Local and 2-Local Derivations of Finite-Dimensional Zinbiel Algebras
Abstract
We study local and $2$-local derivations of finite-dimensional Zinbiel algebras. Over any field of characteristic zero, every Zinbiel algebra A with A2 ≠ 0 and dim A ≤ 4 admits a proper local derivation. The proof uses a filtration argument. We exhibit two five-dimensional Zinbiel algebras with nonzero multiplication whose derivations are all central. Every local derivation on either algebra is therefore a derivation, so the minimum dimension for local rigidity with nonzero multiplication is five. Annihilator-valued interpolation gives a general construction of proper local derivations. When A3 = 0, the multiplication tensor describes all derivations and local derivations. For a finite-dimensional nilpotent algebra A over any field, we show that every $2$-local derivation is a derivation if and only if dim(A/A2) ≤ 1. The proof combines the annihilator interpolation of Sh. Ayupov, K. Kudaybergenov and B. Omirov with a one-generator argument. When all derivations are central, every $2$-local derivation is obtained by composing a homogeneous map A/A2 → Ann(A) with the quotient map, and every such composition is a $2$-local derivation. The five-dimensional examples extend to locally rigid Zinbiel algebras with nonlinear $2$-local derivations in every dimension at least five. We also show that derivations given by left multiplication form a Lie ideal of nilpotency class at most two and give sufficient conditions for the existence of left outer derivations. This ideal is zero in our locally rigid examples.
Related
- Local Derivations and Local Automorphisms of Arbitrary Finite-Dimensional Nilpotent Lie and Leibniz Algebras
- Local and 2-local derivations of Block Lie algebras B(q)
- Lie Algebras of Locally Finite Derivations over Fields of Positive Characteristic
- Locally finite solvable Lie algebras of derivations
- On the Isotropy Groups of Locally Finite Derivations of Laurent Polynomial Algebras
BibTeX
@misc{abdireymov2026local,
title = {{Local and 2-Local Derivations of Finite-Dimensional Zinbiel Algebras}},
author = {Arislanbay Abdireymov and Aloberdi Sattarov and Bakhtiyor Yusupov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07863},
eprint = {2611.07863},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07863},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Local and 2-Local Derivations of Finite-Dimensional Zinbiel Algebras
AU - Arislanbay Abdireymov
AU - Aloberdi Sattarov
AU - Bakhtiyor Yusupov
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07863
UR - https://arxiv.org/abs/2611.07863
AB - We study local and $2$-local derivations of finite-dimensional Zinbiel algebras. Over any field of characteristic zero, every Zinbiel algebra A with A^2 ≠ 0 and dim A ≤ 4 admits a proper local derivation. The proof uses a filtration argument. We exhibit two five-dimensional Zinbiel algebras with nonzero multiplication whose derivations are all central. Every local derivation on either algebra is therefore a derivation, so the minimum dimension for local rigidity with nonzero multiplication is five. Annihilator-valued interpolation gives a general construction of proper local derivations. When A^3 = 0, the multiplication tensor describes all derivations and local derivations. For a finite-dimensional nilpotent algebra A over any field, we show that every $2$-local derivation is a derivation if and only if dim(A/A^2) ≤ 1. The proof combines the annihilator interpolation of Sh. Ayupov, K. Kudaybergenov and B. Omirov with a one-generator argument. When all derivations are central, every $2$-local derivation is obtained by composing a homogeneous map A/A^2 → Ann(A) with the quotient map, and every such composition is a $2$-local derivation. The five-dimensional examples extend to locally rigid Zinbiel algebras with nonlinear $2$-local derivations in every dimension at least five. We also show that derivations given by left multiplication form a Lie ideal of nilpotency class at most two and give sufficient conditions for the existence of left outer derivations. This ideal is zero in our locally rigid examples.
ER -