| This paper mainly investigates weak Hopf group algebras and group corings.In order to construct new examples of crossedπ-category,the notions of weak Hopf group algebras and weak T-algebras are introduced.We explain weak Hopf group algebras from a categorial point of view.Doi-Hopf modules,Yetter-Drinfeld modules and entwined modules over weak T-algebras are discussed.We prove that the functor forgettingπ-action or coaction has an adjoint,provide a necessary and sufficient condition concerning weak Yetter-Drinfeld modules,and study nice relationship between weak Hopfπ-algebras and weak group coentwining structures.In order to continue to develop the theory of group corings,we give some interesting and non-trivial examples of group corings,and prove the relationship between group corings and weak group coentwining structures and invertible weak group coentwining structures.Dually,we introduce the notion ofπ-C-rings,and prove the fact that many theories of Hopf group algebras and weak Hopf group algebras can be unified by the theory ofπ-C-rings.As a generalization of group corings and weak group corings, the notion of lax group corings is introduced.We show the fact that the category of lax group entwined modules is isomorphic to that of comodules over a suitable lax group coring,which unified the corresponding results of group corings and weak group corings. Therefore,a lax group coring is an effective tool to study Hopf group coalgebras and weak Hopf group coalgebras. |
