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

TL;DR

This paper classifies simple modules over the Lie algebra of divergence zero vector fields on an n-dimensional torus, providing criteria for simplicity and explicit constructions for certain modules, advancing understanding of their structure.

Contribution

It introduces a framework to determine when tensor product modules over the Lie algebra are simple and classifies all nonminuscule simple modules, including explicit submodule constructions.

Findings

All nonminuscule modules are simple and pairwise nonisomorphic.

Necessary and sufficient conditions for simplicity of tensor product modules.

Explicit construction of minimal and maximal submodules for minuscule modules.

Abstract

Let $n\ge2$ be an integer, $\mathcal{K}_n$ the Weyl algebra over the Laurent polynomial algebra $A_n=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_n^{\pm1}]$, and $\mathbb{S}_n$ the Lie algebra of divergence zero vector fields on an $n$-dimensional torus. 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 constructed.