{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:MHPMR3CISDLO6O4FVISBFJOH3E","short_pith_number":"pith:MHPMR3CI","schema_version":"1.0","canonical_sha256":"61dec8ec4890d6ef3b85aa2412a5c7d9229e94b2ebe69ded90e7d83b06ff0140","source":{"kind":"arxiv","id":"1907.04411","version":1},"attestation_state":"computed","paper":{"title":"Split Hopf algebras, quasi-shuffle algebras, and the cohomology of Omega Sigma X","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.RA","authors_text":"Nicholas J. Kuhn","submitted_at":"2019-07-09T20:57:09Z","abstract_excerpt":"Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are isomorphic as graded k-vector spaces equipped with a Frobenius (pth-power) map.\n  For the hardest part of this analysis, we work with the dual construction, and are led to study connected graded cocommutative Hopf algebras H with two additional properties: H is free as an associative algebra, and the projection onto the indecomposables is split"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.04411","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2019-07-09T20:57:09Z","cross_cats_sorted":["math.AT","math.CO"],"title_canon_sha256":"ad9e0b04fb033a44970a8c14a81bc93f5c138cf40786d7344c1c42cd01ba7824","abstract_canon_sha256":"70b521a00794b0629d2231d5bdd759d4020843c1c3dc7c3c021eee82d698ee5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:01.829587Z","signature_b64":"K0OsLYfgfQiMhc/6QgmSI5IRlTfYfJWBkFrAdOUaa/11xMCQFP+U5j9boWqXZJyNqiituThUwr/lizuvLN7QCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61dec8ec4890d6ef3b85aa2412a5c7d9229e94b2ebe69ded90e7d83b06ff0140","last_reissued_at":"2026-05-17T23:41:01.828938Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:01.828938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Split Hopf algebras, quasi-shuffle algebras, and the cohomology of Omega Sigma X","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.RA","authors_text":"Nicholas J. Kuhn","submitted_at":"2019-07-09T20:57:09Z","abstract_excerpt":"Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are isomorphic as graded k-vector spaces equipped with a Frobenius (pth-power) map.\n  For the hardest part of this analysis, we work with the dual construction, and are led to study connected graded cocommutative Hopf algebras H with two additional properties: H is free as an associative algebra, and the projection onto the indecomposables is split"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04411","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1907.04411","created_at":"2026-05-17T23:41:01.829034+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.04411v1","created_at":"2026-05-17T23:41:01.829034+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04411","created_at":"2026-05-17T23:41:01.829034+00:00"},{"alias_kind":"pith_short_12","alias_value":"MHPMR3CISDLO","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"MHPMR3CISDLO6O4F","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"MHPMR3CI","created_at":"2026-05-18T12:33:21.387695+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E","json":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E.json","graph_json":"https://pith.science/api/pith-number/MHPMR3CISDLO6O4FVISBFJOH3E/graph.json","events_json":"https://pith.science/api/pith-number/MHPMR3CISDLO6O4FVISBFJOH3E/events.json","paper":"https://pith.science/paper/MHPMR3CI"},"agent_actions":{"view_html":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E","download_json":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E.json","view_paper":"https://pith.science/paper/MHPMR3CI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.04411&json=true","fetch_graph":"https://pith.science/api/pith-number/MHPMR3CISDLO6O4FVISBFJOH3E/graph.json","fetch_events":"https://pith.science/api/pith-number/MHPMR3CISDLO6O4FVISBFJOH3E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E/action/storage_attestation","attest_author":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E/action/author_attestation","sign_citation":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E/action/citation_signature","submit_replication":"https://pith.science/pith/MHPMR3CISDLO6O4FVISBFJOH3E/action/replication_record"}},"created_at":"2026-05-17T23:41:01.829034+00:00","updated_at":"2026-05-17T23:41:01.829034+00:00"}