Anatoli Klimyk Konrad Schmiidgen

Quantum GroupsandTheir Representations

Springer

Table of Contents

Part I. An Introduction to Quantum Groups

1. Hopf Algebras 31.1 Prolog: Examples of Hopf Algebras of Functions on Groups . . 31.2 Coalgebras, Bialgebras and Hopf Algebras 6

1.2.1 Algebras 61.2.2 Coalgebras 81.2.3 Bialgebras 111.2.4 Hopf Algebras 131.2.5* Dual Pairings of Hopf Algebras 161.2.6 Examples of Hopf Algebras 181.2.7 *-Structures 201.2.8* The Dual Hopf Algebra A° 221.2.9* Super Hopf Algebras 231.2.10*/i-Adic Hopf Algebras 25

1.3 Modules and Comodules of Hopf Algebras 271.3.1 Modules and Representations 271.3.2 Comodules and Corepresentations 291.3.3 Comodule Algebras and Related Concepts 321.3.4* Adjoint Actions and Coactions of Hopf Algebras 341.3.5* Corepresentations and Representations

of Dually Paired Coalgebras and Algebras 351.4 Notes 36

2. q-Calculus 372.1 Main Notions on q-Calculus 37

2.1.1 qr-Numbers and (/-Factorials 372.1.2 q-Binomial Coefficients 392.1.3 Basic Hypergeometric Functions 402.1.4 The'Function npo(a; q,z) 412.1.5 The Basic Hypergeometric Function 2</?i • • 422.1.6 Transformation Formulas for 3</?2 and 4<̂ 3 432.1.7 q-Analog of the Binomial Theorem 44

2.2 ^-Differentiation and ^-Integration 442.2.1 ^-Differentiation 44

X Table of Contents

2.2.2 g-Integral 462.2.3 q-Analog of the Exponential Function 472.2.4 gr-Analog of the Gamma Function 48

2.3 ^-Orthogonal Polynomials 492.3.1 Jacobi Matrices and Orthogonal Polynomials 492.3.2 g-Hermite Polynomials 502.3.3 Little g-Jacobi Polynomials 512.3.4 Big q-Jacobi Polynomials 52

2.4 Notes 52

3. The Quantum Algebra Uq(sl2) and Its Representations . . . 533.1 The Quantum Algebras C/,(sl2) and Uh(sl2) 53

3.1.1 The Algebra Uq(s\2) 533.1.2 The Hopf Algebra Uq(sl2) 553.1.3 The Classical Limit of the Hopf Algebra Uq(sl2) 573.1.4 Real Forms of the Quantum Algebra Uq(sl2) 583.1.5 The h-Adic Hopf Algebra Uh(s\2) 60

3.2 Finite-Dimensional Representations of Uq(sl2)for q not a Root of Unity 613.2.1 The Representations Tul 613.2.2 Weight Representations and Complete Reducibility . . . 633.2.3 Finite-Dimensional Representations of Uq{s\2)

and Uh{s\2) 653.3 Representations of Uq(sl2) for q a Root of Unity 66

3.3.1 The Center of Uq\s\2) 663.3.2 Representations of Uq(s\2) 673.3.3 Representations of E/£es(sl2) 71

3.4 Tensor Products of Representations.Clebsch-Gordan Coefficients 723.4.1 Tensor Products of Representations Tj 723.4.2 Clebsch-Gordan Coefficients 743.4.3 Other Expressions for Clebsch-Gordan Coefficients . . . 783.4.4 Symmetries of Clebsch-Gordan Coefficients 81

3.5 Racah Coefficients and 6j Symbols of Uq(su2) 823.5.1 Definition of the Racah Coefficients 823.5.2 Relations Between Racah

and Clebsch-Gordan Coefficients 843.5.3 Symmetry Relations 843.5.4 Calculation of Racah Coefficients 853.5.5 The Biedenharn-Elliott Identity 883.5.6 The Hexagon Relation 903.5.7 Clebsch-Gordan Coefficients

as Limits of Racah Coefficients 903.6 Tensor Operators and the Wigner-Eckart Theorem 92

3.6.1 Tensor Operators for Compact Lie Groups 92

Table of Contents XI

3.6.2 Tensor Operators and the Wigner-Eckart Theoremfor Uq(su2) 93

3.7 Applications 943.7.1 The Uq{sl2) Rotator Model of Deformed Nuclei 943.7.2 Electromagnetic Transitions in the Uq(sl2) Model 95

