罗巴算子与自由模上的项重写系统

The Rota-Baxter operator is the abstraction and generation of the integral operator. Although the Rota-Baxter operator has a close relationship with the integral operator, its study originated from the floating theory of probability studied by American mathematician G.Baxter in 1960. It was also independently discovered by mathematical physicists as the operator form of the classical Yang-Baxter equation in the 1980s. Ever since the turning of the century, the study of basic theory of Rota-Baxter algebra, and important applications to broad areas, such as quantum field theory, number theory, integrable systems and so on, promote the development of this filed. Hence, this demonstrates the far-reaching significance in the further study of Rota-Baxter operators. This project is going to carry out in three aspects. The first one is to consider the Rota-Baxter operator, the integral operator and the averaging operator on the polynomial algebra, and the relationship between them. This may reveal the relation and distinction between Rota-Baxter operators and integral operators; The second one is to study the term rewriting systems on free modules, and make use of it to construct the Rota-Baxter term rewriting system. It can provide a great help for handling the classification problem of Rota-Baxter operators;The last one is to consider the classification problem of generalized Rota-Baxter type operators by using Rota-Baxter term rewriting system and Groebner-Shirshov bases theory. It is very significant for solving the open problem of the classification of linear operators posed by Rota.

罗巴算子是积分算子的抽象和推广.尽管罗巴算子与积分算子有着密切的联系,但其研究来源于美国数学家G.Baxter于1960年概率论中的浮动理论研究上世纪八十年代,罗巴算子又作为经典杨-巴克斯特方程解的算子形式被数学物理学家独立发现.本世纪以来,罗巴代数的基础理论研究和罗巴代数在量子场论,数论,可积系统等方面的应用促进了该领域的迅速发展. 因此,表明研究罗巴算子是有深远意义的.本项目拟开展三个方面的研究.第一方面是研究多项式代数上的罗巴算子,积分算子和平均算子及它们之间的关系,这可揭示罗巴算子和积分算子联系和区别; 第二方面是研究自由模上的项重写系统理论,并由此构造罗巴项重写系统,为解决罗巴算子分类问题提供了一定的帮助; 最后是利用罗巴项重写系统和Groebner-Shirshov 基研究广义罗巴型算子分类,对解决Rota提出的关于线性算子分类公开问题很有意义.

罗巴代数来自于积分算子的代数抽象和推广。由于罗巴代数在数学和理论物理中有着广泛的应用，因此它已发展成为一个公认的新兴领域。本项目首先借助于计算机代数(Mathematica)给出了2和3阶半群代数上权为0的罗巴算子分类，这对于进一步研究一般半群代数上的罗巴算子和广义罗巴型算子分类问题是有意义的。其次，项目研究了自由模上的项重写系统及其相关性质，刻画汇合单的项重写系统，同时开发子模上的项重写系统理论，丰富了项重写系统和计算机代数相关理论。紧接着，项目研究了罗巴型代数的自由对象，如自由交换罗巴代数和自由交换Nijenhuis代数，并与项重写系统理论和Groebner-Shirshov基(GS基)理论建立了联系，并获得了罗巴型算子的若干等价刻画。此结果对于构造广义自由罗巴型代数具有一定借鉴意义的。此外，项目构造了Hom-模上的自由对合Hom-结合代数，并建立了对合Hom-李代数的泛包络Hom-结合代数，由此得到了对合Hom-李代数的Poincare-Birkhoff-Witt定理(PBW定理)，因此，经典李代数中的PBW定理可看成是对合Hom-型李代数的一个特殊情形。最后，项目利用cocycle条件研究了自由交换Nijenhuis代数上的Hopf代数结构，发现了自由交换Nijenhuis代数上具有一个左余单位右对极Hopf代数结构。这些研究理论的构建为进一步研究自由Hom-罗巴型代数上的Hom-Hopf代数结构奠定了基础。

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