Quantum groups in action

对称性首先出现在几何学中，例如正多边形的旋转和反射对称性，或多面体的三维对称性。在高等数学中，对称性是核心指导原则，从伽罗瓦理论中的场对称性到组合学中使用对称性来计算论点。这个项目既植根于代数，通过群的概念，抽象出对称的概念，又植根于分析，通过研究对称的连续性。这可以通过思考圆的旋转对称性来看出，例如，在这种情况下，谈论两个旋转是否“接近”是有意义的。量子群几乎同时来自量子物理学(在量子逆散射方法的研究中)和抽象的调和分析，作为将庞特雷金对偶性推广到非阿贝尔群的探索的一部分。在现代数学中，人们通常不直接研究数学对象，而是研究数学对象之间的映射。这就引出了对代数的研究，在我们的例子中，群代数。当群是可交换的(对称顺序无关紧要，这在几何学的例子中很少见)时，两个主要的群代数通过傅立叶变换联系在一起。当群是圆群时，这是支撑信号处理的思想，信号处理是数字通信中普遍存在的工具，它允许一个人将连续的信号转换为离散数据，这反映在数学事实上，即对圆的“对偶群”是整数。对于非交换群，人们不得不放弃实际存在的“对偶群”。相反，它由一个不同的群代数来表示。通过对Kac代数、紧量子群的研究，在量子物理学的各种例子的启发下，我们得到了局部紧量子群的定义，它包含了所有已知的例子。虽然这项研究的基础是简单地理解更好的实际群，但现在有许多真正的“量子群”的例子：不是真正的群，但显然与对称性有关的物体。这个项目探索了量子群论中的广泛问题。我们将研究局部紧量子群的结构性质：它们的“量子子群”。几何中的对称性是对群“作用”的研究，其中群“作用”在几何对象上。我们将研究量子群的作用，特别是一类直接由量子化产生的量子群(复半单量子群)，以及经典群，而不是其他例子。我们的目的是利用这项研究来研究此类量子群的Baum-Connes猜想。群的一个重要的数学用途是提供代数对象的例子。量子群产生了各种各样的算子代数，我们将研究这些代数的性质，特别是各种“逼近性质”。最后，我们将研究量子信息论中的一些问题。正如数学中常见的那样，不同领域之间的关系以意想不到的方式出现。虽然量子群最初部分起源于量子物理学，但近年来，量子群的不同方面(例如，乍一看与物理几乎没有什么关系的“组合量子群”)出现在量子信息论(本身与量子计算的基础有关)的研究中。正是这些联系，以及与我们其他工作的联系，我们将予以探索。

Symmetry is first encountered in geometry, for example the rotational and reflectional symmetry of regular polygons, or the three dimensional symmetries of polyhedra. In higher mathematics, symmetry is a core guiding principle, from symmetries of fields in Galois theory through to the use of symmetry in counting arguments in Combinatorics. This project has its roots in both algebra, through the notion of a group, an abstraction of the notion of a symmetry, and in analysis, through the study of continuity of symmetries. This can be seen by thinking about the rotational symmetries of a circle, for example, where it makes sense to speak of two rotations being "close together" or not.Quantum groups arose almost simultaneously from quantum physics (in the study of the quantum inverse scattering method) and from abstract harmonic analysis, as part of a quest to generalise Pontraygin duality to non-abelian groups. In modern mathematics, one often does not study a mathematical object directly, but instead studies maps between mathematical objects. This leads to the study of algebras, in our case, the group algebra. When the group is commutative (the order of symmetry does not matter, which is rare in examples from geometry) the two main group algebras are linked through the Fourier Transform. When the group is the circle group, this is the idea underpinning signal processing, a ubiquitous tool in digital communication, which allows one to transform a continuous signal into discrete data, reflected in the Mathematical fact that the "dual group" to the circle is the integers.For non-commutative groups, one has to give up on the "dual group" actually existing. Instead, it is represented by a different group algebra. Through the study of Kac algebras, Compact Quantum Groups, and motivated by various examples motivated from Quantum Physics, we have now arrived at the definition of a Locally Compact Quantum Group, which encompasses all known examples. While this study has its roots in simply understanding better actual groups, there are now many examples of genuinely "quantum groups": objects which are not real groups, but clearly have something to do with symmetry.This project explores a wide gamut of questions from quantum group theory. We will study structural properties of locally compact quantum groups: their "quantum subgroups". Symmetries in geometry are the study of group "actions" where a group "acts" on a geometric object. We will study actions of quantum groups, in particular, a class of quantum groups (the complex semisimple quantum groups) which arise directly from quantization, and classical groups, than other examples. Our aim is to use this study to work on the Baum-Connes conjecture for such quantum groups. An important mathematical use of groups is in providing examples of algebraic objects. Quantum groups give rise to a wide variety of Operator Algebras, and we will study properties of these algebras, in particular various "approximation properties". Finally, we will study some questions from Quantum Information Theory. As is common in mathematics, relations between diverse areas arise in unexpected ways. While quantum groups originally arose in part from Quantum Physics, in recent years quite different aspects of quantum groups (for example, the "combinatorial quantum groups" which at first glance have little to do with physics) have appeared in the study of Quantum Information Theory (itself related to the foundations of Quantum Computation). It is these links, and links with our other work, which we shall explore.