An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond
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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Multiplier Hopf algebras and their duality
Algebraic compact quantum groups
Quantum groups and Cvon Neumann bialgebras
Examples of compact quantum groups
Locally compact quantum groups
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algebraic compact quantum algebraic quantum group antipode associated AW(V bialgebra bijective bimodule C*-algebraic compact quantum C*-algebraic quantum group C*-family C*-module Co(G coaction coalgebra commutes compact matrix quantum compact quantum group comultiplication consider construction corepresentation matrix corepresentation operator Corollary counit crossed product defined Definition denote elements Example exists a unique finite finite-dimensional follows function given group G groupoid Haar measure Hilbert modules Hilbert N-module Hilbert space Ik(G internal tensor product irreducible corepresentation isomorphism left Haar Lemma linear map linearly dense locally compact group locally compact quantum matrix quantum group module morphism multiplier Hopf algebra n.s.f. weight Neumann algebras notation PAut(B Proof Proposition pseudo-multiplicative unitary reduced crossed product relation relative tensor product Remark Rep(G representation respectively S. L. Woronowicz satisfies Section space H subspace Theorem theory unitary corepresentation V-module vector space von Neumann algebras weak Kac system well-behaved Woronowicz x e G