3.8 Notes 96

4. The Quantum Group SLg(2) and Its Representations 974.1 The Hopf Algebra O(SLq{2)) 97

4.1.1 The Bialgebra O{Mq{2)) 974.1.2 The Hopf Algebra O{SLq{2)) 994.1.3 A Geometric Approach to SLq{2) 1014.1.4 Real Forms of O{SLq{2)) 1024.1.5 The Diamond Lemma 103

4.2 Representations of the Quantum Group SLq{2) 1044.2.1 Finite-Dimensional Corepresentations of O(SLq(2)):

Main Results 1044.2.2 A Decomposition of O(SLq(2)) 1054.2.3 Finite-Dimensional Subcomodules of O(SLq(2)) : 1064.2.4 Calculation of the Matrix Coefficients 1084.2.5 The Peter-Weyl Decomposition of O{SLq{2)) 1104.2.6 The Haar Functional of O{SLq{2)) I l l

4.3 The Compact Quantum Group SUq{2)and Its Representations 1134.3.1 Unitary Representations

of the Quantum Group SUq(2) 1134.3.2 The Haar State

and the Peter-Weyl Theorem for O(SUq(2)) 1144.3.3 The Fourier Transform on SUq(2) 1174.3.4 ^Representations and the C*-Algebra of O(SUq(2)) . . 117<

4.4 Duality of the Hopf Algebras Uq(sl2)and O(SLq{2)) .' 1194.4.1 Dual Pairing of the Hopf Algebras Uq{s\2)

and O(SLq(2)) 1194.4.2 Corepresentations of O{SLq(2))

and Representations of Uq(s\2) 1234.5 Quantum 2-Spheres 124

4.5.1 A Family of Quantum Spaces for SLq(2) 1244.5.2 Decomposition of the Algebra O(S^p) 1264.5.3 Spherical Functions on S%p 1294.5.4 An Infinitesimal Characterization of O(S*p) 129

4.6 Notes 132

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5. The g-Oscillator Algebras and Their Representations 1335.1 The g-Oscillator Algebras Ac

q and Aq 1335.1.1 Definitions and Algebraic Properties 1335.1.2 Other Forms of the g-Oscillator Algebra 1365.1.3 The g-Oscillator Algebra

and the Quantum Algebra Uq(sl2) 1375.1.4 The q-Oscillator Algebras

and the Quantum Space Mqi{2) 1405.2 Representations of g-Oscillator Algebras 140

5.2.1 AT-Finite Representations 1405.2.2 Irreducible Representations

with Highest (Lowest) Weights 1415.2.3 Representations Without Highest and Lowest Weights 1435.2.4 Irreducible Representations of Aq

for q a Root of Unity 1455.2.5 Irreducible *-Representations of Aq and Aq 1475.2.6 Irreducible ^Representations

of Another q-Oscillator Algebra 1485.3 The Fock Representation of the g-Oscillator Algebra 149

5.3.1 The Fock Representation 1495.3.2 The Bargmann-Fock Realization 1505.3.3 Coherent States 1525.3.4 Bargmann-Fock Space Realization

of Irreducible Representations of Uq(s\2) 1535.4 Notes 154

Part II. Quantized Universal Enveloping Algebras

6. Drinfeld-Jimbo Algebras 1576.1 Definitions of Drinfeld-Jimbo Algebras 157

6.1.1 Semisimple Lie Algebras 1576.1.2 The Drinfeld-Jimbo Algebras Uq(g) 1616.1.3 The h-Adic Drinfeld-Jimbo Algebras Uh(e) 1656.1.4 Some Algebra Automorphisms

of Drinfeld-Jimbo Algebras 1676.1.5 Triangular Decomposition of Uq(o) 1686.1.6 Hopf Algebra Automorphisms of Uq(o) 1716.1.7 Real Forms of Drinfeld-Jimbo Algebras 172

6.2 Poincare-BirkhofFWitt Theorem and Verma Modules 1736.2.1 Braid Groups 1736.2.2 Action of Braid Groups on Drinfeld-Jimbo Algebras .. 1746.2.3 Root Vectors and Poincare-Birkhoff-Witt Theorem . . . 1756.2.4 Representations with Highest Weights 1776.2.5 Verma Modules 179

Table of Contents XIII

6.2.6 Irreducible Representations with Highest Weights . . . . 1806.2.7 The Left Adjoint Action of Uq(g) 181

6.3 The Quantum Killing Form and the Center of UQ(Q) 1846.3.1 A Dual Pairing of the Hopf Algebras (7,(6+)

and Uq{b-)op ,--., 1846.3.2 The Quantum Killing Form on Uq(g) 1876.3.3 A Quantum Casimir Element 1896.3.4 The Center of Uq(g)

and the Harish-Chandra Homomorphism 1926.3.5 The Center of C/,(g) for q a Root of Unity 194

