{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:WGO4U2QKXVZYHOVV6LVODMLJUX","short_pith_number":"pith:WGO4U2QK","canonical_record":{"source":{"id":"2106.00246","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2021-06-01T05:56:04Z","cross_cats_sorted":["math.DG","math.GR","math.RA"],"title_canon_sha256":"264cc559c976a316a9016600763655280f7ca8490168266c9b21ccddf5d80d31","abstract_canon_sha256":"17533903b51aecaa494873cafe81a418d4d584f559f95a6538b5bd867d991111"},"schema_version":"1.0"},"canonical_sha256":"b19dca6a0abd7383bab5f2eae1b169a5e7d6bd88882dec8bded1469197ecefb4","source":{"kind":"arxiv","id":"2106.00246","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.00246","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"arxiv_version","alias_value":"2106.00246v1","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.00246","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"pith_short_12","alias_value":"WGO4U2QKXVZY","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"pith_short_16","alias_value":"WGO4U2QKXVZYHOVV","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"pith_short_8","alias_value":"WGO4U2QK","created_at":"2026-07-05T02:45:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:WGO4U2QKXVZYHOVV6LVODMLJUX","target":"record","payload":{"canonical_record":{"source":{"id":"2106.00246","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2021-06-01T05:56:04Z","cross_cats_sorted":["math.DG","math.GR","math.RA"],"title_canon_sha256":"264cc559c976a316a9016600763655280f7ca8490168266c9b21ccddf5d80d31","abstract_canon_sha256":"17533903b51aecaa494873cafe81a418d4d584f559f95a6538b5bd867d991111"},"schema_version":"1.0"},"canonical_sha256":"b19dca6a0abd7383bab5f2eae1b169a5e7d6bd88882dec8bded1469197ecefb4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:45:02.381286Z","signature_b64":"EcjL9Zdm31lpnyi++fSOcUfL+IYB0pcVBfRdzi/LYLP2HPIlk0SbqXqrV/bsacZLKlp/3lgNOKHlbUhzlsIwAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b19dca6a0abd7383bab5f2eae1b169a5e7d6bd88882dec8bded1469197ecefb4","last_reissued_at":"2026-07-05T02:45:02.380908Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:45:02.380908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2106.00246","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:45:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B1/Uqxyui3PtwTdAffDDyQmxJUsMaENCyd7k8Oac+mZ0ORWhGDatjA8QIBeiTVMWnUUZl+Dgu3+rFhwAa2xmCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T06:12:47.799628Z"},"content_sha256":"d24376f918b689b854f92e885e8691dc376c0893bb19bdc979f3a233fd4f5c12","schema_version":"1.0","event_id":"sha256:d24376f918b689b854f92e885e8691dc376c0893bb19bdc979f3a233fd4f5c12"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:WGO4U2QKXVZYHOVV6LVODMLJUX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Real graded Lie algebras, Galois cohomology, and classification of trivectors in R^9","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.GR","math.RA"],"primary_cat":"math.RT","authors_text":"H\\^ong V\\^an L\\^e, Mikhail Borovoi, Willem A. de Graaf","submitted_at":"2021-06-01T05:56:04Z","abstract_excerpt":"In this paper we classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was done by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology to obtain the cl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.00246","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.00246/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:45:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eOJJLAF7X5YMmXYUIPgJpyyiTNga6vntkKaobUCdvdF9SqbLU48Ojl/JhRHKPxPDC/s3gKI/kuUfIiQILRj3Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T06:12:47.800347Z"},"content_sha256":"2ef6c986a19643d212d2e572be05c33c18e72632e0fbf15ef483d21e5f4fd437","schema_version":"1.0","event_id":"sha256:2ef6c986a19643d212d2e572be05c33c18e72632e0fbf15ef483d21e5f4fd437"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WGO4U2QKXVZYHOVV6LVODMLJUX/bundle.json","state_url":"https