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

Abstract

Abstract. In this paper, we find the fusion rules for the free wreath product quantum groups G * 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 * S + N . The combinatorial properties of the intertwiner spaces in G * S + N then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between G * S + N and a compact quantum group whose dual is a discrete quantum subgroup of the free product G * SUq(2), 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 Haagerup property, weak amenability and exactness.

Torsion and K-theory for Some Free Wreath Products

“…Once again, we will not give the definition of these objects since it is not necessary for our purpose. All we need is to understand their fusion ring, which was described by F. Lemeux and P. Tarrago in [14]. Consider the free monoid F over Irr(G).…”

“…The idea to study torsion in G ≀ * S + N is to embed it into a compact quantum group whose torsion is better understood, namely a free product. Such an embedding need not exist in general, but it always does once we consider a monoidally equivalent compact quantum group, thanks to the following result of F. Lemeux and P. Tarrago in [14]. From now on, let us denote by u 1 the fundamental representation of SU q (2).…”

Quantum groups, from a functional analysis perspective

“…There are many other interesting things that can be said about the reflection groups constructed above. We refer here to the survey paper [6], and to [2], [3], [34].…”

Free wreath product quantum groups and standard invariants of subfactors

“…When F = S + n , the representation theory of G * S + n has been studied in [LT16]. Moreover, Bichon's construction was partially generalized in [FP16] to the situation in which the right input F is replaced by the universal compact quantum group G aut (A, tr) acting on a finite-dimensional C * -algebra A in a Markov trace-preserving way.…”