AFFINE LIE ALGEBRASAND AFFINE ROOT SYSTEMS
A Killing-Cartan type classification of affine Lie algebras, reduced irreducible affine rootsystems and affine Cartan matrices
Jan S. Nauta
MSc Thesis
under supervision of
Dr. J.V. Stokman
UNIVERSITEIT VAN AMSTERDAM
AFFINE LIE ALGEBRAS AND AFFINE ROOT SYSTEMSA Killing-Cartan type classification of affine Lie algebras, reduced irreducible affine root systems
and affine Cartan matrices
MSc Thesis
Author: Jan S. NautaSupervisor: Dr. Jasper Stokman
Second reader: Dr. Hessel PosthumaMember of examination board: Prof. Dr. Gerard van der Geer
Date: April 20, 2012Cover Illustration: Figure 3.6 on page 79.
Abstract
In this thesis we will construct a commutative triangle of canonical bijec-tions between the isomorphism classes of Lie algebras isomorphic to affineLie algebras, the similarity classes of reduced irreducible affine root systemsand the affine Cartan matrices up to simultaneous permutations of rowsand columns. Together with a classification of affine Cartan matrices upto simultaneous permutations of rows and columns this classifies all threetypes of mathematical objects. The construction of the triangle will be basedon the Killing-Cartan classification of semisimple Lie algebras. After settingup the axiomatic theory of affine root systems from scratch, we will explicitlyshow that each irreducible affine root system gives rise to an affine Cartanmatrix and that the set of real roots of an affine Lie algebra can be naturallyviewed as the associated affine root system of the affine Lie algebra.
Korteweg-de Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdam
He exhibited the characteristic equation of an arbitrary element of the Weylgroup when Weyl was 3 years old and listed the orders of the Coxeter trans-formation 19 years before Coxeter was born!
A.J. Coleman about W.K.J. Killing, 1989
Contents
Introduction vii
1 The classification of semisimple Lie algebras 11.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Root systems of semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 An axiomatic approach of root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 The classification of reduced irreducible root systems . . . . . . . . . . . . . . . . . . 61.2.4 Nonreduced irreducible root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Serre’s Theorem, a commutative triangle and the classification of simple Lie algebras 9
2 Kac-Moody algebras, Affine Lie algebras, and the set of real roots 112.1 Kac-Moody algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Realizations of matrices and their associated Lie algebras . . . . . . . . . . . . . . . 122.1.2 The invariant nondegenerate symmetric bilinear form . . . . . . . . . . . . . . . . . . 152.1.3 Kac-Moody algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.4 The Weyl group and the set of real roots of a Kac-Moody algebra . . . . . . . . . . 19
2.2 A classification of generalized Cartan matrices and some Kac-Moody algebras . . . . . 202.2.1 Three types of generalized Cartan matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Kac-Moody algebras associated to finite Cartan matrices . . . . . . . . . . . . . . . . 212.2.3 A classification of affine Cartan matrices and affine Lie algebras . . . . . . . . . . . 22
2.3 Affine Lie algebras, the set of real roots and affine root systems . . . . . . . . . . . . . . . . . 252.3.1 The normalized invariant form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 The real roots of an affine Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.3 The Weyl group of an affine Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.4 The set of real roots as an affine root system . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.5 An explicit construction of untwisted affine Lie algebras . . . . . . . . . . . . . . . . 33
3 Affine Root Systems 353.1 Affine linear functions and orthogonal reflections in affine Euclidean space . . . . . . . 35
3.1.1 Affine space and affine linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2 Affine Euclidean space and affine linear functions . . . . . . . . . . . . . . . . . . . . . 383.1.3 Orthogonal reflections in affine Euclidean space . . . . . . . . . . . . . . . . . . . . . . 403.1.4 Orthogonal reflections and affine linear automorphisms . . . . . . . . . . . . . . . . 42
3.2 Affine root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.1 Affine root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.2 Similar affine root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
v
3.2.3 Affine root subsystems and irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.4 The direct sum of affine root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 The geometry of affine root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.1 Affine hyperplanes and alcoves in affine Euclidean space . . . . . . . . . . . . . . . . 653.3.2 Alcoves of irreducible affine root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4 Bases of irreducible affine root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.5 A classification of reduced irreducible affine root systems . . . . . . . . . . . . . . . . . . . . . 82
3.5.1 Affine Cartan matrices of irreducible affine root systems . . . . . . . . . . . . . . . . 823.5.2 Explicit constructions of reduced irreducible affine root systems . . . . . . . . . . 853.5.3 The naming of affine Dynkin diagrams explained . . . . . . . . . . . . . . . . . . . . . 893.5.4 A new commutative triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Summary (in Dutch) 93
Bibliography 97
vi
Introduction
The classification of semisimple finite-dimensional Lie algebras over the complex numbers wassettled by Killing and Cartan at the end of the nineteenth century. This celebrated classification isbased on classifying objects that are directly related to semisimple Lie algebras, namely reducedroot systems and Cartan matrices. It is considered a milestone in the history of mathematics. Theclassification can be summarized by the following commutative triangle of bijections∆, A and g
L ∆ → R
C
A
←
g
←
(0.0.1)
together with a classification of indecomposable Cartan matrices up to simultaneous permuta-tions of rows and columns. In (0.0.1) we have denoted byL the isomorphism classes of simple Liealgebras, by R the similarity classes of reduced irreducible root systems and by C the indecom-posable Cartan matrices up to simultaneous permutations of rows and columns. The maps ∆, Aand g will be made explicit in Chapter 1.
In 1966, Serre showed that each semisimple Lie algebra is defined up to isomorphism by finitelymany generators and relations that only depend on the entries of the corresponding Cartan matrix(this actually leads to the map g :C →L ). This inspired Kac and Moody in 1967 to independentlystart the study of Lie algebras on generators that satisfy a natural generalization of the relationsof Serre with slightly weaker conditions on the corresponding Cartan matrix. The most importantof such Lie algebras are now called Kac-Moody algebras, and apart from semisimple Lie algebrasKac-Moody algebras are infinite-dimensional. One particularly interesting Kac-Moody algebra isthe affine Lie algebra which is a Kac-Moody algebra associated to an affine Cartan matrix. Anaffine Cartan matrix has all the properties of the Cartan matrix from the case of simple Lie alge-bras, except that is has determinant zero. This makes an affine Lie algebra an infinite-dimensionalgeneralization of a simple Lie algebra.
In this thesis we want to establish a commutative triangle similar to (0.0.1) for affine Lie alge-bras. In this new triangle Cartan matrices will be replaced by affine Cartan matrices and reducedirreducible root systems by reduced irreducible affine root systems. Furthermore, we will give aclassification of affine Cartan matrices up to simultaneous permutations of rows and columnsanalogous to the classification of indecomposable Cartan matrices up to simultaneous permu-tations of rows and columns. Together with the new triangle this also classifies similarity classes ofreduced irreducible affine root systems and isomorphism classes of affine Lie algebras.
In 1983 Peterson and Kac solved the ’isomorphism problem’ for symmetrizable Kac-Moody al-gebras. They showed in particular that affine Lie algebras up to isomorphism correspond bijec-tively to affine Cartan matrices up to simultaneous permutations of rows and columns. This takes
vii
care of the map g for affine Cartan matrices and affine Lie algebras. In the seventies Macdonaldhas independently studied affine root systems. We will study a slightly adapted version of the the-ory of affine root systems as was known to Macdonald in full detail. In this manuscript we willexplicitly construct the map A for reduced irreducible affine root systems and affine Cartan matri-ces. Finally, we establish a reduced irreducible affine root system R related to an affine Lie algebraL in a canonical way. We will show that if we take ∆ := (g ◦A)−1, then the triangle commutes suchthat ∆(L) = R . Here L is the isomorphism class of L and R is the similarity class of R . This showsthat every affine Lie algebra up to isomorphism is uniquely determined by its corresponding re-duced irreducible affine root system up to similarity. Finally I would like to note that since affineLie algebras are explicitly constructed as Lie algebras on generators and relations, we will showhow to generalize the structures that we define on affine Lie algebras to all Lie algebras isomorphicto affine Lie algebras. However for simplicity we will most of the time only work with affine Liealgebras.
Outline of this thesis
Chapter 1 consists of a brief summary of the classification of all semisimple finite-dimensional Liealgebras over the complex numbers up to isomorphism through the use of root systems and Cartanmatrices as originated in the works of Killing and Cartan. We will establish the canonical bijectivecorrespondence between simple Lie algebras, reduced irreducible root systems and indecompos-able Cartan matrices up to appropriate isomorphism equivalences in the form of a commutativetriangle (0.0.1). Classifying the indecomposable Cartan matrices then finishes the classification ofsemisimple Lie algebras. Along this Chapter we will establish some definitions and results aboutLie algebras and root systems that will be useful in the remaining Chapters.
In Chapter 2 we will introduce Kac-Moody algebras, affine Lie algebras and the necessary toolsto study root systems of Kac-Moody algebras such as the Weyl group, the invariant nondegeneratesymmetric bilinear form and real and imaginary roots. We show that Kac-Moody algebras are ageneralization of semisimple Lie algebras, and show that all finite-dimensional Kac-Moody alge-bras are semisimple Lie algebras. We give the classification of affine Cartan matrices, and showthat affine Lie algebras up to isomorphism correspond bijectively to affine Cartan matrices up tosimultaneous permutations of rows and columns. Finally, we will study the invariant bilinear form,the Weyl group and the real roots of affine Lie algebras. We will observe that the real roots of anaffine Lie algebra satisfy axioms that generalize the axioms of a reduced irreducible finite root sys-tem. This will justify the study of reduced irreducible affine root systems in Chapter 3.
Chapter 3 consists of a detailed axiomatic study of affine root systems with the goal of classify-ing them. First we introduce useful notions as reducedness, irreducibility and similarity that areanalogous to the case of finite root systems. Then we study the geometry of affine root systems andthe affine Weyl group to obtain a special set of generators of an irreducible affine root system. Us-ing these generators we will relate affine Cartan matrices up to simultaneous permutations of rowsand columns bijectively to the similarity classes of reduced irreducible affine root systems. Theclassification of affine Cartan matrices then also classifies the reduced irreducible root systems upto similarity. Along the way we will explicitly realize a complete set of representatives for the sim-ilarity classes of reduced irreducible affine root systems. We end the Chapter by putting togetherthe commutative triangle that gives the canonical bijections between isomorphism classes of Liealgebras that are isomorphic to affine Lie algebras, similarity classes of reduced irreducible affineroot systems and affine Cartan matrices up to simultaneous permutations of rows and columns.
viii
Acknowledgments
First and foremost I would like to thank my supervisor Jasper Stokman for introducing me to thisbeautiful subject, for his inspiring explanations and for his quick and adequate support on ques-tions from my side. Further, I want to thank him for giving me the opportunity to live in Lima,Peru, for 6 months, while I was working on my thesis and communicating with him through Skypeand email. Next, I would like to thank Hessel Posthuma for being the second reader of my thesis.Further I would like to thank the reason I was in Lima, my ever beautiful and lovely Natalia Parodi,for always supporting the work on this thesis, and for listening to my problems so that I could toorganize my mind. I would like to thank Chris van Dorp for the inspirational conversations thatwe had together. Also, I would like to thank Lodewijk Nauta, Ido Niesen, Dirk Broersen and Apo forthe not too sporadic social moments in the masters room. Finally, I would like to thank my parentsRudi Nauta and Elselien van Gils for their support during all the stages of my Master’s program.
Notations and conventions
N,Z,Q,R,C denote the natural numbers, integers, rationals, reals and complex numbers respec-tively.R6=0 denotes the real numbers apart from zero.∐
denotes a disjoint union of sets.A n B denotes the semidirect product of (sub)groups A and B (of a group G ) with B normal.
δi j :=
(
1 i = j ,
0 i 6= j ,is the Kronecker delta.
K ◦ is the interior and K is the closure of the subset K in a topological space.V ∗ denotes the dual of a vector space V .
ix
Chapter 1
The classification of semisimple Liealgebras
In this Chapter we want to give a brief exposition of the classification of all semisimple Lie alge-bras over C up to isomorphism through the use of root systems and Cartan matrices as originatedin the works of Killing and Cartan (see [4]). After setting up the necessary theory we will establisha canonical bijective correspondence between simple Lie algebras, irreducible root systems andindecomposable Cartan matrices up to appropriate isomorphism equivalences in the form of acommutative triangle. Classifying the indecomposable Cartan matrices then finishes the classi-fication of semisimple Lie algebras. Along this Chapter we will go through some definitions andresults on Lie algebras and root systems that will be useful in the remaining of this manuscript.All results quoted in this Chapter can be found in [6] except for the results on nonreduced rootsystems which come from [13] and the axiomatic definition of a Cartan matrix which comes from[8].
1.1 Lie algebras
A Lie algebra arises naturally as a vector space of linear transformations with the commutatorof two linear transformations as a product on it. This product is bilinear but in general neithercommutative nor associative. In this Section we introduce the abstract notion of a Lie algebra, andwe go through some basic algebraic tools that will be useful throughout this manuscript.
Definition 1.1.1. A vector space g overC together with a bilinear operation [., .] : g×g→ g, (x , y ) 7→[x , y ] is called a Lie algebra if the operation [., .] satisfies the following two conditions:
(1) [x ,x ] = 0 for all x ∈ g;(2) [x , [y , z ]]+ [y , [z ,x ]]+ [z , [x , y ]] = 0 for all x , y , z ∈ g (Jacobi identity).
The operation [., .] is called the (Lie) bracket of g.
Note that condition (1) of Definition 1.1.1 together with the bilinearity of the bracket impliesthat [x , y ] = −[y ,x ] for all x , y ∈ g, hence the bracket of a Lie algebra is anticommutative. A Liealgebra g is called abelian if [x , y ] = 0 for all x , y ∈ g. Using the anticommutativity of [., .] this isequivalent to [x , y ] = [y ,x ] for all x , y ∈ g.
Example 1.1.2. Let V be a finite-dimensional complex vector space, and let End(V ) denote the setof linear endomorphisms of V . Then End(V ) is a complex vector space of dimension dim(V )2 and
1
a ring with composition of maps as multiplication. Furthermore, End(V ) becomes a Lie algebrawith the bracket [x , y ] = x y −y x and is called the general linear algebra which is denoted by gl(V ).
A linear mapφ : g1→ g2 between Lie algebras g1 and g2 is called a (Lie algebra) homomorphismif φ is compatible with the Lie brackets of g1 and g2 respectively, i.e. [φ(x ),φ(y )] =φ([x , y ]) for allx , y ∈ g1. In particular, if g1 = g2 then φ is called a (Lie algebra) endomorphism. Two Lie algebrasg1 and g2 are said to be isomorphic if there exist a linear isomorphism φ : g1 → g2 such that φ iscompatible with the Lie brackets of g1 and g2 respectively. Then the mapφ is called a (Lie algebra)isomorphism. Write ad x : g→ g for the map y 7→ [x , y ] for some fixed x ∈ g, then it follows from theaxioms of a Lie algebra that ad x is actually a Lie algebra endomorphism of g.
A (Lie) subalgebra is a vector subspace h of a Lie algebra g such that [x , y ] ∈ h for all x , y ∈ h.A subspace h ⊂ g such that [x , y ] ∈ h for all x ∈ h and y ∈ g is called an ideal of g. Notice that anideal is also a subalgebra, but not the other way around. A nontrivial example of an ideal of a Liealgebra g is the center Z (g) := {x ∈ g : [x , y ] = 0 for all y ∈ g}. Let τ1, . . . ,τr be ideals of a Lie algebrag, then g is said to be the direct sum of τ1, . . . ,τr , and we write g=τ1⊕· · ·⊕τr , if g is the direct sumof τ1, . . . ,τr as vector spaces. This condition forces [x i ,x j ] = 0 for all x i ∈ τi and x j ∈ τj for i 6= j ,hence the bracket of g acts componentwise on the direct sum.
Analogous to other algebraic theories, we have the notion of the quotient Lie algebra g/τ of a Liealgebra g and an ideal τ ⊂ g. Here g/τ coincides with the quotient space together with a bilinearoperation [x+τ, y +τ]τ := [x , y ]+τ for all x+τ, y +τ∈ g/τ. It is not hard to see that this operationis well defined, and that it turns g/τ into a Lie algebra with bracket [., .]τ in a natural way.
A Lie algebra g is said to be free on a subset X ⊂ g if for any given Lie algebra g′ together with aninjectionφ : X ,→ g′ there exists a unique Lie algebra homomorphismψ : g→ g′ such thatψ|X =φ.If g is a free Lie algebra on X , then it follows straightforwardly that g is the unique free Lie algebraon X up to isomorphism. For the existence of a free Lie algebra on a set X consider the complexvector space V having X as basis. Let T 0V = C, put T m V = V ⊗ · · · ⊗V (m copies of V ) for m ∈ Nand define the vector space T(V ) :=
⊕∞i=0 T i V . Further, introduce an associative product on the
homogeneous generators of T(V ) by
(v1⊗ · · ·⊗vk )(w1⊗ · · ·⊗wm ) = v1⊗ · · ·⊗vk ⊗w1⊗ · · ·⊗wm ∈ T k+m V
and extend this bilinearly to an associative product on all of T(V ). Then T(V ) is an associativegraded algebra with 1 which is generated by 1 along with X , and it is called the tensor algebra on V .Consider T(V ) as Lie algebra with the bracket [x , y ] := x ⊗ y − y ⊗x and let g be the Lie subalgebraof T(V ) generated by X , then it turns out that g is a free Lie algebra on X .
Let gbe a free Lie algebra on X and letτbe an ideal of g generated as subalgebra of gby elementsk i for i in an index set I . Consider the canonical Lie algebra homomorphismφ : g→ g/τ, then thequotient Lie algebra g/τ is said to be the Lie algebra with generators φ(x ) for x ∈ X and relationsφ(k i ) = 0 for i ∈ I .
1.2 Root systems
In the following we will define semisimple Lie algebras, Cartan subalgebras and the Killing form.We show that a semisimple Lie algebra together with a choice of a Cartan subalgebra gives rise to afinite set of vectors in a Euclidean space that is invariant under a reflection group that they generatethemselves called the Weyl group. This set of vectors is known as a root system. We will treat thetheory of root systems independent of semisimple Lie algebras, and classify reduced irreducible
2
root systems according to their Cartan matrix. Finally, we will discuss the class of nonreducedirreducible root systems.
1.2.1 Root systems of semisimple Lie algebras
A Lie algebra g over a finite-dimensional complex vector space is called simple if the only idealsof g are {0} and itself, and if g is not abelian. A Lie algebra that can be written as a direct sum offinitely many simple Lie algebras is called semisimple1.
Example 1.2.1. Consider the general linear algebra gl(V ) for a finite-dimensional complex vectorspace V . Any subalgebra of gl(V ) is called a linear Lie algebra. There are four important families oflinear Lie algebras A l (l ≥ 1), Bl (l ≥ 1), C l (l ≥ 1) and Dl (l ≥ 2) which turn out to be simple andtogether they are called the classical Lie algebras:
A l : Let dim(V ) = l +1 and let sl(l +1,C) denote the subalgebra of gl(V ) of linear transformationswith trace 0. This Lie algebra is called the special linear algebra, and is of dimension (l +1)2−1.
Bl : Let dim(V ) = 2l + 1 and choose a basis {v1, . . . , v2l+1} of V . Let f be the nondegenerate
symmetric bilinear form on V defined by the matrix s =
1 0 00 0 I l
0 I l 0
. Define o(2l +1,C) to be the
subalgebra of gl(V ) of linear transformations x ∈ gl(V ) such that f (x (v ), w ) = − f (v,x (w )) for allv, w ∈ V . The Lie algebra o(2l + 1,C) is called the orthogonal algebra (with V of odd dimension!)and is of dimension 2l 2+ l .
C l : Let dim(V ) = 2l and choose a basis {v1, . . . , v2l } of V . Let f be the nondegenerate symmetric
bilinear form on V defined by the matrix s =
�
0 I l
−I l 0
�
. Define sp(2l ,C) to be the subalgebra of
gl(V ) of linear transformations x ∈ gl(V ) such that f (x (v ), w ) =− f (v,x (w )) for all v, w ∈V . The Liealgebra sp(2l ,C) is called the the symplectic algebra and is of dimension 2l 2+ l .
Dl : Let dim(V ) = 2l and choose a basis {v1, . . . , v2l } of V . Let f be the nondegenerate symmetric
bilinear form on V defined by the matrix s =
�
0 I l
I l 0
�
. Define o(2l ,C) to be the subalgebra of
gl(V ) of linear transformations x ∈ gl(V ) such that f (x (v ), w ) =− f (v,x (w )) for all v, w ∈V . The Liealgebra o(2l ,C) is called the orthogonal algebra (with V of even dimension!) and is of dimension2l 2− l .
It turns out that one can define an invariant nondegenerate symmetric bilinear C-valued form(., .)g on a semisimple Lie algebra g called the Killing form by (x , y )g = tr(ad x ad y ) for all x , y ∈ g.Here invariant means that ([x , y ], z )g = (x , [y , z ])g for all x , y , z ∈ g, nondegenerate means that thereexists no nonzero x ∈ g such that (x , y )g = 0 for all y ∈ g and ’tr’ denotes the trace of a linearendomorphism of g.
Consider a semisimple Lie algebra g. An element x ∈ g is called ad-semisimple if ad x is a diag-onalizable endomorphism of g with respect to a suitable basis of g as vector space. Fix a maximalsubalgebra h ⊂ g consisting only of ad-semisimple elements. Such a subalgebra h is said to be aCartan subalgebra of g. One can show that h 6= {0} and that h is an abelian subalgebra of g. Hencethe Jacobi identity yields that H := {ad h : h ∈ h} is a commutating set of endomorphisms of g.By a standard result of linear algebra one observes that the endomorphisms of H are simultane-ously diagonalizable. This leads to the root space decomposition of g with respect to h, i.e. as vector
1This is actually not the definition of a semisimple Lie algebra from [6], but an equivalent characterization which canalso be found in [6].
3
spaces
g= h⊕�
⊕
α∈∆(g,h)
gα
�
,
wheregα = {x ∈ g : [h,x ] =α(h)x for all h ∈ h}
is called the root space associated to α∈ h∗ and
∆(g,h) := {α∈ h∗ \ {0} : gα 6= {0}}
is called the root system with respect to h. The α ∈∆ are called roots. When there is no ambiguityabout g and h we will write∆ for∆(g,h).
The Killing form (., .)g turns out to be is nondegenerate when restricted to h, hence we can definethe linear isomorphism ν : h→ h∗ depending on (., .)g by
ν (h)(h ′) = (h, h ′)g
for h, h ′ ∈ h. This leads to a nondegenerate symmetric bilinear form (., .) on h∗ defined by
(α,β ) := (ν−1(α),ν−1(β ))g
for α,β ∈ h∗.
Proposition 1.2.2. The root system∆ of of semisimple Lie algebra g relative to a Cartan subalgebrah satisfies the following four conditions:
(1) ∆ is finite, does not contain 0 and spans an R-vector space V ⊂ h∗ such that dimR(V ) =dimCh∗ and the form (., .) on h∗ restricted to V defines an inner product on V ;
(2) β −2 (α,β )(α,α)α∈∆ for all α,β ∈∆;
(3) 2 (α,β )(α,α) ∈Z for all α,β ∈∆;
(4) If α∈∆, then Rα∩∆= {α,−α}.
1.2.2 An axiomatic approach of root systems
To obtain a classification of semisimple Lie algebras it turns out to be useful to study root systemsindependently of their semisimple Lie algebra, and to classify them. We will briefly discuss the ax-iomatic theory of root systems in this Subsection, and do the classification in the next Subsection.
Call an R-vector space V of dimension l <∞ endowed with a positive definite symmetric bi-linear form, or inner product, (., .) a Euclidean space.
Definition 1.2.3. A subset∆ of a Euclidean space V that is endowed with the inner product (., .) iscalled a root system if the following three conditions are satisfied:
(1)∆ is finite, does not contain 0 and spans V ;(2) β −2 (α,β )
(α,α)α∈∆ for all α,β ∈∆;
(3) 2 (α,β )(α,α) ∈Z for all α,β ∈∆.
If the following condition is also satisfied we say that ∆ is a reduced root system, otherwise ∆ iscalled nonreduced:
(4) If α∈∆, then Rα∩∆= {α,−α}.The dimension of V is called the rank of∆.
4
From the Proposition 1.2.2 we observe that a root system of a semisimple Lie algebra is a re-duced root system in the sense of Definition 1.2.3. Furthermore, we will see in Section 1.3 thateach reduced root system ∆ gives rise to a semisimple Lie algebra that has ∆ as its correspondingroot system. In the following we will assume root systems to satisfy conditions (1)-(3) of Definition1.2.3, and state explicitly when a result only holds for a reduced root system.
For a nonzero vector α ∈ V the orthogonal reflection in the hyperplane Hα = {v ∈ V : (v,α) = 0}orthogonal to α is the map wα : V →V defined by
wα(β ) =β −2(α,β )(α,α)
α.
Clearly, wα is a linear isometry that fixes Hα and sends α to −α. Let W0(∆) denote the group oflinear transformations of V generated by the orthogonal reflections wα for α ∈∆. We will call thisgroup the Weyl group of∆. Condition (2) of Definition 1.2.3 is then equivalent to saying that W0(∆)stabilizes∆, and together with condition (3) we notice that theZ-span of∆ is a W0(∆)-stable latticein V .
We call two root systems ∆ ⊂ V and ∆′ ⊂ V ′ similar, and write ∆ ' ∆′, if there exists a linearisomorphismψ : V
∼−→V ′ such that
2(ψ(α),ψ(β ))V ′(ψ(α),ψ(α))V ′
= 2(α,β )V(α,α)V
for all α,β ∈ ∆ which restricts to a bijection of ∆ onto ∆′. Here (., .)V (resp. (., .)V ′) is the innerproduct on V (resp. V ′). Similarity yields an equivalence relation on the collection of all rootsystems of which we call the equivalence classes similarity classes of root systems.
A root system ∆ is said to be irreducible if ∆ can not be written as the disjoint union of twononempty subsets ∆1,∆2 ⊂∆ such that (α,β ) = 0 for all α ∈∆1 and β ∈∆2, otherwise ∆ is calledreducible. For example, the root system of a simple Lie algebra is irreducible (and reduced as wealready noted). Let∆ be reducible and let∆=
∐ri=1∆i be a partition of∆ such that (∆i ,∆j ) = 0 for
all i 6= j , then it turns out that each∆i is a root system in spanR(∆i )⊂V . Moreover, each reducibleroot system ∆ can be decomposed into subsets ∆1, . . . ,∆s such that (∆i ,∆j ) = 0 for all i 6= j andeach ∆i is an irreducible root system in spanR(∆i ) ⊂ V . This decomposition is unique up to anordering of the∆i .
Example 1.2.4. As an example we will give here for each classical Lie algebra of Example 1.2.1 an’abstract’ root system that is similar to its own root system. These root systems are called classicalroot systems. We will consider these classical root systems in various spaces Rn with the standardinner product and standard orthonormal unit basis {ε1, . . . ,εn}.
A l : Let V =Rl+1, then∆= {εi − εj : i 6= j } is a reduced root system in V .Bl : Let V =Rl , then∆= {±εi }∪ {±(εi ± εj ) : i 6= j } is a reduced root system in V .C l : Let V =Rl , then∆= {±(εi ± εj ) : i 6= j }∪ {±2εi } is a reduced root system in V .Dl : Let V =Rl , then∆= {±(εi ± εj ) : i 6= j } is a reduced root system in V .
It turns out that all these root systems are irreducible except D2 which decomposes into two or-thogonal root systems A1. Furthermore, B1 and C1 are similar to A1, B2 is similar to C2 and D3 issimilar to A3. In Figure 1.2 we have depicted the classical root systems of rank 2 up to similarity.
A subset Π ⊂ ∆ is called a basis of the root system ∆ if Π is a basis of the vector space V , andif each root β ∈ ∆ can be written as
∑
α∈Π cαα, where the coefficients cα are all nonpositive or allnonnegative integers. SinceΠ is a basis of V we can writeΠ= {α1, . . . ,αl }, and we call the elements
5
Figure 1.1: The classical root systems of rank 2 up to similarity.
of Π simple roots. Furthermore the expression∑
α∈Π cαα =∑l
i=1 cβi αi for β ∈ ∆ is unique, so we
can define the number ht(β ) =∑l
i=1 cβi called the height of β . It turns out that every root system∆
has a unique rootφ with respect to Π such that ht(α)< ht(φ) for all α∈∆ \ {φ}, andφ is called thehighest root of∆ with respect to Π. If all cα ≥ 0 (resp. all cα ≤ 0), then β is said to be positive (resp.negative) with respect to Π. The positive (resp. negative) roots of ∆ are denoted by ∆+ (resp. ∆−),
and∆=∆+∐
∆−. Finally, introduce the partial ordering ≥ on∆ by setting α≥ β if cαi ≥ cβi for all
1≤ i ≤ l where α,β ∈∆.The roots of an irreducible reduced root system can have at most 2 lengths with respect to
the norm induced by (., .). If there are two root lengths in an irreducible root system we will callthe roots having the lowest norm short roots and those having the highest norm long roots. Withrespect to a chosen basis Π of ∆ there exists a unique highest short (resp. long) root θ ∈ ∆ (resp.φ ∈∆). This means that θ ∈∆ (resp. φ ∈∆) is a short (resp. long) root such that ht(α)< ht(θ ) (resp.ht(β ) < ht(φ)) for all short (resp. long) roots α ∈∆ \ {θ } (resp. β ∈∆ \ {φ}). Here the highest longroot coincides with the highest root of∆with respect to Π.
Put
α∨ :=2α
(α,α)
to be the dual root of α ∈∆, then the dual ∆∨ := {α∨ : α ∈∆} of ∆ is a root system in V having thesame Weyl group as∆. It turns out that if {α1, . . . ,αl } is a basis for the reduced root system∆, then{α∨1 , . . . ,α∨l } is a basis for the dual root system∆∨.
1.2.3 The classification of reduced irreducible root systems
To classify root systems it suffices to consider irreducible root systems, since each root system canbe decomposed into a finite disjoint union of irreducible root systems that are mutually orthogo-nal. In this Subsection we consider the case that an irreducible root system ∆ is reduced. We willclassify all similarity classes of reduced irreducible root systems using Cartan matrices and Dynkindiagrams.
First, we want to relate a similarity invariant matrix to a root system. The notion of irreducibilitywill translate to a property of such a matrix called indecomposability. Let A1 and A2 be matrices,then the matrix of the form
�A1 00 A2
�
is called the direct sum of A1 and A2. A matrix A is said tobe indecomposable if there do not exist matrices A1 and A2 such that any matrix obtained fromA by simultaneous permutations of its rows and columns is a direct sum of A1 and A2. After si-
6
multaneously permuting rows and columns any matrix A can be decomposed into a direct sum ofindecomposable matrices.
Definition 1.2.5. A Cartan matrix is a rational integral l × l -matrix A = (a i j )1≤i ,j≤l that satisfiesthe following four conditions:
(1) a i i = 2 for 1≤ i ≤ l ;(2) a i j ≤ 0 if i 6= j ;(3) a i j = 0 implies a j i = 0;(4) All principal minors of A are strictly positive.
Choose a basis Π⊂∆ and fix an ordering Π= {α1, . . . ,αl } of simple roots. Then the matrix
A(∆,Π) := ((α∨i ,αj ))1≤i ,j≤l (1.2.1)
is called the Cartan matrix of ∆ with respect to the ordered basis Π, and it is actually a Cartanmatrix as defined in Definition 1.2.5. Since we consider ∆ to be irreducible, the Cartan matrixA(∆,Π) is indecomposable.
For any choice of basis Π of ∆ it turns out that the Weyl group W0(∆) is generated by the or-thogonal reflections wα for α ∈Π. Furthermore W0(∆) acts transitively on the bases of∆. Now thegenerators of W0(∆) are linear isometries of V , so all elements of W0(∆) are. This implies that theCartan matrix of ∆ does not depend on the choice of basis Π, but only on the ordering of Π. Heredifferent orderings of the same basis Π yield Cartan matrices that coincide up to simultaneouspermutations of rows and columns. Furthermore, one can show that up to simultaneous permu-tations of rows and columns A(∆,Π) determines∆ up to similarity, and one can show that for eachindecomposable Cartan matrix A there exists a corresponding reduced irreducible root system ∆(and some ordered basis Π of∆) such that A = A(∆,Π).
Write A for the equivalence class of the indecomposable Cartan matrix A under the equivalencerelation of simultaneous permutations of rows and columns of matrices, and putC for the collec-tion of indecomposable Cartan matrices up to simultaneous permutations of rows and columns.Further, put∆ for the similarity class of∆ andR for the collection of similarity classes of reducedirreducible root systems. Then we can summarize the above as follows.
Theorem 1.2.6. The map A :R→C defined by∆ 7→ A(∆,Π)=: A(∆) is a bijection.
We can associate a graph to each Cartan matrix up to simultaneous permutations of rows andcolumns. Let A = (a i j )1≤i ,j≤l be a Cartan matrix, then define the graph S(A) called the Dynkindiagram of A as follows. The graph S(A) has l nodes, and for i 6= j the i -th and j -th node are joinedby a i j a j i edges. Furthermore, these edges are equipped with an arrow pointing towards the i -thnode if |a i j |> 1. Now it turns out that a i j a j i ≤ 3 for all i 6= j for any Cartan matrix A = (a i j )1≤i ,j≤l ,so there is no ambiguity about the factorization of the number of edges between node i and node jof S(A) to obtain a i j a j i again. Thus given a Dynkin diagram D one can reconstruct the associatedCartan matrix A = (a i j )i ,j∈I up to a permutation of the index set I of A. It must be noted that theimplicit ordering of the nodes of a Dynkin diagram that we used here is not part of the definitionof a Dynkin diagram. It is only used here to describe its construction.
Each indecomposable Cartan matrix up to simultaneous permutations of rows can be repre-sented by a unique connected Dynkin diagram of which the whole list of possibilities can be foundin Figure 1.2. This gives a classification of the indecomposable Cartan matrices and reduced irre-ducible root systems using the bijection of Theorem 1.2.6.
7
A l (l ≥ 1) :α1 α2
. . .αl−1 αl
Bl (l ≥ 3) :α1 α2
. . .αl−1 αl
⇒
C l (l ≥ 2) :α1 α2
. . .αl−1 αl
⇐
Dl (l ≥ 4) :α1 α2
. . .αl−2 αl−1
αl
E6 :α1 α2 α3 α4 α5
α6
E7 :α1 α2 α3 α4 α5 α6
α7
E8 :α1 α2 α3 α4 α5 α6 α7
α8
F4 :α1 α2 α3 α4
⇒
G2 :α1 α2Ö
Figure 1.2: All possible Dynkin diagrams corresponding to finite Cartan matrices. The labels of the nodessolely serve the purpose of enumerating the nodes, and are not part of the definition of a Dynkin diagram.
The left column of Figure 1.2 contains the name X l of each Dynkin diagram S(A) that is depictedin the right column where l is the rank of A and X l is called the type of S(A). So we can define thetype of a reduced irreducible root system and an indecomposable Cartan matrix as the type of thecorresponding Dynkin diagram. Each classical root system X l of Example 1.2.4 is of type X l forX = A, B ,C , D up to similarity. Further, the labels of the nodes of the Dynkin diagrams in Figure 1.2are not part of the Dynkin diagram, but only serve the purpose now of enumerating the nodes.
1.2.4 Nonreduced irreducible root systems
In this Subsection we would like to give some useful details on nonreduced irreducible root sys-tems and classify them up to similarity. Since the root systems of semisimple Lie algebras arereduced, nonreduced root systems will not play a role in the classification of semisimple Lie alge-bras. However they will give rise to certain affine root systems, and therefore become an integralpart of the classification of affine root systems in Chapter 3.
Let ∆ be a nonreduced irreducible root system, and suppose α ∈ ∆. Then condition (3) ofDefinition 1.2.3 yields thatRα∩∆= {±α},Rα∩∆= {±α,± 1
2α} andRα∩∆= {±α,±2α} are the only
8
possibilities for multiples of α in∆. Hence we can define the indivisible roots∆i nd := {α∈∆ : 12α /∈
∆} of ∆ and the unmultipliable roots ∆u m n := {α ∈ ∆ : 2α /∈ ∆} of ∆. So ∆ = ∆i nd ∪∆u nm wherewe note that the union is not disjoint because of the case that Rα ∩∆ = {±α} for a root α ∈ ∆.Furthermore, ∆i nd and ∆u nm are itself reduced irreducible root systems both having the sameWeyl group as∆. Since the unmultipliable roots all follow as integral multiples of indivisible roots,it turns out that all bases for∆ are also bases for∆i nd . So also∆ has the same Dynkin diagram as∆i nd .
One can observe that both∆i nd and∆u nm have two root lengths. The long roots of∆i nd are theshort roots of ∆u nm , and the long roots of ∆u nm are the short root of ∆i nd multiplied by a factor2. Therefore a nonreduced irreducible root system has 3 root lengths. Furthermore, the long rootsof ∆u nm are the roots of ∆ having the highest root length. After choosing a basis for ∆ we have aunique highest rootφ ∈∆which is a long root of∆u nm .
Finally, one can show that there is only one similarity class of nonreduced irreducible root sys-tems for a fixed rank l . Therefore a nonreduced irreducible root system is similar to its dual rootsystem. Moreover, (∆i nd )∨ = (∆∨)u nm and (∆u nm )∨ = (∆∨)i nd where the short (resp. long) roots of∆i nd (resp. ∆u nm ) become the long (resp. short) roots of (∆∨)u nm (resp. (∆∨)i nd ). A nonreducedirreducible root system∆ is said to be of type BC l . This terminology comes from the fact that∆i nd
is always of type Bl and∆u nm is always of type C l (see Figure 1.2).
1.3 Serre’s Theorem, a commutative triangle and the classification ofsimple Lie algebras
To classify semisimple Lie algebras it suffices to consider simple Lie algebras, since every semisim-ple Lie algebra is a direct sum of simple Lie algebras. For the classification of simple Lie algebraswe will use a Theorem of Serre to construct a corresponding simple Lie algebra for each reducedirreducible root system or indecomposable Cartan matrix. This will give us a bijection between theisomorphism classes of simple Lie algebras, the similarity classes of reduced irreducible root sys-tems and indecomposable Cartan matrices up to simultaneous permutations of rows and columnsin the form of a commutative triangle. Together with the classification of the similarity classes ofreduced irreducible root systems from the previous Section this finishes the classification of simpleLie algebras.
In Subsections 1.2.1 and 1.2.2 we mentioned that the choice of a Cartan subalgebra h in a simpleLie algebra g gives rise to a reduced irreducible root system∆(g,h). It turns out that up to similarity∆(g,h) does not depend on the choice of Cartan subalgebra h⊂ g. Furthermore, the isomorphismclass of g determines∆(g,h) uniquely up to similarity. Put g for the isomorphism class of the simpleLie algebra g and L for the set of isomorphism classes of simple Lie algebras. Then we have thewell defined injective map∆ :L →R given by g 7→∆(g,h) =:∆(g).
Due to Serre we have the following Theorem that shows that for each reduced root system (orindecomposable Cartan matrix) there actually exists a corresponding semisimple Lie algebra.
Theorem 1.3.1. Let h be a complex vector space with basis {h1, . . . , h l }, and let ∆ be a reduced rootsystem in h∗ with basis Π= {α1, . . . ,αl } and corresponding Cartan matrix A = (a i j )1≤i ,j≤l . Consider
9
the Lie algebra g generated by 3n elements e i , f i , h i (1≤ i ≤ l ) that satisfy the following relations
[e i , f j ] =δi j h i ,
[h i , h j ] = 0,
[h i , e j ] = a i j e j ,
[h i , f j ] =−a i j f j ,
(ad e i )1−a i j e j = 0 i 6= j ,
(ad f i )1−a i j f j = 0 i 6= j .
where i , j = 1, . . . , l . Then g is a semisimple Lie algebra with h a Cartan subalgebra and correspond-ing root system∆.
Proof. See §18 of [6].
In Serre’s Theorem it turns out that if∆ is irreducible, then g is simple. This leads to the follow-ing canonical bijection.
Theorem 1.3.2. The injective map∆ :L →R defined by g 7→∆(g,h) =:∆(g) is a bijection.
Furthermore, if A is an indecomposable Cartan matrix and g(A) its corresponding simple Lie al-gebra from Serre’s Theorem, then we obtain a map g : C →L defined by g(A) := g(A). This mapis well defined and bijective by Theorem 1.3.2 and 1.2.6. Hence we obtain the following classicalidentification (see [4]) ofL ,R andC
L ∆ → R
C
A
←
g
←
(1.3.1)
through the bijections ∆, A and g which make the diagram commute. This gives a classificationof all isomorphism classes of simple Lie algebras and similarity classes of reduced irreducible rootsystems using the classification of indecomposable Cartan matrices up to simultaneous permuta-tions of rows columns according to their Dynkin diagrams (see Figure 1.2).
Remark 1.3.1. In this Chapter we have defined root systems, Weyl groups and Cartan matrices.However in the remaining of this manuscript we will also encounter other types of root systems,Weyl groups and Cartan matrices. To distinguish the former from the latter we will call a rootsystem a finite root system, a Weyl group a finite Weyl group and an indecomposable Cartan matrixa finite Cartan matrix from this point on. This terminology will be explained in the next Chapter.
10
Chapter 2
Kac-Moody algebras, Affine Lie algebras,and the set of real roots
In this Chapter we start with the construction of the Lie algebra g(A) associated to any complexmatrix A. This definition is modeled after a suitable generalization of Serre’s Theorem from theprevious Chapter. Although g(A) has a root space decomposition it is does not admit an invariantnondegenerate symmetric bilinear form in general, and it does not easily generalize the notion ofa Weyl group that acts on the root system. Letting A be a generalized Cartan matrix we call g(A)a Kac-Moody (Lie) algebra. Kac-Moody algebras do allow a natural definition of a Weyl group.Also, Kac-Moody algebra’s admit an invariant nondegenerate symmetric bilinear form if and onlyif their generalized Cartan matrix is symmetrizable. Furthermore, each symmetrizable Kac-moodyalgebra has a Serre presentation in terms of generators and relations, and therefore generalize thenotion of a semisimple Lie algebra in a very nice way. Finally, each symmetrizable Kac-moodyalgebra up to isomorphism corresponds to a symmetrizable generalized Cartan matrix up to si-multaneous permutations of rows and columns.
