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

Abstract

In this paper, we find the fusion rules for the free wreath product quantum\ngroups $\\mathbb{G}\\wr_*S_N^+$ for all compact matrix quantum groups of Kac type\n$\\mathbb{G}$ and $N\\ge4$. This is based on a combinatorial description of the\nintertwiner spaces between certain generating representations of\n$\\mathbb{G}\\wr_*S_N^+$. The combinatorial properties of the intertwiner spaces\nin $\\mathbb{G}\\wr_*S_N^+$ then allows us to obtain several probabilistic\napplications. We then prove the monoidal equivalence between\n$\\mathbb{G}\\wr_*S_N^+$ and a compact quantum group whose dual is a discrete\nquantum subgroup of the free product $\\widehat{\\mathbb{G}}*\\widehat{SU_q(2)}$,\nfor some $0<q\\le1$. We obtain as a corollary certain stability results for the\noperator algebras associated with the free wreath products of quantum groups\nsuch as Haagerup property, weak amenability and exactness.\n