{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:DAVQPXH5E6CFOINI3JKER3BAYC","short_pith_number":"pith:DAVQPXH5","canonical_record":{"source":{"id":"0910.0177","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-10-01T13:59:07Z","cross_cats_sorted":[],"title_canon_sha256":"c7d5f6f7347cb4d43c68b1cdda05d9a8ee0febefce724a34df8164e7aa2106a7","abstract_canon_sha256":"7ef84663bd60bd70c8a5477a789c5680065a987a042f4becc53bce46de45c415"},"schema_version":"1.0"},"canonical_sha256":"182b07dcfd27845721a8da5448ec20c08952904b61888af1564fe31243af65ee","source":{"kind":"arxiv","id":"0910.0177","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.0177","created_at":"2026-05-18T00:29:49Z"},{"alias_kind":"arxiv_version","alias_value":"0910.0177v1","created_at":"2026-05-18T00:29:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.0177","created_at":"2026-05-18T00:29:49Z"},{"alias_kind":"pith_short_12","alias_value":"DAVQPXH5E6CF","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"DAVQPXH5E6CFOINI","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"DAVQPXH5","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:DAVQPXH5E6CFOINI3JKER3BAYC","target":"record","payload":{"canonical_record":{"source":{"id":"0910.0177","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-10-01T13:59:07Z","cross_cats_sorted":[],"title_canon_sha256":"c7d5f6f7347cb4d43c68b1cdda05d9a8ee0febefce724a34df8164e7aa2106a7","abstract_canon_sha256":"7ef84663bd60bd70c8a5477a789c5680065a987a042f4becc53bce46de45c415"},"schema_version":"1.0"},"canonical_sha256":"182b07dcfd27845721a8da5448ec20c08952904b61888af1564fe31243af65ee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:49.686813Z","signature_b64":"3pTjZAFnrSCAdQmheChr2+3EIcY2lBowA2DSR+6e1ETcmB3fDJxFW6T8fbpNBrFZEXduSR5w9/qYleTosi+9BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"182b07dcfd27845721a8da5448ec20c08952904b61888af1564fe31243af65ee","last_reissued_at":"2026-05-18T00:29:49.686241Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:49.686241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0910.0177","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dl16NNd6yfZiuOKMgMLf6SbXZGH2El0fuqA6b3bU3zvPZq7f6fAsJofv+VliLfn0JwvNerZlGCNqVF87QY7xBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:45:09.408764Z"},"content_sha256":"d46fa8d84fb90349cf5c3ccc3b4cceba8f0d68d893a03243d738747193b1637d","schema_version":"1.0","event_id":"sha256:d46fa8d84fb90349cf5c3ccc3b4cceba8f0d68d893a03243d738747193b1637d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:DAVQPXH5E6CFOINI3JKER3BAYC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Analytic factorization of Lie group representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Bernhard Kr\\\"otz, Christoph Lienau, Heiko Gimperlein","submitted_at":"2009-10-01T13:59:07Z","abstract_excerpt":"For every moderate growth representation of a real Lie group G on a Frechet space E, we prove a factorization theorem of Dixmier--Malliavin type for the space of analytic vectors E^{\\omega}. There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that E^{\\omega} = A(G) * E^{\\omega}. As a corollary we obtain that E^\\omega coincides with the space of analytic vectors for the Laplace--Beltrami operator on G."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.0177","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CEbStBEyH3kY42iRFcU8xf0Sllq1yloY4nVipKa5soQuHrzj+k3SgwiXaLz9koLWPHdBsRHln94SG27XNuYQDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:45:09.409109Z"},"content_sha256":"ab4217706034e6e14ee228a8f7c9e13ab2b6b1883f85260c53c213f073daca8f","schema_version":"1.0","event_id":"sha256:ab4217706034e6e14ee228a8f7c9e13ab2b6b1883f85260c53c213f073daca8f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DAVQPXH5E6CFOINI3JKER3BAYC/bundle.json","state_url":"https://pith.science/pith/DAVQPXH5E6CFOINI3JKER3BAYC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