# Representations of the Quantum Toroidal Algebra on Highest Weight Modules of the Quantum Affine Algebra of Type <strong>gl</strong><sub><em>N</em></sub>

### 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'*g*(**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 <strong>gl</strong><sub><em>N</em></sub>. Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, pp. 407–450

DOI 10.2977/PRIMS/1195143609