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Given an integral weight $\\xi$, let $\\Psi=\\Psi(\\xi)$ be the subset of roots which have maximal scalar product with $\\xi$. Given a dominant integral weight $\\lambda$ and $\\xi$ such that $\\Psi$ is a subset of the positive roots we construct a finite-dimensional subalgebra $\\bs^{\\lie g}_\\Psi(\\le_\\Psi\\lambda)$ "},"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":"0808.1463","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2008-08-11T08:22:55Z","cross_cats_sorted":["math.QA","math.RA"],"title_canon_sha256":"2257dc27cbef5ac33abc120346cbf889755ac4e81d092ad148fa0b436123466e","abstract_canon_sha256":"9d36fa6383ea5093aa094a23759f01a045289cac7fd6a21cfd35e97dd438ade3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:18.841712Z","signature_b64":"DF3FESHjIuEp8rHVh7Qfj5H2drcC+hvAzEV6qPItZbaJDxJLQnDdcU91xgxcb+ih85eWKZs654hN8lZY7dgrDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7875bed2c40ba53ab58ee3ce072e65259b4d603a4375d91c68aa8e9c54f272a6","last_reissued_at":"2026-05-18T03:46:18.840982Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:18.840982Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA","math.RA"],"primary_cat":"math.RT","authors_text":"Jacob Greenstein, Vyjayanthi Chari","submitted_at":"2008-08-11T08:22:55Z","abstract_excerpt":"Let $\\lie g$ be a simple Lie algebra and let $\\bs^{\\lie g}$ be the locally finite part of the algebra of invariants $(_\\bc\\bv\\otimes S(\\lie g))^{\\lie g}$ where $\\bv$ is the direct sum of all simple finite-dimensional modules for $\\lie g$ and $S(\\lie g)$ is the symmetric algebra of $\\lie g$. Given an integral weight $\\xi$, let $\\Psi=\\Psi(\\xi)$ be the subset of roots which have maximal scalar product with $\\xi$. Given a dominant integral weight $\\lambda$ and $\\xi$ such that $\\Psi$ is a subset of the positive roots we construct a finite-dimensional subalgebra $\\bs^{\\lie g}_\\Psi(\\le_\\Psi\\lambda)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.1463","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":"0808.1463","created_at":"2026-05-18T03:46:18.841086+00:00"},{"alias_kind":"arxiv_version","alias_value":"0808.1463v1","created_at":"2026-05-18T03:46:18.841086+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.1463","created_at":"2026-05-18T03:46:18.841086+00:00"},{"alias_kind":"pith_short_12","alias_value":"PB235UWEBOST","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"PB235UWEBOSTVNMO","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"PB235UWE","created_at":"2026-05-18T12:25:57.157939+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/PB235UWEBOSTVNMO4PHAOLTFEW","json":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW.json","graph_json":"https://pith.science/api/pith-number/PB235UWEBOSTVNMO4PHAOLTFEW/graph.json","events_json":"https://pith.science/api/pith-number/PB235UWEBOSTVNMO4PHAOLTFEW/events.json","paper":"https://pith.science/paper/PB235UWE"},"agent_actions":{"view_html":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW","download_json":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW.json","view_paper":"https://pith.science/paper/PB235UWE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0808.1463&json=true","fetch_graph":"https://pith.science/api/pith-number/PB235UWEBOSTVNMO4PHAOLTFEW/graph.json","fetch_events":"https://pith.science/api/pith-number/PB235UWEBOSTVNMO4PHAOLTFEW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW/action/storage_attestation","attest_author":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW/action/author_attestation","sign_citation":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW/action/citation_signature","submit_replication":"https://pith.science/pith/PB235UWEBOSTVNMO4PHAOLTFEW/action/replication_record"}},"created_at":"2026-05-18T03:46:18.841086+00:00","updated_at":"2026-05-18T03:46:18.841086+00:00"}