关于罗巴族代数、叶型族代数和罗巴族operad的研究

罗巴代数起源于20世纪60年代Baxter的概率和Rota的组合研究。它在数论、叶型代数、Hopf代数、operad和组合方面有重要应用。Operad起源于20世纪70年代初代数拓扑的研究。罗巴族代数是罗巴代数的推广，其理论有待完善。本项目拟研究罗巴族代数、叶型族代数和罗巴族operad相关的问题:（1）利用型装饰根树构造自由罗巴族代数，这和K.Ebrahimi-Fard和郭锂在装饰根森林上的构造有很大不同。（2）刻画叶型族代数的泛包络代数；根树和括号字之间存在双射，作为应用，在带有装饰的置换群上构造自由叶型族代数。（3）研究operad上的罗巴族算子作用，把带算子的罗巴族operad转化为不带算子的叶型族operad。用operad语言，建立其他族代数的统一框架。本项目将丰富罗巴族代数、叶型族代数和罗巴族operad的内容。

Rota-Baxter algebras originated from the research of Baxter's probability theory and Rota's combinatoric theory in the 1960s. It has important applications in number theory, dendriform algebras, Hopf algebras, operad and combinatorics. Operad originated from the study of algebraic topology in the early 1970s.The Rota-Baxter family algebra is a generalization of the Rota-Baxter algebra, and its theory needs to be improved. This project intends to study the problems related to Rota-Baxter family algebras、dendriform family algebras and Rota-Baxter family operad: First, we construct the free Rota-Baxter family algebras with the help of typed decorated rooted trees, which is very different from the construction of K. Ebrahimi-Fard and Guo on typed decorated rooted trees. Second, we characterize the universal enveloping algebra of dendriform family algebras; There is a bijection map between the rooted trees and the bracketed words, as an application, and we construct the free dendriform family algebraic structures in terms of decorated permutation group. Third, we study the role of Rota-Baxter family on the operad and transform the Rota-Baxter family operad into the dendriform family operad. Then we apply the operad language to establish a unified framework for other family algebras.This project will enrich the research content of Rota-Baxter family algebras、dendriform family algebras and Rota-Baxter family operad.