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CONTEMPORARY 442

Lie Algebras, Vertex Operator Algebras and Their Applications International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina

Yi-Zhi Huang Kailash C. Misra Editors http://dx.doi.org/10.1090/conm/442

Lie Algebras, Vertex Operator Algebras and Their Applications In honor of James Lepowsky and Robert Wilson on their sixtieth birthdays CoNTEMPORARY MATHEMATICS

442

Lie Algebras, Vertex Operator Algebras and Their Applications

International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina

Yi-Zhi Huang Kailash C. Misra Editors

American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Andreas Blass Abel Klein

2000 Mathematics Subject Classification. Primary 17810, 17837, 17850, 17865, 17867, 17868, 17869, 81T40, 82823.

Photograph of James Lepowsky and Robert Wilson is courtesy of Yi-Zhi Huang.

Library of Congress Cataloging-in-Publication Data Lie algebras, vertex operator algebras and their applications : an international conference in honor of James Lepowsky and Robert L. Wilson on their sixtieth birthdays, May 17-21, 2005, North Carolina State University, Raleigh, North Carolina / Yi-Zhi Huang, Kailash Misra, editors. p. em. ~(Contemporary mathematics, ISSN 0271-4132: v. 442) Includes bibliographical references. ISBN-13: 978-0-8218-3986-7 (alk. paper) ISBN-10: 0-8218-3986-1 (alk. paper) 1. Lie algebras~Congresses. 2. Vertex operator algebras. 3. Representations of algebras~ Congresses. I. Leposwky, J. (James). II. Wilson, Robert L., 1946- III. Huang, Yi-Zhi, 1959- IV. Misra, Kailash, 1954- V. Title. QA252.3 .L555 2007 512'.482~dc22 2007060784

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission®ams. or g. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 12 11 10 09 08 07 Contents

Preface vii

Biographies of James Lepowsky and Robert Wilson XI List of Ph.D. Students xiii

List of Talks XV List of Participants xix Lie algebras and related topics

A Class of Gradings of Simple Lie Algebras KARIN BAUR AND NOLAN WALLACH 3

Conjugacy Results for the s(2 over an Algebra Which is a UFD STEPHEN BERMAN AND JUN MORITA 17

Kirillov-Reshetikhin Modules Associated to VYJAYANTHI CHARI AND ADRIANO MOURA 41 Support Spaces and Auslander-Reiten Components ROLF FARNSTEINER 61 On the Cohomology of Modular Lie Algebras J ORG FELDVOSS 89 Eisenstein Series on Loop Groups: Maass-Selberg Relations 4 HOWARD GARLAND 115

Generalized Littlewood-Richardson Rule for Exceptional Lie Algebras and AYUMU HOSHINO 159 An Introduction to Qn and Its Graph Related Quotients DAVID NACIN 171 Affine Geometric Crystal of Type G~ 1 ) TOSHIKI NAKASHIMA 179 New Constructions of Yang-Baxter Systems FLORIN F. NICHITA AND DEEPAK PARASHAR 193 Construction of Some Algebras Associated to Directed Graphs and Related to Factorizations of Noncommutative Polynomials VLADIMIR RETAKH, SHIRLEI SERCONEK AND ROBERT LEE WILSON 201

v vi CONTENTS

Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras ALISTAIR SAVAGE 221 Lie algebras of Small Dimensiou H. STRADE 233

