{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:4XCUEKROOBOZZNBM6SIDVBU6GR","short_pith_number":"pith:4XCUEKRO","canonical_record":{"source":{"id":"1811.05123","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-11-13T06:23:46Z","cross_cats_sorted":["hep-th","math.RT"],"title_canon_sha256":"248e5a231610fea20b929e4a041cb0ad9fd2cf5f188940aceb46539b21484b0b","abstract_canon_sha256":"f21646cea54531e627ade9cf6e660287c05c33fea22befaf4876b351f8064b64"},"schema_version":"1.0"},"canonical_sha256":"e5c5422a2e705d9cb42cf4903a869e3479298d4810dd3a1c5e8e47e1db25f211","source":{"kind":"arxiv","id":"1811.05123","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05123","created_at":"2026-05-18T00:01:01Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05123v1","created_at":"2026-05-18T00:01:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05123","created_at":"2026-05-18T00:01:01Z"},{"alias_kind":"pith_short_12","alias_value":"4XCUEKROOBOZ","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"4XCUEKROOBOZZNBM","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"4XCUEKRO","created_at":"2026-05-18T12:32:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:4XCUEKROOBOZZNBM6SIDVBU6GR","target":"record","payload":{"canonical_record":{"source":{"id":"1811.05123","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-11-13T06:23:46Z","cross_cats_sorted":["hep-th","math.RT"],"title_canon_sha256":"248e5a231610fea20b929e4a041cb0ad9fd2cf5f188940aceb46539b21484b0b","abstract_canon_sha256":"f21646cea54531e627ade9cf6e660287c05c33fea22befaf4876b351f8064b64"},"schema_version":"1.0"},"canonical_sha256":"e5c5422a2e705d9cb42cf4903a869e3479298d4810dd3a1c5e8e47e1db25f211","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:01.154694Z","signature_b64":"ba+o/jQWDKD14LMI1Fd6EUpuSZC9uWd58I1kosw75ItSvGQfBdqEdjGmg0DW4HeMYN1eXPDYInh8vE5oxHGTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5c5422a2e705d9cb42cf4903a869e3479298d4810dd3a1c5e8e47e1db25f211","last_reissued_at":"2026-05-18T00:01:01.154062Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:01.154062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.05123","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f6dG0Mn5cuiI39l9CoCGLvg5yGniUkmpVjIxzpQ1PMnigtpE7Ljo/Xhdv0n2YFy3TZME3fWEA/ckcP6PqDW2Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T18:58:41.359792Z"},"content_sha256":"8eec3099356fbb506602ce8f55dba97c815914dbe574766f2eaead548b59db40","schema_version":"1.0","event_id":"sha256:8eec3099356fbb506602ce8f55dba97c815914dbe574766f2eaead548b59db40"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:4XCUEKROOBOZZNBM6SIDVBU6GR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Affine Lie algebras and tensor categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.RT"],"primary_cat":"math.QA","authors_text":"Yi-Zhi Huang","submitted_at":"2018-11-13T06:23:46Z","abstract_excerpt":"We review briefly the existing vertex-operator-algebraic constructions of various tensor category structures on module categories for affine Lie algebras. We discuss the results first conjectured in the work of Moore and Seiberg that led us to the construction of the modular tensor category structure in the positive integral level case. Then we review the existing constructions and results in the following three cases: (i) the level plus the dual Coxeter number is not a nonnegative rational number, (ii) the level is a positive integer and (iii) the level is an admissible number. We also presen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05123","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ylbAiVzE18Kkff6ofx16mB8K3JV3aaJ+VNrKlursyijIN5r6vpzRS1JnRkeDph6XO71UIE4sGG6aCXAA5FpLAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T18:58:41.360529Z"},"content_sha256":"fb03a11f0407495c3f5560c1ed6fbd4b5f97e71cfb569ee84f13edc0c0cccd15","schema_version":"1.0","event_id":"sha256:fb03a11f0407495c3f5560c1ed6fbd4b5f97e71cfb569ee84f13edc0c0cccd15"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4XCUEKROOBOZZNBM6SIDVBU6GR/bundle.json","state_url":"https://pith.science/pith/4XCUEKROOBOZZNBM6SIDVBU6GR