Quantum Subgroups of the Low Rank Lie Algebras

低阶李代数的量子子群

• 批准号: 2055105
• 负责人: Cain Edie-Michell
• 金额: $ 11.24万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055105&HistoricalAwards=false
• 关键词: Quantum; Subgroups; Low; Rank; Lie

The notion of symmetries plays a key role in our understanding of various physical theories. A basic example can be seen in the undergraduate classroom, where symmetries of a system are used to solve difficult flux integrals. Classically the symmetries of a physical system are described by a mathematical object known as a group. One of the major advances of modern physics has been the revelation that groups are not sufficient to capture the symmetries of a quantum system. It is now understood that the symmetries of a quantum system can be captured on its algebra of observables (a von Neumann algebra). The mathematical tool which describes these generalized symmetries is known as a tensor category. This award will support the proposer's plans to use the theory of tensor categories to answer several long-standing questions in mathematical physics and von Neumann algebras. The award also will contribute to US workforce development through the training of undergraduate students via undergraduate research projects.This project has two main goals. The first is to classify type II quantum subgroups for many of the low rank Lie algebras. This question is motivated by mathematical physics, where it is equivalent to extending the Wess-Zumino-Witten conformal field theories. This work builds on recent progress made by Terry Gannon. The second is to construct and classify many new examples of bi-finite bimodules of von Neumann algebras. This question is inspired by the classification of small index subfactors. As in the subfactor classification, PI will uncover exotic examples.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.

对称性的概念在我们理解各种物理理论中起着关键作用。本科生课堂上可以看到一个基本的例子，其中系统的对称性用于解决困难的通量积分。传统上，物理系统的对称性由称为群的数学对象来描述。现代物理学的主要进步之一是揭示了群不足以捕捉量子系统的对称性。现在人们知道，量子系统的对称性可以通过其可观测量代数（冯诺依曼代数）来捕获。描述这些广义对称性的数学工具称为张量范畴。该奖项将支持提议者的计划，即利用张量范畴理论来回答数学物理和冯·诺依曼代数中几个长期存在的问题。该奖项还将通过本科研究项目培训本科生，为美国劳动力发展做出贡献。该项目有两个主要目标。第一个是对许多低阶李代数的 II 型量子子群进行分类。这个问题是由数学物理学引发的，它相当于扩展韦斯-祖米诺-维滕共形场论。这项工作建立在 Terry Gannon 最近取得的进展的基础上。第二个是构造冯诺依曼代数的双有限双模的许多新例子并进行分类。这个问题的灵感来自于小指标子因素的分类。与子因素分类一样，PI 将发现奇异的例子。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。