Characterization of simple highest weight modules

TL;DR

This paper characterizes simple highest weight modules for certain Lie algebras by their local nilpotency under positive root elements, highlighting differences in behavior among various algebra types.

Contribution

It establishes a characterization criterion for simple highest weight modules in specific Lie algebras, and shows the criterion does not hold for higher rank Virasoro and Heisenberg algebras.

Findings

Simple highest weight modules are characterized by local nilpotency of positive root elements.

This property holds for finite-dimensional Lie algebras, affine Kac-Moody, Virasoro, and Heisenberg-Virasoro algebras.

The property does not hold for higher rank Virasoro and Heisenberg algebras.

Abstract

We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.