DELTA-IDEALS OF LIE COLOR ALGEBRAS

Let L = +(g is an element of G) L(g) be a Lie color algebra (possibly restricted) over the field K and graded by the finite abelian group G. If Delta(x)(L) = {l is an element of L \ dim(K)[l, L] is countable}, then Delta(x)(L) is the (restricted) Lie color ideal of L. generated b y all (restricted) countable-dimensional Lie color ideals of L. We use Delta(x)(L) to examine the symmetric Martindale quotient ring of the enveloping algebra U(L) (or the restricted enveloping algebra when cha r K = p > 0). Specifically, we prove THEOREM. If Delta(x)(L) = 0, then U(L) is symmetrically closed. We also examine the Lie color ideal Del ta(L) = (l is an element of L \ dim(K)[l, L] is finite) and the possib ly smaller ideal Delta(L), which is the join of all finite-dimensional Lie color ideals of L. Note that Delta(L) = Delta(L) when char K = p > 0, but that Delta(L) can be considerably larger than a, when char K = 0. Nevertheless, we prove THEOREM. [Delta(L), Delta(L)] subset of or equal to Delta(L). We remark that these results are new and of intere st even when L is an ordinary or super Lie algebra. In fact, we consid er Lie color algebras here only because we can obtain the more general facts with little additional work. (C) 1995 Academic Press, Inc.