Faithful compact quantum group actions on connected compact metrizable spaces

Abstract

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.

Half-Liberated Manifolds and Their Quantum Isometries

“…More specifically, we will be interested in computing the quantum isometry groups of its various half-liberations. Our approach here will be based on the affine quantum isometry group formalism [11], [13], [15], [16], with various technical ingredients from [1], [6], [10], [17]. We will prove that the affine quantum isometry groups of the 6 half-liberated spheres are as follows: In other words, our result will state that, for the sphere X = S N −1 C itself, the quantum isometry groups of the half-liberations are the half-liberations of the usual isometry group.…”

Existence and examples of quantum isometry groups for a class of compact metric spaces

“…On the other hand, we get from [13] an action of S + n on C(T ) which is clearly isometric in our sense, hence (by the universality of Q) a surjective CQG morphism in the reverse direction. In other words,…”

“…shown in the figures below. These are particular types of examples considered in [13] (see also [9]). …”

“…Indeed, it is proved in [11] that there cannot be any faithful smooth action of a genuine compact quantum group on a compact connected smooth manifold, which in particular implies that the quantum isometry group for any such manifold M must coincide with C(ISO(M )). However, Huang [13] constructed several examples of faithful actions of genuine compact quantum groups on compact connected spaces which are obtained by topologically gluing smooth manifolds, e.g. topological join of compact intervals.…”

“…For metric spaces (X, d) which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on (X, d). In fact, our existence theorem applies to a larger class, namely for any compact metric space (X, d) which admits a one-to-one continuous map f : X → R n for some n such that d 0 (f (x), f (y)) = φ(d(x, y)) (where d 0 is the Euclidean metric) for some homeomorphism φ of R + .As concrete examples, we obtain Wang's quantum permutation group S + n and also the free wreath product of Z 2 by S + n as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13]. …”