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

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 Podleś spheres are amenable.