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.

非交换性几何形状是一个纯数学领域，它的起源是由于发现量子现象而动机的数学物理学问题。典型的数学框架（例如，基于相对论的一般理论的4维时空）可以通过代数进行扩大和概括，如果要与量子现象进行一致，则需要扩大和概括。在最终的设置中，有时需要丢弃物理系统的对称性（从结构的转换意义上），并用新的和更奇特的对称性概念替换。这些异国情调对称性的数学实施例被称为“量子群”，它们是该项目的主要重点。在“普通”的几何形状和普通转换在普通空间上的行动的背景下，可以理所当然地认为是有问题的。该研究项目的目的是阐明许多此类问题，即在较大的非交通几何范围内的一系列子场中。该项目研究了围绕量子对称性概念的非交通性几何形状的几个方面。这涉及研究量子群，其表示理论以及它们对代数和几何结构的作用，以及非共同代数几何形状中的伴随问题。一个目标是进一步了解量子刚度的现象，某些结构不承认真正的量子对称性。这意味着，每当以结构性的方式进行量子群的表现足够良好的HOPF代数（这是量子组的代数实施例）时，普通组上的函数代数通过一个函数代数通过一个函数来取代。许多特殊情况是已知的（积分仿射代数品种，某些平滑的非交通性射击代数品种，紧凑的连接平滑歧管，某些类别的度量空间等），但总体现象知之甚少。另一个目标是尝试将特定于离散或还原的代数群（例如Borel亚组，最大托架，重量系统，压缩，残留有限，线性性，线性性）的工具和概念运输到量子组中，以便进一步阐明其结构和表示理论。最后，对称考虑因素允许构建新的平滑非共同投影方案的示例，在某种意义上说，这些方案在某种意义上是在对此类方案进行分类的模量空间中的一般行为。然后，相应的代数的表示理论将散发到目前尚不清楚这些模量空间的性质中。