罗巴型代数,无穷小双代数和杨-巴克斯特方程

罗巴代数和杨-巴克斯特方程分别是数学和数学物理中两个重要且活跃的研究方向.上世纪八十年代,物理学家发现李代数上的罗巴方程恰好是经典杨-巴克斯特方程的算子形式,而结合杨-巴克斯特方程的解又可诱导一个无穷小双代数.因此,罗巴代数,杨-巴克斯特方程和无穷小双代数是三个相对独立又紧密联系的分支.罗巴型代数是指一类带算子代数,起源于Rota提出的算子代数方程分类问题.本项目主要研究罗巴型代数的双代数结构及与杨-巴克斯特方程之间的联系,将分两个部分展开.第一部分研究在自由带算子代数,特别是自由罗巴型代数,上建立各种双代数结构及Hopf代数结构;第二部分将研究罗巴型代数上的结合杨-巴克斯特方程理论及与罗巴型反对称无穷小双代数,O-算子和叶型代数等方面的联系.研究结果将丰富罗巴型代数的双代数理论,同时揭示罗巴型代数在数学物理方向应用的广泛性.

Rota-Baxter algebras and Yang-Baxter equations are two very important and extremely active research directions in mathematics and mathematical physics, respectively. In 1980s, physicists discovered that in the Lie algebra context, the Rota-Baxter equation is precisely the operator form of the classical Yang-Baxter equation, and any solution of the associative Yang-Baxter equation can induce an infinitesimal bialgebra. Thus, Rota-Baxter algebras, Yang-Baxter equations and infinitesimal bialgebras are three relatively independent and closely related branches. Rota-Baxter type algebras are a class of operated algebras, which originate from the open problem on the classification of operator algebraic equations proposed by Rota. This project will focus on bialgebra structures of Rota-Baxter type algebras and the relationships between them and Yang-Baxter equations. It will be divided into two parts. The first topic will study bialgebra structures and Hopf algebra structures on free operated algebras, especially on free Rota-Baxter type algebras; The second will develop the theory of associative Yang-Baxter equations on Rota-Baxter type algebras and their relations with Rota-Baxter type antisymmetric infinitesimal bialgebras, O-operators and dendriform algebras, etc. The results of this project will enrich the bialgebra theory of Rota-Baxter type algebras, and reveal the wide applications of Rota-Baxter type algebras in mathematical physics.