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 \(\mathbb{A}\) and an H-bimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules \({}^C_H{\cal M}_{\mathbb{A}}^H\) is equivalent to the category of right–left generalized Yetter–Drinfeld modules \({}^C{\cal YD}(H)_{\mathbb{A}}\). Using alternative versions of this result we will recover the category isomorphism between the categories of left–left and left–right Yetter–Drinfeld modules over a quasi-Hopf algebra.