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

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.