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

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 Added by Evgeny Mukhin
 Publication date 2018
  fields
and research's language is English




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



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The affine evaluation map is a surjective homomorphism from the quantum toroidal ${mathfrak {gl}}_n$ algebra ${mathcal E}_n(q_1,q_2,q_3)$ to the quantum affine algebra $U_qwidehat{mathfrak {gl}}_n$ at level $kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=kappa^2$. We discuss ${mathcal E}_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${mathcal E}_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $widehat{mathfrak {gl}}_n/widehat{mathfrak {gl}}_{n-1}$.
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