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arXiv 1705.03393math.RTmath.QAmath.RA
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Cartan子代数上的单Witt模

Simple Witt modules that are finitely generated over the cartan subalgebra

Xiangqian Guo, Genqiang Liu, Rencai Lu, Kaiming Zhao

备注：
25 pages
发表信息：
Moscow Mathematical Journal, Vol.20 (1), 43-65 (2020)
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摘要：

设$d\ge1$是整数，$W_d$和$\mathcal{K}_d$分别是Laurent多项式代数$A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_d^{\pm1}]$上的Witt代数和Weyl代数.对广义Witt代数$\widetilde W_d$上的任意$\mathfrak{gl}_d$ -模$M$和任意容许模$P$，定义了张量积$P\otimes M$上的$W_d$ -模结构.本文证明了在Cartan子代数上生成的任何单$W_d$ -模对于在Cartan子代数上生成的有限维单$\mathfrak{gl}_d$-模$M$和单$\mathcal{K}_d$ -模$P$都是$W_d$-模$P \otimes M$的商模.我们还刻画了Cartan子代数上的所有单$\mathcal{K}_d$ -模和所有单容许$\widetilde W_d$ -模.

英文摘要：

Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $M$ and any admissible module $P$ over the extended Witt algebra $\widetilde W_d$, we define a $W_d$-module structure on the tensor product $P\otimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P \otimes M$ for a finite dimensional simple $\mathfrak{gl}_d$-module $M$ and a simple $\mathcal{K}_d$-module $P$ that are finitely generated over the cartan subalgebra. We also characterize all simple $\mathcal{K}_d$-modules and all simple admissible $\widetilde W_d$-modules that are finitely generated over the cartan subalgebra.

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