The $$K$$ -theory of free quantum groups

Abstract

Abstract. In this paper we study the K-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are K-amenable and establish an analogue of the Pimsner-Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the K-theory of free quantum groups. Our approach relies on a generalization of methods from the Baum-Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a γ-element and that γ = 1. As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting.

Torsion and K-theory for Some Free Wreath Products

“…Then, Irr(H) embeds into Irr(G) and we can define an equivalence relation on Irr(G) by setting β ∼ β ′ if there exists γ ∈ Irr(H) such that β ′ ⊂ β ⊗ γ. It was proven in [19,Prop 4.3…”

“…Before studying the behaviour of torsion actions under this construction, let us prove a result concerning divisibility. The notion of divisible quantum subgroup was introduced in [19] to study the Baum-Connes conjecture for the free unitary quantum groups U + F . There are several equivalent definitions but we only give the one which will be useful for us.…”

“…Let us therefore assume that G ev = G and describe the irreducible representations of G inside G * S 1 . Following the proof of [19,Prop 4.3], we set, for ǫ ∈ {−1, 1}, [ǫ] − = min(ǫ, 0) and [ǫ] + = max(ǫ, 0). Let W be the subset of Irr(G * S 1 ) consisting in words of the form…”

Quantum groups, from a functional analysis perspective

“…Finally, many interesting questions arise in relation with Connes' noncommutative geometry [24], and we refer here to [18], [23], [26], [31]. Also, we refer to [20], [22], [43], [44] for more specialized analytic aspects, and to [32] and subsequent papers for free de Finetti theorems [32], in the spirit of Voiculescu's free probability theory [45].…”

Quantum reflections, random walks and cut-off

“…It then follows from the description of the representation theory of free products in [19] that the characters of U + N are products of characters of O + N and powers of z. A more precise description was given in [18,Prop 4.3], which we reproduce here :…”