Irreducible Modules for the Lie Algebra of Divergence Zero Vector Fields on a Torus

TL;DR

This paper studies the irreducibility of certain modules for the Lie algebra of divergence zero vector fields on a torus, extending previous results on polynomial vector fields to divergence-free cases.

Contribution

It extends Rao's irreducibility conditions from polynomial vector fields to divergence zero vector fields on a torus.

Findings

Irreducibility conditions transfer from polynomial to divergence-free vector fields.

Modules restricted to divergence zero subalgebra remain irreducible under similar conditions.

Results generalize Rao's earlier work to a new class of Lie algebra modules.

Abstract

This paper investigates the irreducibility of certain representations for the Lie algebra of divergence zero vector fields on a torus. In "Irreducible Representations of the Lie-Algebra of the Diffeomorphisms of a d-Dimensional Torus," S. Eswara Rao constructs modules for the Lie algebra of polynomial vector fields on a d-dimensional torus, and determines the conditions for irreducibility. The current paper considers the restriction of these modules to the subalgebra of divergence zero vector fields. It is shown here that Rao's results transfer to similar irreducibility conditions for the Lie algebra of divergence zero vector fields.