论Ore扩张中的局部有限导子
On Locally Finite Derivations in Ore Extensions
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中文总结 AI 辅助
将van den Essen的多项式代数局部有限导子分类推广到非交换Ore扩张情形，完成三类Ore扩张的局部有限导子分类并证明相关子代数性质。
中文摘要 AI 辅助
设𝕜为特征零的代数闭域，我们对任意𝕜[x]的Ore扩张的局部有限导子进行分类，从而将van den Essen对多项式代数𝕜[x,y]的分类结果推广到非交换情形。更确切地说，我们考虑𝕜[x]的Ore扩张分类中出现的三类代数：量子平面、第一量子Weyl代数以及微分Ore扩张A_h=𝕜[x][t;h(x)∂_x]。对于量子平面和第一量子Weyl代数，我们确定其局部有限导子，并说明所得分类与Suárez-Alvarez和Vivas关于广义Weyl代数的工作的关联。对于h非恒定的代数A_h，我们在无平方因子与有平方因子两种情形下均得到完整分类。由此，我们证明LFD(A_h)是Der(A_h)的可解且弱局部有限李子代数，尽管作为导子集它并非局部有限。
英文摘要
Let $\Bbbk$ be an algebraically closed field of characteristic zero. We classify the locally finite derivations of arbitrary Ore extensions of $\Bbbk[x]$, thus extending van den Essen's \cite{V92} classification for the polynomial algebra $\Bbbk[x,y]$ to this noncommutative setting. More precisely, we consider the three families arising in the classification of Ore extensions of $\Bbbk[x]$: the quantum plane, the first quantum Weyl algebra, and the differential Ore extensions \[ A_h=\Bbbk[x][t;h(x)\partial_x]. \] For both the quantum plane and the first quantum Weyl algebra, we determine the locally finite derivations and explain how the resulting classifications are related to the work of Suárez-Alvarez and Vivas \cite{SuarezVivas} on generalized Weyl algebras. For the algebras $A_h$, with $h$ nonconstant, we obtain a complete classification in both the square-free and non-square-free cases. As a consequence, we show that $\LFD(A_h)$ is a solvable and weakly locally finite Lie subalgebra of $\Der(A_h)$, although it is not locally finite as a set of derivations.
