The $({\mathfrak{gl}}_m,{\mathfrak{gl}}_n$) duality in the quantum toroidal setting

Abstract

On a Fock space constructed from $mn$ free bosons and lattice ${\mathbb{Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra ${\mathcal{E}}_m$ associated to ${\mathfrak{gl}}_m$, together with a level $m$ action of the quantum toroidal algebra ${\mathcal{E}}_n$ associated to ${\mathfrak{gl}}_n$. We prove that the ${\mathcal{E}}_m$ transfer matrices commute with the ${\mathcal{E}}_n$ transfer matrices after an appropriate identification of parameters.

Latex, 28 pages

References in corpus (3)

• Quantum toroidal gl(1) and Bethe ansatz
• Representations of quantum toroidal $g{l}_n$
• The Integrals of Motion for the Deformed W-Algebra $W_{qt}(s{l}_N^{)}$ II: Proof of the commutation relations

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• Twisted Fock module of toroidal algebra via DAHA and vertex operators