Quantum Groups

by Vladimir K. Dobrev

Overview

With applications in quantum field theory, general relativity and elementary particle physics, this three-volume work studies the invariance of differential operators under Lie algebras, quantum groups and superalgebras. This second volume covers quantum groups in their two main manifestations: quantum algebras and matrix quantum groups. The exposition covers both the general aspects of these and a great variety of concrete explicitly presented examples. The invariant q-difference operators are introduced mainly using representations of quantum algebras on their dual matrix quantum groups as carrier spaces. This is the first book that covers the title matter applied to quantum groups.

Contents
Quantum Groups and Quantum Algebras
Highest-Weight Modules over Quantum Algebras
Positive-Energy Representations of Noncompact Quantum Algebras
Duality for Quantum Groups
Invariant q-Difference Operators
Invariant q-Difference Operators Related to GLq(n)
q-Maxwell Equations Hierarchies

Product Details

ISBN-13:	9783110427783
Publisher:	De Gruyter
Publication date:	07/10/2017
Series:	De Gruyter Studies in Mathematical Physics , #39
Sold by:	Barnes & Noble
Format:	eBook
Pages:	406
File size:	19 MB
Note:	This product may take a few minutes to download.
Age Range:	18 Years

About the Author

Vladimir K. Dobrev, Bulgarian Academy of Sciences, Bulgaria.

