yn)) with coefficients which are polynomial in x = (x1, . . . , xn2). The setof x such that they are linearly dependent everywhere in Kn is an analytic spaceof complex codimension strictly greater than 1 (when K = C).

Proof. The fact that the Xi are y-linear with x-polynomial coefficients followsdirectly from Relations (4.25). Since Fi are Casimir functions for gl(n), we haveXi(xk) = Fi, xk = 0, and [Xi, Xj] = XFi,Fj = 0.

One checks that, for a given x, X1 ∧ · · · ∧Xn = 0 identically on Kn if andonly if x is a singular point for the map (F1, . . . , Fn) from gl∗(n) to Kn. The setof singular points of the map (F1, . . . , Fn) in the complex case is of codimensiongreater than 1 (in fact, it is of codimension 3).


Lemma 4.4.5. Write the Poisson structure Π in the form Π = Π(1) + Π, whereΠ(1) is the linear part and Π denote the higher-order terms. Then Π is a Poissonstructure which can be written in the form

Π =∑i<j

fijXi ∧Xj , (4.26)

where the functions fij are analytic functions which depend only on the variablesx, and they are Casimir functions for gl∗(n) (if we consider the variables x aslinear functions on gl∗(n)).

Proof. We work first locally near a point (x, y) where the vector fields Xk arelinearly independent point-wise. As Π is a 2-vector field in Kn = y (with coeffi-cient depending on x) we have a local formula Π =

∑i<j fijXi∧Xj where fij are

analytic functions in variables (x, y). Since Xk are Hamiltonian vector fields for Πand also for Π(1), we have [Xk, Π] = [Xk, Π]− [Xk, Π(1)] = 0 for k = 1, . . . , n. Thisleads to Xk(fij) = 0 ∀ k, i, j. Hence, because the Xk generate Kn, the functionsfij are locally independent of y. Using analytic extension, Hartog’s theorem andthe fact that the set of x such that X1, . . . , Xn are linearly dependent point-wiseeverywhere in Kn is of complex codimension greater than 1, we obtain that fij arelocal analytic functions in a neighborhood of 0 which depend only on the variablesx. The fact that Π is a Poisson structure, i.e., [Π, Π] = 0, is now evident, becauseXk(fij) = 0 and [Xi, Xj] = 0.

Relations [Xxk, Π] = [Xxi , Π] − [Xxi , Π(1)] = 0 imply that Xxk

(fij) = 0,which means that fij are Casimir functions for gl∗(n). Remark 4.4.6. Lemma 4.4.5 is still valid in the formal case. In fact, every homo-geneous component of Π satisfies a relation of type (4.26).

Lemma 4.4.7. There exists a vector field Y of the form Y =∑n

i=1 αiXi, where theanalytic functions αi depend only on the variables x and are Casimir functionsfor gl∗(n), such that

[Y, Π(1)] = −Π , [Y, Π] = 0. (4.27)

Proof. Since the functions fij of Lemma 4.4.5 are analytic Casimir functions forgl(n), we have fij = φij(F1, . . . , Fn) where φij(z1, . . . , zn) are analytic functionsof n variables. On the other hand, since Π(1), Π and Π = Π(1) + Π are Poissonstructures, they are compatible, i.e., we have [Π(1), Π] = 0. Decomposing thisrelation, we get ∂φij

∂zk+ ∂φjk

∂zi+ ∂φki

∂zj= 0 ∀ i, j, k. This is equivalent to the fact that

the 2-form φ :=∑

ij φijdzi∧dzj is closed. By Poincare’s lemma we get φ = dα witha 1-form α =

∑i αidzi. Then we put Y :=

∑i αi(F1, . . . , Fn)Xi. An elementary

calculation proves that Y is the desired vector field. Return now to the proof of Theorem 4.4.1. Consider a path of Poisson struc-

tures given by Πt := Π(1) + tΠ. As we have [Y, Πt] = Π = ddtΠt, the time-1 map

of the vector field Y moves Π(1) = Π0 into Π = Π1. This shows that Π is locallyanalytically linearizable, thus proving our theorem.


Remark 4.4.8. For any n ∈ N, the algebra aff(n) is a Frobenius Lie algebra, in thesense that its coadjoint representation has an open orbit. In other words, its corre-sponding linear Poisson structure has rank equal to the dimension of the algebraalmost everywhere. One may think that there must be some links between thenondegeneracy and the property of being a Frobenius Lie algebra. Unfortunately,the search for new nondegenerate Lie algebras among Frobenius Lie algebras, car-ried out by Wade and Zung [343], has not brought up any new nondegenerateexample so far, though some of the degenerate Frobenius Lie algebras turn out tobe finitely determined (see Subsection 4.5.5).

In the above proof of Theorem 4.4.1, we implicitly showed that

H2CE(aff(n),Sk(aff(n))) = 0 ∀ k ≥ 2, (4.28)

where Sk denotes the symmetric product of order k and HCE denotes the Cheval-ley–Eilenberg cohomology (it is hidden in the last two lemmas). A purely algebraicproof of this fact was obtained independently by Bordemann, Makhlouf and Petitin [39], who showed that aff(n) is infinitesimally strongly rigid, i.e.,

H2CE(aff(n),Sk(aff(n))) = 0 ∀ k ≥ 0. (4.29)

They also verified that

H1CE(aff(n),Sk(aff(n))) = 0 ∀ k ≥ 1 (4.30)

andH1

CE(aff(n), K) = K. (4.31)

Since aff(n) is a Frobenius Lie algebra, we also have

H0CE(aff(n),Sk(aff(n))) = 0 ∀ k ≥ 1 (4.32)

(geometrically, it means that aff∗(n) can’t admit a homogeneous Casimir functionof degree k ≥ 1, because it has an open symplectic leaf), and

H0CE(aff(n), K) = K. (4.33)

Theorem 4.4.9. Any finite direct sum l =⊕

li, where each li is either simple orisomorphic to aff(ni) for some ni ∈ N, is formally nondegenerate.

Proof (sketch). Using the above formulas, Whitehead’s lemmas, and Hochschild–Serre spectral sequence, one can show that H2

CE(l, Skl) = 0 for any k ≥ 1. Remark 4.4.10. The Lie algebra aff(n)⊕ g (n ∈ N, g semisimple) is infinitesimallystrongly rigid, but the Lie algebra aff(n1) ⊕ aff(n2) (n1, n2 ∈ N) is not infinitesi-mally strongly rigid, because H2

CE(aff(n1)⊕ aff(n2), K) = K (see [39]).

Conjecture 4.4.11. Any finite direct sum l =⊕

li, where each li is either simpleor isomorphic to aff(ni) for some ni ∈ N, is analytically nondegenerate.


Theorem 4.4.1 shows that the above conjecture is true when l = aff(n). It isalso true when l = g⊕ aff(n), g being semisimple, with the same proof. Anothercase where we know that the conjecture is true is the following:

Theorem 4.4.12 (Dufour–Molinier [113]). The direct product

aff(1, K)× · · · × aff(1, K)

of n copies of aff(1, K) is formally and analytically nondegenerate for any naturalnumber n.

Proof (sketch). Denote l = aff(1) × · · · × aff(1) (n times). As discussed above,simple direct computations show that H2

CE(l,Skl) = 0 ∀ k ≥ 2, so l is formallynondegenerate.

Consider now an analytic Poisson structure Π whose linear part Π(1) corre-sponds to l. Consider the set

Σ = x ∈ (K2n, 0) | rankΠ(x) < 2n

of singular points of Π. This set is given by the analytic equation

det(Πij(x))2ni,j=1 = 0 ,

where Πij are the coefficients of Π in a coordinate system.When Π is linear, Σ is just a union of n hyperplanes in K2n in generic po-

sition. Since Π is formally linearizable, there are n formal hyperplanes in genericposition which are formal solutions of det(Πij(x))2n

i,j=1 = 0. Applying Artin’s the-orem [18] about approximation of formal solutions of analytic equations by ana-lytic solutions, we obtain that the equation det(Πij(x))2n

i,j=1 = 0 admits n localhypersurfaces in generic position near 0 as its solutions. Thus, locally, Σ is aunion of n analytic hypersurfaces. So there is a local analytic coordinate system(x1, y1, . . . , xn, yn) such that

Σ =n⋃

i=1

xi = 0 .

It is easy to see that in such a coordinate system we have

xi, xj = xixjσij(x, y), xi, yj = xiβij(x, y) .

A first change of coordinates of the type x′i = xiνi(x, y), y′

j = yj leads to xi, xj =0. Then another change of coordinates of the type x′

i = xi, y′j = yj + bj(x, y) gives

xi, yj = δijxi. Finally, we obtain yi, yj = 0 after a change of coordinates ofthe type x′

i = xi, y′j = yj + cj(x).

Remark 4.4.13. It is also shown in [113] that aff(1, R)×aff(1, R) is C∞-nondegen-erate. We don’t know if other Lie algebras of the type

⊕i aff(ni) are smoothly

nondegenerate or not.


4.5 Some other linearization results

In this section we briefly discuss, mostly without proofs, some other known resultsabout linearization of Poisson structures.

4.5.1 Equivariant linearization

Let Π be a Poisson structure on a manifold P which vanishes at a point p ∈ P , andG be a compact Lie group which acts on P in such a way that the action preservesthe Poisson structure Π and fixes the point p. One is interested in G-equivariantlinearization of Π, i.e., local coordinate transformations near p which linearize Πand the action of G at the same time. In this direction, we have:

Theorem 4.5.1 (Ginzburg [143]). Assume that the Poisson structure Π is smoothlylinearizable near p. Then it also admits a local smooth G-equivariant linearizationnear p.

The above theorem is a consequence of the rigidity principle for structure-preserving actions of compact Lie groups. An example of this rigidity principle isthe following:

Theorem 4.5.2 ([143]). Let (P, Π) be a compact Poisson manifold, G a compactLie group, and ρt : G × (P, Π) → (P, Π) a smooth family of Π-preserving actionsof G (t ∈ [0, 1]). Then there exists a smooth family of Poisson diffeomorphismsφt : P → P which sends ρt to ρ0, i.e., such that ρt(g, x) = φt(ρ0(g, φ−1

t (x)) for allx ∈ P , g ∈ G, and φ0 = Id.

The proof of Theorem 4.5.2 uses Moser’s path method and is based on thevanishing of the first differentiable cohomology group H1(G,Z1

Π(P )), where Z1Π(P )

is the Frechet space of Poisson vector fields on P . The G-module structure onZ1

Π(P ) is induced from the action of G on P and depends smoothly on the pa-rameter t. See the last section of [143] for details.

4.5.2 Linearization of Poisson–Lie tensors

We refer to Sections 5.1, 5.2 and 5.3 for some basic facts about Poisson–Lie groupsand Lie bialgebras used in this subsection.

A Poisson–Lie tensor is a Poisson tensor Π on a Lie group G such that themultiplication map G × G → G is a Poisson morphism. If Π is a Poisson–Lietensor on G and e denotes the neutral element of G then we automatically haveΠ(e) = 0, so one can talk about the linearization problem for Π et e. What makesthe problem of linearization of Poisson–Lie tensors different from general Poissonstructures is the fact that a Poisson–Lie tensor Π is completely determined by itslinear part p at the neutral element e. More precisely, we have:

4.5. Some other linearization results 123

Lemma 4.5.3. A Poisson–Lie tensor Π on a Lie group G is obtained from thecorresponding cocycle p : g → g ∧ g by the following formula:

Π(expx) = (Rexp x)∗exp(adx)− 1

adxp(x) ∀ x ∈ g , (4.34)

where Rexpx denotes the right translation by exp x.

In the above lemma, (G, Π) denotes a Poisson–Lie group, g is the Lie algebraof G, and p : g → g ∧ g is the associated 1-cocycle. The cocycle p may be viewedas a map which assigns to each element x ∈ g the 2-vector p(x) ∈ Λ2Txg ∼= g ∧ g,and as such it is the linear part of Π at the neutral element e of G. The dualmap p∗ : g∗ ∧ g∗ → g∗ is a Lie algebra structure on g∗, turning (g, g∗) into a Liebialgebra.

In fact, Formula (4.34) is true for any multiplicative tensor Π, i.e., it is aconsequence of the multiplicativity condition

Π(gh) = (Lg)∗Π(h) + (Rh)∗Π(g) . (4.35)

It is immediate from Lemma 4.5.3 that if p = 0 then Π also vanishes, and ifg is Abelian then Π is linear.

For Poisson–Lie tensors with a reductive linear part, we have:

Theorem 4.5.4 (Chloup [79]). If (G, Π) is a Poisson–Lie group such that the cor-responding Lie algebra structure on g∗ is reductive, then Π is locally analyticallylinearizable near the neutral element e of G.

Chloup’s proof of Theorem 4.5.4 follows from Lemma 4.5.3, Conn’s Theorem4.1.3, and the following relatively simple proposition.

Proposition 4.5.5 ([79]). Consider a Lie bialgebra (g, g∗) such that g = g1 ⊕ g2

is the direct sum of a semisimple Lie algebra g1 with a Lie algebra g2. Writeg∗ = g∗1 ⊕ g∗2 the dual decomposition of g∗ as a vector space, i.e., 〈g1, g

∗2〉 = 0 and

〈g2, g∗1〉 = 0. Then g∗1 and g∗2 are Lie subalgebras of g∗.

It is known that in the case when g∗ is semisimple compact, the linearizationof Π can be made global:

Theorem 4.5.6 (Ginzburg–Weinstein [148]). If (G, Π) is a simply connectedPoisson–Lie group with a compact semisimple dual Lie algebra g∗, then as a Pois-son manifold (G, Π) is globally smoothly Poisson-isomorphic to g (with the linearPoisson structure corresponding to the semisimple Lie algebra structure of g∗).

Of course, if we replace the word “globally” by the word “locally” in theabove theorem, then it becomes a special case of Conn’s Theorem 4.1.3. Ginzburgand Weinstein’s proof of Theorem 4.5.6 is based on geometric arguments: theyshow that (G, Π) and g have diffeomorphic symplectic foliations, and then useMoser’s path method to show the existence of a Poisson isomorphism. Another


proof of Theorem 4.5.6 was found by Alekseev in [5], where he showed a natu-ral correspondence between Hamiltonian K-actions and Poisson K-actions whichadmit a momentum map in the sense of Lu, where K is a simply-connected com-pact semisimple Poisson–Lie group. (A Poisson action of a Poisson–Lie groupK on a Poisson manifold M is an action of K on M such that the action mapK ×M → M is a Poisson morphism. A momentum map in the sense of Lu fora Poisson K-action on M is a map from M to the dual group K∗ which satisfiessome natural properties, see Section 5.4.) An explicit Poisson isomorphism fromg∗ to G∗ in Ginzburg–Weinstein’s theorem is given by Boalch [35] who interpretedg as a moduli space of meromorphic connections with an irregular singularity andG∗ as the space of monodromy data of these connections.

In particular, let K be a simply-connected compact semisimple Lie group.Denote by G the complexification of K. Then K admits a unique natural Poisson–Lie structure, called the Iwasawa Poisson–Lie structure, induced from the Iwasawadecomposition G = KAN of G (see [224] and Section 5.3). The correspondingPoisson–Lie structure on K∗ = AN is globally linearizable according to Theorem4.5.6. On the other hand, the Lie–Poisson structure on K is not locally linearizablein general:

Theorem 4.5.7 (Cahen–Gutt–Rawnsley [55]). If K is a simply-connected compactsimple Lie group different from SU(2), then the Iwasawa Poisson–Lie structureon K is not locally linearizable at the neutral element.

Cahen, Gutt, and Rawnsley proved the above theorem by showing that thesymplectic foliation on K is not locally homeomorphic to the symplectic foliationof the corresponding linear Poisson structure.

Some other results about linearization of Poisson–Lie tensors, including ex-amples of linearizable and nonlinearizable Poisson–Lie tensors on nilpotent Liegroups, can be found in [79].

4.5.3 Poisson structures with a hyperbolic Rk-action

The following result on the smooth linearizability of a Poisson structure with alinear part of the type Rp Rn and a hyperbolic Hamiltonian Rp-action may beviewed as a generalization of Theorem 4.2.2.

Theorem 4.5.8 ([111]). Let Π be a smooth Poisson structure of maximal rank 2pin a neighborhood of 0 in Rn+p, n ≥ p ≥ 1, Π(0) = 0. Suppose that there are pfunctions f1, . . . , fp, fi(0) = 0, such that fi, fj = 0 ∀ i, j = 1, . . . , p, df1 ∧ · · · ∧dfp(0) = 0. Suppose moreover that the infinitesimal Rp-action ρ generated by theHamiltonian vector fields Xi = Xfi is hyperbolic on the submanifold K = f1 =· · · = fp = 0 in the following sense: the weights λ1, . . . , λn of the linear part ρ(1)

of ρ (i.e., linear maps λi : Rp −→ C which assign to each a ∈ Rp an eigenvalue ofthe linear automorphism ρ(1)(a) of Rn), satisfy the following two properties:


a) The real parts of λi are p by p independent (we don’t distinguish the real partsof λi and λi).

b) There is no resonance relation of the type λi =∑n

j=1 kjλj , where kj arenon-negative integers with k1 + · · ·+ kn ≥ 2.

Then Π is smoothly linearizable in a neighborhood of 0 in Rn+p. Moreover, thereare local smooth coordinates x1, . . . , xn, u1, . . . , up such that

xi, xj = ur, us = 0, xi, ur =n∑

l=1

clirxl ∀ i, j, r, s ,

and the functions fi depend only on the variables u1, . . . , up.

We refer the curious reader to [111] for a proof of the above theorem. Akey ingredient in the proof is Chaperon’s smooth linearization theorem [73] forhyperbolic actions of commutative groups. A generalization of Theorem 4.5.8 inthe analytic case can be found in [324].

4.5.4 Transverse Poisson structures to coadjoint orbits

In [346], Weinstein erroneously claimed that the transverse Poisson structure toany coadjoint orbit in the dual of a finite-dimensional Lie algebra is lineariz-able. Molino [259] found out that Weinstein’s proof needs an additional condition,namely the orbit must be reductive. More precisely, we have:

Theorem 4.5.9 ([259]). Let g be a Lie algebra, and µ ∈ g∗ be a point in the dual ofg equipped with the corresponding linear Poisson structure. Denote by gµ = x ∈g | ad∗

xz = 0 ⊂ the isotropy algebra of µ. If there is a linear complement m of gµ

in g, i.e., g = gµ + m as a vector space, such that [gµ, m] ⊂ m, then the transversePoisson structure at µ is linearizable.

The above theorem follows directly from Dirac’s formula (1.56): just takethe affine space N = ν ∈ g∗ |〈x, ν〉 = constant ∀x ∈ m. N is transverse tothe coadjoint orbit at µ, and the transverse Poisson structure on N is linear withrespect to the affine functions on N which vanish at µ.

If one drops the condition [gµ, m] ⊂ m from Theorem 4.5.9 then it is false.The following simple example of a nonlinearizable transverse Poisson structure toa coadjoint orbit is provided by Givental (see the discussion in [347]): Identifyg = sl(3, R) with its dual via the Killing form, and consider the nilpotent element

µ0 =

⎛⎝0 0 10 0 00 0 0

⎞⎠ .

The adjoint orbit of µ0 is four-dimensional. The transverse Poisson structure atµ0 is a Poisson structure of rank 2 living in a four-dimensional space, with two


Casimir functions inherited from sl(3, R). In an appropriate linear coordinate sys-tem (p, q, r, s) centered at µ0, these Casimir functions have the form 3p2 + r and−2p3 + qs + 2pr; their common zero level set is a surface with an isolated singu-larity (of type A2 in Arnold’s classification). On the other hand, the linear part ofthis transverse structure corresponds to the centralizer gµ0 = x ∈ g | [x, µ0] = 0of µ0, which consists of the matrices⎛⎝α β γ

0 −2α δ0 0 α

⎞⎠ .

Thinking of (α, β, γ, δ) as a basis of g∗µ0and denoting by (a, b, c, d) the dual basis

of gµ0 considered as linear functions on g∗µ0, the common zero level set of the two

Casimir functions c and ac + 3bd on g∗µ0is a union of two 2-planes which intersect

along a line. Just, for topological reasons, the transverse Poisson structure at µ0

can’t be isomorphic to its linear part.An even simpler example is given by M. Duflo (see [347]): Consider the fol-

lowing four-dimensional linear Poisson structure:

∂

∂x1∧ (x2

∂

∂x2+ x3

∂

∂x3+ 2x4

∂

∂x4) + x4

∂

∂x2∧ ∂

∂x3.

The corresponding Lie algebra is a Frobenius Lie algebra. The singular locus isx4 = 0. The points with x4 = 0 and x2

2 + x23 > 0 have Abelian isotropy. But of

course the transverse Poisson structure is not trivial.Remark 4.5.10. Damianou [92] proved in some special cases, and then Cushmanand Roberts [89] proved in the general case, that the transverse Poisson structureto an arbitrary coadjoint orbit in the dual of a semisimple Lie algebra is polynomial,i.e., the Poisson tensor has polynomial coefficients in a linear system of coordinates.Probably, most of these polynomial Poisson structures (except the ones that fitinto Theorem 4.5.9) are nonlinearizable.Remark 4.5.11. Some results about transverse Poisson structures to coadjoint or-bits of infinite-dimensional Lie algebras, e.g., the Virasoro algebra and the Adler–Gelfand–Dickey bracket, can be found in [289, 235, 236]

4.5.5 Finite determinacy of Poisson structures

Given a formal (resp. analytic, resp. smooth) Poisson structure Π = Π(1) + Π(2) +· · · , we will say that it is formally (resp. analytically, resp. smoothly) finitelydetermined if there is a natural number k such that any other formal (resp. an-alytically, resp. smoothly) Poisson structure Π1 = Π(1)

1 + Π(2)1 + · · · such that

Π(l)1 = Π(l) ∀ l ≤ k is formally (resp. analytically, resp. smoothly) locally isomor-

phic to Π.In particular, if a Lie algebra g is formally degenerate, one may still ask if it is

finitely determined, in the sense that the space of formal Poisson structures which


have g as their linear part, modulo formal isomorphisms, is of finite dimension. Ifg is finitely determined, then there is a natural number k (depending on g) suchthat any formal Poisson structure

Π = Π(1) + Π(l) + Π(l+1) + · · ·

with l > k, where Π(1) corresponds to g, is formally linearizable. More gener-ally, any two formal Poisson structures with the same linear part Π(1) and whichcoincide up to degree k are formally isomorphic.

It is clear that a sufficient condition for a Lie algebra g to be finitely deter-mined is the inequality

dimH2CE(g,Sg) < ∞ . (4.36)

Example 4.5.12. The linear three-dimensional Poisson structure

Λ =∂

∂x∧ (y

∂

∂y+ 2z

∂

∂z)

does not satisfy the conditions of Theorem 4.2.2 because of a resonance. In fact,it is a degenerate, but finitely determined Poisson structure: any formal (resp.analytic, resp. smooth) Poisson structure whose linear part is Λ is formally (resp.analytically, resp. smoothly) isomorphic to a Poisson structure of the type

Π =∂

∂x∧ (y

∂

∂y+ 2z

∂

∂z+ cy2 ∂

∂z),

where c is a constant.

Example 4.5.13. Consider a Poisson structure Π = Π(1) + Π(2) + · · · , whose linearpart is of the type

Π(1) =∂

∂x0∧( n∑

i,j=1

aijxi∂

∂xj

), (4.37)

where the matrix A = (aij) is nonresonant in the sense that its eigenvaluesγ1, . . . , γn do not satisfy any nontrivial relation of the type

λi + λj =n∑

k=1

αkλk (4.38)

with 1 ≤ i ≤ j ≤ n, αk ∈ N ∪ 0 (i.e., except the relations λi + λj = λi + λj).Such a Poisson structure Π is called nonresonant in [116]. Simple homologicalcomputations show that Π admits a formal nonhomogeneous quadratic normalform. See [116] for more details, and also a smooth nonhomogeneous quadratizationresult.


Example 4.5.14 ([343]). Consider the following six-dimensional linear Poissonstructure:

Π1 = ∂x1 ∧ Y1 + ∂x2 ∧ Y2 + Λ (4.39)

where

Y1 = y1∂y1 + 2y2∂y2 + 3y3∂y3 + 4y4∂y4,

Y2 = y2∂y2 + y3∂y3 + y4∂y4,

Λ = ∂y1 ∧ (y3∂y2 + y4∂y3).

The corresponding solvable Lie algebra p = K2 L4, where L4 is the nilpotentLie algebra corresponding to Λ, is a six-dimensional Frobenius rigid solvable Liealgebra (see [151]) which is not strongly rigid. Lengthy direct computations, usingthe Hochschild–Serre formula and with the aid of MAPLE, show that H2(p,Skp) =K for k = 3, 4, 5 and H2(p,Skp) = 0 for all other values of k ≥ 1. Thus p is finitelydetermined. It is degenerate: the nonlinear Poisson tensor

Π = Π1 + y21y2

∂

∂y1∧ ∂

∂y3− y1y

22

∂

∂y2∧ ∂

∂y3

is not equivalent to its linear part, because the singular loci of Π and Π1 are notlocally isomorphic.

Chapter 5

Multiplicative and QuadraticPoisson Structures

After studying linearization and normal forms for Poisson structures with a non-trivial linear part in the previous two chapters, it is a logical next step to talkabout Poisson structures which begin with a quadratic term. This leads us tothe study of (homogeneous) quadratic Poisson structures, and then to anotherimportant subject of Poisson geometry, namely Poisson–Lie groups. The reasonis that both Poisson–Lie groups and quadratic Poisson structures are in a sensemultiplicative, and they both arise, most of the times, from classical r-matrices.For example, if we have a multiplicative Poisson structure on a finite-dimensionalunital associative algebra (say the algebra Matn(R) of n× n-matrices), then it isa quadratic Poisson structure, and at the same time a Poisson–Lie structure onthe group of reversible elements of the algebra. In this chapter, we will give a briefintroduction to the theory of Poisson–Lie groups, and then discuss some resultson quadratic Poisson structures and the quadratization problem.

5.1 Multiplicative tensors

Let G be a Lie group. A multi-vector field Λ ∈ V(G), or more generally, a tensorfield on G, is called a multiplicative tensor field , if Λ satisfies the following equation:

Λ(gh) = (Lg)∗Λ(h) + (Rh)∗Λ(g) ∀ g, h ∈ G, (5.1)

where Lg denotes the left translation h → gh and Rh denotes the right translationg → gh.

Exercise 5.1.1. Show that, if Λ is a multiplicative 2-vector field on a Lie group G,then the inversion map g → g−1 of G sends Π to −Π.

130 Chapter 5. Multiplicative and Quadratic Poisson Structures

If Λ is multiplicative, then it is clear that Λ(e) = 0, where e denotes theneutral element of G (put g = h = e in Equation (5.1)). In particular, we can talkabout the linear part Λ(1) of Λ at e.

Theorem 5.1.2 ([107, 346, 224]).

a) A multi-vector field Λ on a connected Lie group G is multiplicative if andonly if Λ(e) = 0 and LXΛ is left-invariant for any left-invariant vector fieldX on G.

b) The Schouten bracket of two multiplicative multi-vector fields is a multiplica-tive multi-vector field.

Proof. a) Let X be a left-invariant vector field. Then its flow is the one-dimensionalgroup of right translations Rexp(tX). If Λ is multiplicative then we have

LXΛ(g) =ddt

∣∣∣∣t=0

Rexp(−tX)Λ(g. exp(tX))

=ddt

∣∣∣∣t=0

Rexp(−tX)[Rexp(tX)Λ(g) + LgΛ(exp(tX))]

=ddt

∣∣∣∣t=0

Rexp(−tX)LgΛ(exp(tX))

=ddt

∣∣∣∣t=0

LgRexp(−tX)Λ(exp(tX))

= Lg

(ddt

∣∣∣∣t=0

Rexp(−tX)Λ(exp(tX)))

= Lg(LXΛ(e)).

Thus LXΛ is left-invariant if Λ is multiplicative and X is left-invariant. Conversely,if L(e) = 0 and LXΛ is left-invariant for any left-invariant X , then

ddt

(Rexp(−tX)Λ(g. exp(sX + tX)) + Lg. exp(sX+tX)Λ(exp(−tX))

)= 0,

implying (by evaluating the above expression at t = 0 and t = −s) that

Λ(g. exp(sX)) = Rexp(sX)Λ(g) + LgΛ(exp(sX)).

Since G is connected, the last equation implies that Λ is multiplicative. Recall thatleft translations are generated by right-invariant vector fields. Hence Assertion a)of the lemma may be restated as follows: Λ is multiplicative if and only if

LY LXΛ = LXLY Λ = 0 (5.2)

for any left-invariant vector field X and right-invariant vector field Y .

b) Let Λ1 and Λ2 be two multiplicative multi-vector fields. We have to show that

LXLY [Λ1, Λ2] = 0 (5.3)

for any left-invariant vector field X and right-invariant vector field Y .

5.1. Multiplicative tensors 131

Note that LXΛ1 is left-invariant and LY Λ2 is right-invariant. Since righttranslations commute with left translations, left-invariant vector fields (which aregenerators of right translations) commute with right-invariant vector fields. Bythe Leibniz rule, it follows that the Schouten bracket of a left-invariant multi-vector field with a right-invariant multi-vector field vanishes. In particular, wehave [LXΛ1,LY Λ2] = 0. Similarly, [LY Λ1,LXΛ2] = 0. Hence

LXLY [Λ1, Λ2]= [LXLY Λ1, Λ2] + [LXΛ1,LY Λ2] + [LY Λ1,LXΛ2] + [Λ1,LXLY Λ2]= 0.

By left translations, we can identify ∧TG with G × ∧g, and if Λ is a k-vector field on G, we may view Λ as a map from G to ∧kg via this identification.Then Λ(1) may be identified with the differential of Λ at e:

Λ(1) = deΛ : g → ∧kg. (5.4)

Theorem 5.1.3 ([224]). If Λ is a multiplicative k-vector field on G, then deΛ is a1-cocycle of the Chevalley–Eilenberg complex of the adjoint action of g on ∧kg.Conversely, if G is a connected simply-connected Lie group, then for any 1-cocyclep : g → ∧kg, there is a unique multiplicative k-vector field Λ on G such thatdeΛ = p.

Proof. Assume that Λ is a multiplicative k-vector field on G. Let x, y ∈ g be twoelements of g, considered as left-invariant vector fields on G. Denote p = deΛ. Wehave to show that p([x, y]) = adxp(y)− adyp(x).

By virtue of Theorem 5.1.2, we have:

adxp(y) = adxLyΛ(e) =ddt

∣∣∣∣t=0

Adexp(tx)(LyΛ(e))

=ddt

∣∣∣∣t=0

Rexp(−tx)Lexp(tx)(LyΛ(e))

=ddt

∣∣∣∣t=0

Rexp(−tx)(LyΛ(exp(tx))) = (LxLyΛ)(e),

hence adxp(y) − adyp(x) = (LxLyΛ)(e) − (LyLxΛ)(e) = (L[x,y]Λ)(e) = p([x, y]),which means that p is a 1-cocycle.

Conversely, assume that p : g → ∧kg is a 1-cocycle, and that G is connectedsimply-connected. The following construction of Λ from p is taken from [198]:

Define the following 1-form p on G with values in ∧kg:

p(X) = Adg(p(Lg−1X)) ∀g ∈ G, X ∈ TgG. (5.5)

It is clear that p is an equivariant 1-form:

(Lg)∗p = Adg p. (5.6)


One verifies directly that, because p is a cocycle, p is a closed 1-form: dp = 0.Since G is simply-connected, p is exact, i.e., there is a unique ∧kg-valued functionP : G→ ∧kg such that dP = p and P (e) = 0. Define the k-vector field Λ as

Λ(g) = (Rg)∗P (g). (5.7)

It is clear that Λ is multiplicative if P satisfies the equation

P (gh)− P (g) = AdgP (h) ∀ g, h ∈ G. (5.8)

But this last equation follows from Equation (5.6), which is its infinitesimal version(for h near e), and the connectedness of G.

5.2 Poisson–Lie groups and r-matrices

Definition 5.2.1. A Lie group G equipped with a Poisson structure Π is called aPoisson–Lie group if the multiplication map (G, Π) × (G, Π) → (G, Π), (g, h) →gh, is a Poisson morphism.

Equivalently, Π is a Poisson–Lie structure on a Lie group G if it is a multi-plicative Poisson tensor on G, i.e., Π satisfies the equation

Π(gh) = (Lg)∗Π(h) + (Rh)∗Π(g) ∀ g, h ∈ G. (5.9)

Poisson–Lie groups form a category: a morphism between two Poisson–Liegroups is a group homomorphism which is a Poisson morphism at the same time.

A Poisson–Lie group (G, Π) is called exact or coboundary if

Π = r+ − r− (5.10)

for some r ∈ g ∧ g, where g is the Lie algebra of G. Here, as usual, r+ (resp. r−)denotes the left-invariant (resp. right-invariant) tensor field whose value at theneutral element e of G is r ∈ g ∧ g ∼= ∧2TeG. (See Notation 1.8.10.)

Lemma 5.2.2. Any 2-vector field of the type Π = r+ − r− with r ∈ g ∧ g is multi-plicative, i.e., satisfies Equation (5.9). Conversely, if G is a connected semisimpleLie group, then any multiplicative 2-vector field Π on G is of the type Π = r+−r−

for some r ∈ g ∧ g.

Proof. Notice that Rgh = RhRg and Lgh = LgLh. Moreover, left translationscommute with right translations, i.e., RgLh = LhRg. Hence we have: Π(gh) =Lghr −Rghr = (RhLgr −Rghr) + (Lghr − LgRhr) = Rh(Lgr −Rgr) + Lg(Lhr −Rhr) = RhΠ(g) + LgΠ(h). Notice that the linear part of r+ − r− at e is given bythe 1-coboundary δr : x → [x, r] (x ∈ g) of the Chevalley–Eilenberg complex ofthe g-module g ∧ g.

Conversely, if g is semisimple, Π is multiplicative and p = deΠ : g → g ∧ gits corresponding 1-cocycle, then by Whitehead’s lemma, H1(g, g ∧ g) = 0, andthis 1-cocycle is a coboundary. In other words, ∃ r ∈ g ∧ g such that p = δr. Byuniqueness (see Theorem 5.1.3), Π must be equal to r+ − r−.

5.2. Poisson–Lie groups and r-matrices 133

Denote by (∧3g)g the set of elements of ∧3g which are invariant under theadjoint action of g. In other words,

(∧3g)g = α ∈ ∧3g | [x, α] = 0 ∀ x ∈ g,

where [x, α] is the Schouten bracket of x with α.

Theorem 5.2.3 ([107]). A multiplicative 2-vector field Π = r+ − r− with r ∈ g ∧ gon a connected Lie group G with Lie algebra g is a Poisson–Lie structure if andonly if r satisfies the following condition:

[r, r] ∈ (∧3g)g. (5.11)

Proof. Recall that left-invariant multi-vector fields commute with right-invariantmulti-vector fields. In particular, we have [r+, r−] = 0.

By Lemma 1.8.9, we have [r+, r+] = [r, r]+. Similarly, [r−, r−] = −[r, r]−.(The sign minus is due to the fact that x → x− is a Lie algebra anti-homomorphismfrom g to V1(G).) Hence we have

[Π, Π] = [r+ − r−, r+ − r−] = [r+, r+] + [r−, r−]− 2[r+, r−] = [r, r]+ − [r, r]−.

Thus, [Π, Π] = 0 if and only if Lg[r, r] = Rg[r, r] ∀g ∈ G. The last equation meansthat [r, r] is invariant under the adjoint action of g.

In particular, if r satisfies the CYBE (classical Yang–Baxter equation)

[r, r] = 0, (5.12)

then Π = r+ − r− is an exact Poisson–Lie structure. For example, if x, y ∈ g aretwo commuting elements, [x, y] = 0, then r = x ∧ y ∈ g ∧ g satisfies the CYBE(5.12) and gives rise to a coboundary Poisson–Lie structure on G.

If an element r ∈ g ∧ g satisfies Condition (5.11), then it is called a classicalr-matrix of g. Condition (5.11) is called the generalized classical Yang–Baxterequation (gCYBE ). If r satisfies the CYBE (5.12), then it is called a triangularr-matrix . The Poisson bracket of an exact Poisson–Lie structure Π = r+ − r−,where r is a classical r-matrix, is often called the Sklyanin bracket .

Remark that the CYBE (5.12) is preserved under homomorphisms: if r is aclassical r-matrix of g, and ρ : g → l is a Lie algebra homomorphism, then ρ(r) isa classical r-matrix of l.

The CYBE (5.12) is also often written as

[r12, r13] + [r12, r23] + [r13, r23] = 0. (5.13)

In the above equation, g may be viewed as a subalgebra of End(V ) where Vis a vector space, and r is viewed as an anti-symmetric element of End(V ) ⊗End(V ), which acts on V ⊗ V . r12 is the endomorphism of V1 ⊗ V2 ⊗ V3, whereV1 = V2 = V3 = V , which acts on V1 ⊗ V2 by r and on V3 by the identity map,i.e., r12(v1 ⊗ v2 ⊗ v3) = r(v1 ⊗ v2) ⊗ v3. r13 and r23 are defined similarly. More


intrinsically, one can replace End(V ) by the universal enveloping algebra Ug of g.Equation (5.13) makes sense also for non-antisymmetric elements r ∈ g ⊗ g, anda non-antisymmetric solution of (5.13) is called a quasi-triangular r-matrix .

Exercise 5.2.4. Show that, for an element r ∈ g ∧ g, Equation (5.13) is equivalentto Equation (5.12).

Suppose now that g is a simple Lie algebra, and identify it with its dual g∗ viathe Killing form 〈x, y〉 = trace(adxady). On ∧3g, we have the induced ad-invariantscalar product: if (e1, . . . , em) is an orthonormal basis of g then (ei ∧ ej ∧ ek, i <j < k) is an orthonormal basis of ∧3g. The equation

〈η, x ∧ y ∧ z〉 = 〈[x, y], z〉 ∀ x, y, z ∈ g (5.14)

defines an element η ∈ ∧3g, which is obviously ad-invariant, i.e., η ∈ (∧3g)g. Infact, (∧3g)g is one-dimensional and is generated by η. Thus, an element r ∈ g∧ g,where g is simple, is a classical r-matrix if and only if it satisfies the followingequation:

〈[r, r], x ∧ y ∧ z〉 = λ〈[x, y], z〉 ∀ x, y, z ∈ g, (5.15)

where λ is a constant. Equation (5.15) is called the modified classical Yang–Baxterequation (mCYBE) of coefficient λ (when λ = 0; when λ = 0 it is the CYBE).If g is simple and r ∈ g ∧ g is a solution of the mCYBE, then there is a uniquenumber c such that r + cΓ is a quasi-triangular r-matrix, where Γ ∈ Ug denotesthe Casimir element of g, and vice versa.

A complete list of solutions of the mCYBE (5.15) for a complex simple Liealgebra was obtained by Belavin and Drinfeld [25].

Theorem 5.2.5 (Belavin–Drinfeld). Let g be a complex simple Lie algebra, andr ∈ g∧g be a solution of the mCYBE (5.15), with λ = 0. Then r can be written as

r = r0 +√−λ

( ∑α∈∆+

E−α ∧ Eα + 2∑

α∈Γ+, β>α

E−β ∧ Eα

)(5.16)

where:(1) ∆+ is a system of positive roots of g with respect to a Cartan subalgebra

h of g.(2) (Hα, Eα) is a Weyl basis of g with 〈Eα, E−α〉 = 1.(3) Γ+ is a subset of the set Φ of simple roots corresponding to ∆+, Γ+ is the set

of positive roots which can be written as a linear combination with integralcoefficients of simple roots in Γ+.

(4) There is another subset Γ− of the set Φ of simple roots, and a map τ : Γ+ →Γ− such that 〈τ(α), τ(β)〉 = 〈α, β〉 ∀ α, β ∈ Γ+ where 〈α, β〉 = 〈Hα, Hβ〉.For any α ∈ Γ+ there is a positive integer k such that r(τ(α), β) = r(α, β)−k(〈α, β〉 + 〈τ(α), β〉) ∀ α ∈ Γ+, β ∈ Φ. The notation β > α means thatβ = τ l(α) for some l ≥ 1.

(5) r0 ∈ ∧2h is determined by Q(α, β), α, β ∈ Φ, such that

Q(τ(α), β) −Q(α, β) = −√−λ(〈α, β〉 + 〈τ(α), β〉). (5.17)

5.2. Poisson–Lie groups and r-matrices 135

The proof of the above theorem is not very difficult, but lengthy. See, e.g.,[56, 78], for a proof, where the case of real simple Lie algebras was also studied. Example 5.2.6. r =

∑α∈∆+ E−α ∧ Eα is a classical r-matrix, called the standard

r-matrix.

There is another way to express the Yang–Baxter equations. To show it, weneed the following lemma:

Lemma 5.2.7. Let g be an arbitrary Lie algebra, r ∈ g ∧ g and α, β, γ ∈ g∗. Thenwe have:

12〈[r, r], α ∧ β ∧ γ〉 = 〈[rα, rβ], γ〉 + 〈[rβ, rγ], α〉 + 〈[rγ, rα], β〉, (5.18)

where rα ∈ g denotes the interior product of r with α: 〈rα, ζ〉 = 〈r, α ∧ ζ〉.Proof. Direct verification using a basis of g.

Suppose that g admits an ad-invariant nondegenerate scalar product. Thenan element r ∈ g ∧ g can be identified with an element R ∈ End(g) such that〈Rx, y〉 = 〈r, x ∧ y〉 ∀ x, y ∈ g. Note that R is antisymmetric: 〈Rx, y〉 = −〈Ry, x〉.Formula (5.18) can then be rewritten as follows:

12〈[r, r], x ∧ y ∧ z〉 = 〈[Rx, Ry], z〉+ 〈[Ry, Rz], x〉+ 〈[Rz, Rx], y〉

= 〈[Rx, Ry]−R([Rx, y] + [x, Ry]), z〉. (5.19)

Hence, in terms of R ∈ End(g), the CYBE (5.12) can be rewritten as

[Rx, Ry]−R([Rx, y] + [x, Ry]) = 0, (5.20)

while the mCYBE (5.15) can be rewritten as

[Rx, Ry]−R([Rx, y] + [x, Ry]) =λ

2[x, y]. (5.21)

Equation (5.21) was first studied by Semenov–Tian–Shansky [313] in relationwith integrable systems, and its solutions (not necessarily anti-symmetric) are alsocalled classical r-matrices. A remarkable property of these matrices is the following:

Theorem 5.2.8 ([313]). Suppose that a matrix R ∈ End(g) satisfies the mCYBE

[Rx, Ry]−R([Rx, y] + [x, Ry]) = −[x, y]. (5.22)

Put[x, y]R =

12([Rx, y] + [x, Ry]). (5.23)

Then [x, y]R is a Lie bracket. Denote by gR the corresponding Lie algebra, andR± = 1

2 (R ± Id). Then R± : gR → g are Lie algebra homomorphisms.

Proof. Direct verification.


Example 5.2.9. If g = g1+g2 as a vector space, where g1 and g2 are Lie subalgebrasof g, and p1 : g → g1 and p2 : g → g2 are the corresponding projections, thenR = p1 − p2 satisfies the mCYBE (5.22). The corresponding Lie bracket is

[x1 + x2, y1 + y2]R = [x1, x2]− [y1, y2], (5.24)

where x1, y1 ∈ g1, x2, y2 ∈ g2. There is a family of (integrable) Hamiltonian sys-tems on g∗R (with respect to the new bracket) generated by Casimir functions of g∗

(with respect to the original bracket), called Adler–Kostant–Symes systems, see,e.g., [2].

5.3 The dual and the double of a Poisson-Lie group

Theorem 5.3.1 ([107]). A multiplicative 2-vector field Π on a connected Lie groupG is a Poisson–Lie structure if and only if its corresponding 1-cocycle p = deΠ :g→ ∧2g defines on the dual g∗ of g a Lie algebra structure [, ]∗ by the formula

〈[α, β]∗, x〉 = 〈p(x), α ∧ β〉 ∀ α, β ∈ g∗, x ∈ g. (5.25)

Proof. The necessity of the above condition is clear: If Π is Poisson, then its linearpart Π(1) at e is a linear Poisson structure on g = TeG, hence it corresponds to aLie algebra structure on g∗, which is given exactly by Formula (5.25). Conversely,by Theorem 5.1.2 and Theorem 5.1.3, [Π, Π] is a multiplicative 3-vector field andis determined uniquely by its linear part at e. The linear part of [Π, Π] at e is[Π(1), Π(1)]. If Formula (5.25) defines a Lie algebra structure on g∗, then Π(1) isthe linear Poisson structure corresponding to this Lie algebra structure, hence[Π(1), Π(1)] = 0, and therefore [Π, Π] = 0, i.e., Π is a Poisson structure, which ismultiplicative by assumptions.

Definition 5.3.2. A Poisson–Lie algebra is a Lie algebra g together with a 1-cocyclep : g → g ∧ g which defines on g∗ a Lie algebra structure by Formula (5.25).

It is clear from Theorem 5.1.3 and Theorem 5.3.1 that a Poisson–Lie algebrais the infinitesimal version of a Poisson–Lie group, and there is a natural one-to-one correspondence between Poisson–Lie algebras and connected simply-connectedPoisson–Lie groups.

Lemma 5.3.3. If (g, [, ]) and (g∗, [, ]∗) are Lie algebras, and p : g → g∧ g is definedby Formula (5.25), then p is a 1-cocycle if and only if the following equation issatisfied for any x, y ∈ g, α, β ∈ g∗:

〈[α, β]∗, [x, y]〉 = −〈ad∗xα, ad∗

βy〉+ 〈ad∗xβ, ad∗

αy〉+ 〈ad∗yα, ad∗

βx〉 − 〈ad∗yβ, ad∗

αx〉.(5.26)

The proof of the above lemma is a straightforward computation based onFormula (5.25).

5.3. The dual and the double of a Poisson-Lie group 137

Definition 5.3.4. A pair (g, g∗) of Lie algebra, where g∗ is the dual vector space ofg, which satisfies Equation (5.26), is called a Lie bialgebra.

Notice that Equation (5.26) is symmetric with respect to g and g∗. Hence, byLemma 5.3.3, if (g, p) is a Poisson–Lie algebra, then (g, g∗) is a Lie bialgebra, andg∗ is also a Poisson–Lie algebra, called the dual Poisson–Lie algebra of (g, p), andvice versa. (The 1-cocycle on g∗ is the one which defines the Lie algebra structureon g.) If (g, g∗) is a Lie bialgebra, then (g∗, g) is also a Lie bialgebra.

At the group level, each simply-connected Poisson–Lie group (G, Π) has aunique simply-connected dual Poisson–Lie group, denoted by (G∗, Π∗).

Theorem 5.3.5 ([108]). If (g1, g2, [, ]1, [, ]2) is a Lie bialgebra, g2 = g∗1, then thebracket

[x + α, y + β] = ([x, y]1 + ad∗αy − ad∗

βx, [α, β]2 + ad∗xβ − ad∗

yα), (5.27)

where x, y ∈ g1, α, β ∈ g2, defines a Lie algebra structure on g1 + g2. Denote thisLie algebra by d = g1 g2. Then

〈x + α, y + β〉 := 〈x, β〉 + 〈y, α〉 (5.28)

is a nondegenerate ad-invariant scalar product on d, with respect to which g1 andg2 are isotropic subspaces, i.e., 〈g1, g1〉 = 0 and 〈g2, g2〉 = 0.

Conversely, if a Lie algebra d is a vector-space direct sum of its two subalge-bras g1, g2, and d admits a nondegenerate ad-invariant scalar product with respectto which g1 and g2 are isotropic subspaces, then (g1, g2) admits a unique naturalLie bialgebra structure.

The proof of the above theorem is a straightforward verification. See, e.g.,Section 10.5 of [333]. Let us verify, for example, the Jacobi identity

[α, [x, y]] = [[α, x], y] + [x, [α, y]]

for α ∈ g2, x, y ∈ g1. If z ∈ d = g1 g2, we will denote by z1 ∈ g1 and z2 ∈ g2 itscomponents in g1 and g2: z = z1 + z2. Then we have

([[α, x], y] + [x, [α, y]])2 = ([−ad∗xα, y])2 + ([x,−ad∗

yα])2= ad∗

y(ad∗xα)− ad∗

x(ad∗yα) = −ad∗

[x,y]α = ([α, [x, y]])2.

On the other hand, for any β ∈ g2 we have

〈([[α, x], y] + [x, [α, y]])1, β〉= 〈([ad∗

αx− ad∗xα, y])1, β〉+ 〈([x, ad∗

αy − ad∗yα])1, β〉

= 〈[ad∗αx, y], β〉 − 〈ad∗

ad∗xαy, β〉+ 〈[x, ad∗

αy], β〉+ 〈ad∗ad∗

yαx, β〉= 〈ad∗

yβ, ad∗αx〉+ 〈ad∗

xα, ad∗βy〉 − 〈ad∗

xβ, ad∗αy〉 − 〈ad∗

yα, ad∗βx〉

= −〈[α, β], [x, y]〉 = 〈ad∗α[x, y], β〉 = 〈([α, [x, y]])1, β〉.

The rest is similar.


A triple (d, g1, g2), where (g1, g2) is a Lie bialgebra and d = g1 g2 as inthe above theorem, is called a Manin triple [108], and d is called the double ofg1 and g2.

Example 5.3.6. Consider d = sl(n, C) with its real invariant scalar product〈X, Y 〉 := (trace(XY )) (the imaginary part of trace(XY )). Put g1 = su(n),and g2 = the subalgebra of gl(n) consisting of traceless upper-triangular matriceswhose diagonal entries are real. Then d = g1 + g2 and g1 and g2 are isotropicsubspaces of d, hence (d, g1, g2) is a Manin triple. As a consequence, the Lie groupSU(n) of Lie algebra su(n) has a natural Poisson–Lie structure corresponding tothis Manin triple, called the Iwasawa Poisson–Lie structure of SU(n). More gen-erally, given a compact Lie group K of Lie algebra k, its complexification G = KC

has a natural Iwasawa decomposition G = KAN where A is Abelian and N isnilpotent. At the algebra level, we have d = k+ l where l = a+n. The triple (d, k, l)is a Manin triple, and the corresponding Poisson–Lie structure on K is called theIwasawa Poisson–Lie structure [224].

Theorem 5.3.7 ([220, 224]). Given a Manin triple (d, g1, g2 = g∗1), denote by D thesimply-connected Lie group of d, and by G1 and G2 the connected Lie subgroupsof D generated by g1 and g2 respectively. Then the corresponding Poisson–Liestructures Π1 on G1 and Π2 on G2 can be given as follows:

(Rg−1Π1(g))(α, β) = −〈p1Adg−1α, p2Adg−1β〉 ∀ g ∈ G1, α, β ∈ g2, (5.29)(Ru−1Π2(u))(X, Y ) = 〈p1Adu−1X, p2Adu−1Y 〉 ∀ u ∈ G2, X, Y ∈ g1, (5.30)

where p1 : d → g1 and p2 : g → g2 are the two natural projections.

Proof. We will prove Formula (5.29). Formula (5.30) is absolutely similar. Definetensor field Λ on G1 by

(Rg−1Λ(g))(α, β) = 〈p1Adg−1α, p2Adg−1β〉. (5.31)

We must show that Λ coincides with −Π1, i.e., it is a multiplicative Poisson struc-ture on G1 whose linear part at the neutral element defines the minus of Liebracket on g2 = g∗1. First notice that Λ is anti-symmetric, i.e., it is a 2-vector field,because

0 = 〈α, β〉 = 〈Adg−1α, Adg−1β〉 = (Rg−1Λ(g))(α, β) + (Rg−1Λ(g))(β, α).

By putting g = exp(tx) (x ∈ g1) in Formula (5.31) and deriving both sides withrespect to t at t = 0, we obtain that the linear part of Λ at the neutral elementdefines the minus of the Lie bracket on g2. It remains to show that Λ is mul-tiplicative. Notice that p2Adgα = Ad∗

gα, which is a consequence of the formula[x, α] = ad∗

xα− ad∗αx (x ∈ g1, α ∈ g2).

5.4. Actions of Poisson–Lie groups 139

We have:

(R(gh)−1Λ(gh))(α, β) = 〈p1Adh−1Adg−1α, p2Adh−1Adg−1β〉= 〈p1Adh−1(p1Adg−1α + p2Adg−1α), p2Adh−1p2Adg−1β〉= 〈Adh−1p1Adg−1α, Adh−1p2Adg−1β〉

+ 〈p1Adh−1p2Adg−1α, p2Adh−1p2Adg−1β〉= (Rg−1Λ(g))(α, β) + (Rh−1Λ(h))(p2Adg−1α, p2Adg−1β)= (Rg−1Λ(g))(α, β) + (Rh−1Λ(h))(Ad∗

g−1α, Ad∗g−1β)

=(Rg−1Λ(g) + Adg(Rh−1Λ(h))

)(α, β),

which means that R(gh)−1Λ(gh) = Rg−1Λ(g) + Adg(Rh−1Λ(h)), i.e., Λ is multi-plicative.

The double of a Poisson–Lie group also admits a Poisson–Lie structure. Moreprecisely, we have:

Theorem 5.3.8 ([313, 220]). With the notations of Theorem 5.3.7, put

Π(d) = Rdπ − Ldπ ∀ d ∈ D, (5.32)

where π ∈ d∗ ∧ d∗ is defined by

π(α + x, β + y) = 〈x, β〉 − 〈y, α〉 (5.33)

∀ (α, x), (β, y) ∈ g2 + g1∼= g∗1 + g∗2 ∼= d∗. Then (D, Π) is a Poisson–Lie group

whose corresponding Lie algebra structure on d∗ is isomorphic to the direct sumg2⊕(−g1), where the Lie bracket on the component g1 is taken with the minus sign.Moreover, the inclusions (G1, Π1) → (D, Π) and (G2, Π2) → (D, Π) are Poissonand anti-Poisson maps respectively.

The proof of the above theorem will be left as an exercise.

5.4 Actions of Poisson–Lie groups

5.4.1 Poisson actions of Poisson–Lie groups

When G is a Lie group equipped with a (nontrivial) Poisson–Lie structure Π,then it is interesting to consider actions of G on Poisson manifolds, which do notpreserve Poisson structures, but rather twist them by the Poisson–Lie structureof G:

Definition 5.4.1. A left (or right) action of a Poisson–Lie group (G, Π) on a Poissonmanifold (M, Λ) is called a Poisson action if the corresponding action map G ×M →M is a Poisson map.


In particular, when a Poisson action (G, Π)× (M, Λ)→ (M, Λ) is transitive,then (M, Λ) is called a Poisson homogeneous space [109]. See, e.g., [109, 222, 216,191, 126], for results about these Poisson homogeneous spaces.Remark 5.4.2. To turn a left Poisson action of G into a right Poisson action orvice versa, we have to change the sign of the Poisson–Lie structure on G.

Example 5.4.3. If (G, Π) is a Poisson–Lie group, then the actions of G on itself byleft and right multiplications are Poisson actions.

The following result provides a family of examples of Poisson actions.

Example 5.4.4 (Semenov–Tian–Shansky [314]). Suppose that G is a Lie group withLie algebra g, and r1, r2 ∈ ∧2g satisfy the mCYBE (5.15) with the same coefficientλ. Then

Πr1,r2 = r+2 − r−1 (5.34)

is a Poisson structure on G. If we denote by Gr1,r2 the group G equipped withthe Poisson structure Πr1,r2 = r+

1 − r−2 , then the multiplication map G×G → Ginduces Poisson morphisms Gr1,r1 ×Gr1,r2 → Gr1,r2 and Gr1,r2 ×Gr2,r2 → Gr1,r2 .The proof of these facts is similar to the proof of Theorem 5.2.3 and Lemma 5.2.2.

Lemma 5.4.5 ([224]). Let ρ : G ×M → M be a left (right) action of a connectedPoisson–Lie group (G, Π) on a Poisson manifold (M, Λ), whose Lie algebra actionis ξ : g → V1(M), extended to a morphism ξ : ∧g → V(M). Then ρ is a Poissonaction if and only if one of the following three equivalent conditions is satisfied:

1) ∀ g ∈ G, z ∈M one has

Λ(ρ(g, z)) = (ρg)∗(Λ(z)) + (ρz)∗(Π(g)), (5.35)

where ρg(z) = ρz(g) = ρ(g, z).2) ∀ X ∈ g one has

Lξ(X)Λ = ∓ ξ((deΠ)(X)), (5.36)

where the sign on the right-hand side is minus if ρ is a left action, and plusif ρ is a right action.

3) ∀ z ∈M, α, β ∈ Ω1(M), X ∈ g one has

(Lξ(X)Λ)z(α, β) = ∓ [(ρz)∗α(z), (ρz)∗β(z)]∗(X). (5.37)

Proof (sketch). Equation (5.35) is just a restatement of the condition that themap ρ : G×M →M is a Poisson map. Equation (5.36) is the infinitesimal versionof Equation (5.35). Indeed, putting g = exp(tX) in (5.35), we get

1t

(Λ(ρ(exp(tX), z))− (ρexp(tX))∗(Λ(z))

)=

1t(ρz)∗(Π(exp(tX))). (5.38)

The limit of (5.38) when t → 0 is (5.36), with the sign minus if ρexp(tX) =exp(−tξ(X)), and the sign plus if ρexp(tX) = exp(tξ(X)). Equation (5.37) is justthe value of (5.36) on the two arguments α, β.


If we have an action ξ : g → V1(M) of a Poisson–Lie algebra (g, p), where pis a 1-cocycle p : g → g∧ g, on a Poisson manifold (M, Λ), such that the equation

Lξ(X)Λ = ∓ξ(p(X)) (5.39)

is satisfied ∀ X ∈ g, then we say it is a left (resp. right) Poisson action if thesign on the right-hand side is minus (resp. plus). It is the infinitesimal version ofPoisson actions of Poisson–Lie groups.

Theorem 5.4.6 (Lu [221]). Let (M, Λ) be a Poisson manifold, (g, p) a Poisson–Liealgebra, and ξ : g → V1(M) a Lie algebra homomorphism which can be lifted to alinear map ξ : g → Ω1(M), i.e., ξ(X) = (ξ(X)) ∀ X ∈ g. Define the g∗-valued1-form Θ on M by

〈Θ(z)(v), X〉 = 〈v, ξ(X)(z)〉 ∀ z ∈M, v ∈ TzM, X ∈ g. (5.40)

Then ξ is a left (resp. right) Poisson action if and only if dΘ + [Θ, Θ] = 0 (resp.dΘ− [Θ, Θ] = 0) on each symplectic leaf of M .

Remark 5.4.7. In the above theorem, [Θ, Θ] is defined by

[Θ, Θ](X, Y ) = [Θ(X), Θ(Y )] ∀ X, Y ∈ TzM,

using the Lie bracket of g∗. The equation dΘ + [Θ, Θ] = 0 is called the (left)Maurer–Cartan equation. It is also written as dΘ + 1

2 [Θ ∧Θ] = 0, and sometimeseven as dΘ + 1

2 [Θ, Θ] = 0 (with a different convention on the definition of [Θ, Θ]).

