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For a form ideal $(I,\\Gamma)$ of the form ring $(\\FormR)$ we denote by $\\EU(2n,I,\\Gamma)$ and $\\GU(2n,I,\\Gamma)$ the relative elementary group and the principal congruence subgroup of level $(I,\\Gamma)$, respectively. Now, let $(I_i,\\Gamma_i) $, $i=0,...,m$, be form ideals of the form ring $(A,\\Lambda)$. The main result of the present paper is the following multiple commutator formula\n"},"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":"1205.6866","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-05-31T01:36:40Z","cross_cats_sorted":[],"title_canon_sha256":"acb02d340a6131b057b0547ff8fc74bd090336a430d8c1c75a01944f906be454","abstract_canon_sha256":"e4ce5f69b219e9a5549c6d87f43123a3c9e59ae01decdef3b138670cbfd536e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:59.508040Z","signature_b64":"JzGLRwhMM57jpn4xSZK2MLiTiz0lbsqACH2aMRAZPz2O3CV6GssYSrj29XzTA4b2fZnr/Px8rQPJOBan8py+Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"080c3dc057e4d3dd736344ea0f11c9005e15ee5f37ce2b527e4dc9d4b06cc799","last_reissued_at":"2026-05-18T03:49:59.507589Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:59.507589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple Commutator Formulas for Unitary Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Nikolai Vavilov, Roozbeh Hazrat, Zuhong Zhang","submitted_at":"2012-05-31T01:36:40Z","abstract_excerpt":"Let $(\\FormR)$ be a form ring such that $A$ is quasi-finite $R$-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak's unitary groups $\\GU(2n,\\FormR)$, $n\\ge 3$. For a form ideal $(I,\\Gamma)$ of the form ring $(\\FormR)$ we denote by $\\EU(2n,I,\\Gamma)$ and $\\GU(2n,I,\\Gamma)$ the relative elementary group and the principal congruence subgroup of level $(I,\\Gamma)$, respectively. Now, let $(I_i,\\Gamma_i) $, $i=0,...,m$, be form ideals of the form ring $(A,\\Lambda)$. The main result of the present paper is the following multiple commutator formula\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.6866","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":"1205.6866","created_at":"2026-05-18T03:49:59.507657+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.6866v1","created_at":"2026-05-18T03:49:59.507657+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.6866","created_at":"2026-05-18T03:49:59.507657+00:00"},{"alias_kind":"pith_short_12","alias_value":"BAGD3QCX4TJ5","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_16","alias_value":"BAGD3QCX4TJ5243D","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_8","alias_value":"BAGD3QCX","created_at":"2026-05-18T12:26:58.693483+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/BAGD3QCX4TJ5243DITVA6EOJAB","json":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB.json","graph_json":"https://pith.science/api/pith-number/BAGD3QCX4TJ5243DITVA6EOJAB/graph.json","events_json":"https://pith.science/api/pith-number/BAGD3QCX4TJ5243DITVA6EOJAB/events.json","paper":"https://pith.science/paper/BAGD3QCX"},"agent_actions":{"view_html":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB","download_json":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB.json","view_paper":"https://pith.science/paper/BAGD3QCX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.6866&json=true","fetch_graph":"https://pith.science/api/pith-number/BAGD3QCX4TJ5243DITVA6EOJAB/graph.json","fetch_events":"https://pith.science/api/pith-number/BAGD3QCX4TJ5243DITVA6EOJAB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB/action/storage_attestation","attest_author":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB/action/author_attestation","sign_citation":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB/action/citation_signature","submit_replication":"https://pith.science/pith/BAGD3QCX4TJ5243DITVA6EOJAB/action/replication_record"}},"created_at":"2026-05-18T03:49:59.507657+00:00","updated_at":"2026-05-18T03:49:59.507657+00:00"}