{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:4XHIQKWMVFPZX45LSRO3UFUXXD","short_pith_number":"pith:4XHIQKWM","canonical_record":{"source":{"id":"2207.05511","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-07-12T13:08:47Z","cross_cats_sorted":["math-ph","math.DS","math.MP"],"title_canon_sha256":"792e692eca5666a5ee0ba4857cb3ba8c3901ab7022575656129d677a4d707a6c","abstract_canon_sha256":"0701bd862a6d625fb5e9d1a993aeb1284d61eddd7d464d7e640c31eb0429ad50"},"schema_version":"1.0"},"canonical_sha256":"e5ce882acca95f9bf3ab945dba1697b8e466c8bd21749a6f087c138e327a40a9","source":{"kind":"arxiv","id":"2207.05511","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2207.05511","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"arxiv_version","alias_value":"2207.05511v2","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.05511","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"pith_short_12","alias_value":"4XHIQKWMVFPZ","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"pith_short_16","alias_value":"4XHIQKWMVFPZX45L","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"pith_short_8","alias_value":"4XHIQKWM","created_at":"2026-07-05T05:31:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:4XHIQKWMVFPZX45LSRO3UFUXXD","target":"record","payload":{"canonical_record":{"source":{"id":"2207.05511","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-07-12T13:08:47Z","cross_cats_sorted":["math-ph","math.DS","math.MP"],"title_canon_sha256":"792e692eca5666a5ee0ba4857cb3ba8c3901ab7022575656129d677a4d707a6c","abstract_canon_sha256":"0701bd862a6d625fb5e9d1a993aeb1284d61eddd7d464d7e640c31eb0429ad50"},"schema_version":"1.0"},"canonical_sha256":"e5ce882acca95f9bf3ab945dba1697b8e466c8bd21749a6f087c138e327a40a9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:31:51.599643Z","signature_b64":"e/kOa+27SjP8ySMGC+Q9Lvdm2k2jD5tPcG3/bdyVHrfQ8vpo9E99fyZUl54bV5atpYlfSIJK+JeQZIT7QtmVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5ce882acca95f9bf3ab945dba1697b8e466c8bd21749a6f087c138e327a40a9","last_reissued_at":"2026-07-05T05:31:51.599133Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:31:51.599133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2207.05511","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":"2026-07-05T05:31:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F8h0pErpBu93G+Yqcs35t9ckH9RjQrScmxEj5LUl5D9UdcwluRdC3vXY7H82Oqp0bS1FlDGpOcJy8SNcAX7GDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T19:20:07.185343Z"},"content_sha256":"f3e0478b86d8e87f6ed58129f86cd2f8495e01dccf584e798da1d43c3dc8acaa","schema_version":"1.0","event_id":"sha256:f3e0478b86d8e87f6ed58129f86cd2f8495e01dccf584e798da1d43c3dc8acaa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:4XHIQKWMVFPZX45LSRO3UFUXXD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.DS","math.MP"],"primary_cat":"math.DG","authors_text":"D. Iglesias Ponte, E. Padr\\'on, I. Gutierrez-Sagredo, J. C. Marrero, Z. Ravanpak","submitted_at":"2022-07-12T13:08:47Z","abstract_excerpt":"In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.05511","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2207.05511/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":"2026-07-05T05:31:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uibrFykcRuDv872mVeJ1F9KhwSnI7lIMVwLf5I8hgHXc9vuBO1G6cyu6ZjJRUWWO1V4Ej/qt8Ljq3rUNQTtPAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T19:20:07.185993Z"},"content_sha256":"a0a6a43e387f87b03ccefddce93d25a2a9b70d8dadd59fb01843cc942c26ab23","schema_version":"1.0","event_id":"sha256:a0a6a43e387f87b03ccefddce93d25a2a9b70d8dadd59fb01843cc942c26ab23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4XHIQKWMVFPZX45LSRO3UFUXXD/bundle.json","state_url":"https://pith.science/pith/4XHIQKWMVFPZX45LSRO3