{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:K5GFGJ4HYWAZZXSL3AKIRRHVXG","short_pith_number":"pith:K5GFGJ4H","canonical_record":{"source":{"id":"1206.5867","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-06-26T00:55:42Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"7b03933e54c02e5295c0aaa6153f931f775d7ce0e6f3cb79a2e9c177b17bf2dd","abstract_canon_sha256":"ccbeb5869cc9b8ff96a5c71ae6b184f91b0fcca977e30ec7672249476050f49f"},"schema_version":"1.0"},"canonical_sha256":"574c532787c5819cde4bd81488c4f5b98b2c17d8273db5cd27f3f2becc87a839","source":{"kind":"arxiv","id":"1206.5867","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5867","created_at":"2026-05-18T02:25:10Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5867v2","created_at":"2026-05-18T02:25:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5867","created_at":"2026-05-18T02:25:10Z"},{"alias_kind":"pith_short_12","alias_value":"K5GFGJ4HYWAZ","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"K5GFGJ4HYWAZZXSL","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"K5GFGJ4H","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:K5GFGJ4HYWAZZXSL3AKIRRHVXG","target":"record","payload":{"canonical_record":{"source":{"id":"1206.5867","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-06-26T00:55:42Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"7b03933e54c02e5295c0aaa6153f931f775d7ce0e6f3cb79a2e9c177b17bf2dd","abstract_canon_sha256":"ccbeb5869cc9b8ff96a5c71ae6b184f91b0fcca977e30ec7672249476050f49f"},"schema_version":"1.0"},"canonical_sha256":"574c532787c5819cde4bd81488c4f5b98b2c17d8273db5cd27f3f2becc87a839","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:10.730438Z","signature_b64":"fHn8hl6H3TQQM+UEBLYgk1I+sRDloFXWYjZFKnkwAlNuwCw5GYrMwWV8bqCVuXB3SIg6JpzBlDDoMitSBIRlAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"574c532787c5819cde4bd81488c4f5b98b2c17d8273db5cd27f3f2becc87a839","last_reissued_at":"2026-05-18T02:25:10.729978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:10.729978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.5867","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yGs7GRAiuXWPZ6udKAKKbaufZr/WLpzIDDaTk56ZUiRV9+IEJksaY8qpxKROz29HgBOW65EpleW/3HCSTX8eCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T15:37:18.730807Z"},"content_sha256":"6df21e90682b1159e727443f197d461025ec219a60c08db1976016357c062a09","schema_version":"1.0","event_id":"sha256:6df21e90682b1159e727443f197d461025ec219a60c08db1976016357c062a09"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:K5GFGJ4HYWAZZXSL3AKIRRHVXG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Nadina Elizabeth Rojas","submitted_at":"2012-06-26T00:55:42Z","abstract_excerpt":"For a finite dimensional Lie algebra $\\g$ over a field $\\k$ of characteristic zero, the $\\mu$-function (respectively $\\mu_{nil}$-function) is defined to be the minimal dimension of $V$ such that $\\g$ admits a faithful representation (respectively a faithful nilrepresentation) on $V$. Let $\\h_m$ be the Heisenberg Lie algebra of dimension $2m + 1$ and let $\\mathfrak{a}_n$ be the abelian Lie algebra of dimension $n$. The aim of this paper is to compute $\\mu(\\h_m \\oplus \\mathfrak{a}_n)$ and $\\mu_{nil}(\\h_m \\oplus \\mathfrak{a}_n)$ for all $m,n \\in \\mathbb{N}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5867","kind":"arxiv","version":2},"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":"2026-05-18T02:25:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2slmsJziqysrZPOFC18MpsXNqGlfVtH+7YrsCgjhDxhdjDOvysxYQMyhzn3au0OL+yw1BXoIv79NJ1k49yhpCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T15:37:18.731139Z"},"content_sha256":"32d8573de11946ee3c2e2150146f797c060b590accab759fed6117367a070889","schema_version":"1.0","event_id":"sha256:32d8573de11946ee3c2e2150146f797c060b590accab759fed6117367a070889"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K5GFGJ4HYWAZZXSL3AKIRRHVXG/bundle.json","state_url":"https://pith.science/pith/K5GFGJ4HYWAZZXSL3AKIRRHVXG