Indecomposable generalized Cartan matrices can be either of finite, affine or indefinite type.We will discuss the classification of generalized Cartan matrices of finite and affine type which turnout to be both symmetrizable. This will show that generalized Cartan matrices of finite type arefinite Cartan matrices and that their associated Kac-Moody algebra is simple. For the remainingof this Chapter we will be considering the next interesting class of Kac-Moody algebras, namelyaffine Lie algebras which are Kac-Moody algebras associated to generalized Cartan matrices ofaffine type. Contrary to simple Lie algebras these Lie algebras are infinite dimensional. Howeverthey do contain a simple subalgebra and can be thought of as infinite-dimensional generalizationsof simple Lie algebras. We will study the set of real roots of an affine Lie algebra which generalizessome of the properties of its ’underlying’ finite root system. Moreover, the set of real roots willgive rise to a new kind of root system called an affine root system which was defined and analyzedindependently by Macdonald (see [10]). We will study affine root systems in full detail in a generalsetting in the next Chapter. This Chapter consists of a summary of results on Kac-Moody algebrasfrom [8], [12] and [15].
11
2.1 Kac-Moody algebras
2.1.1 Realizations of matrices and their associated Lie algebras
We ended the classification of simple Lie algebras in the previous Chapter with Serre’s Theoremwhich yields a simple Lie algebra in terms of generators and relations that only depends on a finiteCartan matrix. We will now start out with any complex n ×n-matrix A, and construct a Lie algebrag(A) using a suitable generalization of Serre’s Theorem. Such a Lie algebra will turn out to have aroot space decomposition which gives rise to a new kind of root system that we will from now oncall a root system. Further, up to a special type of Lie algebra isomorphism g(A) only depends on Aup to simultaneous permutations of rows and columns. To handle the case that det(A) = 0 we firstintroduce the notion of a realization of a complex matrix A.
Definition 2.1.1. A realization of a rank l complex n×n-matrix A = (a i j )1≤i ,j≤n is a triple (h,Π,Π∨),where h is a complex vector space, and Π= {α1, . . . ,αn} ⊂ h∗ and Π∨ = {α∨1 , . . . ,α∨n} ⊂ h are indexedsubset in h∗ and h respectively, satisfying the following three conditions:
(1) dimC(h) = 2n − l ;(2) both Π and Π∨ are linearly independent sets;(3) a i j = ⟨αj ,α∨i ⟩ for 1≤ i , j ≤ n ,
where ⟨., .⟩ : h∗×h→C denotes the pairing ⟨α, h⟩=α(h).
Example 2.1.2. Let A be a complex n×n-matrix of rank l . Reordering the indices of A, if necessary,we may assume that A =
�A1A2
�
with A1 an l ×n submatrix of rank l . Then take h = C2n−l , αi thelinear functional that returns the i -th coordinate on h and α∨i the i -th row of the matrix
C =
�
A1 0A2 In−l
�
for i = 1, . . . , n , where In−l is the (n−l )×(n−l ) identity matrix. Then (h,Π,Π∨)withΠ= {α1, . . . ,αn}and Π∨ = {α∨1 , . . . ,α∨n} is a realization of A.
For a realization (h,Π,Π∨) of a matrix A we callΠ (resp. Π∨) the root basis (resp. coroot basis), andelements of Π (resp. Π∨) are called simple roots (resp. simple coroots). Define the root lattice (resp.coroot lattice) Q :=
∑ni=1Zαi (resp. Q∨ :=
∑ni=1Zα
∨i ), and also set Q+ :=
∑ni=1Z≥0αi . Introduce the
partial ordering ≥ on h∗ by setting α≥β if α−β ∈Q+ for α,β ∈ h∗, hence λ> 0 for all λ∈Q+ \ {0}.There is a natural notion of isomorphism between realizations: Two realizations (h1,Π1,Π∨1 )
and (h2,Π2,Π∨2 ) (of not necessarily the same matrix) are said to be isomorphic if there exists a linearisomorphism φ : h1 → h2 such that φ(Π∨1 ) = Π
∨2 and φ∗(Π2) = Π1, where φ∗ is the pull-back map
φ∗(α) =α ◦φ ∈ h∗1 for α∈ h∗2.
Proposition 2.1.3. Up to isomorphism there exists a unique realization for every complex n × n-matrix A. Moreover, realizations of matrices A and A ′ are isomorphic if and only if A ′ can be obtainedfrom A by simultaneous permutation of the rows and columns of A.
Proof. The existence follows from Example 2.1.2. For the rest of the proof we refer to Proposition1.1 of [8].
Let A = (a i j )1≤i ,j≤n be a complex matrix of rank l together with a realization (h,Π,Π∨). Further,choose hn+1, . . . , h2n−l ∈ h such that {h1 :=α∨1 , . . . , hn :=α∨n , hn+1 . . . , h2n−l } is a basis of h. Introduce
12
the Lie algebra g(A,h,Π,Π∨)with the generators e i , f i (1≤ i ≤ n) and h j (1≤ j ≤ 2n− l ) that satisfythe following relations:
[e i , f j ] =δi j h i (i , j = 1, . . . , n ),
[h i , h j ] = 0 (i , j = 1, . . . , 2n − l ),
[h i , e j ] = ⟨αj , h i ⟩e j (i = 1, . . . , 2n − l ; j = 1, . . . , n ),
[h i , f j ] =−⟨αj , h i ⟩ f j (i = 1, . . . , 2n − l ; j = 1, . . . , n ).
(2.1.1)
Then the construction of g(A,h,Π,Π∨) does not depend on the choice of hn+1, . . . , h2n−l , and h ⊂g(A,h,Π,Π∨) is a commutative subalgebra.
Denote by n+ (resp. n−) the subalgebra of g(A,h,Π,Π∨) generated by e1, . . . , en (resp. f 1, . . . , f n ),then we have the following results on g(A,h,Π,Π∨).
Theorem 2.1.4. (i) n+ (resp. n−) is freely generated by e1, . . . , en (resp. f 1, . . . , f n ).(ii) The Lie algebra g(A,h,Π,Π∨) is Q− g r a d e d . In particular, one has the root space decomposi-
tion
g(A,h,Π,Π∨) =�
⊕
α∈Q+\{0}
g−α
�
⊕ g0⊕�
⊕
α∈Q+\{0}
gα
�
(2.1.2)
with respect to h, where gα = {x ∈ g(A,h,Π,Π∨) : [h,x ] = ⟨α, h⟩x for all h ∈ h} for all α ∈ Q and[gα, gβ ]⊆ gα+β for all α,β ∈Q. Furthermore, dim(g±α)<∞ for all α∈Q+ \ {0} and g0 = h.
(iii) Ifα> 0 (resp. α< 0) then gα is the linear span of elements of the form [. . . [[e i 1 , e i 2 ], e i 3 ] . . . , e i r ](resp. [. . . [[ f i 1 , f i 2 ], f i 3 ] . . . , f i r ]) such that αi 1 + · · ·+αi r =α (resp. −α).
(iv) In (2.1.2) we have⊕
α∈Q+\{0}
g±α = n±,
so that g(A,h,Π,Π∨) decomposes as a vector space as n−⊕h⊕ n+.(v) There exists a unique maximal ideal τ of g(A,h,Π,Π∨) that has a trivial intersection with h.
Furthermore, τ is Q-graded andτ= (τ∩ n−)⊕ (τ∩ n+). (2.1.3)
Proof. This follows from Theorem 1.2 of [8] and the same Theorem of [15].
Now define the quotient Lie algebra
g(A,h,Π,Π∨) := g(A,h,Π,Π∨)/τ
which is called the Lie algebra associated to the matrix A. We keep the same notation for the imagesof e i , f i (1≤ i ≤ n) and h in g(A,h,Π,Π∨) under the canonical map. Notice that h⊂ g(A,h,Π,Π∨) isa commutative subalgebra by (v) of Theorem 2.1.4. This is usually called the Cartan subalgebra ofg(A,h,Π,Π∨), and as we will see in Corollary 2.2.7 it generalizes the notion of a Cartan subalgebraas we introduced it in the previous Chapter for semisimple Lie algebras.
Denote by n+ (resp. n−) the subalgebra of g(A,h,Π,Π∨) generated by e1, . . . , en (resp. f 1, . . . , f n ).The following result then follows from Theorem 2.1.4.
Theorem 2.1.5. (i) The Lie algebra g(A,h,Π,Π∨) is Q−g r a d e d . In particular, one has the root spacedecomposition
g(A,h,Π,Π∨) =⊕
α∈Q
gα
13
with respect to h where gα = gα/(τ∩ gα), and [gα,gβ ]⊆ gα+β for all α,β ∈Q.Moreover, gα = {x ∈ g(A,h,Π,Π∨) : [h,x ] = ⟨α, h⟩x for all h ∈ h} is finite-dimensional and g0 = h.
(ii) If gα 6= {0}, then α = 0, α > 0 or α < 0. Furthermore, if α > 0 (resp. α < 0) then gα is thelinear span of elements of the form [. . . [[e i 1 , e i 2 ], e i 3 ] . . . , e i r ] (resp. [. . . [[ f i 1 , f i 2 ], f i 3 ] . . . , f i r ]) such thatαi 1+ · · ·+αi r =α (resp. −α). In particular, gαi =Ce i , g−αi =C f i and gcαi = 0 for |c |> 1 (i = 1, . . . , n).
(iii) The Lie algebra g(A,h,Π,Π∨) decomposes as n−⊕h⊕n+ as a vector space, where
n− =⊕
α∈Q+\{0}
g−α = n−/(τ∩ n−), n+ =⊕
α∈Q+\{0}
gα = n+/(τ∩ n+).
The subspace gα ⊂ g(A,h,Π,Π∨) is called the root space attached to α. Define the numbermult(α) := dim(gα) which shall be called the multiplicity of α. An element α ∈Q is called a rootif α 6= 0 and mult(α) 6= 0. We let ∆ denote the set of all roots of g(A,h,Π,Π∨) and call it the root sys-tem of g(A,h,Π,Π∨). Corollary 2.2.7 will show that∆ generalizes the notion of a finite root system.
Let g(A1,h1,Π1,Π∨1 ) and g(A2,h2,Π2,Π∨2 ) be Lie algebras associated to matrices A1 and A2 re-spectively. A Lie algebra isomorphism φ : g(A1,h1,Π1,Π∨1 )→ g(A2,h2,Π2,Π∨2 ) is called a realizationpreserving isomorphism if φ(h1) = h2, φ(Π∨1 ) = Π
∨2 , and φ∗(Π2) = Π1, where φ∗ sends α ∈ h∗2 to its
pull-backφ∗(α) =α◦φ ∈ h∗1. Notice that in this situationφ induces an isomorphism of realizations(h1,Π1,Π∨1 ) of A1 and (h2,Π2,Π∨2 ) of A2.
We have the following characterization of the Lie algebra g(A,h,Π,Π∨).
Proposition 2.1.6. Let g be a Lie algebra, h ⊂ g be a commutative subalgebra, e1, . . . , en , f 1, . . . , f n
elements of g, and let Π∨ = {α∨1 , . . . ,α∨n} ⊂ h and Π = {α1, . . . ,αn} ⊂ h∗ be linearly independent setssuch that the relations (2.1.1) are satisfied. Suppose that {h1 := α∨1 , . . . , hn := α∨n , hn+1, . . . , h2n−l } isa basis of h, that e i , f i (1 ≤ i ≤ n) and h j (1 ≤ j ≤ 2n − l ) generate g as a Lie algebra, and that ghas no nonzero ideals which intersect h trivially. Finally, put A = (⟨αj ,α∨i ⟩)1≤i ,j≤n , and assume thatdim(h) = 2n − rank(A). Then there exists a Lie algebra isomorphism φ : g(A,h,Π,Π∨)→ g such thatφ(h) = h,φ(Π∨) =Π∨ andφ∗(Π) =Π.
Proof. For the proof we refer to Proposition 1.4 of [15].
Furthermore, we have that A up to simultaneous permutation of rows and columns determinesits g(A,h,Π,Π∨) uniquely up to realization preserving isomorphism.
Theorem 2.1.7. Let A (resp. A ′) be a complex n × n-matrix together with a realization (h,Π,Π∨)(resp. (h′,Π′,Π′∨)). There exists a realization preserving isomorphism between g(A,h,Π,Π∨) andg(A ′,h′,Π′,Π′∨) if and only if A ′ can be obtained from A by simultaneous permutation of the rowsand columns of A.
Proof. For the proof we refer to Proposition 1.4 of [15] and Proposition 2.1.3.
From Theorem 2.1.7 we observe that the Lie algebra g(A,h,Π,Π∨) does not depend on any chosenrealization for A up to realization preserving isomorphism and is uniquely determined by A up tosimultaneous permutation of rows and columns. From here on we will write g(A) for g(A,h,Π,Π∨) ifthere is no ambiguity about the chosen realization (h,Π,Π∨) for the construction of g(A,h,Π,Π∨). InCorollary 2.1.15 we will see that Theorem 2.1.7 still holds if the condition of a realization preservingisomorphism is be weakened to any Lie algebra isomorphism for a special choice of the matrix A.
Let (h1,Π1,Π∨1 ) (resp. (h2,Π2,Π∨2 )) be a realization of the matrix A1 (resp. A2), then clearly thedirect sum of realizations (h,Π,Π∨) := (h1⊕h2,Π1×{0}∪{0}×Π2,Π∨1 ×{0}∪{0}×Π
∨2 ) is a realization
14
of the direct sum of matrices A :=�A1 0
0 A2
�
. In this case Proposition 2.1.6 tells us that the Lie algebrag(A,h,Π,Π∨) is isomorphic to the direct sum of the ideals g(A1,h1,Π1,Π∨1 ) and g(A2,h2,Π2,Π∨2 ).
A matrix A together with a corresponding realization is said to be indecomposable if there donot exist matrices A1 and A2 such that any matrix obtained from A by simultaneous permutationsof its rows and columns is of the form
�A1 00 A2
�
. After simultaneously permuting rows and columns amatrix A can be decomposed into a direct sum of indecomposable matrices, and the correspond-ing realization into a direct sum of indecomposable realizations (see Proposition 1.9C of [15]). Thisimplies that we can restrict our study to Lie algebras g(A) associated to an indecomposable com-plex matrix A.
2.1.2 The invariant nondegenerate symmetric bilinear form
Similar to the Killing form on semisimple Lie algebras we would like to define an invariant nonde-generate symmetric bilinear form on the Lie algebra g(A). We will observe that we need an extrasymmetry condition on the matrix A to achieve this. Then the desired form turns out to restrict toa nondegenerate symmetric bilinear form on h. We will use this fact to obtain a canonical linearisomorphism between h and h∗, and to induce a nondegenerate symmetric bilinear form on h∗.
Definition 2.1.8. A complex n ×n-matrix A = (a i j )1≤i≤n is said to be symmetrizable if there existsan invertible diagonal matrix D and a symmetric matrix B such that A = D B . In this case the Liealgebra g(A) is called a symmetrizable Lie algebra and the matrix B is called a symmetrization of A.
Next we will see that an invariant nondegenerate symmetric bilinear C-valued form (., .) ong(A,h,Π,Π∨) exists if and only if A is symmetrizable. Furthermore, the restriction of (., .) on h deter-mines (., .) uniquely on g(A,h,Π,Π∨), and different forms (., .) coincide on the subspace
⊕ni=1Cα
∨i ⊂
h up to a factor in C if A is indecomposable.
Theorem 2.1.9. Let A be a complex n×n-matrix with realization (h,Π,Π∨), and put h′ :=⊕n
i=1Cα∨i .
(i) If A is symmetrizable with symmetrization B such that A = diag(ε1, . . . ,εn )B, then any choiceof subspace h′′ ⊂ h such that h= h′⊕h′′ gives rise to an invariant nondegenerate symmetric bilinearC-valued form (., .) on g(A,h,Π,Π∨)) such that (., .)|h is nondegenerate and defined by
(
(α∨i , h) = ⟨αi , h⟩εi (h ∈ h, i = 1, . . . , n ),
(h ′, h ′′) = 0 (h, h ′ ∈ h′′);(2.1.4)
(ii) If there exists an invariant nondegenerate symmetric bilinearC-valued form (., .) on g(A,h,Π,Π∨),then A is symmetrizable and there exists a symmetrization B of A and a subspace h′′ ⊂ h such thatA = diag(ε1, . . . ,εn )B, h= h′⊕h′′ and (., .)|h is nondegenerate and defined by (2.1.4).
Moreover, if (1) or (2) is satisfied then(iii) if (., .)′ is a another invariant nondegenerate symmetric bilinearC-valued form on g(A,h,Π,Π∨)
and A is indecomposable, then there exists a nonzero µ∈C such that (x , y )′ =µ(x , y ) for all x , y ∈ h′;(iv) if (., .)′ is a nondegenerate symmetric bilinearC-valued form on g(A,h,Π,Π∨) such that (., .)′|h =
(., .)|h, then (., .)′ = (., .) on g(A,h,Π,Π∨).
Proof. (1) follows from Theorem 2.2 of [8], (2) follows from Theorem 3.1 of [15] and (4) follows fromTheorem 3.2 of [15]. To prove (3) let (., .) be the invariant nondegenerate symmetric bilinear formon g(A,h,Π,Π∨) of (2) induced by A =D B with D an invertible diagonal matrix and B a symmetricmatrix. If (., .)′ is another invariant nondegenerate symmetric bilinear form on g(A,h,Π,Π∨), then
15
by (2) this form is induced by A = D ′B ′ with D ′ an invertible diagonal matrix and B ′ a symmetricmatrix. Thus B = E B ′ for some invertible diagonal matrix E . Since A is indecomposable, it alsofollows from A =D B =D ′B ′ that B and B ′ are indecomposable. But for B = E B ′ to hold under thecondition that both B and B ′ are symmetric and indecomposable with E an invertible diagonalmatrix, it must be true that E = µI where µ ∈ C is nonzero and I is the n × n identity matrix.This implies that B = µB ′, so µD = D ′. Then (2.1.4) and bilinearity of (., .) and (., .)′ show that(x , y )′ =µ(x , y ) for all x , y ∈ h′.
Remark 2.1.1. At first sight it is not clear from the first equation of (2.1.4) why (α∨i ,α∨j ) = (α∨j ,α∨i ) for
1≤ i , j ≤ n , even though (., .) is a symmetric form. Consider the decomposition A =diag(ε1, . . . ,εn )Bthat induces (., .) in (i) of Theorem 2.1.9. Put A = (a i j )1≤i ,j≤n and B = (b i j )1≤i ,j≤n , then a i j = εi b i j .Further a j i = εj b j i = εj b i j , since B is a symmetric matrix. Recall that a i j = ⟨αj ,α∨i ⟩, then we finallyobtain
(α∨i ,α∨j ) = ⟨αi ,α∨j ⟩εi = a j i εi
= εj b i j εi = εj a i j = (α∨j ,α∨i )
for 1≤ i , j ≤ n .
Since the bilinear form (., .) from ((i) or (ii) of) Theorem 2.1.9 is nondegenerate on h we candefine the linear isomorphism ν : h→ h∗ depending on (., .) by
⟨ν (h), h ′⟩= (h, h ′)
for h, h ′ ∈ h. This leads to a nondegenerate symmetric bilinear form on h∗ defined by
(α,β ) := (ν−1(α),ν−1(β ))
for α,β ∈ h∗. Furthermore, we obtain
⟨α, h⟩= ⟨ν (ν−1(α)), h⟩= (ν−1(α), h) = (α,ν (h)) (2.1.5)
for α∈ h∗ and h ∈ h.Write A = (a i j )1≤i ,j≤n , then from (2.1.4) we obtain (α∨i ,α∨j ) = εj a i j for 1≤ i , j ≤ n and
ν (α∨i ) = εiαi (2.1.6)
for 1≤ i ≤ n . This leads to(αi ,αj ) = a i j /εi (2.1.7)
for 1≤ i , j ≤ n . By an argument similar to that in Remark 2.1.1 one observes that indeed (αi ,αj ) =(αj ,αi ).
Finally, if (., .)′ is another invariant nondegenerate symmetric bilinear form on g(A,h,Π,Π∨),then there exists a nonzero µ ∈ C such that (x , y )′ = µ(x , y ) for all x , y ∈ h′ :=
⊕ni=1Cα
∨i by (iii)
of Theorem 2.1.9. Notice that ν (h′) =⊕n
i=1Cαi by (2.1.6). Putting ν ′ for the linear isomorphismfrom h to h∗ induced by (., .)′ we observe that
ν ′|h′ =µν |h′ . (2.1.8)
Then it follows that(α,β ) =µ(α,β )′ (2.1.9)
for all α,β ∈⊕n
i=1Cαi .
16
2.1.3 Kac-Moody algebras
In the following we introduce the notion of a generalized Cartan matrix. This is a matrix A thatsatisfies the properties of a finite Cartan matrix, except that A does not necessarily have all itsprincipal minors positive. A Lie algebra associated to such a matrix is called a Kac-Moody algebra.As we will see in this Subsection, the generators of a Kac-Moody algebra g(A) satisfy the naturalgeneralization of the relations of Serre’s Theorem. Furthermore, if A is also symmetrizable theng(A) is defined by those relations, and up to isomorphism only depends on A up to simultaneouspermutations of rows and columns. Finally, if g(A) is a symmetrizable Kac-Moody algebra then Acan be expressed using the invariant bilinear form (., .) in a similar way as a Cartan matrix of a finiteroot system is defined using the Killing form.
Definition 2.1.10. A generalized Cartan matrix is a rational integral n ×n-matrix A = (a i j )1≤i ,j≤n
satisfying the following three conditions(1) a i i = 2 for 1≤ i ≤ n ;(2) a i j ≤ 0 if i 6= j ;(3) a i j = 0 implies a j i = 0.
Example 2.1.11. (i) A finite Cartan matrix is a generalized Cartan matrix.(ii) If A is a generalized Cartan matrix, then the transposed AT of A is also a generalized Cartan
matrix.(iii) If A = (a i j )i ,j∈I is a generalized Cartan matrix with I a finite index set, then every permuta-
tion of I gives another generalized Cartan matrix.
The Lie algebra g(A) associated to a generalized Cartan matrix A is called a Kac-Moody alge-bra. Kac-Moody algebras are especially interesting because they admit Serre type relations on thegenerators e1, . . . , en , f 1, . . . , f n of g(A).
Proposition 2.1.12. Let g(A) be a Kac-Moody algebra associated to the n × n generalized Cartanmatrix A, then
(ad e i )1−a i j e j = 0, (ad f i )1−a i j f j = 0
for 1≤ i , j ≤ n and i 6= j .
Proof. See §3.3 of [8].
Recall that by (i) of Theorem 2.1.4 the subalgebra n+ (resp. n−) of g(A) is freely generated bye1, . . . , en (resp. f 1, . . . , f n ). Furthermore, by (v) of Theorem 2.1.4 the Lie algebra g(A) has a uniquemaximal ideal τ that intersects h trivially. Since g(A) = g(A)/τ, Proposition 2.1.12 shows that
(ad e i )1−a i j e j , (ad f i )1−a i j ∈τ
for 1≤ i , j ≤ n and i 6= j .Together with (2.1.1) Proposition 2.1.12 shows that the generators e i , f i (1≤ i ≤ n) and h i (1≤
i ≤ 2n − l ) of a Kac-Moody algebra g(A) satisfy a natural generalization of the relations of Serre’sTheorem for semisimple Lie algebras (see Theorem 1.3.1). Using representation theory of Kac-Moody algebras it is possible to obtain a much stronger result, namely that every symmetrizableKac-Moody algebra is defined by these relations.
17
Theorem 2.1.13. Let A = (a i j )1≤i ,j≤n be a symmetrizable generalized Cartan matrix, then g(A) isthe Lie algebra generated by e i , f i (1≤ i ≤ n) and h i (1≤ i ≤ 2n − l ) with relations
[e i , f j ] =δi jα∨i (i , j = 1, . . . , n ),
[h i , h j ] = 0 (i , j = 1, . . . , 2n − l ),
[h i , e j ] = ⟨αj , h i ⟩e j (i = 1, . . . , 2n − l ; j = 1, . . . , n ),
[h i , f j ] =−⟨αj , h i ⟩ f j (i = 1, . . . , 2n − l ; j = 1, . . . , n ),
(ad e i )1−a i j e j = 0 (i , j = 1, . . . , n ; i 6= j ),
(ad f i )1−a i j f j = 0 (i , j = 1, . . . , n ; i 6= j ).
Proof. Follows from Theorem 9.11 of [8].
Using representation theory of Kac-Moody algebras it is also possible to generalize the state-ment of Theorem 2.1.7 to arbitrary Lie algebra isomorphisms when considering symmetrizableKac-Moody algebras, and thereby establishing a criterion for isomorphism of two symmetrizableKac-Moody algebras.
Theorem 2.1.14. Let A and A ′ be symmetrizable generalized Cartan matrices such that g(A) is iso-morphic to g(A ′), then A ′ can be obtained from A by simultaneous permutation of the rows andcolumns of A.
Proof. This follows from Theorem 2 (b) of [12].
Corollary 2.1.15. Let A and A ′ be symmetrizable generalized Cartan matrices, then g(A) and g(A ′)are isomorphic as Lie algebras if and only if A ′ can be obtained from A by simultaneous permutationof the rows and columns of A.
Proof. Follows from Theorem 2.1.7 and 2.1.14.
This shows that up to isomorphism a symmetrizable Kac-Moody algebra only depends on its asso-ciated symmetrizable generalized Cartan matrix up to simultaneous permutation of the rows andcolumns.
Finally, if A is a symmetrizable generalized Cartan matrix then one can show that
A = diag(ε1, . . . ,εn )B
with ε1, . . . ,εn ∈Q>0 and B a symmetrization with rational coordinates. Fix an invariant nondegen-erate symmetric bilinear form (., .) on g(A) associated to this decomposition of A using Theorem2.1.9. Then from (2.1.7) we obtain
(αi ,αi )> 0 (2.1.10)
for i = 1, . . . , n . The form (., .) on the symmetrizable Kac-Moody algebra g(A) provided by Theorem2.1.9 and satisfying (2.1.10) is called a standard invariant form. In this setting (2.1.6) and (2.1.7)imply
A =�
2(αi ,αj )(αi ,αi )
�
1≤i ,j≤n. (2.1.11)
This expression coincides with the expression of the Cartan matrix of a root system of a semisimpleLie algebra (see (1.2.1)).
18
2.1.4 The Weyl group and the set of real roots of a Kac-Moody algebra
In this Subsection we want to get a hint that Kac-Moody algebras also carry a nice structure ontheir root systems. We will observe that the root basisΠ of a root system∆ of a Kac-Moody algebrag(A,h,Π,Π∨) gives rise to a reflection group on h∗ that we will call the Weyl group W of the Kac-Moody algebra. It turns out that W leaves∆ invariant and that it induces an action on∆. This willlead us to defining the real roots ∆r e of the root system of a Kac-Moody algebra as the roots thatare W -equivalent to Π under the action of W on∆. These are the roots that interest us the most inthe remaining of this Chapter, since turn out to generalize finite root systems nicely for a certainclasses of infinite-dimensional Kac-Moody algebras.
Let A be a generalized Cartan matrix with associated Kac-Moody algebra g(A,h,Π,Π∨). For i =1, . . . , n define the fundamental reflection ri of the space h∗ by
ri (λ) =λ−⟨λ,α∨i ⟩αi (2.1.12)
for all λ ∈ h∗. Clearly ri fixes the subspace {λ ∈ h∗ : ⟨λ,α∨i ⟩ = 0} ⊂ h∗ and ri (αi ) = −αi , so ri is areflection of h∗. Define the Weyl group W of g(A) as the subgroup of the group of linear automor-phisms GL(h∗) of h∗ generated by all fundamental reflections. We will see in Corollary 2.2.7 that theWeyl group generalizes the finite Weyl group of a finite root system.
Proposition 2.1.16. (i)The root system∆ of g(A) is W -invariant, and W acts faithfully on∆;(ii) If A is symmetrizable, then a standard invariant form (., .) considered on h∗ is W -invariant for
g(A) (i.e. (λ,µ) = (w (λ), w (µ)) for all λ,µ∈ h∗ and w ∈W ).
Proof. (i) follows from Proposition 3.7 b) of [8] and (3.12.1) in the proof of Proposition 3.12 of [8],and (ii) follows from Proposition 3.9 of [8].
Let g(A) be a Kac-Moody algebra with root system ∆ and Weyl group W . A root α ∈ ∆ is saidto be real if there exists w ∈W such that w (α) is a simple root, otherwise α is called imaginary.Write∆r e (resp. ∆i m ) for the subset of∆ of real roots (resp. imaginary roots), then∆=∆r e q∆i m .Clearly∆r e and∆i m are W -invariant.
Next, we assume that g(A) is symmetrizable and consider a standard invariant form (., .) on g(A).Then we have the following insightful Lemma on the terminology of ’real’ and ’imaginary’ root.
Lemma 2.1.17. Let α∈∆r e and β ∈∆i m , then
(α,α)> 0, and (β ,β )≤ 0.
Proof. This follows from Proposition 5.1 and 5.2 of [8].
Lemma 2.1.17 shows that real roots have a positive ’squared length’ with respect to any stan-dard invariant form (., .), while imaginary roots have a vanishing or negative ’squared length’ withrespect to (., .). This makes ∆r e more useful for a geometric description than ∆i m . It will actu-ally turn out that ∆r e generalizes the notion of a finite root system in a very nice way for a certainclasses of infinite-dimensional Kac-Moody algebras. Describing such a root system will be themain focus of the Chapter 3.
To end this Section we will show the useful fact that each real root induces a reflection in h∗ thatis contained in W . From (2.1.6) and (2.1.7) we obtain α∨i = 2 ν
−1(αi )(αi ,αi )
∈ h. We generalize this formulaas follows. For α∈∆r e define the dual root
α∨ := 2ν−1(α)(α,α)
∈ h.
19
It follows from (2.1.8) and (2.1.9) that this definition is independent of the choice of standard in-variant form. Further, for α∈∆r e define the reflection rα by
rα(λ) :=λ−⟨λ,α∨⟩α=λ−2(λ,α)(α,α)
α (2.1.13)
for all λ ∈ h∗. Since α ∈ ∆r e there exists w ∈W and i ∈ {1, . . . , n} such that α = w (αi ). Then oneobtains from Proposition 2.1.16 (ii) and (2.1.13)
rα =w ◦ ri ◦w−1 ∈W. (2.1.14)
Hence W contains the reflection of h∗ induced by all real roots.
2.2 A classification of generalized Cartan matrices and some Kac-Moodyalgebras
We proceed to giving a classification of generalized Cartan matrices. First we will observe that anindecomposable generalized Cartan matrix is of finite, affine or indefinite type. For this thesis wewill only be interested in the first two types which turn out to be symmetrizable. It turns out thata generalized Cartan matrix of finite type is actually a finite Cartan matrix. We will observe thatthe corresponding Kac-Moody algebras are simple Lie algebras. Furthermore, every semisimpleLie algebra is isomorphic to a finite-dimensional Kac-Moody algebra, and a Kac-Moody algebrasis finite-dimensional if and only if it is a semisimple Lie algebra. Finite Cartan matrices and simpleLie algebras have been fully classified in Chapter 1.
Affine Cartan matrices are closely related to finite Cartan matrices, and give rise to a class ofinfinite-dimensional Kac-Moody algebras called affine Lie algebras. Similar to the case of simpleLie algebras in Chapter 1 we will be using Dynkin diagrams to classify generalized Cartan matrixof affine type up to simultaneous permutations of rows and columns. Since affine Cartan matricesup to simultaneous permutations of rows and columns are in bijective correspondence with affineLie algebras up to isomorphism, this will immediately classify the latter.
2.2.1 Three types of generalized Cartan matrices
Recall that any matrix can be decomposed into a direct sum of indecomposable matrices. It there-fore suffices to study only the indecomposable generalized Cartan matrices. Further, for a vector vin Rn we will write v > 0 if all coordinates of v (with respect to the stadard basis of Rn ) are strictlypositive. Let A be an indecomposable generalized Cartan matrix, then one and only one of thefollowing three possibilities holds for A by Theorem 4.3 of [8]:
(Fin) there exists v > 0 such that Av > 0;(Aff) there exists v > 0 such that Av = 0;(Ind) there exists v > 0 such that Av < 0.
Definition 2.2.1. We will say that an indecomposable generalized Cartan matrix A is of finite typeif A satisfies (Fin), of affine type if A satisfies (Aff), and of indefinite type if A satisfies (Ind).
Remark 2.2.1. Kac-Moody algebras associated to indecomposable generalized Catran matrices ofindefinite type are not very well understood. For example, it is an open problem in general whatthe root multiplicities are of these Kac-Moody algebras, although some special cases have been
20
solved (see e.g. [5], [9]). Since these Kac-Moody algebras are not in the scope of this manuscriptwe will not consider the case of A being an indecomposable generalized Cartan matrix of indefinitetype anymore.
It turns out that an indecomposable generalized Cartan matrix A of finite or affine type can becharacterized by its principal minors.
Proposition 2.2.2. Let A be an indecomposable generalized Cartan matrix.(i) A is of finite type if and only if all principal minors of A are positive.(ii) A is of affine type if and only if all proper principal minors of A are positive anddet(A) = 0.(iii) A is of affine type if and only if there exists a real-valued vector δ > 0 that is unique up to aconstant factor such that Aδ= 0.
Proof. This follows from Proposition 4.7 of [8].
We introduce the following Definition.
Definition 2.2.3. A rank l affine Cartan matrix is an rational integral square (l +1)× (l +1)-matrixA = (a i j )0≤i ,j≤l satisfying the following five conditions
(1) a i i = 2 for 0≤ i ≤ l ;(2) a i j ≤ 0 if i 6= j ;(3) a i j = 0 implies a j i = 0;(4) det(A) = 0 and all the proper principal minors of A are strictly positive;(5) A is indecomposable.
Now from Proposition 2.2.2 and the Definitions of a finite and affine Cartan matrix we observe thefollowing.
Corollary 2.2.4. The indecomposable generalized Cartan matrices of finite (resp. affine ) type coin-cide with the finite (resp. affine) Cartan matrices.
Notice that if we change det(A) = 0 to det(A) 6= 0 in Definition 2.2.3 we obtain the definitionof a finite Cartan matrix. In that sense affine Cartan matrices can be considered within the gen-eralized Cartan matrices as the matrices that are the closest related to finite Cartan matrices. Inmathematical terms we have the following.
Corollary 2.2.5. Every proper principal submatrix of a finite (resp. affine) Cartan matrix is a finiteCartan matrix.
2.2.2 Kac-Moody algebras associated to finite Cartan matrices
Next we want to characterize Kac-Moody algebras corresponding to finite Cartan matrices.Let hR denote a real vector space such that Π∨ ⊂ hR, Π⊂ h∗R and (C⊗R hR,Π,Π∨) is a realization
of a generalized Cartan matrix A. Then we have the following equivalences.
Proposition 2.2.6. Let A be an indecomposable generalized Cartan matrix. Then the following con-ditions are equivalent:
(1) A is of finite type;(2) A is symmetrizable and any standard invariant form (., .) restricted to hR is positive definite;(3) The Kac-Moody algebra g(A) is a simple Lie algebra;
21
(4) The Weyl group W of the Kac-Moody algebra g(A) is a finite group;(5) The root system∆ of the Kac-Moody algebra g(A) is a finite set;(6) The root system ∆ of the Kac-Moody algebra g(A) does not contain any imaginary roots, i.e.
∆=∆r e and∆i m = ;.
Proof. This follows from Proposition 4.9 and Theorem 5.6 of [8].
Notice that the term ’finite’ in our terminology of generalized Cartan matrix of finite type, finiteroot system and finite Weyl group is now justified by characterizations (3), (4) and (5) of Proposition2.2.6.
Corollary 2.2.7. If A is a finite Cartan matrix, then g(A,h,Π,Π∨) is a simple Lie algebra with h aCartan subalgebra, ∆ a reduced irreducible finite root system with finite Weyl group W and Π is abasis for∆with corresponding finite Cartan matrix A(∆,Π)= A.
Proof. This follows from combining the results of Proposition 2.2.6, Theorem 1.3.1 and 2.1.13 to-gether with the appropriate definitions of Chapter 1.
From the classification (1.3.1) of simple Lie algebras it follows that each simple Lie algebra isisomorphic to a Kac-Moody algebra associated to a finite Cartan matrix. So indeed the Lie algebrasg(A) and in particular (symmetrizable) Kac-Moody algebras generalize semisimple Lie algebras.However the only other Kac-Moody algebras are infinite-dimensional.
Corollary 2.2.8. A Kac-Moody algebra g(A) is finite-dimensional if and only if g(A) is a semisimpleLie algebra.
Proof. This follow from (3) and (5) of Proposition 2.2.6.
2.2.3 A classification of affine Cartan matrices and affine Lie algebras
In the previous Subsection we saw that Kac-Moody algebras associated to finite Cartan matricesare just simple Lie algebras which we have completely classified in Chapter 1. Now we want to shiftour focus to classifying affine Cartan matrices, and their corresponding Kac-Moody algebras whichare known as affine Lie algebras. From Corollary 2.2.8 it follows that they are infinite-dimensionalLie algebras. First we will establish a canonical bijection between affine Cartan matrices up tosimultaneous permutations of rows and columns and affine Lie algebras up to isomorphism. Thenwe give a classification of affine Cartan matrices up to simultaneous permutations of rows andcolumns in a similar way as the classification of finite Cartan matrices in Subsection 1.2.3 usingDynkin diagrams.
In Section 2.1 we saw that symmetrizable generalized Cartan matrices bring a lot of structurewith them on their corresponding Kac-Moody algebras. It turns out that affine Cartan matrices liein this rich class of matrices.
Proposition 2.2.9. Let A be an affine Cartan matrix, then A is symmetrizable.
Proof. This follows from Lemma 4.6 of [8].
Write A for the equivalence class of the affine Cartan matrix A under the equivalence relation ofsimultaneous permutations of rows and columns of matrices. Further, letCa denote the collectionof affine Cartan matrices up to simultaneous permutations of rows and columns. Next, considerthe set ÓLa of all Lie algebras g that are isomorphic to an affine Lie algebra g(A). Write g for the
22
isomorphism class of the g in ÓLa , and put La for the collection of isomorphism classes of ÓLa .Using Proposition 2.2.9 we deduce from Corollary 2.1.15 the following canonical bijection.
Theorem 2.2.10. The map g :Ca →La defined by A 7→ g(A) := g(A) is a bijection.
Remark 2.2.2. For completeness reasons we included all Lie algebras isomorphic to affine Lie al-gebras in the definition ofLa . In the remaining of this thesis however we want to work only withaffine Lie algebras g(A) (apart from Subsection 2.3.5), and leave arbitrary isomorphic Lie algebrasout of the picture to simplify things. Therefore we will summarize here how the most importantstructures of g(A) are inherited by an isomorphic Lie algebra g.
Let g∈ g(A) for an affine Cartan matrix A, then there exists an isomorphismφ : g(A)→ g. Now g
inherits some structures of g(A) throughφ. Assume g(A) = g(A,h,Π,Π∨), then (φ(h),φ∗−1(Π),φ(Π∨))is a realization of A with φ(h) ⊂ g a commuting subalgebra that leads to a root space decom-position of g. The corresponding root system of g is then given by φ∗−1(∆) ⊂ φ(h)∗. Further-more, each α ∈ φ∗−1(∆) can be expressed as a linear combination of elements in φ∗−1(Π) withcoefficients all positive or all negative integers. If (., .) is a standard invariant form on g(A), then(x , y ) := (φ−1(x ),φ−1(y )) for all x , y ∈ g defines an invariant nondegenerate symmetric bilinearform on g such that (α,α) > 0 for all α ∈ φ∗−1(Π). Finally, we can define the Weyl group of g as{φ∗−1 ◦w ◦φ∗ : w ∈W } and the real roots asφ∗−1(∆r e )with respect to g(A) andφ.
In (i) of Example 3.2.3 we will consider ∆r e as an ’affine root system’. In a similar fashion onecan show that φ∗−1(∆r e ) is an affine root system. Then analogous to Subsection 3.5.4 it turns outthatφ∗−1(∆r e ) as an affine root system does not depend on the isomorphismφ up to an appropri-ate equivalence relation called ’similarity’. Furthermore, ∆r e and φ∗−1(∆r e ) are similar affine rootsystems (compare with (3.5.6)).
Now let us classify the affine Cartan matrices up to simultaneous permutations of rows andcolumns. Let A be an affine Cartan matrix, then the Dynkin diagram S(A) still makes sense (seeSubsection 1.2.3). To distinguish between Dynkin diagrams of the different generalized Cartanmatrices we will call a Dynkin diagram S(A) finite (resp. affine) if A is a finite (resp. affine) Cartanmatrix. It turns out that a i j a j i ≤ 4 holds for any affine Cartan matrix A = (a i j )1≤i ,j≤n , so there isno ambiguity about the factorization of the number of edges between node i and node j of S(A)to obtain a i j a j i again. Thus given an affine Dynkin diagram D one can reconstruct the associatedaffine Cartan matrix A = (a i j )i ,j∈I up to a permutation of the index set I of A.
To classify the affine Cartan matrices up to simultaneous permutation of rows and columns, itsuffices to classify all possible affine Dynkin diagrams. This has been done in [8], and we state theresult below.
Theorem 2.2.11. (i) All possible affine Dynkin diagrams are listed in Figure 2.1 (where the labels atthe nodes are not part of the definition of a Dynkin diagram).
(ii) Let D be a Dynkin diagram with all the labels as depicted in Figure 2.1. Then there exists anaffine Cartan matrix A such that the following two conditions hold
(1) D =S(A)(2) the numerical label a i of the node αi of the Dynkin diagram D in Figure 2.1 is the i -th
coordinate of the unique column vector δ > 0 such that Aδ = 0 where the a i are positive relativelyprime integers.
Proof. See Theorem 4.8 in [8].
23
Au1 = A t
1 :1
α0
1
α1
⇐⇒
Aul = A t
l (l ≥ 2) :1
α1
1
α2
. . .1
αl−1
1
αl
1α0
B ul (l ≥ 3) :
1
α1 α2
2 2
α3
. . .2
αl−1
2
αl
α0
1
⇒
B tl (l ≥ 2) :
1
α0
1
α1
. . .1
αl−1
1
αl
⇐ ⇒
C ul (l ≥ 2) :
1
α0
2
α1
. . .2
αl−1
1
αl⇒ ⇐
C tl (l ≥ 3) :
1
α1 α2
2 2
α3
. . .2
αl−1
1
αl
⇐
α0
1
Dul =D t
l (l ≥ 4) :1
α1 α2
2 2
α3
. . .αl−2
2 1
αl−1
α0
1αl
1
E u6 = E t
6 :1
α1
2
α2 α3
3 2
α4
1
α5
α6
2α0
1
E u7 = E t
7 :1
α0
2
α1
3
α2 α3
4 3
α4
2
α5
1
α6
α7
2
E u8 = E t
8 :1
α0
2
α1
3
α2
4
α3
5
α4 α6
6 4
α6
2
α7
α8
3
24
F u4 :
1
α0
2
α1
3
α2
4
α3
2
α4
⇒
F t4 :
1
α0
2
α1
3
α2
2
α3
1
α4
⇐
G u2 :
1
α0
2
α1
3
α2Ö
G t2 :
1
α0
2
α1
1
α2×
BC m1 :
2
α0
1
α1
BC ml (l ≥ 2) :
2
α0
2
α1
. . .2
αl−1
1
αl
⇐ ⇐
Figure 2.1: All possible affine Dynkin diagrams.