6.4 Notes 196

7. Finite-Dimensional Representationsof Drinfeld-Jimbo Algebras 1977.1 General Properties of Finite-Dimensional Representations

of Uq(g) 1977.1.1 Weight Structure and Classification 1977.1.2 Properties of Representations 2007.1.3 Representations of ft-Adic Drinfeld-Jimbo Algebras . . . 2027.1.4 Characters of Representations and Multiplicities

of Weights 2037.1.5 Separation of Elements of Uq(o) 2047.1.6 The Quantum Trace

of Finite-Dimensional Representations 2057.2 Tensor Products of Representations 207

7.2.1 Multiplicities in Tensor Products of Representations . . 2087.2.2 Clebsch-Gordan Coefficients 211

7.3 Representations of Uq(gln) for q not a Root of Unity 2127.3.1 The Hopf Algebra Uq(gln) 2127.3.2 Finite-Dimensional Representations of Uq(g\n) 2137.3.3 GePfarid-Tsetlin Bases and Explicit Formulas

for Representations 2147.3.4 Representations of Class 1 2177.3.5 Tensor Products of Representations 2187.3.6 Tensor Operators and the Wigner-Eckart Theorem .. . 2197.3.7 Clebsch-Gordan Coefficients

for the Tensor Product T m <g> Tx ..., 2207.3.8 Clebsch-Gordan Coefficients

for the Tensor Product Tm <g> Tp 2217.3.9 The Tensor Product T m <g> Tx for q±x -* 0 224

7.4 Crystal Bases 2257.4.1 Crystal Bases of Finite-Dimensional Modules 2267.4.2 Existence and Uniqueness of Crystal Bases 2277.4.3 Crystal Bases of Tensor Product Modules 228

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7.4.4 Globalization of Crystal Bases 2297.4.5 Crystal Bases of Uq(x\-) 230

7.5 Representations of Uq(g) for q a Root of Unity 2327.5.1 General Results 2327.5.2 Cyclic Representations 2347.5.3 Cyclic Representations of the Algebra !7e(sl;+i) 2357.5.4 Representations of Minimal Dimensions 2377.5.5 Representations of [/^(slj+i) in Gel'fand-Tsetlin Bases 238

7.6 Applications 2407.7 Notes 242

8. Quasitriangularity and Universal U-Matrices 2438.1 Quasitriangular Hopf Algebras 243

8.1.1 Definition and Basic Properties , 2438.1.2 .R-Matrices for Representations 2468.1.3 Square and Inverse of the Antipode 247

8.2 The Quantum Double and Universal .R-Matrices 2508.2.1 The Quantum Double of Skew-Paired Bialgebras 2508.2.2 Quasitriangularity of Quantum Doubles

of Finite-Dimensional Hopf Algebras 2548.2.3 The Rosso Form of the Quantum Double 2578.2.4 Drinfeld-Jimbo Algebras as Quotients

of Quantum Doubles 2588.3 Explicit Form of Universal .R-Matrices 259

8.3.1 The Universal fl-Matrix for Uh(s\2) •; 2598.3.2 The Universal ft-Matrix for Uh(g) 2618.3.3 .R-Matrices for Representations of Uq(o) 264

8.4 Vector Representations and H-Matrices 2678.4.1 Vector Representations of Drinfeld-Jimbo Algebras . . . 2678.4.2 .R-Matrices for Vector Representations 2698.4.3 Spectral Decompositions of .R-Matrices

for Vector Representations 2728.5 L-Operators and L-Functionals 275

8.5.1 L-Operators and .L-Functionals 2758.5.2 .L-Functionals for Vector Representations 2778.5.3 The Extended Hopf Algebras U^xt{g) 2818.5.4 .L-Functionals for Vector Representations of Uq(g) . . . . 2838.5.5 The Hopf Algebras U{R) and U^Q) 285

8.6 An Analog of the Brauer-Schur-Weyl Duality 2888.6.1 The Algebras Uq(soN) 2888.6.2 Tensor Products of Vector Representations 2898.6.3 The Brauer-Schur-Weyl Duality

for Drinfeld-Jimbo Algebras 2918.6.4 Hecke and Birman-Wenzl-Murakami Algebras 293

8.7 Applications 294

Table of Contents XV

8.7.1 Baxterization 2958.7.2 Elliptic Solutions

of the Quantum Yang-Baxter Equation 2978.7.3 il-Matrices and Integrable Systems 298Notes ;.-.....-.-. . 300

