Quantum Groups and Their Representations (Theoretical and Mathematical Physics)

✍ Scribed by Anatoli Klimyk

• Publisher: Springer Verlag
• Year: 1997
• Tongue: English
• Leaves: 571
• Edition: 1
• Category: Library

✦ Synopsis

This book provides a treatment of the theory of quantum groups (quantized universal enveloping algebras and quantized algebras of functions) and q-deformed algebras (q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The theory of the simplest and most important quantum groups and their representations is presented in detail. A number of topics and results from the more advanced general theory are developed and discussed. Many applications in mathematical and theoretical physics are indicated. The book starts as an introduction for the beginner and continues at a textbook level for graduate students in physics and in mathematics. It may serve as a reference for more advanced readers.

✦ Table of Contents

QUANTUM GROUPS AND THEIR REPRESENTATIONS
Title Page
About
Preface
Table of Contents
Part I: An Introduction to Quantum Groups
Chapter 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[sup(o)]
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
Chapter 2. q-Calculus
2.1 Main Notions of 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 [sub(1)]phisub(0)
2.1.5 The Basic Hypergeometric Function [sub(2)]phi[sub(1)]
2.1.6 Transformation Formulas for [sub(3)]phi[sub(2)] and [sub(4)]phi[sub(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
Chapter 3. The Quantum Algebra Usub(q) and Its Representations
3.1 The Quantum Algebras Usub(q) and Usub(h)
3.1.1 The Algebra Usub(q)
3.1.2 The Hopf Algebra Usub(q)
3.1.3 The Classical Limit of the Hopf Algebra Usub(q)
3.1.4 Real Forms of the Quantum Algebra Usub(q)
3.1.5 The h-Adic Hopf Algebra Usub(h)
3.2 Finite-Dimensional Representations of Usub(q) for q not a Root of Unity
3.2.1 The Representations T[sub(wl)]
3.2.2 Weight Representations and Complete Reducibility
3.2.3 Finite-Dimensional Representations of Usub(q) and Usub(h)
3.3 Representations of Usub(q) for q a Root of Unity
3.3.1 The Center of Usub(q)
3.3.2 Representations of Usub(q)
3.3.3 Representations of U[sub(q)]sup(res)
3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients
3.4.1 Tensor Products of Representations T[sub(l)]
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 Usub(q)
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 Usub(q)
3.7 Applications
3.7.1 The Usub(q) Rotator Model of Deformed Nuclei
3.7.2 Electromagnetic Transitions in the Usub(q) Model
3.8 Notes
Chapter 4. The Quantum Group SLsub(q) and Its Representations
4.1 The Hopf Algebra O(SLsub(q))
4.1.1 The Bialgebra O(Msub(q))
4.1.2 The Hopf Algebra O(SLsub(q))
4.1.3 A Geometric Approach to SLsub(q)
4.1.4 Real Forms of O(SLsub(q))
4.1.5 The Diamond Lemma
4.2 Representations of the Quantum Group SLsub(q)
4.2.1 Finite-Dimensional Corepresentations of O(SLsub(q)): Main Results
4.2.2 A Decomposition of O(SLsub(q))
4.2.3 Finite-Dimensional Subcomodules of O(SLsub(q))
4.2.4 Calculation of the Matrix Coefficients
4.2.5 The Peter–Weyl Decomposition of O(SLsub(q))
4.2.6 The Haar Functional of O(SLsub(q))
4.3 The Compact Quantum Group SUsub(q) and Its Representations
4.3.1 Unitary Representations of the Quantum Group SUsub(q)
4.3.2 The Haar State and the Peter-Weyl Theorem for O(SUsub(q))
4.3.3 The Fourier Transform on SUsub(q)
4.3.4 -Representations and the C-Algebra of O(SUsub(q))
4.4 Duality of the Hopf Algebras Usub(q) and O(SLsub(q))
4.4.1 Dual Pairing of the Hopf Algebras Usub(q) and O(SLsub(q))
4.4.2 Corepresentations of O(SLsub(q)) and Representations of Usub(q)
4.5 Quantum 2-Spheres
