arXiv 1210.18152012-10-05math.RA

微分型算子与Gröbner-Shirshov基

Differential Type Operators and Gröbner-Shirshov Bases

Li Guo, William Y. Sit, Ronghua Zhang

备注：: 31 pages; Journal of Symbolic Computation
发表信息：: J. Symbolic Computation, 52 (2013), 97-123
DOI：: 10.1016/j.jsc.2012.05.014

摘要：

Gian-Carlo Rota关于结合代数的一个长期存在的问题是所有可定义在其上的线性算子的分类。在20世纪70年代，只有少数已知的算子，例如，导数算子，差分算子，平均算子和Rota-Baxter算子。在罗塔提出他的问题后，又出现了几个。然而，在解决这一问题方面总体上进展甚微。部分原因是人们对这个问题的确切含义还没有很好地理解。在本文中，我们提出了一个制定的问题，使用的框架操作代数和查看一个结合代数与线性算子作为一个满足一定的操作多项式单位。这个框架还允许我们应用重写系统和Gröbner-Shirshov基的理论。为了缩小我们的重点更多的运营商，罗塔感兴趣的，我们进一步考虑两个特殊类别的运营商，即那些推广微分或Rota-Baxter运营商。事实证明，这两类算子对应于那些在两种不同的单项序下拥有Gröbner-Shirshov基的算子。在这个框架下工作，并借助计算机代数，我们能够得出这两类运算符的列表，并提供一些证据证明这些列表可能是完整的。我们的搜索已经发现了相当多的新的运营商的这些类型，其性质预计类似的微分算子和Rota-Baxter运营商分别。最近，在相关领域出现了一个更统一的方法，如差代数和微分代数，Rota-Baxter代数和Nijenhuis代数。这些理论中的相似之处可以通过罗塔问题的进展来更有效地探索。

英文摘要：

A long standing problem of Gian-Carlo Rota for associative algebras is the classification of all linear operators that can be defined on them. In the 1970s, there were only a few known operators, for example, the derivative operator, the difference operator, the average operator, and the Rota-Baxter operator. A few more appeared after Rota posed his problem. However, little progress was made to solve this problem in general. In part, this is because the precise meaning of the problem is not so well understood. In this paper, we propose a formulation of the problem using the framework of operated algebras and viewing an associative algebra with a linear operator as one that satisfies a certain operated polynomial identity. This framework also allows us to apply theories of rewriting systems and Gröbner-Shirshov bases. To narrow our focus more on the operators that Rota was interested in, we further consider two particular classes of operators, namely, those that generalize differential or Rota-Baxter operators. As it turns out, these two classes of operators correspond to those that possess Gröbner-Shirshov bases under two different monomial orderings. Working in this framework, and with the aid of computer algebra, we are able to come up with a list of these two classes of operators, and provide some evidence that these lists may be complete. Our search has revealed quite a few new operators of these types whose properties are expected to be similar to the differential operator and Rota-Baxter operator respectively. Recently, a more unified approach has emerged in related areas, such as difference algebra and differential algebra, and Rota-Baxter algebra and Nijenhuis algebra. The similarities in these theories can be more efficiently explored by advances on Rota's problem.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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