Rota-baxter Operators, Averaging Operators And Their Related Constructions Of Operads

A Rota-Baxter operator is introduced as an algebraic abstraction and a general-ization of the integration operator. This thesis focuses on Rota-Baxter algebra and its related averaging algebra. It covers three aspects, operator aspect, algebraic aspect and operad aspect of Rota-Baxter algebra and averaging algebra. This thesis consists of six chapters.Chapter one introduces the background and motivations of this thesis. The main results of this thesis are also presented. Chapter two is preliminaries, some basic notations, definitions and terminologies that will be used in this thesis are presented.Chapter three first determines all Rota-Baxter operators (of weight zero) on sl(2,C). Then we use three approaches to derive solutions of CYBE in sl(2, C) and the other two kinds of six dimensional Lie algebras from Rota-Baxter operators on sl(2,C). Finally, all pre-Lie structures derived from sl(2,C) and its Rota-Baxter operators are determined.Chapter four studies averaging operators from an algebraic point of view. Firstly, we give an explicit construction of free nonunitary averaging algebra on a nonempty set. Secondly, we study the enumeration for subsets of averaging words on singleton when the averaging operator is taken to be idempotent. We obtain the generating function of averaging words in two variables parameterizing the number of appearances of the variable and the operator respectively and find a connection of this kind of averaging words and large Schroder numbers, leading us to give interpretations of large Schroder numbers in terms of bracketed words and rooted trees, as well as a new recursive formula for these numbers.Chapter five focuses on Rota-Baxter algebras and constructions of operads. This chapter is a subsequent study of [6]. A new concept "C-splitting" is introduced to generalize [6] in two directions. In one direction, we generalize the arity of the operads under consideration from binary to any arities. In the other direction, we introduce the concept of a configurationto give a uniform treatment of different splitting pat-terns that include the bisuccessor and trisuccessor. C-splitting gives a systematic characterization of splitting for any algebra and the relationship between C-splitting and dendriform-type algebras. It also gives a "rule" to construct new dendriform-type algebras. Finally, we study the Rota-Baxter operator on any algebra and introduce relative C-Rota-Baxter operator to describe the relationship between C-splitting and Rota-Baxter actions.Chapter six focuses on averaging algebras and constructions of operads. A new concept "replicator" on binary operad is introduced to study Di-type algebras. It generalizes the "doubles" of associative algebra to any binary algebras. It relates many algebras which seem to be independent together and gives a general "rule" to construct new Di-type algebras. As desired, we also prove the Koszul duality of replicator of (?) and splitting of (?)P! for binary quadratic operads. Then we can use Manin white product to describe replicator. As an application, we give a convenient way to compute Manin white product of (?) and certain quadratic operads. Finally, we relate the averaging operator actions with replicators corresponding to Rota-Baxter operator and splitting. It is another connection of Rota-Baxter operator and averaging operator.