Infinite Dimensional Lie algebras, Quantum Groups and their Applications

提出的研究背后的概念可以被描述为对量子理论和统计力学中对称性的数学方面的研究。本课题的第一部分重点研究了二维时空中具有特殊边界条件的量子场论模型分析中出现的对象。在第二部分，PI将研究为什么一些机械系统可以精确解析解的原因，以及与之相关的对称性论证。第三和第五部分涉及对称性统计方面的问题。项目的第四部分是将量子力学与一种特殊的二维量子场论联系起来。这一部分有望给现在相对陈旧的量子力学学科带来新的见解。在项目的第一部分，PI提议在有限维单李群、仿射李群及其量子对应物的广泛背景下研究harich - chandra系列。第二部分可以看作是第一部分的经典对应物。看来，一个经典的哈密顿对应的哈里什-钱德拉理论是一个超可积系统。第三和第五部分概述了表征理论与统计力学相结合的项目。这里的一个具体方向是研究统计力学的可积模型的极限形状。项目的第四部分建立在PI的先前结果之上，主要目标是给出相应泊松模型的费曼图中可积系统特征函数的全半经典渐近描述。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The concept behind proposed research can be characterized as the study of mathematical aspects of symmetry properties in quantum theory and statistical mechanics. The first part of the project is focused on the objects which appear in the analysis of models of quantum field theory in two-dimensional space time with special conditions on the boundary. In the second part the PI will study the reasons why some mechanical systems can be solved exactly analytically and the symmetry arguments that go with this. The third and the fifth parts are related to questions involving statistical aspects of symmetry. The fourth part of the project is to connect quantum mechanics and a special two dimensional quantum field theory. This part is expected to bring new insight into now relatively the old subject of quantum mechanics.In the first part of the project the PI proposes to investigate Harish-Chandra series in a broad setting of finite dimensional simple Lie groups, affine Lie groups and their quantum counterparts. The second part can be viewed as a classical counterpart of the first part. It appears that a classical Hamiltonian counterpart of Harish-Chandra theory is a superintegrable system. The third and the fifth parts outline projects at the interface of representation theory and statistical mechanics. One of the specific directions here is the study of of limit shapes in integrable models of statistical mechanics. The fourth part of the project is built of prior results of the PI and the main goal is to give a description of a full semiclassical asymptotic of eigenfunctions of integrable systems in terms of Feynman diagrams for the corresponding Poisson sigma model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.