Proof. We will make use of the following formula: If ζ1, ζ2 are two 1-forms on aPoisson manifold (M, Λ), then

[ζ1, ζ2] = (Lζ1ζ2 − Lζ2ζ1 − d (Λ(ζ1, ζ2))) . (5.41)

This formula corresponds to the fact that the bracket

ζ1, ζ2 = Lζ1ζ2 − Lζ2ζ1 − d(Λ(ζ1, ζ2))= d(Λ(ζ1, ζ2)) + iζ1dζ2 − iζ2dζ1

(5.42)

is a Lie bracket on the space Ω1(M) of 1-forms on M , and the anchor map :Ω1(M) → V1(M) is a Lie algebra homomorphism with respect to this bracketand the usual Lie bracket on V1(M). This bracket is actually the natural Liealgebroid bracket on T ∗M associated to a Poisson structure on M , and it will bediscussed in more detail in Chapter 8. Formula (5.41) can be proved by a simpledirect verification in the symplectic case using Cartan’s formula, and the generalPoisson case can be reduced to the symplectic case by restriction to symplecticleaves.


Using (5.41) and (5.42), we get, for any ζ1, ζ2 ∈ Ω1(M) and X ∈ g,

Lξ(X)Λ(ζ1, ζ2) = 〈ξ(X), ζ1, ζ2〉 − ζ1(〈ξ(X), ζ2〉) + ζ2(〈ξ(X), ζ1〉)= −〈Θ([ζ1, ζ2]), X〉+ ζ1(〈Θ(ζ2), X〉)− ζ2(〈Θ(ζ1), X〉)= 〈dΘ(ζ1, ζ2), X〉.

On the other hand,

ξ(p(X))(ζ1, ζ2) = 〈[Θ, Θ](ζ1, ζ2), X〉. (5.43)

Therefore ξ is left (right) Poisson if and only if (dΘ± [Θ, Θ]) (ζ1, ζ2) = 0 forarbitrary 1-forms ζ1, ζ2 on M , i.e., dΘ± [Θ, Θ] = 0 on each symplectic leaf of M .

5.4.2 Dressing transformations

Given a Poisson–Lie group (G, Π), there are natural Poisson actions of its dualLie algebra g∗ on G, which generate (local) Poisson actions of G∗ on G, calleddressing transformations of G [314, 351, 224], which can be constructed as follows.

For each element α ∈ g∗, we denote by α+ (resp. α−) the left-invariant (resp.right-invariant) 1-form on G whose value at the neutral element e is α. Denote by = Π : T ∗G → TG the usual anchor map of Π.

Proposition 5.4.8 ([351]). The maps α → (α+) and α → (α−) are Lie algebrahomomorphisms from g∗ to V1(G).

Proof. Using Formula (5.41), in order to prove Proposition 5.4.8, it is enough tocheck that

α+, β+ = ([α, β]∗)+ and α−, β− = ([α, β]∗)−. (5.44)

It is clear from the second line of (5.42) and the definition of the Lie bracketof g∗ that the above equalities hold at the neutral element e of G. The rest of theproof is now a direct consequence of the following characterization of Poisson-Liegroups:

Theorem 5.4.9 ([351, 97]). Let G be a connected Lie group with a Poisson tensorΠ. Then Π is Poisson–Lie if and only if Π(e) = 0 and for any two left-invariant(resp., right-invariant) 1-forms ζ1, ζ2 on G, their bracket ζ1, ζ2 defined by (5.42)is also left-invariant (resp., right-invariant).

Proof. Denote by ζ1 and ζ2 two arbitrary left-invariant 1-forms. Denote by X aleft-invariant vector field and by Y a right-invariant vector field. Then ζ1, ζ2 isleft-invariant if and only if

(LY ζ1, ζ2)(X) = 0 ∀ X, Y.


Using Formula (5.42), Cartan’s formula, and the fact that X(ζ1) and X(ζ2) areconstant, we get

(LY ζ1, ζ2)(X) = Y (ζ1, ζ2(X))= Y [X(Π(ζ1, ζ2)) + (dζ2)(ζ1, X)− (dζ1)(ζ2, X)]= Y [(LXΠ)(ζ1, ζ2)) + Π(LXζ1, ζ2) + Π(ζ1,LXζ2)

+ (dζ2)(ζ1, X)− (dζ1)(ζ2, X)]= Y [(LXΠ)(ζ1, ζ2)] = (LY LXΠ)(ζ1, ζ2).

Thus, (LY ζ1, ζ2)(X) = 0 ∀X, Y, ζ1, ζ2 if and only if LY LXΠ = 0 for any left-invariant X and right-invariant Y , i.e., Π is multiplicative. The case of right-invariant 1-forms is completely similar.

In view of Proposition 5.4.8, we have the following definition:

Definition 5.4.10. The actions of g∗ on (G, Π) given by the maps α → (α+) andα → (α−), and their corresponding (local or global) actions of G∗ on G, are calleddressing actions or dressing transformations .

The word local in the above definition means that maybe the action mapG∗×G→ G is not well defined globally, but is well defined for a neighborhood of(e∗, e) ∈ G∗ ×G.

It is clear from the definition that the symplectic leaves of (G, Π) are orbitsof the dressing actions. In particular, if Π = 0 then the dressing actions are alsotrivial.

Example 5.4.11. Consider the case when G is simply-connected Abelian, i.e., G = gis a vector space with a trivial Lie bracket. Then the two dressing actions coincide,and they are just the coadjoint action of g∗ on g = (g∗)∗.

The action given by (α+) is called the left dressing action. The correspondingleft dressing action of G∗ on G is defined by ρ(exp(α), z) = exp(−α+)(z) forα ∈ g∗, z ∈ G. The action given by (α−) is called the right dressing action. Thesetwo dressing actions are intertwined by the inversion map S : g → g−1 of G. Inother words, we have

S∗(α+) = (α−). (5.45)

This equality follows directly from S∗Π = −Π and S∗α+ = −α−.Let (d = g1 g2, g1, g2) be a Manin triple. Suppose that there are connected

Lie groups D, G1, G2 with Lie algebras d, g1, g2 such that D = G1G2, in the sensethat G1, G2 ⊂ D, and any element d ∈ D can be factorized in a unique way intoa product d = g1g2 with g1 ∈ G1, g2 ∈ G2. Then the dressing actions have thefollowing simple geometric interpretation: for any g1 ∈ G1, g2 ∈ G2, there is aunique element gg2

1 ∈ G1 and a unique element gg12 ∈ G2 such that

g2.g1 = gg21 .gg1

2 . (5.46)


Then the map (g2, g1) → gg21 is a left action of G2 on G1, called the left dressing

transformation. One can verify directly that this action is generated by the Liealgebra action g2 → V1(G1) : α → (α+). Similarly, (g1, g2) → gg1

2 is the rightdressing action of G1 on G2.

Theorem 5.4.12 ([314, 224]). The left and right dressing actions of G∗ on G arePoisson actions.

Proof. We will prove that the left dressing action is Poisson. The case of rightdressing action is similar. Denote by Θ the left-invariant Maurer–Cartan form onG, i.e., the g-valued 1-form on G defined by

Θ(X) = Lg−1X ∈ g ∀ X ∈ TgG. (5.47)

This form satisfies the Maurer–Cartan equation

dΘ + [Θ, Θ] = 0. (5.48)

(This classical fact can be checked easily by evaluating dΘ on left-invariant vectorfields.) Having this equation in mind, Theorem 5.4.12 is now just a special case ofTheorem 5.4.6.

5.4.3 Momentum maps

Momentum maps for Poisson actions of Poisson–Lie groups were introduced byLu [221], in analogy with momentum maps for Hamiltonian actions of Lie groups.

Let : G ×M → M be a left (resp. right) Poisson action of a Poisson–Liegroup G on a Poisson manifold M . Let G∗ be the dual Poisson-Lie group. Denoteby ξ : g → V1(M) the corresponding Lie algebra homomorphism. For each X ∈ g,denote by X+ (resp. X−) the left- (resp. right-) invariant 1-form on G∗ with valueX at the neutral element.

Definition 5.4.13. A smooth map J : M → G∗ is called a momentum map for theleft (resp. right) Poisson action : G×M →M if for each X ∈ g,

ξ(X) = (J∗(X+)) (resp. ξ(X) = (J∗(X−))). (5.49)

If, moreover, J is a Poisson map, then it is called an equivariant momentum map.

Example 5.4.14. For both the left and the right dressing action of G∗ on G, theidentity map of G is an equivariant momentum map.

Remark 5.4.15. In general, it is not clear whether a given Poisson action of aPoisson–Lie group admits an equivariant momentum map or not, see [143]. One canview Lu’s theory of momentum maps of Poisson–Lie group actions as a particularcase of Xu’s theory of momentum maps for quasi-symplectic groupoid actions [361].

5.5. r-matrices and quadratic Poisson structures 145

5.5 r-matrices and quadratic Poisson structures

In this section, we will discuss some known constructions of quadratic Poissonstructures from r-matrices.

Theorem 5.5.1 ([297]). Let r ∈ g ∧ g be a triangular r-matrix of a Lie algebra g,and ρ : g → End(V ) be a linear action of g on a vector space V . Identify End(V )with the space of linear vector fields on V . Then the image rV of r under the map∧2ρ : g ∧ g → End(V ) ∧ End(V ) is a quadratic Poisson structure on V .

Proof. The fact that ∧2ρ(r) is quadratic is clear, because the wedge product oftwo linear vector fields is a quadratic 2-vector field. The fact that ∧2ρ(r) is Poissonis a particular case of the following more general result:

Theorem 5.5.2. If r ∈ g ∧ g is a triangular r-matrix and g acts on a manifoldM , then the image of r under the induced map g ∧ g → Γ(∧2TM) = V2(M) isa Poisson tensor on M , which we will denote by rM . If Π is a Poisson structureon M which is invariant under the action of g, then rM is compatible with Π :[rM , Π] = 0.

Proof. If ρ : g → V1(M) is a Lie algebra homomorphism, then ∧ρ : ∧g → V(M)preserves the Schouten bracket. Hence if [r, r] = 0 then [∧2ρ(r),∧2ρ(r)] = 0, i.e.,rM = ∧2ρ(r) is a Poisson structure. Similarly, if Π is invariant under the action of g,then [ρ(x), Π] = 0 ∀ x ∈ g, and by Leibniz’s rule we have [∧ρ(α), Π] = 0 ∀ α ∈ ∧g;in particular [∧2ρ(r), Π] = 0. Corollary 5.5.3 ([161]). If r ∈ g ∧ g is a triangular r-matrix, then its inducedquadratic Poisson structure on g∗ via the coadjoint action is compatible with thelinear Poisson structure on g∗.

Example 5.5.4. When g = gl(n) then by taking the standard linear action of gl(n)on Rn or Cn, we get an n-dimensional quadratic Poisson structure associated toeach triangular r-matrix r ∈ gl(n) ∧ gl(n). On the other hand, the adjoint actionof gl(n) on itself will give rise to n2-dimensional quadratic Poisson structures.

When g acts on V or M and r ∈ g∧ g is an r-matrix which is not triangular,i.e., [r, r] ∈ (∧3g)g and [r, r] = 0, then sometimes rV or rM is still a Poissonstructure. For example, Donin and Gurevich [104] showed that if M is a symmetricspace of a semisimple Lie group G (i.e., M = G/H such that g has a lineardecomposition g = h + m with [h, m] ⊂ m, [m, m] ⊂ h), and r is an r-matrix ofg, then rM is a Poisson structure on M . The following result of Zakrzewski [365]gives several series of quadratic Poisson structures on Rn related to r-matrices.

Theorem 5.5.5 ([365]). In the following three cases, for any classical r-matrix ing ∧ g the induced quadratic 2-vector field rV on V is Poisson:

1. g = so(n, R), V = Rn (the action of g on V is the natural one);2. g = sl(n, R), V = Rn;3. g = sp(2n, R), V = R2n.


For g = su(n) and V = Cn = R2n, the induced 2-vector field is Poisson if andonly if r is triangular.

Proof (sketch). Recall that, if g is a simple Lie algebra, then (∧3g)g is one-dimensional and is generated by an element η whose Killing-transported versionto ∧3g∗ is η† defined by η†(X, Y, Z) = 〈[X, Y ], Z〉 ∀ X, Y, Z ∈ g. Let M be amanifold on which g acts. If [r, r] ∈ (∧3g)g and [r, r] = 0, then rM is Poisson ifand only if ηM = 0, where ηM is the image of η in V3(M) under the extension ofthe homomorphism g → V1(M).

Now, ηM (x) = 0 for a point x ∈ M if and only if the image of η under theprojection ∧3g → ∧3(g/gx) is zero, where gx denotes the isotropy algebra of theaction of g on M at x. Equivalently, for any X, Y, Z ∈ g⊥x we have 〈[X, Y ], Z〉 =〈ν, X ∧ Y ∧ Z〉 = 0, where g⊥x denotes the orthogonal complement of gx in g withrespect to the Killing form. It means that [g⊥x , g⊥x ] ⊂ gx. In other words, we havethe following criterion:

[rM , rM ](x) = 0 ⇐⇒ [g⊥x , g⊥x ] ⊂ gx . (5.50)

The rest of the proof is a straightforward case by case analysis, based on the abovecriterion. Example 5.5.6 ([365]). Consider the standard r-matrix for sl(n, R), which isgiven by

r = c∑j<k

ekj ∧ ej

k (c ∈ R), (5.51)

where ekj denotes the matrix whose entry at line j column k is 1 and all the other

entries are 0. Direct calculations show that r induces the following bracket on Rn

via the natural action of sl(n, R) on Rn:

xj , xk = cxjxk ∀ 1 ≤ j < k ≤ n. (5.52)

Let A be a (finite-dimensional) associative algebra. Then, similarly to thecase of Lie groups, we can talk about multiplicative tensors on A. In particular, amultiplicative Poisson structure on A is a Poisson structure Π on A which satisfiesEquation (5.9), i.e.,

Π(ab) = (La)∗Π(b) + (Rb)∗Π(a) ∀ a, b ∈ A, (5.53)

where La denotes the left multiplication by a, La(x) = a.x, and Rb denotes theright multiplication by b. In other words, a Poisson structure Π on A is multiplica-tive if the product map A×A→ A o A is a Poisson morphism.

Example 5.5.7. If Π = r+−r− is an exact Poisson–Lie structure on G = GL(n, K),where r ∈ gl(n, K)∧gl(n, K) is a classical r-matrix of gl(n, K), then Π can naturallybe extended to a multiplicative Poisson structure on the algebra Matn(K) of n×nmatrices by the same formula: Π(M) = (LM )∗r − (RM )∗r ∀ M ∈ Matn(K). Byconstruction, it is a quadratic Poisson structure on Matn(K). The correspondingPoisson bracket on Matn(K) is called the Sklyanin bracket [318].

5.6. Linear curl vector fields 147

In fact, as was shown by Balinsky and Burman [21], multiplicative Poissonstructures on associative algebras are usually automatically quadratic.

Theorem 5.5.8 ([21]). If A is a finite-dimensional associative algebra with unit,then any smooth multiplicative Poisson structure on A is quadratic.

Proof. Denote by I the unit element of A, by (ei) a linear basis of A, and by(xi) the dual basis of A∗, considered as a linear coordinate system of A. LetΠ =

∑i<j Πij

∂∂xi

∧ ∂∂xj

be a smooth multiplicative Poisson tensor on A.Note that, for each t ∈ R, the left and right multiplication maps LtI and

RtI are the dilatation by a factor of t in A, considered as a vector space. Puttinga = b = tI in Equation (5.53), we get

Πij(t2I) = 2t2Πij(tI).

The only smooth function g(t) of one variable which satisfies the functional equa-tion g(t2) = 2t2g(t) is the trivial function g(t) = 0. Thus we have Π(tI) = 0 ∀ t ∈ R.

Now, putting a = tI in Equation (5.53) and taking into account the fact thatΠ(tI) = 0, we get

Πij(tb) = t2Πij(b) ∀ b ∈ A, t ∈ R.

In other words, Πij are homogeneous functions of degree 2 on A. Since they aresmooth, they are homogeneous quadratic functions. Remark 5.5.9. Quadratic Poisson structures play an important role in integrablesystems, and many other examples of them (all related to r-matrices) can be foundin the literature. See, e.g., [208, 283, 326, 136, 282]. In [208, 283], natural cubicPoisson structures arising from r-matrices are also given.

5.6 Linear curl vector fields

LetΠ =

14

∑i,j,k,l

Πklijxkxl

∂

∂xi∧ ∂

∂xj(5.54)

be a quadratic Poisson structure (Πklij = Πlk

ij = −Πklji). Its curl vector field X =

DΩΠ with respect to the volume form Ω = dx1 ∧ · · · ∧ dxn is a linear vector fieldwhich has the following expression:

X =∑i,j,k

Πkjij xk

∂

∂xi. (5.55)

Recall that, if we change the linear coordinate system linearly, the corre-sponding volume form will be multiplied by a constant function, and thereforethe curl vector field will not be changed. So we may write the curl vector fieldas X = DΠ, without reference to Ω. Recall from Lemma 2.6.9 that X = DΠpreserves Π and the volume form Ω: LXΠ = 0, LXΩ = 0.


Definition 5.6.1. If Π is a quadratic Poisson structure, then the eigenvalues of itslinear curl vector field X = DΠ will be called the eigenvalues of Π.

Recall that an n-dimensional linear vector field X is isochore, i.e., it preservesthe standard volume form, if and only if its eigenvalues λ1, . . . , λn satisfy theresonance relation

n∑j=1

λj = 0. (5.56)

In particular, the eigenvalues of a quadratic Poisson structure Π will satisfy theabove relation.

Definition 5.6.2. A quadratic Poisson structure will be called nonresonant if itseigenvalues λ1, . . . , λn do not satisfy any relation of resonance other than (5.56).In other words, if

∑nj=1 cjλj = 0 with cj ∈ Z then c1 = · · · = cn.

Definition 5.6.3. A quadratic Poisson structure of the type

Π =∑

1≤i<j≤n

cijxixj∂

∂xi∧ ∂

∂xj, (5.57)

where cij are constants, is called a diagonal quadratic Poisson structure.

Remark 5.6.4. Any quadratic 2-vector field of the form (5.57) is automatically aPoisson structure, and can be given by a triangular r-matrix. But quadratic Pois-son structures arising from r-matrices (via representations) need not be diagonalin general.

One may argue that a “generic” (in a disputable sense) quadratic Poissonstructure is nonresonant. The following theorem says that “generic” quadraticPoisson structures are diagonalizable.

Theorem 5.6.5 (Dufour–Haraki [118]). If the eigenvalues λ1, . . . , λn of a quadraticPoisson structure Π does not verify any relation of the type

λi + λj = λr + λs (5.58)

with i < j and r, s = i, j, then Π is diagonalizable, i.e., there is a linearcoordinate system in which Π is diagonal. In particular, nonresonant quadraticPoisson structures are diagonalizable.

Proof. Notice that the condition of the above theorem implies that the eigenvaluesλ1, . . . , λn are pairwise different (λ1 = λ2 leads to λ1 + λ2 = λ1 + λ1). Thus thelinear curl vector field X = DΠ is diagonalizable, i.e., there is a linear coordinatesystem in which X is diagonal:

X =∑

λixi∂

∂xi. (5.59)

5.6. Linear curl vector fields 149

Then the equation [X, Π] = 0 can be written as

0 =[∑

i

λixi∂

∂xi,∑rsuv

Πuvrs xuxv

∂

∂xr∧ ∂

∂xs

]=

∑rsuv

Πuvrs (λu + λv − λr − λs)xuxv

∂

∂xr∧ ∂

∂xs,

whence Πuvrs = 0 for r, s = u, v, hence the result.

When the conditions of the above theorem are not satisfied, but the curlvector field is nontrivial, then the following decomposition theorem may be useful.

Theorem 5.6.6 (Liu–Xu [217]). If Π is an n-dimensional quadratic Poisson struc-ture, then it can be decomposed in a unique way as a sum of two terms:

Π = Π0 +1n

X ∧ I , (5.60)

where Π0 is a quadratic Poisson structure with a trivial curl vector field, I denotesthe radial linear vector field

∑xi∂/∂xi, and X is a linear vector field such that

[X, Π0] = 0.

Proof. If Π = Π0+ 1nX∧I with D(Π0) = 0, then D(Π) = D(Π0)+ 1

nD(X∧I) = X ,and we must have Π0 = Π − 1

nX ∧ I where X is the curl vector field of Π.Conversely, we have [X, Π] = 0, and [I, Π] = 0 (because P is a quadratic 2-vectorfield), therefore [Π, X ∧I] = 0, i.e., the Poisson structure P is compatible with thePoisson structure X ∧ I. Hence Π0 = Π− 1

nX ∧ I is also a Poisson structure.

Definition 5.6.7. A quadratic Poisson structure is called exact if its curl vectorfield is zero.

Example 5.6.8. A quadratic Poisson structure Π in R3 is exact if and only if

iΠ(dx1 ∧ dx2 ∧ dx3) = df

where f is a homogeneous polynomial function of degree 3.

In a sense, Theorem 5.6.6 reduces the problem of classification of quadraticPoisson structures to the problem of classification of exact quadratic Poisson struc-tures and linear vector fields which preserve them.

For example, in the case of three-dimensional quadratic Poisson structures,the curl vector field and Theorem 5.6.6 lead to the following classification:

Theorem 5.6.9 ([118, 217]). Any quadratic Poisson structure in R3 can be writtenin a linear system of coordinates (x, y, z) as(

∂f

∂x

∂

∂y∧ ∂

∂z+

∂f

∂y

∂

∂z∧ ∂

∂x+

∂f

∂z

∂

∂x∧ ∂

∂y

)+X∧

(x

∂

∂x+ y

∂

∂y+ z

∂

∂z

)(5.61)


where f is a homogeneous cubic polynomial and X is a linear vector field withtrace 0 such that X(f) = 0, and the pair (X, f) belongs to one of the followingnine cases:

(1) X = 0, f any cubic polynomial;(2) X = αx ∂

∂x + βy ∂∂y + γz ∂

∂z with α = β = γ = 0, f = axyz;

(3) X = αx ∂∂x − αy ∂

∂y with α = 0, f = axyz + bz3;

(4) X = αx ∂∂x + αy ∂

∂y − 2αz ∂∂z with α = 0, f = axyz + bx2z + cy2z;

(5) X = y ∂∂x + z ∂

∂y , f = az3 + 2bz2x− bzy2;

(6) X = y ∂∂x , f = ay3 + by2z + cyz2 + dz3;

(7) X = αx ∂∂x + y ∂

∂xαy ∂∂y − 2αz ∂

∂z with α = 0, f = ay2z;

(8) X = βy ∂∂x − βx ∂

∂y with β = 0, f = az(x2 + y2) + bz3;

(9) X = αx ∂∂x +βy ∂

∂xαy ∂∂y −βx ∂

∂y − 2αz ∂∂z with α, β = 0, f = az(x2 + y2).

The problem of classification of four-dimensional quadratic Poisson structureswas studied by El Galiou [124].

5.7 Quadratization of Poisson structures

In this section, we will discuss the problem of quadratization, or more generally,normal forms for Poisson structures with zero 1-jet at a point. So let

Π = Π(2) + Π(3) + · · · (5.62)

be the Taylor expansion of a Poisson structure Π in a local system of coordinates(x1, . . . , xn), which begins with terms of degree 2. Each term Π(k) is a homogeneous2-vector field of degree k. Then the quadratic part Π(2) of Π is a quadratic Poissonstructure: the equation [Π, Π] = 0 implies that [Π(2), Π(2)] = 0.

Denote byX = X(1) + X(2) + · · · (5.63)

the Taylor expansion of the curl vector field of X = DΩΠ of Π with respect to thevolume form Ω = dx1 ∧ · · · ∧ dxn. Then X(k) = DΩΠ(k+1). In particular, X(1) isthe curl vector field of Π(2).

Definition 5.7.1. The eigenvalues of the linear vector field X(1) are called theeigenvalues of Π. If Π(2) is nonresonant in the sense of Definition 5.6.2, then wesay that Π is nonresonant .

The following theorem provides a normal form a la Poincare–Dulac for Pois-son structures with zero 1-jet.

5.7. Quadratization of Poisson structures 151

Theorem 5.7.2. Let Π be a Poisson structure on a neighborhood of 0 in Cn with zero1-jet at 0, and eigenvalues λ1, . . . , λn. Then there is a formal change of variableswhich puts Π in the following form:

Π =∑

λi+λj=〈λ,I〉|I|≥2

aIijx

I ∂

∂xi∧ ∂

∂xj, (5.64)

where I = (I1, . . . , In) denotes a multi-index, aIij are constants, 〈λ, I〉 =

∑λjIj,

|I| = ∑Ij, xI =

∏x

Ij

j .

Proof. According to the isochore version of the classical Poincare–Dulac theorem,there is a formal volume-preserving transformation of variables which puts X inPoincare–Dulac normal form. In other words, there is a formal system of coordi-nates (x1, . . . , xn) in which the curl vector field X of Π with respect to the volumeform dx1 ∧ · · · ∧ dxn has the following property:

[X, Xs] = 0. (5.65)

where Xs denotes the semisimple part of the linear part X(1) of X (see AppendixA.5). The semisimple linear vector field Xs is diagonalizable, i.e., we may assumefurthermore that

Xs =n∑

j=1

λjxj∂

∂xj. (5.66)

Then we have the following formula:⎡⎣Xs,∑i,j,I

aIijx

I ∂

∂xi∧ ∂

∂xj

⎤⎦ =∑i,j,I

(〈λ, I〉 − λi − λj) aIijx

I ∂

∂xi∧ ∂

∂xj. (5.67)

This formula implies that, if Π =∑

i,j,I aIijx

I ∂∂xi

∧ ∂∂xj

and [Xs, Π] = 0, thenaI

ij = 0 unless 〈λ, I〉 − λi − λj = 0. Thus the theorem is proved modulo thefollowing lemma. Lemma 5.7.3. With the above notations, if X is in Poincare–Dulac normal form,then we have [Xs, Π] = 0.

Proof. For each q ∈ N we denote by A[q]1 (resp., A[q]

2 ) the space of polynomialvector fields (resp., polynomial 2-vector fields) of degree at most q in the variablesx1, . . . , xn (which normalize X and puts Xs in diagonal form). Denote by X [q],S [q]

and N [q] the maps from A[q]2 to itself, induced respectively from the maps P →

[X, P ], P → [Xs, P ] and P → [X −Xs, P ] by truncating at degree q. It followsdirectly from the formula [X, A ∧ B] = [X, A] ∧ B + A ∧ [X, B] and [X, Xs] = 0that N [q] and S [q] commute. N [q] is nilpotent because the map from A[q]

1 to itselfdefined by Z → [X −Xs, Z] is nilpotent. Finally, Formula (5.67) shows that S [q]


is semisimple (with the eigenvalues λI−λi−λj and eigenvectors xI ∂

∂xi∧ ∂

∂xjfor

|I| ≤ q).Return now to the relation [X, Π] = 0 (property of the curl vector field). It

induces the relationsX [q]

(Π[q]

)= 0,

where Π[q] = Π(2) + · · ·+Π(q) denotes the q-jet of Π. Thus Π[q] lies in the kernel ofX [q]. Since X [q] admits the Jordan decomposition X [q] = S [q] +N [q], we also have

S [q](Π[q]

)= 0

for any q ∈ N. The result then follows by taking limit q →∞. As a corollary of Theorem 5.7.2, we have:

Proposition 5.7.4. If Π = Π(2) + · · · is a Poisson structure with zero 1-jet andnonresonant quadratic part, then it admits a formal normal form of the type

Π =∑i<j

αij(ρ)Yi ∧ Yj =12

∑i=j

αij(ρ)Yi ∧ Yj , (5.68)

where Yi = xi∂

∂xi, ρ =

∏nk=1 xk and αij(ρ) = −αji(ρ). If, moreover, the eigen-

values of Π satisfies the ω-condition of Bruno, and Π is analytic, then there is alocal analytic system of coordinates which puts Π in the above normal form.

The ω-condition of Bruno in the above theorem is the following small-divisorcondition on the eigenvalues λ1, . . . , λn: denote

ωk = min|I|≤2k

∣∣∣∑ Ijλj

∣∣∣ ; I ∈ Zn admissible ,∑

Ijλj = 0

, (5.69)

where |I| =∑

Ij , I = (Ij) ∈ Zn is called admissible if there is an index j suchthat qi ≥ 0 if i = j and Ij ≥ −1. We will say that the ω-condition of Bruno issatisfied if ∞∑

k=1

12k

log1ωk

< ∞. (5.70)

Proof. Since X is volume-preserving and its eigenvalues λ1, . . . , λn do not admitany resonance relation other than

∑λj = 0, the isochore version of the classical

Poincare–Dulac theorem (see Appendix A.5) implies that X has a formal normalform of the type

X = b(ρ)( n∑

i=1

λixi∂

∂xi

).

According to Bruno’s theorem [48] (see Appendix A.5) that, under the ω-condition,there is a local complex analytic system of coordinates which puts X in the above


normal form. (Because X preserves the volume form, the required coordinate trans-formation can be chosen to preserve the volume form too.) In this analytic coor-dinate system Π will automatically have the form Π =

∑i<j αij(ρ)Yi ∧ Yj , where

Yi = xi∂/∂xi.

In fact, in the nonresonant case, we can improve Proposition 5.7.4 to obtainthe following quadratization result.

Theorem 5.7.5. If Π = Π(2) + · · · is a Poisson structure with zero 1-jet at 0,whose quadratic part is nonresonant, then Π is formally quadratizable, i.e., thereis a formal coordinate system in which Π is quadratic. If, moreover, the eigenvaluesof Π satisfies the ω-condition of Bruno, and Π is analytic, then it admits a localanalytic quadratization.

Theorem 5.7.5 follows directly from Proposition 5.7.4 and the following twolemmas, which are valid in the formal, analytic, C∞, as well as C1, categories.

Lemma 5.7.6. Consider a Poisson structure Π of the type

Π =12

∑i,j

αij(ρ)Yi ∧ Yj

as in Proposition 5.7.4, with αij = −αji. Put Γi =∑n

j=1 αij . Then we have therelation

dαij

dρΓk +

dαjk

dρΓi +

dαki

dρΓj = 0 (5.71)

for every triplet (i, j, k). Moreover, if at least one of the eigenvalues of Π is non-trivial, then there is a function Γ = Γ(ρ) such that Γ(0) = 1 and

Γi = λiΓ ∀ i = 1, . . . , n. (5.72)

Proof. The first relation follows directly from the identities [Π, Π] = 0, Yi(ρ) = ρ,[Yi, Yj ] = 0, and the formula

[A∧B, C ∧D] = [A, C]∧B ∧D− [A, D]∧B∧C− [B, C]∧A∧D+ [B, D]∧A∧C,

where A, B, C, D are arbitrary vector fields. In turn, it follows from the first rela-tion that

n∑k=1

(dαij

dρΓk +

dαjk

dρΓi +

dαki

dρΓj) = 0,

or dΓj

dρ Γi − dΓi

dρ Γj = 0 (we have∑

k Γk = 0 because the matrix (αij) is antisym-metric). It implies that the ratios Γi/Γj are constants. But we have Γi(0) = λi,which leads to the second relation.


Lemma 5.7.7. Under the assumptions and notations of the previous lemma, thereis a change of variables of the form

(x1, . . . , xn) → (x1θ1(ρ), . . . , xnθn(ρ)) (5.73)

with θi(0) = 1, which quadratizes the Poisson structure Π.

Proof. Put x′i = xiθi(ρ), where θi are to be determined. We have

x′i, x

′j = θiθj xi, xj+ θixj xi, θj+ θjxi θi, xj

= θiθjxixjαij + ρθidθj

dρxixj

∑e

αie − ρθjdθi

dρxixj

∑e

αje

= xixj

[αijθiθj + ρ

(θi

dθj

dρλi − θj

dθi

dρλj

)Γ]

= x′ix

′jαij(0) + xixj

[(αij − αij(0))θiθj + ρ

(θi

dθj

dρλi − θj

dθi

dρλj

)Γ]

.

We want to find the θi such thatx′

i, x′j

= x′

ix′jαij(0) = xixjθiθjαij(0),

or equivalently,

αij − αij(0) = ρΓ

(dθi

dρ

θiλj −

dθj

dρ

θjλi

). (5.74)

In view of Relation (5.72), Relation (5.71) becomes

dαij

dρλk +

dαjk

dρλi +

dαki

dρλj = 0,

therefore

(αij − αij(0))λk + (αjk − αjk(0))λi + (αki − αki(0))λj = 0,

and in particular (supposing that λ1 = 0)

αij − αij(0) = µiλj − µjλi

with

µi =α1i − α1i(0)

λ1.

We deduce from this that the system of equations (5.74) can be solved, provingthe lemma.

Let us mention here another result, about normal forms of Poisson structurewith a diagonal quadratic part. We will use the notation Yi = xi

∂∂xi

.


Theorem 5.7.8 (Dufour–Wade [114]). Let Π = Π(2) be a Poisson structure withzero 1-jet, whose quadratic part Π(2) =

∑i<j aijYi ∧ Yj is diagonal and satisfies

the following hypothesis (H):

(H) If (∑

ij aijIjYi) ∧ Yi ∧ Yj = 0 for a multi-index I with Ij = Ij = −1, Ik ≥0 ∀ k = i, j and

∑nk=1 Ik ≥ 1, then A.I = 0 where A = (aij), i.e.,

∑j aijIj =

0 ∀i. Then Π is formally isomorphic to a Poisson structure of the type

Π =∑i<j

A.I=0

xIαIijYi ∧ Yj (5.75)

Example 5.7.9. Let us interpret hypothesis (H) in the case n = 3. Consider adiagonal quadratic Poisson structure on a three-dimensional linear space given by

Π(2) = cxy∂

∂x∧ ∂

∂y+ ayz

∂

∂y∧ ∂

∂z+ bzx

∂

∂z∧ ∂

∂x,

where at least one of the three coefficients a, b, c is nonzero. Any multi-index Iwith |I| ≥ 1 containing two negative components has one of the following forms:

1) I = (I1,−1,−1) with I1 > 2;2) I = (−1, I2,−1) with I2 > 2;3) I = (−1,−1, I3) with I3 > 2.

So hypothesis (H) is equivalent to the combination of the following three condi-tions:

1) If c− b = 0 then cI1 + a = 0 and bI1 + a = 0 for any I1 > 2;2) If c− a = 0 then cI2 + b = 0 and aI2 + b = 0 for any I2 > 2 ;3) If a− b = 0 then bI3 + c = 0 and aI3 + c = 0, for any I3 > 2.

Since at least one of the three numbers a, b, c is non-zero, it follows that hypothesis(H) is equivalent to the condition a = b = c.

We refer to [114] for a proof of Theorem 5.7.8, where the following smoothquadratization theorem for Poisson structures with a diagonal quadratic part isalso obtained.

Let Π(2) =∑

i<j aijxixj∂

∂xi∧ ∂

∂xjbe a diagonal quadratic Poisson structure

on Rn. Denote by A1 = (a1j), . . . , An = (anj) the rows of A = (aij), and by λ1 =∑j a1j , . . . , λn =

∑j anj the sums of the components of A1, . . . , An respectively.

Theorem 5.7.10 ([114]). Suppose that the following three conditions are satisfied:

i) ∀ I = (I1, . . . , In) ∈ Zn such that Ii ≥ 0,∑n

i=1 Ii > 1, we have

Ai =n∑

j=1

AjIj ∀ i.


ii) ∀ I = (I1, . . . , In) ∈ Zn such that Ii = Ij = −1, Ik ≥ 0 if k = i, j, and∑ni=1 Ii > 2, there is an index k = i, j such that

aik + ajk =∑l =i,j

aklIl.

iii) There is an index i such that the coefficients aik, for all k = i, are of samesign and such that λjλi < 0, ∀j = i.

Then any C∞-smooth Poisson structure Π = Π(2) + Π′ on Rn, where the 2-jet ofΠ′ at 0 is zero, is locally C∞-smoothly isomorphic to Π(2).

Theorem 5.7.10 follows from Theorem 5.7.8 and a parameterized version ofSternberg’s smooth linearization theorem [321], due to Roussarie [304].Remark 5.7.11. Some generalizations of improvements of Theorem 5.7.8 and The-orem 5.7.5 were obtained recently by Lohrmann [219]. See also [341, 169] for someother results on smooth normal forms and quadratization of Poisson structures indimension 3 with a trivial 1-jet.

5.8 Nonhomogeneous quadratic Poisson structures

In this section, we will briefly discuss nonhomogeneous quadratic, or polynomialof degree 2, Poisson structures of the type

Π = Π(1) + Π(2), (5.76)

where Π(1) is linear and Π(2) is quadratic. Note that, in this case, the Jacobiidentity [Π, Π] = 0 is equivalent to [Π(1), Π(1)] = [Π(2), Π(2)] = [Π(1), Π(2)] = 0.Thus, if we have a nonhomogeneous quadratic Poisson structure then we have aquadratic Poisson structure which is compatible with a linear Poisson structure,and vice versa.

If Π is a homogeneous quadratic structure on a vector space Kn, and z ∈ Kn

is a point at which Π(z) = 0, then in the affine coordinate system centered atz, Π will become a nonhomogeneous quadratic Poisson structure (without theconstant term). For example, the Sklyanin structure Π = r+ − r− on Matn(K),where r is a classical r-matrix of gl(n, K), is homogeneous quadratic near 0, andis nonhomogeneous quadratic near the identity element Id ∈ Matn(K). Moregenerally, we have:

Proposition 5.8.1. If A is an associative algebra with a unit element e and a smoothmultiplicative Poisson structure Π, then in an affine coordinate system centeredat e, Π is the sum of a quadratic Poisson structure Π2 with a linear Poissonstructure Π(1).

Proof. It is a direct corollary of Theorem 5.5.8. Remark 5.8.2. In the above theorem, if g is the Lie algebra of A, G ⊂ A is theLie group of invertible elements of A, then (G, Π) is a Poisson–Lie group, and the

5.8. Nonhomogeneous quadratic Poisson structures 157

linear part Π(1) of Π at the unit element e actually corresponds to the dual Liealgebra structure on g∗.

A large family of nonhomogeneous quadratic Poisson structures is given bythe following result of Oh [284].

Theorem 5.8.3 ([284]). Let g be a Lie algebra, and µ a point in g∗. If the isotropysubalgebra gµ = x ∈ g | ad∗

xµ = 0 has a complementary Lie subalgebra kin g (i.e., g is the vector-space direct sum of its subalgebras gµ and k), thenthe transverse Poisson structure at µ of the linear Poisson structure of g∗ is(nonhomogeneous) quadratic.

Proof. Denote by (xi) a basis of gµ, and (yk) a basis of k. Take N = µ + k⊥ to bethe affine subspace which is transverse at µ to the coadjoint orbit of µ. Accordingto Dirac’s formula (1.56), the transverse Poisson structure on a neighborhood ofµ in N is given as follows:

xi|N , xj |NN (z) = xi, xj(z)−∑k,h

xi, yk(z)rkl(z)yl, xj(z), (5.77)

where (rkl(z)) is the inverse of the matrix (yk, yl(z)).In the case when g = gµ + k and (yk) is a basis of the Lie subalgebra k, the

matrix (yk, yh) is a constant matrix on N , and therefore the functions rkh areconstant on N . On the other hand, the functions xi, xj, xi, yk and yh, xj areaffine functions on N . Hence the Dirac formula shows that the Poisson structureon N is nonhomogeneous quadratic with respect to the coordinate system (xi). Remark 5.8.4. As observed by Bhaskara and Rama [29], the matrix (rkl) in theabove proof is actually a triangular r-matrix. More precisely, r =

∑i<j rijyi∧yj ∈

k ∧ k ⊂ g ∧ g satisfies the CYBE [r, r] = 0.Another interesting result about nonhomogeneous quadratic Poisson struc-

tures, which seems to be strongly related to Proposition 5.8.1 and Theorem 5.8.3,is the following:

Theorem 5.8.5 (Diatta–Medina [101]). Let (G, ω+) be a Lie group with a left-invariant symplectic form ω+. Denote by r ∈ g ∧ g the triangular r-matrix as-sociated to ω+, and r+ (resp. r−) the corresponding left- (resp. right-) invariantPoisson tensor (r+ is the Poisson tensor dual to ω+). Then the Poisson–Lie ten-sor Π = r+ − r− is polynomial of degree 2 with respect to the flat affine structureon G defined by ω+.

The flat affine structure ∇ on the symplectic Lie group (G, ω+) is defined asfollows [212]:

ω+(∇x+y+, z+) := −ω+(y+, [x+, z+]), (5.78)

where x+, y+, z+ denote left-invariant vector fields.The proof of Theorem 5.8.5, given in [101], is based on direct algebraic cal-

culations. It would be nice to find a more conceptual proof.

Chapter 6

Nambu Structures andSingular Foliations

6.1 Nambu brackets and Nambu tensors

Nambu structures were first introduced by Nambu [275] in a special case, and thengeneralized and formalized by Takhtajan [328].

Definition 6.1.1 ([328]). Let q be a natural number. A smooth Nambu structureof order q on a smooth manifold M is a skew-symmetric q-linear operator, calledthe Nambu bracket and denoted by . . ., from C∞(M)×· · ·×C∞(M) to C∞(M),which satisfies the following conditions:

a) Leibniz rule:

f1, . . . , fq−1, g1g2 = f1, . . . , fq−1, g1g2 + g1f1, . . . , fq−1, g2 , (6.1)

b) fundamental identity:

f1,...,fq−1,g1,...,gq=q∑

i=1

g1,...,gi−1,f1,...,fq−1,gi,gi+1,...,gq (6.2)

∀ fi, gi ∈ C∞(M).

It is clear from the above definition that a Nambu structure of order 2 isnothing but a Poisson structure. In general, Nambu brackets may be viewed asq-ary generalizations of Poisson brackets.

Given a Nambu bracket of order q and a (q − 1)-tuple of functions (f1, . . . ,fq−1), one defines the Hamiltonian vector field of (f1, . . . , fq−1) by the followingformula:

Xf1,...,fq−1 (g) = f1, . . . , fq−1, g. (6.3)

160 Chapter 6. Nambu Structures and Singular Foliations

It follows from the Leibniz rule that the vector field Xf1,...,fq−1 is well defined, andfrom the anti-symmetricity of the Nambu bracket that the functions f1, . . . , fq−1

are first integrals of Xf1,...,fq−1 .

Exercise 6.1.2. Show that, given a Nambu bracket of order q > 1 and a smoothfunction g, the following formula will give a Nambu bracket of order q − 1:

f1, . . . , fq−1g := f1, . . . , fq−1, g . (6.4)

Similarly to the case of Poisson manifolds, the Leibniz rule together with theanti-symmetricity means that Nambu brackets can be characterized by q-vectorfields: for each Nambu bracket . . . of order q there is a unique q-vector field Λ,called its Nambu tensor , such that

f1, . . . , fq = Λ(f1, . . . , fq) := 〈df1 ∧ · · · ∧ dfq, Λ〉 . (6.5)

Exercise 6.1.3. Show that the fundamental identity (6.2) is equivalent to the con-dition that every Hamiltonian vector field preserves the Nambu tensor:

LXf1 ,...,fqΛ = 0 . (6.6)

Exercise 6.1.4. Show that a q-vector field Λ on a manifold M is a Nambu tensorif and only if its restriction to the open subset M0 = x ∈ M | Λ(x) = 0 is aNambu tensor.

When q = 1, any vector field is a Nambu tensor. Nambu tensors of orderq = 2 are the same as Poisson tensors, as mentioned earlier. Nambu tensors oforder q ≥ 3 are characterized by the following decomposition theorem.

Theorem 6.1.5 ([140, 9, 274]). Suppose that q ≥ 3. Then a q-vector field Λ onan m-dimensional manifold M is a Nambu tensor if and only if for every pointz ∈M such that Λ(z) = 0, there is a local system of coordinates (x1, . . . , xm) in aneighborhood of z, in which Λ has the following form:

Λ =∂

∂x1∧ · · · ∧ ∂

∂xq. (6.7)

In such a local coordinate system, the Nambu bracket has the form

f1, . . . , fq =∂(f1, . . . , fq)∂(x1, . . . , xq)

. (6.8)

Proof. Since Λ(z) = 0, there is a Hamiltonian vector field X = Xf1,...,fq−1 suchthat X(z) = 0. Let fq be a smooth function defined in a neighborhood of z suchthat X(fq) = 1. For i = 1, . . . , q, we denote by Xi the Hamiltonian vector field of

Fi = (f1, . . . , fi−1, fi+1, . . . , fq) .

6.1. Nambu brackets and Nambu tensors 161

The vector fields Xi are linearly independent, because Xi(fi) = ±1 andXi(fj) = 0 ∀ i = j. Moreover, they commute pairwise. Indeed, we have LXi(dfj) =d(Xi(fj)) = 0 ∀ i, j, and LXiΛ = 0. In other words, Xi preserves df1, . . . ,dfq andΛ. So it also preserves anything which can be created from these entities:

[Xi, Xj ] = LXiXj = LXi〈df1 ∧ · · · ∧ dfj−1 ∧ dfj+1 ∧ · · · ∧ fq, Λ〉 = 0 .

Applying Frobenius Theorem 1.5.3, we get a local regular q-dimensional foliationgenerated by X1, . . . , Xq, and a local coordinate system

(x1, . . . , xm) := (f1, . . . , fq, y1, . . . , ym−q) (6.9)

such that y1, . . . , ym−q are first integrals of the foliation (i.e., Xi(yj) = 0 ∀i, j). Inthis coordinate system we have

x1, . . . , xq = 1 (6.10)

andx1, . . . , xi−1, xi+1, . . . , xq, yj = Xi(yj) = 0 ∀ i, j . (6.11)

To prove that Λ = ∂∂x1

∧ · · · ∧ ∂∂xq

, it remains to show that for any s ≥ 2 and anyindices i1 = · · · = iq−s, j1 = · · · = js we have

xi1 , . . . , xiq−s , yj1 , . . . , yjs = 0 .

Consider the case q − s ≥ 1. By rearranging the indices if necessary, we mayassume that q = i2, . . . , iq−2. Then using (6.10) and (6.11) repetitively togetherwith the Leibniz rule and the fundamental identity, we get:

xi1 , . . . , xiq−s , yj1 , . . . , yjs= xi1x1, . . . , xq, xi2 , . . . , xiq−s , yj1 , . . . , yjs

=12x1, . . . , xi1−1, x

2i1 , xi1+1, . . . , xq, xi2 , . . . , xiq−s , yj1 , . . . , yjs

=12x1, . . . , xi1−1, x

2i1 , xi1+1, . . . , xq−1, xq, xi2 , . . . , xiq−s , yj1 , . . . , yjs

= xi1x1, . . . , xq−1, xq, xi2 , . . . , xiq−s , yj1 , . . . , yjs= xi1x1, . . . , xq, xi2 , . . . , xiq−s , yj1 , . . . , yjs= xi11, xi2 , . . . , xiq−s , yj1 , . . . , yjs = 0 .

Similarly, when s = q we have

yj1 , . . . , yjq = yj1x1, . . . , xq, yj2 , . . . , yjq= yj1x1, x2, . . . , xq, yj2 , . . . , yjq .

The fundamental identity (6.2) allows us to decompose the last bracket intothe sum of brackets which contain at least one entry of type xi and one entry oftype yj . So the previous equalities imply that this bracket is also equal to 0.


Exercise 6.1.6. Find the place(s) in the above proof where we use the fact thatq ≥ 3.

Exercise 6.1.7. Show that if Λ is a Nambu structure of order q ≥ 3 and x1, . . . , xr

are r functions (r < q) such that idx1∧···∧dxrΛ(z) = 0 at a point z, then in aneighborhood of z these functions can be completed into a canonical coordinatesystem (x1, . . . , xm), i.e., a coordinate system in which Λ = ∂

∂x1∧ · · · ∧ ∂

∂xq.

A q-vector field Λ is called decomposable if it can be written as the wedgeproduct of q vector fields: Λ = X1∧· · · ∧Xq. According to Theorem 6.1.5, Nambutensors of order q ≥ 3 are locally decomposable near non-vanishing points. This isone of the differences between Nambu and Poisson structures: a Poisson tensor isnot decomposable at a point of rank ≥ 4.

Example 6.1.8. (Direct product). If Λi is a Nambu tensor of order qi on Mi (i =1, 2) then Λ1 ∧ Λ2 is a Nambu tensor of order q1 + q2 on M1 ×M2.

Remark 6.1.9. The q-vector field ∂∂x1

∧ · · · ∧ ∂∂xq

+ ∂∂y1

∧ · · · ∧ ∂∂yq

is not a Nambutensor when q ≥ 3 because it is not decomposable. So the direct sum of two Nambutensor is not a Nambu tensors in general, unlike the Poisson case.

Exercise 6.1.10. Show that if Λ is a Nambu tensor of order q ≥ 3 and g is asmooth function, then gΛ is a Nambu tensor. In particular, any m-vector field onan m-dimensional manifold is a Nambu tensor.

Exercise 6.1.11. Show that if Λ is a Nambu tensor then the Schouten bracket ofΛ with itself vanishes: [Λ, Λ] = 0. More generally, if Λ is a Nambu tensor of orderq and f1, . . . , fr are smooth functions, 0 ≤ r ≤ q − 1, then

[idf1∧···∧dfrΛ, Λ] = 0 . (6.12)

(Hint: use Exercise 6.1.7).

A consequence of Theorem 6.1.5 is that a nontrivial Nambu structure of orderq ≥ 3 is essentially just a singular q-dimensional foliation with a leaf-wise volumeform. Let us explain this statement:

Consider the open subset M0 = x ∈ M | Λ(x) = 0 of M consisting of theregular points of Λ. (Generically, M0 is dense in M .) Locally near a point x ∈M0

we have Λ = ∂∂x1

∧ · · · ∧ ∂∂xq

. As a consequence, the characteristic distribution CΛof Λ, which by definition is the distribution on M generated by the Hamiltonianvector fields of Λ, is a regular distribution, which near x is simply spanned by

∂∂x1

, . . . , ∂∂xq

. This distribution is clearly integrable and is the tangent distribu-tion of a regular q-dimensional foliation in M0. Locally, in a coordinate system(x1, . . . , xm) where Λ = ∂

∂x1∧ · · · ∧ ∂

∂xq, the leaves of this associated foliation is

given simply by the equations xq+1 = constant, . . . , xm = constant. Λ can berestricted to the leaves of this foliation. On each leaf, Λ becomes a non-vanishingmulti-vector field of top order, so it is dual to a volume form η on the leaf, i.e.,

6.1. Nambu brackets and Nambu tensors 163

〈η, Λ〉 = 1. Conversely, given a regular q-dimensional foliation and a leaf-wisevolume form, the q-vector field dual to the volume form will be a Nambu tensor.

When considered over M , the characteristic distribution CΛ of Λ is still inte-grable, and the associated singular foliation FΛ has two types of leaves: singularzero-dimensional leaves (i.e., points), which are the points of the set M\M0 =x ∈M | Λ(x) = 0, and regular q-dimensional leaves lying in M0.Remark 6.1.12. The discussion above shows that, although their definition pre-sents them as generalizations of general Poisson structures, it is more realistic toconsider Nambu structures as generalizations of Poisson structures of maximalrank 2.

At first sight, the class of singular foliations arising from Nambu structureslooks rather poor: they are allowed to have only two types of leaves, namely regularleaves and zero-dimensional leaves. This is however a wrong impression. We willsay that two singular foliationsF1 and F2 on a manifold M are essentially the same(or essentially coincide) if for almost every point x ∈ M we have TxF1 = TxF2.Then any singular foliation can be essentially given by a Nambu structure:

Proposition 6.1.13. Let F be a smooth singular foliation of dimension q (i.e.,dimTxF = q almost everywhere) on a smooth manifold M . Then F can be es-sentially generated by a Nambu tensor, i.e., there is a smooth Nambu tensor Λ oforder q on M such that FΛ essentially coincides with F .

Proof (sketch). We can construct q smooth vector fields X1, . . . , Xq on M , whichare tangent to F , and which are linearly independent almost everywhere. Now putΛ = X1 ∧ · · · ∧Xq.

A priori, the foliation FΛ in the above proposition may contain many moresingularities than F . There is a process, which we call saturation, for removing“unnecessary” singularities:

Let F1 and F2 be two singular foliations on a manifold M . We will say thatF2 contains F1 if TxF2 ⊃ TxF1 ∀ x ∈M .

Definition 6.1.14. The saturation of a singular foliation F is a singular foliation,denoted by Sat(F), which essentially coincides with F almost everywhere, andwhich contains any singular foliation which essentially coincides with F . WhenF = Sat(F) we will say that F is a saturated singular foliation.

The above definition does not say that the saturation of a singular foliationalways exists. What is obvious is that if Sat(F) exists then it is unique. If F is aregular foliation then it is clear that Sat(F) = F .

The process of saturation (when it works) makes a singular foliation“smoother”, leaving out some “details” from it. Sometimes these “details” arejust cumbersome and it’s best to forget about them. Sometimes they are whatmake a singular foliation interesting.

Example 6.1.15. Consider the symplectic foliation FΠ of the linear Poisson struc-ture Π = x ∂

∂x ∧ ∂∂y in R2. It has two regular two-dimensional leaves, and a line of


singular 0-dimensional leaves (the line x = 0). Sat(FΠ) consists of just one leaf,i.e., the whole plane.

Example 6.1.16. Consider the following (rather pathological) singular foliation onR2 = (x, y): the half-plane x < 0 is a two-dimensional leaf, and each point(x, y) with x ≥ 0 is a 0-dimensional leaf. This singular foliation does not admit asaturation.

Excluding pathological situations such as in the above example, then thesaturation of a singular foliation really exists. More precisely, we have:

Theorem 6.1.17. If F is a singular foliation of dimension q such that dimTxF = qalmost everywhere, then Sat(F) exists.

The proof of Theorem 6.1.17 is postponed to the next section, where we willshow another way of defining Sat(F).

Exercise 6.1.18. Show that, if a compact Lie group G acts on a manifold M ,then the associated singular foliation on M given by the orbits of G is saturated.More generally, any orbit-like singular foliation is saturated. (An orbit-like foliation[261] is a singular foliation which near each point is locally isomorphic to a singularfoliation given by the orbits of an action of a compact Lie group on a manifold).

Similarly to the case of vector fields, we will say that two Nambu structuresΛ1, Λ2 (not necessarily of the same order) on a manifold commute if their Schoutenbracket vanishes: [Λ1, Λ2] = 0. This leads to the notion of commuting singularfoliations , i.e., foliations which can be essentially generated by commuting Nambutensors. This seems to be an interesting geometric notion, which has been verylittle studied, to our knowledge. We will list here a few basic facts about commutingNambu structures.

Lemma 6.1.19. If a Nambu structure Λ1 of degree q1 on a manifold M commuteswith a Nambu structure Λ2 of degree q2, and Λ1 ∧ Λ2(m) = 0 at a point m, thenthere is a local system of coordinates (x1, . . . , xn) such that in a neighborhood ofm we have

Λ1 = ∂/∂x1 ∧ · · · ∧ ∂/∂xq1 , (6.13)Λ2 = ∂/∂xq1+1 ∧ · · · ∧ ∂/∂xq1+q2 . (6.14)

Lemma 6.1.20. If two Nambu structures Λ1, Λ2 commute, then their exterior prod-uct Λ1 ∧ Λ2 is again a Nambu structure.

Lemma 6.1.21. If two Nambu structures Λ1, Λ2 commute, and X is a vector fieldsuch that LXΛ1 = [X, Λ1] = 0 and X ∧ Λ1 = 0, then LX(Λ1 ∧ Λ2) = 0.

Lemma 6.1.22. Let Λ1, Λ2 be two commuting Nambu structures, and let L be a leafof the associated foliation of Λ1. If Λ1 ∧ Λ2(x0) = 0 for some point x0 ∈ L, thenΛ1 ∧ Λ2(x) = 0 for any point x ∈ L. If Λ1 ∧ Λ2(x0) = 0 for some point x0 ∈ L,then Λ1 ∧ Λ2(x) = 0 for any point x ∈ L.

The proofs of the above lemmas will be left as exercises.

6.2. Integrable differential forms 165

6.2 Integrable differential forms

Let Ω be a smooth volume form on a smooth m-dimensional manifold M . Recallthat the map Λ → iΛΩ is a linear isomorphism from the space Vq(M) of smoothq-vector fields on M to the space Ωm−q of smooth differential (m−q)-forms on M .The following proposition characterizes Nambu structures in terms of differentialforms.

Proposition 6.2.1. Suppose that Λ is a q-vector field, and q ≥ 3, Ω is a volumeform on an m-dimensional manifold M , m − q = p ≥ 0. Then Λ is a Nambutensor if and only if the p-form ω = iΛΩ satisfies the following conditions:

iAω ∧ ω = 0 ∀ (p− 1)-vector field A (6.15)

andiAω ∧ dω = 0 ∀ (p− 1)-vector field A. (6.16)

Proof. Suppose that Λ is a Nambu tensor, and let x be an arbitrary point of M .If Λ(x) = 0 then ω(x) = 0 and Relations (6.15) and (6.16) are obviously satisfiedat point x. Suppose now that Λ(x) = 0. Then there is a local coordinate system(x1, . . . , xm) near x in which we have Λ = ∂

∂x1∧ · · · ∧ ∂

∂xq. Write the volume form

as Ω = fdx1 ∧ · · · ∧ dxm, where f is a smooth function. Then we have

ω = fdxq+1 ∧ · · · ∧ dxm and dω = df ∧ dxq+1 ∧ · · · ∧ dxm .

These formulae imply (6.15) and (6.16) in a neighborhood of x.Conversely, suppose that ω satisfies (6.15) and (6.16). We will assume that

p > 0 (the case p = 0 is obvious: any m-vector field is a Nambu tensor, see Theorem6.1.5 and Exercise 6.1.10). Let x be a point such that Λ(x) = 0. Then ω(x) = 0.Condition (6.15) together with ω(x) = 0 means that ω is locally decomposable in aneighborhood of x. In other words, there are differential 1-forms α1, . . . , αp, whichare linearly independent in a neighborhood of x, such that

ω = α1 ∧ · · · ∧ αp . (6.17)

(See, e.g., [43].) Choose p vector fields X1, . . . , Xp in a neighborhood of x, whichsatisfy 〈Xi, αj〉 = δij (where δij is the Kronecker symbol: δii = 1 and δij = 0 ifi = j). Take Aj = X1 ∧ · · · ∧ Xj ∧ · · · ∧Xp. (The hat means that the term Xj ismissing in the product.) We have iAj ω = ±αj for every j = 1, . . . , p. Condition(6.16) gives αj ∧ dω = 0. As we have

dω =p∑

k=1

α1 ∧ · · · ∧ αk−1 ∧ dαk ∧ αk+1 ∧ · · · ∧ αp ,

we obtain thatdαj ∧ α1 ∧ · · · ∧ αp = 0 ∀ j = 1, . . . , p. (6.18)


The above condition (6.18) is known as Frobenius integrability condition for a p-tuple of 1-forms (α1, . . . , αp). This condition is equivalent to the condition thatthe annulator distribution of α1, . . . , αp, i.e., the distribution consisting of tangentvectors whose pairings with α1, . . . , αp vanish, is a (regular) involutive distributionof rank q = m−p. (Exercise: show this equivalence.) Frobenius Theorem 1.5.3 thenimplies that there is a local system of coordinates (x1, . . . , xm) in a neighborhoodof x (such that x1, . . . , xp are first integrals for the associated foliation of the aboveannulator distribution), such that

αi =p∑

j=1

aijdxj ∀ i = 1, . . . , p , (6.19)

where aij are some smooth functions in a neighborhood of x. The above expressionfor αi under the integrability condition (6.18) is also known as the Frobeniustheorem.

