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Assume that $u,v$ are invertible elements in $M(B)$ so that $E(vu\\otimes 1)E=E$. This last condition is obviously fulfilled if $u$ and $v$ are each other inverses, but there are also other cases. Now modify $\\Delta$ and define $\\Delta'(a)=(u\\otimes 1)\\Delta(a)(v\\otimes 1)$ for all $a"},"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":"1407.0513","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-07-02T10:40:22Z","cross_cats_sorted":[],"title_canon_sha256":"39512f84c13388737b29bee5fa7c036703a5be560afab74633d7514d9d688914","abstract_canon_sha256":"696ecccb3dcf05c9dfe7208a19728f36bb65b10695efa87d341bb138aca37f3a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:26.709245Z","signature_b64":"YFQyaxk/1lA2d+HHAV2M2FReXe6x1lHoBDhMRzxuokvTMgjkXpL3zFGyzmjDE3uaDldtZE+OqpxZR+KFKe+zAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"376eeeabfc18c3a9a119bdca6ef4bda6fbc3ec918a47057e8afbe734bf10bac2","last_reissued_at":"2026-05-18T02:48:26.708811Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:26.708811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Modified weak multiplier Hopf algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alfons Van Daele","submitted_at":"2014-07-02T10:40:22Z","abstract_excerpt":"Let $(A,\\Delta)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A,\\Delta)$ and by $B$ the image of the source map. Recall that $B$ is a non-degenerate algebra, sitting nicely in the multiplier algebra $M(A)$ of $A$ so that also $M(B)$ can be viewed as a subalgebra of $M(A)$. Assume that $u,v$ are invertible elements in $M(B)$ so that $E(vu\\otimes 1)E=E$. This last condition is obviously fulfilled if $u$ and $v$ are each other inverses, but there are also other cases. Now modify $\\Delta$ and define $\\Delta'(a)=(u\\otimes 1)\\Delta(a)(v\\otimes 1)$ for all $a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0513","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":"1407.0513","created_at":"2026-05-18T02:48:26.708872+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.0513v1","created_at":"2026-05-18T02:48:26.708872+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.0513","created_at":"2026-05-18T02:48:26.708872+00:00"},{"alias_kind":"pith_short_12","alias_value":"G5XO5K74DDB2","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_16","alias_value":"G5XO5K74DDB2TIIZ","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_8","alias_value":"G5XO5K74","created_at":"2026-05-18T12:28:28.263976+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/G5XO5K74DDB2TIIZXXFG55F5U3","json":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3.json","graph_json":"https://pith.science/api/pith-number/G5XO5K74DDB2TIIZXXFG55F5U3/graph.json","events_json":"https://pith.science/api/pith-number/G5XO5K74DDB2TIIZXXFG55F5U3/events.json","paper":"https://pith.science/paper/G5XO5K74"},"agent_actions":{"view_html":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3","download_json":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3.json","view_paper":"https://pith.science/paper/G5XO5K74","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.0513&json=true","fetch_graph":"https://pith.science/api/pith-number/G5XO5K74DDB2TIIZXXFG55F5U3/graph.json","fetch_events":"https://pith.science/api/pith-number/G5XO5K74DDB2TIIZXXFG55F5U3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3/action/storage_attestation","attest_author":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3/action/author_attestation","sign_citation":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3/action/citation_signature","submit_replication":"https://pith.science/pith/G5XO5K74DDB2TIIZXXFG55F5U3/action/replication_record"}},"created_at":"2026-05-18T02:48:26.708872+00:00","updated_at":"2026-05-18T02:48:26.708872+00:00"}