Noncommutative Geometry Seminar

Date: March 29, 2024

Time: 11:10AM - 12:10PM

Location: BLOC 306

Speaker: Fan Lu , Tsinghua University

Title: Classification of exchange relation planar algebras of rank 5

Abstract: Exchange relation planar algebras are natural generalizations of Kac algebras from skein theoretical point of view. We show that its classification is essentially solving a system of algebraic equations, but too complicated to solve directly. Then we introduce a key concept, the type of the fusion rule, which completely detects exchange relations as forest types. According to types, the system of equations reduces to exponentially many subsystems, which are solvable individually. In addition, we propose new analytic criteria to rule out most types from being subfactor planar algebras. Eventually, we are able to classify exchange relation planar algebras of rank 5. This method recovers the previous classification up to rank 4 of Bisch, Jones, and Liu with quick proofs. This is joint work with Zhengwei Liu.