Two-sided Two-cosided Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

Abstract

For a quasi-Hopf algebra H, an H-bicomodule algebra A and an Hbimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules C H M H A is equivalent to the category of right-left generalized YetterDrinfeld modules C Y D(H) A . Using alternative versions of this result we will recover the category isomorphism between the categories of left-left and left-right YetterDrinfeld modules over a quasi-Hopf algebra.

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

“…For a right H-comodule algebra (A, ρ, φ ρ ), we show that the tensor functor − ⊗ k H is a comonad on the category A M H and we consider the category of two-sided Hopf modules Other forms of adjoint functors to −⊗ k H are obtained by defining Hausser-Nill and BulacuCaenepeel type coinvariants for this category (following [6,5], [9]). The relationship between these is explicitly described.…”