Vertex (operator) algebras and related topics

Symmetric Polynomials and HD-Quantum Vertex Algebras I. I. ANGUELOVA 269 Hr-Vertex Algebras MAARTEN J. BERGVELT 279 On Intertwining Operators and Recursions CORINA CALINESCU 289 Representations of Vertex Operator Algebras CHONGYING DONG AND CUIPO JIANG 303 On Non-Semisimple Fusion Rules and Tensor Categories J URGEN FUCHS 315 The Duality between Vertex Operator Algebras and Coalgebras, Modules and Co modules KEITH HUBBARD 339 Some Developments in Theory, Old and New JAMES LEPOWSKY 355 Twisted Modules and Quasi-Modules for Vertex Operator Algebras HAISHENG LI 389 Chiral Algebras and Partition Functions GEOFFREY MASON AND MICHAEL P. TUITE 401 Modular Forms and Almost Linear Dependence of Graded Dimensions ANTUN MILAS 411 ( k, r )-Admissible Configurations and Intertwining Operators MIRKO PRIMC 425 Hilbert Schemes of Points on the Minimal Resolution and Soliton Equations ZHENBO QIN AND WEIQIANG WANG 435 Twining Characters and Picard Groups in Rational CHRISTOPH SCHWEIGERT, JURGEN FUCHS, AND INGO RUNKEL 463 Preface

The of finite- and infinite-dimensional Lie algebras has been an important area of mathematical research with numerous applications in many areas of mathematics and . In the last few decades, the classification of the finite-dimensional simple modular Lie algebras has been completed (except for low characteristics) and infinite-dimensional Lie algebras such as Kac-Moody Lie algebras, the and their generalizations have been discovered and studied extensively, with exciting connections to many other fields of mathematics, including combinatorics, , number theory, partial differential equations, topology, conformal field theory, statistical mechanics and integrable systems. The interaction of an important class of infinite-dimensional Lie algebras known as affine Lie algebras with integrable systems led Drinfeld and Jimbo to introduce quantized universal enveloping algebras, also known as quantum groups, associ- ated with symmetrizable Kac-Moody Lie algebras. The representation theory of quantum groups given by Lusztig exhibits similarity with Kac-Moody Lie algebras. In fact Kazhdan and Lusztig showed that the category of modules for a associated with a finite-dimensional at a root of unity is equivalent as a rigid braided tensor category to a suitable category of modules for the corresponding . The representation theory of affine Lie alge- bras together with the theory of the "moonshine module" constructed by Frenkel, Lepowsky and Meurman also led Borcherds to a mathematical definition of a new called a vertex (operator) algebra, which is a mathematically precise algebraic counterpart of the concept of what physicists came to call a "chi- ral algebra" in two-dimensional conformal field theory as formalized by Belavin, Polyakov and Zamolodchikov. These algebras and their representations play im- portant roles in or have deep connections with a number of areas in mathematics and physics, including, in particular, the representation theory of the Fischer-Griess Monster finite and the phenomena of "monstrous moonshine," the rep- resentation theory of the Virasoro algebra and affine Lie algebras, two-dimensional conformal field theory, modular functions, the theory of Riemann surfaces and al- gebraic curves, the geometric Langlands program, knot invariants and invariants of three-manifolds, quantum groups, monodromy associated with differential equa- tions, mirror symmetry, elliptic genera and elliptic cohomology, topological field theories, and theory. During May 17-21, 2005, an international conference on "Lie algebras, vertex operator algebras and their applications" was held in North Carolina State Univer- sity, in honor of James Lepowsky and Robert Wilson on their sixtieth birthdays. James Lepowsky and Robert Wilson have made enormous contributions, individu- ally and jointly, to the development of both the theory of Lie algebras and the theory

