Structures Of Rota-Baxter Bialgebras And Their Module Actions

The aim of this dissertation is to carry out a series of researches on Rota-Baxter Bialgebras and the related Hopf quasigroups from the following parts.Firstly,we introduce and discuss the notions of Rota-Baxter bialgebra equation systems and Rota-Baxter Hopf algebras.Then we construct a lot of examples based on Hopf quasi-groups.Secondly,the Galois linear maps play an important role in discussing properties of Hopf algebras and Hopf(co)quasigroups.For a bialgebra H,if it is unital and associative as an algebra and counital coassociative as a coalgebra,then the Galois linear maps T1 and T2 could be defined.In this paper,using the Galois linear maps,the efficient and necessary conditions of H to be a Hopf algebra are presented.On the other side,for a unital algebra(no need to be associative),and a counital coassociative coalgebra A,if the coproduct and counit are both algebra morphisms,then the efficient and necessary conditions of A to be a Hopf quasigroup is also given by the Galois linear maps.Furthermore,as a corollary,the quasigroups case is also considered.Thirdly,let H be a Hopf coquasigroup over a field k possessing an adjoint quasicoaction.We first show that if M is any right H-module and N is any right H-quasicomodule such that τMM,N○τN,M=idN(?)M,where τN,M:N(?)M→M(?)N is a favourable map,then we have H=k.As an application of this result,we get that symmetric category HHYDQ of Yetter-Drinfeld quasicomodules over H is trivial,as a generalization of Pareigis,Theorem.Furthermore,let(H,R)be a quasitriangular Hopf coquasigroup and(B,σ)coquasitriangular Hopf coquasigroup.Then we show that the category of generalized Long quasicomodules BHLQ is a braided monoidal subcategory of Yetter-Drinfeld category(?)YDQ.We also give a new approach to a braided monoidal category by generalizing one of Schauenburg’s main results in the setting of Hopf coquasigroups introduced by Klim and Majid.This yields new sources of braidings which provide solutions to the Yang-Baxter equation playing an important role in various areas of mathematics.Finally,let H be a triangular Hopf coquasigroup with bijective antipode and B a cotri-angular Hopf coquasigroup with bijective antipode.Under some favourable conditions,the aim of this paper is to find some objects satisfying the double centralizer property for any Hopf coquasigroup A which can be written as a tensor product Hopf coquasigroup H(?)B.As consequence of our theory,both Schur’s double centralizer theorems for triangular and cotriangular Hopf algebras can be obtained.Our main result provides a new approach to construct more objects which have double centralizer property too.