罗巴代数的表示和罗巴代数在operad中的应用

罗巴(Rota-Baxter)算子是积分算子的代数抽象和推广，是分析问题代数化研究的一个重要组成部分。它的研究始于G.Baxter 对Spitzer等式的研究。 罗巴代数与代数组合、交换代数、 Hopf代数、代数operad理论、数论、Yang-Baxter方程、量子场论（QFT）等都有着很多非常重要的联系。其深入应用到数学以及理论物理中的很多分支。. 罗巴代数表示论以及罗巴代数在operad中的应用是罗巴代数研究的重要内容。本课题首先研究一些特殊代数的罗巴结构进一步考虑其上的模的性质和分类并详细研究罗巴结合代数与罗巴李代数之间的关系，进一步用于Yang-Baxter方程的研究。其次研究与罗巴算子作用对偶的平均算子作用以及与元分裂相关的operad等式。

A Rota-Baxter operator is introduced as an algebraic abstraction and a generalization of the integration operator. It is a very important branch of the algebraic view study of analysis problems. It is originated from G.Baxter's study of Spitzer identity. Rota-Baxter algebra has a wide range of applications in algebraic combinatorics、commutative algebras、Hopf algebras、algebraic operad、number theory、Yang-Baxter equations、quantum field theory (QFT) .. The representations of Rota-Baxter algebra and applications of Rota-Baxter algebra in operad are important research fields in Rota-Baxter algebras. In this research, the Rota-Baxter structures of some typical algebras and their Rota-Baxter modules will be studied. we also want to explore the relationship between Rota-Baxter associative algebras and Rota-Baxter Lie algebras and try to find its applications in the research of Yang-Baxter equations. Then we study the action of averaging operator which is Koszul dual to the action of Rota-Baxter operator. Finally, we try to establish some operadic identities about the arity-splitting.