Nilpotent orbits and quiver representation theory

这个项目探索了箭图表示理论(QRT)和李氏理论(LT)之间的新联系，并潜在地应用于量子理论(QT)。QRT和LT之间的联系可以追溯到著名的Gabriel-Kac关于根系和箭图表示的定理，自那以后吸引了人们对该领域的巨大关注和努力。通过Ringel(比勒费尔德，德国)、Lusztig(美国麻省理工学院)和Nakajima(京都，日本)的一系列工作，QRT对Qt做出了实质性贡献。Jensen，Su和Yu最近在双抛物代数/海藻的开放轨道方面的工作加强了这种联系，并开辟了一个新的研究领域，这是拟议中的项目所要研究的。在他学习的第一年，他将学习与该项目相关的基础课程，箭图表示理论，同调代数和仿射李代数。在今年晚些时候，他还应该继续讨论李代数和箭图表示中的Richardson元素/一般轨道的更具体的主题，并致力于很好地理解文献中关于这一主题的主要结果。他将开始进行一些计算，探索研究对象之间可能存在的新联系。

This project explores new connections between quiver representation theory (QRT) and Lie theory (LT) with potential applications to quantum theory (QT). The connection between QRT and LT dates back to the famous Gabriel-Kac Theorem on root systems and quiver representations and have since drawn enormous attention and effort into the area. It led to a substantial contribution of QRT to QT, via a sequence of work by Ringel (Bielefeld, Germany), Lusztig (MIT, USA) and Nakajima (Kyoto, Janpan). Jensen, Su and Yu's recent work on open orbits in biparabolic algebras/seaweeds strengthens the connection and opens up a new research field, which the proposed project is to study.In the first year of his study, he will learn the basic courses related to the project, quiver representation theory, homological algebra and affine Lie algebras. Later in the year, he should also move on to more specific topics on Richardson elements/generic orbits in Lie algebras and quiver representations, and aim for a good understanding of the main results on the subject in the literature. He will start to do some computation and explore the possible new connections between the subjects.