Algebry operatorów i ich zastosowania w fizyce

The notion of quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this talk I will define a quantum family of homomorphisms of locally compact quantum groups. I will show that, roughly speaking, such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras will be introduced and the algebraic counterpart of the aforementioned result will be proved. If time permits I will discuss the cases when the quantum group of inner automorphisms of a locally compact quantum group G acts on the dual of G by classical automorphisms. Joint work with Mariusz Budziński.