{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:K4E2P3PQHXK3VDIJNS2HXPP7J6","short_pith_number":"pith:K4E2P3PQ","canonical_record":{"source":{"id":"1111.5958","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-25T12:13:06Z","cross_cats_sorted":[],"title_canon_sha256":"53c3b7cc573204f64c53948751b268f08cc3bfbaeabdcf15b13c0d09da54b65b","abstract_canon_sha256":"65773bd26c2c46b478bd6c309805d44efac7d3c1d137f295cdc924b541e91110"},"schema_version":"1.0"},"canonical_sha256":"5709a7edf03dd5ba8d096cb47bbdff4faf4ed9cb8fa7898a26adc0085d7df38e","source":{"kind":"arxiv","id":"1111.5958","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.5958","created_at":"2026-05-18T04:04:07Z"},{"alias_kind":"arxiv_version","alias_value":"1111.5958v2","created_at":"2026-05-18T04:04:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5958","created_at":"2026-05-18T04:04:07Z"},{"alias_kind":"pith_short_12","alias_value":"K4E2P3PQHXK3","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"K4E2P3PQHXK3VDIJ","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"K4E2P3PQ","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:K4E2P3PQHXK3VDIJNS2HXPP7J6","target":"record","payload":{"canonical_record":{"source":{"id":"1111.5958","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-25T12:13:06Z","cross_cats_sorted":[],"title_canon_sha256":"53c3b7cc573204f64c53948751b268f08cc3bfbaeabdcf15b13c0d09da54b65b","abstract_canon_sha256":"65773bd26c2c46b478bd6c309805d44efac7d3c1d137f295cdc924b541e91110"},"schema_version":"1.0"},"canonical_sha256":"5709a7edf03dd5ba8d096cb47bbdff4faf4ed9cb8fa7898a26adc0085d7df38e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:07.768169Z","signature_b64":"e2o0ZRQb0C/CLLU0mpXM97mTIsTcfdoIS7VMt6HC/cI8ZlPEGtQsoad77DgNEOB1Fl+75b9mMO+2mmGLsY6MBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5709a7edf03dd5ba8d096cb47bbdff4faf4ed9cb8fa7898a26adc0085d7df38e","last_reissued_at":"2026-05-18T04:04:07.767658Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:07.767658Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.5958","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":"2026-05-18T04:04:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IKXlTBRDBQekSvyfseOmuyylMRDUaR6c8NWPMiLvz4bB91XjJTEDoS7lImMg5HsS/+3+brzX1olOrox4dlbgBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T02:36:58.466386Z"},"content_sha256":"44c2e45e4b65258f520d664df75835aa135a50d30f18089973d895f9170448a4","schema_version":"1.0","event_id":"sha256:44c2e45e4b65258f520d664df75835aa135a50d30f18089973d895f9170448a4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:K4E2P3PQHXK3VDIJNS2HXPP7J6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cohomological properties of unimodular six dimensional solvable Lie algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Maura Macr\\`i","submitted_at":"2011-11-25T12:13:06Z","abstract_excerpt":"In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture of Guan about step of nilpotency of a symplectic solvmanifold finding that it is true for all six dimensional unimodular solvable Lie algebras. Finally, we consider some cohomologies for symplectic manifolds introduced by Tseng and Yau in the context of symplectic Hogde theory an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5958","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":"2026-05-18T04:04:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kIJCoNORudF6bN2UkJhZIKZpCf/WPQ4M8JXTgkrWzhrap2/1iq3fyuKrgwDaf23HG0cfA1xVxrJ1PcCxUXpADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T02:36:58.466741Z"},"content_sha256":"f781c5527aff1f8d6fa01033fd8ae28d8a4b86ffa3716df58b80303b69ad3dea","schema_version":"1.0","event_id":"sha256:f781c5527aff1f8d6fa01033fd8ae28d8a4b86ffa3716df58b80303b69ad3dea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K4E2P3PQHXK3VDIJNS2HXPP7J6/bundle.json","state_url":"https://pith.science/