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

Let $\mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,ψ)$ and $Γ$ a discrete group. We want to compute the fusion rules of $\widehatΓ\wr_* \mathbb{G}$. 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 $ψ$. We will also prove the simplicity of the reduced C*-algebra, when $ψ$ is a trace, as well as the Haagerup property of $L^\infty(\widehatΓ\wr_* \mathbb{G})$, when $Γ$ is moreover finite.

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