π-Twisted Smash Coproducts, π-L-R Smash Coproducts And Smash Product Over L-R Dimodule Algebras

The notions of Hopf 7r-coalgebras,whereπis a discrete group,generalize of Hopf algebras.The thesis introduces the notions ofπ-smash coproducts different in[8],π-twisted smash coproducts,π-L-R smash coproducts,and studies their properties and relationship.The thesis consists of three chapters.In chapter 1,we review the basic definitions,notions and examples used in this thesis.In chapter 2,for Hopf 7r-coalgebras H and leftπ-H comodule coalgebras C,we define C×H=C(?) H as & a K-space and its comultiplication by. Moreover,we prove C x H is aπ-coalgebras,and introduce the concept of theπ-smash coproducts(Theorem 2.1.1).Using it,we give a necessary condition forπ-smash coproducts to be a Hopfπ-coalgebras(Theorem 2.1.5):C×H is a Hopfπ-coalgebras when C isπ-H co-module bialgebras and there exists a family of linear maps (antipode) if every Ha is commutative.Next,we introduce the concepts of theπ-twisted smash coproducts andπ-L-R smash coproducts respectively. Then our main result is now the following (Theorem 2.2.9):If H is a finite type Hopfπ-coalgebras, then there exists aπ-coalgebras iso-morphism betweenπ-twisted smash coproducts andπ-L-R smash coproducts.In chapters 3,firstly,we review the basic definitions ofⅡ-module algebras,H-comodule coalgebras,twisted H-dimodule and left-right H-dimodule.Motivated by [13-16], we introduce the concept of L-R-H dimodule and L-R-H dimodule algebras. Moreover,any left-right H-dimodule is a L-R-H dimodule with the trivial right action and left coaction.Let H be a Hopf algebra, A a two-sided H-module algebra and X a two-sided H-comodule algebra. We define smash coproduct A(?)X= A(?)X as a K-space and its multiplication by for any a,b∈A and x, y∈X. A(?) X is an associative algebra with the unit 1(?)1.Finally, we give the necessary conditions for the smash product A(?)X to be a L-R-H dimodule algebras (Theorem 3.2.4) and the smash product A(?) X be a bialgebra (Theorem 3.2.5).