Automorphism Groups of Three-Dimensional Leibniz Algebras: A Complete Description
Abstract
Automorphism groups are among the most natural structural invariants of an algebra, and their explicit determination is a basic problem in the structure theory of Leibniz algebras. In a series of earlier papers, automorphism groups were obtained for several classes of low-dimensional, one-generated, nilpotent, and non-nilpotent Leibniz algebras. In the present paper we complete this picture for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are not recomputed; instead, they are incorporated by reference, with changes of basis recorded when necessary. For the remaining types we determine the automorphism groups explicitly through a direct analysis of the automorphism conditions. Particular attention is paid to exceptional parameter values and to characteristic two. A final table summarizes the automorphism group of every type in the classification.
Related
- Derivation Algebras of Three-Dimensional Leibniz Algebras: A Complete Description
- Local Derivations and Local Automorphisms of Arbitrary Finite-Dimensional Nilpotent Lie and Leibniz Algebras
- The structure of automorphism groups of zero-dimensional monomial algebras
- Kupershmidt--Nijenhuis structures on pre-Leibniz algebras
- Omni-representations of Leibniz conformal algebras
BibTeX
@misc{kurdachenko2026automorphism,
title = {{Automorphism Groups of Three-Dimensional Leibniz Algebras: A Complete Description}},
author = {Leonid A. Kurdachenko and Oleksandr O. Pypka and Mykola M. Semko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24320},
eprint = {2609.24320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24320},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Automorphism Groups of Three-Dimensional Leibniz Algebras: A Complete Description
AU - Leonid A. Kurdachenko
AU - Oleksandr O. Pypka
AU - Mykola M. Semko
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24320
UR - https://arxiv.org/abs/2609.24320
AB - Automorphism groups are among the most natural structural invariants of an algebra, and their explicit determination is a basic problem in the structure theory of Leibniz algebras. In a series of earlier papers, automorphism groups were obtained for several classes of low-dimensional, one-generated, nilpotent, and non-nilpotent Leibniz algebras. In the present paper we complete this picture for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are not recomputed; instead, they are incorporated by reference, with changes of basis recorded when necessary. For the remaining types we determine the automorphism groups explicitly through a direct analysis of the automorphism conditions. Particular attention is paid to exceptional parameter values and to characteristic two. A final table summarizes the automorphism group of every type in the classification.
ER -