CONTEMPORARY MATHEMATICS

297

Recent Developments in Infinite-Dimensional Lie Algebras

and Conformal Field Theory Proceedings of an International Conference on Infinite-

Dimensional Lie Theory and Conformal Field Theory May 23-27, 2000

University of Virginia, Charlottesville, Virginia

Stephen Berman Paul Fendley Yi-Zhi Huang Kailash Misra Brian Parshall

Editors

CoNTEMPORARY MATHEMATICS

297

Recent Developments in Infinite-Dimensional Lie Algebras

and Conformal Field Theory Proceedings of an International Conference on Infinite-

Dimensional Lie Theory and Conformal Field Theory May 23-27, 2000

University of Virginia, Charlottesville, Virginia

Stephen Berman Paul Fendley Yi-Zhi Huang Kailash Misra Brian Parshall

Editors

American Mathematical Society Providence, Rhode Island

http://dx.doi.org/10.1090/conm/297

Editorial Board

Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius This volume contains the proceedings of an International Conference on Infinite-

Dimensional Lie Theory and Conformal Field Theory held at the University of Virginia, Charlottesville, on May 23-27, 2000.

2000 Mathematics Subject Classification. Primary 17B10, 17B37, 17B65, 17B69, 17B80, 17B81; Secondary 81 T40, 82B23.

Library of Congress Cataloging-in-Publication Data Recent developments in infinite-dimensional Lie algebras and conformal field theory / Stephen Berman ... [et al.], editors.

p. em. -(Contemporary mathematics; ISSN 0271-4132; 297) Includes bibliographical references. ISBN 0-8218-2716-2 (softcover : alk. paper) 1. Infinite-dimensional Lie algebras-Congresses. 2. Conformal invariants-Congresses.

3. Quantum field theory-Congresses. I. Berman, Stephen, 1944- II. Contemporary mathematics (American Mathematical Society) ; v. 297.

QA252.3 .R42 2002 5121.55--dc21 2002071170

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Table of Contents

Talks Presented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Vertex Operator Algebras and the Representation Theory of Toroidal Algebras S. Berman, Y. Billig, and J. Szmigielski ....................................... 1

Symmetric Functions and Representations of Quantum Affine Algebras V. Chari and M. Kleber ...................................................... 27

Two Realizations of Toroidal st2(1C) Ben Cox ..................................................................... 47

Vertex Lie Algebras, Vertex Poisson Algebras and Vertex Algebras C. Dong, H. Li, and G. Mason ............................................... 69

Type A Fusion Rules from Elementary Group Theory A. Feingold and M. Weiner .................................................. 97

Lie Algebra Automorphisms in Conformal Field Theory J. Fuchs and C. Schweigert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

On Lepowsky-Wilson's Z-Algebra Y. Hara, M. Jimbo, H. Konno, S. Odake, and J. Shiraishi 143

Scattering Rules in Soliton Cellular Automata Associated with Crystal Bases G. Hatayama, A. Kuniba, M. Okado, T. Takagi, andY. Yamada ............ 151

Algebra Versus Analysis in Statistical Mechanics and Quantum Field Theory B. McCoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Weak Modules and Logarithmic Intertwining Operators for Vertex Operator Algebras A. Milas .................................................................... 201

Conjugate Bailey Pairs A. Schilling and S. 0. Warnaar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

iii

iv TABLE OF CONTENTS

Irreducibility of Affine Heeke Algebra Modules Induced from Specht Modules M. Vazirani ................................................................. 257

Algebraic Structures behind Hilbert Schemes and Wreath Products W. Wang ................................................................... 271

Some Generalizations of Genus Zero Two-Dimensional Conformal Field Theory W. Zhao .................................................................... 297

