{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:B7NPVY63BO5MJQ6DAUUPY4TUAS","short_pith_number":"pith:B7NPVY63","canonical_record":{"source":{"id":"1609.01549","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-06T13:45:08Z","cross_cats_sorted":["math-ph","math.MP","math.RA"],"title_canon_sha256":"26b99fb9af09d2869669fae2f386844be2610c947899268abf071b2d1eb4b8b4","abstract_canon_sha256":"bf8a62d9803f265a342d01729071ba874cc9b5053c2f4252a56ea15ccf1cc24e"},"schema_version":"1.0"},"canonical_sha256":"0fdafae3db0bbac4c3c30528fc727404a41fe841435c33dd3e73e94fa6ba4bfb","source":{"kind":"arxiv","id":"1609.01549","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01549","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01549v1","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01549","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"B7NPVY63BO5M","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"B7NPVY63BO5MJQ6D","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"B7NPVY63","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:B7NPVY63BO5MJQ6DAUUPY4TUAS","target":"record","payload":{"canonical_record":{"source":{"id":"1609.01549","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-06T13:45:08Z","cross_cats_sorted":["math-ph","math.MP","math.RA"],"title_canon_sha256":"26b99fb9af09d2869669fae2f386844be2610c947899268abf071b2d1eb4b8b4","abstract_canon_sha256":"bf8a62d9803f265a342d01729071ba874cc9b5053c2f4252a56ea15ccf1cc24e"},"schema_version":"1.0"},"canonical_sha256":"0fdafae3db0bbac4c3c30528fc727404a41fe841435c33dd3e73e94fa6ba4bfb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:39.544966Z","signature_b64":"RtS3Jo9DTn4Z+m3Yu2ZFolwEJ+6AQebzg/h8/6ONw/onFQoiEEicKzLx511gwIXHIjBRwgZZTjLeuqfWJHlFAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fdafae3db0bbac4c3c30528fc727404a41fe841435c33dd3e73e94fa6ba4bfb","last_reissued_at":"2026-05-18T00:29:39.544517Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:39.544517Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.01549","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":"2026-05-18T00:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Due6P9rMKNjJ8wXZB1Zh0N1I2r9CuCte77R2qswayftmM98ThORSm03IVLC4Jzl0gQjg+q1kH7RgcRwd/zNAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:56:20.462397Z"},"content_sha256":"1aad2aa68851bed1f64e6df90a67cc2a82be2bc2caf557d7863c4c04a4bfd3e2","schema_version":"1.0","event_id":"sha256:1aad2aa68851bed1f64e6df90a67cc2a82be2bc2caf557d7863c4c04a4bfd3e2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:B7NPVY63BO5MJQ6DAUUPY4TUAS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A new approach to representations of $3$-Lie algebras and abelian extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.RA"],"primary_cat":"math.RT","authors_text":"Abdenacer Makhlouf, Jiefeng Liu, Yunhe Sheng","submitted_at":"2016-09-06T13:45:08Z","abstract_excerpt":"In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01549","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":"2026-05-18T00:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WkXWqWnygvXL4LwA4PAVJ/CkWbeRVm+YBFSoKikLvAE4FSVoNAMom6fzi9y9VNMAN9c17aMBHZMeO0r5mtMLBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:56:20.462783Z"},"content_sha256":"16c501c4965629ac683027f82768df436f2c9919e5235fc8af1da148b6670af7","schema_version":"1.0","event_id":"sha256:16c501c4965629ac683027f82768df436f2c9919e5235fc8af1da148b6670af7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B7NPVY63BO5MJQ6DAUUPY4TUAS/bundle.json","state_url":"https://pith.science/pith/B7NPVY63BO5MJQ6DAUUPY4TUAS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B7NPVY63BO5MJQ6DAUUPY4TUAS/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":"2026-06-02T07:56:20Z","links":{"resolver":"https://pith.science/pith/B7NPVY63BO5MJQ6DAUUPY4TUAS","bundle":"https://pith.science/pith/B7NPVY63BO5MJQ6DAUUPY4TUAS/bundle.json","state":"https://pith.science/pith/B7NPVY63BO5MJQ6DAUUPY4TUAS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B7NPVY63BO5MJQ6DAUUPY4TUAS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:B7NPVY63BO5MJQ6DAUUPY4TUAS","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":"bf8a62d9803f265a342d01729071ba874cc9b5053c2f4252a56ea15ccf1cc24e","cross_cats_sorted":["math-ph","math.MP","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-06T13:45:08Z","title_canon_sha256":"26b99fb9af09d2869669fae2f386844be2610c947899268abf071b2d1eb4b8b4"},"schema_version":"1.0","source":{"id":"1609.01549","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01549","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01549v1","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01549","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"B7NPVY63BO5M","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"B7NPVY63BO5MJQ6D","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"B7NPVY63","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:16c501c4965629ac683027f82768df436f2c9919e5235fc8af1da148b6670af7","target":"graph","created_at":"2026-05-18T00:29:39Z","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":"In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.","authors_text":"Abdenacer Makhlouf, Jiefeng Liu, Yunhe Sheng","cross_cats":["math-ph","math.MP","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-06T13:45:08Z","title":"A new approach to representations of $3$-Lie algebras and abelian extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01549","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":"sha256:1aad2aa68851bed1f64e6df90a67cc2a82be2bc2caf557d7863c4c04a4bfd3e2","target":"record","created_at":"2026-05-18T00:29:39Z","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":"bf8a62d9803f265a342d01729071ba874cc9b5053c2f4252a56ea15ccf1cc24e","cross_cats_sorted":["math-ph","math.MP","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-09-06T13:45:08Z","title_canon_sha256":"26b99fb9af09d2869669fae2f386844be2610c947899268abf071b2d1eb4b8b4"},"schema_version":"1.0","source":{"id":"1609.01549","kind":"arxiv","version":1}},"canonical_sha256":"0fdafae3db0bbac4c3c30528fc727404a41fe841435c33dd3e73e94fa6ba4bfb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fdafae3db0bbac4c3c30528fc727404a41fe841435c33dd3e73e94fa6ba4bfb","first_computed_at":"2026-05-18T00:29:39.544517Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:39.544517Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RtS3Jo9DTn4Z+m3Yu2ZFolwEJ+6AQebzg/h8/6ONw/onFQoiEEicKzLx511gwIXHIjBRwgZZTjLeuqfWJHlFAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:39.544966Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.01549","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1aad2aa68851bed1f64e6df90a67cc2a82be2bc2caf557d7863c4c04a4bfd3e2","sha256:16c501c4965629ac683027f82768df436f2c9919e5235fc8af1da148b6670af7"],"state_sha256":"3f96396b845af11374bf71a5b533bd0a71ba633d5a64de77f0ac39d0e6693268"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nQAFlNcyBY+VvtJlVKbYgN+PDTuT+i6qj61aNG8ARy8JyQKH3o6qY7sf4ElOB3VdmJylodRu2gHKIXq6dR27AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T07:56:20.465232Z","bundle_sha256":"ebd8160ebd4fca16298657c8c76101940a9171bd2fad1c23e7a0df173cad7e58"}}