Operated Algebras,Left Counital Bialgebras And DN-bialgebras

In this thesis,we mainly study left counital bialgebras,generalized Rota-Baxter algebras,DN-bialgebras and the universal enveloping differential algebras of differential Lie algebras from the view of operated algebras.This thesis consists of six chapters.In Chapter 1,we introduce the related backgrounds,motivations of our research.For the completeness of this paper,we also recall some related definitions and facts.In Chapter 2,we first recall the definition and some basic properties of left counital bialgebras.Using an extended 1-cocycle condition,we define a coproduct on the space of bi-decorated planar rooted forests and equip the space with a left counital bialgebra structure.Combining the concepts of operated algebras and left counital bialgebras,we introduce the concept of left counital(?,?)-cocyle bialgebras and show that the constructed left counital bialgebra is the free object in the category of left counital(?,?)-cocyle bialgebras.We prove that a connected graded left counital bialgebra is a left counital Hopf algebra.Having this fact in hand,the constructed left counital bialgebra is proved to be a left counital Hopf algebra.Finally,we construct a Rota-Baxter system on the space of bi-decorated planar rooted forests and a left counital Hopf algebra on the Rota-Baxter system.In Chapter 3,we first introduce the concepts and basic properties of -extended triassociative semigroups and ?-Rota-Baxter algebras.Then using -extended triassociative semigroup,we construct an ?-Rota-Baxter algebra on the space of typed angularly decorated planar rooted trees.Finally,using commutative -extended triassociative semigroup,a commutative ?-Rota-Baxter algebra is constructed on the space of typed words.In Chapter 4,we first introduce some basic properties of Dendriform-Nijenhuis bialgebras(abbr.DN-bialgebras)and the connection between DN-bialgebras and Lie algebras.Using a symmetric 1-cocycle condition,we construct a DN-bialgebra on the space of decorated planar rooted forests.Combining the concepts of operated algebras and DN-bialgebras,we propose the concept of ?-cocycle DN-bialgebras and show that the constructed DN-bialgebra is a free object in the category of ?-cocycle DN-bialgebras.In Chapter 5,we first introduce the concept of DN-associative Yang-Baxter equations(abbr.DN-AYBE).Then using a primary derivation,we show that the solutions of DN-AYBE on an algebra induce the DN-bialgebra structures on this algebra.Moreover,we prove that the solutions of DN-AYBE on an algebra induce TD operators on this algebra.We also propose the concept of quasitriangular DN-bialgebras and give the relationship between quasitriangular DN-bialgebras with tridendriform algebras,Post Lie algebras and Lie algebras.Finally,we get the solutions of DN-AYBE on the unitary algebras of dimension two and three over the field of complex numbers.In Chapter 6,we introduce the concepts and some basic properties of(modified)?-differential algebras and(modified)?-differential Lie algebras.Then we construct the free(modified)?-differential algebra on a(modified)?-differential module.Using the above construction,we get the universal enveloping(modified)?-differential algebra of a(modified)?-differential Lie algebra.We also prove the corresponding Poincar?e-Birkhoff-Witt theorem.Finally,the Wronskian envelope of a modified -differential Lie algebra is constructed.