arXiv 2007.022602020-07math.RTmath.RA

$\mathbb{S}^1\times \mathbb{C}$上向量场李代数的不可约Jet模

Irreducible Jet modules for the vector field Lie algebra on $\mathbb{S}^1\times \mathbb{C}$

Mengnan Niu, Genqiang Liu

发表信息：: Communications in Algebra,2021
DOI：: 10.1080/00927872.2020.1862857

摘要：

对于$\mathbb{C}$上的交换代数$A$，记为$\mathfrak{g}=\text{Der}(A)$。Smash积$A\# U(\mathfrak{g})$上的模称为jet $\mathfrak{g}$ -模，其中$U(\mathfrak{g})$是$\mathfrak{g}$的泛包络代数.本文研究了$A=\mathbb{C}[t_1^{\pm 1},t_2]$情形下的jet模，证明了$A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$，其中$\mathcal{D}$是Weyl代数$\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$，并且$L$是$A\# U(\mathfrak{g})$的李子代数，称为对应于$\mathfrak{g}$的jet李代数。使用李代数同构$θ:L \rightarrow \mathfrak{m}_{1,0}Δ$，其中$\mathfrak{m}_{1,0}Δ$是在点$(1,0)$消失的向量场的子代数，我们证明了任何不可约的有限维$L$ -模同构于不可约的$\mathfrak{gl}_2$ -模.作为应用，我们给出了$\mathfrak{g}$上具有一致有界权空间的不可约jet模的张量积实现。

英文摘要：

For a commutative algebra $A$ over $\mathbb{C}$,denote $\mathfrak{g}=\text{Der}(A)$. A module over the smash product $A\# U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.In the present paper, we study jet modules in the case of $A=\mathbb{C}[t_1^{\pm 1},t_2]$.We show that $A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the Weyl algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of $A\# U(\mathfrak{g})$ called the jet Lie algebra corresponding to $\mathfrak{g}$.Using a Lie algebra isomorphism $θ:L \rightarrow \mathfrak{m}_{1,0}Δ$, where $\mathfrak{m}_{1,0}Δ$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional $L$-module is isomorphic to an irreducible $\mathfrak{gl}_2$-module. As an application, we give tensor product realizations of irreducible jet modules over $\mathfrak{g}$ with uniformly bounded weight spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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