vii viii PREFACE of vertex operator algebra~_In 1978, in their joint seminal work on the construc- tion ofrepresentations of sl(2), they discovered what are now called "twisted vertex operators," which were used in their vertex-operator-theoretic proof of the famous Rogers-Ramanujan identities. Subsequently this led them to the discovery of what they called "Z-algebras," which became an important tool for vertex-operator- theoretic proofs of combinatorial identities. Also, this sequence of ideas played a role in leading Lepowsky, jointly with Frenkel and Meurman, to a construction of the "moonshine module vertex operator algebra," whose automorphism group is the Monster. Most of Lepowsky's recent work involves the representation theory of vertex operator algebras and applications. On the other hand Wilson (largely in joint work with Block) made major contributions towards the classification of the finite-dimensional simple modular Lie algebras. In his most recent work, growing out of the theory of quasideterminants, Wilson has collaborated with Israel Gelfand, Vladimir Retakh and Shirlei Serconek in studying algebras related to polynomial equations and to graphs. In addition to being excellent researchers, Lepowsky and Wilson are also wonderful teachers and mentors. Jointly and individually they have supervised 27 Ph.D. students, and many of these students are now established researchers in their fields. The conference and these proceedings are dedicated to them for their important contributions to these fields and their efforts devoted to the training and mentoring of younger generations of mathematicians. This conference brought together researchers from all over the world, including some former students and post-doctoral associates of James Lepowsky and Robert Wilson working on various aspects of Lie algebras, vertex operator algebras and their applications. Some of the speakers gave inspiring expository talks on the development and status of their respective research areas. Others outlined and explored challenges as well as future directions of research for the twenty-first cen- tury. The focus of the conference was mainly 6n Lie algebras, quantum groups, vertex operator algebras and their applications to number theory, combinatorics, integrable systems, conformal field theory and statistical mechanics. About one hundred researchers from Argentina, Australia, Brazil, Canada, China, Croatia, France, Germany, India, Israel, Italy, Japan, Korea, , United Kingdom and USA participated in this conference. The participation of a number of graduate students and junior researchers made the conference very lively and inspiring and the informal interactions among experts, junior researchers and graduate students made the conference a great success. There were about fifty invited and contributed talks given by senior, mid-career and junior researchers working on different aspects of Lie algebras, quantum groups, vertex operator algebras and their applications. The present volume is the proceedings of this conference. The list of talks given at this conference is included in this volume. All the speakers were invited to contribute to these proceedings and many of them did so. For the convenience of the readers we have grouped these contributions into two broad categories: 1. Lie algebras and related topics. 2. Vertex (operator) algebras and related topics. Of course some of the papers could have been in either category. We hope that the papers in this volume will be beneficial to senior researchers as well as beginners and will inspire more research activities in these directions. We are very grateful to the National Science Foundation and the College of Physical and Mathematical Sciences and the Mathematics Department at North PREFACE ix

Carolina State University for the funding and support of this conference. We ap- preciate the Mathematics Department staff and the Conference Housing staff at North Carolina State University for their help during the conference. We thank all the participants, the speakers and especially the authors whose papers are included in this volume. Our special thanks to the anonymous referees for their careful reviews of the papers included in this volume.

Yi-Zhi Huang and Kailash C. Misra Biographies of James Lepowsky and Robert Wilson

James Lepowsky was born on July 5, 1944 in City. He attended and then Harvard College, 1961-65, where Shlomo Stern- berg introduced him to Lie groups and encouraged him to go to M.I.T. and learn about Lie theory from Bertram Kostant. At M.I.T., Lepowsky decided very soon to study with Kostant, and he received his Ph.D. in 1970 under the joint direction of Sigurdur Helgason and Bertram Kostant. His thesis involved representations of real rank one groups, branching laws and minimal K-types. Lepowsky was a lecturer and research associate at Brandeis University from 1970 to 1972. Around this time, he collaborated with Gerald McCollum and with Nolan Wallach on problems in Lie theory and representation theory. He was an assistant professor at from 1972 to 1977, including a year as a Yale Junior Faculty Fellow at the Institute for Advanced Study, where he has also been a mem- ber during four other periods. At Yale he initiated a study of the structure of "generalized Verma modules" and he developed generalizations of the Bernstein- Gelfand-Gelfand resolution. Howard Garland stimulated Lepowsky's interest in Macdonald's identities and homology. Using the results of collaborations with Gar- land, Stephen Milne and Alex Feingold at Yale, Lepowsky began collaborating with Robert Wilson, who was on leave from Rutgers in 1976-77, on what became twisted vertex operator realizations of affine Lie algebras and structures they called "Z-algebras," with applications to partition identities. After lecturing at Universite VI in 1978 during the second year of a Sloan Fellowship, Lepowsky began teaching at Rutgers, where he continued collaborating with Wilson, and where, with Arne Meurman, his first Ph.D. student, he began working on what became a collaboration with Igor Frenkel and Meurman on the construction of a "moonshine module" for the Monster group, based on a new algebra of vertex operators. Some of this work was done when Frenkel, Lepowsky and Meurman were at the Mathematical Sciences Research Institute in 1983-84, and their monograph on this work was completed while the authors were at the Institute for Advanced Study in 1987-88, at which time Lepowsky was a Guggenheim Fellow. Lepowsky has also written monographs with Mirko Prime; with Frenkel and Yi-Zhi Huang, his Rutgers colleague and former student; with Chongying Dong; and with his and Wilson's former student Haisheng Li. Huang and Lepowsky have developed a tensor category theory for suitable classes of modules for a vertex operator alge- bra. Most of Lepowsky's recent work involves the representation theory of vertex operator algebras and applications.

