Hom-Tensor Relations for Two-Sided Hopf Modules Over Quasi-Hopf Algebras

Abstract

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