THE FREE WREATH PRODUCT OF A DISCRETE GROUP BY A QUANTUM AUTOMORPHISM GROUP

Let 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 (Gamma) over cap (sic)* G. First of all, we will revise the representation theory of 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-infinity((Gamma) over cap (sic)* G), when Gamma is moreover finite.