The affine Yangian of revisited

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• 作者：Alexander Tsymbaliuk ^{otsymbaliuk@scgp.stonybrook.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Affine Yangian ; Yangian of gl1gl1 ; Toroidal of gl1gl1 ; Representation theory ; Shuffle algebra
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：583-645
• 全文大小：931 K

文摘

The affine Yangian of gl₁${\mathfrak{gl}}_1$ 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 gl₁${\mathfrak{gl}}_1$ in the same way as the Yangian Y_h(g)$Y_h(\mathfrak{g})$ is an “additivization” of the quantum loop algebra U_q(Lg)$U_q(L\mathfrak{g})$ for a simple Lie algebra g$\mathfrak{g}$. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of gl₁${\mathfrak{gl}}_1$ by generalizing the main result of [10] to the current settings.