Quivers in quantum symmetry: a path algebra framework for algebras in tensor categories

量子对称性中的颤动：张量范畴代数的路径代数框架

• 批准号: 2303334
• 负责人: Ryan Kinser
• 金额: $ 29.8万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2303334&HistoricalAwards=false
• 关键词: Quivers; quantum; symmetry; path; algebra

The concept of "symmetry" has been recognized since time immemorial, because of its ubiquitous presence in the natural world. Symmetries can be immediately visually apparent, as with mirror images or circles, but also may be more abstract. For example, solutions to an equation may have symmetries that help us better understand them, or molecules may have symmetries that affect how they interact with one another. There are a number of ways to use the language of mathematics to formalize the concept of symmetry, and some of these lie within the field of Abstract Algebra. Informally, this project will work with symmetries of an object computationally, somewhat like familiar number systems used in everyday life, but more complicated. The project lies within the newly evolving field of "quantum symmetry" which allows for greater flexibility in the concept, at the expense of losing some intuition. However, it can be formalized just as rigorously using mathematics, specifically in the language of Hopf algebras and tensor categories. The proposed work broadens participation in mathematics by having specific subprojects for a diverse group of doctoral students, which will prepare them for their own research careers, in academics, business, industry, or government.More precisely, this project extends the foundations of representation theory of finite dimensional associative algebras to the setting of algebras in tensor categories. The PI will particularly focus on finite tensor categories and expects the strongest results for fusion categories. The proposed extension of quivers and their representations to the setting of finite tensor categories will accommodate richer underlying structures than just vector spaces, such as the action of a group by automorphisms or a Lie algebras by derivations, as well as vector spaces graded by groups, and even non-classical structures where the objects have fractional (Frobenius-Perron) dimension, and thus cannot be considered to have an underlying vector space in any natural way. A specific class of tensor categories which will receive detailed attention are representation categories of Hopf algebras. This includes many examples of interest across broad areas of mathematics, such as quantum groups. This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

自远古时代以来，“对称性”的概念由于其在自然世界中的普遍存在而得到认可。 与镜像或圆圈一样，对称性可以立即在视觉上明显，但也可能更加抽象。 例如，方程解决方案可能具有对称性，可以帮助我们更好地理解它们，或者分子可能具有影响它们彼此相互作用的对称性。有多种方法可以使用数学语言正式化对称性的概念，其中一些位于抽象代数的领域内。从非正式的角度来看，该项目将与对象计算的对称性合作，有点像日常生活中使用的熟悉的数字系统，但更复杂。该项目属于“量子对称性”的新发展的领域，该领域允许在概念上更加灵活，以失去一些直觉为代价。 但是，可以使用数学，特别是在Hopf代数和张量类别的语言中，可以同样严格地将其形式化。拟议的工作通过为多样化的博士生拥有特定的副标题，从而扩大了数学的参与，这将为自己的研究职业做好准备，在学者，商业，工业或政府中为自己的研究职业做好准备。更确切地说，该项目将有限的二级辅助性代数理论的基础扩展到了Algebras of Algebras类别的环境。 PI将特别关注有限的张量类别，并期望融合类别的结果最强。 The proposed extension of quivers and their representations to the setting of finite tensor categories will accommodate richer underlying structures than just vector spaces, such as the action of a group by automorphisms or a Lie algebras by derivations, as well as vector spaces graded by groups, and even non-classical structures where the objects have fractional (Frobenius-Perron) dimension, and thus cannot be considered to have an underlying矢量空间以任何自然的方式。一类特定的张量类别将受到详细关注，这是HOPF代数的表示类别。这包括许多在数学广泛领域（例如量子组）中引起的兴趣的例子。该项目由代数和数理论计划和启发竞争性研究的既定计划共同资助（EPSCOR）。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的审查标准的评估来获得支持的。