The notion of a "quantum group" was introduced by V.G. Dinfeld́ and M. Jimbo, independently, in their study of the quantum Yang-Baxter equation arising from 2-dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody 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 Kac-Moody 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.

Readership

Graduate students and research mathematicians interested in nonassociative rings and algebras.

Reviews

"Book by Hong and Kang is the first expository text which gives a detailed account on the relationship between crystal bases and combinatorics. This book provides an accessible and "crystal clear" introduction and overview of the relatively new subject of quantum groups and crystal bases, ... It will be an indispensable companion to the research papers."

-- Mathematical Reviews

Table of Contents

• Lie algebras and Hopf algebras
• Kac-Moody algebras
• Quantum groups
• Crystal bases
• Existence and uniqueness of crystal bases
• Global bases
• Young tableaux and crystals
• Crystal graphs for classical Lie algebras
• Solvable lattice models
• Perfect crystals
• Combinatorics of Young walls
• Bibliography
• Index of symbols
• Index