Table of Contents

1 Quantum Groups and Quantum Algebras 1

1.1 Hopf Algebras and Quantum Groups 1

1.2 Quantum Algebras 4

1.2.1 Drinfeld's Definition 4

1.2.2 Universal R-Matrix and Casimirs 7

1.2.3 Jimbo's Definition 11

1.3 Drinfeld Second Realization of Quantum Affine Algebras 14

1.4 Drinfeld's Realizations of Yangians 17

1.4.1 The First Drinfeld Realization of Yangians 17

1.4.2 The Second Drinfeld Realization of Yangians 18

1.5 q-Deformations of Noncompact Lie Algebras 19

1.5.1 Preliminaries 19

1.5.2 q-Deformation of the Real Forms 21

1.5.3 Example so(p,r) 25

1.5.4 Example so(2,1) 26

1.5.5 q-Deformed Lorentz Algebra U_q(so(3,1)) 26

1.5.6 q-Deformed Real Forms of so(5) 27

1.5.7 q-Deformed de Sitter Algebra so(4,1) 27

1.5.8 q-Deformed Anti de Sitter Algebra so(3,2) 28

1.5.9 q-Deformed Algebras U_q(sl(4,C)) and U_q(su(2,2)) 28

1.5.10 q-deformed Poincaré and Weyl Algebras 32

2 Highest-Weight Modules over Quantum Algebras 34

2.1 Verma Modules, Singular Vectors, and Irreducible Subquotients 34

2.2 q-Fock Type Representations 37

2.3 Vertex Operators 39

2.4 Singular Vectors in Chevalley Basis 40

2.4.1 U_q(A_e) 42

2.4.2 U_q(D_e) 43

2.4.3 U_q(E_e) 43

2.4.4 U_q(B_e) 44

2.4.5 U_q(C_e) 44

2.4.6 U_q(F₄) 45

2.4.7 U_q(G₂) 46

2.5 Singular Vectors in Poincaré-Birkhoff-Witt Basis 46

2.5.1 PBW Basis 46

2.5.2 Singular Vectors for U_q(A_e) in PBW Basis 47

2.5.3 Singular Vectors for U_q(D_e) in PBW Basis 51

2.6 Singular Vectors for Nonstraight Roots 56

2.6.1 Bernstein-Gel'fand-Gel'fand Resolution 56

2.6.2 Case of U_q(D_e) in PBW Basis 61

2.6.3 Case of U_q(De) in the Simple Roots Basis 63

2.7 Representations at Roots of Unity 64

2.7.1 Generalities 64

2.7.2 The Example of U_q(sl(2)) at Roots of Unity 67

2.7.3 Classification in the U_q(sl(3, C)) Case 68

2.7.4 Cyclic Representations of U_q(G) 76

2.8 Characters of Irreducible HWMs 79

2.8.1 Generalities 79

2.8.2 U_q(sl(3,C)) 80

2.8.3 Uq(sl(3, C)) at Roots of Unity 81

2.8.4 Conjectures 83

3 Positive-Energy Representations of Noncompact Quantum Algebras 86

3.1 Preliminaries 86

3.2 Quantum Anti de Sitter Algebra 87

3.2.1 Representations 87

3.2.2 Roots of Unity Case 91

3.2.3 Character Formulae 95

3.3 Conformal Quantum Algebra 97

3.3.1 Generic Case 97

3.3.2 Roots of 1 Case 100

3.3.3 Massless Case 102

3.3.4 Character Formulae 104

4 Duality for Quantum Groups 107

4.1 Matrix Quantum Groups 107

4.1.1 Differential Calculus on Quantum Planes 111

4.2 Duality between Hopf Algebras 113

4.3 Matrix Quantum Group GL_p,q(2) 113

4.4 Duality for GL_p,q(2) 115

4.5 Duality for Multiparameter Quantum GL(n) 119

4.5.1 Multiparameter Deformation of GL(n) 120

4.5.2 Commutation Relations of the Dual Algebra 122

4.5.3 Hopf Algebra Structure of the Dual Algebra 125

4.5.4 Drinfeld-Jimbo Form of the Dual Algebra 128

4.5.5 Special Cases of Hopf Algebra Splitting 130

4.6 Duality for a Lorentz Quantum Group 131

4.6.1 Matrix Lorentz Quantum Group 131

4.6.2 Dual Algebras to the Algebras L_q and L_g 133

4.6.3 Coalgebra Structure of the Dual Algebras 137

4.7 Duality for the Jordanian Matrix Quantum Group GL_g,h(2) 139

4.7.1 Jordanian Matrix Quantum Group GL_g,h(2) 140

4.7.2 The Dual of GL_g,h(2) 141

4.7.3 Algebra Structure of the Dual 143

4.7.4 Coalgebra Structure of the Dual 146

4.7.5 One-Parameter Cases 148

4.7.6 Application of a Nonlinear Map 149

4.8 Duality for Exotic Bialgebras 151

4.8.1 Exotic Bialgebras: General Setting 151

4.8.2 Exotic Bialgebras: Triangular Case 1 152

4.8.3 Exotic Bialgebras: Triangular Case 2 158

4.8.4 Exotic Bialgebras: Triangular Case 3 161

4.8.5 Higher-Order R-matrix Relations and Quantum Planes 164

4.8.6 Exotic Bialgebras: Nontriangular Case SO3 167

4.8.7 Exotic Bialgebras: Nontriangular Case S14 176

4.8.8 Exotic Bialgebras: Nontriangular Case S14o 183

4.8.9 Exotic Bialgebras: Higher Dimensions 187

Conclusions and Outlook 198

5 Invariant q-Difference Operators 199

5.1 The Case of GL_p,q(2) 199

5.1.1 Left and Right Action of U_p,q(gl(2)) on GL_p,q(2) 199

5.1.2 Induced Representations of U_p,q and Intertwining Operators 203

5.1.3 The Case U_q(sl(2)) 208

5.2 The Case of GL_g,h(2) 209

5.2.1 Left and Right Action of U_g,h(gl(2)) on GL_g,h(2) 209

5.2.2 Induced Representations of U_g,h and Intertwining Operators 212

5.2.3 Representations of the Jordanian Algebra U_h(sl(2)) 218

5.2.4 Highest-Weight Modules over U_h(sl(2)) 219

5.2.5 Singular Vectors of U_h(sl(2)) Verma Modules 221

5.3 q-Difference Intertwining Operators for a Lorentz Quantum Algebra 223

5.3.1 A Matrix Lorentz Quantum Group 223

5.3.2 The Lorentz Quantum Algebra 224

5.3.3 Representations of the Lorentz Quantum Algebra 226

5.3.4 q-Difference Intertwining Operators 234

5.3.5 Classification of Reducible Representations 235

5.3.6 The Roots of Unity Case 236

5.4 Representations of the Generalized Lie Algebra sl(2)_q 238

5.4.1 Preliminaries 238

5.4.2 The Quantum Lie Algebra sl(2)_q 239

5.4.3 Highest-Weight Representations 240

5.4.4 Highest-Weight Representations of the Restricted Algebra 244

5.4.5 Highest-Weight Representations at Roots of Unity 245

5.4.6 Highest-Weight Representations at Roots of Unity of the Restricted Algebra 248

5.5 Representations of U_q(so(3)) of Integer Spin Only 250

5.5.1 Preliminaries 250

5.5.2 Matrix Quantum Group SO_q (3) and the Dual U_q(G) 250

5.5.3 Representations of U_q(so(3)) 253

6 Invariant q-Difference Operators Related to GL_q(n) 261

6.1 Representations Related to GL_q(n) 261

6.1.1 Actions of U_q(gl(n)) and U_q(sl(n)) 263

6.1.2 Representation Spaces 266

6.1.3 Reducibility and Partial Equivalence 272

6.2 The Case of U_q(sl(3)) 274

6.3 Polynomial Solutions of q-Difference Equations in Commuting Variables 280

6.3.1 Procedure For the Construction of the Representations 281

6.3.2 Reducibility of the Representations and Invariant Subspaces 284

6.3.3 Newton Diagrams 290

6.4 Application of the Gelfand-(Weyl)-Zetlin Basis 292

6.4.1 Correspondence with the GWZ Basis 292

6.4.2 q-Hypergeometric Realization of the GWZ Basis 296

6.4.3 Explicit Orthogonality of the GWZ Basis 300

6.4.4 Normalized GWZ basis 303

6.4.5 Scalar Product and Normalized GWZ States 307

6.4.6 Summation Formulae 309

6.4.7 Weight Pyramid of the SU(3) UIRs 311

6.4.8 The Irregular Irreps in Terms of GWZ States 316

6.5 The Case of U_q(sl(4)) 321

6.5.1 Elementary Representations 321

6.5.2 Intertwining Operators 324

7 q-Maxwell Equations Hierarchies 327

7.1 Maxwell Equations Hierarchy 327

7.2 Quantum Minkowski Space-Time 331

7.2.1 q-Minkowski Space-Time 331

7.2.2 Multiparameter Quantum Minkowski Space-Time-333

7.3 q-Maxwell Equations Hierarchy 335

7.4 q-d'Alembert Equations Hierarchy 340

7.4.1 Solutions of the q-d'Alembert Equation 340

7.4.2 q-Plane-Wave Solutions 341

7.4.3 q-Plane-Wave Solutions for Non-Zero Spin 344

7.5 q-Plane-Wave Solutions of the Potential q-Maxwell Hierarchy 347

7.6 q-Plane-Wave Solutions of the Full q-Maxwell Equations 350

7.7 q-Weyl Gravity Equations Hierarchy 354

7.7.1 Linear Conformal Gravity 355

7.7.2 Plane-Wave Solutions of g-Weyl Gravity 358

Bibliography 361

Author Index 391

Subject Index 392