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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 Ah=Bbbk[x][t;h(x)∂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 Ah, 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(Ah) is a solvable and weakly locally finite Lie subalgebra of Der(Ah), although it is not locally finite as a set of derivations.

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Cite

BibTeXRIS
BibTeX
@misc{baltazar2026locally,
  title = {{On Locally Finite Derivations in Ore Extensions}},
  author = {R. Baltazar and A. Bianchi and M. Veloso and J. Schwarz},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28257},
  eprint = {2608.28257},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28257},
}
RIS
TY  - UNPB
TI  - On Locally Finite Derivations in Ore Extensions
AU  - R. Baltazar
AU  - A. Bianchi
AU  - M. Veloso
AU  - J. Schwarz
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28257
UR  - https://arxiv.org/abs/2608.28257
AB  - 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)∂_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.
ER  -