Hom-tensor Relations for Two-sided Hopf Modules over Quasi-hopf Algebras

For a Hopf algebra H over a commutative ring k, the category MH of right Hopf modules is equivalent to the category Mk of k-modules, that is, the comparison functor −⊗k H : Mk → MH is an equivalence (Fundamental theorem of Hopf modules). This was proved by Larson and Sweedler via the notion of coinvariants McoH for any M ∈ MH . The coinvariants functor (−) coH : MH → Mk is right adjoint to the comparison functor and can be understood as the Hom-functor HomH(H,−) (without referring to an antipode). For a quasi-Hopf algebra H, the category HMH of quasi-Hopf H-bimodules has been introduced by Hausser and Nill and coinvariants are defined to show that the functor −⊗k H : Mk → HMH is an equivalence. It is the purpose of this paper to show that the related coinvariants functor, right adjoint to the comparison functor, can be seen as the functor HHom H H(H ⊗k H,−) More generally, let H be a quasi-bialgebra and A an H-comodule algebra A (as introduced by Hausser and Nill). Then − ⊗k H is a comonad on the category AMH of (A, H)-bimodules and defines the Eilenberg-Moore comodule category (AMH) which is just the category AMH of two-sided Hopf modules. Following ideas of Hausser, Nill, Bulacu, Caenepeel and others, two types of coinvariants are defined to describe right adjoints of the comparison functor − ⊗k H : AM → AMH and to establish an equivalence between the categories AM and AMH provided H has a quasi-antipode. As our main results we show that these coinvariants functors are isomorphic to the functor AHom H H(A⊗k H,−) : AMH → AM and give explicit formulas for these isomorphisms.