Modified Rota-Baxter Algebras And Modified Tridendriform Algebras

In this paper,we mainly study(associative)modified Rota-Baxter algebras and modified tridendriform algebras,with emphasis on the relationship between the categories of modified Rota-Baxter algebras and the categories of modified tridendriform algebras.As a generalization of Rota-Baxter operator,the form of the modified RotaBaxter operator is similiar to the modified classical Yang-Baxter equation.By generalizing the well-known fact that every Rota-Baxter algebra gives a dendriform algebra,we obtain that every modified Rota-Baxter algebra gives a modified tridendriform algebra,further we construct the universal enveloping modified Rota-Baxter algebra of a modified tridendriform algebra.Applying the relationship between operads and modified tridendriform algebras,we further obtain all the quadratic nonsymmetric relations that modified tridendriform algebra satisfy.And this thesis consists of four chapters.The first chapter introduces the background and its recent development,and gives some basic notions and related notations.The second chapter reviews an explicit construction of free modified Rota-Baxter algebras by bracketed words.We consider both the cases when the free modified RotaBaxter algebra is generated by a set and by another algebra.The third chapter first reviews the definition and related conclusion of modified tridendriform algebras,we obtain the relationship between the modified Rota-Baxter algebras and the modified tridendriform algebras,further we obtain the natural functor from the categories of modified Rota-Baxter algebras and the categories of modified tridendriform algebras.Then we introduce the definition of universal enveloping algebra,and apply the construction of free modified Rota-Baxter algebras to construct the universal enveloping modified Rota-Baxter algebra of a modified tridendriform algebra.The fourth chapter first reviews the definition and related conclusion of operads,and applies the construction of free modified Rota-Baxter algebras to obtain the relationship between the modified tridendriform algebras and operads,we further obtain all the quadratic nonsymmetric relations that modified tridendriform algebra satisfy.