Quantum groups and noncommutative geometry

量子群和非交换几何

• 批准号: EP/L013916/1
• 负责人: Christian Voigt
• 金额: $ 12.4万
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL013916%2F1
• 关键词: Quantum; groups; noncommutative; geometry

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.

量子群是描述数学和物理中对称性的数学对象，包括与空间和时间的基本问题有关的现象。该理论与数学中的许多领域都有联系，包括表示论、组合数学和算子代数。非常值得注意的是，量子群也可以用来研究低维拓扑中的问题，比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，在当前的研究重点的主题。一个主要的目标是研究如果从“远处”观察一个量子群会发生什么。从技术上讲，这相当于将思想从粗糙几何转移到量子群领域，并考虑后者的大尺度性质。为了理解粗糙几何的含义，我们可以把整数集合想象成真实的直线的子集。整数的局部结构与真实的直线的结构非常不同--整数是一个离散的集合，而真实的直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，无限远的观察者将不会注意到整数和真实的线之间的任何差异。从大尺度的角度来看，这两个空间仍然可以从一个点上区分开来-这意味着即使从“遥远”的地方也保留了一些关于空间维度的信息。除此之外，我们将研究问题的交叉点表示理论的量子群和运营商K理论。李群的经典表示论是一个庞大的学科，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（时空的对称群）的表示决定的。如果一个经典对称群变形，那么通常会出现一些新的和意想不到的现象。我们将特别研究由德林费尔德双元组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旗流形的几何和经典量子群的算子K-理论。粗略地说，算子K-理论是一个不变量，它可以用来从量子群中提取同调信息，并区分量子群。我们的方法联合收割机技术从不同领域的数学，最显着的粗糙几何，算子代数，和表示理论，但也微分几何和范畴理论，这个项目的总体目标是提供这些领域之间的新的联系。

项目成果

The spatial Rokhlin property for actions of compact quantum groups

• DOI: 10.1016/j.jfa.2016.09.023
• 发表时间: 2016-05
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Selccuk Barlak;G'abor Szab'o;Christian Voigt

Compact quantum metric spaces from quantum groups of rapid decay

快速衰变量子群的紧致量子度量空间

• DOI: 10.4171/jncg/220
• 发表时间: 2016
• 期刊: Journal of Noncommutative Geometry
• 影响因子: 0.9
• 作者: Bhowmick J

Equivariant Fredholm modules for the full quantum flag manifold of SUq(3)

SUq(3) 的全量子标志流形的等变 Fredholm 模块

• 发表时间: 2015
• 期刊: Documenta Mathematica
• 影响因子: 0.9
• 作者: Voigt C

On the structure of quantum automorphism groups

• DOI: 10.1515/crelle-2014-0141
• 发表时间: 2014-11
• 期刊: arXiv: Operator Algebras
• 作者: Christian Voigt

EQUIVARIANT FREDHOLM MODULES FOR THE FULL QUANTUM FLAG MANIFOLD OF SU

SU全量子标志流形的等变FREDHOLM模块

• 发表时间: 2015
• 期刊: DOCUMENTA MATHEMATICA
• 影响因子: 0.9
• 作者: Voict Christian