$\W_n^+$- and $W_n$-module structures on $U(h)$

Abstract

Let $\h_n$ be the Cartan subalgebra of the Witt algebras $\W_n^+=\text{Der}\C[t_1, t_2, ..., t_n]$ and $\W_n=\text{Der}\C[t_1^{\pm 1},t_2^{\pm 1},\cdots,t_n^{\pm1}]$ where $1\le n\le \infty$. In this paper, we classify the modules over $\W_n^+$ and over $\W_n$ which are free $U(\h_n)$-modules of rank $1$. These are the $\W_n^+$-modules $\Omega ({\Lambda }_n,a,S)$ for some $Λ_n=(λ_1,\cdots,λ_n) \in (\C^*)^n, a\in \C$, and $S\subset \{1,2,...,n\}$; and $\W_n$-modules $\Omega ({\Lambda }_n,a)$ for some $Λ_n\in (\C^*)^n$ and some $a\in \C.$

16 pages

References in corpus (3)

• Simple sl_{n+1}-module structures on U(h)
• Classification of simple $W_n$-modules with finite-dimensional weight spaces
• Irreducible modules over Witt algebras ${\mathcal{W}}_n$ and over ${\mathfrak{sl}}_{n+1}(\mathbb{C})$

Cited by in corpus (2)

• Modules over the Heisenberg-Virasoro and $W(2,2)$ algebras
• Irreducible weight modules over Witt algebras with infinite dimensional weight spaces