{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:LHJVWC2RIBYHUYKQ3Q2MJNDPXU","short_pith_number":"pith:LHJVWC2R","canonical_record":{"source":{"id":"0711.4624","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2007-11-28T23:42:00Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"dbf6eb5f9ae7c0ef4b70741206a1e3dffff7f032a690b9f935e40d152cdd00bc","abstract_canon_sha256":"4c3d1c91b269f10c803d241459d2c6cc9bd6508490e02d3c965f8d2001a71f39"},"schema_version":"1.0"},"canonical_sha256":"59d35b0b5140707a6150dc34c4b46fbd239ec397dbfb5dcc8535659ea0690428","source":{"kind":"arxiv","id":"0711.4624","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.4624","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"0711.4624v1","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.4624","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"LHJVWC2RIBYH","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"pith_short_16","alias_value":"LHJVWC2RIBYHUYKQ","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"pith_short_8","alias_value":"LHJVWC2R","created_at":"2026-07-04T15:05:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:LHJVWC2RIBYHUYKQ3Q2MJNDPXU","target":"record","payload":{"canonical_record":{"source":{"id":"0711.4624","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.QA","submitted_at":"2007-11-28T23:42:00Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"dbf6eb5f9ae7c0ef4b70741206a1e3dffff7f032a690b9f935e40d152cdd00bc","abstract_canon_sha256":"4c3d1c91b269f10c803d241459d2c6cc9bd6508490e02d3c965f8d2001a71f39"},"schema_version":"1.0"},"canonical_sha256":"59d35b0b5140707a6150dc34c4b46fbd239ec397dbfb5dcc8535659ea0690428","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:05:55.998713Z","signature_b64":"Jz4FwGn1ASFOg/UORc4DEg57nqdhya7ZpORJ35d4Otk+rTojqYzPdyqmUWe/l1AyfpJ3Og2ZrZ0RJlEWqlCeBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59d35b0b5140707a6150dc34c4b46fbd239ec397dbfb5dcc8535659ea0690428","last_reissued_at":"2026-07-04T15:05:55.998340Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:05:55.998340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0711.4624","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":"2026-07-04T15:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ait+8KGHCJFvw4fKtNuBDAnQeelOBHSUbQ2RwFGuyqJA5p3MFIa1IN3UVDJFvdqI1iromyVhX+pN2UK3kzIpBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:10:45.046203Z"},"content_sha256":"20c9943a3adaf1de7ebef72a701c4ff19609bc58a0481a02d0e4f8785b3006b3","schema_version":"1.0","event_id":"sha256:20c9943a3adaf1de7ebef72a701c4ff19609bc58a0481a02d0e4f8785b3006b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:LHJVWC2RIBYHUYKQ3Q2MJNDPXU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"W-algebra W(2,2) and the vertex operator algebra L(1/2,0)\\otimes L(1/2,0)","license":"","headline":"","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"C. Dong, W. Zhang","submitted_at":"2007-11-28T23:42:00Z","abstract_excerpt":"In this paper the W-algebra W(2,2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to a highest irreducible W(2,2)-module or a tensor product of two irreducible Virasoro vertex operator algebras. Furthermore, any rational, C_2-cofinite simple vertex operator algebra whose weight 1 subspace is zero and weight 2 subspace is 2-dimensional, and with central charge c=1 is isomorphic to L(1/2,0)\\otimes L(1/2,0)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4624","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/0711.4624/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":"2026-07-04T15:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kHfFb23q8640Y9pSKsvu6GbamToOEHoL/H5IHV14IVUv0QgyVAACQto9hcnz8y8gQPGAhkLzlmHVc2CB0VyvCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:10:45.047161Z"},"content_sha256":"b5d4136e3ecb0159beedae37cfc347ac924c0bbb9889f5fca446da13552cabaf","schema_version":"1.0","event_id":"sha256:b5d4136e3ecb0159beedae37cfc347ac924c0bbb9889f5fca446da13552cabaf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LHJVWC2RIBYHUYKQ3Q2MJNDPXU/bundle.json","state_url":"https://pith.science/pith