Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

Abstract

In this paper we study non-nilpotent non-Lie Leibniz F-algebras with one-dimensional derived subalgebra, where F is a field with char(F) ≠ 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 Ln, where n = dimF Ln. This generalizes the result found in [11], which is only valid when F = C . Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of Ln. Eventually, we solve the coquecigrue problem for Ln by integrating it into a Lie rack.