The affine Yangian of gl1 revisited
The affine Yangian of gl1 revisited
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DOI：
10.1016/j.aim.2016.08.041
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发表时间：
2014-04
影响因子：
1.7
通讯作者：
A. Tsymbaliuk
中科院分区：
文献类型：
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作者：
A. Tsymbaliuk
gl 1的仿射Yangian最近同时出现在Maulik-Okounkov [11]和Schiffmann-Vasserot [20]与Alday-Gaiotto-Tachikawa猜想有关的工作中。虽然[11]的表示是纯几何的，但[20]中的代数表示是相当复杂的。在这篇文章中，我们提供了这个代数的一个简单的循环实现，它可以被看作是gl 1的量子环形代数的一个“加法化”，就像Yangian Yh（g）是一个简单李代数g的量子循环代数Uq（Lg）的“加法化”一样。我们还通过将[10]的主要结果推广到当前的设置来解释仿射Yangian和gl 1的量子环面代数的表示理论之间的相似性。
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.