Supports of weight modules over Witt algebras

TL;DR

This paper advances the understanding of weight modules over Witt algebras by explicitly describing their supports, aiding the classification of simple modules with finite-dimensional weight spaces.

Contribution

It provides the first explicit description of supports for simple weight modules over Witt algebras and extends support analysis to a broader class of graded Lie algebras.

Findings

Supports of simple weight modules over Witt algebras are explicitly characterized.

Support descriptions are extended to certain $ extbf{Z}^n$-graded Lie algebras with root space decompositions.

Results facilitate classification of simple weight modules with finite-dimensional weight spaces.

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 $\Z^n$-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\oplus_{\alpha\in\Z^n}\mathfrak {g}_\alpha$ with respect to the abelian subalgebra $\mathfrak {g}_0$, with the property $[\mathfrak{g}_\alpha,\mathfrak {g}_\beta]= \mathfrak {g}_{\alpha+\beta}$ for all $\alpha,\beta\in\Z^n$, $\alpha\neq \beta$ (this class contains the algebra $W_n$).