Classification of quasifinite$\mathcal{W}_\infty$-modules-modules

Abstract

It is proved that an irreducible quasifinite$\mathcal{W}_\infty$-module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight$\mathcal{W}_\infty$-module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ^m₁ …D ^m₁ foru ∈F[Γ] (the group algebra), andm _i≥0 with$\sum {_{i = 1}^n m_i } \geqslant 1$, whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ.