DELTA-METHODS IN ENVELOPING-ALGEBRAS OF LIE-SUPERALGEBRAS .2.

[摘要] Let 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 DELTA = DELTA(L) = {l is-an-element-of L\dim(K)[L, l] < infinity}, considered in earlier work, can be replaced by DELTA(L), the join of all finite-dimensional superideals of L. Since DELTA(L) can be appreciably smaller than DELTA 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 DELTA 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 DELTA(L) = 0, then U(L) is prime, the natural automorphism sigma of order 2 of L is X-outer when L1 not-equal 0, and the adjoint representation of U(L) on U(L) is faithful. (C) 1994 Academic Press, Inc.

[发布日期] 1994-06-15