報告人：譚海軍 副教授
報告題目：The polynomial modules over the symplectic Lie algebras
報告時間：2025年12月23日（周二）14:30-15:30
報告地點：云龍校區6號樓304報告廳
主辦單位：數學與統計學院、數學研究院、科學技術研究院
報告人簡介：
東北師范大學數學與統計學院副教授，主要研究領域為李代數和結合代數的表示理論，主持中國博士后基金，吉林省青年基金等項目。在Algebr. Represent. Theory, J. Algebra等著名SCI雜志上發表學術論文多篇。
報告摘要：
If a polynomial algebra $\C[x_1,\cdots, x_n]$ is equipped with a module structure over a Lie algebra $\mathfrak{a}$, then we call it a polynomial module over $\mathfrak{a}$. In this talk, I will introduce some new polynomial module structures over the symplectic Lie algebra $\sp_{2l}(\C)$.
Let $\p$ be a maximal parabolic subalgebra of $\sp_{2l}(\C)$ with a nonzero abelian nilradical $\n$. There exist the $\sp_{2l}(\C)$-module structures on the polynomial algebra $\UU(\n)$ as a free $\UU(\n)$-module of rank one. Firstly, the corresponding $\sp_{2l}(\C)$-module structure is determined by two parameters $C\in\C$ and $\Phi\in\UU(\n)$, and so is denoted by $\tau(C,\Phi)$. Secondly, the parameter $C$ determines the simplicity of $\tau(C,\Phi)$. More precisely, $\tau(C,\Phi)$ is simple if and only if $C\notin\frac{l+1}{2}-\frac{1}{2}\Z_+$. And the parameter $\Phi$ determines whether $\tau(C,\Phi)$ is a weight module, that is, $\tau(C,\Phi)$ is a weight module if and only if $\Phi\in\C$. Thirdly, if $C\in\frac{l+1}{2}-\frac{1}{2}\Z_+$, then $\tau(C,\Phi)$ is both Noetherian and Artinian, and whether the composition factor is a weight module depends on whether a system of equations relative to the parameter $\Phi$ has solutions. This is a joint work with Chen Yan.