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The main difference with multiplier Hopf algebras is that now, the canonical maps $T_1$ and $T_2$ on $A\\otimes A$, defined by $$T_1(a\\otimes b)=\\Delta(a)(1\\otimes b) \\qquad\\quad\\text{and}\\qquad\\quad T_2(c\\otimes a)=(c\\otimes 1)\\Delta(a),$$ are no longer assumed to be bijective. Also recall that a weak multiplier Hopf algebra is called regular if its antipode is a bijective map from $A$ to itself.\n  In this pa"},"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":"1701.04951","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-01-18T05:28:44Z","cross_cats_sorted":[],"title_canon_sha256":"81718fe71d1a8e909eb75eac1e937471076a022250d4857108bec1ad2fcecfcb","abstract_canon_sha256":"5ee61e495cd5ee9b8af1713b0e1d50bc4ff5f43c4ba65234bec91b7c67aa3014"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:10.063565Z","signature_b64":"RVXHWYs6uyUqlFefAvfr+mss3PMtGt4Rd2dqB0Yt4+PnkxmMfLGTcNDvkJXP6L4KQnXdn4Snvwqcdi4L9nEhAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"521c2a79a6e6700e9ef78c2bf5c7c6376e65baff6b0e912692ca497ae68b1d99","last_reissued_at":"2026-05-18T00:34:10.062816Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:10.062816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak multiplier Hopf algebras III. 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Also recall that a weak multiplier Hopf algebra is called regular if its antipode is a bijective map from $A$ to itself.\n  In this pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04951","kind":"arxiv","version":4},"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":"1701.04951","created_at":"2026-05-18T00:34:10.062953+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.04951v4","created_at":"2026-05-18T00:34:10.062953+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04951","created_at":"2026-05-18T00:34:10.062953+00:00"},{"alias_kind":"pith_short_12","alias_value":"KIOCU6NG4ZYA","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"KIOCU6NG4ZYA5HXX","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"KIOCU6NG","created_at":"2026-05-18T12:31:24.725408+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/KIOCU6NG4ZYA5HXXRQV7LR6GG5","json":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5.json","graph_json":"https://pith.science/api/pith-number/KIOCU6NG4ZYA5HXXRQV7LR6GG5/graph.json","events_json":"https://pith.science/api/pith-number/KIOCU6NG4ZYA5HXXRQV7LR6GG5/events.json","paper":"https://pith.science/paper/KIOCU6NG"},"agent_actions":{"view_html":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5","download_json":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5.json","view_paper":"https://pith.science/paper/KIOCU6NG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.04951&json=true","fetch_graph":"https://pith.science/api/pith-number/KIOCU6NG4ZYA5HXXRQV7LR6GG5/graph.json","fetch_events":"https://pith.science/api/pith-number/KIOCU6NG4ZYA5HXXRQV7LR6GG5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5/action/storage_attestation","attest_author":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5/action/author_attestation","sign_citation":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5/action/citation_signature","submit_replication":"https://pith.science/pith/KIOCU6NG4ZYA5HXXRQV7LR6GG5/action/replication_record"}},"created_at":"2026-05-18T00:34:10.062953+00:00","updated_at":"2026-05-18T00:34:10.062953+00:00"}