{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:XIW5IWKK5IAEVFEXJIVOV2UQBN","short_pith_number":"pith:XIW5IWKK","canonical_record":{"source":{"id":"1104.1415","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-07T19:47:51Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"d9f584d15dd8a1d026200305e063fe03a827282dc1495af31db44cfd00dbe5ed","abstract_canon_sha256":"dd99b1db5f146840deb363eced714c734d85cd096d1e8ad5d912d725e832a5bd"},"schema_version":"1.0"},"canonical_sha256":"ba2dd4594aea004a94974a2aeaea900b5be9d711c636bbad8cb0c222cbe579a8","source":{"kind":"arxiv","id":"1104.1415","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.1415","created_at":"2026-05-18T04:21:12Z"},{"alias_kind":"arxiv_version","alias_value":"1104.1415v2","created_at":"2026-05-18T04:21:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.1415","created_at":"2026-05-18T04:21:12Z"},{"alias_kind":"pith_short_12","alias_value":"XIW5IWKK5IAE","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"XIW5IWKK5IAEVFEX","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"XIW5IWKK","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:XIW5IWKK5IAEVFEXJIVOV2UQBN","target":"record","payload":{"canonical_record":{"source":{"id":"1104.1415","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-07T19:47:51Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"d9f584d15dd8a1d026200305e063fe03a827282dc1495af31db44cfd00dbe5ed","abstract_canon_sha256":"dd99b1db5f146840deb363eced714c734d85cd096d1e8ad5d912d725e832a5bd"},"schema_version":"1.0"},"canonical_sha256":"ba2dd4594aea004a94974a2aeaea900b5be9d711c636bbad8cb0c222cbe579a8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:12.039609Z","signature_b64":"EVfsU6ZvGYd3tHv7AH1M/MoicH0Y2Si5nVsu6MZ1VU4ImEGXo4wLqQwWJeye5BgrFx7sBWJ0qO0fzfNXMO9rCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba2dd4594aea004a94974a2aeaea900b5be9d711c636bbad8cb0c222cbe579a8","last_reissued_at":"2026-05-18T04:21:12.038684Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:12.038684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.1415","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":"2026-05-18T04:21:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x8zZE9T64GkmUawdCtlVA803YDaDSiglq8bwnmbO4YTqmvqigUxMJ87TjZDkz26LzeFrreZNDDrvpbIvOjK9CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:19:39.638928Z"},"content_sha256":"3125492df848f717817cd57e9fddbc1aa463dec80761eb1109b4bf6bf0a415fd","schema_version":"1.0","event_id":"sha256:3125492df848f717817cd57e9fddbc1aa463dec80761eb1109b4bf6bf0a415fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:XIW5IWKK5IAEVFEXJIVOV2UQBN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Belkale-Kumar cup product and relative Lie algebra cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.AG","authors_text":"Sam Evens, William Graham, with an appendix jointly written with Edward Richmond","submitted_at":"2011-04-07T19:47:51Z","abstract_excerpt":"We study the Belkale-Kumar family of cup products on the cohomology of a generalized flag variety. We give an alternative construction of the family using relative Lie algebra cohomology, and in particular, identify the Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every value of the parameter. As a consequence, we extend a fundamental disjointness result of Kostant to a family of Lie algebras. In an appendix, written jointly with Edward Richmond, we extend a Levi movability result of Belkale and Kumar to arbitrary parameters."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1415","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":"2026-05-18T04:21:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dayYkZB7qPVFEn010QSTfr2zeYZgwtQ2Oyu8U2K0OYWNSpr5qfvvyAPiJzgU836FsmuwsQ9J5910RrMRLT8pCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:19:39.639595Z"},"content_sha256":"332755d739b546976342bd3c806f5a03f1c698139eda5546b6008fb1af30c5bd","schema_version":"1.0","event_id":"sha256:332755d739b546976342bd3c806f5a03f1c698139eda5546b6008fb1af30c5bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XIW5IWKK5IAEVFEXJIVOV2UQBN/bundle.json","state_url":"https://pith