Leibniz algebras with an abelian subalgebra of codimension tw0

Abstract

A characterization of the finite-dimensional Leibniz algebras with an abelian subalgebra of codimension two over a field $\mathbb{F}$ of characteristic $p\ne 2$ 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 $\mathfrak{h}(\mathbb{F})\oplus {\mathbb{F}}^{n-4}$.