So locally we have ω = α1 ∧ · · · ∧ αp = gdx1 ∧ · · · ∧ dxp for some smoothfunction g. If the local expression of Ω is fdx1 ∧ · · · ∧ dxm (f = 0) then

Λ = ± g

f

∂

∂xp+1∧ · · · ∧ ∂

∂xm.

Now we can use Theorem 6.1.5 (together with Exercises 6.1.4 and 6.1.10) to deducethat Λ is a Nambu tensor. Exercise 6.2.2. Show that Proposition 6.2.1 is also true in the case q = 2 underthe additional assumption that rankΛ(x) ≤ 2 ∀ x ∈ M.

Definition 6.2.3. An integrable differential form of order p is a differential p-formω which satisfies Conditions (6.15) and (6.16).

Exercise 6.2.4. Show that if ω is an integrable p-form and f is a smooth function,then fω is also an integrable p-form, and dω is an integrable (p + 1)-form.

In particular, if ω is a 1-form, then Condition (6.15) is empty, and Condition(6.16) is written as

ω ∧ dω = 0 (6.20)

and is known as the integrability condition for 1-forms.Proposition 6.2.1 means that Nambu tensors are dual to integrable forms

in a natural way. Hence each integrable differential p-form (which is nontrivialalmost everywhere) defines a singular foliation of codimension p, and that’s whyit is called integrable. In the regular region M0 = x ∈ M | ω(x) = 0, theannulator distribution of ω, generated by smooth vector fields X which annulateω, i.e., iXω = 0, is a regular involutive distribution of codimension p, so it is thetangent distribution to a regular foliation of codimension p. We can extend thisfoliation in a stupid way to the singular set of ω, by declaring that each singularpoint of ω is a zero-dimensional leaf. This is exactly what we get by considering a

6.2. Integrable differential forms 167

dual Nambu structure Λ of ω (i.e., iΛΩ = ω where Ω is a volume form). We willshow how to define the saturation of this singular foliation.

It is easy to see that if ω is integrable then its annulator distribution, definedas above, is still involutive when considered over the whole manifold M . Unfortu-nately, as we mentioned in Chapter 1, for singular foliations, involutivity does notguarantee integrability, though under some additional assumptions it does.

Example 6.2.5. Consider a 1-form α defined on R3 = (x1, x2, x3) by the followingformula: α = exp(−1

x1)dx2 when x1 > 0, α = exp( 1

x1)dx3 when x1 < 0, and α = 0

when x1 = 0. Then α is a smooth integrable 1-form. Its annulator distribution isinvolutive but not integrable.

To avoid pathologies such as in the above example, we will introduce anotherdistribution:

Definition 6.2.6. Let ω be a differential p-form on a manifold M . Then the maximalinvariant distribution of ω, denoted by Dω, is the singular distribution on M whichis generated by smooth vector fields X which satisfy the following properties:

a) X annulates ω: iXω = 0.b) X preserves ω projectively in the following weak sense: denote by φt

X the localflow of X , then for any point x ∈M and τ ∈ R such that φτ

X(x) is well definedand ω(x) = 0, there is a number f(τ, x) such that ((φτ

X)∗ω)(x) = f(τ, x)ω(x).

Exercise 6.2.7. Show that, for an arbitrary given smooth p-form ω, the set ofvector fields which satisfy Conditions a) and b) in the above definition is a C∞(M)-module.

Exercise 6.2.8. Show that, if f is a function such that f(x) = 0 almost everywhere,then Dfω = Dω for any differential form ω.

Exercise 6.2.9. Show that if ω is an integrable p-form and x is a regular point,i.e., ω(x) = 0, then (Dω)x = (Annω)x and dim(Dω)x = m − p where m is thedimension of the manifold.

Proposition 6.2.10. If ω is an arbitrary smooth p-form which is non-vanishingalmost everywhere, then its maximal invariant distribution Dω is an integrabledistribution.

Proof. Let X be an arbitrary vector field which preserves ω projectively and suchthat iXω = 0. We have to show that X preserves Dω. Then we can apply Theorem1.5.1 to conclude that Dω is integrable. Let x ∈ M be an arbitrary point, Yx be anarbitrary vector in (Dω)x. Suppose that ϕτ

X(x) is well defined for some τ > 0, whereϕt

X denotes the local flow of X . We have to show that (ϕτX)∗(Yx) ∈ (Dω)ϕτ

X(x).Let Y be a vector field which annulates ω and preserves ω projectively, and suchthat the value of Y at x is Yx. We may assume that Y vanishes outside a smallneighborhood of x (see Exercise 6.2.8), so that Y τ := (ϕτ

X)∗Y is well defined andis supported in a small neighborhood of ϕτ

X(x). Since ϕτX preserves ω projectively,

and since iY ω = 0, we obtain that i(Y τ )ω = 0. (We use the fact that ω = 0


almost everywhere to show that this equality holds almost everywhere, but thenby continuity it holds everywhere.) Similarly, one can show that Y τ preserves ωprojectively, since both X and Y preserves ω projectively. Hence, by definition,Y τ lies in the family of vector fields which generate Dω. In particular, we have(ϕτ

X)∗(Yx) = Y τϕτ

X(x) ∈ (Dω)ϕτX(x).

Definition 6.2.11. If ω is an integrable differential form, then the associated singu-lar foliation Fω of its maximal invariant distribution Dω will be called the maximalinvariant foliation of ω, or simply the associated foliation of ω.

Theorem 6.1.17 is now a direct consequence of Proposition 6.1.13 and thefollowing result:

Proposition 6.2.12. If ω is an integrable differential form which is non-vanishingalmost everywhere, then its maximal invariant foliation Fω is saturated (i.e., it isthe saturation of itself ).

Proof. Let F be any singular foliation which essentially coincides with Fω. Wemust show that F ⊂ Fω. Let X be any smooth vector field tangent to F . We willshow that X satisfies the conditions of Definition 6.2.6. Note that X is tangentto Fω at least almost everywhere, so by definition of Fω we have iXω = 0 almosteverywhere, which implies by continuity that iXω = 0 everywhere. Let us verifythat X preserves ω projectively in the weak sense. Assume that x, y are twopoints such that y = φτ

X(x) for some τ and ω(x), ω(y) = 0. We have to showthat ((φτ

X)∗ω)(x) = cω(x) for some number c. By continuity, it is enough to provethis in the case x is a generic point. In particular, we may assume that x andy are two regular points of Fω, and F coincides with Fω at these points. Butthen φτ

∗(TxFω) = TyFω. Since ω is integrable, ω(x) (resp. ω(y)) is determineduniquely by TxFω (resp. TyFω) up to a multiplicative constant, and we also havethat (φτ

X)∗(ω(y)) is equal to ω(x) up to a multiplicative constant.

6.3 Frobenius with singularities

Let (α1, . . . , αp) be a p-tuple of germs at 0 of holomorphic 1-forms in (Cn, 0). Wewill say that α = (α1, . . . , αp) is completely integrable (resp. formally completelyintegrable) if there are germs of holomorphic (resp. formal) functions fi, gij (1 ≤i, j ≤ p) such that det(gij(0)) = 0 and αi =

∑pj=1 gijdfj ∀ i.

Clearly, the Frobenius integrability condition

dαj ∧ α1 ∧ · · · ∧ αp = 0 ∀ j = 1, . . . , p (6.21)

is a necessary condition for complete integrability. The p-form ω = α1 ∧ · · · ∧ αp,under the Frobenius condition, is an integrable form. If, moreover, ω(0) = 0, then(α1, . . . , αp) is also completely integrable: there is a local holomorphic coordinatesystem (x1, . . . , xn) in which ω has the form ω = hdx1∧· · ·∧dxp. Since αi∧ω = 0,

6.3. Frobenius with singularities 169

we also have αi ∧ dx1 ∧ · · · ∧ dxp = 0, which implies that αi can be written asαi =

∑pj=1 gijdxj .

Note that, if αi =∑p

j=1 gijdfj ∀ i, and df1∧· · ·∧dfp ≡ 0, det(gij) ≡ 0, then

ω = α1 ∧ · · · ∧ αp = det(gij) df1 ∧ · · · ∧ dfp ≡ 0,

and the functions f1, . . . , fp are functionally independent first integrals of theassociated foliation of ω, which is of codimension p. In other words, we havea complete set of first integrals (f1, . . . , fp), and the associated foliation of ωessentially coincides with the singular fibration given by the level sets of the map(f1, . . . , fp) : (Cn, 0)→ Cp.

The Frobenius with singularity problem may be formulated as follows: underwhich additional conditions a p-tuple (α1, . . . , αp) of germs at 0 of holomorphic 1-forms in (Cn, 0), which satisfies the Frobenius condition, but with α1∧· · ·∧αp(0) =0, is completely integrable. More generally, under which additional conditions agerm of singular foliation essentially coincides with a germ of singular fibration,or admits at least a nontrivial first integral.

Several conjectures concerning the Frobenius with singularities problem (forcodimension 1 singular foliations) were formulated by Rene Thom in the beginningof the 1970s, and later proved or disproved by Malgrange [233, 234], Mattei–Moussu [242], Cerveau–Mattei [70] and other people (see, e.g., [269]). We willpresent here the result of Malgrange, which will be used later in this chapter, inparticular in the problem of linearization of Nambu structures.

Theorem 6.3.1 (Malgrange [233, 234]). Let α = (α1, . . . , αp) be a p-tuple of germsat 0 of holomorphic 1-forms in (Cn, 0), which satisfies the Frobenius condition(6.21). Denote by S = x ∈ Cn | α1 ∧ · · · ∧ αp(x) = 0 the singular set of α.Suppose that one of the following two conditions is satisfied:

i) codim S ≥ 3;ii) codim S = 2 and α is formally completely integrable.

Then α is completely integrable.

We will give here a sketch of the proof of the above theorem for the casep = 1, following Malgrange [233].

We will first show that α = α1 is formally completely integrable, i.e., it canbe written as α = fdg where f and g are formal functions, and then show thatf, g can be chosen convergent. A trick is to add one dimension to the space, andfind a (formal) integrable 1-form β in (Cn+1, 0) of the type

β = dt +∞∑

k=0

tk

k!ωk, (6.22)

where t is the additional variable, ωk are germs of holomorphic 1-forms in (Cn, 0),and ω0 = α. Since β is integrable and regular at 0, by the classical theorem of


Frobenius we have β = FdG where F and G are functions in (Cn+1, 0), whichimplies that α = fdg where f and g are the restrictions of F and G to t = 0.

The integrability condition dβ ∧ β = 0 is equivalent to an infinite sequenceof equations in ωk:

α ∧ ωk+1 = ψk, k = 0, 1, 2, . . . . (6.23)

where ψk = dωk −∑k

l=1

(kl

)ωl ∧ ωk−l+1. (6.24)

The above equations can be solved inductively. For each k, assuming that ωk

is already found (recall that ω0 = α), it follows from the equation dα ∧ α that wehave

ψk ∧ α = 0. (6.25)

If the codimension of the singular set S = x ∈ (Cn, 0), α(x) = 0 is at least 3,then by de Rham’s division theorem (see Appendix A.2), ψk is dividable by α, i.e.,there exists ωk+1 which satisfies Equation (6.23). If codimS = 2 and α is formallycompletely integrable, then one verifies directly that ψk is also dividable by α.The formal part of the proof is finished.

The next step is to show that∑∞

k=0tk

k! ωk converges. Fix a local coordinatesystem (x1, . . . , xn) in (Cn, 0), and denote by O = Cx1, . . . , xn the ring of germsof holomorphic functions in (Cn, 0). For each n-tuple ρ = (ρ1, . . . , ρn) of smallpositive numbers (called a polyradius) and a function f ∈ O, f =

∑I aIx

I whereI = (I1, . . . , In) means a multi-index and xI means the monomial

∏nj=1 x

Ij

j , put

|f |ρ =∑

I

|aI |ρI . (6.26)

Note that |fg| ≤ |f |ρ.|g|ρ ∀ f, g ∈ O. We will extend this pseudo-norm toON , N ∈ N, by the formula |F |ρ = sup1≤i≤N |Fi|ρ, F = (F1, . . . , Fn) ∈ ON . Inparticular, for a 1-form ω =

∑fidxi we will write |ω|ρ =

∑i |fi|ρ, and similarly

for 2-forms. In order to make estimates on |ωk|ρ, we will use the following twolemmas:

Lemma 6.3.2 ([233]). For any O-linear map Φ : ON → OM , where N, M ∈ N,there is a C-linear map Ψ : OM → ON such that ΦΨΦ = Φ (such a Ψ is called asplitting of Φ), with the following additional property: for any positive polyradiusρ′ there is a polyradius ρ ≤ ρ′ (i.e., ρi ≤ ρ′i ∀ i) and a constant c = c(ρ) > 0(which may depend on ρ) such that

|Ψ(F )|τρ ≤ c|F |τρ ∀ 12≤ τ ≤ 1, F ∈ OM . (6.27)

Lemma 6.3.3. For any polyradius ρ there is a positive constant c1 = c1(ρ) suchthat for any 1/2 < s < τ < 1 and any germ of holomorphic 1-form ω on (Cn, 0)we have

|dω|sρ ≤c1

τ − s|ω|τρ. (6.28)

6.4. Linear Nambu structures 171

Lemma 6.3.3 is elementary, while Lemma 6.3.2 is a fine version of the “privi-leged neighborhoods theorem” in complex analysis which goes back to Henri Car-tan. See [233] for a nice proof of Lemma 6.3.2. Applying the above lemmas, weobtain from Equation (6.23) the following inequality:

|ωk+1|sρ ≤c′

τ − s|ωk|τρ + c

∑1≤l≤k

(kl

)|ωl|τρ.|ωk−l+1|τρ

∀ k = 0, 1, 2, . . . ,12≤ s < τ ≤ 1, (6.29)

where ρ is a multiradius such that |α|ρ < ∞ and which satisfies Lemma 6.3.3 withrespect to the linear map ω → α ∧ ω; c and c′ are some positive constants (ρ, cand c′ don’t depend on k).

Define a sequence of positive numbers vk by the following recursive formula:

v0 = |α|ρ, vk+1 = c′evk + c

k∑l=1

vlvk+1−l (e = exp(1)). (6.30)

Then the series F (t) =∑

vktk converges because it is a solution of the equationF (t) = c′ev0t + c′etF (t) + cF (t)2. By recurrence, one can verify directly that itfollows from Equation (6.30) and Inequality (6.29) that we have

|ωk|sρ ≤k!vk

(1− s)k∀ k ∈ N, 1/2 ≤ s < 1. (6.31)

This last inequality, together with the convergence of∑

vktk, implies that∑

tk

k! ωk

converges. The case p > 1 of Malgrange’s Theorem 6.3.1 is similar to the case p = 1,

though technically much more involved, especially the part about convergence,see [234].Remark 6.3.4. The method, used in the above proof, of dividing repetitively by a 1-form, is called the Godbillon–Vey algorithm, because it was first used by Godbillonand Vey in the definition of a characteristic class (the Godbillon–Vey class) ofcodimension 1 foliations [149].

6.4 Linear Nambu structures

A Nambu tensor Λ =∑

i1<···<iqΛi1...iq

∂∂xi1

∧ · · · ∧ ∂∂xi1

in some linear coordinatesystem (x1, . . . , xm) is called a linear Nambu structure, if its coefficients Λi1...iq arelinear functions. More generally, if Λi1...iq are homogeneous functions of a givendegree then we say that Λ is a homogeneous Nambu tensor of that degree. Linearand homogeneous differential forms are defined in an obvious similar way. It is


clear that if Λ is a linear Nambu tensor of order q = 2 (if q = 2, i.e., Λ is Poisson,then we assume in addition that rankΛ ≤ 2), and Ω is a constant volume form,then iΛΩ is a linear integrable differential form, and vice versa.

In this section we will classify linear integrable differential forms and Nambustructures. We first state the result for differential forms:

Theorem 6.4.1 ([119, 245]). Let ω be a linear integrable differential p-form onan m-dimensional linear space V (over R or C – when considered over C, ω isholomorphic). Then there exists a linear coordinate system (x1, . . . , xm) such thatω belongs to (at least ) one of the following two types:

Type I: ω = dx1 ∧ · · · ∧ dxp−1 ∧ α, where α is an exact 1-form of the typeα = d(

∑p+rj=p ±x2

j/2 +∑s

i=1 xixp+r+i), with −1 ≤ r ≤ q = m − p,0 ≤ s ≤ q − r.

Type II: ω =∑p+1

i=1 aidx1∧· · ·∧dxi−1∧dxi+1∧· · ·∧dxp+1 with ai =∑p+1

j=1 ajixj,

where aij are constant. The matrix (aj

i ) can be chosen to be in Jordanform.

Proof. We will assume that p > 0 (the case p = 0 is obvious and belongs to TypeII). Put ω =

∑nj=1 xjωj where ωj are constant p-forms. Then ω = ωj at points

(x1 = 0, . . . , xj = ε, . . . , xn = 0). At any point ω is either decomposable (i.e., awedge product of covectors) or zero, so is ωj since it is constant. Denote by Ej

the span of ωj , i.e.,

Ej = Span(ωj)def= SpaniAωj , A is a (p− 1)-vector

= Annulatorx ∈ V, ixωj = 0 ⊂ V ∗.

Then dimEj = p if ωj = 0, because of decomposability. We have:

Lemma 6.4.2. If ωi = 0 and ωj = 0 for some indices i and j, then dim(Ei ∩Ej) ≥p− 1.

Proof of Lemma 6.4.2. Putting xk = 0 for every k = i, j, we obtain that xiωi +xjωj = ω is decomposable or null for any xi, xj . In particular, ωi + ωj is decom-posable. If dim(Ei ∩ Ej) = d < p, then there is a basis (e1, . . . , ed, f1, . . . , fp−d,g1, . . . , gp−d) of Ei + Ej such that ωi = e1 ∧ · · · ∧ ed ∧ f1 ∧ · · · ∧ fp−d, ωj =e1 ∧ · · · ∧ ed ∧ g1 ∧ · · · ∧ gp−d and

ωi + ωj = e1 ∧ · · · ∧ ed ∧ [f1 ∧ · · · ∧ fp−d + g1 ∧ · · · ∧ gp−d].

It follows easily that if p−d ≥ 2, then Span(ωi+ωj) = Ei+Ej , dimSpan(ωi+ωj) >p and ωi + ωj is not decomposable.

Return now to Theorem 6.4.1. We can assume that E1, . . . , Eh = 0 andEh+1, . . . , En = 0 for some number h. Put E = E1 ∩E2 ∩ · · · ∩Eh. Then there aretwo alternative cases: dim E ≥ p− 1 and dimE < p− 1.


Case I. dimE ≥ p − 1. Then denoting by (x1, . . . , xp−1) a set of p − 1 linearlyindependent covectors contained in E, and which are considered as linear functionson V , we have

ωi = dx1 ∧ dx2 ∧ · · · ∧ dxp−1 ∧ αi ∀ i = 1, . . . , h

for some constant 1-forms αi, and hence

ω = dx1 ∧ dx2 ∧ · · · ∧ dxp−1 ∧ α (6.32)

where α =∑

xiαi is a linear 1-form.

Case II. In this case, without loss of generality, we can assume that dim(E1 ∩E2 ∩ E3) < p − 1. Then Lemma 6.4.2 implies that dim(E1 ∩ E2 ∩ E3) = p − 2.For an arbitrary index i, 3 < i ≤ h, put F1 = E1 ∩ Ei, F2 = E2 ∩ Ei, F3 =E3 ∩ Ei. Recall that dimF1, dimF2, dimF3 ≥ p − 1 according to Lemma 6.4.2,but dim(F1 ∩ F2 ∩ F3) = dim(E1 ∩ E2 ∩ E3 ∩ Ei) < p− 1, hence we cannot haveF1 = F2 = F3. Thus we can assume that F1 = F2. Then either F1 and F2 are twodifferent hyperplanes in Ei, or one of them coincides with Ei. In any case we haveEi = F1 + F2 ⊂ E1 + E2 + E3. It follows that

m∑1

Ei =h∑1

Ei = E1 + E2 + E3 .

On the other hand, we have dim(E1 + E2 + E3) = dimE1 + dimE2 + dimE3 −dim(E1 ∩E2)− dim(E1 ∩E3)− dim(E2 ∩E3) + dim(E1 + E2 + E3) = 3p− 3(p−1) + (p− 2) = p + 1. Thus

dim(E1 + E2 + · · ·+ Em) = p + 1 .

It follows that there is a system of linear coordinates (x1, . . . , xm) on V such that(x1, . . . , xp+1) span E1 + · · ·+ Em and therefore

ωi =p+1∑j=1

γji dx1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dxp+1 .

Hence we have

ω =∑

xiωi =p+1∑j=1

ajdx1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ xp+1 , (6.33)

where aj are linear functions on V .To finish the proof of Theorem 6.4.1, we still need to normalize further the

obtained forms (6.32) and (6.33).

Return now to Case I and suppose that ω = dx1 ∧ · · · ∧ dxp−1 ∧ α whereα =

∑αjdxj with αj being linear functions. We can put αj = 0 for j = 1, . . . , p−1


since it will not affect ω. Then we have α =∑

j≥p,i=1,...,m αijxidxj . Equation

(6.16) implies that α∧dx1 ∧ · · · ∧dxp−1 ∧dα = 0. If we consider (x1, . . . , xp−1) asparameters and denote by d′ the exterior derivation with respect to the variables(xp, . . . , xm), then the last equation means α∧d′α = 0. That is, α can be consideredas an integrable 1-form in the space of variables (xp, . . . , xm), parametrized by(x1, . . . , xp−1). We will distinguish two subcases: d′α = 0 and d′α = 0.

Subcase a). Suppose that d′α = 0. Then according to Poincare’s Lemma we haveαj =

∑p−1i=1 αi

jxi + ∂/∂xjq(2), where q(2) is a quadratic function in the variables

(xp, . . . , xm). By a linear change of coordinates on (xp, . . . , xm), we have q(2) =∑p+rj=p ±x2

j/2, for some number r ≥ −1, and accordingly

α =p+r∑j=p

(±xj +

p−1∑i=1

αijxi

)dxj +

∑i=1,...,p−1

j=p+r+1,...,m

αijxidxj .

By a linear change of coordinates on (x1, . . . , xp−1) on the one hand, and on(xp+r+1, . . . , xm) on the other hand, we can normalize the second part of theabove expression to obtain

α =p+r∑j=p

(±xj +

p−1∑i=1

αijxi

)+

s∑j=1

xjdxp+r+j

for some number s (0 ≤ s ≤ min(p− 1, m− p− r)).Replacing xj (j = p, . . . , p + r) by new xj = xj ∓ αi

jxi we get

ω = dx1 ∧ · · · ∧ dxp−1 ∧ α ,

where

α = d( p+r∑

j=p

±x2j/2 +

s∑i=1

xixp+r+i

)(with −1 ≤ r ≤ q = m − p, 0 ≤ s ≤ q − r). These are the linear integrable formsof Type I in Theorem 6.4.1.

Subcase b). Suppose that d′α = 0. Then since d′α is of constant coefficients, wecan change the coordinates (xp, . . . , xn) linearly so that d′α = dxp ∧dxp+1 + · · ·+dxp+2r ∧ dxp+2r+1 in these new coordinates, for some r ≥ 0.

If r ≥ 1, then considering the coefficients of the term dxp ∧ dxp+1 ∧ dxi (i >p + 1), dxp ∧ dxp+2 ∧ dxp+3 and dxp+1 ∧ dxp+2 ∧ dxp+3 in 0 = α ∧ d′α, we obtainthat all the coefficients of α are zero, i.e., α = 0, which is absurd. Thus d′α =dxp∧dxp+1, and the condition α∧d′α = 0 implies α = α1dxp+α2dxp+1 with linearfunctions α1 and α2 depending only on x1, . . . , xp−1, xp, xp+1. In this Subcase b),ω = dx1 ∧ · · · ∧ dxp−1 ∧ α also has the form (6.33), as in Case II.


Suppose now that ω has the form (6.33), as in Case II or Subcase b) of Case 1:

ω =∑

xiωi =p+1∑j=1

ajdx1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dxp+1.

There are also two subcases:

a) ∂aj/∂xi = 0 for j = 1, . . . , p + 1, i = p + 2, . . . , m. In other words,

ω =p+1∑

i,j=1

aijxidx1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dxp+1

with constant coefficients aij .

To see that (aij) can be put in Jordan form, notice that the linear Nambu

tensor Λ dual to ω (i.e., ω = iΛΩ where Ω is a constant volume form) is, up tomultiplication by a constant:

Λ =( p+1∑

i,j=1

±aijxi∂/∂xj

)∧ ∂/∂xp+2 ∧ · · · ∧ ∂/∂xm .

The first term in Λ is a linear vector field, which is uniquely defined by a lineartransformation Rp+1 → Rp+1 given by the matrix (ai

j), so this matrix can be putin Jordan form.

b) There is j ≤ p + 1 and i ≥ p + 2 such that ∂aj/∂xi = 0. We can assume that∂a1/∂xm = 0. Putting A = ∂/∂x3 ∧ · · · ∧ ∂/∂xp+1 in 0 = iAω ∧ dω, we obtain

0 = (a1dx2 + a2dx1)∧∑

i=1,...,m

j=1,...,p+1

dxi ∧∂aj

∂xidx1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dxp+1.

Considering the coefficient of dx1 ∧ · · · ∧ dxp+1 ∧ dxm in the above equation, weget

a1∂a2/∂xm − a2∂a1/∂xn = 0 .

Since ∂a1/∂xm = 0, it follows that a2 is linearly dependent on a1. Similarly, aj

is linearly dependent on a1 for any j = 1, . . . , p + 1. Thus ω = a1ω1 where ω1 isdecomposable and constant: ω1 = dx1 ∧ · · · ∧ dxp in some new linear system ofcoordinates. If a1 is linearly independent of (x1, . . . , xp), then we can also assumethat xp+1 = a1 in this new coordinate system. Thus we get back to Subcase a) ofCase II by a linear change of coordinates.

Remark 6.4.3. Theorem 6.4.1 first appeared in [119], and later in an independentwork of de Medeiros (see Theorem A of [245]). The proof given in [245] is verynice, though less detailed than the one given here.


Rewriting Theorem 6.4.1 in terms of Nambu tensors, we have:

Theorem 6.4.4 ([119]). Every linear Nambu tensor Λ of order q ≥ 3, or of orderq = 2 but under the additional assumption that rankΛ ≤ 2 if q = 2, on an m-dimensional linear space, belongs to (at least ) one of the following two types:

Type I: Λ =∑r+1

j=1 ±xj∂/∂x1 ∧ · · · ∧ ∂/∂xj−1 ∧ ∂/∂xj+1 ∧ · · · ∧ ∂/∂xq+1 +∑sj=1±xq+1+j∂/∂x1 ∧ · · · ∧ ∂/∂xr+j ∧ ∂/∂xr+j+2 ∧ ∂/∂xq+1

(with −1 ≤ r ≤ q, 0 ≤ s ≤ min(p− 1, q − r)).Type II: Λ = ∂/∂x1 ∧ · · · ∧ ∂/∂xq−1 ∧ (

∑mi,j=q bi

jxi∂/∂xj).

Remark 6.4.5. Strictly speaking, since linear Nambu tensors are determined bytheir dual linear integrable forms only up to a non-zero multiplicative constant,Theorem 6.4.1 implies Theorem 6.4.4 only up to a multiplicative constant. Butthat constant in the expression of Λ can be killed by a homothety when q = 1.Remark 6.4.6. There is an obvious and very interesting duality between Type Iand Type II in the above theorems: the expression for Type I linear integrabledifferential forms looks very similar to the expression for Type II linear Nambustructures, and vice versa.Remark 6.4.7. The case q = 2 of Theorem 6.4.4 gives us a classification of linearPoisson structures of rank ≤ 2. They correspond to Lie algebras whose coadjointorbits are at most two-dimensional. So we recover a classification, obtained byArnal, Cahen and Ludwig [12], of these Lie algebras: Type II tensors give semi-direct products of Kn−1 with K (K = R or C), where K acts on Kn−1 linearly bythe matrix (bi

j). Type I tensors give, according to values of r and s, the followingcases:

1) ±x1∂

∂x2∧ ∂

∂x3± x2

∂

∂x3∧ ∂

∂x1± x3

∂

∂x1∧ ∂

∂x2,

2) ±x1∂

∂x2∧ ∂

∂x3± x2

∂

∂x3∧ ∂

∂x1+ ε1x4

∂

∂x1∧ ∂

∂x2,

3) ±x1∂

∂x2∧ ∂

∂x3+ ε1x4

∂

∂x3∧ ∂

∂x1+ ε2x5

∂

∂x1∧ ∂

∂x2,

4) ε1x4∂

∂x2∧ ∂

∂x3+ ε2x5

∂

∂x3∧ ∂

∂x1+ ε3x6

∂

∂x1∧ ∂

∂x2,

where εi take values 0 or 1. These Type I tensors give us direct products of someLie algebras of dimension ≤ 6 with Abelian Lie algebras.

Theorems 6.4.1 and 6.4.4 give us a clear picture about the singular foliationsassociated to linear Nambu structures and integrable forms. A Type I foliation hasp = m − q first integrals, namely x1, . . . , xp−1 and

∑p+rj=p ±x2

j +∑s

j=1 xjxp+r+j .Hence a Type 1 foliation looks like a cabbage pile. A Type II foliation is a Carteseanproduct of a one-dimensional foliation given by a linear vector field in a linear spacewith (a 1-leaf foliation on) another linear space. Hence a Type II foliation lookslike a book . So a cabbage pile is dual to a book (?!).


It is natural to call singular foliations given by linear Nambu structures lin-ear foliations. There is, however, another kind of linear foliations: those given bylinear vector fields. To distinguish these two different cases, we make the followingdefinition:

Definition 6.4.8. A Nambu-linear foliation is a singular foliation Fλ associated toa linear Nambu tensor Λ of order q = 3, or of order q = 2 but with the additionalcondition that rankΛ ≤ 2. A Lie-linear foliation is a singular foliation generatedby a Lie algebra of linear vector fields on a linear space.

Exercise 6.4.9. Show that any Nambu-linear foliation is a Lie-linear foliation, butthe inverse is not true.

An advantage of Nambu tensors and integrable forms is that we can usethem to define homogeneous foliations on a linear space: a homogeneous foliationis a singular foliation which can be given, up to saturation, by a homogeneousintegrable differential form. Note that we can’t use Lie algebras of homogeneousvector fields to define homogeneous foliations: a Lie bracket of two vector fields ofdegree k will be a vector field of degree 2k − 1, which is greater than k if k > 1.

Exercise 6.4.10. Show that a Lie-linear foliation is a homogeneous foliation.

The problem of classification of homogeneous foliations (or homogeneousNambu structures) is, to our knowledge, an interesting and largely open problem.In Section 6.8 we will consider the case of quadratic foliations of codimension 1.

Remark 6.4.11. Similarly to the fact that linear Poisson structures correspond toLie algebras, linear Nambu structures correspond to the so-called n-ary Lie alge-bras, also known as Filippov algebras because they were first studied by Filippov[134]. By definition, an n-ary Lie algebra structure of order n on a vector space Vis a skew-symmetric n-linear mapping [. . . ] : V × · · · × V → V which satisfies therelation

[u1, . . . , un−1, [v1, . . . , vn]] =n∑

i=1

[v1, . . . , vi−1, [u1, . . . , un−1, vi], vi+1, . . . , vn]

(6.34)∀ ui, vj ∈ V . A warning though: when n ≥ 3, not every n-ary Lie algebra corre-sponds to a linear Nambu structure. For example, consider two nontrivial n-aryalgebras of order n ≥ 3 on two vector spaces V1, V2. Then the formula

[(a1, b1), . . . , (an, bn)] := ([a1, . . . , an]1, [b1, . . . , bn]2)

defines an n-ary Lie algebra on the direct sum V1 ⊕ V2. However, this n-ary Liealgebra structure on V1⊕V2 can never correspond to a linear Nambu structure onV ∗

1 ⊕V ∗2 , because the corresponding linear n-vector field on V ∗

1 ⊕V ∗2 is not locally

decomposable.


Exercise 6.4.12. Show that any n-ary algebra of dimension 3 or 4 comes from alinear Nambu structure. Find an n-ary algebra of dimension 5 and degree 3 whichdoes not come from a linear Nambu structure.

Remark 6.4.13. Similarly to the case of Poisson structures, one can talk aboutNambu–Lie structures (= multiplicative Nambu structures) on Lie groups, andview linear Nambu structures as a special case (when the Lie algebra is freeAbelian). See, e.g., [152, 334].

6.5 Kupka’s phenomenon

The so-called Kupka’s phenomenon, first observed by Kupka [204], is the following:if ω is an integrable 1-form such that dω(z) = 0 at some point z, then there is alocal coordinate system (x1, . . . , xm) in a neighborhood of z such that ω dependsonly on the first two coordinates, i.e., we have

ω = a(x1, x2)dx1 + b(x1, x2)dx2 . (6.35)

This phenomenon is important because it reduces the study of singularities ofintegrable 1-forms ω near points where dω = 0 to the study of 1-forms in dimension2, and implies the local stability of integrable 1-forms under certain conditions[204]. (Singularities of 1-forms and vector fields in dimension 2 can already behighly complicated).

In this section, we will explain Kupka’s phenomenon in a more general setting,and translate it into the language of Nambu tensors.

Recall from Section 2.6 that, given a volume form Ω, the curl operator DΩ

is defined by DΩ = Ω d Ω, where Ω is the map Λ → iΛΩ and Ω = (Ω)−1.

Lemma 6.5.1. If Λ is a Nambu tensor of order q, then its curl DΩΛ with respectto an arbitrary volume form Ω is a Nambu tensor of order q − 1. If q = 3, thenDΩΛ is a Poisson structure of rank ≤ 2.

Proof. Λ is Nambu of order = 2 ⇒ Ω(Λ) is integrable ⇒ d Ω(Λ) is integrable⇒ Ω d Ω(Λ) is Nambu.

Near a point z where Λ(z) = 0, Λ = ∂∂x1

∧ · · · ∧ ∂∂xp

, Ω = fdx1 ∧ · · · ∧ dxm,we have (see Proposition 2.6.5):

DΩΛ = [Λ, ln |f |] = (−1)q+1id ln |f |Λ . (6.36)

Lemma 6.5.2. Let Ω be a volume form and Λ a Nambu tensor of order q > 2 on amanifold M. Then for every r = 0, 1, . . . , q − 2, and smooth functions g1, . . . , gr,we have:

a) i(dg1∧···∧dgr)DΩΛ ∧ Λ = 0. In particular, the exterior product of any Hamil-tonian vector field of DΩΛ with Λ vanishes.

b) [i(dg1∧···∧dgr)DΩΛ, Λ] = 0. In particular, any Hamiltonian vector field of DΩΛis also an infinitesimal automorphism of Λ.

6.5. Kupka’s phenomenon 179

Proof. Assertion a) follows from Equation (6.36) and the local decomposabilityof Λ near a non-singular point. Assertion b) follows from Equation (6.36) andEquation (6.12). Theorem 6.5.3. Let Λ be a Nambu tensor of order q on a manifold M , and Ω bea volume form on M . When q = 2 we will assume in addition that rankΛ ≤ 2.Suppose that DΩΛ(z) = 0 at a point z ∈ M . Then in a neighborhood of z there isa local coordinate system (x1, . . . , xq−1, y1, . . . , yp+1), and a vector field X whichdepends only on (y1, . . . , yp+1), such that

Λ =∂

∂x1∧ · · · ∧ ∂

∂xq−1∧X. (6.37)

Proof. Consider the case q > 2. Since DΩΛ is a Nambu tensor of order q − 1 ≥ 2(of rank ≤ 2 when q − 1 = 2) and DΩΛ(z) = 0, in a neighborhood of z there is alocal coordinate system in which we have

DΩΛ =∂

∂x1∧ · · · ∧ ∂

∂xq−1.

Then ∂∂xi

are local Hamiltonian vector fields for DΩΛ. Assertion a) of Lemma 6.5.2implies that ∂

∂xi∧Λ = 0 (∀ i = 1, . . . , q − 1), which means that we can write Λ as

Λ =∂

∂x1∧ · · · ∧ ∂

∂xq−1∧X ,

where X is a vector field which does not contain the terms ∂∂x1

, . . . , ∂∂xq−1

. Asser-tion b) of Lemma 6.5.2 implies that [ ∂

∂xi, Λ] = 0 ∀ i = 1, . . . , q − 1. Hence X is

also invariant with respect to ∂∂x1

, . . . , ∂∂xq−1

.The case q = 2 is similar. Translating the above theorem into the language of integrable differential

forms by duality, we obtain the following result, due to de Medeiros [244], whichmay be called Kupka’s phenomenon for integrable p-forms:

Theorem 6.5.4 ([244]). Let ω be an integrable p-form on an m-dimensional mani-fold M , p < m, such that dω(z) = 0 at a point z. Then in a neighborhood of z thereis a coordinate system (x1, . . . , xm) in which ω depends only on the (p + 1) firstcoordinates (i.e., locally ω is the pull-back of a p-form on Kp+1 by a projection).

Proof. Fix a volume form Ω, and put Λ = Ω(ω). Then Theorem 6.5.3 impliesthat there is a coordinate system (x1, . . . , xq−1, y1, . . . , yq+1) in which we haveΩ = fdx1 ∧ · · · ∧ dxq−1 ∧ dy1 ∧ · · · ∧ dyp+1 (for some non-zero function f), dω =fdy1 ∧ · · · ∧ dyp+1, and ω = fiX(dy1 ∧ · · · ∧ dyp+1) where X is a vector fieldwhich depends only on the variables (y1, . . . , yp+1). Since df ∧dy1 ∧ · · · ∧ dyp+1 =d(dω) = 0, f is a function which depends only on the variables y1, . . . , yp+1.


We will now present some generalizations of Kupka’s phenomenon to the casewhen dω(z) = 0 (or DΩΛ = 0).

Generalizing the Type II linear Nambu structures, we will say that a q-vectorfield A is of Type IIr at a point z if in a neighborhood of z there is a local coordinatesystem (x1, . . . , xm) such that

A = ∂/∂x1 ∧ · · · ∧ ∂/∂xr ∧B ,

where B is a (q−r)-vector field which is independent of the coordinates x1, . . . , xr.Note that if A is Nambu then B will be automatically Nambu.

Theorem 6.5.5. Let Λ be a Nambu tensor and Ω a volume form. If the curl DΩΛis non-vanishing almost everywhere and is of Type IIr at a point z, then Λ is alsoof Type IIr at z.

Proof. WriteDΩΛ = ∂/∂x1 ∧ · · · ∧ ∂/∂xr ∧ Y ,

where Y is a (q − 1 − r)-tensor independent of (x1, . . . , xr). Then for each i =1, . . . , r, we have ∂/∂xi ∧DΩΛ = 0 and L∂/∂xi

(DΩΛ) = 0. Since DΩΛ = 0 almosteverywhere, it implies that ∂/∂xi is a locally Hamiltonian vector field for DΩΛalmost everywhere (near points where DΩΛ = 0). Applying Assertion a) of Lemma6.5.2, we obtain that ∂/∂xi ∧ Λ = 0 (almost everywhere, hence everywhere in aneighborhood of z). Since this fact is true for any i = 1, . . . , r, we can write

Λ = ∂/∂x1 ∧ · · · ∧ ∂/∂xr ∧B

for some (q − r)-vector field B such that idxiB = 0 ∀ i = 1, . . . , r. Again, since∂/∂xi is locally Hamiltonian for DΩΛ almost everywhere, applying Assertion b)of Lemma 6.5.2, we get that L∂/∂xi

Λ = 0 ∀ i = 1, . . . , r. It means that B does notdepend on x1, . . . , xr. Theorem 6.5.6. Let ω be an integrable differential form on Rn or Cn, and s be anatural number, s ≤ n. If dω is non-zero almost everywhere and depends on atmost s coordinates in a neighborhood of 0, then the same holds true for ω.

Proof. Suppose that, in a local system of coordinates (x1, . . . , xn), dω depends onlyon x1, . . . , xs. We denote by Λ the Nambu tensor associated to ω with respect tothe volume form Ω = dx1 ∧ · · · ∧ dxn. We have

DΩΛ = Y ∧ ∂/∂xs+1 ∧ · · · ∧ ∂/∂xn

where Y is a multi-field independent of xs+1, . . . , xn. By the proof of the precedingtheorem we get

Λ = P ∧ ∂/∂xs+1 ∧ · · · ∧ ∂/∂xn

where P is independent of xs+1, . . . , xn. Returning to ω, we get the desired result.

6.5. Kupka’s phenomenon 181

Theorem 6.5.7 (de Medeiros [245]). Let ω be a holomorphic integrable p-form in(Cn, 0) with coordinates (x1, . . . , xn), 2 ≤ p ≤ n− 1.

i) If codimiΠω = 0 ≥ 3, where Π = ∂∂x1

∧ · · · ∧ ∂∂xp−1

and iΠω = 0 meansthe set of zeroes of iΠω in Cn, then there is a local holomorphic coordinatesystem (y1, . . . , yn) in which we have

ω = fdg ∧ dy1 ∧ · · · ∧ dyp−1, (6.38)

where f, g are holomorphic functions in (Cn, 0), f(0) = 0.ii) If codimV ω|V = 0 ≥ 3, where V ∈ Cn is the p + 1-dimensional linear

subspace xp+2 = · · · = xn = 0, then the dual Nambu tensor Λ (ω =iΛ(dx1 ∧ · · · ∧ dxn)) is of Type II(q − 1) at 0, where q = n− p.

Proof (sketch). i) Put α = iΠω and βj = iΠjω, where Πj = ∂∂x1

∧ · · · ∧ ∂∂xj

∧· · · ∧ ∂

∂xp−1. It follows from the integrability of ω that βi ∧ α = 0. By de Rham’s

division theorem [98] (see Appendix A.2), we can write βj = γj ∧ α, and we alsohave ω ∧ γj = 0 by integrability of ω. Note that 〈γj ,

∂∂xi〉 = ±δij . It follows that

ω = ±α ∧ γ1 ∧ · · · ∧ γp−1. Now apply Malgrange’s Theorem 6.3.1 to the p-tuple(α, γ1, . . . , γp−1) to conclude.

ii) Similarly to the proof of the first part, we can write

Λ = Z ∧ Y1 ∧ · · · ∧ Yq−1,

where Y1, . . . , Yi−1 are locally linearly independent everywhere. Let x1 be a func-tion such that Y1(x1) = 1. By fixing Y2 on x1 = 0 and changing it at otherpoints by the flow of Y1, we can arrange it so that [Y1, Y2] = 0. The above decom-position of Λ still holds with respect to the new Y2 (with a new Z). Similarly, wecan arrange it so that [Yi, Yj ] = 0 for any i, j ≤ q − 1. So we can write Yi = ∂

∂yi

andΛ = Z ∧ ∂

∂y1∧ · · · ∧ ∂

∂yq−1

in some system of coordinates. We may assume that Z =∑

j≥p fi∂

∂yi. The involu-

tivity of the characteristic distribution of Λ (plus the fact that the set where Z = 0is of codimension ≥ 2) implies that there are functions hi such that [Yi, Z] = hiZ.We have ∂hi

∂yj= ∂hj

∂yi, and by Poincare’s lemma we can write hi = ∂g

∂yifor some

function g. Then

Λ = exp(g)Z ∧ ∂

∂y1∧ · · · ∧ ∂

∂yq−1,

where Z = exp(−g)Z is independent of y1, . . . , yp. Finally, the multiplicative factorexp(g) can be killed, say, by changing the coordinate function y1 (and leaving theother coordinate functions intact).

The second part of the above theorem together with Theorem 6.5.6 impliesthe following result:


Corollary 6.5.8 ([59, 245]). If ω is a holomorphic integrable p-form in (Cn, 0), 1 ≤p ≤ n− 2, such that codim

Wdω|

W= 0 ≥ 3, where W = xp+3 = · · · = xn = 0

is a linear subspace of dimension p+2, then there is a local holomorphic coordinatesystem in which ω depends on at most p + 2 variables.

The case p = 1 of the above corollary was known to Camacho and Lins Neto[59], the general case p ≥ 1 is due to de Medeiros [245].

6.6 Linearization of Nambu structures

Consider a singular point of a Nambu tensor Λ of order q. In a local coordinatesystem centered at that point, we will write the Taylor expansion of Λ as follows:

Λ = Λ(1) + Λ(2) + · · · (6.39)

where Λ(k) is a homogeneous q-vector field of degree k. Then Λ(1) is also a linearNambu tensor, called the linear part of Λ. (The easiest way to see that Λ(1) isNambu is by using the dual picture: if a differential form is integrable then itslinear part is also integrable.) Up to linear automorphisms, Λ(1) is well defined,i.e., it does not depend on the choice of local coordinates.

In this section we present some results about the linearization of Λ, i.e., theexistence of a local diffeomorphism (smooth, analytic, or formal) which transformsΛ into its linear part Λ(1). As usual, in order to get positive results, we will have toimpose some conditions on Λ. And we will consider two separate cases, dependingon whether Λ(1) belongs to Type I or Type II. A singular point of a Nambustructure is called of Type I (respectively, Type II) if its linear part is of Type I(respectively, Type II).

First let us consider the Type II case.

Proposition 6.6.1. Let z be a nondegenerate singular point of Type II of a Nambutensor λ of order q ≥ 3 (or of order q = 2 but under the additional condition thatrankΛ ≤ 2), on an m-dimensional manifold M , whose linear part has the formΛ(1) = ∂/∂x1 ∧ · · · ∧ ∂/∂xq−1 ∧ (

∑mi,j=q bi

jxi∂/∂xj). If the matrix (bij)

mi,j=q has a

non-zero trace, then there is a local system of coordinates (x1, . . . , xm) in which Λcan be written as

Λ =∂

∂x1∧ · · · ∧ ∂

∂xq−1∧X .

where X =∑m

i=q ci(xq, . . . , xm)∂/∂xi is a vector field which does not depend on(x1, . . . , xq−1).

Proof. One checks that, with respect to any volume form Ω, we have

(DΩΛ)(z) =( m∑

i=q

bii

)∂/∂x1 ∧ · · · ∧ ∂/∂xq−1(z) = 0 .

So the above proposition is a direct consequence of Theorem 6.5.3.

6.6. Linearization of Nambu structures 183

Remark 6.6.2. The above proposition remains true in the case when∑

bii = 0 but

det(bij) = 0 and q ≤ m − 2. (See Theorem 6.1 of [119] – there is a multiplicative

factor in that theorem, but one can kill it by another coordinate transformationwhich leaves xq, . . . , xm unchanged.)

The above proposition reduces the problem of linearization of Λ to the prob-lem of linearization of a vector field X . Applying Sternberg’s linearization theorem[321] for vector fields in the smooth case, and Bruno’s theorem [48] for the analyticcase (see Appendix A.5), we get the following theorem:

Theorem 6.6.3 ([119]). Let z be a singular point of Type II of a Nambu tensor Λ oforder q ≥ 3 (or of order q = 2 under the additional assumption that rankΛ ≤ 2) onan m-dimensional manifold M , whose linear part at z has the form Λ(1) = ∂/∂x1∧· · · ∧ ∂/∂xq−1 ∧ (

∑mi,j=q bi

jxi∂/∂xj). If the matrix (bij) is nonresonant, i.e., if its

eigenvalues (λ1, . . . , λp+1) do not satisfy any relation of the form λi =∑p+1

j=1 mjλj

with mj being nonnegative integers and∑

mi ≥ 2, then Λ is smoothly linearizable,i.e., there is a local smooth system of coordinates (x1, . . . , xn) in a neighborhoodof O, in which Λ coincides with its linear part:

Λ = ∂/∂x1 ∧ · · · ∧ ∂/∂xq−1 ∧( n∑

i,j=q

bijxi∂/∂xj

).

The above linearization can be made analytic (i.e., real analytic or holomorphic) ifΛ is analytic and the eigenvalues λ1, . . . , λp+1 of (bi

j) satisfy Bruno’s ω-condition.

Remark 6.6.4. It is easy to show that two Nambu tensors of order q ≥ 2 of thetype

Λ =∂

∂x1∧ · · · ∧ ∂

∂xq−1∧ Z

(where Z is a vector field which does not depend on x1, . . . , xq−1) are locally iso-morphic if and only if corresponding vector fields Z are locally orbitally equivalent,i.e., they are equivalent up to multiplication by a non-vanishing function. (Hint:consider coordinate transformations which change only x1 and leave other coordi-nates unchanged.) So the right theory to use in the study of Type II singularitiesof Nambu structures is the theory of orbital classification of vector fields (see, e.g.,[17], for results on this theory).

We now turn to singularities of Type I. We will call a singular point z ofType I of a Nambu structure Λ of order q ≥ 2 nondegenerate if the linear partΛ(1) of Λ at z has the form

Λ =q+1∑j=1

±xj∂/∂x1 ∧ · · · ∧ ∂/∂xj−1 ∧ ∂/∂xj+1 ∧ · · · ∧ ∂/∂xq+1 . (6.40)

Equivalently, z is nondegenerate if the dual linear integrable p-form ω(1) (p + q =m) of Λ(1) has the form

ω(1) = dx1 ∧ · · · ∧ dxp−1 ∧ dQ(2) , (6.41)


where Q(2) = 12

∑mi=p±x2

i is a nondegenerate quadratic function of q+1 variables.(Here we changed the ordering of indices.) When Q(2) = ± 1

2 (∑m

i=p x2i ) is a positive

or negative definite function, then we say that z is a nondegenerate elliptic singularpoint of Type I.

Theorem 6.6.5 ([119]). Let z be a nondegenerate singular point of Type II of asmooth Nambu tensor Λ of order q ≥ 3 (or of order q = 2 under the additionalassumption that rankΛ ≤ 2) on an m-manifold M . Then Λ is formally linearizableat z. If Λ is analytic (real or complex) then it is, up to multiplication by a non-vanishing function, analytically linearizable. If z is elliptic, then Λ is smoothlylinearizable.

The rest of this section is devoted to the proof of the above theorem. In theformal case, we will first show how to linearize the associated singular foliationformally, i.e., linearize Λ up to multiplication by a non-vanishing function. Then wewill kill that function by another change of variables. In the analytic case, we willuse the formal linearization together with Malgrange’s Theorem 6.3.1 to show theexistence of analytic first integrals of the associated foliation. These analytic firstintegrals allow us to linearize the foliation analytically. In the smooth case, we willuse blowing-up techniques to linearize the associated singular foliation smoothly,then use Moser’s path method to linearize Λ. Most of the time, we will work withω = Ω(Λ). Λ is determined by ω up to multiplication by a non-vanishing function.

6.6.1 Decomposability of ω

In this subsection, we will show an auxiliary result which says that, in the neigh-borhood of a nondegenerate singular point of Type I, ω can be decomposed intoa wedge product:

ω = γ1 ∧ · · · ∧ γp−1 ∧ α , (6.42)

where γi = dxi + θi with θi(z) = 0. In particular, we deduce from this decompo-sition that the set of singular points of ω forms a submanifold Σ of dimension p.

According to the definition of nondegenerate singularities of Type I, we cansuppose that ω has a Taylor expansion ω = ω(1) + ω(2) + · · · , with ω(1) = dx1 ∧· · · ∧ dxp−1 ∧ dQ(2), where Q(2) = 1

2

∑mj=p±x2

j . Express ω as a polynomial indx1, . . . ,dxp−1:

ω = dx1 ∧ · · · ∧ dxp−1 ∧ α +p−1∑j=1

dx1 ∧ · · · dxj · · · ∧ dxp−1 ∧ βj

+∑

1≤i<j≤p−1

dx1 ∧ · · · dxi · · · dxj · · · ∧ dxp−1 ∧ γij + · · · .

Here α, βi, γij , . . . are differential forms which, when written in coordinates(x1, . . . , xm), do not contain the terms dx1, . . . ,dxp−1. Applying the equationiAω ∧ ω = 0 to A = ∂/∂x1 ∧ · · · ∧ ∂/∂xp−1, we have α ∧ ω = 0. It follows


that α ∧ βj = 0, α ∧ γij = 0, etc. We can consider α and βj as differential formson the space of variables (xp, . . . , xm), parametrized by x1, . . . , xp−1, and by ourassumption of nondegeneracy, we can apply de Rham division theorem [98] (seeAppendix A.2), which says that, since the number of variables is q + 1 > 2 andthe order of βj is 2, βj is divisible by α: βj = α∧ θj where θj are smooth 1-forms.

Applying the equation iAω∧ω = 0 to A = ∂/∂x1∧· · ·∧∂/∂xj−1∧∂/∂xj+1∧· · · ∧ ∂/∂xp−1 ∧ ∂/∂xp, we get

0 = ω ∧(〈α, ∂/∂xp〉((−1)p−jdxj + θj)− 〈θj , ∂/∂xp〉α

).

Since 〈α, ∂/∂xp〉 = 〈α(1), ∂/∂xp〉 + · · · = ±xp + · · · = 0, and we already haveω ∧ α = 0, we get that ω ∧ γj = 0 where γj = dxj + (−1)p−jθj . Since γj donot vanish and are linearly independent at z, it follows that ω is divisible by theproduct of γj :

ω = γ1 ∧ · · · ∧ γp−1 ∧ α′

for some 1-form α′. By adding a combination of γj to α′, we can assume thatα′ does not contain the terms dx1, . . . ,dxp−1 when written in the coordinates(x1, . . . , xn). Then considering the terms containing dx1 ∧ · · · ∧ dxp−1 on the twosides of the equation ω = γ1∧· · ·∧γp−1∧α′, it follows that in fact we have α′ = α.

6.6.2 Formal linearization of the associated foliation

We will apply the Godbillon–Vey algorithm twice to arrive at a formal linearizationof the associated foliation.

A consequence of the expression ω = γ1 ∧ · · · ∧ γp−1 ∧α, with γi = dxi + · · · ,α =

∑mi=p±xidxi + · · · , and the implicit function theorem, is that the set of

singular points of ω near z is a (p − 1)-dimensional submanifold, which we willdenote by Σ. By rectifying Σ, we may assume that Σ is given by

Σ = xp = · · · = xm = 0. (6.43)

To simplify the formulas which follow, we will change the name of the last q + 1coordinates and denote them by y:

y := (y1, . . . , yq+1) := (xp, . . . , xm) .

In this subsection, if F is a differential form, then we will denote by

F = F (0) + F (1) + F (2) + · · ·

the Taylor expansion of F with respect to the variables y, i.e., F (k) is homogeneousof degree k in y. In particular, we have α(0) = 0 and

α(1) =∑

αij(x)yjdyi

where αij(x) = ±δij+ higher-order terms.


By a linear change of variables leaving y unchanged, we may assume that

γi = dxi + γ(1)i + γ

(2)i + · · · ∀ i = 1, . . . , p− 1 .

We will use transformations of coordinates to kill the terms γ(k)i in the above

expression consecutively. Suppose that we have already killed (k − 1) first terms(where k ≥ 1), so we can write

γi = dxi + γ(k)i + γ

(k+1)i + · · · ∀ i = 1, . . . , p− 1 .

Let us show how to kill the term γ(k)i . The integrability of ω implies that

γ1 ∧ · · · ∧ γp−1 ∧ α ∧ dγi = 0 ∀ i = 1, . . . , p− 1 .

The terms of degree k (in y) in the above equation give

dx1 ∧ · · · ∧ dxp ∧ α(1) ∧ dyγki = 0 ,

where dy denotes the exterior derivation with respect to y. This gives α(1)∧dyγki =

0, and, by the de Rham division theorem, we can write

dyγki = α(1) ∧ β

(k−2)i . (6.44)

Similarly, γ1 ∧ · · · ∧ γp−1 ∧α∧dα = 0, which leads to α(1) ∧ dyα(1) = 0, thento dyα(1) = α(1) ∧ β for some β, hence dyα(1) = 0 (because dyα(1) is of degree 0in y).

So the equation dy dy = 0 applied to Equation (6.44) leads to α(1) ∧dyβ

(k−2)i = 0, hence we can write

dyβ(k−2)i = α(1) ∧ β

(k−4)i . (6.45)

Iterating the above procedure, we obtain a finite sequence β(k−2)i , β

(k−4)i , etc.

of 1-forms which are homogeneous in y. So there is an r (r = 0 or r = 1) such thatdyβ

(r)i = 0 and β

(r)i = dyφ

(r+1)i , where φ

(r+1)i is an (r + 1)-homogeneous function

in y. From the equation dyβ(r+2)i = α(1)∧β

(r)i , we get dyβ

(r+2)i = −dy[φ(r+1)

i α(1)],hence β

(r+2)i = −φ

(r+1)i α(1) + dyφ

(r+3)i for some function φ

(r+3)i which is homoge-

neous of degree r+3 in y. Similarly, from the equation dyβ(r+4)i = α(1)∧β

(r+2)i , we

get dyβ(r+4)i = −dy

(φ

(r+3)i α(1)

), hence β

(r+4)i = −φ

(r+3)i α(1) + dyφ

(r+5)i . Going

back this way, we arrive at the following expression for γ(k)i :

γ(k)i = −φ

(k−1)i α(1) + dyφ

(k+1)i . (6.46)

Replacing γi by γ′i = γi + φ(k)α (it won’t change ω), and xi by x′

i = xi + φ(k+1)i ,

we getω = γ′

1 ∧ · · · ∧ γ′p−1 ∧ α


withγ′

i = dx′i + (γ′)(k+1)

i + · · · .

In other words, we have killed the terms of degree k (in y) in the expressionof γ. Repeating the above process for k going from 1 to ∞ and taking the (formal)limit, we arrive at a formal coordinate system (x1, . . . , xp−1, y1, . . . , yq+1) near zin which ω has the following form:

ω = dx1 ∧ · · · ∧ dxp−1 ∧ α ,

where we may assume that α does not contain the terms dx1, . . . ,dxp−1. Underthis form, the integrability condition on ω can be written as

α ∧ dyα = 0 .

