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-prim e 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 characteri stic not equal 2. Specifically; we show that if i = root-1 epsilon K a nd if U(L) has a unique minimal graded-prime ideal, then this ideal is : necessarily prime. As will be apparent, the latter result follows qu ickly from the existence of an anti-automorphism of U(L) whose square is the automorphism of the enveloping algebra associated with its Z(2) -grading. The second section, which is independent of the first, studi es more general Lie color algebras and shows that if U(L) is graded-pr ime and if most homogeneous components L-g of L are infinite dimension al 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 gen erated by the infinite support of L, then we prove that C = C-1 is hom ogeneous.