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 (sic)(*). S-N(+) for all compact matrix quantum groups of Kac type G and N >= 4. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of G (sic)(*) S-N(+). The combinatorial properties of the intertwiner spaces in G (sic)(*) S-N(+) allow us to obtain several probabilistic applications. We prove also the monoidal equivalence between G(sic)(*) S-N(+) and a compact quantum group whose dual is a discrete quantum subgroup of the free product (G) over cap* (<(SUq(2))over cap>), for some 0 < q <= 1. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as the AOPAP property and exactness. (C) 2015 Published by Elsevier Inc.