So we may view α as a family, parametrized by x1, . . . , xp−1, of formal integrable1-forms in variables (y1, . . . , yq+1). with a nondegenerate linear part. It is a well-known result that such an integrable 1-form α is formally linearizable up to mul-tiplication by a non-vanishing function (see, e.g., [267]). Let us show how to do it,in a way similar to the above normalization of γi.

By a change of coordinates leaving (x1, . . . , xp−1) unchanged, we may assumethat α(1) = f(

∑q+11 ±yidyi), where f is a non-vanishing function. Since we want

to linearize α up to multiplication by a non-vanishing function, we will write α as

α = (ζ(1) + ζ(2) + ζ(3) + · · · )h

where ζ(k) is homogeneous of degree k in y, ζ(1) = dQ(2) =∑q+1

1 ±yidyi, and h isa non-vanishing function. Now we will show how to kill the nonlinear terms ζ(k)

in the expression of α consecutively. Suppose that we already have

ζ(2) = · · · = ζ(k−1) = 0

for some k ≥ 2. Then the terms of degree k in the equation α∧dyα = 0 gives ζ(1)∧dyζ(k) = 0. So by the de Rham division theorem we can write dyζ(k) = ζ(1)∧β(k−2).Then we have 0 = dy(dyζ(k)) = ζ(1) ∧ dyβ(k−2), which implies that dyβ(k−2) =ζ(1) ∧ β(k−4), and so on. Repeat this process until we get dyβ(k−2r) = 0, whichimplies that β(k−2r) = dyφ(k−2r+1). Now go back (the same way that we did withγi): dβ(k−2r+2) = ζ(1) ∧ dyφ(k−2r+1), β(k−2r+2) = −φ(k−2r+1)ζ(1) + dyφ(k−2r+3),and so on, until we get

ζ(k) = −φ(k−1)ζ(1) + dyφ(k+1) .

This last equation gives

ζ(1) + ζ(k) + · · · = dhQ(2) − φ(k−1)dhQ(2) + dyφ(k+1) + · · ·= (1− φ(k−1))dh(Q(2) + φ(k+1)) + terms of degree ≥ (k + 1) .


Putting Q′ = Q(2) + φ(k+1) and f ′ = (1− φ(k−1))f , we get

α = (dhQ′ + terms of degree ≥ (k + 1)) f ′ .

Since Q′ is a Morse function in y with the quadratic term Q(2), we can transformQ′ into Q by a coordinate transformation of the type y′ = y + terms of degree ≥(k + 1). This is the coordinate transformation which kills the term ζ(k) in theexpression of α.

Repeating the above process, we arrive at a linearization of α up to multi-plication by a non-vanishing function.

Summarizing, in this subsection we have shown that there is a formal coor-dinate system (x1, . . . , xp−1, y1, . . . , yq+1) in which we have

ω = hdx1 ∧ · · · ∧ dxp−1 ∧ d(12

q+1∑i=1

±y2i )

where h is a non-vanishing function. In these coordinates, Λ = Ωf lat(ω) is alsolinearized up to multiplication by a non-vanishing function (which depends on hand the volume form). The associated singular foliation does not depend on h,and is formally linearized in these coordinates.

6.6.3 The analytic case

The above formal linearization implies that the associated singular foliation admitsformal first integrals x1, . . . , xp−1 and Q = 1

2

∑q+1i=1 ±y2

i . In the analytic (i.e., realanalytic or holomorphic case), the decomposability of ω into the wedge productof γ1, . . . , γp−1, α, together with the existence of the above formal first integrals,allows us to apply Malgrange’s Theorem 6.3.1, which tells us that these formal firstintegrals x1, . . . , xp−1 and Q = 1

2

∑q+1i=1 ±y2

i can in fact be chosen locally analytic.Thus we obtain a local analytic linearization of the associated singular foliation,implying that Λ (and ω) can be linearized analytically up to multiplication by ananalytic non-vanishing function.

6.6.4 Formal linearization of Λ

We may assume that

Λ = fΛ1 = f( q+1∑

i=1

±xi∂

∂x1∧ · · · ∧ ∂

∂xi−1∧ ∂

∂xi+1∧ · · · ∧ ∂

∂xn

)where f is a non-vanishing function.

We want to change the coordinates (x1, . . . , xq+1) (and leave xq+2, . . . , xm un-changed) so as to make f = 1. We will forget about the parameters (xq+2, . . . , xm),and may assume for simplicity that m = q + 1.


Write f =∑

f (k), where f (k) is homogeneous of degree k in (x1, . . . , xq+1).By a change of coordinates of the type x′

1 = gx1, . . . , x′q+1 = gxq+1, we can make

f (0) = 1. We assume now that we already have f (1) = · · · = f (k−1) = 0 for somek ≥ 1. We will show that there is a change of coordinates which changes xi byterms of degree ≥ r, and which kills f (k). It amounts to finding a vector field Xsuch that

LXΛ1 = f (k)Λ1

where L denotes the Lie derivative. Consider the volume form Ω = dx1∧· · ·∧dxq+1.Then it is easy to see, by contracting Λ1 with Ω, that the equation LXΛ1 = f (k)Λ1

is equivalent to the equation

dX(Q) = (f (k) + divΩX)dQ,

where Q = (1/2)∑

i εix2i , εi = ±1. In turn, this equation is equivalent to the

following system of equations:

divΩX + f (r) = d(2QF (Q))dQ ,

X(Q) = 2QF (Q) ,

where F is an unknown function. Write X = A + Y , where A = F (Q)∑

xi∂

∂xi,

and Y is a vector field such that Y (Q) = 0. Then the above system of equationsis equivalent to a system of the type

Y (Q) = 0 ,

β(Q) + divΩY = f (k) ,

where β is an unknown function. The equation Y (Q) = 0 is equivalent to the factthat Y =

∑i<j fijYij where Yij = εixj∂/∂xi − εjxi∂/∂xj. For such a Y , we have

divΩY =∑

i<j Yij(fij). Denote by J the set of homogeneous polynomials of degreek. The solvability of the above system of equation follows from the following facts,which can be verified easily by choosing appropriate fij :

1. If a monomial xI = xI11 . . . x

Iq+1q+1 has one of Ii to be an odd number, then it

belongs to J .2. Qs is equivalent to λx2s

1 modulo J for some non-zero number λ.3. Any monomial xI = xI1

1 . . . xIq+1q+1 , with all Ii even, is equivalent to λx

∑Ii

1

modulo J for some number λ.

Thus, the above system of equations can always be solved, and therefore the non-constant terms in f can be killed one by one.

Remark 6.6.6. We don’t know if in the analytic case we can also linearize Λanalytically (without a multiplicative factor) or not. Our guess would be that theanswer is yes.


6.6.5 The smooth elliptic case

When Λ is smooth and z is a nondegenerate elliptic singular point of Type I, thenwe can use geometric arguments instead of formal power series in order to linearizeΛ smoothly. First let us show (in a sketchy way) how to linearize the associatedfoliation smoothly by the blowing-up method.

We may assume that

ω = dx1 ∧ · · · ∧ dxp−1 ∧ dQ + · · · ,

in a local coordinate system (x, y) = (x1, . . . , xp−1, y1, . . . , yq+1), where Q =∑p+11 y2

i , and that the local singular set of ω is Σ = y = 0.We will perform an oriented blow-up along the singular set Σ, i.e., we will

consider the mapping

E : Rp−1 × R× Sq −→ Rp−1 × Rq+1 = Rm ,

given by E(x, ρ, u) = (x, ρu), where Sq is the unit sphere in Rq+1.

Lemma 6.6.7. Under the above map E, the local associated singular foliation ofω on Rm lifts to a regular smooth q-dimensional foliation on a neighborhood of0 × 0 × Sq in Rp−1 × R × Sq, which admits the spheres x × 0 × Sq asleaves.

We will leave the above lemma as an exercise. (Hint: lift the tangent vectorfields.)

Since q ≥ 2, Sq is compact simply-connected, and we can use Reeb’s stabilitytheorem [356] (see Appendix A.3) to conclude that the induced foliation on Rp−1×R × Sq is a trivial fibration. In particular, it admits p smooth independent firstintegrals f1, . . . , fp.

Notice that, since the map E is generically 2-to-1, it induces an involutionσ : (x, ρ, u) → (x,−ρ,−u). If f is a first integral, then fσ := f σ is also a firstintegral. In order to get the first integrals for the associated singular foliation Fω,we have to look for first integrals on Rp−1×R×Sq which are σ-invariant, and whichproject to smooth functions on Rm. To turn a first integral f into a σ-invariantfirst integral is easy: just take f + fσ. But not every σ-invariant smooth functionon Rp−1×R×Sq projects to a smooth function on Rm. (For example, the functiony1y2/(

∑y2

i ) is not smooth on Rm but its pull-back is smooth on Rp−1 ×R× Sq.)So we need the following lemma.

Lemma 6.6.8. Every σ-invariant smooth first integral of the induced foliation onRp−1×R×Sq projects to a smooth first integral of the associated singular foliationFω on Rm.

Proof. Using results from the previous subsections, and Borel’s theorem about theexistence of smooth functions with an arbitrary given Taylor expansion, we canassume that

ω = hdx1 ∧ · · · ∧ dxp−1 ∧ dQ + µ


where h is a smooth non-vanishing function, and µ is a smooth differential formwhich is flat along the local singular submanifold Σ (and not just at one point z– by looking more carefully at the formal linearization process, the reader will seethat we do it along Σ). In particular, the vector fields

Yij = ±Ω(dy1 ∧ · · · dyi · · · dyj · · · ∧ dyq+1 ∧ ω/h) ,

which are tangent to the associated foliation Fω, has the form

Yij = yi∂

∂yj− yj

∂

∂yi+ Zij ,

where Zij are vector fields which are flat along Σ. Lifting the vector fields Yij byE, we get vector fields

Yij = ui∂

∂uj− uj

∂

∂ui+ Zij

on Rp−1 × R × Sq, which are tangent to the induced foliation, where Zij is flatwith respect to ρ. (ui

∂∂uj

− uj∂

∂uiare infinitesimal rotations of Sq.)

Let f be a σ-invariant first integral on Rp−1 × R × Sq. Write the Taylorexpansion of f with respect to the variable ρ as follows:

f = f0(x, u) + f1(x, u)ρ + f2(x, u)ρ2 + · · · .

The idea is to show that, ∀ i = 0, 1, 2, . . . , fi(x, u) does not depend on u. Thenfi(x, u) = 0 if i is odd (because f is σ-invariant), and the projection of f on Rm

will be smooth. It is clear that f0 does not depend on u, because f is constant onthe spheres x×0×Sq. For the other fi, use the equations Yij(f) = 0 and thefact that Yij = ui

∂∂uj

− uj∂

∂uiup to a flat term in ρ.

Using the above lemma, we can find appropriate smooth first integrals forFω, which allow us to linearize Fω smoothly.

Let us now linearize the Nambu tensor Λ. Linearizing the associated singularfoliation smoothly, we may assume that

Λ = f( q+1∑

i=1

(−1)i ∂

∂y1∧ · · · ∂

∂yi· · · ∧ ∂

∂yq+1

),

where f is a smooth non-vanishing function. Using the formal linearizability of Λ(see Subsection 6.6.4), we may assume that f = 1 + g, where g is a smooth flatfunction along Σ. As we said in the beginning of this chapter, a Nambu structureis essentially just a foliation plus a leaf-wise volume form. The volume of the leafSx,ρ = (x, y) | ∑ y2

i = ρ2 with respect to Λ is Cρq−1(1+ε), where C is a positiveconstant, and ε is a smooth function which is flat along Σ (which depends on g).By a change of variables of the type (x′, y′) = (x, h(ρ, x)y), where h(ρ, x) − ρ


is a flat function in ρ, we may kill the term ε in the above volume form, i.e.,make the volume form of Ss,ρ with respect to Λ equal to the volume form of the

linear Nambu tensor Λ0 =∑q+1

i=1 (−1)i ∂∂y1

∧ · · · ∂∂yi· · · ∧ ∂

∂yq+1. Now we can use

Moser’s path method, and even Moser’s result about the equivalence of volumeforms with the same volume on a compact manifold (see Appendix A.1), to findan isomorphism from Λ to Λ0 which preserves each leaf of the associated singularfoliation. In fact, we can choose this isomorphism in such a way that it is a localdiffeomorphism whose ∞-jet at points on Σ is Identity. The smooth linearizationof Λ is proved. Remark 6.6.9. When the quadratic function Q in ω(1) = dx1 ∧ · · · ∧dxp−1 ∧dQ isof the type Q = ±(y2

1 +y22−

∑q+1i=3 y2

i ), then the associated singular foliation of ω(1)

has local leaves of type S1 ×Dq−1. In this case, the associated singular foliationof ω and Λ = Ω(ω) near a nondegenerate singular point of Type I and with theabove type of Q is not smoothly linearizable in general. The reason is similar tothe reason why the Lie algebra sl(2, R) is smoothly degenerate: the local leavesof type S1 × Dq−1 are unstable under a small perturbation and can be made tospiral. When Q =

∑r1 y2

i −∑q+1

r+1 y2i with r = 2, 0 and q+1−r = 2, 0, we conjecture

that the associated singular foliation is still smoothly linearizable. (The results ofMoussu [267, 268] on the existence of smooth first integrals for integrable 1-formsimply that this conjecture is true in the case of integrable 1-forms and Nambutensors of order m− 1.)

6.7 Integrable 1-forms with a non-zero linear part

In this section we will give formal normal forms for Nambu tensors of order m− 1and integrable 1-forms near a singular point z where they have a nontrivial linearpart.

We will distinguish two cases, depending on whether the curl of the Nambustructure (or the differential of the dual 1-form) vanishes at z or not. The easy caseis when the curl does not vanish: in that case, Kupka’s phenomenon (see Section6.5) says that our Nambu tensor has the local form

∂/∂x1 ∧ · · · ∧ ∂/∂xm−2 ∧X

where X is a vector field which depends only on two variables xm−1, xm. The localclassification of such Nambu tensors is then reduced to the classification of vectorfields in dimension 2 up to orbital equivalence.

When the curl of Λ vanishes at z, we have the following formal normal form:

Theorem 6.7.1. Let Λ be a (smooth or formal ) Nambu tensor of order m − 1 onan m-dimensional manifold with m ≥ 3. Suppose that Λ(z) = 0 and DΩΛ(z) = 0where Ω is some volume form, and that the linear part λ(1) of Λ at z is nontrivial.

6.7. Integrable 1-forms with a non-zero linear part 193

Then there is a formal coordinate system (x1, . . . , xm) at z in which we have:

Λ = xm∂

∂x1∧· · ·∧ ∂

∂xm−1+

m−1∑i=1

(−1)m−i

(∂f

∂xi+ xm

∂g

∂xi

)∂

∂x1∧· · · ∂

∂xi· · ·∧ ∂

∂xm,

(6.47)where f and g are two formal functions which are independent of xm and suchthat df ∧ dg = 0.

A dual integrable 1-form of Λ in the above coordinates will have the form

ω = h.

(m−1∑i=1

(∂f

∂xi+ xm

∂g

∂xi

)dxi + xmdxm

)= h.(df+xmdg+xmdxm) , (6.48)

where h is a formal function such that h(0) = 0.

Proof. It follows from the assumptions of the above theorem and the classificationof linear Nambu structures, that we can find a coordinate system in which we have

x1, . . . , xm−1(1) = xn (6.49)

for the linear Nambu bracket corresponding to the linear part Λ(1) of Λ.Remark 6.7.2. In fact, the above theorem remains true if we replace the conditionthat DΩΛ(z) = 0 by Equation (6.49).

In this section, we will use following notations:

x := (x1, . . . , xm−1), y := xm .

For each function h = h(x, y), we will denote by

h(0) + h(1) + · · ·+ h(k) + · · ·

the Taylor expansion of h with respect to the variables x1, . . . , xm−1 (i.e., considery as a parameter in this expansion).

Lemma 6.7.3. Suppose that there are coordinates x = (x1, . . . , xm−1) and y suchthat

x1, . . . , xm−1 = y + c(r+2)(x, y) + c(r+3)(x, y) + · · ·and

x1, . . . , xi−1, xi+1, . . . , xm−1, y = (−1)m−i(a(0)i (x, y) + a

(1)i (x, y) + · · · ) ,

where a(0)i , . . . , a

(r−1)i are affine functions in y, for some number r ≥ 0. Then there

is a coordinate transformation of the form

x′1 = x1 + µ(r+2)(x, y)

x′2 = x2, . . . , x

′m−1 = xm−1

y′ = y + γ(r+1)(x, y) + γ(r+2)(x, y) ,


which gives

x′1, . . . , x

′n−1 = y′ + C(r+3)(x′, y′) + C(r+4)(x′, y′) + · · ·

and

x′1, . . . , x

′i−1, x

′i+1, . . . , x

′n−1, y

′= (−1)n−i(a(0)

i (x′, y′) + · · ·+ a(r−1)i (x′, y′) + A

(r)i (x′, y′) + · · · ) ,

where A(r)i is an affine function in y′.

Proof of Lemma 6.7.3. Making a coordinate transformation of the form x = x,y = y(1 + e(r+1)(x, y)), we get

x1, . . . , xn−1 = y + y(c(r+1)(x, y)) + c(r+2)(x, y) + · · ·

and

x1, . . . , xi−1, xi+1, . . . , xn−1, y= (−1)n−i(a(0)

i (x, y) + · · ·+ a(r−1)i (x, y) + A

(r)i (x, y) + · · · ) ,

with

A(r)i = a

(r)i − ∂er+1

∂xiy2. (6.50)

Let us show how to choose e(r+1) in such a way that A(r)i becomes an affine

function in y. Putting Ω = dx1 ∧ · · · ∧ dxm−1 ∧ dy, we have ω := iΛΩ = Γdy +∑i ∆idxi with

Γ = x1, . . . , xm−1, ∆i = (−1)m−ix1, . . . , xi−1, xi+1, . . . , xm.

Recall that we have ω ∧ dω = 0. The terms containing dxi ∧ dxj ∧ dy in thisequation give

Γ(∂∆i/∂xj − ∂∆j/∂xi) + ∆i(∂∆j/∂y − ∂Γ/∂xj)−∆j(∂∆i/∂y − ∂Γ/∂xi) = 0.(6.51)

We write ∆k = αk(x) + yβk(x) + y2δk(x, y). The hypothesis of the lemmasays that the Taylor expansion of δk in the variables x begins with terms of degree≥ k: δk = δ

(r)k + δ

(r+1)k + · · · . The terms in Equation (6.51) which are cubic in y

and of degree r − 1 in x give:

∂δ(r)i /∂xj − ∂δ

(r)j /∂xi = 0. (6.52)

Using Poincare’s lemma, we can choose e(r+1) such that

δ(r)i = ∂er+1/∂xi .

6.7. Integrable 1-forms with a non-zero linear part 195

Equation (6.50) now gives

A(r)i = α

(r)i + β

(r)i + y2(δ(r)

i − ∂er+1/∂xi) = α(r)i + β

(r)i , (6.53)

which means that A(r)i is an affine function in y.

After the above coordinate transformation turning A(r)i into an affine function

in y, we can suppose

Γ := x1, . . . , xm−1 = y + yc(r+1)(x, y) + c(r+2)(x, y) + · · ·

and

∆i := (−1)m−ix1, . . . xi . . . , xm−1, y = a(0)i (x, y) + a

(1)i (x, y) + · · ·

where a(s)i are affine in y for s = 0, . . . , r.

In the second step we use a coordinate transformation of the type x1 =x1 + θ(r+2)(x, y), x2 = x2, . . . , xn−1 = xn−1, y = y with ∂θ(r+2)/∂x1 = −c(r+1).Then we obtain

x1, . . . , xn−1 = y + c(r+2)(x, y) + · · ·and

x1, . . . , xi−1, xi+1, . . . , xn−1, y= (−1)n−i(a(0)

i (x, y) + · · ·+ a(r)i (x, y) + A

(r+1)i (x, y) + · · · ) .

So after this second step we can suppose that Γ = y + c(r+2) + · · · and ∆i =a(0)i + · · · , where the functions a

(s)i are affine in y for s = 0, . . . , r. Finally, change

y by y = y + c(r+2) to finish the proof of Lemma 6.7.3. Now we return to the proof of Theorem 6.7.1. Using Equation (6.49), we can

take x1, . . . , xm−1 as the new variable y to get x1, . . . , xm−1 = y. Then applyLemma 6.7.3 consecutively for r = 0, 1, 2, . . . . Taking the formal limit, we obtain aformal coordinate system (x1, . . . , xm−1, y) in which we have x1, . . . , xm−1 = yand x1, . . . xi . . . , xm−1, y = (−1)m−i(αi(x) + yβi(x)) for i = 1, . . . , m− 1.

In these coordinates, the dual integrable 1-form ω = Ω(Λ) of Λ with respectto the volume form Ω = dx1 ∧ · · · ∧ dxm−1 ∧ dy is

ω =m−1∑i=1

(αi(x) + yβi(x))dxi + ydy .

The terms containing dxi ∧ dxj ∧ dy in the equation ω ∧ dω = 0 give

(αiβj − αjβi) + y(∂αi/∂xj − ∂αj/∂xi) + y2(∂βi/∂xj − ∂βj/∂xi) = 0.

So we get, for every i and j,

∂αi/∂xj = ∂αj/∂xi, ∂βi/∂xj = ∂βj/∂xi, αiβj = αjβi .


Poincare’s lemma gives∑

i αidxi = df and∑

i βidxi = dg for some formal func-tions f and g which do not depend on y. Relations αiβj = αjβi mean thatdf ∧ dg = 0. Theorem 6.7.1 is proved. Corollary 6.7.4. Let ω be an integrable 1-form, which vanishes at a point z, with anontrivial linear part at z. Then, up to multiplication by a non-vanishing function,ω is formally the pull-back of an integrable 1-form on a two-dimensional space.

Proof. The case dω(z) = 0 is just Kupka’s phenomenon. Consider now the casedω(z) = 0. Then we can apply Theorem 6.7.1 to write

ω = h

(m−1∑i=1

(∂f

∂xi+ xm

∂g

∂xi

)dxi + ydy

).

So, up to multiplication by a non-vanishing function h, we may assume that ω =df + ydg + ydy. Since df ∧ dg = 0, i.e., f and g are functionally dependent,they can be written as f(x) = a(ν(x)) and g(x) = b(ν(x)), where a and b areformal functions of one variable, and ν is a formal function. Then we have ω =(a′(ν) + yb′(ν))dν + ydy = φ∗ω2, with ω2 = (a′(u) + vb′(u))du + vdv and φ :(x, y) → (u, v) = (ν(x), y). This ends the proof of the corollary.

In fact, Corollary 6.7.4 can be proven directly, without using Nambu struc-tures. The crucial point of the proof is that, up to multiplication by a non-vanishingfunction, an integrable 1-form of the type ydy+

∑m−1i=1 Aidxi is formally equivalent

to a form of the type ydy + α0 + yα1 where α0 and α1 are 1-forms which do notdepend on y. This last result has the following generalization.

Theorem 6.7.5. Let ω = yrdy +∑m−1

i=1 Aidxi be a formal integrable 1-form onRm or Cm, r ≥ 1. Then, up to multiplication by a non-vanishing function, ω isformally equivalent to an integrable 1-form ω0 of the type

ω0 = yrdy +r∑

i=0

yiαi ,

where αi are 1-forms which depend only on x1,. . . , xm−1.

The proof of the above theorem, written in terms of a dual Nambu tensor, isabsolutely similar to the proof of Lemma 6.7.3 and Theorem 6.7.1. Remark 6.7.6. It is easy to show that a formal integrable 1-form of the typey2dy +

∑2i=0 yiαi, where αi (i = 0, 1, 2) do not depend on y, is a pull-back of a

1-form on K2 (K = R or C) by a formal map from Km to K2. We don’t knowwhether or not the same is true for ω0 = yrdy +

∑ri=0 yiαi when r > 2.

Remark 6.7.7. One may view the results of this section as generalizations ofBogdanov–Takens normal forms (for two-dimensional vector fields with a non-trivial nilpotent linear part). Due to recent results on the existence of analyticBogdanov–Takens normal forms [325], it is natural to ask if there is an analytic

6.8. Quadratic integrable 1-forms 197

version for the above normal forms of integrable 1-forms and Nambu tensors oforder m− 1 as well. It turns out that Frank Loray did just that in a recent paper[218], i.e., he obtained the holomorphic version of the results presented in thissection, and some other results, by using some beautiful geometric arguments (ex-tension of the foliation to a neighborhood of CP1 in CP1 × Cm−1). See [218] fordetails. We don’t know whether there is a smooth version.

6.8 Quadratic integrable 1-forms

In this section we will give a classification of quadratic integrable 1-forms or, equiv-alently, a classification of quadratic Nambu tensors of order m − 1 in dimensionm, up to multiplication by a constant. The study of quadratic integrable 1-formsand of their perturbations was initiated by Lins Neto [214], who obtained a partialclassification of them (see Remark 6.8.1).

Let Λ be a quadratic Nambu tensor of order m− 1 in dimension m. Then itscurl DΩΛ (with respect to a constant volume form Ω) is a linear Nambu tensor.According to the classification of linear Nambu tensors, we have the following twocases:

1) DΩΛ is of Type II: In this case we will also say that Λ and the dual quadraticintegrable 1-form are of Type II. Then DΩΛ is of Type II(n− 3) (see Section 6.5).According to the generalized Kupka’s phenomenon (Theorem 6.5.5), Λ is also ofType II(n− 3). Hence we have

Λ = ∂/∂x4 ∧ · · · ∧ ∂/∂xn ∧ Λ3 ,

where Λ3 is a quadratic Poisson structure which depends only on the variablesx1, x2 and x3. In other words, Λ3 is a quadratic Poisson structure on a three-dimensional space. The classification of these three-dimensional quadratic Poissonstructures was done in [118] and [217] (see Section 5.6).Remark 6.8.1. The quadratic integrable 1-forms studied by Lins Neto [214] are ofType II. More precisely, he showed that, if there is a three-dimensional subspaceV 3 of Kn, such that the linear curl vector field of the “restriction” of Λ to V 3

(= the dual quadratic 2-vector field of the restriction of iΛΩ to V 3) is hyperbolic,then Λ can be decomposed as above (this is a particular case of Corollary 6.5.8),and moreover its dual 1-form is stable under small perturbations. See [214] fordetails, and also [59, 72] for some generalizations, in particular to the case ofhomogeneous integrable 1-forms of higher degrees.

2) DΩΛ is of Type I: In this case it is more convenient to work with a dual quad-ratic integrable 1-form ω.

Theorem 6.8.2. Let ω be a quadratic integrable 1-form such that dω is of Type I.Then in an appropriate linear system of coordinates (x1, . . . , xm) we have

ω = (βx21 + θQ)dx1 + γx1dQ ,


where β, θ, γ are constants and Q is a homogeneous quadratic function. In partic-ular, ω can be written as a pull-back of a 1-form on a two-dimensional space by anonhomogeneous quadratic map.

According to the definition of Type I, we have dω = dx ∧ dQ where Q is ahomogeneous quadratic function. We will consider the case when Q is of the typeQ =

∑ri=1±y2

i /2 + xz in a system of coordinates x, y1, . . . , yr, z, t1, . . . , ts withr ≥ 2, r + s = m− 2. (The other cases, i.e., r = 0, 1 or Q does not depend on x,are similar and simpler.)

As we have dω = −d(Qdx), we can put ω = −qdx + df where f is a homo-geneous function of degree 3. Denote Q =

∑ri=1±y2

i /2. We have

0 = ω ∧ dω = df ∧ dx ∧ dQ

=(∑

i

∂f

∂yidyi +

∂f

∂zdz +

∑j

∂f

∂tjdtj

)∧ dx ∧ (dQ + xdz) .

The terms in dtj ∧ dx ∧ dyi in this formula give ∂f/∂tj = 0, so f is independentof the tj . The terms in dyj ∧ dx ∧ dyi give

(∑i

∂f

∂yidyi

)∧ dQ = 0 ,

and an elementary calculation leads to

f = (λx + µz)Q + b(x, z) ,

where λ and µ are constant and b is a homogeneous cubic function. Putting thislast expression into the equation df ∧ dx ∧ dQ = 0, we get

0 = ((µQ + ∂b/∂z)dz + (λx + µz)dQ) ∧ dx ∧ (dq + xdz)

= ((λx + µz)x− µQ− ∂b/∂z)dQ ∧ dx ∧ dz.

The terms in yi in this last equation lead first to µ = 0, then to ∂b/∂z = λx2, thento b = λx2z + αx3, and finally to f = λxQ + αx3, where α is a constant. Puttingthis in the expression of ω, we get ω = θQdx + γxdQ + βx2dx, where θ, β and γare constants such that γ − θ = 1. Remark 6.8.3. The results in this section and the previous section lend support tothe following conjecture: every integrable 1-form on Rn or Cn with a non-trivial2-jet at 0 is, up to multiplication by a non-vanishing function, a pull-back of anintegrable 1-form in dimension 3. More generally we can ask whether or not everyintegrable 1-form on Rn or Cn with a non-zero q-jet at 0 is, up to multiplication bya non-vanishing function, a pull-back of an integrable 1-form in dimension q + 1.

6.9. Poisson structures in dimension 3 199

6.9 Poisson structures in dimension 3

In Chapter 4 we have given the list of real Lie algebras of dimension 3: besides thesemisimple algebras sl(2) and so(3), we also have semi-direct products R A R2

where A is a 2× 2 matrix which can be put in Jordan form by an isomorphism.In this section we will denote by Π a Poisson structure on a three-dimensional

manifold. It is the same as a Nambu structure of order 2 in dimension 3, and isdual to an integrable 1-form in dimension 3 via the contraction with a volumeform.

Definition 6.9.1. Assume that Π vanishes at a point p0. We will say that Π has asingularity of type V (resp. B, resp. N ) if the linear part of Π at p0 correspondsto an algebra R A R2 where the trace of A is non-zero (resp. the trace of A iszero but A is invertible, resp. A is nilpotent non-zero).

The following results concerning the above three types of singularities wereobtained by Dufour and Zhitomirskii in [117] (see also [225]). We refer to [117] fordetails and proofs.

Theorem 6.9.2. If Π admits a singularity of type V at a point p0, then Π is locallyequivalent to ∂/∂x∧Z where Z is a vector field which depends only on the variablesy and z and which vanishes at the origin. Moreover, if Z has an algebraicallyisolated singularity at the origin, then we have for Z, up to local isomorphisms,the following list of models:

Z = z∂

∂y+ (θy + z)

∂

∂z;

Z = (ny + δzn)∂

∂y+ z

∂

∂z;

Z = (−m

ny + γyn+1zm + γay2n+1z2m)

∂

∂y+ z

∂

∂z;

Z = ((±1)n+1 + ay2n+1)∂

∂y+ z

∂

∂z.

Here θ is an arbitrary number in R \ 0, n and m are positive integers, δ is equalto 0 or 1, and a is an arbitrary real number.

The hypothesis that the trace of A is non-zero is equivalent to the non-vanishing of the curl of Π at m. So the above theorem is just a consequence ofTheorem 6.5.3 and Remark 6.6.4 in the case q = 2. The singularities of type Vare non-isolated: they appear along the curve (y = z = 0).

We will say that p0 is an algebraically isolated singularity of Π if, in a localsystem of coordinates (x, y, z), the ideal generated by the germs of the functionsx, y, y, z and z, x is of finite codimension in the algebra of germs of functionsof three variables (x, y, z). This condition is a generic condition, and it assures thatthe singular point p0 is topologically isolated (see [268]).


Theorem 6.9.3. If Π admits an algebraically isolated singularity of type B, then Πis locally formally isomorphic to a model

H(x, y, z)(

z∂

∂x∧ ∂

∂y± x

∂

∂y∧ ∂

∂z± yk ∂

∂z∧ ∂

∂x

)where k is an integer greater than or equal to 2 and H is a function which doesnot vanish at p0.

Theorem 6.9.3 can be proved by studying the dual integrable 1-form andusing the results of Moussu [268]. When the symplectic leaves are locally closed,then the above normal form is also a smooth normal form [117]. Using Malgrange’sresults (Theorem 6.3.1), it is also shown in [117] that in the analytic case the abovenormal form is an analytic one. We can also normalize further to a formal normalform in which H depends only on y.

Theorem 6.9.4. If Π admits a singularity of type N , then Π is locally formallyequivalent to a model

z∂

∂x∧ ∂

∂y+ (

∂f(x, y)∂x

+ z∂g(x, y)

∂x)

∂

∂y∧ ∂

∂z+ (

∂f(x, y)∂y

+ z∂g(x, y)

∂y)

∂

∂z∧ ∂

∂x

where f and g are two functionally dependent functions (i.e., df ∧ dg ≡ 0) whose1-jets at 0 are zero.

Theorem 6.9.4 is in fact the three-dimensional case of Theorem 6.7.1. Thislast case of type N singularities is by far the most difficult to study.

The above normal forms can be extended to families of Poisson structures(or integrable 1-forms). This allows one to study the bifurcations of singularitiesof Poisson structures in dimension 3. Without entering into the details, we willmention that singularities of type V are stable (they do not disappear under a smallperturbation). Singularities of type B are not stable, but they appear generically in1-parameter families of Poisson structures. The surprising fact is that singularitiesof type N also appear generically in 1-parameter families (while algebras of thetype R A R2 with A nilpotent appear generically only in 2-parameter families ofLie algebras).Remark 6.9.5. Some results on global topological stability of integrable 1-forms,and hence of Poisson structures, on 3-manifolds, were obtained by Camacho [58].We will assume that M is an orientable 3-manifold with a given volume form. Wesay that an integrable 1-form o M ω (resp. a Poisson structure Π) is topologicallystable if, for any other integrable 1-form ω′ (resp. Poisson structure Π′), closeenough to ω (resp. Π) in Ck-topology for some k, there is a homeomorphism whichexchanges the regular leaves of the associated foliation of ω (resp. Π) with thoseof ω′ (resp. Π′). Camacho [58] obtains topological stability with respect to C2-topology by imposing a set of local and semi-local conditions (too technical to becompletely reproduced here).

6.9. Poisson structures in dimension 3 201

The local conditions are that the only permitted singularities must be (inPoisson structures’ terms):• those with a non-zero linear part which corresponds, up to isomorphisms, to

so(3), sl(2) or RA R2, such that A has eigenvalues with non-zero real parts,• those with trivial linear part but with a quadratic part isomorphic, up to

complexification, to a diagonal quadratic Poisson structure

ayz∂y ∧ ∂z + bzx∂z ∧ ∂x + cxy∂x ∧ ∂y,

where a = b = c.

Chapter 7

Lie Groupoids

In this chapter we will give a quick introduction to the theory of Lie groupoids,and then present some normal form theorems for Lie groupoids and symplecticgroupoids in the proper case. For a more comprehensive introduction, the readermay consult other references, in particular the book by Mackenzie [228], and also[60, 254].

The relations between Lie groupoids and Poisson structures will become clearnear the end of this chapter when we discuss symplectic groupoids, and in Chap-ter 8 where it is shown that Lie algebroids, i.e., the infinitesimal versions of Liegroupoids, are nothing but fiber-wise linear Poisson manifolds.

7.1 Some basic notions on groupoids

7.1.1 Definitions and first examples

Formally speaking, a groupoid is a (small) category in which each morphism isinvertible. Let us spell it out, and fix some notations:

Definition 7.1.1. A groupoid over a set M is a set Γ together with the followingstructure maps:

1) Two maps s, t : Γ→ M , called the source map and the target map.2) A product map m : Γ(2) → Γ, (g, h) → g.h, where

Γ(2) = (g, h) ∈ Γ× Γ | s(g) = t(h), (7.1)

which satisfies the following conditions:a) Compatibility with s and t:

s(g.h) = s(h), t(g.h) = t(g) ∀ (g, h) ∈ Γ(2). (7.2)

204 Chapter 7. Lie Groupoids

b) Associativity:

(g.h).k = g.(h.k) ∀ (g, h, k) ∈ Γ(3) (7.3)

where, for k ≥ 1,

Γ(k) = (g1, . . . , gk) | gi ∈ Γ, s(gi) = t(gi+1) ∀ i. (7.4)

3) An embedding ε : M → Γ, called the identity section, such that

g.ε(s(g)) = g = ε(t(g)).g. (7.5)

In particular α ε = β ε is the identity map on M.4) An inversion map ı : Γ → Γ such that s(ı(g)) = t(g), t(ı(g)) = s(g), and

ı(g).g = ε(s(g)), g.ı(g) = ε(t(g)). (7.6)

The inversion ı(g) of an element g ∈ Γ is denoted by g−1.

If Γ is a groupoid over M then M is also denoted by Γ(0) and is called theset of objects, or base points, and is often identified with the set ε(M) of identityelements of Γ. Γ, also denoted by Γ(1), is called the set of arrows. An arrow g ∈ Γfrom x = s(g) to y = t(g) will be denoted as g : x→ y. A groupoid Γ over M willoften be denoted by Γ ⇒ M .

A groupoid morphism from a groupoid Γ1 ⇒ M1 to a groupoid Γ2 ⇒ M2 isa map φ : Γ1 → Γ2 which preserves the structure maps. In other words, if we view(M1, Γ1) and (M2, Γ2) as categories, then φ is a functor. In particular, φ inducesa map between the base spaces, which we will also denote by φ : M1 → M2. Agroupoid morphism is also called a homomorphism.

Example 7.1.2. A group is a groupoid over a point.

Example 7.1.3. For any set M , put G = M ×M , with the structure maps definedas follows: s(x, y) = x, t(x, y) = y, (x, y).(y, z) = (x, z), ε(x) = (x, x), (x, y)−1 =(y, x). Then G is a groupoid over M , called the pair groupoid . If Γ ⇒ M is anarbitrary groupoid over M , then the map (s, t) : Γ →M ×M , which is sometimescalled the anchor of Γ, is a homomorphism from Γ to the pair groupoid of M .

Example 7.1.4. Associated to each left action ρ : G×M → M of a group G on aset M , there is a transformation groupoid defined as follows: Γ = G×M, s(g, x) =x, t(g, x) = ρ(g, x), ε(x) = (e, x), (g, x)−1 = (g−1, ρ(g, x)), where g ∈ G, x ∈ M ,and e is the neutral element of G. The transformation groupoid will be denotedby G ×M ⇒ M , or G M , and viewed as a semi-direct product of G with M .For right actions, there are similar transformation groupoids denoted by M G.

If Γ ⇒ M is a groupoid and x is a point of M , then

Γx = g ∈ Γ | s(g) = t(g) = x = s−1(x) ∩ t−1(x) (7.7)

7.1. Some basic notions on groupoids 205

is a group: if g, h ∈ Γx then g.h ∈ Γx and g−1 ∈ Γx. This group is called theisotropy group of x. The set

O(x) = t(g) | g ∈ Γ, s(g) = x = s(g) | g ∈ Γ, t(g) = x (7.8)

is called the orbit of x, or of Γ through x. For example, when Γ = G M is atransformation groupoid, then the orbit of Γ through x is the same as the orbitof x under the action of G. If O(x) = x, or equivalently, s−1(x) = t−1(x) = Γx,then x is called a fixed point . The orbit space of Γ is, as its name indicates, thespace of orbits of Γ on M , i.e., the quotient space of M by the equivalence relationinduced by Γ: two points of M are equivalent iff they lie on the same orbit.

In practice, M = Γ(0) is often a space of structures of some kind (for example,the space of flat connections on a given vector bundle), and the set of arrows Γ(1)

consists of isomorphisms among the structures. Then the orbit space is called amoduli space.

A subset B of M is called invariant if it is saturated by the orbits of Γ; inother words, if x ∈ B then O(x) ⊂ B.

A groupoid Γ ⇒ M is called transitive if it has only one orbit, i.e., O(x) = Mfor some (hence any) x ∈ M . In other words, for any x1, x2 ∈ M there is g ∈ Γsuch that s(g) = x1, t(g) = x2. A totally intransitive groupoid is a groupoid, eachorbit of which consists of just one point. A totally intransitive groupoid is just afamily of groups.

If Γ ⇒ M is a groupoid and U is a subset of M , then the restriction ΓU of Γto U ,

ΓU = g ∈ Γ | s(g), t(g) ∈ U, (7.9)

is a subgroupoid of Γ: the inclusion map ΓU → Γ is a groupoid morphism.More generally, if U, V are subsets of M , we will denote by

ΓUV = t−1(U) ∩ s−1(V ) (7.10)

the set of arrows from V to U . If U = x is a point then we will write ΓxV forΓxV and so on. Note that ΓDD = ΓD, and Γxx = Γx.

One may impose various topological and geometrical structures on agroupoid, depending on what one wants. We will be mainly interested in Liegroupoids, also known as differentiable groupoids, to be defined in the next sub-section.

Example 7.1.5. Let Γq be the groupoid (over Rq) of germs of local diffeomorphismsof Rq (q ∈ N). If φ : U → V , where U, V are open subsets of Rq, is a diffeomor-phism, then denote by Uφ the set of germs of φ at points x ∈ U . These sets Uφ

form a basis of open subsets of a topology of Γq. With respect to this topology, Γq

is a topological groupoid in a natural sense, and it plays an important role in thetheory of Γ-structures (generalized foliations) of Haefliger [164]. A Γ-structure ona topological space Y is represented by a 1-cocycle on Y with values in Γ: a familyof continuous maps φij : Uij = Ui ∩Uj → Γ, where (Ui) is an open covering of Y ,


is called a 1-cocycle if ∀ i, j, k,∀ x ∈ Ui ∩Uj ∩Uk we have φik(x) = φij(x).φjk(x).Two 1-cocycles on Y with values in Γ define the same Γ-structure if they differ bya coboundary, and the set of Γ-structures on Y is H1(Y, Γ). A regular codimensionq foliation on a manifold Y may be viewed as a Γq-structure, though not everyΓq-structure on Y arises from a regular foliation.

7.1.2 Lie groupoids

Definition 7.1.6 ([123]). A Lie groupoid is a groupoid Γ ⇒ M such that Γ is asmooth not necessarily separated manifold, M is a smooth separated (i.e., Haus-dorff) closed submanifold of Γ, the structure maps are smooth, and the maps s, tare submersions. A Lie groupoid morphism is a groupoid morphism between Liegroupoids, which is smooth.

Remark 7.1.7. A non-separated smooth manifold is a non-separated topologicalspace, which admits a smooth atlas. For example, take two copies R1 and R2 ofR, and glue their negative parts together: x ∈ R1 is glued to x ∈ R2 if x < 0. Theresulting space (R1∪R2)/ ∼ is a non-separated one-dimensional smooth manifold,which is the leaf space of the foliation of R2 \ (x, 0) | x ≥ 0 by vertical lines.The reason why one allows Γ to be non-separated is that there are many naturalexamples which are so. Except total spaces of groupoids, all other manifolds inthis book are assumed to be Hausdorff paracompact.Remark 7.1.8. In the definition of a Lie groupoid, the maps are usually assumedto be C∞-smooth. When they are only Ck-smooth (k ∈ N), we will say that it isa Ck-smooth groupoid.

Example 7.1.9. The transformation groupoid of a smooth Lie group action is aLie groupoid.

Example 7.1.10. Let F be a regular smooth foliation on a manifold M . We candefine its fundamental groupoid as follows: Γ is the space of triples (x, y, γ), wherex, y ∈M lie in a same leaf of F , and γ is a path in M lying on F connecting x to y,considered only up to homotopy (among such paths). The source and target mapsare given by the origin and the end of γ, and the product is given by concatenation.The holonomy groupoid of a foliation F is defined similarly by the same triples(x, y, γ) as in the case of the fundamental groupoid, except that two paths γ1 andγ2 are considered to be equivalent if they are not necessarily homotopic but havethe same holonomy, i.e., the holonomy of the foliation along the loop γ1 − γ2 istrivial (see Appendix A.3 for a definition of holonomy). These are Lie groupoidswhose total spaces are not necessarily separated. For example, the fundamentalgroupoid and holonomy groupoid of the Reeb foliation of the three-dimensionalsphere S3 [356] are non-separated.

Example 7.1.11. In the previous example, if F consists of just one leaf, i.e., theconnected manifold M itself, then its fundamental groupoid is called the funda-mental groupoid of M . It is clear that the fundamental groupoid of M is transitive


(provided that M is connected), and its isotropy groups are isomorphic to thefundamental group of M .

Lemma 7.1.12. If Γ ⇒ M is a Lie groupoid, then the partition of M into orbits ofΓ is a smooth singular foliation.

Proof. The local foliation property follows directly from the fact that the sourceand target maps are submersions.

Lemma 7.1.13. If Γ ⇒ M is a Lie groupoid, then for any x ∈ M , the isotropygroup Γx = s−1(x) ∩ t−1(x) is a Lie group.

The preimages of the source map s : Γ → M of a Lie groupoid are calleds-fibers . Those of the target map t : Γ →M are called t-fibers . They are subman-ifolds of Γ.

Lemma 7.1.14. Each s-fiber (t-fiber) of a Lie groupoid Γ ⇒ M is a Hausdorffclosed submanifold of Γ.

A submanifold S ⊂ Γ is called a (smooth) bisection of a Lie groupoid Γ ⇒ Mif s|S : S →M and t|S : S → M are diffeomorphisms.

Lemma 7.1.15. The space of bisections of a groupoid is a group.

Proof. The product S1.S2 of two bisections S1 and S2 is defined as follows:

S1.S2 = g.h | (g, h) ∈ (S1 × S2) ∩ Γ(2). (7.11)

One verifies directly that the right-hand side of the above formula is a bisection,andthat the usual axioms of a group are satisfied.

The group of smooth bisections of a Lie groupoid admits a natural topologywhich makes it into a Frechet Lie group, but we know almost nothing about thetopological structure of such groups.

Example 7.1.16. The group of bisections of a pair groupoid M×M ⇒ M is nothingbut the group Diff(M) of diffeomorphisms of M .

Example 7.1.17. Let P → M be a principal bundle over a manifold M withstructural group G, i.e., G acts freely on P on the right and P/G = M. Then thequotient (P × P )/G of the pair groupoid P × P by the diagonal action of G hasa unique natural Lie groupoid structure for which the projection map P × P →(P × P )/G is a Lie groupoid morphism. (P × P )/G is a transitive groupoid overM . Its bisections can be identified with graphs of gauge transformations of P , i.e.,diffeomorphisms from P to itself which preserve the action of G. For this reason,(P × P )/G ⇒ M is called the gauge groupoid of P .

Exercise 7.1.18. Show that any transitive Lie groupoid is the gauge groupoid ofsome principal bundle.


7.1.3 Germs and slices of Lie groupoids

Let Γ ⇒ M be a Lie groupoid, and U an open subset of M . Then the restrictionΓU of Γ to U is again a Lie groupoid. In particular, we can talk about the germof Γ over a point x ∈M .

On the other hand, let D be a submanifold of M , which is transversal to theorbital foliation (= singular foliation of M by the orbits of Γ), i.e., TxD+TxO(x) =TxM ∀x ∈ D, where O(x) = t(s−1(x)) is the orbit through x. Then ΓD is also aLie groupoid. In particular, if D intersects an orbit O transversally at a point x,i.e., TxD + TxO(x) = TxM and TxD ∩ TxO(x) = 0, then D is called a slice to Oat x, ΓD is called a slice groupoid (or slice for short) of Γ at x, and the germ ofΓD at x is called the transverse groupoid structure of Γ at x.

Exercise 7.1.19. Show that, up to isomorphisms, the transverse groupoid of a Liegroupoid at a point x on an orbit O does not depend on the choice of x on O noron the slice.

7.1.4 Actions of groupoids

Definition 7.1.20. A left groupoid action of a groupoid Γ ⇒ M on a space Nconsists of a map µ : N → M called the momentum map, or also moment map,and a map

: Γ ∗N → N, (g, x) = g.x (7.12)

called the action map, where

Γ ∗N = Γ ∗µ N = (g, x) ∈ Γ×N | s(g) = µ(x), (7.13)

such that the following compatibility conditions are satisfied:

µ(g.x) = t(g) ∀ (g, x) ∈ Γ ∗N, (7.14)

(g.(h.x)) = (g.h).x ∀ (h, x) ∈ Γ ∗N, (g, h) ∈ Γ(2), (7.15)

(εµ(x)).x = x ∀ x ∈ N. (7.16)

The action is called free if g.x = x ∀ (g, x) ∈ Γ ∗N such that g /∈ M .

When Γ acts on N , we will say that N is a Γ module, or Γ space. In thecase when µ : N → µ(N) = M is a vector bundle over M and the action of Γon N is fiber-wise linear, then we say that N is a linear Γ module, or a linearrepresentation of Γ.

Exercise 7.1.21. Let Γ ⇒ M be a Lie groupoid, and E → M a vector bundleover M . Denote by GL(E) the Lie groupoid whose arrows are linear isomorphismsamong the fibers of E. Show that a linear representation of Γ on E is the sameas a Lie groupoid morphism from Γ to GL(E) which projects to the identity mapon M .


Right groupoid actions are defined similarly. When Γ is a group, i.e., M =Γ(0) is a point, we recover the usual definition of a group action. In the case ofa Lie groupoid action, one requires the moment map and the action map to besmooth.

Similarly to the case of group actions, if : Γ ∗ N → N is a left groupoidaction, then Γ∗N itself has a groupoid structure: its target map is , and its sourcemap is the projection to N . The multiplication of (g, x) ∈ Γ∗N with (h, y) ∈ Γ∗Nis defined when x = h.y and the result is

(g, x).(h, y) = (g.h, y). (7.17)

The inversion map is(g, x)−1 = (g−1, g.x), (7.18)

and the identity section isx → (εµ(x), x). (7.19)

This groupoid is called the semi-direct product of Γ with N and can be denotedby Γ N ⇒ N .

Example 7.1.22. Γ acts on itself on the left and on the right. The moment mapfor the left (resp. action) of Γ on itself is the target map (resp. the source map).Γ also acts on its base space Γ(0) on the left and on the right, with the identitymap of Γ(0) as the moment map.

Example 7.1.23. Consider a left action of a transformation groupoid G M on Nwith moment map µ : N → M . (g, x) ∈ G M acts on y ∈ N if µ(y) = x. Wecan forget about x and define g.y = (g, µ(y)).y. Then it is a left action of G on N ,which projects to the action of G on M via the map µ : N → M . One can showthat the groupoid (G M) N is isomorphic to G N (exercise).

7.1.5 Haar systems

A smooth Haar system on a Lie groupoid Γ ⇒ M is a smooth family of measureson the t-fibers of Γ, which are invariant under left translations. In other words, foreach t-fiber t−1(x) of Γ, we have a smooth measure µx on t−1(x), which is given by asmooth nowhere vanishing density form on t−1(x). This density form also dependssmoothly on x. The invariance under left translations means that, for each g ∈ Γ,the corresponding left translation map Lg : t−1(s(g)) → t−1(t(g)), Lg(h) = g.h, isa measure-preserving diffeomorphism from t−1(s(g)) to t−1(t(g)).

Haar systems on groupoids are analogs of left-invariant measures on groups.To define a Haar system on Γ ⇒ M , it suffices to define it at points of M , thenextend it to other points of Γ by left translations.

Given a Haar system (µx) on Γ ⇒ M , one can turn the space Cc(Γ) ofcompactly supported continuous functions on Γ into an associative algebra with


the following convolution product :

(φ1 ∗ φ2)(g) =∫

k∈t−1(t(g))

φ1(k)φ2(k−1.g)dµt(g). (7.20)

The above convolution product is associative, and the reason is precisely be-cause (µx) is invariant under left translations. The closure of Cc(Γ) in a suitablenorm is a C∗-algebra. These C∗-algebras play an important role in noncommuta-tive geometry [82]. We refer to [299, 82, 207, 60] for the theory of these groupoidalgebras.

7.2 Morita equivalence

Let φ : Γ → G and ψ : Γ′ → G be Lie groupoid morphisms. Then the fibered productΓ×GΓ′ is the groupoid whose objects are triples (y, g, y′) where y ∈ Γ(0) (the spaceof objects of Γ), y′ ∈ Γ′

(0), and g : φ(y)→ ψ(y′) in G. Arrows (y, g, y′) → (z, g′, z′)in Γ×G Γ′ are pairs of arrows (h, k), h : y → z in Γ and k : y′ → z′ in Γ′, with theproperty that g′.φ(h) = ψ(k).g:

φ(y)g−−−−−→ ψ(y′)

φ(h)

$$ ψ(k)

φ(z)g′

−−−−−→ ψ(z′)

(7.21)

The product map in Γ ×G Γ′ is defined in an obvious way. The fibered productΓ×G Γ′ is a Lie groupoid as soon as its space of objects (Γ ×G Γ′)(0) = Γ(0) ×G(0)

G ×G(0)Γ′

(0) is a manifold.A homomorphism φ from a Lie groupoid Γ1 over M1 to a Lie groupoid Γ2

over M2 is called a (weak) equivalence if it satisfies the following two conditions:

i) φ induces a surjective submersion

t π1 : Γ2 ×M2 M1 →M2, (7.22)

where Γ2×M2 M1 = (g, x) ∈ Γ2×M1 | s(g) = φ(x), and π1 is the projectionto the first component (i.e., to Γ2). In particular, every orbit of Γ2 containsa point in the image φ(M1) of M1.

ii) The square

Γ1φ−−−−−→ Γ2

(s, t)

$$ (s, t)

M1 ×M1φ×φ−−−−−→ M2 ×M2

(7.23)

7.2. Morita equivalence 211

is a fibered product. (Here M1 × M1 and M2 × M2 are viewed as pairgroupoids.) In particular, for any x, y ∈ M1, the set Γ1,xy = t−1(x) ∩ s−1(y)of arrows from y to x is diffeomorphic to the set Γ2,φ(x)φ(y) = t−1(φ(x)) ∩s−1(φ(y)) of arrows from φ(y) to φ(x).

Example 7.2.1. The identity map is clearly a weak equivalence from a groupoidto itself.

Example 7.2.2. Let F be a smooth foliation on a manifold M and let T be acomplete transversal of F in M . In other words, T is an open submanifold of M(i.e., T is the interior of a submanifold with boundary), which may be disconnected,whose dimension is equal to the codimension of F , and which is transversal to Fand intersects each leaf of F at at least one point. Let Γ be the holonomy groupoidof F (see Example 7.1.10). Then the restriction ΓT of Γ to T is a Lie groupoid,called the transversal holonomy groupoid of F with respect to T . The inclusionmap ΓT → Γ is a weak equivalence.

Example 7.2.3. Let Γ ⇒ M be a transitive Lie groupoid, i.e., the gauge groupoidof a principal bundle over a connected manifold M , and x be a point of M . Thenthe inclusion map Γx → Γ is a weak equivalence from the isotropy group Γx toΓ. On the other hand, there is a weak equivalence from Γ to Γx if and only if thecorresponding principal bundle over M is trivial.

Lemma 7.2.4. If φ : Γ1 → Γ2 and ψ : Γ2 → Γ3 are weak equivalences of Liegroupoids, then the composition ψ φ is also a weak equivalence.


Lemma 7.2.5. If φ : G → Γ is a homomorphism and ψ : H → Γ is a weakequivalence of Lie groupoids, then G ×Γ H is a Lie groupoid and the projection tothe first component π1 : G ×Γ H → G is a weak equivalence.


Lemma 7.2.6. If G → Γ1, G → Γ2, H → Γ2, H → Γ3 are weak equivalences ofLie groupoids, then there exists a Lie groupoid K and weak equivalences K → Γ1,K → Γ3.

Proof. Put K = G×Γ2H. According to the previous lemmas, π1 : K → G is a weakequivalence, hence φπ1 : K → Γ1 is a weak equivalence. Similarly, ψπ2 : K → Γ3

is also a weak equivalence.

The above lemmas and examples show that the weak equivalence is not anequivalence relation (it is transitive but not reflexive), but it can be turned intoan equivalence relation, called Morita equivalence, as follows:

Definition 7.2.7. Two Lie groupoids Γ1 and Γ2 are called Morita equivalent if thereexists a third Lie groupoid G and weak equivalences from G to Γ1 and Γ2.


Roughly speaking, the Morita equivalence class of a Lie groupoid reflects itstransverse structure. In particular, we have:

Proposition 7.2.8. If two Lie groupoids Γ1 and Γ2 are Morita equivalent, thenthere is a homeomorphism from the orbit space of Γ1 to the orbit space of Γ2 (withinduced topology), such that if O1 is an orbit of Γ1 and O2 is the correspondingorbit of Γ2, then the transverse groupoid structure of Γ1 at O1 is isomorphic tothe transverse groupoid structure of Γ2 at O2.

The proof of the above proposition is another direct verification, based onthe definition of weak equivalence. We will leave it as an exercise.

Exercise 7.2.9. Show that if G ⇒ G(0) and H ⇒ H(0) are two Lie groups which areMorita equivalent, then dimG − 2 dimG(0) = dimH− 2 dimH(0).

Let φ : Γ → G and ψ : Γ → H be weak equivalences of Lie groupoids. LetPGH = PGH/ ∼ be the quotient space of the space

PGH = (g, γ, h) ∈ G × Γ×H | s(g) = t(φ(γ)), t(h) = s(ψ(γ)) (7.24)

by the equivalence relation

(g, γ1.γ2.γ3, h) ∼ (g.φ(γ1), γ2, ψ(γ3).h). (7.25)

One verifies directly that PGH is a manifold of dimension dimPGH = 12 (dimG +

dimH), and the projections p1 : (g, γ, h) → t(g) and p2 : (g, γ, h) → s(h) aresubmersions from PGH to G(0) and H(0) respectively. G acts freely on PGH fromthe left by the formula g′.(g, γ, h) = (g′.g, γ, h) with the moment map p1. Similarly,H acts freely on PGH from the right by the formula (g, γ, h).h′ = (g, γ, h.h′). Thesetwo actions obviously commute, i.e., PGH is a free (G,H)-bimodule1. Moreover, theorbits of G (resp., H) on PGH are precisely the fibers of the submersion p2 : P →H(0) (resp., p1 : PGH → G(0)), and we have PGH/G = H(0) (resp. PGH/H = G(0)).Conversely, we have:

Proposition 7.2.10. Let G and H be two Lie groupoids. Suppose that there is afree (G,H)-bimodule PGH such that the moment maps p1 : PGH → G(0) and p2 :PGH → H(0) of the actions of G and H on PGH are submersions, and the orbitsof G (resp. H) on PGH are precisely the fibers of p2 (resp. p1). Then G is Moritaequivalent to H.

Proof. Put Γ = G PGH H. In other words, Γ = (g, x, h) ∈ G × PGH ×H | s(g) = p1(x), t(h) = p2(x). The source map is (g, x, h) → x, the target mapis (g, x, h) → g.x.h, and the product map is (g′, g.x.h, h′).(g.x.h) = (g′.g, x, h.h′).One verifies directly that the projections from Γ to G and H are weak Moritaequivalences.

1A (G,H)-bimodule is a manifold together with a left action of G and a right action of H,such that the two actions commute.

