Contents. Introduction. Part I From groups to quantum groups 1

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1 Preface Introduction vii xv Part I From groups to quantum groups 1 1 Hopf algebras Motivation: Pontrjagin duality The concept of a Hopf algebra Definition Examples related to groups Axiomatics of Hopf algebras Coalgebras and bialgebras Convolution Properties of the antipode Another characterization of Hopf algebras Hopf -algebras The duality of Hopf algebras The duality of finite-dimensional Hopf algebras Dual pairings of Hopf algebras The restricted dual of a Hopf algebra Multiplier Hopf algebras and their duality Definition of multiplier Hopf algebras Multipliers of algebras Multiplier bialgebras Multiplier Hopf algebras Integrals and their modular properties The concept of an integral Existence and uniqueness The modular element of an integral The modular automorphism of an integral Duality The duality of regular multiplier Hopf algebras The duality of algebraic quantum groups... 63

2 x 3 Algebraic compact quantum groups Corepresentations of Hopf -algebras Definition and examples Reformulation of the concept of a corepresentation Construction of new corepresentations Corepresentation theory and structure theory Decomposition into irreducible corepresentations Schur s orthogonality relations Characterization of compact quantum groups Characters of corepresentations Modular properties of the Haar state Discrete algebraic quantum groups Part II Quantum groups and C -/von Neumann bialgebras 95 4 First definitions and examples C -bialgebras and von Neumann bialgebras Bialgebras associated to groups Approaches to quantum groups in the setting of von Neumann algebras and C -algebras C -algebraic compact quantum groups Definition and examples Corepresentations of C -bialgebras Unitary corepresentations of C -algebraic compact quantum groups Corepresentation operators of C -bialgebras Constructions related to corepresentation operators Corepresentation theory and structure theory Decomposition into irreducible corepresentations Schur s orthogonality relations Characterization of C -algebraic compact quantum groups The relation to algebraic compact quantum groups From C -algebraic to algebraic CQGs From algebraic to C -algebraic CQGs Examples of compact quantum groups Compact matrix quantum groups The compact quantum group SU.2/ Definition and first properties Corepresentations and their weights Corepresentations and differential calculi

3 xi Modular properties of the Haar state Products of compact quantum groups The free unitary and the free orthogonal quantum groups Multiplicative unitaries The concept of a multiplicative unitary Motivation Definition and examples The legs of a multiplicative unitary Definition and first properties Well-behaved multiplicative unitaries Examples The dual pairing, counit, and antipode of the legs Classes of well-behaved multiplicative unitaries Regular multiplicative unitaries Manageable and modular multiplicative unitaries Locally compact quantum groups The concept of a locally compact quantum group Weights Locally compact quantum groups in the setting of von Neumann algebras The modular automorphism group of a weight Reduced C -algebraic quantum groups Additional prerequisites Main properties The multiplicative unitary The antipode and modular properties The duality of locally compact quantum groups Passage between the different levels Examples of locally compact quantum groups C -algebras generated by unbounded elements The quantum groups E.2/ and ye.2/ The quantum az C b group Part III Selected topics Coactions on C -algebras, reduced crossed products, and duality Actions of groups and Takesaki Takai duality Coactions of C -bialgebras on C -algebras Weak Kac systems

4 xii Balanced multiplicative unitaries Weak Kac systems Examples of weak Kac systems Reduced crossed products and dual coactions The reduced crossed product of a coaction of A.V / The dual coaction of a coaction of A.V / The dual coaction of a coaction of ya.v / Comparison with the reduced crossed product of an action Kac systems and the Baaj Skandalis duality theorem Kac systems The Baaj Skandalis duality theorem Pseudo-multiplicative unitaries on Hilbert spaces The relative tensor product of Hilbert modules Hilbert modules over von Neumann algebras Outline of the construction Bounded elements of a Hilbert module Construction of the relative tensor product Properties of the relative tensor product Hopf von Neumann bimodules The fiber product of von Neumann algebras Hopf von Neumann bimodules Pseudo-multiplicative unitaries on Hilbert spaces Definition The legs of a pseudo-multiplicative unitary The pseudo-multiplicative unitary of a groupoid Pseudo-multiplicative unitaries on C -modules Pseudo-multiplicative unitaries on C -modules The flipped internal tensor product of C -modules Definition and examples Obstructions to the construction of the legs Semigroup grading techniques on right C -bimodules Homogeneous operators and C -families Homogeneous elements of right C -bimodules Examples related to groupoids Hopf C -families The internal tensor product of C -families Morphisms of C -families Hopf C -families The legs of a decomposable pseudo-multiplicative unitary Coactions of Hopf C -families

5 xiii 12 Appendix C -algebras C -modules Von Neumann algebras Slice maps Auxiliary results Bibliography 385 Symbol Index 397 Index 401