{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:EEFWEARMB7MXYWFXGGU4RVQQ35","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":"2c431779c1389c35283cd643d491be7d7b3ce0612c3773e9482c65777f4b16e2","cross_cats_sorted":["math.SG"],"license":"","primary_cat":"math.DG","submitted_at":"2007-12-27T12:53:04Z","title_canon_sha256":"fc7b89e1e56eea3d05db2c080fdcf5fc27dcba6f5cf2b76ecebbf3f57321ffde"},"schema_version":"1.0","source":{"id":"0712.4228","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.4228","created_at":"2026-05-18T00:37:38Z"},{"alias_kind":"arxiv_version","alias_value":"0712.4228v1","created_at":"2026-05-18T00:37:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.4228","created_at":"2026-05-18T00:37:38Z"},{"alias_kind":"pith_short_12","alias_value":"EEFWEARMB7MX","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"EEFWEARMB7MXYWFX","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"EEFWEARM","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:84c1a0fbcdc58c620692e98331e509b91454ebf1ba2f980f66b77d2493a54155","target":"graph","created_at":"2026-05-18T00:37:38Z","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":"For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F, we study the localization map Y^1: H^1(A,F)-> H^1(L_x,F_x), where L_x is the adjoint algebra at x in M. The main result in this paper is that: Ker Y^1_x=Ker(p^{1*})=H^1_{deR}(M,F_0). Here p^{1*} is the lift of H^1(\\huaA,F) to its counterpart over the universal covering space of M and H^1_{deR}(M,F_0) is the F_0=H^0(L,F)-coefficient deRham cohomology. We apply these results to study the associated vector bundles to principal fiber bundles and the structure of transitive Lie bialgebroids.","authors_text":"Z. Chen, Z.-J. Liu","cross_cats":["math.SG"],"headline":"","license":"","primary_cat":"math.DG","submitted_at":"2007-12-27T12:53:04Z","title":"The Cohomology of Transitive Lie Algebroids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.4228","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":"sha256:1c961a70a3128eccfd6a25c35cdaabbfb8964fe8b74ac70b4731ec6f90227c20","target":"record","created_at":"2026-05-18T00:37:38Z","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":"2c431779c1389c35283cd643d491be7d7b3ce0612c3773e9482c65777f4b16e2","cross_cats_sorted":["math.SG"],"license":"","primary_cat":"math.DG","submitted_at":"2007-12-27T12:53:04Z","title_canon_sha256":"fc7b89e1e56eea3d05db2c080fdcf5fc27dcba6f5cf2b76ecebbf3f57321ffde"},"schema_version":"1.0","source":{"id":"0712.4228","kind":"arxiv","version":1}},"canonical_sha256":"210b62022c0fd97c58b731a9c8d610df44cf2f3a816979d7a5a31dbc02397cea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"210b62022c0fd97c58b731a9c8d610df44cf2f3a816979d7a5a31dbc02397cea","first_computed_at":"2026-05-18T00:37:38.922559Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:37:38.922559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IYKsVSUAHCmAijxDTBE0oXyKusdGdtBFF0r/jzyYj3oshu55HhcRSF/x2YxZE4cNSwOC+7WlxUUvRZPNnJN1Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:37:38.923211Z","signed_message":"canonical_sha256_bytes"},"source_id":"0712.4228","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c961a70a3128eccfd6a25c35cdaabbfb8964fe8b74ac70b4731ec6f90227c20","sha256:84c1a0fbcdc58c620692e98331e509b91454ebf1ba2f980f66b77d2493a54155"],"state_sha256":"26644cbfbfeea24c760dda80207c6350b9c360979ae8a73dc555bc43763ae209"}