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An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond
Thomas Timmermann, University of Münster, Mathematisches Institut, Germany
A publication of the European Mathematical Society.
EMS Textbooks in Mathematics
2008; 427 pp; hardcover
Volume: 5
ISBN-10: 3-03719-043-4
ISBN-13: 978-3-03719-043-2
List Price: US$78
Member Price: US$62.40
Order Code: EMSTEXT/5
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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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.


Graduate students and research mathematicians interested in quantum groups.

Table of Contents

Part I. From groups to quantum groups
  • Hopf algebras
  • Multiplier Hopf algebras and their duality
  • Algebraic compact quantum groups
Part II. Quantum groups and \(C^*\)-/von Neumann bialgebras
  • First definitions and examples
  • \(C^*\)-algebraic compact quantum groups
  • Examples of compact quantum groups
  • Multiplicative unitaries
  • Locally compact quantum groups
Part III. Selected topics
  • Coactions on \(C^*\)-algebras, reduced crossed products, and duality
  • Pseudo-multiplicative unitaries on Hilbert spaces
  • Pseudo-multiplicative unitaries on \(C^*\)-modules
  • Appendix
  • Bibliography
  • Symbol Index
  • Index
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