An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond

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European Mathematical Society, 2008 - Mathematics - 407 pages
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This book provides an introduction to the theory of quantum groups with emphasis on their duality and on the setting of operator algebras. Part I of the text presents the basic theory of Hopf algebras, Van Daele's duality theory of algebraic quantum groups, and Woronowicz's compact quantum groups, staying in a purely algebraic setting. Part II focuses on quantum groups in the setting of operator algebras. Woronowicz's compact quantum groups are treated in the setting of $C *$-algebras, and the fundamental multiplicative unitaries of Baaj and Skandalis are studied in detail. An outline of Kustermans' and Vaes' comprehensive theory of locally compact quantum groups completes this part. Part III leads to selected topics, such as coactions, Baaj-Skandalis-duality, and approaches to quantum groupoids in the setting of operator algebras. The book is addressed to graduate students and non-experts from other fields. Only basic knowledge of (multi-) linear algebra is required for the first part, while the second and third part assume some familiarity with Hilbert spaces, $C *$-algebras, and von Neumann algebras
 

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Contents

Hopf algebras
3
Multiplier Hopf algebras and their duality
40
Algebraic compact quantum groups
65
Quantum groups and Cvon Neumann bialgebras
95
Examples of compact quantum groups
135
Multiplicative unitaries
166
Locally compact quantum groups
203
Coactions on C algebras reduced crossed products and duality
251
Pseudomultiplicative unitaries on Hilbert spaces
289
Pseudomultiplicative unitaries on Cmodules
328
Appendix
369
Bibliography
385
Symbol Index
397
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