Algebry operatorów i ich zastosowania w fizyce

In the theory of classical groups, given a (closed) subset of a group one may ask about the (closed) subgroup it generates. The quantum version of this generation procedure is what was called Hopf Image by Banica and Bichon in the context of Hopf algebras. The aim of this talk is to present a uniform method for constructing a closed quantum subgroup generated by a "quantum subset" in the context of locally compact quantum groups. The method we will present is (essentially) consistent with previous constructions: the one for CQG algebras due to Banica and Bichon, the one for compact quantum groups due to Skalski and Sołtan, and with the notion of (finite) generating set of a discrete quantum group due to Vergnioux. We will sketch some of the properties and applications of this construction and, hopefully, present some examples.