On Levi’s Theorem for Leibniz Algebras

Abstract

A Lie algebra over a field of characteristic 0 splits over its soluble radical and all complements are conjugate. I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not. The Levi-Malcev theorem asserts that, if L is a finite-dimensional Lie algebra over a field of characteristic 0, then L splits over its soluble radical and that all complements are conjugate. In this note, I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not. As for Lie algebras, the splitting theorem achieves some reduction of the investigation of representations of Leibniz algebras to problems of Leibniz representations of semi-simple Lie algebras and of representations of soluble Leibniz algebras. T 1. Let L be a finite-dimensional Leibniz algebra over a field of characteristic 0 and let R be its soluble radical. There exists a semi-simple subalgebra S of L such that S + R = L and S ∩ R = 0.

Leibniz Algebras Associated to Extensions ofsl2

“…From the analogue of Levi's decomposition for Leibniz algebras (see [4]), we conclude that the above construction is universal for a simple Leibniz algebra assuming G = L/I M = I Q = L, that is, any simple Leibniz algebra is realized by this construction.…”

“…Recently, Barnes proved an analogue of Levi's Theorem for Leibniz algebras [4]. Namely, a Leibniz algebra is decomposed into a semidirect sum of its solvable radical and a semisimple Lie algebra.…”

Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type L4

“…We follow the steps in Theorems 5.2.1, 5.2.2 and 5.2.3 to find codimension one left solvable extensions. We notice in Theorem 5.2.3, that we have eight of them as well: l n+1,1 , l n+1,2 , l n+1, 3 , g n+1, 4 , l 5,5 , l 5,6 , g 5,7 and g 5,8 , such that l n+1,1 is right when a = 0 and l 5,5 is right when b = −1. We find four solvable indecomposable left Leibniz algebras with a codimension two nilradical as well: l n+2,1 , (n ≥ 5), l 6,2 , l 6,3 and l 6,4 stated in Theorem 5.2.4, where none of them is right.…”

“…Since then many analogs of important theorems in Lie theory were found to be true for Leibniz algebras, such as the analogue of Levi's theorem which was proved by Barnes [3]. He showed that any finite-dimensional complex Leibniz algebra is decomposed into a semidirect sum of the solvable radical and a semisimple Lie algebra.…”

Algorithmic method to obtain combinatorial structures associated with Leibniz algebras

“…An example of this fact is the still-unsolved classification of Lie and Leibniz algebras. According to Levi's and Malcev's theorems (see [13,15], respectively) for Lie algebras and the analogue for Leibniz algebras (see [1]), the classification of these algebras only requires classifying semisimple and solvable algebras when considering fields of characteristic zero. Semisimple Lie algebras were classified by Killing and Cartan in 1890 over the complex number field and semisimple Leibniz algebras were studied in [11] and in some cases they can be decomposed into the direct sum of simple Lie ideals for fields of characteristic zero.…”