Smooth representations of the extended Schrödinger–Virasoro algebra

The Schrödinger–Virasoro algebra was originally introduced during the process of investigating the free Schrödinger equations. In order to investigate vertex representations of the Schrödinger–Virasoro algebra, Unterberger introduced a new infinite-dimensional Lie algebra [Formula: see text] called the extended Schrödinger–Virasoro algebra in the context of two-dimensional conformal field theory and statistical physics. In this paper, we study simple smooth modules over [Formula: see text], which can be viewed as an extension of the Schrödinger–Virasoro algebra by a conformal current with conformal weight 1. More specifically, these modules are determined by simple modules over finite-dimensional solvable Lie algebras. Moreover, we present several equivalent descriptions for simple smooth modules over [Formula: see text].