{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ODDCJHFPKNO2X3ENLZLKSGMD3F","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":"fbc3ab70ab7192761cc38b796d8336916e12b886f8269dc029c237dbd808fef0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-11-15T22:56:02Z","title_canon_sha256":"f361cd8bade294f44b26f2b7493a28ee90c0697530334f60d1f069a4c35fdf73"},"schema_version":"1.0","source":{"id":"1211.3773","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.3773","created_at":"2026-05-18T01:38:54Z"},{"alias_kind":"arxiv_version","alias_value":"1211.3773v5","created_at":"2026-05-18T01:38:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3773","created_at":"2026-05-18T01:38:54Z"},{"alias_kind":"pith_short_12","alias_value":"ODDCJHFPKNO2","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"ODDCJHFPKNO2X3EN","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"ODDCJHFP","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:6b36abb28e58d38eba14cce26efc003f4a3563d8894813ecb2eb8da37e844fe8","target":"graph","created_at":"2026-05-18T01:38:54Z","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"},"paper":{"abstract_excerpt":"We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, \"Quantum groupoids\", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of \"quantum groupoids\". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of \"quantum groupoids\". On the other hand, we develop a suitable version of a \"quantum du","authors_text":"Fabio Gavarini, Sophie Chemla","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-11-15T22:56:02Z","title":"Duality functors for quantum groupoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3773","kind":"arxiv","version":5},"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:c0fe688f800fb00022e2bdbf7577c4f49163833f4f6cf81fbf7b44b5b149a6d2","target":"record","created_at":"2026-05-18T01:38:54Z","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":"fbc3ab70ab7192761cc38b796d8336916e12b886f8269dc029c237dbd808fef0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-11-15T22:56:02Z","title_canon_sha256":"f361cd8bade294f44b26f2b7493a28ee90c0697530334f60d1f069a4c35fdf73"},"schema_version":"1.0","source":{"id":"1211.3773","kind":"arxiv","version":5}},"canonical_sha256":"70c6249caf535dabec8d5e56a91983d97c381fd2cb795e58a0256581bc830db5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70c6249caf535dabec8d5e56a91983d97c381fd2cb795e58a0256581bc830db5","first_computed_at":"2026-05-18T01:38:54.489277Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:54.489277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6vazrZgT9B7w9or+4HT0AB3eNuVDz1ptjIuDskxKirFfDjcjmeNOGBx6MZtw2a8jMWe+NQbx4/t9KSQC+sENCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:54.489951Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.3773","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c0fe688f800fb00022e2bdbf7577c4f49163833f4f6cf81fbf7b44b5b149a6d2","sha256:6b36abb28e58d38eba14cce26efc003f4a3563d8894813ecb2eb8da37e844fe8"],"state_sha256":"3acced8c009c8fe757f34784b6614f64caf27c7958fea6b9f1e50d1c41f7fe27"}