The affine Yangian of gl1 revisited

By A. Tsymbaliuk. 2014

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Abstract 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 ( L g ) 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/bcce3dfb4bcac495b1b61450d814a88f2087e463