The fusion rules of some free wreath product quantum groups and applications

In this article we find the fusion rules of the free wreath product quantum groups Γˆ≀⁎SN+ for any discrete group Γ  . To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations. We then apply the knowledge of the fusion rules to prove, in most cases, several operator algebraic properties of the associated reduced C⁎C⁎-algebras such as simplicity and uniqueness of the trace. We also prove that the associated von Neumann algebra is a full type II1II1-factor and that the dual of Γˆ≀⁎SN+ has the Haagerup approximation property for all finite groups Γ.