Annihilators of highest weight $\frak{sl}(\infty)$-modules

TL;DR

This paper characterizes the annihilators of simple highest weight modules over the infinite-dimensional Lie algebra rak{sl}(infty), showing they are always integrable and establishing a classification analogous to Duflo's theorem for primitive ideals.

Contribution

It provides a criterion for nonzero annihilators in rak{sl}(infty) modules and classifies prime integrable ideals as annihilators of simple modules over ideal Borel subalgebras.

Findings

Annihilators of simple highest weight modules are always integrable.

Established a criterion for nonzero annihilators in rak{sl}(infty).

Proved that prime integrable ideals are annihilators of modules over ideal Borel subalgebras.

Abstract

We give a criterion for the annihilator in U$(\frak{sl}(\infty))$ of a simple highest weight $\frak{sl}(\infty)$-module to be nonzero. As a consequence we show that, in contrast with the case of $\frak{sl}(n)$, the annihilator in U$(\frak{sl}(\infty))$ of any simple highest weight $\frak{sl}(\infty)$-module is integrable, i.e., coincides with the annihilator of an integrable $\frak{sl}(\infty)$-module. Furthermore, we define the class of ideal Borel subalgebras of $\frak{sl}(\infty)$, and prove that any prime integrable ideal in U$(\frak{sl}(\infty))$ is the annihilator of a simple $\frak b^0$-highest weight module, where $\frak b^0$ is any fixed ideal Borel subalgebra of $\frak{sl}(\infty)$. This latter result is an analogue of the celebrated Duflo Theorem for primitive ideals.