{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:4J3CD7LSNHBYJQIMUI2YZ77YJU","short_pith_number":"pith:4J3CD7LS","canonical_record":{"source":{"id":"1103.1241","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-03-07T11:13:56Z","cross_cats_sorted":["math.CT","math.QA"],"title_canon_sha256":"f6e97f4835cddb68fe6bc9430b40916c52ea82ff193f3e5d362f28ce9e0efc36","abstract_canon_sha256":"4462c104a08fc8bdab6e433ea77dbfb6c717b151c06950b4838736b052e4c41a"},"schema_version":"1.0"},"canonical_sha256":"e27621fd7269c384c10ca2358cfff84d29d9f7e3776359f7e29d574dbdda3b2e","source":{"kind":"arxiv","id":"1103.1241","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1241","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1241v2","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1241","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"4J3CD7LSNHBY","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4J3CD7LSNHBYJQIM","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4J3CD7LS","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:4J3CD7LSNHBYJQIMUI2YZ77YJU","target":"record","payload":{"canonical_record":{"source":{"id":"1103.1241","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-03-07T11:13:56Z","cross_cats_sorted":["math.CT","math.QA"],"title_canon_sha256":"f6e97f4835cddb68fe6bc9430b40916c52ea82ff193f3e5d362f28ce9e0efc36","abstract_canon_sha256":"4462c104a08fc8bdab6e433ea77dbfb6c717b151c06950b4838736b052e4c41a"},"schema_version":"1.0"},"canonical_sha256":"e27621fd7269c384c10ca2358cfff84d29d9f7e3776359f7e29d574dbdda3b2e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:01.155456Z","signature_b64":"bH32xQl3KtzPn5AD8UqCxItKLblGk5MbU3Cdek7oTA7e8xQsWsvY9ZVy3TJuFPcF0K1p3nYxNiljsV1oS0N3BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e27621fd7269c384c10ca2358cfff84d29d9f7e3776359f7e29d574dbdda3b2e","last_reissued_at":"2026-05-18T02:58:01.154931Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:01.154931Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.1241","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":"2026-05-18T02:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k0qAoEBF1ueVbL4FuYVNjZHzDPZ+U8G5NX0WrhBbvy4/I/wP+WgrNRbzmVZmqSgoKyyTaOP2gOWz5pebenPPDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:24:58.802234Z"},"content_sha256":"f867d9ea1ab26590d640f8b984f6c51368fa0d3777a2371b0912a84de5d3c7df","schema_version":"1.0","event_id":"sha256:f867d9ea1ab26590d640f8b984f6c51368fa0d3777a2371b0912a84de5d3c7df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:4J3CD7LSNHBYJQIMUI2YZ77YJU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Ringel--Hall Lie algebra of a spherical object","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.QA"],"primary_cat":"math.RT","authors_text":"Changjian Fu, Dong Yang","submitted_at":"2011-03-07T11:13:56Z","abstract_excerpt":"For an integer $w$, let $\\cs_w$ be the algebraic triangulated category generated by a $w$-spherical object. We determine the Picard group of $\\cs_w$ and show that each orbit category of $\\cs_w$ is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. When $n=2$, the orbit category $\\cs_w/\\Sigma^2$ is 2-periodic triangulated, and we characterize the associated Ringel--Hall Lie algebra in the sense of Peng and Xiao."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1241","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":"2026-05-18T02:58:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IdJLtISur41zt7SYDSqYYH73HxJRyMNDZ22MiElYlnQZlojyGH0MdY87Xb6Z5vRBCxeapdqdDBcyJuHPH60ADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:24:58.802878Z"},"content_sha256":"8d70e180c04d22e31fc25c65b35730e2a813b5abd9f664e711099e1ab70d0bf4","schema_version":"1.0","event_id":"sha256:8d70e180c04d22e31fc25c65b35730e2a813b5abd9f664e711099e1ab70d0bf4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4J3CD7LSNHBYJQIMUI2YZ77YJU/bundle.json","state_url":"https://pith.science/pith/4J3CD