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Working paper

Homomorphisms between different quantum toroidal and affine Yangian algebras

math. arxive. Cornell University, 2015. No. 1512.09109.
Bershtein M., Tsymbaliuk A.
This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by  $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively.  Our motivation arises from the milestone work Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra U_q(L\\mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established by constructing an  isomorphism of $\mathbb{C}[[\hbar]]$-algebras $\Phi:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to \widehat{Y}_\hbar(\mathfrak{g})$ (with $\ \widehat{}\ $ standing for the appropriate completions).  These two completions model the behavior of the algebras in the formal neighborhood of $h=0$.  The same construction can be applied to the toroidal setting with  $q_i=\exp(\hbar_i)$ for $i=1,2,3$.    In the current paper, we are interested in the more general relation: $\mathrm{q}_1=\omega_{mn}e^{h_1/m}, \mathrm{q}_2=e^{h_2/m}, \mathrm{q}_3=\omega_{mn}^{-1}e^{h_3/m}$, where $m,n\in \mathbb{N}$ and $\omega_{mn}$ is an $mn$th root of $1$.   For any such choice of $m,n,\omega_{mn}$ and the corresponding values $\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3$, we construct a homomorphism $\Phi^{\omega_{mn}}_{m,n}$ from the completion of the formal version of $\mathcal{U}^{(m)}_{\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3}$ to the completion of the formal version of $\mathcal{Y}^{(mn)}_{h_1/mn,h_2/mn,h_3/mn}$.  We also construct homomorphisms $\Psi^{\omega',\omega}_{m,n}$  between the completions of the formal versions of $\mathcal{U}^{(m)}_{\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3}$  with different parameters $m$ and $\omega_{mn}$.