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K5GFGJ4HYWAZZXSL3AKIRRHVXG/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":"2026-06-28T15:37:18Z","links":{"resolver":"https://pith.science/pith/K5GFGJ4HYWAZZXSL3AKIRRHVXG","bundle":"https://pith.science/pith/K5GFGJ4HYWAZZXSL3AKIRRHVXG/bundle.json","state":"https://pith.science/pith/K5GFGJ4HYWAZZXSL3AKIRRHVXG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K5GFGJ4HYWAZZXSL3AKIRRHVXG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:K5GFGJ4HYWAZZXSL3AKIRRHVXG","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":"ccbeb5869cc9b8ff96a5c71ae6b184f91b0fcca977e30ec7672249476050f49f","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-06-26T00:55:42Z","title_canon_sha256":"7b03933e54c02e5295c0aaa6153f931f775d7ce0e6f3cb79a2e9c177b17bf2dd"},"schema_version":"1.0","source":{"id":"1206.5867","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5867","created_at":"2026-05-18T02:25:10Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5867v2","created_at":"2026-05-18T02:25:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5867","created_at":"2026-05-18T02:25:10Z"},{"alias_kind":"pith_short_12","alias_value":"K5GFGJ4HYWAZ","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"K5GFGJ4HYWAZZXSL","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"K5GFGJ4H","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:32d8573de11946ee3c2e2150146f797c060b590accab759fed6117367a070889","target":"graph","created_at":"2026-05-18T02:25:10Z","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 a finite dimensional Lie algebra $\\g$ over a field $\\k$ of characteristic zero, the $\\mu$-function (respectively $\\mu_{nil}$-function) is defined to be the minimal dimension of $V$ such that $\\g$ admits a faithful representation (respectively a faithful nilrepresentation) on $V$. Let $\\h_m$ be the Heisenberg Lie algebra of dimension $2m + 1$ and let $\\mathfrak{a}_n$ be the abelian Lie algebra of dimension $n$. The aim of this paper is to compute $\\mu(\\h_m \\oplus \\mathfrak{a}_n)$ and $\\mu_{nil}(\\h_m \\oplus \\mathfrak{a}_n)$ for all $m,n \\in \\mathbb{N}$.","authors_text":"Nadina Elizabeth Rojas","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-06-26T00:55:42Z","title":"Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5867","kind":"arxiv","version":2},"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":"sha256:6df21e90682b1159e727443f197d461025ec219a60c08db1976016357c062a09","target":"record","created_at":"2026-05-18T02:25:10Z","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":"ccbeb5869cc9b8ff96a5c71ae6b184f91b0fcca977e30ec7672249476050f49f","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-06-26T00:55:42Z","title_canon_sha256":"7b03933e54c02e5295c0aaa6153f931f775d7ce0e6f3cb79a2e9c177b17bf2dd"},"schema_version":"1.0","source":{"id":"1206.5867","kind":"arxiv","version":2}},"canonical_sha256":"574c532787c5819cde4bd81488c4f5b98b2c17d8273db5cd27f3f2becc87a839","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"574c532787c5819cde4bd81488c4f5b98b2c17d8273db5cd27f3f2becc87a839","first_computed_at":"2026-05-18T02:25:10.729978Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:10.729978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fHn8hl6H3TQQM+UEBLYgk1I+sRDloFXWYjZFKnkwAlNuwCw5GYrMwWV8bqCVuXB3SIg6JpzBlDDoMitSBIRlAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:10.730438Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5867","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6df21e90682b1159e727443f197d461025ec219a60c08db1976016357c062a09","sha256:32d8573de11946ee3c2e2150146f797c060b590accab759fed6117367a070889"],"state_sha256":"9e8cc158660ce0a6adb7ec7e3c8bc609b5d1747d9b317b89aeb097b4a5a1f3a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uyNH0Uz1Twx/WoGTBPLJe9jKI4KbZKd24H0l6xM1piQ7a1Fsgycpi2ryrDeTKLpw/tZ1Vv3CTEqU+QSLYahlBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T15:37:18.732962Z","bundle_sha256":"49c6e3dd64119653b9104a67ca8b352aed032102f80399b6a83e2fe7b4a8762a"}}