The affine Yangian of gl1 revisited

The affine Yangian of gl1 has recently appeared simultaneously in the work of Maulik-Okounkov [11] and Schiffmann-Vasserot [20] in connection with the Alday-Gaiotto-Tachikawa conjecture. While the presentation from [11] is purely geometric, the algebraic presentation in [20] 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 gl1 in the same way as the Yangian Yh(g) is an “additivization” of the quantum loop algebra Uq(Lg) for a simple Lie algebra g. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of gl1 by generalizing the main result of [10] to the current settings.