UFUXXD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4XHIQKWMVFPZX45LSRO3UFUXXD/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":"2026-08-14T19:20:07Z","links":{"resolver":"https://pith.science/pith/4XHIQKWMVFPZX45LSRO3UFUXXD","bundle":"https://pith.science/pith/4XHIQKWMVFPZX45LSRO3UFUXXD/bundle.json","state":"https://pith.science/pith/4XHIQKWMVFPZX45LSRO3UFUXXD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4XHIQKWMVFPZX45LSRO3UFUXXD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:4XHIQKWMVFPZX45LSRO3UFUXXD","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":"0701bd862a6d625fb5e9d1a993aeb1284d61eddd7d464d7e640c31eb0429ad50","cross_cats_sorted":["math-ph","math.DS","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-07-12T13:08:47Z","title_canon_sha256":"792e692eca5666a5ee0ba4857cb3ba8c3901ab7022575656129d677a4d707a6c"},"schema_version":"1.0","source":{"id":"2207.05511","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2207.05511","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"arxiv_version","alias_value":"2207.05511v2","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.05511","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"pith_short_12","alias_value":"4XHIQKWMVFPZ","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"pith_short_16","alias_value":"4XHIQKWMVFPZX45L","created_at":"2026-07-05T05:31:51Z"},{"alias_kind":"pith_short_8","alias_value":"4XHIQKWM","created_at":"2026-07-05T05:31:51Z"}],"graph_snapshots":[{"event_id":"sha256:a0a6a43e387f87b03ccefddce93d25a2a9b70d8dadd59fb01843cc942c26ab23","target":"graph","created_at":"2026-07-05T05:31:51Z","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/2207.05511/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Po","authors_text":"D. Iglesias Ponte, E. Padr\\'on, I. Gutierrez-Sagredo, J. C. Marrero, Z. Ravanpak","cross_cats":["math-ph","math.DS","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-07-12T13:08:47Z","title":"Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.05511","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:f3e0478b86d8e87f6ed58129f86cd2f8495e01dccf584e798da1d43c3dc8acaa","target":"record","created_at":"2026-07-05T05:31:51Z","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":"0701bd862a6d625fb5e9d1a993aeb1284d61eddd7d464d7e640c31eb0429ad50","cross_cats_sorted":["math-ph","math.DS","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-07-12T13:08:47Z","title_canon_sha256":"792e692eca5666a5ee0ba4857cb3ba8c3901ab7022575656129d677a4d707a6c"},"schema_version":"1.0","source":{"id":"2207.05511","kind":"arxiv","version":2}},"canonical_sha256":"e5ce882acca95f9bf3ab945dba1697b8e466c8bd21749a6f087c138e327a40a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5ce882acca95f9bf3ab945dba1697b8e466c8bd21749a6f087c138e327a40a9","first_computed_at":"2026-07-05T05:31:51.599133Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:31:51.599133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e/kOa+27SjP8ySMGC+Q9Lvdm2k2jD5tPcG3/bdyVHrfQ8vpo9E99fyZUl54bV5atpYlfSIJK+JeQZIT7QtmVCA==","signature_status":"signed_v1","signed_at":"2026-07-05T05:31:51.599643Z","signed_message":"canonical_sha256_bytes"},"source_id":"2207.05511","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3e0478b86d8e87f6ed58129f86cd2f8495e01dccf584e798da1d43c3dc8acaa","sha256:a0a6a43e387f87b03ccefddce93d25a2a9b70d8dadd59fb01843cc942c26ab23"],"state_sha256":"0d4e7d804db8012083118f3918de2fdb4ea2fc9968f5f967ad1b88eafd3dac6f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4BW/nyf1gQ9ikHncBugaQ1b+Tn8k4IHoHMGxhZXRCkI07RmTuCTRvcnaOUk4DUmWxwEE24x0vPlqzOWtql6bAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T19:20:07.189120Z","bundle_sha256":"ffd57274a51d4a5e53d134461f74e8fe7d2a574f3818681ae3e9d7cf43aa288e"}}