
Jin Hong,SeokJin Kang

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Author: Jin Hong,SeokJin Kang
Publisher: American Mathematical Soc.
ISBN: 0821828746
Category: Mathematics
Page: 307
View: 7472
The notion of a ``quantum group'' was introduced by V.G. Dinfeld and M. Jimbo, independently, in their study of the quantum YangBaxter equation arising from 2dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of KacMoody algebras. And over the past 20 years, they have turned out to be the fundamental algebraic structure behind many branches of mathematics and mathematical physics, such as solvable lattice models in statistical mechanics, topological invariant theory of links and knots, representation theory of KacMoody algebras, representation theory of algebraic structures, topological quantum field theory, geometric representation theory, and $C^*$algebras. In particular, the theory of ``crystal bases'' or ``canonical bases'' developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.