Maschke Theorem Over Hopf Algebras And Fundamental Theorem Over The Relative Yetter-Drinfel’d Hopf Algebras

This paper mainly gives Maschke theorem for the two-sided L-R Smash products, and the fundamental theorem and Maschke theorem over the relative Yetter-Drinfel’d Hopf algebras.Meanwhile, we study braided products of twisted quantum Yang-Baxter module algebras and Smash coproducts of twisted dimodule coalgebras.The whole paper consists of four chapters.In chapter1, we introduce the recent development of Hopf algebras, relative Yetter-Drinfel’d Hopf algebras and the theory of twisting, and the contents we studied and the main results in this paper.In chapter2, we introduce the concept of two-sided L-R Smash products, and give a sufficient and necessary condition for two-sided L-R Smash products to be bialgebras(Hopf algebras), and give a new description of Maschke theorem for two-sided L-R Smash products.In chapter3, we give the definition of the twisted quantum Yang-Baxter module algebras, study its property and give a sufficient and necessary condition for their braided products to be bialgebras. Meanwhile, we introduce the concept of the dimodule coalgebras, and give a sufficient and necessary condition for their Smash coproducts to be bialgebras.In chapter4, we mainly give the fudamental theorem of relative Hopf modules over the relative Yetter-Drinfel’d Hopf algebras and Maschke theorem over it.