7.3. Proper Lie groupoids 213

An important fact about Morita equivalent groupoids is that they have equiv-alent “categories of modules”: if N is a left H space then PGH ×H N is a left Gspace and PHG ×G PGH ×H N ∼= N .

The existence of a free bimodule which satisfies the conditions of Proposi-tion 7.2.10 is often used as the definition of Morita equivalence. The precedingdiscussion shows that this definition is equivalent to Definition 7.2.7. A free bi-module which satisfies the conditions of Proposition 7.2.10 is called a generalizedisomorphism from H to G.

More generally, a (G,H)-bimodule P is called a generalized morphism fromHto G if the moment map p2 : P → H(0) is a principal G-bundle, i.e., the action of Gis free and its orbits are precisely the fibers of p2. The reason is that, if φ : H → G isa morphism, then its “graph”, namely the quotient space of (g, h) ∈ G×H | s(g) =t(φ(h)) by the equivalence relation (g, h1.h2) ∼ (g.φ(h1), h2), together with thenatural actions of G and H, satisfies the conditions of a generalized morphism.

Exercise 7.2.11. Show that the graph of a Lie groupoid morphism defined aboveis a manifold which can also be written as (g, q) ∈ G × H(0) | s(g) = t(φ(1q)).Define the composition of two generalized morphisms, and show that Lie groupoidstogether with generalized morphisms form a category.

Remark 7.2.12. One can define Morita equivalence for topological groupoids in asimilar way (see, e.g., [271]). Morita equivalence of groupoids was probably firstused by Haefliger in his theory of Γ-structures (see [165]). As stressed by Haefliger[165], many important invariants and properties of foliations are preserved underMorita equivalence (of their holonomy groupoids). Probably, the same may besaid about general Lie (or topological) groupoids. The name Morita comes fromthe analogy with a natural equivalence relation in algebra, first studied by Morita[264]: two unital rings A and B are called Morita-equivalent if the category ofA-modules is isomorphic to the category of B-modules. Since then, the notion ofMorita equivalence has been generalized to and studied in many other situationsin algebra and geometry. For example, Muhly, Renault and Williams [271] provedthat if two locally compact topological groupoids are Morita equivalent then theircorresponding C∗-algebras (with respect to given Haar systems) are “stronglyMorita equivalent”. For Morita equivalence of Poisson manifolds see, e.g., [54, 85,87, 145, 357, 358].

7.3 Proper Lie groupoids

7.3.1 Definition and elementary properties

Recall that a continuous map φ : X → Y between two Hausdorff topological spacesis called proper if the preimages of compact subsets of Y under φ are compactsubsets of X . A (left) action of a Lie group G on a manifold M is called a properaction if the map G ×M → M ×M, (g, x) → (g.x, x) is a proper map. If G is


compact, then any smooth action of G is automatically proper. But non-compactgroups can also act properly. For example, the action of any Lie group on itself bymultiplication on the left is proper.

The following definition generalizes the notion of proper actions of Lie groupsto the case of Lie groupoids.

Definition 7.3.1. A Lie groupoid Γ ⇒ M is called a proper Lie groupoid if Γ isHausdorff and the map (s, t) : Γ →M ×M is proper.

Example 7.3.2. A smooth action of a Lie group G on a manifold M is proper ifand only if the corresponding transformation groupoid G M is proper.

The above properness condition has some immediate topological conse-quences, which we put together into a proposition:

Proposition 7.3.3 ([254, 354]). Let G⇒M be a proper Lie groupoid. Then we have:i) The isotropy group Gm = p ∈ G | s(p) = t(p) = m of any point m ∈ M is

a compact Lie group.ii) Each orbit O of G on M is a closed submanifold of M .iii) The orbit space M/G together with the induced topology is a Hausdorff space.iv) If H is a Hausdorff Lie groupoid which is Morita-equivalent to G, then H is

also proper.v) If N is a submanifold of M which intersects an orbit O transversally at a

point m ∈ M , and B is a sufficiently small open neighborhood of m in N ,then the slice GB = s−1(B) ∩ t−1(B) is a proper Lie groupoid over B whichhas m as a fixed point.

Proof. Points i) and v) follow directly from the definition. A sketchy proof ofpoint iv) can be found in [254], the chapter on Lie groupoids, and will be left as anexercise. The proof of point ii) can be found in [354] and is similar to the followingproof of point iii). Let us give here a proof of point iii): Let x, y ∈ M such thattheir orbits are different: O(x) ∩ O(y) = ∅, or equivalently, s−1(y) ∩ t−1(x) = ∅.Denote by Dz

1 ⊃ Dz2 ⊃ . . . z a series of compact neighborhoods (i.e., compact

sets which contain open neighborhoods) of z in M , where z = x or y, such that⋂n∈N

Dzn = z. We have

⋂n∈N

t−1(Dxn)∩s−1(Dy

n) = t−1(x)∩s−1(y) = ∅. Since Γis proper, the sets t−1(Dx

n) ∩ s−1(Dyn) are compact. It follows that there is n ∈ N

such that t−1(Dxn) ∩ s−1(Dy

n) = ∅, or equivalently, O(Dxn) ∩ O(Dy

n) = ∅, whereO(Dx

n) is the union of orbits through Dxn. But the orbit space of O(Dx

n) (resp.O(Dy

n)) is a (compact) neighborhood of x (resp., y) in the orbit space of M . Thusthe orbit space of M is Hausdorff. Definition 7.3.4. A smooth action of a Lie groupoid Γ on a manifold N is calleda proper action if the corresponding semi-direct product Γ N is a proper Liegroupoid.

Exercise 7.3.5. Show that if Γ is a proper Lie groupoid, then its smooth actionson manifolds are automatically proper.

7.3. Proper Lie groupoids 215

Exercise 7.3.6. Show that the adjoint action of a non-compact simple Lie groupG on its Lie algebra g is not proper.

7.3.2 Source-local triviality

A Lie groupoid Γ ⇒ M is called source-locally trivial , if the source map s : Γ →Mis a locally trivial fibration.

For example, the transformation groupoid of a Lie group action is source-locally trivial.

Recall that the inversion map exchanges s-fibers with t-fibers, hence a source-locally trivial groupoid is also target-locally trivial and vice versa.

While the condition of properness is preserved under Morita equivalence ofgroupoids, source-local triviality is not, as the following simple example shows.

Example 7.3.7 ([354]). Let X1 be the plane R2 with the origin removed. Let thegroupoid G1 be the equivalence relation on X1, with quotient space R, consistingof all the pairs of points lying on the same vertical line. (In other words, thespace of objects is X1, and there is a unique arrow from a point x ∈ X1 to a pointy ∈ X1 if and only if they lie on the same vertical line.) The anchor of G1 is proper,but the source map is not locally trivial over any point lying on the vertical linepassing through the origin. Let G2 be the restriction of G1 to a horizontal line notpassing through the origin. G1 is just a trivial groupoid over X2, so it is properand source-locally trivial. The inclusion map from G1 to G2 is a weak equivalence,so these two groupoids are Morita-equivalent.

See [354] for more examples of groupoids which are not source-locally trivial.

Proposition 7.3.8. If Γ ⇒ B is a proper Lie groupoid with a fixed point m ∈ B,then any neighborhood of m in B contains an open neighborhood U such that ΓU

is a source-locally trivial proper Lie groupoid.

In particular, a slice of a proper Lie groupoid is a proper Lie groupoid whichcan be chosen source-locally trivial.

Proof. Since the isotropy group Γm = s−1(m) ⊂ Γ is compact and s is a submer-sion, there is a neighborhood U of Γm in Γ such that s|U : U → s(U) is a trivialfibration, and s(U) is a neighborhood of m in B which can be chosen arbitrarilysmall. Fix a local coordinate system on B near x, and for each n ∈ N denote byBn the open ball in B of radius 1/n centered at x with respect to this coordinatesystem. We can assume that B1 ⊂ s(U) is relatively compact. By properness ofΓ, the sets Vn = s−1(Bn) ∩ t−1(Bn) are relatively compact, and

⋂∞n=1 Vn = Γm,

where Vn means the closure of Vn. Since Vn ⊃ Vn+1 ∀n and U is a neighborhoodof Γm, there is n ∈ N such that Vn ⊂ U . We may assume that V1 ⊂ U .

Define Un = t−1(Bn) ∩ U and Dn = s(Un). Then for n large enough, Dn ⊂B1 is a small open neighborhood of m in B, and ΓDn = s−1(Dn) ∩ t−1(Dn) =s−1(Dn) ∩ U ⊂ U . Indeed, since the target map t is a submersion, Un is a small


neighborhood of Γm, and since s(Γm) = m, Dn = s(Un) is a small neighborhoodof m, ∩∞

n=1Dn = m, so Dn ⊂ B1 provided that n is large enough. Since V1 =s−1(B1) ∩ t−1(B1) ⊂ U , we also have s−1(Dn) ∩ t−1(Dn) ⊂ U , hence s−1(Dn) ∩t−1(Dn) ⊂ s−1(Dn)∩U. Conversely, let g be an arbitrary element of s−1(Dn)∩U ,we will show that g ∈ t−1(Dn), i.e., t(g) ∈ Dn (provided that n is large enough).By definition, g ∈ U and s(g) ∈ Dn = s(t−1(Bn) ∩ U), and there is an elementh ∈ U such that t(h) ∈ Bn and s(h) = s(g) ∈ Dn. Since Dn is small, g and h liein a small neighborhood of Γm in Γ, and by continuity of the product map andthe inversion map, we can assume that h.g−1 ∈ U . Then t(h.g−1) = t(h) ∈ Bn, soh.g−1 ∈ t−1(Bn) ∩ U , and t(g) = s(h.g−1) ∈ s(t−1(Bn) ∩ U) = Dn.

Since ΓDn = s−1(Dn) ∩ U , we have that s : ΓDn → Dn is a trivial bundle,i.e., ΓDn is source-locally trivial. The fact that ΓDn is proper is automatic. Notethat t : ΓDn → Dn is also a trivial fibration for n large enough.

Proposition 7.3.9 ([354]). If Γ ⇒ B is a source-locally trivial proper Lie groupoidwith a fixed point m ∈ B, then any neighborhood of m in B contains an invariantopen neighborhood of m.

Proof. Use the neighborhoods Dn constructed in the proof of the previous propo-sition. In the source-locally trivial case, they are invariant.

7.3.3 Orbifold groupoids

Orbifold groupoids form an interesting class of proper groupoids. They were in-troduced by Moerdijk and Pronk [255, 256] as a convenient setting in which tostudy structures on orbifolds.

Definition 7.3.10. An orbifold groupoid is a proper Lie groupoid whose isotropygroups are finite.

Recall that a smooth orbifold is a space V which is locally “diffeomorphic”to the quotient of a smooth manifold by a finite group action. More precisely,there is an open covering (Ui) of V , open subsets Ui of Rn, finite groups Gi whichact on Ui, and projections πi : Ui → Ui = Ui/Gi. If xi and xj are points ofUi and Uj such that πi(xi) = πj(xj), then there is an open neighborhood Wi

of xi, an open neighborhood Wj of xj and an (automatically unique) smoothdiffeomorphism φij : Wi → Wj such that πi|Wi

= πj |Wj φij . The above open

covering (Ui = Ui/Gi) is called an atlas of V , and each open set Ui = Ui/Gi iscalled a defining chart . Orbifolds are natural generalizations of manifolds. Theywere introduced by Satake2, who proved an analogue of the Gauss–Bonnet formulafor them [309].

2Satake used the term V -manifold. The term orbifold was probably coined by Thurston, see[329].

7.4. Linearization of Lie groupoids 217

Given an orbifold V with an atlas (Ui = Ui/Gi) as above, the disjoint unionP =

∐i Ui is the base space of a Lie groupoid ΓV , which consists of germs of

diffeomorphisms generated by the elements of Gi, and of germs of the above lo-cal diffeomorphisms φij . The orbit space of ΓV is precisely V . Notice that theisotropy groups of ΓV are finite (they are subgroups of the groups Gi). It is alsoclear that ΓV is a proper Lie groupoid. Thus, any smooth orbifold can be repre-sented as the orbit space of an orbifold groupoid [165]. The converse is also true:if Γ is an orbifold groupoid then its orbit space is an orbifold [255]. It is a conse-quence of the etale case of the local linearization theorem for proper Lie groupoids(Theorem 7.4.7).

7.4 Linearization of Lie groupoids

7.4.1 Linearization of Lie group actions

Probably the most well-known result about local linearization of Lie group actionsnear a fixed point is the following theorem of Bochner:

Theorem 7.4.1 (Bochner [36]). Any Cn-action (n = 1, 2, . . . ,∞) of a compactLie group G on a manifold V with a fixed point x is locally Cn-isomorphic in aneighborhood of x to a linear action.

Proof. The proof of Bochner’s theorem, based on the averaging method, is verysimple [36]: Denote by ρ : G ×M → M an action of G on M with a fixed pointz ∈M . The linear part ρ(1) of ρ, i.e., the differential of ρ at z, is a linear action of Gon TzM . By a local coordinate system (x1, . . . , xm) centered at z, we will identifya neighborhood of z in M with a neighborhood of z in TzM , and understand bothρ and ρ(1) as actions of G on (a neighborhood of 0 in) V = TzM . Denote by µ theHaar probability measure on G. Then the following map φ,

φ(x) =∫

G

ρ(1)(g−1, ρ(g, x))dµ, (7.26)

is a local diffeomorphism of V (whose differential at 0 is the identity) which inter-twines ρ with ρ(1):

ρ(1) φ = φ ρ. (7.27)

In other words, φ−1 is a linearization of the action ρ, and it has the same smooth-ness class as ρ. Remark 7.4.2. Of course, the above theorem also holds for analytic actions ofcompact Lie groups (then we will have a local analytic linearization), with thesame proof. Using the so-called unitary trick to turn the action of a non-compactsemisimple Lie group (or Lie algebra) to that of a compact Lie group (in thecomplexified space), Guillemin and Sternberg showed that an analytic action ofa non-compact semisimple Lie group (or Lie algebra) is also locally linearizable[160], a result which was also obtained independently by Kushnirenko [205].


Remark 7.4.3. Smooth actions of non-compact Lie groups are often locally non-linearizable. See, e.g., [57] for some results in the non-compact semisimple case.

Example 7.4.4. Consider an action of R on the plane generated by the vector fieldX = x∂/∂y − y∂/∂x − (sin2 x)(x∂/∂x − y∂/∂y). Then the linearized action hasclosed orbits, while the orbits of X are spiralling towards 0. So for topologicalreasons, this action of R is locally nonlinearizable.

Consider now an orbitO(z) = G.z of an action of a Lie group G on a manifoldM , where z ∈M is not a fixed point of the action. Then the isotropy group Gz =g ∈ G, g.z = z acts naturally on the normal vector space W = TzM/TzO(z)(via the derivation of the action of Gz in the neighborhood of z). Similarly, Gacts on the normal bundle NO of O in M in a fiber-wise linear fashion, andwe can identify NO with G ×Gz W . A natural question arises: does there exista tubular neighborhood of O in M which is diffeomorphic, by a G-equivariantdiffeomorphism, to a neighborhood of the zero section in NO ∼= G ×Gz W? Apositive answer to this linearization question was obtained by Koszul [200] inthe case when G is compact, and then by Palais [291] in the case when G isnot compact but its action on M is proper. More precisely, we have the followingtheorem, called the slice theorem. (Imagine the fibers of a neighborhood of O, afterthe identification of this neighborhood with a neighborhood of the zero section ofthe normal bundle NO, as slices.)

Theorem 7.4.5 (Slice theorem [200, 291]). Let a Lie group G act properly on amanifold M , and z be a point of M . Then there is a G-equivariant diffeomorphismfrom a neighborhood of the orbit O(z) = G.z of z in M to a neighborhood of thezero section of G×Gz W , which sends O(z) to the zero section, where Gz = g ∈G, g.z = z is the isotropy group at z, and W = TzM/TzO(z) is the normal spaceto O(z) at z (on which Gz acts linearly).

The proof of Theorem 7.4.5 follows from Bochner’s Theorem 7.4.1 (applied tothe action of the compact isotropy group) and some relatively simple topologicalarguments. See, e.g., Chapter 2 of [121] and Appendix B of [158] for details andsome applications.

7.4.2 Local linearization of Lie groupoids

Let Γ ⇒ B be a Lie groupoid with a fixed point x ∈ B. Then the isotropy groupG = Γx = s−1(x) = t−1(x) acts naturally on TxB as follows. For g ∈ G, v ∈TxM , denote by γ(v), r ∈ [0, 1] a smooth path in Γ such that γ(0) = g andddr |r=0s(γ(r)) = v, and put

g.v =ddr

∣∣∣∣r=0

t(γ(r)). (7.28)

One can check that this definition is independent of the choice of γ, and gives alinear action of G on TxM . The corresponding transformation groupoid G M


is called the linear part of Γ at x. The general local linearization problem forLie groupoids is: given a Lie groupoid with a fixed point, is it locally isomorphicto its linear part? The infinitesimal version of this linearization problem for Liegroupoids is the linearization problem for Lie algebroids, which will be discussed inChapter 8. To make the problem more precise, we will need the following definition:

Definition 7.4.6. A Lie groupoid Γ ⇒ B with a fixed point x ∈ B is said to belocally isomorphic to a Lie groupoid G ⇒ D with a fixed point y ∈ D if there isan open neighborhood U (resp., V ) of x (resp., y) in B (resp., D) such that ΓU isisomorphic to GV .

The following local linearization theorem generalizes Bochner’s Theorem7.4.1 to the case of proper Lie groupoids.

Theorem 7.4.7 ([370]). Any Cn-smooth (n ∈ N∪∞) proper Lie groupoid Γ ⇒ Bwith a fixed point x0 ∈ B is Cn-smoothly locally linearizable, i.e., it is Cn-smoothlylocally isomorphic to the transformation groupoid G V of a linear action of Gon V , where G = Gx0 is the isotropy group of x0 and V = Tx0B.

Remark 7.4.8. The etale case of Theorem 7.4.7 (i.e., the case when the isotropygroup Gx0 is finite) is relatively simple and was obtained by Moerdijk and Pronk[255] (see also [354]). The general case was conjectured by Weinstein [353, 354].

We will give here a sketch of the proof of Theorem 7.4.7, referring the readerto [370] for the details.

By Proposition 7.3.8, we can assume that the proper groupoid Γ ⇒ B islocally-source trivial, and by Proposition 7.3.9, we may shrink B to an arbitrarilysmall neighborhood of x0 in B.

Note that Theorem 7.4.7 is essentially equivalent to the existence of a smoothsurjective homomorphism φ from Γ to G (after shrinking B to a sufficiently smallinvariant neighborhood of x0), i.e., a smooth map φ : Γ → G which satisfies

φ(p.q) = φ(p).φ(q) ∀ (p, q) ∈ Γ(2) := (p, q) ∈ Γ× Γ, s(p) = t(q) , (7.29)

and such that the restriction of φ to G = s−1(x0) ⊂ Γ is an automorphism of G.We may assume that this automorphism is identity.

Indeed, if there is an isomorphism from Γ ⇒ B to a transformation groupoidGU , then the composition of the isomorphism map Γ → G×U with the projec-tion G×U → G is such a homomorphism. Conversely, if we have a homomorphismφ : Γ → G, whose restriction to G = s−1(x0) ⊂ Γ is the identity map of G, thenshrinking B to a sufficiently small invariant neighborhood of z in B if necessary,we get a diffeomorphism

(φ, s) : Γ → G×B. (7.30)

Denote by θ the inverse map of (φ, s). Then there is an action of G on B definedby g.x = t(θ(g, x)), and the map (φ, s) will be an isomorphism from Γ ⇒ B to thetransformation groupoid G B. This transformation groupoid is linearizable byBochner’s Theorem 7.4.1, implying that the groupoid Γ ⇒ B is linearizable.


In order to find such a homomorphism from Γ to G, we will use the averagingmethod. The idea is to start from an arbitrary smooth map φ : Γ → G such thatφ|G = Id. (Recall that G = Γx0 = s−1(x0) = t−1(x0).) Then Equality (7.29) is notsatisfied in general, but it is satisfied for p, q ∈ G. Hence it is “nearly satisfied”in a small neighborhood of G = s−1(x0) in Γ. In other words, if the base B issmall enough, then φ(p.q)φ(q)−1 is near φ(p) for any (p, q) ∈ Γ(2). We will replaceφ(p) by the average value of φ(p.q)φ(q)−1 for q running on t−1(s(p)) (it is to bemade precise how to define this average value). This way we obtain a new mapφ : Γ → G, which will be shown to be “closer” to a homomorphism than theoriginal map φ. By iterating the process and taking the limit, we will obtain atrue homomorphism φ∞ from Γ to G.

Notice that the t-fibers of Γ ⇒ B are compact and diffeomorphic to G =t−1(x0) by assumptions. As a consequence, there exists a smooth Haar probabilitysystem (µx) on Γ, i.e., a smooth Haar system such that for each x ∈ B, thevolume of t−1(x) with respect to µx is 1. Such a Haar probability system (µx) canbe constructed as follows: begin with an arbitrary Haar system (µ′

x) on Γ, thendefine µ = µ′/I where I is the left-invariant function I(g) =

∫t−1(t(g)) dµ′

t(g). Wewill fix a Haar probability system µ = (µx) on Γ.

We fix a bi-invariant metric on the Lie algebra g of G and the induced bi-invariant metric d on G itself. Denote by 1G the neutral element of G. For eachnumber ρ > 0, denote by Bg(ρ) (resp., BG(ρ)) the closed ball of radius ρ in g(resp., G) centered at 0 (resp., 1G). By resizing the metric if necessary, we willassume that the exponential map

exp : Bg(1)→ BG(1) (7.31)

is a diffeomorphism. Denote by

log : BG(1)→ Bg(1) (7.32)

the inverse of exp. Define the distance ∆(φ) of φ : Γ → G from being a homomor-phism as follows:

∆(φ) = sup(p,q)∈Γ(2)

d(φ(p.q).φ(q)−1.φ(p)−1, 1G

). (7.33)

Let φ : Γ → G be a smooth map such that φ|G is identity. We will assumethat ∆(φ) ≤ 1, so that the following map φ : Γ → G is clearly well defined:

φ(p) = exp

(∫q∈t−1(s(p))

log(φ(p.q).φ(q)−1 .φ(p)−1)dµs(p)

).φ(p) . (7.34)

It is clear that φ is a smooth map from Γ to G, and its restriction to G =s−1(x0) ⊂ Γ is also identity. The proof of the following lemma, which says thatwhen G is Abelian we are done, is straightforward:


Lemma 7.4.9. With the above notations, if G is essentially commutative (i.e., theconnected component of the neutral element of G is commutative) then φ is ahomomorphism.

In particular, if G is finite then φ is a homomorphism, and we obtain a proofof Theorem 7.4.7 in the etale case.

In general, due to the noncommutativity of G, φ is not necessarily a homo-morphism, but ∆(φ) (the distance of φ from being a homomorphism) is of theorder of ∆(φ)2. More precisely, we have:

Lemma 7.4.10. There is a positive constant C0 > 0, C0 ≤ 1 (which depends onlyon G and on the choice of the bi-invariant metric on it ) such that if ∆(φ) ≤ C0

then φ is well defined and

∆(φ) ≤ (∆(φ))2/C0 ≤ ∆(φ) . (7.35)

The above lemma implies that we have the following fast convergent iterativeprocess: starting from an arbitrary given smooth map φ : Γ → G, such thatφ|G = Id, construct a sequence of maps φn : Γ → G by the recurrence formulaφ1 = φ, φn+1 = φn. Then this sequence is well defined (after shrinking B once toa smaller invariant neighborhood of x0 if necessary), and

φ∞ = limn→∞ φn (7.36)

exists (in C0-topology), and is a continuous homomorphism from Γ to G.More elaborate estimates using Ck-norms (where k does not exceed the

smoothness class of Γ) show that in fact we have φ∞ = limn→∞ φn in Ck-topology,and hence φ∞ is a Ck-smooth homomorphism. This concludes the proof of Theo-rem 7.4.7. See [370] for the details.Remark 7.4.11. The above iterative averaging process is inspired by a similarprocess which was deployed by Grove, Karcher and Ruh in [155] to prove thefollowing theorem about approximation of near-homomorphisms between compactLie groups by homomorphisms. The idea of using Grove–Karcher–Ruh’s iterativeaveraging method was proposed by Weinstein [352, 353].

Theorem 7.4.12 (Grove–Karcher–Ruh [155]). If G and K are two given compactLie groups, then any map φ : G → K which is a near-homomorphism (i.e., the mapG×G→ K : (g, h) → φ(g).φ(h).φ(h−1g−1) is sufficiently close to the constant map(g, h) → eK in C0-topology) can be approximated by a homomorphism φ0 : G→ K(i.e., the map G→ K : g → φ(g)−1.φ0(g) is close to the constant map g → eK).

An immediate consequence of Theorem 7.4.7 is the following:

Corollary 7.4.13. The characteristic foliation of a proper Lie groupoid is an orbit-like foliation in the sense of Molino, with closed leaves. In particular, it is a sin-gular Riemannian foliation with closed leaves.


Recall that a Riemannian foliation [298] is a foliation which admits a trans-verse Riemannian metric, i.e., a Riemannian metric on a complete transversalwhich is invariant under holonomy. We refer to [260] for an introduction to Rie-mannian foliations, including singular Riemannian foliations. An orbit-like folia-tion [261] is a singular Riemannian foliation which is locally linearizable.

Another immediate consequence of Theorem 7.4.7 is that, if G ⇒ M is aproper groupoid, then the orbit space M/G (together with the induced topologyand smooth structure from M) locally looks like the quotient of a vector space bya linear action of a compact Lie group. (Locally, the orbit space M/G is the sameas the orbit space of a slice B/GB .) In analogy with the fact that orbifolds areorbit spaces of etale proper groupoids (see Subsection 7.3.3), it would be naturalto call the orbit space of a proper Lie groupoid a (smooth) orbispace.

7.4.3 Slice theorem for Lie groupoids

Consider a Lie groupoid G ⇒ M , and an orbit O of G on M . Then the restrictionGO := p ∈ G | s(p), t(p) ∈ O of G to O is a transitive Lie groupoid over O.Similarly to the case of Lie group actions and the case of Lie groupoids with afixed point, the structure of G induces a linear action of GO on the normal vectorbundle NO of O in M . The corresponding semi-direct product GO NO is thelinear model for G in the neighborhood of O.

The following theorem, which generalizes Koszul–Palais’ slice Theorem 7.4.5to the case of Lie groupoids, and which can also be called a slice theorem, describesthe structure of a proper Lie groupoid in the neighborhood of an orbit, under somemild conditions.

Theorem 7.4.14 (Slice theorem). Let G ⇒ M be a source-locally trivial proper Liegroupoid, and let O be an orbit of G which is a manifold of finite type. Then thereis a neighborhood U of O in M such that the restriction of G to U is isomorphic tothe restriction of the transformation groupoid GO×ONO ⇒ NO to a neighborhoodof the zero section of NO.

In the above theorem, the condition that O is of finite type means that thereis a proper map from O to R with a finite number of critical points. Theorem7.4.14 was obtained by Weinstein [354] modulo Theorem 7.4.7. The original resultof Weinstein [354] may be formulated as follows:

Theorem 7.4.15 ([354]). Let Γ ⇒ M be a proper groupoid, and let O be an orbit ofΓ which is a manifold of finite type. Suppose that the restriction ΓD of Γ to a sliceD through x ∈ O is isomorphic to the restriction of the transformation groupoidΓx × NxO ⇒ NxO to a neighborhood of zero. Then there is a neighborhood U ofO in M such that the restriction of Γ to U is isomorphic to the restriction of thetransformation groupoid GO×O NO ⇒ NO to a neighborhood of the zero section.

We refer to [354] for the proof of Theorem 7.4.15. It makes use of the followingauxiliary topological result to treat the case when the orbit is not compact:

7.5. Symplectic groupoids 223

Proposition 7.4.16 ([354]). Let f : X → Y be a submersion. For any y ∈ Y , ifO = f−1(y) is a manifold of finite type, then there is a neighborhood U of O in Xsuch that f |U : U → f(U) is a trivial fibration. In other words, there is a retractionρ : U → O such that (ρ, f) : U → O × f(U) is a diffeomorphism.

If f is equivariant with respect to actions of a compact group K on X andY , with y a fixed point, and f−1(y) is of finite type as a K-manifold, then U canbe chosen to be K-invariant and ρ to be K-equivariant.

7.5 Symplectic groupoids

7.5.1 Definition and basic properties

Symplectic groupoids were introduced independently by Karasev [189], Weinstein[349], and Zakrzewski [364], in connection with symplectic realization and quanti-zation of Poisson manifolds. One of the main motivations is the following question:Global objects which integrate Lie algebras are Lie groups. General Poisson man-ifolds may be viewed as non-linear analogs of Lie algebras. So what are the globalobjects which are analogous to Lie groups and which have Poisson manifolds astheir infinitesimal objects? The answer is, they are symplectic groupoids.

Definition 7.5.1. A symplectic groupoid is a Lie groupoid Γ ⇒ P , equipped witha symplectic form σ on Γ, such that the graph of the multiplication map

∆ = (g, h, g.h) ∈ Γ× Γ× Γ | (g, h) ∈ Γ(2) (7.37)

is a Lagrangian submanifold of Γ×Γ×Γ, where Γ means the manifold Γ with theopposite symplectic form −σ.

The condition that ∆ is a Lagrangian submanifold of Γ× Γ× Γ means thatdim∆ = 3

2 dim Γ and ∆ is isotropic. The condition that ∆ is isotropic may beexpressed more visually as follows: If φ1, φ2, φ3 : D2 → Γ are three maps from atwo-dimensional disk D2 to Γ such that φ1(x).φ2(x) = φ3(x) for any x ∈ D2, then∫

D2φ∗

1ω +∫

D2φ∗

2ω =∫

D2φ∗

3ω. (7.38)

In other words, roughly speaking, we have

Area(D1) + Area(D2) = Area(D3), (7.39)

where Area means the symplectic area, and Di = φi(D2).

Example 7.5.2. If (M, ω) is a symplectic manifold, then the pair groupoid M ×M ⇒ M is a symplectic groupoid, where M means M with the opposite symplecticform −ω.


Example 7.5.3. The cotangent bundle T ∗N of a manifold N has a natural sym-plectic groupoid structure over N : the source map coincides with the target mapand is the projection map from T ∗N to N , the symplectic structure on T ∗N is thestandard one, and the multiplication map is given by the usual sum of covectorswith the same base points.

Example 7.5.4. Consider the transformation groupoid G×g∗ ⇒ g∗ of the coadjointaction of a Lie group G. Identify G× g∗ with T ∗G via left translations, and equipit with the standard symplectic form. Then it becomes a symplectic groupoid,which we will call a standard symplectic groupoid and denote by T ∗G ⇒ g∗.The corresponding Poisson structure on g∗ is the standard linear (Lie-) Poissonstructure.

We will put together some basic facts about symplectic groupoids into thefollowing theorem:

Theorem 7.5.5 ([189, 349, 83, 3]). If (Γ, σ) ⇒ P is a symplectic groupoid withsymplectic form σ, source map s, target map t, and inversion map ı, then wehave:

a) dim P = 12 dimΓ.

b) P is a Lagrangian submanifold of Γ (P is identified with the image ε(P ) ofthe identity section ε : P → Γ).

c) The foliation by s-fibers is symplectically dual to the foliation by t-fibers. Inother words, for any g ∈ Γ, Tgs

−1(s(g)) = (Tgt−1(t(g)))⊥.

d) The inversion map ı : Γ → Γ is an anti-symplectomorphism:

ı∗σ = −σ. (7.40)

e) For any two functions φ, ψ on P ,

s∗φ, t∗ψ = 0. (7.41)

f) For any function f on P , the Hamiltonian vector field Xs∗f is tangent to t-fibers and is invariant under left translations in Γ. Similarly, Xt∗f is tangentto s-fibers and is invariant under right translations.

g) There is a unique Poisson structure Π on P such that the source map s is aPoisson map, and the target map t is anti-Poisson.

h) If B is a sufficiently small open neighborhood of a point x in a submanifold inP which intersects the symplectic leaf O = O(x) of x transversally at x, thenthe slice (ΓB , ω|ΓB ) ⇒ B is a symplectic groupoid, and the correspondingPoisson structure on B is the transverse Poisson structure of P at x.

i) For any point x ∈ P , the Lie algebra of the isotropy group Γx corresponds tothe linear part of the transverse Poisson structure of (P, Π) at x.

j) If x ∈ P is a regular point (with respect to the characteristic foliation on P ),then the isotropy group Γx is Abelian.


k) For each g ∈ Γ, the set Kg = s−1(s(g)) ∩ t−1(t(g)) = g.Gs(g) = Gt(g).g is anisotropic submanifold of Γ, and if h ∈ Kg then Kh = Kg.

l) Denote by Preg the set of regular elements of P , and by Γreg the set of g ∈ Γsuch that s(g) ∈ Preg. Then the pull-back of the characteristic foliation on Pby s or t is a coisotropic foliation on Γreg whose dual isotropic foliation has asleaves connected components of the submanifolds Kg = s−1(s(g))∩ t−1(t(g)).

m) Conversely, suppose that (Γ, σ) ⇒ P is a Lie groupoid equipped with a sym-plectic structure σ and a Poisson structure Π such that the inversion map ıis anti-symplectic, and the foliation by s-fibers is symplectically dual to thefoliation by t-fibers; then it is a symplectic groupoid.

Proof. a) The graph ∆ given by (7.37) has dimension dim ∆ = 2 dim Γ − dimP .If it is a Lagrangian submanifold in Γ × Γ × Γ, then its dimension is half thedimension of Γ × Γ × Γ, i.e., dim∆ = 3

2 dimΓ, implying that dimP = 12 dimΓ.

Remark that, as a consequence, the dimension of each s-fiber and each t-fiber isequal to the dimension of P , i.e., half the dimension of Γ.

b) If X, Y are two tangent vectors to P and a point x ∈ P , then (X, X, X) and(Y, Y, Y ) are two tangent vector fields to ∆ at (x, x, x) ∈ ∆. Since ∆ is Lagrangian,we have 0 = (σ⊕ σ⊕−σ)((X, X, X), (Y, Y, Y )) = σ(X, Y ) + σ(X, Y )− σ(X, Y ) =σ(X, Y ). It means that P is an isotropic submanifold of Γ. Since dim P = 1

2 dimΓ,it is a Lagrangian submanifold.

c) If X ∈ Tgs−1(s(g)) and Y ∈ Tgt

−1(t(g)) where g ∈ Γ, then (X, ı∗X, 0)and (Y, 0, (Lg−1)∗Y ) are two tangent vector fields to ∆ at (g, g−1, g.g−1) ∈ ∆,where Lg−1 : t−1(t(g)) → t−1(s(g)) is the left translation by g−1. Again, since∆ is Lagrangian, we have 0 = σ(X, Y ) + σ(ı∗X, 0) − σ(0, (Lg−1)∗Y ) = σ(X, Y ).Since dimTgs

−1(s(g)) = dimTgt−1(t(g)) = 1

2 dim σ, they are symplectically dualto each other.

d) If X, Y ∈ TgΓ then (X, ı∗X, X ′) and (Y, ı∗, Y ′) are tangent to ∆ at(g, g−1, s(g)), where X ′ = s∗X, Y ′ = s∗Y ∈ Ts(g)P . Using Assertion b) and thefact that ∆ is Lagrangian, we get σ(X, Y ) + σ(ı∗X, ı∗Y ) = 0.

e) It is a direct consequence of c) that Xs∗φ(g) ∈ Tgt−1(t(g)) and Xt∗ψ(g) ∈

Tgs−1(s(g)), therefore s∗φ, t∗ψ(g) = ω(Xs∗φ, Xt∗ψ)(g) = 0.

f) It follows directly from e).g) If φ, ψ are two smooth functions on P , then it follows from d) and the

Jacobi identity that s∗φ, s∗ψ, t∗ξ = 0 for any function ξ on P . It implies thats∗φ, s∗ψ is invariant on s-fibers, so there is a unique function on P , which we willdenote by φ, ψ, such that s∗φ, s∗ψ = s∗φ, ψ. This defines a unique Poissonstructure on P , which we will denote by Π, such that the map s : Γ → P is aPoisson map. Since t = s ı and ı is anti-symplectic, t is anti-Poisson.

h) Since the graph ∆B of the multiplication map in ΓB is a subset of ∆, itis isotropic, and since dim ΓB = dimΓ− 2 dimO and dimB = dimP − dimO, westill have that dim∆B = (3/2) dimΓB. In order to show that ΓB is a symplecticgroupoid, it remains to verify that the restriction of ω to ΓB is nondegenerate. Letf1, . . . , f2s be a family of independent local functions on P such that fi|B = 0,


where 2s = dimO. Let g be an arbitrary point of ΓB. We have s(g), t(g) ∈ B. Con-sider 4s tangent vectors to Γ at g, Xi = Xs∗fi(g) and Yi = Xt∗fi(g), i = 1, . . . , 2s.It follows from the previous points that we have ω(Xi, Yj) = 0 ∀ i, j, while thematrix ω(Xi, Xj) = fi, fj(s(g)) and the matrix ω(Yi, Yj) = −fi, fj(t(g)) arenondegenerate provided that B is small enough. Thus the space W spanned byX1, . . . , X2s, Y1, . . . , Y2s is a symplectic 4s-dimensional subspace of TgΓ. It also fol-lows from the previous points of the theorem that W is symplectically orthogonalto TgΓB, and dimTgΓB + dim W = dimΓ. Hence TgΓB is a symplectic subspaceof TgΓ. In other words, the restriction of ω to ΓB is nondegenerate.

Consider now a local coordinate system ψ1, . . . , ψm on B, where m = dim B,and extend them to functions ψ1, . . . , ψm in a neighborhood of x in P such thatfi, ψj(y) = 0 ∀ y ∈ B, ∀ i, j. Then one verifies directly that for any g ∈ ΓB wehave Xi(s∗ψj) = Yi(ψj) = 0 ∀ i, j, where Xi, Yi are defined as above, which impliesthat s∗ψi, s

∗ψjω|ΓB(g) = s∗ψi, s

∗ψj(g). Since the source map is a Poissonmap, we have ψi, ψjB(s(g)) = ψi, ψj(s(g)), where the Poisson structure on Bis the one induced from the symplectic structure on ΓB. But this last formula isalso a special case of Dirac’s formula for the restriction of the Poisson structurefrom P to B.

i) In view of point h), it is enough to prove point i) in the case when x isa fixed point of the symplectic groupoid Γ ⇒ P . Consider a coordinate systemψ1, . . . , ψm in a neighborhood of x in P . Then the vector fields Xs∗ψ1 , . . . , Xs∗ψm

are tangent to the isotropy group Γx = s−1(x) = t−1(x), and form a basis ofleft-invariant vector fields there. So we have

Xs∗ψi,s∗ψj|Γx= Xs∗ψi , Xs∗ψj|Γx

=∑

k

ckijXs∗ψk

|Γx, (7.42)

where ckij are structural constants of the Lie algebra g of Γx. Since the source map

is a Poisson map, it follows that

ψi, ψj =∑

k

ckijψk + O(2) (7.43)

on P , i.e., ckij are structural constants are also structural constants for the linear

part of the Poisson structure on P at x.j) If x is a regular point of P then the transverse Poisson structure at x is

trivial. Now apply point i).k) It follows directly from point c).l) It follows directly from point c) and point k).m) The proof of the converse part will be left as an exercise. We will also denote a symplectic groupoid by (Γ, σ) ⇒ (P, Π) to emphasize

the fact that Γ is a symplectic manifold and P is a Poisson manifold. In particular,Assertions b) and g) of the above theorem say that s : (Γ, σ, ε(P )) → (P, Π) is amarked symplectic realization of P with the marked Lagrangian submanifold ε(P ).


Exercise 7.5.6. Show that any Poisson vector field on the base space (P, Π) of asymplectic groupoid (Γ, σ) ⇒ (P, Π) with connected s-fibers can be naturally liftedto a symplectic vector field on (Γ, σ) which preserves the groupoid structure of Γ.Find an example when this symplectic vector field is not (globally) Hamiltonian.

A Poisson manifold is called integrable [95] if it can be realized by a symplecticgroupoid. For example, any symplectic manifold is an integrable Poisson manifold(see Example 7.5.2). The integrability and non-integrability of Poisson manifoldswill be discussed in Section 8.8.

Example 7.5.7. Let G be a Lie group with a free proper Hamiltonian action ona symplectic manifold (M, ω) with an equivariant momentum map µ : M → g∗.Then the quotient M/G, together with the reduced Poisson structure from M , isan integrable Poisson manifold. A symplectic groupoid integrating M/G is

M ∗M/G ⇒ M/G, (7.44)

whereM ∗M = (x, y) ∈ M ×M, µ(x) = µ(y), (7.45)

i.e., M ∗ M/G is the Marsden–Weinstein reduction of the symplectic manifold(M, ω) × (M,−ω) with respect to the diagonal action of G. The structure mapsof this groupoid are induced from the structure maps of the pair groupoid M ×M ⇒ M in an obvious way. This example is a symplectic groupoid version ofreduction by Lie group actions: while the symplectic manifold M is reduced to thePoisson manifold M/G, the corresponding symplectic groupoid M ×M is reducedto M ∗M/G.

Exercise 7.5.8. What are the isotropy groups of the above symplectic groupoidM ∗M/G ⇒ M/G?

7.5.2 Proper symplectic groupoids

Definition 7.5.9. A symplectic groupoid (Γ, σ) ⇒ (P, Π) is called proper if it isproper as a Lie groupoid.

Exercise 7.5.10. Find necessary and sufficient conditions for the symplecticgroupoid M ∗M/G ⇒ M/G given in Example 7.5.7 to be proper.

It follows point h) of Theorem 7.5.5 and point v) of Proposition 7.3.3 that aslice of a proper symplectic groupoid is a proper symplectic groupoid with a fixedpoint. The local structure of proper symplectic groupoids is given by the followingtheorem:

Theorem 7.5.11 ([370]). If Γ ⇒ P is a proper symplectic groupoid with a fixedpoint x ∈ P , then it is locally symplectically isomorphic to the standard symplecticgroupoid T ∗G ⇒ g∗, where G = Γx is the isotropy group of x. In other words,there is an open neighborhood U of x in P and an open neighborhood V of 0 in g∗


such that the restriction of Γ to U is isomorphic to the restriction of T ∗G ⇒ g∗

to V by a symplectic Lie groupoid isomorphism.

Proof. Recall from point i) of Theorem 7.5.5 that the linear part of the Poissonstructure Π at x is isomorphic to the Lie–Poisson structure on g∗. Theorem 7.4.7allows us to linearize Γ near x without the symplectic structure. The correspondinglinear action of G must be (isomorphic to) the coadjoint action, so without losinggenerality we may assume that P is a neighborhood of x = 0 in g∗, and the orbitson P near 0 are nothing but the coadjoint orbits (though the symplectic formon each orbit may be different from the standard one). But then, as was shownby Ginzburg and Weinstein [148] using a standard Moser’s path argument (seeAppendix A.1), since G is compact, the Poisson structure on P is actually locallyisomorphic to the Lie–Poisson structure of g∗. We can now apply the followingproposition to finish the proof of Theorem 7.5.11:

Proposition 7.5.12. If G is a (not necessarily connected ) compact Lie group andg is its Lie algebra, then any proper symplectic groupoid (Γ, ω) ⇒ U with a fixedpoint 0 whose base Poisson manifold is a neighborhood U of 0 in g∗ with theLie–Poisson structure and whose isotropy group at 0 is G is locally isomorphic toT ∗G ⇒ g∗.

Proof of Proposition 7.5.12. Without loss of generality, we can assume that Γ issource-locally trivial.

We will first prove the above proposition for the case when G is connected.The Lie algebra g can be written as a direct sum g = s⊕l, where s is semisimple andl is Abelian. Denote by (f1, . . . , fn, h1, . . . , hm) a basis of linear functions on g∗,where f1, . . . , fn correspond to s and h1, . . . , hm correspond to l. Then the vectorfields Xs∗fi , Xs∗hj generate a Hamiltonian action of g on (Γ, ω). When restrictedto the isotropy group G = Γ0 over the origin of g∗, the vector fields Xs∗fi , Xs∗hj

become left-invariant vector fields on G, and the action of g integrates to theright action of G on itself by multiplication on the right. Assume that the aboveHamiltonian action of g integrates to a right action of G on Γ. Then we are done.Indeed, since the action is free on Γ0, we may assume, by shrinking the basespace U , that the action is free on Γ. Then one can verify directly that the map(g, y) → ε(Ad∗

gy) g, g ∈ G, y ∈ U , where ε : U → Γ denotes the identity sectionand g denotes the right action by g, is a symplectic isomorphism between therestriction of the standard symplectic groupoid G × g∗ ∼= T ∗G ⇒ g∗ to U ⊂ g∗

and Γ.In general, the action of g on Γ integrates to an action of the universal

covering of G on Γ, which does not factor to an action of G on Γ if the Abelianpart of G is nontrivial, i.e., l = 0. So we may have to change the generators ofthis g action, by changing h1, . . . , hm to new functions h′

i which are still Casimirfunctions of g∗. Such a change of variables (leaving fi intact) will be a local Poissonisomorphism of g∗. We want to choose h′

i so that the Hamiltonian vector field Xs∗h′i

are periodic, i.e., they generate Hamiltonian T1-actions.


Note that for each y ∈ U ⊂ g∗, the isotropy group Γy = s−1(y)∩t−1(y) of Γ aty admits a canonical injective homomorphism to G (via an a priori non-symplecticlocal linearization of Γ using Theorem 7.4.7). Denote by Tm

0 the Abelian torus ofdimension m in the center of G (the Lie algebra of Tm

0 is l). The coadjoint actionof Tm

0 on g∗ is trivial. It follows that each isotropy group Γy contains a torus Tmy

whose image under the canonical injection to G is Tm0 . For each q ∈ Γ, denote

Tmq = q.Tm

s(q) = Tmt(q).q. Note that if r ∈ Tm

q then Tmr = Tm

q .Choose a basis γ1, . . . , γm of one-dimensional sub-tori Tm. Translate them to

each point q ∈ Γ as above, we get m curves γ1,q, . . . , γm,q ⊂ Tmq ∀ q ∈ Γ. Recall

that, due to the fact that Tm0 lies in the center of G, these curves are well defined

and depend continuously on q.Since G = s−1(0) = t−1(0) is a Lagrangian submanifold of Γ, the symplectic

form ω of Γ is exact (near G) and we can write ω = dα. Define m functions Hi,i = 1, . . . , m on Γ via the following integral formula, known as Arnold–Mineur’sformula for action functions of integrable Hamiltonian systems (see Appendix A.4):

Hi(q) =∫

γi,q

α. (7.46)

Recall from Theorem 7.5.5 that the “regular” part Γreg of Γ admits a nat-ural symplectically complete foliation by isotropic submanifolds Kq = q.Gs(q),and since Γ is proper, these submanifolds are compact. So this foliation is aproper non-commutatively integrable Hamiltonian system (see Appendix A.4).Since γi,q ⊂ Kq ∀i, it follows from the classical Arnold–Liouville–Mineur theoremon action-angle variables of integrable Hamiltonian systems (Theorem A.4.5) thatHi are action functions, i.e., the Hamiltonian vector fields XHi are periodic (ofperiod 1) and generate T1-actions, and they are tangent to the isotropic subman-ifolds Kq, q ∈ Γ. This fact is true in Γreg, which is dense in Γ, so by continuityit’s true in Γ.

By construction, the action functions Hi are invariant on the leaves of thedual coisotropic foliation of the foliation by Kg, g ∈ Γreg, so they project to(independent) Casimir functions on P . In other words, we have m independentCasimir functions h′

1, . . . , h′m such that s∗h′

i = t∗h′i = Hi.

The infinitesimal action of g on Γ generated by Hamiltonian vector fieldsXs∗fi , Xs∗h′

j, where the functions f1, . . . , fn are as before, now integrates into an

action of S × Tm on Γ, where S is the connected simply-connected semisimpleLie group with Lie algebra s. The group S × Tm is a finite covering of G, i.e., wehave an exact sequence 0 → G → S × Tm → G → 0, where G is a finite group.Indeed, by construction, for every element g ∈ G ⊂ S × Tm, the action φ(g) of gon Γ is identity on the isotropy group G, and its differential at the neutral elemente ∈ G ⊂ Γ is also the identity map of TeΓ. Since a finite power of φ(g) is theidentity map on Γ, it follows that φ(g) itself is the identity map. Hence the actionof G on Γ is trivial, and the action of S×Tm on Γ factors to a Hamiltonian actionof G on Γ. The proposition is proved for the case when G is connected.


Consider now the case G is disconnected. Denote by G0 the connected com-ponent of G which contains the neutral element, and by Γ0 the corresponding con-nected component of Γ0 (we assume that the base U is connected and sufficientlysmall). Then Γ0 is a proper symplectic groupoid over U whose isotropy group at0 is G0. According to the above discussion, Γ0 can be locally symplectically lin-earized, i.e., we may assume that Γ0 is symplectically isomorphic to (T ∗G0 ⇒ g∗)U

with the standard symplectic structure. Consider a map φ : Γ → G, whose restric-tion to the isotropy group G = s−1(0)∩ t−1(0) is identity, and whose restriction toΓ0 is given by the projection T ∗G0 ∼= G0×g∗ → G0 after the above symplectic iso-morphism from Γ0 to (G0×g∗ ⇒ g∗)U . We can arrange it so that φ(p) = φ(p−1)−1

for any p ∈ Γ, and also φ(p).φ(q) = φ(p.q) for any p ∈ Γ0, q ∈ Γ. (This is possiblebecause φ|Γ0 : Γ0 → G0 is a homomorphism.) Then the averaging process used inthe proof of Theorem 7.4.7 does not change the value of φ on Γ. By repeating theproof of Theorem 7.4.7, we get a homomorphism φ∞ : Γ → G, which coincideswith φ on Γ0.

Identifying Γ with G × U via the isomorphism p → (φ∞(p), s(p)) as in theproof of Theorem 7.4.7, and then with (T ∗G ⇒ g∗)U , we will assume that Γ,as a Lie groupoid, is nothing but the restriction (T ∗G ⇒ g∗)U of the standardsymplectic groupoid T ∗G ⇒ g∗ to U ⊂ g∗, and the symplectic structure ω on(T ∗G)U

∼= G × U coincides with the standard symplectic structure ω0 on theconnected component (T ∗G0)U

∼= G0 ×U . For each θ ∈ G/G0, we will denote thecorresponding connected component of G by Gθ and the corresponding connectedcomponent of Γ by Γθ. We will use Moser’s path method to find a groupoidisomorphism of Γ which moves ω to ω0.

Let f : U → R be a function on U . Then according to Theorem 7.5.5, theHamiltonian vector fields Xω

s∗f and Xω0s∗f of s∗f with respect to ω and ω0 are both

invariant under left translations in Γ, and since they coincide in Γ0 they mustcoincide in Γ, because any element in Γ can be left-translated from an elementin Γ0. So we have a common Hamiltonian vector field Xs∗f for both ω and ω0.Similarly, we have a common Hamiltonian vector field Xt∗f for both ω and ω0. Itmeans that iX(ω − ω0) = 0 for any X ∈ Tps

−1(s(p)) + Tpt−1(t(p)), which implies

that ω−ω0 is a basic closed 2-form with respect to the coisotropic singular foliationwhose leaves are connected components of the sets s−1(s(t−1(t(p))), p ∈ Γ. Inparticular, for any connected component Γθ of Γ, where θ ∈ G/G0, there is aunique closed 2-form βθ on U , which is basic with respect to the foliation by theorbits of the coadjoint action of G0 on U , such that

ω − ω0 = s∗βθ on Γθ. (7.47)

The coadjoint action of G on U induces an action ρ of G/G0 on the space ofconnected coadjoint orbits (orbits of G0) on U : if O is a connected coadjoint orbiton U , then ρ(θ)(O) is the orbit Ad∗

GθO. Since Γ is a symplectic groupoid withrespect to both ω and ω0, the closed 2-form ω − ω0 is also compatible with theproduct map in Γ, i.e., Equation (7.38) is still satisfied if we replace ω by ω− ω0.


By projecting this compatibility condition to U , we get the following equality:

βθ1θ2 = βθ2 + ρ(θ2)∗βθ1 ∀ θ1, θ2 ∈ G/G0. (7.48)

Since the 2-forms βθ are closed on U which are basic with respect to thefoliation by connected coadjoint orbits (i.e., orbits of the coadjoint action of G0),we can write

βθ = dαθ, (7.49)

where αθ are 1-forms on U which are also basic with respect to the foliation byconnected coadjoint orbits. Indeed, write βθ = dαθ, then define αθ by the averagingformula

αθ =∫

G0(Ad∗

g)∗αθdµG0 , (7.50)

where µG0 is the Haar measure on G0. Then αθ is invariant with respect to thecoadjoint action of G0, and dαθ = βθ. One verifies easily that αθ must automati-cally vanish on vector fields tangent to the coadjoint orbits, or otherwise βθ wouldnot be a basic 2-form.

Moreover, by averaging αθ with respect to the action of G/G0 via the formula

αnewθ =

1|G/G0|

∑θ′∈G/G0

(αθ′θ − ρ(θ)∗αθ), (7.51)

we may assume that the 1-forms αθ satisfy the equation

αθ1θ2 = αθ2 + ρ(θ2)∗αθ1 ∀ θ1, θ2 ∈ G/G0. (7.52)

Consider the vector field Z on Γ defined by

s∗αθ = iZω = iZω0 on Γθ. (7.53)

One verifies directly that the flow φtZ of Z preserves the groupoid structure of Γ,

and φ1Z moves ω to ω0.

Corollary 7.5.13 ([370]). If (Γ ⇒ P, ω + Ω) is a proper symplectic groupoid whoseisotropy groups are connected, then the orbit space P/Γ is a manifold with locallypolyhedral boundary. Moreover, this orbit space admits a natural integral affinestructure, and near each point is locally affinely isomorphic to a Weyl chamber(with the standard affine structure) of a compact Lie group (namely the corre-sponding isotropy group).

The reason is that the quotient of the dual of the Lie algebra of a connectedcompact Lie group by its coadjoint action can be naturally identified to a Weylchamber. The affine structure is provided by action functions defined by Formula7.46. In fact, the foliation of the “regular” part Γreg of Γ by tori s−1(x) ∩ t−1(y)turns it to a proper noncommutatively integrable system, the interior of the orbit


space P/Γ is the reduced base space of this integrable system, and the action func-tions provide an integral affine structure on this reduced base space (see AppendixA.4 and [370]). The fact that this intrinsically defined affine structure coincideswith the affine structures coming from the Weyl chambers via local linearizationsof Γ will be left as an exercise.

When the isotropy groups are not connected, then locally P/Γ is isomorphicto a quotient of the Weyl chamber by a finite group action. The reason is that, ifG is a disconnected group, then its coadjoint action on g∗ may mix the connectedcoadjoint orbits (orbits of the connected part G0 of G) by an action of G/G0.

Example 7.5.14. Consider the following disconnected double covering G = T2 "θ.T2 of T2, where θ is an element such that θ.g.θ−1 = g−1 ∀ g ∈ T2. Then thecoadjoint action of G0 = T2 on R2 = Lie(G)∗ is trivial, but the coadjoint actionof θ on R2 is given by the map (x, y) → (−x,−y). The quotient space of R2 bythe coadjoint action of G is the orbifold R2/Z2.

7.5.3 Hamiltonian actions of symplectic groupoids

Definition 7.5.15. A Hamiltonian action of a symplectic groupoid (Γ, σ) ⇒ (P, Π)on a Poisson manifold (M, Λ) is a Lie groupoid action such that its graph(g, y, z) | (g, y) ∈ Γ ∗M, z = g.y is a coisotropic submanifold of Γ ×M ×M ,where M denotes M with the opposite Poisson structure −Λ.

Remark that the dimension of the graph (g, y, z) | (g, y) ∈ Γ ∗M, z = g.yis half the dimension of Γ×M ×M . In the case when M is symplectic, to say thatthis graph is coisotropic is the same as to say that it is isotropic (or Lagrangian).The coisotropic condition is a compatibility condition which may be expressed bya formula similar to Equation (7.38).

Example 7.5.16. The action of a symplectic groupoid (Γ, σ) on itself by multipli-cation is Hamiltonian. The natural action of Γ on its Poisson base space is alsoHamiltonian.

Exercise 7.5.17. Show that if a symplectic groupoid (Γ, σ) ⇒ (P, Π) acts Hamil-tonianly on a Poisson manifold (M, Λ), then the corresponding momentum mapµ : (M, Λ)→ (P, Π) is a Poisson map.

Theorem 7.5.18 ([248]). Let G be a connected Lie group. Then there is a naturalone-to-one correspondence between Hamiltonian actions of the standard symplecticgroupoid T ∗G ⇒ g∗ and Hamiltonian actions of G.

Proof. The identification between Hamiltonian G-actions and Hamiltonian(T ∗G ⇒ g∗)-actions on a Poisson manifold (M, Λ) is given by the formula

(Lgα).x = g.x, (7.54)

where g ∈ G, x ∈ (M, Λ) such that µ(x) = α ∈ g∗. The rest of the proof is a directverification.


The above theorem means that one may view the theory of Hamiltonianactions of Lie groups as a particular case of the theory of Hamiltonian actions ofsymplectic groupoids.

Exercise 7.5.19. Describe Hamiltonian actions of a symplectic pair groupoid(M, σ)× (M,−σ) ⇒ (M, σ) on Poisson manifolds.

Remark 7.5.20. Lu’s theory of equivariant momentum maps for Poisson actions ofa compact Poisson–Lie group can also be embedded into the theory of Hamiltonianactions of symplectic groupoids, see [223]. In fact, Lu’s theory is in a sense equiv-alent to the usual theory of equivariant momentum maps of Hamiltonian actionsof a compact Lie group, see [5].

7.5.4 Some generalizations

Motivated in part by the theories of quasi-Hamiltonian spaces and twisted Poissonstructures (see, e.g., [8, 6, 7, 315, 197]), Xu [361] and Bursztyn–Crainic–Weinstein–Zhu [53] introduced the notion of quasi-symplectic groupoids, also called twistedpresymplectic groupoids. A quasi-symplectic groupoid is a Lie groupoid Γ ⇒ P ,equipped with a 2-form ω on Γ and a 3-form Ω on P , which satisfy the followingfour conditions:

i) dω = t∗Ω− s∗Ω.ii) dΩ = 0.iii) The graph ∆ = (p, q, p.q) | p, q ∈ Γ, s(p) = t(q) of the product operation

of Γ is isotropic with respect to the 2-form ω ⊕ ω ⊕ (−ω) on Γ× Γ× Γ.iv) Identify P with its unity section ε(P ) in Γ. Due to condition iii), for each

point m ∈ P , the differential t∗ of the target map t can be restricted to a map

t∗ : kerωm ∩ Tms−1(m) → kerωm ∩ TmP , (7.55)

where kerωm denotes the kernel of ω at m, and the condition is that thisrestricted map is bijective.The first three conditions mean that ω is a twisted presymplectic form and

Ω is the twisting term, and the last condition is a weak nondegeneracy conditionon ω. If ω is nondegenerate and Ω = 0 then one gets back to the notion ofsymplectic groupoids. The base space P of a quasi-symplectic groupoid is a twistedDirac manifold, and if the 2-form ω is nondegenerate then P is a twisted Poissonmanifold, see [53, 65, 52]. For Dirac structures, see Appendix A.8.

