Simple lie color algebras of weyl type

Abstract

A class of graded simple associative algebras are constructed, and from them, simple Lie color algebras are obtained. The structure of these simple Lie color algebras is explicitly described. More precisely, for an (ε, Γ)-color-commutative associative algebraA with an identity element over a fieldF of characteristic not 2, and for a color-commutative subalgebraD of color-derivations ofA, denote byA[D] the associative subalgebra of End (A) generated byA (regarded as operators onA via left multiplication) andD. It is easily proved that, as an associative algebra,A[D] is Γ-graded simple if and only ifA is Γ-gradedD-simple. SupposeA is Γ-gradedD-simple. Then, (a)A[D] is a free leftA-module; (b) as a Lie color algebra, the subquotient [A[D],A[D]]/Z(A[D])∩[A[D],A[D]] is simple (except one minor case), whereZ(A[D]) is the color center ofA[D].