報(bào)告人：譚海軍 副教授

報(bào)告題目：The polynomial modules over the symplectic Lie algebras

報(bào)告時(shí)間：2025年12月23日（周二）14:30-15:30

報(bào)告地點(diǎn)：云龍校區(qū)6號(hào)樓304報(bào)告廳

主辦單位：數(shù)學(xué)與統(tǒng)計(jì)學(xué)院、數(shù)學(xué)研究院、科學(xué)技術(shù)研究院

報(bào)告人簡(jiǎn)介：

東北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院副教授，主要研究領(lǐng)域?yàn)槔畲鷶?shù)和結(jié)合代數(shù)的表示理論，主持中國(guó)博士后基金，吉林省青年基金等項(xiàng)目。在Algebr. Represent. Theory, J. Algebra等著名SCI雜志上發(fā)表學(xué)術(shù)論文多篇。

報(bào)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.