It is easy to check that a sufficiently small slice of a (proper) quasi-symplecticgroupoid is again a (proper) quasi-symplectic groupoid. A result of Xu (Proposi-tion 4.8 of [361]) says that if (Γ ⇒ P, ω + Ω) is a quasi-symplectic groupoid, andβ is an arbitrary 2-form on P , then (Γ ⇒ P, ω′ + Ω′), where ω′ = ω + t∗β − s∗βand Ω′ = Ω+dβ, is again a quasi-symplectic groupoid, and moreover it is Morita-equivalent in a natural sense to (Γ ⇒ P, ω+Ω). This result together with Theorem7.5.11 lead to the following proposition.


Proposition 7.5.21 ([370]). If (Γ ⇒ P, ω+Ω) is a proper quasi-symplectic groupoidwith a fixed point m, then it is locally isomorphic to a quasi-symplectic groupoid ofthe type (T ∗G ⇒ g∗, ω0 + t∗β − s∗β + dβ), where (T ∗G ⇒ g∗, ω0) is the standardsymplectic groupoid of the isotropy group G = Γm of m, and β is a 2-form on g∗.If, moreover, the isotropy groups of Γ are connected, then the orbit space P/Γ isan integral affine manifold with locally convex polyhedral boundary which locallylooks like a Weyl chamber.

The (local) convexity of orbit spaces P/Γ of symplectic and quasi-symplecticgroupoids is very closely related to convexity properties of momentum maps insymplectic geometry, see [370].

Example 7.5.22. Consider the AMM (Alekseev–Malkin–Meinrenken) groupoid[361]: it is the transformation groupoid G ×G ⇒ G of the conjugation action ofa compact Lie group G, equipped with a natural quasi-symplectic structure aris-ing from Alekseev–Malkin–Meinrenken’s theory of group-valued momentum maps[8]. Xu [361] developed a theory of quasi-Hamiltonian actions of quasi-symplecticgroupoids, and showed a natural equivalence between quasi-Hamiltonian spaceswith G-valued momentum maps and quasi-Hamiltonian spaces of the AMMgroupoid. The orbit space of the AMM groupoid is naturally affine-equivalentto a Weyl alcove of G. In particular it is a convex affine polytope.

There is another interesting class of groupoids, called Poisson groupoids ,introduced by Weinstein [350]. The definition of a Poisson groupoid is similarto the definition of a symplectic groupoid, except for the fact that the arrowspace Γ is now equipped with a Poisson instead of a symplectic structure, andthe graph of the multiplication map is now required to be coisotropic instead ofLagrangian. For example, the pair groupoid of a Poisson manifold is a Poissongroupoid. Poisson groupoids generalize at the same time symplectic groupoidsand Poisson–Lie groups, and play an important role in Poisson geometry. Theyare, unfortunately, out of the scope of this book. See, e.g., [229, 230, 231, 228] andreferences therein. Just as Poisson–Lie groups are related to r-matrices, Poissongroupoids are very closely related to so-called dynamical r-matrices, see, e.g., [125]and references therein.

Chapter 8

Lie Algebroids

8.1 Some basic definitions and properties

8.1.1 Definition and some examples

Lie algebroids were introduced by Pradines [294] as infinitesimal versions of Liegroupoids.

Definition 8.1.1. A Lie algebroid (A → M, [, ], ) is a (finite-dimensional) vectorbundle A→M over a manifold M , equipped with a linear bundle map : A→ TMcalled the anchor map, and a Lie bracket [, ] on the space Γ(A) of sections of A,such that the following Leibniz rule is satisfied:

[α, fβ] = (α(f))β + f [α, β] (8.1)

for any sections α, β of A and function f on M .

We will often denote a Lie algebroid simply by a letter A, or by (A, [, ], ).

Remark 8.1.2. The above definition makes sense in many categories: formal,smooth, real analytic, holomorphic, etc. (In the real analytic and holomorphiccases, Γ(A) should be replaced by the sheaf of local analytic sections of A.) It alsomakes sense when M is a manifold with boundary.

Remark 8.1.3. Due to the Leibniz rule, the Lie bracket is a bi-differential operatorof first order in each variable. In other words, the value of [α, β] at a point xdepends only on the value of α, β and their first derivatives at x. As a consequence,the restriction of a Lie algebroid over M to an open subset of M is again a Liealgebroid. We can also talk about a germ of a Lie algebroid at a point x ∈ M .

Lemma 8.1.4. If (A, [, ], ) is a Lie algebroid, then the anchor map is a Lie algebrahomomorphism:

[α, β] = [α, β] ∀ α, β ∈ Γ(A). (8.2)

236 Chapter 8. Lie Algebroids

Proof. By the Jacobi identity and the Leibniz rule, we have

0 = [[α, β], fγ] + [[β, fγ], α] + [[fγ, α], β]= f [[α, β], γ] + ([α, β](f))γ

+f [[β, γ], α]− (α(f))[β, γ] + (β(f))[γ, α] − (α(β(f)))γ+f [[γ, α], β]− (β(f))[γ, α] − (α(f))[γ, β] + (β(α(f)))γ

= (([α, β] − [α, β])(f))γ.

Since this is true for any α, β, γ ∈ Γ(A) and function f , one concludes that [α, β] =[α, β]. Remark 8.1.5. Condition (8.2) is a fundamental property of Lie algebroids, andis often considered as a part of the definition of a Lie algebroid, though it isa consequence of the other conditions. Lemma 8.1.4 is implicit in [173]. See, e.g.,[154] for a discussion on the axioms of Lie algebroids. If, instead of the Lie bracket,one imposes other algebraic structures on Γ(A) and other compatibility conditions,then one arrives at other kinds of algebroids.Remark 8.1.6. Isomorphisms of Lie algebroids can be defined in an obvious way.It is more tricky to define morphisms of Lie algebroids, which will be consideredin Section 8.3

Example 8.1.7. A Lie algebra can be thought of as a Lie algebroid over a point.

Example 8.1.8. If M is a manifold then TM is a Lie algebroid: the anchor mapis the identity map, and the Lie bracket is the usual Lie bracket of vector fields.This is called the tangent algebroid of M . More generally, if F is a regular foliationin M , then the tangent algebroid of F is the vector sub-bundle of TM consistingof tangent spaces to F , with the usual Lie bracket, and the inclusion map asthe anchor. Any Lie algebroid whose anchor map is injective is isomorphic to thetangent algebroid of some regular foliation.

Example 8.1.9. If Y is a hypersurface in a manifold M , then according to Melrose[246], there is a vector bundle T (M, Y ) over M , called the Y -tangent bundle ofM , such that the space V1

Y (M) of smooth vector fields on M which are tangentto Y is isomorphic to the space of smooth sections of T (M, Y ). The Lie bracketon V1

Y (M) turns T (M, Y ) into a Lie algebroid.

Example 8.1.10. If ξ : g → V1(M) is an action of a Lie algebra g on a manifold M ,then we can associate to it the following transformation algebroid : the vector bun-dle is the trivial bundle g×M → M , the anchor map is (X, z) = ξ(X)(z) ∀ X ∈g, z ∈M, and the Lie bracket on the sections of g×M →M , considered as mapsfrom M to g, is defined as

[α, β](z) = [α(z), β(z)] + (ξ(α(z)))z(β)− (ξ(β(z)))z(α). (8.3)

In particular, if α, β are two constant sections then their bracket is a constantsection given by the Lie bracket of g. The last two terms in the above formula are

8.1. Some basic definitions and properties 237

due to the Leibniz rule. We will denote the transformation algebroid of an actionof g on M as g M (it is a kind of semi-direct product in the category of Liealgebroids). For a linear action of g on a vector space V , to avoid confusion withthe Lie algebra g V , we will denote the corresponding transformation algebroidas gV .

Example 8.1.11. If the anchor map is identically zero, then the Lie bracket onΓ(A) is a point-wise Lie bracket: due to the equality [α, fβ] = f [α, β], the valueof [α, β] over a point z ∈ M depends only on the value of α and β over z. This isthe case of a bundle of Lie algebras over M .

Example 8.1.12. Attached to each Poisson manifold (M, Π), there is a natural Liealgebroid structure (T ∗M, [, ], ) on the cotangent bundle of M , whose anchor map : T ∗M → TM is the usual anchor map of Π, 〈α, β〉 = Π(α, β) ∀ α, β ∈ Ω1(M),and whose Lie bracket is

[α, β] = Lαβ − Lβα− dΠ(α, β)= d(Π(α, β)) + iαdβ − iβdα.

(8.4)

The above Lie bracket on differential 1-forms of (M, Π) probably first appearedin the work of Fuchssteiner [139]. It is immediate that this bracket satisfies theLeibniz rule:

[α, fβ] = d(f(Π(α, β))) + iαd(fβ)− fiβdα

= f [α, β] + Π(α, β)df + iα(df ∧ β)= f [α, β] + ((α)(f))β.

Let us verify that this bracket satisfies the Jacobi identity. If α = df, β = dg, γ =dh are exact 1-forms, then by definition [df, dg] = df, g and so on, and theJacobi identity for the triple (df, dg, dh) follows from the Jacobi identity for thetriple (f, g, h) with respect to the Poisson bracket. For more general 1-forms, onecan use the Leibniz rule to reduce to the case of exact 1-forms. (T ∗M, [, ], ) iscalled the cotangent algebroid of (M, Π).

Exercise 8.1.13. Show that a Lie algebroid structure ([, ], ) on T ∗M comes from aPoisson structure on M if and only if is anti-symmetric (i.e., 〈α, β〉 = −〈β, α〉for any two 1-forms α, β on M) and the bracket of two arbitrary closed 1-forms isagain a closed 1-form.

8.1.2 The Lie algebroid of a Lie groupoid

Let Γ ⇒ M be a Lie groupoid over a manifold M, with source map s and targetmap t. A tangent vector field X on Γ is called a left-invariant vector field if itsatisfies the following two properties: i) X is tangent to t-fibers t−1(x) (x ∈ M).ii) X is invariant under the left action of Γ on itself: for each g ∈ Γ, the lefttranslation Lg : h → g.h, which maps t−1(s(g)) to t−1(t(g)), preserves X .


Denote the space of smooth left-invariant vector fields on Γ by VL(Γ). Itis clear the Lie bracket of two left-invariant vector fields is again a left-invariantvector field:

[VL(Γ),VL(Γ)] ⊂ VL(Γ). (8.5)

Denote by A the vector bundle over M consisting of tangent spaces to t-fibersat M :

Ax = Tx(t−1(x)). (8.6)

By left translations, each section of A gives rise to a unique left-invariant vectorfield on G. In other words, the space of smooth sections Γ(A) of A may be identifiedwith VL(Γ), and therefore it inherits the Lie bracket from VL(Γ). The anchor map : A→ TM is defined to be the (restriction to A of the) differential of the sourcemap s : Γ →M. This anchor map can also be defined as follows: identify C∞(M)with the space C∞

L (Γ) of left-invariant functions on Γ. For each X ∈ VL(Γ) andf ∈ C∞

L (Γ), one has X(f) ∈ C∞L (Γ). This way sections of A can be mapped to

derivations of C∞(M), i.e., vector fields on M . One verifies directly that, equippedwith the above Lie bracket and anchor map, A becomes a Lie algebroid, called theLie algebroid of the Lie groupoid Γ ⇒ M , and sometimes denoted by Lie(Γ).Remark 8.1.14. Some authors use right-invariant vector fields tangent to s-fibersto define the Lie algebroid of a Lie groupoid. The resulting Lie algebroid is thesame as the one given by left-invariant vector fields, up to an isomorphism.

Example 8.1.15. If G M is the transformation groupoid of a smooth actionof a Lie group G on a manifold M , then its Lie algebroid is the transformationalgebroid g M of the corresponding Lie algebra action g → V1(M).

Exercise 8.1.16. Show that the Lie algebroid of a smooth pair groupoid M×M ⇒M is isomorphic to the tangent algebroid of M .

Exercise 8.1.17. Show that the Lie algebroid of a symplectic groupoid (Γ, σ) ⇒(P, Π) is isomorphic to the cotangent algebroid of (P, Π).

8.1.3 Isotropy algebras

Let (A, [, ], ) be a Lie algebroid over a manifold M . For a point z ∈ M , we willdenote by Az the fiber of A over z, and by ker (z) the kernel of the anchor map

z : Az → TzM. (8.7)

The kernel ker (z) has a natural Lie algebra structure, defined as follows.For any αz , βz ∈ ker (z), denote by α, β arbitrary sections of A whose value at zis αz and βz respectively, and put

[αz , βz] = [α, β](z). (8.8)

The above bracket on ker (z) is well defined. Indeed, if α is another section ofA with α(z) = αz , then locally there are functions f1, . . . , fn on M which vanish

8.1. Some basic definitions and properties 239

at z and a basis α1, . . . , αn of A such that α − α =∑

fiαi. By the Leibniz rule,we have [α, β](z)− [α, β](z) =

∑fi(z)[αi, β](z)−∑

(β)(fi)(z)αi(z) = 0, becausefi(z) = 0 and β(z) = 0.

The kernel ker (z) together with its natural Lie bracket is called the isotropyalgebra of A at z.

8.1.4 Characteristic foliation of a Lie algebroid

The distribution C : x ∈ M → Cx = Imx, where is the anchor map of a Liealgebroid (A, [, ], ), is called the characteristic distribution of A. The dimensionof Imx is called the rank of (or of A) at x. If A is smooth then its characteristicdistribution is a smooth distribution generated by vector fields of the type α,where α is a smooth section of A. Similarly to the case of Poisson manifolds (seeRemark 8.5.6), it is an integrable distribution. The corresponding singular foliationis called the characteristic foliation of A. Its leaves are called leaves or orbits of A.

Exercise 8.1.18. Show that, if A is the Lie algebroid of a Lie groupoid Γ ⇒ M,then the orbits of A are the same as the orbits of Γ, and the isotropy algebra ofA at a point x ∈ M is the Lie algebra of the isotropy group of Γ at x.

If O is an orbit of a Lie algebroid A, then the restriction AO of A to Oalso is a Lie algebroid: If α, β are two sections of A over O, and α, β are theirextensions to sections of A over M , then we can define the bracket of α with βas the restriction of [α, β] to O. The Leibniz rule implies that this bracket doesnot depend on the choice of the extensions α, β. Note that AO is a transitive Liealgebroid , i.e., its anchor map is surjective.

Remark 8.1.19. There is a natural question: can every singular foliation be real-ized, at least locally, as the characteristic foliation of a Lie algebroid? We don’tknow the answer to this question.

8.1.5 Lie pseudoalgebras

The purely algebraic version of a Lie algebroid is often called a Lie pseudoalgebra.By definition, a Lie pseudoalgebra is a pair (L, C), where C is a commutativealgebra over a commutative ring R, and L is a Lie algebra over R, which is aC-module and which acts on C by derivations, i.e., ∀α, β ∈ L, f, g ∈ C,

α.(fg) = f(α.g) + g(α.f) (8.9)

and[α, β].f = α.(β.f) − β.(α.f), (8.10)

such that the following compatibility condition (Leibniz rule) is satisfied

[α, fβ] = f [α, β] + (α.f)β. (8.11)


Clearly, if A is a smooth Lie algebroid over M then (Γ(A), C∞(M)) is a Liepseudoalgebra. Many algebraic constructions involving Lie algebroids can in factbe made for Lie pseudoalgebras.Remark 8.1.20. Lie pseudoalgebras are also called by many other names by dif-ferent authors (see [227]): Lie d-ring [290], Lie–Cartan pair [192], Lie–Rinehartalgebra [180], differential Lie algebra [199], etc. They were probably first studiedby Herz [173] and Rinehart [301], before the appearance of Lie algebroids.

8.2 Fiber-wise linear Poisson structures

Let π : E →M be a vector bundle. A basic function on E is a function of the typeg π where g is a function on M . A fiber-wise linear function on E is a functionwhose restriction to each fiber of E is linear.

Definition 8.2.1. Let π : E → M be a vector bundle. A Poisson structure Π onE is called a fiber-wise linear Poisson structure if it satisfies the following threeconditions:

(i) The Poisson bracket of any two basic functions is zero.(ii) The Poisson bracket of a basic function and a fiber-wise linear function is a

basic function.(iii) The Poisson bracket of two fiber-wise linear functions is a fiber-wise linear

function.

Recall that there is a natural one-to-one correspondence between finite-dimensional Lie algebras and linear Poisson structures. In this section, we will showa similar natural one-to-one correspondence between Lie algebroids and fiber-wiselinear Poisson structures.

Let (A, [, ], ) be a Lie algebroid over a manifold M . We will construct afiber-wise linear bracket on the total space of the dual bundle A∗ of A as follows:

f, g = 0, (8.12)α, f = (α)(f), (8.13)α, β = [α, β]. (8.14)

Here f, g are functions on M , considered as basic functions on A∗, α, β are sectionsof A, considered as fiber-wise linear functions on A∗. On the right-hand side,(α)(f) is considered as a basic function on A∗, and [α, β] is considered as a fiber-wise linear function. This bracket can be extended to other functions on A∗ by theLeibniz rule. In a local coordinate system (α1, . . . , αn, x1, . . . , xm) on A∗, where(x1, . . . , xm) is a local coordinate system on M and (α1, . . . , αn) is a basis of localsections of A, the corresponding 2-vector field Π on A∗ can be written as follows:

Π =12

∑i,j

[αi, αj ]∂

∂αi∧ ∂

∂αj+∑i,j

(αi)(xj)∂

∂αi∧ ∂

∂xj. (8.15)

8.2. Fiber-wise linear Poisson structures 241

Let us show that the above bracket satisfies the Jacobi identity, i.e., it isa Poisson bracket. By the Leibniz rule, the verification of the Jacobi identity isreduced to a verification in the following four cases: 1) all the three functions arefiber-wise linear; 2) two of them are fiber-wise linear, the third one is basic; 3)two of them are basic, the third one is fiber-wise linear; 4) all three functions arebasic. In the first case, the Jacobi identity is nothing else but the Jacobi identityfor the Lie bracket [, ] on Γ(A). In the second case, it is a consequence of the factthat the anchor map is a Lie algebra homomorphism:

α, β, f = [α, β], f = [α, β](f) = α, β, f − β, α, f.The third and the forth cases are trivial. Thus, we have shown that the abovebracket is a Poisson bracket. And by construction, this Poisson bracket is fiber-wise linear on A∗.

Conversely, given a fiber-wise linear Poisson structure Π on A∗, Formulas(8.13) and (8.14) will define a Lie algebroid structure on A: the Jacobi identity forΓ(A) and the Leibniz rule are just particular cases of the Jacobi identity for Π. Inother words, we have proved the following result.

Theorem 8.2.2. There is a natural one-to-one correspondence between Lie alge-broids and fiber-wise linear Poisson structures, given by Formulas (8.12)–(8.14).

Example 8.2.3. The linear Poisson structure on the dual of a Lie algebra is aparticular case of the preceding construction.

When studying isomorphisms and infinitesimal automorphisms of Lie alge-broids, it may be convenient to view them as fiber-wise linear Poisson structures. APoisson isomorphism φ : E1 → E2 between a fiber-wise linear Poisson structure Π1

on a vector bundle E1 and a fiber-wise linear Poisson structure Π2 on a vector bun-dle E1 is called fiber-wise linear if it is a vector bundle isomorphism. Similarly, aPoisson vector field on a vector bundle E with a fiber-wise linear Poisson structureΠ is called fiber-wise linear if it is an infinitesimal vector bundle isomorphism of E.

A vector field on (the total space of) a Lie algebroid A is called an infinitesi-mal automorphism of A, if its local flow maps fibers to fibers linearly (i.e., it is aninfinitesimal vector bundle isomorphism), and preserves the Lie bracket on Γ(A)and the anchor map.

Exercise 8.2.4. Show that there is a natural one-to-one correspondence betweenLie algebroid isomorphisms and fiber-wise linear Poisson isomorphisms. Similarly,there is a natural one-to-one correspondence between infinitesimal automorphismsof a Lie algebroid A and fiber-wise linear Poisson vector fields on A∗.

Example 8.2.5. Let ξ be a section of a Lie algebroid A. Then ξ may be viewedas a fiber-wise linear function on A∗. Its corresponding Hamiltonian vector fieldXξ is fiber-wise linear, and generates a (local) fiber-wise linear flow of Poissonisomorphisms of A∗, and hence a flow (of automorphisms) on A by duality. Moregenerally, ξ may be a time-dependent section (i.e., a family of sections dependenton a time parameter), then it still generates a flow of automorphisms on A.


One may also be tempted to define morphisms (and not just isomorphisms)from a Lie algebroid A1 to a Lie algebroid A2 via fiber-wise linear Poisson mor-phisms from A∗

2 to A∗1. This actually leads to the notion of comorphisms of Lie

algebroids [174]. Note that in general these comorphisms are not the infinitesimalversion of Lie groupoid morphisms.

8.3 Lie algebroid morphisms

Definition 8.3.1. A Lie algebroid morphism from a Lie algebroid (A → M, [, ], )to a Lie algebroid (A′ →M ′, [, ]′, ′) is a vector bundle morphism φ : A→ A′ (i.e.,a smooth fiber-wise linear map), which is compatible with the anchor maps andthe Lie brackets.

Denote the projection of φ to M by the same letter φ : M → M ′. Thecompatibility of φ with the anchor maps means that

(φ(α)) = (φ)∗(α) ∀ α ∈ A. (8.16)

The compatibility of φ with the Lie brackets means the following, according toHiggins and Mackenzie [174]: for any smooth sections α, β of A with decomposi-tions

φ α =∑

i

fi(α′i φ), φ β =

∑i

gi(β′i φ), (8.17)

where fi, gi are functions on M and α′i, β

′i are sections of A′, we have

φ [α, β] =∑ij

figj([α′i, β

′j ]φ)+

∑j

(α(gj))(β′j φ)−

∑j

(β(fj))(α′j φ). (8.18)

In particular, if there are smooth sections α′, β′ of A′ such that

φ α = α′ φ, φ β = β′ φ, (8.19)

then we also haveφ [α, β] = [α′, β′] φ. (8.20)

Exercise 8.3.2. Show that, if α and β are given, then the right-hand side of (8.18)does not depend on the choice of decompositions of φ α and φ β in (8.17).

Theorem 8.3.3 ([174]). There is a natural functor from the category of Liegroupoids to the category of Lie algebroids, which associates to each Lie groupoidmorphism a Lie algebroid morphism.

Proof. Let φ : (Γ1 ⇒ M1)→ (Γ2 ⇒ M2) be a Lie groupoid morphism. For i = 1, 2,identify the Lie algebroid Ai = Lie(Γi) of Γi with the vector bundle over Mi whosefiber over x ∈Mi is the tangent space to t−1(x) at x (see Subsection 8.1.2). Then

8.4. Lie algebroid actions and connections 243

the map φ∗ : TΓ1 → TΓ2 induces a map, denoted by the same symbol φ∗, fromLie(Γ1) to Lie(Γ2). One verifies directly that

φ∗ : Lie(Γ1)→ Lie(Γ2)

satisfies the conditions of a Lie algebroid morphism. The functoriality of the con-struction is obvious. Example 8.3.4. If (A, [, ], ) is a Lie algebroid over M , then the anchor map :A→ TM is a Lie algebroid morphism from A to the tangent algebroid of M .

Example 8.3.5. Let (A, [, ], ) be a Lie algebroid over M and z a point of M . Thenthe inclusion map from the isotropy algebra ker z of z to A is a Lie algebroidmorphism.

Example 8.3.6. Let L be a leaf of the characteristic foliation of a Lie algebroid A.Then there is a short exact sequence of Lie algebroid morphisms

0 → kerL → AL → TL→ 0, (8.21)

where AL is the restriction of A to L; kerL denotes the totally intransitive Liealgebroid consisting of isotropy Lie algebras of A over L (ker means the kernel ofthe anchor map).

Example 8.3.7. Given two Lie algebroids A1 over M1 and A2 over M2, we can definetheir direct product to be the Lie algebroid A1 × A2 which is dual to the directproduct A∗

1 × A∗2 of fiber-wise linear Poisson manifolds. The natural projections

from A1 ×A2 to A1 and A2 are Lie algebroid morphisms.

Example 8.3.8. A smooth A-path of a Lie algebroid A is a Lie algebroid morphismfrom TI to A, where TI means the tangent algebroid of the interval I = [0, 1].It can be characterized as a path γ : I → A in A, which projects to a base pathπ γ : I →M in M , where π : A→ M denotes the projection, such that

(γ(t)) =ddt

π(γ(t)) ∀ t ∈ I. (8.22)

Exercise 8.3.9. Show that a Lie algebroid morphism between two tangent al-gebroids TN1 and TN2 is the lifting of a smooth map between the manifoldsN1 and N2.

Remark 8.3.10. There is also a notion of comorphisms of Lie groupoids, which isa global version of comorphisms of Lie algebroids, see [175].

8.4 Lie algebroid actions and connections

Actions of Lie algebroids of manifolds are defined similarly to actions of Liegroupoids.


Definition 8.4.1. A Lie algebroid action of a Lie algebroid (A → M, [, ], ) on amanifold N consists of a map µ : N →M called the moment map (or momentummap), and a Lie algebra homomorphism ρ : Γ(A) → V1(N) from the Lie algebra ofsections of A to the Lie algebra of vector fields on N , which satisfies the followingcompatibility conditions: if f is a function on M and α is a section of A then

ρ(fα) = (µ∗f)ρ(α) (8.23)

andα(µ(y)) = µ∗(ρ(α)(y)) ∀ y ∈ N. (8.24)

Exercise 8.4.2. Show that if a groupoid G acts on a manifold N then it inducesan action of the Lie algebroid A = Lie(G) on N .

Example 8.4.3. (A →M, [, ], ) acts on M in an obvious way: the moment map isthe identity map, and the action map ρ : Γ(A) → V1(M) is the anchor.

Exercise 8.4.4. (See [174].) Let ρ : Γ(A) → V1(N) be an action of A on N withmoment map µ : N →M . Show that the pull-back µ∗A of the bundle A→ M bythe map µ has a natural Lie algebroid structure over N .

A special case of Lie algebroid actions is when N = E is a vector bundleover M , the moment map µ : E →M is the projection map, and the vector fieldsρ(α), α ∈ Γ(A), are fiber-wise linear (i.e., they generate local linear isomorphismsof the vector bundle E). In this case we say that we have a linear representationof A, or a linear A-module. It is clear that a linear module of a Lie groupoid G isalso a linear module of its Lie algebroid Lie(G).

Exercise 8.4.5. Let E →M be a vector bundle, and denote by gl(E) = Lie(GL(E))the Lie algebroid of the Lie groupoid of linear isomorphisms among the fibers of E(see Exercise 7.1.21). Show that a linear action of a Lie algebroid (A → M, [, ], )on E is the same as a Lie algebroid morphism from A to gl(E) which projects tothe identity map on M .

Exercise 8.4.6. Show that a linear representation of a tangent algebroid TM on avector bundle E over M is the same as a flat linear connection on E.

The above exercise suggests that linear representations of general Lie alge-broids may be viewed as flat connections in an appropriate sense.

Definition 8.4.7 ([131]). If E is a vector bundle over the base space M of a Liealgebroid A, then by a A-connection on E we mean a linear map (the covariantderivative map)

Γ(A)× Γ(E) → Γ(E), (α, ξ) → ∇αξ (8.25)

such that ∇fαξ = f∇αξ and ∇α(fξ) = f∇αξ + α(f)ξ for any function f on M ,section α of A and section ξ of E.

8.4. Lie algebroid actions and connections 245

Remark 8.4.8. The above notion of A-connections generalizes the usual notion oflinear connections on vector bundles (the case A = TM). When A = T ∗M is thecotangent algebroid of a Poisson manifold, a T ∗M -connection is also known as acontravariant connection, see, e.g., [333, 130]. One can also define A-connectionson principal bundles in a similar way [131].

Similarly to the case of usual linear connections, each A-connection is deter-mined by a corresponding A-horizontal lifting: it is a C∞(M)-linear map

hor : Γ(A)→ V1linear(E) (8.26)

from the space of sections of A to the space of fiber-wise linear vector fields on Esuch that

π∗ hor(α) = α (8.27)

for any α ∈ Γ(A). The word horizontal is not totally correct, because if an elementp ∈ A lies in the kernel of , then hor(p) is actually a vertical vector in E whichmay be nontrivial, but we will use it anyway. Given an A-horizontal lifting, thecorresponding A-covariant derivative can be defined by the usual formula

∇αξ = limδ→0

φδhor(α)ξ − ξ

δ, (8.28)

where φδhor(α) is the (local) fiber-wise linear flow of time δ generated by the vector

field hor(α).Another equivalent way to define a connection is via parallel transport. Given

an A-path a with base bath γ : I →M (see Example 8.3.8), and a point u ∈ Eγ(0),the parallel transport of u along a is the path u(t) ∈ Eγ(t) defined by the formula

u(t) = φthor(αt)

u , (8.29)

where αt is a time-dependent section of A such that αt(γt) = α(t) ∀ t, and φthor(αt)

is the flow of the time-dependent vector field hor(αt). Equivalently, u(t) is thesolution of the ordinary differential equation

∇αu(t) = 0 (8.30)

with initial boundary value u(0) = u, where ∇αu(t) is the covariant derivative ofu(t) along α, which can be defined by the formula

∇αu(t) = ∇αξt(x) +dξt

dt(x) at x = γ(t), (8.31)

where ξt is a time-dependent section of E such that ξt(γ(t)) = u(t). (The aboveformulas do not depend on the choice of αt and ξt.)

Given an A-connection, its curvature is a 2-form R∇ on A with values inE∗ ⊗ E, given by the usual formula

R∇(α, β) = ∇α∇β −∇β∇α −∇[α,β]. (8.32)


(One verifies directly that R∇(α, β)ξ is C∞(M)-linear in α, β and ξ.) When thecurvature vanishes, i.e., R∇ ≡ 0, one says that ∇ is a flat connection. In terms ofhorizontal lifting, the flatness condition can be written as

[hor(α), hor(β)] = hor([α, β]) ∀ α, β ∈ Γ(A), (8.33)

which means that the map hor is a linear representation of A. In other words, aflat linear A-connection is nothing but a linear representation of A.

Example 8.4.9. LetO ⊂ M be a leaf of the characteristic foliation of a Lie algebroid(A, [, ], ) over M . Then ker O is a vector bundle over O, and the formula ∇αβ :=[α, β], α ∈ Γ(AO), β ∈ Γ(ker O) defines a flat AO-connection on ker O (which iscalled the Bott connection). The holonomy of this flat connection, i.e., the linearmaps defined by parallel transport along AO-paths, is called the linear holonomyof A [146, 131].

8.5 Splitting theorem and transverse structures

The following splitting theorem for Lie algebroids first appeared in a weaker formin the work of Fernandes [131] and Weinstein [352], and was then refined in [112].

Theorem 8.5.1. Let (A, [, ], ) be a (smooth or analytic) Lie algebroid over a man-ifold M . Then in a neighborhood of each point of M the algebroid A can be de-composed locally into the direct product of a tangent algebroid and a Lie algebroidwhose rank at the origin is 0. More precisely, if the rank of at a point z of M isq, then there is a system of local coordinates (x1, . . . , xq, y1, . . . , ys) on M centeredat z, and a basis of local sections (α1, . . . , αq, β1, . . . , βr) of A over a neighborhoodof z, such that the following conditions are satisfied for all possible indices:

[αi, αj ] = 0, αi =∂

∂xi, (8.34)

[βi, αj ] = 0, βi(xj) = 0, L∂/∂xjβi = 0, (8.35)

[βi, βj ] =∑

k

fkij(y)βk, (8.36)

where fkij(y) are functions which depend only on the variables y = (y1, . . . , ys).

Proof. We will prove by induction on the rank q = rank z . When q = 0 thereis nothing to prove, so we will assume q > 0. Suppose by induction that, for aninteger d with 0 ≤ d < q, we have found a local coordinate system (x1, . . . , xd,y1, . . . , ys+q−d) of M , and a basis of local sections (α1, . . . , αd, β1, . . . , βr+q−d)of A, which satisfy Equations (8.34), (8.35), (8.36). We will now try to replace dby d + 1.

Since d < q, there must be an index i, 1 ≤ i ≤ r+q−d, such that βi(z) = 0.We may assume that β1(z) = 0. By a change of variables (which leaves Equations

8.5. Splitting theorem and transverse structures 247

(8.34), (8.35), (8.36) intact), we may assume that

β1 =∂

∂y1. (8.37)

For each i = 2, . . . , r + q − 1, we will replace βi by βi =∑r+q−1

j=1 gji βj , where

(gji ) is an invertible matrix whose entries are functions of variables (y1, . . . , ys+q−d).

By such a change, we want to achieve the following condition:

[β1, βi] = 0 ∀ i = 2, . . . , r + q − d. (8.38)

By the Leibniz rule, the above condition is equivalent to the following system ofordinary differential equations:

∂(gji )

∂y1+ (gk

i )(f j1k) = 0, (8.39)

where (f j1k) is a matrix of functions of variables (y1, . . . , ys+q−r) which appear in

Equation (8.14), i.e.,

[β1, βk] =r+q−d∑

j=1

f j1kβj . (8.40)

Equation (8.39) is just an ordinary differential equation, which can be solvedlocally, say by fixing the following boundary condition: when y1 = 0 we pose gi

i = 1and gj

i = 0 if i = j. Thus we may assume that Condition (8.38) is satisfied. Wemay rename βi by βi, and forget about the tilde, so we have

[β1, βi] = 0 ∀ i = 2, . . . , r + q − d. (8.41)

This last equation implies in particular that βi is invariant with respect to ∂/∂y1.Indeed, we have

L∂/∂y1βi = [β1, βi] = [β1, βi] = 0. (8.42)

Note that [β1, (βi(y1))β1] = 0 because βi(y1) is invariant with respect to y1.Replacing βi by βi − (βi(y1))β1 for each i ≥ 2, we may also assume that

βi(y1) = 0 ∀ i = 2, . . . , r + q − d. (8.43)

Finally, look at the equation

[βi, βj] = f1ijβ1 +

r+q−d∑k=2

fkijβk (8.44)

for i, j ≥ 2. We want to show that f1ij = 0. Applying the anchor map to the

above equation, and valuing it on y1, we have 0 = [βi, βj ](y1) = (f1ijβ1 +


∑r+q−dk=2 fk

ijβk)(y1) = f1ij , whence f1

ij = 0. Thus we have

[βi, βj ] =r+q−d∑

k=2

fkijβk. (8.45)

Now we can rename y1 by xd+1 and β1 by αd+1 to achieve our inductionprocedure. Remark 8.5.2. The above splitting theorem implies that, similarly to the case ofPoisson manifolds, to study the local structure of a Lie algebroid, it is enough toconsider the case when the anchor map vanishes at a point.Remark 8.5.3. A similar splitting theorem for the so-called Koszul–Vinberg alge-broids was obtained by Nguiffo Boyom and Wolak in [278].

Corollary 8.5.4. If the characteristic distribution Im of a Lie algebroid A hasconstant rank q near a point z ∈M, then A is locally isomorphic to the product ofthe tangent algebroid TKq with a bundle of Lie algebras.

Proof. The constant rank condition means that the vector fields βi, where βi aresections of A given by the splitting theorem 8.5.1, must vanish. This leads to theresult. Example 8.5.5. If A is a transitive Lie algebroid, then locally A is isomorphic tothe direct product of a tangent algebroid with a finite-dimensional Lie algebra. Inparticular, the isotropy Lie algebras of a transitive Lie algebroid are isomorphic.More generally, if A is a Lie algebroid, then the isotopy algebras of points on asame orbit of A are isomorphic.

Remark 8.5.6. Similarly to the case of Poisson manifolds (see Remark 1.5.10), adirect consequence of Theorem 8.5.1 is that to each Lie algebroid (A, [, ], ) over amanifold M there is a natural associated singular foliation on M whose tangentdistribution is the characteristic distribution of A.

Let (A, [., .], ) be a Lie algebroid over a manifold M , and N be a submanifoldof M which is transversal to the characteristic foliation, i.e., for any point x ∈ Nwe have

TxN + Im(x) = TxM. (8.46)

(We don’t require that TxN ∩ Im(x) = 0.) Then we can define the algebroidrestriction AN of A to N as follows:

AN = α ∈ A | α ∈ TN. (8.47)

Due to the transversality of N to the characteristic foliation, AN is a subbundleof A|N (whose codimension in A|N is equal to the codimension of N in M). Thereis a unique natural Lie algebroid structure on AN , for which the inclusion mapAN → A is a Lie algebroid morphism. The Lie bracket on Γ(AN ) can be defined by

[α, β] = [α, β]|N , (8.48)

8.6. Cohomology of Lie algebroids 249

where α, β are sections of AN , and α, β are sections of A which extend them. Oneverifies directly that the above formula does not depend on the choice of α and β.

In particular, when N is transversal to the characteristic foliation at a pointx ∈M , and TN ∩ Im(x) = 0, then the germ of the Lie algebroid AN at x is calledthe transverse Lie algebroid of A at x. Given any such a transversal submanifoldN , by repeating the proof of Theorem 8.5.1 but taking N into account, we canchoose a local coordinate system (x1, . . . , xq, y1, . . . , ys) on M centered at x, anda basis of local sections (α1, . . . , αq, β1, . . . , βr) of A, which satisfy the conditionsof Theorem 8.5.1, and such that locally N is given by x1 = · · · = xq = 0, andAN is spanned by β1, . . . , βr. Theorem 8.5.1 says that the germ of A at x can bewritten as the direct product of the germ of AN at x with the germ of a tangentalgebroid.

Let N ′ be another submanifold transversal to the characteristic foliation ata point x′ which lies on the same orbit as x, such that T ′

N ∩ Im(x′) = 0. Tojustify the words transverse Lie algebroid, we will show that the germ of AN at xis isomorphic to the germ of A′

N at x′. It is a direct consequence of the following:

Proposition 8.5.7 ([131]). With the above notations, there is an isomorphism of Awhich sends a neighborhood of x in N to a neighborhood of x′ in N ′.

Proof. We can assume that x = x′. There is a smooth section α of A with compactsupport, such that the time-1 flow exp(α) of α sends x to x′ and a neighborhoodof x in N to a neighborhood of x′ in N ′. The time-1 flow exp(Xα) of the Hamil-tonian vector field Xα of α on A∗ is then a fiber-wise linear Poisson isomorphism,which projects to the map exp(α) on N . By duality, this fiber-wise linear Poissonisomorphism of A∗ corresponds to a Lie algebroid isomorphism of A.

8.6 Cohomology of Lie algebroids

Associated to each Lie algebroid (A → M, [, ], ) there is a natural differentialcomplex, called the de Rham complex of A:

· · · −→ Γ(∧p−1A∗) dA−→ Γ(∧pA∗) dA−→ Γ(∧p−1A∗) −→ · · · , (8.49)

where the space Γ(∧pA∗) of p-cochains is the space of smooth p-forms on A, i.e.,C∞(M)-multilinear antisymmetric maps

Γ(A)× · · · × Γ(A) −→ C∞(M),

and the differential operator dA is given by Cartan’s formula:

dAω(s1, . . . , sp+1) =∑

i

(−1)i+1(si)ω(s0, . . . si . . . , sp)

+∑i<j

(−1)i+jω([si, sj ], s1, . . . si . . . sj . . . , sp). (8.50)


Exercise 8.6.1. Show that the anchor map and the Lie bracket of the Lie algebroidA are completely determined by the above differential operator dA, and dAdA =0.

Definition 8.6.2. The cohomology of the above de Rham complex of a Lie algebroidA is called the algebroid cohomology of A and denoted by H(A).

For example, if A = TM is the tangent algebroid of a manifold M , thenH(A) is nothing but the de Rham cohomology H

dR(M)of M . If A = T ∗M is thecotangent algebroid of a Poisson manifold (M, Π), then the above de Rham com-plex coincides with the Lichnerowicz complex of (M, Π), and H(A) = H(M, Π)is the Poisson cohomology of A.

Remark 8.6.3. One may view Γ(∧A∗) = ⊕pΓ(∧pA∗) as the space of superfunc-tions on a supermanifold A associated to the vector bundle A → M , and thedifferential dA of the de Rham complex of A as a vector field of degree 1 on A.This leads to an equivalent definition of a Lie algebroid as a supermanifold witha vector field dA of degree 1 such that (dA)2 = 0 [331, 10].

More generally, given a linear module E of a Lie algebroid A, one has anatural differential complex

· · · −→ Γ(∧p−1A∗ ⊗ E) d−→ Γ(∧pA∗ ⊗ E) d−→ Γ(∧p−1A∗ ⊗ E) −→ · · · (8.51)

which generalizes the Chevalley–Eilenberg complex. The differential d is againgiven by Cartan’s formula

dω(s1, . . . , sp+1) =∑

i

(−1)i+1∇siω(s0, . . . si . . . , sp)

+∑i<j

(−1)i+jω([si, sj ], s1, . . . si . . . sj . . . , sp), (8.52)

where ∇ denotes the covariant derivative of the corresponding flat A-connectionon E. The corresponding cohomology is called the algebroid cohomology of A withcoefficients in E. See, e.g., [85, 127, 131, 153, 181, 202, 203, 360] and referencestherein for some results on Lie algebroid cohomology and its relations with ho-mology, characteristic classes of Lie algebroids, and differentiable cohomology ofLie groupoids.

Here we want to discuss another cohomology of Lie algebroids, called defor-mation cohomology [88], which is more directly related to the problems of normalforms and deformations of Lie algebroids.

Let E → M be a vector bundle over a manifold M. A vector field X onE is called fiber-wise constant vertical (resp., fiber-wise linear) if the derivationby X of any fiber-wise linear function is basic (resp., fiber-wise linear) and thederivation of any basic function vanishes (resp., is basic). In a fibered local systemof coordinates (x1, . . . , xn, y1, . . . , yq), where xi are coordinates on M and yj arelinear coordinates on the fiber, a fiber-wise constant vertical vector field has the

8.6. Cohomology of Lie algebroids 251

form ∑j

Xj(x)∂yj ,

and a fiber-wise linear vector field has the form∑kj

Xkj (x)yk∂yj +

∑i

Bi(x)∂xi.

For p > 0, we denote by Vplin(E) the C∞(M)-module of p-vectors on E

generated by the exterior products X1 ∧ · · · ∧ Xp where X1, . . . , Xp−1 are fiber-wise constant vertical vector fields and Xp is fiber-wise linear. Elements of Vp

lin(E)are called fiber-wise linear p-vector fields ; in the fibered coordinates (x1, . . . , xn,y1, . . . , yq), they have the local form∑i1...ipi

aii1...ip

(x)yi∂yi1 ∧ · · · ∧ ∂yip +∑

j1...jp−1k

bkj1...jp−1

(x)∂yj1 ∧ · · · ∧ ∂yjp−1 ∧ ∂xk.

In particular, V1lin(E) is the set of fiber-wise linear vector fields on E. We denote

by V0lin(E) the C∞(M)-module of functions on E which is generated by basic

functions and fiber-wise linear functions. The space Vlin(E) = ⊕p≥0Vp

lin(E) iscalled the space of fiber-wise linear multi-vectors on E.

For example, if A is a Lie algebroid, then the corresponding Poisson structureon A∗ is a fiber-wise linear 2-vector field on A∗.

The following fundamental lemma, whose proof is straightforward, allows usto specialize Poisson cohomology to the fiber-wise linear world.

Lemma 8.6.4. The Schouten bracket of two fiber-wise linear multi-vector fields ona vector bundle is again a fiber-wise linear multi-vector field.

Let Π be a fiber-wise linear Poisson tensor (i.e., an element of V2lin(E) with

[Π, Π] = 0). Then the differential operator δ : A → [Π, A] of the Lichnerowiczcomplex (see Section 2.1) gives, by restriction, a differential complex (Vlin(E), δ),i.e.,

· · · δ−→ Vp−1lin (E) δ−→ Vp

lin(E) δ−→ Vp+1lin (E) δ−→ · · · . (8.53)

We will denote the cohomology of this complex by Hlin(E, Π) and call it the fiber-

wise linear Poisson cohomology of (E, Π). If A is a Lie algebroid and (A∗, Π)its dual fiber-wise linear Poisson manifold, then H

lin(A∗, Π) is also called thedeformation cohomology of A, see Crainic and Moerdijk [88] who also gave somerelations between usual algebroid cohomology (with coefficients in A-modules) anddeformation cohomology.

It is clear that, just as Poisson cohomology governs deformations of Poissonstructures, fiber-wise linear Poisson cohomology governs fiber-wise linear defor-mations of fiber-wise linear Poisson structures. In other words, deformation co-homology of Lie algebroids govern their deformations. In particular, similarly to


Section 2.2, deformation cohomology may be used to study local normal forms ofLie algebroids.

Due to the local splitting theorem for Lie algebroids, in the study of lo-cal structure of a Lie algebroid (A → M, [, ], ) near a point 0 ∈ M , we mayassume that the rank of #0 : A0 → T0M is zero. Fix a local coordinate system(x1, . . . , xm, y1, . . . , yn) on the dual bundle A∗ of A, where y1, . . . , yn are fiber-wiselinear functions and x1, . . . , xm are basic function which vanish at 0 ∈ M . Supposethat the Taylor expansion of the associated fiber-wise linear Poisson structure Πon A∗ begin with terms of degree k,

Π = Π(k) + Π(k+1) + · · · , (8.54)

where, for every r > 0, Π(r) is homogeneous of degree r; it is a fiber-wise linear2-vector field of the form

∑jli ai

jl(x)yi∂yj ∧ ∂yl +∑

su bus (x)∂ys ∧ ∂xu, where the

coefficients aijl(x) are homogeneous polynomials of degree r − 1 in variables x =

(x1, . . . , xm) and bus (x) are homogeneous polynomials of degree r. In particular,

Π(k) is the “principal part” of Π, i.e., the non-zero homogeneous part of lowestdegree. When k = 1, we recover the linear part. The lowest degree terms of theequation [Π, Π] = 0 give [Π(k), Π(k)] = 0, so the principal part Π(k) is again afiber-wise Poisson structure. In the language of Lie algebroids, Π(k) correspondsto a homogeneous Lie algebroid of degree k, which will be called the homogeneouspart of Π at 0, and which doesn’t depend on the choice of local coordinates.Similarly, one can define quasi-homogeneous Lie algebroids, exactly as in Section2.2. Moreover, similarly to the Poisson case, the deformation cohomology also hasquasi-homogeneous versions Hp

lin,(r)(A∗, Π(k)) when we restrict our attention to

fiber-wise linear quasi-homogeneous multi-vector fields of degree r. So Proposition2.2.1 and Theorem 2.2.2 have versions with the suffix lin, i.e., in the fiber-wiselinear world. This gives the theoretic framework for the study of (formal) normalformal norms and, in particular, homogenization of Lie algebroids near a singularpoint. The problem of linearization of Lie algebroids will be considered in detailin the next section.

8.7 Linearization of Lie algebroids

Recall that, due to the local splitting theorem for Lie algebroids, in the study oflocal structure of a Lie algebroid (A, [, ], ) near a point x, we may assume thatthe rank of #x : Ax → TxM is zero.

Consider a Lie algebroid (A, [, ], ), over a neighborhood U of the origin inKN with a local system of coordinates x = (x1, . . . , xN ), such that 0 = 0. Denoteby (α1, . . . , αn) a basis of sections of A over U . Write

[αi, αj ] =n∑

k=1

ckijαk + h.o.t. (8.55)

8.7. Linearization of Lie algebroids 253

and

αi =∑j,k

bkijxk

∂

∂xj+ h.o.t., (8.56)

where ckij and bij are constants, and h.o.t. means higher-order terms. If we for-

get about the higher-order terms in the above expressions, then we get an n-dimensional Lie algebra (namely the isotropy algebra of A at 0) with structuralcoefficients ck

ij , which acts on the linear space V = KN via linear vector fields∑j,k bk

ijxk∂/∂xj . The transformation algebroid of this linear Lie algebra action iscalled the linear part of A and denoted by A(1).

Of course, A(1) does not depend on the choice of local coordinates and localsections of A. It is the transformation algebroid lV of a linear action of l onV = KN , where l = A0 is the isotropy algebra of A at 0.

The local linearization problem for Lie algebroids is: does there exist a local(formal, analytic or smooth) isomorphism from A to A(1). If it is the case, thenwe say that A is locally linearizable (formally, analytically or smoothly).

Similarly to the problem of formal linearization of Poisson structures, theformal linearization of a Lie algebroid A whose anchor vanishes at a point isgoverned by a cohomology group, namely the group

H2(l,⊕k≥1Sk(V ∗)⊗ l

)⊕H1

(l,⊕k≥2Sk(V ∗)⊗ V

),

where Sk(V ∗) denotes the k-symmetric power of the l-module V ∗ (the dual ofl-module V ), and l acts on itself by the adjoint action. In other words, we havethe following result, first mentioned in [353] (in a not very precise way).

Theorem 8.7.1. With the above notations, if

H2(l,⊕p≥1Sp(V ∗)⊗ l) = 0, (8.57)

then A is formally isomorphic to the transformation algebroid of an action of lwith a fixed point. If, moreover,

H1(l,⊕p≥2Sp(V ∗)⊗ V ) = 0, (8.58)

then A is formally linearizable.

Proof. Suppose that H2(l,⊕p≥1Sp(V ∗) ⊗ l) = 0. We will find a formal basis ofsections α∞

1 , . . . , α∞n of A such that

[α∞i , α∞

j ] =n∑

k=1

ckijα

∞k . (8.59)

If such a formal basis of sections can be found, then A is formally isomorphic tothe transformation algebroid of the formal (nonlinear) action of l on V generated


by α∞i . We will do it by induction. Suppose that, for an integer p ≥ 0, we found

a basis of sections αp1, . . . , α

pn of A such that

[αpi , α

pj ] =

n∑k=1

ckijα

pk + β

(p)ij + h.o.t., (8.60)

where β(p)ij =

∑nk=1 b

k(p)ij αp

k are sections whose coefficients bk(p)ij are homogeneous

polynomials of degree p with respect to a given system of coordinates (x1, . . . , xN )on V = KN . (When p = 0, such a basis of sections exists, because Equation (8.60)is the same as Equation (8.55).) We can identify β

(p)ij with an element of Sp(V ∗)⊗l,

by mapping (αpi ) to a basis (ξi) of l such that [ξi, ξj ] =

∑ckijξk. One checks directly

that, due to the Jacobi identity for the sections of A, βp : ξi ∧ ξj →∑n

k=1 bk(p)ij ξk

is a 2-cocycle of the l-module Sp(V ∗)⊗ l. By our assumptions, it is a coboundary.Denote by ζp : ξi →

∑nk=1 z

k(p)i ξk a 1-cochain whose coboundary is βp. Put

αp+1i = αp

i −∑n

k=1 zk(p)i αp

k. Then we have

[αp+1i , αp+1

j ] =n∑

k=1

ckijα

p+1k + O(p + 1), (8.61)

where O(p + 1) means terms of degree ≥ p + 1 in variables (x1, . . . , xn).Repeating the above process for each p ∈ N, we get a series of bases αp

i ,which converges formally. The formal limit α∞

i = limp→∞ αpi satisfies Equation

(8.59). The first part of Theorem 8.7.1 is proved.The second part of Theorem 8.7.1 follows from the first part and the following

linearization result for formal Lie algebra actions.

Theorem 8.7.2 ([172]). Suppose that a finite-dimensional Lie algebra l acts formallyon a manifold KN with the origin 0 as a fixed point, and that the linear part of theaction of l at 0 corresponds to an l-module V = KN such that H1(l,⊕p≥2Sp(V ∗)⊗V ) = 0. Then this action of l is formally linearizable.

We will leave the proof of Theorem 8.7.2, which is implicitly given in [172],as an exercise (see the proof of Theorem 3.1.6). It is very similar to the proof ofthe first part of Theorem 8.7.1, except for the fact that we will have to deal withsome 1-cocycles instead of 2-cocycles.

Exercise 8.7.3. Write down the relation between the cohomology groups that ap-pear in Theorem 8.7.1 and the deformation cohomology of the transformationalgebroid lV.

A particular case of Theorem 8.7.1 is when the isotropy algebra l is semisim-ple. In that case, due to Whitehead’s lemma, the cohomological conditions inTheorem 8.7.1 are automatically satisfied, and we get:

8.7. Linearization of Lie algebroids 255

Theorem 8.7.4 ([353, 112]). Let A be a formal Lie algebroid over a space KN

(K = R or C), whose anchor vanishes at the origin 0 and whose isotropic algebraat 0 is semisimple, then A is formally linearizable.

In analogy with Poisson structure, one may say that a finite-dimensionalmodule V of a finite-dimensional Lie algebra l is formally (resp. analytically,resp. smoothly) nondegenerate, if the corresponding linear transformation alge-broid lV is formally (resp. analytically, resp. smoothly) nondegenerate in thesense that any Lie algebroid, whose anchor vanishes at a point and whose lin-ear part at that point is isomorphic to lV , is formally (resp. analytically, resp.smoothly) linearizable. Theorem 8.7.4 says that any finite-dimensional module ofa semisimple Lie algebra is formally nondegenerate.

In fact, Theorem 8.7.4 also holds in the analytic case (i.e., any finite-dimen-sional module of a semisimple Lie algebra is analytically nondegenerate), and itis a special case of the following Theorem 8.7.5, which is the formal and analyticLevi decomposition theorem for Lie algebroids.

Theorem 8.7.5 ([369]). Let A be a local analytic (resp. formal ) Lie algebroid over(KN , 0), whose anchor map #x : Ax → TxKN vanishes at 0: #0 = 0. Denoteby l = A0 the isotropy algebra A at 0, and by l = g r its Levi decompo-sition. Then there exists a local analytic (resp. formal ) system of coordinates(x∞

1 , . . . , x∞N ) of (KN , 0), and a local analytic (resp. formal ) basis of sections

(α∞1 , . . . , α∞

m , β∞1 , . . . , β∞

n−m) of A, where n = dim l and m = dim g, such thatwe have:

[α∞i , α∞

j ] =∑

k ckijα

∞k ,

[α∞i , β∞

j ] =∑

k akijβ

∞k ,

#α∞i =

∑j,k bk

ijx∞k ∂/∂x∞

j ,

(8.62)

where ckij , a

kij , b

kij are constants, with ck

ij being the structural coefficients of thesemi-simple Lie algebra g.

The above form (8.62) is called a Levi normal form or Levi decomposition of A.

Proof. Theorem 8.7.5 may be viewed as a special case of Theorems 3.2.3 and3.2.6, due to the characterization of Lie algebroids as fiber-wise linear Poissonstructures. We will indicate here how to get a proof of Theorem 8.7.5 from theproof of Theorems 3.2.3 and 3.2.6 given in Chapter 3.

Denote by (α1, . . . , αn) a basis of local analytic (resp. formal) sections onA, and (x1, . . . , xN ) a local system of coordinates on KN . Together they form alocal fiber-wise linear coordinate system on the dual bundle A∗ of A. Recall fromSection 8.2 that A∗ is equipped with the corresponding fiber-wise linear Poissonstructure Π whose Poisson bracket is given as follows:

αi, αj = [αi, αj ],αi, xj = αi(xj),xi, xj = 0.

(8.63)


Since 0 = 0, Π also vanishes at the origin of A∗ (i.e., the null point in the fiberA∗

0 of A∗ over 0). The linear part of Π at the origin is l V , and its Levi factor isthe same as the Levi factor of l.

It is easy to see that, to find a Levi decomposition of A is the same as to finda Levi decomposition of Π, given by a coordinate system

(α∞1 , . . . , α∞

m , β∞1 , . . . , β∞

n−m, x∞1 , . . . , x∞

N )

on A∗, such that (α∞1 , . . . , α∞

m ) form a Levi factor, α∞i , β∞

i are fiber-wise linearfunctions, and x∞

i are base functions. Such a Levi decomposition of Π on A∗ willbe called a fiber-wise linear Levi decomposition.

The existence of a Levi decomposition of Π on A∗ is provided by (the proof of)Theorem 3.2.6 given in Chapter 3 (which includes a proof of Theorem 3.2.3). Butwe have to modify that proof a little bit to make sure that this Levi decomposition,in the case of fiber-wise Poisson structures, can be chosen fiber-wise linear. Hereare the few modifications to be made to the construction of Levi decompositiongiven in Section 3.3:

i) After Step l (l ≥ 0), we will get a local coordinate system

(αl1, . . . , α

lm, βl

1, . . . , βln−m, xl

1, . . . , xlN )

of A∗ with the following properties: xl1, . . . , x

ln are base functions (i.e., functions

on (KN , 0)); αl1, . . . , α

lm, βl

1, . . . , βln−m are fiber-wise linear functions (i.e., they are

sections of A); αli, α

lj −

∑k ck

ijαlk = O(|x|2l

); αli, β

lj −

∑k ak

ijβlk = O(|x|2l

);αl

i, xlj −

∑k bk

ijxlk = O(|x|2l+1). Here ck

ij , akij , b

kij are structural constants as ap-

peared in the statement of Theorem 8.7.5.

ii) Replace the space O of all local analytic functions by the subspace of localanalytic functions which are fiber-wise linear. Similarly, replace the space Ol oflocal analytic functions without terms of order ≤ 2l by the subspace of fiber-wiselinear analytic functions without terms of order ≤ 2l.

iii) Replace Y l by the subspace of vector fields of the following form:

n−m∑i=1

pi∂/∂vli +

N∑i=1

qi∂/∂xli

where pi are fiber-wise linear functions and qi are base functions. For the replace-ment of Y l

k, we require that pi do not contain terms of order ≤ 2k − 1 in variables(x1, . . . , xN ), and qi do not contain terms of order ≤ 2k.

One checks that the above subspaces are invariant under the g-actions in-troduced in Section 3.3, and the cocycles introduced there will also live in thecorresponding quotient spaces of these subspaces. Details are left to the reader.

Similarly, a slight modification of the proof of the smooth Levi decompo-sition Theorem 3.2.9 for Poisson structures leads to the following smooth Levidecomposition theorem for Lie algebroids.

