Free wreath product quantum groups: The monoidal category, approximation properties and free probability

In this paper, we find the fusion rules of the free wreath product quantum groups G≀⁎SN+ for all compact matrix quantum groups of Kac type GG and N≥4N≥4. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of G≀⁎SN+. The combinatorial properties of the intertwiner spaces in G≀⁎SN+ allow us to obtain several probabilistic applications. We prove also the monoidal equivalence between G≀⁎SN+ and a compact quantum group whose dual is a discrete quantum subgroup of the free product Gˆ⁎SUq(2)ˆ, for some 0