{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ENFRH5QNJCFMOETARSU46UWFDU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"40783a4930e115a7ebed945b149cb4dabffcf7159e7eb46a4381cb0d5bfe547a","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-05-22T15:58:42Z","title_canon_sha256":"5bbf9a61d587e265e4d51d1e88b3b011a8102f72dad148282fa5f3bcce39a4a2"},"schema_version":"1.0","source":{"id":"0905.3714","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0905.3714","created_at":"2026-05-17T23:53:20Z"},{"alias_kind":"arxiv_version","alias_value":"0905.3714v3","created_at":"2026-05-17T23:53:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.3714","created_at":"2026-05-17T23:53:20Z"},{"alias_kind":"pith_short_12","alias_value":"ENFRH5QNJCFM","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"ENFRH5QNJCFMOETA","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"ENFRH5QN","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:e8cd3157e14df901d3791c2f0109500ac43cae643ab0584167a4ed558af47da2","target":"graph","created_at":"2026-05-17T23:53:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider the finite $W$-algebra $U(\\g,e)$ associated to a nilpotent element $e \\in \\g$ in a simple complex Lie algebra $\\g$ of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a conjecture of Premet, that $U(\\g,e)$ always has a 1-dimensional representation, when $\\g$ is of type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduc","authors_text":"Gerhard Roehrle, Glenn Ubly, Simon M. Goodwin","cross_cats":["math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-05-22T15:58:42Z","title":"On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.3714","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:57d64eea7d958ef98de98e35ce1bc2eeb7f59ed56e24e3a1cc0a18203dcd478e","target":"record","created_at":"2026-05-17T23:53:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"40783a4930e115a7ebed945b149cb4dabffcf7159e7eb46a4381cb0d5bfe547a","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-05-22T15:58:42Z","title_canon_sha256":"5bbf9a61d587e265e4d51d1e88b3b011a8102f72dad148282fa5f3bcce39a4a2"},"schema_version":"1.0","source":{"id":"0905.3714","kind":"arxiv","version":3}},"canonical_sha256":"234b13f60d488ac712608ca9cf52c51d3c963752fd1bed791c2aa00253e8bedd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"234b13f60d488ac712608ca9cf52c51d3c963752fd1bed791c2aa00253e8bedd","first_computed_at":"2026-05-17T23:53:20.012582Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:20.012582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8nZDqEk/qH1f/LqJYbxIhF73YNjRgJd2WSMBACVFcWqT/q5Ltx8AFD/BZ2f1CB+P2yJluNZYiXxX9WZ/x39tAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:20.013186Z","signed_message":"canonical_sha256_bytes"},"source_id":"0905.3714","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57d64eea7d958ef98de98e35ce1bc2eeb7f59ed56e24e3a1cc0a18203dcd478e","sha256:e8cd3157e14df901d3791c2f0109500ac43cae643ab0584167a4ed558af47da2"],"state_sha256":"677487ac07570b1b7c4099cc2c40f87984996b81a39761e148e48b2293da7023"}