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4XCUEKROOBOZZNBM6SIDVBU6GR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T18:58:41Z","links":{"resolver":"https://pith.science/pith/4XCUEKROOBOZZNBM6SIDVBU6GR","bundle":"https://pith.science/pith/4XCUEKROOBOZZNBM6SIDVBU6GR/bundle.json","state":"https://pith.science/pith/4XCUEKROOBOZZNBM6SIDVBU6GR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4XCUEKROOBOZZNBM6SIDVBU6GR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:4XCUEKROOBOZZNBM6SIDVBU6GR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f21646cea54531e627ade9cf6e660287c05c33fea22befaf4876b351f8064b64","cross_cats_sorted":["hep-th","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-11-13T06:23:46Z","title_canon_sha256":"248e5a231610fea20b929e4a041cb0ad9fd2cf5f188940aceb46539b21484b0b"},"schema_version":"1.0","source":{"id":"1811.05123","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05123","created_at":"2026-05-18T00:01:01Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05123v1","created_at":"2026-05-18T00:01:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05123","created_at":"2026-05-18T00:01:01Z"},{"alias_kind":"pith_short_12","alias_value":"4XCUEKROOBOZ","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"4XCUEKROOBOZZNBM","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"4XCUEKRO","created_at":"2026-05-18T12:32:05Z"}],"graph_snapshots":[{"event_id":"sha256:fb03a11f0407495c3f5560c1ed6fbd4b5f97e71cfb569ee84f13edc0c0cccd15","target":"graph","created_at":"2026-05-18T00:01:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We review briefly the existing vertex-operator-algebraic constructions of various tensor category structures on module categories for affine Lie algebras. We discuss the results first conjectured in the work of Moore and Seiberg that led us to the construction of the modular tensor category structure in the positive integral level case. Then we review the existing constructions and results in the following three cases: (i) the level plus the dual Coxeter number is not a nonnegative rational number, (ii) the level is a positive integer and (iii) the level is an admissible number. We also presen","authors_text":"Yi-Zhi Huang","cross_cats":["hep-th","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-11-13T06:23:46Z","title":"Affine Lie algebras and tensor categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05123","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8eec3099356fbb506602ce8f55dba97c815914dbe574766f2eaead548b59db40","target":"record","created_at":"2026-05-18T00:01:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f21646cea54531e627ade9cf6e660287c05c33fea22befaf4876b351f8064b64","cross_cats_sorted":["hep-th","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-11-13T06:23:46Z","title_canon_sha256":"248e5a231610fea20b929e4a041cb0ad9fd2cf5f188940aceb46539b21484b0b"},"schema_version":"1.0","source":{"id":"1811.05123","kind":"arxiv","version":1}},"canonical_sha256":"e5c5422a2e705d9cb42cf4903a869e3479298d4810dd3a1c5e8e47e1db25f211","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5c5422a2e705d9cb42cf4903a869e3479298d4810dd3a1c5e8e47e1db25f211","first_computed_at":"2026-05-18T00:01:01.154062Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:01.154062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ba+o/jQWDKD14LMI1Fd6EUpuSZC9uWd58I1kosw75ItSvGQfBdqEdjGmg0DW4HeMYN1eXPDYInh8vE5oxHGTAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:01.154694Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.05123","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8eec3099356fbb506602ce8f55dba97c815914dbe574766f2eaead548b59db40","sha256:fb03a11f0407495c3f5560c1ed6fbd4b5f97e71cfb569ee84f13edc0c0cccd15"],"state_sha256":"9e9f81c6ba4fc9eef9bca63daf80859d4e34bc38500db6026ff7d94d5228c371"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tPt+7nUsB4rRhIP0Kg7q5bK3rH8I2U1oRDWr0q8swzw2ipjARCB7zWlJuZLsqqX49sJf0yQjzFdzr4nPYD5tAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T18:58:41.365636Z","bundle_sha256":"6c5fc3f98b6443cbd8aee9705162a915878c2cb65ff3bb498db28e39b7341fa0"}}