Cyclotomic Integers, Fusion Categories, and Subfactors
Cyclotomic Integers, Fusion Categories, and Subfactors
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分圆整数、融合范畴和子因子
DOI：
10.1007/s00220-010-1136-2
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发表时间：
2010
影响因子：
2.4
通讯作者：
Noah Snyder
中科院分区：
文献类型：
--
作者：
Frank Calegari;S. Morrison;Noah Snyder
融合范畴内对象的维数是环整整数，因此数论结果对融合范畴和有限深度子因子的研究具有重要意义。我们给出两个这样的应用。第一个应用是确定区间（2,76 /33）内的完整数字列表，这些数字可以作为融合类别中对象的Frobenius-Perron维度。这个列表中最小的数字是在V. Ostrik编写的附录中构建的一个新的融合类别中实现的，而其他的都是通过已知的例子实现的。第二个应用证明了在固定图上添加二价树得到的任何图族中，要么只有有限多个图是子因子的主图，要么该族由An或Dn Dynkin图组成。该结果是有效的，并将其应用于指数小于5的子因子分类中出现的几个族。
Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the Appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the An or Dn Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less than 5.