Quantum Groups and Noncommutative Spaces : Perspectives on Quantum Geometry ; [the present volume is based on an activity organized at the Max Planck Institute for Mathematics in Bonn, during the days August 6-8, 2007, dedicated to the topic of Quantum Gr

In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory. This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system - harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincaré symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest 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 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. Contributions byTomasz Brzezinski, Branimir Cacic, Rita Fioresi, Rita Fioresi and Fabio Gavarini, Debashish Goswami, Christian Kassel, Avijit Mukherjee, Alfons Van Daele, Robert Wisbauer, Alessandro Zampini The volume is aimed as introducing techniques and results on Quantum Groups and Noncommutative Geometry, in a form that is accessible to other researchers in related areas as well as to advanced graduate students. The topics covered are of interest to both mathematicians and theoretical physicists. Prof. Dr. Matilde Marcolli, Department of Mathematics, California Institute of Technology, Pasadena, California, USA. Dr. Deepak Parashar, Cambridge Cancer Trials Centre and MRC Biostatistics Unit, University of Cambridge, United Kingdom.