{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:GFRAJ7APZ3UA4QBIAVW7VC7VR6","short_pith_number":"pith:GFRAJ7AP","canonical_record":{"source":{"id":"1208.5693","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-08-28T15:48:54Z","cross_cats_sorted":[],"title_canon_sha256":"5305c847b9a00b2f87bafa4d707018bd5d565491d312867ba5efa212181fccf7","abstract_canon_sha256":"8f7f28493ee5c5806cf84158b7ac54f7ab7013e7dc71846b20a6d7c1dd7c2870"},"schema_version":"1.0"},"canonical_sha256":"316204fc0fcee80e4028056dfa8bf58f983412375f6f7e7ec1ac5419d0d2952b","source":{"kind":"arxiv","id":"1208.5693","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.5693","created_at":"2026-05-18T03:46:58Z"},{"alias_kind":"arxiv_version","alias_value":"1208.5693v1","created_at":"2026-05-18T03:46:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5693","created_at":"2026-05-18T03:46:58Z"},{"alias_kind":"pith_short_12","alias_value":"GFRAJ7APZ3UA","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GFRAJ7APZ3UA4QBI","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GFRAJ7AP","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:GFRAJ7APZ3UA4QBIAVW7VC7VR6","target":"record","payload":{"canonical_record":{"source":{"id":"1208.5693","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-08-28T15:48:54Z","cross_cats_sorted":[],"title_canon_sha256":"5305c847b9a00b2f87bafa4d707018bd5d565491d312867ba5efa212181fccf7","abstract_canon_sha256":"8f7f28493ee5c5806cf84158b7ac54f7ab7013e7dc71846b20a6d7c1dd7c2870"},"schema_version":"1.0"},"canonical_sha256":"316204fc0fcee80e4028056dfa8bf58f983412375f6f7e7ec1ac5419d0d2952b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:58.002707Z","signature_b64":"Wcv/ft7jhv8ML3SvUYAJKY/qqeAs29i4SjxvbW28RBNQl1ee9YcuUB8IN9OuekB8MPAh//DmqF3AXYLzghYSDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"316204fc0fcee80e4028056dfa8bf58f983412375f6f7e7ec1ac5419d0d2952b","last_reissued_at":"2026-05-18T03:46:58.001925Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:58.001925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.5693","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":"2026-05-18T03:46:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FO/w5NIh5LgsDzBLMQebxD764fZPM4Poq7l/ySLU7a54JdUM2sHhooaJvX7HW3wImi79o+kSF6A1i8C2RDZPAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:14:08.044457Z"},"content_sha256":"448e5ffd35e48b3bdc036a2bf79f395e833f4958c01424cb1c07f0fcad697e89","schema_version":"1.0","event_id":"sha256:448e5ffd35e48b3bdc036a2bf79f395e833f4958c01424cb1c07f0fcad697e89"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:GFRAJ7APZ3UA4QBIAVW7VC7VR6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The doubles of a braided Hopf algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Alain Brugui\\`eres, Alexis Virelizier","submitted_at":"2012-08-28T15:48:54Z","abstract_excerpt":"Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) of A, which is a quasitriangular Hopf algebra in B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. Here, quasitriangular means endowed with an R-matrix (our notion of R-matrix for a Hopf algebra in B involves the coend C of B). In general, i.e. when B does not necessarily admit a coend, we construct a quasitriangular Hopf monad d_A on the center Z(B) of B whose category of modules is iso"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5693","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":"2026-05-18T03:46:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q0IYRViQC8nDJLr43Th9ABp14rQ16zqeYnVH2Nri10M3OQ5lokGmmFqLylSKp5YGAUmLJQpls7dbW+yEy1IsCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:14:08.044817Z"},"content_sha256":"e962737f74492f02664505ebf8708eb8e8bd541f8aafbce6961643ebea3cc107","schema_version":"1.0","event_id":"sha256:e962737f74492f02664505ebf8708eb8e8bd541f8aafbce6961643ebea3cc107"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GFRAJ7APZ3UA4QBIAVW7VC7VR6/bundle.json","state_url":"https://pith.science/pith