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 \({\text {char}}(\mathbb {F}) \ne 2\). 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 Demir et al. (Algebras and Representation Theory 19:405-417, 2016), 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.