组合Hopf代数及其罗巴代数结构

本项目研究组合Hopf代数的分类与表示意义以及其中的罗巴代数结构，属于代数学与组合数学的交叉课题。拟继续申请人有关组合Hopf代数，尤其是Rota猜想的前期工作，以Coxeter群的元素为基底构造Hopf代数，系统地推广在代数组合中起核心作用的几类Hopf代数，并研究它们之间的关系；用代数塔研究对偶组合Hopf代数的分类及其代数表示理论解释；同时，利用组合Hopf代数给出自由罗巴代数的一般性构造，并将结果应用于多重zeta函数值和弱拟对称函数重整化的研究之中。本项目使用的关键手段为根系、代数表示论中的组合方法和罗巴代数理论，目标是得到一些重要类型的组合Hopf代数，取得完整的分类结果，并利用罗巴代数为组合Hopf代数的研究建立一个可行的理论框架。

The present project is devoted to investigating the classification, representation theory interpretations and Rota-Baxter algebra structures of combinatorial Hopf algebras, and lies in the intersection of algebra and combinatorics. On the basis of our research on combinatorial Hopf algebras, especially the prior work on Rota’s conjecture, we proceed to construct Hopf algebras with linear bases parameterized by elements of Coxeter groups and study the relationships among them, so that we can obtain a systemic generalization of some important Hopf algebras in algebraic combinatorics. We also plan to establish the classification and representation theory interpretations for dual combinatorial Hopf algebras by using towers of algebras, and construct general free Rota-Baxter algebras in terms of combinatorial Hopf algebras which will be applied to multiple zeta values and the renormalization of weak quasisymmetric functions. The main idea of our study is based on root systems, combinatorial methods in representations and Rota-Baxter algebra theory. The main aim of the project is to obtain some important combinatorial Hopf algebras and their classification results, and establish a feasible framework of Rota-Baxter algebras for the study of combinatorial Hopf algebras.