Homomorphisms between different quantum toroidal and affine Yangian algebras

Abstract

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of ${\mathfrak{sl}}_n$, denoted by ${\mathcal{U}}_{q_1,q_2,q_3}^{(n)}$ and ${\mathcal{Y}}_{h_1,h_2,h_3}^{(n)}$, 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 $\mathbb{C}[[\hslash ]]$-algebras $\Phi :{\hat{U}}_{\exp (\hslash )}(L\mathfrak{g})\to {\hat{Y}}_{\hslash }(\mathfrak{g})$ (with $\hat{}$ 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 $q_i=\exp ({\hslash }_i)$ 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 ${\Phi }_{m,n}^{{\omega }_{mn}}$ from the completion of the formal version of ${\mathcal{U}}_{{\mathrm{q}}_1,{\mathrm{q}}_2,{\mathrm{q}}_3}^{(m)}$ to the completion of the formal version of ${\mathcal{Y}}_{h_1/mn,h_2/mn,h_3/mn}^{(mn)}$. 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.

v2: 30 pages, significant modifications from the previous version, minor mistakes corrected. v3: Published version, 30 pages, minor corrections, some details added

References in corpus (6)

• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• Bethe subalgebras of quantum affine gl(n) via shuffle algebras
• Representations of quantum toroidal $g{l}_n$
• Affine Yangian action on the Fock space
• Classical limits of quantum toroidal and affine Yangian algebras

Cited by in corpus (12)

• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• Ding-Iohara-Miki symmetry of network matrix models
• Explicit examples of DIM constraints for network matrix models
• Coproduct for the Yangian of an affine Kac-Moody algebra
• Webs of W-algebras
• Anomaly in RTT relation for DIM algebra and network matrix models
• A slow review of the AGT correspondence
• On Extensions of $\widehat{\mathfrak{gl}(m|n)}$ Kac-Moody algebras and Calabi-Yau Singularities
• Braid Group Action on Affine Yangian
• Vertex Representations for Yangians of Kac-Moody algebras
• The formal shift operator on the Yangian double
• Classical limits of quantum toroidal and affine Yangian algebras