Enveloping algebras of lie color algebras: primeness versus graded-primeness

Let $G$ be a finite abelian group and let $L$ be a, possibly restricted, $G$-graded 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 $\neq 2$. Specifically, we show that if $i=\sqrt {-1}\in K$ 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 anti-automorphism of $U(L)$ whose square is the automorphism of the enveloping algebra associated with its ${\mathbb{Z}}_{2}$-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 $L_{g}$ of $L$ are infinite dimensional over $K$, then $U(L)$ is prime. Here we use $\Delta$-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 $C=C_{1}$ is homogeneous.