4.5.1 A Family of Quantum Spaces for SLsub(q)
4.5.2
4.5.3 Spherical Functions on S[sup(2)][sub(qp)]
4.5.4 An Infinitesimal Characterization of O(S[sup(2)][sub(qp)])
4.6 Notes
Chapter 5. The q-Oscillator Algebras and Their Representations
5.1 The q-Oscillator Algebras A[sup(c)][sub(q)] and A[sub(q)]
5.1.1 Definitions and Algebraic Properties
5.1.2 Other Forms of the q-Oscillator Algebra
5.1.3
5.1.4 The q-Oscillator Algebras and the Quantum Space Msub(q)
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 A[sup(c)][sub(q)] for q a Root of Unity
5.2.5 Irreducible -Representations of A[sup(c)][sub(q)] and A[sub(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 Usub(q)
5.4 Notes
Part II: Quantized Universal Enveloping Algebras
Chapter 6. Drinfeld–Jimbo Algebras
6.1 Definitions of Drinfeld–Jimbo Algebras
6.1.1 Semisimple Lie Algebras
6.1.2 The Drinfeld–Jimbo Algebras Usub(q)
6.1.3 The h-Adic Drinfeld–Jimbo Algebras Usub(h)
6.1.4 Some Algebra Automorphisms of Drinfeld–Jimbo Algebras
6.1.5 Triangular Decomposition of Usub(q)
6.1.6 Hopf Algebra Automorphisms of Usub(q)
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 Usub(q)
6.3 The Quantum Killing Form and the Center of Usub(q)
6.3.1 A Dual Pairing of the Hopf Algebras Usub(q) and Usub(q)[sup(op)]
6.3.2 The Quantum Killing Form on Usub(q)
6.3.3 A Quantum Casimir Element
6.3.4 The Center of Usub(q) and the Harish-Chandra Homomorphism
6.3.5 The Center of Usub(q) for q a Root of Unity
6.4 Notes
Chapter 7. Finite-Dimensional Representations of Drinfeld–Jimbo Algebras
7.1 General Properties of Finite-Dimensional Representations of Uq(g)
7.1.1 Weight Structure and Classification
7.1.2 Properties of Representations
7.1.3 Representations of h-Adic Drinfeld–Jimbo Algebras
7.1.4 Characters of Representations and Multiplicities of Weights
7.1.5 Separation of Elements of Usub(q)
7.1.6 The Quantum Trace of Finite-Dimensional Representations
7.2 Tensor Products of Representations
7.2.1 Multiplicities in Tensor Products of Representations
7.2.2 Clebsch–Gordan Coefficients
7.3 Representations of Usub(q) for q not a Root of Unity
7.3.1 The Hopf Algebra Usub(q)
7.3.2 Finite-Dimensional Representations of Usub(q)
7.3.3 Gel'fand–Tsetlin Bases and Explicit Formulas for Representations
7.3.4 Representations of Class 1
7.3.5 Tensor Products of Representations
7.3.6 Tensor Operators and the Wigner–Eckart Theorem
7.3.7 Clebsch–Gordan Coefficients for the Tensor Product T[sub(m)]X T[sub(1)]
7.3.8 Clebsch–Gordan Coefflcients for the Tensor Product T[sub(m)]X T[sub(p)]
7.3.9
7.4 Crystal Bases
7.4.1 Crystal Bases of Finite-Dimensional Modules
7.4.2 Existence and Uniqueness of Crystal Bases
7.4.3 Crystal Bases of Tensor Product Modules
7.4.4 Globalization of Crystal Bases
7.4.5 Crystal Bases of U'sub(q)
7.5 Representations of Usub(q) for q a Root of Unity
7.5.1 General Results
7.5.2 Cyclic Representations
7.5.3
7.5.4 Representations of Minimal Dimensions
7.5.5
7.6 Applications
7.7 Notes
Chapter 8. Quasitriangularity and Universal R-Matrices
8.1 Quasitriangular Hopf Algebras
8.1.1 Definition and Basic Properties
8.1.2 R-Matrices for Representations
8.1.3 Square and Inverse of the Antipode
8.2 The Quantum Double and Universal R-Matrices
8.2.1 The Quantum Double of Skew-Paired Bialgebras
8.2.2 Quasitriangularity of Quantum Doubles of Finite-Dimensional Hopf Algebras
8.2.3 The Rosso Form of the Quantum Double
8.2.4 Drinfeld–Jimbo Algebras as Quotients of Quantum Doubles
8.3 Explicit Form of Universal R-Matrices
8.3.1 The Universal R-Matrix for Usub(h)
8.3.2 The Universal R-Matrix for Usub(h)
8.3.3 R-Matrices for Representations of Usub(q)
8.4 Vector Representations and R-Matrices
