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

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