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

Abstract

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

References in corpus (1)

• Cartan subalgebras of root-reductive Lie algebras