Locally Compact Quantum Groups and Groupoids

These are the proceedings of the workshop “Quantum Groups, Hopf Algebras and their Applications”, which was held in Strasbourg in February 2002. They contain an “Introduction of the editor” and 7 original articles. All of them represent substantial contributions to the theory of quantum groups and groupoids, and every specialist in the field should be familiar with them. The introduction describes the development of the theory of quantum groups and groupoids in a very concise and deep way, so that even a mathematician who is not a specialist on this topic can adequately understand why the theory has proceeded in this way or that. For the non-specialist, let us also mention that the paper by S. Vaes and L. Vainerman (On low dimensional locally compact quantum groups) contains Preliminaries, where we can find basic information about these groups. This paper represents the continuation of the research of both authors on extensions of locally compact quantum groups. J. Kustermans and E. Koelink (Quantum SUq ~(1,1) and its Pontryagin dual), present an overview of the quantum group SUq~ (1,1) and study its Pontryagin dual. A. Van Daele, in his paper (Multiplier Hopf*-algebras with positive integrals: A laboratory for locally compact quantum groups), gives a survey of the theory of algebraic quantum groups and their relations with locally compact quantum groups. The remaining four papers deal with quantum groupoids (M. Enock: Quantum groupoids and pseudo-multiplicative unitaries; P. Schauenberg: Morita base change in quantum groupoids; K. Szlachányi: Galois actions by finite quantum groupoids; J.-M. Vallin: Multiplicative partial isometries and finite quantum groupoids).

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