{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:KINO7IULHRK6OJ67EBBEKT53OX","short_pith_number":"pith:KINO7IUL","canonical_record":{"source":{"id":"1607.00527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-02T16:01:44Z","cross_cats_sorted":[],"title_canon_sha256":"d41c04d350d847e04244b581f2de59e7d2f47717af5a2b2d28a905b128c2ce59","abstract_canon_sha256":"c22d92433f5fe5523773f409df5bd04e0b406b56cfd601514067700f2beaf291"},"schema_version":"1.0"},"canonical_sha256":"521aefa28b3c55e727df2042454fbb75c1c7b3d5ff4edcb2c1cf00e52fc0ea3f","source":{"kind":"arxiv","id":"1607.00527","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.00527","created_at":"2026-05-18T01:11:36Z"},{"alias_kind":"arxiv_version","alias_value":"1607.00527v1","created_at":"2026-05-18T01:11:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.00527","created_at":"2026-05-18T01:11:36Z"},{"alias_kind":"pith_short_12","alias_value":"KINO7IULHRK6","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KINO7IULHRK6OJ67","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KINO7IUL","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:KINO7IULHRK6OJ67EBBEKT53OX","target":"record","payload":{"canonical_record":{"source":{"id":"1607.00527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-02T16:01:44Z","cross_cats_sorted":[],"title_canon_sha256":"d41c04d350d847e04244b581f2de59e7d2f47717af5a2b2d28a905b128c2ce59","abstract_canon_sha256":"c22d92433f5fe5523773f409df5bd04e0b406b56cfd601514067700f2beaf291"},"schema_version":"1.0"},"canonical_sha256":"521aefa28b3c55e727df2042454fbb75c1c7b3d5ff4edcb2c1cf00e52fc0ea3f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:36.852858Z","signature_b64":"0G6Rfw5+1iGc+21297xYylaMuHy23Cv3wiWDqAfqcPghoT9CYgLD2BbKA/B2GPQRiIkbzDNShJ9dTYIQS1qDBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"521aefa28b3c55e727df2042454fbb75c1c7b3d5ff4edcb2c1cf00e52fc0ea3f","last_reissued_at":"2026-05-18T01:11:36.852515Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:36.852515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.00527","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":"2026-05-18T01:11:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nv/TIWWJkZDo1faFlLMAcOx2BR7tT3iYFdQbG9nbFjahBtvlAk6/H+PmVVdH0hVE0DufPwBKu0eTL1zJIEoZCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T00:34:46.608281Z"},"content_sha256":"cfdfc8ef370e38f44b9f3f0c5ce75cd13e6fd02ab708b23e45f5afec93ce1f53","schema_version":"1.0","event_id":"sha256:cfdfc8ef370e38f44b9f3f0c5ce75cd13e6fd02ab708b23e45f5afec93ce1f53"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:KINO7IULHRK6OJ67EBBEKT53OX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Double Bruhat cells and symplectic groupoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jiang-Hua Lu, Victor Mouquin","submitted_at":"2016-07-02T16:01:44Z","abstract_excerpt":"Let $G$ be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure $\\pi_{{\\rm st}}$ determined by a pair of opposite Borel subgroups $(B, B_-)$. We prove that for each $v$ in the Weyl group $W$ of $G$, the double Bruhat cell $G^{v,v} = BvB \\cap B_-vB_-$ in $G$, together with the Poisson structure $\\pi_{{\\rm st}}$, is naturally a Poisson groupoid over the Bruhat cell $BvB/B$ in the flag variety $G/B$. Correspondingly, every symplectic leaf of $\\pi_{{\\rm st}}$ in $G^{v,v}$ is a symplectic groupoid over $BvB/B$. For $u, v \\in W$, we show that the double"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00527","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":"2026-05-18T01:11:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UDBhwad0GyCo4X9dcY7mqeEoU9hXdIfnJ1x5o7jHrB6cvoa5fDKJQ5pWmIMowGVVplgFk9wbgM/YZFxtCQDHAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T00:34:46.608649Z"},"content_sha256":"a66a375da3a5d30221e788a48d5ab49819dc6fded66952edb75d9999e2b158f8","schema_version":"1.0","event_id":"sha256:a66a375da3a5d30221e788a48d5ab49819dc6fded66952edb75d9999e2b158f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KINO7IULHRK6OJ67EBBEKT53OX/bundle.json","state_url":"https://pith.science