罗巴代数和operad

罗巴代数是积分算子的代数化，始于20世纪60年代Baxter的概率研究和Rota的组合研究，后来又被物理学家独立发现是经典杨-巴克斯特方程的算子形式，在数学和物理中都有着广泛的应用，现已成为公认的新兴学科。Operad系统地研究代数运算的合成，起源于20世纪60年代代数拓扑的研究。特别地，Koszul对偶理论在operad研究中起着重要的作用。本项目围绕罗巴代数和operad开展工作，研究多重罗巴代数，建立它与多重叶型代数、多重预李代数和极化杨-巴克斯特方程的关系；研究线性算子分类的Rota分类问题，发展OPI(算子多项式恒等式)-代数理论，并研究operad版本的Rota分类问题;研究罗巴operad和一般的高次operad的Koszul的对偶理论，并在此基础上，研究它们编码的代数对应的形变、同伦代数和同调代数。本项目将丰富罗巴代数和operad的研究内容。

Rota-Baxter algebra is the algebraization of the integral operator, originated from the probabilistic study of Baxter and the combinatorial study of Rota in the 1960s. It was also independently discovered by physicists as the operator form of the classical Yang-Baxter equation, with broad applications in mathematics and physics. Now Rota-Baxter algebra has become a new subject. Operad is a systematic study of composition of algebraic operations, started from the algebraic topology in the 1960s.In particular, Koszul dual theory is crucial in the study of operad. This project will do some researches around Rota-Baxter algebras and operads. We study multiple Rota-Baxter algebras, and erect the relationship between multiple Rota-Baxter algebras, multiple dendriform algebras, multiple pre-Lie algebras and polarized Yang-Baxter equations. We study the Rota’s classification problem on linear operators and develop the theory of OPI-algebras. We characterize the operadic version of Rota’s classification problem. We study the Koszul dual theory of Rota-Baxter operads and general operads with high degree, and further devote ourself to the corresponding deformations, homotopy algebras and homology algebras of the algebras encoded by these operads. This project will enrich the research contents of the theory of Rota-Baxter algebras and operads.