Tensor modules over the Lie algebras of divergence zero vector fields on $\mathbb{C}^n$

TL;DR

This paper investigates the conditions under which tensor modules over the divergence-zero vector field Lie algebra are simple or reducible, providing a classification of their subquotients.

Contribution

It establishes necessary and sufficient conditions for the irreducibility of tensor modules over the Lie algebra of divergence-zero vector fields on ^n.

Findings

Criteria for irreducibility of tensor modules

Classification of simple subquotients when reducible

Connection between divergence-zero vector fields and Weyl algebra modules

Abstract

Let $n\geq 2$ be an integer, $S_n$ be the Lie algebra of vector fields on $\mathbb{C}^n$ with zero divergence, and $D_n$ be the Weyl algebra over the polynomial algebra $A_n=\mathbb{C}[t_1,t_2,\cdots,t_n]$. In this paper, we study the simplicity of the tensor $S_n$-module $F(P,M)$, where $P$ is a simple $D_n$-module and $M$ is a simple $\mathfrak{sl}_n$-module. We obtain the necessary and sufficient conditions for $F(P,M)$ to be an irreducible module, and determine all simple subquotients of $F(P,M)$ when it is reducible.