Affine Yangians as Limits of Quantum Toroidal Algebras

TL;DR

This paper proves a fundamental isomorphism between affine Yangians and quantum toroidal algebras, establishing structural properties and confirming conjectures about their relationship in the context of affine Kac-Moody Lie algebras.

Contribution

It demonstrates that affine Yangians are isomorphic to the associated graded algebra of quantum toroidal algebras, extending finite-dimensional results to the affine case.

Findings

Established a PBW basis for affine Yangians in all untwisted affine types.

Identified the classical limit of affine Yangians as the universal enveloping algebra of polynomial currents.

Constructed a PBW basis for quantum toroidal algebras using a new torsion-freeness argument.

Abstract

We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is isomorphic, as a $\mathbb{C}[\hbar]$-algebra, to the associated graded algebra of the quantum toroidal algebra $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ with respect to a canonical filtration. This result constitutes the affine analogue of Drinfeld's conjecture on the relationship between Yangians and quantum loop algebras, previously established in the finite-dimensional setting by Gautam--Toledano Laredo and by Guay--Ma. As principal applications of this isomorphism, we derive two fundamental structural properties of affine Yangians: a Poincar\'e--Birkhoff--Witt (PBW) basis for $Y_\hbar(\mathfrak{g})$ in all untwisted affine types, and the identification of its classical limit as the universal enveloping algebra $U(\mathfrak{g}[u])$ of the polynomial current Lie algebra. A key ingredient of independent interest is our construction of a PBW basis for $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ itself, which relies on a new torsion-freeness argument for the quantum toroidal algebra and the topological Nakayama lemma.