The left column of Figure 2.1 contains the name Xjl of each affine Dynkin diagram S(A) in the
right column where l is the rank of A and Xjl is called the type of A and S(A). The naming of types
of affine Dynkin diagrams as listed in Figure 2.1 differs substantially from [8]. This is because Kacclassifies the affine Dynkin diagrams according to the construction of their corresponding Kac-Moody algebra, while we will do this according to their corresponding affine root system. Thelatter will be explained in detail in Section 3.5. For now remember that in our naming it is possiblefor an affine Dynkin diagrams to be of two types at the same time. For example, in Figure 2.1 thereis a Dynkin diagram that is of both type Au
2 and type A t2, although type G u
2 and G t2 correspond to
two different affine Dynkin diagrams.For later purposes we have fixed an enumeration of the nodes of the affine Dynkin diagrams in
Figure 2.1 as follows. In Figure 2.1 the nodes of S(A) are enumerated by α0, . . . ,αl for a rank l affineCartan matrix A. If S(A) is of type X
jl where j ∈ u , t and X ∈ {A, . . . ,G }, then the Dynkin subdiagram
with nodes α1, . . . ,αl is the finite Dynkin diagram of the type X l . If j = m , then Xjl = BC m
l andthe Dynkin subdiagram with nodes α1, . . . ,αl is the finite Dynkin diagram of the type C l whichrepresents the finite root system of unmultipliable roots of BC l .
2.3 Affine Lie algebras, the set of real roots and affine root systems
In the previous Section we distinguished two important classes of symmetrizable generalized Car-tan matrices, namely finite and affine Cartan matrices. We saw that Kac-Moody algebras corre-sponding to finite Cartan matrices are just simple finite-dimensional Lie algebras which we treatedand classified in Chapter 1. Further, we ended the previous Section with a classification of affineLie algebras by classifying all affine Cartan matrices up to simultaneous permutation of rows andcolumns. In this Section we want to discuss affine Lie algebras in more detail. First we will explicitlychoose a standard invariant form on a specifically chosen affine Lie algebra within its isomorphism
25
class. Then it will turn out that affine Lie algebras contain a simple Lie algebra as subalgebra andin that sense are infinite-dimensional generalizations of simple Lie algebras. Using our knowledgeof simple Lie algebras we will describe the real roots of an affine Lie algebra explicitly, and studyits Weyl group. This will lead us to the notion of an affine root system which we will discuss in fulldetail in the next Chapter. As an example we end this Chapter with an explicit construction of anuntwisted affine Lie algebra.
2.3.1 The normalized invariant form
In this Subsection we will fix a standard invariant form for an affine Lie algebra g(A).Let A = (a i j )0≤i j≤l be a rank l affine Cartan matrix, and consider its Dynkin diagram S(A) as
depicted in Figure 2.1. Assume that the index set of A is ordered such that A satisfies (ii) of Theorem2.2.11. Put a i for the numerical label of the node with label αi of S(A) in Figure 2.1 for i = 0, . . . , l .Notice that the transposed AT of A is also a rank l affine Cartan matrix. Then S(A) coincides withS(AT ), except that the directions of possible arrows of S(AT ) are reversed in comparison with S(A).Consider the Dynkin diagram S(AT ) in Figure 2.1 with the same enumeration of nodes as S(A). Puta∨i for the numerical label of the node of S(AT ) with label αi for i = 0, . . . , l . From Figure 2.1 itfollows that
a∨0 = 1.
The matrix A is symmetrizable by Proposition 2.2.9, hence there exist ε0, . . . ,εl ∈ C and a sym-metrization B such that A = diag(ε0, . . . ,εl )B . Moreover, one can show that there exists a sym-metrization B such that
A = diag(a 0a∨0−1, . . . , a l a∨l
−1)B.
Consider the affine Lie algebra g := g(A,h,Π,Π∨) associated to A with Π := {α0, . . . ,αl } and Π∨ :={α∨0 , . . . ,α∨l }. Fix an element d ∈ h such that ⟨αi , d ⟩=δ0i for i = 0, . . . , l , then {α∨0 , . . . ,α∨l , d } is a basisof h. By Theorem 2.1.9 there exists an invariant nondegenerate symmetric bilinear C-valued form(., .) on g such that (., .) is nondegenerate on h and uniquely defined by
(α∨i ,α∨j ) = ⟨αi ,α∨j ⟩εi = a j a∨j−1a j i (i , j = 0, . . . , l );
(α∨i , d ) = 0 (i = 1, . . . , l );
(α∨0 , d ) = a 0;
(d , d ) = 0.
Remark 2.1.1 shows that indeed (α∨i ,α∨j ) = (α∨j ,α∨i ) = a i a∨i
−1a i j for 0≤ i , j ≤ l .
Consider the vector space isomorphism ν : h→ h∗ induced by (., .), and put Λ0 := a−10 ν (d ). Then
⟨Λ0,α∨i ⟩=δ0i for i = 0, . . . , l and ⟨Λ0, d ⟩= 0, and {α0, . . . ,αl ,Λ0} is a basis of h∗. The induced bilinearform (., .) on h∗ is given by
(αi ,αj ) = a∨i a−1i a i j (i , j = 0, . . . , l );
(αi ,Λ0) = 0 (i = 1, . . . , l );
(α0,Λ0) = a−10 ;
(Λ0,Λ0) = 0.
(2.3.1)
Put δ=∑l
i=0 a iαi , then we obtain from (ii) of Theorem 2.2.11 and (2.3.1)
(αi ,δ) = 0 (i = 0, . . . , l ), (δ,δ) = 0, (Λ0,δ) = 1. (2.3.2)
26
Since a i and a∨i−1 are positive integers (see Figure 2.1), and a i i = 2 we observe from (2.3.1) that
(αi ,αi ) > 0 for i = 0, . . . , l . Thus (., .) is a standard invariant form on g, and this specific choice of(., .) is called the normalized invariant form on g. It is normalized in the sense that we fixed onespecific standard invariant form, while others coincide on
⊕li=0Cαi up to multiplication with a
complex number by (2.1.9). Also notice that restricted to⊕l
i=0Rαi the normalized invariant formis R-valued.
2.3.2 The real roots of an affine Lie algebra
As we will see in this Subsection the affine Lie algebra g that was defined in the previous Subsection
contains a simple Lie algebra◦g as subalgebra, and the root system ∆ of g contains the finite root
system◦∆ of
◦g. We will explicitly describe the set of real roots∆r e of g in terms of δ and
◦∆. Further,
we introduce an element θ related to◦∆ to describe the root basis Π in terms of
◦∆.
First, let us introduce some notation. Denote by◦h (resp.
◦hR) the linear span over C (resp. R) of
α∨1 , . . . ,α∨l . Further denote by◦h∗ (resp.
◦h∗R) the linear span over C (resp. R) of α1, . . . ,αl . Then we
have the direct sum of subspaces
h∗ =◦h∗⊕Cδ⊕CΛ0.
By (2.3.1), (2.3.2), (2.1.5) and (2.1.6) one observes that δ and Λ0 vanish on◦h. Hence the set
◦h∗ ⊂ h∗
(resp.◦h∗R ⊂ h∗) can actually be identified with the dual of
◦h (resp.
◦hR) by restriction to
◦h (resp.
◦hR).
Consider the matrix◦
A = (a i j )1≤i j≤l which is obtained from A by removing the row and column
with index 0. By Corollary 2.2.5 this is a finite Cartan matrix. Let◦g denote the subalgebra of the
affine Lie algebra g generated by e1, . . . , e l , f 1, . . . , f l , then by Theorem 1.3.1◦g is a simple Lie algebra
corresponding to the finite Cartan matrix◦
A. Now◦g∩h=
◦h and (
◦h,◦Π,◦Π∨) := (
◦h,{α1, . . . ,αl },{α∨1 , . . . ,α∨l })
is a realization of the matrix◦
A, so we observe that◦g = g(
◦A,◦h,◦Π,◦Π∨). Notice that the finite Dynkin
diagram S(◦
A) is obtained from S(A) by removing the node α0 and the vertices attached to it.
Put◦
Q := spanZ◦Π, then the finite set
◦∆ := {α ∈ ∆ : α ∈
◦Q} is the root system of
◦g. Hence by
Corollary 2.2.7◦∆ is a finite root system with
◦Π a basis for
◦∆. Further, put
◦∆l (resp.
◦∆s ) for the long
(resp. short) roots of◦∆with respect to (., .), and let
◦W be the finite Weyl group of
◦∆.
Consider the real roots ∆r e of g= g(A) and its corresponding Dynkin diagram S(A) as depictedin Figure 2.1. Then S(A) is of type X
jl with j ∈ {u , t , m }, X ∈ {A, . . . ,G }, and l ∈N. If j = u we will say
that∆r e is of untwisted type, if j = t we will say that∆r e is of twisted type and if j =m then we willsay that∆r e is of mixed type. Notice that it is possible for∆r e to be of untwisted and twisted type.Using this terminology we want to describe the real and imaginary roots of the affine Lie algebra g
explicitly. The set of imaginary roots ∆i m turns out to be generated by δ which, together with the
root system◦∆ of the simple Lie algebra
◦g, will be used to describe∆r e .
Proposition 2.3.1. (i)∆i m = {nδ : n ∈Z};(ii)∆r e = {α+nδ :α∈
◦∆, n ∈Z} if∆r e is of untwisted type;
(iii)∆r e = {α+n (α,α)2 δ :α∈
◦∆, n ∈Z} if∆r e is of twisted type but not of untwisted type;
(iv)∆r e = {α+n (α,α)2 δ : α ∈
◦∆, n ∈Z} ∪ { 2
(α,α)α+(2n +1) 2(α,α)δ) : α ∈
◦∆l , n ∈Z} if∆r e is of mixed
type.
27
Proof. This follows from Theorem 5.6 and Proposition 6.3 of [8].
Remark 2.3.1. One can compute explicitly the squared root length (α,α) for eachα∈∆r e as follows.Choose w ∈ W such that w (α) = αi for some αi ∈ Π. Then by Proposition 2.1.16 (ii) we have(α,α) = (αi ,αi ). Now the first equation of (2.3.1) we obtain
(αi ,αi ) = a∨i a−1i a i i = 2a∨i a−1
i .
For ∆r e that is only of untwisted type put r = 1, for ∆r e of twisted type (and possibly untwisted)but not type G t
2 put r = 2 and for∆r e of type G t2 put r = 3. Checking all possibilities for a i and a∨i
in Figure 2.1 we observe the following squared root lengths occurring in∆r e .
∆r e type squared root lengths (α0,α0)untwisted 2
r , 2 2twisted, not untwisted type 2, 2r 2
mixed
(
1, 4 (l = 1)
1, 2, 4 (l > 1)1
Recall that if ∆r e is of mixed type then◦∆ is a finite root system of type C l . It follows that if α ∈
◦∆l
(resp. β ∈◦∆s ), then α∈∆r e and (α,α) = 4 (resp. (β ,β ) = 2). Then for all α∈
◦∆l we have 2
(α,α)α=12α
and for all α ∈◦∆s we have 2
(α,α)α = α. In other words,◦∆∪ 1
2
◦∆l is a nonreduced irreducible finite
root system with◦∆ the unmultipliable roots and 1
2
◦∆l the indivisible short roots.
Introduce the following element
θ :=δ−a 0α0 =l∑
i=1
a iαi ∈◦
Q , (2.3.3)
then by (2.3.1) and (2.3.2)(θ , θ ) = 2a 0. (2.3.4)
Proposition 2.3.2. (i) Let∆r e of untwisted type, and letφ be the highest root of◦∆with respect to the
basis◦Π= {α1, . . . ,αl }. Then θ =φ and Π= {α0,α1, . . . ,αl }with α0 =δ−φ.
(ii) Let∆r e of twisted but not untwisted type, and let θ be the highest short root of◦∆with respect
to the basis◦Π= {α1, . . . ,αl }. Then θ = θ and Π= {α0,α1, . . . ,αl }with α0 =δ−θ .
(iii) Let ∆r e of mixed type, and let φ be the highest root of◦∆ with respect to the basis
◦Π =
{α1, . . . ,αl }. Then θ =φ and Π= {α0,α1, . . . ,αl }with α0 = 12 (δ−φ).
Proof. This follows from Proposition 6.4 of [8].
2.3.3 The Weyl group of an affine Lie algebra
In this Subsection we study the Weyl group W of g in more detail. It turns out that W is the semidi-
rect product of the finite Weyl group◦
W of◦∆ and a translation group over a lattice. Furthermore,
we can interpret the action of W on certain real subspaces of h∗ modulo δ as an affine Weyl groupaction.
28
First, notice that (δ,αi ) = 0 for i = 0, . . . , l by (2.3.2). Since W is generated by the fundamentalreflections ri for i = 0, . . . , l , we observe from (2.1.13) that w (δ) =δ for all w ∈W . Now let W ′ ⊂Wbe the subgroup generated by the fundamental reflections ri for i = 1, . . . , l . Then w (Λ0) = Λ0 for
all w ∈W ′, hence W ′ acts trivially on Cδ+CΛ0. Also, W ′ leaves◦h∗ =
⊕li=1Cαi invariant and W ′
acts faithfully on◦h∗ since W acts faithfully on h∗ ((i) of Proposition 2.1.16). Hence we can identify
W ′ with◦
W as groups, and we will write◦
W ⊂W .After a direct (but perhaps tedious) computation using (2.3.3) and (2.3.4) one can observe that
for λ∈ h∗ we have
r0rθ (λ) =λ+(λ,δ)a−10 θ − ((λ, a−1
0 θ )+1
2(a−1
0 θ , a−10 θ )(λ,δ))δ. (2.3.5)
In general it is interesting to consider for α∈◦h∗ the linear endomorphism tα of h∗ defined by
tα(λ) =λ+(λ,δ)α− ((λ,α)+1
2(α,α)(λ,δ))δ. (2.3.6)
Then (Λ0,δ) = 1 and (Λ0,◦h∗) = 0 imply
tα(Λ0) = Λ0+α−1
2(α,α)δ (2.3.7)
and for λ∈ h∗ such that (λ,δ) = 0 we have
tα(λ) =λ− (λ,α)δ. (2.3.8)
This describes tα completely on h∗ by (2.3.2). It follows now from a straightforward calculation that
tα ◦ tβ = tα+β ,
andw ◦ tα ◦w−1 = tw (α)
for w ∈◦
W , and from (2.3.5) and (2.3.6) we have
r0 = ta−10 θ
rθ
for the fundamental reflection r0. These identities inspire the idea that the Weyl group W is the
semidirect product of the finite Weyl group◦
W and a lattice that is generated by the◦
W -orbit of
a−10 θ . It turns out that this lattice can also be described in terms of the coroot lattice
◦Q∨ or the root
lattice◦
Q of◦g.
Proposition 2.3.3. W =◦
W nT with T = {tα :α∈M } an abelian group and
M =Z(◦
W (a−10 θ )) =
ν (◦
Q∨) if∆r e is of untwisted type,◦
Q if∆r e is of twisted or mixed type,
a lattice that spans◦h∗R. In particular, ν (
◦Q∨) =
◦Q if∆r e is of both untwisted and twisted type.
29
Proof. See Proposition 6.5 of [8].
Let us introduce some extra notation so that we can describe the action of W on real subspaces
of g∗. Recall that◦h∗R =
⊕li=1Rαi and put
h∗R :=◦h∗R⊕Rδ⊕RΛ0,
then ∆ ⊂ h∗R and◦∆ ⊂
◦h∗R since ∆ ⊂ Q =
∑li=0Zαi . Notice that the restriction of the normalized
invariant form (., .) on◦h∗R (resp.
◦h∗R ⊕Rδ) is R-valued and even positive definite (resp. positive
semidefinite and vanishing on Rδ) by (2.1.10) and (2.3.2).It turns out that W acts on certain hyperplanes of h∗R relative to δ in a very special way. For
s ∈ {0, 1} consider the following subsets of h∗R
h∗s := {λ∈ h∗R : (λ,δ) = s }.
Then h∗0 =◦h∗R⊕Rδ=
∑li=0Rαi , h∗1 = h∗0+Λ0 and both h∗0 and h∗1 are W -invariant. By (i) of Proposition
2.1.16 we also observe that W acts faithfully on h∗0.Consider the R-vector space h∗R and its subspace Rδ, then the relation x ∼ y if and only if
x − y ∈ Rδ on h∗R yields the quotient space h∗R/Rδ. Next, consider the subset h∗1 =◦h∗R ⊕Rδ+Λ0
of h∗R. Then for x , y ∈ h∗1 there exist unique µ,ν ∈◦h∗R and ξ,ξ′ ∈ R such that x = µ+ξδ+Λ0 and
y = ν+ξ′δ+Λ0. Hence the equivalence relation∼ restricted to h∗1 becomesµ+ξδ+Λ0 ∼ ν+ξ′δ+Λ0
if and only if µ= ν . Write µ+Λ0 for the equivalence class of µ+ξδ+Λ0 ∈ h∗1, and put h∗1/Rδ for theset of equivalence classes of h∗1 under ∼.
Consider W =◦
W nT with T = {tα : α ∈M } as in Proposition 2.3.3, then we obtain an induced
action of W on h∗1/Rδ as follows. Since◦
W fixes δ and Λ0 we get
w (µ+Λ0) =w (µ)+Λ0 (2.3.9)
for w ∈◦
W , and◦
W acts faithfully on h∗1/Rδ since◦
W acts faithfully on◦h∗ . Further, (2.3.7) and (2.3.8)
implytα(µ+Λ0) =µ+α+Λ0 (2.3.10)
for α ∈M ⊂◦h∗R, and it immediately follows that T acts faithfully on h∗1/Rδ. Thus we conclude by
Proposition 2.3.3 that W acts faithfully on h∗1/Rδ.
Notice that◦h∗R considered as an abelian group acts faithfully and transitively on h∗1/Rδ by
◦h∗R×
h∗1/Rδ→ h∗1/Rδ, (µ,ν +Λ0) 7→µ+ν +Λ0. In the language of Chapter 3 we call h∗1/Rδ an affine space
with space of translations◦h∗R. Further, we can identify h∗1/Rδ with
◦h∗R by the projection π(ν +Λ0) =
ν . This gives a bijective correspondence π : h∗1/Rδ→◦h∗R. Forcing the vector space structure of
◦h∗R
onto h∗1/Rδ throughπwe can view h∗1/Rδ as a vector space with origin Λ0. Since we identify h∗1/Rδ
with◦h∗R by the means of projection it is clear that this induces a symmetric bilinear form on
◦h∗R that
is just the restriction of (., .) onto◦h∗R. This form on
◦h∗R is positive definite as we noted earlier.
The action of W on h∗1/Rδ translates to a (nonlinear) action of W on◦h∗R which we denote by ’af’
through the group isomorphism af : W → af(W ) defined by af(w ) =π ◦w ◦π−1. This leads to
af(w )(µ) =w (µ)
30
for w ∈◦
W and µ∈◦h∗R by (2.3.9), and
af(tα)(µ) =µ+α
for α ∈M and µ ∈◦h∗R by (2.3.10). Hence af(T ) acts as a group on
◦h∗R by translation over the lattice
M ⊂◦h∗R. This shows that if∆r e is of untwisted type, then af(W ) =
◦W naf(T ) is the affine Weyl group
of the finite root system◦∆ in the language of [2] and [7] which explains the word ’affine’ in our
terminology.
2.3.4 The set of real roots as an affine root system
In this Subsection we will compare some important properties of◦∆ and ∆r e . We want to observe
that∆r e satisfies axioms analogous to the axioms of a reduced irreducible finite root system. Thiswill give rise to a new root system called an affine root system which we will rigorously introducein Chapter 3.
A root β ∈◦∆ defines a linear functional on
◦h∗R by the mapping µ 7→ (β ,µ). Further rβ is a re-
flection of◦h∗R in the kernel this linear functional which is the hyperplane orthogonal to β . As we
will see in Example 3.2.3, we can interpret ∆r e as a set of so called ’affine linear functions’ on the
’affine Euclidean space’◦h∗R. This means that real roots are linear functionals on
◦h∗R composed with
a translation of◦h∗R, and that
◦h∗R is a vector space where we have ’forgotten the origin’. Then af(rα+nδ)
for α+nδ ∈∆r e can be thought of as the reflection of◦h∗R in the kernel of α+nδ. In the following
calculations we want to show that this generalizes the action of rβ for β ∈◦∆ on
◦h∗R.
Let α ∈◦h∗R and n ∈ R such that α+nδ ∈∆r e and let λ ∈ h∗1, then by (2.1.13), (2.3.1) and (2.3.2)
we have
rα+nδ(λ) =λ−2(λ,α+nδ)
(α+nδ,α+nδ)(α+nδ)
=λ−2(λ,α)+n
(α,α)(α+nδ)
Since λ∈ h∗1 there exist µ∈◦h∗R and ξ∈R such that λ=µ+ξδ+Λ0. So on the class of λ in h∗1/Rδ we
have
rα+nδ(λ) := rα+nδ(λ) =µ−2(µ,α)+n
(α,α)α+Λ0.
This shows that
af(rα+nδ)(µ) =µ−2(µ,α)+n
(α,α)α (2.3.11)
defines a reflection of◦h∗R in the translated hyperplane {µ ∈
◦h∗R : (µ,α) = −n}. In other words, the
action of af(rα+nδ) on◦h∗R coincides with a translation of the action of rα on
◦h∗R that is uniquely
defined by n and α.Although ∆r e is an infinite set, which one directly observes from Proposition 2.3.1, we will see
in the following Theorem that ∆r e satisfies some axioms that are analogous to the axioms of areduced irreducible finite root system.
31
Theorem 2.3.4. (i)∆r e spans h∗0 =◦h∗R⊕Rδ, and (α,α)> 0 for all α∈∆r e ;
(ii) rα(β )∈∆r e for all α,β ∈∆r e ;(iii) 2 (β ,α)
(α,α) ∈Z for all α,β ∈∆r e ;
(iv) af(W ) considered as a discrete group acts properly on◦h∗R;
(v) Let p : h∗0 =◦h∗R⊕Rδ→
◦h∗R be the projection along the direct sum. For each α ∈ ∆r e the fiber
p (α)−1 contains at least 2 distinct elements;(vi) Let α∈∆r e , then Rα∩∆r e = {±α};(vii) There do not exist nonempty subsets∆1,∆2 ⊂∆r e such that∆1q∆2 =∆r e and (α,β ) = 0 for
all α∈∆1, β ∈∆2.
Proof. (i) Trivially,Π= {α0, . . . ,αl } spans Q =∑l
i=0Zαi overZ. FurtherΠ⊂∆r e ⊂Q , thus∆r e spans
h∗0 =∑l
i=0Rαi overR. Further by Lemma 2.1.17 (or Proposition 2.3.1) we observe that (α,α)> 0 forall α∈∆r e .
(ii) Let α ∈ ∆r e , then rα ∈W by (2.1.14). So for β ∈ ∆r e we have rα(β ) ∈ ∆r e since ∆r e is W -invariant.
(iii) Let α,β ∈ ∆r e , then there exists w ∈ W and αi ∈ Π such that w (α) = αi . Hence by (ii) ofProposition 2.1.16 we have
2(β ,α)(α,α)
= 2(w (β ),αi )(αi ,αi )
.
Since w (β )∈∆⊂Q there exist c0, . . . , c l ∈Z such that w (β ) =∑l
j=0 c jαj , hence
2(w (β ),αi )(αi ,αi )
=l∑
j=0
c j ·2(αj ,αi )(αi ,αi )
But 2(αj ,αi )(αi ,αi )
= a i j ∈Z by (2.1.11), so clearly 2 (β ,α)(α,α) ∈Z.
(iv) We have af(W ) =◦
W n af(T ) by Proposition 2.3.3 and the definition of af. Here◦
W is a finite
group by Proposition 2.2.6, and af(T ) considered as a discrete group acts properly on◦h∗
R since the
lattice M is a discrete set in◦h∗R. This implies that af(W ) considered as a discrete group acts properly
on◦h∗R.(v) This follows directly from the explicit descriptions of∆r e in Proposition 2.3.1.(vi) First, the root basis Π is contained in ∆r e by definition of ∆r e . Second, ri (αi ) = −αi ∈ ∆r e
for i = 0, . . . , l since∆r e is W -invariant. Then by (ii) of Theorem 2.1.5 we observe that Rαi ∩∆r e ={±αi } for i = 0, . . . , l . Now let α be any real root and assume that cα ∈ ∆r e for some c ∈ R, thenc 6= 0 and there exist w ∈W and αi ∈ Π such that w (cα) = αi . Hence by linearity of w on h∗ weobserve that w (α) = c−1αi ∈∆r e . Thus we observe that c =±1 and Rα∩∆r e = {±α}.
(vii) This follows pretty straightforwardly from the fact that A = (a i j )0≤i ,j≤l is indecomposable:Assume that there exist nonempty subsets∆1,∆2 ⊂∆r e such that∆1q∆2 =∆r e and (α,β ) = 0 forall α ∈∆1, β ∈∆2. Also assume that Π ⊂∆1. Let β ∈∆2 then there exists αi ∈ Π and w ∈W suchthat w (β ) = αi . Since w is a composition of finitely many fundamental reflections, there exists
γ ∈∆2 and αj ∈ Π such that rj (γ) = γ− 2(γ,αj )(αj ,αj )
αj ∈∆1. But (γ,αj ) = 0 since γ ∈∆2 and αj ∈∆1, so
rj (γ) = γ ∈∆2. This is a contradiction with the fact that rj (γ) ∈∆1. Hence we obtain that Π *∆1,and similarly Π*∆2.
Recall that by (2.1.11) we have a i j = 2(αi ,αj )(αi ,αi )
. Now choose αi ∈ ∆1 ∩Π and αj ∈ ∆2 ∩Π, then(αi ,αj ) = 0 so also a i j = 0. This shows that A is decomposable which contradicts the assumption
32
that it is not. Hence there do not exist nonempty subsets∆1,∆2 ⊂∆r e such that∆1q∆2 =∆r e and(α,β ) = 0 for all α∈∆1, β ∈∆2.
Let us briefly analyze the statements of Theorem 2.3.4 with regards to the definition of a reducedirreducible finite root system. Statement (i) says that ∆r e spans h∗0, and it implies that ∆r e doesnot contain 0 or any element of h∗0 with vanishing squared length with respect to (., .). Although∆r e is an infinite set, statement (iv) makes sure that ∆r e is not too ’large’. Now (ii) and (iii) arestraightforward generalizations of (2) and (3) of Definition 1.2.3. Notice that (2.1.9) implies thatthe integer 2 (β ,α)
(α,α) does not depend on the choice of (., .). Further, (v) is an extra axiom of whichits utility will be explained in Remark 3.2.3. Finally, (vi) captures the ’reducedness’ and (vii) the’irreducibility’ of∆r e .
In the language of Chapter 3, Theorem 2.3.4 asserts that∆r e is an reduced irreducible affine root
system on the affine Euclidean space◦h∗R together with the form (., .). In that Chapter we will define
these notions rigorously, and analyze them in a general setting. Our main goal will be to showthat reduced irreducible affine root systems are in bijective correspondence with affine Cartanmatrices. Together with the results of this Chapter, this will give us a commutative triangle like(1.3.1) but then for affine Lie algebras, affine root systems and affine Cartan matrices.
2.3.5 An explicit construction of untwisted affine Lie algebras
Kac realizes a complete set of representatives for the isomorphism classes of affine Lie algebras(see [8]). He distinguishes between untwisted and twisted affine Lie algebras. This terminologycoincides with our terminology of untwisted and twisted type sets of real roots in the sense thateach (un)twisted set of real roots corresponds to an (un)twisted affine Lie algebra. What we callmixed type, Kac also considers to be of twisted type. To give an example of Lie algebras isomorphicto affine Lie algebras we will now briefly show how to construct the untwisted affine Lie algebra
L (◦g) associated to the simple Lie algebra◦g. Here we still consider
◦g as subalgebra of the affine Lie
algebra g= g(A,h,Π,Π∨) with all the associated notation as introduced earlier in this Section suchthat∆r e is untwisted.
Let L := C[t , t −1] be the algebra of Laurent polynomials P =∑
k∈Z ck t k (where only finitelymany ck ∈C are nonzero). Consider the loop algebra
L (◦g) :=L ⊗C◦g
which is an infinite-dimensional Lie algebra with bracket [., .]0 defined by
[t m ⊗x , t n ⊗ y ]0 := t m+n ⊗ [x , y ]
for all m , n ∈Z and x , y ∈ ◦g. Introduce an element K , and consider the vector space
L (◦g) :=L (◦g)⊕CK .
Defining for all m , n ∈Z, x , y ∈ ◦g and λ,µ∈C the bracket
[t m ⊗x +λK , t n ⊗ y +µK ]1 := [t m ⊗x , t n ⊗ y ]0+mδm ,−n (x , y )K
turns L (◦g) into a Lie algebra with center CK . Finally, introduce the element d and consider
L (◦g) := L (◦g)⊕Cd
33
with bracket
[(t m ⊗x )+λK +µd , (t n ⊗y )+λ1K +µ1d ]2 := [t m ⊗x +λK , t n ⊗y +λ1K ]1+µnt n ⊗y −µ1m t m ⊗x
for all m , n ∈Z, x , y ∈ ◦g and λ,λ1,µ,µ1 ∈C. Then L (◦g) is a Lie algebra with center CK such that dacts onL as t d
d t in the bracket [., .]2.
We can identify◦g with the subalgebra 1⊗ ◦g⊂ L (◦g) by the map x 7→ 1⊗x . Furthermore,
h′ :=◦h⊕CK ⊕Cd
is an (l + 2)-dimensional commutative subalgebra of L (◦g). Let δ′ ∈ h′∗ such that δ′|◦h⊕CK
= 0,
⟨δ′, d ⟩ = 1. Further, extend λ ∈◦h∗ to a linear function on h′ by putting ⟨λ, K ⟩ = ⟨λ, d ⟩ = 0. In this
way we can identify◦h∗ with a subspace of h′∗.
Put
∆′ := {jδ′+γ : j ∈Z,γ∈◦∆}∪ {jδ′ : j ∈Z} ⊂ h′∗,
then L (◦g) has a root space decomposition with respect to h′ namely
L (◦g) = h′⊕
⊕
α∈∆′L (◦g)α
!
whereL (◦g)jδ′+γ = t j ⊗ ◦gγ andL (◦g)jδ′ = t j ⊗◦h for all j ∈Z and γ∈
◦∆.
Finally, letφ denote the highest root of◦∆with respect to the basis
◦Π= {α1, . . . ,αl }. Put
Π′ := {α′0 :=δ′−φ,α′1 :=α1, . . . ,α′l :=αl }
and
Π′∨ := {α′0∨ :=
2
(φ,φ)K −1⊗φ∨,α′1
∨ := 1⊗α∨1 , . . . ,α′l∨ := 1⊗α∨l }.
Using Proposition 2.1.6 one can now show that there exists an isomorphismφ : g(A,h′,Π′,Π′∨)→L (◦g) such that φ(h′) = h′, φ(Π′∨) = Π′∨ and φ∗(Π′) = Π′. By Theorem 2.1.7 there exists a realizationpreserving isomorphism between g(A,h′,Π′,Π′∨) and g(A,h,Π,Π∨). Hence there exists an isomor-
phism ψ : g(A,h,Π,Π∨) → L (◦g) such that φ(h) = h′, ψ(Π∨) = Π′∨ and ψ∗(Π′) = Π. Using Remark
2.2.2 and Proposition 2.3.1 we observe that we can consider {jδ′+γ : j ∈ Z,γ ∈◦∆}=ψ∗−1(∆r e ) as
the real roots of L (◦g)with respect to g(A,h,Π,Π∨) andψ.
34
Chapter 3
Affine Root Systems
In this chapter we will exhibit the abstract theory of affine root systems based on the axioms wefound in Theorem 2.3.4 for ∆r e . Similarly to the abstract theory of finite root systems, no knowl-edge of Lie algebras is required. Although affine root systems have already been studied some timeago (see [10]), this thesis contains a slightly adapted and more detailed version of the theory ob-tained from these first explorations. First we describe the landscape of affine root systems, namelyaffine linear functions on affine Euclidean space and reflections of affine Euclidean space. Thenwe define affine root systems and notions as their duals, affine root subsystems, irreducibility, re-ducedness and similarity. We will use the geometry of affine root systems to define a special set ofgenerators called the basis of an affine root system, which differs substantially from the definitionof a basis of a finite root system. We classify all reduced irreducible affine root systems up to simi-larity by obtaining a bijective correspondence between reduced irreducible affine root systems upto similarity and affine Cartan matrices up to simultaneous permutations of rows and columns.This leads to an explicit construction of a reduced irreducible affine root system for each similarityclass, which allows us to explain the naming of the classification of affine Dynkin diagrams. We fin-ish the Chapter by relating the obtained correspondence to the bijective correspondence betweenaffine Cartan matrices up to simultaneous permutations of rows and columns and isomorphismclasses of Lie algebras that are isomorphic to affine Lie algebras to finally obtain an analogue of(1.3.1).
3.1 Affine linear functions and orthogonal reflections in affine Euclideanspace
In this section we define Euclidean spaces without a specified origin, so called affine Euclideanspaces. Natural maps between such spaces arise by composing linear maps with translations.Maps that have special interest to us are functions to the reals R, and so called orthogonal re-flections in affine hyperplanes which generalize orthogonal reflections in hyperplanes in vectorspaces. This Section serves to set up all the formalities to define and study affine root systemsrigorously in the next Sections.
3.1.1 Affine space and affine linear maps
Definition 3.1.1. Let E be a set and let V be a finite-dimensional vector space (over a field K )acting as abelian group faithfully and transitively on E . We call E an affine space (over K ), V its
35
space of translations and the elements of V translations of E. The dimension of V over K is alsosaid to be the dimension of E over K .
Example 3.1.2. Let V be a finite-dimensional vector space over a field K , and consider the actionof V as an abelian group on itself by addition of vectors in V . Then V is an affine space over K in anatural way with itself as its space of translations.
Since V is an abelian group, we have v (w (x )) = (v +w )(x ) = (w +v )(x ) =w (v (x )) for all v, w ∈Vand x ∈ E . Now assume v (x ) = x for some v ∈ V and x ∈ E , then v (w (x )) = w (v (x )) = w (x ) forany w ∈ V . Transitivity of the group action of V on E then implies v (y ) = y for all y ∈ E , hence byfaithfulness of the group action v must be the zero vector of V . In other words, the group action ofV on E is simply transitive.
Fix a point x ∈ E , then the simply transitive group action of V on E gives a bijective correspon-dence between the translations of E and the points of E by sending v ∈ V to v (x ) ∈ E . Write theaction of v on x as x + v , and if y = x + v for y ∈ E we write v = y − x which is well definedbecause the group action is simply transitivity. We can now let E inherit the K -vector space struc-ture of V by defining addition in E by (x + v ) + (x +w ) = x + (v +w ) and scalar multiplication byλ(x + v ) = x +λv for all v, w ∈ V and λ ∈ K . Let Ex denote the vector space E with x fixed, thenwe have a natural linear isomorphism x + v 7→ v between Ex and V . Notice that x plays the roleof the origin in Ex , so in some sense an affine space E is a vector space where we forgot what theorigin was. In the remaining of this Chapter we will identify an affine space E with its space oftranslations V as the vector space Ex in the way just described, when an origin x ∈ E is chosen.
Remark 3.1.1. The simply transitive group action of V on E guarantees that each translation v ∈Vof E generates a unique translation map tv : E → E defined by tv (y ) = y + v . Choosing an originx ∈ E and identifying E as Ex with the vector space V we obtain the unique translation map t ′v :V → V by tv (w ) = w + v for all w ∈ V . Since V is a vector space the map tv is a bijection of V .Furthermore, tv is linear if and only if v = 0. In the remaining of this chapter we will make nodistinction in the notation of the maps tv and t ′v , and we will write tv for both of them specifyingthe domain clearly.
Definition 3.1.3. A nonempty subset L of an affine space E with space of translations V is said tobe an affine subspace of E if there exists x ∈ L and a subspace U ⊂ V such that L is the U-orbitof x under the action of V on E . The subspace U is called the space of translations of L, and inparticular if U ⊂ V is of codimension 1 (resp. dimension 1), or a hyperplane (resp. line) in V , thenL is said to be an affine hyperplane (resp. affine line) in E . Two distinct affine hyperplanes are saidto be parallel if their spaces of translations coincide, and nonparallel otherwise.
Note that in the above Definition the subspace U ⊂ V does not depend on the choice of x ∈ L.Further, V acts simply transitively on E , hence U acts simply transitively on L. This shows that anaffine subspace L of an affine space E , as in the above definition, is an affine space of itself withspace of translations U .
Proposition 3.1.4. Two distinct affine hyperplanes H resp. H ′ in an affine space E with spaces oftranslations U resp. U ′ in V have a nonempty intersection if and only if H and H ′ are nonparallelaffine hyperplanes in E . Furthermore, this nonempty intersection H ∩H ′ is an affine subspace in Eand an affine hyperplane in H and H ′ with space of translations U ∩U ′.
Proof. If dim(V ) = 1, then {0} ⊂ V is the only subspace of codimension 1. Hence all affine hyper-planes in E are one-point sets, and all pairs of distinct affine hyperplanes in E are parallel and have
36
an empty intersection. So the Proposition holds in this case and we may assume in the followingthat dim(V )> 1.
Let H resp. H ′ be two distinct affine hyperplanes in E with spaces of translations U resp. U ′
in V , and choose x ∈ H and y ∈ H ′. Let H and H ′ be parallel affine hyperplanes , then U = U ′,H = x+U and H ′ = y +U . Assume now that H ∩H ′ 6= ; and let z ∈H ∩H ′, then by transitivity of thegroup action of U on H and H ′ we get H = x+U = z+U = y +U =H ′. This contradicts the fact thatH and H ′ are distinct. Hence we conclude that H ∩H ′ = ; and that two distinct affine hyperplaneswith nonempty intersection are nonparallel.
On the other hand, if H and H ′ are nonparallel, then U and U ′ are two distinct hyperplanes inV . Hence U ∩U ′ is of codimension 2 in V , and U ∩U ′ 6= ; since dim(V )> 1. Let x ∈H and y ∈H ′,then by transitivity of the action of U (resp. U ′) on H (resp. H ′) there exist v ∈U and v ′ ∈U ′ suchthat x +v = y +v ′ ∈H ∩H ′. Hence H ∩H ′ 6= ;, so the first statement of the Proposition is proved.
Define L :=H∩H ′ and W :=U∩U ′, then for x ∈ L it is clear by simple transitivity that L = x+W .Now the second statement of the Proposition follows.
Next, we proceed with the description of natural maps between affine spaces. On the one handthese maps should have a linear character on the underlying space of translation. On the otherhand, as an affine space does not have a specified origin these maps may also carry a translation.
Definition 3.1.5. A map a : E → E ′ between affine spaces E resp. E ′ over the same field K withspaces of translations V resp. V ′ is called affine linear is there exists a K -linear map Da : V → V ′
such that for all x ∈ E and v ∈Va (x +v ) = a (x )+Da (v ).
Let Hom(E , E ′) denote the set of affine linear maps from E to E ′.
Example 3.1.6. (i) For v ∈ V the translation map tv of E by v (see Remark 3.1.1) is an affine linearmap of E to itself such that Dtv is the identity map idV on V . On the other hand, if t is an affinelinear map of E to itself such that Dt = idV , then t = tv for some v ∈ V . In particular, the identitymap idE on E coincides with t0.
(ii) Let E , E ′ and E ′′ be affine spaces with spaces of translations V , V ′ and V ′′ respectively, allover the same field K . Consider a ∈Hom(E , E ′), b ∈Hom(E ′, E ′′), x ∈ E and v ∈V , then we observefrom affine linearity of both a and b that
(b ◦a )(x +v ) =b (a (x +v ))
=b (a (x )+Da (v ))
=b (a (x ))+Db (Da (v ))
= (b ◦a )(x )+ (Db ◦Da )(v ).
Since Da ∈ HomK (V, V ′) and Db ∈ HomK (V ′, V ′′) we have Db ◦Da ∈ HomK (V, V ′′), so b ◦ a ∈Hom(E , E ′′)with
D(b ◦a ) =Db ◦Da . (3.1.1)
(iii) Let a : E → E ′ be an affine linear map such that a (x ) = y for some x ∈ E and y ∈ E ′, thena : Ex → E ′y is a linear map sending x + v to y +Da (v ). Conversely, let a : Ex → E ′y be a linearmap, and let φ : Ex → V (resp. φ′ : E ′y → V ′) be the canonical linear isomorphism determinedby x + v 7→ v (resp. y +w 7→ w ). Then α := φ′ ◦ a ◦φ−1 : V → V ′ is the linear map such that
37
a (x +v ) = y +α(v ) for all v ∈V . Now a (x ) = y , and for z ∈ E and v ∈V there exists a unique w ∈Vsuch that z = x +w . Therefore
a (z +v ) = a (x +w +v )
= a (x )+α(w +v )
= a (x )+α(w )+α(v ) (3.1.2)
= a (x +w )+α(v )
= a (z )+α(v ).
which implies that a : E → E ′ is an affine linear map satisfying a (x ) = y .Fix x ∈ E and y ∈ E ′, and let HomK (Ex , E ′y ) denote the K -vector space of K -linear maps from
Ex to E ′y , here considered as the set of affine linear maps from E to E ′ sending x to y . For a ∈HomK (Ex , E ′y ) it is clear from (ii) that tv ′ ◦ a ∈ Hom(E , E ′) for all v ′ ∈ V ′. On the other hand, ifa ∈Hom(E , E ′) such that a (x ) = z , then there exists a unique v ′ ∈ V ′ such that tv ′ (z ) = z + v ′ = y .This implies that (tv ′ ◦a )(x ) = y , so tv ′ ◦a ∈HomK (Ex , E ′y ) by (i). We conclude that
Hom(E , E ′) =∐
v ′∈V ′(tv ′ HomK (Ex , E ′y )) (3.1.3)
where tv ′ acts on each a ∈ HomK (Ex , E ′y ) by tv ′ a = tv ′ ◦ a . Also, we observe that a : E → E ′ isan affine linear bijection if and only if a : Ex → E ′y is a linear isomorphism. Moreover, its inversea−1 : E ′y → Ex is also a linear isomorphism with D(a−1) = (Da )−1, so a−1 ∈ Hom(E ′, E ). We callsuch affine linear bijections affine linear isomorphisms. The translation maps tv of E by v ∈ V areexamples of affine linear isomorphisms.
