Representations of the Quantum Toroidal Algebra on Highest Weight Modules of the Quantum Affine Algebra of Type $\mathfrak{gl}_N$

Abstract

A representation of the quantum toroidal algebra of type slN is con- structed on every integrable irreducible highest weight module of the the quantum affine algebra of typegl N. The q-version of the level-rank duality giving the reciprocal decomposition of the q-Fock space with respect to mutually commutative actions of U ' ( b glN) of level L and U ' ( b slL) of level N is described.