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

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*Thomas Timmermann*

A publication of the European Mathematical Society

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.

#### Readership

Graduate students and research mathematicians interested in quantum groups.