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

对称性是指在进行某些更改时重复或保持不变的模式，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，从贝壳的螺旋到行星围绕太阳的轨道。他们也隐藏在数学对象和物理定律背后。量子群和李代数是数学家用来研究这些对称性的工具。这个项目是深入了解这些模式的底层结构，即使它们略有改变或扭曲，以及它们如何影响我们周围一切的行为。该项目还将为研究生提供研究培训机会。更详细地说，PI将在量子对称对产生的i-量子群中开发新兴方向，并在A型之外的各种经典类型的设置中开发应用。主要内容包括i-量子群的辫群作用，仿射i-量子群和扭Yangians的Drinfeld表示及其在W-代数中的应用，有限W-代数模的抛物范畴中的特征标公式;和i-量子群的分类，以及对Hecke的应用，该奖项反映了NSF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。

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.