/pith/KINO7IULHRK6OJ67EBBEKT53OX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KINO7IULHRK6OJ67EBBEKT53OX/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":"2026-05-28T00:34:46Z","links":{"resolver":"https://pith.science/pith/KINO7IULHRK6OJ67EBBEKT53OX","bundle":"https://pith.science/pith/KINO7IULHRK6OJ67EBBEKT53OX/bundle.json","state":"https://pith.science/pith/KINO7IULHRK6OJ67EBBEKT53OX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KINO7IULHRK6OJ67EBBEKT53OX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KINO7IULHRK6OJ67EBBEKT53OX","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":"c22d92433f5fe5523773f409df5bd04e0b406b56cfd601514067700f2beaf291","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-02T16:01:44Z","title_canon_sha256":"d41c04d350d847e04244b581f2de59e7d2f47717af5a2b2d28a905b128c2ce59"},"schema_version":"1.0","source":{"id":"1607.00527","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.00527","created_at":"2026-05-18T01:11:36Z"},{"alias_kind":"arxiv_version","alias_value":"1607.00527v1","created_at":"2026-05-18T01:11:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.00527","created_at":"2026-05-18T01:11:36Z"},{"alias_kind":"pith_short_12","alias_value":"KINO7IULHRK6","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KINO7IULHRK6OJ67","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KINO7IUL","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:a66a375da3a5d30221e788a48d5ab49819dc6fded66952edb75d9999e2b158f8","target":"graph","created_at":"2026-05-18T01:11:36Z","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":"Let $G$ be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure $\\pi_{{\\rm st}}$ determined by a pair of opposite Borel subgroups $(B, B_-)$. We prove that for each $v$ in the Weyl group $W$ of $G$, the double Bruhat cell $G^{v,v} = BvB \\cap B_-vB_-$ in $G$, together with the Poisson structure $\\pi_{{\\rm st}}$, is naturally a Poisson groupoid over the Bruhat cell $BvB/B$ in the flag variety $G/B$. Correspondingly, every symplectic leaf of $\\pi_{{\\rm st}}$ in $G^{v,v}$ is a symplectic groupoid over $BvB/B$. For $u, v \\in W$, we show that the double","authors_text":"Jiang-Hua Lu, Victor Mouquin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-02T16:01:44Z","title":"Double Bruhat cells and symplectic groupoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00527","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:cfdfc8ef370e38f44b9f3f0c5ce75cd13e6fd02ab708b23e45f5afec93ce1f53","target":"record","created_at":"2026-05-18T01:11:36Z","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":"c22d92433f5fe5523773f409df5bd04e0b406b56cfd601514067700f2beaf291","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-02T16:01:44Z","title_canon_sha256":"d41c04d350d847e04244b581f2de59e7d2f47717af5a2b2d28a905b128c2ce59"},"schema_version":"1.0","source":{"id":"1607.00527","kind":"arxiv","version":1}},"canonical_sha256":"521aefa28b3c55e727df2042454fbb75c1c7b3d5ff4edcb2c1cf00e52fc0ea3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"521aefa28b3c55e727df2042454fbb75c1c7b3d5ff4edcb2c1cf00e52fc0ea3f","first_computed_at":"2026-05-18T01:11:36.852515Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:36.852515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0G6Rfw5+1iGc+21297xYylaMuHy23Cv3wiWDqAfqcPghoT9CYgLD2BbKA/B2GPQRiIkbzDNShJ9dTYIQS1qDBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:36.852858Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.00527","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfdfc8ef370e38f44b9f3f0c5ce75cd13e6fd02ab708b23e45f5afec93ce1f53","sha256:a66a375da3a5d30221e788a48d5ab49819dc6fded66952edb75d9999e2b158f8"],"state_sha256":"2319914d498d2f19fcb45857f9080188a11e2d0860a23ba5d339ccbfc5e6c82d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gB0/iQqesnrfLCu/x+Iv3e7wYvrEnHZ/e9T+Bq8C8rE1zYtUGDa30tkd1Eao9R6MksRMioh/pOBYqwgS1AuuDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T00:34:46.610679Z","bundle_sha256":"dbcb1a1c454ac0f921bcf6e61010efb6bdc71776104e854ffb1991ab0c79fb48"}}