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

TL;DR

This paper characterizes ideals in the universal enveloping algebra of an infinite-dimensional union of simple Lie algebras, proving a conjecture about the uniqueness of the non-zero proper ideal in non-diagonal cases.

Contribution

It provides an explicit description of the zero-sets of graded ideals and classifies prime integrable ideals in the enveloping algebra of certain infinite-dimensional Lie algebras.

Findings

In non-diagonal cases, the enveloping algebra admits only the augmentation ideal as a proper ideal.

Zero-sets of radical Poisson ideals are classified for $rak{sl}_ty$, $rak{so}_ty$, and partially for $rak{sp}_ty$.

The study links Poisson ideals with integrable ideals, providing a detailed classification.

Abstract

We study (two-sided) ideals $I$ in the enveloping algebra $\U(\frak g_\infty)$ of an infinite-dimensional Lie algebra $\frak g_\infty$ obtained as the union (equivalently, direct limit) of an arbitrary chain of embeddings of simple finite-dimensional Lie algebras $\frak g_1\to\frak g_2\to...\to\frak g_n\to...$ with $\lim\limits_{n\to\infty}\dim\frak g_n=\infty$. 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 $\frak g_\infty$ 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 ${\bf S}^\cdot(\frak g_\infty)$ 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 $\frak g_\infty\cong\frak{sl}_\infty, \frak{so}_\infty$, all zero-sets of radical Poisson ideals of ${\bf S}^\cdot(\frak g_\infty)$ arise from prime integrable ideals of $\U(\frak g_\infty)$. For $\frak g_\infty\cong\frak{sp}_\infty$ only "half" of the zero-sets of Poisson ideals ${\bf S}^\cdot(\frak g_\infty)$ arise from integrable ideals of $\U(\frak g_\infty)$.