A Characterization of Local Nilpotence for Derivations of Ore Extensions
Abstract
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 Ah=Bbbk[x][t;h(x)∂x], where h∈Bbbk[x]∖Bbbk. For the first Weyl algebra A1, 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⟨ x,y⟩: 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.
Related
- Local Derivations on Conformal Galilei Algebras and Their Central Extensions
- 2-Local Derivations on Conformal Galilei Lie Algebras and Their Central Extensions
- A Characterization of Functions on the Integer Lattice That Are L-Extendable through Local Convex Extensions
- Derivations of Lie p-rings with applications on ring extensions and computational methods
- n-Derivations of Lie Superalgebras: Structure, Decomposition, and Central Extensions
BibTeX
@misc{baltazar2026characterization,
title = {{A Characterization of Local Nilpotence for Derivations of Ore Extensions}},
author = {Rene Baltazar and Samuel A. Lopes and Oscar Morales},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19470},
eprint = {2609.19470},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19470},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A Characterization of Local Nilpotence for Derivations of Ore Extensions
AU - Rene Baltazar
AU - Samuel A. Lopes
AU - Oscar Morales
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.19470
UR - https://arxiv.org/abs/2609.19470
AB - 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)∂_x], where h∈Bbbk[x]∖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⟨ x,y⟩: 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.
ER -