Quantum Groups and Noncommutative Spaces: Perspectives on Quantum Geometry

✍ Scribed by Matilde Marcolli, Deepak Parashar

• Publisher: Vieweg+Teubner Verlag
• Year: 2010
• Tongue: English
• Leaves: 250
• Edition: 2011
• Category: Library

✦ Synopsis

This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.

✦ Table of Contents

Cover......Page 1
Quantum Groups
and Noncommutative
Spaces......Page 3
ISBN 9783834814425......Page 4
Contents......Page 6
Preface......Page 8
2. Contramodules......Page 11
3. Anti-Yetter-Drinfeld contramodules......Page 12
4. Hopf-cyclic homology of module coalgebras......Page 14
5. Hopf-cyclic cohomology of module algebras......Page 15
6. Anti-Yetter-Drinfeld contramodules and hom-connections......Page 16
References......Page 18
1. Introduction......Page 19
2. Preliminaries and Definitions......Page 21
3. Bimodules and Multiplicity Matrices......Page 30
4. Dirac Operators and their Structure......Page 45
5. Applications to the Recent Work of Chamseddine and Connes......Page 64
References......Page 77
1. Introduction......Page 79
2. Basic definitions......Page 80
3. Summary and observations on results by Berele and Regev......Page 84
4. Tensor representations of the general linear supergroup......Page 86
References......Page 88
1. Introduction......Page 90
2. The Poisson Lie group GLn(k) and its quantum deformation......Page 93
3. The quantum Grassmannian and its big cell......Page 95
4. The Quantum Duality Principle for quantum Grassmannians......Page 98
References......Page 105
1. Introduction......Page 106
2. Preliminaries......Page 107
3. C∗-action of QISO+R(D)......Page 109
References......Page 112
1. Introduction......Page 114
2. Principal fiber bundles and Hopf Galois extensions......Page 116
3. Twisted algebras......Page 117
4. The generic cocycle......Page 119
5. The Sweedler algebra......Page 121
6. The generic Galois extension......Page 123
7. The integrality condition......Page 124
8. How to construct elements of BαH......Page 127
References......Page 130
1. Introduction......Page 131
2. Preliminaries......Page 132
3. Quantization......Page 134
4. Quantization of ΩH......Page 136
References......Page 139
1. Introduction......Page 140
3. Compact and discrete quantum groups......Page 143
4. Algebraic quantum groups......Page 145
5. Locally compact quantum groups......Page 149
References......Page 154
1. Introduction......Page 156
2. Algebras......Page 157
3. Category of A-modules......Page 158
4. Coalgebras and comodules......Page 161
5. Bialgebras and Hopf algebras......Page 164
6. General categories......Page 167
References......Page 173
1. Introduction......Page 174
2. The classical Hopf bundle......Page 176
3. The quantum principal Hopf bundle......Page 189
4. A -Hodge duality on Ω(SUq(2)) and a Laplacian on SUq(2)......Page 208
5. A -Hodge structure on Ω(S2q) and a Laplacian operator on A(S2q)......Page 214
6. Connections on the Hopf bundle......Page 219
7. A gauged Laplacian on the quantum Hopf bundle......Page 230
8. An algebraic formulation of the classical Hopf bundle......Page 233
9. Back on a covariant derivative on the exterior algebra Ω(SUq(2))......Page 244
References......Page 248