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 Podle\'s spheres are amenable.