Generalized Polynomial modules over the Virasoro algebra

Abstract

Let ${\mathcal{B}}_r$ be the $(r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $\mathcal{V}$. Irreducible ${\mathcal{B}}_r$-modules were used to construct irreducible Whittaker modules in [MZ2] and irreducible weight modules with infinite dimensional weight spaces over $\mathcal{V}$ in [LLZ].In the present paper, we construct non-weight Virasoro modules $F(M,\Omega (\lambda ,\beta ))$ from irreducible ${\mathcal{B}}_r$-modules $M$ and $(\mathcal{A},\mathcal{V})$-modules $\Omega (\lambda ,\beta )$. We give necessary and sufficient conditions for the Virasoro module $F(M,\Omega (\lambda ,\beta ))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also we also give the isomorphism criterion for two $F(M,\Omega (\lambda ,\beta ))$.

The introduction was revised