Talks Presented

M. Kashiwara Crystal Bases for Representations of Quantized Affine Lie Algebras, I, II, III

B. McCoy The Dominance of Algebra in Statistical Mechanics and Quantum Field Theory, I, II, III

N. Read Lie Superalgebras and Disordered Systems, I, II, III

P. Abramenko On Commutator Relations in Kac-Moody Groups

K. Barron Geometric Aspects of Neveu-Schwarz Lie Superalgebras

Y. Billig VOAs Arising in the Representation Theory of Toroidal Lie Algebras

V. Chari Representations of Quantum Affine Algebras

C. Dong Reductive Lie Algebras and Holomorphic VOAs

J. Fuchs Lie Algebra Automorphisms and Conformal Field Theory

V. Gurarie Logarithmic Algebras at c = 0

P. H. Hai A Formula for Computing the Integral on the Quantum Supergroup GLq(mjn)

Y. Hara q-Deformations of Superconformal Field Theory

R. Kedem Structure of Coinvariants of Integrable s[2-Modules

A. Kuniba Crystals and Soliton Cellular Automata, I

M. Kleber Quantum Affine Algebra Representations and Discrete Dynamical Systems

J. Lepowsky Vertex Operator Algebras and the Zeta Function

M. Okado Crystals and Soliton Cellular Automata, II

v

vi TALKS PRESENTED

A. Schilling Expressions for Type A Branching Functions

E. Vasserot On the Action of the Dual Group on the Cohomology of Perverse Sheaves on the Affine Grassmanian

M. Vazirani Irreducible Modules of Heeke Algebras

W. Wang The McKay Correspondence

W. Zhao Generalizations of the Genus Zero CFT to Locally Trivialized G-Bundles over CP2

Preface

This volume constitutes the proceedings of an international conference on "In-finite dimensional Lie theory and conformal field theory" held at the University of Virginia from May 23 to May 27, 2000. The conference aimed to provide an opportunity for mathematicians and physicists to interact in an active research area of mutual interest. The talks focused on recent developments in the represen-tation theory of affine, quantum affine, and extended affine Lie algebras and Lie superalgebras. In addition, they highlighted applications to conformal field theory, integrable and disordered systems.

Because of its many applications to mathematics and mathematical physics, the representation theory of infinite dimensional Lie and quantized enveloping al-gebras comprises an important area of current research. Many of these applications concern, in fact, the representation theory of affine and quantum affine Lie alge-bras. The crystal basis theory introduced independently by Kashiwara and Lusztig around 1990 provides a method to study the combinatorics of the integrable repre-sentations of affine and quantum affine Lie algebras. The theory of crystal bases is still in a formative stage. During the conference, Kashiwara delivered three lectures on the crystal basis theory, while presenting his vision for the future. One hopes that crystal basis theory will play a role in the representation theory of symmetriz-able Kac-Moody Lie algebras similar to the role played by Young tableaux in the representation theory of symmetric groups. In fact, the semistandard tableaux form the crystal basis for the irreducible rational representations of sln(C). The theory of crystal basis is closely related to solvable lattice models. Solvable vertex models in a ferromagnetic regime give rise to soliton cellular automata at q = 0. In the paper by Hatayama, Kuniba, Okado, Takagi and Yamada a class of such automata associated with classical quantum affine algebras are studied using crystal basis theory. The one-dimensional configuration sums of certain solvable lattice models and the representation theory of the affine Lie algebra A~ 1 ) are used in the paper by Schilling and Warnaar to derive interesting new q-series identities. The paper by Vazirani shows an interesting connection between the crystal bases of certain quantum affine algebras and the irreducible representations of the affine Heeke al-gebra of type A while Chari and Kleber's article explores the relationship between the quantum affine algebras and symmetric functions.

In 1983, the first conference on infinite-dimensional Lie theory and confor-mal field theory titled "Vertex operators in mathematics and physics" was held in MSRJ. In that conference, topics covered ranged from representations of infinite-dimensional Lie algebras and the construction of the moonshine module to string and conformal field theories. Since then, researches in this direction have broadened and deepened greatly. Comparing the papers in the proceedings of the conference mentioned above and those in this volume, one can see clearly the rapid evolution of the field.

