Maximal torus theory for compact quantum groups

Abstract

Associated to any compact quantum group G ⊂ U + N is a canonical family of group dual subgroups Γ Q ⊂ G, parametrized by unitaries Q ∈ U N , playing the role of "maximal tori" for G. We present here a series of conjectures, relating the various algebraic and analytic properties of G to those of the family { Γ Q |Q ∈ U N }.2000 Mathematics Subject Classification. 46L65.

Quantum permutation groups

“…Let us discuss now some applications of all this material, following [31], [37], to general structure and classification questions for the compact quantum groups.…”

“…In order to deal with this issue, the idea, from [31], will be that of using: Proposition 10.20. Given a closed subgroup G ⊂ U + N and a matrix Q ∈ U N , we let T Q ⊂ G be the diagonal torus of G, with fundamental representation spinned by Q:…”

“…The various statements in Theorem 10.25 and Theorem 10.26 are conjectured to hold for any compact quantum group. We refer to [31] and to subsequent papers for a number of verifications, notably covering many basic examples of easy quantum groups.…”

Quantum groups, from a functional analysis perspective

“…There are as well several explicit conjectures regarding the maximal tori, the general idea being that the knowledge of T solves most of the problems regarding G. See [13].…”

Affine noncommutative geometry

“…In the remainder of this section we discuss the integration over G L M N , with a number of explicit formulae. Our main result will be the fact that the operations of type G L M N → G L+ M N are indeed "liberations", in the sense of the Bercovici-Pata bijection [24]. The integration over G L M N is best introduced as follows: Definition 10.15.…”