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Quantum Symmetry

量子对称性

基本信息

批准号：
1903192
负责人：
Lee DeVille
金额：
$ 16.2万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-05-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1903192&HistoricalAwards=false
关键词：
Quantum Symmetry

项目摘要

Investigation of the symmetries of an object is a central question in mathematics and its applications. Mathematically, this is the study of invertible, property-preserving transformations from an object to itself. It is understood that the symmetries of objects we can visualize (for instance, classical objects such as spaces or manifolds or the functions on such objects) form mathematical structures known as groups. On the other hand, only recently has an appropriate notion of symmetry been developed for quantum objects and their noncommutative algebras of functions, which have been ubiquitous in mathematics and physics since the origin of quantum mechanics. It has been discovered that replacing group actions with actions of Hopf algebras is a natural and effective approach. The goal of this research project is to deepen and extend understanding of such quantum symmetries.This project will advance comprehensively the analysis and applications of quantum symmetry, including tackling the basic question: for a given algebra A, when do genuine Hopf algebra actions on A exist? Moreover, ring-theoretic, homological, and representation-theoretic properties of the Hopf algebras (or quantum groups) that "coact universally" on A will be studied. Beyond the setting of Hopf algebra actions, the investigator will employ the framework of tensor categories to study the occurrence of quantum symmetry, as they serve as a natural categorification of Hopf algebras; one benefit of this framework is that it handles actions of generalized (e.g., weak, quasi) Hopf algebras as well.

研究物体的对称性是数学及其应用的核心问题。从数学上讲，这是对从对象到自身的可逆、属性保持变换的研究。据了解，我们可以想象的对象的对称性（例如，空间或流形等经典对象或此类对象上的函数）形成称为群的数学结构。另一方面，直到最近才为量子物体及其非交换函数代数发展出适当的对称概念，自量子力学起源以来，这些概念在数学和物理学中就无处不在。人们发现，用 Hopf 代数的作用代替群作用是一种自然且有效的方法。本研究项目的目标是加深和扩展对这种量子对称性的理解。本项目将全面推进量子对称性的分析和应用，包括解决一个基本问题：对于给定的代数A，什么时候A上存在真正的Hopf代数作用？此外，还将研究在 A 上“普遍作用”的 Hopf 代数（或量子群）的环论、同调和表示论性质。除了 Hopf 代数作用的设置之外，研究者还将利用张量范畴的框架来研究量子对称性的出现，因为它们是 Hopf 代数的自然分类；该框架的一个好处是它还可以处理广义（例如弱、拟）Hopf 代数的操作。