数学与统计学院知识库

本文中, 我们研究了（结合）修正罗巴代数和修正三叶型代数, 重点在于研究修正罗巴代数范畴与修正三叶型代数范畴之间的关系. 修正罗巴代数算子的形式类似于修正杨-巴克斯特方程, 是罗巴代数算子的推广. 作为每个罗巴代数可以诱导出一个叶型代数这一著名事实的推广, 我们得出了每个修正罗巴代数可以诱导出一个修正三叶型代数, 进一步构造出修正三叶型代数的泛包络罗巴代数. 运用operads 与修正三叶型代数的联系, 给出修正三叶型代数满足的所有的二元关系. 全文共分为四章.
第一章介绍了本文研究课题的背景及其进展, 并给出本文需要的基本概念和一些相关的记号.   第二章回顾了借助括号字构造自由修正罗巴代数的详细过程. 我们分别考虑了集合上和代数上的自由修正罗巴代数两种情形.   第三章首先回顾了修正三叶型代数相关的概念及相关结论, 得出修正三叶型代数与修正罗巴代数之间的关系, 进一步得出修正三叶型代数范畴与修正罗巴代数范畴之间存在着自然函子. 然后回顾泛包络代数的概念, 并构造出修正三叶型代数的泛包络修正罗巴代数.   第四章首先回顾了operads 的相关概念和结论, 并利用自由修正罗巴代数的构造, 得出修正三叶型代数和operads 之间的联系, 并进一步给出修正三叶型代数满足的所有的二元关系.

In this paper, we mainly study (associative) modified Rota-Baxter algebras and modified tridendriform algebras, with emphasis on the relationship between the categories of modified Rota-Baxter algebras and the categories of modified tridendriform algebras. As a generalization of Rota-Baxter operator, the form of the modified Rota-Baxter operator is similiar to the modified classical Yang-Baxter equation. By generalizingthe well-known fact that every Rota-Baxter algebra gives a dendriform algebra, we obtain that every modified Rota-Baxter algebra gives a modified tridendriform algebra, further we construct the universal enveloping modified Rota-Baxter algebra of a modified tridendriform algebra. Applying the relationship between operads and modified tridendriform algebras, we further obtain all the quadratic nonsymmetric relations that modified tridendriform algebra satisfy. And this thesis consists of four chapters.

The first chapter introduces the background and its recent development, and gives some basic notions and related notations.  The second chapter reviews an explicit construction of free modified Rota-Baxter algebras by bracketed words. We consider both the cases when the free modified Rota-Baxter algebra is generated by a set and by another algebra.  The third chapter first reviews the definition and related conclusion of modified tridendriform algebras, we obtain the relationship between the modified Rota-Baxter algebras and the modified tridendriform algebras, further we obtain the natural functor
from the categories of modified Rota-Baxter algebras and the categories of modified tridendriform algebras. Then we introduce the definition of universal enveloping algebra, and apply the construction of free modified Rota-Baxter algebras to construct the universal enveloping modified Rota-Baxter algebra of a modified tridendriform algebra.  The fourth chapter first reviews the definition and related conclusion of operads, and applies the construction of free modified Rota-Baxter algebras to obtain the relationship between the modified tridendriform algebras and operads, we further obtain all the quadratic nonsymmetric relations that modified tridendriform algebra satisfy.