Quantum Groups and Their Representations

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 $$\mathcal{A}{\!^c_q\!}$$ 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 $$\mathcal{A}{\!^c_q\!}$$ for q a Root of Unity.- 5.2.5 Irreducible *-Representations of $$\mathcal{A}{\!^c_q\!}$$ 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.- 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 Uq(g).- 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 ?q(gln) for q not a Root of Unity.- 7.3.1 The Hopf Algebra ?q(gln).- 7.3.2 Finite-Dimensional Representations of ?q(gln).- 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 Tm ? T1.- 7.3.8 Clebsch-Gordan Coefficients for the Tensor Product Tm ? Tp.- 7.3.9 The Tensor Product Tm ? T1 for q±1? 0.- 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 $$U{^\prime_q}(\!\text{n}\_)$$.- 7.5 Representations of Uq(g) for q a Root of Unity.- 7.5.1 General Results.- 7.5.2 Cyclic Representations.- 7.5.3 Cyclic Representations of the Algebra U?(sll+1).- 7.5.4 Representations of Minimal Dimensions.- 7.5.5 Representations of U?(sll+1) in Ge?fand-Tsetlin Bases.- 7.6 Applications.- 7.7 Notes.- 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 Uh(sl2).- 8.3.2 The Universal R-Matrix for Uh(g).- 8.3.3 R-Matrices for Representations of Uq(g).- 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 $${U}^\text{ext}_q (\!\text{g}\!)$$.- 8.5.4 L-Functionals for Vector Representations of Uq(g).- 8.5.5 The Hopf Algebras $$\mathcal{U}(\!\text{R}\!)$$ and $${U}^L_q (\!\text{g}\!)$$.- 8.6 An Analog of the Brauer-Schur-Weyl Duality.- 8.6.1 The Algebras ?q(soN).- 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.- III. Quantized Algebras of Functions.- 9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces.- 9.1 The Approach of Faddeev-Reshetikhin-Takhtajan.- 9.1.1 The FRT Bialgebra $$\mathcal{A}(\!{R}\!)$$.- 9.1.2 The Quantum Vector Spaces ?L(f; R) and ?R(f; R).- 9.2 The Quantum Groups GLq(N) and SLq(N).- 9.2.1 The Quantum Matrix Space Mq(N) and the Quantum Vector Space $$\mathbb{C}{^N_q}$$.- 9.2.2 Quantum Determinants.- 9.2.3 The Quantum Groups GLq(N) and SLq(N).- 9.2.4 Real Forms of GLq(N) and SLq(N) and *-Quantum Spaces.- 9.3 The Quantum Groups Oq(N) and Spq(N).- 9.3.1 The Hopf Algebras $$\mathcal{O}(\!O_{\!q}(\!N)\!)$$ and $$\mathcal{O}(\!Sp_{\!q}(\!N)\!)$$.- 9.3.2 The Quantum Vector Space for the Quantum Group Oq(N).- 9.3.3 The Quantum Group SOq(N).- 9.3.4 The Quantum Vector Space for the Quantum Group Spq(N).- 9.3.5 Real Forms of Oq(N) and Spq(N) and *-Quantum Spaces.- 9.4 Dual Pairings of Drinfeld-Jimbo Algebras and Coordinate Hopf Algebras.- 9.5 Notes.- 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 $$\mathcal{A}(R)$$ and Coordinate Hopf Algebras $$\mathcal{O}(\!G{_q}\!)$$.- 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.- 11. Corepresentation Theory and Compact Quantum Groups.- 11.1 Corepresentations 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 GLq(N).- 11.5.1 Some Quantum Subgroups of GLq(N).- 11.5.2 Submodules of Relative Invariant Elements.- 11.5.3 Irreducible Representations of GLq(N).- 11.5.4 Peter-Weyl Decomposition of $$\mathcal{O}(G{{L}_{}}_{q}(N))$$.- 11.5.5 Representations of the Quantum Group Uq(N).- 11.6 Quantum Homogeneous Spaces.- 11.6.1 Definition of a Quantum Homogeneous Space.- 11.6.2 Quantum Homogeneous Spaces Associated with Quantum Subgroups.- 11.6.3 Quantum Gel'fand Pairs.- 11.6.4 The Quantum Homogeneous Space Uq(N-1)\Uq(N).- 11.6.5 Quantum Homogeneous Spaces of Infinitesimally Invariant Elements.- 11.6.6 Quantum Projective Spaces.- 11.7 Notes.- IV. Noncommutative Differential Calculus.- 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 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 $$\mathbb{C}{^N_q}$$ and the Quantum Weyl Algebra.- 12.3.4 Covariant Differential Calculi on the Quantum Hyperboloid.- 12.4 Notes.- 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.- 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 SLq(2).- 14.1.4 Another Left-Covariant Differential Calculus on SLq(2).- 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+- and the 4D_-Calculus on SLq(2).- 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 ?±,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.