/GFRAJ7APZ3UA4QBIAVW7VC7VR6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GFRAJ7APZ3UA4QBIAVW7VC7VR6/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":"2026-05-28T06:14:08Z","links":{"resolver":"https://pith.science/pith/GFRAJ7APZ3UA4QBIAVW7VC7VR6","bundle":"https://pith.science/pith/GFRAJ7APZ3UA4QBIAVW7VC7VR6/bundle.json","state":"https://pith.science/pith/GFRAJ7APZ3UA4QBIAVW7VC7VR6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GFRAJ7APZ3UA4QBIAVW7VC7VR6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GFRAJ7APZ3UA4QBIAVW7VC7VR6","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":"8f7f28493ee5c5806cf84158b7ac54f7ab7013e7dc71846b20a6d7c1dd7c2870","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-08-28T15:48:54Z","title_canon_sha256":"5305c847b9a00b2f87bafa4d707018bd5d565491d312867ba5efa212181fccf7"},"schema_version":"1.0","source":{"id":"1208.5693","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.5693","created_at":"2026-05-18T03:46:58Z"},{"alias_kind":"arxiv_version","alias_value":"1208.5693v1","created_at":"2026-05-18T03:46:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5693","created_at":"2026-05-18T03:46:58Z"},{"alias_kind":"pith_short_12","alias_value":"GFRAJ7APZ3UA","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GFRAJ7APZ3UA4QBI","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GFRAJ7AP","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:e962737f74492f02664505ebf8708eb8e8bd541f8aafbce6961643ebea3cc107","target":"graph","created_at":"2026-05-18T03:46:58Z","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":"Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) of A, which is a quasitriangular Hopf algebra in B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. Here, quasitriangular means endowed with an R-matrix (our notion of R-matrix for a Hopf algebra in B involves the coend C of B). In general, i.e. when B does not necessarily admit a coend, we construct a quasitriangular Hopf monad d_A on the center Z(B) of B whose category of modules is iso","authors_text":"Alain Brugui\\`eres, Alexis Virelizier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-08-28T15:48:54Z","title":"The doubles of a braided Hopf algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5693","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":"sha256:448e5ffd35e48b3bdc036a2bf79f395e833f4958c01424cb1c07f0fcad697e89","target":"record","created_at":"2026-05-18T03:46:58Z","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":"8f7f28493ee5c5806cf84158b7ac54f7ab7013e7dc71846b20a6d7c1dd7c2870","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-08-28T15:48:54Z","title_canon_sha256":"5305c847b9a00b2f87bafa4d707018bd5d565491d312867ba5efa212181fccf7"},"schema_version":"1.0","source":{"id":"1208.5693","kind":"arxiv","version":1}},"canonical_sha256":"316204fc0fcee80e4028056dfa8bf58f983412375f6f7e7ec1ac5419d0d2952b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"316204fc0fcee80e4028056dfa8bf58f983412375f6f7e7ec1ac5419d0d2952b","first_computed_at":"2026-05-18T03:46:58.001925Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:58.001925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Wcv/ft7jhv8ML3SvUYAJKY/qqeAs29i4SjxvbW28RBNQl1ee9YcuUB8IN9OuekB8MPAh//DmqF3AXYLzghYSDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:58.002707Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.5693","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:448e5ffd35e48b3bdc036a2bf79f395e833f4958c01424cb1c07f0fcad697e89","sha256:e962737f74492f02664505ebf8708eb8e8bd541f8aafbce6961643ebea3cc107"],"state_sha256":"7908f6c3e844e184fe6530648719e4b5be17f4d0471d16efbfa920cf38009669"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k5qeF+NGuCsIOcqIC4+t3in+nlT2QwoUkjVkKTfTUdfllb2o2sg9aTNJZHnlfakLYsVuzsXsdOHq/ylvfz6ZDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T06:14:08.046889Z","bundle_sha256":"9fc420d4ac439c4a2470a11eed1a3d50f0bdff0eccb5c04707e49edff34b0173"}}