# Moscow Mathematical Journal

Volume 20, Issue 1, January–March 2020  pp. 43–65.

Simple Witt Modules that are Finitely Generated over the Cartan Subalgebra

Authors:  Xiangqian Guo (1), Genqiang Liu (2), Rencai Lu (3), and Kaiming Zhao (4)
Author institution:(1) School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 730000 P. R. China
(2) School of Mathematics and Statistics, and Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, P. R. China
(3) Department of Mathematics, Soochow University, Suzhou, P. R. China
(4) School of Mathematical Science, Hebei Normal (Teachers) University, Shijiazhuang, Hebei, 050016 P. R. China and
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5

Summary:

Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the Weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, \dots, x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $V$ and any admissible module $P$ over the extended Witt algebra $\widetilde{W}_d$, we define a $W_d$-module structure on the tensor product $P\otimes V$. In this paper, we classify all simple $W_d$-modules that are finitely generated over the Cartan subalgebra. They are actually the $W_d$-modules $P \otimes V$ for a finite-dimensional simple $\mathfrak{gl}_d$-module $V$ and a simple $\mathcal{K}_d$-module $P$ that is a finite-rank free module over the polynomial algebra in the variables $x_1\frac{\partial}{\partial x_1},\dots,x_d\frac{\partial}{\partial x_d}$, except for a few cases which are also clearly described. We also characterize all simple $\mathcal{K}_d$-modules and all simple admissible $\widetilde{W}_d$-modules that are finitely generated over the Cartan subalgebra.

2010 Math. Subj. Class. 17B10, 13C10, 17B20, 17B65, 17B66, 17B68.

Keywords: Witt algebra, weight module, irreducible module, de Rham complex, Quillen–Suslin Theorem