标题： 通过筛法森林融合图对交换关系平面代数的分类 显示英文标题

标题： Classification of exchange relation planar algebras through sieving forest fusion graphs

摘要： 我们建议一种子因子可分解融合双代数的分类方案，特别是针对交换关系平面代数的分类方案。 该方案首先将交换关系的无限图示一致性方程转化为最多三次的有限代数方程组。 然后我们引入一个关键概念，即融合双代数的融合图，并证明对于任何最小投影，其融合图是一棵树当且仅当平面代数具有交换关系。 对于每个融合图，三次方程组可简化为线性和二次方程，这些方程可以高效求解。 为了处理指数级的融合图，我们提出了两个新的分析标准，以筛选出大多数不能成为子因子平面代数的候选者。 基于这些结果，我们对具有交换关系的五维子因子可分解融合双代数进行了分类。 该方案通过快速证明恢复了Bisch、Jones和第二作者之前的四维以下分类结果。 我们开发了一个计算机程序，使用我们的标准筛选树状融合图而无需求解方程。 效率为100%，因为所有剩余的图在五维以内都可以由交换关系平面代数实现。 显示英文摘要

摘要： 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.