The affine Yangian of $\mathfrak{gl}_1$ revisited

TL;DR

This paper introduces a simplified loop realization of the affine Yangian of gl_1, connecting geometric and algebraic perspectives, and explores its representation theory similarities with quantum toroidal algebras.

Contribution

It provides a new, simpler loop realization of the affine Yangian of gl_1, bridging geometric and algebraic approaches, and extends representation theory comparisons.

Findings

A loop realization of the affine Yangian of gl_1 is constructed.

The representation theories of the affine Yangian and quantum toroidal algebras are shown to be similar.

The work generalizes Gautam and Toledano Laredo's results to this setting.

Abstract

The affine Yangian of $\mathfrak{gl}_1$ has recently appeared simultaneously in the work of Maulik-Okounkov and Schiffmann-Vasserot in connection with the Alday-Gaiotto-Tachikawa conjecture. While the former presentation is purely geometric, the latter algebraic presentation is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an "additivization" of the quantum toroidal algebra of $\mathfrak{gl}_1$ in the same way as the Yangian $Y_h(\mathfrak{g})$ is an "additivization" of the quantum loop algebra $U_q(L\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of $\mathfrak{gl}_1$ by generalizing the milestone result of Gautam and Toledano Laredo to the current settings.