Quantum automorphism groups and SO(3)-deformations

TL;DR

This paper characterizes quantum automorphism groups with SO(3) fusion rules as automorphism groups of finite-dimensional C*-algebras with homogeneous states, extending known results and exploring classification of related Hopf algebras.

Contribution

It establishes a correspondence between SO(3)-like quantum groups and automorphism groups of algebra-state pairs, generalizes representation theory results, and discusses classification of certain Hopf algebras.

Findings

Quantum groups with SO(3) fusion rules are automorphism groups of algebra-state pairs.

Representation categories are studied beyond positive states.

Discussion on classification of cosemisimple Hopf algebras with SO(3) fusion rules.

Abstract

We show that any compact quantum group having the same fusion rules as the ones of $SO(3)$ is the quantum automorphism group of a pair $(A, \varphi)$, where $A$ is a finite dimensional $C^*$-algebra endowed with a homogeneous faithful state. We also study the representation category of the quantum automorphism group of $(A, \varphi)$ when $\varphi$ is not necessarily positive, generalizing some known results, and we discuss the possibility of classifying the cosemisimple (not necessarily compact) Hopf algebras whose corepresentation semi-ring is isomorphic to that of $SO(3)$.