The affine Yangian of ${\mathfrak{gl}}_1$ revisited

Abstract

The affine Yangian of ${\mathfrak{gl}}_1$ 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 ${\mathfrak{gl}}_1$ in the same way as the Yangian $Y_h(\mathfrak{g})$ is an "additivization" of the quantum loop algebra $U_q(L\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of ${\mathfrak{gl}}_1$ by generalizing the milestone result of Gautam and Toledano Laredo to the current settings.

v2: 43 pages, minor corrections, some details added

References in corpus (5)

• Bethe subalgebras of quantum affine gl(n) via shuffle algebras
• Homomorphisms between different quantum toroidal and affine Yangian algebras
• More lectures on Hilbert schemes of points on surfaces
• A presentation of the deformed W_{1+\infty} algebra
• Classical limits of quantum toroidal and affine Yangian algebras

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