8.4.1 Vector Representations of Drinfeld–Jimbo Algebras
8.4.2 R-Matrices for Vector Representations
8.4.3 Spectral Decompositions of R-Matrices for Vector Representations
8.5 L-Operators and L-Functionals
8.5.1 L-Operators and L-Functionals
8.5.2 L-Functionals for Vector Representations
8.5.3 The Extended Hopf Algebras U[sub(q)]sup(ext)
8.5.4 L-Functionals for Vector Representations of Usub(q)
8.5.5 The Hopf Algebras U(R) and U[sub(q)]sup(L)
8.6 An Analog of the Brauer–Schur–Weyl Duality
8.6.1 The Algebras Usub(q)
8.6.2 Tensor Products of Vector Representations
8.6.3 The Brauer–Schur–Weyl Duality for Drinfeld–Jimbo Algebras
8.6.4 Hecke and Birman–Wenzl–Murakami Algebras
8.7 Applications
8.7.1 Baxterization
8.7.2 Elliptic Solutions of the Quantum Yang–Baxter Equation
8.7.3 R-Matrices and Integrable Systems
8.8 Notes
Part III: Quantized Algebras of Functions
Chapter 9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces
9.1 The Approach of Faddeev–Reshetikhin–Takhtajan
9.1.1 The FRT Bialgebra A(R)
9.1.2
9.2 The Quantum Groups GLsub(q) and SLsub(q)
9.2.1 The Quantum Matrix Space Msub(q) and the Quantum Vector Space C[sub(q)][sup(N)]
9.2.2 Quantum Determinants
9.2.3 The Quantum Groups GLsub(q) and SLsub(q)
9.2.4 Real Forms of GLsub(q) and SLsub(q) and -Quantum Spaces
9.3 The Quantum Groups Osub(q) and Spsub(q)
9.3.1 The Hopf Algebras O(Osub(q)) and O(Spsub(q))
9.3.2 The Quantum Vector Space for the Quantum Group Osub(q)
9.3.3 The Quantum Group SOsub(q)
9.3.4 The Quantum Vector Space for the Quantum Group Spsub(q)
9.3.5 Real Forms of Osub(q) and Spsub(q) and -Quantum Spaces
9.4 Dual Pairings of Drinfeld–Jimbo Algebras and Coordinate Hopf Algebras
9.5 Notes
Chapter 10. Coquasitriangularity and Crossed Product Constructions
10.1 Coquasitriangular Hopf Algebras
10.1.1 Definition and Basic Properties
10.1.2 Coquasitriangularity of FRT Bialgebras A(R) and Coordinate Hopf Algebras O(G[sub(q)])
10.1.3 L-Functionals of Coquasitriangular Hopf Algebras
10.2 Crossed Product Constructions of Hopf Algebras
10.2.1 Crossed Product Algebras
10.2.2 Crossed Coproduct Coalgebras
10.2.3 Twisting of Algebra Structures by 2-Cocycles and Quantum Doubles
10.2.4 Twisting of Coalgebra Structures by 2-Cocycles and Quantum Codoubles
10.2.5 Double Crossed Product Bialgebras and Quantum Doubles
10.2.6 Double Crossed Coproduct Bialgebras and Quantum Codoubles
10.2.7 Realifications of Quantum Groups
10.3 Braided Hopf Algebras
10.3.1 Covariantized Products for Coquasitriangular Bialgebras
10.3.2 Braided Hopf Algebras Associated with Coquasitriangular Hopf Algebras
10.3.3 Braided Hopf Algebras Associated with Quasitriangular Hopf Algebras
10.3.4 Braided Tensor Categories and Braided Hopf Algebras
10.3.5 Braided Vector Algebras
10.3.6 Bosonization of Braided Hopf Algebras
10.3.7 -Structures on Bosonized Hopf Algebras
10.3.8 Inhomogeneous Quantum Groups
10.3.9 -Structures for Inhomogeneous Quantum Groups
10.4 Notes
Chapter 11. Corepresentation Theory and Compact Quantum Groups
11.1 Corepresentat ions of Hopf Algebras
11.1.1 Corepresentations
11.1.2 Intertwiners
11.1.3 Constructions of New Corepresentations
11.1.4 Irreducible Corepresentations
11.1.5 Unitary Corepresentations
11.2 Cosemisimple Hopf Algebras
11.2.1 Definition and Characterizations
11.2.2 The Haar Functional of a Cosemisimple Hopf Algebra
11.2.3 Peter–Weyl Decomposition of Coordinate Hopf Algebras
11.3 Compact Quantum Group Algebras
11.3.1 Definitions and Characterizations of CQG Algebras
11.3.2 The Haar State of a CQG Algebra
11.3.3 C-Algebra Completions of CQG Algebras
11.3.4 Modular Properties of the Haar State
11.3.5 Polar Decomposition of the Antipode