8.8. Integrability of Lie brackets 257

Theorem 8.7.6 ([263]). For each p ∈ N∪∞ and n, N ∈ N there is p′ ∈ N∪∞,p′ < ∞ if p < ∞, such that the following statement holds: Let A be an n-dimensional, Cp′

-smooth Lie algebroid over a neighborhood of the origin 0 in RN

with anchor map # : A → TRN , such that #0 = 0. Denote by l the isotropyLie algebra of A at 0, and by l = g + r a decomposition of L into a direct sumof a semisimple compact Lie algebra g and a linear subspace r which is invari-ant under the adjoint action of g. Then there exists a local Cp-smooth systemof coordinates (x∞

1 , . . . , x∞N ) of (RN , 0), and a local Cp-smooth basis of sections

(s∞1 , s∞2 , . . . , s∞m , v∞1 , . . . , v∞n−m) of A, where m = dim g, such that we have:

[s∞i , s∞j ] =∑

k ckijs

∞k ,

[s∞i , v∞j ] =∑

k akijv

∞k ,

#s∞i =∑

j,k bkijx

∞k ∂/∂x∞

j ,

(8.64)

where ckij , a

kij , b

kij are constants, with ck

ij being the structural constants of the com-pact semisimple Lie algebra g.

In particular, when r = 0, Theorem 8.7.6 gives a local smooth linearization ofa Lie algebroid whose isotropy algebra is semisimple compact. This may be viewedas the infinitesimal version of Theorem 7.4.7 in the compact semisimple case.

8.8 Integrability of Lie brackets

The so-called Lie’s third theorem says that any (finite-dimensional) Lie algebra canbe integrated into a Lie group. A natural question arises: does Lie’s third theoremhold for Lie algebroids, i.e., is it true that any (smooth real) Lie algebroid can beintegrated into a Lie groupoid? This question was studied by many people, startingwith Pradines [295] who claimed, erroneously, that the answer is positive. WhatPradines actually obtained is that any Lie algebroid can be integrated into a localgroupoid : the product map is not necessarily globally defined on Γ(2), but onlylocally defined near the identity section. In the case of transitive Lie algebroids,an abstract theory of homological obstructions to integrability was developed byMackenzie [226], and a concrete non-integrable example was found by Almeidaand Molino [11]. Recently, a necessary and sufficient condition for smooth Liealgebroids (and Poisson manifolds) to be integrable was found by Crainic andFernandes [86, 87]. We will briefly present their results here, referring the readerto [86, 87] for the details.

8.8.1 Reconstruction of groupoids from their algebroids

Assume that a Lie algebroid A over a connected manifold M is integrable, i.e.,it is the Lie algebroid of a Lie groupoid G ⇒ M . Similarly to the case of Liegroups, by taking a covering of the identity component of G, we may assume


that G is connected and its s-fibers are simply-connected, and then G is uniquelydetermined by A [257]. Remark that the inversion map of G sends s-fibers tot-fibers, so the t-fibers of G are also simply-connected.

We will try to reconstruct G from A. By a G-path, we will mean a map

g : I = [0, 1]→ G (8.65)

of smoothness class C2, such that g(0) = ε(x), the identity element of x for somex ∈ M , and t(g(τ)) = x ∀ τ ∈ I. (In [86] it is written as s(g(τ)) = x, becausethey adopt a different convention: they use right -invariant vector fields on a Liegroupoid to define its Lie algebroid.)

Differentiating a G-path g, we will get a A-path a : [0, 1] → A (i.e., a Liealgebra homomorphism from TI to A, where I = [0, 1], see Example 8.3.8) of classC1 by the formula

a(τ) = g(τ)−1 dg

dτ(τ). (8.66)

Conversely, any A-path of class C1 can be integrated to a G-path.If two G-paths g and g′ have the same end point q = g(1) = g′(1), then they

are homotopic in the corresponding t-fiber by a homotopy gρ (g0 = g, g1 = g′) of G-paths with fixed end points, because we have assumed that the t-fibers are simply-connected. We may assume that this homotopy is of class C2. By differentiatingthis homotopy, we get a family of A-paths aρ which are equivalent in the sensethat their corresponding G-paths have the same end point. We can write

G = P (A)/ ∼ , (8.67)

where P (A) is the space of A-paths of class C1 and ∼ is the above equivalence.This equivalence can be characterized infinitesimally, i.e., in terms of A, as follows:

Let aρ be a family of A-paths of class C1 which depends on a parameterρ ∈ [0, 1] in a C2 fashion. Denote by γρ = π aρ the corresponding family of basepaths, where π : A → M is the projection. If aρ is the differentiation of gρ thenγρ(τ) = s(gρ(τ)). We will assume that these base paths have the same end points:γρ(0) = γ0(0) and γρ(1) = γ0(1) for all ρ. Let ξρ be a family of time-dependentsections of A such that

ξρ(τ, γρ(τ)) = aρ(τ). (8.68)

Putb(ρ, τ) =

∫ τ

0

φτ,rξρ

dξρ

dρ(r, γρ(r))dr, (8.69)

where φτ,rξρ

denotes the flow in A from time r to time τ generated by the time-dependent section ξρ (see Example 8.2.5). One verifies directly that b depends onlyon the family aρ but not on the choice of time-dependent sections ξρ. Moreover,if gρ is the family of G-paths integrating aρ, then we have

b(ρ, τ) = gρ(τ)−1 dgρ(τ)dρ

. (8.70)


In other words, b(ρ, τ) is the differentiation of g(ρ, τ) = gρ(τ) in the ρ-direction.We stress here the fact that b(ρ, τ) is well defined by Formula (8.69) even if thealgebroid A is not integrable. In the integrable case, the condition that the familyof G-paths gρ (starting from a common point ε(x)) have the same end point maybe expressed infinitesimally as

b(ρ, 1) = 0 ∀ ρ ∈ [0, 1]. (8.71)

Definition 8.8.1. We say that two A-paths a and a′ of class C1, whose base pathsγ and γ′ have the same end points, are equivalent, and write a ∼ a′, if they can beconnected by a C2-family of A-paths aρ whose base paths have the same end pointsand such that b(ρ, τ) defined by Formula (8.69) satisfies the equation β(ρ, 1) = 0for all ρ ∈ [0, 1].

Clearly, this definition of equivalence coincides with the previous equivalencenotion in the integrable case. Even if A is not integrable, the following naturalfact still holds true:

Proposition 8.8.2 ([86]). Let a1 and a1 be two equivalent A-paths from x to y,where A is a Lie algebroid over a manifold M and x, y ∈ M . Then for any flatA-connection on a vector bundle E over M , the parallel transports Ex → Ey alonga0 and a1 coincide.

In general, for any Lie algebroid A, the quotient

G(A) = P (A)/ ∼ (8.72)

is called the Weinstein groupoid of A (the construction is due to Weinstein and isinspired by a similar construction for the case of Lie algebras [121]). It coincideswith the connected t-simply connected groupoid integrating A in the integrablecase. In general, it is a topological groupoid: the groupoid structure comes fromthe concatenation of A-paths. (The concatenation of two A-paths of class C1 isonly piecewise differentiable in general, but it can be made C1 by a homotopyusing a cut-off function trick, see [86].) The algebroid A is integrable if and only ifG is smooth. In fact, P (A) is a Banach manifold, the equivalence relation ∼ definesa smooth foliation on P (A) of finite codimension, and G(A) is the leaf space of thisfoliation. It is a smooth (not necessarily Haussdorff) manifold if and only if eachleaf admits a transversal section which intersects every leaf at at most one point.

8.8.2 Integrability criteria

For any point x ∈ M , denote by gx = ker the isotropy Lie algebra of A atx, considered as a Lie algebroid over x, and by G(gx) the simply-connected Liegroup integrating gx, considered as the group of gx-paths modulo equivalence.Then there is a natural homomorphism from G(gx) to the isotropy group G(A)x

of x in G(A) (consider gx-paths as A-paths). Denote by Nx(A) the kernel of this


homomorphism: it is the subgroup of G(gx) consisting of equivalence classes ofgx-paths which are equivalent in A to the trivial A-path over x. If G(A) is smooththen in particular Nx(A) must be discrete in G(gx), i.e., the quotient G(gx)/Nx(A)must be a Lie group of Lie algebra gx.

We can characterize Nx(A) as the image of the second monodromy of theleaf O(x) through x in M as follows: Let [γ] ∈ π2(O(x), x) be represented by asmooth map γ : I × I → O(x) which maps the boundary into x. We can choose amorphism of Lie algebroid

adτ + bdρ : TI × TI → A (8.73)

(which maps u(ρ, τ)∂/∂τ+v(ρ, τ)∂/∂ρ to u(ρ, τ)a(ρ, τ)+v(ρ, τ)b(ρ, τ) ∈ A), whichlifts dγ : TI×TI → TO(x) via the anchor, and such that a(0, τ), b(ρ, 0) and b(ρ, 1)vanish. Since γ is constant on the boundary, a1(τ) = a(1, τ) stays inside gx, i.e.,a1 is a gx-path. Denote the equivalence class of a1 in G(gx) by ∂(γ).

Lemma 8.8.3. The map γ → ∂(γ) is well defined and depends only on the homotopyclass [γ] of γ. It induces a homomorphism

∂ : π2(O(x), x) −→ G(gx), (8.74)

whose image is exactly Nx(A).

In fact, the image ∂([γ]) of any [γ] ∈ π2(O(x), x) lies in the center of G(gx).To see it, recall that there is a natural flat AO-connection on the isotropy bundleker O where O is the leaf through x (see Example 8.4.9). By construction, theA-path a1 which represents ∂(γ) is equivalent in A (hence in AO) to the trivialpath, so by Proposition 8.8.2 the parallel transport in ker O along a1 with respectto this flat AO-connection, Ta1 : gx → gx, is the identity map of gx. On the otherhand, since a1(τ) ∈ gx ∀ τ , it is easy to see that this parallel transport map Ta1 isnothing but the adjoint action map Ad[a1] : gx → gx, where [a1] is the element inG(gx) represented by a1. Hence Ad[a1] = Id, which means that ∂([γ]) = [a1] liesin the center of G(gx).

Denote by Nx(A) the subset of gx formed by those elements v of gx withthe property that the constant A-path v is equivalent to the trivial A-path overx. Then we have:

Lemma 8.8.4. For any Lie algebra A over M and any x ∈M , the group Nx(A) liesin the center Z(G(gx)) of G(gx), and its intersection with the connected componentZ(G(gx))0 of the center is isomorphic to Nx(A). In particular, Nx(A) is discretein G(gx) if and only if Nx(A) is discrete in gx.

In particular, G(A)x is a Lie group of Lie algebra gx if and only if Nx(A) isdiscrete. For G(A) to be integrable, we will need a stronger condition, namely thatNx(A) is locally “uniformly” discrete when x varies. More precisely, fix an arbitrarycontinuous fiber-wise metric d on A, and put r(x) = ∞ if Nx(A) = 0 and

r(x) = d(0, Nx(A) \ 0) (8.75)


otherwise. Then the condition that Nx(A) is discrete is equivalent to the conditionr(x) > 0, and the local uniform discreteness means that lim infy→x r(y) > 0. Acareful analysis of the smoothness of G(A) leads to the following theorem.

Theorem 8.8.5 (Crainic–Fernandes [86]). A Lie algebroid A over a manifold Mis integrable if and only if for every x ∈ A, the group Nx(A) is discrete andlim infy→x r(y) > 0.

For example, when the Lie algebroid is totally intransitive (i.e., the anchormap is trivial), then r(x) = ∞ for all x ∈M , and we recover from Theorem 8.8.5the following integrability result of Douady and Lazard [106]: any Lie algebrabundle can be integrated into a Lie group bundle.

What makes Theorem 8.8.5 effective is the fact that, in many cases, the setsNx(A) can be computed explicitly via Proposition 8.8.6 below. Suppose thatO xis a leaf of A on M , and σ : TL → AO is a splitting of the surjective linear mapO : AO → TO. The curvature of σ is a 2-form on O with values in gO = ker O,defined by the formula

Ωσ(X, Y ) = [σ(X), σ(Y )]− σ([X, Y ]). (8.76)

The vector bundle Z(gO), whose fiber over a point y ∈ O is the center Z(gy)of the isotropy algebra gy, admits a canonical flat linear connection given by∇Xα = [σ(X), α] for X ∈ V1(O), α ∈ Γ(Z(gO)) (this connection does not dependon the choice of σ).

Proposition 8.8.6 ([86]). With the above notations, suppose that the image of thecurvature form Ωσ lies in Z(gO), then we have

Nx(A) =∫

γ

Ωσ | γ ∈ π2(O(x), x)

. (8.77)

In the above proposition, the values of Ωσ are translated to Z(gx) by thecanonical flat connection on Z(gO) (via parallel transport along the paths lyingin a representative of γ) before taking the integral.

Example 8.8.7. (Almeida–Molino’s non-integrable example [11].) If ω is an arbi-trary closed 2-form on a connected manifold M , then it induces a transitive Liealgebroid Aω = TM⊕L, where L is the trivial line bundle, with anchor (X, λ) → Xand Lie bracket

[(X, f), (Y, g)] = ([X, Y ], X(g)− Y (f) + ω(X, Y )). (8.78)

(See, e.g., [226].) Applying Proposition 8.8.6 to the obvious splitting of Aω, weobtain that

Nx(Aω) =∫

γ

ω | γ ∈ π2(M, x)⊂ R (8.79)

is the group of periods of ω [86]. If for example, this group of periods is densein R (e.g., it has two incommensurable generators), then the corresponding Liealgebroid is not integrable [11].


Exercise 8.8.8. Recover from Theorem 8.8.5 the following result of Dazord [96]:every transformation Lie algebroid (of a smooth action of a Lie algebra on amanifold) is integrable.

8.8.3 Integrability of Poisson manifolds

Recall that a Poisson manifold (M, Π) is integrable if there is a symplectic groupoidover it. The problem of integrability of Poisson manifolds is very similar to theproblem of integrability of Lie algebroids, due to the following result:

Theorem 8.8.9 (Mackenzie–Xu [231]). A Poisson manifold is integrable if and onlyif its cotangent Lie algebroid is integrable.

Conversely, a Lie algebroid A is integrable if and only if A∗ with the corre-sponding fiber-wise linear symplectic structure is an integrable Poisson manifold(see, e.g., [228]).

In particular, one can use Theorem 8.8.5 and Proposition 8.8.6 (rewritten interms of Poisson structures) to verify integrability and non-integrability of Poissonmanifolds. See [87] and references therein for details and other results concern-ing integrability of Poisson structures. In particular, let us mention the followingresult:

A symplectic realization (M, ω) of a Poisson manifold (P, Π) is called completeif for any function f on P such that the Hamiltonian vector field of f on P iscomplete, the Hamiltonian vector field of the lifting of f on M is also complete.

Theorem 8.8.10 ([87]). A Poisson manifold admits a complete symplectic realiza-tion if any only if it is integrable.

Example 8.8.11. Any two-dimensional Poisson manifold is integrable: if the Poissonstructure is nondegenerate then it is obviously integrable, if not then the corre-sponding cotangent algebra is integrable by Theorem 8.8.5, because the secondhomotopy groups of the symplectic leaves are trivial.

Remark 8.8.12. The problem of integrability of regular Poisson manifolds wasstudied by Alcalde-Cuesta and Hector [4]. In the regular case, the groups Nx(A)which control integrability can be given in terms of variations of symplectic areasof 2-spheres on symplectic leaves, see [4, 87].Remark 8.8.13. In the case when A is the cotangent algebroid of a Poisson man-ifold, the Weinstein groupoid G(A) was also constructed by Cattaneo and Felder[68] by a symplectic reduction process. Their original motivation was not the in-tegrability problem, but rather Poisson sigma-models and quantization of Poissonmanifolds (see Appendix A.9).

Appendix

A.1 Moser’s path method

A smooth time-dependent vector field X = (Xt)t∈]a,b[ on a manifold M is a smoothpath of smooth vector fields Xt on M , parametrized by a parameter t (the timeparameter) taken in some interval ]a, b[. Any smooth path (φt)t∈]a,b[ of diffeomor-phisms determines a time-dependent vector field X by the formula

Xt(φt(x)) =∂φt

∂t(x). (A.1)

Conversely, the classical theory of ordinary differential equations says that,given a smooth time-dependent vector field X = (Xt)t∈]a,b[, in a neighborhoodof any (x0, t0) in M×]a, b[, we can define a unique smooth map (x, t) → φt(x),called the flow of X starting at time t0, with φt0(x) ≡ x and which satisfiesEquation (A.1). Because φt0 is locally the identity, φt are local diffeomorphismsfor t sufficiently near t0. In some circumstances, for example when X has compactsupport, these local diffeomorphisms extend to global diffeomorphisms. Takingthen t0 = 0, we can get by this procedure a path of (local) diffeomorphisms (φt)t

with φ0 = Id.Suppose that we want to prove that two tensors Λ and Λ′, on a manifold M,

are isomorphic, i.e., we want to show the existence of a diffeomorphism φ of theambient manifold such that

φ∗(Λ′) = Λ. (A.2)

Sometimes, this isomorphism problem can be solved with the help of Moser’spath method , which consists of the following:

• First, construct an adapted smooth path (Λt)t∈[0,1] of such tensors, withΛ0 = Λ and Λ1 = Λ′.

• Second, try to construct a smooth path (φt)t∈[0,1] of diffeomorphisms of Mwith φ0 =Id and

φ∗t (Λt) = Λ0 ∀ t ∈ [0, 1], (A.3)

or equivalently,∂(φ∗

t (Λt))/∂t = 0 ∀ t ∈ [0, 1]. (A.4)

264 Appendix

One tries to construct a time-dependent vector field X = (Xt) whose flow(starting at time 0) gives φt. Equation (A.4) is then translated to the followingequation on (Xt):

LXt(Λt) = −∂Λt

∂t. (A.5)

If this last equation can be solved, then one can define φt by integrating X , andφ = φ1 will solve Equation (A.2).

Moser’s path method works best for local problems because time-dependentvector fields can always be integrated locally. In global problems, one usually needsan additional compactness condition to assure that φ = φ1 is globally defined.

The method is named after Jurgen Moser, who first used it to prove thefollowing result:

Theorem A.1.1 ([265]). Let ω and ω′ be two volume forms on a manifold M whichcoincide everywhere except on a compact subset K. If we have ω − ω′ = dα wherethe form α has its support in K, then there is a diffeomorphism φ of M, which isidentity on M \K, such that φ∗(ω′) = ω.

Proof. We have ω′ = fω where f is a strictly positive function on M (f = 1 onM \K). Then ωt = ftω, with ft = tf + 1− t, t ∈ [0, 1], is a path of volume formson M. Now Equation (A.5) reduces to

diXtωt = ω − ω′. (A.6)

The hypothesis of the theorem allows us to replace Equation (A.6) by

iXtωt = α. (A.7)

But this last equation has a unique solution Xt. Since the support of Xt lies inK, we can integrate this time-dependent vector field to a path of diffeomorphismsφt on M which is identity on M \K, and such that φ∗

1(ω′) = ω.

In particular, when M = K we get the following corollary:

Corollary A.1.2 ([265]). Two volume forms on a compact manifold are isomorphicif and only if they have the same total volume.

It was pointed out by Weinstein [344, 345] that the path method works verywell in the local study of symplectic manifolds. A basic result in that direction isthe following.

Theorem A.1.3. Let (ωt)t∈[0,1] be a smooth path of symplectic forms on a manifoldM. If we have ∂ωt/∂t = dγt for a smooth path γt of 1-forms with compact support,then there is a diffeomorphism φ of M with φ∗ω1 = ω0.

Proof. Equation (A.5) follows from the equation iXtωt = −γt, which has a solution(Xt)t because ωt is nondegenerate, so the path method works in this case.

A.1. Moser’s path method 265

Theorem A.1.4 (see [243]). Let Kbe a compact submanifold of a manifold M.Suppose that ω0 and ω1 are two symplectic forms on M which coincide at eachpoint of K. Then there exist neighborhoods N0 and N1 of K and a diffeomorphismφ : N0 −→ N1, which fixes K, such that φ∗ω1 = ω0.

Proof. Consider the path ωt = (1−t)ω0+tω1 of symplectic forms in a neighborhoodof K. Similarly to the proof of the previous theorem, it is sufficient to find a 1-formγ such that γ(x) = 0 for any x ∈ K and

dγ = ∂ωt/∂t = ω1 − ω0. (A.8)

The existence of such a γ is a generalization of Poincare’s lemma which saysthat a closed form is exact on any contractible open subset of a manifold. It canbe proved by the following method, inspired by the Moser path method. Choosea sufficiently small tubular neighborhood T of K in M and denote by ψt themapping from T into T which is the linear contraction v → tv along the fibers ofT. With θ = ω1 − ω0, we have

θ = ψ∗1(θ)− ψ∗

0(θ) =∫ 1

0

∂ψ∗t θ

∂tdt, (A.9)

because ψ1 is the identity and ψ0 has its range in K. Now, if we denote by Yt thetime-dependent vector field associated by Formula (A.1) to the path (ψt)t∈]0,1],we get

∂ψ∗t θ

∂t= ψ∗

tLYtθ = dγt, (A.10)

with γt = ψ∗t ιYtθ. The path γt is, a priori, defined only for t > 0, but it extends

clearly to t = 0; also it vanishes on K. So Equation (A.9) gives θ = dγ withγ =

∫ 1

0γtdt, and leads to the conclusion.

A direct corollary of Theorem A.1.4 is the following result of Weinstein:

Theorem A.1.5 ([344]). Let L be a compact Lagrangian submanifold of the sym-plectic manifold (M, ω). There is a neighborhood N1 of L in M, a neighborhood N0

of L (identified with the zero section) in T ∗L and a diffeomorphism φ : N0 −→ N1,which fixes L, such that φ∗ω is the canonical symplectic form on T ∗L.

Proof (sketch). Choose a Lagrangian complement Ex to each TxL in TxM inorder to get a fiber bundle E over L complement to TL in TM |L. We constructa fiber bundle isomorphism f : E −→ T ∗L by f(v)(w) = ω(v, w). As E realizes anormal bundle to L, we can consider that f gives a diffeomorphism from a tubularneighborhood of L in M to a neighborhood of L in T ∗L which sends ω|L to thecanonical symplectic form of T ∗L (restricted to L). So we can suppose that ω isdefined on a neighborhood of L in T ∗L and is equal to the canonical symplecticform at every point on L. Then we achieve our goal using Theorem A.1.4.

When the compact submanifold K is just one point, we recover from TheoremA.1.4 the classical Darboux’s theorem:

266 Appendix

Theorem A.1.6 (Darboux). Every point of a symplectic manifold admits a neigh-borhood with a local system of coordinates (p1, . . . , pn, q1, . . . , qn) (called Darbouxcoordinates or canonical coordinates) in which the symplectic form has the stan-dard form ω =

∑ni=1 dpi ∧ dqi.

Proof. We can suppose that we work with a symplectic form ω near the origin 0in R2n. Moreover we can suppose, up to a linear change, that a first system ofcoordinates is chosen such that ω(0) = ω0(0) where ω0 =

∑ni=1 dpi∧dqi. Then we

apply Theorem A.1.4 to the case K = 0.

In fact, Moser’s path method gives a simple proof of the following equivariantversion of Darboux’s theorem [345], which would be very hard (if not impossible)to prove by the classical method of coordinate-by-coordinate construction.

Theorem A.1.7 (Equivariant Darboux theorem). Let G be a compact Lie groupwhich acts symplectically on a symplectic manifold (M, ω) and which fixes a pointz ∈M . Then there is a local canonical system of coordinates (p1, . . . , pn, q1, . . . , qn)in a neighborhood of z in M , with respect to which the action of G is linear.

Proof. One first linearizes the action of G near z using Bochner’s Theorem 7.4.1.Then, after a linear change, one arrives at a system of coordinates (p1, . . . , pn,q1, . . . , qn) in which the action of G is linear, and such that ω(0) = ω0(0) whereω0 =

∑ni=1 dpi ∧ dqi. One then uses the path method to move ω to ω0 in a G-

equivariant way. To do this, one must find a 1-form γ such that ω−ω0 = dγ as inthe proof of Theorem A.1.3, and moreover γ must be G-invariant in order to assurethat the resulting flow φt is G-invariant (i.e., it preserves the action of G). Notethat both ω and ω0 are G-invariant. In order to find such a G-invariant 1-form γ,one starts with an arbitrary 1-form γ such that ω−ω0 = dγ, and then averages itby the action of G:

γ =∫

G

ρ(g)∗γdµ, (A.11)

where ρ denotes the action of G, and dµ denotes the Haar measure on G.

Moser’s path method has also become an essential tool in the study of singu-larities of smooth maps. In that domain we often have to construct equivalencesbetween two maps f0 and f1, e.g., relations g φ = f (right equivalence), org φ = ψ f (right-left equivalence), or more general equivalences (contact equiv-alence, etc.), which are given by some diffeomorphisms (φ, ψ, etc.). To use thepath method we first construct an appropriate path (ft) which connects f0 tof1. Then we try to find a path of diffeomorphisms which gives the correspondingequivalence between ft and f0. Differentiating the equation with respect to t, weget a version of Equation (A.5). For example, in the case of right-left equivalencewe find the equation

dft(Xt(x)) + Yt(ft(x)) = −∂ft(x)∂t

, (A.12)

A.1. Moser’s path method 267

where the unknowns are time-dependent vector fields Xt and Yt on the source andthe target spaces respectively. The singularists have developed various methodsto solve these equations, such as the celebrated preparation theorem (see, e.g.,[150, 16]). The Tougeron theorem that we present below is typical of the use ofthe path method in this domain.

Let C∞0 (Rn) be the algebra of germs at the origin of smooth functions f :Rn → R. We denote by the same letter such a function and its germ at the origin.Let ∆(f) ⊂ C∞0 (Rn) be the ideal generated by the partial derivatives ∂f

∂x1, . . . ,

∂f∂xn

of f. The codimension of f ∈ C∞0 (Rn) is, by definition, the dimension of thereal vector space C∞0 (Rn)/∆(f).

We say that f ∈ C∞0 (Rn) is k-determinant if every g ∈ C∞0 (Rn), such thatf and g have the same Taylor expansion at the origin up to order k, is rightequivalent to f.

Theorem A.1.8 (Tougeron [330]). If f ∈ C∞0 (Rn) has finite codimension k, then itis (k + 1)-determinant.

Proof. Denote by M ⊂ C∞0 (Rn) the ideal of germs of functions vanishing at theorigin. Consider the following sequence of inequalities:

dim C∞0 (Rn)/(M + ∆(f)) ≤ dim C∞0 (Rn)/(M2 + ∆(f)) ≤ · · ·≤ dim C∞0 (Rn)/(Mm + ∆(f)) ≤ dim C∞0 (Rn)/(Mm+1 + ∆(f)) ≤ · · · ≤ k.

(A.13)

It follows that there is a number q ∈ Z, 0 ≤ q ≤ k, such that Mq + ∆(f) =Mq+1 + ∆(f), and hence Mq ⊂ Mq+1 + ∆(f). Applying Nakayama’s lemma(Lemma A.1.9) to this relation, we get Mq ⊂ ∆(f), which implies that

Mk+2 ⊂ M2∆(f). (A.14)

Let g ∈ C∞0 (Rn) be a function with the same (k + 1)-Taylor expansion asf. We consider the path ft = (1 − t)f + tg and, following the path method, tryto construct a path of local diffeomorphisms (φt)t which fixes the origin and suchthat ft φt = f. Here the equation to solve (Equation (A.5)) becomes

n∑i=1

∂ft

∂xi(x)Xi(t, x) = (f − g)(x), (A.15)

where the unknown functions Xi(t, x) must be such that Xi(t, 0) = 0. In fact, wewill try to find Xi such that Xi(t, .) ∈ M2 for any t, so the differential of theresulting diffeomorphisms φt at 0 will be equal to identity.

By compactness of the interval [0, 1], we need only be able to construct (φt)t

for t near any fixed t0 ∈ [0, 1],, i.e., we need to solve Equation (A.15) only for tnear t0 (and x near 0). We denote by A the ring of germs at (t0, 0) of smoothfunctions of (t, x) ∈ R × Rn; C∞0 (Rn) is considered naturally as a subring of A.

268 Appendix

Then Equation (A.15) (with Xi ∈ M2, where M is now the ideal of A generatedby germs of functions f such that f(t, 0) = 0 for any t near t0) can be replaced by

Mk+2 ⊂ M2∆(ft), (A.16)

where ∆(ft) is now the ideal of A generated by the functions h(t, x) = ∂ft(x)∂xi

(recall that f − g belongs to Mk+2).Now, because ∂ft

∂xi= ∂f

∂ximodulo Mk+1A, we have ∆(f) ⊂ ∆(ft) + Mk+1A,

and Relation (A.14) leads to

Mk+2A ⊂ M2∆(ft) + Mk+3A, (A.17)or

Mk+2A ⊂ M2∆(ft) + I ·Mk+2A, (A.18)

where I is the ideal of A consisting of germs of functions vanishing at (t0, 0).Applying again Nakayama’s lemma, we get Mk+2A ⊂ M2∆(ft), i.e., Relation(A.16), which leads to the result. Lemma A.1.9 (Nakayama’s lemma). Let R be a commutative ring with unit and Ian ideal of R such that, for any x in I, 1 + x is invertible. If M and N are twoR-modules, M being finitely generated, then the relation M ⊂ N + IM impliesthat M ⊂ N.

Proof. Choose a system of generators m1, . . . , mq for M. Then M ⊂ N +IM givesa linear system

∑j=1,...,q aj

imj = ni where ni are elements of N, i = 1, . . . , q andaj

i are elements of R of the form aji = δj

i + νji , where δj

i is the Kronecker symboland νj

i are in I. As the matrix (aji ) is invertible, we can write the mi as linear

combinations of the nj and obtain M ⊂ N. A special case of the preceding Tougeron’s theorem with k = 1 is the Morse

lemma which says that, near any singular point with invertible Hessian matrix, asmooth function is right equivalent to (x1, . . . , xn) → c +

∑i=1,...,n±x2

i .Moser’s path method is also useful in the study of contravariant tensors,

e.g., vector fields and Poisson structures. In the case of vector fields, we find theequation

[Xt, Zt] = −∂Zt

∂t, (A.19)

where (Zt)t is a given path of vector fields and (Xt)t is the unknown (see,e.g., [304]).

For Poisson structures, we get a similar equation:

[Xt, Πt] = −∂Πt

∂t, (A.20)

where (Πt)t is a path of Poisson structures. However, the use of Moser’s pathmethod in the study of Poisson structures is rather tricky, because it is not easy

A.2. Division theorems 269

to find a path of Poisson structures which connects two given Poisson structures,even locally, due to the Jacobi condition. One needs to do some preparatory workfirst, for example to make the two original Poisson structures have the same char-acteristic foliation (then it will become easier to find a path connecting the twostructures). The equivariant splitting theorem for Poisson structures can be provedusing this approach, see [251].

A.2 Division theorems

We will give here a version of Saito’s division theorem [308] which works not onlyfor germs of analytic forms but also for germs of smooth forms.

Let R be a commutative ring with unit. A regular sequence of R is a sequence(a1, a2, . . . , aq) of elements of R such that

i) a1 is not a zero divisor of R, andii) the class of ai is not a zero divisor in R/(a1R + · · ·+ ai−1R) for i = 2, . . . , q.

Let M be a free module over R of finite rank n. Let ω1, . . . , ωk be givenelements of M and e1, . . . , en a basis of M,

ω1 ∧ · · · ∧ ωk =∑

1≤i1<···<ik≤n

ai1···ikei1 ∧ · · · ∧ eik

.

We call A the ideal generated by the family of coefficients ai1···ik.

Theorem A.2.1. With the above notations, let ω be an element of the exteriorproduct

∧pM with p strictly smaller than the maximal length of regular sequences

of R with all elements lying in A. Then the relation

ω ∧ ω1 ∧ · · · ∧ ωk = 0 (A.21)

implies the existence of k elements η1, . . . , ηk of∧p−1

M such that

ω =k∑

i=1

ηi ∧ ωi. (A.22)

When k = 1 this theorem is de Rham’s division theorem [98].

Proof. We will use a double induction on (p, k).For p = 0, ω is just an element of R and the hypothesis implies that it is 0.

So the result is proved in that case.We suppose that either k = 1 and the result is proved for (p− 1, 1), or k > 1

and the result is proved for (p, k − 1) and (p− 1, k).Let (a1, a2, . . . , aq) be a regular sequence of R with all elements in A. Fix an

element ω ∈ ∧pM with 0 < p < q. We extend R to Ra1 , the localization of R by

the powers of a1; this means that we add 1/a1 (the inverse of a1). We can consider

270 Appendix

that ω and the ωi have coefficients in Ra1 . With this trick we obtain that theideal generated by the coefficients of ω1 ∧ · · · ∧ ωk contains an invertible element.It means that ω1, . . . , ωk are independent and can be seen as a part of a basis ofM over Ra1 . The hypothesis ω ∧ ω1 ∧ · · · ∧ ωk = 0 implies that ω =

∑ki=1 ηi ∧ ωi

over Ra1 . Since this relation is valid only after the preceding localization, we get

am1 ω =

k∑i=1

ηi ∧ ωi (A.23)

for an appropriate natural number m, when we return to the initial ring R.Now, when we tensorize with R/am

1 R, this last relation gives simply

0 =k∑

i=1

ηi ∧ ωi, (A.24)

where the image of α ∈ ∧rM in

∧rM⊗R/am

1 R is written α. For any j = 1, . . . , kwe have

ηj ∧ω1∧· · ·∧ωk = ±( k∑

i=1

ηi∧ωi

)∧ω1∧· · ·∧ωj−1 ∧ωj+1 ∧· · ·∧ωk = 0. (A.25)

Denote by a the class of an element a of R modulo am1 . Then the sequence

(a2, a3, . . . , aq) is a regular sequence in R/am1 R (see Exercise 11 of Chapter X of

[44]). So we can apply to ηj the induction hypothesis: we obtain ξi

j ∈∧p−2

M ⊗R/am

1 R (i = 1, . . . , k) such that ηj =∑k

i=1 ξi

j∧ωi. Returning to R, these relationsgive

ηj =k∑

i=1

ξij ∧ ωi + am

1 νj . (A.26)

So Equation (A.23) becomes

am1

(ω −

k∑j=1

νj ∧ ωj

)=

k∑i,j=1

ξij ∧ ωi ∧ ωj. (A.27)

Multiplying by ω2 ∧ · · · ∧ ωk, we get

am1

(ω −

k∑j=1

νj ∧ ωj

)ω2 ∧ · · · ∧ ωk = 0 (A.28)

(in the case k = 1 we skip this step). As a1 is not a zero divisor, we can eraseam1 in this last equation. In the case k = 1 this gives the result. If k > 1 we first

A.3. Reeb stability 271

remark that the ideal generated by the coefficients of ω2 ∧ · · · ∧ ωk contains A.Hence we can apply our induction hypothesis (case (p, k− 1)) to get the equation

ω −k∑

j=1

νj ∧ ωj =k∑

i=2

βi ∧ ωi (A.29)

which leads to the result.

Let the 1-form α =∑n

i=1 aidxi be analytic (resp. smooth) on Rn or holo-morphic on Cn. We say that the origin is an algebraically isolated singularity ofα if the ideal generated by the germs at 0 of a1, . . . , an has finite R-codimensionin the algebra of germs at 0 of analytic (resp. smooth) or holomorphic functionson Rn or on Cn.

We say that such a 1-form has the division property at 0 if the germ relationω ∧ α = 0 implies a germ relation ω = β ∧ α for any germ of p-form ω with p < n(β being the germ of a p− 1-form).

The following theorem is a consequence of de Rham’s division theorem (i.e.,the case k = 1 of Theorem A.2.1).

Theorem A.2.2. Let the 1-form α =∑n

i=1 aidxi be analytic or smooth on a neigh-borhood of 0 in Rn, or holomorphic on a neighborhood of 0 in Cn. If 0 is analgebraically isolated singularity of α, then α has the division property at 0.

More generally, if α is a holomorphic 1-form on (Cn, 0) such that the sin-gular set S = x ∈ Cn | α(x) = 0 has codimension m ≥ 2, then any germ ofholomorphic p-form ω such that p < m and ω ∧α = 0, is dividable by α. See, e.g.,[267, 233].

A.3 Reeb stability

Recall that the holonomy of a leaf L of regular foliation F on a manifold M at apoint x ∈ L is a homomorphism from the fundamental group π1(L, x) to the groupDiff(Rp, 0) of germs of diffeomorphisms of (Rp, 0), where p is the codimension ofF . It can be defined as follows: Let γ : [0, 1]→ L be a loop in L, γ(0) = γ(1) = x.Denote by U a tubular neighborhood of γ in M , and π : U → L a projection (whichis identity on L∩U), such that the preimage of each point of y ∈ L∩U is a smallp-dimensional disk Dp

y. If z ∈ Dx is sufficiently close to x, then there is a uniqueloop γz : [0, 1]→ Lz ∩ U , where Lz is the leaf through z, such that γz(0) = z andπ(γz(t)) = γ(t) ∀ t ∈ [0, 1]. The map φγ : z → γz(1) is a local diffeomorphismof (Dp

x, x). Its germ at x is called the holonomy around γ and depends only onthe homotopy class of the loop γ, so we can assign to each element [γ] of π1(L, x)a germ of diffeomorphism φγ of (Dp

x, 0). The concatenation of two loops clearlyleads to the composition of two corresponding germs of diffeomorphisms, so weget a group homomorphism from π1(L, x) to Diff(Dp

x, 0). This is, by definition,

272 Appendix

the holonomy of F at x. Its image in Diff(Dpx, 0) is called the holonomy group of

F at x. Up to isomorphisms, this holonomy depends only on F and L, and noton the choice of x, U and π. We will say that a leaf L has finite (resp. trivial)holonomy if its holonomy group is finite (resp., trivial). For example, when L issimply connected, then it automatically has trivial holonomy.

The following theorem, called the Reeb stability theorem, says that if theholonomy of a leaf is trivial, then the foliation is “stable” near that leaf.

Theorem A.3.1 (Reeb [356]). Let L be a compact leaf of a regular foliation F ofcodimension p on a manifold M . Assume that L has trivial holonomy (this condi-tion is automatically satisfied if L is simply-connected ). Then there is a neighbor-hood U of L in M in which the foliation F is diffeomorphic to a trivial fibrationF × Rp → Rp.

Proof (sketch). Since L is compact, there is a tubular neighborhood U of L inM and a projection π : U → L, such that the preimage of a point x ∈ L is ap-dimensional disk Dp

x. Since the holonomy of L is trivial, for each point y ∈ Uclose enough to L, there is a unique point ρ(y) ∈ Dp

x such that y can be connectedto ρ(y) by a path which lies on a leaf of F and which is close to L. The map

y → (π(y), ρ(y)) (A.30)

is a diffeomorphism from a neighborhood of L in U to a neighborhood of L× xin L × Dp

x. Under this map, leaves of F are sent to fibers of the trivial fibrationL×Dp

x → Dpx, hence the conclusion of the theorem.

Reeb’s stability theorem can be used to give a simple proof of the followingfolklore theorem, which is somehow difficult to find in the literature (see, e.g.,[132]):

Theorem A.3.2. Let g be the Lie algebra of a connected simply-connected compactLie group G. Then any smooth action of g on a manifold M with a fixed pointp ∈ M can be integrated, in a neighborhood of p, to an action of G.

Proof (sketch). Following Palais [290], let us consider the action of g on G ×Mgiven by x → x+ + ξx, where x+ denotes the left-invariant vector field on Ggenerated by x ∈ g, and ξx is the vector field on M generated by x via theaction of g. The image of this action spans a regular involutive distribution onG × M (of rank equal to the dimension of G), so by Frobenius’ theorem thisdistribution is integrable. Since p is a fixed point of the action of g, the submanifoldG×p is a leaf of the corresponding foliation on G×M . Since this leaf is compactsimply-connected, the nearby leaves are also diffeomorphic to G by Reeb’s stabilitytheorem. This implies that the action of g on M integrates to a free (right) actionof G in a neighborhood of G × p in G × M (which projects to the action ofG on itself by multiplication from the right). By projecting the action of G tothe second component of the product, one gets an action of G in a neighborhoodof p in M .

A.4. Action-angle variables 273

A.4 Action-angle variables

There is a huge amount of literature on integrable systems. See, e.g., [19, 38, 20,102, 2] for an introduction to the subject and many concrete examples. In thissection, we will discuss mainly only one particular aspect of integrable Hamilto-nian systems, namely the existence of action-angle variables, with an emphasis onaction variables. We need these action variables in the study of proper symplecticgroupoids (Section 7.5) and convexity of momentum maps (see [370]).

Recall that a function H (or the corresponding Hamiltonian vector field XH)on a 2n-dimensional symplectic manifold (M2n, ω) is called Liouville integrable, ifit admits n functionally independent first integrals in involution. In other words,there are n functions F1 = H, F2, . . . , Fn on M2n such that dF1 ∧ · · · ∧ dFn = 0almost everywhere and Fi, Fj = 0 ∀ i, j. The map

F = (F1, . . . , Fn) : (M2n, ω)→ Kn (A.31)

(where K = R or C) is called the momentum map of the system. It is actually themomentum map of a Hamiltonian action of the Abelian Lie algebra of dimension non (M2n, ω). The (regular) level sets of F are Lagrangian submanifolds of (M2n, ω),and they are invariant with respect to XH . A classical result attributed to Liouville[215] says that, in the smooth case, if a connected level set N is compact and doesnot intersect with the boundary of M2n (of if M2n has no boundary), then it isdiffeomorphic to a standard torus Tn (= quotient of Rn by a co-compact lattice),and the Hamiltonian system XH is quasi-periodic on N . In other words, there is aperiodic coordinate system (q1, . . . , qn) on N with respect to which the restrictionof XH to N has constant coefficients: XH =

∑γi∂/∂qi, γi being constants. For

this reason, N is called a Liouville torus.In practice, one often deals with Hamiltonian systems which admit a non-

Abelian algebra of symmetries. For such systems, it is more convenient to workwith a generalized notion of Liouville integrability. The following definition isessentially due to Nekhoroshev [276] and Mischenko and Fomenko [252]:

Definition A.4.1. A Hamiltonian vector field XH on a Poisson manifold (M, Π) iscalled integrable in generalized Liouville sense if there are positive integers p, q, aq-tuple F = (F1, . . . , Fq) of functions on M and a p-tuple X = (XG1 , . . . , XGp)of Hamiltonian vector fields on (M, Π), such that the following conditions aresatisfied:

i) Fi, Gj = Gi, Gj = Fi, H = Gi, H = 0 ∀ i, j,ii) dF1 ∧ · · · ∧ dFq = 0 and XG1 ∧ · · · ∧XGp = 0 almost everywhere,iii) p + q = dim M .

We may put G1 = H in the above definition. When (M, Π) is symplectic andp = q = dimM/2, the functions in F automatically Poisson-commute and we getback to the usual Liouville integrability. The above notion of integrability is alsocalled noncommutative integrability due to the fact that the components of F do

274 Appendix

not Poisson-commute in general, and in many cases one may choose them from afinite-dimensional non-Abelian Lie algebra of functions under the Poisson bracket.

By abuse of language, we will also say that (X,F) is an integrable Hamiltoniansystem in generalized Liouville sense, and call (p, q) the bi-degree of the system.We will say that the system (X,F) is proper if the generalized momentum mapF : M → Rq is a proper map from M to its image, and the image of the singular setx ∈M, XG1∧· · ·∧XGp(x) = 0 of the commuting Hamiltonian vector fields underthe map F : M → Rq is nowhere dense in Rq. A connected component N of a levelset of F will be called a regular if it is regular with respect to F, and the vector fieldsof X are independent everywhere on N . We have the following obvious generaliza-tion of Liouville’s theorem: a regular connected level set N of a proper noncom-mutatively integrable Hamiltonian system (X,F) is diffeomorphic to a torus Tp onwhich the flow of any vector field XH such that H, Fi = 0 ∀i is quasi-periodic;in analogy with the Liouville-integrable case, we will call such N a Liouville torus.

We will restrict our attention to the case when (M, Π) is a symplectic man-ifold, i.e., Π is nondegenerate. There is a very close relation between noncommu-tatively integrable Hamiltonian systems and symplectically complete foliations inthe sense of Libermann (see Section 1.9). Indeed, if (X,F) is an integrable sys-tem on a symplectic manifold (M, ω), then the regular part of M (consisting ofpoints x such that dF1 ∧ · · · ∧ dFq(x) = 0 and XG1 ∧ · · · ∧ XGp(x) = 0) is asymplectically complete foliation by isotropic submanifolds (= level sets of F):the dual foliation is given by the level set of the map G = (G1, . . . , Gp). (Thepair (F,G) is a so-called dual pair , see [288].) Conversely, assume that W is asymplectically complete foliation on M . Then the set of points x ∈ M such thatdim(TxW ∩ (TxW)⊥) is locally constant is a dense open subset of M on whichTxW ∩ (TxW)⊥ is an integrable distribution. The corresponding foliation F isisotropic symplectically complete, its dual foliation G is given by the coisotropicdistribution TxW∪ (TxW)⊥. Assume that we can realize the leaves of F (resp., G)as the connected components of the level set of a map F = (F1, . . . , Fq) from Mto Rq (resp., of a map G = (F1, . . . , Fp) from M to Rp) almost everywhere. Then(XG1 , . . . , XGp , F1, . . . , Fq) is an integrable Hamiltonian system of bi-degree (p, q).

Example A.4.2. A Hamiltonian Tp-action on a Poisson manifold can be seen asa proper integrable system: X is generated by the components of the momentummap of the Tp-action, and F consists of an appropriate number Tp-invariant func-tion. One can associate to each Hamiltonian action of a compact Lie group G on aPoisson manifold a proper integrable system: H is the composition of the momen-tum map µ :M→g∗ with a generic Ad∗-invariant function h :g∗→R (see [371]).

Example A.4.3. If (Γ, ω) ⇒ (P, Π) is a proper symplectic groupoid, then it givesrise to a natural proper integrable Hamiltonian system (X,F) on (Γ, ω): F consistsof functions of the types s∗f and t∗f , f : P → R, and X consists of Hamiltonianvector fields of the type Xs∗f , where f is a Casimir function on P . The corre-sponding symplectically complete isotropic foliation is given by the submanifoldss−1(x) ∩ t−1(y), x, y ∈ P .

A.4. Action-angle variables 275

Remark A.4.4. Proper integrable Hamiltonian systems in generalized Liouvillesense can also be made into Liouville integrable systems (by changing the set offirst integrals), see [137].

The description of a Hamiltonian system near a Liouville torus is given by thefollowing theorem, called the Liouville–Mineur–Arnold theorem, about the exis-tence of action-angle variables. It was obtained by Henri Mineur in 1935 [249, 250]for Liouville-integrable systems, was forgotten, then rediscovered by V.I. Arnoldand other people in 1960s, and extended in a straightforward way to the noncom-mutatively integrable case by Nekhoroshev [276].

Theorem A.4.5 (Liouville–Mineur–Arnold). Let N be a Liouville torus of a properHamiltonian system (X,F) of bi-degree (p, q) on a symplectic manifold (M, ω).Then there is a neighborhood U(N) of N and a symplectomorphism

Ψ : (U(N), ω) → (D2r ×Dp × Tp,r∑1

dxi ∧ dyi +p∑1

dνi ∧ dφi), (A.32)

where 2r = q−p = dimM −2p, (xi, yi) are coordinates on D2r, νi are coordinatesof Dp, φi (mod 1) are periodic coordinates of Tp, such that F does not depend onθi = Ψ∗φi, and the functions which generate the Hamiltonian vector fields in Xdepend only on I1, . . . , Ip, where Ii = Ψ∗νi.

The variables (Ii, θi) in the above theorem are called action-angle variables .In the Liouville-integrable case (p = dimM/2), they form a complete system ofvariables in a neighborhood of N . If p < dimM/2, we need dimM −2p additionalvariables. The map

(I1, . . . , In) : (U(N), ω) → Rp (A.33)

is the momentum map of a free Hamiltonian torus Tp-action on (U(N), ω) whichpreserves F. The existence of this Hamiltonian torus action is essentially equiv-alent to the Liouville–Mineur–Arnold theorem. Indeed, once the action variables(I1, . . . , Ip) which generate a free torus action are found, the corresponding an-gle variables (θ1, . . . , θp) can also be found easily by the following method: fix acoisotropic section to the foliation by Liouville tori in U(N), put θi = 0 on thiscoisotropic section, and extend them to the rest of U(N) in a unique way suchthat XIi(θj) = 0 if i = j and XIi(θi) = 1. The fact that the section is coisotropicwill ensure that θi, θj = 0. When p < dimM/2, we can find additional variablesin a way similar to the classical proof of Darboux’s theorem. Remark that thequasi-periodicity of the system on N also follows immediately from the existenceof this torus action.

The existence of action-angle variables is very important, both for the the-ory of near-integrable systems (K.A.M. theory), and for the quantization of inte-grable systems (Bohr–Sommerfeld rule). Actually, Mineur was an astrophysicist,and Bohr–Sommerfeld quantization was his motivation for finding action-anglevariables.

276 Appendix

Mineur [250] also wrote down the following simple formula, which we willcall the Mineur–Arnold formula, for action functions:

Ii(z) =∫

Γi(z)

β, (A.34)

where z is a point in U(N), β is a primitive of the symplectic form ω, i.e., dβ = ω,and Γi(z) is a 1-cycle on the Liouville torus which contains z and which depends onz continuously (the cycle given by the orbits of the periodic vector field XIi = ∂

∂θi).

Of course, this formula is an immediate consequence of Theorem A.4.5.Since Formula (A.34) depends on the choice of Γi and β, the momentum

map (I1, . . . , Ip) is not unique, but it is unique only up to an integral affine trans-formation. As a consequence, they induce a well-defined integral affine structure.This affine structure lives on the base space (= space of Liouville tori) in theLiouville-integrable case, and on the “reduced” base space (= leaf space of thecoisotropic foliation dual to the foliation by Liouville tori) in the noncommuta-tively integrable case.Remark A.4.6. To a large extent, torus actions (of appropriate dimensions) and(partial) action-angle variables also exist near singularities of proper integrableHamiltonian systems, see [371] and references therein.

A.5 Normal forms of vector fields

A.5.1 Poincare–Dulac normal forms

Let X be an analytic or formal vector field in a neighborhood of 0 in Km, whereK = R or C, with X(0) = 0. When K = R, we may also view X as a holomorphic(i.e., complex analytic) vector field by complexifying it. Denote by

X = X(1) + X(2) + X(3) + · · · (A.35)

the Taylor expansion of X in some local system of coordinates, where X(k) is ahomogeneous vector field of degree k for each k ≥ 1. The algebra of linear vectorfields on Km, under the standard Lie bracket, is nothing but the reductive algebragl(m, K) = sl(m, K)⊕K. In particular, we have

X(1) = Xs + Xnil, (A.36)

where Xs (resp., Xnil) denotes the semisimple (resp., nilpotent) part of X(1).There is a complex linear system of coordinates (xj) in Cm which puts Xs intodiagonal form:

Xs =m∑

j=1

γjxj∂/∂xj, (A.37)

where γj are complex coefficients, called eigenvalues of X (or X(1)) at 0.

A.5. Normal forms of vector fields 277

Definition A.5.1. The vector field X is said to be in Poincare–Dulac normal form,or normal form for short, if it commutes with the semisimple part of its linearpart:

[X, Xs] = 0. (A.38)

A transformation of coordinates which puts X in Poincare–Dulac normal form iscalled a Poincare–Dulac normalization.

For each natural number k ≥ 1, the vector field Xs acts linearly on the spaceof homogeneous vector fields of degree k by the Lie bracket, and the monomialvector fields are the eigenvectors of this action:[ m∑

j=1

γjxj∂/∂xj , xb11 xb2

2 . . . xbnn ∂/∂xl

]=( n∑

j=1

bjγj − γl

)xb1

1 xb22 . . . xbn

n ∂/∂xl.

(A.39)When an equality of the type

m∑j=1

bjγj − γl = 0 (A.40)

holds for some nonnegative integer m-tuple (bj) with∑

bj ≥ 2, we will say thatthe monomial vector field xb1

1 xb22 . . . xbm

m ∂/∂xl is a resonant term, and that them-tuple (b1, . . . , bl − 1, . . . , bm) is a resonance relation for the eigenvalues (γi).Equation (A.38) means that a vector field is in Poincare–Dulac normal form ifand only if its Taylor expansion does not contain any nonresonant nonlinear term.In particular, if X is nonresonant, i.e., its eigenvalues do not satisfy any resonancerelation, then X becomes linear when it is normalized. In general, Poincare–Dulacnormalization may be viewed as a partial linearization (see Subsection A.5.3). Ofcourse, when the semisimple linear part Xs of X is trivial or very resonant, thenthe Poicare–Dulac normal form does not give much information, and more refinednormal forms are be needed.

Theorem A.5.2 (Poincare–Dulac). Any analytic or formal vector field admits aformal Poincare–Dulac normalization.

The proof of the above theorem is rather simple and is given by the (formal)composition of an (infinite) sequence of local coordinate transformations whichkill all the nonresonant nonlinear terms, term by term. Each of these coordinatetransformations is provided by the following simple lemma (which is similar toProposition 2.2.1 of Chapter 2):

Lemma A.5.3. Let X = X(1) + X(2) + · · · be a vector field such as above. Supposethat we have X(q) = [Z, X(1)] + T (q), where Z is a homogeneous vector field ofdegree q with q > 1. Then the time-1 flow φ1

Z of Z transforms X to a vector fieldadmitting a Taylor expansion

(φ1Z)∗X = X(1) + · · ·+ X(q−1) + T (q) + · · · , (A.41)

278 Appendix

i.e., this transformation eliminates the term [Z, X(1)] in X without changing theterms of degree smaller than q.

It is much more difficult to show the existence, or lack thereof, of a convergent(i.e., local analytic) normalization for a given analytic vector field. One of the bestresults in this direction is a theorem of A. Bruno, which we use several times inthis book:

Theorem A.5.4 (Bruno [48]). If a local holomorphic vector field X on (Cm, 0) withX(0) = 0 satisfies the ω-condition, and admits a formal normal form of the typea(x)

∑mj=1 γixi∂/∂xj where a(x) is a formal first integral of

∑mj=1 γixi∂/∂xj, then

it admits a local analytic normalization. In particular, if X is nonresonant andsatisfies the ω-condition, then it is locally analytically linearizable.

The ω-condition in the above theorem is the following Diophantine condition,invented by Bruno, on the eigenvalues of X :

∞∑k=1

− logωk

2k< +∞, (A.42)

where

ωk = min∣∣∣ m∑

j=1

cjγj − γl

∣∣∣ ; (ci) ∈ Zn+, 2 ≤

∑cj ≤ 2k, l = 1, . . . , m

. (A.43)

Remark that the ω-condition is weaker than the following Diophantine con-dition of Siegel: there are positive constants C, q > 0 such that∣∣∣∑

i

ciγi − γl

∣∣∣ ≤ C(∑

i

|ci|)−q

(A.44)

for all (ci) ∈ Zn+ such that 2 ≤

∑cj ≤ 2k and all l = 1, . . . , m.

The proof of Theorem A.5.4 is based on Kolmogorov’s fast convergencemethod. See, e.g., [49, 122, 239, 323] for details and some generalizations.

A.5.2 Birkhoff normal forms

Consider now a local analytic or formal Hamiltonian vector field X = XH on asymplectic space (K2n, 0) with a standard symplectic structure, such that X(0) =0. We may assume that H(0) = 0 and dH(0) = 0. Then the Taylor expansion ofX corresponds to the Taylor expansion

H = H(2) + H(3) + H(4) + · · · (A.45)

of H , X(j) = XH(j+1) . In particular, X(1) = XH(2) ∈ sp(2n, K), which is a simpleLie algebra, and the decomposition X(1) = Xs + Xnil lives in this algebra andcorresponds to a decomposition of H(2),

H(2) = Hs + Hnil. (A.46)


There is a complex canonical linear system of coordinates (xj , yj) in C2n in whichHs has diagonal form:

Hs =n∑

j=1

λjxjyj, (A.47)

where λj are complex coefficients, called frequencies of H (or XH) at 0. Thesefrequencies are uniquely determined by H only up to a sign (i.e., up to multi-plication by ±1), and the eigenvalues of XH are λ1,−λ1, . . . , λn,−λn. In partic-ular, a Hamiltonian vector field will have auto-resonance relations of the typeλi + (−λi) = 0.

Equation (A.38) in the Hamiltonian case can also be written as

H, Hs = 0, (A.48)

and one says that H (or XH) is in Birkhoff normal form if H satisfies the aboveequation. A symplectic transformation of coordinates which puts XH in Birkhoffnormal form is called a Birkhoff normalization. In other words, a Birkhoff normal-ization is a Poincare–Dulac normalization with the additional property of beingsymplectic.

The theory of Birkhoff normal forms is very similar to the theory of Poincare–Dulac normal forms, and one may regroup the two theories together and call it thetheory of Poincare–Birkhoff normal forms. In particular, the proof of the followingtheorem is absolutely similar to the proof of Theorem A.5.2.

Theorem A.5.5 (Birkhoff et al.). Any analytic or formal Hamiltonian vector fieldon a symplectic manifold admits a formal Birkhoff normalization.

Moreover, it follows from the toric characterization of Poincare–Birkhoff nor-mal forms (see Subsection A.5.3) that we have:

Proposition A.5.6. An analytic Hamiltonian vector field XH on a symplectic man-ifold which vanishes at a point admits a local analytic Birkhoff normalization ifand only if it admits a local analytic Poincare–Dulac normalization (when oneforgets about the symplectic structure).

For example, Theorem A.5.4 together with Proposition A.5.6 imply the fol-lowing positive result of Russmann [307] and Bruno [48]: if H is an analytic Hamil-tonian function on a symplectic manifold which admits a formal normal form of thetype h(

∑nj=1 λjxjyj) for some formal function h of one variable, and the numbers

±λ1, . . . ,±λn satisfy Bruno’s ω-condition, then H admits a local analytic Birkhoffnormalization. (However, this result is not very applicable in practice, because itis very rare for a Hamiltonian to have a normal form of the type h(

∑nj=1 λjxjyj).)

A resonance relation in the Hamiltonian case is a relation of the type

n∑j=1

cjλj = 0 with (cj) ∈ Zn, (A.49)

280 Appendix

where λj are the frequencies. They correspond to monomial functions (resonanceterms)

∏nj=1 x

aj

j ybj

j such that Hs,∏n

j=1 xaj

j ybj

j = 0. Here cj = aj−bj. In particu-lar, the terms

∏nj=1(xjyj)aj are always resonant (we will call them auto-resonant).

A Hamiltonian vector field XH is called nonresonant at 0 if its frequencies do notadmit any nontrivial resonance relation

∑nj=1 cjλj = 0 with (cj) = 0. Due to auto-

resonant terms, a nonresonant Hamiltonian vector field is not formally linearizablein general, and the best one can do is to put a nonresonant Hamiltonian functionin the form H = h(x1y1, . . . , xnyn) via a Birkhoff normalization.

Due to auto-resonances, it is even more difficult for a Hamiltonian vector fieldto admit a convergent normalization than a general vector field. Nevertheless, indimension 4 there is a positive result by Moser [266]: if λ1, λ2 are two non-zero com-plex numbers such that λ1/λ2 /∈ R, then any local holomorphic Hamiltonian vectorfield in C4 with frequencies λ1, λ2 at 0 admits a convergent Birkhoff normalization.If λ1/λ2 ∈ R then a convergent normalization doesn’t exist in general [316].

