The Structure Of Hopf π-COALGEBRA With A Projection

In this paper,we mainly discuss the following questions:one is the biproduct structure together with its properties over Hopfπ-coalgebra; the other is the factorization of Hopfπ-coalgebra with a projection.First,we introduce the background of Hopfπ—coalgebra and present some basic prop-erties,the motivation and meaning of this thesis are introduced.Second,byπ-smash products andπ-smash coproducts,we give some sufficient and necessary conditions for aπ-smash product and aπ-smash coproduct to be a biproduct over Hopfπ-coalgebra,and we present its properties.Finally, by utilizing Hopfπ-comodule structure theorem,we can get the factorization theorem for any Hopfπ-coalgebras with a projection:Let A=({Aα,mα,1Aα},△,ε,S) and H=({Hα,mα,1Hα),△,ε,S)be Hopfπ-coalgebras.If there are two families of mapsτ={τα}α∈πand i={iα)α∈π:A(?)H,then we can get B=AcoH,such that there is an algebra structure and a coalgebra structure on B(?)H,and B(?)H is isomorphic with A as semi-Hopfπ-coalgebras,where for anyα∈π,ταοiα=idAα,B={Ba}α∈π={AαcoH)a∈π, B(?)H={AαcoH(?)Hα)α∈π,τ={τα:Aα→Hα}α∈πand i={iα:Hα→Aα)α∈πare semi-Hopfπ-coalgebra morphism.