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

Abstract

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.