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 $α\in\C$, 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 $α_1, α_2, λ_1, λ_2\in\C$ 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 $β\in\C[t,t^{-1}]$.

15 pages

References in corpus (5)

• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• Irreducible Virasoro modules from irreducible Weyl modules
• Whittaker categories for the Virasoro algebra
• Simple Virasoro modules induced from codimension one subalgebras of the positive part
• Simple Virasoro modules which are locally finite over a positive part

Cited by in corpus (2)

• A family of simple weight modules over the Virasoro algebra
• Irreducible modules over Witt algebras ${\mathcal{W}}_n$ and over ${\mathfrak{sl}}_{n+1}(\mathbb{C})$