标题： 类型$A$中的矩阵空间坐标环的大代数 显示英文标题

标题： Big algebra in type $A$ for the coordinate ring of the matrix space

摘要： 在本文中，我们考虑Hausel最近为矩阵空间坐标环上的$\mathrm{GL}_n$作用引入的大代数$\operatorname{Mat}(n,r)$。特别地，我们得到了大代数生成元的显式公式，这些公式用具有多项式系数的微分算子表示。我们证明类型$A$的大代数是交换的，并将其与Yangian$\mathrm{Y}(\mathfrak{gl}_{n})$中的Bethe子代数相关联。我们将这些结果应用于标准表示的对称幂的大代数$\mathrm{GL}_n$。 显示英文摘要

摘要： In this note we consider the big algebra recently introduced by Hausel for the $\mathrm{GL}_n$-action on the coordinate ring of the matrix space $\operatorname{Mat}(n,r)$. In particular, we obtain explicit formulas for the big algebra generators in terms of differential operators with polynomial coefficients. We show that big algebras in type $A$ are commutative and relate them to the Bethe subalgebra in the Yangian $\mathrm{Y}(\mathfrak{gl}_{n})$. We apply these results to big algebras of symmetric powers of the standard representation of $\mathrm{GL}_n$.