{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:3ZWG6QZLNZ2M7NDF44ERTK2AQC","short_pith_number":"pith:3ZWG6QZL","schema_version":"1.0","canonical_sha256":"de6c6f432b6e74cfb465e70919ab40809e79fcc2c0b513eb1eb91859c1b1ad55","source":{"kind":"arxiv","id":"1602.06534","version":2},"attestation_state":"computed","paper":{"title":"Non-degeneracy conditions for braided finite tensor categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.QA","authors_text":"Kenichi Shimizu (Shibaura Institute of Technology)","submitted_at":"2016-02-21T14:46:00Z","abstract_excerpt":"For a braided finite tensor category $\\mathcal{C}$ with unit object $1 \\in \\mathcal{C}$, Lyubashenko considered a certain Hopf algebra $\\mathbb{F} \\in \\mathcal{C}$ endowed with a Hopf pairing $\\omega: \\mathbb{F} \\otimes \\mathbb{F} \\to 1$ to define the notion of a `non-semisimple' modular tensor category. We say that $\\mathcal{C}$ is non-degenerate if the Hopf pairing $\\omega$ is non-degenerate. In this paper, we show that $\\mathcal{C}$ is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its M\\\"uger center is trivial, if and only if t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1602.06534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-02-21T14:46:00Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"aa6bc6804e9cf278afb3f13fe2cc1823ef547d6ad630f4b7115a928cd938a5ad","abstract_canon_sha256":"69ff578a6b20f0aef7467706b46c8aadbbed9560d7f82aecef926d141e3c8aea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:59.425579Z","signature_b64":"wsMCEduhaZBxTx1a2xqJ8HWyueCt0tH87nPBWx6eb+m6fqPeJIO6nY0hRtbDSmipjj3nnuMF9eReie5sfvnpDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de6c6f432b6e74cfb465e70919ab40809e79fcc2c0b513eb1eb91859c1b1ad55","last_reissued_at":"2026-05-18T01:03:59.424829Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:59.424829Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-degeneracy conditions for braided finite tensor categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.QA","authors_text":"Kenichi Shimizu (Shibaura Institute of Technology)","submitted_at":"2016-02-21T14:46:00Z","abstract_excerpt":"For a braided finite tensor category $\\mathcal{C}$ with unit object $1 \\in \\mathcal{C}$, Lyubashenko considered a certain Hopf algebra $\\mathbb{F} \\in \\mathcal{C}$ endowed with a Hopf pairing $\\omega: \\mathbb{F} \\otimes \\mathbb{F} \\to 1$ to define the notion of a `non-semisimple' modular tensor category. We say that $\\mathcal{C}$ is non-degenerate if the Hopf pairing $\\omega$ is non-degenerate. In this paper, we show that $\\mathcal{C}$ is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its M\\\"uger center is trivial, if and only if t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06534","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1602.06534","created_at":"2026-05-18T01:03:59.424976+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.06534v2","created_at":"2026-05-18T01:03:59.424976+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.06534","created_at":"2026-05-18T01:03:59.424976+00:00"},{"alias_kind":"pith_short_12","alias_value":"3ZWG6QZLNZ2M","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_16","alias_value":"3ZWG6QZLNZ2M7NDF","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_8","alias_value":"3ZWG6QZL","created_at":"2026-05-18T12:29:58.707656+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.16241","citing_title":"Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories","ref_index":52,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC","json":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC.json","graph_json":"https://pith.science/api/pith-number/3ZWG6QZLNZ2M7NDF44ERTK2AQC/graph.json","events_json":"https://pith.science/api/pith-number/3ZWG6QZLNZ2M7NDF44ERTK2AQC/events.json","paper":"https://pith.science/paper/3ZWG6QZL"},"agent_actions":{"view_html":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC","download_json":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC.json","view_paper":"https://pith.science/paper/3ZWG6QZL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.06534&json=true","fetch_graph":"https://pith.science/api/pith-number/3ZWG6QZLNZ2M7NDF44ERTK2AQC/graph.json","fetch_events":"https://pith.science/api/pith-number/3ZWG6QZLNZ2M7NDF44ERTK2AQC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC/action/storage_attestation","attest_author":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC/action/author_attestation","sign_citation":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC/action/citation_signature","submit_replication":"https://pith.science/pith/3ZWG6QZLNZ2M7NDF44ERTK2AQC/action/replication_record"}},"created_at":"2026-05-18T01:03:59.424976+00:00","updated_at":"2026-05-18T01:03:59.424976+00:00"}