Quantum Subgroups of the Low Rank Lie Algebras

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.