# Topological automorphism groups of compact quantum groups

## TL;DR

This paper investigates the topological structure of automorphism groups of compact quantum groups, revealing parallels with classical results, characterizing inner automorphisms, and constructing examples with non-finitely generated fusion rings.

## Contribution

It establishes the topological properties of automorphism groups of compact quantum groups, including the structure of inner and outer automorphisms, and provides a novel example of a quantum group with a non-finitely generated fusion ring.

## Key findings

- Connected components of automorphism groups coincide with inner automorphisms.
- Inner automorphism groups of compact matrix quantum groups are compact Lie groups.
- Constructed a compact matrix quantum group with a non-finitely generated fusion ring.

## Abstract

We study the topological structure of the automorphism groups of compact quantum groups showing that, in parallel to a classical result due to Iwasawa, the connected component of identity of the automorphism group and of the "inner" automorphism group coincide. For compact matrix quantum groups, which can be thought of as quantum analogues of compact Lie groups, we prove that the inner automorphism group is a compact Lie group and the outer automorphism group is discrete. Applications of this to the study of group actions on compact quantum groups are highlighted. We end with the construction of a compact matrix quantum group whose fusion ring is not finitely generated, unlike the classical case.