pith/K4E2P3PQHXK3VDIJNS2HXPP7J6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K4E2P3PQHXK3VDIJNS2HXPP7J6/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":"2026-06-22T02:36:58Z","links":{"resolver":"https://pith.science/pith/K4E2P3PQHXK3VDIJNS2HXPP7J6","bundle":"https://pith.science/pith/K4E2P3PQHXK3VDIJNS2HXPP7J6/bundle.json","state":"https://pith.science/pith/K4E2P3PQHXK3VDIJNS2HXPP7J6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K4E2P3PQHXK3VDIJNS2HXPP7J6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:K4E2P3PQHXK3VDIJNS2HXPP7J6","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":"65773bd26c2c46b478bd6c309805d44efac7d3c1d137f295cdc924b541e91110","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-25T12:13:06Z","title_canon_sha256":"53c3b7cc573204f64c53948751b268f08cc3bfbaeabdcf15b13c0d09da54b65b"},"schema_version":"1.0","source":{"id":"1111.5958","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.5958","created_at":"2026-05-18T04:04:07Z"},{"alias_kind":"arxiv_version","alias_value":"1111.5958v2","created_at":"2026-05-18T04:04:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5958","created_at":"2026-05-18T04:04:07Z"},{"alias_kind":"pith_short_12","alias_value":"K4E2P3PQHXK3","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"K4E2P3PQHXK3VDIJ","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"K4E2P3PQ","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:f781c5527aff1f8d6fa01033fd8ae28d8a4b86ffa3716df58b80303b69ad3dea","target":"graph","created_at":"2026-05-18T04:04:07Z","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 the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture of Guan about step of nilpotency of a symplectic solvmanifold finding that it is true for all six dimensional unimodular solvable Lie algebras. Finally, we consider some cohomologies for symplectic manifolds introduced by Tseng and Yau in the context of symplectic Hogde theory an","authors_text":"Maura Macr\\`i","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-25T12:13:06Z","title":"Cohomological properties of unimodular six dimensional solvable Lie algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5958","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":"sha256:44c2e45e4b65258f520d664df75835aa135a50d30f18089973d895f9170448a4","target":"record","created_at":"2026-05-18T04:04:07Z","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":"65773bd26c2c46b478bd6c309805d44efac7d3c1d137f295cdc924b541e91110","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-25T12:13:06Z","title_canon_sha256":"53c3b7cc573204f64c53948751b268f08cc3bfbaeabdcf15b13c0d09da54b65b"},"schema_version":"1.0","source":{"id":"1111.5958","kind":"arxiv","version":2}},"canonical_sha256":"5709a7edf03dd5ba8d096cb47bbdff4faf4ed9cb8fa7898a26adc0085d7df38e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5709a7edf03dd5ba8d096cb47bbdff4faf4ed9cb8fa7898a26adc0085d7df38e","first_computed_at":"2026-05-18T04:04:07.767658Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:07.767658Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e2o0ZRQb0C/CLLU0mpXM97mTIsTcfdoIS7VMt6HC/cI8ZlPEGtQsoad77DgNEOB1Fl+75b9mMO+2mmGLsY6MBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:07.768169Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.5958","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44c2e45e4b65258f520d664df75835aa135a50d30f18089973d895f9170448a4","sha256:f781c5527aff1f8d6fa01033fd8ae28d8a4b86ffa3716df58b80303b69ad3dea"],"state_sha256":"f1bae1c70348ab9ae3659ee2fe8f92739d8a070d1c17ef4ae358415a2e8c2352"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+WGJWUwIa94KirjOejhTs6vRAQzXPwH2EJSBq2EieGlmk96P6sPZJDLlZv4IWeD5JzJP8HIsuIUluGI43tmSCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T02:36:58.468760Z","bundle_sha256":"27a12f33f8d79e2f75e52ebb4f9b1f53d480498e724af0cba8798c56f2b42bd8"}}