A.5.3 Toric characterization of normal forms

In this subsection we will explain a simple but important fact about Poincare–Birkhoff normal forms, namely that they are governed by torus actions.

Denote by R the set of resonance relations for the eigenvalues (γ1, . . . , γm)of a given vector field X which vanishes at 0. In other words, (cj) ∈ R if and onlyif (cj) ∈ Zn,

∑j cjγj = 0,

∑j cj ≥ 1, cj ≥ −1 for all j, and at most one of the cj

is negative. The numberr = dimZ(R⊗ Z) (A.50)

is called the degree of resonance of X . If X = XH is a Hamiltonian vector fieldon a symplectic space of dimension m = 2n, then we always have r ≥ n dueto n auto-resonance relations λj + (−λj) = 0, and we will call r′ = r − n theHamiltonian degree of resonance of XH .

Denote by Q ⊂ Zm the integral sublattice of Zm consisting of m-dimensionalvectors (ρj) ∈ Zm which satisfy the following properties:

m∑j=1

ρjcj = 0 ∀ (cj) ∈ R , and ρj = ρk if γj = γk. (A.51)

The numberd = dimZQ (A.52)

is called the toric degree of X at 0. In general, we have d + r ≤ m. In the Hamil-tonian case, we have d + r = m = 2n, or d + r′ = n.

Let (ρ1j), . . . , (ρ

dj ) be a basis of Q. For each k = 1, . . . , d define the following

diagonal linear vector field Zk:

Zk = 2π√−1

m∑j=1

ρkj xj∂/∂xj. (A.53)


The above vector fields Z1, . . . , Zd have the following remarkable properties:They commute pairwise and commute with Xs and Xnil. They are linearly in-dependent almost everywhere. Each Zk, k = 1, . . . , d, is periodic of real period1, and together they generate a linear action of the real torus Td in Cm, whichpreserves Xs and Xnil. In the Hamiltonian case, this action also preserves thestandard symplectic structure. The semisimple linear part Xs is a linear combi-nation of Z1, . . . , Zd. A simple calculation shows that X is in Poincare–Birkhoffnormal form, i.e., [X, Xs] = 0, if and only if we have

[X, Zk] = 0 ∀ k = 1, . . . , d. (A.54)

The above commutation relations mean that if X is in normal form, then itis preserved by the effective d-dimensional torus action generated by Z1, . . . , Zd.Conversely, if there is a torus action which preserves X , then because the torus isa compact group we can linearize this torus action (using Bochner’s linearizationTheorem 7.4.1 in the non-Hamiltonian case, and the equivariant Darboux theoremA.1.7 in the Hamiltonian case), leading to a normalization of X . In other words, wehave the following simple result, probably first written down explicitly in [372, 368],which says that a Poincare–Birkhoff normalization for a vector field is nothing buta linearization of a corresponding torus action:

Theorem A.5.7. A holomorphic (Hamiltonian) vector field X in a neighborhood of0 in Cm (or C2n with a standard symplectic form), which vanishes at 0, admitsa convergent Poincare–Birkhoff normalization if and only if it is preserved by aneffective holomorphic (Hamiltonian) action of a real torus of dimension d, whered is the toric degree of X at 0, in a neighborhood of 0 in Cm (or C2n), which has0 as a fixed point and whose linear part at 0 has appropriate weights (given by thelattice Q defined in (A.51)).

The above theorem is true in the formal category as well. But of course, anyvector field admits a formal Poincare–Birkhoff normalization, and a formal torusaction. Theorem A.5.7 has many direct implications. One of them is that, a realanalytic vector field admits a local real analytic Poincare–Birkhoff normalizationif and only if it its complexification admits a local holomorphic normalization, see[372, 368]. Another one is Proposition A.5.6 mentioned in the previous subsection.In fact, if there is a torus action which preserves a Hamiltonian vector field XH

and whose linear part is generated by the linear vector fields Z1, . . . , Zd definedabove, then one can show easily that this torus action must automatically preservethe symplectic structure.

Theorem A.5.7 leads to another method for finding a convergent Poincare–Birkhoff normalization: find a torus action. This method works well for integrablesystems, due to Liouville–Mineur–Arnold Theorem A.4.5 and its generalizationsabout the existence of torus actions in integrable systems. In particular, we have:

Theorem A.5.8. Any integrable analytic vector field which vanishes at a pointadmits a convergent Poincare–Birkhoff normalization.

282 Appendix

In the above theorem, integrable means integrable in generalized Liouvillesense in the Hamiltonian case, or a similar notion of integrability in the generalcase (existence of a complete set of analytic commuting vector fields and commonfirst integrals, see [368]). For the proof, see [372, 368]. A particular important caseof the above theorem, when X is a nonresonant Hamiltonian vector field, wasobtained by Ito [183] (Ito’s proof is based on the fast convergence method andis very technical). Some other partial cases of Theorem A.5.8 were obtained in[336, 337, 50, 188], using different methods.

If a dynamical system near an equilibrium point is invariant under a compactgroup action which fixes the equilibrium point, then this compact group actioncommutes with the (formal) torus action of the Poincare–Birkhoff normalization.Together, they form a bigger compact group action, whose linearization leads toa simultaneous Poincare–Birkhoff normalization and linearization of the compactsymmetry group, i.e., we can perform the Poincare–Birkhoff normalization in aninvariant way. This is a known result in dynamical systems, see, e.g., [367], butthe toric point of view gives a simple proof of it. The case of equivariant vectorfields is similar. For example, one can speak about Poincare–Dulac normal formsfor time-reversible vector fields, see, e.g., [206].

One can probably use the toric point of view to study normal forms of Hamil-tonian vector fields on Poisson manifolds as well. For example, let g∗ be the dualof a semisimple Lie algebra, equipped with the standard linear Poisson structure,and let H : g∗ → K be a regular function near the origin 0 of g∗. The correspondingHamiltonian vector field XH will vanish at 0, because the Poisson structure itselfvanishes at 0. Applying Poincare–Birkhoff normalization techniques, we can killthe “nonresonant terms” in H (with respect to the linear part of H , or dH(0)). Thenormalized Hamiltonian will be invariant under the coadjoint action of a subtorusof a Cartan torus of the (complexified) Lie group of g. In the “nonresonant” case,we have a Cartan torus action which preserves the system.

A.5.4 Smooth normal forms

Let X be a C∞ vector field on Rn, vanishing at 0. We decompose its linear part at0 in the form X(1) = Xs +Xnil like in the analytic case (semisimple plus nilpotentpart), and say that X is in normal form if we have [Xs, X ] = 0. A smooth localtransformation of coordinates which puts X in normal form is called a smoothnormalization of X. The smooth normalization is substantially different, and in asense simpler, than the analytic normalization problem. In particular, according toa famous theorem of Sternberg [321] generalized by Chen [76], if X is a hyperbolicvector field (i.e., its eigenvalues have non-zero real parts), then X admits a smoothnormalization. See, e.g., [304, 73, 182, 46, 22, 47, 26, 74, 75] for results and meth-ods on problems of smooth normalization and Ck normalization of vector fields.The usual methods for proving the existence of smooth normalization of a vectorfield extend readily to the case with parameters, i.e., families of vector fields. Forexample, let X(u) be a smooth vector field on Rn, depending smoothly on a pa-

A.6. Normal forms along a singular curve 283

rameter u belonging to a compact manifold P. We suppose that X(u) vanishes atthe origin of Rn, for every u and has a linear part X(1) which is independent ofu. If the eigenvalues λ1, . . . , λn of X(1) have all non-zero real parts and have noresonance relations, then there is a smooth family φ(u) of local diffeomorphismsof Rn, fixing the origin and all defined in a fixed neighborhood of the origin suchthat φ(u)∗X(u) = X(1) for every u.

A.6 Normal forms along a singular curve

Let (M, Π) be a C∞-smooth Poisson manifold, and m be a point where Π vanishes.We will assume that the curl vector field DωΠ of Π with respect to some smoothdensity ω (see Section 2.6) doesn’t vanish at m; this requirement is independentof the chosen density, and the Poisson structure has to vanish along the wholeorbit of DωΠ through m. In this section, we are interested in the case where thisorbit is closed, and we denote it by Γ. As we want to study the germ of Π alongΓ we may consider that M is a neighborhood of Γ = S1 × 0 in S1 × Rn withcoordinates (θ, x1, . . . , xn), and that Π vanishes on Γ.

The first invariant attached to this situation is the period c of the curl vectorfield DωΠ on Γ: it doesn’t depend on the choice of ω because Π vanishes on Γ.

The second thing we have to do is to take care of the linear part of Π atthe points of Γ. If we choose local coordinates (x0, x1, . . . , xn) vanishing at m, andsuch that DωΠ is ∂/∂x0, then the linear part , (1) of the Poisson bracket satisfiesrelations of the type

x0, xi(1) =∑n

j=1 dij xj ,

xr, xi(1) =∑n

j=1 cjrixj ,

(A.55)

for r and i varying from 1 to n. This means that it corresponds to a semi-directproduct R L of R with an n-dimensional Lie algebra L. Since DωΠ is an in-finitesimal automorphism of Π, the isomorphism class of this Lie algebra R Ldepends on Γ but does not depend on the choice of the point m on Γ, so it is alsoan invariant of Π along Γ.

We will impose the following additional hypothesis on the matrix (dij): itseigenvalues, denoted by λ1, . . . , λn are real and do not satisfy any resonance rela-tion of the types

λi = p1λ1 + · · ·+ pnλn

or λi + λj = p1λ1 + · · ·+ pnλn ,(A.56)

where (p1, . . . , pn) is a non-zero multi-index with pk ≥ 0 and∑n

i=1 pi ≥ 2, excepttrivial relations λi + λj = λi + λj . In particular, the relations λi + λj = 2λi, withi = j, i.e., λi = λj , are forbidden, so the matrix (dij) is diagonalizable, so bya linear change of coordinates, we may assume that (dij) is diagonal: dij = 0 ifi = j, and dii = λi. It follows from the relations x0, xi(1) = λixi and the Jacobiidentity that we have x0, xi, xj(1)(1) = (λi + λj)xi, xj(1). Since λi + λj is

284 Appendix

not an eigenvalue due to the nonresonance condition, we have xi, xj(1) = 0, i.e.,L is an Abelian algebra.

Using the standard normalization procedure, one can show that, in this sit-uation, Π is formally nonhomogeneously quadratizable near every point of Γ, i.e.,we can find formal coordinates in which our Poisson structure has only linearand quadratic terms. Here we will present a result of Brahic [45] which says thatthis quadratization extends to a neighborhood of the curve Γ. Moreover we willdescribe the invariants attached to the isomorphism class of the germ of Π along Γ.

Theorem A.6.1 ([45]). Under the above hypotheses, up to a double covering, thereis a smooth system of coordinates (θ, x1, . . . , xn) in a neighborhood of Γ, whereθ ∈ S1 = R/Z is a periodic coordinate, such that

xi, xj = aijxixj + o∞(x) i, j = 1, . . . , n,

θ, xi = µixi i = 1, . . . , n,

where o∞(x) stands for a smooth function flat along Γ, and µi and aij are constant.

Proof. We will divide the proof into several small steps.Step 1. The brackets induced by Π can, in a neighborhood of Γ, be written as

xi, xj = O2(x) ,

θ, xi =∑n

j=1 hi,j(θ)xj + O2(x) ,

where hi,j are smooth functions on S1 = R/Z, O2(x) denotes smooth functionson S1 ×Rn which are of order 2 in the variables x1, . . . , xn, and the matrix Hθ =(hi,j(θ))i,j=1,...,n has eigenvalues k(θ)λi, i = 1, . . . , n for some k ∈ C∞(S1).Step 2. Up to a double covering of a neighborhood of Γ, in an appropriate coordi-nate system we have:

xi, xj = O2(x) ,

θ, xi = k(θ)λixi + O2(x).

Indeed, since for each θ ∈ S1, the eigenvalues k(θ)λi, i = 1, . . . , n, of Hθ are differ-ent, the space Rn = θ ×Rn decomposes into a direct sum of n one-dimensionaleigenspaces Ei of H . By varying θ, we get n line bundles over S1. By taking adouble covering of the circle if necessary, we may assume that these line bundlesare trivial. Then we can diagonalize Hθ by a smooth change of variables which islinear on each θ×Rn. (It is easy to construct examples where a double coveringis really necessary.)Step 3. Reparametrizing S1 by the formula

θnew =

∫ θ

0k−1(t)dt∫ 1

0 k−1(t)dt,

we can writexi, xj = O2(x), θ, xi = µixi + O2(x),

where µi are non-zero constants.

A.6. Normal forms along a singular curve 285

Step 4. The Hamiltonian vector field Xθ is tangent to each subspace θ×Rn andhas the type Xθ =

∑ni=1(µixi +pi) ∂

∂xi, where pi are O2(x). Since, by our hypothe-

ses, there is no resonance relation among the eigenvalues µi, the parametrized ver-sion of Sternberg’s smooth linearization theorem (see [304]) implies that there is asmooth coordinate transformation which fixes θ and which linearizes Xθ. Applyingthis coordinate transformation, we get

xi, xj = O2(x), θ, xi = µixi.

Step 5. Denote uij = xi, xj. The Jacobi identity∮

θ,xi,xjθ, xi, xj = 0 now

becomes(µi + µj)uij = Xθ(uij).

Denote by∑

I uijI(θ)xI the Taylor expansion of uij in the variables (x1, . . . , xn).Then the above equation implies

(µi + µj)uijI (θ) =( n∑

j=1

Ijµj

)uijI(θ)

for every multi-index I = (I1, . . . , In) with∑n

j=1 Ij ≥ 2. The non-resonance hy-pothesis says that uijI = 0 unless Ii = Ij = 1 and Ik = 0 ∀ k = i, j. In otherwords, we have uij = vij(θ)xixj + o∞(x). Finally, to turn vij(θ) into constants,we may apply the diffeomorphism

(θ, x1, . . . , xn) −→ (θ, x1, χ2(θ)x2, . . . , χn(θ)xn),

where

χj(θ) = exp

(1µ1

∫ θ

0

(u1,j(t)− u1,j)dt

), j = 2, . . . , n,

with u1,j =∫ 1

0 u1,j(t)dt. Remark A.6.2. In the case n = 1 the same proof shows that we have a smoothnormal form

Π = µx∂

∂θ∧ ∂

∂x,

where µ is a constant, in a neighborhood of the curve Γ. A more elaborate analysis[45] shows that we also have a smooth nonhomogeneous quadratic normal formwithout the flat term in the case when n = 2 and when n > 2 but all the µi havethe same sign. When Π is analytic and the eigenvalues satisfy some Diophantineconditions, then there is also an analytic normalization.

The constants µi and aij , which appear in Theorem A.6.1, are invariantsof the Poisson structure. Moreover, we can give a geometrical meaning to theseinvariants as follows. The sum of the µi is the inverse of the period of the curlvector field along the singular curve; the n-tuple (µ1, . . . , µn), up to multiplication

286 Appendix

by a scalar, is determined by the Lie algebra RL which corresponds to the linearpart of Π at a point on Γ; and the numbers aij can be read off the holonomy ofsingular symplectic foliation of Π along Γ.

To see the geometrical meaning of aij , let us first look at what happens inthe case n = 2: in that case, the normal form is

Π =∂

∂θ∧(

µ1x1∂

∂x1+ µ2x2

∂

∂x1

)+ ax1x2

∂

∂x1∧ ∂

∂x1,

and the regular symplectic leaves are surfaces which intersect each plane θ =constant along trajectories of the vector field Xθ = µ1x1∂/∂x1 + µ2x2∂/∂x1;after going around Γ once, a symplectic leaf which intersects θ = 0 at the Xθ-trajectory of a point (x1, x2) ∈ R2 will intersect θ = 0 again at the Xθ-trajectoryof the point (x1, x2 exp(−a/µ1)).

When n > 2, remark that restriction of the normal form

Π =∂

∂θ∧( n∑

i=1

µixi∂

∂xi

)+

n∑i,j=1

ai,jxixj∂

∂xi∧ ∂

∂xj

to any (foliation invariant) subspace xk = 0 ∀ k = i, j is again a Poissonstructure in normal form with n = 2 as above but with (i, j) in place of (1, 2) andaij in place of a. So the preceding paragraph shows that we can read aij off thebehavior of the symplectic leaves. This works also in the smooth case of TheoremA.6.1, when there are flat terms (a nontrivial exercise).

A.7 The neighborhood of a symplectic leaf

In this section, following Vorobjev [340], we will give a description of a Poissonstructure in the neighborhood of a symplectic leaf in terms of geometric data, andthen use these geometric data to study the problem of linearization of Poissonstructures along a symplectic leaf.

A.7.1 Geometric data and coupling tensors

First let us recall the notion of an Ehresmann (nonlinear) connection. Let p : E −→S be a submersion over a manifold S. Denote by TV E the vertical subbundle ofthe tangent bundle TE of E, and by V1

V (E) the space of vertical tangent vectorfields (i.e., vector fields tangent to the fibers of the fibration) of E. An Ehresmannconnection on E is a splitting of TE into the direct sum of TV E and anothertangent subbundle THE, called the horizontal subbundle of E. It can be definedby a V1

V (E)-valued 1-form Γ ∈ Ω1(E) ⊗ V1V (E) on E such that Γ(Z) = Z for

every Z ∈ TV E. Then the horizontal subbundle is the kernel of Γ: THE := X ∈TE, Γ(X) = 0. For every vector field u ∈ V1(S) on S, there is a unique liftingof u to a horizontal vector field Hor(u) ∈ V1

H(E) on E (whose projection to Sis u). The curvature of an Ehresmann connection is a V1

V (E)-valued 2-form on S,

A.7. The neighborhood of a symplectic leaf 287

CurvΓ ∈ Ω2(S)⊗ V1V (E), defined by

CurvΓ(u, v) := [Hor(u), Hor(v)]−Hor([u, v]), u, v ∈ V1(S), (A.57)

and the associated covariant derivative ∂Γ : Ωi(S)⊗C∞(E) −→ Ωi+1(S)⊗C∞(E)is defined by an analog of Cartan’s formula:

∂ΓK(u1, . . . , uk+1) =∑

i

(−1)i+1LHor(ui)(K(u1, . . . , ui, . . . , uk+1))

+∑i<j

(−1)i+jK([ui, uj ], u1, . . . , ui, . . . , uj, . . . , uk+1).(A.58)

Remark that ∂Γ ∂Γ = 0 if and only if Γ is a flat connection, i.e., CurvΓ = 0.Suppose now that S is a symplectic leaf in a Poisson manifold (M, Π), and

E is a small tubular neighborhood of S with a projection p : E −→ S. Then thereis a natural Ehresmann connection Γ ∈ Ω1(E) ⊗ V1

V (E) on E, whose horizontalsubbundle is spanned by the Hamiltonian vector fields Xfp, f ∈ C∞(S). ThePoisson structure Π splits into the sum of its horizontal part and its vertical part,

Π = V +H, (A.59)

where V = ΠV ∈ V2V (E) and H = ΠH ∈ V2

H(E) (there is no mixed part). Thehorizontal 2-vector field H is nondegenerate on THE. Denote by F its dual 2-form;it is a section of ∧2T ∗

HE which can be defined by the following formula:

F(Xfp, Xgp) = 〈Π, p∗df ∧ p∗dg〉, f, g ∈ C∞(S), (A.60)

(recall that Xfp, Xgp ∈ V1H(E)). Via the horizontal lifting of vector fields, F may

be viewed as a nondegenerate C∞(E)-valued 2-form on S, F ∈ Ω2(S)⊗ C∞(E).The above triple (V , Γ, F) is called a set of geometric data for (M, Π) in a

neighborhood of S.Conversely, given a set of geometric data (V , Γ, F), one can define a 2-vector

field Π on E by the formula Π = V +H, where H is the horizontal 2-vector fielddual to F. A natural question arises: how to express the condition [Π, Π] = 0, i.e.,Π is a Poisson structure, in terms of geometric data (V , Γ, F)? The answer to thisquestion is given by the following theorem:

Theorem A.7.1 (Vorobjev [340]). A triple of geometric data (V , Γ, F) on a fibrationp : E −→ S, where Γ is an Ehresmann connection on E, V ∈ V2

V (E) is a vertical2-vector field, and F ∈ Ω2(S) ⊗ C∞(E) is a nondegenerate C∞(E)-valued 2-formon S, determines a Poisson structure on E (by the above formulas) if and only ifit satisfies the following four compatibility conditions:

[V ,V ] = 0, (A.61)

LHor(u)V = 0 ∀ u ∈ V1(S), (A.62)∂ΓF = 0, (A.63)

CurvΓ(u, v) = V(d(F(u, v))) ∀ u, v ∈ V1(S), (A.64)

where V means the map from T ∗E to TE defined by 〈V(α), β〉 = 〈V , α ∧ β〉.

288 Appendix

Remark A.7.2. Equations (A.61) and (A.62) mean that the vertical part V of Π is aPoisson structure (on each fiber of E) which is preserved under parallel transport.This gives another proof of Theorem 1.6.1 which says that the transverse Poissonstructure to a symplectic leaf is unique up to local isomorphisms.Remark A.7.3. In the above theorem, E is not necessarily a tubular neighborhoodof S. The symplectic case (E is a symplectic manifold) of the above theoremwas obtained by Guillemin, Lerman and Sternberg in [159]. In fact, the proof ofthe symplectic case can be easily adapted to the Poisson case because a Poissonmanifold is just a singular foliation by symplectic manifolds. The Poisson structureΠ is called the coupling tensor of (V , Γ, F) (it couples a horizontal tensor with avertical tensor via a connection).

Proof. Consider a local system of coordinates (x1, . . . , xm, y1, . . . , yn−m) on E,where y1, . . . , yn−m are local functions on a fiber and x1, . . . , xm are local functionson S (m = dim S is even). Denote the horizontal lifting of the vector field ∂xi :=∂/∂xi from S to E by ∂xi. Then we can write Π = V +H, where

V =12

∑ij

aij∂yi ∧ ∂yj (aij = −aji), (A.65)

and

H =12

∑ij

bij∂xi ∧ ∂xj (bij = −bji) (A.66)

is the dual horizontal 2-vector field of F.The condition [Π, Π] = 0 is equivalent to

0 = [V ,V ] + 2[V ,H] + [H,H] = A + B + C + D, (A.67)

where

A = [V ,V ], (A.68)

B = 2∑

i

[V , ∂xi] ∧Xi, where Xi =∑

j

bij∂xj , (A.69)

C =∑ij

[V , bij ] ∧ ∂xi ∧ ∂xj +∑ij

∂xi ∧ ∂xj ∧(∑

kl

bikbjl[∂xk, ∂xl]), (A.70)

D =∑ijkl

bij∂xj(bkl) ∂xi ∧ ∂xk ∧ ∂xl. (A.71)

Notice that A, B, C, D belong to complementary subspaces of V3(E), so thecondition A + B + C + D = 0 means that A = B = C = D = 0.

The equation A = 0 is nothing but Condition (A.61): [V ,V ] = 0.The equation B = 0 means that [V , ∂xi] = 0 ∀ i, i.e., L∂xi

V = 0 ∀ i, whichis equivalent to Condition (A.62).


The equation D = 0 means that∮

ikl

∑j bij∂xj(bkl) = 0 ∀ i, k, l, where

∮ikl

denotes the cyclic sum. Let us show that this condition is equivalent to Condition(A.63). Notice that F(∂xi, ∂xj) = cij , where (cij) is the inverse matrix of (bij),and ∂ΓF(∂xi, ∂xj , ∂xk) =

∮ijk ∂xi(cjk). By direct computations, we have

∂ΓF

(∑α

biα∂xα,∑

β

bjβ∂xβ ,∑

γ

bkγ∂xγ

)= 2

∮ijk

∑l

bil∂xl(bjk). (A.72)

Thus the condition D = 0 is equivalent to the condition

∂ΓF

(∑α

biα∂xα,∑

β

bjβ∂xβ ,∑

γ

bkγ∂xγ

)= 0 ∀ i, j, k. (A.73)

Since the matrix (bij) is invertible, the last condition is equivalent to ∂ΓF = 0.Similarly, by direct computations, one can show that the condition C = 0 is

equivalent to Condition (A.64). Theorem A.7.4 (Vorobjev [340]). Let E be a sufficiently small neighborhood E of asymplectic leaf S of a Poisson manifold (M, Π), together with a given projection p :E −→ S. Denote by (V , Γ, F) the associated geometric data in E. Consider an arbi-trary tensor field φ ∈ Ω1(S)⊗C∞(E) whose restriction to Ω1(S) = Ω1(S)⊗C∞(S)via the inclusion S → E is trivial, and the following new set of geometric data:

V ′ = V , (A.74)

Γ′ = Γ− V(dp∗φ), (A.75)F′ = F− ∂Γφ− φ, φV . (A.76)

Then the coupling tensor Π′ of (V ′, Γ′, F′) is also a Poisson tensor, and there isa diffeomorphism f between neighborhoods of S, which fixes every point of S andsuch that f∗Π = Π′.

In the above theorem, V(dp∗φ) means an element of Ω1(E)⊗V1V (E) defined

by the formula V(dp∗φ)(w) = V(d(φ(p∗w)), w ∈ TE, where p∗w is the projectionof w to S, φ(p∗w) is viewed as a function on the fiber TxE over the origin x ofp∗w, and V(d(φ(p∗w)) is the Hamiltonian vector field with respect to V on TxEof the function φ(p∗w). Similarly, φ, φV means an element of Ω2(S) ⊗ C∞(E)defined by φ, φV(u, v) = φ(u), φ(v)V , where u, v ∈ V1(S), and the bracket istaken with respect to V .

Proof (sketch). We will use Moser’s path method. Consider the following familyof geometric data,

Vt = V ,

Γt = Γ− tV(dp∗φ),

Ft = F− t∂Γφ− t2φ, φV ,

290 Appendix

and the corresponding family of coupling tensors Πt, t ∈ [0, 1]. Define a time-dependent vector field X = (Xt)t∈[0,1] as follows: Xt is the unique horizontalvector field with respect to Γt which satisfies the equation

XtFt = −φ

(where φ and Ft are considered as differential forms on E by lifting). One verifiesdirectly that we have

[Xt, Πt] = −∂Πt

∂t. (A.77)

It implies that the time-1 flow ϕ1X of X = (Xt) moves Π = Π0 to Π′ = Π1. As a

consequence, Π′ is automatically a Poisson tensor. Note that X vanishes on S, soϕ1

X fixes every point of S. Remark A.7.5. The flow ϕt

X in the above proof preserves the symplectic leaves ofΠ (so Π and Π′ have the same foliation though not the same symplectic forms onthe leaves). What the flow does is to change the projection map p. It also allowsus to compare different geometric data of the same Poisson structure but withrespect to different projection maps.

A.7.2 Linear models

Consider the geometric data (V , Γ, F) of a Poisson structure in a neighborhood Eof a symplectic leaf S with respect to a projection p : E → S. We will embed Ein the normal bundle NS of S by a fiber-wise embedding which maps S to thezero section in NS and which projects to the identity map on S. Then we canview (V , Γ, F) as geometric data in a neighborhood of S (identified with the zerosection) in NS.

Denote by V(1) the fiber-wise linear part of V , Γ(1) the fiber-wise linear part ofΓ, and F(1) the fiber-wise affine part of F in NS. For example, if X, Y ∈ TxS, thenF(X, Y ) is a function on a neighborhood of zero in NxS, and F(1)(X, Y ) is the sumof the constant part and the linear part of F(X, Y ) on Nx. By looking at the fiber-wise linear terms of the equations in Theorem A.7.1, we obtain immediately that(V(1), Γ(1), F(1)) also satisfy these equations, which implies that the coupling tensorΠ(1) of (V(1), Γ(1), F(1)) is also a Poisson structure, defined in a neighborhood of Sin NS. We will call Π(1) the Vorobjev linear model of Π along the symplectic leaf S.

Theorem A.7.6. Up to isomorphisms, the Vorobjev linear model of a Poisson struc-ture Π along a symplectic leaf S is uniquely determined by Π and S (and does notdepend on the choice of the projection).

Proof. We will fix a projection p : E → S, and linearize the fibers of E by anembedding from E to NS which is compatible with p. This way we may considerthe linear model Π(1) of Π with respect to p as living in E. Consider now anotherarbitrary projection p1. We can find a smooth path of projections pt with p0 =p and p1 = p1. There is a unique time-dependent vector field Y = (Yt) in a


neighborhood of S which satisfies the following properties: Yt is tangent to thesymplectic leaves of Π, is symplectically orthogonal to the intersections of thefibers of pt with the symplectic leaves, vanishes on S, and the flow ϕt

Y of Y movesthe fibers of p0 to the fibers of pt: pt ϕt

Y = p0. Denote Πt = (ϕtY )−1Π. Denote by

Π(1)t the Vorobjev linear model of Πt with respect to the projection p (Π(1)

t alsolives in E via the fixed linearization of E). To prove the theorem, it is sufficient toshow that Π(1)

1 is isomorphic to Π(1) by a diffeomorphism in a neighborhood of S.Denote by (Vt, Γt, Ft) the geometric data of Πt with respect to p. Note that

∂Πt

∂t = −[Yt, Πt] by construction. Similar to the proof of Theorem A.7.4, we have

Vt = V ,∂Γt

∂t= −V(dp∗φt),

∂Ft

∂t= −∂Γtφt,

where φt is a family of elements of Ω1(S)⊗C∞(E) defined by φt = −YtFt. Lookingonly at the fiber-wise linear terms of the above equations, we get

V(1)t = V(1),

∂Γ(1)t

∂t= −(V(1))(dp∗φ(1)

t ),∂F

(1)t

∂t= −∂

Γ(1)t

φ(1)t ,

which implies that∂Π(1)

t

∂t= −[Zt, Π

(1)t ],

where Z = (Zt) is the time-dependent vector field defined by the formula φ(1)t =

−ZtF(1)t . As a consequence, the time-1 flow of (Zt) moves Π(1) to Π(1)

1 . Remark A.7.7. The linear model of a Poisson structure along a symplectic leafcan also be constructed from the transitive Lie algebroid which is the restrictionof the cotangent algebroid to the symplectic leaf, see [340]. We will leave it as anexercise for the reader to show that Vorobjev’s original construction via transitiveLie algebroids is equivalent to the above construction.

The following simple example shows that, in general, one can’t hope to finda local isomorphism between a Poisson structure and its Vorobjev linear modelalong a symplectic leaf; even if the leaf is simply-connected, the normal bundle istrivial and the transverse Poisson structure is linearizable.

Example A.7.8. Put M = S2×R3 with Poisson structure Π = fΠ1+Π2, where Π1

is a nondegenerate Poisson structure on S2, Π2 = x∂y ∧ ∂z + y∂z ∧ ∂x + z∂x∧ ∂yis the Lie–Poisson structure on R3 corresponding to so(3), and f = f(x2 +y2 +z2)is a Casimir function on (R3, Π2). Since the linear part of f on R3 is trivial, thelinear model of Π is f(0)Π1 +Π2. If f is not a constant then Π can’t be isomorphicto Π(1) near S for homological reasons: the regular symplectic leaves are S2 × S2,the integral of the symplectic form over the first component S2 is a constant (doesnot depend on the leaf) in the linear model Π(1), but is not a constant when thesymplectic form comes from Π.

If one wants to linearize only V and Γ but not F, then the situation becomesmore reasonable. See [45] for some results in that direction.

292 Appendix

A.8 Dirac structures

An almost Dirac structure on a manifold M is a subbundle L of the bundle TM ⊕T ∗M , which is isotropic with respect to the natural indefinite symmetric scalarproduct on TM ⊕ T ∗M ,

〈(X1, α1), (X2, α2)〉 :=12(〈α1, X2〉+ 〈α2, X1〉) (A.78)

for (X1, α1), (X2, α2) ∈ Γ(TM ⊕ T ∗M), and such that the rank of L is maximalpossible, i.e., equal to the dimension of M .

For example, if ω is an arbitrary differential 2-form on M , then its graphLω = (X, iXω) | X ∈ TM is an almost Dirac structure. Furthermore, an almostDirac structure L is the graph of a 2-form if and only if Lx ∩ (0 ⊕ T ∗

x M) = 0for any x ∈ M . Similarly, if Λ is an arbitrary 2-vector field on M , then the setLΛ = (iαΛ, α) | α ∈ T ∗M is also an almost Dirac structure.

A Dirac structure is an almost Dirac structure plus an integrability condition.To formulate this condition, consider the following bracket on Γ(TM ⊕ T ∗M),called the Courant bracket [84]:

[(X1, α1), (X2, α2)]C = ([X1, X2],LX1α2 − iX2dα1). (A.79)

An almost Dirac structure L is called a Dirac structure if it is closed under theCourant bracket, i.e., [(X1, α1), (X2, α2)]C ∈ Γ(L) for any (X1, α1), (X2, α2) ∈Γ(L). In this case, the pair (M, L) is called a Dirac manifold .

Example A.8.1. If ω is a 2-form on M , then the almost Dirac structure Lω =(X, Xω) | X ∈ TM is a Dirac structure if and only if ω is closed. Similarly, ifΛ is a 2-vector field on M , then the almost Dirac structure LΛ = (αΛ, α) | α ∈T ∗M is a Dirac structure if and only if Λ is a Poisson structure. In other words,Dirac structures generalize both presymplectic structures and Poisson structures.

Example A.8.2. If L is a Dirac structure on M such that its canonical projectionpr1 : L → M to M vanishes at a point x0 ∈ M , then for x near x0 we haveLx ∩ (TxM ⊕ 0) = 0, which implies that locally L = LΛ is the graph of a 2-vector field Λ, and the integrability of L means that Λ satisfies the Jacobi identity.Thus locally a Dirac structure whose projection to TM vanishes at a point is thesame as a Poisson structure which vanishes at that point.

The Courant bracket (A.79) is not anti-symmetric nor does it satisfy theJacobi identity on Γ(TM ⊕ T ∗M). But if L is a Dirac structure, then one canverify easily that the restriction of the Courant bracket to Γ(L) is anti-symmetricand satisfies the Jacobi identity, and it turns L into a Lie algebroid over M whoseanchor map is the canonical projection pr1 : L→ TM from L to TM , see [84]. Forexample, when L = LΛ comes from a Poisson structure, then this Lie algebroid isnaturally isomorphic to the cotangent algebroid associated to Λ.

A.8. Dirac structures 293

In particular, if L is a Dirac structure, then its associated distribution DL

on M, (DL)x = pr1(Lx), is integrable and gives rise to the associated foliation FL

on M . Furthermore, there is a 2-form ωL defined on each leaf of this foliation bythe formula

ΩL(X, Y ) = 〈α, Y 〉 ∀ (X, α), (Y, β) ∈ Lx. (A.80)

The fact that L is isotropic assures that ΩL is well defined and skew-symmetric.Moreover, we have:

Theorem A.8.3 ([84]). If L is a Dirac structure on M , then dΩL

= 0 on any leafof the associated singular foliation FL of L.

The meaning of the above proposition is that, roughly speaking, a Diracstructure is a singular foliation by presymplectic leaves. Its proof is a straightfor-ward verification similar to the Poisson case. Note that L is completely determinedby DL and ΩL.

A submanifold of a Poisson manifold is not a Poisson manifold in general,but is a Dirac manifold under some mild assumptions. More generally, we have:

Proposition A.8.4 ([84]). Let Q be a submanifold of a Dirac manifold (M, L). IfLq ∩ (TqQ ⊕ T ∗

q M) has constant dimension (i.e., its dimension does not dependon q ∈ Q), then there is a natural induced Dirac structure LQ on Q defined by theformula

(LQ)q =Lq ∩ (TqQ⊕ T ∗

q M)Lq∩(0 ⊕ (TQ)0)

. (A.81)

A special case of the above proposition is when Q = N is a slice, i.e., a localtransversal to a presymplectic leaf O at a point x0. Then the condition of thetheorem is satisfied, so N admits an induced Dirac structure, whose projection toTN vanishes at x0, thus in fact N admits a Poisson structure which vanishes atx0, and one can talk about the transverse Poisson structure to a presymplecticleaf in a Dirac manifold – provided that it is unique up to local isomorphisms.

Vorobjev’s (semi)local description of Poisson structures via coupling tensors(see Subsection A.7.1) can be naturally extended to the case of Dirac structures.More precisely, given a triple of geometric data (V , Γ, F) on a manifold E witha submersion p : E → S, where Γ is an Ehresmann connection, V is a vertical2-vector field, and F is a (maybe degenerate) C∞(E)-valued 2-form on S, denoteby L = L(V , Γ, F) the associated subbundle of TE ⊕ T ∗E, which is generated bysections of the types (α, iαV) and (X, iXF), where X ∈ V1

HE is a horizontal vectorfield and α is a vertical 1-form, i.e., α|THE = 0. Here iXF means the contractionof F, considered as a horizontal 2-form on E, with X . Then L is an almost Diracstructure on E.

Theorem A.8.5 ([115]). Given a set of geometric data (V , Γ, F) for a submersionp : E → S such as above, the corresponding almost Dirac structure L(V , Γ, F)is a Dirac structure if and only if the following four conditions (the same as in

294 Appendix

Theorem A.7.1) are satisfied:

[V ,V ] = 0, (A.82)

LHor(u)V = 0 ∀ u ∈ V1(S), (A.83)∂ΓF = 0, (A.84)

CurvΓ(u, v) = V(d(F(u, v))) ∀ u, v ∈ V1(S). (A.85)

Conversely, if E is a sufficiently small tubular neighborhood of a presymplectic leafS with a projection map p : E → S in a Dirac manifold (M, L), then there is aunique triple of geometric data (V , Γ, F) on (E, p) such that L = L(V , Γ, F) on E.Moreover, the vertical Poisson structure V vanishes on S, and the restriction of F

to S is the presymplectic form of S induced from L.

The proof of Theorem A.8.5 is absolutely similar to the Poisson case. (Theonly difference between the Dirac case and the Poisson case is that the horizontal2-form F may be degenerate in the Dirac case.) A direct consequence of TheoremA.8.5 is that, similarly to the Poisson case, the transverse Poisson structure to apresymplectic leaf in a Dirac manifold is well defined, i.e., up to local isomorphismsit does not depend on the choice of the slice. Another simple consequence is that thedimensions of the presymplectic leaves of a Dirac structure have the same parity.

Dirac and almost Dirac structures provide a convenient setting in which tostudy dynamical systems with constraints (holonomic and non-holonomic) andcontrol theory, and there is a theory of symmetry and reduction of (almost) Diracstructures, which generalizes the theory for symplectic and Poisson structures.See, e.g., [84, 105, 91, 32, 33, 34] and references therein.

For a generalization of the notion of Dirac structures to Lie algebroids, see[216]. In a different development, the complex version of Dirac structures (L is acomplex subbundle of (TM ⊕ T ∗M)⊗C which satisfies the same conditions as inthe real case) leads to generalized complex structures, see, e.g., [176, 156].

A.9 Deformation quantization

A product operation

: C∞(M)[[ν]] × C∞(M)[[ν]] −→ C∞(M)[[ν]], (A.86)

on the space C∞(M)[[ν]] of formal series in ν with coefficients in the space C∞(M)of smooth functions on a smooth manifold M , which turns C∞(M)[[ν]] into anassociative algebra, is called a differential star product on M , or star product forshort, if it satisfies the following conditions:

i) 1 F = F 1 = F for any F ∈ C∞(M)[[ν]],ii) is ν-linear, i.e.,

(νF ) G = F (νG) = ν(F G) ∀ F, G ∈ C∞(M)[[ν]] , (A.87)

A.9. Deformation quantization 295

iii) there are bi-differential operators Cn(., .) on M with smooth coefficients suchthat for any f, g ∈ C∞(M) we have

f g = fg +∞∑

n=1

νnCn(f, g). (A.88)

Given a star product on M with Taylor expansion (A.88), put

f, g =12(C1(f, g)− C1(g, f)) (A.89)

for f, g ∈ C∞(M). Then ., . is a Poisson bracket on M . Indeed, denote [f, g] =12ν (f g − g f). It follows from the associativity of that [., .] is a Lie bracket,i.e., it satisfies the Jacobi identity. Since f, g is the zeroth term of [f, g] in itsTaylor expansion with respect to ν, the zeroth term of the Taylor expansion of theJacobi identity for [., .] is nothing but the Jacobi identity for ., .. Similarly, byassociativity of we have [f1 f2] g = f1 [f2, g] + [f1, g] f2, and the zerothterm of this equality yields the Leibniz equality f1f2, g = f1f2, g + f1, gf2

for ., .. The anti-commutativity of ., . is obvious.Conversely, given a Poisson manifold (M, Π) one is interested in finding a

star product on M , which satisfies Equation (A.89), where the Poisson bracketcomes from Π. If such a star product exists, then one says that (M, Π) admits adeformation quantization.

Deformation quantization was proposed by Bayen, Flato, Frønsdal, Lich-nerowicz and Sternheimer [24] as a tool to study quantum physics, based on thephilosophy that a higher level (quantum) physical theory is a deformation of alower level (classical) one. The first example comes from the Weyl quantizationrule, which associates to each symbol1 a(x1, . . . , xn, ξ1, . . . , ξn) on R2n = T ∗Rn apseudo-differential operator W (a) on Rn defined by the oscillatory integral formula

(W (a)u)(x) =1

(2π)n

∫R2n

ei〈x−y,ξ〉/a

(x + y

2, ξ

)u(y)dydξ. (A.90)

For example, W (ξj)u =

i∂

∂xju, W (xj)u = xj .u, and one has the usual Heisenberg

commutation relations [Pj , Qk] =

i δjkId where Pj = W (ξj), Qk = W (xk). If aand b are two symbols on R2n then one may write

W (a)W (b) = W (c), (A.91)

1A symbol a on R2n = T ∗Rn is a function which satisfies the following asymptotic condition:there is a real number N such that for any compact subset K ⊂ Rn and any multi-indices

α, β ∈ Zn+ there is a constant CK,α,β such that supx∈K |∂α

x ∂βξ a(x, ξ)| ≤ CK,α,β(1 + |ξ|)N−|α|.

296 Appendix

where c has the following asymptotic expansion when → 0 (see, e.g., [179]):

c(x, ξ) ∼∑

α,β∈Zn+

(−1)|β|(/2i)|α+β|

α!β!(∂α

ξ ∂βxa(x, ξ))(∂α

x ∂βξ b(x, ξ))

= exp(

2i

∑j

(∂ξj .∂yj − ∂xj .∂ζj ))a(x, ξ)b(y, ζ)

∣∣∣y=x,ζ=ξ

. (A.92)

Considering ν = /i in the above formula as a formal variable and writingc = a b, one gets the following star product on the standard symplectic space(R2n,

∑j ∂ξj ∧ ∂xj), called the Moyal product [270]:

f g(x, ξ) = exp(ν

2

∑j

(∂ξj .∂yj − ∂xj .∂ζj ))f(x, ξ)g(y, ζ)

∣∣∣y=x,ζ=ξ

. (A.93)

The existence of a star product for an arbitrary symplectic manifold wasfirst established by De Wilde and Lecomte [99] in 1983 using algebraic methods.A more geometric proof, using Weyl quantization and symplectic connections, wasfound by Fedosov [128] and Omori–Maeda–Yoshioka [286]. The problem of classifi-cation of star products on symplectic manifolds was studied by Nest–Tsygan [277],Deligne [100], Bertelson–Cahen–Gutt [28], and other people. The result is that,given a symplectic manifold (M, ω), there is a one-to-one correspondence betweenthe set of equivalence classes of star products for (M, ω) and H2(M, R)[[ν]]. See[163] for a very nice exposition of this result.

The existence and classification up to equivalence of star products on anarbitrary Poisson manifold was obtained in 1997 by Kontsevich:

Theorem A.9.1 (Kontsevich [195]). For any smooth Poisson manifold (M, Π), theset of equivalence classes of differential star products on (M, Π) can be naturallyidentified with the set of equivalence classes of formal Poisson deformation of Π:

Πν = Π + νΠ1 + ν2Π2 + · · · ∈ V2(M)[[ν]], [Πν , Πν ] = 0. (A.94)

In particular, one can take Πν = Π, so a deformation quantization for (M, Π)always exists. If the Poisson structure is rigid (i.e., does not admit any nontrivialformal deformation), then the corresponding star product is unique up to equiva-lence, and vice versa.

Kontsevich obtained the above theorem as a corollary of another very deepresult, called the formality theorem. Consider the algebra Dpoly(M) of poly-differ-ential operators on a manifold M , equipped with the Hochschild differential andthe Gerstenhaber bracket coming from the associative algebra of linear operatorson C∞(M) (see [142]), and the algebra Tpoly(M) =

⊕k Vk(M) of multi-vector

fields on M with the trivial differential and the Schouten bracket. According toa smooth version of Hochschild–Kostant–Rosenberg’s theorem [177], Tpoly(M) isthe Hochschild cohomology of Dpoly(M). Kontsevich’s formality theorem says that

A.9. Deformation quantization 297

Dpoly(M) is formal, i.e., it is quasi-isomorphic, as a differential graded Lie algebra(or more precisely, as a L∞-algebra), to its cohomology Tpoly(M). We will nottry to explain what it means here, referring the reader to [195, 196, 103, 64]instead. As a particular case of an explicit construction of a quasi-isomorphismfrom Tpoly(Rn) to Dpoly(Rn), Kontsevich [195] gave the following explicit formulafor a star product on (Rn, Π):

f g =∞∑

k=0

νk∑

Γ∈Gk

wΓCΓ(Π)(f, g), (A.95)

where:• Gk is the set of oriented graphs Γ such that Γ has k + 2 vertices VΓ =1, . . . , k, L, R and 2k labelled edges (with no multiple edges and no edgesof the form (v, v) for v ∈ VΓ), EΓ = (e1

1, e2,1 , . . . , e1

k, e2k) where e1

j and e2j

start at vertex j.• CΓ(Π) is a bi-differential operator defined by the following formula, where

Πij = xi, xj are the coefficients of Π on Rn:

CΓ(Π)(f, g) =∑

I:EΓ→1,...,m

[ k∏j=1

( ∏e∈EΓ; e=(∗,j)

∂xI(e)

)ΠI(e1

j )I(e2j )

]×( ∏

e∈EΓ; e=(∗,L)

∂xI(e)

)f ×

( ∏e∈EΓ; e=(∗,R)

∂xI(e)

)g. (A.96)

• wΓ is a real number defined as follows. Denote by H = p ∈ C; Im(p) > 0the upper half-plane, and Hk = (p1, . . . , pk) | pj ∈ H, pi = pj ∀ i = j. Forp ∈ H, q ∈ H ∪ R, define φ(p, q) = 1

2i log((q − p)(q − p)(q − p)−1(q − p)−1).Assign a point pj ∈ H to each vertex j of Γ, 1 ≤ j ≤ k, point 0 ∈ R ⊂ C

to the vertex L, and point 1 ∈ R ⊂ C to the vertex R. Every edge e ∈ EΓ

defines an ordered pair (p, q) of points on H∪R, thus a function φe = φ(p, q)on H with values in R/2πZ. Now put

wΓ =1

k!(2π)2k

∫Hk

∧kj=1(dφe1

j∧ dφe2

j). (A.97)

Kontsevich [195] obtained the formality theorem (and hence the existenceof deformation quantization) for a general manifold from the formality theoremfor Rn and abstract arguments. A more explicit globalization of Kontsevich’s starproduct formula (A.95) from Rn to an arbitrary Poisson manifold (M, Π) wasobtained by Cattaneo–Felder–Tomassini [69, 67] by a method similar to Fedosov’smethod [128] for the symplectic case.

For people familiar with quantum field theory, Formula (A.95) looks like theexpansion of a Feynman path integral. In fact, it seems that a field theory thatgives rise to star products on Poisson manifolds exists and is known as the Poisson

298 Appendix

sigma model , first studied by Shaller–Strobl [310] and other physicists. A field ina Poisson sigma model is a pair (X, η), where X : Σ → M is a map from atwo-dimensional surface Σ to a given Poisson manifold (M, Π), and η is a sectionof T ∗Σ ⊗ X∗(T ∗M) with the boundary condition that η(u)v = 0 ∀ u ∈ ∂Σ, v ∈Tu(∂Σ). The action functional is

S(X, η) =∫

Σ

〈η, dX〉+ 12〈η, (Π X)η〉. (A.98)

Cattaneo and Felder [66] found the following path integral formula for a starproduct on an arbitrary given Poisson manifold (M, Π):

f g =∫

X(∞)=x

f(X(1))g(X(0))eiS(X,η)/dXdη, (A.99)

where Σ is now a two-dimensional disk, 0, 1,∞ are three distinct points on theboundary of Σ, S is the above functional, and f, g and functions on (M, Π). Theinterested reader may consult, e.g., [66, 67, 68, 69] for details and relations to for-mal symplectic groupoids. Unfortunately, the authors of this book are not familiarwith quantum field theory, and the above formulas look mysterious to them.

Deformation quantization has many other aspects and is related to manyother subjects, e.g., quantum groups, representation theory, index theory, otherquantization theories, etc. See, e.g., the survey articles [322, 162] and referencestherein.

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Index

A-connection, 244A-path, 243action map, 208action-angle variables, 275affine connection, 73algebroid cohomology, 250almost Dirac structure, 292almost Levi factor, 85almost linearized, 85anchor, 204anchor map, 13, 41, 235annulator distribution, 166anti-Poisson map, 12associated foliation, 168associated singular foliation, 163atlas, 216

basic function, 240bi-degree, 274bi-Hamiltonian system, 34bimodule, 212Birkhoff normal form, 279Birkhoff normalization, 279bisection, 207Bochner’s theorem, 217book, 176bundle of Lie algebras, 237

cabbage pile, 176canonical coordinates, 3, 14, 266Cartan decomposition, 112Cartan involution, 112Cartan motion algebra, 115Cartan’s formula, 2Cartan–Chevalley–Eilenberg

construction, 50Casimir element, 89Casimir functions, 40characteristic distribution, 162, 239

characteristic foliation, 239characteristic space, 13chart, 216Chevalley–Eilenberg complex, 49classical r-matrix, 133classical Yang–Baxter equation, 33, 133classical Yang–Mills–Higgs setup, 26coboundary Poisson-Lie structure, 132codimension, 267coisotropic, 12, 13commuting singular foliations, 164compact Levi factor, 81compatible Poisson structures, 33complete symplectic realization, 262complete transversal, 211completely integrable, 168configuration space, 4conservation of energy, 4contravariant connection, 245contravariant tensors, 9convolution product, 210cotangent algebroid, 237coupling tensor, 288Courant bracket, 292covariant derivation, 73covariant derivative, 244curl, 70curl operator, 70, 73, 178curl vector field, 72curvature, 245, 261, 286CYBE, 33, 133

Darboux coordinates, 3, 266Darboux theorem, 3de Rham complex, 42, 249decomposable, 162, 165deformation cohomology, 251deformation quantization, 295

318 Index

degree of resonance, 280density, 71derivation, 3diagonal quadratic Poisson structure,

148differential operator, 39differential star product, 294Dirac manifold, 292Dirac structure, 292Dirac’s formula, 22dressing action, 143dressing transformation, 143dual pair, 37, 274dual Poisson–Lie algebra, 137dual Poisson–Lie group, 137

Ehresmann connection, 286eigenvalue, 276eigenvalues, 148, 150elliptic singular point, 184equivariant momentum map, 25, 144Euler equation, 24Euler top, 24exact Poisson structure, 41exact Poisson-Lie structure, 132

fiber-wise linear, 241fiber-wise linear p-vector fields, 251fiber-wise linear function, 240fiber-wise linear Poisson cohomology,

251fiber-wise linear Poisson structure, 240fibered product, 210filtration, 54finite codimension, 63finitely determined, 126finitely generated distribution, 19first integral, 4fixed point, 205flat connection, 246flow, 263formal completion, 78formal Levi factor, 79formality theorem, 296formally completely integrable, 168free action, 208Frobenius integrability condition, 166Frobenius Lie algebra, 120Frobenius theorem, 18, 166

Frobenius with singularity, 169fundamental groupoid, 206fundamental identity, 159

Γ-structure, 205gauge groupoid, 207gCYBE, 133generalized classical Yang–Baxter

equation, 133generalized isomorphism, 213generalized morphism, 213geometric data, 287germ, 42Godbillon–Vey algorithm, 171graded anti-commutativity, 28, 32graded Jacobi identity, 28, 32graded Leibniz rule, 28groupoid, 203groupoid action, 208groupoid morphism, 204

Haar system, 209Hamiltonian action, 25, 232Hamiltonian equation, 3Hamiltonian system, 3Hamiltonian vector field, 2, 3, 159Hochschild–Serre spectral sequence, 57holonomy, 271holonomy groupoid, 206homogeneous foliation, 177homogeneous Lie algebroid, 252homogeneous part, 45homotopy operator, 89horizontal lifting, 245horizontal subbundle, 286

identity section, 204induced symplectic form, 14infinitesimal automorphism, 10, 241infinitesimally strongly rigid Lie

algebra, 107inner product, 30, 69integrability condition, 166integrable differential form, 166integrable distribution, 17integrable in generalized Liouville

sense, 273integrable Poisson manifold, 227integral submanifold, 17

Index 319

interpolation inequality, 102invariant density, 72inversion map, 204involutive distribution, 18isochore vector field, 148isolated singularity, 63isotropic, 12isotropy algebra, 125, 239isotropy group, 205Iwasawa decomposition, 112Iwasawa Poisson–Lie structure, 124,

138

Jacobi identity, 31Jacobian determinant, 7

k-determinant, 267Kirillov–Kostant–Souriau form, 20Kupka’s phenomenon, 179

Lagrangian, 12leaves, 16left action, 23left dressing action, 143left-invariant vector field, 237Leray spectral sequence, 56Levi decomposition, 77, 81, 255Levi factor, 77Levi normal form, 81, 255Levi–Malcev theorem, 77Lichnerowicz complex, 39Lie algebra action, 23Lie algebroid, 235Lie algebroid action, 244Lie algebroid morphism, 242Lie bialgebra, 137Lie groupoid, 206Lie groupoid morphism, 206Lie pseudoalgebra, 239Lie’s third theorem, 257Lie–Poisson structure, 20Lie-linear foliation, 177linear A-module, 244linear connection, 73linear holonomy, 246linear Nambu structure, 171linear part, 45, 182, 219, 253linear Poisson structure, 8linear representation, 208, 244

Liouville 1-form, 4Liouville form, 72Liouville integrable, 273Liouville torus, 273, 274Liouville–Mineur–Arnold theorem, 275local foliation property, 16

Manin triple, 138marked symplectic realization, 35Marsden–Ratiu reduction, 26Marsden–Weinstein–Meyer reduction,

26Maurer–Cartan equation, 141, 144Maurer–Cartan form, 144maximal invariant distribution, 167maximal invariant foliation, 168Mayer–Vietoris sequence, 44mCYBE, 134Mineur–Arnold formula, 276modified classical Yang–Baxter

equation, 134modular class, 72modular vector field, 72moduli space, 205moment map, 25, 208, 244momentum map, 25, 124, 144, 208,

244, 273Morita equivalent, 211Moser’s path method, 263Moyal product, 296multi-derivation, 6multi-vector, 5multiplicative Poisson structure, 146multiplicative tensor field, 129multiplicity, 63

n-ary Lie algebra, 177Nambu bracket, 159Nambu structure, 159Nambu tensor, 160Nambu-linear foliation, 177non-equivariant momentum map, 25noncommutative integrability, 273nondegenerate, 183nondegenerate Lie algebra, 105nondegenerate module, 255nonresonant, 280nonresonant Poisson structure, 150

320 Index

nonresonant quadratic Poissonstructure, 148

normal form, 45normalization, 45normed vanishing of cohomology, 90

ω-condition, 278obstructions to formal deformation, 40orbifold, 216orbifold groupoid, 216orbispace, 222orbit, 205orbit space, 205orbit-like foliation, 164

pair groupoid, 204pairing, 5parallel transport, 245pencil of Poisson structures, 33phase space, 4Poincare–Dulac normal form, 277Poincare–Dulac normalization, 277Poisson action, 124, 139, 141Poisson algebra, 24Poisson bracket, 1Poisson cohomology, 39Poisson groupoid, 234Poisson harmonic form, 75Poisson homogeneous space, 140Poisson homology, 74Poisson isomorphism, 10Poisson manifold, 1Poisson map, 9Poisson morphism, 9Poisson sigma model, 297, 298Poisson structure, 1, 6Poisson tensor, 6Poisson theorem, 4Poisson vector field, 10Poisson–Lie algebra, 136Poisson–Lie group, 132Poisson–Lie structure, 132Poisson–Lie tensor, 122pro-finite Lie algebra, 78pro-solvable radical, 78product map, 203product Poisson structure, 10proper action, 24, 213, 214proper Lie groupoid, 214

proper map, 24, 213proper symplectic groupoid, 227

quasi-homogeneous, 46quasi-homogeneous part, 46quasi-homogenization, 46quasi-radial vector field, 46quasi-symplectic groupoid, 233quasi-triangular r-matrix, 134

radial vector field, 46radical, 57rank, 13real rank, 112reduced Hamiltonian system, 24reduced Poisson structure, 24, 26Reeb stability, 272regular distribution, 17regular foliation, 16regular Poisson structure, 14regular sequence, 269regularized volume, 69resonance relation, 277, 279Riemannian foliation, 222right action, 23right dressing action, 143rigid Lie algebra, 54

s-fiber, 207saturated singular foliation, 163saturation, 163Schouten bracket, 30, 32second monodromy, 260simple singularity, 61singular distribution, 16singular foliation, 16Sklyanin bracket, 133, 146slice, 208slice groupoid, 208slice theorem, 218, 222smoothing operator, 101source map, 203source-locally trivial, 215spectral sequence, 55splitting theorem, 13, 246standard symplectic groupoid, 224star product, 294strongly rigid Lie algebra, 107symbol, 295

Index 321

symmetric space, 112symplectic foliation, 19symplectic form, 2symplectic groupoid, 223symplectic manifold, 2symplectic morphism, 10symplectic orthogonal, 12symplectic realization, 34symplectically complete foliation, 37

t-fiber, 207tame Frechet space, 101tangent algebroid, 236tangent distribution, 17target map, 203toric degree, 280totally intransitive groupoid, 205transformation algebroid, 236transformation groupoid, 204transitive groupoid, 205

transitive Lie algebroid, 239transversal holonomy groupoid, 211transverse groupoid structure, 208transverse Lie algebroid, 249transverse Poisson structure, 21triangular r-matrix, 133Type I, 172, 176Type II, 172, 176Type IIr, 180

unimodular Lie algebra, 73unimodular Poisson manifold, 72

volume form, 70Vorobjev linear model, 290

Weinstein groupoid, 259Weyl quantization, 295Whitehead’s lemmas, 50

Yang–Mills–Higgs phase space, 26

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Poisson Structures and Their Normal Forms (Progress in Mathematics)

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