Non-weight modules over the Heisenberg–Virasoro algebra and the W algebra W(2,2)

In this paper, we construct and study some non-weight modules for the Heisenberg–Virasoro algebra and the [Formula: see text] algebra [Formula: see text]. We determine the modules, whose restriction to the universal enveloping algebra of the degree-[Formula: see text] part (modulo center) are free of rank [Formula: see text] for these two algebras. In the most interesting case, this degree-[Formula: see text] part is not the Cartan subalgebra. We also determine the simplicity of these modules, which provide new simple modules for the [Formula: see text] algebra [Formula: see text].