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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
Lie Algebras, Vertex Operator Algebras
and Their Applications
http://dx.doi.org/10.1090/conm/442
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
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except those granted to the United States 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.
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10987654321 12 11 10 09 08 07
Contents
Preface
Biographies of James Lepowsky and Robert Wilson
List of Ph.D. Students
List of Talks
List of Participants
Lie algebras and related topics
A Class of Gradings of Simple Lie Algebras
vii
XI
xiii
XV
xix
KARIN BAUR AND NOLAN WALLACH 3
Conjugacy Results for the Lie Algebra s(2 over an Algebra Which is a UFD STEPHEN BERMAN AND JUN MORITA 17
Kirillov-Reshetikhin Modules Associated to G2
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 E6 and F4
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
Vertex (operator) algebras and related topics
Symmetric Polynomials and HD-Quantum Vertex Algebras I. I. ANGUELOVA
Hr-Vertex Algebras MAARTEN J. BERGVELT
On Intertwining Operators and Recursions CORINA CALINESCU
Representations of Vertex Operator Algebras CHONGYING DONG AND CUIPO JIANG
On Non-Semisimple Fusion Rules and Tensor Categories J URGEN FUCHS
The Duality between Vertex Operator Algebras and Coalgebras, Modules and Co modules
233
269
279
289
303
315
KEITH HUBBARD 339
Some Developments in Vertex Operator Algebra 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 Conformal Field Theory CHRISTOPH SCHWEIGERT, JURGEN FUCHS, AND INGO RUNKEL 463
Preface
The representation theory of finite- and infinite-dimensional Lie algebras has been an important area of mathematical research with numerous applications in many areas of mathematics and physics. 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 Virasoro algebra and their generalizations have been discovered and studied extensively, with exciting connections to many other fields of mathematics, including combinatorics, group theory, 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 quantum group associated with a finite-dimensional simple Lie algebra at a root of unity is equivalent as a rigid braided tensor category to a suitable category of modules for the corresponding affine Lie algebra. 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 algebraic structure 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 simple group 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 string 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, Sweden, 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 New York City. He attended Stuyvesant High School 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 Yale University 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 Paris 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 Nathan Jacobson, 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 Quantum Vertex Algebras
Katrina Barron Deformations of the N = 2 Neveu-Schwarz algebra and even and odd spectral flow on the worldsheet geometry of N = 2 superconformal field theory
Karin Baur Admissible characters in the sense of Lynch
Georgia Benkart Perfect crystals
Stephen Berman Covering Algebras
Richard E. Block Dual coalgebras and cofree coalgebras
Corina Calinescu Principal subspaces of representations of affine Lie algebras and vertex operator algebras
Vyjayanthi Chari Weyl, Demazure and Kirillov-Reshetikhin modules
Chongying Dong On the uniqueness of the moonshine vertex operator algebra
Rolf Farnsteiner Affine quivers, polyhedral groups, and representation type
Jorg Feldvoss Existence of triangular Lie bialgebra structures
Igor Frenkel Instanton and quiver constructions of affine Lie algebra representations and special bases
A vital Frumkin A probability approach to Klyachko theorem
J iirgen Fuchs An analogue of the Verlinde formula for non-rational vertex algebras
Howard Garland Counting the number of rational points in certain moduli spaces of vector bundles on surfaces
Dimitar Grantcharov On the automorphisms of Kac-Moody Lie superalgebras
Ayumu Hoshino Polyhedral realizations of crystal bases for modified quantum algebras of arbitrary rank 2 cases
Keith Hubbard Vertex operator coalgebras and their relations to VOAs
Dijana Jakelic Branched crystals and the category 0
XV
Seok-Jin Kang Nakajima's monomials and crystal bases
Rinat Kedem Kostka polynomials and representations of affine algebras
xvi TALKS
Apoorva Khare Toshiki Nakashima The EGG category 0 over a skew group Affine Geometric Crystals and Tropical ring R
Alexander Kirillov, Jr. Wess-Zumino- Witten model as an orbifold theory
Stefan Kolb Quantum symmetric pairs and the reection equation
Bertram Kostant Powers of the Dedekind-ry function and representation theory
Michael Lau Bosonic and fermionic representations
James Lepowsky Some developments in vertex operator algebra theory, old and new
Haisheng Li Quantum vertex algebras associated with Zamolodchikov-Faddeev algebras
Yiqiang Li A description of affine canonical bases
Geoffrey Mason Vertex algebras and partition functions
Olivier Mathieu Growth for groups and Lie algebras
Arne Meurman On generators and relations for vertex operator algebras
Antun Milas Number theoretic properties of Andrews-Gordon series
