{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:Z65MQBQAENL7W7CBND5JZHOHMI","short_pith_number":"pith:Z65MQBQA","canonical_record":{"source":{"id":"1103.1924","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-03-10T01:22:42Z","cross_cats_sorted":[],"title_canon_sha256":"28e9575afbc87ac10c7d58e614025182e1601421f5058a98bf51f9cfe3d2d7e6","abstract_canon_sha256":"b87a8bea36154ed9dbd0e5135139a36058884974ce0542e4b1fff22dadae9059"},"schema_version":"1.0"},"canonical_sha256":"cfbac806002357fb7c4168fa9c9dc762214f612b784bc9a2231acee4113c5afb","source":{"kind":"arxiv","id":"1103.1924","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1924","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1924v2","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1924","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"pith_short_12","alias_value":"Z65MQBQAENL7","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"Z65MQBQAENL7W7CB","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"Z65MQBQA","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:Z65MQBQAENL7W7CBND5JZHOHMI","target":"record","payload":{"canonical_record":{"source":{"id":"1103.1924","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-03-10T01:22:42Z","cross_cats_sorted":[],"title_canon_sha256":"28e9575afbc87ac10c7d58e614025182e1601421f5058a98bf51f9cfe3d2d7e6","abstract_canon_sha256":"b87a8bea36154ed9dbd0e5135139a36058884974ce0542e4b1fff22dadae9059"},"schema_version":"1.0"},"canonical_sha256":"cfbac806002357fb7c4168fa9c9dc762214f612b784bc9a2231acee4113c5afb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:02.793569Z","signature_b64":"/WMoXipFz7Oo9J/YWB+lMOdPQWR6SmSliH5hw9OplTpKn8yaqwRYHkOzcKLCCC0Nv9mkNChBATWhnTwaoC2NAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfbac806002357fb7c4168fa9c9dc762214f612b784bc9a2231acee4113c5afb","last_reissued_at":"2026-05-18T04:00:02.792720Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:02.792720Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.1924","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":"2026-05-18T04:00:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sNA6PY4z0ScLlnLXuSKSeOV5OHudFNIPxZpj9jLmo2nEfC5+hvLuBODEdDKmR+vtOXIOWQGzvEAvgb1j+2AEAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T04:07:56.247710Z"},"content_sha256":"00dd97135477ec505e3a5fbb76a54078e46d6051d9270df646b1086d2946672b","schema_version":"1.0","event_id":"sha256:00dd97135477ec505e3a5fbb76a54078e46d6051d9270df646b1086d2946672b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:Z65MQBQAENL7W7CBND5JZHOHMI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Ke Liang, Mingming Ren, Zhiqi Chen","submitted_at":"2011-03-10T01:22:42Z","abstract_excerpt":"In this paper, we discuss the decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness. In fact, it is a decomposition of a Lie group into totally geodesic sub-manifolds which is different from the De Rham decomposition on a Lie group. As an application, we give a decomposition of a Lie group with a left invariant pseudo-Riemannian Einstein metric, and prove that the decomposition is unique up to the order of the parts in the decomposition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1924","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":"2026-05-18T04:00:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HnNCYxdl7tjEChWUvSJtXrNyp4XKTILKm4vXPlXwdYpkV15uI7n61zcU5uVWu7AvakrVygef8TQP1O5Uo2iUDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T04:07:56.248406Z"},"content_sha256":"d661168c90da8f420b5cc5cf9c6c63ba637db196670a082f87fe469dbbabcf14","schema_version":"1.0","event_id":"sha256:d661168c90da8f420b5cc5cf9c6c63ba637db196670a082f87fe469dbbabcf14"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z65MQBQAENL7W7CBND5JZHOHMI/bundle.json","state_url":"https://pith.science/pith/Z65MQBQAENL7W7CBND5JZHOHMI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z65MQBQAENL7W7CBND5JZHOHMI/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":"2026-06-04T04:07:56Z","links":{"resolver":"https://pith.science/pith/Z65MQBQAENL7W7CBND5JZHOHMI","bundle":"https://pith.science/pith/Z65MQBQAENL7W7CBND5JZHOHMI/bundle.json","state":"https://pith.science/pith/Z65MQBQAENL7W7CBND5JZHOHMI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z65MQBQAENL7W7CBND5JZHOHMI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:Z65MQBQAENL7W7CBND5JZHOHMI","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":"b87a8bea36154ed9dbd0e5135139a36058884974ce0542e4b1fff22dadae9059","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-03-10T01:22:42Z","title_canon_sha256":"28e9575afbc87ac10c7d58e614025182e1601421f5058a98bf51f9cfe3d2d7e6"},"schema_version":"1.0","source":{"id":"1103.1924","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1924","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1924v2","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1924","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"pith_short_12","alias_value":"Z65MQBQAENL7","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"Z65MQBQAENL7W7CB","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"Z65MQBQA","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:d661168c90da8f420b5cc5cf9c6c63ba637db196670a082f87fe469dbbabcf14","target":"graph","created_at":"2026-05-18T04:00:02Z","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":"In this paper, we discuss the decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness. In fact, it is a decomposition of a Lie group into totally geodesic sub-manifolds which is different from the De Rham decomposition on a Lie group. As an application, we give a decomposition of a Lie group with a left invariant pseudo-Riemannian Einstein metric, and prove that the decomposition is unique up to the order of the parts in the decomposition.","authors_text":"Ke Liang, Mingming Ren, Zhiqi Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-03-10T01:22:42Z","title":"The decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1924","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":"sha256:00dd97135477ec505e3a5fbb76a54078e46d6051d9270df646b1086d2946672b","target":"record","created_at":"2026-05-18T04:00:02Z","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":"b87a8bea36154ed9dbd0e5135139a36058884974ce0542e4b1fff22dadae9059","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-03-10T01:22:42Z","title_canon_sha256":"28e9575afbc87ac10c7d58e614025182e1601421f5058a98bf51f9cfe3d2d7e6"},"schema_version":"1.0","source":{"id":"1103.1924","kind":"arxiv","version":2}},"canonical_sha256":"cfbac806002357fb7c4168fa9c9dc762214f612b784bc9a2231acee4113c5afb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cfbac806002357fb7c4168fa9c9dc762214f612b784bc9a2231acee4113c5afb","first_computed_at":"2026-05-18T04:00:02.792720Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:02.792720Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/WMoXipFz7Oo9J/YWB+lMOdPQWR6SmSliH5hw9OplTpKn8yaqwRYHkOzcKLCCC0Nv9mkNChBATWhnTwaoC2NAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:02.793569Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.1924","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00dd97135477ec505e3a5fbb76a54078e46d6051d9270df646b1086d2946672b","sha256:d661168c90da8f420b5cc5cf9c6c63ba637db196670a082f87fe469dbbabcf14"],"state_sha256":"58978d4a6dc2e55a535b2bc1e1a12908f7ae5371178733aeacfc02fda9bb2d41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mU8G+fU2M79r8qbl/6zCnMEeFlrQcvk+g2sAF9woEsjEL9uEx70IJeVvpkizpUlxaJjaRgfZUF3Jd4sy4QS8Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T04:07:56.251872Z","bundle_sha256":"df3797000fae102b7a7f13ad8b622196a6af35edc7e4e06c5ff45f3a916f5f58"}}