Quantum Symmetry

量子对称性

基本信息

批准号：
1663775
负责人：
Chelsea Walton
金额：
$ 16.2万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2018-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1663775&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.

对对象的对称性的调查是数学及其应用中的核心问题。从数学上讲，这是对从对象到自身的可逆性，保护财产的转换的研究。可以理解的是，我们可以可视化对象的对称性（例如，经典对象，例如空间，歧管或此类对象上的函数）形成了称为组的数学结构。另一方面，直到最近才针对量子对象及其非交通性代数开发了适当的对称性概念，这些函数的函数代数在数学和物理学上一直无处不在，因为量子力学的起源。已经发现，用HOPF代数的行动代替小组行动是一种自然而有效的方法。该研究项目的目的是加深和扩展对此类量子对称性的理解。该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，真正的Hopf代数何时存在？此外，将研究“普遍共同”的HOPF代数（或量子组）的环理论，同源性和表示理论特性。除了设定HOPF代数行动外，研究人员还将采用张量类别的框架来研究量子对称性的发生，因为它们是HOPF代数的自然分类；该框架的一个好处是，它处理了广义（例如弱，准）霍普夫代数的行动。