Representations of non-finitely graded Lie algebras related to Virasoro algebra

In this paper, we study representations of non-finitely graded Lie algebras 𝒲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} related to Virasoro algebra, where ϵ = ± 1 {\epsilon=\pm 1} . Precisely speaking, we completely classify the free 𝒰 ⁢ ( 𝔥 ) {\mathcal{U}(\mathfrak{h})} -modules of rank one over 𝒲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} , and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules.