Geometric Structures in Poisson Geometry and Quantization

泊松几何和量子化中的几何结构

• 批准号: 0072171
• 负责人: Ping Xu
• 金额: $ 7.4万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072171&HistoricalAwards=false
• 关键词: Geometric; Structures; Poisson; Geometry; Quantization

DMS-0072171Ping XuThis project involves the study of Poisson structuresusing the theory of Lie groupoids and Lie algebroids, in particular, Poisson groupoids and Lie bialgebroids.The theory of Poisson groupoids was developed as a unification of both Drinfel'd's Poisson group theory and the theory of symplectic groupoids of Karasev-Weinstein.The investigator aims to apply this theory to study integrable systems such as Calogero-Moser systems. He will also continue his study on Courant algebroids and Dirac structures from the viewpoint of Dirac generating operators,as applied to objects in Poisson geometry such as moment maps and equivariant cohomology. This project also involves the study of deformation quantization, in particular on quantum groupoids. More specifically, it includes the study of universal enveloping algebras of Courant algebroids, Kontsevich's formalitytype conjecture for Lie algebroids, and cohomology theory of deformation of Hopf algebroids, all of which are components in quantization of Lie bialgebroids. An important application is to study quantization of classical dynamical r-matrices.Poisson geometry is largely motivated by physics, which isin fact a mathematical tool used to give a theoretical framework encompassing large parts of classical mechanics. Quantizationis developed in order to gain a better understanding betweenclassical mechanics and quantum mechanics. At present, Poisson geometry finds various applications including control theory, machining automation and robotic manipulation.

DMS-0072171徐平本项目涉及使用李群胚和李代数胚理论研究泊松结构，特别是，Poisson群胚和李双代数胚。Poisson群胚理论是Drinfel'd的Poisson群理论和Karasev-Weinstein的辛群胚理论的统一。研究者的目的是将这一理论应用于研究可积系统，如Calogero，莫泽系统。 他还将继续他的研究柯朗代数和狄拉克结构的观点狄拉克生成运营商，适用于对象的泊松几何，如矩映射和等变上同调。 该项目还涉及变形量子化的研究，特别是量子群胚。 更具体地说，它包括对Courant代数胚的泛包络代数的研究，Kontsevich对李代数胚的形式型猜想，以及Hopf代数胚的变形的上同调理论，所有这些都是李双代数胚量子化的组成部分。 泊松几何的一个重要应用是研究经典动力学r-矩阵的量子化，泊松几何在很大程度上受到物理学的启发，它实际上是一种数学工具，用来给出一个包含经典力学大部分内容的理论框架。量子化的发展是为了更好地理解经典力学和量子力学。目前，泊松几何发现各种应用，包括控制理论，加工自动化和机器人操作。