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

Jinxin Hu, Rencai Lü

TL;DR

This work addresses the problem of classifying the simplicity of tensor modules $F(P,M)$ for the Lie algebra of divergence-zero vector fields $S_n$, with $P$ a simple module over the Weyl algebra $D_n$ and $M$ a simple $ rak{sl}_n$-module. Using Shen's monomorphism to realize $F(P,M)$ as an $S_n$-module, the authors prove a dichotomy: if $M$ is not isomorphic to any fundamental $ rak{sl}_n$-module $V( abla_k)$, then $F(P,M)$ is simple; otherwise, they construct and analyze submodules $L_n(P,r)$ and $ ilde L_n(P,r)$ to describe all simple subquotients, including explicit quotients and special cases for $P riangleleft A_n$ or $A_n^F$. The results yield a detailed lattice of simple subquotients for weight modules, culminating in a concrete classification in terms of the $L_n(P,r)$ family and related quotients, with implications for the broader program of classifying simple Harish-Chandra modules for related algebras. The work broadens understanding of tensor constructions over Cartan-type Lie algebras and provides tools for Harish-Chandra module classifications.

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.

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

TL;DR

This work addresses the problem of classifying the simplicity of tensor modules for the Lie algebra of divergence-zero vector fields , with a simple module over the Weyl algebra and a simple -module. Using Shen's monomorphism to realize as an -module, the authors prove a dichotomy: if is not isomorphic to any fundamental -module , then is simple; otherwise, they construct and analyze submodules and to describe all simple subquotients, including explicit quotients and special cases for or . The results yield a detailed lattice of simple subquotients for weight modules, culminating in a concrete classification in terms of the family and related quotients, with implications for the broader program of classifying simple Harish-Chandra modules for related algebras. The work broadens understanding of tensor constructions over Cartan-type Lie algebras and provides tools for Harish-Chandra module classifications.

Abstract

Let be an integer, be the Lie algebra of vector fields on with zero divergence, and be the Weyl algebra over the polynomial algebra . In this paper, we study the simplicity of the tensor -module , where is a simple -module and is a simple -module. We obtain the necessary and sufficient conditions for to be an irreducible module, and determine all simple subquotients of when it is reducible.
Paper Structure (4 sections, 16 theorems, 84 equations)

This paper contains 4 sections, 16 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Tensor modules of $S_n$
  4. Example: Weight modules

Key Result

Lemma 2.1

(i) Any simple weight $D_{(i)}$ module is isomorphic to one of the following simple weight $D_{(i)}$ modules: where $\lambda_i\in\mathbb{C}\backslash\mathbb{Z}$. (ii) Let P be any simple weight $D_n$ module. Then $P\cong V_1\otimes V_2\otimes \cdots\otimes V_n$, where $V_i$ is a simple $D_{(i)}$ module. Therefore, the support set of any simple weight $D_n$ module is of the form $X=X_1\times X_2\t

Theorems & Definitions (29)

  • Lemma 2.1: (FGM)
  • Lemma 2.2: (LLZ)
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 19 more