Simple Witt modules that are finitely generated over the cartan subalgebra

Abstract

Let $d\ge 1$ be an integer, $W_d$ and ${\mathcal{K}}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C}[x_1^{\pm 1},x_2^{\pm 1},...,x_d^{\pm 1}]$, respectively. For any ${\mathfrak{gl}}_d$-module $M$ and any admissible module $P$ over the extended Witt algebra ${\tilde{W}}_d$, we define a $W_d$-module structure on the tensor product $P\otimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P\otimes M$ for a finite dimensional simple ${\mathfrak{gl}}_d$-module $M$ and a simple ${\mathcal{K}}_d$-module $P$ that are finitely generated over the cartan subalgebra. We also characterize all simple ${\mathcal{K}}_d$-modules and all simple admissible ${\tilde{W}}_d$-modules that are finitely generated over the cartan subalgebra.

25 pages

References in corpus (2)

• Simple sl_{n+1}-module structures on U(h)
• Irreducible modules over Witt algebras ${\mathcal{W}}_n$ and over ${\mathfrak{sl}}_{n+1}(\mathbb{C})$