://pith.science/pith/WGO4U2QKXVZYHOVV6LVODMLJUX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WGO4U2QKXVZYHOVV6LVODMLJUX/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":"2026-08-04T06:12:47Z","links":{"resolver":"https://pith.science/pith/WGO4U2QKXVZYHOVV6LVODMLJUX","bundle":"https://pith.science/pith/WGO4U2QKXVZYHOVV6LVODMLJUX/bundle.json","state":"https://pith.science/pith/WGO4U2QKXVZYHOVV6LVODMLJUX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WGO4U2QKXVZYHOVV6LVODMLJUX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:WGO4U2QKXVZYHOVV6LVODMLJUX","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":"17533903b51aecaa494873cafe81a418d4d584f559f95a6538b5bd867d991111","cross_cats_sorted":["math.DG","math.GR","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2021-06-01T05:56:04Z","title_canon_sha256":"264cc559c976a316a9016600763655280f7ca8490168266c9b21ccddf5d80d31"},"schema_version":"1.0","source":{"id":"2106.00246","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.00246","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"arxiv_version","alias_value":"2106.00246v1","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.00246","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"pith_short_12","alias_value":"WGO4U2QKXVZY","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"pith_short_16","alias_value":"WGO4U2QKXVZYHOVV","created_at":"2026-07-05T02:45:02Z"},{"alias_kind":"pith_short_8","alias_value":"WGO4U2QK","created_at":"2026-07-05T02:45:02Z"}],"graph_snapshots":[{"event_id":"sha256:2ef6c986a19643d212d2e572be05c33c18e72632e0fbf15ef483d21e5f4fd437","target":"graph","created_at":"2026-07-05T02:45:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2106.00246/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was done by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology to obtain the cl","authors_text":"H\\^ong V\\^an L\\^e, Mikhail Borovoi, Willem A. de Graaf","cross_cats":["math.DG","math.GR","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2021-06-01T05:56:04Z","title":"Real graded Lie algebras, Galois cohomology, and classification of trivectors in R^9"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.00246","kind":"arxiv","version":1},"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:d24376f918b689b854f92e885e8691dc376c0893bb19bdc979f3a233fd4f5c12","target":"record","created_at":"2026-07-05T02:45:02Z","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":"17533903b51aecaa494873cafe81a418d4d584f559f95a6538b5bd867d991111","cross_cats_sorted":["math.DG","math.GR","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2021-06-01T05:56:04Z","title_canon_sha256":"264cc559c976a316a9016600763655280f7ca8490168266c9b21ccddf5d80d31"},"schema_version":"1.0","source":{"id":"2106.00246","kind":"arxiv","version":1}},"canonical_sha256":"b19dca6a0abd7383bab5f2eae1b169a5e7d6bd88882dec8bded1469197ecefb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b19dca6a0abd7383bab5f2eae1b169a5e7d6bd88882dec8bded1469197ecefb4","first_computed_at":"2026-07-05T02:45:02.380908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:45:02.380908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EcjL9Zdm31lpnyi++fSOcUfL+IYB0pcVBfRdzi/LYLP2HPIlk0SbqXqrV/bsacZLKlp/3lgNOKHlbUhzlsIwAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:45:02.381286Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.00246","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d24376f918b689b854f92e885e8691dc376c0893bb19bdc979f3a233fd4f5c12","sha256:2ef6c986a19643d212d2e572be05c33c18e72632e0fbf15ef483d21e5f4fd437"],"state_sha256":"94c2961af6f15195ec5278ea465ae7227bd5fb6174d97f461395981b5f8bb248"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4FUoj1rN6P8ocmT58Q8TwDEOoi242pvsAfIIPMtPpZ0/zIO/FPZCUOmUSUq1kBaKOBefi+2jVuzdo50kUHPhCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T06:12:47.811682Z","bundle_sha256":"57415678a4d84c77778daecd6311725516f1cb704d108ccbcb81f075c0055b9f"}}