Non-Nilpotent Leibniz Algebras with One-Dimensional Derived Subalgebra

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) different from 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=dimL_n. This generalizes the result found in Demir et al. (Algebras and Representation Theory 19:405-417, 2016), 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 L_n. Eventually, we solve the coquecigrue problem for L_n by integrating it into a Lie rack.