Symmetries of a generic coaction

TL;DR

This paper investigates the symmetries of a generic coaction on finite-dimensional C*-algebras, revealing their quantum automorphism groups share fusion rules with SO(3) and exploring their structural and dual properties.

Contribution

It establishes the fusion rules for the quantum automorphism group of finite-dimensional C*-algebras and derives structural and duality results for these groups.

Findings

Fusion rules match those of SO(3) for the automorphism groups.

Duals of these groups are non-amenable when dimension exceeds 4.

Fixed point subfactors have index n and principal graph A_infinity.

Abstract

If B is C*-algebra of finite dimension n>3 then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say G, have the same fusion rules as the ones of SO(3). As consequences, we get (1) a structure result for G in the case where B is a matrix algebra (2) if n>4 then the dual of G is not amenable (3) the fixed point subfactor P^G\subset (B\otimes P)^G has index n and principal graph A_\infty.