Characterization of simple smooth modules (Q6054758)

The authors characterize simple smooth modules over some infinite-dimensional \(\mathbb{Z}\)-graded Lie algebras. They prove that if one fixed positive root element of a \(\mathbb{Z}\)-graded Lie algebra \(\mathfrak{g}\) locally finitely acts on a simple \(\mathfrak{g}\)-module \(V\), then \(V\) is a smooth \(\mathfrak{g}\)-module. These infinite-dimensional \(\mathbb{Z}\)-graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, etc. This result for untwisted affine Kac-Moody algebras holds unless one changes the condition from ``locally finitely'' to ``locally nilpotently''. The authors also show that these are not the case for the Heisenberg algebra.