Simple modules over the Lie algebras of divergence zero vector fields on a torus

Abstract

Abstract Let {n\geq 2} be an integer, {\mathbb{S}_{n}} the Lie algebra of divergence zero vector fields on an n-dimensional torus, and {\mathcal{K}_{n}} the Weyl algebra over the Laurent polynomial algebra {A_{n}=\mathbb{C}[x_{1}^{\pm 1},x_{2}^{\pm 1},\dots,x_{n}^{\pm 1}]} . For any {\mathfrak{sl}_{n}} -module V and any module P over {\mathcal{K}_{n}} , we define an {\mathbb{S}_{n}} -module structure on the tensor product {P\otimes V} . In this paper, necessary and sufficient conditions for the {\mathbb{S}_{n}} -modules {P\otimes V} to be simple are given, and an isomorphism criterion for nonminuscule {\mathbb{S}_{n}} -modules is provided. More precisely, all nonminuscule {\mathbb{S}_{n}} -modules are simple, and pairwise nonisomorphic. For minuscule {\mathbb{S}_{n}} -modules, minimal and maximal submodules are concretely determined.