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arXiv · 2608.28257

On Locally Finite Derivations in Ore Extensions

R. Baltazar·A. Bianchi·M. Veloso·J. Schwarz
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Abstract

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 Su\'arez-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.

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R. Baltazar, A. Bianchi, M. Veloso, J. Schwarz. 2026-08-28. On Locally Finite Derivations in Ore Extensions. https://arxiv.org/abs/2608.28257

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