Simple Quantum Groups, Quantum Isometry Groups and Applications

王树洲这个项目致力于紧量子群及其应用的研究。本研究的主要目的是分类简单紧量子群及其表示，并研究非对易空间中的量子等距群。该项目的一部分是使简单紧量子群和量子等距群的概念变得精确。作为第一个重要的应用，我们将研究Cones有限空间中粒子物理标准模型的量子等距群。作为量子群思想的另一种应用，我们将考察构造紧单群在Murray-von Neumann因子上的遍历作用这一众所周知的问题的新方法。自然界中有各种各样的对称性。正方形或星形的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，数学家们发现了一个概念，称为群，它提供了对感兴趣的对称性进行编码的最有效的方法。李群是群中最重要的一类。它们在数学和物理中无处不在。然而，开始明显的是，在许多重要的情况下，群体是不够的。在80年代中期，数学家和物理学家S发现了一个新的概念，称为量子群，以应对这一挑战。量子群不是一般的群，但它比群编码更一般的对称性。量子群在量子力学和量子场论以及非对易几何等数学领域中都是非常有用的。量子群最自然的例子是李群的变形(在技术意义上)。最近，这个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，简单量子群的分类成为一个首要问题，简单量子群是量子群的基石。这个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密切相关的分支中的各种问题。我们相信，这个项目中问题的成功解决不仅将把我们带入量子群理论的一个新的地平线，而且它们将对数学和物理的其他领域产生广泛的影响。

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.