Leibniz algebras with an abelian subalgebra of codimension tw0

TL;DR

This paper classifies finite-dimensional Leibniz algebras with an abelian subalgebra of codimension two over fields of characteristic not two, revealing their structure as solvable or specific direct sums involving Heisenberg and simple Lie algebras.

Contribution

It provides a complete characterization of such Leibniz algebras, detailing their structure and classification in terms of solvability and direct sum decompositions.

Findings

Leibniz algebras with abelian subalgebra of codimension two are solvable or decomposable.

They include extensions of the Heisenberg algebra and simple Lie algebras.

The classification depends on the algebra's dimension and structure.

Abstract

A characterization of the finite-dimensional Leibniz algebras with an abelian subalgebra of codimension two over a field $\mathbb{F}$ of characteristic $p\neq2$ is given. In short, a finite-dimensional Leibniz algebra of dimension $n$ with an abelian subalgebra of codimension two is solvable and contains an abelian ideal of codimension at most two or it is a direct sum of a Lie one-dimensional solvable extension of the Heisenberg algebra $\mathfrak{h}(\mathbb{F})$ and $\mathbb{F}^{n-4}$ or a direct sum of a $3$-dimensional simple Lie algebra and $\mathbb{F}^{n-3}$ or a Leibniz one-dimensional solvable extension of the algebra $\mathfrak{h}(\mathbb{F}) \oplus \mathbb{F}^{n-4}$.