Quantum groups and noncommutative geometry

量子群是描述数学和物理中的对称性的数学对象，包括与空间和时间基本问题相关的现象。该理论与数学中的许多领域都有联系，包括表示理论、组合学和算子代数。值得注意的是，量子群还可以用于研究低维拓扑中的问题，如区分结和寻找三维流形的不变量。在这个项目中，我们研究了一系列问题，这些问题是当前该学科研究的重点。一个主要目的是研究如果从“远处”观察量子群会发生什么。从技术上讲，这相当于将思想从粗糙几何转移到量子群的领域，并考虑后者的大尺度性质。为了更好地理解粗糙几何，我们可以把整数集合想象成实数直线的子集。整数的局部结构与实线的结构有很大的不同，整数是一个离散的集合，而实线是一个连续的连通空间。然而，如果我们“缩小”线上的积分点似乎越来越近，无限远的观察者不会注意到整数和实线之间的任何区别。从大尺度的角度来看，这两个空间仍然可以从一个点区分开来——这意味着即使从“很远”的地方也可以保留一些关于空间维度的信息。除此之外，我们还将研究量子群的表示理论和k -算子理论的交叉问题。李群的经典表示理论是一个广泛的学科，其应用范围从数论到物理学。例如，基本粒子的性质是由庞加莱群的表示决定的，庞加莱群是时空的对称群。如果一个人使一个经典对称群变形，通常会出现一些新的意想不到的现象。我们将特别研究由德林菲尔德双元表示的变形半单复李群的主级数表示的结构。这将有助于理解量子标志流形的几何和经典量子群的算子k理论。粗略地说，k -算子理论是一个不变量，可以用来从量子群中提取同调信息，并区分量子群。我们的方法结合了数学各个领域的技术，最著名的是粗几何、算子代数和表示理论，还有微分几何和范畴论，这个项目的总体目标是在这些领域之间提供新的联系。

Quantum groups are mathematical objects that describe symmetries in mathematics and physics, including phenomena which are related to fundamental questions about space and time. The theory has connections to a large range of fields in mathematics, including representation theory, combinatorics, and operator algebras. Quite remarkably, quantum groups can also be used to study problems in low-dimensional topology, like distinguishing knots and finding invariants of 3-dimensional manifolds. In this project we study a range of questions at the current focus of research in the subject. One main aim is to study what happens if one looks at a quantum group from "far away". Technically, this amount to transport ideas from coarse geometry to the realm of quantum groups, and to consider the large scale properties of the latter. To get an idea of what coarse geometry is about one can imagine the set of integers as a subset of the real line. The local structure of the integers is very different from the structure of the real line - the integers are a discrete set, whereas the real line is a continuous and connected space. However, if we "zoom out" the integral points on the line appear to get closer and closer, and an infinitely far observer will not notice any difference between the integers and the real line. On a large scale perspective, both spaces can still be distinguished from a single point - which means that even from "far away" some amount of information about the dimension of spaces is retained. Apart from this we shall study problems at the intersection of representation theory of quantum groups and operator K-theory. Classical representation theory of Lie groups is a vast subject, with applications ranging from number theory to physics. For instance, the properties of elementary particles are determined by representations of the Poincar\'e group, the symmetry group of space-time. If one deforms a classical symmetry group then typically some new and unexpected phenomena show up. We will investigate in particular the structure of principal series representations of deformed semisimple complex Lie groups represented by Drinfeld doubles. This will help to understand the geometry of quantum flag manifolds and the operator K-theory of classical quantum groups. Roughly speaking, operator K-theory is an invariant which can be used to extract homological information from a quantum group and to distinguish among quantum groups. Our methods combine techniques from various fields in mathematics, most notably coarse geometry, operator algebras, and representation theory, but also differential geometry and category theory, and an overall objective of this project is to provide new links between these areas.