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Let G be a finite abelian group and let L be a, possibly restricted, Ggraded Lie color algebra. Then the enveloping algebra U(L) is also G-graded, and we consider the question of whether U(L) being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field K of characteristic 6= 2. Specifically, we show that if i = √ −1 ∈ K and if U(L) has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an antiautomorphism of U(L) whose square is the automorphism of the enveloping algebra associated with its Z2-grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if U(L) is graded-prime and if most homogeneous components Lg of L are infinite dimensional over K, then U(L) is prime. Here we use ∆-methods to study the grading on the extended centroid C of U(L). In particular, if G is generated by the infinite support of L, then we prove that C = C1 is homogeneous.

Enveloping algebras of Lie color algebras: Primeness versus graded-primeness