Maschke functors, semisimple functors and separable functors of the second kind: applications

We introduce separable functors of the second kind (or H-separable functors) and H-Maschke functors. H-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be H-separable or H-Maschke, in terms of generalized (co)Casimir elements (integrals, in the case of Hopf algebras), are given. An H-separable functor is always H-Maschke, but the converse holds in particular situations. A special role will be played by Frobenius functors and their relations to H-separability. Our concepts are applied to modules, comodules, entwined modules, quantum Yetter–Drinfeld modules, relative Hopf modules.