Quantum Symmetry

研究物体的对称性是数学及其应用中的一个中心问题。从数学上讲，这是研究从一个对象到它自己的可逆的、保持性质的变换。可以理解，我们可以可视化的对象的对称性(例如，诸如空间或流形之类的经典对象或此类对象上的函数)形成了称为群的数学结构。另一方面，直到最近才为量子对象及其非对易函数代数提出了适当的对称性概念，自量子力学起源以来，它们在数学和物理中一直无处不在。研究发现，用Hopf代数的作用来代替群作用是一种自然而有效的方法。这个研究项目的目的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，包括解决这样一个基本问题：对于给定的代数A，A上的真Hopf代数作用何时存在？此外，还将研究在A上“全余”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，研究者还将使用张量范畴的框架来研究量子对称性的发生，因为它们是Hopf代数的自然范畴；这个框架的一个好处是它也处理广义的(例如，弱的、准的)Hopf代数的作用。

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.