Quantum families of invertible maps and related problems

TL;DR

This paper explores quantum families of invertible maps, strengthening the understanding of universal quantum groups acting on finite quantum spaces and providing new analytic frameworks for their subgroups.

Contribution

It introduces a new perspective on quantum families of invertible maps, showing Wang's quantum automorphism groups are universal, and reformulates the Hopf image construction analytically.

Findings

Wang's quantum automorphism groups are universal for quantum families of invertible maps.

The Hopf image construction is reformulated in an analytic language.

Defines quantum subgroups generated by families of quantum subgroups or invertible maps.

Abstract

The notion of families of quantum invertible maps ($C^*$-algebra homomorphisms satisfying Podle\'s condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular Wang's quantum automorphism groups are shown to be universal with respect to quantum families of invertible maps. Further the construction of the Hopf image of Banica and Bichon is phrased in the purely analytic language and employed to define the quantum subgroup generated by a family of quantum subgroups or more generally a family of quantum invertible maps.