Tetsuji Miwa Sklyanin's algebra and trace functional
Adriano Moura q-characters of fundamental representations of quantum affine algebras
David Nacin Partially commuting algebras and their connections to Qn
Erhard Neher Central extensions of Lie tori
Deepak Parashar Coloured Yang-Baxter operators
Brian Parshall Some results on Specht modules and algebraic groups
Mirko Prime Combinatorial bases of Feigin-Stoyanovsky type subspaces and interwining operators
Alexander Retakh Modular conformal algebras
Vladimir Retakh Factorizations of polynomials over noncommutative rings and algebras associated with quivers
Eric Rowell Modular tensor categories: towards classification
N atasha Rozhkovskaya Central elements of the universal enveloping algebra of g[( n, q and Yangians
Alistair Savage Branching rules and quiver varieties
Christoph Schweigert Twining characters and Picard groups in rational conformal field theories
Shirlei Serconek Distributivity of lattices of defining relations
Bin Shu Cartan invariants and blocks for Zassenhaus algebras
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
TALKS xvii
List of participants
lana Anguelova University of Illinois, Urbana-Champaign
Bojko Bakalov North Carolina State University
Katrina Barron University of Notre Dame
Karin Baur University of California, San Diego
Georgia Benkart University of Wisconsin, Madison
Jennifer Berg University of California, Berkeley
Maarten Bergvelt University of Illinois, Urbana-Champaign
Stephen Berman University of Saskatchewan, Canada
Richard Block Univeristy of California, Riverside
Julius Borcea Stockholm University, Sweden
Geoffrey Buhl Rutgers University
Carina Calinescu Rutgers University
Stefano Capparelli Universita di Roma, Italy
Vyjayanthi Chari University of California, Riverside
Konstantina Christodoulopoulou University of Wisconsin, Madison
William Cook North Carolina State University
Will Davis North Carolina State University
Chongying Dong University of California, Santa Cruz
Rolf Farnsteiner University of Bielefeld, Germany
Alex Feingold State University of New York, Binghamton
Jorg Feldvoss University of South Alabama
John Francis Massachusetts Institute of Technology
Igor Frenkel Yale University
Avital Frumkin Tel Aviv University, Israel
Jiirgen Fuchs Karlstads Universitet, Sweden
Howard Garland Yale University
Lucy Gow University of Sydney, Australia
Dimitar Grantcharov San Jose State University
Thomas Gregory Ohio State University, Mansfield
xix
XX PARTICIPANTS
Terrell Hodge Western Michigan University
Gerald Hohn Kansas State University
Nora Hopkins Indiana State University
Ayumu Hoshino Sophia University, Japan
Yi-Zhi Huang Rutgers University
Keith Hubbard University of Notre Dame
Dijana Jakelic University of Virginia
Naihuan Jing North Carolina State University
Elizabeth Jurisich College of Charleston
Seok-Jin Kang Seoul National University, Korea
Rinat Kedem University of Illinois, Urbana-Champaign
Apoorva Khare University of Chicago
Alexander Kirillov Jr. State University of New York, Stony Brook
Stefan Kolb Virginia Polytechnic Institute and State University
Liang Kong Rutgers University
Bertram Kostant Massachusetts Institute of Technology
Jonathan Kujawa University of Georgia
Shrawan Kumar University of North Carolina at Chapel Hill
Michael Lau University of Ottawa, Canada
Jim Lepowsky Rutgers University
Wanglai Li SBS International, a Boeing Company
Haisheng Li Rutgers University, Camden
Yiqiang Li Kansas State University
Jose Liberati Universidad Nacional de Cordoba, Argentina
Geoffrey Mason University of California, Santa Cruz
Olivier Mathieu Universite Claude Bernard, France
Thomas McAvoy North Carolina State University
Hayk Melikyan North Carolina Central University
Arne Meurman Lund University, Sweden
Antun Milas State University of New York, Albany
Stephen Milne Ohio State Univeristy
Kailash C. Misra North Carolina State University
Tetsuji Miwa Kyoto University, Japan
Adriano Moura University of California, Riverside
David Nacin Rutgers University
Toshiki Nakashima Sophia University, Japan
Erhard Neher University of Ottawa, Canada
PARTICIPANTS xxi
Deepak Parashar University of Wales, Swansea
Brian Parshall Univeristy of Virginia
Mirko Prime University of Zagreb, Croatia
Eswara Rao Tata Institute of Fundamental Research
Alexander Retakh Massachusetts Institute of Technology
Vladmir Retakh Rutgers University, New Brunswick
Michael Roitman University of Illinois
Natalia Rojkovskaia University of Wisconsin-Madison
Eric Rowell Indiana University
Barbara Sanborn Arizona State 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
Darren Thompson Elon University
Kathleen Thompson North Carolina State University
Imre Thba Virginia Polytechnic Institute and State University
Monica Vazirani University of California, Davis
Weiqiang Wang University of Virginia
Robert Wilson Rutgers Univeristy
Gaywalee Yamskulna Illinois State University
Doron Zeilberger Rutgers University
Wei Zhang University of California
Lin Zhang Rutgers University
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 mathematical physics, 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 Pavel Etingof, 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 orbifolds, 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 kernel, 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
9 78 0 821 839867