Quantum bounded symmetric domains /

by Vaksman, L. L. , Kvantovye ogranichennye simmetricheskie oblasti. Series: Translations of mathematical monographs ; . v. 238 Published by : American Mathematical Society, (Providence, R.I. :) Physical details: xi, 256 p. : ill. ; 27 cm. ISBN:9780821849095; 0821849093 (alk. paper). Year: 2010
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Includes bibliographical references and index.

Ch. 1. Quantum Disc -- 1.1. A caution to pedants -- 1.2. Topology -- 1.3. Symmetry -- 1.4. An invariant integral -- 1.5. Differential calculi -- 1.6. Integral representations -- 1.7. On kernels of intertwining integral operators -- Ch. 2. Basic quantum theory of bounded symmetric domains -- 2.1. Summary on quantum universal enveloping algebras -- 2.2. Summary on algebras of functions on compact quantum groups -- 2.3. Quantum vector spaces and Harish-Chandra modules -- 2.4. Spaces of functions in quantum bounded symmetric domains -- 2.5. The canonical embedding -- 2.6. Covariant differential calculi and invariant differential operators -- Ch. 3. Conclusion -- 3.1. Boundaries and spherical principal series -- 3.2. The analytic continuation of holomorphic discrete series and the Penrose transform.

This book provides exposition of the basic theory of quantum bounded symmetric domains. The area became active in the late 1990s at a junction of noncommutative complex analysis and extensively developing theory of quantum groups. It is well known that the classical bounded symmetric domains involve a large number of nice constructions and results of the theory of C*-algebras, theory of functions and functional analysis, representation theory of real reductive Lie groups, harmonic analysis, and special functions. In a surprising advance of the theory of quantum bounded symmetric domains, it turned out that many classical problems admit elegant quantum analogs. Some of those are expounded in the book. Anyone with an interest in the subject will welcome this unique treatment of quantum groups. Table of Contents: Quantum disc; Basic quantum theory of bounded symmetric domains; Conclusion; Bibliography; Index. About the book: The book is written by a leading expert in a very clear, careful, and stimulating way. I strongly recommend it to graduate students and research mathematicians interested in noncommutative geometry, quantum groups, C*-algebras, or operator theory.

Translated from the Russian.

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