Classification of exchange relation planar algebras through sieving forest fusion graphs

We suggest a classification scheme for subfactorizable fusion bialgebras, particularly for exchange relation planar algebras. This scheme begins by transforming infinite diagrammatic consistency equations of exchange relations into a finite set of algebraic equations of degree at most 3. We then introduce a key concept, the fusion graph of a fusion bialgebra, and prove that the fusion graph for any minimal projection is a forest if and only if the planar algebra has an exchange relation. For each fusion graph, the system of degree 3 equations reduces to linear and quadratic equations that are efficiently solvable. To deal with exponentially many fusion graphs, we propose two novel analytic criteria to sieve most candidates from being subfactor planar algebras. Based on these results, we classify 5-dimensional subfactorizable fusion bialgebras with exchange relations. This scheme recovers the previous classification up to 4-dimension by Bisch, Jones, and the second author with quick proof. We developed a computer program to sieve forest fusion graphs using our criteria without solving the equations. The efficiency is 100%, as all remaining graphs are realized by exchange relation planar algebras up to 5-dimension.