Preface

Lie theory represents a major area of mathematical research. Besides its in-

creasing importance within mathematics (to geometry, combinatorics, ﬁnite and

inﬁnite groups, etc.), it has important applications outside of mathematics (to

physics, computer science, etc.).

During the twentieth century, the theory of Lie algebras, both ﬁnite and inﬁnite

dimensional, has been a major area of mathematical research with numerous ap-

plications. In particular, during the late 1970s and early 1980s, the representation

theory of Kac-Moody Lie algebras (analogs of ﬁnite dimensional semisimple Lie al-

gebras) generated intense interest. In part, the subject was driven by its interesting

connections with such topics as combinatorics, group theory, number theory, partial

diﬀerential equations, topology and with areas of physics such as conformal ﬁeld

theory, statistical mechanics, and integrable systems. The representation theory of

an important class of inﬁnite dimensional Lie algebras known as aﬃne Lie algebras

led to the discovery of Vertex Operator Algebras (VOAs) in the 1980s. VOAs are

precise algebraic counterparts to “chiral algebras” in two-dimensional conformal

ﬁeld theory as formalized by Belavin, Polyakov, and Zamolodchikov. These alge-

bras and their representations play important roles in a number of areas, including

the representation theory of the Fischer-Griess Monster ﬁnite simple group and the

connection with the phenomena of “monstrous moonshine,” the representation the-

ory of the Virasoro algebra and aﬃne Lie algebras, and two-dimensional conformal

ﬁeld theory.

In 1985, the interaction of aﬃne Lie algebras with integrable systems led Drin-

feld and Jimbo to introduce a new class of algebraic objects known as quantized

universal enveloping algebras (also called quantum groups) associated with sym-

metrizable Kac-Moody Lie algebras. These are q-deformations of the universal en-

veloping algebras of the corresponding Kac-Moody Lie algebras, and, like universal

enveloping algebras, they carry an important Hopf algebra structure. The abstract

theory of integrable representations of quantum groups, developed by Lusztig, il-

lustrates the similarity between quantum groups and Kac-Moody Lie algebras. The

quantum groups associated with ﬁnite dimensional simple Lie algebras also have

strong connections with the representations of aﬃne Lie algebras. The theory of

canonical bases for quantum groups has provided deep insights into the represen-

tation theory of quantum groups. More recently, the theory of geometric crystals

introduced by Berenstein and Kazhdan has opened new doors in representation

theory. In particular, canonical bases at q = 0 (crystal bases) provide a beautiful

combinatorial tool for studying the representations of quantum groups. The quan-

tized universal enveloping algebra associated with an aﬃne Lie algebra is called

a quantum aﬃne algebra. Quantum aﬃne algebras quickly became an interesting

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