Quantum Isometry Groups [electronic resource] / by Debashish Goswami, Jyotishman Bhowmick.

Format:

Book

Author/Creator:

Goswami, Debashish., Author.

Bhowmick, Jyotishman., Author.

Series:

Infosys Science Foundation Series in Mathematical Sciences, 2364-4036

Language:

English

Subjects (All):

Global analysis (Mathematics).

Manifolds (Mathematics).

Mathematical physics.

Geometry, Differential.

Functional analysis.

Quantum theory.

Global Analysis and Analysis on Manifolds.

Mathematical Physics.

Differential Geometry.

Functional Analysis.

Quantum Physics.

Local Subjects:

Global Analysis and Analysis on Manifolds.

Mathematical Physics.

Differential Geometry.

Functional Analysis.

Quantum Physics.

Physical Description:

1 online resource (254 pages).

Edition:

1st ed. 2016.

Place of Publication:

New Delhi : Springer India : Imprint: Springer, 2016.

Summary:

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.

Contents:

Chapter 1. Introduction

Chapter 2. Preliminaries

Chapter 3. Classical and Noncommutative Geometry

Chapter 4. Definition and Existence of Quantum Isometry Groups

Chapter 5. Quantum Isometry Groups of Classical and Quantum

Chapter 6. Quantum Isometry Groups of Discrete Quantum Spaces

Chapter 7. Nonexistence of Genuine Smooth CQG Actions on Classical Connected Manifolds

Chapter 8. Deformation of Spectral Triples and Their Quantum Isometry Groups

Chapter 9. More Examples and Computations

Chapter 10. Spectral Triples and Quantum Isometry Groups on Group C*-Algebras.

Notes:

Includes bibliographical references at the end of each chapters.