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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 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 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. 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 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 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 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
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