Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

TL;DR

This paper classifies a specific class of non-nilpotent Leibniz algebras with one-dimensional derived subalgebra over fields with characteristic not equal to 2, and explores their automorphisms, derivations, and integration into Lie racks.

Contribution

It generalizes previous results to arbitrary fields with characteristic not 2 and explicitly describes the structure, automorphisms, derivations, and integration of these Leibniz algebras.

Findings

Algebras are isomorphic to a direct sum of a 2D non-nilpotent Leibniz algebra and an abelian algebra.

Explicit description of derivations, automorphisms, and biderivations for these algebras.

Solution of the coquecigrue problem by integrating into a Lie rack.

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 $\operatorname{char}(\mathbb{F}) \neq 2$. 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.