7LSNHBYJQIMUI2YZ77YJU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4J3CD7LSNHBYJQIMUI2YZ77YJU/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":"2026-05-31T15:24:58Z","links":{"resolver":"https://pith.science/pith/4J3CD7LSNHBYJQIMUI2YZ77YJU","bundle":"https://pith.science/pith/4J3CD7LSNHBYJQIMUI2YZ77YJU/bundle.json","state":"https://pith.science/pith/4J3CD7LSNHBYJQIMUI2YZ77YJU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4J3CD7LSNHBYJQIMUI2YZ77YJU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:4J3CD7LSNHBYJQIMUI2YZ77YJU","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":"4462c104a08fc8bdab6e433ea77dbfb6c717b151c06950b4838736b052e4c41a","cross_cats_sorted":["math.CT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-03-07T11:13:56Z","title_canon_sha256":"f6e97f4835cddb68fe6bc9430b40916c52ea82ff193f3e5d362f28ce9e0efc36"},"schema_version":"1.0","source":{"id":"1103.1241","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1241","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1241v2","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1241","created_at":"2026-05-18T02:58:01Z"},{"alias_kind":"pith_short_12","alias_value":"4J3CD7LSNHBY","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4J3CD7LSNHBYJQIM","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4J3CD7LS","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:8d70e180c04d22e31fc25c65b35730e2a813b5abd9f664e711099e1ab70d0bf4","target":"graph","created_at":"2026-05-18T02:58:01Z","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 an integer $w$, let $\\cs_w$ be the algebraic triangulated category generated by a $w$-spherical object. We determine the Picard group of $\\cs_w$ and show that each orbit category of $\\cs_w$ is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. When $n=2$, the orbit category $\\cs_w/\\Sigma^2$ is 2-periodic triangulated, and we characterize the associated Ringel--Hall Lie algebra in the sense of Peng and Xiao.","authors_text":"Changjian Fu, Dong Yang","cross_cats":["math.CT","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-03-07T11:13:56Z","title":"The Ringel--Hall Lie algebra of a spherical object"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1241","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:f867d9ea1ab26590d640f8b984f6c51368fa0d3777a2371b0912a84de5d3c7df","target":"record","created_at":"2026-05-18T02:58:01Z","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":"4462c104a08fc8bdab6e433ea77dbfb6c717b151c06950b4838736b052e4c41a","cross_cats_sorted":["math.CT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-03-07T11:13:56Z","title_canon_sha256":"f6e97f4835cddb68fe6bc9430b40916c52ea82ff193f3e5d362f28ce9e0efc36"},"schema_version":"1.0","source":{"id":"1103.1241","kind":"arxiv","version":2}},"canonical_sha256":"e27621fd7269c384c10ca2358cfff84d29d9f7e3776359f7e29d574dbdda3b2e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e27621fd7269c384c10ca2358cfff84d29d9f7e3776359f7e29d574dbdda3b2e","first_computed_at":"2026-05-18T02:58:01.154931Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:01.154931Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bH32xQl3KtzPn5AD8UqCxItKLblGk5MbU3Cdek7oTA7e8xQsWsvY9ZVy3TJuFPcF0K1p3nYxNiljsV1oS0N3BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:01.155456Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.1241","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f867d9ea1ab26590d640f8b984f6c51368fa0d3777a2371b0912a84de5d3c7df","sha256:8d70e180c04d22e31fc25c65b35730e2a813b5abd9f664e711099e1ab70d0bf4"],"state_sha256":"c75de32e7ea1edc909f29be6914b0b6c6f92270dc8b4d2fc52bfd4a08f9c306f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M9YVpVmvuz38bgLWPqKTENl5kmcgHStozDZ7YsBppgt8n9fiV1pSwZC/BJChDiovnIwxzdc8EDc0j8cdVrc9Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T15:24:58.806551Z","bundle_sha256":"a4684e745e364c062864cf8da549038c2c4bc0144b4d907854bb02ae94731a8c"}}