The affine Yangian of gl1 revisited

The affine Yangian of gl(1) 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 gl(1) in the same way as the Yangian Y-h(g) is an additivization of the quantum loop algebra U-q(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 gl(1) by generalizing the main result of [10] to the current settings. (C) 2016 Elsevier Inc. All rights reserved.