Characterization of simple highest weight modules

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.

Cited by in corpus (12)

• Whittaker modules for the affine Lie algebra $A_1^{(1)}$
• Irreducible representations of untwisted affine Kac-Moody algebras
• Simple Virasoro modules which are locally finite over a positive part
• Module structure on $U(\mathfrak{h})$ for Kac-Moody algebras
• $U(\fh)$-free modules over the Block algebra $\BB(q)$
• A class of simple weight Virasoro modules
• A family of new simple modules over the Schrödinger-Virasoro algebra
• The Heisenberg-Virasoro Lie conformal superalgebra
• Simple restricted modules for Neveu-Schwarz algebra
• Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules
• Weight modules over infinite dimensional Weyl algebras
• Simple modules over the Lie algebras of divergence zero vector fields on a torus