Hopf Hopf Pi On Pi - Algebra - H - Model Of The Structure

In the middle of20th century,Hopf algebra is an intresting subject for many mathematicians.So it has been studied extensively and deeply,and many results have been got by mathematicians.It promoted effectively the study and development of the quantum group.In2002,V G Turave introduced a new algebraic structure which was the Hopf π-coalgebra.Hopf π-coalgebra was a generalization of Hopf algebra.Virelizier studied detailedly Hopf π-coalgebra. Meanwhile the notions of π-algebra and Hopf π-algebra were introduced and studied.Upon the backgroud above,in this paper we mainly talk about some properties of Hopf π-algebra and Hopf π-H-module on the Hopf π-algebra.We give and prove the structure theorem of the Hopf π-H-module on the Hopf π-algebra,and we show that the tensor product of Hopf π-H-module is also Hopf vH-module.The paper is organized as follows.We introduce firstly the basic notions about π-algebra,Hopf π-algebra,π-H-module,Hopf π-H-module and so on.Then, we give both basic lemmas and prove some basic properties of Hopf π-algebra.See,one is that if H=({Ha,Δa,εa}a∈π,m,u,S)is a Hopf π-algebra,then for any α,β∈π, mα,β,u are coalgebra homomorphisms if and only if△={△α}α∈π,ε={εα)α∈π are π-algebra homomorphisms. The other is that the antipode S is a π-algebra anti-homomorphism)hism and a family of coalgebra anti-homomorphisms.Secondly,we give the notion of the coinvariant subcomodule MIcoH of the Hopf π-H-module M, and show that MIcoH H is also a Hopf π-H-module. Then we prove the structure theorem of Hopf π-H-module on the Hopf π-H-algebra.That is, let H=({Hα,△α,εα}α∈π,m,u,S) be a Hopf π-algebra,M=({Mα)α∈π Ψ={Ψα,β}α,β∈π},ρ={ρα}α∈π)be a Hopf π-H-module.Considering the coinvariant subcomodule MIcoH of M and a trivial Hopf π-H-module MIcoH H,we get a Hopf π-H-module isomorphism M≌MIcoH.As a corollary,let π={1} a goup,we get the fundamental theorem of usual Hopf H-module. Finall y, we discuss the tensor product of Hopf π-H-module, and give π-H-module structure and comodule structure of the tensor product.We show that the tensor product M(?)N of both Hopf π-H-modules(M,ψM,ρM) and (N,ψN,ρN)is also a Hopf π-H-module.