Quantum families of quantum group homomorphisms

TL;DR

This paper introduces the concept of quantum families of homomorphisms for locally compact quantum groups and Hopf algebras, demonstrating that such families are essentially classical and exploring their algebraic properties and relations.

Contribution

It defines quantum families of homomorphisms in both operator algebra and algebraic frameworks, proving their classical nature and analyzing their structural properties.

Findings

Quantum families of homomorphisms are classical in nature.

The algebraic counterpart of quantum families preserves counits and coinverses.

The concept aligns with weak coactions and the adjoint coaction analysis.

Abstract

The notion of a 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 paper we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.