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

TL;DR

This paper establishes a duality between two quantum toroidal algebras associated with rak{gl}_m and rak{gl}_n on a Fock space, showing their transfer matrices commute under certain conditions, revealing deep algebraic symmetries.

Contribution

It constructs explicit actions of quantum toroidal algebras rak{gl}_m and rak{gl}_n on a Fock space and proves their transfer matrices commute, demonstrating a duality in the quantum toroidal setting.

Findings

Constructed level n action of rak{gl}_m quantum toroidal algebra.

Constructed level m action of rak{gl}_n quantum toroidal algebra.

Proved transfer matrices commute after parameter identification.

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.