This book start with an introduction to quantum groups for the beginner and continues as a textbook for graduate students in physics and in mathematics. It can also be used as a reference by more advanced readers.
The authors cover a large but well-chosen variety of subjects from the theory of quantum groups (quantized universal enveloping algebras, quantized algebras of functions) and q-deformed algebras (q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The book is written with potential applications in physics and mathematics in mind. The basic quantum groups and quantum algebras and their representations are given in detail and accompanied by explicit formulas. A number of topics and results from the more advanced general theory are developed and discussed.

Inhalt

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 Aº.- 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 Representations of Uq(sl2) for q a Root of Unity.- 3.3.1 The Center of Uq(sl2).- 3.3.2 Representations of Uq(sl2).- 3.3.3 Representations of $$U^{res}Q(!\text{sl}2!)$$.- 3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients.- 3.4.1 Tensor Products of Representations Tl.- 3.4.2 Clebsch-Gordan Coefficients.- 3.4.3 Other Expressions for Clebsch-Gordan Coefficients.- 3.4.4 Symmetries of Clebsch-Gordan Coefficients.- 3.5 Racah Coefficients and 6j Symbols of Uq(su2).- 3.5.1 Definition of the Racah Coefficients.- 3.5.2 Relations Between Racah and Clebsch-Gordan Coefficients.- 3.5.3 Symmetry Relations.- 3.5.4 Calculation of Racah Coefficients.- 3.5.5 The Biedenharn-Elliott Identity.- 3.5.6 The Hexagon Relation.- 3.5.7 Clebsch-Gordan Coefficients as Limits of Racah Coefficients.- 3.6 Tensor Operators and the Wigner-Eckart Theorem.- 3.6.1 Tensor Operators for Compact Lie Groups.- 3.6.2 Tensor Operators and the Wigner-Eckart Theorem for ?q(sl2).- 3.7 Applications.- 3.7.1 The Uq(sl2) Rotator Model of Deformed Nuclei.- 3.7.2 Electromagnetic Transitions in the Uq(sl2) Model.- 3.8 Notes.- 4. The Quantum Group SLq(2) and Its Representations.- 4.1 The Hopf Algebra $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$.- 4.1.1 The Bialgebra $$\mathcal{O}({{M}{}}{q}\left( 2 \right))$$.- 4.1.2 The Hopf Algebra $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$.- 4.1.3 A Geometric Approach to SLq(2).- 4.1.4 Real Forms of $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$.- 4.1.5 The Diamond Lemma.- 4.2 Representations of the Quantum Group SLq(2).- 4.2.1 Finite-Dimensional Corepresentations of $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$: Main Results.- 4.2.2 A Decomposition of $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$.- 4.2.3 Finite-Dimensional Subcomodules of $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$.- 4.2.4 Calculation of the Matrix Coefficients.- 4.2.5 The Peter-Weyl Decomposition of $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$.- 4.2.6 The Haar Functional of $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$.- 4.3 The Compact Quantum Group SUq(2) and Its Representations.- 4.3.1 Unitary Representations of the Quantum Group SUq(2).- 4.3.2 The Haar State and the Peter-Weyl Theorem for $$\mathcal{O}(S{{U}_{}}q\left( 2 \right))$$.- 4.3.3 The Fourier Transform on SUq(2).- 4.3.4 Representations and the C-Algebra of $$\mathcal{O}(S{{U}{}}q\left( 2 \right))$$.- 4.4 Duality of the Hopf Algebras Uq(sl2) and $$\mathcal{O}(S{{L}{q}}\left( 2 \right))$$.- 4.4.1 Dual Pairing of the Hopf Algebras Uq(sl2) and $$\mathcal{O}(S{{L}{q}}\left( 2 \right))$$.- 4.4.2 Corepresentations of $$\mathcal{O}(S{{L}{}}{q}\left( 2 \right))$$ and Representations of Uq(sl2).- 4.5 Quantum 2-Spheres.- 4.5.1 A Family of Quantum Spaces for SLq(2).- 4.5.2 Decomposition of the Algebra $$\mathcal{O}(!S^2{q\rho}!)$$.- 4.5.3 Spherical Functions on $$S^2{q\rho}$$.- 4.5.4 An Infinitesimal Characterization of.- 4.6 Notes.- 5. The q-Oscillator Algebras and Their Representations.- 5.1 The q-Oscillator Algebras $$\mathcal{A}{!^cq!}$$ and $$\mathcal{A}{q}$$.- 5.1.1 Definitions and Algebraic Properties.- 5.1.2 Other Forms of the q-Oscillator Algebra.- 5.1.3 The q-Oscillator Algebra and the Quantum Algebra ?q(sl2).- 5.1.4 The q-Oscillator Algebras and the Quantum Space $$\mathop M\nolimits{\mathop q\nolimits^2 } (2)$$.- 5.2 Representations of q-Oscillator Algebras.- 5.2.1 N-Finite Representations.- 5.2.2 Irreducible Representations with Highest (Lowest) Weights.- 5.2.3 Representations Without Highest and Lowest Weights.- 5.2.4 Irreducible Representations of $$\mathcal{A}{!^cq!}$$ for q a Root of Unity.- 5.2.5 Irreducible *-Representations of $$\mathcal{A}{!^cq!}$$ and $$\mathcal{A}{_q}$$.- 5.2.6 Irreducible *-Representations of Another q-Oscillator Algebra.- 5.3 The Fock Representation of the q-Oscillator Algebra.- 5.3.1 The Fock Representation.- 5.3.2 The Bargmann-Fock Realization.- 5.3.3 Coherent States.- 5.3.4 Bargmann-Fock Space Realization of Irreducible Representations of ?q(sl2).- 5.4 Notes.- II. Quantized Universal Enveloping Algebras.- 6. Drinfeld-Jimbo Algebras.- 6.1 Definitions of Drinfeld-Jimbo Algebras.- 6.1.1 Semisimple Lie Algebras.- 6.1.2 The Drinfeld-Jimbo Algebras Uq(g).- 6.1.3 The h-Adic Drinfeld-Jimbo Algebras Uh(g).- 6.1.4 Some Algebra Automorphisms of Drinfeld-Jimbo Algebras.- 6.1.5 Triangular Decomposition of Uq(g).- 6.1.6 Hopf Algebra Automorphisms of Uq(g).- 6.1.7 Real Forms of Drinfeld-Jimbo Algebras.- 6.2 Poincaré-Birkhoff-Witt Theorem and Verma Modules.- 6.2.1 Braid Groups.- 6.2.2 Action of Braid Groups on Drinfeld-Jimbo Algebras.- 6.2.3 Root Vectors and Poincaré-Birkhoff-Witt Theorem.- 6.2.4 Representations with Highest Weights.- 6.2.5 Verma Modules.- 6.2.6 Irreducible Representations with Highest Weights.- 6.2.7 The Left Adjoint Action of Uq(g).- 6.3 The Quantum Killing Form and the Center of Uq(g).- 6.3.1 A Dual Pairing of the Hopf Algebras Uq(b+) and Uq(b+)op.- 6.3.2 The Quantum Killing Form on Uq(g).- 6.3.3 A Quantum Casimir Element.- 6.3.4 The Center of Uq(g) and the Harish-Chandra Homomorphism.- 6.3.5 The Center of Uq(g) for q a Root of Unity.- 6.4 Notes.…