Preface

This volume is the Proceedings of the Ramanujan International Symposium

on " Kac-Moody Lie algebras and applications " which was held in the Ramanu-

jan Institute for Advanced Study in Mathematics, University of Madras, Chennai,

India during January 28-31, 2002. All the papers in this volume are refereed and

all of them except three are based on the talks delivered at the symposium. The

other three papers were contributed by those who could not participate in the sym-

posium. The main aim of the symposium was to provide a forum to researchers,

both in mathematics and physics, working in Kac-Moody Lie algebras and related

topics, to discuss the new developments in this rapidly growing area of research as

well as to stimulate further developments. The symposium also served its purpose

by attracting many young research scholars from different parts of India and intro-

ducing them to many areas of current research related to Kac-Moody Lie algebras

which is one of the research areas of current interest at the Ramanujan Institute

for Advanced Study in Mathematics.

Kac-Moody Lie algebras were independently introduced by Victor Kac and

Robert Moody around 1967. These algebras can be thought of as infinite dimen-

sional analogs of finite dimensional semisimple Lie algebras. For more details on

the history of the early developments of this important class of Lie algebras, the

reader can refer to a recent article " Victor Kac and Robert Moody: Their paths

to Kac-Moody Lie algebras " by S. Berman and K. H. Parshall, which appeared

in " The Mathematical Intelligencer ", Vol. 24, No.1, Winter 2002. During the

last three decades, the theory of Kac-Moody Lie algebras has attracted researchers

from different areas of mathematics and physics because of its close connections

with combinatorics, differential equations, group theory, modular forms, singulari-

ties, knot theory, statistical mechanics, quantum field theory, and string theory, to

name a few. In fact many of these connections are with an important class of Kac-

Moody Lie algebras known as affine Lie algebras. For example, in early 1970's, Kac

proved the Weyl-Kac character formula for representations of general Kac-Moody

Lie algebras and showed that in the special case of !-dimensional representation

of affine Lie algebras, this formula turns into the celebrated Macdonald identi-

ties. Building on this, in the later part of 1970's, it was discovered by Lepowsky

and Milne that the characters of certain representations of affine Lie algebras are

closely related to the product sides of the Rogers-Ramanujan identities which were

independently discovered first by L. J. Rogers in 1894 and later rediscovered by the

Indian Mathematical genius, Srinivasa Ramanujan in 1913. In 1978, while trying to

give a Lie theoretic proof of these intriguing identities, Lepowsky and Wilson gave

the vertex operator realizations of certain affine Lie algebras representations which

are now known as the principal realizations. Around the same time Frenkel and

Kac, also independently Segal used the homogeneous vertex operators which appear

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