{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:JU2SIJVI44ESCI4HVZS7O4NQ5T","short_pith_number":"pith:JU2SIJVI","canonical_record":{"source":{"id":"1901.02559","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-01-08T23:38:24Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"35bb84b60e4a6cd237492f6a3c1c97de0103fe0a6c7ebeeebc8e0fde1111b934","abstract_canon_sha256":"683624fa64851cf9d911655c8e64b7e7528009fc3511a664e2348a07b3c71e31"},"schema_version":"1.0"},"canonical_sha256":"4d352426a8e709212387ae65f771b0ecedb9bef21c9e8f45d9a231e399b435a9","source":{"kind":"arxiv","id":"1901.02559","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.02559","created_at":"2026-05-17T23:43:50Z"},{"alias_kind":"arxiv_version","alias_value":"1901.02559v2","created_at":"2026-05-17T23:43:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.02559","created_at":"2026-05-17T23:43:50Z"},{"alias_kind":"pith_short_12","alias_value":"JU2SIJVI44ES","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"JU2SIJVI44ESCI4H","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"JU2SIJVI","created_at":"2026-05-18T12:33:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:JU2SIJVI44ESCI4HVZS7O4NQ5T","target":"record","payload":{"canonical_record":{"source":{"id":"1901.02559","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-01-08T23:38:24Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"35bb84b60e4a6cd237492f6a3c1c97de0103fe0a6c7ebeeebc8e0fde1111b934","abstract_canon_sha256":"683624fa64851cf9d911655c8e64b7e7528009fc3511a664e2348a07b3c71e31"},"schema_version":"1.0"},"canonical_sha256":"4d352426a8e709212387ae65f771b0ecedb9bef21c9e8f45d9a231e399b435a9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:50.130837Z","signature_b64":"vEHswwnFfRdIk1MWTUSaSB0goKn+4Jycb/LmvdCabJlP3FuVhA3DIkmna3wJnmerKdwk2zupof4j9JeGjo/9CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d352426a8e709212387ae65f771b0ecedb9bef21c9e8f45d9a231e399b435a9","last_reissued_at":"2026-05-17T23:43:50.130224Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:50.130224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.02559","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":"2026-05-17T23:43:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rRZHEA3SHVwwrIO33jsmnVS7LFLVAjDP1ohhoT5Wd1x0iN8ORSIabR+qhZyJDuSuwHggVYbNQRD5UDUM1u6JDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:04:01.110039Z"},"content_sha256":"bbd601458e22db4d395eb2095ebd1e619308b5d13f997a95991901817f7f804a","schema_version":"1.0","event_id":"sha256:bbd601458e22db4d395eb2095ebd1e619308b5d13f997a95991901817f7f804a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:JU2SIJVI44ESCI4HVZS7O4NQ5T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Metric Lie groups admitting dilations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.MG","authors_text":"Enrico Le Donne, Sebastiano Nicolussi Golo","submitted_at":"2019-01-08T23:38:24Z","abstract_excerpt":"We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \\infty)\\rightarrow\\mathtt{Aut}(G)$, $\\lambda\\mapsto\\delta_\\lambda$, so that $ d(\\delta_\\lambda x,\\delta_\\lambda y) = \\lambda d(x,y)$, for all $x,y\\in G$ and all $\\lambda>0$.\n  First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.02559","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":"2026-05-17T23:43:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U8W0l5AdlHSBI8XOwc/KhZURAMxZy8nwn+xuTiBjvBNxwC7OaNu713h8v6eZZ2Jz90o5ZtW9ZXSEvqiFar8vAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:04:01.110405Z"},"content_sha256":"a32c50e78eae8174f12012cc34e0dc7bb3b8cd6c372ef8ae011eaf3d6d63aa75","schema_version":"1.0","event_id":"sha256:a32c50e78eae8174f12012cc34e0dc7bb3b8cd6c372ef8ae011eaf3d6d63aa75"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JU2SIJVI44ESCI4HVZS7O4NQ5T/bundle.json","state_url":"https://pith.science