On the structure of split Lie color algebras

Abstract

The class of Lie color algebras contains the one of Lie superalgebras and so the one of Lie algebras. In order to begin an approach to the structure of arbitrary Lie color algebras, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Lie color algebras as the natural extension of the classes of split Lie algebras and split Lie superalgebras. By developing techniques of connections of roots for this kind of algebra, we show that any such algebra L is of the form L = U + ∑ j I j with U a subspace of the abelian (graded) subalgebra H and eachx I j a well described (graded) ideal of L satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal (graded) ideals, each one being a simple split Lie color algebra.