Ore扩张的导子的局部幂零性刻画
A Characterization of Local Nilpotence for Derivations of Ore Extensions
AI总结：
本文研究Pan关于局部幂零导子迷向群刻画的非交换推广，证明其对微分Ore扩张成立，对第一Weyl代数局部有限导子成立，并构造自由结合代数中反例。
AI中文摘要：
设$\Bbbk$是特征为零的代数闭域。I. Pan的一个近期定理刻画了$\Bbbk[x,y]$的非零局部幂零导子，其刻画基于它们的迷向群：这样的导子是局部幂零的当且仅当其迷向群包含任意大次数的自同构。我们研究了与仿射平面相关的几个非交换代数的这种刻画的类似物。我们的主要结果表明，Pan的刻画可以推广到微分Ore扩张$A_h=\Bbbk[x][t;h(x)\partial_x]$，其中$h\in\Bbbk[x]\setminus\Bbbk$。对于第一Weyl代数$A_1$，我们为每个非零局部有限导子建立了相同的等价性。对于量子平面和第一量子Weyl代数，已知结果意味着相应的等价性对非零导子空泛地成立。相反，对于自由结合代数$\Bbbk\langle x,y\rangle$，逆命题不成立：我们构造了一个非局部幂零的内导子，其迷向群具有无界次数。我们进一步通过自同构和导子在交换化下的行为来分析这一失败。
英文摘要：
Let $\Bbbk$ be an algebraically closed field of characteristic zero. A recent theorem of I. Pan characterizes the nonzero locally nilpotent derivations of $\Bbbk[x,y]$ in terms of their isotropy groups: such a derivation is locally nilpotent if and only if its isotropy group contains automorphisms of arbitrarily large degree. We study analogues of this characterization for several noncommutative algebras related to the affine plane. Our main result shows that Pan's characterization extends to the differential Ore extensions $A_h=\Bbbk[x][t;h(x)\partial_x]$, where $h\in\Bbbk[x]\setminus\Bbbk$. For the first Weyl algebra $A_1$, we establish the same equivalence for every nonzero locally finite derivation. For the quantum plane and the first quantum Weyl algebra, known results imply that the corresponding equivalence holds vacuously for nonzero derivations. In contrast, the converse fails for the free associative algebra $\Bbbk\langle x,y\rangle$: we construct a non-locally-nilpotent inner derivation whose isotropy group has unbounded degree. We further analyze this failure through the behavior of automorphisms and derivations under abelianization.
