The affine Yangian of \mathfrakgl₁ revisited

2014/04/30 by Alexander Tsymbaliuk · 2 citations
Mathematics · #math.RT

paper · pdf · doi:10.1016/j.aim.2016.08.041

published as Advances in Mathematics 304 (2017), 583-645 · v2: 43 pages, minor corrections, some details added

arxiv created 2016/09/16 · arxiv updated 2019/02/12

Abstract

The affine Yangian of \mathfrakgl₁ 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 \mathfrakgl₁ in the same way as the Yangian Y_h(\mathfrakg) is an "additivization" of the quantum loop algebra U_q(L\mathfrakg) for a simple Lie algebra \mathfrakg. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of \mathfrakgl₁ by generalizing the milestone result of Gautam and Toledano Laredo to the current settings.

Cited by

• Classical limits of quantum toroidal and affine Yangian algebras
• Free field realization of the quantum toroidal algebra of \mathfrakgl₁ with general levels