Some Studies On Rota-baxter Algebras,Nijenhuis Algebras And Hopf Algebras,and Reynolds Algebras

In this thesis,we study the relationships between free Rota-Baxter algebras and Hopf algebras,free Nijenhuis algebras and left counity right antipode Hopf algebras respectively,and also construct the free Reynolds algebras on a set.There are four chapters.In Chapter 1,we introduce the background and its recent development,and give some basic notions and related results.In the left three Chapters,we construct the Hopf algebras,cocycle Hopf algebras on free Rota-Baxter algebras,the left counity right antipode Hopf algebras on free Nijenhuis algebras,and the free Reynolds algebras on a set respectively.As two important algebraic structures in quantum field theory,Rota-Baxter algebras and Hopf algebras should be further studied on their relationships.It is wellknown that planar rooted trees bridge both of the two algebraic structures.The Rota-Baxter type operated algebras,as a quotient algebra of the free operated algebra,can be constructed by using the bracketed words.In Chapter 2 of this thesis,we first discuss the factorization properties in free Rota-Baxter algebras.We then obtain the Hopf algebra structure on free Rota-Baxter algebras constructed by using the bracketed words.Finally,the cocycle Hopf algebra structure is studied on free Rota-Baxter algebras constructed by using the planar rooted trees.Nijenhuis operator is one of the Rota-Baxter type operators and Nijenhuis algebras are the “homogenous”version of Rota-Baxter algebras.The study methods of the Rota-Baxter algebras have good reference value for Nijenhuis algebras.In Chapter3,we first study the free unitary Nijenhuis algebra on a set by using the alternating words,and then give the notions of the so-called left co-bialgebra and left counity right antipode Hopf algebra.Finally,we construct the left counity right antipode Hopf algebras on free Nijenhuis algebras.The concept of a Reynolds operator was introduced in the study of fluid dynamics,and has wide range applications.So far,there were only a few papers about this subject from analytic point of view.In Chapter 4 of this thesis,we study Reynolds operators from an algebraic point of view.The construction of free Reynolds algebras is useful.On the one hand,the free object is certainly important to the category of Reynolds algebra itself.On the other hand,In Rota's view,Reynolds operator is an infinitesimal version of the Rota-Baxter operators.The study of Reynolds algebras may give a new perspective on Rota-Baxter algebras.The outline of this chapter is as follows.We first give some elementary properties of the Reynolds algebra.Then we construct the so-called Reynolds words,which are as a linear basis of free Reynolds algebras.Finally,we check the universal property of the free Reynolds algebra on a set.