标题： 与量子环面分叉代数相关的交换子代数$\mathfrak{gl}_{m|n}$ 显示英文标题

标题： Commutative subalgebra of a shuffle algebra associated with quantum toroidal $\mathfrak{gl}_{m|n}$

摘要： We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal B_{m|n}(s)$ depending on parameters $s=(s_1,\dots,s_{m+n})$, $\prod_i s_i=1$, given by appropriate regularity conditions. 我们证明$\mathcal B_{m|n}(s)$是一个自由多项式代数，并给出显式的生成元，这些生成元推测对应于在$\mathcal E_{m|n}$模的零次部分上计算的$s$加权$R$矩阵的迹，这些模属于级别$\pm 1$。 显示英文摘要

摘要： We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal B_{m|n}(s)$ depending on parameters $s=(s_1,\dots,s_{m+n})$, $\prod_i s_i=1$, given by appropriate regularity conditions. We show that $\mathcal B_{m|n}(s)$ is a free polynomial algebra and give explicit generators which conjecturally correspond to the traces of the $s$-weighted $R$-matrix computed on the degree zero part of $\mathcal E_{m|n}$ modules of levels $\pm 1$.