Quantum Isometry Groups

Description

This book offers an up-to-date overview of the recently proposed theory of quantum isometry groups. Written by the founders, it is the first book to present the research on the “quantum isometry group”, highlighting the interaction of noncommutative geometry and quantum groups, which is a noncommutative generalization of the notion of group of isometry of a classical Riemannian manifold. The motivation for this generalization is the importance of isometry groups in both mathematics and physics. The framework consists of Alain Connes’ “noncommutative geometry” and the operator-algebraic theory of “quantum groups”. The authors prove the existence of quantum isometry group for noncommutative manifolds given by spectral triples under mild conditions and discuss a number of methods for computing them. One of the most striking and profound findings is the non-existence of non-classical quantum isometry groups for arbitrary classical connected compact manifolds and, by using this, the authors explicitly describe quantum isometry groups of most of the noncommutative manifolds studied in the literature. Some physical motivations and possible applications are also discussed. Front Matter....Pages i-xxviii Preliminaries....Pages 1-35 Classical and Noncommutative Geometry....Pages 37-67 Definition and Existence of Quantum Isometry Groups....Pages 69-95 Quantum Isometry Groups of Classical and Quantum Spheres....Pages 97-127 Quantum Isometry Groups of Discrete Quantum Spaces....Pages 129-147 Nonexistence of Genuine Smooth CQG Coactions on Classical Connected Manifolds....Pages 149-162 Deformation of Spectral Triples and Their Quantum Isometry Groups....Pages 163-177 Spectral Triples and Quantum Isometry Groups on Group \(C^{*}\) -Algebras....Pages 179-198 An Example of Physical Interest....Pages 199-219 More Examples and Open Questions....Pages 221-235