Supports of weight modules over Witt algebras

Abstract

In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras $W_n$, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a ${\mathbb{Z}}^n$-graded Lie algebra $\mathfrak{g}$ having a root space decomposition ${\oplus }_{\alpha \in {\mathbb{Z}}^n}{\mathfrak{g}}_{\alpha }$ with respect to the abelian subalgebra ${\mathfrak{g}}_0$, with the property $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]={\mathfrak{g}}_{\alpha +\beta }$ for all $\alpha ,\beta \in {\mathbb{Z}}^n$, $\alpha \ne \beta$ (this class contains the algebra $W_n$).

17 pages