The (glm,gln) Duality in the Quantum Toroidal Setting

On a Fock space constructed from mn free bosons and lattice Z mn , we give a level n action of the quantum toroidal algebra E m associated to gl m , together with a level m action of the quantum toroidal algebra E n associated to gl n . We prove that the E m transfer matrices commute with the E n transfer matrices after an appropriate identification of parameters.