Some remarks on semisimple Leibniz algebras

Abstract

From Levi's Theorem it is known that every finite-dimensional Lie algebra over a field of characteristic zero is decomposed into semidirect sum of its solvable radical and semisimple subalgebra. Moreover, semisimple part is the direct sum of simple ideals. In Barnes (preprint) [6] 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 we find a condition under which they decompose into direct sum of simple ideals.