{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:GLRJQYKHHJ5FFXOTMEK2WUR5RY","short_pith_number":"pith:GLRJQYKH","canonical_record":{"source":{"id":"math/0410621","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2004-10-29T14:10:27Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"25d6abc31576011441bd29baaf9a135dad0675a2e7dd2056bb6c8cdff9976fdd","abstract_canon_sha256":"c5a6fc1a77260108916b98d49c0b713e53ddfe1965eadeeb5255e2bab2a61c66"},"schema_version":"1.0"},"canonical_sha256":"32e29861473a7a52ddd36115ab523d8e3d1475b7fa730acddbfd2d7e74d4e164","source":{"kind":"arxiv","id":"math/0410621","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410621","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410621v1","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410621","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"pith_short_12","alias_value":"GLRJQYKHHJ5F","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"pith_short_16","alias_value":"GLRJQYKHHJ5FFXOT","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"pith_short_8","alias_value":"GLRJQYKH","created_at":"2026-07-04T14:39:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:GLRJQYKHHJ5FFXOTMEK2WUR5RY","target":"record","payload":{"canonical_record":{"source":{"id":"math/0410621","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2004-10-29T14:10:27Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"25d6abc31576011441bd29baaf9a135dad0675a2e7dd2056bb6c8cdff9976fdd","abstract_canon_sha256":"c5a6fc1a77260108916b98d49c0b713e53ddfe1965eadeeb5255e2bab2a61c66"},"schema_version":"1.0"},"canonical_sha256":"32e29861473a7a52ddd36115ab523d8e3d1475b7fa730acddbfd2d7e74d4e164","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:39:19.012459Z","signature_b64":"E2h9SM1J8bdZHVxO352FSi0/5DhlWzRhakB47Jg08F0IgAxdSXAKbf6+YKNrzcg/lkUEjKkOl7bQW9WkBfe/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32e29861473a7a52ddd36115ab523d8e3d1475b7fa730acddbfd2d7e74d4e164","last_reissued_at":"2026-07-04T14:39:19.012051Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:39:19.012051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0410621","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":"2026-07-04T14:39:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1BNx9FGIta7C07eWLz4RBIU4Fv6mY6mJeKGOTo5yMo3VIxlQqWKgpUjAt7Hj6/yaB9hU9dFx35Vh3JMyz49dAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T21:40:59.614689Z"},"content_sha256":"5c5d57960241a31cf17639e6a532fa3732d7ef37d258739e731be4658e7ddd6a","schema_version":"1.0","event_id":"sha256:5c5d57960241a31cf17639e6a532fa3732d7ef37d258739e731be4658e7ddd6a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:GLRJQYKHHJ5FFXOTMEK2WUR5RY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homotopy algebras and noncommutative geometry","license":"","headline":"","cross_cats":["math.AT"],"primary_cat":"math.QA","authors_text":"Alastair Hamilton, Andrey Lazarev","submitted_at":"2004-10-29T14:10:27Z","abstract_excerpt":"We study cohomology theories of strongly homotopy algebras, namely $A_\\infty, C_\\infty$ and $L_\\infty$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_\\infty$-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a $C_\\infty$-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\\infty$-algebra (an $\\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' oper"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410621","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0410621/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T14:39:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gAWE1I1ndm33DqAvF9uVYM+WlHegHju/3jaLwfgyt646OCxx0Tc9VsvmDnZ2FF/S5OIOXW+jC3CwFXBKEGvuAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T21:40:59.615612Z"},"content_sha256":"987f1443294aae27c8f9583b00a31e60e328eb0a5a99173d2fcf959bb63edf44","schema_version":"1.0","event_id":"sha256:987f1443294aae27c8f9583b00a31e60e328eb0a5a99173d2fcf959bb63edf44"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GLRJQYKHHJ5FFXOTMEK2WUR5RY/bundle.json","state_url":"https://pith.science/pith/GLRJQYKHHJ5FFXOTMEK2WUR5RY