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Quantum Groups, Quantum Symmetries, and Non-Commutative Geometry

量子群、量子对称性和非交换几何

基本信息

批准号：
1565226
负责人：
Alexandru Chirvasitu
金额：
$ 10.2万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2017-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1565226&HistoricalAwards=false
关键词：
Quantum Groups Symmetries Non Commutative

项目摘要

Noncommutative geometry is a field of pure mathematics that traces its origins to problems in mathematical physics motivated by the discovery of quantum phenomena. The typical mathematical framework in which geometric entities (such as the 4-dimensional space-time that underlies the general theory of relativity) can be studied algebraically needs to be enlarged and generalized if it is to be reconciled with quantum phenomena. In the resulting setup, the symmetries of a physical system (in the sense of structure-preserving transformations) sometimes need to be discarded and replaced with new and more exotic notions of symmetry. The mathematical embodiment of these exotic symmetries are known as "quantum groups," and they are the main focus of this project. Much of what can be taken for granted in the context of "plain" geometry and actions of ordinary transformations on ordinary spaces becomes problematic in the noncommutative setting. This research project aims to shed light on a number of these problems, in a range of subfields within the larger realm of noncommutative geometry. This project investigates several aspects of noncommutative geometry that revolve around the notion of quantum symmetry. This involves studying quantum groups, their representation theory, and their actions on algebraic and geometric structures, as well as attendant problems in non-commutative algebraic geometry. One goal is to further understand the phenomenon of quantum rigidity, whereby certain structures admit no truly quantum symmetries. What this means is that whenever a sufficiently well-behaved Hopf algebra (which is the algebraic embodiment of a quantum group) coacts in a structure-preserving manner, the coaction factors through one by the function algebra on an ordinary group. Many special cases of this are known (integral affine algebraic varieties, certain smooth non-commutative projective algebraic varieties, compact connected smooth manifolds, certain classes of metric spaces, etc.), but the general phenomenon is poorly understood. Another goal is to attempt to transport tools and concepts specific to discrete or reductive algebraic groups (such as Borel subgroups, maximal tori, weight systems, compactifications, residual finiteness, linearity) over to quantum groups in order to further elucidate their structure and representation theory. Finally, symmetry considerations allow for the construction of new examples of smooth noncommutative projective schemes that in some sense behave generically within the moduli spaces that classify such schemes. The representation theory of the corresponding algebras would then shed light into the nature of these moduli spaces that are at the moment not well understood.

非对易几何是纯数学的一个领域，其起源可以追溯到由量子现象的发现所激发的数学物理问题。几何实体（如作为广义相对论基础的四维时空）可以用代数方法研究的典型数学框架需要扩大和推广，如果它要与量子现象相协调的话。在这样的设置中，物理系统的对称性（在结构保持变换的意义上）有时需要被丢弃，并被新的和更奇特的对称性概念所取代。这些奇异对称性的数学体现被称为“量子群”，它们是这个项目的主要焦点。许多在“普通”几何和普通空间上的普通变换的作用的上下文中可以被认为是理所当然的东西，在非对易的环境中就成了问题。该研究项目旨在阐明非对易几何更大领域内一系列子领域的许多此类问题。本计画研究非对易几何的几个方面，这些方面都是围绕量子对称性的概念。这涉及研究量子群，它们的表示理论，它们对代数和几何结构的作用，以及非交换代数几何中的伴随问题。一个目标是进一步理解量子刚性现象，即某些结构不承认真正的量子对称性。这意味着，只要一个行为足够好的霍普夫代数（量子群的代数体现）以结构保持的方式相互作用，相互作用就会被普通群上的函数代数分解为1。许多特殊的情况是已知的（积分仿射代数簇，某些光滑的非交换射影代数簇，紧致连通光滑流形，某些类的度量空间等），但人们对这一普遍现象知之甚少。另一个目标是试图将离散或约化代数群（如Borel子群、极大环面、权系统、紧化、剩余有限性、线性）的工具和概念转移到量子群上，以进一步阐明它们的结构和表示理论。最后，对称性的考虑，允许建设新的例子，光滑的非交换投影计划，在某种意义上表现一般的模空间内，这样的计划进行分类。相应代数的表示论将揭示这些模空间的本质，这些模空间目前还没有得到很好的理解。