The \({(\mathfrak{gl}_{m},\mathfrak{gl}_{n})}\) Duality in the Quantum Toroidal Setting

On a Fock space constructed from mn free bosons and lattice \({\mathbb{Z}_{mn}}\), we give a level n action of the quantum toroidal algebra \({\mathcal{E}_{m}}\) associated to \({\mathfrak{gl}_{m}}\), together with a level m action of the quantum toroidal algebra \({\mathcal{E}_n}\) associated to \({\mathfrak{gl}_n}\). We prove that the \({\mathcal{E}_m}\) transfer matrices commute with the \({\mathcal{E}_n}\) transfer matrices after an appropriate identification of parameters.