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

TL;DR

This paper determines the fusion rules and monoidal equivalences for free wreath product quantum groups, leading to new insights into their probabilistic properties and operator algebra stability.

Contribution

It provides a combinatorial description of intertwiner spaces and establishes monoidal equivalence with certain free product quantum groups, advancing understanding of their structure.

Findings

Fusion rules for all compact matrix quantum groups of Kac type

Monoidal equivalence with quantum groups related to free products

Stability results for associated operator algebras

Abstract

In this paper, we find the fusion rules for the free wreath product quantum groups $\mathbb{G}\wr_*S_N^+$ for all compact matrix quantum groups of Kac type $\mathbb{G}$ and $N\ge4$. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of $\mathbb{G}\wr_*S_N^+$. The combinatorial properties of the intertwiner spaces in $\mathbb{G}\wr_*S_N^+$ then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between $\mathbb{G}\wr_*S_N^+$ and a compact quantum group whose dual is a discrete quantum subgroup of the free product $\widehat{\mathbb{G}}*\widehat{SU_q(2)}$, for some $0<q\le1$. 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.