数学和物理中的组合代数

In recent years, two classed of algebraic structures with combinatorial significance have been studied extensively, showing their importance in theory and applications. One class is Hopf algebras including the Hopf algebras of trees from algebra, combinatorics and physics studies. The other class is algebras with linear operators, called operated algebras, especially Rota-Baxter algebras. Often they come together as in the case of Hopf algebra and Rota-Baxter algebra in their applications in quantum field theory (algebraic Birkhoff decomposition), multiple zeta values and quasi-symmetric functions. More recently, the algebraic structures of convex cones have also emerged in the related studies. ..Based on previous work of the applicant and coauthors, this project gives a systematic study of these three classes of algebraic structures and the interaction among them. The project consists of three independent yet related parts. ..The first part continues the study of Rota’s conjecture relating Rota-Baxter algebras and Hopf algebras of symmetric type functions. It also considers Hopf algebra structures on the free objects of several types of operated algebras and their relationship with the existing Hopf algebras of similar type，such as the one of Connes-Kreimer. It further deals with bialgebra structures on free objects of non-associative algebras, from splitting of the operations. ..The second part continues our research on an open problem of Rota, asking for the classification of algebraic identities that a linear operator on algebras can satisfy. Based our completed works on the solution of Rota’s Problem in special cases of differential type and Rota-Baxter type operators, we establish close relationship between Roto’s Problem and rewriting systems in computer science and Groebner-Shirshov bases in computational algebra. We thus plan to solve the problem in several more cases with the goal of solving the problem in general. We also explore Rota’s Problem for non-associative algebras. Constructions of Hopf algebras will also be carried out for solutions of Rota’s Problem, preferably by a general procedure. ..The third part establishes the connection between operated Hopf algebras, especially the rooted tree Hopf algebra, and the coalgebra of convex cones, both of which play a pivot role in the algebraic Birkhoff decomposition. It applies the universal property of rooted tree algebra to explore possible bialgebra properties of convex cones. It also applies the close relationship between convex cones and multi-variable meromorphic functions to compare the traditional algebraic Birkhoff decomposition centered on rooted tree Hopf algebra and the new algebraic Birkhoff decomposition centered on convex cones, with the view of their applications in renormalization problems.

Hopf代数和罗巴代数得到广泛的组合研究，并应用于数论，组合和量子场论。最近凸锥的代数结构也类似出现。..基于申请人等的前期工作，本课题拟研究这些代数结构的三个独立而又相关的部分。..第一部分探讨Rota有关罗巴代数与对称函数关系的猜测，建立几种与自由罗巴代数相关的广义拟对称函数及其Hopf代数结构，并考虑带算子自由代数的Hopf代数结构和非结合代数自由对象的双代数结构。..第二部分考察Rota的公开问题，即对代数上线性算子所能满足的算子代数方程进行分类。基于对Rota问题以重写系统及GS基的刻画，拟在多型算子代数中对此问题求解，并探讨此问题的一般情形和非结合代数版本。..第三部分建立根树Hopf代数和凸锥余代数的关系。利用根树Hopf代数的泛性质探讨凸锥的可能双代数结构，并利用凸锥和多元亚纯函数的密切关系，综合基于根树Hopf代数的代数伯克霍夫分解和基于凸锥的广义代数伯克霍夫分解。

罗巴代数(也称为巴克斯特代数)的研究起源于1960年G.Baxter对概率论的研究,其目的是为了更好剖析浮动理论中的Spitzer等式.罗巴代数是一类带算子的结合代数,它可看成是分析中积分算子的代数推广,是Rota和他的学派于20世纪60年代建立的.作为经典杨-巴克斯特方程的算子形式,李代数上的罗巴算子于上世纪80年代被物理学家独立地发现.作为一名杰出的组合学家,Rota推动了这项研究并于上世纪90年代提出罗巴代数上的若干公开问题..本世纪之交,罗巴代数的几个重要应用于被发现,特别是研究量子场论重整化的Connes和Kreimer代数方法.此后,罗巴代数及其相关的各个数学领域和理论物理, 如Hopf代数,Operads,组合学,数论,量子场论和杨-巴克斯特方程,得到了迅猛发展.在2020年数学学科分类(MSC2020)中,"Yang-Baxter方程和Rota-Baxter算子"被添加为17B38,表明该学科作为数学领域中的主流研究领域得到了正式认可..本项目的研究目标在于解决物理学、组合学和积分方程中一些长期存在的问题以及处理最新进展中出现的新现象.首先,Rota很早以前就预测罗巴代数可看成是对称函数推广的一个理想的载体.虽然在一段时间这个问题进展不大,但是罗巴代数的新发展为我们解答Rota问题提供了新的思路.最近一类带有多重线性算子的新结构出现于Martin Hairer(2014年菲尔兹奖获得者)关于正则结构的代数重整化的工作中.我们发现这些多重算子与带多重罗巴算子的代数有关.在李代数方面,通过对李代数的分解得到了Semenov-Tian-Shansky关于李群的经典基本分解定理.而李代数的分解可由罗巴算子给出.自然会问,在李群上是否有一个罗巴算子可以直接进行李群分解?不幸的是,这样的算子以前是不知道的..另一个重要的问题是,如何在罗巴李(结合)代数上建立一个恰当的形变和上同调理论.事实上,罗巴代数的operad是具有非平凡的一元运算,且不是二元二次的,这意味着已有的形变和上同调一般理论不适用于罗巴代数..本项目的研究为这些问题以及其他相关的结构,问题和应用提供了解决方案,特别是在Hopf代数,表示和重整化方面.