Thoma type results for discrete quantum groups

Abstract

Abstract. Thoma's theorem states that a group algebra C * (Γ) is of type I if and only if Γ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group Γ, we have a stationary model of type π :, with F being a finite quantum group, and with L being a compact group. We discuss then some refinements of these results in the quantum permutation group case, Γ ⊂ S + N , by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank ≤ 1.

Quasi-flat representations of uniform groups and quantum groups

“…We are of course mostly interested in understanding when these models satisfy the various notions of faithfulness from Definition 1.5. The subject here is non-trivial, and the above results, together with some other results from [6], [8], [9], which are more technical and will be explained later on, suggest the following conjecture: Conjecture 1.9. Assume that G ⊂ S + N is quasi-transitive, with orbits having size K, and consider the universal flat model π : C(G) → M K (C(X G )).…”

“…Regarding the evidence, in the classical case, G ⊂ S N , everything about π is of course known from Proposition 1.6 above. The question left is that of understanding the precise meaning of the "transitivity type" condition found there, as well as its interpretation as an "virtual abelianity" condition regarding the discrete dual G. See [6], [8].…”

“…For such a quantum group the simplest possible models are those which are "quasi-flat" in the sense that the standard coordinates u ij ∈ C(G), known to be projections, are mapped into projections of rank ≤ 1. Some work on the quasi-flat models was recently done in [2,4,6,8,9].…”

Quantum groups, from a functional analysis perspective

“…This follows indeed from a routine Haar measure computation, which enhances the algebraic considerations from the proof of Theorem 8.2. See [7].…”

“…There are many other interesting examples of stationary models, including the Pauli matrix model for the algebra C(S + 4 ), discussed in [6]. We refer to [7] and to subsequent papers for more on this subject, and for some recent results on the non-stationary case as well. There might be actually a relation here with lattice models too [27].…”