Quantum isometries and group dual subgroups

Abstract

We study the discrete groups Λ whose duals embed into a given compact quantum group, Λ ⊂ G. In the matrix case G ⊂ U + n the embedding condition is equivalent to having a quotient map Γ U → Λ, where F = {Γ U |U ∈ U n } is a certain family of groups associated to G. We develop here a number of techniques for computing F , partly inspired from Bichon's classification of group dual subgroups Λ ⊂ S + n . These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.2000 Mathematics Subject Classification. 58J42 (46L87).

Maximal torus theory for compact quantum groups

“…We conclude that the embeddings Λ ⊂ U + N come from the quotient maps C(U + N ) → C * (Λ) of type u → QwQ * , and so the subgroups Λ ⊂ G ⊂ U + N must appear as in the statement. See [2], [37]. We have as well the following related result, from [9]:…”

“…As already mentioned in the introduction, while the general philosophy for these conjectures is quite old, going back to [2], [9], the statements are new, based on a quite substantial amount of recent work in the area, that we will explain here.…”

“…As for the considerations in [2], these were motivated by a key rigidity conjecture in noncommutative geometry [21], solved in the meantime by Goswami and Joardar [22]. The main observation in [2] was the fact that a non-classical group dual cannot act on a compact connected Riemannian manifold. We will be back to this in section 6 below.…”

“…Following [2], we first have the following result: Proposition 2.3. Given a discrete group Γ =< g 1 , .…”

“…Here (1) was obtained in [2], via a computation that we will generalize later on, and (2) was obtained as well in [2], via a direct computation. As for (3), the result here goes back to Bichon's work in [10], and can be obtained as well from (1,2). See [2].…”