xi xii

Robert Wilson was born on January 16, 1946 in Washington, D.C. and attended public schools in Maryland. While he was an undergraduate at The American Uni- versity in Washington, D.C. from 1962 to 1965, Professor Irving Katz, his professor and mentor, introduced him to abstract algebra and, in particular, to the theory of Lie algebras (by supervising a reading course using Jacobson's recently published Lie Algebras). While a graduate student at Yale University from 1965 to 1969, Wil- son studied under , George Seligman and Robert Steinberg and was present when Alexei Kostrikin gave lectures in May 1967 describing his seminal work with Igor Shafarevich introducing Lie algebras of Cartan type in prime char- acteristic. Wilson's thesis, Nonclassical Simple Lie Algebras, was written under the direction of George Seligman. Wilson, a Courant Instructor at New York University from 1969 to 1971, came to Rutgers in 1971 and has remained there since. For two decades he worked extensively on the structure and classification theory of simple Lie algebras in prime characteristic. Large parts of this work, including the classification of the simple restricted Lie algebras over algebraically closed fields of characteristic > 7, were done jointly with Richard Block. While visiting Yale University (during a sabbatical leave from Rutgers in 1976-77), Wilson was introduced to the theory of affine Lie algebras by Howard Garland, and began a collaboration with James Lepowsky in which they originated the theory of vertex operator representations of affine Lie algebras and used this theory and generalizations to give a vertex-operator theoretic proof of the Rogers-Ramanujan identities and vertex-operator theoretic interpretations of the Gordon-Andrews- Bressoud generalizations of the RR identities. Some of this work was done while Wilson was a Member at the Institute for Advanced Study in fall 1980 and again in 1987-88, and during visits to MSRI during 1983-84. Wilson also collaborated with Earl Taft and David Radford on the construction of certain families of Hopf algebras. In his most recent work, growing out of the theory of quasideterminants, he has collaborated with Israel Gelfand, Vladimir Retakh and Shirlei Serconek in studying algebras related to polynomial equations and to graphs. Between 1990 and 2C03, Wilson devoted much of his time to academic adminis- tration at Rutgers, serving as Chair of the Mathematics Department from 1990 to 1993 and then holding several positions (ultimately Vice-Dean) in the office of the Dean of the Faculty of Arts and Sciences. List of Ph.D. students advised by James Lepowsky and Robert Wilson

Ph.D. students advised by James Lepowsky:

Arne Meurman, 1981 David Mitzman, 1983 Richard Pfister, 1984 Leila Figueiredo, 1986 Haruo Tsukada, 1988 (co-advised with Igor Frenkel) Cristiano Husu, 1990 Yi-Zhi Huang, 1990 Hong Guo, 1995 Katrina Barron, 1996 (co-advised with Yi-Zhi Huang) Galin Georgiev, 1996 Antun Milas, 2001 Lin Zhang, 2004 Carina Calinescu, 2006

Ph.D. students advised by Robert Wilson:

David Kopcso, 1974 Mark Hunacek, 1978 Shirlei Serconek, 1980 Kailash Misra, 1982 Marly Mandia, 1986 Shari Prevost, 1989 Yasmine Sanderson, 1995 (co-advised with Olivier Mathieu) David N acin, 2005