Although investigation of various generalizations of the affine Lie algebras be-gan in the late 1980s, the pace of this study has increased considerably in recent years. Toroidal Lie algebras and their quantum counterparts, both of which have attracted much attention from physicists and mathematicians, provide one example. These algebras fit into the larger context of extended affine Lie algebras. Applica-tions of these algebras to integrable systems have been studied, and some talks at the conference touched on this work along with the relevant representation theory.

vii

viii PREFACE

The paper by Berman, Billig and Szmigielski shows how to give realizations of an infinite class of irreducible modules for some toroidal Lie algebras in terms of vertex algebras, while the paper by Cox presents two free field realizations of a toroidal sl2 (C) algebra.

The mathematicians and physicists, although having quite different perspec-tives on the questions with which the conference dealt, made efforts to understand each other's work. Because of their real world based point of view, the physicists perceive results and directions that mathematicians often do not expect. On the other hand, mathematicians bring rigor to results which would otherwise only have been understood intuitively. Clearly, both points of view have importance. McCoy's article provides an interesting physics perspective about analysis and algebra.

Vertex operator algebras (VOAs) give another important generalization of affine Lie algebras and their representations. VOAs have deep connections with conformal field theories which play an important role in both condensed matter physics and string theory. Mathematicians in different fields now study these theories as a natural mathematical structure. Several papers in this volume study various aspects of vertex operator algebras and their connections with conformal field theories. The paper by Dong, Li and Mason study vertex Lie algebras and vertex poisson algebras which can be viewed as classical analogues of VOAs. Taking a combinatorial point of view, Feingold and Weiner's article presents results concerning fusion rules in the case of type A affine algebras. Specializing certain Wakimoto type represenations of deformed Virasoro algebras, Hara, Jimbo, Konno, Odake and Shiraishi give a new way of obtaining the Lepowsky-Wilson Z-algebras. The paper by Fuchs and Schweigert discusses various results and conjectures concerning Lie algebras, VOAs and conformal field theories from the physicists' point of view. The paper by Milas introduces logarithm intertwining operators and initiates a study of logarithm conformal field theories in terms of the representation theory of vertex operator algebras. Zhao establishes an equivalence between VOAs which are also integrable representations of affine Lie algebras and certain projective algebras over a partial operad. The paper by Wang establishes connections between Hilbert schemes and vertex algebras.

During the conference, M. Kashiwara, B. McCoy, and N. Read each presented a three-lecture mini-course, providing an excellent opportunity for graduate students and other non-experts to obtain a better grasp of the material. In addition, there were nineteen invited talks delivered at the conference. Some of these talks provided an overview of the subjects, while others focused on current research on these subjects. A majority of these speakers have contributed articles for this proceedings.

We thank all the participants, the speakers and especially the authors whose papers are included here. We also thank the enormous contribution from the anony-mous referees who spent much time reviewing these papers. We express here our gratitude to the National Science Foundation (through grant DMS-0070590), the Dean of the Faculty of the University of Virginia, and the Departments of Mathe-matics and Physics of the University of Virginia for the funding and support of this conference. Finally, we thank Ms. Julie Riddleberger who provided considerable technical assistance in the preparation of this volume.

The Editors

Titles in This Series

297 Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, and Brian Parshall, Editors, Recent developments in infinite-dimensional Lie algebras and conformal field theory, 2002

296 Sean Cleary, Robert Gilman, Alexei G. Myasnikov, and Vladimir Shpilrain, Editors, Combinatorial and geometric group theory, 2002

295 Zhangxin Chen and Richard E. Ewing, Editors, Fluid flow and transport in porous media: Mathematical and numerical treatment, 2002

294 Robert Coquereaux, Ariel Garcia, and Roberto Trinchero, Editors, Quantum symmetries in theoretical physics and mathematics, 2002

293 Donald M. Davis, Jack Morava, Goro Nishida, W. Stephen Wilson, and Nobuaki Yagita, Editors, Recent progress in homotopy theory, 2002