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GLRJQYKHHJ5FFXOTMEK2WUR5RY/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":"2026-08-10T21:40:59Z","links":{"resolver":"https://pith.science/pith/GLRJQYKHHJ5FFXOTMEK2WUR5RY","bundle":"https://pith.science/pith/GLRJQYKHHJ5FFXOTMEK2WUR5RY/bundle.json","state":"https://pith.science/pith/GLRJQYKHHJ5FFXOTMEK2WUR5RY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GLRJQYKHHJ5FFXOTMEK2WUR5RY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:GLRJQYKHHJ5FFXOTMEK2WUR5RY","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":"c5a6fc1a77260108916b98d49c0b713e53ddfe1965eadeeb5255e2bab2a61c66","cross_cats_sorted":["math.AT"],"license":"","primary_cat":"math.QA","submitted_at":"2004-10-29T14:10:27Z","title_canon_sha256":"25d6abc31576011441bd29baaf9a135dad0675a2e7dd2056bb6c8cdff9976fdd"},"schema_version":"1.0","source":{"id":"math/0410621","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0410621","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"arxiv_version","alias_value":"math/0410621v1","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0410621","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"pith_short_12","alias_value":"GLRJQYKHHJ5F","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"pith_short_16","alias_value":"GLRJQYKHHJ5FFXOT","created_at":"2026-07-04T14:39:19Z"},{"alias_kind":"pith_short_8","alias_value":"GLRJQYKH","created_at":"2026-07-04T14:39:19Z"}],"graph_snapshots":[{"event_id":"sha256:987f1443294aae27c8f9583b00a31e60e328eb0a5a99173d2fcf959bb63edf44","target":"graph","created_at":"2026-07-04T14:39:19Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0410621/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study cohomology theories of strongly homotopy algebras, namely $A_\\infty, C_\\infty$ and $L_\\infty$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_\\infty$-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a $C_\\infty$-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\\infty$-algebra (an $\\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' oper","authors_text":"Alastair Hamilton, Andrey Lazarev","cross_cats":["math.AT"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2004-10-29T14:10:27Z","title":"Homotopy algebras and noncommutative geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0410621","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":"sha256:5c5d57960241a31cf17639e6a532fa3732d7ef37d258739e731be4658e7ddd6a","target":"record","created_at":"2026-07-04T14:39:19Z","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":"c5a6fc1a77260108916b98d49c0b713e53ddfe1965eadeeb5255e2bab2a61c66","cross_cats_sorted":["math.AT"],"license":"","primary_cat":"math.QA","submitted_at":"2004-10-29T14:10:27Z","title_canon_sha256":"25d6abc31576011441bd29baaf9a135dad0675a2e7dd2056bb6c8cdff9976fdd"},"schema_version":"1.0","source":{"id":"math/0410621","kind":"arxiv","version":1}},"canonical_sha256":"32e29861473a7a52ddd36115ab523d8e3d1475b7fa730acddbfd2d7e74d4e164","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32e29861473a7a52ddd36115ab523d8e3d1475b7fa730acddbfd2d7e74d4e164","first_computed_at":"2026-07-04T14:39:19.012051Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:39:19.012051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E2h9SM1J8bdZHVxO352FSi0/5DhlWzRhakB47Jg08F0IgAxdSXAKbf6+YKNrzcg/lkUEjKkOl7bQW9WkBfe/CQ==","signature_status":"signed_v1","signed_at":"2026-07-04T14:39:19.012459Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0410621","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c5d57960241a31cf17639e6a532fa3732d7ef37d258739e731be4658e7ddd6a","sha256:987f1443294aae27c8f9583b00a31e60e328eb0a5a99173d2fcf959bb63edf44"],"state_sha256":"230bf73923eb04739415a8aeda0ee47f5b9d2064c4cbf5e0ae668c2634331089"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"10095NdDdkxDi5aPu5bJIrMPS5QHp2nCwHtc1WPOAqf7WEBbui/A309x8m+D6lAnnsR7C5b0TLqivOx2B0snDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T21:40:59.621902Z","bundle_sha256":"7e263f7f5a8e9af6988000be55ffb60ede0f356882bde09b4b508f9510e7628a"}}