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

Abstract

We provide a review of results on two-sided ideals in the enveloping algebra $\operatorname{U}(\mathfrak g(\infty))$ of a locally simple Lie algebra $\mathfrak g(\infty)$. We pay special attention to the case when $\mathfrak g(\infty)$ is one of the finitary Lie algebras $\mathfrak{sl}(\infty), \mathfrak o(\infty), \mathfrak{sp}(\infty)$. The main results include a description of all integrable ideals in $\operatorname{U}(\mathfrak g(\infty))$, 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 $\mathfrak g(\infty)=\mathfrak o(\infty), \mathfrak{sp}(\infty)$. All annihilators of simple highest weight modules are integrable ideals for $\mathfrak g(\infty)=\mathfrak{sl}(\infty),$ $\mathfrak o(\infty)$. Finally, we prove that the lattices of ideals in $\operatorname{U}(\mathfrak o(\infty))$ and $\operatorname{U}(\mathfrak{sp}(\infty))$ are isomorphic.