On Ideals in the Enveloping Algebra of a Locally Simple Lie Algebra

We study (two-sided) ideals I in the enveloping algebra U(g infinity) of an infinite-dimensional Lie algebra g infinity obtained as the union (equivalently, direct limit) of an arbitrary chain of embeddings of simple finite-dimensional Lie algebras g(1) -> g(2) -> ... -> g(n) -> ... with lim(n ->infinity) dim g(n)= infinity. Our main result is an explicit description of the zero-sets of the corresponding graded ideals grI. We use this description and results of Zhilinskii to prove Baranov's conjecture that, if g(infinity) is not diagonal in the sense of Baranov and Zhilinskii, then U(g(infinity)) has a single nonzero proper ideal: the augmentation ideal. Our study is based on a complete description of the radical Poisson ideals in S (g(infinity)) and their zero- sets. We then discuss in detail integrable ideals of U (g(infinity)). These are certain ideals I subset of U (g(infinity)) for which I n U(g(n)) is an intersection of ideals of finite codimension in U(gn) for any n >= 1. We present a classification of prime integrable ideals based on work of Zhilinskii. For g(infinity)congruent to sl infinity, so infinity, all zero-sets of radical Poisson ideals of S (g(infinity)) arise from prime integrable ideals of U(g(infinity)). For g(infinity)congruent to sp(infinity) only half of the zerosets of Poisson ideals of S . (g(infinity)) arise from integrable ideals of U(g(infinity)).