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

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.