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首页> 外文会议>Proceedings of the 36th international symposium on symbolic and algebraic computation. >On Rota's Problem for Linear Operators in Associative Algebras
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On Rota's Problem for Linear Operators in Associative Algebras

机译：关联代数中线性算子的Rota问题

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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. 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. 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 have 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.
机译：对于关联代数，Gian-Carlo Rota长期存在的问题是可以在其上定义的所有线性算子的分类。在1970年代，只有很少的已知算子，例如导数算子，差算子，平均值算子和Rota-Baxter算子。罗塔提出他的问题后，又出现了一些情况。但是，总体上解决该问题的进展很小。在某种程度上，这是因为对该问题的确切含义没有很好的理解。在本文中，我们提出了一个使用可操作代数框架的问题的表述，并将带有线性算子的关联代数视为满足某个可操作多项式恒等式的代数。为了使我们的注意力集中在Rota感兴趣的运算符上，我们进一步考虑两类特殊的运算符，即广义差分运算符或Rota-Baxter运算符。借助计算机代数，我们可以列出这两种运算符的列表，并提供一些证据证明这些列表可能是完整的。我们的搜索发现了许多这类新的算子，它们的特性分别期望与微分算子和Rota-Baxter算子相似。

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