2012/10/01 by Ivan Penkov, Penkov, Ivan, Alexey Petukhov +1 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.AG #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.1210.0466
arxiv created 2012/10/01 · arxiv updated 2012/10/02
We study (two-sided) ideals I in the enveloping algebra \U(\frak g_∞) of an infinite-dimensional Lie algebra \frak g_∞ obtained as the union (equivalently, direct limit) of an arbitrary chain of embeddings of simple finite-dimensional Lie algebras \frak g1→\frak g2→...→\frak gn→... with limn→∞dim\frak gn=∞. Our main result is an explicit description of the zero-sets of the corresponding graded ideals \gr I. We use this description and results of A. Zhilinskii to prove Baranov's conjecture that, if \frak g_∞ is not diagonal in the sense of A. Baranov and A. Zhilinskii, then \U(\frak g_∞) admits a single non-zero proper ideal: the augmentation ideal. Our study is based on a complete description of the radical Poisson ideals in \bf S^⋅(\frak g_∞) and their zero-sets. We then discuss in detail integrable ideals of \U(\frak g_∞), i.e. ideals I⊂\U (\frak g_∞) for which I∩\U (\frak gn) is an intersection of ideals of finite-codimension in \U(\frak gn) for any n≥ 1. We present a classification of prime integrable ideals based on work of A. Zhilinskii. For \frak g_∞≅\fraksl_∞, \frakso_∞, all zero-sets of radical Poisson ideals of \bf S^⋅(\frak g_∞) arise from prime integrable ideals of \U(\frak g_∞). For \frak g_∞≅\fraksp_∞ only "half" of the zero-sets of Poisson ideals \bf S^⋅(\frak g_∞) arise from integrable ideals of \U(\frak g_∞).