Weight modules over generalized Witt algebras with 1-dimensional weight spaces

In this paper, indecomposable and irreducible weight representations with 1-dimensional weight spaces for simple generalized Witt algebras over any field of characteristic 0 are classified. There are five classes of such nontrivial indecomposable modules

Weight modules over exp-polynomial Lie algebras

In this paper, we generalize a result by Berman and Billig on weight modules over Lie algebras with polynomial multiplication. More precisely, we show that a highest weight module with an exp-polynomial ``highest weight'' has finite dimensional weight spaces. We also get a class of irreducible weight modules with finite dimensional weight spaces over generalized Virasoro algebras which do not occur over the classical Virasoro algebra

Irreducible weight modules over Witt algebras with infinite dimensional weight spaces

Let $d>1$ be an integer. In 1986, Shen defined a class of weight modules $F^\alpha_b(V)$ over the Witt algebra $\mathcal{W}_d$ for \a\in\C^d, b\in\C, and an irreducible module $V$ over the special linear Lie algebra \sl_d. In 1996, Eswara Rao determined the necessary and sufficient conditions for these modules to be irreducible when $V$ is finite dimensional. In this note, we will determine the necessary and sufficient conditions for all these modules $F^\alpha_b(V)$ to be irreducible where $V$ is not necessarily finite dimensional. Therefore we obtain a lot of irreducible $\mathcal{W}_d$-modules with infinite dimensional weight spaces