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

In this paper, we find the fusion rules for the free wreath product quantum groups $\mathbb{G}{\wr }_{\ast }{S}_N^{+}$ for all compact matrix quantum groups of Kac type $\mathbb{G}$ and $N\ge 4$. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of $\mathbb{G}{\wr }_{\ast }{S}_N^{+}$. The combinatorial properties of the intertwiner spaces in $\mathbb{G}{\wr }_{\ast }{S}_N^{+}$ then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between $\mathbb{G}{\wr }_{\ast }{S}_N^{+}$ and a compact quantum group whose dual is a discrete quantum subgroup of the free product $\widehat{\mathbb{G}}\ast \widehat{S{U}_q(2)}$, for some $0<q\le 1$. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as Haagerup property, weak amenability and exactness.