.science/pith/XIW5IWKK5IAEVFEXJIVOV2UQBN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XIW5IWKK5IAEVFEXJIVOV2UQBN/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":"2026-05-26T00:19:39Z","links":{"resolver":"https://pith.science/pith/XIW5IWKK5IAEVFEXJIVOV2UQBN","bundle":"https://pith.science/pith/XIW5IWKK5IAEVFEXJIVOV2UQBN/bundle.json","state":"https://pith.science/pith/XIW5IWKK5IAEVFEXJIVOV2UQBN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XIW5IWKK5IAEVFEXJIVOV2UQBN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XIW5IWKK5IAEVFEXJIVOV2UQBN","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":"dd99b1db5f146840deb363eced714c734d85cd096d1e8ad5d912d725e832a5bd","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-07T19:47:51Z","title_canon_sha256":"d9f584d15dd8a1d026200305e063fe03a827282dc1495af31db44cfd00dbe5ed"},"schema_version":"1.0","source":{"id":"1104.1415","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.1415","created_at":"2026-05-18T04:21:12Z"},{"alias_kind":"arxiv_version","alias_value":"1104.1415v2","created_at":"2026-05-18T04:21:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.1415","created_at":"2026-05-18T04:21:12Z"},{"alias_kind":"pith_short_12","alias_value":"XIW5IWKK5IAE","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"XIW5IWKK5IAEVFEX","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"XIW5IWKK","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:332755d739b546976342bd3c806f5a03f1c698139eda5546b6008fb1af30c5bd","target":"graph","created_at":"2026-05-18T04:21:12Z","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 study the Belkale-Kumar family of cup products on the cohomology of a generalized flag variety. We give an alternative construction of the family using relative Lie algebra cohomology, and in particular, identify the Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every value of the parameter. As a consequence, we extend a fundamental disjointness result of Kostant to a family of Lie algebras. In an appendix, written jointly with Edward Richmond, we extend a Levi movability result of Belkale and Kumar to arbitrary parameters.","authors_text":"Sam Evens, William Graham, with an appendix jointly written with Edward Richmond","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-07T19:47:51Z","title":"The Belkale-Kumar cup product and relative Lie algebra cohomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1415","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":"sha256:3125492df848f717817cd57e9fddbc1aa463dec80761eb1109b4bf6bf0a415fd","target":"record","created_at":"2026-05-18T04:21:12Z","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":"dd99b1db5f146840deb363eced714c734d85cd096d1e8ad5d912d725e832a5bd","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-07T19:47:51Z","title_canon_sha256":"d9f584d15dd8a1d026200305e063fe03a827282dc1495af31db44cfd00dbe5ed"},"schema_version":"1.0","source":{"id":"1104.1415","kind":"arxiv","version":2}},"canonical_sha256":"ba2dd4594aea004a94974a2aeaea900b5be9d711c636bbad8cb0c222cbe579a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba2dd4594aea004a94974a2aeaea900b5be9d711c636bbad8cb0c222cbe579a8","first_computed_at":"2026-05-18T04:21:12.038684Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:21:12.038684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EVfsU6ZvGYd3tHv7AH1M/MoicH0Y2Si5nVsu6MZ1VU4ImEGXo4wLqQwWJeye5BgrFx7sBWJ0qO0fzfNXMO9rCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:21:12.039609Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.1415","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3125492df848f717817cd57e9fddbc1aa463dec80761eb1109b4bf6bf0a415fd","sha256:332755d739b546976342bd3c806f5a03f1c698139eda5546b6008fb1af30c5bd"],"state_sha256":"85f341f871beb1b7cbfb36afc7c0314932b5fa27799237b347463459d6ebd98b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mp6kCvvuhnLjHIji5rVK9IiD+XinB5/nsJNNehTfeEzRi9DA/KH6ZCOqb8o8JGChOCCus4olPCxtiz++bNQLBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T00:19:39.643224Z","bundle_sha256":"299b7ccc5ca56f3040782263e141bd034f374050894f7e53444a50f2c266e9ee"}}