Quantum Groups and Their Representations

ISBN-13: 9783642646010 / Angielski / Miękka / 2011 / 552 str.

Anatoli Klimyk; Konrad Schm Dgen

Opis

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.

Recenzje

From the reviews
"Klimyk and Schmüdgen are kind to their readers. Proofs are given in full, and there are helpful explanations of the basic concepts ... the book has the virtue of comprehensivness in its chose range of topics. It is easy to dip into and use as a reference book." (A. Sudbery, Bulletin of the London Mathematical Society, 2000)

Spis treści

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 $$U^_Q(\!\text_2\!)$$.- 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 $$\mathcal(S{_{}}_\left( 2 \right))$$.- 4.1.1 The Bialgebra $$\mathcal({_{}}_\left( 2 \right))$$.- 4.1.2 The Hopf Algebra $$\mathcal(S{_{}}_\left( 2 \right))$$.- 4.1.3 A Geometric Approach to SLq(2).- 4.1.4 Real Forms of $$\mathcal(S{_{}}_\left( 2 \right))$$.- 4.1.5 The Diamond Lemma.- 4.2 Representations of the Quantum Group SLq(2).- 4.2.1 Finite-Dimensional Corepresentations of $$\mathcal(S{_{}}_\left( 2 \right))$$: Main Results.- 4.2.2 A Decomposition of $$\mathcal(S{_{}}_\left( 2 \right))$$.- 4.2.3 Finite-Dimensional Subcomodules of $$\mathcal(S{_{}}_\left( 2 \right))$$.- 4.2.4 Calculation of the Matrix Coefficients.- 4.2.5 The Peter-Weyl Decomposition of $$\mathcal(S{_{}}_\left( 2 \right))$$.- 4.2.6 The Haar Functional of $$\mathcal(S{_{}}_\left( 2 \right))$$.- 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 $$\mathcal(S{_{}}q\left( 2 \right))$$.- 4.3.3 The Fourier Transform on SUq(2).- 4.3.4 Representations and the C*-Algebra of $$\mathcal(S{_{}}q\left( 2 \right))$$.- 4.4 Duality of the Hopf Algebras Uq(sl2) and $$\mathcal(S\left( 2 \right))$$.- 4.4.1 Dual Pairing of the Hopf Algebras Uq(sl2) and $$\mathcal(S\left( 2 \right))$$.- 4.4.2 Corepresentations of $$\mathcal(S{_{}}_\left( 2 \right))$$ 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 $$\mathcal(\!S^2_{q\rho}\!)$$.- 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 $$\mathcal{\!^c_q\!}$$ and $$\mathcal$$.- 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 $$\mathcal{\!^c_q\!}$$ for q a Root of Unity.- 5.2.5 Irreducible *-Representations of $$\mathcal{\!^c_q\!}$$ and $$\mathcal$$.- 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 $$U{^\prime_q}(\!\text\_)$$.- 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 $$^\text_q (\!\text\!)$$.- 8.5.4 L-Functionals for Vector Representations of Uq(g).- 8.5.5 The Hopf Algebras $$\mathcal(\!\text\!)$$ and $$^L_q (\!\text\!)$$.- 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 $$\mathcal(\!\!)$$.- 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 $$\mathbb{^N_q}$$.- 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 $$\mathcal(\!O_{\!q}(\!N)\!)$$ and $$\mathcal(\!Sp_{\!q}(\!N)\!)$$.- 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 $$\mathcal(R)$$ and Coordinate Hopf Algebras $$\mathcal(\!G\!)$$.- 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 $$\mathcal(G{_{}}_(N))$$.- 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 $$\mathbb{^N_q}$$ 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.

Blurb

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.