Algebraic Quantum Symmetry

对称是数学中最古老的概念之一。从19世纪中期的群开始，许多代数结构被引入来公理化这个概念。可以说，在每个领域，捕捉对称性的数学工具都有一个代数的底层结构；被称为双代数、Hopf代数或Hopf型代数的结构。如今，这些设置之间的联系是通过范畴论的透镜来检查的，范畴论允许使用被称为单一性范畴的特殊结构，它有许多应用，包括量子信息，量子场论和弦理论。本课题的主要目标是研究一元代数范畴内代数对象的对称性，特别是hopf型代数的共/表示范畴。该项目将资助本科生的研究，PI将继续为数学科学中代表性不足的群体成员开展宣传工作。给定一个对象X， X的对称是X到自身的保性质变换，X的可逆对称的集合形成一个群：X的自同构群。此后，群的推广被引入，不仅捕获对象的对称性，而且捕获不能被观察到的对象的函数代数的对称性（例如，量子物理中的对象）。这种从“经典对称”到“量子对称”的转变起源于量子力学，并出现在诸如共形场论、低维拓扑和算子代数等活跃的研究领域。在检验“经典对称”框架之外的代数的各种对称设置，由交换代数上的群作用组成，PI将继续他们在“量子对称”方面的工作，涉及双代数或Hopf代数在具有平凡基的非交换代数上的共/作用。PI还将进一步研究“弱量子对称”，涉及弱双代数或弱霍普夫代数在具有非平凡基的非交换代数上的协同作用。此外，PI将通过“范畴量子对称”来检查代数：这涉及到一般单一性范畴的代数研究，而不一定是（弱）bi/Hopf代数的共/表示范畴，包括几种类型的半单一性范畴，例如融合范畴和模张量范畴，包括半单一性和非半单一性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Symmetry is one of the oldest notions in mathematics. Many algebraic structures have been introduced to axiomatize this notion, starting with groups in the mid-19th century. Arguably, in each area, the mathematical tools that capture symmetry have an underlying structure that is algebraic; structures that are known as a bialgebra, a Hopf algebra, or a Hopf-type algebra. Nowadays, connections between these settings are examined through the lens of category theory that allows for the use of special structures known as monoidal categories, which have numerous applications including quantum information, quantum field theory and string theory. The main goal of the project is to study symmetries of algebra objects within monoidal categories, especially co/representation categories of Hopf-type algebras. This project will fund undergraduate research and the PI will continue their advocacy work for members of underrepresented groups in the mathematical sciences. Given an object X, a symmetry of X is a property-preserving transformation from X to itself, and the collection of invertible symmetries of X forms a group: the automorphism group of X. Since then, generalizations of groups have been introduced to capture the symmetries of not only objects, but also of function algebras of objects that cannot be observed (e.g., objects in quantum physics). This move from “classical symmetry” to “quantum symmetry” has its origins in quantum mechanics, and arises in active research areas such as conformal field theory, low-dimensional topology, and operator algebras. In examining the various settings of symmetry of algebras beyond the framework of "classical symmetry", comprised of groups actions on commutative algebras, the PI will continue their work in "quantum symmetry" involving co/actions of bialgebras, or of Hopf algebras, on noncommutative algebras with a trivial base. The PI will also delve further into "weak quantum symmetry" involving co/actions of weak bialgebras, or of weak Hopf algebras, on noncommutative algebras with a non-trivial base. Moreover, the PI will examine algebras via "categorical quantum symmetry”: This pertains to studying algebras in general monoidal categories, not necessarily in co/representation categories of (weak) bi/Hopf algebras, including several types of semisimple monoidal categories, for example, fusion categories, and modular tensor categories, both semisimple and nonsemisimple.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.