Quantum Subgroups of the Low Rank Lie Algebras

对称的概念在我们理解各种物理理论中起着关键作用。一个基本的例子可以在本科课堂上看到，一个系统的对称性被用来解决困难的通量积分。经典地，物理系统的对称性是用数学对象群来描述的。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。现在人们了解到，量子系统的对称性可以用它的可观测代数（冯·诺伊曼代数）来描述。描述这些广义对称性的数学工具被称为张量范畴。该奖项将支持申请人计划使用张量范畴理论来回答数学物理和冯·诺伊曼代数中几个长期存在的问题。该奖项还将通过本科生研究项目培训本科生，为美国劳动力发展做出贡献。这个项目有两个主要目标。首先是对许多低秩李代数的II型量子子群进行分类。这个问题是由数学物理引起的，它相当于扩展了Wess-Zumino-Witten共形场论。这项工作建立在特里·甘农最近取得的进展的基础上。二是构造和分类von Neumann代数双有限双模的新实例。这个问题的灵感来自于小指数子因子的分类。与子因子分类一样，PI将揭示一些奇异的例子。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。

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.