Hopf images in locally compact quantum groups

TL;DR

This paper investigates the concept of Hopf images in locally compact quantum groups, extending previous theories from compact and discrete cases to a broader non-compact setting, with new characterizations and insights.

Contribution

It provides a unified framework for Hopf images in locally compact quantum groups and introduces new characterizations of fullness using partial actions and representation theory.

Findings

Extended Hopf image theory to non-compact quantum groups.

Characterized fullness of Hopf images via partial actions.

Connected Hopf images with representation $C^*$-categories.

Abstract

This manuscript is devoted to the study of the concept of a generating subset (a.k.a. Hopf image of a morphism) in the setting of locally compact quantum groups. The aim of this paper is to provide an accurate description of the Hopf image of a given morphism. We extend and unify the previously existing approaches for compact and discrete quantum groups and present some results that can shed light on some local perspective in the theory of quantum groups. In particular, we provide a characterization of fullness of Hopf image in the language of partial actions as well as in representation-theoretic terms in the spirit of representation $C^*$-categories, extending some known results not only to the broader setting of non-compact quantum groups, but also encompassing a broader setting of generating subsets.