An affine linear map a : E → E ′ in the context of Definition 3.1.5 respects affine subspaces inthe sense that an affine subspace L ⊂ E gets mapped onto an affine subspace L′ ⊂ E ′. Indeed,write L = x +U for some x ∈ L with U ⊂ V the space of translations of L, then a (L) = a (x +U ) =a (x ) +Da (U ) where U ′ := Da (U ) ⊂ V ′ is a linear subspace. By Definition 3.1.3, L′ := a (L) ⊂ E ′
is an affine linear subspace with space of translations U ′, and dim(U ′) ≤ dim(U ). In particular, ifa : E → E ′ is an affine linear isomorphism, then Da : V → V ′ is a linear isomorphism. Hence inthis case affine hyperplanes get mapped onto affine hyperplanes. Furthermore, no two distinctaffine hyperplanes in E get mapped to the same affine hyperplane in E ′ and we exhaust all affinehyperplanes in E ′ this way.
3.1.2 Affine Euclidean space and affine linear functions
From now on we will fix K to be the field of real numbersR, so that the space of translations V be-comes an l -dimensional real vector space. Further, we turn V into a Euclidean space by equipping
it with a real positive definite symmetric bilinear form (., .)V . Let |v |V :=p
(v, v )V for all v ∈V , thenthe affine space E becomes a metric space with the distance function |x − y |E := |v |V for x , y ∈ Eand v ∈V such that y = x +v . We call a real affine space E endowed with the metric |.|E in this wayan affine Euclidean space. If we choose an origin x in an affine Euclidean space E and identify Ewith the vector space V as Ex , then we can let Ex inherit the inner product space structure of V .Define a real positive definite symmetric bilinear form (., .)Ex on Ex by (x+v,x+w )Ex := (v, w )V forall v, w ∈ V . Then the canonical identification of Ex with V defined by x + v 7→ v preserves (., .)Ex
and (., .)V respectively as defined in the following. In the remaining of this Chapter we will consider
38
Ex to be equipped with the inner product (., .)Ex . Notice that the metric space E with metric |.|Ecoincides with Ex considered as a metric space with metric induced by (., .)Ex . This implies that Ecan be considered as a topological space having the Euclidean topology induced by |.|E .
Remark 3.1.2. In the remaining of this manuscript we will view an affine Euclidean space E astopological space always as being equipped with the Euclidean topology induced by |.|E .
Proposition 3.1.7. If a : E → E ′ is an affine linear map between affine Euclidean spaces E and E ′,then a : E → E ′ is also a continuous map.
Proof. Choose an origin x ∈ E and y ∈ E ′, then a : Ex → E ′y is a linear map between finite-dimensional vector spaces by (iii) of Example 3.1.6. Hence a : Ex → Ey is a continuous map. SinceEx (resp. E ′y ) and E (resp. E ′) coincide as topological spaces we conclude that a : E → E ′ is acontinuous map.
Corollary 3.1.8. If a : E → E ′ is an affine linear isomorphism between affine Euclidean spaces Eand E ′, then a : E → E ′ is also a homeomorphism.
Proof. This follows directly from Proposition 3.1.7.
In affine Euclidean space we have the notion of convexness from the structure of the space oftranslations over the totally ordered field R.
Definition 3.1.9. A subset U of an affine Euclidean space E with space of translation V is calledconvex if for every x , y ∈U and for all 0≤λ≤ 1 we have x+λv ∈U where v ∈V is the unique vectorsuch that y = x + v . The convex subset {x +λv : 0≤ λ≤ 1} (resp. {x +λv : 0< λ < 1}) of E is saidthe be the line segment (resp. open line segment) between points x , y ∈ E .
Example 3.1.10. Any affine subspace H of an affine Euclidean space E is a convex subset by lin-earity of the space of translations of H . In particular, this holds for affine hyperplanes and affinelines. Also notice that any convex subset of an affine Euclidean space is arc-connected, hence alsoconnected.
Recall from Example 3.1.2 that a vector space has the natural structure of an affine space, hencewe can consider the reals R as an affine Euclidean space. Let bE :=Hom(E ,R) denote the set of allaffine linear functions a : E → R, and choose 0 ∈ R as origin to turn R into an R-vector space.For a ,b ∈ bE and λ ∈ R define a + b as a function on E by (a + b )(x ) = a (x ) + b (x ) for all x ∈ E ,and λa by (λa )(x ) = λ · a (x ). Clearly both a + b and λa are affine linear functions on E withD(a +b ) =Da +Db and D(λa ) = λDa respectively. Then the vector space properties of R turn bEinto an R-vector space.
Example 3.1.11. The constant functions on E with values in R are contained in bE and have allgradient 0. On the other hand, an affine linear function on E is constant if it has has gradient 0.Further, choose an origin x ∈ E and consider the set of R-linear functionals on Ex denoted by E ∗x(where 0 is the origin of R). From (iii) of Example 3.1.6 we observe that E ∗x ⊂ bE .
Remark 3.1.3. It will happen more than once that we want to say something about the constantfunction c ∈ bE that is identically one on the affine Euclidean space E . For the simplicity of thestatements we will refer to this function as the constant one function on E .
39
Now, identify V with its dual space V ∗ by means of the scalar product (., .)V . The gradient of anaffine linear function a ∈ bE is the unique vector Da ∈V such that a (x +v ) = a (x )+ (Da , v )V for allx ∈ E and v ∈V . We define a real positive semidefinite symmetric bilinear form on bE by
(a ,b )bE := (Da , Db )V
for all a ,b ∈ bE . Two affine linear functions a ,b ∈ bE are called orthogonal if (a ,b )bE = 0. Further,
a nonzero a ∈ bE is called isotropic if (a , a )bE = 0, and nonisotropic otherwise. Note that a ∈ bE is
isotropic if and only if a is a nonzero constant function on E .
Proposition 3.1.12. The vector space bE of affine linear functions on E is of dimension dimR(E )+1.Moreover, if we let c denote the constant one function on E , then bE can be identified with V ⊕Rc bythe linear isomorphismφx : a 7→ (Da , a (x )) after fixing an origin x ∈ E .
Proof. Fix a point x ∈ E and let a ∈ bE , then a (x + v ) = a (x ) + (Da , v )V for each v ∈ V with Da thegradient of a . Hence every a ∈ bE is the sum of a unique constant function with value a (x ) on Eand a unique linear functional (x +Da , .)Ex on Ex sending x + v to (Da , v )V for all v ∈ V . Thusφ′x : bE → E ∗x ⊕Rc , a 7→ ((x +Da , .)Ex , a (x )) is a well defined injective linear map. Moreover φ′x issurjective, because the constant functionsRc and the linear functionals E ∗x on Ex are contained inbE as we have seen in Example 3.1.11. Identify E ∗x with V by the linear isomorphism (x +Da , .)Ex 7→Da , where the inverse map sends a vector α∈V to the linear functional (x +α, .)Ex on Ex such thatx + v 7→ (α, v )V for all v ∈ V . This yields a linear isomorphism φx : bE → V ⊕Rc , a 7→ (Da , a (x ))with inverse map sending α+µc ∈ V ⊕Rc to the affine linear function a with gradient α definedby a (x + v ) = µ+ (α, v )V for all v ∈ V (affine linearity follows analogous to the computation donein (3.1.2)).
Corollary 3.1.13. Choose an origin x ∈ E and identify bE with V ⊕Rc , then the map D : bE → V thatsends a ∈ bE to its gradient Da is a projection onto V along the direct sum bE =V ⊕Rc .
Define the real positive semidefinite symmetric bilinear form
(α+λc ,β +µc )V⊕Rc := (α,β )V (3.1.4)
on V ⊕Rc for α,β ∈V and λ,µ∈R. Then we immediately obtain the following Corollary.
Corollary 3.1.14. The linear isomorphismφx preserves the forms on bE and V ⊕Rc respectively.
3.1.3 Orthogonal reflections in affine Euclidean space
Next, we want to consider orthogonal reflections in affine hyperplanes in the affine Euclideanspace E . We first recall orthogonal reflections in hyperplanes in Euclidean spaces (see for example[6] Chapter 3, §9). For each nonzero v ∈V define
v ∨ :=2v
(v, v )V,
then the orthogonal reflection wv : V →V in the hyperplane orthogonal to v is given by
wv (u ) = u − (v, u )V v ∨ (3.1.5)
for all u ∈ V . Indeed, wv is a linear isometry of V fixing the hyperplane orthogonal to v andmapping v to −v .
Let H be an affine hyperplane in E with space of translations U ⊂ V , then E = H +U⊥ whereU⊥ = {z ∈V : (z , u )V = 0 for all u ∈U} is the orthoplement of U in V .
40
Definition 3.1.15. The orthogonal reflection in the affine hyperplane H (with space of translationsU ) is the map wH : E → E defined by wH (h +w ) = h −w for all h ∈H and w ∈U⊥.
Clearly, wH is a bijection of E with itself by simply transitivity of the group action of V on E ,and wH fixes the affine hyperplane H ⊂ E . Moreover, we have the following observation.
Proposition 3.1.16. The orthogonal reflection wH : E → E in the affine hyperplane H ⊂ E is a metricpreserving affine linear isomorphism.
Proof. Let h ∈H be an origin of E . We will proof that wH : Eh → Eh is an orthogonal reflection inthe hyperplane orthogonal to h + v for some nonzero v ∈U⊥ as described by (3.1.5). This impliesthat wH preserves the metric on E , and by (iii) of Example 3.1.6 we obtain that wH : E → E is anaffine linear isomorphism.
Let v ′ ∈V then there are unique u ∈U and u ′ ∈U⊥ such that v ′ = u +u ′. Hence
wH (h +v ′) =wH (h +u +u ′) = h +u −u ′ = h +v ′−2u ′. (3.1.6)
If u ′ = 0 we are done, else notice that for v ∈U⊥ \ {0} there exists λ ∈R6=0 such that v = λu ′. Then(v, u ′)V = sgn(λ)|v |V |u ′|V where sgn(λ)∈ {±1} denotes the sign of λ. Since (v, u )V = 0 we have
(v, v ′)V v ∨ = (v, u ′)V v ∨ = 2(v, u ′)V(v, v )V
v = 2 sgn(λ)v|u ′|V|v |V
= 2u ′. (3.1.7)
Substituting the left-hand side of the equations of (3.1.7) into the right-hand side of (3.1.6) showsthat wH : Eh → Eh is an orthogonal reflection in the hyperplane orthogonal to h + v as describedby (3.1.5).
Now for each nonisotropic a ∈ bE the gradient Da ∈ V is nonvanishing. This implies thatker((Da , .)V ) = {v ∈ V : (Da , v )V = 0} has codimension 1 in V , hence ker((Da , .)V ) is a hyperplanein V . So for x ∈ E such that a (x ) = 0, we have {y ∈ E : a (y ) = 0}= x +ker((Da , .)V ). Defining
Ha := {y ∈ E : a (y ) = 0}, (3.1.8)
we conclude that Ha is an affine hyperplane in E with space of translations ker((Da , .)V )⊂V .
Definition 3.1.17. Let E be an affine Euclidean space with space of translation V , then we call anonzero vector v ∈ V a normal vector to an affine hyperplane H ⊂ E with space of translationsU ⊂V if (v, w )V = 0 for all w ∈U .
Example 3.1.18. The normal vectors of Ha are λDa for λ∈R6=0.
Further, for each nonisotropic a ∈ bE define
a∨ :=2a
(a , a )bE=
2a
(Da , Da )V,
and the map wa : E → E ,wa (x ) = x −a∨(x )Da = x −a (x )Da∨. (3.1.9)
Proposition 3.1.19. The map wa : E → E coincides with orthogonal reflection wHa . Furthermore,we have Dwa =wDa .
41
Proof. For x ∈ E we have the unique expression x = h +w for some h ∈Ha and w ∈U⊥, where Uis the space of translations of the affine hyperplane Ha ⊂ E . Thus
wa (x )(3.1.9)= x −a (x )Da∨
=h +w −a (h +w )Da∨
=h +w − (Da , w )V Da∨.
But Da ∈U⊥, so (Da , w )V Da∨ = 2w by a similar argument as (3.1.7). Thus wa coincides with theorthogonal reflection wHa in the affine hyperplane Ha by definition of wHa .
From affine linearity of a and wa we obtain for x ∈ E and v ∈V
wa (x )+Dwa (v ) =wa (x +v )(3.1.9)= x +v − (a (x +v ))Da∨
= x +v − (a (x )+ (Da , v )V )Da∨
= x −a (x )Da∨+wDa (v ) (3.1.10)(3.1.9)= wa (x )+wDa (v ),
so we immediately observeDwa =wDa . (3.1.11)
Propositions 3.1.16 and now imply the following Corollary.
Corollary 3.1.20. The map wa : E → E for a ∈ bE is a metric preserving affine linear isomorphism.
3.1.4 Orthogonal reflections and affine linear automorphisms
By (iii) of Example 3.1.6, End(E ) := Hom(E , E ) =∐
v∈V (tv EndR(Ex )) for a fixed x ∈ E whereEndR(Ex ) is theR-vector space ofR-linear maps from Ex to itself, here considered as the set affinelinear maps from E to itself fixing x . Let GL(E )⊂ End(E ) be the group of affine linear isomorphismsof E to itself, or affine linear automorphisms of E , with idE as identity element and the composi-tion of maps as group operation. Composition of maps is a well defined associative operation onGL(E ) by (ii) of Example 3.1.6, and the existence of inverse elements follows from (ii) of that sameExample. Further, let t (V ) ⊂ GL(E ) denote the subgroup of translation maps of E , then t (V ) iscanonically isomorphic to the group of translation maps of V which we will also denote by t (V )(see Remark 3.1.1). Finally, let GLR(V ) (resp. GLR(Ex )) denote the group of linear automorphismsof V (resp. Ex ) with the composition of maps as group operation. Notice that
GLR(Ex )∼=GLR(V ) (3.1.12)
by conjugation with the linear isomorphism Ex∼−→V, x +v → v .
Proposition 3.1.21. The group GL(E ) is isomorphic to the semidirect product t (V )oGLR(V ).
Proof. Let x ∈ E , let t (V ) ⊂ GL(E ) be the subgroup of translation maps of E , and consider thesubgroup GL(E )x ⊂ GL(E ) of affine linear automorphisms of E that keep x fixed. Then t (V ) ∩GL(E )x = {idE } since idE = t0 ∈ t (V ) is the only translation map that leaves x fixed. Now, let v ∈ Vand let y ∈ E , then there exists a unique w ∈V such that y = x+w . Further, let g ∈GL(E )x , then by
42
(iii) of Example 3.1.6 we have that D g ∈ GLR(V ) with (D g )−1 = D(g −1). Furthermore, we observethat
(g ◦ tv ◦ g −1)(y ) = (g ◦ tv ◦ g −1)(x +w )
= g (x +(D g )−1(w )+v )
= x +w +D g (v )
= tD g (v )(x +w )
= tD g (v )(y ),
so g ◦ tv ◦ g −1 = tD g (v ) ∈ t (V ). Because GL(E ) ⊂ End(E ) =∐
v∈V (tv EndR(Ex )) we have GL(E ) =t (V )GL(E )x := {t ◦ g : t ∈ t (V ) and g ∈ GL(E )x }, so we conclude that t (V ) ⊂ GL(E ) is a normalsubgroup, and that GL(E ) = t (V )oGL(E )x .
Now consider t (V ) canonically as the group of translation maps of V . By (ii) and (iii) of Example3.1.6 in combination with the isomorphism of (3.1.12) we can also identify GL(E )x canonicallywith the group GLR(V ) by the group isomorphism a 7→Da . This implies that that GL(E ) ∼= t (V )oGLR(V ).
For a ∈ bE the orthogonal reflection wa in the affine hyperplane Ha is a metric preserving affinelinear automorphism of E by Corollary 3.1.20, hence wa ∈ GL(E ). Then we observe from (3.1.10)together with Proposition 3.1.12 that for x ∈ E we have
wa = t−a (x )Da∨ ◦wx+Da , (3.1.13)
where wx+Da ∈GLR(Ex ). Furthermore, we observe by the proof of Proposition 3.1.12 that
wx+C1 ◦ · · · ◦wx+Ck ◦ tv ◦w−1x+Ck
◦ · · · ◦w−1x+C1
= twC1◦···◦wCk (v )(3.1.14)
for all v ∈V and nonzero C1, . . . ,Ck ∈V .Now GL(E ) acts canonically on the affine Euclidean space E as a group of affine linear auto-
morphisms of E . Furthermore, this action implies a linear action of GL(E ) on the vector space bE .For g ∈ GL(E ) and a ∈ bE put g (a ) := a ◦ g −1, then a ◦ g −1 ∈ bE follows from the fact a ∈ bE andg −1 ∈ End(E ) together with (ii) of Example 3.1.6. It follows easily by evaluation in E that this actionis linear. Also, because g ∈GL(E )we observe that a 6=b in bE implies a ◦ g −1 6=b ◦ g −1, and if a ∈ bEis a constant function then a ◦ g −1 = a . Thus for g ∈GL(E ) and a ∈ bE the mapping a 7→ a ◦ g −1 is alinear automorphism of bE that fixes the constant functions.
In particular, for a ,b ∈ bE the orthogonal reflection wa acts on bE by wa (b ) =b ◦w−1a =b ◦wa , or
explicitlywa (b ) =b − (a∨,b )
bE a (3.1.15)
which follows from (3.1.9). Also, for v ∈ V it follows easily that the action of the translation map tv
on b ∈ bE becomestv (b ) =b − (Db , v )V c , (3.1.16)
where c ∈ bE is the constant one function on E .Let GLR,c ( bE ) denote the group of linear automorphisms of bE that fix the constant functions with
composition of maps as group operation, then we observe the following.
Proposition 3.1.22. For g ∈ GL(E ) and a ∈ bE , the mapping g 7→ {a 7→ a ◦ g −1} defines a groupisomorphism between GL(E ) and GLR,c ( bE ).
43
Proof. We already saw that the mappingφ : g 7→ {a 7→ a ◦ g −1} is well defined, and it is follows in astraightforward way that φ is a group homomorphism. Let f 6= g in GL(E ), then there exists x ∈ Esuch that f −1(x ) 6= g −1(x ), say y = f −1(x ) and z = g −1(x ). Choose y as origin of E and identify Ewith V as the vector space Ey . By this identification there exists a unique vector v ∈ V such thatz = y +v . Consider anR-linear functional a on Ey that sends z = y +v to 1∈R. By (iii) of Example3.1.6, a can be viewed as an affine linear function on E , and we note 0 = a (y ) 6= a (z ) = 1. Thusa ◦ f −1 6= a ◦ g −1, and we observe thatφ is injective.
Next, we proof surjectivity ofφ. Choose an origin x ∈ E to identify E as Ex with the vector spaceV . Also idenfity bE with E ∗x ⊕Rc as in the proof of Proposition 3.1.12, where c is the constant onefunction on E and E ∗x is the vector space of R-linear functionals on Ex . Let T ∈ GLR,c ( bE ), thenT can be viewed as a linear automorphism of E ∗x ⊕Rc such that T restricted to Rc is the identitymap on Rc . Consider the restriction T ′ := T |E ∗x : E ∗x → E ∗x ⊕Rc , and let px : E ∗x ⊕Rc → E ∗x (resp.pr : E ∗x ⊕Rc → Rc ) be the linear projection onto E ∗x (resp. Rc ) along the direct sum. Since T isan isomorphism that fixes Rc the image T ′(E ∗x ) has trivial intersection with Rc , and T ′ must be of
rank dim(E ∗x ) = dim(Ex ) = dim(V ) = l . Hence S := px ◦T ′ : E ∗x∼−→ E ∗x is a linear automorphism and
γ := pr ◦T ′ : E ∗x → R is a linear map such that T ′(α) = S(α) + γ(α)c for α ∈ E ∗x . Thus there exists a
linear automorphism σ : Ex∼−→ Ex such that S(α) = α ◦σ−1 for α ∈ E ∗x . By (iii) of Example 3.1.6, σ
is an affine linear automorphism of E , so σ ∈GL(E ). Further, there exists a vector η ∈ V such thatγ(α) = (Dα,η)V for all α ∈ E ∗x ⊂ bE , where Dα ∈ V is the gradient of α. Let a ∈ bE , then a =α+λc forsome λ∈R and α∈ E ∗x . This leads to
T (a ) = T (α+λc )
= T ′(α)+λc
=S(α)+γ(α)c +λc
=α ◦σ−1+(Dα,η)V c +λc
Sinceσ ∈GL(E )we have that c ◦σ−1 = c , so
α ◦σ−1+(Dα,η)V c +λc =α ◦σ−1+(Dα,η)V c ◦σ−1+λc ◦σ−1
= (α+λc +(Dα,η)V c ) ◦σ−1.
Then (3.1.16) implies that
(α+λc +(Dα,η)V c ) ◦σ−1 = (α+λc ) ◦ t−η ◦σ−1
= a ◦ t−η ◦σ−1,
where t−η ◦σ−1 ∈ GL(E ). Hence φ−1(T ) = (t−η ◦σ−1)−1 = σ ◦ tη which proves the surjectivity ofφ.
We end this Section with a general result on metric preserving affine linear automorphismsof E . This will prove to be useful in the next Section when we want to consider compositions oforthogonal reflections of affine Euclidean space.
Proposition 3.1.23. Let w ∈ GL(E ) be metric preserving on E and let a ∈ bE , then Dw : V → V isa linear isometry and we have the gradient D(w (a )) = (Dw )(Da ). Furthermore, for nonisotropica ,b ∈ bE we have (w (a )∨, w (b ))
bE = (a∨,b )
bE .
44
Proof. Let w ∈ GL(E ) be a metric preserving affine linear isomorphism, then by (iii) of Example3.1.6 we observe that Dw : V → V is linear isometry. By Proposition 3.1.22 w acts on bE by w (a ) =a ◦w−1, so we obtain for x ∈ E and v ∈V
(D(w (a )), v )V =w (a )(x +v )−w (a )(x )
= a (w−1(x +v ))−a (w−1(x ))
= a (w−1(x )+ (Dw )−1(v ))−a (w−1(x )))
= (Da , (Dw )−1(v ))V= ((Dw )(Da ), v )V .
In other words,D(w (a )) = (Dw )(Da ). (3.1.17)
Further, the action of w on bE fixes the constant functions, hence it sends nonisotropic vectorsto nonisotropic vectors. This implies that for nonisotropic a ,b ∈ bE the expression (w (a )∨, w (b ))
bE
is well defined. Now, by definition of (., .)bE we have
(w (a )∨, w (b ))bE = 2
(Dw (a ), Dw (b ))V(Dw (a ), Dw (a ))V
,
so by (3.1.17)
2(Dw (a ), Dw (b ))V(Dw (a ), Dw (a ))V
= 2((Dw )(Da ), (Dw )(Db ))V((Dw )(Da ), (Dw )(Da ))V
.
Finally, since Dw is a linear isometry of V , hence
(w (a )∨, w (b ))bE = 2
((Dw )(Da ), (Dw )(Db ))V((Dw )(Da ), (Dw )(Da ))V
= 2(Da , Db )V(Da , Da )V
= (a∨,b )bE .
3.2 Affine root systems
In this section we will define the main objects that we would like to investigate in this Chapter,namely affine root systems. Analogously to the theory of finite root systems we will introduceconcepts like the dual affine root system, reducedness, similarity, and irreducibility. The latterconcept is very subtle in the theory of affine root systems, therefore we will use two Subsectionsto fully study it. We let E be an affine Euclidean space with space of translations V of dimensionl > 0, and bE the vector space of affine linear functions on E . The Euclidean metric on E inducesthe Euclidean topology on E , making it into a locally compact space.
3.2.1 Affine root systems
We define our main object of interest.
Definition 3.2.1. A subset R of nonisotropic vectors in bE (i.e. nonconstant functions on E ) is calledan affine root system on E if the following five conditions are satisfied
(1) R spans bE ;(2) wa (b )∈R for all a ,b ∈R ;
45
(3) (a∨,b )bE ∈Z for all a ,b ∈R ;
(4) the subgroup W (R)⊂GL(E ) generated by the orthogonal reflections wa for a ∈R consideredas a discrete group acts properly on E ;
(5) for each gradient C ∈ {Da : a ∈R} there are at least two distinct b ,b ′ ∈R such that C =Db =Db ′.The elements of R are called affine roots and the dimension of V is said to be the rank of R .
Remark 3.2.1. Condition (4) of the above Definition is equivalent to: for any two compact subsetsK and K ′ of E there are only finitely many w ∈W (R) such that w (K )∩K ′ 6= ; (see (D’2) in ChapterV §3 on p.77 of [2] and the Remark of Chapter III §4.5 of [3]).
Definition 3.2.2. For an affine root system R ⊂ bE the group W (R) is called the affine Weyl group ofR , and its generators wa for a ∈R are called reflections.
By condition (2) of Definition 3.2.1 we observe that that the affine Weyl group W (R) acts on Ras a group. Then condition (2) of Definition 3.2.1 can also be read as: W (R) stabilizes R .
Example 3.2.3. (i) Let A be an affine Cartan matrix, and consider the set of real roots ∆r e ⊂ h∗0 =◦h∗R ⊕Rδ corresponding to the affine Lie algebra g(A) in the context of Section 2.3. Then we can
consider the vector space◦h∗R with positive definite normalized invariant form (., .) as an affine Eu-
clidean space by Example 3.1.2. Now we want to identify h∗0 with the space of affine linear functionsc◦h∗R on
◦h∗R in a natural way preserving the forms on both spaces.
First,c◦h∗R can be identified with
◦h∗R ⊕Rc by Proposition 3.1.12 and Corollary 3.1.14, where c is
the constant one function on◦h∗R after choosing the origin 0 ∈
◦h∗R. Then (α+ nc ,β +m c )◦
h∗R⊕Rc=
(α,β ) by Corollary 3.1.14. Hence from the proof of Proposition 3.1.12 it follows that it is natural
to consider each α ∈◦h∗R ⊂ h∗0 as a linear functional on
◦h∗R defined by µ 7→ (α,µ). Further, notice
that it is natural to identify δ with c because in Section 2.3 we have chosen to work in the affinehyperplane h∗1 = {λ∈ h
∗R : (λ,δ) = 1} of h∗0. If we would have started out with h∗s = {λ∈ h
∗R : (λ,δ) = s }
for any other nonzero s ∈ R, then it would be natural to identify δ with s c . This would turn ∆r e
in an affine root system that is ’similar’ to the one that we have discussed here starting out with h∗1(see (ii) of Example 3.2.9).
We conclude that it is natural to identifyc◦h∗R =
◦h∗R⊕Rc with h∗0 =
◦h∗R⊕Rδ by the linear isomor-
phism that is the identity map on◦h∗R and sends c to δ. This means that α+nδ ∈ h∗0 acts on µ ∈
◦h∗R
by(α+nδ)(µ) = (α,µ)+n .
Clearly the form (., .)◦h∗R⊕Rc
and the normalized invariant form (., .) on◦h∗R⊕Rδ are preserved by this
identification. Now by (3.1.9) we have
wα+nδ(µ) =µ− (α+nδ)(µ) ·2α
(α,α)=µ−2
(µ,α)+n
(α,α)α
for the reflection of µ∈◦h∗R in the affine hyperplane Hα+nδ. For real roots this expression coincides
by (2.3.11) withaf(rα+nδ)(µ).
46
Further af(W ) is generated by af(rαi ) forαi ∈Π, so surely by af(rα+nδ) forα+nδ ∈∆r e . Also, W (∆r e )is generated by wα+nδ for α+nδ ∈∆r e . Hence af(W ) and W (∆r e ) are isomorphic by the obviousmapping af(rα+nδ) 7→ wα+nδ. Then it follows from Theorem 2.3.4 and Corollary 3.1.13 that ∆r e is
an affine root system on◦h∗R with affine Weyl group af(W ) =
◦W naf(T ).
(ii) Let R0 be a reduced irreducible finite root system in V with finite Weyl group W0(R0) ⊂GLR(V ). Let {α1, . . . ,αl } be a basis of R0, then {α∨1 , . . . ,α∨l } is a basis of R∨0 . Furthermore, R∨0 is
contained in the coroot lattice Q∨ :=∑l
i=1Zα∨i ⊂ V . Write t (Q∨) := {tv : v ∈Q∨} for subgroup of
translations of t (V ) over the lattice Q∨. Consider the vector space V as affine Euclidean space Ewith V as space of translations (see Example 3.1.2). Let c be the constant one function on E , andidentify bE with V ⊕Rc after choosing the origin 0 ∈ E (see Prop. 3.1.12 and Cor. 3.1.14). Then thesubset
RuR0
:= {m c +α}m∈Z,α∈R0
of V⊕Rc turns out to be an affine root system on E with affine Weyl group W (RuR0) = t (Q∨)oW0(R0),
where t (Q∨), W0(R0)⊂W (RuR0) are subgroups. In Section 3.5 we will give an extensive proof of this
result (see Proposition 3.5.5).
Analogous to the theory of finite root systems as described in Chapter 1 we have the notions ofindivisible and unmultipliable affine roots in an affine root system R . Condition (3) of Definition3.2.1 yields thatRa∩R = {±a },Ra∩R = {±a ,± 1
2 a } andRa∩R = {±a ,±2a } are the only possibilities
for multiples of an affine root a ∈ R . Define the indivisible affine roots R i nd := {a ∈ R : 12 a /∈ R} of
R and the unmultipliable affine roots Ru nm := {a ∈ R : 2a /∈ R} of R , then R = R i nd ∪Ru nm (unionnot disjoint). From a straightforward check of the five conditions of Definition 3.2.1 it follows thatboth R i nd and Ru nm are affine root systems on E with affine Weyl group W (R).
Definition 3.2.4. An affine root system R is said to be reduced if each a ∈R is indivisible. It is callednonreduced otherwise.
Example 3.2.5. (i) From the possibilities of multiples of affine roots in an affine root system R onE as noted in the previous paragraph, we observe that both R i nd and Ru nm are reduced affine rootsystems on E .
(ii) Example 3.2.3 (i) made clear that the set of real roots ∆r e corresponding to the affine Liealgebra g(A) is an affine root system. By (vi) of Theorem 2.3.4 it then follows that ∆r e is a reducedaffine root system.
(iii) The affine root system RuR0
of Example 3.2.3 is reduced since R0 is reduced.
For our purposes we will only be interested in reduced affine root systems, but not until the lastSection of this Chapter will we make this distinction.
Analogous to the theory of finite root systems, we also have the notion of the dual of an affineroot system.
Definition 3.2.6. If R is an affine root system, then we define the dual of R to be the set R∨ := {a∨ :a ∈R}.
Proposition 3.2.7. The dual R∨ of an affine root system R on E is an affine root system on E withthe same affine Weyl group as R.
Proof. We will check all conditions of Definition 3.2.1 for R∨. Since R∨ is obtained from R by pos-itively scaling each affine root in R , the dual R∨ ⊂ bE is a subset of nonisotropic vectors satisfying
47
condition (1) and (5). Further, (a∨)∨ = a for all a ∈ R , so R∨ also satisfies condition (3). Next, weobserve from (3.1.15) that wa∨ =wa in GLR,c ( bE ) for all a ∈ R . This implies that W (R∨) =W (R), sosurely condition (4) is satisfied. Thus to proof condition (2), it is enough to show that wa (R∨)⊂R∨
for all a ∈ R . Let a ∈ R and b ′ ∈ R∨, then there exists b ∈ R such that b ′ = b∨, so wa (b ′) =wa (b∨) =b∨ − (a∨,b∨)
bE a = wa (b ) · 2(b ,b )
bE. Now by 3.1.17, D(wa (b )) = (Dwa )(Db ), where Dwa : V → V is a
linear isometry. But then (b ,b )bE = (Db , Db )V = (Dwa (Db ), Dwa (Db ))V = (Dwa (b ), Dwa (b ))V =
(wa (b ), wa (b )) bE . Hence wa (b ′) =wa (b ) · 2(wa (b ),wa (b )) bE
=wa (b )∨ ∈R∨.
3.2.2 Similar affine root systems
Next, we define an appropriate equivalence relation called ’similarity’ on the set of affine root sys-tems analogous to the notion of similarity for finite root systems. We will show that similarity pre-serves important structures on affine root systems like the affine Weyl group, the integers (a∨,b )
bE
of Definition 3.2.1 and the collections of affine hyperplanes induced by the affine roots.
Definition 3.2.8. Let R ⊂ bE and R ′ ⊂cE ′ be two affine root systems. We call R and R ′ similar, andwrite R 'R ′, if there exists a linear isomorphism T : bE
∼−→cE ′ such that
(T (a )∨, T (b ))bE = (a
∨,b )bE
for all a ,b ∈R , which restricts to a bijection of R onto R ′.
The equivalence classes under the equivalence relation similarity on the collection of affineroot systems are called similarity classes.
Example 3.2.9. (i) Every w ∈ W (R) realizes a similarity of R with itself. Indeed, w ∈ W (R) actson bE as linear automorphism that fixes the constant functions by Proposition 3.1.22. Then (ii) ofDefinition 3.2.1 implies that w maps R bijectively onto itself, and Proposition 3.1.23 tells us that(w (a )∨, w (b ))
bE = (a∨,b )
bE for all a ,b ∈R .(ii) Recall from (i) of Example 3.2.3 that we identified δ ∈ h∗0 with the constant one function c on
◦h∗R. However if we interpretδ as nc for some n ∈R, then it follows easily that the mapφ :
◦h∗R⊕Rc →
◦h∗R⊕Rc that is the identity on
◦h∗R and sends c to nc realizes a similarity of∆r e with itself.
Proposition 3.2.10. If the linear isomorphism T : bE∼−→cE ′ realizes a similarity between affine root
systems R ⊂ bE and R ′ ⊂cE ′, then(i) R is (non)reduced implies R ′ is (non)reduced(ii) T sends constant functions to constant functions(iii) W (R)∼=W (R ′).
Proof. (i) Clearly, similarity respects reducedness since T is linear.(ii) Since R is nonempty, we can choose distinct affine roots b and b ′ such that Db =Db ′ =: C
by (5) of Definition 3.2.1. By affine linearity of b and b ′ we have b (x + v )−b ′(x + v ) = b (x )−b ′(x )for all x ∈ E and v ∈V , hence b −b ′ ∈ bE is a constant function on E . Then for all a ∈R we have
(T (a )∨, T (b −b ′))bE = (T (a )
∨, T (b )−T (b ′))bE
= (T (a )∨, T (b ))bE − (T (a )
∨, T (b ′))bE
= (a∨,b )bE − (a
∨,b ′)bE
= (Da∨,C )V − (Da∨,C )V = 0.
48
Since R spans bE , and R ′ spanscE ′, and T |R : R→R ′ is a bijection, we observe that (a ′, T (b−b ′))bE = 0
for all a ′ ∈cE ′. This implies that T (b −b ′) ∈cE ′ is a contant function on E ′. Hence by linearity Tmaps constant function to constant functions.
(iii) Now if R 'R ′, realized by the linear isomorphism T , then it is easy to check that
T ◦wa ◦T−1 =wT (a ) (3.2.1)
on cE ′ for all a ∈ R by evaluating both sides of the equation at T (b ) for b ∈ R . Since T ◦wa ◦wb ◦T−1 = T ◦wa ◦ T ◦ T−1 ◦wb ◦ T−1 = wT (a ) ◦wT (b ) and T |R : R → R ′ is a bijection, we have thesurjective group homomorphism eT : W (R) → W (R ′), w 7→ T ◦w ◦ T−1. On the other hand, thesurjective group homomorphism W (R ′)→W (R), w 7→ T−1 ◦w ◦T is clearly the inverse of eT , henceW (R)∼=W (R ′).
Let c ∈ bE and c ′ ∈cE ′ be constant one functions, then by (ii) of Proposition 3.2.10 T (c )∈R6=0c ′.
Definition 3.2.11. A linear isomorphism T : bE∼−→cE ′ that realizes a similarity between affine root
systems R ⊂ bE and R ′ ⊂cE ′ such that T (c )∈R>0c ′ is said to be a similarity transformation betweenR and R ′. Furthermore, if T (c ) = c ′ then we will call T a normalized similarity transformation.
Example 3.2.12. (i) From (i) of Example 3.2.9 we observe that w ∈W (R) is a normalized similaritytransformation of R with itself.
(ii) For an affine root system R and λ ∈ R6=0 define λR := {λa }a∈R , then R ' λR realized by the
similarity transformation T : bE∼−→ bE , a 7→ |λ|a . The affine root system λR is called a rescaling of
R .(iii) If the linear isomorphism T : bE
∼−→ cE ′ realizes a similarity between affine root systemsR ⊂ bE and R ′ ⊂cE ′, then −T also realizes a similarity between R and R ′. Furthermore, either T or−T must be a similarity transformation. We conclude that if R ' R ′, then there exists a similaritytransformation realizing a similarity between R and R ′.
Example 3.2.12 shows that we may always assume without loss of generality that a similaritybetween two affine root systems is realized by a similarity transformation. Furthermore, it turnsout that each similarity transformation T : bE
∼−→cE ′ can be given in terms of a unique affine linearisomorphismψ : E ′→ E .
Proposition 3.2.13. Let T : bE∼−→cE ′ be a similarity transformation realizing a similarity between
the affine root systems R ⊂ bE and R ′ ⊂cE ′, then there exists λ∈R>0 and an affine linear isomorphismψ : E → E ′ such that T (a ) =λ(a ◦ψ−1) for all a ∈ bE .
Proof. Consider two affine root systems R ⊂ bE and R ′ ⊂cE ′ such that R ' R ′ realized by the sim-ilarity transformation T . Let c (resp. c ′) denote the constant one function on E (resp. E ′), thenT (c ) = λc ′ for some λ ∈ R>0. Write T = λ eT where eT : bE
∼−→cE ′ is the linear isomorphism definedby eT (a ) = λ−1T (a ) that sends c to c ′. Choose and origin x ∈ E (resp. y ∈ E ′) and identify E (resp.E ′) with V (resp. V ′) as Ex (resp. E ′y ). Further, identify bE (resp. cE ′) with E ∗x ⊕Rc (resp. E ′y
∗⊕Rc ′)
as in the proof of Proposition 3.1.12. Next, we consider the restriction T ′ := eT |E ∗x : E ∗x → E ′y∗⊕Rc ′,
and let py : E ′y∗ ⊕Rc ′ → E ′y
∗ (resp. pr : E ′y∗ ⊕Rc ′ → Rc ′) be the linear projection onto E ′y
∗ (resp.Rc ′) along the direct sum. Similarly to the proof of surjectivity in Proposition 3.1.22, we obtain thatS := py ◦T ′ : E ∗x
∼−→ E ′y∗ is a linear isomorphism and γ := pr ◦T ′ : E ∗x →R is a linear map such that
T ′(α) = S(α) + γ(α)c ′ for all α ∈ E ∗x . Hence there exists a linear automorphism σ : Ex∼−→ E ′y such
that S(α) = α ◦σ−1 for α ∈ E ∗x . By (iii) of Example 3.1.6, σ is an affine linear automorphism of E .
49
Further, there exists a vector η ∈ V such that γ(α) = (Dα,η)V for all α ∈ E ∗x ⊂ bE . Analogously to theproof of surjectivity in Proposition 3.1.22, we find that eT (a ) = a ◦ψ−1 for a ∈ bE , whereψ : E → E ′
is defined to be the affine linear isomorphism σ ◦ tη. Thus T (a ) = λ eT (a ) = λ(a ◦ψ−1) for all a ∈ bEwhich concludes the proof.
Let H := {Ha : a ∈ R} (resp. H ′ := {Ha ′ : a ′ ∈ R ′}) be the collection of affine hyperplanes inE (resp. E ′) induced by R ⊂ bE (resp. R ′ ⊂cE ′). Then a similarity transformation realizing R ' R ′
induces a bijection betweenH andH ′.
Corollary 3.2.14. Let T : bE∼−→cE ′ be a similarity transformation realizing a similarity between the
affine root systems R ⊂ bE and R ′ ⊂cE ′, and let λ > 0 and ψ : E → E ′ the affine linear isomorphismsuch that T (a ) = λ(a ◦ψ−1) for all a ∈ bE . Then T induces a bijection fromH ontoH ′ given by themapping Ha 7→ψ(Ha ) =HT (a ).
Proof. Let H ∈H , then there exists a ∈ R (not unique) such that H =Ha . For a ∈ R there exists aunique affine root a ′ ∈ R ′ such that a ′ = T (a ), because T |R : R → R ′ is a bijection. We observe thatthe mapping Ha 7→HT (a ) fromH to is well defined and yields a bijection fromH ontoH ′. Finally,let λ > 0 andψ : E → E ′ the affine linear isomorphism such that T (a ) = λ(a ◦ψ−1) for all a ∈ bE asin Proposition 3.2.13, then
ψ(Ha ) =ψ({x ∈ E : a (x ) = 0})= {y ∈ E ′ :λ(a ◦ψ−1) = 0}=HT (a ).
3.2.3 Affine root subsystems and irreducibility
Next we will introduce the notion of an affine root subsystem of an affine root system, and wewill show that each affine root subsystem can be viewed as affine root system in a natural way.Further, we show how affine root systems decompose uniquely into elementary affine root sub-systems called irreducible affine root subsystems.
Definition 3.2.15. A nonempty subset R ′ of an affine root system R on an affine Euclidean space Eis called an affine root subsystem of R if wa (R ′)⊂ R ′ for all a ∈ R ′ and if for each gradient C ∈ {Db :b ∈R ′} there are at least two distinct affine roots in R ′ with C as gradient.
Example 3.2.16. Trivially R is an affine root subsystem of itself. Further, it is not difficult to seethat both R i nd and Ru nm are reduced affine root subsystems of the affine root system R .
Remark 3.2.2. Let T : bE →cE ′ be a linear isomorphism realizing a similarity between the affine rootsystems R ⊂ bE and R ′ ⊂ cE ′. For eR ⊂ R an affine root subsystem we define eR ′ := T (eR) ⊂ R ′. Leta ′,b ′ ∈ eR ′, then there exist unique a ,b ∈ eR such that a ′ = T (a ) and b ′ = T (b ). Thus by (3.2.1) wehave
wa ′ (b ′) =wT (a )(T (b )) = (T ◦wa ◦T−1)(T (b )) = T (wa (b )).
Since wa (b )∈ eR by Definition 3.2.15, we observe that wa ′ (b ′) = T (wa (b ))∈ eR ′.Let C ∈ {Da ′ : a ′ ∈ eR ′}, then there exists b ′ ∈ eR ′ such that Db ′ = C . Furthermore, there exists a
unique b ∈ eR such that b ′ = T (b ). By Definition 3.2.15 we observe that there exists a ∈ eR distinctof b such that Da =Db , therefore b −a is a constant function on E . This implies that T (b −a ) =
50
T (b )−T (a ) is a constant function onfE ′ because T sends constant function to constant functions.We conclude that DT (a ) = DT (b ) = C , hence eR ′ ⊂ R ′ is an affine root subsystem. In this sensesimilarity respects affine root subsystems.