DAVQPXH5E6CFOINI3JKER3BAYC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T06:45:09Z","links":{"resolver":"https://pith.science/pith/DAVQPXH5E6CFOINI3JKER3BAYC","bundle":"https://pith.science/pith/DAVQPXH5E6CFOINI3JKER3BAYC/bundle.json","state":"https://pith.science/pith/DAVQPXH5E6CFOINI3JKER3BAYC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DAVQPXH5E6CFOINI3JKER3BAYC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:DAVQPXH5E6CFOINI3JKER3BAYC","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":"7ef84663bd60bd70c8a5477a789c5680065a987a042f4becc53bce46de45c415","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-10-01T13:59:07Z","title_canon_sha256":"c7d5f6f7347cb4d43c68b1cdda05d9a8ee0febefce724a34df8164e7aa2106a7"},"schema_version":"1.0","source":{"id":"0910.0177","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.0177","created_at":"2026-05-18T00:29:49Z"},{"alias_kind":"arxiv_version","alias_value":"0910.0177v1","created_at":"2026-05-18T00:29:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.0177","created_at":"2026-05-18T00:29:49Z"},{"alias_kind":"pith_short_12","alias_value":"DAVQPXH5E6CF","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"DAVQPXH5E6CFOINI","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"DAVQPXH5","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:ab4217706034e6e14ee228a8f7c9e13ab2b6b1883f85260c53c213f073daca8f","target":"graph","created_at":"2026-05-18T00:29:49Z","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":"For every moderate growth representation of a real Lie group G on a Frechet space E, we prove a factorization theorem of Dixmier--Malliavin type for the space of analytic vectors E^{\\omega}. There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that E^{\\omega} = A(G) * E^{\\omega}. As a corollary we obtain that E^\\omega coincides with the space of analytic vectors for the Laplace--Beltrami operator on G.","authors_text":"Bernhard Kr\\\"otz, Christoph Lienau, Heiko Gimperlein","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-10-01T13:59:07Z","title":"Analytic factorization of Lie group representations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.0177","kind":"arxiv","version":1},"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:d46fa8d84fb90349cf5c3ccc3b4cceba8f0d68d893a03243d738747193b1637d","target":"record","created_at":"2026-05-18T00:29:49Z","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":"7ef84663bd60bd70c8a5477a789c5680065a987a042f4becc53bce46de45c415","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-10-01T13:59:07Z","title_canon_sha256":"c7d5f6f7347cb4d43c68b1cdda05d9a8ee0febefce724a34df8164e7aa2106a7"},"schema_version":"1.0","source":{"id":"0910.0177","kind":"arxiv","version":1}},"canonical_sha256":"182b07dcfd27845721a8da5448ec20c08952904b61888af1564fe31243af65ee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"182b07dcfd27845721a8da5448ec20c08952904b61888af1564fe31243af65ee","first_computed_at":"2026-05-18T00:29:49.686241Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:49.686241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3pTjZAFnrSCAdQmheChr2+3EIcY2lBowA2DSR+6e1ETcmB3fDJxFW6T8fbpNBrFZEXduSR5w9/qYleTosi+9BA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:49.686813Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.0177","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d46fa8d84fb90349cf5c3ccc3b4cceba8f0d68d893a03243d738747193b1637d","sha256:ab4217706034e6e14ee228a8f7c9e13ab2b6b1883f85260c53c213f073daca8f"],"state_sha256":"6a8450d8e5e881990492106a22dee038231263934ad7f1d20abab870665684d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gzkVh3CW3tAm7qj1GDMeMcOW6Es7+TNyEGBVajGJMVu+LebPIgVnTkIflU6ucQQFeecYQ5Np9XT6EWrjWc5BAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T06:45:09.411166Z","bundle_sha256":"845fcdb8be1bd94c233d56eae7b9087958bd00b82afb6a36962ae22ac48b0f6d"}}