11.3.6 Multiplicative Unitaries of CQG Algebras
11.4 Compact Quantum Group C-Algebras
11.4.1 CQG C-Algebras and Their CQG Algebras
11.4.2 Existence of the Haar State of a CQG C-Algebra
11.4.3 Proof of Theorem 39
11.4.4 Another Definition of CQG C-Algebras
11.5 Finite-Dimensional Representations of GLsub(q)
11.5.1 Some Quantum Subgroups of GLsub(q)
11.5.2 Submodules of Relative Invariant Elements
11.5.3 Irreducible Representations of GLsub(q)
11.5.4 Peter–Weyl Decomposition of O(GLsub(q))
11.5.5 Representations of the Quantum Group Usub(q)
11.6 Quantum Homogeneous Spaces
11.6.1 Definition of a Quantum Homogeneous Space
11.6.2 Quant um Homogeneous Spaces Associated with Quantum Subgroups
11.6.3 Quantum Gel'fand Pairs
11.6.4 The Quantum Homogeneous Space Usub(q)Usub(q)
11.6.5 Quantum Homogeneous Spaces of Infinitesimally Invariant Elements
11.6.6 Quantum Projective Spaces
11.7 Notes
Part IV: Noncommutative Differential Calculus
Chapter 12. Covariant Differential Calculus on Quantum Spaces
12.1 Covariant First Order Differential Calculus
12.1.1 First Order Differential Calculi on Algebras
12.1.2 Covariant First Order Calculi on Quantum Spaces
12.2 Covariant Higher Order Differential Calculus
12.2.1 Differential Calculi on Algebras
12.2.2 The Differential Envelope of an Algebra
12.2.3 Covariant Differential Calculi on Quantum Spaces
12.3 Construction of Covariant Differential Calculi on Quantum Spaces
12.3.1 General Method
12.3.2 Covariant Differential Calculi on Quantum Vector Spaces
12.3.3 Covariant Differential Calculus on C[sub(q)][sup(N)] and the Quantum Weyl Algebra
12.3.4 Covariant Differential Calculi on the Quantum Hyperboloid
12.4 Notes
Chapter 13. Hopf Bimodules and Exterior Algebras
13.1 Covariant Bimodules
13.1.1 Left-Covariant Bimodules
13.1.2 Right-Covariant Bimodules
13.1.3 Bicovariant Bimodules (Hopf Bimodules)
13.1.4 Woronowicz' Braiding of Bicovariant Bimodules
13.1.5 Bicovariant Bimodules and Representations of the Quantum Double
13.2 Tensor Algebras and Exterior Algebras of Bicovariant Bimodules
13.2.1 The Tensor Algebra of a Bicovariant Bimodule
13.2.2 The Exterior Algebra of a Bicovariant Bimodule
13.3 Notes
Chapter 14. Covariant Differential Calculus on Quantum Groups
14.1 Left-Covariant First Order Differential Calculi
14.1.1 Left-Covariant First Order Calculi and Their Right Ideals
14.1.2 The Quantum Tangent Space
14.1.3 An Example: The 3D-Calculus on SLsub(q)
14.1.4 Another Left-Covariant Differential Calculus on SLsub(q)
14.2 Bicovariant First Order Differential Calculi
14.2.1 Right-Covariant First Order Differential Calculi
14.2.2 Bicovariant First Order Differential Calculi
14.2.3 Quantum Lie Algebras of Bicovariant First Order Calculi
14.2.4 The 4D[sub(+)]- and the 4D[sub(–)]-Calculus on SLsub(q)
14.2.5 Examples of Bicovariant First Order Calculi on Simple Lie Groups
14.3 Higher Order Left-Covariant Differential Calculi
14.3.1 The Maurer–Cartan Formula
14.3.2 The Differential Envelope of a Hopf Algebra
14.3.3 The Universal DC of a Left-Covariant FODC
14.4 Higher Order Bicovariant Differential Calculi
14.4.1 Bicovariant Differential Calculi and Differential Hopf Algebras
14.4.2 Quantum Lie Derivatives and Contraction Operators
14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras
14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups
14.6.1 A Family of Bicovariant First Order Differential Calculi
14.6.2 Braiding and Structure Constants of the FODC Ga[sub(+- ,z)]
14.6.3 A Canonical Basis for the Left-Invariant 1-Forms
14.6.4 Classification of Bicovariant First Order Differential Calculi
14.7 Notes
Bibliography
1. Books
2. Articles
Index
Subject Index
Index of Symbols