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On a Fock space constructed from $mn$ free bosons and lattice ${\Bbb{Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra$\mathscr {E}_m$ associated to $\mathfrak{gl}_m$, together with a level $m$action of the quantum toroidal algebra ${\mathscr E}_n$ associated to${\mathfrak {gl}}_n$. We prove that the $\mathscr {E}_m$ transfer matricescommute with the $\mathscr {E}_n$ transfer matrices after an appropriateidentification of parameters.

The $(\mathfrak{gl}_m,\mathfrak{gl}_n$) duality in the quantum toroidal setting