PreprintCC BY-NC-ND 4.0arXiv
Local Derivations and Local Automorphisms of Arbitrary Finite-Dimensional Nilpotent Lie and Leibniz Algebras
Abstract
We study local derivations and local automorphisms of finite-dimensional nilpotent complex Lie and right Leibniz algebras. We prove that every non-abelian Lie algebra in this class admits both a local derivation which is not a derivation and a local automorphism which is not an automorphism. The constructions use a derivation D satisfying 0≠ D([L,L])⊆ Z(L). For non-abelian right Leibniz algebras, both existence statements hold whenever the dimension is at most seven or the nilpotency class is at most four. We construct an eight-dimensional non-Lie algebra of class five satisfying LocDer(A)=Der(A) and LocAut(A)=Aut(A). Consequently, eight and five are the least possible dimension and nilpotency class, respectively, for either equality.
Related
- 1/2 derivations and biderivations of 3-dimensional nilpotent Lie-Yamaguti algebras
- An eight-dimensional nilpotent Leibniz algebra without outer derivations
- Local and 2-Local Derivations of Finite-Dimensional Zinbiel Algebras
- The algebraic classification of five-dimensional nilpotent right alternative algebras
- On the Structure of Low-Dimensional Poisson Algebras over Arbitrary Fields
BibTeX
@misc{atajonov2026local,
title = {{Local Derivations and Local Automorphisms of Arbitrary Finite-Dimensional Nilpotent Lie and Leibniz Algebras}},
author = {Khusainboy Atajonov and Abror Khudoyberdiyev and Bakhtiyor Yusupov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40348},
eprint = {2610.40348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40348},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Local Derivations and Local Automorphisms of Arbitrary Finite-Dimensional Nilpotent Lie and Leibniz Algebras
AU - Khusainboy Atajonov
AU - Abror Khudoyberdiyev
AU - Bakhtiyor Yusupov
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40348
UR - https://arxiv.org/abs/2610.40348
AB - We study local derivations and local automorphisms of finite-dimensional nilpotent complex Lie and right Leibniz algebras. We prove that every non-abelian Lie algebra in this class admits both a local derivation which is not a derivation and a local automorphism which is not an automorphism. The constructions use a derivation D satisfying 0≠ D([L,L])⊆ Z(L). For non-abelian right Leibniz algebras, both existence statements hold whenever the dimension is at most seven or the nilpotency class is at most four. We construct an eight-dimensional non-Lie algebra of class five satisfying LocDer(A)=Der(A) and LocAut(A)=Aut(A). Consequently, eight and five are the least possible dimension and nilpotency class, respectively, for either equality.
ER -