Crossed Coproducts Over Hopf Algebras And Weak Hopf Algebras

The purpose of this paper is to study the properties of crossed coproducts. Including aboutcocleft extensions, inner coactions and equivalence of crossed coproducts, the conditions of whencrossed coproducts and crossed products can be Hopf algebras, and the conditions of whencrossed coproducts coalgebras can be weak bialgebras, futher more, can be weak Hopf algebras.It is divided into three chapters.In Chapter 1, we present some preliminary notions, properties and correlative lemmas.In Chapter 2, according to the notion of crossed coproducts, we discuss the questions aboutcocleft extentions, inner coactions and equivalence of crossed coproducts, we get the equivalenttheorem of crossed coproducts coalgebras. And through inner coactions, we obtain the relation-ship bewteen two kinds of crossed coproducts, such as comultiplication, counit, linear mapαandα′.In Chapter 3, on the base of the notions of crossed coprducts and crossed products, under theconditions ofσandρare trivial, we get the result of crossed products and crossed coproducts canbe Hopf algebras as a special case. Then we discuss the conditions of when crossed coproductscoalgebras can be weak bialgebras and weak Hopf algebras.