Non-Nilpotent Leibniz Algebras with One-Dimensional Derived Subalgebra

Abstract

In this paper we study non-nilpotent non-Lie Leibniz $$\mathbb {F}$$ F -algebras with one-dimensional derived subalgebra, where $$\mathbb {F}$$ F is a field with $${\text {char}}(\mathbb {F}) \ne 2$$ 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 $$L_n$$ L n , where $$n=\dim _\mathbb {F}L_n$$ n = dim 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 $$\mathbb {F}=\mathbb {C}$$ F = C . Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of $$L_n$$ L n . Eventually, we solve the coquecigrue problem for $$L_n$$ L n by integrating it into a Lie rack.