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对称和反射对称，或者多面体的三维对称。在高等数学中，对称性是一个核心指导原则，从伽罗瓦理论中的场对称性到组合数学中计数参数的对称性。这个项目有它的根源在两个代数，通过一个组的概念，一个抽象的概念的对称性，并在分析，通过研究连续性的对称性。这可以通过考虑一个圆的旋转对称性来理解，例如，在那里，两个旋转是“靠近”还是不靠近是有意义的。量子群几乎同时从量子物理学（在量子逆散射方法的研究中）和抽象调和分析中产生，作为将庞特赖金对偶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。这导致了代数的研究，在我们的例子中，群代数。当群是可交换的（对称性的阶数无关紧要，这在几何学的例子中很少见），两个主群代数通过傅里叶变换联系起来。当组是圆组时，这是支撑信号处理的思想，数字通信中无处不在的工具，它允许将连续信号转换为离散数据，反映在数学事实中，圆的“对偶组”是整数。对于非交换组，必须放弃实际存在的“对偶组”。相反，它由不同的群代数表示。通过对Kac代数、紧量子群的研究，并受到量子物理学中各种例子的启发，我们现在已经得到了局部紧量子群的定义，它包含了所有已知的例子。虽然这项研究的根源在于更好地理解实际的群，但现在有许多真正的“量子群”的例子：不是真实的群，但显然与对称性有关的对象。我们将研究局部紧量子群的结构性质：它们的“量子子群”。几何学中的对称性是研究群的“作用”，即群对几何对象的“作用”。我们将研究量子群的作用，特别是一类直接从量子化产生的量子群（复半单量子群）和经典群，而不是其他例子。我们的目标是利用这项研究工作的鲍姆-康纳斯猜想这样的量子群。群在数学上的一个重要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“近似性质”。最后，我们将研究量子信息理论中的一些问题。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地起源于量子物理学，但近年来量子群的不同方面（例如，乍一看与物理学几乎没有关系的“组合量子群”）已经出现在量子信息论（本身与量子计算的基础有关）的研究中。我们将探讨的正是这些联系，以及与我们其他工作的联系。

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.