Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

Abstract

In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with $\mathrm{char}(\mathbb{F})\ne 2$. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by $L_n$, where $n={\dim }_{\mathbb{F}}{L}_n$. This generalizes the result found in [11], which is only valid when $\mathbb{F}=\mathbb{C}$. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of $L_n$. Eventually, we solve the coquecigrue problem for $L_n$ by integrating it into a Lie rack.

arXiv commentFinal version, accepted for publication

References in corpus (5)

• Two-step nilpotent Leibniz algebras
• Derivations of two-step nilpotent algebras
• Biderivations of low-dimensional Leibniz algebras
• On the representability of actions of Leibniz algebras and Poisson algebras
• Classification of some Solvable Leibniz Algebras

Cited by in corpus (1)

• On Lie-holomorphs of Leibniz algebras