Categorical Studies On Rota-Baxter Algebras,Differential Algebras And Dendriform Algebras

In this thesis,we study Rota-Baxter algebras,differential algebras and dendriform algebras from a categorical point of view.A Rota-Baxter operator and a differential operator are algebraic abstractions and generalizations of the integral calculus and differential calculus,respectively.The studies of Rota-Baxter algebra and differential algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus,resulting in differential Rota-Baxter algebras.By the ideas of liftings of monads and mixed distributive laws,we give categorical explanations of differential Rota-Baxter algebras.Further we study the relationships among extensions of operators,liftings of monads,and mixed distributive laws.Using the concepts of internal categories,we define strict Rota-Baxter 2-algebras and dendriform 2-algebras,and characterize them by crossed modules.This thesis consists of four chapters.Chapter one introduces the background of the project,and then gives the motivations and main results.Finally,some basic terminologies and notations that will be used in this thesis are presented.Chapter two first obtains the monad giving Rota-Baxter algebras and the comonad giving differential algebras by the constructions of a free Rota-Baxter algebra and a cofree differential algebra,respectively.Then the construction of a free differential Rota-Baxter algebra on a differential algebra gives a lifting of the monad on the category of differential algebras.Dually,we construct a cofree differential Rota-Baxter algebra on a Rota-Baxter algebra,and obtain a lifting of the comonad on the category of Rota-Baxter algebras.At last,a mixed distributive law of the monad giving RotaBaxter algebras over the comonad giving differential algebras is established,and we study the structure of differential Rota-Baxter algebras by the properties of a mixed distributive law.Chapter three focuses on mixed algebraic structures constructed by a RotaBaxter operator and a differential operator.By a canonical way,this chapter introduces coextensions of operators to cofree differential algebras,and extensions of operators to free Rota-Baxter algebras.In order to better understand the interrelationships among extensions of Rota-Baxter and differential operators,lifting monads and comonads,and mixed distributive laws,we define a set of polynomials in two noncommutative variables,and obtain a class of constraint conditions between a differential operator and a Rota-Baxter operator.Consequently,the existences of certain extensions of operators and these categorical properties are equivalent.Furthermore,given a constraint,we distinguish whether the extension of each Rota-Baxter operator to a cofree differential algebra is still a Rota-Baxter operator,and then provide a classification of constraints that satisfy these equivalent conditions.Chapter four defines strict Rota-Baxter 2-algebras and dendriform 2-algebras.By generalizing the concepts of the modules of Rota-Baxter algebras and dendriform algebras,we introduce Rota-Baxter crossed modules and dendriform crossed modules,and prove that they are equivalent to the corresponding 2-algebras.As a categorical lifting of the well-known fact that every Rota-Baxter algebra gives a dendriform algebra,we obtain the transformation from Rota-Baxter 2-algebras to dendriform2-algebras,and then the transformation of their crossed modules follows.