/LHJVWC2RIBYHUYKQ3Q2MJNDPXU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LHJVWC2RIBYHUYKQ3Q2MJNDPXU/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":"2026-08-03T20:10:45Z","links":{"resolver":"https://pith.science/pith/LHJVWC2RIBYHUYKQ3Q2MJNDPXU","bundle":"https://pith.science/pith/LHJVWC2RIBYHUYKQ3Q2MJNDPXU/bundle.json","state":"https://pith.science/pith/LHJVWC2RIBYHUYKQ3Q2MJNDPXU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LHJVWC2RIBYHUYKQ3Q2MJNDPXU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:LHJVWC2RIBYHUYKQ3Q2MJNDPXU","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":"4c3d1c91b269f10c803d241459d2c6cc9bd6508490e02d3c965f8d2001a71f39","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.QA","submitted_at":"2007-11-28T23:42:00Z","title_canon_sha256":"dbf6eb5f9ae7c0ef4b70741206a1e3dffff7f032a690b9f935e40d152cdd00bc"},"schema_version":"1.0","source":{"id":"0711.4624","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.4624","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"0711.4624v1","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.4624","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"LHJVWC2RIBYH","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"pith_short_16","alias_value":"LHJVWC2RIBYHUYKQ","created_at":"2026-07-04T15:05:55Z"},{"alias_kind":"pith_short_8","alias_value":"LHJVWC2R","created_at":"2026-07-04T15:05:55Z"}],"graph_snapshots":[{"event_id":"sha256:b5d4136e3ecb0159beedae37cfc347ac924c0bbb9889f5fca446da13552cabaf","target":"graph","created_at":"2026-07-04T15:05:55Z","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/0711.4624/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper the W-algebra W(2,2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to a highest irreducible W(2,2)-module or a tensor product of two irreducible Virasoro vertex operator algebras. Furthermore, any rational, C_2-cofinite simple vertex operator algebra whose weight 1 subspace is zero and weight 2 subspace is 2-dimensional, and with central charge c=1 is isomorphic to L(1/2,0)\\otimes L(1/2,0).","authors_text":"C. Dong, W. Zhang","cross_cats":["math.RT"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2007-11-28T23:42:00Z","title":"W-algebra W(2,2) and the vertex operator algebra L(1/2,0)\\otimes L(1/2,0)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4624","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":"sha256:20c9943a3adaf1de7ebef72a701c4ff19609bc58a0481a02d0e4f8785b3006b3","target":"record","created_at":"2026-07-04T15:05:55Z","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":"4c3d1c91b269f10c803d241459d2c6cc9bd6508490e02d3c965f8d2001a71f39","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.QA","submitted_at":"2007-11-28T23:42:00Z","title_canon_sha256":"dbf6eb5f9ae7c0ef4b70741206a1e3dffff7f032a690b9f935e40d152cdd00bc"},"schema_version":"1.0","source":{"id":"0711.4624","kind":"arxiv","version":1}},"canonical_sha256":"59d35b0b5140707a6150dc34c4b46fbd239ec397dbfb5dcc8535659ea0690428","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"59d35b0b5140707a6150dc34c4b46fbd239ec397dbfb5dcc8535659ea0690428","first_computed_at":"2026-07-04T15:05:55.998340Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:05:55.998340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Jz4FwGn1ASFOg/UORc4DEg57nqdhya7ZpORJ35d4Otk+rTojqYzPdyqmUWe/l1AyfpJ3Og2ZrZ0RJlEWqlCeBw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:05:55.998713Z","signed_message":"canonical_sha256_bytes"},"source_id":"0711.4624","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20c9943a3adaf1de7ebef72a701c4ff19609bc58a0481a02d0e4f8785b3006b3","sha256:b5d4136e3ecb0159beedae37cfc347ac924c0bbb9889f5fca446da13552cabaf"],"state_sha256":"8806b82ecf0ad6f9bd99f3ea11b9c71740529619991d0ec4cd1a5655c4467304"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ce8A+/PH1Trrz+bp7r0aya45n2PXWxZuIh0YkQUzatSr/rjSxqEcREqLhP/H5K7asPP4T/7pP0MXI3kNmfqUBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T20:10:45.061003Z","bundle_sha256":"93d808409724bf44f20533b7bc37731b6a6cec84f6978f272d90a29ba1e22c30"}}