Quantizations and Double Affine Representation Theory

量化和双仿射表示理论

基本信息

批准号：
2001139
负责人：
Ivan Loseu
金额：
$ 56万
依托单位：
Yale University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001139&HistoricalAwards=false
关键词：
Quantizations Double Affine Representation Theory

项目摘要

Representation theory is broadly understood as a study of symmetry. More precisely, it is a study of ways for a given algebraic object to be realized concretely via linear symmetries. Quantization means a passage from Classical Physics to Quantum Physics. An interplay between Representation theory and Quantization has been, and still is, of enormous importance for both mathematics and physics. This is a project to both solve some long-tanding classical problems at the interface of Representation theory and Quantization and to uncover and study a new kind of "double affine" symmetry. The principal goals are to fully describe special kinds of symmetries arising from Quantization and determine their crucial numerical characteristics. This project provides research training opportunities for graduate students. This project consists of two parts. The first part centers around the Orbit method, an idea going back to Kirillov in the 1960's, that suggests that interesting representations should be constructed from geometric data related to suitable group actions. The PI plans to classify quantizations of equivariant covers of nilpotent orbits in classical Lie algebras and use this classification to solve several important problems in Lie representation theory. The PI also plans to classify certain interesting Harish-Chandra modules. The second part concentrates on the study of the representation theory of double affine type including that of the rational Cherednik algebras over fields of large positive characteristic and of quantum affine algebras of type A. The primary goal of this part of the project is to obtain character formulas for the corresponding categories.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.

代表理论被广泛理解为对对称的研究。更确切地说，这是对通过线性对称性具体实现给定代数对象的方法的研究。量化是指从古典物理学到量子物理学的通过。表示理论与量化之间的相互作用已经并且仍然对数学和物理学都非常重要。这是一个项目，既可以在表示理论的界面和量化的界面上解决一些长期的经典问题，又可以揭示和研究一种新型的“双重仿射”对称性。主要目标是充分描述量化引起的特殊类型的对称性，并确定其至关重要的数值特征。该项目为研究生提供了研究培训机会。该项目由两个部分组成。第一部分以轨道方法为中心，这是一个回到基里洛夫（Kirillov）在1960年代的想法，这表明应从与合适的小组动作相关的几何数据中构建有趣的表示形式。 PI计划对经典谎言代数中的nilpotent轨道的封面进行分类，并使用此分类来解决谎言代表理论中的几个重要问题。 PI还计划对某些有趣的Harish-Chandra模块进行分类。第二部分集中于对双重仿射类型的表示理论的研究，包括在较大积极特征和A型量子仿射代数的领域的理性Cherednik代数的研究。该项目的主要目标是获得相应类别的性格公式。这些奖励反映了NSF的合法任务和范围的范围。标准。