Part III. Quantized Algebras of Functions

9. Coordinate Algebras of Quantum Groupsand Quantum Vector Spaces 3039.1 The Approach of Faddeev-Reshetikhin-Takhtajan 303

9.1.1 The FRT Bialgebra A(R) 3039.1.2 The Quantum Vector Spaces XL(f;R) and XR(f; R) . . 307

9.2 The Quantum Groups GLq{N) and SLq(N) 3099.2.1 The Quantum Matrix Space Mq(N)

and the Quantum Vector Space C^ 3109.2.2 Quantum Determinants 3119.2.3 The Quantum Groups GLq(N) and SLq(N) 3139.2.4 Real Forms of GLq{N) and SLq(N)

and *-Quantum Spaces 3169.3 The Quantum Groups Oq(N) and Spq(N) 317

9.3.1 The Hopf Algebras O(Oq(N)) and O(Spq(N)) 3189.3.2 The Quantum Vector Space

for the Quantum Group Oq(N) 3209.3.3 The Quantum Group SOq(N) 3239.3.4 The Quantum Vector Space

for the Quantum Group Spq(N) 3249.3.5 Real Forms of Oq(N) and Spq(N)

and *-Quantum Spaces 3259.4 Dual Pairings of Drinfeld-Jimbo Algebras

and Coordinate Hopf Algebras 3279.5 Notes 330

10. Coquasitriangularity and Crossed Product Constructions . 33110.1 Coquasitriangular Hopf Algebras 331

10.1.1 Definition and Basic Properties 33110.1.2 Coquasitriangularity of FRT Bialgebras A(R)

and Coordinate Hopf Algebras O(Gq) 33710.1.3 L-Functionals of Coquasitriangular Hopf Algebras . . . . 342

10.2 Crossed Product Constructions of Hopf Algebras 34910.2.1 Crossed Product Algebras 34910.2.2 Crossed Coproduct Coalgebras 35210.2.3 Twisting of Algebra Structures by 2-Cocycles

and Quantum Doubles 354

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10.2.4 Twisting of Coalgebra Structures by 2-Cocyclesand Quantum Codoubles 357

10.2.5 Double Crossed Product Bialgebrasand Quantum Doubles 359

10.2.6 Double Crossed Coproduct Bialgebrasand Quantum Codoubles 362

10.2.7 Realifications of Quantum Groups 36310.3 Braided Hopf Algebras 365

10.3.1 Covariantized Productsfor Coquasitriangular Bialgebras 365

10.3.2 Braided Hopf AlgebrasAssociated with Coquasitriangular Hopf Algebras . . . . 370

10.3.3 Braided Hopf AlgebrasAssociated with Quasitriangular Hopf Algebras 376

10.3.4 Braided Tensor Categories and Braided Hopf Algebras 37710.3.5 Braided Vector Algebras 38010.3.6 Bosonization of Braided Hopf Algebras 38210.3.7 *-Structures on Bosonized Hopf Algebras 38610.3.8 Inhomogeneous Quantum Groups 38810.3.9 *-Structures for Inhomogeneous Quantum Groups . . . . 390

10.4 Notes . 394

11. Corepresentation Theory and Compact Quantum Groups. 39511.1 Corepresentations of Hopf Algebras 395

11.1.1 Corepresentations , 39511.1.2 Intertwiners 39711.1.3 Constructions of New Corepresentations 39711.1.4 Irreducible Corepresentations 39811.1.5 Unitary Corepresentations 401

11.2 Cosemisimple Hopf Algebras 40211.2.1 Definition and Characterizations 40211.2.2 The Haar Functional of a Cosemisimple Hopf Algebra. 40411.2.3 Peter-Weyl Decomposition

of Coordinate Hopf Algebras 40811.3 Compact Quantum Group Algebras 415

11.3.1 Definitions and Characterizations of CQG Algebras . . . 41511.3.2 The Haar State of a CQG Algebra 41911.3.3 C*-Algebra Completions of CQG Algebras 42011.3.4 Modular Properties of the Haar State 42211.3.5 Polar Decomposition of the Antipode ...- 42611.3.6 Multiplicative Unitaries of CQG Algebras 427

11.4 Compact Quantum Group C*-Algebras 42911.4.1 CQG C*-Algebras and Their CQG Algebras 42911.4.2 Existence of the Haar State of a CQG C*-Algebra 43111.4.3 Proof of Theorem 39 433

