Casimir connections, Yangians and quantum loop algebras

量子群在八十年代中期被发现，它是一维和二维统计力学模型的对称性。经过一段时间的紧张发展，它们最近通过Nekrasov-Shatashvilii的工作重新出现为四维超对称量子规范理论的对称性，并通过Maulik-Okounkov的工作重新成为支配Nakajima箭图簇的计数几何的约束。本提议涉及到两个这样的无限维量子群，它们与复半单李代数有关：杨安和量子循环代数。该提议的一个重要组成部分是PI和S.Gautam之间的联合，并建立在他们最近发现的这些量子群之间的精确联系之上。它一方面致力于将上述联系推广到有限维表示范畴之间的等价性，另一方面，利用这种等价性，以一种让人想起Kohno-Drinfeld定理的方式，通过量子循环代数的量子Weyl群算子来计算杨氏三角Casimir联系。这一提议对延安和量子循环代数的表示理论、量子可积系统和计数几何都有潜在的意义。量子群是自然界最基本的对称性的变形。它们在80年代中期被发现是一维和二维统计力学模型的对称性，令人惊讶的是，最近又出现了四维量子规范理论的对称性，以及几何中一类计数问题的约束。对它们内在和外在结构的数学研究往往是理解和解决它们所支配的物理和数学系统的关键，因为这些对称性的存在极大地制约了这些系统。这个项目的目标是更好地理解两类这样的无限维量子群之间的关系，并使用这种关系来描述由第一类量子群控制的系统的演化，因为它的一些物理参数(例如质量)根据第二类量子群进行了调整。

Quantum groups were discovered in the mid--eighties as symmetries of 1 and 2-dimensional Statistical Mechanical models. After a period of intense development, they reemerged more recently as the symmetries of 4-dimensional supersymmetric quantum gauge theories, through the work of Nekrasov--Shatashvilii, and as the constraints governing the enumerative geometry of Nakajima quiver varieties, through the work of Maulik-Okounkov. The present proposal bears upon two such infinite-dimensional quantum groups which are associated to a complex semisimple Lie algebra: the Yangian and the quantum loop algebra. An important component of the proposal is joint between the PI and S. Gautam, and builds upon a precise link they recently discovered between these quantum groups. It endeavors on the one hand to promote the above link to an equivalence between categories of finite-dimensional representations and, on the other, to use this equivalence to compute the monodromy the trigonometric Casimir connection of the Yangian in terms of the quantum Weyl group operators of the quantum loop algebra, in a way reminiscent of the Kohno--Drinfeld theorem. The proposal has potential implications for the representation theory of Yangians and quantum loop algebras, quantum integrable systems and enumerative geometry.Quantum groups are deformations of the most basic symmetries of Nature. They were discovered in the mid-eighties as symmetries of 1 and 2-dimensional statistical mechanical models and, amazingly, reemerged recently as the symmetries of 4-dimensional quantum gauge theories, as well as the constraints of a class of enumerative problems in geometry. The mathematical study of their intrinsic and extrinsic structures is often key in understanding, and solving, the physical and mathematical systems they govern, since the presence of these symmetries greatly constrains these systems. The goal of this project is to better understand the relationship between two such classes of infinite-dimensional quantum groups, and to use this relation to describe the evolution of the systems governed by the first as some of its physical parameters (masses for example) are tuned, in terms of the second.