{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TYXWNR7FIYIPOP5OF65E5MHKKB","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":"4ae002dbe923fe2833dc83732c0695dbe44ea57bf85373f8d0a5b9192674327f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-02-19T01:19:31Z","title_canon_sha256":"2f2435f078b89dddb2336182e1dbec39bd8dd61abe909cee848c13f29c5ff07d"},"schema_version":"1.0","source":{"id":"1202.4121","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.4121","created_at":"2026-05-18T03:40:36Z"},{"alias_kind":"arxiv_version","alias_value":"1202.4121v2","created_at":"2026-05-18T03:40:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4121","created_at":"2026-05-18T03:40:36Z"},{"alias_kind":"pith_short_12","alias_value":"TYXWNR7FIYIP","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TYXWNR7FIYIPOP5O","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TYXWNR7F","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:05072913ab515ece2a856eaee62e742525a67c28d00d915b630229763bee7da2","target":"graph","created_at":"2026-05-18T03:40:36Z","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":"Let $H$ be a pointed Hopf algebra. We show that under some mild assumptions $H$ and its associated graded Hopf algebra $\\gr H$ have the same Gelfand-Kirillov dimension. As an application, we prove that the Gelfand-Kirillov dimension of a connected Hopf algebra is either infinity or a positive integer. We also classify connected Hopf algebras of GK-dimension three over an algebraically closed field of characteristic zero.","authors_text":"Guangbin Zhuang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-02-19T01:19:31Z","title":"Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4121","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:dd5478b8c8d96acc969cd2e2842abd1c0c9c7aa89f8705869e5315c3090b8db6","target":"record","created_at":"2026-05-18T03:40:36Z","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":"4ae002dbe923fe2833dc83732c0695dbe44ea57bf85373f8d0a5b9192674327f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-02-19T01:19:31Z","title_canon_sha256":"2f2435f078b89dddb2336182e1dbec39bd8dd61abe909cee848c13f29c5ff07d"},"schema_version":"1.0","source":{"id":"1202.4121","kind":"arxiv","version":2}},"canonical_sha256":"9e2f66c7e54610f73fae2fba4eb0ea506cfaab05f14dcef9879f50e34eddfac7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e2f66c7e54610f73fae2fba4eb0ea506cfaab05f14dcef9879f50e34eddfac7","first_computed_at":"2026-05-18T03:40:36.531299Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:36.531299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CixxNhKD2SNOBv9cs530NxTKviRmstu1cES6Rl73CoVK3McakRo1mjxf2ESH39eORYWkrDTmQgouYMdCathbBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:36.531901Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.4121","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd5478b8c8d96acc969cd2e2842abd1c0c9c7aa89f8705869e5315c3090b8db6","sha256:05072913ab515ece2a856eaee62e742525a67c28d00d915b630229763bee7da2"],"state_sha256":"8e9adc81ea66175a3783a63b789ebc9221d6a19764213b87dd13ae1b051c5947"}