{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:JAKV45UFMAS6PS4F5NA2FSEHA6","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":"4bb705855e7414db3c92f96d4e1e987d2a8ff18c1aa20dd68416642cb5fdd2a6","cross_cats_sorted":["math.CT"],"license":"","primary_cat":"math.QA","submitted_at":"2006-10-09T07:52:50Z","title_canon_sha256":"72a79c7da17a45cdbff352b6dc89c2a482264ff003c846272c1207c6186c69c6"},"schema_version":"1.0","source":{"id":"math/0610273","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610273","created_at":"2026-05-18T04:41:45Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610273v1","created_at":"2026-05-18T04:41:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610273","created_at":"2026-05-18T04:41:45Z"},{"alias_kind":"pith_short_12","alias_value":"JAKV45UFMAS6","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"JAKV45UFMAS6PS4F","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"JAKV45UF","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:0390c5e0e092d1b040448844e514f026c79fd3b7e5bffcb12e27a1d352e12d16","target":"graph","created_at":"2026-05-18T04:41:45Z","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":"We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra $A$ is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that $A$ is a double cross product, or biproduct in Madjid's terminology. The last result is based on a universal pro","authors_text":"A. Ardizzoni, C. Menini, D. Stefan","cross_cats":["math.CT"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2006-10-09T07:52:50Z","title":"Weak Projections onto a Braided Hopf Algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610273","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:e4d2542f8862dd255755c86fb89c60335f3e3829787cd05e00e1ea05d594558f","target":"record","created_at":"2026-05-18T04:41:45Z","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":"4bb705855e7414db3c92f96d4e1e987d2a8ff18c1aa20dd68416642cb5fdd2a6","cross_cats_sorted":["math.CT"],"license":"","primary_cat":"math.QA","submitted_at":"2006-10-09T07:52:50Z","title_canon_sha256":"72a79c7da17a45cdbff352b6dc89c2a482264ff003c846272c1207c6186c69c6"},"schema_version":"1.0","source":{"id":"math/0610273","kind":"arxiv","version":1}},"canonical_sha256":"48155e76856025e7cb85eb41a2c88707a89cf336f2608e2c9ba61933a3c96ccb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48155e76856025e7cb85eb41a2c88707a89cf336f2608e2c9ba61933a3c96ccb","first_computed_at":"2026-05-18T04:41:45.019987Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:45.019987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q1l7B7KMkKN+AUBsB6qIVbfmqrTSwotlxocRNeuXTnV9LPN5RqUT26xSUg4K02wImOq5z3+sYiquG0kJQWSuCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:45.020452Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0610273","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e4d2542f8862dd255755c86fb89c60335f3e3829787cd05e00e1ea05d594558f","sha256:0390c5e0e092d1b040448844e514f026c79fd3b7e5bffcb12e27a1d352e12d16"],"state_sha256":"08f638fd4321270ff0bf16464ad54aeca2e9df004eef7cc3dd1e8448784a72bf"}