Delta Methods in Enveloping Algebras of Lie Superalgebras, II

AbstractLet L = L0⊕L1 be a Lie superalgebra over a field K of characteristic 0 with enveloping algebra U(L) or let L be a restricted Lie superalgebra over a field K of characteristic p > 2 with restricted enveloping algebra U(L). In this paper we continue our study of linear identities in U(L) and sharpen the previously known results in several ways. Specifically, we show that the Lie superideal Δ = ΔL = {l ∈ L|dimK[L, l] < ∞}, considered in earlier work, can be replaced by ΔL, the join of all finite-dimensional superideals of L. Since ΔL can be appreciably smaller than Δ when K has characteristic 0, these new results are correspondingly stronger than the older ones. Next, when L1 was allowed to be infinite dimensional, the earlier results on linear identities required that Δ be contained in L0, the even part of L. Here we are able to totally eliminate this annoying hypothesis. Finally, we show that the results obtained are in fact independent of the special nature of any basis used in the course of the proof. As a consequence, we conclude that the center and the semi-invariants of U(L) are supported by the finite-dimensional superideals of L. Furthermore, if ΔL = 0, then U(L) is prime, the natural automorphism σ of order 2 of L is X-outer when L1 ≠ 0, and the adjoint representation of U(L) on U(L) is faithful