Commutative subalgebra of a shuffle algebra associated with quantum toroidal ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$

We define and study the shuffle algebra $S{h}_{m\mathrm{\mid }n}$ of the quantum toroidal algebra ${\mathcal{E}}_{m\mathrm{\mid }n}$ associated to Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$. We show that $S{h}_{m\mathrm{\mid }n}$ contains a family of commutative subalgebras ${\mathcal{B}}_{m\mathrm{\mid }n}(s)$ depending on parameters $s=(s_1,\dots ,s_{m+n})$, ${\prod }_i{s}_i=1$, given by appropriate regularity conditions. We show that ${\mathcal{B}}_{m\mathrm{\mid }n}(s)$ 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 ${\mathcal{E}}_{m\mathrm{\mid }n}$ modules of levels $\pm 1$.