{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:IB2X5V6E4ZH34V7DBZT4N5PDRE","short_pith_number":"pith:IB2X5V6E","canonical_record":{"source":{"id":"1303.3231","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-13T17:48:05Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"d4dbcfe5f02bf4ef5136ab73b3ae0a1be54d9ab317243b2daff7d6fc265c418b","abstract_canon_sha256":"4f3b1339657385a34b9939014daf6cc92b682de70c02d1c648d7a6fd02c4f981"},"schema_version":"1.0"},"canonical_sha256":"40757ed7c4e64fbe57e30e67c6f5e38935922a7464083019fa270bd095a4ce7c","source":{"kind":"arxiv","id":"1303.3231","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3231","created_at":"2026-05-18T02:45:21Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3231v2","created_at":"2026-05-18T02:45:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3231","created_at":"2026-05-18T02:45:21Z"},{"alias_kind":"pith_short_12","alias_value":"IB2X5V6E4ZH3","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IB2X5V6E4ZH34V7D","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IB2X5V6E","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:IB2X5V6E4ZH34V7DBZT4N5PDRE","target":"record","payload":{"canonical_record":{"source":{"id":"1303.3231","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-13T17:48:05Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"d4dbcfe5f02bf4ef5136ab73b3ae0a1be54d9ab317243b2daff7d6fc265c418b","abstract_canon_sha256":"4f3b1339657385a34b9939014daf6cc92b682de70c02d1c648d7a6fd02c4f981"},"schema_version":"1.0"},"canonical_sha256":"40757ed7c4e64fbe57e30e67c6f5e38935922a7464083019fa270bd095a4ce7c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:21.682469Z","signature_b64":"Rx9UoX0TtlrM0zz1l/rB/JJSe4fg98S3nFjok16AlWh1+j0fp2vjFIUDAst9Z2QbwEqOG+vY3YJp/aBfovXTBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40757ed7c4e64fbe57e30e67c6f5e38935922a7464083019fa270bd095a4ce7c","last_reissued_at":"2026-05-18T02:45:21.681906Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:21.681906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.3231","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:45:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IQupofpccb8+ECpORKf95F86EfoTZTX8uSqmHDGO2m51GJclIeWyEFHPh4Gh93H3ZnFJOCyZXXwd8ZtogKx+Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T14:46:10.387977Z"},"content_sha256":"2cb94c8e2f33de9653297083761f5e71ebba7d658d3c0138cbf65189a731bcd1","schema_version":"1.0","event_id":"sha256:2cb94c8e2f33de9653297083761f5e71ebba7d658d3c0138cbf65189a731bcd1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:IB2X5V6E4ZH34V7DBZT4N5PDRE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A compendium of Lie structures on tensor products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.RA","authors_text":"Pasha Zusmanovich","submitted_at":"2013-03-13T17:48:05Z","abstract_excerpt":"We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3231","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:45:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tM/AxBRzhClAEt2EHn8RZv4BWNNiLGEQAUhEk2Xqgo/xfPnD8HFz1kQxxluHpSIaBOH4HCcCPUFqykbFE4aYCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T14:46:10.388332Z"},"content_sha256":"d1abcbc63831fea8ccd9d5cea01443d88f9d3ac30b29c9f0704d8ed5858a9e78","schema_version":"1.0","event_id":"sha256:d1abcbc63831fea8ccd9d5cea01443d88f9d3ac30b29c9f0704d8ed5858a9e78"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IB2X5V6E4ZH34V7DBZT4N5PDRE/bundle.json","state_url":"https://pith.science/pith/IB2X5V6E4ZH34V7DBZT4N5PDRE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IB2X5V6E4ZH34V7DBZT4N5PDRE/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":"2026-06-26T14:46:10Z","links":{"resolver":"https://pith.science/pith/IB2X5V6E4ZH34V7DBZT4N5PDRE","bundle":"https://pith.science/pith/IB2X5V6E4ZH34V7DBZT4N5PDRE/bundle.json","state":"https://pith.science/pith/IB2X5V6E4ZH34V7DBZT4N5PDRE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IB2X5V6E4ZH34V7DBZT4N5PDRE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:IB2X5V6E4ZH34V7DBZT4N5PDRE","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":"4f3b1339657385a34b9939014daf6cc92b682de70c02d1c648d7a6fd02c4f981","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-13T17:48:05Z","title_canon_sha256":"d4dbcfe5f02bf4ef5136ab73b3ae0a1be54d9ab317243b2daff7d6fc265c418b"},"schema_version":"1.0","source":{"id":"1303.3231","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.3231","created_at":"2026-05-18T02:45:21Z"},{"alias_kind":"arxiv_version","alias_value":"1303.3231v2","created_at":"2026-05-18T02:45:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3231","created_at":"2026-05-18T02:45:21Z"},{"alias_kind":"pith_short_12","alias_value":"IB2X5V6E4ZH3","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IB2X5V6E4ZH34V7D","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IB2X5V6E","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:d1abcbc63831fea8ccd9d5cea01443d88f9d3ac30b29c9f0704d8ed5858a9e78","target":"graph","created_at":"2026-05-18T02:45:21Z","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 demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.","authors_text":"Pasha Zusmanovich","cross_cats":["math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-13T17:48:05Z","title":"A compendium of Lie structures on tensor products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3231","kind":"arxiv","version":2},"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":"sha256:2cb94c8e2f33de9653297083761f5e71ebba7d658d3c0138cbf65189a731bcd1","target":"record","created_at":"2026-05-18T02:45:21Z","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":"4f3b1339657385a34b9939014daf6cc92b682de70c02d1c648d7a6fd02c4f981","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-13T17:48:05Z","title_canon_sha256":"d4dbcfe5f02bf4ef5136ab73b3ae0a1be54d9ab317243b2daff7d6fc265c418b"},"schema_version":"1.0","source":{"id":"1303.3231","kind":"arxiv","version":2}},"canonical_sha256":"40757ed7c4e64fbe57e30e67c6f5e38935922a7464083019fa270bd095a4ce7c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"40757ed7c4e64fbe57e30e67c6f5e38935922a7464083019fa270bd095a4ce7c","first_computed_at":"2026-05-18T02:45:21.681906Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:45:21.681906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rx9UoX0TtlrM0zz1l/rB/JJSe4fg98S3nFjok16AlWh1+j0fp2vjFIUDAst9Z2QbwEqOG+vY3YJp/aBfovXTBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:45:21.682469Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.3231","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2cb94c8e2f33de9653297083761f5e71ebba7d658d3c0138cbf65189a731bcd1","sha256:d1abcbc63831fea8ccd9d5cea01443d88f9d3ac30b29c9f0704d8ed5858a9e78"],"state_sha256":"36eaeb223c3861207d779ad563588efda61c77d75692659de68cf8bcb1f7d8ca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z/rcePwAukz58A0DFsf1WgyZfZS17BWSoWNqnUp4gdmz3XXcjxb6cLJpl6bVBeAnrp4g3d2btIS569ezmWCPDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T14:46:10.390170Z","bundle_sha256":"3b0380ce3aa5deaaae9050cee7788c6fbbc5252280d177adc58501a90ec8c75a"}}