In the following we want to show that we can identify an affine root subsystem R ′ of an affineroot system R on E with an affine root system eR ′ on an affine Euclidean space E ′ in a natural way. IfR ′ ⊂R is an affine root subsystem such that R ′ spans bE , then the gradients {Db : b ∈R ′} of R ′ spanthe space of translation V of E by Proposition 3.1.12. Furthermore, it follows straightforwardlyfrom Definition 3.2.15 that R ′ itself is an affine root system on E . Now assume that R ′ does notspan bE , then the gradients {Db : b ∈ R ′} of R ′ do not span V . Moreover, because R ′ ⊂ R is anonempty set of nonisotropic vector contained in bE , the gradients of R ′ span a nontrivial propersubspace V ′ of V .
Let V ′ ⊂ V be a nontrivial proper subspace, and let V ′⊥ = {v ∈ V : (v, w ) = 0 for all w ∈ V ′} bethe orthogonal complement of the subspace V ′ in V . Then V decomposes as the direct sum ofsubspaces V ′⊕V ′⊥. The group action of V on E induces an action of the subgroup V ′⊥ (resp. V ′)of V on E . Define E ′ := {x +V ′⊥ : x ∈ E } (resp. E ′′ := {x +V ′ : x ∈ E }) to be the set of orbits of Eunder the action of the subgroup V ′⊥ ⊂V (resp. V ′ ⊂V ) on E .
Proposition 3.2.17. The set E ′ (resp. E ′′) can be viewed as affine Euclidean space with space oftranslation V ′ (resp. V ′⊥) in a canonical way such that the map p ′ : E → E ′, x 7→ x + V ′⊥ (resp.p ′⊥ : E → E ′′, x 7→ x +V ′) is affine linear with Dp ′ : V → V ′ (resp. Dp ′⊥ : V → V ′⊥) the orthogonalprojection onto V ′ (resp. V ′⊥).
Proof. Let x , y ∈ E such that x and y lie in the same V ′⊥-orbit, hence there exists v ′ ∈V ′⊥ such thatx +v ′ = y . Then also (x +v )+v ′ = (x +v ′)+v = y +v for any v ∈V , hence x +v and y +v lie in thesame V ′⊥-orbit. Thus the action of V on E implies an action of V on E ′, namely for an orbit O ∈ E ′
with representative x ∈ E and any v ∈V we have O +v = (x +V ′⊥)+v := (x +v )+V ′⊥. In a similarfashion, one observes that V acts on E ′′, namely for an orbit O ′ ∈ E ′′ with representative y ∈ E andany v ′ ∈V we have O ′+v ′ = (y +V ′)+v ′ := (y +v ′)+V ′.
Now let O,O ′ ∈ E ′ be distinct orbits, then we can choose representatives x , y ∈ E such thatO = x +V ′⊥ and O ′ = y +V ′⊥ with y −x = v ′ for some v ′ ∈V ′. Hence O +v ′ =O ′ which shows thatthe action of V on E ′ is transitive, moreover the restricted action of the subgroup V ′ ⊂ V on E ′ istransitive. Further, for v ∈V we have v = v1+v2 for unique v1 ∈V ′ and v2 ∈V ′⊥, so
O +v = (x +V ′⊥)+v
= ((x +V ′⊥)+v2)+v1
= ((x +v2)+V ′⊥)+v1 (3.2.2)
= (x +V ′⊥)+v1
=O +v1.
Since V ′ ⊂ V is a nontrivial proper subspace this implies that the group action of V on E ′ is notfaithful. Now let w , w ′ ∈ V ′ such that O +w = (x + V ′⊥) +w = (x + V ′⊥) +w ′ = O +w ′, then(x +w ) + V ′⊥ = (x +w ′) + V ′⊥. Thus there exists a vector z ∈ V ′⊥ such that x +w = x +w ′ + z .Since the action of V on E is simply transitive we have w =w ′+ z , hence z = 0. We conclude thatthe action of V ′ on E ′ is simply transitive. Analogously one observes that the action of V ′⊥ on E ′′
induced by the action of V on E ′′ is simply transitive. Also for x ∈ E and v ∈V such that v = v1+v2
51
for v1 ∈V ′ and v2 ∈V ′⊥ we have
(x +V ′)+v = ((x +V ′)+v1)+v2
= ((x +v1)+V ′)+v2 (3.2.3)
= (x +V ′)+v2.
Let x , y ∈ E , then there exists a unique v ∈V such that y −x = v . Now v = v1+v2 for unique v1 ∈V ′ and v2 ∈ V ′⊥. We turn E ′ into a metric space with distance function d ((x +V ′⊥), (y +V ′⊥))E ′ :=p
(v1, v1)V ′ , where (., .)V ′ is the positive definite symmetric bilinear form on the vector space V ′
induced by the by the scalar product on V ′ ⊂ V . Notice that the metric d (., .)E ′ is well definedon the pairs orbits of E ′, because different representatives of the same orbit only differ by vectorsin V ′⊥. Similarly, we turn E ′′ into a metric space with distance function d ((x +V ′), (y +V ′))E ′′ :=p
(v2, v2)V ′⊥ , where (., .)V ′⊥ is the positive definite symmetric bilinear form on the vector space V ′⊥
induced by the by the scalar product on V ′⊥ ⊂V . In this setting E ′ (resp. E ′′) is an affine Euclideanspace with space of translations V ′ (resp. V ′⊥). Furthermore, from (3.2.2) (resp. (3.2.3)) we observethat the mapping p ′ : E → E ′, x 7→ x + V ′⊥ (resp. p ′⊥ : E → E ′′, x 7→ x + V ′) is affine linear withDp ′ ∈HomR(V, V ′) (resp. Dp ′⊥ ∈HomR(V, V ′⊥)) the orthogonal projection of V = V ′⊕V ′⊥ onto V ′
(resp. V ′⊥) along the direct sum.
Next we turn the set E ′× E ′′ into an affine Euclidean space with space of translations V ′⊕V ′⊥
by defining the action of V ′⊕V ′⊥ on E ′×E ′′ componentwise by
(x1,x2)+ (v1, v2) = (x1+v1,xn +vn ) (3.2.4)
where x1 ∈ E ′, x2 ∈ E ′′, v1 ∈V ′ and v2 ∈V ′⊥. Since the componentwise actions are simply transitivethe action of V ′⊕V ′⊥ on E ′× E ′′ is simply transitive. Further, define the inner product (., .)V ′⊕V ′⊥
on V ′⊕V ′⊥ by((u 1, u 2), (v1, v2))V ′⊕V ′⊥ = (u 1, v1)V ′ +(u 2, v2)V ′⊥ (3.2.5)
where u 1, v1 ∈ V ′ and u 2, v2 ∈ V ′⊥. This turns E ′× E ′′ into an affine Euclidean space with V ′⊕V ′⊥
as space of translations.
Lemma 3.2.18. The map p : E → E ′×E ′′ sending x ∈ E to its orbits (p ′(x ), p ′⊥(x )) = (x +V ′⊥,x +V ′)is an affine linear isomorphism.
Proof. Since both p ′ : E → E ′ and p ′⊥ : E → E ′′ are affine linear maps, it follows straightforwardly
that p is an affine linear map with Dp : V∼−→ V ′⊕V ′⊥, v 7→ (Dp ′(v ), Dp ′⊥(v )) = v a linear isomor-
phism. Because Dp is a linear isomorphism (iii) of Example 3.1.6 shows that p is an affine linearisomorphism.
LetcE ′ be the space of affine linear functions on E ′ endowed with the bilinear form (., .)cE ′ defined
by (a ′,b ′)cE ′ := (Da ′, Db ′)V ′ for all a ′,b ′ ∈cE ′. Then for a ′ ∈cE ′, x ∈ E and v = v1+ v2 ∈ V ′⊕V ′⊥ = V
we obtain using (3.2.2) that
(a ′ ◦p ′)(x +v ) = a ′((x +v )+V ′⊥)
= a ′((x +V ′⊥)+v )
= a ′((x +V ′⊥)+v1)
= a ′(x +V ′⊥)+ (Da ′, v1)V ′ (3.2.6)
52
where the last equation follows from affine linearity of a ′ on E ′. Then a ′(x +V ′⊥) + (Da ′, v1)V ′ =(a ′ ◦p ′)(x ) + (Da ′, v )V . Hence (a ′ ◦p ′) is an affine linear function on E with gradient D(a ′ ◦p ′) =Da ′. Furthermore (a ′ ◦ p ′) 6= (b ′ ◦ p ′) for distinct a ′,b ′ ∈cE ′. So we can define the injective linearhomomorphism π : cE ′ ,→ bE by π(a ′) = a ′ ◦ p ′. Notice that the subspace π(cE ′) of bE contains allconstant functions on bE . Indeed, the constant one function c ′ : E ′ → R gets sent to the constantone function π(c ′) = c ′ ◦p ′ : E →R on E which we will denote by c . As usual we write (., .)
bE for thebilinear form on bE , and (., .)V for the inner product on V . Then for a ′,b ′ ∈cE ′ we have
(π(a ′),π(b ′))bE = (Dπ(a
′), Dπ(b ′))V= (Da ′, Db ′)V ′
= (a ′,b ′)cE ′ . (3.2.7)
Hence the map π preserves the forms oncE ′ and bE respectively.We return to the specific situation that V ′ ⊂ V is subspace spanned by the gradients of the
affine roots contained in the affine root subsystem R ′ ⊂ R . By Proposition 3.1.12, cE ′ is a vectorspace of dimension dim(V ′)+1, hence the injectivity of the linear mapπ guarantees thatπ(cE ′)⊂ bEis a subspace of the same dimension. Furthermore one observes that {a ∈ bE : Da ∈ V ′} ⊂ bE isa subspace, and by the isomorphism of Proposition 3.1.12 it is clear that {a ∈ bE : Da ∈ V ′} is ofdimension dim(V ′) + 1. We observe from (3.2.6) that π(cE ′)⊂ {a ∈ bE : Da ∈ V ′}, thus we may evenconclude that
π(cE ′) = {a ∈ bE : Da ∈V ′}. (3.2.8)
Finally, R ′ ⊂ {a ∈ bE : Da ∈V ′} by definition of V ′, so R ′ ⊂π(cE ′).
Proposition 3.2.19. The set eR ′ :=π−1(R ′)⊂cE ′ is an affine root system on E ′.
Proof. Since π preserves the forms on cE ′ and bE it is clear that eR ′ is a nonempty subset of non-isotropic vectors ofcE ′. Now we will check the five conditions of Definition 3.2.1.
(1) To show that eR ′ spans cE ′ it suffices to show that R ′ spans π(cE ′) = {a ∈ bE : Da ∈ V ′}. Fora ′ ∈ R ′ there exists a ∈ R ′ distinct from a ′ such that Da =Da ′ by Definition 3.2.15. Choose x ∈ Eand identify {a ∈ bE : Da ∈ V ′} with V ′⊕Rc using Proposition 3.1.12. Then clearly a and a ′ spanRDa ⊕Rc , hence R ′ spans V ′⊕Rc . We observe that eR ′ satisfies criterion (1) of Definition 3.2.1.
(2) Let a ′,b ′ ∈ eR ′, then there exist a ,b ∈ R ′ such that a = π(a ′) and b = π(b ′). Hence looking atorthogonal reflections incE ′ we observe
wa ′ (b ′)(3.1.15)= b ′− (a ′∨,b ′)
cE ′ a′
=π−1(b )− (π−1(a )∨,π−1(b ))bE π−1(a )
=π−1(b − (π−1(a )∨,π−1(b ))bE a ) (3.2.9)
(3.2.7)= π−1(b − (a∨,b )bE a )
(3.1.15)= π−1(wa (b )).
Since wa (b ) ∈ R ′ by Definition 3.2.15, we observe that wa ′ (b ′) = π−1(wa (b )) ∈ bR ′. Therefore crite-rion (2) of Definition 3.2.1 is satisfied
(3) Since π preserves the forms oncE ′ and bE it is immediate that criterion (3) of Definition 3.2.1is satisfied.
(4) Let W ′ ⊂ W (R) be the subgroup generated by the reflections wa such that a ∈ R ′, and letW ( eR ′) be the group generated by the orthogonal reflections wa ′ of E ′ for a ′ ∈ eR ′. Let wa ∈W ′ for
53
some a ∈ R ′ and consider wa ∈ GL(E ) (see Proposition 3.1.22), then by (3.1.11) we have Dwa =wDa . Now Da ∈ V ′ since a ∈ R ′, thus by (3.1.5) Dwa |V ′ = wDa |V ′ : V ′ → V ′ and Dwa |V ′⊥ =wDa |V ′⊥ = idV ′⊥ . Let w ′ ∈ W ′, then w ′ = w1 ◦ · · · ◦wk with reflections w1, . . . wk ∈ W ′. Hence (ii)of Example 3.1.6 guarantees that Dw ′ = Dw1 ◦ · · · ◦Dwk which implies that Dw ′|V ′ : V ′ → V ′ andDw ′|V ′⊥ = idV ′⊥ .
First we want to establish an isomorphism between W ′ and W ( eR ′), so that we can later exploitthe fact that W (R), and hence W ′, acts properly on E to prove that W ( eR ′) acts properly on E ′. UsingProposition 3.1.22 we can consider W ′ (resp. W ( eR ′)) as subgroup of GLR,c ( bE ) (resp. GLR,c ′ (cE ′)). Letw ′ ∈W ′, then w ′ ◦π :cE ′→ bE is a linear map that sends c ′ to c . Now π(cE ′) = {a ∈ bE : Da ∈ V ′} by(3.2.8), and by Proposition 3.1.23 we have the gradient D(w ′(a )) = (Dw ′)(Da ) for all a ∈ bE . But weknow that Dw ′ ∈ GL(V ) satisfies Dw ′|V ′ : V ′ → V ′ and Dw ′|V ′⊥ = idV ′⊥ , so the image (w ′ ◦π)(cE ′)is contained in {a ∈ bE : Da ∈ V ′} = π(cE ′). Let w ∈ GLR,c ′ (cE ′) be the unique linear automorphismsuch that π ◦w =w ′ ◦π, where we consider π :cE ′
∼−→ π(cE ′) as linear isomorphism onto its image.Then it follows straightforwardly from the uniqueness of w that w ′ 7→w =:φ(w ′) defines a grouphomomorphism φ : W ′ → W ( eR ′) . Furthermore, (3.2.9) implies that φ satisfies φ(wa ) = wπ−1(a )
for all a ∈ R ′. Thus we observe from the definition of eR ′ and W ( eR ′) that the homomorphism φ issurjective.
Next we want to prove injectivity of φ. Suppose w ′ ∈ W ′ such that φ(w ′) = idcE ′ , then for all
a ′ ∈cE ′ we have π(a ) = π(φ(w ′)(a )) = w ′(π(a )). This implies that w ′|π(cE ′) = idπ(cE ′). Choosing an
origin x ∈ E and using Proposition 3.1.12, we can identify bE with V ⊕Rc = V ′ ⊕V ′⊥ ⊕Rc , wherethe subspace π(cE ′) = {a ∈ bE : Da ∈V ′} of bE is identified with V ′⊕Rc . Further, we identify E as thevector space Ex with V through the linear isomorphism Ex → V,x + v 7→ v . Then for a ∈ bE thereexist λ∈R, α∈V ′ and β ∈V ′⊥ such that a =α+β +λc and a (x +v1+v2) = (α, v1)V ′+(β , v2)V ′⊥+λfor all v1 ∈ V ′ and v2 ∈ V ′⊥. We already know that w ′|π(cE ′) = idπ(cE ′) = idV ′⊕Rc , so by linearity of w ′
we only need to show that w ′|V ′⊥ = idV ′⊥ to prove injectivity ofφ.By Proposition 3.1.21 we have the identification of GL(E ) with t (V )oGLR(V ) using x ∈ E as
origin. Hence there exists a unique u ∈ V such that w ′−1 = tu ◦Dw ′−1. For the proof that w ′|V ′⊥ =idV ′⊥ we need to establish that u ∈ V ′. Since w ′−1 ∈W ′ we can write w ′−1 = wa 1 ◦ · · · ◦wa r witha i ∈ R ′ for 1 ≤ i ≤ r . Then by (3.1.13) we have wa i = t−a i (x )Da∨i
◦wDa i for each i . Notice that−a i (x )Da∨i ∈V ′ because a i ∈R ′. Put u i :=−a i (x )Da∨i ∈V ′, then
w ′−1 = tu 1 ◦wDa 1 ◦ tu 2 ◦wDa 2 ◦ · · · ◦ tu r−1 ◦wDa r−1 ◦ tu r ◦wDa r .
We can rewrite this as
w ′−1 = tu 1 ◦ (wDa 1 ◦ tu 2 ◦w−1Da 1) ◦ (wDa 1 ◦wDa 2 ◦ tu 3 ◦w−1
Da 2◦w−1
Da 1) ◦ . . .
◦ (wDa 1 ◦ · · · ◦wDa r−1 ◦ tu r ◦w−1Da r−1
◦ · · · ◦w−1Da 1) ◦ (wDa 1 ◦wDa 2 ◦ · · · ◦wDa r )
Using (3.1.14) and the fact that Dw ′−1 =wDa 1 ◦wDa 2 ◦ · · · ◦wDa r we can write
w ′−1 = tu 1 ◦ twDa 1 (u 2) ◦ · · · ◦ twDa 1 (wDa 2 (...wDa r−1 (u r )... )) ◦Dw ′−1
= tu 1+wDa 1 (u 2)+···+wDa 1 (wDa 2 (...wDa r−1 (u r )... )) ◦Dw ′−1.
Since u ∈ V such that w ′−1 = tu ◦ Dw ′−1 is unique, we must have u = u 1 + wDa 1 (u 2) + · · · +wDa 1 (wDa 2 (. . . wDa r−1 (u r ) . . . )). As we have already seen a i ∈ R ′ implies that wDa i : V ′ → V ′, sobecause u 1, . . . , u r ∈V ′ we observe that
u = u 1+wDa 1 (u 2)+ · · ·+wDa 1 (wDa 2 (. . . wDa r−1 (u r ) . . . ))∈V ′.
54
Now let us show that w ′(β ) =β forβ ∈V ′⊥, then by linearity of w ′we will have w ′ = idV ′⊕V ′⊥⊕Rc =idbE . Notice that Dw ′|V ′ : V ′ → V ′ and Dw ′|V ′⊥ = idV ′⊥ implies that Dw ′−1|V ′ : V ′ → V ′ and
Dw ′−1|V ′⊥ = idV ′⊥ . For v ∈ V write v = v1 + v2 for unique v1 ∈ V ′ and v2 ∈ V ′⊥, then we havew ′−1(v ) = (tu ◦Dw ′−1)(v ) =Dw ′−1(v1)+v2+u . So
w ′(β )(v ) = (β ◦w ′−1)(v )
= (β , Dw ′−1(v1)+v2+u )V .
Further, β ∈ V ′⊥ and Dw ′−1(v1), u ∈ V ′, so w ′(β )(v ) = (β , v2)V = (β , v )V = β (v ). This shows thatw ′ = idV ′⊕V ′⊥⊕Rc = id
bE which implies thatφ is injective. Thusφ is a group isomorphism.Before we can conclude the proof we want to observe how the characterizing formula w ′ ◦π=
π◦φ(w ′) ofφ for w ′ ∈GLR,c ( bE ) translates to a formula for w ′ considered in GL(E ). Let a ′ ∈ eR ′ thenthere exists a unique a ∈ R ′ such that a =π(a ′) = a ′ ◦p ′. By (3.1.9) and the properties of the affinelinear map p ′ : E → E ′ we have for x ∈ E
p ′(wa (x )) = p ′(x −a (x )Da∨)
= p ′(x )−a (x )Dp ′(Da∨)
= p ′(x )−a (x )Da ′∨
= p ′(x )−a ′(p (x ))Da ′∨
= x +V ′⊥−a ′(x +V ′⊥)Da ′∨
=wa ′ (x +V ′⊥) =wa ′ (p ′(x )),
where we used that Da ∈V ′ since a ∈R ′. Then for p ′⊥ we obtain
p ′⊥(wa (x )) = p ′⊥(x −a (x )Da∨)
= p ′⊥(x )−a (x )Dp ′⊥(Da∨)
= p ′⊥(x ).
Thus p ′ ◦wa =wa ′ ◦p ′ and p ′⊥ ◦wa = p ′⊥. Moreover, one can directly compute in a similar fashion
that p ′ ◦φ(w )−1 =w ◦p ′ and p ′⊥ ◦φ(w )−1 = p ′⊥ for all w ∈W ( eR ′).Finally, we can deduce condition (4): Let K ′1, K ′2 ⊂ E ′ and K ′′ ⊂ E ′′ be compact. Suppose w ∈
W ( eR ′) such that w (K ′1)∩K ′2 6= ;, then also (w (K ′1)×K ′′)∩ (K ′2×K ′′) 6= ;. The map p : E → E ′× E ′′
is an affine linear isomorphism, hence Corollary 3.1.8 implies that p is a homeomorphism. PutK1 := p−1(K ′1×K ′′)⊂ E and K2 := p−1(K ′2×K ′′)⊂ E , then both K1 and K2 are compact in E since pis an affine linear isomorphism. Further
p ′(φ−1(w )(K1)) =w (p ′(K1)) =w (K ′1)
andp ′⊥(φ
−1(w )(K1)) = p ′⊥(K1) = K ′′,
hence φ−1(w )(K1) ∩ K2 6= ;. This only holds for only finitely many w ∈ W ( eR ′) since W (R) actsproperly on E and φ is an isomorphism. Thus there exist only finitely many w ∈W ( eR ′) such thatw (K ′1)∩K ′2 6= ;. We conclude that W ( eR ′) acts properly on E ′, hence criterion (4) of Definition 3.2.1is holds for eR ′.
(5) Let a ′ ∈ eR ′, then a := π(a ′) ∈ R ′ so by (5) of Definition 3.2.15 there exists b ∈ R ′ such thatDa =Db . This implies that a−b is a constant function on E . Put b ′ :=π−1(b ), then a ′ 6=b ′ in eR ′ by
55
injectivity ofπ. Furtherπ−1(a−b ) =π−1(a )−π−1(b ) = a ′−b ′must also be a constant function sinceπ sends constant functions to constant functions. Thus Da ′ =Db ′ which proves that criterion (5)of Definition 3.2.1 holds for eR ′.
This shows that eR ′ is an affine root system on E ′ with affine Weyl group isomorphic to the sub-group W ′ ⊂W (R).
Definition 3.2.20. We will call the affine root system eR ′ on E ′ as constructed in this Subsection theaffine root system associated to the affine root subsystem R ′. The form preserving injective linearmap π :cE ′ ,→ bE that sends eR ′ bijectively onto R ′ as in the previous is called the realizing map of eR ′.If R ′ ⊂R ⊂ bE is already an affine root system on E , then R ′ is its own associated affine root system.
Lemma 3.2.21. If there exists a nonempty subset R ′ of an affine root system R on E such that(a ,b )
bE = 0 for all a ∈R ′ and b ∈R \R ′, then both R ′ and R \R ′ are affine root subsystems of R.
Proof. Let a ′,b ′ ∈R ′, then (a ′, a )bE = (b
′, a )bE = 0 for all a ∈R \R ′. Since R ⊂ bE is a nonempty subset
of nonisotropic vectors which span bE there must exist an affine root b ∈R such that (wa ′ (b ′),b )bE 6=
0, else wa ′ (b ′)∈R would be isotropic. Now wa ′ (b ′) =b ′−(a ′∨,b ′)bE a ′, so we obtain that (wa ′ (b ′), a )
bE =(b ′, a )
bE − (a ′∨,b ′)bE (a
′, a )bE = 0 for all a ∈ R \ R ′. This implies that there exists b ∈ R ′ such that
(wa ′ (b ′),b )bE 6= 0, hence wa ′ (b ′)∈R ′ for all a ′,b ′ ∈R ′.
Further, let a ′ ∈R ′, then there exists an affine root a ∈R distinct from a ′ such that Da =Da ′ by(5) of Definition 3.2.1. Because R consists only of nonisotropic vectors contained in bE we obtain(a , a ′)
bE = (Da , Da ′)V = (Da , Da )V = (a , a )bE 6= 0. The definition of R ′ now implies that a ∈R ′, so for
each gradient of an affine root of R ′ there exist at least two distinct affine roots in R ′ with the samegradient. This shows that R ′ ⊂ R is an affine root subsystem. Furthermore, since this argument issymmetric in R ′ and R \R ′ we observe that R \R ′ is also an affine root subsystem of R .
Lemma 3.2.21 invokes the following Definition.
Definition 3.2.22. An affine root system R on E is said to be reducible if there exist nonemptysubsets R ′, R ′′ ⊂ R such that R = R ′ q R ′′, and (a ,b )
bE = 0 for all a ∈ R ′ and b ∈ R ′′. It is calledirreducible otherwise. An affine root subsystem R ′ ⊂ R is said to be (ir)reducible if the affine rootsystem eR ′ associated to R ′ is (ir)reducible.
Example 3.2.23. (i) Let R be an affine root system on E of rank 1 and assume that R is reducible.Hence there exist nonempty subsets R ′, R ′′ ⊂ R such that R = R ′qR ′′ and (a ,b )
bE = 0 for all a ∈ R ′
and b ∈ R ′′. Let a ∈ R ′ and b ∈ R ′′ then Da , Db 6= 0, because R consists of nonisotropic vectorscontained in bE . Since R is of rank 1 the space of translations V of E is of dimension 1. This impliesthat (a ,b )
bE = (Da , Db )V 6= 0 which contradicts the assumption that R is reducible. We concludethat an affine root system R on E of rank 1 is always irreducible.
(ii) Let R (resp. R ′) be an affine root system on E (resp. E ′), and let T : bE → cE ′ be a lin-ear isomorphism realizing a similarity between R and R ′. Assume that R is irreducible, and R ′
is reducible. Then R ′ = R ′1 q R ′2 such that (a ′,b ′)cE ′ = 0 for all a ′ ∈ R ′1 and b ′ ∈ R ′2. Now R =
T−1(R ′1)qT−1(R ′2), and for a ′ ∈ R ′1 and b ′ ∈ R ′2 we have (T−1(a ′)∨, T−1(b ′))bE = (a
′∨,b ′)cE ′ = 0. This
implies that (T−1(a ′), T−1(b ′))bE = 0, or (a ,b )
bE = 0 for all a ∈ T−1(R ′1) and b ∈ T−1(R ′2). This impliesthat R is reducible which contradicts our assumptions. We conclude that both R and R ′ must beeither reducible or irreducible, so similarity respects the notion (ir)reducibility.
(iii) Example 3.2.5 (ii) made clear that the set of real roots ∆r e corresponding to the affine Liealgebra g(A) is a reduced affine root system. By (vii) of Theorem 2.3.4 it follows that ∆r e is alsoirreducible.
56
(iv) The reduced affine root system RuR0
of Example 3.2.5 (iii) is irreducible since R0 is irreducibleas we will see in Proposition 3.5.5 in slightly more detail.
Proposition 3.2.24. For an affine root system R on E there exist irreducible affine root subsystemsR1, . . . , Rm ⊂R such that R can be written uniquely as the disjoint union
∐mi=1 Ri with (a ,b )
bE = 0 foreach a ∈Ri , b ∈R j and i 6= j up to a reordering of the indices.
Proof. We proceed with induction on the rank l of R . Affine root systems of rank one must beirreducible by (i) of Example 3.2.23. Now assume we are done up to some l > 1. If R is reduciblewrite R = R ′qR ′′ such that (a ,b )
bE = 0 for all a ∈ R ′ and b ∈ R ′′. By construction of the affine rootsystem eR ′ on E ′ (resp. fR ′′ on E ′′) associated to R ′ (resp. R ′′) we observe that the rank of eR ′ (resp.fR ′′) is strictly smaller than l . Hence the induction hypothesis holds for both eR ′ and fR ′′. This meansthat we can write eR ′ = R ′1q · · ·qR ′k (resp. fR ′′ = R ′′1 q · · ·qR ′′n ) for irreducible affine root subsystems
R ′1, . . . , R ′k ⊂ eR ′ (resp. R ′′1 , . . . , R ′′n ⊂ fR ′′) such that (a ,b )cE ′ = 0 (resp. (a ,b )
ÓE ′′ = 0) for each a ∈ R ′i ,b ∈R ′j (resp. a ∈R ′′i , b ∈R ′′j ) and i 6= j .
Consider the injective linear map π′ : cE ′ ,→ bE (resp. π′′ : ÓE ′′ ,→ bE ) that realizes the affine rootsystem eR ′ (resp. fR ′′) associated to R ′ (resp. R ′′). We want to show that the decomposition we are
looking for is R =∐k
i=1π′(R ′i )q
∐nj=1π
′′(R ′′j ), where R ′ = π′( eR ′) =∐k
i=1π′(R ′i ) and R ′′ = π′′(fR ′′) =
∐nj=1π
′′(R ′′j ). Notice that the injectivity of both π′ and π′′ justify the disjoint unions. Furthermore,
since the realizing mapsπ′ andπ′′ are form preserving we observe that (a ,b )bE = 0 for all a ∈π′(R ′i ),
b ∈ π′(R ′j ) (resp. a ∈ π′′(R ′′i ), b ∈ π′′(R ′′j )) and i 6= j . Because R = R ′qR ′′ such that (a ′,b ′)bE = 0 for
a ′ ∈R ′ and b ′ ∈R ′′ we also have (a ,b )bE = 0 for all a ∈π′(R ′i ) and b ∈π′′(R ′′j ).
Since both π′ and π′′ are bijections onto their image with similar arguments (but with reversedrealizing maps) as in the proofs of (2) and (5) from the proof of Proposition 3.2.19 that π′(R ′i ) fori = 1, . . . , k and π′′(R ′′j ) for j = 1, . . . , n are affine root subsystems of R . Consider the affine root
system fR ′i on E ′i associated to R ′i realized by the injective linear map π′i :cE ′i ,→cE ′ for i ∈ {1, . . . , k }.Also, consider the affine root systemâπ′(R ′i ) on Eπ
′
i associated toπ′(R ′i )⊂R realized byπi :dEπ′
i ,→ bE .
ThenfR ′i gets bijectively mapped onto π′(R ′i )⊂R by π′ ◦π′i , and âπ′(R ′i ) gets bijectively mapped ontoπ′(R ′i )⊂R by πi (see the following diagram).
cE ′i ⊃fR′i
π′i−−−→ R ′i ⊂ eR ′ ⊂cE ′
yπ′
dEπ′
i ⊃âπ′(R′i )
πi−−−→ π′(R ′i )⊂R ′ ⊂ bE
BecausefR ′i ⊂cE′i (resp. âπ′(R ′i )⊂
dEπ′
i ) is an affine root system, itsR-span is all ofcE ′i (resp. dEπ′
i ). Then
by linearity we notice that π′(π′i (cE ′i )) = πi (dEπ
′
i ). So considering πi as a linear isomorphism on its
image, we obtain the linear isomorphism π−1i ◦π′ ◦π
′i : cE ′i
∼−→dEπ′
i that maps fR ′i bijectively ontoâπ′(R ′i ). Furthermore, π−1
i , π′ and π′i are form preserving, so we observe that π−1i ◦π′ ◦π
′i realizes
a similarity between fR ′i and âπ′(R ′i ). Since fR ′i is irreducible by assumption, we observe from (ii)
of Example 3.2.23 that âπ′(R ′i ) is also irreducible for i ∈ {1, . . . , k }. In a similar fashion one obtains
that äπ′′(R ′′j ) is also irreducible for j ∈ {1, . . . , n}. We conclude that R has a decomposition intoirreducible root subsystems as stated in the Proposition.
Next, we want to prove the uniqueness of this decomposition. Assume we can write R =∐m
i=1 Ri
and R =∐n
j=1 R ′j for irreducible affine root subsystems Ri , R ′j ⊂ R (i ∈ {1, . . . , m }, j ∈ {1, . . . , n})
57
such that (a ,b )bE = 0 for all a ∈ Ri ,b ∈ Rk (resp. a ∈ R ′i ,b ∈ R ′k ) and i 6= k . Further, assume
that after reordering the indices there exists q > 1 such that R1 ⊂ R ′1 q · · · q R ′q and R1 ∩ R ′j 6= ;for j = 1, . . . ,q . Consider the affine root system fR1 on E1 associated to R1 with realizing mapπ : cE1 ,→ bE . Since π preserves the bilinear forms on cE1 and bE respectively, we observe thatfR1 = π−1(R ′1 ∩ R1)q · · · qπ−1(R ′q ∩ R1) with π−1(R ′k ∩ R1) 6= ; for k = 1, . . . ,q and (a ′,b ′)
cE1= 0 for
all a ′ ∈π−1(R ′i ∩R1),b ′ ∈π−1(R ′j ∩R1) and i 6= j . This implies that fR1 is reducible which contradicts
the assumption that fR1 is irreducible. We conclude that for each i there exists a unique j such thatRi ⊂ R ′j . Since the argument is symmetric in Ri and R ′j , we also have that for every j there exists aunique k such that R ′j ⊂ Rk . But then for each i there exist unique j and k such that Ri ⊂ R ′j ⊂ Rk
which implies that i = k and Ri = R ′j . Furthermore we obtain m = n , and after reordering theindices it becomes clear that Ri =R ′i for all i = 1, . . . , n .
The unique decomposition of R into∐m
i=1 Ri as in Proposition 3.2.24 is called the orthogonaldecomposition of R .
3.2.4 The direct sum of affine root systems
In this Subsection we construct new affine root systems by joining affine root systems together ina natural way called a direct sum. Further, we will show that each affine root system is similar toa direct sum of irreducible affine root systems, and that one only needs to study a complete set ofrepresentatives of the similarity classes of irreducible affine root systems to understand affine rootsystems up to similarity.
Let E1 (resp. E2) be an affine Euclidean space with a finite-dimensional space of translationV1 (resp. V2) that is endowed with the inner product (., .)V1 (resp. (., .)V2 ). Also for i = 1, 2 letcE i be the space of affine linear functions on E i , endowed with the bilinear form (., .)
cE isuch that
(a i ,b i )cE i= (Da i , Db i )Vi for all a i ,b i ∈cE i . Put E := E1× E2 and V := V1⊕V2, then E turns into an
affine Euclidean space with V as space of translations as follows: The action of V on E is definedcomponentwise as
(x1,x2)+ (v1, v2) = (x1+v1,x2+v2)
where x i ∈ E i and vi ∈ Vi for i = 1, 2, and the inner product (., .)V on V is defined componentwiseas
((u 1, u 2), (v1, v2))V = (u 1, v1)V1 +(u 2, v2)V2
where x i ∈ E i and u i , vi ∈Vi for i = 1, 2. Thus E has a Euclidean metric again.For each i ∈ {1, 2} let p i : E → E i be the projection map that sends (x1,x2) to x i . It follows in
a straightforward way that p i ∈ Hom(E , E i ) for i = 1, 2. Furthermore, Dp i ∈ HomR(V, Vi ) is theorthogonal projection on the i -th coordinate of V defined by Dp i (v1, v2) = vi , where each v j ∈ Vj .For x ∈ E and v ∈ V there exist x i ∈ E i and vi ∈ Vi for i = 1, 2 such that x = (x1,x2) and v = (v1, v2).Hence for a i ∈cE i we have
(a i ◦p i )(x +v ) = (a i ◦p i )((x1,x2)+ (v1, v2))
= a i (p i ((x1+v1,x2+v2)))
= a i (x i +vi )
= a i (x i )+ (Da i , vi )Vi ,
where the last equation follows from affine linearity of a i on E i . For i = 1, 2, define ιi : Vi ,→V to be the injective linear homomorphism that embeds Vi canonically into V = V1 ⊕ V2. Then
58
a i (x i )+(Da i , vi )Vi = (a i ◦πi )(x )+(ιi (Da i ), v )V , hence (a i ◦πi ) is an affine linear function on E withgradient D(a i ◦πi ) = ιi (Da i ). Furthermore, (a i ◦πi ) 6= (b i ◦πi ) for distinct a i ,b i ∈cE i . So if we let bEdenote the space of affine linear functions on E , we can define the injective linear homomorphismπi :cE i ,→ bE by π(a i ) = a i ◦p i for i = 1, 2.
For a i ∈cE i and a ′j ∈cE j , we have
(πi (a i ),πj (a ′j )) bE = (Dπi (a i ), Dπj (a ′j ))V
= (ιi (Da i ), ιj (Da ′j ))V
=
(
(a i , a ′j )cE i(i = j ),
0 (i 6= j ).(3.2.10)
Hence the maps πi preserve the forms on cE i and bE respectively, and (π1(cE1),π2(cE2)) bE = 0. Noticethat bothπ1(cE1) andπ2(cE2) contain all constant functions on bE , because the constant one functionc i : E i → R gets sent to the constant one function c := π(c i ) = c i ◦ p i : E → R for i = 1, 2. On theother hand, if a ∈ π1(cE1) ∩π2(cE2), then (a , a )
bE must vanish by (3.2.10) which implies that a is aconstant function on E . Thus π1(cE1)∩π2(cE2) =Rc . Finally, let a ∈ bE , x i ∈ E i and vi ∈Vi for i = 1, 2,then we have
a ((x1,x2)+ (v1, v2)) = a ((x1,x2))+ (Da , (v1, v2))V= a ((x1,x2))+ (Da , ι1(v1))V +(Da , ι2(v2))V= a ((x1,x2))+ (Dp1(Da ), v1)V1 +(Dp2(Da ), v2)V2
By Proposition 3.1.12, we can choose a i ∈ cE i such that Da i = Dp i (Da ) for i = 1, 2 and a 1(x1) +a 2(x2) = a ((x1,x2)). Hence a =π1(a 1)+π2(a 2), and we observe that π1(cE1) and π2(cE2) generate bE .
Now let Ri be an affine root system on E i for i = 1, 2.
Proposition 3.2.25. The set R :=π1(R1)∪π2(R2) is an affine root system on E .
Proof. As we have seen, the injective linear maps π1 : cE1 ,→ bE and π2 : cE2 ,→ bE send constantfunctions to constant functions. Since R1 ⊂cE1 and R2 ⊂cE2 are affine root systems we observe thatR = π1(R1)∪π2(R2)⊂ bE is a subset of nonisotropic vectors. Now we will check the five conditionsof Definition 3.2.1.
(1) Since R1 ⊂ cE1 and R2 ⊂ cE2 are affine root systems we know that R1 (resp. R2) spans cE1
(resp. cE2). Recall that the subspaces π1(cE1) and π2(cE2) of bE generate bE . We observe that R =π1(R1)∪π2(R2) spans bE .
(2) For a i ∈cE i and a ′j ∈cE j we have the orthogonal reflection in bE
wπi (a i )(πj (a ′j )) =πj (a ′j )− (πi (a i )∨,πj (a ′j )) bE πi (a i )
=πj (a ′j )− (Dπi (a i )∨, Dπj (a ′j ))V πi (a i )
=πj (a ′j )− (ιi (Da i )∨, ιj (Da ′j ))V πi (a i )
=
(
πj (a ′j )− (Da∨i , Da ′j )Vi πi (a i ) (i = j ),
πj (a ′j ) (i 6= j ),
=
(
πj (wa i (a′j )) (i = j ),
πj (a ′j ) (i 6= j ),
59
where wa i is a reflection in W (Ri ). Using the fact that both R1 and R2 satisfy condition (2) ofDefinition 3.2.1 we observe that wa (b )∈R for all a ,b ∈R =π1(R1)∪π2(R2).
(3) From (3.2.10) it follows that (a∨,b )bE ∈ Z for all a ∈ π1(R1) and b ∈ π2(R2), because both
R1 ⊂cE1 and R2 ⊂cE2 satisfy condition (3) of Definition 3.2.1.(4) We want to exploit the fact that W (R1) (resp. W (R2)) acts properly on E1 (resp. E2) to prove
that W (R) acts properly on E = E1×E2. Therefore we first establish a group isomorphism betweenW (R1)×W (R2) and W (R). Consider wa ∈W (R) for some a ∈ R , then a = πi (a i ) for some a i ∈ Ri
and i ∈ {1, 2}, hence Da =Dπi (a i ) = ιi (Da i ). Let x = (x1,x2)∈ E1×E2 = E , then by (3.1.9) we have
wa (x ) = x −a (x )Da∨
= (x1,x2)−a i (p i ((x1,x2)))ιi (Da i )∨
= (x1,x2)−a i (x i )ιi (Da i )∨.
We observe that wa |E j coincides with the identity function idE j on E j ⊂ E for j 6= i , and that wa |E i
coincides with wa i ∈ W (Ri ) on E i . In other words, either wa = (wa 1 , idE2 ) or wa = (idE1 , wa 2 )where these expressions acts componentwise on E = E1 × E2. Consider the injective group ho-momorphisms ∂1 : GL(E1) ,→ GL(E ) (resp. ∂2 : GL(E2) ,→ GL(E )) given by ∂1(w1) = (w1, idE2 ) (resp.∂2(w2) = (idE1 , w2)where the last expression acts componentwise on E = E1×E2. Then ∂i restrictsto an injective group homomorphism W (Ri ) ,→ W (R) for i = 1, 2 such that ∂i (wa i ) = wπi (a i ) for
a i ∈ Ri . Thus we obtain the group isomorphism ∂ : W (R1)×W (R2)∼−→W (R) sending (w1, w2) to
∂1(w1) ◦ ∂2(w2).Assume that W (R) does not act properly on E , then there exist compact subsets K1 and K2 of E
such that w (K1)∩K2 6= ; for infinitely many w ∈W (R). Notice that E with its Euclidean topology ishomeomorphic to E1× E2 with the to the product topology of the Euclidean topologies on the E i .For i = 1, 2, this turns p i : E → E i into a continuous function by definition of the product topologyof E = E1× E2, hence the projected sets p i (K1) and p i (K2) in E i are compact. Let w ∈W (R) suchthat w (K1)∩K2 6= ;, then by ∂ we have the unique expression w = ∂1(w1) ◦ ∂2(w2) with w i ∈W (Ri )such that w i (p i (K1)) ∩ p i (K2) 6= ; for i = 1, 2. Since W (Ri ) acts properly on E i , there only existfinitely many w i ∈ W (Ri ) such that w i (p i (K1)) ∩ p i (K2) 6= ; for i = 1, 2. Using the isomorphism∂ we observe that this can only yield finitely many w ∈ W (R) such that w (K1) ∩ K2 6= ;. Thiscontradiction tells us that W (R) acts properly on E .
