Characterization of simple smooth modules

Abstract

In this paper, we characterize simple smooth modules over some infinite-dimensional Z-graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z-graded Lie algebra g locally finitely acts on a simple g-module V, then V is a smooth g-module. These infinite-dimensional Z-graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from “locally finitely” to “locally nilpotently”. We also show that these are not the case for the Heisenberg algebra.