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 \mathfrak{sl}_N is defined on every integrable irreducible highest weight module of the quantum affme algebra of type \mathfrak{gl}_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'q(\mathfrak{gl}_N) of level L and U'_q(\mathfrak{sl}_L) of level N is described.