{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:GHCFT4LTTUOT5W7FZZUGKARFEM","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":"6ae127e93ddeecda5b09cb04d1af2a033d92b1a1821358d798e0f03d38e8accd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2011-08-25T09:31:36Z","title_canon_sha256":"e50414e716a32beabe80de1f3cd5061174c2ebb29374a28be30123c0b5ddba31"},"schema_version":"1.0","source":{"id":"1108.5047","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5047","created_at":"2026-05-18T04:04:06Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5047v2","created_at":"2026-05-18T04:04:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5047","created_at":"2026-05-18T04:04:06Z"},{"alias_kind":"pith_short_12","alias_value":"GHCFT4LTTUOT","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"GHCFT4LTTUOT5W7F","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"GHCFT4LT","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:e9feb21266992e0a465416d791b005a7f4c4cccba227f2e3cc9aabc6740dcea7","target":"graph","created_at":"2026-05-18T04:04:06Z","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 consider differential operators over a noncommutative algebra $A$ generated by vector fields. These are shown to form a unital associative algebra of differential operators, and act on $A$-modules $E$ with covariant derivative. We use the repeated differentials given in the paper to give a definition of noncommutative Sobolev space for modules with connection and Hermitian inner product. The tensor algebra of vector fields, with a modified bimodule structure and a bimodule connection, is shown to lie in the centre of the bimodule connection category ${}_A\\mathcal{E}_A$, and in fact to be an","authors_text":"Edwin Beggs, Tomasz Brzezinski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2011-08-25T09:31:36Z","title":"Noncommutative differential operators, Sobolev spaces and the centre of a category"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5047","kind":"arxiv","version":2},"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:c6511f59b7287324117e0a00c6cc4599fed1184f7d265a0024d5371acda17b39","target":"record","created_at":"2026-05-18T04:04:06Z","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":"6ae127e93ddeecda5b09cb04d1af2a033d92b1a1821358d798e0f03d38e8accd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2011-08-25T09:31:36Z","title_canon_sha256":"e50414e716a32beabe80de1f3cd5061174c2ebb29374a28be30123c0b5ddba31"},"schema_version":"1.0","source":{"id":"1108.5047","kind":"arxiv","version":2}},"canonical_sha256":"31c459f1739d1d3edbe5ce6865022523000ad1023ae598c13585ae95dc6860bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31c459f1739d1d3edbe5ce6865022523000ad1023ae598c13585ae95dc6860bc","first_computed_at":"2026-05-18T04:04:06.609555Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:06.609555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Auh/vEnM2BzpknrBMD1PA239eXDcU6ysbdMgZdcv5rL8NR13DOXn2Jv+el6UsW2tly/NXbUNJ9OtqAHO28HSDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:06.610269Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.5047","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6511f59b7287324117e0a00c6cc4599fed1184f7d265a0024d5371acda17b39","sha256:e9feb21266992e0a465416d791b005a7f4c4cccba227f2e3cc9aabc6740dcea7"],"state_sha256":"f05313a1aadbb2a91f9b96a1c1d55ae8fdffa8cc7c36c5047f6c825c8eea69fe"}