# Representations of the Quantum Toroidal Algebra on Highest Weight Modules of the Quantum Affine Algebra of Type $gl_{N}$

### Kouichi Takemura

Kyoto University, Japan### Denis Uglov

Kyoto University, Japan

## Abstract

A representation of the quantum toroidal algebra of type $sl_{N}$ is defined on every integrable irreducible highest weight module of the quantum affme algebra of type $gl_{N}$. The $q$-version of the level-rank duality giving the reciprocal decomposition of the $q$-Fock space with respect to mutually commutative actions of $U_{′}q(gl_{N})$ of level $L$ and $U_{q}(sl_{L})$ of level $N$ is described.

## Cite this article

Kouichi Takemura, Denis Uglov, Representations of the Quantum Toroidal Algebra on Highest Weight Modules of the Quantum Affine Algebra of Type $gl_{N}$. Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, pp. 407–450

DOI 10.2977/PRIMS/1195143609