Representations of the Quantum Toroidal Algebra on highest weight modules of the Quantum Affine Algebra of type gl(N)

TL;DR

This paper constructs a representation of the Quantum Toroidal Algebra on certain modules of the Quantum Affine Algebra of type gl(N), revealing a quantum level-rank duality and reciprocal decompositions.

Contribution

It introduces a novel representation of the Quantum Toroidal Algebra on highest weight modules of the Quantum Affine Algebra of type gl(N), and establishes a quantum analogue of the classical level-rank duality.

Findings

Constructed a representation of the Quantum Toroidal Algebra on highest weight modules.

Derived a quantum level-rank duality relating different algebra actions.

Described the reciprocal decomposition of the q-Fock space.

Abstract

A representation of the Quantum Toroidal Algebra of type sl(N) is constructed on every irreducible integrable highest weight module of the Quantum Affine Algebra of type gl(N). As an intermediate step in the construction, we obtain a quantum analogue of the classical level-rank duality, describing 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.