Algebraic Quantum Symmetry

代数量子对称性

• 批准号: 2100756
• 负责人: Chelsea Walton
• 金额: $ 22.5万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-04-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100756&HistoricalAwards=false
• 关键词: Algebraic; Quantum; Symmetry

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.

对称性是数学中最古老的概念之一。从 19 世纪中叶的群开始，许多代数结构被引入来公理化，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基础结构。 ; 被称为双代数、霍普夫代数或霍普夫型代数的结构如今通过范畴论的视角来检验。使用被称为幺半群范畴的特殊结构，它有许多应用，包括量子信息、量子场论和弦理论。该项目的主要目标是研究幺半群范畴内代数对象的对称性，特别是 Hopf 的共/表示范畴。该项目将资助本科生研究，PI 将继续为数学科学中代表性不足的群体的成员开展宣传工作。给定一个对象 X，X 的对称性是从 X 到 X 的属性保持变换。本身，并且 X 的可逆对称性的集合形成一个群：X 的自同构群。从那时起，群的推广被引入，不仅可以捕获对象的对称性，还可以捕获无法观察到的对象的函数代数的对称性（从“经典对称性”到“量子对称性”的转变起源于量子力学，并出现在共形场论、低维拓扑和量子力学等活跃的研究领域。在检查“经典对称”框架之外的代数对称性的各种设置（包括交换代数的群作用）时，PI 将继续他们在涉及双代数或 Hopf 的共同作用的“量子对称”方面的工作。代数，关于具有平凡基础的非交换代数 PI 还将进一步深入研究弱的“弱量子对称性”相互作用。此外，PI 将通过“分类量子对称性”检查代数：这涉及研究一般幺半群范畴中的代数，而不一定是共同/表示范畴中的代数。 （弱）bi/Hopf 代数，包括几种类型的半单幺半群范畴，例如融合范畴和模数张量类别，包括半简单和非半简单。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。