Primeness Criteria for Universal Enveloping Algebras of Lie Color Algebras

Let = + − be a Lie color algebra with dim − < ∞. We write det ≠ 0 if the matrix formed by brackets between elements of a basis of − is nonsingular. Unlike Lie super algebras, a Lie color algebra may have det ≠ 0 and a universal enveloping algebra U( ) which is not prime. We will provide examples and show that U( ) is semiprime whenever det ≠ 0. Our main theorem is a criterion for U( ) to be prime. As a corollary, we prove that U( ) is prime whenever det ≠ 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 2l of G the base field does not contain a primitive (2l)th-root of unity.