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 $U^{\prime }q({\mathfrak{gl}}_N)$ of level $L$ and $U_q^{\prime }({\mathfrak{sl}}_L)$ of level $N$ is described.