Classification of simple $W_n$-modules with finite-dimensional weight spaces

Abstract

We classify all simple $W_n$-modules with finite-dimensional weight spaces. Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao. This generalizes the classical result of Mathieu on simple weight modules for the Virasoro algebra. In our proof of the classification we construct a functor from the category of cuspidal $W_n$-modules to the category of $W_n$-modules with a compatible action of the algebra of functions on a torus. We also present a new identity for certain quadratic elements in the universal enveloping algebra of $W_1$, which provides important information about cuspidal $W_1$-modules.

References in corpus (1)

• Partial classification of modules for Lie-algebra of diffeomorphisms of d-dimensional torus

Cited by in corpus (9)

• Partial classification of cuspidal simple modules for Virasoro-like algebra
• Irreducible modules over Witt algebras ${\mathcal{W}}_n$ and over ${\mathfrak{sl}}_{n+1}(\mathbb{C})$
• Classification of irreducible bounded weight modules over the derivation Lie algebras of quantum tori
• $\W_n^+$- and $W_n$-module structures on $U(h)$
• Classification of simple cuspidal modules for solenoidal Lie algebras
• Irreducible weight modules over Witt algebras with infinite dimensional weight spaces
• Jet modules for the centerless Virasoro-like algebra
• local derivations on Witt algebras
• Exponentiation and Fourier transform of tensor modules of $\mathfrak{sl}(n+1)$