292 A. Chenciner, R. Cushman, C. Robinson, and Z. Xia, Editors, Celestial Mechanics, 2002

291 Bruce C. Berndt and Ken Ono, Editors, q-series with applications to combinatorics, number theory, and physics, 2001

290 Michel L. Lapidus and Machiel van Frankenhuysen, Editors, Dynamical, spectral, and arithmetic zeta functions, 2001

289 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, Second summer school in analysis and mathematical physics: Topics in analysis: Harmonic, complex, nonlinear and quantization, 2001

288 Marisa Fernandez and Joseph A. Wolf, Editors, Global differential geometry: The mathematical legacy of Alfred Gray, 2001

287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001

286 Edward L. Green, Serkan Ho§ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001

285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001

284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001

283 Paul A. Milewski, Leslie M. Smith, Fabian Waleffe, and Esteban G. Tabak, Editors, Advances in wave interaction and turbulence, 2001

282 Arlan Ramsay and Jean Renault, Editors, Groupoids in analysis, geometry, and physics, 2001

281 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering II, 2001

280 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering I, 2001

279 Alejandro Adem, Gunnar Carlsson, and Ralph Cohen, Editors, Topology, geometry, and algebra: Interactions and new directions, 2001

278 Eric Todd Quinto, Leon Ehrenpreis, Adel Faridani, Fulton Gonzalez, and Eric Grinberg, Editors, Radon transforms and tomography, 2001

277 Luca Capogna and Loredana Lanzani, Editors, Harmonic analysis and boundary value problems, 2001

276 Emma Previato, Editor, Advances in algebraic geometry motivated by physics, 2001 275 Alfred G. Noel, Earl Barnes, and Sonya A. F. Stephens, Editors, Council for

African American researchers in the mathematical sciences: Volume III, 2001 274 Ken-ichi Maruyama and John W. Rutter, Editors, Groups of homotopy

self-equivalences and related topics, 2001

TITLES IN THIS SERIES

273 A. V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, and C. Vinsonhaler, Editors, Abelian groups, rings and modules, 2001

272 Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki, Editors, Quadratic forms and their applications, 2000

271 J. P. C. Greenlees, Robert R. Bruner, and Nicholas Kuhn, Editors, Homotopy methods in algebraic topology, 2001

270 Jan Denef, Leonard Lipschitz, Thanases Pheidas, and Jan Van Geel, Editors, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000

269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations and foliations in dynamics, geometry and topology, 2001

268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential geometric methods in the control of partial differential equations, 2000

267 Nicolas Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen Schneider, Editors, New trends in Hopf algebra theory, 2000

266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and analytic geometry, 2000

265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique: Homotopy theory in the Swiss Alps, 2000

264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Efim I. Zelmanov, Editors, Combinatorial and computational algebra, 2000

263 Yan Guo, Editor, Nonlinear wave equations, 2000 262 Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors,

Crystallographic groups and their generalizations, 2000 261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability

on algebraic structures, 2000 260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in

analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000

259 D. V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and its applications, 2000

258 Karsten Grove, Ib Henning Madsen, and Erik Kjrer Pedersen, Editors, Geometry and topology: Aarhus, 2000

257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000

256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000

255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear PDE's, dynamics and continuum physics, 2000

254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000

253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej.

Because of its many applications to mathematics and mathematical physics, the represen-tation theory of infinite-dimensional Lie and quantized enveloping algebras comprises an important area of current research. This volume includes articles from the proceedings of an international conference, "Infinite-Dimensional Lie Theory and Conformal Field Theory", held at the University of Virginia. Many of the contributors to the volume are prominent researchers in the field.

This conference provided an opportunity for mathematicians and physicists to interact in an active research area of mutual interest. The talks focused on recent developments in the representation theory of affine, quantum affine, and extended affine Lie algebras and Lie superalgebras. They also highlighted applications to conformal field theory, integrable and disordered systems.

Some of the articles are expository and accessible to a broad readership of mathematicians and physicists interested in this area; others are research articles that are appropriate for more advanced readers.

ISBN 0-8218-2716-2

9 780821 827161