Tensor modules over the Lie algebras of divergence zero vector fields on $\mathbb{C}^n$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912966531612672 |
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| author | Hu, Jinxin Lü, Rencai |
| author_facet | Hu, Jinxin Lü, Rencai |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05944 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor modules over the Lie algebras of divergence zero vector fields on $\mathbb{C}^n$ Hu, Jinxin Lü, Rencai Representation Theory 17B10, 17B65, 17B66 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. |
| title | Tensor modules over the Lie algebras of divergence zero vector fields on $\mathbb{C}^n$ |
| topic | Representation Theory 17B10, 17B65, 17B66 |
| url | https://arxiv.org/abs/2505.05944 |