Abstract:

The author discussed the reducibility or semisimpleness of Doi Hom-Hopf modules. Given a Doi HomHopf datum (H,A,C), the forgetful functor was used to send the objects in the category M^C_A of Doi Hom-Hopf modules into the category M_A of right (A,β)-Hom-modules, and look for a deformation of a splitting map of a monomorphism in the category M_A so as to lead to the integral for Doi HomHopf datum (H,A,C). With the
 help of the integral introduced, the author  proved the Maschketype theorem for Doi Hom-Hopf modules. As an application, the author defined the category of Hom-Yetter-Drinfeld modules and  proved the category of HomYetterDrinfeld modules being a subcategory of our category of Doi HomHopf modules, then the author  obtained immediately the version of Maschke type theorem for Hom-Yetter-Drinfeld modules.

Key words: Monoidal Hom-Hopf algebra, Doi Hom-Hopf module, Maschke type theorem, Hom-Yetter-Drinfeld module