Quantum Subgroups of the Low Rank Lie Algebras

对称性的概念在我们对各种物理理论的理解中起着关键作用。一个基本的例子可以在本科生的课堂上看到，在那里一个系统的对称性被用来解决困难的通量积分。传统上，物理系统的对称性是由一个称为群的数学对象来描述的。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。现在可以理解的是，量子系统的对称性可以通过它的可观测代数(冯·诺伊曼代数)来捕捉。描述这些广义对称性的数学工具称为张量范畴。该奖项将支持提出者使用张量范畴理论来回答数学物理和冯·诺依曼代数中的几个长期存在的问题的计划。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。该项目有两个主要目标。第一种是对许多低秩李代数的第二类量子子群进行分类。这个问题是由数学物理引起的，它等价于推广了Wess-Zumino-Witten共形场理论。这项工作建立在特里·甘农最近取得的进展的基础上。二是构造和分类了von Neumann代数的双有限双模的许多新的例子。这个问题的灵感来自于对小指数子因素的分类。正如在子因素分类中一样，PI将发现奇异的例子。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.