Homomorphisms between different quantum toroidal and affine Yangian algebras

TL;DR

This paper constructs a homomorphism linking quantum toroidal algebras and affine Yangians, extending previous work to more general roots of unity and providing two proofs for the relation.

Contribution

It introduces a new homomorphism between quantum toroidal algebras and affine Yangians for roots of unity, generalizing prior isomorphisms in the formal neighborhood of parameters.

Findings

Constructed a homomorphism between completed quantum toroidal and affine Yangian algebras.

Provided two proofs: via faithful representations and shuffle algebra approach.

Extended the relation to roots of unity settings.

Abstract

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 of 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 $1$. Assuming $\omega_{mn}^m$ is a primitive $n$-th root of unity, 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 propose two proofs of this result: (1) by constructing the compatible isomorphism between the faithful representations of the algebras; (2) by combining the direct verification of Gautam and Toledano Laredo for the classical setting with the shuffle approach.