Ph.D. students co-advised by James Lepowsky and Robert Wilson:

Stefano Capparelli, 1988 Xiaoping Xu, 1992 Elizabeth Jurisich, 1994 Haisheng Li, 1994 Chuanfu Xie, 1994 Wanglai Li, 1997

xiii List of talks lana Anguelova Igor Frenkel Quantum Vertex Algebras and quiver constructions of affine Lie algebra representations and Katrina Barron special bases Deformations of the N = 2 Neveu-Schwarz algebra and even and A vital Frumkin odd spectral flow on the A probability approach to Klyachko geometry of N = 2 superconformal field theorem theory J iirgen Fuchs Karin Baur An analogue of the Verlinde formula for Admissible characters in the sense of non-rational vertex algebras Lynch Howard Garland Georgia Benkart Counting the number of rational points Perfect crystals in certain moduli spaces of vector Stephen Berman bundles on surfaces Covering Algebras Dimitar Grantcharov Richard E. Block On the automorphisms of Kac-Moody Dual coalgebras and cofree coalgebras Lie superalgebras

Corina Calinescu Ayumu Hoshino Principal subspaces of representations Polyhedral realizations of crystal bases of affine Lie algebras and vertex for modified quantum algebras of operator algebras arbitrary rank 2 cases Vyjayanthi Chari Keith Hubbard Weyl, Demazure and Vertex operator coalgebras and their Kirillov-Reshetikhin modules relations to VOAs Chongying Dong On the uniqueness of the moonshine Dijana Jakelic vertex operator algebra Branched crystals and the category 0 Rolf Farnsteiner Seok-Jin Kang Affine quivers, polyhedral groups, and Nakajima's monomials and crystal representation type bases Jorg Feldvoss Rinat Kedem Existence of triangular Lie bialgebra Kostka polynomials and representations structures of affine algebras

XV xvi TALKS

Apoorva Khare Toshiki Nakashima The EGG category 0 over a skew group Affine Geometric Crystals and Tropical R

Alexander Kirillov, Jr. Erhard Neher Wess-Zumino- Witten model as an Central extensions of Lie tori theory Stefan Kolb Deepak Parashar Quantum symmetric pairs and the Coloured Yang-Baxter operators reection equation Brian Parshall Bertram Kostant Some results on Specht modules and Powers of the Dedekind-ry function and algebraic groups representation theory Mirko Prime Michael Lau Combinatorial bases of Bosonic and fermionic representations Feigin-Stoyanovsky type subspaces and James Lepowsky interwining operators Some developments in vertex operator algebra theory, old and new Alexander Retakh Modular conformal algebras Haisheng Li Quantum vertex algebras associated Vladimir Retakh with Zamolodchikov-Faddeev algebras Factorizations of polynomials over noncommutative rings and algebras Yiqiang Li associated with quivers A description of affine canonical bases Geoffrey Mason Eric Rowell Vertex algebras and partition functions Modular tensor categories: towards classification Olivier Mathieu Growth for groups and Lie algebras N atasha Rozhkovskaya Arne Meurman Central elements of the universal On generators and relations for vertex enveloping algebra of g[( n, q and operator algebras Yangians Antun Milas Alistair Savage Number theoretic properties of Branching rules and quiver varieties Andrews-Gordon series Christoph Schweigert Tetsuji Miwa Twining characters and Picard groups Sklyanin's algebra and trace functional in rational conformal field theories Adriano Moura q-characters of fundamental Shirlei Serconek representations of quantum affine Distributivity of lattices of defining algebras relations David Nacin Bin Shu Partially commuting algebras and their Cartan invariants and blocks for connections to Qn Zassenhaus algebras TALKS xvii

