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

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.