Quantum Groups and Their Representations [Paperback]

I. An Introduction to Quantum Groups.- 1. Hopf Algebras.- 1.1 Prolog: Examples of Hopf Algebras of Functions on Groups.- 1.2 Coalgebras, Bialgebras and Hopf Algebras.- 1.2.1 Algebras.- 1.2.2 Coalgebras.- 1.2.3 Bialgebras.- 1.2.4 Hopf Algebras.- 1.2.5* Dual Pairings of Hopf Algebras.- 1.2.6 Examples of Hopf Algebras.- 1.2.7 *-Structures.- 1.2.8* The Dual Hopf Algebra A?.- 1.2.9* Super Hopf Algebras.- 1.2.10* h-Adic Hopf Algebras.- 1.3 Modules and Comodules of Hopf Algebras.- 1.3.1 Modules and Representations.- 1.3.2 Comodules and Corepresentations.- 1.3.3 Comodule Algebras and Related Concepts.- 1.3.4* Adjoint Actions and Coactions of Hopf Algebras.- 1.3.5* Corepresentations and Representations of Dually Paired Coalgebras and Algebras.- 1.4 Notes.- 2. q-Calculus.- 2.1 Main Notions on q-Calculus.- 2.1.1 q-Numbers and q-Factorials.- 2.1.2 q-Binomial Coefficients.- 2.1.3 Basic Hypergeometric Functions.- 2.1.4 The Function 1?0(a; q, z).- 2.1.5 The Basic Hypergeometric Function 2?1.- 2.1.6 Transformation Formulas for 3?2 and 4?3.- 2.1.7 q-Analog of the Binomial Theorem.- 2.2 q-Differentiation and q-Integration.- 2.2.1 q-Differentiation.- 2.2.2 q-Integral.- 2.2.3 q-Analog of the Exponential Function.- 2.2.4 q-Analog of the Gamma Function.- 2.3 q-Orthogonal Polynomials.- 2.3.1 Jacobi Matrices and Orthogonal Polynomials.- 2.3.2 q-Hermite Polynomials.- 2.3.3 Little q-Jacobi Polynomials.- 2.3.4 Big q-Jacobi Polynomials.- 2.4 Notes.- 3. The Quantum Algebra Uq(sl2) and Its Representations.- 3.1 The Quantum Algebras Uq(sl2) and Uh(sl2).- 3.1.1 The Algebra Uq(sl2).- 3.1.2 The Hopf Algebra Uq(sl2).- 3.1.3 The Classical Limit of the Hopf Algebra Uq(sl2).- 3.1.4 Real Forms of the Quantum Algebra Uq(sl2).- 3.1.5 The h-Adic Hopf Algebra Uh(sl2).- 3.2 Finite-Dimensional Representations of Uq(sl2) for q not a Root of Unity.- 3.2.1 The Representations T?l.- 3.2.2 Weight Representations and Complete Reducibility.- 3.2.3 Finite-Dimensional Representations of ?q(sl2) and Uh(sl2).- 3.3 Represló,