L-R Smash Coproducts And L-R Twisted Coproducts For Comodule Coaigebras

In this paper, we give a Maschke theorem for L-R smash coproducts by a new method, and study the relation between L-R twisted coproducts and lefe(right) twisting datums.This paper is organized as follows:In Chapter1, we introduce the development and background of Hopf algebras, and give main results of this paper.In Chapter2, we construct a new coalgebra, which is called generalized diagonal crossed coproduct. Moreover, we give a coalgebra isomorphism between generalized diagonal crossed coproducts and L-R smash coproducts.In Chapter3, we give the Maschke theorem for L-R smash coproducts.In Chapter4, we define L-R twisting datums and right twisting datums on a coalgebra C, and introduce the concepts of L-R twisted coprodcts and right twisted coproducts. Meanwhile, we give the relations among L-R twisted coproducts, left twisted coproducts and right twisted coproducts.