A class of simple weight Virasoro modules

Abstract

For a simple module $M$ over the positive part of the Virasoro algebra (actually for any simple module over some finite dimensional solvable Lie algebras $\mathfrak{a}_r$) and any $\alpha\in\C$, a class of weight modules $\mathcal {N}(M, \alpha)$ over the Virasoro algebra are constructed. The necessary and sufficient condition for $\mathcal {N}(M, \a)$ to be simple is obtained. We also determine the necessary and sufficient conditions for two such irreducible Virasoro modules to be isomorphic. Many examples for such irreducible Virasoro modules with different features are provided. In particular the irreducible weight Virasoro modules $\Gamma(\alpha_1, \alpha_2, \lambda_1, \lambda_2)$ are defined on the polynomial algebra $\C[x]\otimes \C[t, t^{-1}]$ for any $\alpha_1, \alpha_2, \lambda_1, \lambda_2\in\C$ with $\lambda_1$ or $\lambda_2$ nonzero. By twisting the weight modules $\mathcal {N}(M, \alpha)$ we also obtain nonweight simple Virasoro modules $\mathcal {N}(M, \beta)$ for any $\beta\in\C[t,t^{-1}]$.