{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:TDLYVZ47GV7NRUNTHNTZG77QAX","short_pith_number":"pith:TDLYVZ47","schema_version":"1.0","canonical_sha256":"98d78ae79f357ed8d1b33b67937ff005c18ce581bfb0a3e36afdf3b40e3bb0a8","source":{"kind":"arxiv","id":"1510.07143","version":1},"attestation_state":"computed","paper":{"title":"The commutators of classical groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"N. Vavilov, R. Hazrat, Z. Zhang","submitted_at":"2015-10-24T13:09:16Z","abstract_excerpt":"In his seminal paper, half a century ago, Hyman Bass established a commutator formula in the setting of (stable) general linear group which was the key step in defining the K_1 group. Namely, he proved that for an associative ring A with identity, E(A)=[E(A),E(A)]=[GL(A),GL(A)] where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings, and for a range of classical and algebraic like-groups, mostly in relation to subnormal subgroups of these groups. The major classical theorems "},"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":"1510.07143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-24T13:09:16Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"e049309e34f0ab6ac893d4d2b0417b6225851d2afb6337a0fcb7df1078c0c19e","abstract_canon_sha256":"c3d35ecfca3bd65ccaea0293e9b18fd7698c5cacf4665d5892bef1ecbb013ed6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:57.019527Z","signature_b64":"hYJ9WIbql0Cb3Y8GMGRi2ISOXjdlvfAPdT7D5cOVXufUs19aatXhMurD+iY9fADK2ZyN6GoOBwoPDajDN2KMCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98d78ae79f357ed8d1b33b67937ff005c18ce581bfb0a3e36afdf3b40e3bb0a8","last_reissued_at":"2026-05-18T01:25:57.018907Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:57.018907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The commutators of classical groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"N. Vavilov, R. Hazrat, Z. Zhang","submitted_at":"2015-10-24T13:09:16Z","abstract_excerpt":"In his seminal paper, half a century ago, Hyman Bass established a commutator formula in the setting of (stable) general linear group which was the key step in defining the K_1 group. Namely, he proved that for an associative ring A with identity, E(A)=[E(A),E(A)]=[GL(A),GL(A)] where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings, and for a range of classical and algebraic like-groups, mostly in relation to subnormal subgroups of these groups. The major classical theorems "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07143","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":"1510.07143","created_at":"2026-05-18T01:25:57.018991+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.07143v1","created_at":"2026-05-18T01:25:57.018991+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07143","created_at":"2026-05-18T01:25:57.018991+00:00"},{"alias_kind":"pith_short_12","alias_value":"TDLYVZ47GV7N","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TDLYVZ47GV7NRUNT","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TDLYVZ47","created_at":"2026-05-18T12:29:42.218222+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/TDLYVZ47GV7NRUNTHNTZG77QAX","json":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX.json","graph_json":"https://pith.science/api/pith-number/TDLYVZ47GV7NRUNTHNTZG77QAX/graph.json","events_json":"https://pith.science/api/pith-number/TDLYVZ47GV7NRUNTHNTZG77QAX/events.json","paper":"https://pith.science/paper/TDLYVZ47"},"agent_actions":{"view_html":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX","download_json":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX.json","view_paper":"https://pith.science/paper/TDLYVZ47","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.07143&json=true","fetch_graph":"https://pith.science/api/pith-number/TDLYVZ47GV7NRUNTHNTZG77QAX/graph.json","fetch_events":"https://pith.science/api/pith-number/TDLYVZ47GV7NRUNTHNTZG77QAX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX/action/storage_attestation","attest_author":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX/action/author_attestation","sign_citation":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX/action/citation_signature","submit_replication":"https://pith.science/pith/TDLYVZ47GV7NRUNTHNTZG77QAX/action/replication_record"}},"created_at":"2026-05-18T01:25:57.018991+00:00","updated_at":"2026-05-18T01:25:57.018991+00:00"}