Classification of Category $\mathcal{J}$ Modules for Divergence Zero Vector Fields on a Torus

TL;DR

This paper classifies indecomposable and irreducible modules for divergence zero vector fields on a torus, focusing on modules with compatible Laurent polynomial actions and finite-dimensional weight spaces.

Contribution

It provides a complete classification of modules in this specific category, advancing understanding of their structure and representation theory.

Findings

Classification of indecomposable modules

Classification of irreducible modules

Modules have finite-dimensional weight spaces

Abstract

We consider a category of modules that admit compatible actions of the commutative algebra of Laurent polynomials and the Lie algebra of divergence zero vector fields on a torus and have a weight decomposition with finite dimensional weight spaces. We classify indecomposable and irreducible modules in this category.