Delta Ideals of Lie Color Algebras

Abstract

Let L = ⊕g ∈ GLg be a Lie color algebra (possibly restricted) over the field K and graded by the finite abelian group G. If Δ∞(L) = {l ∈ L |dimK[l, L] is countable}, then Δ∞(L) is the (restricted) Lie color ideal of L generated by all (restricted) countable-dimensional Lie color ideals of L. We use Δ∞(L) to examine the symmetric Martindale quotient ring of the enveloping algebra U(L) (or the restricted enveloping algebra when char K = p > 0). Specifically, we prove THEOREM. If Δ∞(L) = 0, thenU(L) is symmetrically closed. We also examine the Lie color ideal Δ(L) = {l ∈ L |dimK[l, L] is finite} and the possibly smaller ideal ΔL, which is the join of all finite-dimensional Lie color ideals of L. Note that Δ(L) = ΔL when char K = p > 0, but that Δ(L) can be considerably larger than ΔL, when char K = 0. Nevertheless, we prove THEOREM. [Δ(L), Δ(L)] ⊆ ΔL. We remark that these results are new and of interest even when L is an ordinary or super Lie algebra. In fact, we consider Lie color algebras here only because we can obtain the more general facts with little additional work.