{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7CU4NT46DCPDJTTQP4MN6S45QD","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":"5788a708a392aba7516b120128d42bcbc4f5cd374de97d70d1821b7ab88267df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-03-09T15:46:17Z","title_canon_sha256":"430c40040189639461d8e75c88fe5de256177c8db0240a043e5b1665c58fb7d9"},"schema_version":"1.0","source":{"id":"1803.03572","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.03572","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"arxiv_version","alias_value":"1803.03572v1","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03572","created_at":"2026-05-18T00:21:39Z"},{"alias_kind":"pith_short_12","alias_value":"7CU4NT46DCPD","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7CU4NT46DCPDJTTQ","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7CU4NT46","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:4dc443afc90f533ed7f178c4ced790a2bdb2535c4be6574321f369a50f3abe67","target":"graph","created_at":"2026-05-18T00:21:39Z","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":"For a fixed finite group $Q$ and semi-simple finite dimensional algebra $S$, we examine an equivalence between strongly $Q$-graded algebras (extensions) with identity component $S$ and $S^1$-gerbes on action groupoids of $Q$ on the set of isomorphism classes of simple objects of the category of $S$-modules. This clarifies the nature of the map considered in arXiv:1312.7316. Motivated by this and arXiv:0909.3140(2) we suggest and study a notion of extensions suitable to the case when $S$ is replaced by a Hopf algebra, in the sense that there is a bijection between extensions with \"fiber\" $H$ an","authors_text":"Ilya Shapiro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-03-09T15:46:17Z","title":"Extensions and duality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03572","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:fcb6d75de180ba612cdce9c00a1230e351b30aa63f90f67f1b8d7d780e408749","target":"record","created_at":"2026-05-18T00:21:39Z","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":"5788a708a392aba7516b120128d42bcbc4f5cd374de97d70d1821b7ab88267df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2018-03-09T15:46:17Z","title_canon_sha256":"430c40040189639461d8e75c88fe5de256177c8db0240a043e5b1665c58fb7d9"},"schema_version":"1.0","source":{"id":"1803.03572","kind":"arxiv","version":1}},"canonical_sha256":"f8a9c6cf9e189e34ce707f18df4b9d80e31d850d782bf9605f6eaf87f1976d30","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8a9c6cf9e189e34ce707f18df4b9d80e31d850d782bf9605f6eaf87f1976d30","first_computed_at":"2026-05-18T00:21:39.103241Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:39.103241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NeajxTC/eEmbVeKZK2Qa2WqYttWb7Ob0m9/KnNXpPJ9rg/MhMXV4Oh0AiyZC7yWSx7BAHg/6bJA7zgoxJyqiBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:39.104005Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.03572","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fcb6d75de180ba612cdce9c00a1230e351b30aa63f90f67f1b8d7d780e408749","sha256:4dc443afc90f533ed7f178c4ced790a2bdb2535c4be6574321f369a50f3abe67"],"state_sha256":"dfd6468ef680c9d9e332d17f1ca5b7b52ee7bdd59e15db879c77456e6ad29e93"}