The affine Yangian of \(\mathfrak{gl}_1\) revisited

DOI10.1016/J.AIM.2016.08.041zbMATH Open1350.81017arXiv1404.5240OpenAlexW2594149305MaRDI QIDQ329503

Author name not available (Why is that?)

Publication date: 21 October 2016

Published in: (Search for Journal in Brave)

Abstract: The affine Yangian of

m a t h f r a k {g l}_{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

m a t h f r a k {g l}_{1}

in the same way as the Yangian

Y_{h} (m a t h f r a k g)

is an "additivization" of the quantum loop algebra

U_{q} (L m a t h f r a k g)

for a simple Lie algebra

m a t h f r a k g

. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of

m a t h f r a k {g l}_{1}

by generalizing the milestone result of Gautam and Toledano Laredo to the current settings.

Full work available at URL: https://arxiv.org/abs/1404.5240

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