Simple Heisenberg-Virasoro modules from Weyl algebra modules

In this paper, we constructed and studied two classes of modules over the Heisenberg-Virasoro algebra HVir from Weyl algebra modules, parallel to the similar construction for Virasoro algebra in [13,9]. More precisely, given any module P over the degree-1 Weyl algebra, we define the HVir-modules M(P, V ), where V is a restricted module over the nonnegative part a of HVir, and G(P, W, xi, b, c), where W is a restricted module over another subalgebra b C HVir and xi, b, c E C with xi =? 0. We determined the irreducibility of these modules as well as the isomorphisms within each class respectively. Moreover, any two such modules from different classes are never isomorphic although they might be closely related, as we showed in the last section. (c) 2023 Elsevier Inc. All rights reserved.