Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

Abstract

For a quasi-Hopf algebra H, a left H-comodule algebra 𝔅 and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules C ℳ(H)𝔅 in terms of modules. We will also show that for an H-bicomodule algebra 𝔸 and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules 𝔸𝒴𝒟(H) C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.