Irreducible weight modules over Witt algebras with infinite dimensional weight spaces

Abstract

Let $d>1$ be an integer. In 1986, Shen defined a class of weight modules $F_b^{\alpha }(V)$ 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 $F_b^{\alpha }(V)$ 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.

References in corpus (3)

• Simple sl_{n+1}-module structures on U(h)
• Classification of simple $W_n$-modules with finite-dimensional weight spaces
• $\W_n^+$- and $W_n$-module structures on $U(h)$

Cited by in corpus (1)

• Classification of irreducible bounded weight modules over the derivation Lie algebras of quantum tori