{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:2IDSY5NMQB6FDOEMKGU2VEGBID","short_pith_number":"pith:2IDSY5NM","canonical_record":{"source":{"id":"2502.05137","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2025-02-07T18:14:12Z","cross_cats_sorted":["math.MP","math.RA"],"title_canon_sha256":"2010c907f11a06389fab8d8f10a90b90565ac7187298977ab3111c5cb8935e66","abstract_canon_sha256":"7f772e7e66a65fcf30731f84653ef6a46bf8fcf6e24169c758dc070656a0c22f"},"schema_version":"1.0"},"canonical_sha256":"d2072c75ac807c51b88c51a9aa90c140e0bb10377a53f8358958b958724f8b19","source":{"kind":"arxiv","id":"2502.05137","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.05137","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"arxiv_version","alias_value":"2502.05137v1","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.05137","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"pith_short_12","alias_value":"2IDSY5NMQB6F","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"pith_short_16","alias_value":"2IDSY5NMQB6FDOEM","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"pith_short_8","alias_value":"2IDSY5NM","created_at":"2026-07-05T10:11:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:2IDSY5NMQB6FDOEMKGU2VEGBID","target":"record","payload":{"canonical_record":{"source":{"id":"2502.05137","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2025-02-07T18:14:12Z","cross_cats_sorted":["math.MP","math.RA"],"title_canon_sha256":"2010c907f11a06389fab8d8f10a90b90565ac7187298977ab3111c5cb8935e66","abstract_canon_sha256":"7f772e7e66a65fcf30731f84653ef6a46bf8fcf6e24169c758dc070656a0c22f"},"schema_version":"1.0"},"canonical_sha256":"d2072c75ac807c51b88c51a9aa90c140e0bb10377a53f8358958b958724f8b19","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:11:09.839942Z","signature_b64":"yIkpwvO1aRMclWjw0Abg+O+eIk5P6yCh26dHFsKnnKOXX7Q/PYgIo3kORZc0wwPUv3NKhECLZisrqCwXbHZQAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2072c75ac807c51b88c51a9aa90c140e0bb10377a53f8358958b958724f8b19","last_reissued_at":"2026-07-05T10:11:09.839465Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:11:09.839465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.05137","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":"2026-07-05T10:11:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qw75l0BpVWN1UBGHmYYqG53MpHu0N8HHvQFUaie38SCS6rg1YHwJlruRh43XL1HigVD3UjAO2E/QhEffsrM6DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T11:48:52.273027Z"},"content_sha256":"14afc0d5e888f33cb25c13ad6c3d3da0f7617b8eed85780bffb64f93555dec6d","schema_version":"1.0","event_id":"sha256:14afc0d5e888f33cb25c13ad6c3d3da0f7617b8eed85780bffb64f93555dec6d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:2IDSY5NMQB6FDOEMKGU2VEGBID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.RA"],"primary_cat":"math-ph","authors_text":"Emanuele Sgroi, Francesco Oliveri, Giorgio Gubbiotti, Pierandrea Vergallo","submitted_at":"2025-02-07T18:14:12Z","abstract_excerpt":"We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are uniquely determined by a triple composed by a Lie algebra, its most general non-degenerate quadratic Casimir and a 2-cocycle. We present some classes of operators associated to Lie algebras with non-degenerate quadratic Casimirs and we give a description of such operators in low dimensions. Finally, motivated by the example of the KdV equation we discuss the c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.05137","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/2502.05137/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":"2026-07-05T10:11:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lqrafkeiigprkcHz6LxsR/zkKhdq7qmGn1QpXOtraNrbCKauyl7sYvoXL4Ztrkn1nNPeQVom1CKb7VDovLGxCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T11:48:52.273531Z"},"content_sha256":"58fd926b0fabeb573118269c4c49f012bcc82ce59d86602525f047f1f8468912","schema_version":"1.0","event_id":"sha256:58fd926b0fabeb573118269c4c49f012bcc82ce59d86602525f047f1f8468912"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2IDSY5NMQB6FDOEMKGU2VEGBID/bundle.json","state_url":"https://pith.science/pith/2IDSY5NMQB6FDOEMKGU2VEGBID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2IDSY5NMQB6FDOEMKGU2VEGBID/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":"2026-08-10T11:48:52Z","links":{"resolver":"https://pith.science/pith/2IDSY5NMQB6FDOEMKGU2VEGBID","bundle":"https://pith.science/pith/2IDSY5NMQB6FDOEMKGU2VEGBID/bundle.json","state":"https://pith.science/pith/2IDSY5NMQB6FDOEMKGU2VEGBID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2IDSY5NMQB6FDOEMKGU2VEGBID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:2IDSY5NMQB6FDOEMKGU2VEGBID","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":"7f772e7e66a65fcf30731f84653ef6a46bf8fcf6e24169c758dc070656a0c22f","cross_cats_sorted":["math.MP","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2025-02-07T18:14:12Z","title_canon_sha256":"2010c907f11a06389fab8d8f10a90b90565ac7187298977ab3111c5cb8935e66"},"schema_version":"1.0","source":{"id":"2502.05137","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.05137","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"arxiv_version","alias_value":"2502.05137v1","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.05137","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"pith_short_12","alias_value":"2IDSY5NMQB6F","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"pith_short_16","alias_value":"2IDSY5NMQB6FDOEM","created_at":"2026-07-05T10:11:09Z"},{"alias_kind":"pith_short_8","alias_value":"2IDSY5NM","created_at":"2026-07-05T10:11:09Z"}],"graph_snapshots":[{"event_id":"sha256:58fd926b0fabeb573118269c4c49f012bcc82ce59d86602525f047f1f8468912","target":"graph","created_at":"2026-07-05T10:11:09Z","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/2502.05137/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are uniquely determined by a triple composed by a Lie algebra, its most general non-degenerate quadratic Casimir and a 2-cocycle. We present some classes of operators associated to Lie algebras with non-degenerate quadratic Casimirs and we give a description of such operators in low dimensions. Finally, motivated by the example of the KdV equation we discuss the c","authors_text":"Emanuele Sgroi, Francesco Oliveri, Giorgio Gubbiotti, Pierandrea Vergallo","cross_cats":["math.MP","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2025-02-07T18:14:12Z","title":"Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.05137","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":"sha256:14afc0d5e888f33cb25c13ad6c3d3da0f7617b8eed85780bffb64f93555dec6d","target":"record","created_at":"2026-07-05T10:11:09Z","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":"7f772e7e66a65fcf30731f84653ef6a46bf8fcf6e24169c758dc070656a0c22f","cross_cats_sorted":["math.MP","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2025-02-07T18:14:12Z","title_canon_sha256":"2010c907f11a06389fab8d8f10a90b90565ac7187298977ab3111c5cb8935e66"},"schema_version":"1.0","source":{"id":"2502.05137","kind":"arxiv","version":1}},"canonical_sha256":"d2072c75ac807c51b88c51a9aa90c140e0bb10377a53f8358958b958724f8b19","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d2072c75ac807c51b88c51a9aa90c140e0bb10377a53f8358958b958724f8b19","first_computed_at":"2026-07-05T10:11:09.839465Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:11:09.839465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yIkpwvO1aRMclWjw0Abg+O+eIk5P6yCh26dHFsKnnKOXX7Q/PYgIo3kORZc0wwPUv3NKhECLZisrqCwXbHZQAg==","signature_status":"signed_v1","signed_at":"2026-07-05T10:11:09.839942Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.05137","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14afc0d5e888f33cb25c13ad6c3d3da0f7617b8eed85780bffb64f93555dec6d","sha256:58fd926b0fabeb573118269c4c49f012bcc82ce59d86602525f047f1f8468912"],"state_sha256":"1a45e124a5a099d502543b0b136323a2bb84d6050dd6f8e650f658305619cd03"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VHI6BzWJ9dsxRsUI//41GnkUWe/SstXI/qQv1uZ5V1IaKw9rvyY1U0KMIuYiZCuIn3WcdTsLITU1rhq0V6IdAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T11:48:52.277330Z","bundle_sha256":"920110196eec20eee1a0fd2ba83b9e7a5ab64aa40d63b5ff6be2e4668391f083"}}