(5) Let a ∈ R , then there exists i ∈ {1, 2} and a ′ ∈ Ri such that a = πi (a ′). By condition (5)Definition 3.2.1 for Ri there exists b ′ ∈ Ri distinct of a ′ such that Da ′ = Db ′. This implies thata ′ −b ′ is a constant function on E i . Put b := πi (b ′), then a 6= b in R by injectivity of πi . Furtherπi (a ′ − b ′) = πi (a ′)− πi (b ′) = a − b must also be a constant function since πi sends constantfunctions to constant functions. We conclude that Da =Db .
The constructions of this subsection inspire the following definition of a direct sum of affineroot systems.
Definition 3.2.26. The affine root system R =π1(R1)∪π2(R2) on E = E1×E2 as defined in Proposi-tion 3.2.25 is called the direct sum of the affine root systems R1 (resp. R2) on E1 (resp. E2), and wedenote it by R =R1⊕R2.
Corollary 3.2.27. If the affine root system R on E1×E2 is the direct sum of R1 (resp. R2) on E1 (resp.E2), then πi (Ri ) ⊂ R is an affine root subsystem for i = 1, 2 and (a ,b ) = 0 for all a ∈ π1(R1) andb ∈π2(R2).
60
Proof. By definition of R1 and R2, we observe that both π1(R1) and π2(R2) are nonempty subsetsof R . From the proof of condition (2) and (5) of Proposition 3.2.25 we conclude the first statement.The latter statement follows from (3.2.10).
Proposition 3.2.28. Let R be a reducible affine root system, and write R ′ qR ′′ such that (a ,b ) = 0for all a ∈ R ′ and b ∈ R ′′. Let eR ′ (resp. fR ′′) be the affine root system asociated to R ′ (resp. R ′′), thenR ' eR ′⊕fR ′′ realized by a normalized similarity transformation.
Proof. By Lemma 3.2.21 both R ′ and R ′′ are affine root subsystems of R , so the affine root systemeR ′ on E ′ (resp. fR ′′ on E ′′) asociated to R ′ (resp. R ′′) is well defined. Consider the realizing mapπ′ : cE ′ ,→ bE (resp. π′ : ÓE ′′ ,→ bE ) of eR ′ (resp. fR ′′) defined by π′(a ) = a ◦ p ′ (resp. π′′(b ) = b ◦ p ′⊥)with p ′ : E → E ′ (resp. p ′⊥ : E → E ′′) the canonical map that sends E to the space of orbits E ′ asin Proposition 3.2.17. Then p : E → E ′× E ′′, x 7→ (p ′(x ), p ′⊥(x )) is an affine linear isomorphism by
Lemma 3.2.18. Further, the map π : ÛE ′×E ′′→ bE , a 7→ a ◦p is clearly injective since p is bijective,and it follows from a direct computation that π is also linear. The dimensions of that spaces of
translations of E ′× E ′′ and E are the same, hence by Proposition 3.1.12 the dimensions of ÛE ′×E ′′
and bE are the same. This implies that π is a linear isomorphism.
Consider the direct sum eR ′⊕fR ′′ on E ′× E ′′ with canonical embeddings π1 :cE ′ ,→ÛE ′×E ′′ and
π2 : ÓE ′′ ,→ÛE ′×E ′′. Here π1(a ) = a ◦ p1 (resp. π2(b ) = b ◦ p2) where p1 : E ′ × E ′′ → E ′ (resp. p2 :E ′×E ′′→ E ′′) is the affine linear map that projects to the first (resp. second) coordinate of E ′×E ′′.Let a ∈ eR ′⊕fR ′′, say a ∈π1( eR ′), then there exists a unique a ′ ∈ eR ′ such that a =π1(a ′) = a ′ ◦p1. Weobserve that π(a ) = a ◦p = a ′ ◦p1 ◦p = a ′ ◦p ′ =π′(a ′)∈R ′. This yields a bijection of π1( eR ′) onto R ′,and in a similar manner one obtains a bijection of π2(fR ′′) onto R ′′. Thus π|
fR ′⊕ÝR ′′ : eR ′⊕fR ′′→ R is abijection. Finally, π1 and π2 are form preserving maps, so it follows that also π is form preserving.This shows thatπ realizes a similarity between R and eR ′⊕fR ′′. Furthermore, notice thatπ sends theconstant one function on E ′× E ′′ to the constant one function on E which makes it a normalizedsimilarity transformation.
Next, we would like to generalize the notion of the direct sum of two affine root system to thedirect sum of any finite amount of affine root systems. Therefore we first show associativity of thedirect sum of affine root systems, i.e. (R1⊕R2)⊕R3 =R1⊕(R2⊕R3)with Ri ⊂cE i affine root systemsfor i = 1, 2, 3.
Let Ri be an affine root system on E i with space of translations Vi for i = 1, 2, 3. Consider thedirect sum R1⊕R2 (resp. R2⊕R3) on E12 = E1×E2 (resp. E23 = E2×E3) with canonical embeddingsπ1 : cE1 ,→dE12 and π2 : cE2 ,→dE12 (resp. π′2 : cE2 ,→dE23 and π′3 : cE3 ,→dE23). Here π1(a ) = a ◦p1 (resp.π2(b ) = b ◦p2) where p1 : E12→ E1 (resp. p2 : E12→ E2) is the affine linear map that projects to thefirst (resp. second) coordinate of E12. Also π′2(a ) = a ◦p ′2 (resp. π′3(b ) = b ◦p ′3) where p ′2 : E23→ E2
(resp. p ′3 : E23 → E3) is the affine linear map that projects to the first (resp. second) coordinate ofE23. Further consider the direct sum R := (R1⊕R2)⊕R3 (resp. R ′ :=R1⊕ (R2⊕R3)) on E = E12×E3
(resp. E ′ = E1 × E23) with canonical embeddings π12 : dE12 ,→ bE and π3 : cE3 ,→ bE (resp. π′1 : cE1 ,→cE ′ and π′23 : dE23 ,→ cE ′). Here π12(a ) = a ◦ p12 (resp. π3(b ) = b ◦ p3) where p12 : E → E12 (resp.p3 : E → E3) is the affine linear map that projects to the first (resp. second) coordinate of E . Alsoπ′1(a ) = a ◦p ′1 (resp. π′23(b ) =b ◦p ′23) where p ′1 : E ′→ E1 (resp. p ′23 : E ′→ E23) is the affine linear mapthat projects to the first (resp. second) coordinate of E ′. Then R =π12(π1(R1))qπ12(π2(R2))qπ3(R3)and R ′ =π′1(R1)qπ′23(π
′2(R2))qπ′23(π
′3(R3)).
As affine Euclidean spaces we have E = E ′ = E1 × E2 × E3 with space of translations V = V1 ⊕
61
V2⊕V3 acting on E by
(x1,x2,x3)+ (v1, v2, v3) = (x1+v1,x2+v2,x3+v3)
where x i ∈ E i and vi ∈Vi for i = 1, 2, 3, and the inner product (., .)V on V is defined as
((u 1, u 2, u 3), (v1, v2, v3)V = (u 1, v1)V1 +(u 2, v2)V2 +(u 3, v3)V3
where x i ∈ E i and u i , vi ∈Vi for i = 1, 2, 3. This implies that bE =cE ′.Let a ∈ R , say a ∈ π12(π1(R1)), then there exists a unique b ∈ R1 such that a = π12(π1(b )) =
b ◦p1 ◦p12. Thus there also exists a unique a ′ ∈ π′1(R1) ⊂ R ′ such that a ′ = π′1(b ) = b ◦p ′1. Clearly,p1 ◦p12 = p ′1, so a = a ′. In this manner we observe that π12(π1(R1)) = π′1(R1). Similarly on obtainsπ12(π2(R2)) = π′23(π
′2(R2)) and π3(R3) = π′23(π
′3(R3)), so we find that R = R ′. Hence we may write
R1⊕R2⊕R3 := (R1⊕R2)⊕R3 =R1⊕(R2⊕R3). Inductively we can now define the direct sum of affineroot systems Ri for i = 1, . . . , n to be
⊕ni=1 Ri for n > 1. For n = 1 we put
⊕ni=1 Ri =R1.
Lemma 3.2.29. Let Ri and R ′i be affine root systems such that Ri ' R ′i for i = 1, 2, then there existλ1,λ2 ∈R6=0 such that R1⊕R2 'λ1R ′1⊕λ2R ′2.
Proof. Let Ri ⊂ cE i (resp. R ′i ⊂ cE′i ) be affine root systems for i = 1, 2. Consider the direct sum
R1 ⊕ R2 ⊂ÛE1×E2 (resp. R ′1 ⊕ R ′2 ⊂ÛE′1×E ′2) with canonical embeddings πi : cE i ,→ÛE1×E2 (resp.
π′i :cE ′i ,→ÛE′1×E ′2) for i = 1, 2. Here πi (a ) = a ◦p i (resp. π′i (b ) =b ◦p ′i ) where p i : E1×E2→ E i (resp.
p ′i : E ′1× E ′2→ E ′i ) is the affine linear that projects to the i -th coordinate of E1× E2 (resp. E ′1× E ′2)
for i = 1, 2. Assume that Ti : cE i∼−→ cE ′i realizes a similarity between Ri and R ′i for i = 1, 2. By (ii)
of Proposition 3.2.10 the linear isomorphism Ti sends constant functions to constant functionsfor i = 1, 2. Therefore there exists λi ∈ R6=0 such that λi Ti sends the constant one function onE i to the constant one function on E ′i for i = 1, 2. Notice that by (ii) of Example 3.2.12 the linearisomorphism λi Ti is a normalized similarity transformation realizing a similarity between Ri andthe rescaling λi R ′i of R ′i for i = 1, 2.
We will now construct a linear isomorphism T : ÛE1×E2 →ÛE ′1×E ′2 that realizes R1 ⊕ R2 'λ1R ′1⊕λ2R ′2. By Proposition 3.2.13 there exists an affine linear isomorphismψi : E i → E ′i such thatλi Ti (a ) = a ◦ψ−1
i for i = 1, 2. Consider the map p : E ′1×E ′2→ E1×E2, (x1,x2) 7→ (ψ−11 (x1),ψ−1
2 (x2)).Since bothψ1 andψ2 are affine linear isomorphisms it follows from a direct computation that also
p is an affine linear isomorphism. Define the map T : ÛE1×E2 →ÛE ′1×E ′2 by T (a ) = a ◦ p . Now Tis well defined and injective, because p is an affine linear isomorphism. Furthermore, it followsstraightforwardly that T is a linear map. Using Proposition 3.1.12 and the linear isomorphism T1
(resp. T2) we observe that the spaces of translations of E1 and E ′1 (resp. E2 and E ′2) have the same
dimension. Using Proposition 3.1.12 again this implies that ÛE1×E2 and ÛE ′1×E ′2 have the samedimension. We conclude that π is a linear isomorphism.
Let a ∈ R1⊕R2, say a ∈πi (Ri ), then there exists a unique a i ∈ Ri such that a =πi (a i ) = a i ◦p i .We observe that
T (a ) = a ◦p = a i ◦p i ◦p = a i ◦ψ−1i ◦p ′i =π
′i (λi Ti (a i ))∈π′i (λi R ′i ) (3.2.11)
for i = 1, 2. This yields a bijection T |R1⊕R2 : R1⊕R2→ λ1R ′1⊕λ2R ′2. Finally, because πi and π′i areform preserving maps, and λi Ti is a linear isomorphism realizing a similarity for i = 1, 2 it followsfrom (3.2.11) that (T (a )∨, T (b ))
ÚE ′1×E ′2= (a∨,b )
ÚE1×E2for all a ,b ∈ R1⊕R2. This shows that T realizes
a similarity between R1⊕R2 and λ1R ′1⊕λ2R ′2.
62
Now we generalize Lemma 3.2.29.
Proposition 3.2.30. Let Ri and R ′i be affine root systems such that Ri 'R ′i for i = 1, . . . , m , then thereexist λ1, . . . ,λm ∈R6=0 such that
⊕mi=1 Ri '
⊕mi=1λi R ′i .
Proof. Let Ri and R ′i be affine root systems such that Ri ' R ′i for i = 1, . . . , m . We proceed withinduction to m . For m = 1 there is nothing to prove, so assume that we are done up to m > 1.By the induction hypothesis there exist λ1, . . . ,λm−1 ∈ R6=0 such that
⊕m−1i=1 Ri '
⊕m−1i=1 λi R ′i . Put
R :=⊕m−1
i=1 Ri (resp. R ′ :=⊕m−1
i=1 R ′i ), then by Lemma 3.2.29 there exists µ,λ ∈ R \ {0} such that
R ⊕Rm 'µR ′⊕λR ′m . This leads to⊕m
i=1 Ri ' (⊕m−1
i=1 µλi R ′i )⊕λR ′m which finishes the proof.
This leads to the following special case.
Corollary 3.2.31. Let Ri and R ′i be affine root systems such that Ri ' R ′i realized by a normalizedsimilarity transformation Ti for i = 1, . . . , m . Then
⊕mi=1 Ri '
⊕mi=1 R ′i realized by a normalized
similarity transformation.
Proof. Since Ti sends the constant one function to the constant one function we have λi = 1 fori = 1, . . . , n in the proof of Lemma 3.2.29. The Corollary now follows from Proposition 3.2.30.
Finally, we want to observe how affine root systems in general are built up as direct sums ofirreducible affine root systems associated to the affine root subsystems of the orthogonal decom-position.
Proposition 3.2.32. Let R be an affine root system with orthogonal decomposition R1 q · · · q Rm ,and let fRi be the affine root system associated to Ri for i = 1, . . . , m . Then R '
⊕mi=1fRi realized by a
normalized similarity transformation.
Proof. We proceed with induction on the rank l of R . Affine root systems of rank one must be irre-ducible by (i) of Example 3.2.23, hence R is its own associated affine root system which concludesthis case. Now assume we are done up to some l > 1. If R is irreducible we are done again, sosuppose that R is reducible. Write R = R ′ qR ′′ such that (a ,b ) = 0 for all a ∈ R ′ and b ∈ R ′′, andlet eR ′ on E ′ (resp. fR ′′ on E ′′) be the affine root system associated to R ′ (resp. R ′′). By Proposition3.2.28 we have
R ' eR ′⊕fR ′′
realized by a normalized similarity transformation.By Proposition 3.2.24 we have the orthogonal decomposition eR ′ = R ′1 q · · · q R ′k (resp. fR ′′ =
R ′′1 q · · · qR ′′n ). Now let fR ′i (resp. fR ′′j ) be the affine root system associated to R ′i (resp. R ′′j ) realized
by the injective linear map π′i : cE ′i ,→cE ′ (resp. π′j : ÓE ′′j ,→ÓE ′′) for 1 ≤ i ≤ k (resp. 1 ≤ j ≤ n). By
construction of the affine root system eR ′ on E ′ (resp. fR ′′ on E ′′) we observe that the rank of eR ′
(resp. fR ′′) is strictly smaller than l . Therefore the induction hypothesis holds for both eR ′ and fR ′′.
Thus eR ′ '⊕k
i=1fR ′i (resp. fR ′′ '
⊕nj=1fR ′′j ) realized by a normalized similarity transformation. So by
Corollary 3.2.31 we obtain
eR ′⊕fR ′′ 'k⊕
i=1
fR ′i ⊕n⊕
j=1
fR ′′j
realized by a normalized similarity transformation.Next consider the injective linear map π′ : cE ′ ,→ bE (resp. π′′ : ÓE ′′ ,→ bE ) that realizes the affine
root system eR ′ (resp. fR ′′) associated to R ′ (resp. R ′′). Then we find from the proof of Proposition
63
3.2.24 that the orthogonal decomposition of R is∐k
i=1π′(R ′i )q
∐nj=1π
′′(R ′′j ). Let âπ′(R ′i )⊂dEπ
′
i (resp.
äπ′′(R ′′j )⊂ÔEπ
′′
j ) be the affine root system associated to π′(R ′i )⊂ R (resp. π′′(R ′′j )⊂ R) realized by the
injective linear map πi : dEπ′
i ,→ bE (resp. πj : ÔEπ′′
j ,→ bE ) for i = 1, . . . , k (resp. j = 1, . . . , n). Then
we have to show R '⊕k
i=1âπ′(R ′i )⊕
⊕nj=1äπ′′(R ′′j ) realized by a normalized similarity transformation
to prove this Proposition. By the proof of Proposition 3.2.24 we also have that fR ′i 'âπ′(R′i ) (resp.
fR ′′j 'äπ′′(R′′j )) realized by a normalized similarity transformation for i = 1, . . . , k (resp. j = 1, . . . , n).
So by Corollary 3.2.31 we obtain
k⊕
i=1
fR ′i ⊕n⊕
j=1
fR ′′j 'k⊕
i=1
âπ′(R ′i )⊕n⊕
j=1
äπ′′(R ′′j )
realized by a normalized similarity transformation. We conclude that
R 'k⊕
i=1
âπ′(R ′i )⊕n⊕
j=1
äπ′′(R ′′j )
realized by a normalized similarity transformation, as we had to show.
Proposition 3.2.32 leads to the following statement.
Corollary 3.2.33. Every affine root system R is similar to a direct sum⊕n
i=1 Ri of irreducible affineroot systems Ri .
We arrive at the main result of this Subsection, namely that it suffices to study a complete setof representatives of the similarity classes of the irreducible affine root systems to understand thesimilarity classes of all affine root systems.
Theorem 3.2.34. Let {Ri }i∈I be a complete set of representatives of similarity classes of the irre-ducible affine root systems with index set I , then for each affine root system R there exist λ1, . . . ,λm ∈R6=0 such that R '
⊕mj=1λj R j .
Proof. Let R be an affine root system with orthogonal decomposition R1q · · · qRm for m > 1 (seeProposition 3.2.24). Then by Proposition 3.2.32 we have R '
⊕mj=1fR j realized by a normalized
similarity transformation, where fR j is the affine root system associated to R j for j = 1, . . . , m . Bydefinition of the orthogonal decomposition, fR j is an irreducible affine root system for j = 1, . . . , m .This implies that for each 1 ≤ j ≤ m there exists a unique i j ∈ I such that fR j ' Ri j . Then by
Proposition 3.2.30 there exist λ1, . . . ,λm ∈R6=0 such that⊕m
j=1fR j '
⊕mj=1λi Ri j .
This result shows that the irreducible affine root systems are the building blocks of all affineroot systems. Therefore we will work mostly with irreducible affine root systems in the rest of thischapter, unless stated explicitly. Furthermore, we will classify the similarity classes of the reducedirreducible affine root systems in the last Section of this Chapter.
Remark 3.2.3. In the work of Macdonald (see [10]) the definition of an affine root system coincideswith ours, except that condition (5) of Definition 3.2.1 is omitted. It turns out that if an affineroot system in Macdonald’s sense is irreducible, then condition (5) is satisfied and we have anirreducible affine root system in our sense. However, a reducible affine root system in Macdonald’s
64
sense that does not satisfy condition (5) decomposes as a mixture of both irreducible affine rootsystems and irreducible finite root systems. It has been suggested by Mark Reeder to add condition(5) to the definition of an affine root system in order to avoid such decompositions. Stokman (see[14]) proposes an equivalent condition: The additive subgroup VR := {v ∈V : tv ∈W (R)} ⊂V spansV . One proves this equivalence as follows.
Assume R ⊂ bE satisfies Definition 3.2.1 including (5) and consider distinct a ,b ∈ R such thatDa =Db =: C . Then b (x + v )−a (x + v ) = b (x )+ (C , v )V −a (x )− (C , v )V = b (x )−a (x ) for all x ∈ Eand all v in the space of translations V of E . Thus b (x )− a (x ) = µ for all x ∈ E and some fixedµ∈R6=0. Then for any x ∈ E (3.1.13) and (3.1.14) reveal that
wa ◦wb = t−a (x )C∨ ◦wx+C ◦ t−b (x )C∨ ◦wx+C
= t−a (x )C∨ ◦ tb (x )C∨
= t (b (x )−a (x ))C∨
= tµC∨ ,
(3.2.12)
hence we observe that wa ◦wb = tµC∨ ∈W (R) for some µ ∈ R6=0. In other words µC∨ ∈ VR . Sincethe gradients {Da : a ∈ R} of R span V by (1) of Definition 3.2.1 together with Proposition 3.1.12and µC∨ is a nonvanishing scalar multiple of C , we find one implication of the equivalence.
For the other implication assume that R is an affine root system in Macdonald’s sense and thatVR spans V . Choose C ∈ {Da : a ∈R}, then C 6= 0 and there exists a ∈R such that C =Da . Becausedim(V )> 0 and VR spans V we can choose v ∈ VR such that (v,C )V 6= 0. Then tv ∈W (R), tv (a ) ∈ Rand tv (a ) = a−(C , v )V c 6= a . Then we obtain Dtv (a ) =Da ◦Dt−v =Da from (i) and (ii) of Example3.1.6 and Proposition 3.1.23 which shows that (5) of Definition 3.2.1 holds. Thus R is an affine rootsystem.
3.3 The geometry of affine root systems
In this section let R be an affine root system on an affine Euclidean space E of rank l , and let Vdenote the space of translations of E . In the following we are interested in the collection H :={Ha : a ∈ R} of affine hyperplanes in E induced by R and its configuration in E (see (3.1.8)). Theaffine hyperplanes H define a partition of the affine Euclidean space E and give rise to specialsubsets of E called alcoves. Since these alcoves turn out to be closely related to the structure ofaffine root systems, we will give a precise geometrical description of them. Along the way we willobtain an important result on the generators of W (R) and define the gradient root system D(R) ofR .
Because of the geometric flavor of this Section we will sometimes clarify a proof with a sketchof the situation. Furthermore, many statements in this Section are very intuitive when they areconsidered with H explicitly depicted (see for example Figure 3.1). It must be noted that mostproofs of in Section are adapted versions from statements that can be found in [2].
3.3.1 Affine hyperplanes and alcoves in affine Euclidean space
Recall that we consider an affine Euclidean space E as being equipped with the Euclidean topologywhich makes E a locally compact space. Now a collection of subsetsP of a locally compact spaceX is called locally finite if all compact subsets K of X meet with only finitely many distinct elementsofP ([11], Chapter 3, §26, §29 and §39).
65
Figure 3.1: Example of a hyperplane configurationH (of a reduced irreducible affine root system of typeAu
2 ).
Proposition 3.3.1. H is a locally finite collection of subsets of E .
Proof. Let K ⊂ E be a compact subset and assume that H ∈ H meets K , then there exists a ∈ Rsuch that H = Ha . Now the orthogonal reflection wa ∈ W (R) in the affine hyperplane Ha is ahomeomorphism of E by Corollary 3.1.8, hence wa (K ) ⊂ E is also compact. Further wa fixes Ha
in E , so K ∩Ha =wa (K ∩Ha ) =wa (K )∩Ha 6= ; which implies wa (K )∩K 6= ;. This only holds forfinitely many a ∈ R by Remark 3.2.1, so only finitely many H ∈H meet K . We conclude thatH islocally finite.
Until further notice we forget the special nature ofH , and letH denote any locally finite col-lection of affine hyperplanes in the affine Euclidean space E . Further, we develop some notationto study such more general collections. Recall that a nonempty subset U of a topological space Xis a connected component of X if and only if for every connected subset U ′ ⊂X , we have U ′∩U = ;or U ′ ⊂U ([11], Chapter 3, §25). Let H be an affine hyperplane in E , and A be a nonempty con-nected subset of E not meeting H . Let DH (A) denote the connected component of E \H contain-ing A, or in other words, DH (A) is the unique open half-space in E bounded by H containing A. SoDH (A) = DH (B ) means that the connected subsets A and B of E are contained in the same con-nected component of E \H , or that A and B lie on the same side of the affine hyperplane H . For acollection of affine hyperplanesN of E which do not meet A, write DN (A) :=
⋂
H∈N DH (A). If A isa singleton, say A = {x }, we put DH (x ) :=DH (A). Finally write Er e g := E \
⋃
H∈H H for the regularpoints of E relative toH which are the points of E that do not lie on an affine hyperplane ofH . Inthe remaining of this Subsection we will develop some theory aboutH in this general setting. Inthe next Subsection we will apply this theory to (irreducible) affine root systems.
Lemma 3.3.2. Er e g is open in E . Moreover, the connected components of Er e g are open in E .
Proof. We will proof the latter, then the former is implied. Let C be a connected component ofEr e g , and let x ∈ C be a point. Since E is locally compact, there exists a compact subset K of Esuch that x lies in the interior K ◦ of K . The collectionH is locally finite by Proposition 3.3.1, soK only meets finitely many affine hyperplanes ofH . This means that there exists a finite subsetN ⊂ H consisting of affine hyperplanes that meet K . Then K ◦ \H is an open neighborhood ofx in E for all H ∈ N , so A :=
⋂
H∈N K ◦ \H ⊂ Er e g is also an open neighborhood of x . Because Eis equipped with the Euclidean topology E is locally connected. So there exists a connected open
66
neighborhood B of x in E such that B ⊂ A. But x lies in the connected component C , so we musthave B ⊂C . This shows that C is open in E .
Lemma 3.3.3. For every x ∈ E there exist at most finitely many affine hyperplanes passing throughx that belong toH . Also, there exists a connected open neighborhood of x that does not meet anyaffine hyperplane belonging toH , except for those passing through x .
Proof. The first statement is clear sinceH is locally finite. So consider the collectionN of affinehyperplanes contained inH that do not pass through x . SinceH is locally finite, so isN . HenceLemma 3.3.2 applies to Er e g relative toN , in which x is contained. Thus x lies in one of its con-nected components, which is open in E and clearly only meets affine hyperplanes ofH passingthrough x .
Lemma 3.3.4. Let L ∈H be an affine hyperplane, then there exists a point x ∈ L that does not belongto any other affine hyperplane H 6= L ofH .
Proof. SinceH is a locally finite collection of affine hyperplanes in E , the collectionL := {L ∩H :H ∈H \{L}} of affine hyperplanes in the affine Euclidean space L (see Prop. 3.1.4) is locally finite.If y ∈ L does not belong to any affine hyperplane inH \{L}we are done, so suppose the opposite.Since L is locally finite in L, Lemma 3.3.3 yields that there are only finitely many affine hyper-planes ofL passing through y , say H1, . . . , Hm , and y has a connected open neighborhood U ⊂ Lthat does not meet any affine hyperplane ofL other than H1, . . . , Hm . The affine Euclidean space Lis in bijective correspondence with it space of translations, hence finitely many affine hyperplanesdo not exhaust the space L. This means that we can consider an affine line S in L through y thatis not contained in (but does intersect) the affine hyperplanes H1, . . . , Hm (for L of dimension 1 welet S = L). Then S∩U \{y } ⊂ L is nonempty and contains a point x does not meet any affine hyper-plane ofL other than H1, . . . , Hm since x ∈U \{y }. But also x /∈Hi for 1≤ i ≤m because x ∈S\{y }.This shows that there exists a point x ∈ L that does not belong to any other affine hyperplane H 6= LofH .
Introduce a relation∼ on E by saying that x ∼ y if for any affine hyperplane H ∈H either x ∈Hand y ∈H , or DH (x ) =DH (y ). It is a straightforward check that ∼ is an equivalence relation on E .
Definition 3.3.5. An equivalence class F under the equivalence relation ∼ on E is called a facet ofE relative toH .
Let x be a point in a facet F of E relative toH , then an affine hyperplane H ∈H contains F ifand only if x ∈ H . By Lemma 3.3.3 there are only finitely many of such affine hyperplanes, hencetheir intersection defines an affine subspace L of E by Proposition 3.1.4 called the support of F. LetN be the collection of affine hyperplanes inH that do not contain the facet F , then by definitionof ∼we observe that
F = L ∩DN (x ). (3.3.1)
In other words, F ⊂ L and F ⊂DH (x ) for all H ∈N . So if we write F , L and DH (x ) for the closures inE of F , L and DH (x ) respectively, then F ⊂ L and F ⊂DH (x ) for every H ∈H . Now L = L, becauseL is an affine subspace which must be closed in E . This implies that F ⊂ L∩ (
⋂
H∈N DH (x )). On the
other hand consider y ∈ L ∩ (⋂
H∈N DH (x )), then we have y ∈ L. Since DH (x ) =DH (x )∪H we alsoobserve that for each H ∈ N either y ∈ DH (x ) or y ∈H . Since x ∈ F we have by (3.3.1) that x ∈ L,and for each H ∈ N we have x ∈ DH (x ) and x /∈ H . So the open line segment with extremities x
67
and y is contained in L and in DH (x ) for each H ∈ N , hence in F by (3.3.1). Thus certainly y iscontained in F , and we obtain
F = L ∩⋂
H∈NDH (x ). (3.3.2)
Definition 3.3.6. An alcove of E relative toH is a facet of E relative toH that is not contained inany affine hyperplane belonging toH .
Hence the alcoves in Figure 3.1 are the white triangles. It is not a coincidence that they areconnected.
Proposition 3.3.7. The alcoves of E relative toH are the connected components of Er e g .
Proof. First we show that an alcove C ⊂ Er e g is connected. Let x , y ∈ C , then DH (x ) = DH (y ) =DH (C ) for all H ∈H by Definition 3.3.5 and 3.3.6. Consider the line segment S between x and y ,then by linearity DH (z ) = DH (x ) = DH (C ) for all z ∈ S and H ∈ H . This implies S ⊂ C , so C isarc-connected and in particular connected.
Now, assume that there exists a connected subset U ⊂ Er e g such that C ⊆U . Then for x ∈C andy ∈U we can not have that DH (x ) 6=DH (y ) for some H ∈H , else U = (DH (x )∩U )∪ (DH (y )∩U ) isa partition in disjoint open sets of U in Er e g . Thus DH (x ) =DH (y ) for all x ∈C , y ∈U and H ∈H .But this means that y is contained in the same facet as x , which is C . We conclude that C =U andthat C is a connected component of Er e g .
On the other hand, let U be a connected component of Er e g . For x , y ∈U and H ∈H we obtainDH (x ) = DH (y ) by the argument of the last paragraph. By Definition 3.3.6 and the definition ofthe relation ∼, we observe that U is contained in an alcove C . But since all alcoves are connectedcomponents of Er e g , U and C coincide.
Remark 3.3.1. Lemma 3.3.2 combined with the last Proposition reveals that the alcoves of E relativeto H are open subsets of E . This will also be proven in Theorem 3.3.23 when we describe thealcoves explicitly.
Next, we generalize the conclusion of Remark 3.3.1 from alcoves to arbitrary facets.
Lemma 3.3.8. Every facet F of E is an open subset in its support L relative toH .
Proof. From Remark 3.3.1 we observe that alcoves of E relative toH are open in E . Now consideran arbitrary facet F ⊂ E that is not an alcove relative toH . Then by (3.3.1) we have F = L ∩DN (x )for some x ∈ F , where L =H1∩· · ·∩Hm andN =H \{H1, . . . , Hm } for some H1, . . . , Hm ∈H . NoticethatN ⊂H is locally finite, so the collection of affine hyperplanes {H∩L : H ∈N } is a locally finitecollection of affine hyperplanes in the affine subspace L ⊂ E . This implies that in L we can writeF =
⋂
H∈N DH∩L(x ) for some x ∈ F . We observe from (3.3.1) that F is an alcove of L relative to{H ∩ L : H ∈N }, thus Remark 3.3.1 ensures us that F is an open subset in L.
Lemma 3.3.9. Let F and F ′ be facets relative toH such that F ′ ∩ F 6= ;, then F ′ ⊂ F .
Proof. If F ′ meets F ⊂ F then Definition 3.3.5 implies that F = F ′. So assume that F ′ does not meetF but does meet F , and let y ∈ F ′∩(F \F ). Further let x ∈ F , then as in (3.3.1) we have F = L∩DN (x )where L is the support of F and N ⊂H the affine hyperplanes that do not contain F . Write N ′
for subset of affine hyperplanes of N that meet y and put N ′′ :=N \N ′. Then for any H ∈ N ′′
we have y /∈ H and y ∈ DH (x ) by (3.3.2), so y ∈ DH (x ) and DH (x ) = DH (y ). By (3.3.2) F ⊂ L, so in
68
particular y ∈ L. Hence y ∈ L ∩⋂
H∈N ′ H ∩⋂
H∈N ′′ DH (x ), so since y is contained in the facet F ′ weobtain from (3.3.1)
F ′ = L ∩⋂
H∈N ′
H ∩⋂
H∈N ′′
DH (x ).
On the other hand, rewriting (3.3.2) for F we obtain
F = L ∩⋂
H∈N ′
DH (x )∩⋂
H∈N ′′
DH (x ).
This leads indeed to F ′ ⊂ F .
Lemma 3.3.10. Every x ∈ E is in the closure of at least one alcove of E .
Proof. If E is an affine Euclidean space of dimension 0, then E is a singleton and does not haveaffine hyperplanes so the Lemma is clear. Otherwise let x ∈ E , then by Lemma 3.3.3 there are onlyfinitely many affine hyperplanes ofH passing through x , say H1, . . . , Hm , and x has an open neigh-borhood U ⊂ E that does not meet any affine hyperplane ofH other than H1, . . . , Hm . Considerthe affine line S through x that is not contained in any of the affine hyperplanes H1, . . . , Hm (for Lof dimension 1 we let S = L), and let y 6= x be a point on S close enough to x such that the open linesegment S′ with extremities x and y lies in U . Since S′ also does not meet any affine hyperplane Hi
for i = 1, . . . , m it is a connected subset of Er e g , and hence contained in an alcove C by Proposition3.3.7. But x lies in the closure of S′, so clearly x is contained in the closure of C .
Definition 3.3.11. A wall of an alcove C is an affine hyperplane L ∈H such that L is the supportof a facet that is contained in the closure C .
As one can see, each alcove in Figure 3.1 has three walls. Next, we obtain a criterion for H ∈Hto be a wall.
Lemma 3.3.12. Let C be an alcove of E relative toH and let L ∈ H be an affine hyperplane in E .Assume that there exists a point x ∈ L ∩C that does not belong to any of the affine hyperplanes inN :=H \{L}, then L ∩DN (C ) is the unique facet with support L contained in C , hence L is a wallof C .
Proof. By Lemma 3.3.3 there exists a connected open neighborhood U of x in E that does notmeet any of the affine hyperplanes ofN . But since x ∈C , the open set U does meet C . Then for allH ∈N , we have
DH (x ) =DH (U ) =DH (U ∩C ) =DH (C ) =DH (y ) (3.3.3)
for some y ∈ C . By the relation "∼", the facet F containing x has support L. Now by (3.3.1) and(3.3.3), F = L ∩DN (x ) = L ∩DN (C ). Further, (3.3.2) and (3.3.3) tell us,
C =⋂
H∈HDH (y ) =DL(y )∩
⋂
H∈NDH (y ) =DL(y )∩
⋂
H∈NDH (C ). (3.3.4)
So since L ⊂ DL(y ) and DH (C ) ⊂ DH (C ) for all H ∈ N , we observe that F ⊂ C . Also by (3.3.4), anyfacet contained in C with support L must lie on the same side of all affine hyperplanes inN as C .By definition, different facets with support L can not be on the same side for all affine hyperplanesinN , thus only one such facet exists, namely F = L ∩DN (C ).
Proposition 3.3.13. Every L ∈H is the wall of at least one alcove of E relative toH .
69
Proof. Let L ∈H be an affine hyperplane, then by Lemma 3.3.4 there exists a point x ∈ L that doesnot belong to any other affine hyperplane H 6= L ofH . By Lemma 3.3.10 there exists an alcove Csuch that x is contained in its closure. Finally, Lemma 3.3.12 implies that L is a wall of C .
The next Lemma shows that an alcove C and the facets that are contained in its closure relativetoH only depend on the walls of the alcove C .
Lemma 3.3.14. Let C be an alcove of E relative toH and letM ⊂H be the set of walls of C .(i) C =DM (C );(ii) F ⊂C is a facet of E relative toH if and only if then F is also a facet of E relative toM .
Proof. (i) Fix x ∈ C , then DH (C ) = DH (x ) for all H ∈ H since C is an alcove relative toH . Thusto prove C = DM (C ) with M ⊂ H the set of walls of C relative to H , it is enough to show thatC = DM (x ). Now because M ⊂ H and C = DH (x ) by (3.3.1), we observe that C ⊂ DM (x ). Onthe other hand let y ∈ DM (x ), then by linearity the line segment S with extremities x and y isalso contained in DM (x ). This implies that S does not meet any wall of C . Writing N for thesubcollection of affine hyperplanes ofH meeting S we getM ∩N = ;. Furthermore S is compactin E , so the locally finiteness of H leads to N being finite. Now x ∈ C ∩ S = DH (C ) ∩ S, S isconnected and S does not meet any affine hyperplane ofH \N , thus S ⊂DH \N (C ). So if we canshow that DH \N (C ) =C , then y ∈C which implies DM (x )⊂C so we are done. In fact we will showin the following that for each finite subsetN ⊂H such thatN ∩M = ; it holds that C =DH \N (C ).To do that we proceed with induction to the cardinal n ∈Z≥0 ofN .
If n = 0 we haveH \N =H for which we already know that C =DH (C ). So we assume that weare done up to n > 0 including n . PutL :=H \N , thenM ⊂L and by the induction hypothesiswe have C =DL (C ). Notice thatL is a locally finite collection of affine hyperplanes in E . Since Cis an alcove relative toH we have DH (C ) =DH (x ) for all H ∈H . In particular DH (C ) =DH (x ) forall H ∈L , so C =DL (C ) =DL (x )which is an alcove relative toL by (3.3.1).
Next we want to show thatM is also the set of walls of C relative to L . By Lemma 3.3.12,Mis contained in the set of walls of C relative to L . On the other hand, any wall H of C relative toL must contain a point x ∈ C that does not meet any affine hyperplane belonging to L \ {H} byDefinition 3.3.11. Hence by Lemma 3.3.3 there exists an open neighborhood Ux ⊂ E of x such thatUx does not meet any affine hyperplane contained in L apart from H . Moreover using Lemma3.3.3 we can choose Ux small enough such that Ux only meets hyperplanes ofH passing throughx , and by choosing Ux bounded we make sure that the closure Ux is compact. Then the locallyfiniteness ofH guarantees that Ux only meets finitely many affine hyperplanes contained inH . Ifx is not contained in any other affine hyperplane ofH than H , then H is also a wall of C relative toH by definition of x . Otherwise, suppose that there exists an affine hyperplane H ′ ∈H differentfrom H that contains x . Then H ′ induces a partition of H as H = H1 qH2 q (H ∩H ′) such thatDH ′ (H1) 6=DH ′ (H2) (see Figure 3.2). Since the only hyperplane ofL that meets Ux is H , we observethat all points of Ux ∩H are in the same facet as x relative toL . This in turn implies that Ux ∩H ⊂C because H is a wall of C relative to L . Now choose y ∈ DH ′ (H1) and z ∈ DH ′ (H2) such thaty , z ∈ Ux ∩H . Then there exist open neighborhoods Uy of y and Uz of z respectively such thatUy ⊂DH ′ (H1) and Uz ⊂DH ′ (H2) (see Figure 3.2). But Uy ∩C 6= ; and Uz ∩C 6= ; since y , z ∈C . Thisimplies that C ∩DH ′ (H1) 6= ; and C ∩DH ′ (H2) 6= ; which contradicts C being an alcove relative toH . This leads to H being the only affine hyperplane ofH passing through x , so H is also a wall ofC relative toH . We conclude thatM is also the set of walls of C relative toL .
Let L ∈L such that L is not a wall of C relative toL , then L /∈M as we have just seen. BecauseC =DL (C ) by the induction hypothesis we want to show that DL\{L}(C ) =DL (C ) in the following
70
Figure 3.2: Assuming that x ∈C is contained in a wall H of the alcove C relative toL (M but not relative toM leads to a contradiction. Namely, these assumptions imply that x must also lie on H ′ ∈M , which leadsto C not being alcove relative toM .
to complete the induction. We immediately observe that DL (C ) ⊂ DL\{L}(C ). On the other handassume that L meets DL\{L}(C ) and choose z in their intersection, then z /∈ H for any H ∈ L \{L}. Hence z /∈ L ∩C , because otherwise L would be a wall of C relative to L by Lemma 3.3.12.Further by Lemma 3.3.3 there exists a connected open neighborhood U of z not meeting any affinehyperplane H ∈ L \ {L}, so U ⊂ DL\{L}(C ). Notice that ; 6= U ∩DL(C ), so since C = DL (C ) andU ⊂DL\{L}(C ) we observe that ; 6=U ∩DL(C ) =U ∩C ⊂C . But any open ball with center z meetsU ∩DL(C ). In other words z ∈ C which contradicts the fact that z /∈ L ∩C since z ∈ L. Henceour assumption that L meets DL\{L}(C ) was wrong. Now every open halfspace DH (C ) for H ∈ Lis convex, hence the intersection of convex sets DL\{L}(C ) is also convex. It follows directly fromarc-connectedness of a convex set that DL\{L}(C ) is connected in E . Then C =DL (C )⊂DL\{L}(C )together with L ∩DL\{L}(C ) = ; implies that DL\{L}(C ) ⊂ DL(C ), so we obtain that DL\{L}(C ) ⊂DL(C ). We conclude that DL\{L}(C ) =DL (C ), which coincides with C by the induction hypothesis.Hence for each finite subsetN ⊂H such thatN ∩M = ; it holds that C =DH \N (C ). As the firstparagraph of this proof indicates, we obtain now C =DM (C ).
(ii) Let F be a subset of the closure C of C and suppose that F is a facet of E relative toM withsupport L. Then F is open in L by Lemma 3.3.8. Now let H ′ be an affine hyperplane that meets Fbut does not contain F . Since F ⊂ L is open this implies that F 6⊂ DH ′ (x ) for all x ∈ E \H ′. SinceF ⊂ C we observe that also C 6⊂ DH ′ (x ) for all x ∈ E \H ′. But by (3.3.2) we have C =
⋂
H∈H DH (y )for some y ∈C , so the only possibility is that H ′ /∈H . This shows that all H ∈H that meet F alsocontain F , hence F is also a facet of E relative toH .
On the other hand, suppose F ⊂C is a facet of E relative toH . Since the collection of wallsM ofC is part of the locally finite collection of affine hyperplanesH ,M is also a locally finite collectionof affine hyperplanes in E . As we have seen in the previous part of this Lemma it holds that C =DM (C ) =
⋂
H∈M DH (x ) for some x ∈C . Hence (3.3.1) shows that C is also an alcove relative toM .
