On ideals in the enveloping algebra of a locally simple Lie algebra

Abstract

We study (two-sided) ideals $I$ in the enveloping algebra $\U(\frak g_\infty)$ of an infinite-dimensional Lie algebra ${\mathfrak{g}}_{\infty }$ obtained as the union (equivalently, direct limit) of an arbitrary chain of embeddings of simple finite-dimensional Lie algebras ${\mathfrak{g}}_1\to {\mathfrak{g}}_2\to ...\to {\mathfrak{g}}_n\to ...$ with $\underset{n\to \infty }{\lim }\dim {\mathfrak{g}}_n=\infty$. 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 ${\mathfrak{g}}_{\infty }$ is not diagonal in the sense of A. Baranov and A. Zhilinskii, then $\U(\frak g_\infty)$ admits a single non-zero proper ideal: the augmentation ideal. Our study is based on a complete description of the radical Poisson ideals in ${\mathbf{S}}^{\cdot }({\mathfrak{g}}_{\infty })$ and their zero-sets. We then discuss in detail integrable ideals of $\U(\frak g_\infty)$, i.e. ideals $I\subset\U (\frak g_\infty)$ for which $I\cap\U (\frak g_n)$ is an intersection of ideals of finite-codimension in $\U(\frak g_n)$ for any $n\ge 1$. We present a classification of prime integrable ideals based on work of A. Zhilinskii. For ${\mathfrak{g}}_{\infty }\cong {\mathfrak{sl}}_{\infty },{\mathfrak{so}}_{\infty }$, all zero-sets of radical Poisson ideals of ${\mathbf{S}}^{\cdot }({\mathfrak{g}}_{\infty })$ arise from prime integrable ideals of $\U(\frak g_\infty)$. For ${\mathfrak{g}}_{\infty }\cong {\mathfrak{sp}}_{\infty }$ only "half" of the zero-sets of Poisson ideals ${\mathbf{S}}^{\cdot }({\mathfrak{g}}_{\infty })$ arise from integrable ideals of $\U(\frak g_\infty)$.