On ideals in U$(\frak{sl}(\infty))$, U$(\frak o(\infty))$, U$(\frak{sp}(\infty))$

TL;DR

This paper reviews the structure of two-sided ideals in the universal enveloping algebras of infinite-dimensional finitary Lie algebras, providing new criteria for ideal non-vanishing and establishing isomorphisms between ideal lattices.

Contribution

It offers a comprehensive description of integrable ideals and introduces a new criterion for the annihilators of simple highest weight modules in infinite-dimensional Lie algebras.

Findings

All annihilators of simple highest weight modules are integrable ideals for sl(∞) and o(∞).

A criterion for the nonzero annihilator of arbitrary simple modules is established.

Lattices of ideals in U(o(∞)) and U(sp(∞)) are shown to be isomorphic.

Abstract

We provide a review of results on two-sided ideals in the enveloping algebra U$(\frak g(\infty))$ of a locally simple Lie algebra $\frak g(\infty)$. We pay special attention to the case when $\frak g(\infty)$ is one of the finitary Lie algebras $\frak{sl}(\infty), \frak o(\infty), \frak{sp}(\infty)$. The main results include a description of all integrable ideals in U$(\frak g(\infty))$, as well as a criterion for the annihilator of an arbitrary (not necessarily integrable) simple highest weight module to be nonzero. This criterion is new for $\frak g(\infty)=\frak o(\infty), \frak{sp}(\infty)$. All annihilators of simple highest weight modules are integrable ideals for $\frak g(\infty)=\frak{sl}(\infty), \frak o(\infty)$. Finally, we prove that the lattices of ideals in U$(\frak o(\infty))$ and U$(\frak{sp}(\infty))$ are isomorphic.