Helmut Strade The classification of the simple Lie algebras over fields of positive characteristic: history, state of art, outlook David Taylor Trace functions of integrable modules over classical subalgebras of g("' Imre Thba Reconstructing braided semisimple tensor categories Monica Vazirani Vanishing integrals of Macdonald polynomials Weiqiang Wang Hilbert schemes, vertex operators, and integrable hierarchies Robert Wilson Modules and combinatorial identities List of participants lana Anguelova Konstantina Christodoulopoulou University of Illinois, University of Wisconsin, Madison Urbana-Champaign William Cook Bojko Bakalov North Carolina State University North Carolina State University Will Davis Katrina Barron North Carolina State University University of Notre Dame Chongying Dong Karin Baur University of California, Santa Cruz University of California, San Diego Rolf Farnsteiner University of Bielefeld, Germany Georgia Benkart University of Wisconsin, Madison Alex Feingold State University of New York, Jennifer Berg Binghamton University of California, Berkeley Jorg Feldvoss Maarten Bergvelt University of South Alabama University of Illinois, Urbana-Champaign John Francis Institute of Technology Stephen Berman University of Saskatchewan, Canada Igor Frenkel Yale University Richard Block Univeristy of California, Riverside Avital Frumkin Tel Aviv University, Israel Julius Borcea Jiirgen Fuchs University, Sweden Karlstads Universitet, Sweden Geoffrey Buhl Howard Garland Yale University Carina Calinescu Lucy Gow Rutgers University University of Sydney, Australia Stefano Capparelli Dimitar Grantcharov Universita di Roma, Italy San Jose State University Vyjayanthi Chari Thomas Gregory University of California, Riverside Ohio State University, Mansfield

xix XX PARTICIPANTS

Terrell Hodge Michael Lau Western Michigan University University of Ottawa, Canada Gerald Hohn Jim Lepowsky Kansas State University Rutgers University Nora Hopkins Wanglai Li Indiana State University SBS International, a Boeing Company Ayumu Hoshino Haisheng Li Sophia University, Japan Rutgers University, Camden Yi-Zhi Huang Yiqiang Li Rutgers University Kansas State University Keith Hubbard Jose Liberati University of Notre Dame Universidad Nacional de Cordoba, Dijana Jakelic Argentina University of Virginia Geoffrey Mason Naihuan Jing University of California, Santa Cruz North Carolina State University Olivier Mathieu Elizabeth Jurisich Universite Claude Bernard, France College of Charleston Thomas McAvoy Seok-Jin Kang North Carolina State University Seoul National University, Korea Hayk Melikyan Rinat Kedem North Carolina Central University University of Illinois, Arne Meurman Urbana-Champaign , Sweden Apoorva Khare Antun Milas State University of New York, Albany Alexander Kirillov Jr. Stephen Milne State University of New York, Stony Ohio State Univeristy Brook Kailash C. Misra Stefan Kolb North Carolina State University Virginia Polytechnic Institute and State University Tetsuji Miwa Kyoto University, Japan Liang Kong Rutgers University Adriano Moura University of California, Riverside Bertram Kostant Massachusetts Institute of Technology David Nacin Rutgers University Jonathan Kujawa University of Georgia Toshiki Nakashima Sophia University, Japan Shrawan Kumar University of North Carolina at Chapel Erhard Neher Hill University of Ottawa, Canada PARTICIPANTS xxi

Deepak Parashar Darren Thompson University of Wales, Swansea Elon University

Brian Parshall Kathleen Thompson Univeristy of Virginia North Carolina State University

Mirko Prime Imre Thba University of Zagreb, Croatia Virginia Polytechnic Institute and State University Eswara Rao Monica Vazirani Tata Institute of Fundamental Research University of California, Davis Alexander Retakh Weiqiang Wang Massachusetts Institute of Technology University of Virginia Vladmir Retakh Robert Wilson Rutgers University, New Brunswick Rutgers Univeristy Michael Roitman Gaywalee Yamskulna University of Illinois Illinois State University Natalia Rojkovskaia Doron Zeilberger University of Wisconsin-Madison Rutgers University

