Quantum Groups, W-algebras, and Brauer-Kauffmann Categories

量子群、W 代数和布劳尔-考夫曼范畴

• 批准号: 2401351
• 负责人: Weiqiang Wang
• 金额: $ 33万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401351&HistoricalAwards=false
• 关键词: Quantum; Groups; algebras; Brauer; Kauffmann

Symmetries are patterns that repeat or stay the same when certain changes are made, like rotating a shape or reflecting it in a mirror. They are everywhere in nature, from the spirals of a seashell to the orbits of planets around the sun. They also hide behind mathematical objects and the laws of physics. Quantum groups and Lie algebras are tools mathematicians use to study these symmetries. This project is a deep dive into understanding the underlying structure of these patterns, even when they're slightly changed or twisted, and how they influence the behavior of everything around us. The project will also provide research training opportunities for graduate students. In more detail, the PI will develop emerging directions in i-quantum groups arising from quantum symmetric pairs as well as develop applications in various settings of classical types beyond type A. The topics include braid group actions for i-quantum groups; Drinfeld presentations for affine i-quantum groups and twisted Yangians, and applications to W-algebras; character formulas in parabolic categories of modules for finite W-algebras; and categorification of i-quantum groups, and applications to Hecke, Brauer and Schur 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.

对称性是在进行某些更改时重复或保持相同的模式，例如旋转形状或在镜子中反射它。它们在自然界中无处不在，从贝壳的螺旋形到围绕太阳的行星轨道。它们还隐藏在数学对象和物理定律后面。量子组和谎言代数是数学家用于研究这些对称性的工具。该项目深入了解这些模式的基础结构，即使它们略有变化或扭曲，以及它们如何影响我们周围一切事物的行为。该项目还将为研究生提供研究培训机会。更详细地，PI将在由量子对称对引起的I量词组中开发新兴方向，并在A型的各种经典类型的设置中开发应用。 Drinfeld的演讲，用于仿射I-Quantum群和扭曲的Yangians，以及对W-Algebras的应用；有限w- algebras模块的抛物线类别中的字符公式；以及对Hecke，Brauer和Schur类别的I-Quantum组的分类。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来提供支持。