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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