SOLVABLE LEIBNIZ ALGEBRAS WITH ABELIAN NILRADICAL

1. IntroductionLeibniz algebras were defined by Loday in 1993[13, 14]. Recently, there has been a trend toshow how various results from Lie algebras extend to Leibniz algebras [1, 3, 17]. In particular,there has been interest in proving results on nilpotency and related concepts which can beused to help broaden properties of Lie algebras to Leibniz algebras. Specifically, variationsof Engel’s theorem for Leibniz algebras have been proven by different authors [6, 8], andBarnes has proven Levi’s theorem for Leibniz algebras [4] and shown that left-multiplicationby any minimal ideal of a Leibniz algebra is either zero or anticommutative [5].Inanefforttoclassify Lie algebras, many authorsplace variousrestrictions onthenilradical[10, 16, 20, 21]. In [16], Ndogmo and Winternitz study solvable Lie algebras with Abeliannilradical. The goal of this paper is to utilize these results to develop some results in theLeibniz setting.Recent work has been done on classification of certain classes of Leibniz algebras including[2, 7, 11, 12]. We construct a general classification theorem for solvable Leibniz algebras withAbelian nilradical over C. Furthermore, we discuss, in detail, the case of one-dimensionalextensions, and provide an explicit classification of all one-dimensional extensions of one-,two-, and three-dimensional Abelian nilradicals. In [9], Can˜ete and Khudoyberdiyev clas-sify all non-nilpotent 4-dimensional Leibniz algebras over C. Our classification of the one-dimensional extensions of three-dimensional Abelian nilradicals recovers this result. We