Nilpotent orbits and quiver representation theory

本项目探讨了颤振表示理论（QRT）和李理论（LT）之间的新联系，以及在量子理论（QT）中的潜在应用。QRT和LT之间的联系可以追溯到著名的关于根系统和颤振表示的加布里埃尔-卡茨定理，并从此引起了对该领域的巨大关注和努力。它通过Ringel（德国比勒费尔德）、Lusztig（美国麻省理工学院）和Nakajima（日本京都）的一系列工作，导致了QRT对QT的重大贡献。Jensen， Su和Yu最近在双抛物代数/海藻的开放轨道上的工作加强了这种联系，并开辟了一个新的研究领域，这是提议的项目正在研究的。在第一年的学习中，他将学习与项目相关的基础课程，颤振表示理论，同调代数和仿射李代数。今年晚些时候，他还应该继续讨论李代数和颤振表示中的理查森元素/一般轨道等更具体的主题，并致力于更好地理解文献中关于这一主题的主要结果。他将开始做一些计算，并探索学科之间可能的新联系。

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.