$\mathcal{D}$-局部幂零代数、其理想结构及单性判据
$\D$-locally nilpotent algebras, their ideal structure and simplicity criteria
AI总结：
本文引入并研究$\mathcal{D}$-局部幂零代数，刻画其理想结构并给出单性判据，将微分算子代数的已有结果推广到更广泛的代数类。
摘要：
$\mathcal{D}$-局部幂零代数这一类代数（在本文中引入）是对交换代数上微分算子代数的广泛推广。其例子包括：交换代数上的所有微分算子环$\mathcal{D}(A)$（在任意特征下）、$\mathcal{D}(A)$中包含代数$A$的所有子代数、幂零、可解及半单李代数的泛包络代数、任意Poisson代数的Poisson泛包络代数、迭代Ore扩张$A[x_1,\ldots,x_n;\delta_1,\ldots,\delta_n]$、某些广义Weyl代数，以及其他代数。在文献\cite{SimCrit-difop}中，给出了交换代数上微分算子代数的单性判据（这曾是一个长期悬而未决的问题）。本文旨在描述$\mathcal{D}$-局部幂零代数的理想结构，并作为推论给出它们的单性判据（这是对文献\cite{SimCrit-difop}结果的推广）。文中还考虑了若干例子。
英文摘要：
The class of $\D$-locally nilpotent algebras (introduced in the paper) is a wide generalization of the algebras of differential operators on commutative algebras. Examples includes all the rings $\CD (A)$ of differential operators on commutative algebras (in arbitrary characteristic), all subalgebras of $\CD (A)$ that contain the algebra $A$, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions $A[x_1, \ldots , x_n ; \d_1 , \ldots , \d_n]$, certain generalized Weyl algebras, and others. In \cite{SimCrit-difop}, simplicity criteria are given for the algebras differential operators on commutative algebras (it was a long standing problem). The aim of the paper is to describe the ideal structure of $\D$-locally nilpotent algebras and as a corollary to give simplicity criteria for them (it is a generalization of the results of \cite{SimCrit-difop}). Examples are considered.
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。
