{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:CRTGLXWYFCTNZQCUXFEEQXFGXP","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":"1abcc0bd8567e6828c504e7f00cec30ceb73552aa4bbd043fb7599c4fd891983","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-25T19:55:37Z","title_canon_sha256":"77782239da19f9c4309ffedd3d996913b1c2fc62fe63382bbec1015104a2649e"},"schema_version":"1.0","source":{"id":"1908.10207","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.10207","created_at":"2026-07-04T23:59:54Z"},{"alias_kind":"arxiv_version","alias_value":"1908.10207v1","created_at":"2026-07-04T23:59:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.10207","created_at":"2026-07-04T23:59:54Z"},{"alias_kind":"pith_short_12","alias_value":"CRTGLXWYFCTN","created_at":"2026-07-04T23:59:54Z"},{"alias_kind":"pith_short_16","alias_value":"CRTGLXWYFCTNZQCU","created_at":"2026-07-04T23:59:54Z"},{"alias_kind":"pith_short_8","alias_value":"CRTGLXWY","created_at":"2026-07-04T23:59:54Z"}],"graph_snapshots":[{"event_id":"sha256:26c1fa23b8fbfa4b663d43c5c351affc63563cd127382e05334fdc70cc05c7b6","target":"graph","created_at":"2026-07-04T23:59:54Z","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/1908.10207/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In these notes we study left-invariant involutive structures on $\\mathrm{SU}(2)$, the most na\\\"ive non-commutative compact Lie group. We determine closedness of the range (in the smooth topology) of a single complex vector field spanning the standard CR structure of $\\mathrm{SU}(2)$ and also compute the smooth cohomology spaces of a corank $1$ structure. In our approach, it is fundamental to understand concretely the irreducible representations of the ambient Lie group and how left-invariant vector fields operate on their matrix coefficients (which we borrow from the book of Ruzhansky and Turu","authors_text":"Gabriel Ara\\'ujo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-25T19:55:37Z","title":"Computing cohomology spaces of left-invariant involutive structures on $\\mathrm{SU}(2)$: examples"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.10207","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":"sha256:07f6536f7c6d1fba87a09dfda7ee0f0b8b5044df1d254f33854dd09ae99a1d8b","target":"record","created_at":"2026-07-04T23:59:54Z","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":"1abcc0bd8567e6828c504e7f00cec30ceb73552aa4bbd043fb7599c4fd891983","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-25T19:55:37Z","title_canon_sha256":"77782239da19f9c4309ffedd3d996913b1c2fc62fe63382bbec1015104a2649e"},"schema_version":"1.0","source":{"id":"1908.10207","kind":"arxiv","version":1}},"canonical_sha256":"146665ded828a6dcc054b948485ca6bbc8de9b431757c12fb0c10b4678f9089b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"146665ded828a6dcc054b948485ca6bbc8de9b431757c12fb0c10b4678f9089b","first_computed_at":"2026-07-04T23:59:54.863416Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:59:54.863416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e2aySvzinMXq8b4DbNhTb1zxCzKQiqzIO7jlqE1YwoOQ4swr4LZ1fp58je0AfNNZIcSXkN+xk8TCVf1ig/8sCg==","signature_status":"signed_v1","signed_at":"2026-07-04T23:59:54.863831Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.10207","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:07f6536f7c6d1fba87a09dfda7ee0f0b8b5044df1d254f33854dd09ae99a1d8b","sha256:26c1fa23b8fbfa4b663d43c5c351affc63563cd127382e05334fdc70cc05c7b6"],"state_sha256":"717c989f9fa88c9f7adc1cb8c604aa7b2771990e60add1b2de5238472966f060"}