Eric Rowell Wei Zhang Indiana University University of California

Barbara Sanborn Lin Zhang Arizona State University Rutgers University

Alistair Savage Fields Institute and University of Toronto

Christoph Schweigert U niveristy of Hamburg, Germany

Shirlei Serconek Federal University of Goias

Bin Shu University of Virginia and East China Normal University

Oleg Smirnov College of Charleston

Ernie Stitzinger North Carolina State University

Helmut Strade University of Hamburg

David Taylor University of Virginia Titles in This Series

442 Yi-Zhi Huang and Kailash C. Misra, Editors, Lie algebras, vertex operator algebras and their applications, 2007 441 Louis H. Kauffman, David E. Radford, and Fernando J. 0. Souza, Editors, Hopf algebras and generalizations, 2007 440 Fernanda Botelho, Thomas Hagen, and James Jamison, Editors, Fluids and Waves, 2007 439 Donatella Danielli, Editor, Recent developments in nonlinear partial differential equations, 2007 438 Marc Burger, Michael Farber, Robert Ghrist, and Daniel Koditschek, Editors, Topology and robotics, 2007 437 Jose C. Mouriio, Joao P. Nunes, Roger Picken, and Jean-Claude Zambrini, Editors, Prospects in , 2007 436 Luchezar L. Avramov, Daniel Christensen, William G Dwyer, Michael A Mandell, and Brooke E Shipley, Editors, Interactions between homotopy theory and algebra, 2007 435 Krzysztof Jarosz, Editor, Function spaces, 2007 434 S. Paycha and B. Uribe, Editors, Geometric and topological methods for quantum field theory, 2007 433 , Shlomo Gelaki, and Steven Shnider, Editors, Quantum groups, 2007 432 Dick Canery, Jane Gilman, Juha Heinoren, and Howard Masur, Editors, In the tradition of Ahlfors-Bers, IV, 2007 431 Michael Batanin, Alexei Davydov, Michael Johnson, Stephen Lack, and Amnon Neeman, Editors, Categories in algebra, geometry and mathematical physics, 2007 430 Idris Assani, Editor, Ergodic theory and related fields, 2007 429 Gui-Qiang Chen, Elton Hsu, and Mark Pinsky, Editors, Stochastic analysis and partial differential equations, 2007 428 Estela A. Gavosto, Marianne K. Korten, Charles N. Moore, and Rodolfo H. Torres, Editors, Harmonic analysis, partial differential equations, and related topics, 2007 427 Anastasios Mallios and Marina Haralampidou, Editors, Topological algebras and applications, 2007 426 Fabio Ancona, Irena Lasiecka, Walter Littman, and Roberto Triggiani, Editors, Control methods in POE-dynamical systems, 2007 425 Su Gao, Steve Jackson, and Yi Zhang, Editors, Advances in Logic, 2007 424 V.I. Burenko, T. Iwaniec, and S. K. Vodopyanov, Editors, Analysis and geometry in their interaction, 2007 423 Christos A. Athanasiadis, Victor V. Batyrev, Dimitrios I. Dais, Martin Henk, and Francisco Santos, Editors, Algebraic and geometric combinatorics, 2007 422 JongHae Keum and Shigeyuki Kondo, Editors, Algebraic geometry, 2007 421 Benjamin Fine, Anthony M. Gaglione, and Dennis Spellman, Editors, Combinatorial group theory, discrete groups, and number theory, 2007 420 William Chin, James Osterburg, and Declan Quinn, Editors, Groups, rings and algebras, 2006 419 Dinh V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and Its applications, 2006 418 Lothar Gerritzen, Dorian Goldfeld, Martin Kreuzer, Gerhard Rosenberger, and Vladimir Shpilrain, Editors, Algebraic methods in cryptography, 2006 417 Vadim B. Kuznetsov and Siddhartha Sahi, Editors, Jack, Hall-Littlewood and Macdonald polynomials, 2006 416 Toshitake Kohno and Masanori Morishita, Editors, Primes and knots, 2006 TITLES IN THIS SERIES

