Simple Lie color algebras from graded associative algebras

In this paper, a complete generalization of Hersteinʹs theorem to the case of Lie color algebras is obtained. Let G be an abelian group, F a field of characteristic not 2, ε :G×G→F* an anti-symmetric bicharacter. Suppose A= g GAg is a G-graded simple associative algebra over F. In this paper it is proved that [A,A]ε/([A,A]ε∩Zε(A)) is a simple (ε,G)-Lie color algebra if dimZεA>8, where Zε=Zε(A) is the color center of A. If A(3)≠0 and dimZεA=8, then there are two such algebras A such that [A,A]ε/(Zε∩[A,A]ε) is not simple or commutative. This extends a result by Montgomery.