Quantum Isometry Groups

By: Debashish Goswami, Jyotishman Bhowmick

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Paperback | 30 April 2018

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264 Pages

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23.5 x 15.5 x 1.4

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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.

Industry Reviews

"Several examples and applications are provided, including an explicit description of the quantum isometry groups of most of the noncommutative Riemannian manifolds studied in literature. Introductory sections on the basics of Riemannian geometry noncommutative geometry and quantum groups make the book enjoyable to readers with no previous expertise in the subject." (Domenico Fiorenza, Mathematical Reviews, April 2018)

"The book gives an up-to date of the results of an analogue of the group of isometries in the framework of noncommutative geometry and quantum groups. A unique feature of this book is the emphasis on the interaction of C-star algebraic compact quantum groups ... .The book is an interesting book for the researchers working in this area which is still 'hot' enough. The physical motivations and possible applications given in this book are more interesting examples." (Ahmed Hegazi, zbMATH 1381.81004, 2018)

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