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.

“对称”的概念自古以来就被人们所认识，因为它在自然界中无处不在。 对称性可以立即在视觉上明显，如镜像或圆圈，但也可能更抽象。 例如，方程的解可能具有对称性，这有助于我们更好地理解它们，或者分子可能具有对称性，这影响了它们如何相互作用。有许多方法可以使用数学语言来形式化对称性的概念，其中一些属于抽象代数领域。非正式地说，这个项目将在计算上处理对象的对称性，有点像日常生活中使用的熟悉的数字系统，但更复杂。该项目位于新发展的“量子对称性”领域，该领域允许在概念上有更大的灵活性，但代价是失去一些直觉。 然而，它可以严格地使用数学形式化，特别是在霍普夫代数和张量范畴的语言中。拟议的工作扩大了数学的参与，为不同群体的博士生，这将为他们自己的研究生涯，在学术界，商业，工业或政府准备特定的子项目。更确切地说，这个项目扩展了有限维关联代数的表示论的基础，以设置张量范畴中的代数。PI将特别关注有限张量类别，并期待融合类别的最强结果。所提出的将箭图及其表示扩展到有限张量范畴的设置，将容纳比向量空间更丰富的底层结构，例如群的自同构作用或李代数的导子作用，以及由群分级的向量空间，甚至对象具有分数的非经典结构。（Frobenius-Perron）维，因此不能被认为具有任何自然方式的底层向量空间。一个特定的类张量范畴将得到详细的关注是表示范畴的霍普夫代数。这包括许多在广泛的数学领域感兴趣的例子，如量子群。该项目由代数和数论计划和刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了NSF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。