The free wreath product of a discrete group by a quantum automorphism group

TL;DR

This paper computes the fusion rules for the free wreath product of a discrete group with a quantum automorphism group, extending representation theory and analyzing properties like simplicity and the Haagerup property.

Contribution

It provides a detailed description of fusion rules for the free wreath product involving quantum automorphism groups and establishes new structural properties.

Findings

Fusion rules for the free wreath product are explicitly computed.

The reduced C*-algebra is shown to be simple when $ extpsi$ is a trace.

The Haagerup property is proven for the associated von Neumann algebra when $ extGamma$ is finite.

Abstract

Let $\mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi)$ and $\Gamma$ a discrete group. We want to compute the fusion rules of $\widehat{\Gamma}\wr_* \mathbb{G}$. First of all, we will revise the representation theory of $\mathbb{G}$ and, in particular, we will describe the spaces of intertwiners by using noncrossing partitions. It will allow us to find the fusion rules of the free wreath product in the general case of a state $\psi$. We will also prove the simplicity of the reduced C*-algebra, when $\psi$ is a trace, as well as the Haagerup property of $L^\infty(\widehat{\Gamma}\wr_* \mathbb{G})$, when $\Gamma$ is moreover finite.