## 姚裕丰: Simple modules over the Lie algebras of divergence zero vector fields on a torus

：姚裕丰 教授

Let $n\geq 2$ be an integer, $\mathcal{K}_n$ the Weyl algebra over the Laurent polynomial algebra $A_n=\mathbb{C} [x_1^{\pm 1}, x_2^{\pm 1}, ..., x_n^{\pm 1}]$, and $\mathbb{S}_n$ the Lie algebra of divergence zero vector fields on an $n$-dimensional torus. For any $\mathfrak{sl}_n$-module $V$ and any module $P$ over $\mathcal{K}_n$, we define an $\mathbb{S}_n$-module structure on the tensor product $P\otimes V$. In this talk, necessary and sufficient conditions for the $\mathbb{S}_n$-modules $P\otimes V$ to be simple are given, and an isomorphism criterion for nonminuscule $\mathbb{S}_n$-modules is provided. More precisely, all nonminuscule $\mathbb{S}_n$-modules are simple, and pairwise nonisomorphic. For minuscule $\mathbb{S}_n$-modules, minimal and maximal submodules are concretely constructed. This is a joint work with Brendan Frisk Dubsky, Xianqian Guo and Kaiming Zhao.