On split Leibniz algebras

Abstract

In order to begin an approach to the structure of arbitrary Leibniz algebras, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Leibniz algebras as the natural extension of the class of split Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that any of such algebras (L,[·,·]) is of the form L=U+∑jIj with U a subspace of the abelian subalgebra H, (in the sense [H,H]=0), and any Ij a well described ideal of L, satisfying [Ij,Ik]=0 if j≠k. In the case of L being of maximal length we characterize the simplicity of the algebra in terms of connections of roots.