415 Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment, Editors, Quantum graphs and their applications, 2006 414 Deguang Han, Palle E. T. Jorgensen, and David Royal Larson, Editors, Operator theory, operator algebras, and applications, 2006 413 Georgia M. Benkart, Jens C. Jantzen, Zongzhu Lin, Daniel K. Nakano, and Brian J. Parshall, Editors, Representations of algebraic groups, quantum groups and Lie algebras, 2006 412 Nikolai Chernov, Yulia Karpeshina, Ian W. Knowles, Roger T. Lewis, and Rudi Weikard, Editors, Recent advances in differential equations and mathematical physics, 2006 411 J. Marshall Ash and Roger L. Jones, Editors, Harmonic analysis: Calder6n-Zygmund and beyond, 2006 410 Abba Gumel, Carlos Castilla-Chavez, Ronald E. Mickens, and Dominic P. Clemence, Editors, Mathematical studies on human disease dynamics: Emerging paradigms and challenges, 2006 409 Juan Luis Vazquez, Xavier Cabre, and Jose Antonio Carrillo, Editors, Recent trends in partial differential equations, 2006 408 Habib Ammari and Hyeonbae Kang, Editors, Inverse problems, multi-scale analysis and effective medium theory, 2006 407 Alejandro Adem, Jeslls Gonzalez, and Guillermo Pastor, Editors, Recent developments in algebraic topology, 2006 406 Jose A. de Ia Peiia and Raymundo Bautista, Editors, Trends in representation theory of algebras and related topics, 2006 405 Andrew Markoe and Eric Todd Quinto, Editors, Integral geometry and tomography, 2006 404 Alexander Borichev, Hi.kan Hedenmalm, and Kehe Zhu, Editors, Bergman spaces and related topics in complex analysis, 2006 403 Tyler J. Jarvis, Takashi Kimura, and Arkady Vaintrob, Editors, Gromov-Witten theory of spin curves and , 2006 402 Zvi Arad, Mariagrazia Bianchi, Wolfgang Herfort, Patrizia Longobardi, Mercede Maj, and Carlo Scoppola, Editors, Ischia group theory 2004, 2006 401 Katrin Becker, Melanie Becker, Aaron Bertram, PaulS. Green, and Benjamin McKay, Editors, Snowbird lectures on string geometry, 2006 400 Shiferaw Berhanu, Hua Chen, Jorge Hounie, Xiaojun Huang, Sheng-Li Tan, and Stephen S.-T. Yau, Editors, Recent progress on some problems in several complex variables and partial differential equations, 2006 399 Dominique Arlettaz and Kathryn Hess, Editors, An Alpine anthology of homotopy theory, 2006 398 Jay Jorgenson and Lynne Walling, Editors, The ubiquitous heat , 2006 397 Jose M. Munoz Porras, Sorin Popescu, and Rubl E. Rodriguez, Editors, The geometry of Riemann surfaces and Abelian varieties, 2006 396 Robert L. Devaney and Linda Keen, Editors, Complex dynamics: Twenty-five years after the appearance of the Mandelbrot set, 2006 395 Gary R. Jensen and Steven G. Krantz, Editors, 150 Years of Mathematics at Washington University in St. Louis, 2006

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej. The articles in thjs book are based on talks given at the international conference on "Lie algebras, vertex operator algebras and their applications", in honor of James Lepowsky and Robert Wilson on their sixtieth birthdays, held in May of 2005 at North Carolina State University. Some of the papers in tills volume give inspiring expositions on the develop- ment and status of their respective research areas. Others outline and explore the challenges as well as the future directions of research for the twenty-first century. The focus of the papers in thjs volume is mainly on Lie algebras, quantum groups, vertex operator algebras and their applications to number theory, combinatorics and conformal field theory. This book is useful for graduate students and researchers in mathematics and mathematical physics who want to be introduced to different areas of current research or explore the frontiers of research in the areas mentioned above.

ISBN 978-0-8218-3986-7

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