Non-weight representations of Cartan type S Lie algebras

TL;DR

This paper classifies all module structures on the universal enveloping algebra of Cartan subalgebras for certain divergence-zero vector field Lie algebras, detailing their submodules.

Contribution

It provides a complete classification of module structures and submodules for the universal enveloping algebra of Cartan subalgebras in Cartan type S Lie algebras.

Findings

All module structures on the universal enveloping algebra are determined.

All submodules of these modules are explicitly described.

The results enhance understanding of representation theory for divergence-zero vector field Lie algebras.

Abstract

For the two Cartan type S subalgebras of the Witt algebra $\W_n$, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra $\h_n$. We also give all submodules of these modules.