Frobenius and separable functors for the category of entwined modules over cowreaths, II: Applications

Let H be a quasi-Hopf algebra. We apply results obtained in [8] to give necessary and sufficient conditions for the forgetful functor from Doi-Hopf modules, two-sided Hopf modules or Yetter-Drinfeld modules over H to representations of the underlying algebra to be Frobenius (resp. separable). We show that in some situations these conditions reduce to the unimodularity and/or (co)semisimplicity of the quasi-Hopf algebra H.