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We proved that for any Lie algebra $L$ of dimension $n$ over $\\mathbb{K}$ there exists a subalgebra $\\bar{L}$ of $W_n(\\mathbb{K})$ which is isomorphic to $L$ and such that every $\\mathbb{K}$-basis of $\\bar L$ is an $R$-basis of the $"},"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":"1105.4748","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-05-24T12:31:52Z","cross_cats_sorted":[],"title_canon_sha256":"19f4dddf470f562e0f19b337479445b5107438825aff5f7d5068fa4f632dcb23","abstract_canon_sha256":"5a217e49bff7c8985064c05508160b8f1692fc033f17e1c786e6742a064e0c35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:21.858307Z","signature_b64":"XYNCEfG7sM8+BHpgtvQlnlIPkfZSk+RrJQjiBLBDOLnwC5KlsBPosemwNULigm2N0omGPsMunTYjPOdSHeusDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc6b5b2daf059e96e999b8a74c00729aebd77afcca5659d5acb38eb8d6808010","last_reissued_at":"2026-05-18T04:21:21.857884Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:21.857884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On noncommutative bases of the free module $W_n(\\mathbb K)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Ievgen Makedonskyi","submitted_at":"2011-05-24T12:31:52Z","abstract_excerpt":"Let $\\mathbb{K}$ be an algebraically closed field of characteristic zero and $R=\\mathbb{K}[x_1,x_2,...x_n]$ the polynomial ring in $n$ variables over $\\mathbb K.$ We study bases of the free $R$-module $W_n(\\mathbb{K})$ of all $\\mathbb{K}$-derivations of the ring $R$, such that their linear span over $\\mathbb K$ is a subalgebra of the Lie algebra $W_n(\\mathbb{K})$. We proved that for any Lie algebra $L$ of dimension $n$ over $\\mathbb{K}$ there exists a subalgebra $\\bar{L}$ of $W_n(\\mathbb{K})$ which is isomorphic to $L$ and such that every $\\mathbb{K}$-basis of $\\bar L$ is an $R$-basis of the $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.4748","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":"1105.4748","created_at":"2026-05-18T04:21:21.857949+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.4748v1","created_at":"2026-05-18T04:21:21.857949+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.4748","created_at":"2026-05-18T04:21:21.857949+00:00"},{"alias_kind":"pith_short_12","alias_value":"3RVVWLNPAWPJ","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_16","alias_value":"3RVVWLNPAWPJN2MZ","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_8","alias_value":"3RVVWLNP","created_at":"2026-05-18T12:26:18.847500+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/3RVVWLNPAWPJN2MZXCTUYADSTL","json":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL.json","graph_json":"https://pith.science/api/pith-number/3RVVWLNPAWPJN2MZXCTUYADSTL/graph.json","events_json":"https://pith.science/api/pith-number/3RVVWLNPAWPJN2MZXCTUYADSTL/events.json","paper":"https://pith.science/paper/3RVVWLNP"},"agent_actions":{"view_html":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL","download_json":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL.json","view_paper":"https://pith.science/paper/3RVVWLNP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.4748&json=true","fetch_graph":"https://pith.science/api/pith-number/3RVVWLNPAWPJN2MZXCTUYADSTL/graph.json","fetch_events":"https://pith.science/api/pith-number/3RVVWLNPAWPJN2MZXCTUYADSTL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL/action/storage_attestation","attest_author":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL/action/author_attestation","sign_citation":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL/action/citation_signature","submit_replication":"https://pith.science/pith/3RVVWLNPAWPJN2MZXCTUYADSTL/action/replication_record"}},"created_at":"2026-05-18T04:21:21.857949+00:00","updated_at":"2026-05-18T04:21:21.857949+00:00"}