So let us assume that F 6= C . By (3.3.2) we obtain C =⋂
H∈M DH (x ) for some x ∈ C , so F ⊂ C \Cis contained in
⋃
H∈M H . Hence the support of F relative toH is the intersection of finitely manywalls of C , say H1, . . . , Hm ∈ M . By (3.3.1) we obtain F = H1 ∩ · · · ∩Hm
⋂
H∈H \{H1,...,Hm }DH (x ) forsome x ∈ F . Then clearly F ⊂ F ′ := H1 ∩ · · · ∩Hm
⋂
H∈M\{H1,...,Hm }DH (x ) for some x ∈ F where we
note that F ′ is a facet relative to M by (3.3.1). Further it follows that F ′ ⊂⋂
H∈M DH (y ) = C forsome y ∈ C . Now by Lemma 3.3.8 both F and F ′ are open subsets of their support H1 ∩ · · · ∩Hm .Suppose that F ( F ′ then there exists z ∈ F ′ \ F and H ′ ∈H such that DH ′ (z ) 6=DH ′ (F ). Hence weobserve that similarly to the previous paragraph that F ′ 6⊂DH ′ (x ) for all x ∈ E \H ′, hence C 6⊂DH ′ (x )
71
for all x ∈ E \H ′. This gives a contradiction with (3.3.2) which yields C =⋂
H∈H DH (y ) for somey ∈C . Thus we conclude that F = F ′, so F is also a facet relative toM .
3.3.2 Alcoves of irreducible affine root systems
We return to studying the the collection of affine hyperplanesH = {Ha : a ∈ R} generated by anaffine root system R on an affine Euclidean space E with space of translations V . Our main goal isto explicitly describe the alcoves relative toH . Before we can do that we need some results on theaffine Weyl group W (R).
In this Subsection we will consider W (R) as subgroup of GL(E ) since we will work mostly in E(see Proposition 3.1.22). Recall that Ha = {y ∈ E : a (y ) = 0} for a ∈ R , so for b ∈ R the orthogonalreflection wb ∈W (R) acts on Ha by wb (Ha ) =Hwb (a ) by (i) of Example 3.2.12 and Corollary 3.2.14.Hence W (R) acts on the collection of affine hyperplanesH . Now every w ∈W (R) is an affine linearautomorphism of E , so also a homeomorphism of E by Corollary 3.1.8. In particular w permutesthe connected components of Er e g , which are the alcoves of E relative toH by Proposition 3.3.7.Clearly this is a group action of W (R) on the collection of alcoves of E .
Lemma 3.3.15. Let C be an alcove relative toH , let H be a wall of C and let w ∈W (R), then w (H )is a wall of w (C ).
Proof. Because H is a wall of C , H is the support of a facet F that is contained in C . This impliesthat there exists x ∈ F such that x does not meet any affine hyperplane ofH other than H . Sincethe homeomorphism w ∈W (R) of E permutesH and the collection of alcoves relative toH weobserve that w (H ) ∈ H and that w (C ) is an alcove relative to H . Furthermore, w (x ) does notmeet any affine hyperplane ofH other than w (H ). Let U be an open neighborhood of w (x ), thenw−1(U ) is an open neighborhood of x . Since x ∈ C we have w−1(U ) ∩C 6= ;. This implies thatU ∩w (C ) 6= ;, hence w (x ) ∈w (C ). By Lemma 3.3.12 we conclude that w (H ) is a wall of the alcovew (C ).
In the following Theorem we will see that the action of W (R) on the collection of alcoves istransitive, and that the affine Weyl group is generated by the reflections in the walls of any fixedalcove.
Theorem 3.3.16. Fix an alcove C of E relative toH .(i) For any x ∈ E there exists w ∈W (R) such that w (x )∈C ;(ii) W (R) acts transitively on the collection of all alcoves;(iii) W (R) is generated by the orthogonal reflections wa for a ∈R such that Ha is a wall of C .
Proof. Consider the subgroup W ′ of W (R) generated by the orthogonal reflections of E in the wallsof C . We will first proof (i) and (ii) for W ′, and then show that W ′ =W (R) in complete the proof ofthe whole Theorem.
(i) Let x ∈ E and write J for the W ′-orbit of x . Let z ∈C , then there is a closed ball A with centerz meeting J . Since A is a compact set in the locally compact space E , Remark 3.2.1 ensures us thatA has finite intersection with J . Hence there exists a point y in J with minimum distance to z .
We shall prove that y ∈ C . Recall from (i) of Lemma 3.3.14 that C = DM (C ) where M ⊂ His the set of walls of C , then also C = DM (x ′) for all x ′ ∈ C . We observe from (3.3.1) that C is analcove relative toM , hence by (3.3.2) we have C =DM (x ′) =DM (C ). Therefore we will show thaty ∈DH (C ) for all H ∈M .
72
Figure 3.3: The only option for y is to lie on the same side of Ha as z , else wa (y ) lies closer to z than y itself.
Let H ∈ M , then H = Ha for some a ∈ R . Further wa ∈ W ′ so wa (y ) ∈ J which implies that|y − z |E ≤ |wa (y )− z |E by definition of y . Let y ′ ∈H be the point such that there exists v ∈ V withy = y ′ + v and wa (y ) = y ′ − v , then y ′ lies exactly in between y and wa (y ), and v is orthogonalto H . Also consider the vector u ∈ V such that z = y ′ + u . Then |y − z |E ≤ |wa (y )− z |E implies(v −u , v −u )V ≤ (v +u , v +u )V , or (v, u )V ≥ 0 (see Figure 3.3). Because v is orthogonal to H thisinequality show that y ∈ DH (z ) = DH (C ) for H ∈ M . Thus y ∈ C , and since y ∈ J there existsw ∈W ′ such that y =w (x ).
(ii) Let C ′ be an alcove of E relative toH , let x be a point in C ′, and write J for the W ′-orbit of x .Let z ∈C , then similarly to (i) there exists a point y in J with minimum distance to z . Suppose thaty /∈C , then there exists a wall Ha of C for some a ∈ R such that DHa (z ) 6=DHa (y ). Clearly, y , wa (y )and z do not lie on the same line else |y − z |E > |wa (y )− z |E which contradicts the choice of y ∈ J .
Consider the trapezoid with vertices z , wa (z ), y , wa (y ), then z (resp. y ) is a vertex of the long(resp. short) side or of the short (resp. long) side of the pair of parallel sides of the trapezoid (seeFigure 3.4). Choose γ in the first case as the angle of wa (y ) and γ′ in the second case as the angleof z , then π
2 ≤ γ,γ′ < π. This implies that −1 < cos(γ), cos(γ′) ≤ 0. Although it is also possible that
Figure 3.4: The two possible trapezoids with vertices z , wa (z ), y , wa (y ).
both parallel sides have the same length, both cases we are considering here cover that situationsince γ and γ′ possibly equal π2 . In the first case the law of cosines yields
|y − z |2E = |wa (y )− y |2E + |wa (y )− z |2E −2|wa (y )− y |E |wa (y )− z |E cos(γ)
> |wa (y )− z |2E ,
and in the second case it yields
|wa (y )−wa (z )|2E = |wa (y )− z |2E + |wa (z )− z |2E −2|wa (y )− z |E |wa (z )− z |E cos(γ′)
> |wa (y )− z |2E .
73
But wa preserves the metric on E , so |wa (y )−wa (z )|E = |y −z |E . In both cases we obtain |wa (y )−z |E < |y − z |E , and because wa ∈ W ′ we obtain that wa (y ) ∈ J has strictly smaller distance to zthan y . This contradicts the choice of y as a point in J with minimum distance to z . We concludethat y ∈C , hence there exists w ∈W ′ such that w (x ) = y ∈C . Since w permutes the alcoves of E ,we obtain w (C ′) =C .
(iii) Since W ′ ⊂W (R), we only have to prove that wa ∈W ′ for all a ∈R to show that W ′ =W (R).By Proposition 3.3.13, for every a ∈ R there exists an alcove C ′ of E such that Ha is a wall of C ′.By the previous part of this Theorem, there exists an orthogonal reflection w ∈W ′ such that C ′ =w (C ). Furthermore by Lemma 3.3.15 we observe that w−1(Ha ) is a wall of C , hence there existsan affine root b ∈ R such that w−1(Ha ) = Hb . Consequently Ha = w (Hb ) = Hw (b ), so a = λw (b )for some λ ∈ R6=0. By (3.2.1) and (i) of Example 3.2.9 we have w ◦wb ◦w−1 = ww (b ). But ww (b )
and wa are orthogonal reflections in the same affine hyperplane, so ww (b ) =wa . We conclude thatwa =w ◦wb ◦w−1 ∈W ′.
The set of walls of a fixed alcove was an important ingredient of Theorem 3.3.16. We want tounderstand this set a little better by looking at vectors normal to these walls. For a fixed alcove Crelative toH , letM be the set of walls of C . For each wall H ∈M , let tH ∈ V be the unit vectornormal to H such that h+ tH lies on the same side of H as C for all h ∈H . Write N := {tH : H ∈M},then it turns out that the angles between the vectors of N are not acute.
Lemma 3.3.17. For all t , t ′ ∈N it holds that (t , t ′)V ≤ 0.
Proof. Assume that Ha 6=Hb are parallel walls of C for some a ,b ∈ R , then tHa =−tHb or tHa = tHb
by Definition 3.1.3. In the former case we clearly have (tHa , tHb )V ≤ 0, so assume the latter thenDHa (C )∩DHb (C ) = DHb (C ) or DHa (C ). By symmetry of variables we may assume the outcome isDHa (C ), hence Hb ∩DHa (C ) = ;. By (3.3.2) C ⊂DHa (C ), so C ∩Hb = ;. This contradicts Hb being awall of C so we can discard this case.
Now assume Ha 6=Hb are nonparallel walls of C, then Ha ∩Hb 6= ; by Proposition 3.1.4. Choosean origin x ∈ Ha ∩Hb of E and identify E with V as the vector space Ex , then Ha and Hb arehyperplanes in Ex . Moreover Ha ∩Hb ⊂ Ex is of codimension 2, so we can let X be the plane per-pendicular to Ha ∩Hb passing through x . Further, let Γ :=X ∩DHa (C )∩DHb (C ) be the intersectionof two open half planes in X and let Sa := X ∩Ha ∩Γ (resp. Sb := X ∩Hb ∩Γ) be a closed halfline inX contained in X ∩Ha (resp. X ∩Hb ) and the closure of Γ. Clearly the boundary ∂ Γ of Γ is given bySa ∪Sb (see Figure 3.5).
Figure 3.5: The plane X .
74
Now assume the contrary to what we would like to prove, namely that (tHa , tHb )V > 0. Then theangle φ between Sa and Sb in Γ lies strictly between π
2 and π. Consider fSa := (X ∩Ha ) \ (Sa \ {x }),the closure of complement of Sa in the line Ha ∩X , then the angle ψ = π−φ between fSa and Sb
in X ∩DHa (C ) lies strictly between 0 and π2 . Also note that fSa ∩ X ∩DHb (C ) = ;. Consider L :=
wb (Ha ) = Hwb (a ) ∈ H which contains Ha ∩Hb , then S := wb (fSa ) ⊂ L ∩ X is a closed halfline inX ∩DHb (C )making an angleψ′ =ψwith Sb between 0 and π
2 . Hence S ⊂ Γ and S∩Sa =S∩Sb = {x }.This means that in X we have DL∩X (Sa \{x }) 6=DL∩X (Sb \{x }). Then because Ex =X ⊕ (Ha ∩Hb )wenotice that DL(Sa \{x }) 6=DL(Sb \{x }) in E . Since L ∈H we have C ⊂DL(Sa \{x }) or C ⊂DL(Sb \{x }).Without loss of generality we can assume that
C ⊂DL(Sa \ {x }) =DL(C ). (3.3.5)
We want to show that C ∩Hb ⊂Ha ∩Hb , because then there is no facet contained in C with sup-port Hb which is equivalent to Hb not being a wall of C . This contradiction then yields (tHa , tHb )V ≤0. Indeed, let q ∈C ∩Hb then q ∈DL(C ) by (3.3.5), q ∈DHa (C ) by (3.3.2) for C and certainly q ∈Hb .This leads to
q ∈DL(C )∩DHa (C )∩Hb . (3.3.6)
Now choose x and X such that q ∈ X . From the structure of Γ (see also Figure 3.5) we observe thatSb = X ∩DHa (C )∩Hb which contains q by (3.3.6) and the fact that q ∈ X . Also q ∈DL(C ) by (3.3.6),so let us assume that q ∈DL(C ). Since DL(C ) =DL(Sa \{x }) 6=DL(Sb \{x }), we note that q /∈Sb \{x }.Further we have x ∈ Ha ∩Hb ⊂ L, so x /∈ DL(C ) which leads to q 6= x . This leads to q /∈ Sb whichcontradicts our observation that q ∈ Sb . Thus we conclude q ∈ DL(C ) \DL(C ) = L. In particular,q ∈ L ∩Sb = {x } ⊂Ha ∩Hb . This show that C ∩Hb ⊂Ha ∩Hb as we wanted.
This Lemma leads to the first result on the description of the alcoves relative toH induced byR .
Proposition 3.3.18. The set of wallsM ⊂H of an alcove C of E relative toH is finite.
Proof. Let N be the set of normal unit vectors of the walls of a fixed alcove C relative toH lying onthe same side as C as in Lemma 3.3.17. Consider tH and tH ′ with H 6=H ′ inM . Since (tH , tH ′ )V ≤ 0by Lemma 3.3.17 we have
|tH − tH ′ |2V = 2−2(tH , tH ′ )V ≥ 2. (3.3.7)
Next, consider the unit sphere P ⊂ V together with a cover P of open balls with radiusp
2. ThenN ⊂ P and each open ball of P contains at most one vector of N by (3.3.7). Since P is compactthere exists a finite subcoverQ ⊂P of P . This shows that that N is finite, henceM is finite.
Moreover, we obtain a finiteness result on the directions of normal vectors to the affine hyper-planes ofH .
Proposition 3.3.19. There are only finitely many normal unit vectors possible to the affine hyper-planes ofH = {Ha : a ∈R}.
Proof. Let C be an alcove of E relative toH and letM ⊂H its set of walls. By Lemma 3.3.9 everyfacet relative toH (resp. M ) that meets C is contained in C . Furthermore, the facets containedin C relative toH are the same as the facets contained in C relative toM by (ii) of Lemma 3.3.14.By Proposition 3.3.18 the collection of affine hyperplanesM is a finite. Hence C is the union offinitely many facets relative toM (or toH ) by definition of a facet and (3.3.2). But every facet only
75
meets finitely many affine hyperplanes of H , so there are only finitely many hyperplanes of Hmeeting C . Then the set N (C ) of unit vectors normal to the affine hyperplanes inH that meet Cis finite. Furthermore, there exists λ< 1 such that (t , t ′)V ≤λ for all distinct t , t ′ ∈N (C ).
Define N (H ) to be the set of unit vectors such that each vector is normal to an affine hyper-planes H ∈H . Let t , t ′ ∈N (H ) be distinct, then t and t ′ are parallel vectors if and only if t =−t ′.Otherwise, let t (resp. t ′) be a unit vector normal to the affine hyperplane H (resp. H ′) in H .Since t and t ′ are not parallel we have H ∩H ′ 6= ;. Let x ∈ H ∩H ′, then there exists w ∈ W (R)such that w (x ) ∈ C by (i) of Theorem 3.3.16. Since w (H ), w (H ′) ∈ H and w (x ) ∈ w (H ) ∩w (H ′)we observe that the unit vectors normal to w (H ) and w (H ′) are contained in N (C ). By (iii) of Ex-ample 3.1.6 the affine linear map w : E → E can be considered as a linear map w : Ex → Ew (x )
defined by x + v 7→ w (x ) +Dw (v ) for all v in the space of translations V of E . Now w ∈ W (R),hence w = wa 1 ◦ · · · ◦wa m for a 1, . . . a m ∈ R . Then by (ii) of Example 3.1.6 and (3.1.11) we haveDw = wDa 1 ◦ · · · ◦wDa m which is a linear isometry because each wDa i is an orthogonal reflectionin V . This shows that Dw (t ) (resp. Dw (t ′)) is a unit vector normal to w (H ) (resp. w (H ′)), henceDw (t ), Dw (t ′) ∈ N (C ) and (t , t ′)V = (Dw (t ), Dw (t ′))V ≤ λ. Then |t − t ′|V = 2− 2(t , t ′)V ≥ 2− 2λ.Hence covering the unit sphere P in V with open balls of diameter
p2−2λ and using the compact-
ness of P we observe similarly to the proof of Proposition 3.3.18 that N (H ) is a finite set.
Next we need to study the gradients of an affine root system R . Let D(R) := {Da : a ∈ R} ⊂ V bethe set of gradients of affine roots in R . Then D(R) is a finite root system in a natural way called thegradient root system of R .
Proposition 3.3.20. D(R) is a finite root system in V , it is irreducible if R is, and the mapping D :W (R)→W0(D(R)), w 7→Dw is a surjective group homomorphism with kernel the subgroup t (R)⊂W (R) of translation maps of E that are contained in W (R).
Proof. Since R spans bE and only contains nonisotropic vectors of bE one observes from Proposi-tion 3.1.12 that D(R) spans V and does not contain 0. Also, because of condition (3) of Definition3.2.1 we have that RDa ∩ R = {±Da }, RDa ∩ R = {±Da ,± 1
2 Da } and RDa ∩ R = {±Da ,±2Da }are the only possibilities for multiples of the gradient Da of an affine root a ∈ R . Further, thereare only finitely many normal unit vectors possible to the affine hyperplanes of H by Proposi-tion 3.3.19. Since each gradient Da is a normal vector to Ha ∈ H we observe that D(R) mustbe finite. Now, let a ,b ∈ R then wa (b ) ∈ R , so D(wa (b )) ∈ D(R). But by (3.1.17) and (3.1.11) wehave D(wa (b )) = (Dwa )(Db ) = wDa (Db ) ∈ D(R). Finally, for all α,β ∈ D(R) we have (α∨,β )V =(Da∨, Db )V = (a∨,b )
bE ∈ Z for some a ,b ∈ R . Thus we conclude that D(R) is a finite root system inV . It follows directly from the previous argument that D(R) is irreducible if R is.
By (3.1.1), D : W (R)→W0(D(R)) is a group homomorphism. Further, we have Dwa =wDa for alla ∈R from (3.1.11). Since W0(D(R)) is generated by the reflections wDa for a ∈R it follows that D issurjective. Finally, Dw = 1 if and only if w (x+v ) =w (x )+v for all x ∈ E and v ∈V . Since the kernelof a group homomorphism is a subgroup, we observe that it must be the subgroup t (R)⊂W (R) oftranslations.
We will see later on in Remark 3.5.1 that the gradient root system D(R) need not be reduced evenif R is reduced.
From here we will assume that R is an irreducible affine root system for the sake of simplicity ofthe statements of our results. To study the alcoves relative toH in more detail, we need to knowmore about the unit vectors normal to the walls of an alcove. We continue with a general Lemmaabout a finite set of vectors.
76
Lemma 3.3.21. Let {t0, . . . , tn} be a linearly dependent set of vectors spanning V such that(1) (t i , t j )V ≤ 0 for i 6= j ;(2) there is no partition of {0, . . . , n} into two nonempty subsets I and J such that (t i , t j )V = 0 for
i ∈ I and j ∈ J .Then there exist coefficients c0, . . . , cn ∈R>0 such that
∑ni=0 c i t i = 0, and if c ′0, . . . , c ′n ∈R are such that
∑ni=0 c ′i t i = 0, then there exists a constant ξ∈R such that c i = ξc ′i for all 0≤ i ≤ n, i.e. n = dim(V ).
Proof. Let {e0, . . . , en} be a standard basis of Rn+1. Put a i j := (t i , t j )V for 0 ≤ i , j ≤ n , then thematrix A = (a i j )0≤i ,j≤n is a real symmetric square matrix, so we can define a symmetric bilinearform q on Rn+1 by
q (n∑
i=0
x i e i ,n∑
j=0
y j e j ) :=n∑
i ,j=0
x i y j a i j
for x0, . . . ,xn , y0, . . . , yn ∈ R. Also we can define its related quadratic form q (v ) := q (v, v ) for v ∈Rn+1, so
q (n∑
i=0
x i e i ) =n∑
i ,j=0
x i x j a i j = |n∑
i=0
x i t i |2V ≥ 0 (3.3.8)
for x0, . . . ,xn ∈ R. By the Corollary of Proposition 2 §7.1 of [1] the kernel of q , which is definedas N := {v ∈ Rn+1 : q (v, w ) = 0 for all w ∈ Rn+1}, equals the subspace of isotropic vectors withrespect to the quadratic form q . Thus (3.3.8) yields N = {(c0, . . . , cn ) ∈ Rn+1 :
∑ni=0 c i t i = 0}, and
since {t0, . . . , tn} is a linearly dependent set we conclude that dim(N ) ≥ 1. To study N further fix∑n
i=0 c i e i ∈N , then
q (n∑
i=0
|c i |e i ) =n∑
i ,j=0
|c i ||c j |a i j =∑
0≤i 6=j≤n
|c i c j |(t i , t j )V +n∑
i=0
c 2i |t i |2V
≤∑
0≤i 6=j≤n
c i c j (t i , t j )V +n∑
i=0
c 2i |t i |2V =
n∑
i ,j=0
c i c j a i j =q (n∑
i=0
c i e i ) = 0
since (t i , t j )V ≤ 0 for i 6= j . By (3.3.8) we conclude that q (∑n
i=0 |c i |e i ) = 0, so∑n
i=0 |c i |e i ∈N . This inturn shows that 0=q (
∑ni=0 |c i |e i , e j ) =
∑ni=0 |c i |a i j for all j .
Let I = {i : c i 6= 0} and let j /∈ I , then |c i |a i j ≤ 0 for i ∈ I and |c i |a i j = 0 for i /∈ I . Since∑n
i=0 |c i |a i j = 0 for all j , it is immediate that a i j = 0 for i ∈ I and j /∈ I . Assumption (2) of thisLemma implies then that I = ; or I = {0, . . . , n}, hence every nonzero vector of N has nonvanishingcoordinates. If the dimension of N would be greater than 1, then the intersection of N with thehyperplane x i = 0 for some i would be of dimension at least 1. This would contradict that everynonzero vector of N has nonvanishing coordinates, hence dim(N ) = 1. Further, for
∑ni=0 c i e i ∈N
also∑n
i=0 |c i |e i ∈N , so N contains a vector with only positive coordinates. Since dim(N ) = 1, thisvector generates N = {(c0, . . . , cn )∈Rn+1 :
∑ni=0 c i t i = 0}, and we can draw all conclusions stated in
the Proposition.
Fix an alcove C relative toH and letM be its set of walls. Let N = {tH ∈ V : H ∈M} be as inLemma 3.3.17. Hence tH ∈ N ⊂ V is the unit vector normal to H ∈M such that h + tH is on thesame side of H as C for all h ∈H .
Lemma 3.3.22. The set N ⊂V satisfies the conditions of Lemma 3.3.21.
77
Proof. By Proposition 3.3.20 the gradient root system D(R) = {Da : a ∈ R} spans V. Hence theonly fixed point of V under the action of the finite Weyl group W0(D(R)) is 0. Now the affine Weylgroup W (R) is generated by the reflections wa for Ha ∈M by Theorem 3.3.16. Thus the surjectivehomomorphism D : W (R)→W0(D(R)) of Proposition 3.3.20 implies that WD := {Dwa : Ha ∈M}={wDa : Ha ∈M} generates W0(D(R)) as a group. This means that WD only leaves 0 as a fixed pointin V . Then the normal vectors {Da : Ha ∈ M} of the affine hyperplanesM span V . Since eachvector tHa ∈ N coincides with Da up to a nonzero scaling for Ha ∈M we conclude that N spansV .
Now assume that N is a linearly independent set, then the walls of C intersect in exactly onepoint, say x ∈ E . Since the reflections wa ∈W (R) for Ha ∈M generate W (R), we observe that x isa fixed point for the action of W (R) on E . So if a ∈R then wa (x ) = x −a (x )Da∨ = x , which impliesa (x ) = 0. Consequently for v ∈ V it holds that a (x + v ) = (Da , v )V , so after choosing the originx ∈ E we have that a ∈ E ∗x for every a ∈ R . By Proposition 3.1.12 this is in contradiction with thefact that R spans bE , hence N is a linearly dependent set.
Finally, N is finite by Proposition 3.3.18, criterion (1) of Lemma 3.3.21 holds by Lemma 3.3.17and criterion (2) of the Lemma is satisfied since R is irreducible.
This leads us to the main Theorem of this Section which gives a precise geometric descriptionof an alcove relative toH induced by an irreducible affine root system R . This description is im-portant for the remaining of this Chapter since it will give us a way to define the analogue of a basisa finite root system.
Theorem 3.3.23. The alcoves of E relative toH are open l -simplices with l +1 walls.
Proof. LetM be the set of walls of a fixed alcove C and N their unit normals as in Lemma 3.3.22.Since dim(V ) = l Lemma 3.3.21 and lemma:alc.7 yield that N has exactly l + 1 elements, sayN = {t0, . . . , t l }. By the proof of the Lemma 3.3.17 there can not exist distinct parallel walls of Cwith the same unit normal in N , hence M contains exactly l + 1 affine hyperplanes of H . PutM = {H0, . . . , Hl } such that t i ∈N is a normal vector of Hi ∈M for each i ∈ {0, . . . , l }. Assume that{t1, . . . , t l } is a linearly dependent set, then there exist d 1, . . . , d l ∈ R, not all vanishing, such that∑l
i=1 d i t i = 0. On the other hand,∑l
i=0 c i t i = 0 for some c0, . . . , c l ∈R>0 by Lemma 3.3.21. So sub-
tracting a small or large enough multiple of∑l
i=1 d i t i to∑l
i=0 c i t i yields a new vanishing relationbetween t0, . . . , t l with coefficients not all positive or all negative. This contradicts the second state-ment of Lemma 3.3.21. Hence {t1, . . . , t l } forms a basis of V , which implies that the hyperplanesH1, . . . , Hl have a unique intersection point x0.
Let x0 be the origin of E and write Ex0 for this vector space. Choose a basis {t ′1, . . . , t ′l } in Vdual to {t1, . . . , t l }, i.e. (t i , t ′j )V = δi j for all 1 ≤ i , j ≤ l . By Lemma 3.3.22, there exist q1, . . . ,ql ∈R>0 such that t0 = −(q1t1 + · · ·+ ql t l ). Since t0 is orthogonal to H0, there exists q ∈ R such thatH0 = {x0 + t : t ∈ V and (t , t0)V = −q}. Also every x ∈ E can be written uniquely in the formx = x0+ t with t = ξ1t ′1+ . . .ξl t ′l for some real coefficients ξ1, . . . ,ξl , since {t ′1, . . . , t ′l } is a basis ofV . A point x written in this form belongs to C = DM (C ) (see (i) of Lemma 3.3.14) if and only ifDHi (x ) =DHi (x0+ t ) =DHi (C ) for 0≤ i ≤ l . In other words, x = x0+ t is on the same side of Hi ast i for 0≤ i ≤ l , or (t , t i )V > 0 for 1≤ i ≤ l and (t , t0)V >−q . Equivalently there exist ξ1 . . . ,ξl ∈R>0
such that q1ξ1+ · · ·+ql ξl < q . Note that q > 0 since C 6= ;. Write ξi =λiqi
q for 1≤ i ≤ l , then x ∈C
if and only if λi ∈R>0 for 1≤ i ≤ l and∑l
i=1λi < 1. Putting x i = x0+qqi
t ′i for 1≤ i ≤ l , we observe
that x = x0+∑l
i=1λi (x i −x0) = (1−∑l
i=1λi )x0+∑l
i=1λi x i , so C is the open l -simplex with verticesx0, . . . ,x l .
78
Figure 3.6: An alcove C with walls H0, H1, H2 and normal vectors t0, t1, t2 as in Theorem 3.3.23. The hyper-plane configuration corresponds to a reduced irreducible affine root system of type Au
2 .
3.4 Bases of irreducible affine root systems
Let R be an irreducible affine root system of rank l on an affine Euclidean space E . Let H ={Ha : a ∈ R} be the set of affine hyperplanes in E generated by R (see (3.1.8)), and write Er e g =E \⋃
H∈H H for the regular points of E relative toH . Fix an alcove C of E relative toH and letM ⊂H be its set of walls. Let B (C ), or just B if there is no ambiguity about the fixed alcove, be theset of indivisible a ∈R such that Ha ∈M and a (x )> 0 for all x ∈C . The set B is said to be a basis ofR (relative to the alcove C ), its elements are called simple affine roots and the reflections wa ∈W (R)with a ∈ B are called simple reflections. In this Section we will show that B generates R i nd throughthe action of W (R). Since W (R) is generated by the simple reflections (Theorem 3.3.16), this showsthat B contains all the information necessary to construct R i nd . Therefore B will play a centralrole in the next Section when we classify all reduced irreducible affine root systems. We will alsoshow here that B satisfies a natural generalization of the criteria of a basis for a finite root system.
Proposition 3.4.1. B is a basis of the l +1-dimensional R-vector space bE .
Proof. From the proof of Theorem 3.3.23 we haveM = {H0, . . . , Hl }, and the normal vectors t1, . . . , t l
to H1, . . . , Hl respectively form a basis for V . Hence H1, . . . , Hl have the unique intersection pointx0 ∈ E . Identify E with V as a vector space by choosing the origin x0. Put B = {a 0, . . . , a l } such thata i vanishes on Hi for all i . Then a 1, . . . , a l can be considered as linear functionals on V by (iii) ofExample 3.1.6, so for each i ∈ {1, . . . , l } there exists a unique vector Da i ∈V such that a i = (Da i , .)V .Then each Da i is a normal vector to Hi . Thus the linear functionals a 1, . . . , a l must be linearly in-dependent, else we get a contradiction with the normal vectors t1, . . . , t l being a basis of V . Thisimplies that the linear functionals a 1, . . . , a l form a basis of V ∗. Now, there exists Da 0 ∈ V suchthat a 0(x0+v ) = a (x0)+(Da 0, v )V for all v ∈V . Furthermore, C is an l -simplex by Theorem 3.3.23,
so H0 does not contain x0 and a (x0) 6= 0. Hence there exist c1, . . . , c l ∈ R such that a 0+∑l
i=1 c i a i
is the nonzero constant function on E that is identically a 0(x0). By Proposition 3.1.12 B is now abasis of the (l +1)-dimensional space bE .
79
In the following we will use transitive action of W (R) on the collection of alcoves of E relativetoH to show that B generates R i nd by letting the W (R) act on B .
Proposition 3.4.2. For each indivisible a ∈ R there exists w ∈ W (R) such that a = w (b ) for someb ∈ B.
Proof. Let a ∈ R , then the affine hyperplane Ha is a wall of some alcove C ′ by Proposition 3.3.13.Since Ha = {y ∈ E : a (y ) = 0} and since C ′ ⊂ Er e g is connected, we have a (x ) > 0 or a (x ) < 0 forall x ∈C ′. Assume that a is positive on C ′. By Theorem 3.3.16 there exists an orthogonal reflectionw ∈W (R) such that C ′ = w (C ), hence w−1(a ) is positive on C . We observe by Lemma 3.3.15 thatHw−1(a ) =w−1(Ha ) is a wall of C , so if we let a ∈R be indivisible then w−1(a )∈ B .
If a is negative on C ′, then DHa (C ′) 6= DHa (wa (C ′)) are the connected components of E \Ha =E \ {y ∈ E : a (y ) = 0} so a is positive on the alcove wa (C ′). By Lemma 3.3.15 we have that Ha
is a wall of wa (C ′). We can now repeat the argument of the last paragraph with a and the alcovewa (C ′).
Moreover, we observe the following.
Proposition 3.4.3. B and R generate the same Z-lattice in bE of rank l +1.
Proof. Write L(B ) (resp. L(R)) for the lattice generated by B (resp. R). Clearly L(B ) ⊂ L(R), andL(R) is generated by the indivisible affine roots of R . By Proposition 3.4.2 it is enough to prove thatL(B ) is stable under W (R), moreover by Theorem 3.3.16 it is enough to show that wa (L(B ))⊂ L(B )for all a ∈ B . For a ,b ∈ B we have wa (b ) = b − (a∨,b )
bE a by (3.1.15) with (a∨,b )bE ∈Z by Definition
3.2.1, which implies that wa (b ) ∈ L(B ). Therefore wa (L(B )) ⊂ L(B ) for all a ∈ B , which completesthe proof.
Definition 3.4.4. An affine root a ∈ R is called positive (resp. negative) (relative to C ) if a (x ) > 0(resp. a (x )< 0) for all x ∈C .
Since no affine root vanishes on Er e g , every affine root in R is either positive or negative. Let R+
denote the subset of positive roots of R and R− the negative roots, then we have the disjoint unionR = R+ ∪R−. Moreover, R− =−R+ since wa (a ) =−a for all a ∈ R by (3.1.15). This also shows thatH = {Ha : a ∈R}= {Ha : a ∈R+}.
By Theorem 3.3.23 we have that an alcove C is an open l -simplex in E . Let x0, . . . ,x l be thevertices of C , and turn E into a vector space by choosing an origin. Then C is given by the points
x ∈ E of the form x =∑l
i=0λi x i with∑l
i=0λi = 1 and each λi > 0. Put B = B (C ) = {a 0, . . . , a l } suchthat a i (x j ) = 0 for i 6= j . In other words, x j ∈Ha i for all i 6= j , so since C is an l -simplex x i /∈Ha i .But it does hold that x i ∈C and by definition of B we have that a i is positive on C , hence a i (x i )> 0for each i . Also by definition of B , we observe that B ⊂ R+. Moreover, analogous to the theory offinite root systems we have the following decomposition of positive (resp. negative) affine rootsinto positive (resp. negative) sums of simple affine roots.
Proposition 3.4.5. For every positive (resp. negative) affine root a ∈ R there exist positive (resp.
negative) λ0, . . . ,λl ∈Z such that a =∑l
i=0λi a i .
Proof. By Proposition 3.4.3, we can write a ∈ R as a =∑l
i=0λi a i with λi ∈ Z for 0 ≤ i ≤ l . Evalu-ating both sides of this equation at x i yields λi = a (x i )/a i (x i ) for 0 ≤ i ≤ l . Assume a ∈ R+, thena (x )≥ 0 for all a ∈C and in particular a (x i )≥ 0. Because a i (x i )> 0, we obtain λi ≥ 0 for 0≤ i ≤ l .Likewise, if a is negative, we observe that λi ≤ 0 for 0≤ i ≤ l .
80
In the next Proposition we will see that a similarity transformation between two affine rootsystems is compatible with the notion of a basis of an affine root system.
Proposition 3.4.6. Let R ⊂ bE and R ′ ⊂cE ′ be irreducible affine root system such that R ' R ′ realizedby the similarity transformation T : bE
∼−→cE ′. If B is a basis of R, then B ′ := T (B ) is a basis of R ′.
Proof. First, consider two irreducible affine root systems R ⊂ bE and R ′ ⊂ cE ′ such that R ' R ′
realized by a similarity transformation T : bE∼−→cE ′. Then by Proposition 3.2.13 there exists λ∈R>0
and an affine linear isomorphismψ : E → E ′ such that T (a ) =λ(a ◦ψ−1) for all a ∈ bE .Next, letH = {Ha : a ∈R+} (resp.H ′ := {Ha ′ : a ′ ∈R ′+}) be the collection of affine hyperplanes
in E (resp. E ′) induced by R (resp. R ′), and let B = {a 0, . . . , a l } be a basis of R with respect to analcove C ⊂ E relative toH . First we show that ψ(C ) ⊂ E ′ is an alcove relative toH ′, and then weshow that the set B ′ := T (B ) = {λ(a 0 ◦ψ−1), . . . ,λ(a l ◦ψ−1)}=: {a ′0, . . . , a ′l } ⊂ R ′ is a basis of R ′ withrespect toψ(C ).
By Corollary 3.2.14,ψ induces a bijection fromH ontoH ′ given by the mapping Ha 7→ψ(Ha ) =HT (a ). Sinceψ : E → E ′ is an affine linear isomorphism,ψ is also a homeomorphism relative to theEuclidean topologies on E and E ′ respectively by Corollary 3.1.8. Henceψ′ :=ψ|Er e g : Er e g → E ′r e gis a homeomorphism relative the induced topologies on Er e g and E ′r e g respectively. Now C is analcove in E , so Proposition 3.3.7 together with the fact thatψ is a homeomorphism yields thatψ(C )is an alcove in E ′ relative toH ′.
Now B ′ consists of indivisible roots because T is linear and T |R : R → R ′ is a bijection. Further,using Theorem 3.3.23 we can writeM := {Ha 0 , . . . , Ha l } ⊂H for the set of walls of the alcove C ⊂ E .Then by a similar argument as in Lemma 3.3.15 one observes thatM ′ := {ψ(Ha 0 ), . . . ,ψ(Ha l )} ⊂H ′
is the set of walls of ψ(C ). Finally, let y ∈ψ(C ), then for 1 ≤ i ≤ l we have a ′i (y ) = λa i (ψ−1(y )) =λa i (x ) for some x ∈ C . But λ > 0, and a i is positive on C because B is a basis relative to C . Weconclude that all elements in B ′ are positive on the alcoveψ(C ), thus B ′ is a basis for R ′with respectto the alcoveψ(C ).
Remark 3.4.1. Notice that being a similarity transformation is a necessary condition on T in Propo-sition 3.4.6. Otherwise we obtain λ < 0 from the proof of Proposition 3.2.13, and in that case onecan not conclude that the elements in B ′ are positive on the alcoveψ(C ).
We end this section with a result on finite root systems contained in an irreducible affine rootsystem R . For each i let bE i be the subspace of affine linear functions of bE that vanish on the vertexx i of the alcove C , and write Ri := R ∩ bE i . Since bE = Hom(E ,R) (where 0 is the origin in R), weobserve by (3.1.3) that bE i coincides with the vector space HomR(Ex i ,R). Since bE i does not containany nonzero constant functions the symmetric bilinear form (., .)
bE is positive definite on bE i , hencebE i is an inner product space with respect to this form.
Proposition 3.4.7. Ri is a finite root system in bE i , and it is reduced if R is. A basis of Ri is given byB \ {a i }.
Proof. By Lemma 3.3.3 there exist only finitely many affine hyperplanes in H passing throughx i . Furthermore, we know from nonreduced affine root systems that maximally four affine rootsinduce the same affine hyperplane ofH . Hence Ri is finite, and it does not contain 0 since R doesnot contain any constant functions. Further, a j (x i ) = 0 for i 6= j , so B \ {a i } ⊂ Ri ⊂ bE i . Since B is abasis of the (l +1)-dimensional space bE , B \{a i } is a linearly independent set of l vectors in bE i . AsbE i is a strict subspace of bE , we conclude that B \ {a i } and hence Ri spans bE i . Also for a ,b ∈ Ri , itis clear from the definition of R that (a∨,b )
bE ∈Z, and that wa (b ) = b − (a∨,b )bE a ∈ R . In particular
wa (b )∈Ri since a and b vanish at x i . Finally, it is immediate that Ri is reduced if R is.
81
In the last paragraph we already saw that B \ {a i } is a basis of the vector space bE i . Now leta ∈ Ri , then by Proposition 3.4.5 there exist either all positive or all negative λ0, . . . ,λl ∈ Z such
that a =∑l
j=0λj a j . Evaluating both sides of this equation at x i yields λi = 0, hence a =∑
i 6=j λj a j
with either all positive or all negative λj ∈Z for each j 6= i . Thus B \ {a i } is a basis of the finite rootsystem Ri .
3.5 A classification of reduced irreducible affine root systems
In this Section we will proceed with the classification of reduced irreducible affine root systems upto similarity. First, we will relate affine Cartan matrices to irreducible affine root systems using thenotion of a basis. Then we will construct a bijective correspondence between reduced irreducibleaffine root systems up to similarity class, and affine Cartan matrices up to simultaneous permu-tations of rows and columns. Using the earlier developed classification of affine Cartan matrices,this will finish the classification. Along the way we will also explicitly construct a complete set ofrepresentatives for the similarity classes of reduced irreducible affine root systems, which we willuse to explain the naming of the classification of affine Dynkin diagrams. Finally, we will relate theobtained correspondence to the bijective correspondence between affine Cartan matrices up tosimultaneous permutations of rows and columns and isomorphism classes of Lie algebras that areisomorphic to affine Lie algebras to obtain an analogue of (1.3.1).
3.5.1 Affine Cartan matrices of irreducible affine root systems
Fix an ordered basis B = (a 0, . . . , a n ) of an irreducible affine root system R ⊂ bE , and define thematrix A(R , B ) = (a i j )0≤i ,j≤l where a i j := (a∨i , a j ) bE . If there is no ambiguity about R and B , we willjust write A for A(R , B ). The coefficients a i j for 0 ≤ i , j ≤ l are called the affine Cartan integers ofR , and the matrix A is said to be the affine Cartan matrix of R . The following result justifies thisdefinition.
Proposition 3.5.1. For each basis B of R, the (l + 1)× (l + 1)-matrix A = A(R , B ) is a rank l affineCartan matrix.
Proof. We will check all conditions of Definition 2.2.3 explicitly for the (l + 1)× (l + 1)-matrix A =A(R , B ). Firstly, A = (a i j )0≤i ,j≤l is a rational integral matrix such that a i i = 2 for all i by criterion (3)of Definition 3.2.1. Secondly, a i j = 0 implies a j i = 0 by the definition of the affine Cartan integers.Further, recall that each vector Da k for 0≤ k ≤ l is a nonzero multiple of the vector eHa k
as definedbefore Lemma 3.3.21. Since a i and a j are positive on the fixed alcove C , we even have that Da k is apositive multiple of eHa k
for 0≤ k ≤ l . Now a i j = (a∨i , a j ) bE = (Da∨i , Da j )V with Da∨i =2
(a i ,a i ) bEDa i ,
so Lemma 3.3.21 and 3.3.22 give a i j ≤ 0 if i 6= j .Next, {Da 0, . . . , Da l } is a linearly dependent set in V since dim(V ) = l . Since a i j = (Da∨i , Da j )V
for 0 ≤ i , j ≤ l , we observe that det(A) = 0. Further, by Proposition 3.4.7, B \ {a k } is a basis for afinite root system. Hence the proper principal submatrix A ′ := (a i j )i ,j∈{0,...,l }\{k } of A is a Cartanmatrix of a finite root system. This implies that A ′ is decomposable into a direct sum of finiteCartan matrices. From linear algebra we have that A ′ =
�A1 00 A2
�
implies det(A ′) = det(A1)det(A2),hence every principal minor of A ′ is the product of principal minors finite Cartan matrices. Itfollows from Proposition 2.2.2 that every principal minor of A ′ is strictly positive. This implies thatall proper principal minors of A are strictly positive. Finally, from the irreducibility of R it followsdirectly that A is an indecomposable matrix.
82
Next, we are interested in the dependency of the affine Cartan matrix of an irreducible affineroot system on the different choices of representatives of its similarity class and on the choices ofbases.