Table of Contents XVII

11.4.4 Another Definition of CQG C*-Algebras 43411.5 Finite-Dimensional Representations of GLq(N) 435

11.5.1 Some Quantum Subgroups of GLq(N) 43511.5.2 Submodules of Relative Invariant Elements 43611.5.3 Irreducible Representations of GLq(N) 43711.5.4 Peter-Weyl Decomposition of O(GLq(N)) 43911.5.5 Representations of the Quantum Group Uq(N) 441

11.6 Quantum Homogeneous Spaces 44211.6.1 Definition of a Quantum Homogeneous Space 44211.6.2 Quantum Homogeneous Spaces

Associated with Quantum Subgroups 44311.6.3 Quantum Gel'fand Pairs 44511.6.4 The Quantum Homogeneous Space Uq(N-l)\Uq(N) . . 44711.6.5 Quantum Homogeneous Spaces

of Infinitesimally Invariant Elements 45111.6.6 Quantum Projective Spaces 452

11.7 Notes 454

Part IV. Noncommutative Differential Calculus

12. Covariant Differential Calculus on Quantum Spaces 45712.1 Covariant First Order Differential Calculus 457

12.1.1 First Order Differential Calculi on Algebras 45712.1.2 Covariant First Order Calculi on Quantum Spaces . . . . 459

12.2 Covariant Higher Order Differential Calculus 46112.2.1 Differential Calculi on Algebras 46112.2.2 The Differential Envelope of an Algebra 46212.2.3 Covariant Differential Calculi on Quantum Spaces . . . . 463

12.3 Construction of Covariant Differential Calculion Quantum Spaces 46412.3.1 General Method . . . . .\ 46412.3.2 Covariant Differential Calculi

on Quantum Vector Spaces 46712.3.3 Covariant Differential Calculus on C ^

and the Quantum Weyl Algebra 46812.3.4 Covariant Differential Calculi

on the Quantum Hyperboloid 47112.4 Notes 472

13. Hopf Bimodules and Exterior Algebras 47313.1 Covariant Bimodules 473

13.1.1 Left-Covariant Bimodules 47313.1.2 Right-Covariant Bimodules 47713.1.3 Bicovariant Bimodules (Hopf Bimodules) 477

XVIII Table of Contents

13.1.4 Woronowicz' Braiding of Bicovariant Bimodules 48013.1.5 Bicovariant Bimodules and Representations

of the Quantum Double 48313.2 Tensor Algebras and Exterior Algebras

of Bicovariant Bimodules . . . .. 48513.2.1 The Tensor Algebra of a Bicovariant Bimodule 48513.2.2 The Exterior Algebra of a Bicovariant Bimodule 488

13.3 Notes 490

14. Covariant Differential Calculus on Quantum Groups 49114.1 Left-Covariant First Order Differential Calculi 491

14.1.1 Left-Covariant First Order Calculiand Their Right Ideals 491

14.1.2 The Quantum Tangent Space 49414.1.3 An Example: The 3D-Calculus on SLq(2) 49614.1.4 Another Left-Covariant Differential Calculus

on SLq(2) 49814.2 Bicovariant First Order Differential Calculi 498

14.2.1 Right-Covariant First Order Differential Calculi 49814.2.2 Bicovariant First Order Differential Calculi 49914.2.3 Quantum Lie Algebras

of Bicovariant First Order Calculi 50014.2.4 The 4D+- and the 4D_-Calculus on SLq{2) 50414.2.5 Examples of Bicovariant First Order Calculi

on Simple Lie Groups 50514.3 Higher Order Left-Covariant Differential Calculi 506

14.3.1 The Maurer-Cartan Formula 50614.3.2 The Differential Envelope of a Hopf Algebra 50714.3.3 The Universal DC of a Left-Covariant FODC 508

14.4 Higher Order Bicovariant Differential Calculi 51114.4.1 Bicovariant Differential;Calculi

and Differential Hopf Algebras 51114.4.2 Quantum Lie Derivatives and Contraction Operators. . 514

14.5 Bicovariant Differential Calculion Coquasitriangular Hopf Algebras 517

14.6 Bicovariant Differential Calculion Quantized Simple Lie Groups .\ 52114.6.1 A Family of Bicovariant First Order

Differential Calculi .52114.6.2 Braiding and Structure Constants of the FODC F±:Z . 52414.6.3 A Canonical Basis for the Left-Invariant 1-Forms 52514.6.4 Classification of Bicovariant First Order

Differential Calculi 52714.7 Notes 528

Table of Contents XIX

Bibliography 529

Index 545