Some remarks on semisimple Leibniz algebras

TL;DR

This paper examines the structure of semisimple Leibniz algebras, demonstrating that the classical splitting theorem does not hold universally and identifying conditions for their decomposition into simple ideals.

Contribution

It extends the analysis of semisimple Leibniz algebras, showing the failure of the splitting theorem and providing conditions for their decomposition into simple components.

Findings

The splitting theorem for semisimple Leibniz algebras is not generally valid.

Certain classes of semisimple Leibniz algebras can decompose into simple ideals under specific conditions.

The paper extends Levi's theorem to Leibniz algebras and explores their structural properties.

Abstract

From the Levi's Theorem it is known that every finite dimensional Lie algebra over a field of characteristic zero is decomposed into semidirect sum of solvable radical and semisimple subalgebra. Moreover, semisimple part is the direct sum of simple ideals. In \cite{Bar} the Levi's theorem is extended to the case of Leibniz algebras. In the present paper we investigate the semisimple Leibniz algebras and we show that the splitting theorem for semisimple Leibniz algebras is not true. Moreover, we consider some special classes of the semisimple Leibniz algebras and find a condition under which they decompose into direct sum of simple ideals.