Primeness criteria for universal enveloping algebras of Lie color algebras

Let L = L+ circle plus L- be a Lie color algebra with dim L- < <infinity>. We write det L not equal 0 if the matrix formed by brackets between elements of a basis of L- is nonsingular. Unlike Lie super algebras, a Lie color algebra L may have det L not equal 0 and a universal enveloping algebra U(L) which is not prime. We will provide examples and show that U(L) is semiprime whenever det L not equal 0. Our main theorem is a criterion for U(L) to be prime. As a corollary, we prove that U(L) is prime whenever det L not equal 0 and the grading group G is either a finite group whose 2-torsion subgroup is cyclic or a finitely generated group such that for each elementary divisor 2(l) of G the base field does not contain a primitive (2(l))th-root of unity. (C) 2001 Academic Press.