Introduction to quantum groups and crystal bases / Jin Hong, Seok-Jin Kang.

Format:

Book

Author/Creator:

Hong, Jin, 1972- author.

Kang, Seok-Jin, author.

Series:

Graduate studies in mathematics ; v. 42.

Graduate studies in mathematics, 1065-7339 ; volume 42

Language:

English

Subjects (All):

Quantum groups.

Representations of quantum groups.

Physical Description:

1 online resource (327 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Language Note:

English

Summary:

The notion of a "quantum group" was introduced by V.G. Dinfel&dacute; 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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Introduction""; ""Chapter 1. Lie Algebras and Hopf Algebras""; ""Â1.1. Lie algebras""; ""Â1.2. Representations of Lie algebras""; ""Â1.3. The Lie algebra sl(2, F)""; ""Â1.4. The special linear Lie algebra sl(n, F)""; ""Â1.5. Hopf algebras""; ""Exercises""; ""Chapter 2. Kac-Moody Algebras""; ""Â2.1. Kac-Moody algebras""; ""Â2.2. Classification of generalized Cartan matrices""; ""Â2.3. Representation theory of Kac-Moody algebras""; ""Â2.4. The category O[sub(int)]""; ""Exercises""; ""Chapter 3. Quantum Groups""

""Â3.1. Quantum groups""""Â3.2. Representation theory of quantum groups""; ""Â3.3. A[sub(1)]-forms""; ""Â3.4. Classical limit""; ""Â3.5. Complete reducibility of the category O[sup(q)][sub(int)]""; ""Exercises""; ""Chapter 4. Crystal Bases""; ""Â4.1. Kashiwara operators""; ""Â4.2. Crystal bases and crystal graphs""; ""Â4.3. Crystal bases for U[sub(q)](sl[sub(2)]-modules""; ""Â4.4. Tensor product rule""; ""Â4.5. Crystals""; ""Exercises""; ""Chapter 5. Existence and Uniqueness of Crystal Bases""; ""Â5.1. Existence of crystal bases""; ""Â5.2. Uniqueness of crystal bases""

""Chapter 8. Crystal Graphs for Classical Lie Algebras""""Â8.1. Example: U[sub(q)](B[sub(3)]-crystals""; ""Â8.2. Realization of U[sub(q)](A[sub(nâ€?1)]-crystals""; ""Â8.3. Realization of U[sub(q)](C[sub(n)]-crystals""; ""Â8.4. Realization of U[sub(q)](B[sub(n)]-crystals""; ""Â8.5. Realization of U[sub(q)](D[sub(n)]-crystals""; ""Â8.6. Tensor product decomposition of crystals""; ""Exercises""; ""Chapter 9. Solvable Lattice Models""; ""Â9.1. The 6-vertex model""; ""Â9.2. The quantum afRne algebra U[sub(q)](sl[sub(2)]""; ""Â9.3. Crystals and paths""; ""Exercises""

""Chapter 10. Perfect Crystals""""Â10.1. Quantum afflne algebras""; ""Â10.2. Energy functions and combinatorial R-matrices""; ""Â10.3. Vertex operators for U[sub(q)](sl[sub(2)]-modules""; ""Â10.4. Vertex operators for quantum affine algebras""; ""Â10.5. Perfect crystals""; ""Â10.6. Path realization of crystal graphs""; ""Exercises""; ""Chapter 11. Combinatorics of Young Walls""; ""Â11.1. Perfect crystals of level 1 and path realization""; ""Â11.2. Combinatorics of Young walls""; ""Â11.3. The crystal structure""; ""Â11.4. Crystal graphs for basic representations""; ""Exercises""

""Bibliography""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 297-299) and indexes.

Description based on print version record.

ISBN:

1-4704-2093-7