Fusion modules and amenability of coideals of compact and discrete quantum groups

TL;DR

This paper introduces a new framework for understanding amenability in fusion modules over quantum groups, linking coamenability of representations with properties of coideals, and applies it to Podleś spheres.

Contribution

It defines amenable fusion modules over fusion algebras and establishes a duality between coamenability of quasi-regular representations and amenability of fusion modules.

Findings

Characterization of coamenability via fusion modules

Duality result generalizing Tomatsu's theorem

Application to Podleś spheres showing amenability

Abstract

We give a definition of an amenable fusion module over a fusion algebra. A notion of relative integrability for the `coduals' of coideals of compact quantum groups was recently introduced in the joint work of de Commer and Dzokou Talla. We use this property to construct an analogue of the quasi-regular representation. Then, we characterize a certain coamenability property of quasi-regular representations with amenability of their associated fusion modules. Afterwards, we obtain a duality result that generalizes Tomatsu's theorem for this coamenability property and an amenability property of their `codual' coideals (under an additional assumption). As an example, we apply this result to show the fusion modules associated to certain non-standard Podle\'s spheres are amenable.