Simple Quantum Groups, Quantum Isometry Groups and Applications

简单量子群、量子等距群及其应用

• 批准号: 9970745
• 负责人: Marc Rieffel
• 金额: $ 6.58万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970745&HistoricalAwards=false
• 关键词: Simple; Quantum; Groups; Isometry; Applications

AbstractShuzhou Wang This project is devoted to the study of compact quantum groups and their applications. The main goals in this study are to classify simple compact quantum groups and their representations, and to investigate quantum isometry groups of non-commutative spaces. A part of the project is to make precise the notions of simple compact quantum groups and quantum isometry groups. As a first important application, the quantum isometry group of Connes' finite space for the Standard Model of particle physics will be studied. As another application of ideas from quantum groups, a new approach to the well known problem of constructing ergodic actions of compact simple group on the Murray-von Neumann factor will be examined. There are various kinds of symmetries in nature. A square or a star shaped object have the simplest symmetries. To systematically study more complex symmetries, mathematicians discovered a notion, called a group, which provided the most effective way to encode the symmetries of interest. Lie groups form one of the most important classes of groups. They are ubiquitous in mathematics and physics. However, it began to be apparent that groups are inadequate in many important situations. In the mid 80's, mathematicians and physicists discovered a new notion, called a quantum group, to meet this challenge. A quantum group is not a group in general, but it encodes more general symmetries than a group. Quantum groups are very useful in quantum mechanics and quantum field theory, as well as non-commutative geometry and other fields of mathematics. The most natural examples of quantum groups are deformations (in a technical sense) of Lie groups. More recently, the author of this project discovered several new classes of quantum groups that are not deformations of Lie groups. Because of the discoveries of the author, it becomes a paramount problem to classify simple quantum groups, which are the building blocks of quantum groups. The main goals in this project are to tackle this problem and to apply simple quantum groups to various problems in closely related branches of mathematics and physics. We believe that successful solutions of the problems in this project will not only bring us to a new horizon in the theory of quantum groups, but they will also have wide impact in other fields of mathematics and physics.

摘要王树舟该项目致力于紧量子群及其应用的研究。本研究的主要目标是对简单紧量子群及其表示进行分类，并研究非交换空间的量子等距群。该项目的一部分是精确化简单紧量子群和量子等距群的概念。作为第一个重要的应用，将研究粒子物理标准模型的 Connes 有限空间的量子等距群。作为量子群思想的另一个应用，我们将研究一种新方法来解决众所周知的在默里-冯·诺依曼因子上构造紧单群遍历作用的问题。自然界中有多种对称性。 正方形或星形物体具有最简单的对称性。为了系统地研究更复杂的对称性，数学家发现了一个称为群的概念，它提供了编码感兴趣的对称性的最有效方法。 李群是最重要的群类之一。它们在数学和物理学中无处不在。 然而，在许多重要情况下，团体的作用开始变得明显。 八十年代中期，数学家和物理学家发现了一个新概念，称为量子群，来应对这一挑战。 量子群不是一般的群，但它编码了比群更一般的对称性。 量子群在量子力学和量子场论以及非交换几何和其他数学领域非常有用。量子群最自然的例子是李群的变形（在技术意义上）。 最近，该项目的作者发现了几类新的量子群，它们不是李群的变形。由于作者的发现，对简单量子群进行分类成为一个首要问题，简单量子群是量子群的构建块。 该项目的主要目标是解决这个问题，并将简单的量子群应用于数学和物理学密切相关的分支中的各种问题。 我们相信，该项目中问题的成功解决不仅将为我们带来量子群理论的新视野，而且还将对数学和物理的其他领域产生广泛的影响。