Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Banach Center Publications / Wszystkie tomy

Streszczenie

In these notes, we study $C^*$-algebraic quantum groups and quantum groupoids by viewing them as quantized objects of classical locally compact groups and groupoids, as well as frameworks to extend the Pontryagin duality. The discussions are given primarily as a survey of background and known results, and a summary of more recent results. The early sections discuss the bridge between the classical setting and the quantum setting, through deformation quantization of Poisson structures and non-commutative geometric motivations. A survey of results on quantum groups follows, starting from Hopf algebras and compact quantum groups to general locally compact quantum groups. Then finally, we turn to the study of still on-going attempts at developing a theory of quantum groupoids. We describe the purely algebraic theory of weak multiplier Hopf algebras by Van Daele and Wang, then the framework of $C^*$-algebraic quantum groupoids of separable type by Van Daele and the author. Current framework is rather limited, however, and some discussions on formulating a fuller general framework are briefly given.