On the structure of quantum automorphism groups

TL;DR

This paper computes the K-theory of quantum automorphism groups of finite-dimensional C*-algebras, revealing non-isomorphic structures for different quantum permutation groups and exploring torsion phenomena in discrete quantum groups.

Contribution

It provides the first detailed K-theoretic analysis of quantum automorphism groups, highlighting their structural diversity and torsion properties.

Findings

K-theory distinguishes quantum permutation groups for different n

Quantum automorphism groups exhibit genuine quantum torsion phenomena

Duals of these groups are fundamental examples of discrete quantum groups with torsion

Abstract

We compute the $ K $-theory of quantum automorphism groups of finite dimensional $ C^* $-algebras in the sense of Wang. The results show in particular that the $ C^* $-algebras of functions on the quantum permutation groups $ S_n^+ $ are pairwise non-isomorphic for different values of $ n $. Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena.