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

Abstract

We provide a review of results on two-sided ideals in the enveloping algebra $U(\mathfrak{g}(\infty ))$ 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 $U(\mathfrak{g}(\infty ))$, 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 $U(\mathfrak{o}(\infty ))$ and $U(\mathfrak{sp}(\infty ))$ are isomorphic.