Some Representations of Nongraded Divergence-Free Lie Algebras

TL;DR

This paper classifies generalized weight modules with limited multiplicities over nongraded divergence-free Lie algebras, expanding understanding of their structure without relying on toral Cartan subalgebras.

Contribution

It provides a complete classification of such modules, a significant step in understanding nongraded divergence-free Lie algebras.

Findings

Classification of modules with weight multiplicities ≤ 1

Insight into structure without toral Cartan subalgebras

Extension of Lie algebra representation theory

Abstract

Divergence-free Lie algebras are originated from the Lie algebras of volume-preserving transformation groups. Xu constructed a certain nongraded generalization, which may not contain any toral Cartan subalgebra. In this paper, we give a complete classification of the generalized weight modules over these algebras with weight multiplicities less than or equal to one.