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The $(\mathfrak{gl}_m,\mathfrak{gl}_n$) duality in the quantum toroidal setting (1801.08433v1)

Published 25 Jan 2018 in math.QA

Abstract: 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 matrices commute with the $\mathscr {E}_n$ transfer matrices after an appropriate identification of parameters.

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