/pith/JU2SIJVI44ESCI4HVZS7O4NQ5T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JU2SIJVI44ESCI4HVZS7O4NQ5T/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":"2026-06-23T21:04:01Z","links":{"resolver":"https://pith.science/pith/JU2SIJVI44ESCI4HVZS7O4NQ5T","bundle":"https://pith.science/pith/JU2SIJVI44ESCI4HVZS7O4NQ5T/bundle.json","state":"https://pith.science/pith/JU2SIJVI44ESCI4HVZS7O4NQ5T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JU2SIJVI44ESCI4HVZS7O4NQ5T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:JU2SIJVI44ESCI4HVZS7O4NQ5T","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":"683624fa64851cf9d911655c8e64b7e7528009fc3511a664e2348a07b3c71e31","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-01-08T23:38:24Z","title_canon_sha256":"35bb84b60e4a6cd237492f6a3c1c97de0103fe0a6c7ebeeebc8e0fde1111b934"},"schema_version":"1.0","source":{"id":"1901.02559","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.02559","created_at":"2026-05-17T23:43:50Z"},{"alias_kind":"arxiv_version","alias_value":"1901.02559v2","created_at":"2026-05-17T23:43:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.02559","created_at":"2026-05-17T23:43:50Z"},{"alias_kind":"pith_short_12","alias_value":"JU2SIJVI44ES","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"JU2SIJVI44ESCI4H","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"JU2SIJVI","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:a32c50e78eae8174f12012cc34e0dc7bb3b8cd6c372ef8ae011eaf3d6d63aa75","target":"graph","created_at":"2026-05-17T23:43:50Z","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 left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \\infty)\\rightarrow\\mathtt{Aut}(G)$, $\\lambda\\mapsto\\delta_\\lambda$, so that $ d(\\delta_\\lambda x,\\delta_\\lambda y) = \\lambda d(x,y)$, for all $x,y\\in G$ and all $\\lambda>0$.\n  First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties ","authors_text":"Enrico Le Donne, Sebastiano Nicolussi Golo","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-01-08T23:38:24Z","title":"Metric Lie groups admitting dilations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.02559","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:bbd601458e22db4d395eb2095ebd1e619308b5d13f997a95991901817f7f804a","target":"record","created_at":"2026-05-17T23:43:50Z","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":"683624fa64851cf9d911655c8e64b7e7528009fc3511a664e2348a07b3c71e31","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-01-08T23:38:24Z","title_canon_sha256":"35bb84b60e4a6cd237492f6a3c1c97de0103fe0a6c7ebeeebc8e0fde1111b934"},"schema_version":"1.0","source":{"id":"1901.02559","kind":"arxiv","version":2}},"canonical_sha256":"4d352426a8e709212387ae65f771b0ecedb9bef21c9e8f45d9a231e399b435a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d352426a8e709212387ae65f771b0ecedb9bef21c9e8f45d9a231e399b435a9","first_computed_at":"2026-05-17T23:43:50.130224Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:50.130224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vEHswwnFfRdIk1MWTUSaSB0goKn+4Jycb/LmvdCabJlP3FuVhA3DIkmna3wJnmerKdwk2zupof4j9JeGjo/9CA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:50.130837Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.02559","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bbd601458e22db4d395eb2095ebd1e619308b5d13f997a95991901817f7f804a","sha256:a32c50e78eae8174f12012cc34e0dc7bb3b8cd6c372ef8ae011eaf3d6d63aa75"],"state_sha256":"4e889fab754a3c42d57505db994b62230501e018f70631812d1ac7ea641b577e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RVuj6ZTorlU+X9s+Bo12k8a/82yAibAOw21ZwJhPRPoRdPDf/gtIJoRcL2mGwn70VSMOihaJ415ALvHy4tCyCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T21:04:01.112358Z","bundle_sha256":"ebb36de7db84593302be30b72f0ca56cd1ca38ac0414c6384e8c31993fd603d4"}}