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It is easily proved that, as an associative algebra, $A[D]$ is $G$-graded simple if and only if $A$ is $\\G$-graded $D$-simple. Suppose $A$ is $\\G$-graded $D$-simple. Then,\n (a) $A[D]$ is a free left $A$-module;\n (b) as a Lie color algebra, the subquotie"},"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":"math/0304075","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2003-04-05T10:22:40Z","cross_cats_sorted":[],"title_canon_sha256":"a83e247160c146a5d58d275caddbae70ca595f2f05ac230f2fd4d70849ecf812","abstract_canon_sha256":"5a3f31579011d52c2ac01ee1dfa15cc0c08662182403c81a048f5bfa1a0d2a6b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:28.967392Z","signature_b64":"hOrVJFuKmLmkj6dv3g04tVAu4m2XrsdfyS+I3rzESjh3+7JacBqUPiw+hG2iX8GMpd8Qi7K6iZWDKdtk5ItnBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"126f195d74d784bb11353ae574a517d5e051c2a8797cf68cdc23c0d1c2766a37","last_reissued_at":"2026-05-18T01:05:28.966919Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:28.966919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simple Lie Color Algebras of Weyl Type","license":"","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Kaiming Zhao, Linsheng Zhu, Yucai Su","submitted_at":"2003-04-05T10:22:40Z","abstract_excerpt":"For an $(\\epsilon,G)$-color-commutative associative algebra $A$ with an identity element over a field $F$ of characteristic not 2, and for a color-commutative subalgebra $D$ of color-derivations of $A$, denote by $A[D]$ the associative subalgebra of ${\\rm End}(A)$ generated by $A$ (regarding as operators on $A$ via left multiplication) and $D$. It is easily proved that, as an associative algebra, $A[D]$ is $G$-graded simple if and only if $A$ is $\\G$-graded $D$-simple. Suppose $A$ is $\\G$-graded $D$-simple. Then,\n (a) $A[D]$ is a free left $A$-module;\n (b) as a Lie color algebra, the subquotie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0304075","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":"math/0304075","created_at":"2026-05-18T01:05:28.966998+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0304075v1","created_at":"2026-05-18T01:05:28.966998+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0304075","created_at":"2026-05-18T01:05:28.966998+00:00"},{"alias_kind":"pith_short_12","alias_value":"CJXRSXLU26CL","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_16","alias_value":"CJXRSXLU26CLWEJV","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_8","alias_value":"CJXRSXLU","created_at":"2026-05-18T12:25:51.375804+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/CJXRSXLU26CLWEJVHLSXJJIX2X","json":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X.json","graph_json":"https://pith.science/api/pith-number/CJXRSXLU26CLWEJVHLSXJJIX2X/graph.json","events_json":"https://pith.science/api/pith-number/CJXRSXLU26CLWEJVHLSXJJIX2X/events.json","paper":"https://pith.science/paper/CJXRSXLU"},"agent_actions":{"view_html":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X","download_json":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X.json","view_paper":"https://pith.science/paper/CJXRSXLU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0304075&json=true","fetch_graph":"https://pith.science/api/pith-number/CJXRSXLU26CLWEJVHLSXJJIX2X/graph.json","fetch_events":"https://pith.science/api/pith-number/CJXRSXLU26CLWEJVHLSXJJIX2X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X/action/storage_attestation","attest_author":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X/action/author_attestation","sign_citation":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X/action/citation_signature","submit_replication":"https://pith.science/pith/CJXRSXLU26CLWEJVHLSXJJIX2X/action/replication_record"}},"created_at":"2026-05-18T01:05:28.966998+00:00","updated_at":"2026-05-18T01:05:28.966998+00:00"}