{"paper":{"title":"Faithful simple objects, orders and gradings of fusion categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Sonia Natale","submitted_at":"2011-10-08T01:20:56Z","abstract_excerpt":"We establish some relations between the orders of simple objects in a fusion category and the structure of its universal grading group. We consider fusion categories which have a faithful simple object and show that its universal grading group must be cyclic. As for the converse, we prove that a braided nilpotent fusion category with cyclic universal grading group always has a faithful simple object. We study the universal grading of fusion categories with generalized Tambara-Yamagami fusion rules. As an application, we classify modular categories in this class and describe the modularizations"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1686","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}