Proposition 3.5.2. Let R ⊂ bE and R ′ ⊂cE ′ be irreducible affine root systems such that R 'R ′ realizedby the similarity transformation T : R
∼−→R ′, and let B = {a 0, . . . , a l } be an ordered basis of R.(i) Let B ′ := T (B ) = {T (a 0), . . . , T (a l )}=: {a ′0, . . . , a ′l }, then A(R , B ) = A(R ′, B ′).(ii)Let eB be another ordered basis of R, then A(R , B ) coincides with A(R , eB ) up to simultaneous
permutation of rows and columns.
Proof. (i) Consider two irreducible affine root systems R ⊂ bE and R ′ ⊂ cE ′ such that R ' R ′ real-ized by a similarity transformation T : bE
∼−→cE ′. Further, let B = {a 0, . . . , a l } be an ordered basisof R with respect to the alcove C ⊂ E . Then by Proposition 3.4.6 we obtain that B ′ := T (B ) ={T (a 0), . . . , T (a l )} =: {a ′0, . . . , a ′l } is an ordered basis of R ′ with respect to the alcove ψ(C ) ⊂ E ′.Here ψ : E → E ′ is the affine linear isomorphism such that T (a ) = λ(a ◦ψ−1) for some fixedλ > 0 from Proposition 3.2.13. Now it follows from the definition of similarity that (a ′∨i , a ′j )cE ′ =(T (a i )∨, T (a j ))cE ′ = (a
∨i , a j ) bE for all 0≤ i , j ≤ l , hence the matrices A(R , B ) and A(R ′, B ′) coincide.
(ii) Next, let eB be an ordered basis of R with respect to the alcove eC ⊂ E . By Theorem 3.3.16there exists w ∈W (R) such that w ( eC ) =C where w is considered as an element of GL(E ). Then wconsidered as an element of GLR,c ( bE ) is a normalized similarity transformation of R with itself byExample 3.2.12. Hence we observe from the last paragraph that B ′ := {w (a 0), . . . , w (a l )} ⊂ R is anordered basis of R with respect to the alcove w−1(C ) = eC .
Now for each a ∈ eB there exists a unique b ∈ B ′ such that both a and b vanish on a wall ofC . Since both a and b are indivisible roots, they must coincide. Consequently eB = B ′ up to anordering, hence eB is the w -image of B up to an ordering. Since w acting on bE is a similaritytransformation of R with itself we have (w (a i )∨, w (a j )) bE = (a
∨i , a j ) bE for all 0 ≤ i , j ≤ l . Hence we
observe that the matrices A(R , B ) and A(R , B ′) coincide up to simultaneous permutations of rowsand columns.
We conclude the matrix A(R , B ) up to simultaneous permutations of rows and columns does notdepend on the choice of basis B of R nor on the choice of representative of the similarity class ofR .
Recall that we write A for the equivalence class of the affine Cartan matrix A under the equiva-lence relation of simultaneous permutations of rows and columns of matrices. Further, recall thatCa denotes the collection of indecomposable affine Cartan matrices up to simultaneous permu-tations of rows and columns. Now, write R for the similarity class of the irreducible affine rootsystem R , and put Ra for the similarity classes of reduced irreducible affine root systems. ThenProposition 3.5.2 immediately implies the following result.
Corollary 3.5.3. There exists a well defined map A :Ra →Ca given by R 7→ A(R , B ) =: A(R), whereB is any choice of ordered basis of the representative R of the similarity class R.
Moreover, the affine Cartan matrix up to simultaneous permutations of rows and columns of areduced irreducible affine root systems R determines R uniquely up to similarity.
Proposition 3.5.4. The map A :Ra →Ca is injective.
Proof. Let R , R ′ ∈Ra , and assume that we have representatives R ∈ R (resp. R ′ ∈ R ′) together withan ordered basis B ⊂ bE (resp. B ′ ⊂cE ′) such that A(R) = A(R , B ) = A(R ′, B ′) = A(R ′). Then we can
83
fix an new ordering on B = (a 0, . . . , a l ) and B ′ = (a ′0, . . . , a ′l ) such that A(R , B ) = A(R ′, B ′). In the
following we will construct a linear isomorphism T : bE →cE ′ that realizes a similarity between Rand R ′. This will imply that the similarity classes R and R ′ coincide which shows the injectivity ofA.
Since B (resp. B ′) is a basis of bE (resp. cE ′), there exists a unique linear isomorphism T : bE →cE ′defined by T (a i ) = a ′i for 0≤ i ≤ l . For a i , a j ∈ B we obtain
wT (a i )(T (a j )) = T (a j )− (T (a i )∨, T (a j ))cE ′ T (a i )
= T (a j )− (a∨i , a j ) bE T (a i )
= T (a j − (a∨i , a j ) bE a i )
= T (wa i (a j )).
Since B is a basis of bE we have T ◦wa i ◦T−1 =wT (a i ) oncE ′ for all a i ∈ B . Further, T ◦wa i ◦wa j ◦T−1 =T ◦wa i ◦T ◦T−1◦wa j ◦T−1 =wT (a i )◦wT (a j ) for a i , a j ∈ B , and the simple reflections wa i for a i ∈ B ′
generate W (R ′). Hence we have a group isomorphism eT : W (R)∼−→W (R ′), w 7→ T ◦w ◦T−1 with
inverse w ′ 7→ T−1 ◦w ′ ◦T .Next, we want to observe that T |R : R → R ′ is a bijection. By Proposition 3.4.2 we have that
each a ∈ R is W (R)-conjugate to some a i ∈ B , say a = w (a i ) for some w ∈W (R). Consequently,T (a ) = T (w (a i )) = T (w (T−1(T (a i )))) = (T ◦w ◦T−1)(a ′i ) ∈ R ′ since T ◦w ◦T−1 ∈W (R ′) by eT , thusT |R : R → R ′ is a well-defined map. But T is a linear isomorphism, so since each affine root in Ris a unique sum of simple affine roots in B , T |R is also injective. This argument is symmetric in Rand R ′, so we easily observe that also T−1|R ′ : R ′ → R is well defined and injective. Consequentlywe obtain that T |R : R→R ′ is a bijection.
Finally, for all a ,b ∈R we have wT (a )(T (b )) = T (b )−(T (a )∨, T (b ))cE ′ T (a ) and (T ◦wa ◦T−1)(T (b )) =
T (b )− (a∨,b )bE T (a ). If we would know that wT (a ) = T ◦wa ◦ T−1 for all a ∈ R , then we obtain
(a∨,b )bE = (T (a )
∨, T (b ))cE ′ for all a ∈ R . To show that wT (a ) = T ◦wa ◦T−1 for a ∈ R , write a =w (a i )
for some w ∈ W (R) and a i ∈ B . Further, we can write w = w i 1 ◦ · · · ◦w i r for some r ∈ N, whereeach w i j = wa i j
for some i j ∈ {0, . . . , l }. In the following we leave out the ◦-symbol in the formu-
las to improve readability. Firstly, we already know that Twa i T−1 = wT (a i ) = wa ′i, so put w i ′j
:=wa ′i j
=wT (a i j ) = Twa i jT−1 = Tw i j T−1. Secondly, by (i) of Example 3.2.9 and (3.2.1) we obtain that
w wa i w−1 =ww (a i ) and w i ′1. . . w i ′r
wa ′i(w i ′1
. . . w i ′r)−1 =w (w i ′1
...w i ′r)(a ′i )
. This leads directly to
Twa T−1 = Tww (a i )T−1
= Tw wa i w−1T−1
= Tw i 1 . . . w i r wa i w i r . . . w i 1 T−1
= Tw i 1 T−1 . . . Tw i r T−1Twa i T−1Tw i r T−1 . . . Tw i 1 T−1
=w i ′1. . . w i ′r
wa ′iw i ′r
. . . w i ′1
=w i ′1. . . w i ′r
wa ′i(w i ′1
. . . w i ′r)−1
=w (w i ′1...w i ′r
)(a ′i )
=w (Tw i 1 T−1...Tw i r T−1)(T (a i ))
=wTw T−1(T (a i )) =wT (w (a i )) =wT (a ).
Hence we conclude that (a∨,b )bE = (T (a )
∨, T (b ))cE ′ for all a ∈R .
84
Thus we have shown that T realizes a similarity between R and R ′, but then R and R ′ mustcoincide. Since A(R) = A(R ′)we conclude that A :Ra →Ca is injective.
3.5.2 Explicit constructions of reduced irreducible affine root systems
In Proposition 3.5.4 of the previous Subsection we saw that the map A :Ra →Ca from Corollary3.5.3 is injective. Our goal in this Subsection is to show that A is also a surjective map. First we willconstruct some concrete reduced irreducible affine root systems based on the explicit realizationsof∆r e in Proposition 2.3.1. Then we will pick a specific set of these realizations and show that thecorresponding affine Cartan matrices form a complete set of representatives for the classes inCa .This leads to the bijectivity of A :Ra →Ca which implies that we have constructed a complete setof representatives for the similarity classes inRa .
Let R0 be an irreducible finite root system (possibly nonreduced) in V with finite Weyl groupW0(R0)⊂GLR(V ). Let {α1, . . . ,αl } be a basis of R0, then {α∨1 , . . . ,α∨l } is a basis of R∨0 . Furthermore, R0
(resp. R∨0 ) is contained in the root lattice Q :=∑l
i=1Zαi ⊂ V (resp. coroot lattice Q∨ :=∑l
i=1Zα∨i ⊂
V ). Write t (L) := {tv : v ∈ L} for subgroup of translations of t (V ) over the lattice L ⊂ V . Next, con-sider the vector space V as affine Euclidean space E with V as space of translations (see Example3.1.2). Let c be the constant one function on E , and identify bE with V ⊕Rc after choosing the origin0 ∈ E (see Prop. 3.1.12 and Cor. 3.1.14). By Proposition 3.1.21 we have the identification of GL(E )with t (V )oGLR(V ) using the same choice of origin. Hence for each a :=λc +α∈ bE with α∈V \{0}and λ∈Rwe have
wa = t−λα∨ ◦wα (3.5.1)
by (3.1.13) and (3.1.12), where wα ∈ GLR(V ). Here tv (a ) = a − (Da , v )V c for v ∈ V and a ∈ bE by(3.1.16).
Consider the subsets of V ⊕Rc
RuR0
:= {m c +α}m∈Z,α∈R i nd0∪{(2m +1)c +β}m∈Z,β∈R0\R i nd
0
and
R tR0
:= {m|α|2V
2c +α}m∈Z,α∈Ru nm
0∪{(2m +1)
|β |2V2
c +β}m∈Z,β∈R0\Ru nm0
where R i nd0 (resp. Ru nm
0 ) are the indivisible (resp. unmultipliable) roots of R0.We will now see that Ru
R0and R t
R0are reduced irreducible affine root systems that are related by
their dual.
Proposition 3.5.5. RuR0
is a reduced irreducible affine root system on E with gradient root systemD(Ru
R0) = R0 and affine Weyl group W (Ru
R0) = t (Q∨)oW0(R0), where t (Q∨), W0(R0) ⊂ W (Ru
R0) are
subgroups.
Proof. First we will check all conditions of Definition 3.2.1 for RuR0
as introduced before this Propo-sition. From its definition it is clear that Ru
R0considered in V ⊕Rc is a nonempty subset of non-
isotropic elements with respect to the form (3.1.4) on V ⊕Rc .(1) Since R0 spans V it is clear that Ru
R0spans V ⊕Rc .
(2) It follows from a straightforward computation that the reflections generated by RuR0
stabilizeRu
R0.(3) By Corollary 3.1.14 using the form (3.1.4) on V ⊕Rc we obtain ((m c +α)∨, (nc +β ))V⊕Rc =
(α∨,β )V ∈Z for m , n ∈Z and α,β ∈R0.
85
(4) Consider the group W (RuR0) generated by the reflections wa for a ∈ Ru
R0. Then by (3.5.1) we
havewa =wm c+α = t−mα∨wα (3.5.2)
if we write a =m c+α for some m ∈Z and α∈R0. Letting m = 0, we observe that wα ∈W (R) for allα ∈ R0, hence W0(R0) ⊂W (R) is a subgroup. Also, letting m = −1 we observe that tα∨ = wa w−1
α =wa wα ∈W (Ru
R0) for all α ∈ R0. Hence we observe that t (Q∨)⊂W (Ru
R0) is a subgroup. Furthermore,
t (Q∨)∩W0(R0) = {idV }, because t0 = idV is the only translation map that fixes 0. By (3.1.14) we alsohave for w ∈ W0(R0) and γ ∈Q∨ that w tγw−1 = tw (γ). Now R0 is W0(R0)-invariant so Q∨ as well,hence we obtain that tw (γ) ∈ t (Q∨). Further, (3.5.2) implies that W (Ru
R0) = t (Q∨)W0(R0) := {t w :
t ∈ t (Q∨) and w ∈ W0(R0)}. So we conclude that t (Q∨) ⊂ W (RuR0) is a normal subgroup and that
W (RuR0) = t (Q∨)oW0(R0). Then t (Q∨) acts properly on E as a discrete group, because Q∨ ⊂ V is
a discrete set. Furthermore, W0(R0) ⊂ W (RuR0) is a finite group, so W (Ru
R0) = t (Q∨)oW0(R0) acts
properly on E .(5) It follows from the definition of Ru
R0that for each α ∈ R0 there are at least two distinct a ,b ∈
RuR0
such that Da =Db =α.We conclude that Ru
R0is an affine root system on E . Further, by definition of Ru
R0all its affine
roots are indivisible, hence RuR0
is reduced. Also, since R0 is irreducible and ((m c + α)∨, (nc +β ))V⊕Rc = (α∨,β )V for m , n ∈ Z and α,β ∈ R0, the affine root system Ru
R0is irreducible as well.
Finally, it is clear from Corollary 3.1.13 that D(RuR0) =R0.
Remark 3.5.1. The construction of RuR0
gives rise to an example of a reduced irreducible affine rootsystem with nonreduced irreducible gradient root system. Let R0 be a nonreduced irreduciblefinite root system, then the affine root system Ru
R0is reduced with gradient root system R0.
Proposition 3.5.6. R tR∨0
coincides with the dual of RuR0
.
Proof. Let m ∈Z and α∈R0 such that m c +α∈RuR0
, then
(m c +α)∨ =2
(m c +α, m c +α)V(m c +α) =
2
|α|2V(m c +α) =m
2
|α|2Vc +α∨ =m
|α∨|2V2
c +α∨. (3.5.3)
This leads to
(RuR0)∨ = {(m c +α)∨}m∈Z,α∈R i nd
0∪{((2m +1)c +β )∨}m∈Z,β∈R0\R i nd
0
(3.5.3)= {m|α∨|2V
2c +α∨}m∈Z,α∈R i nd
0∪{(2m +1)
|β∨|2V2
c +β∨}m∈Z,β∈R0\R i nd0
= {m|α|2V
2c +α}m∈Z,α∈(R i nd
0 )∨ ∪{(2m +1)|β |2V
2c +β}m∈Z,β∈R∨0 \(R
i nd0 )∨
= {m|α|2V
2c +α}m∈Z,α∈(R∨0 )u nm ∪{(2m +1)
|β |2V2
c +β}m∈Z,β∈R∨0 \(R∨0 )
u nm
=R tR∨0
,
where we used that (R i nd0 )∨ = (R∨0 )
u nm .
Corollary 3.5.7. R tR0
is a reduced irreducible affine root system on E with gradient root systemD(R t
R0) = R0 and affine Weyl group W (R t
R0) = t (Q)oW0(R0), where t (Q), W0(R0) ⊂W (R t
R0) are sub-
groups.
86
Proof. By Proposition 3.5.6, R tR∨0
coincides with the dual of RuR0
. Since R∨0∨ = R0, we observe that
R tR0
coincides with the dual of RuR∨0
. Therefore by Proposition 3.5.5 together with Proposition 3.2.7,
R tR0
is an affine root system with affine Weyl group W (R tR0) = t (Q)oW0(R∨0 ), where t (Q), W0(R∨0 ) ⊂
W (R tR0) are subgroups. But W0(R0) =W0(R∨0 ), so we obtain W (R t
R0) = t (Q)oW0(R0). Furthermore,
similarly to the case of RuR0
we have that R tR0
is irreducible (see proof of Prop. 3.5.5). Also, R tR0
isthe dual of the reduced affine root system Ru
R0, and it follows straightforwardly from the defini-
tion of the dual of an affine root system that R tR0
needs to be reduced too. Finally, it is clear fromProposition 3.1.12 that D(R t
R0) =R0.
It turns out that we have already come across reduced irreducible affine root systems of theform Ru
R0and R t
R0.
Example 3.5.8. Let A be an affine Cartan matrix, and consider the set of real roots ∆r e ⊂ h∗0 =◦h∗R⊕Rδ corresponding to the affine Lie algebra g(A) in the context of Section 2.3. By Example 3.2.3
(i), 3.2.5 (ii), and 3.2.23 (iii) we have that ∆r e is a reduced irreducible affine root system on◦h∗R. By
Proposition 2.3.1 we have
(1)∆r e = {α+nδ :α∈◦∆, n ∈Z} if∆r e is of untwisted type;
(2)∆r e = {α+n (α,α)2 δ :α∈
◦∆, n ∈Z} if∆r e is of twisted type but not of untwisted type;
(3) ∆r e = {α+n (α,α)2 δ : α ∈
◦∆, n ∈ Z} ∪ { (α,α)
2 α+ (2n + 1) (α,α)2 δ) : α ∈
◦∆l , n ∈ Z} if ∆r e is of mixed
type.
Letting V :=◦h∗R, (., .)V = (., .), R0 :=
◦∆ and c := δ we observe that Ru
R0=∆r e if∆r e is of untwisted
type and R tR0= ∆r e if ∆r e is of twisted but not untwisted type. If ∆r e is of mixed type then we
observe from Remark 2.3.1 that◦∆∪ 1
2
◦∆l is a nonreduced irreducible finite root system with
◦∆ the
unmultipliable roots and 12
◦∆l the indivisible short roots. So if we put R0 :=
◦∆∪ 1
2
◦∆l , then R t
R0=∆r e .
Analogue to the case of reduced (irreducible) finite root systems we have the following dualityof bases.
Lemma 3.5.9. If (a 0, . . . , a l ) is a basis of the reduced irreducible affine root system R, then (a∨0 , . . . , a∨l )is a basis of the dual R∨.
Proof. Let {a 0, . . . , a l } be a basis of R with respect to some alcove C . Now a∨ = 2(a ,a )
bEa = 0 if and
only if a = 0, so R∨ and R generate the same collection of affine hyperplanesH in E . This meansthat they also generate the same alcoves. Since a∨ is a positive multiple of a ∈ R and R is reducedwe observe that {a∨0 , . . . , a∨l } is a basis of the reduced irreducible affine root system R∨ with respectto the alcove C .
Finally, we obtain the following important Proposition.
Proposition 3.5.10. (i) The injective map A :Ra →Ca defined by the mapping R 7→ A(R , B ) =: A(R)is also surjective.
(ii) The following reduced irreducible affine root systems form a complete set of representatives ofthe similarity classes inRa :(1) Ru
R0with R0 running through the similarity classes of reduced irreducible finite root systems (i.e.
R0 of type A l (l ≥ 1), Bl (l ≥ 3),C l (l ≥ 2), Dl (l ≥ 4), E6, E7, E8, F4,G2);
87
(2) R tR0
with R0 running through the similarity classes of reduced irreducible finite root systems hav-ing two root lengths (i.e. R0 of type Bl (l ≥ 3),C l (l ≥ 2), F4,G2);(3) Ru
R0with R0 a nonreduced irreducible finite root systems (i.e. R0 of type BC l (l ≥ 1)).
Proof. First we will construct bases for the reduced irreducible affine root systems in each of thethree cases as stated in (ii).
(1) Let B ′ := {α1, . . . ,αl } be a basis of R0. Further, let φ be the unique highest root of R0 withrespect to B ′. We will show that B := {α0 := c −φ,α1, . . . ,αl } is a basis of Ru
R0with respect to a
certain alcove C . Note that B contains only indivisible affine roots since RuR0
is reduced. Nowconsider the set C := {x ∈ V : (αi ,x )V > 0 for 1 ≤ i ≤ l and (φ,x )V < 1}, then by definition of C allelements of B are positive on C . We claim that C = {x ∈ V : 0< (α,x )V < 1 for all α ∈ R+0 }. Clearly,{x ∈ V : 0 < (α,x )V < 1 for all α ∈ R+0 } ⊆ C holds. On the other hand if x ∈ C , then (αi ,x )V > 0 for1 ≤ i ≤ l , so also (α,x )V > 0 for all α ∈ R+0 since B ′ is a basis of R0. Further, φ is the highest rootof R0, so φ −α ≥ 0 for all α ∈ R+0 in the dominance partial order on R0 with respect to B ′. Hence
for α ∈ R+0 we can write φ −α =∑l
i=1 c iαi where c i ≥ 0 for all i . For x ∈ C we have (αi ,x )V > 0
for all i , which leads to (φ−α,x )V = (∑l
i=1 c iαi ,x )V ≥ 0. Thus (α,x )V ≤ (φ,x )V < 1 for all α ∈ R+0which shows that C = {x ∈ V : 0 < (α,x )V < 1 for all α ∈ R+0 }. So every x ∈ C lies in between theaffine hyperplanes Hα and H1−α for all α ∈ R+0 . Further, by definition of Ru
R0there are no affine
roots yielding parallel affine hyperplanes to Hα lying in between Hα and H1−α for α ∈ R+0 . Thisimplies C ⊂ Er e g relative toH = {Ha : a ∈ Ru
R0}, and C is connected in Er e g . Moreover if x ∈ Er e g
but x /∈ C , then there exists α ∈ R+0 such that (α,x )V < 0 or (α,x )V > 1. Thus C is a connectedcomponent of Er e g , and an alcove of E relative toH by Lemma 3.3.7. Then by Theorem 3.3.23,C is an open simplex with l + 1 walls, and it is clear that B = {α0,α1, . . . ,αl } generates the walls ofC = {x ∈ V : (αi ,x )V > 0 for 1 ≤ i ≤ l and (φ,x )V < 1}. Thus B is a basis of Ru
R0with respect to the
alcove C .(2) Since R t
R0= (Ru
R∨0)∨ we deduce from case (1) and Lemma 3.5.9 that B := {α0 := |θ |2V
2 c −θ ,α1, . . . ,αl } is a basis of R t
R0with respect to the alcove C . Here B ′ := {α1, . . . ,αl } is a basis of R0,
and θ ∈R0 is the highest short root of R0 with respect to B ′.(3) Let B ′ = {α1, . . . ,αl } be basis of R0, then B ′ is also a basis of the reduced irreducible finite
root system R i nd0 . Further, let φ be the highest root of R0, hence φ = 2α for some α ∈ R i nd
0 . ThenB := {α0 := c −φ,α1, . . . ,αl } is a set of indivisible affine roots since Ru
R0is reduced. Define C = {x ∈
V : (αi ,x )V > 0 for 1≤ i ≤ l and (φ,x )V < 1}, then the affine roots in B are positive on C . By similararguments as in (1), C = {x ∈ V : 0< (α,x )V < 1 for all α ∈ Ru nm
0+}. So every x ∈C lies between Hα
and H1−α for α ∈ Ru nm0
+, and by definition of RuR0
there are no affine roots yielding parallel affinehyperplanes to Hα lying in between Hα and H1−α, so C ⊂ Er e g and is connected. Then analogousto (1) one observes that B is a basis of Ru
R0with respect to the alcove C .
It is an elementary exercise to compute the affine Cartan matrix and draw the affine Dynkindiagram for each reduced irreducible affine root system with the constructed basis B as describedin this proof. Using the classification of affine Dykin diagrams from Figure 2.1 we get the followingcorrespondence. If Ru
R0is as in (1) with R0 of type X l , then the affine Dynkin diagram S(Ru
R0) is of
type X ul . If R t
R0is as in (2) with R0 of type X l and not of type C2, then the affine Dynkin diagram
S(R tR0) is of type X t
l . If R tR0
is as in (2) with R0 of type C2, then the affine Dynkin diagram S(R tR0) is
of type B t2 . If Ru
R0is as in (3) with R0 of type BC l , then the affine Dynkin diagram S(Ru
R0) is of type
BC ml .We conclude that the reduced irreducible affine root systems as stated in (ii) are in bijective
correspondence with all affine Dynkin diagrams. It follows that the injective map A of Proposition3.5.4 is surjective, and that the reduced irreducible affine root systems of (ii) form a complete set
88
of representatives ofRa .
Remark 3.5.2. (i) Consider case (1) (resp. (2)) of the proof of Proposition 3.5.10. If we equip thenodes of the affine Dynkin diagram S(Ru
R0) (resp. S(R t
R0)) with the ’α’-labeling of Figure 2.1, then
removing node α0 and the edges connected to it yields the finite Dynkin diagram correspondingto R0 (forgetting the labeling of the nodes again). In case (3) we need to remove node αl in S(Ru
R0)
and the edges connected to it to obtain the finite Dynkin diagram for R0 (which is of type Bl ).(ii) The construction of a basis for R t
R0in case (2) of Proposition 3.5.10 (ii) is valid for any reduced
irreducible finite root system R0. In the situation that R0 has only one root length, the highest shortroot coincides with the highest root with respect to a chosen basis for R0. Then it is immediatethat the corresponding affine Cartan matrix coincides with the affine Cartan matrix of Ru
R0, hence
RuR0'R t
R0by Proposition 3.5.4.
In view of Proposition 3.5.10 and (ii) of Remark 3.5.2 we call an affine root system R of untwistedtype if R ' Ru
R0with R0 reduced, of twisted type if R ' R t
R0with R0 reduced and of mixed type if
R 'RuR0
with R0 nonreduced. Notice from (ii) of Remark 3.5.2 that affine roots systems R such thatR 'Ru
R0with R0 of type A l (l ≥ 1), Dl (l ≥ 4), E6, E7 and E8 are both of untwisted and twisted type.
3.5.3 The naming of affine Dynkin diagrams explained
In this Subsection we want to give the rationale behind the naming of the types of affine Dynkindiagrams as can be found in Figure 2.1.
First, this naming is directly linked to the classification of similarity classes of reduced irre-ducible affine root systems. Second, it relies on the classification of similarity classes of reducedirreducible finite root systems. This is possible since the similarity class (or type if you want) of thegradient root system D(R) is an invariant for the similarity class of the affine root system R .
Lemma 3.5.11. If R ⊂ bE and R ′ ⊂cE ′ are similar affine root systems, then D(R) and D(R ′) are similargradient root systems.
Proof. Let V (resp. V ′) be the space of translations of the affine Euclidean space E (resp. E ′), andput c (resp. c ′) for the constant one function on E (resp. E ′). Choose an origin x ∈ E (resp. y ∈ E ′),then by Proposition 3.1.12 we can identify bE (resp. cE ′) with V ⊕Rc (resp. V ′⊕Rc ′) with respect tothe choice of origin. Assume that T : V⊕Rc →V ′⊕Rc ′ realizes the similarity between R and R ′, andlet pV ′ : V ′⊕Rc ′→ V ′ be the projection onto V ′ along the direct sum. Then t := pV ′ ◦T |V : V → V ′
is a linear isomorphism that maps D(R) onto D(R ′) by Corollary 3.1.13. Furthermore, it followsdirectly from the properties of T that (t (α)∨, t (β ))V ′ = (α∨,β )V for all α,β ∈D(R).
If the reduced irreducible affine root system R is of type Xjl , then R is similar to exactly one of
the reduced irreducible affine root systems that is stated in (ii) of Proposition 3.5.10. Then Lemma3.5.11, Proposition 3.5.5, Corollary 3.5.7 and the proof of Proposition 3.5.10 show that D(R) is oftype X l , except if R is of type B t
2 then D(R) is of type C2. The latter exception has to do with the factthat the finite Dynkin diagram of type C2 can also be considered as being of type B2, since it alsonaturally fits into the family Bl as the rank 2 Dynkin diagram. However the affine Dynkin diagramof type C u
2 (resp. B t2 ) fits naturally into the family C u
l (resp. B tl ) as the rank 2 Dynkin diagram.
For the sake of the classification of finite Dynkin diagrams we made the choice of considering thefinite Dynkin diagram of type C2 as part of the C family.
The final ingredient of the naming of a type of affine Dynkin diagram is the untwised, twisted ormixed type of the affine root system. By definition these types are similarity invariants of reduced
89
irreducible affine root systems. If R is of type Xjl , then j ∈ {u , t , m }. If j = u , then R is of untwisted
type. If j = t , then R is of twisted type. If j =m , then R is of mixed type. On the other hand, if R is areduced irreducible affine root system of untwisted, twisted or mixed type respectively with D(R)of type X l but not C2, then R is of type X
jl with j equal to u , t or m respectively. If D(R) is of type
C2 and R is of untwisted (resp. twisted) type, then R is of type C u2 (resp. B t
2 ).
3.5.4 A new commutative triangle
We end this Chapter relating reduced irreducible affine root systems back to affine Lie algebras andreal roots. This will gives us a canonical bijection between affine Lie algebras, reduced irreducibleaffine root systems and affine Cartan matrices up to the appropriate equivalent relations in theform of a commuting triangle. Furthermore, this will prove that every affine Lie algebra up toisomorphism is uniquely determined by its set of real roots up to similarity.
Proposition 3.5.10 (i) together with the classification of affine Cartan matrices from Figure 2.1gives us a classification of reduced irreducible affine root systems. Together with Theorem 2.2.10we obtain the following canonical bijections between the reduced irreducible affine root systemsup to similarity Ra , the affine Cartan matrices up to simultaneous permutations of rows andcolumnsCa and the isomorphism classesLa of Lie algebras isomorphic to an affine Lie algebra
RaA−→Ca
g−→La .
Define∆ := (g ◦A)−1 :La →Ra , then we obtain the commutative diagram
La∆ → Ra
Ca
A
←
g
←
(3.5.4)
Finally, using the map ∆ we want to show that for an affine Lie algebra its set of real roots isits naturally associated affine root system. So consider g′ ∈ La , then there exists an affine Cartanmatrix A such that g′ = g(A). Let the affine Lie algebra g := g(A,h,Π,Π∨) be a representative whereof g′. We consider g as in the context of Section 2.3. So let W denote the Weyl group of g, let (., .)denote the normalized invariant form that is defined on h∗ by 2.3.1, let Π = {α0, . . . ,αl } and put
h∗0 =◦h∗R⊕Rδwith
◦h∗R =
⊕li=1Rαi and δ=
∑li=0 a iαi such that the vector (a 0, . . . , a l ) is in the kernel
of A and has positive relatively prime integer coordinates. Recall that (., .) is positive definite on◦h∗R
and vanishes on Rδ.By (2.1.11) we observe that
A =�
2(αi ,αj )(αi ,αi )
�
0≤i ,j≤l.
Hence by definition of ∆ : La → Ra we must have that Π corresponds to the basis of a reduced
irreducible affine root system in h∗0 =◦h∗R ⊕Rδ with bilinear form (., .). Consider h∗0 as the space
of affine linear function on the affine space◦h∗R as in (i) of Example 3.2.3. Since Π ⊂ h∗0 is a basis
of a reduced irreducible affine root system R , (iii) of Theorem 3.3.16 tells us that the affine Weyl
group W (R) is generated by the simple reflections wαi in◦h∗R. Consider W (R) in GLR,c (h∗0) using
90
Proposition 3.1.22, then for β ∈ h∗0 we have
wαi (β ) =β −2(β ,αi )(αi ,αi )
αi = ri (β ) (3.5.5)
where ri ∈W is the fundamental reflection of h∗ generated by αi ∈ Π. Now W is generated by thefundamental reflections ri for i = 0, . . . , l and the action of W on h∗0 is faithful, so we can identifyW with W (R) using the identification of their generators in (3.5.5). By Proposition 3.4.2 we havethat for each a ∈ R there exists αi ∈Π and w ∈W (R) such that a =w (αi ) and R is W (R)-invariant.But∆r e is a W -invariant subset of h∗0 and for each α∈∆r e there exists αi ∈Π and w ∈W such thatα=w (αi ), so we must have R =∆r e . In other words,
∆(g(A,h,Π,Π∨)) =∆r e . (3.5.6)
In particular, every affine Lie algebra up to isomorphism is uniquely determined by its set of realroots up similarity. We conclude that we can classify affine Lie algebras not only according to theiraffine Cartan matrix, but also according to their associated affine root system which correspondswith the set of real roots of the affine Lie algebra.
91
92
Summary (in Dutch)
Het is een bekend wiskundig feit dat simpele Lie-algebras, gereduceerde irreduciebele wortelsys-temen en eindige Cartan-matrices op de juiste isomorfie-equivalenties na in bijectieve correspon-dentie met elkaar zijn. In deze scriptie trachten we op een analoge wijze zo een bijectieve corre-spondentie te beschrijven tussen affiene Lie-algebras, gereduceerde irreduciebele affiene wortel-systemen en affiene Cartan-matrices. Aangezien de bijectie tussen affiene Lie-algebras en affieneCartan-matrices al duidelijk is, beslaat het merendeel van dit manuscript het bestuderen van affienewortelsystemen met het doel ze te relateren aan affiene Cartan-matrices. Om enig inzicht te gevenin affiene wortelsystemen, zullen we hier de iets eenvoudigere ’gewone’ wortelsystemen bespreken.Aangezien wortelsystemen veel te maken hebben met spiegelingen en spiegelsymmetrieën zullenwe daar eerst iets over vertellen.
Beschouw een eindig-dimensionale reële vectorruimte V met een inproduct (., .). Het inproduct(., .) geeft ons de mogelijkheid om te praten over lengtes van en hoeken tussen vectoren in V . Webrengen in herinnering dat we met een hypervlak in V een lineaire deelruimte van V bedoelen vanéén dimensie lager dan V zelf.
We kunnen de ruimte V orthogonaal spiegelen in een hypervlak H in V . Dit leidt tot een lineairetransformatie van V . Een (orthogonale) spiegeling in V is een lineair automorfisme van V dat depunten van een hypervlak H in V op zijn plaats houdt en een vector α die loodrecht op H staatnaar−α stuurt. Aangezien V wordt opgespannen door het hypervlak H en de vector α, legt dit eenspiegeling compleet vast vanwege lineariteit (zie Figuur 4.1). We noteren deze spiegeling met wα.
Figure 4.1: Een voorstelling van een spiegeling wα in een 2-dimensionale ruimte.
93
Er bestaat een eenvoudige formule voor een spiegeling wα, namelijk
wα(v ) = v −2(v,α)(α,α)
α
voor alle v ∈V . Het is gemakkelijk in te zien dat in de bovenstaande forumule inderdaad de puntenvan het hypervlak
Hα = {w ∈V : (w ,α) = 0}
loodrecht op α op zijn plaats worden gehouden, en dat α naar −α wordt gestuurd. Intuïtief kanHα gezien worden als een ’dubbelzijdige spiegel’ in V , waarbij wα het spiegelbeeld geeft van elkepunt in V ten opzichte van Hα (zie Figuur 4.1). Verder kan worden opgemerkt dat als je dezelfdespiegeling twee keer achter elkaar uitvoert dan worden alle punten van V op zijn plaats gelaten(vergelijk dit met het feit dat jij zelf het spiegelbeeld van jouw eigen spiegelbeeld bent).
We noemen een eindige ondergroep van de groep van lineaire automorfismen van V die wordtvoortgebracht door spiegelingen een eindige reflectiegroep. In deze scriptie beschouwen we eindigerefectiegroepen die aan een extra symmetrie-eis voldoen. Een rooster L in V is een abelse onder-groep van V van de vorm
Zv1+ · · ·+Zvl
met {v1, . . . , vl } een basis van V . In Figuur 4.2 staat een voorbeeld van een rooster in een 2-dimensionaleruimte afgebeeld. We zeggen dat een eindige reflectiegroep W kristallografisch is als W een roosterinvariant laat in V . Deze naamgeving komt uit de kristallografie waar symmetriegroepen vankristalroosters een belangrijke rol spelen.
Figure 4.2: Een rooster Zv1+Zv2.
Een wortelsysteem ∆ is een verzameling vectoren in V die aan de volgende vier condities vol-doet
(1)∆ is eindig, bevat 0 niet en spant V op;(2)∆∩Rα= {α,−α} voor alle α∈∆;(3) wα(β )∈∆ voor alle β ∈∆;(4) 2 (β ,α)
(α,α) ∈Z voor alle α,β ∈∆.De groep W0 voortgebracht door de spiegelingen wα voor α ∈ ∆ is een kristallografische eindige
94
reflectiegroep genaamd de Weyl-groep van ∆. Het blijkt dat elke kristallografische eindige reflec-tiegroep een Weyl-groep van een wortelsysteem is. Echter er bestaan structureel verschillendewortelsystemen met dezelfde Weyl-groep.
Figure 4.3: Drie wortelsystemen in een 2-dimensionale ruimte.
Een wortelsysteem∆ kan gezien worden als de kleinste verzameling vectoren in V die V opspant,zodanig dat∆ invariant is onder de kristallografische eindige reflectiegroep W0, en dat de spiegelin-gen wα voor α∈∆ de groep W0 voortbrengen. Dit impliceert dat∆ een aantal spiegelsymmetrieënheeft. In Figuur 4.3 staan drie wortelsystemen afgebeeld in een 2-dimensionale ruimte V , waarbijelke pijl een vector α ∈ ∆ voorstelt. Het blijkt dat ∆ te partitioneren is in twee verzamelingen ∆+
en ∆−, zodat zowel ∆+ als ∆− precies de normalen bevat van de hypervlakken waarin gespiegeldwordt binnen W0. Bovendien geldt dan∆+ =−∆−.
Indien we een hypervlak in V transleren met een vector krijgen we een deelverzameling vanV die mogelijk niet de oorsprong bevat en geen lineaire deelruimte van V is. We noemen zo eenverschoven hypervlak een affien hypervlak. Zo kan het hypervlak
Hα = {w ∈V : (w ,α) = 0}
loodrecht op α∈V na een translatie met een vector in V beschreven worden als het affiene hyper-vlak
Hα,k := {w ∈V : (w ,α) = k }
voor een zekere k ∈R. Het is nog steeds mogelijk om orthogonaal te spiegelen in een affien hyper-vlak Hα,k . Bovendien wordt de formule voor zo een affiene spiegeling wα,k gegeven door slechtseen kleine aanpassing te maken in de formule voor de spiegeling wα:
wα,k (v ) = v −2(v,α)−k
(α,α)α
voor v ∈ V . Dit is geen lineaire transformatie van V maar een zogeheten affiene transformatie.Affiene transformaties bestaan in het algemeen uit een lineaire transformatie samengesteld meteen translatie.
Beschouw nu een wortelsysteem ∆. De groep Wa van affiene transformaties van V die wordtvoortgebracht door de affiene spiegelingen wα,k met α ∈ ∆ en k ∈ Z heet de affiene Weyl-groepvan ∆. Een affiene Weyl-groep is groep van oneindige orde. Voor het linker wortelsysteem ∆ inFiguur 4.3 hebben we in Figuur 4.4 de hypervlakken Hα,k met α∈∆ en k ∈Z afgebeeld. De affieneWeyl-groep van∆ permuteert deze hypervlakken.
95
Figure 4.4: De affiene hypervlakken gerelateerd aan een affiene Weyl-groep.
In deze scriptie bestuderen we onder andere Macdonald’s generalisatie van de axiomatischedefinitie van een wortelsysteem met een aantal noodzakelijke technische modificaties. Dit noe-men we een affien wortelsysteem. Analoog aan de situatie bij wortelsystemen heeft elk affienewortelsysteem R een gerelateerde affiene Weyl-groep Wa . Een affien wortelsysteem R kan danook, analoog aan het geval van een wortelsysteem, gezien worden als een Wa -invariante verzamel-ing normalen van de affiene hypervlakken waarin gespiegeld wordt binnen de gerelateerde affieneWeyl-groep Wa . Om affiene wortelsystemen te bestuderen blijkt het makkelijk te zijn om te werkenin zogenaamde affiene ruimtes. Dit zijn vectorruimtes waarbij we ’vergeten’ zijn waar de oorsprongzich bevindt. Uiteindelijk relateren we affiene wortelsystemen aan de hand van speciale genera-toren aan affiene Cartan-matrices en laten we zien hoe een affiene Lie-algebra op natuurlijke wijzeeen geassocieerd affien wortelsysteem heeft.
96
Bibliography
[1] N. Bourbaki, Algèbre Chapitre 9: Formes sesquilinéaires et formes quadratiques, Éléments demathématique, Hermann, 1959.
[2] N. Bourbaki, Lie Groups and Lie Algebras Chapter 4-6, Elements of mathematics, Springer-Verlag, 2002.
[3] N. Bourbaki, General Topology Part 1, Elements of mathematics, Hermann, 1966.
[4] A.J. Coleman, The Greatest Mathematical Paper of All Time, The Mathematical Intelligencer11 (1989), no. 3, 29-38.
[5] J. Hontz, K.C. Misra, Root multiplicities of the indefinite Kac-Moody Lie algebras HD (3)4 and
HG (1)2 , Communications in Algebra 30 (2002), no. 6, 2941-2959.
[6] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts inMathematics 9, Springer-Verlag, 1972.
[7] J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in adv. math. 29,Cambridge University Press, 1990.
[8] V.G. Kac, Infinite dimensional Lie algebras, Third edition, Cambridge University Press, 1990.
[9] V.W. Klima, K.C. Misra, Root multiplicities of the indefinite Kac-Moody algebras of symplectictype, Communications in Algebra 36 (2008), no. 2, 764-782.
[10] I.G. Macdonald, Affine Root Systems and Dedekind’s η-Function, Inventiones Mathematicae15 (1972), 91-143.
[11] J.R. Munkres, Topology, Second Edition, Prentice Hall, 2000.
[12] D.H. Peterson, V.G. Kac, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci.USA 80 (1983), 1778-1782.
[13] T.A. Springer, Linear Algebraic Groups, Second Edition, Birkhäuser Boston, 1998.
[14] J.V. Stokman, Macdonald Polynomials, arXiv:1111.6112v1.
[15] Z. Wan, Introduction to Kac-Moody Algebra, World Scientific Publishing, 1991.
97
![Lie Algebras - University of Idahobrooksr/liealgebraclass.pdf · volumes [1], Lie Groups and Lie Algebras, Chapters 1-3, [2], Lie Groups and Lie Algebras, Chapters 4-6, and [3], Lie](https://static.documents.pub/doc/80x56/5ec51f9cde3711693f3d65c7/lie-algebras-university-of-idaho-brooksrliealgebraclasspdf-volumes-1-lie.jpg)
![Chapter 7 Lie Groups, Lie Algebras and the Exponential Mapcis610/cis61005sl8.pdf · Lie Groups, Lie Algebras and the Exponential Map 7.1 Lie Groups and Lie Algebras In Gallier [?],](https://static.documents.pub/doc/80x56/5f0c1a337e708231d433c07b/chapter-7-lie-groups-lie-algebras-and-the-exponential-map-cis610-lie-groups.jpg)
