{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:6JF3S6BARKUQHBRHMKFJHZTYSZ","short_pith_number":"pith:6JF3S6BA","canonical_record":{"source":{"id":"2407.01450","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2024-07-01T16:43:50Z","cross_cats_sorted":["hep-th","math-ph","math.MP","math.QA","nlin.SI"],"title_canon_sha256":"72bb9e11474db7f81e6ff675b426099fd756724b06dafd75ce28001cd5f08be4","abstract_canon_sha256":"f1f97a458301b3fa85530c6629bd5b47fa1fe808351bd506f3207a14c4bf86a3"},"schema_version":"1.0"},"canonical_sha256":"f24bb978208aa9038627628a93e6789677352c9737c4fcdbb511210a395e59c3","source":{"kind":"arxiv","id":"2407.01450","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01450","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01450v2","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01450","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"pith_short_12","alias_value":"6JF3S6BARKUQ","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"pith_short_16","alias_value":"6JF3S6BARKUQHBRH","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"pith_short_8","alias_value":"6JF3S6BA","created_at":"2026-07-05T11:45:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:6JF3S6BARKUQHBRHMKFJHZTYSZ","target":"record","payload":{"canonical_record":{"source":{"id":"2407.01450","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2024-07-01T16:43:50Z","cross_cats_sorted":["hep-th","math-ph","math.MP","math.QA","nlin.SI"],"title_canon_sha256":"72bb9e11474db7f81e6ff675b426099fd756724b06dafd75ce28001cd5f08be4","abstract_canon_sha256":"f1f97a458301b3fa85530c6629bd5b47fa1fe808351bd506f3207a14c4bf86a3"},"schema_version":"1.0"},"canonical_sha256":"f24bb978208aa9038627628a93e6789677352c9737c4fcdbb511210a395e59c3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:45:53.058224Z","signature_b64":"ru69OQ9omdCd+FSDr2zR/qSPaZpC4HI9xOr7ExIJTRhG9Iu5dBbiseEgVePOjW1CEHhGy++k0y0chG+A+R1yDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f24bb978208aa9038627628a93e6789677352c9737c4fcdbb511210a395e59c3","last_reissued_at":"2026-07-05T11:45:53.057702Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:45:53.057702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.01450","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":"2026-07-05T11:45:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oYwPN+FFayMvxNfQsbq3qTmHmwVgLwuEx5lINh3xgDg2o65tl0b4o4asJO43Vbznulf5iuJUqB6AuKDz5ZJdAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T07:10:15.983060Z"},"content_sha256":"54cf1b328ff7cb534c744001d300681719a641bb3d16924580492a298c792a94","schema_version":"1.0","event_id":"sha256:54cf1b328ff7cb534c744001d300681719a641bb3d16924580492a298c792a94"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:6JF3S6BARKUQHBRHMKFJHZTYSZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two-Parameter Quantum Groups and $R$-Matrices: Classical Types","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP","math.QA","nlin.SI"],"primary_cat":"math.RT","authors_text":"Alexander Tsymbaliuk, Ian Martin","submitted_at":"2024-07-01T16:43:50Z","abstract_excerpt":"We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\\mathfrak{g})$ for classical $\\mathfrak{g}$, both through the decomposition of $V\\otimes V$ into irreducibles $U_{r,s}(\\mathfrak{g})$-submodules as well as by evaluating the universal $R$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $U^{\\pm}_{r,s}(\\mathfrak{g})$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01450","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.01450/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":"2026-07-05T11:45:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L84Nzx15yALlliGe7LfEosQJvqBinZRFVItW7kcQZYB2TXsivmdcnKBTCoiYJaQTaBH5r2TELLlrjIPUWee9DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T07:10:15.983563Z"},"content_sha256":"a15198d987c803391e901e5b37dce398dec804431770493fdc1f829737e0bda8","schema_version":"1.0","event_id":"sha256:a15198d987c803391e901e5b37dce398dec804431770493fdc1f829737e0bda8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6JF3S6BARKUQHBRHMKFJHZTYSZ/bundle.json","state_url":"https://pith.science/pith/6JF3S6BARKUQHBRHMKFJHZTYSZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6JF3S6BARKUQHBRHMKFJHZTYSZ/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":"2026-08-23T07:10:15Z","links":{"resolver":"https://pith.science/pith/6JF3S6BARKUQHBRHMKFJHZTYSZ","bundle":"https://pith.science/pith/6JF3S6BARKUQHBRHMKFJHZTYSZ/bundle.json","state":"https://pith.science/pith/6JF3S6BARKUQHBRHMKFJHZTYSZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6JF3S6BARKUQHBRHMKFJHZTYSZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:6JF3S6BARKUQHBRHMKFJHZTYSZ","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":"f1f97a458301b3fa85530c6629bd5b47fa1fe808351bd506f3207a14c4bf86a3","cross_cats_sorted":["hep-th","math-ph","math.MP","math.QA","nlin.SI"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2024-07-01T16:43:50Z","title_canon_sha256":"72bb9e11474db7f81e6ff675b426099fd756724b06dafd75ce28001cd5f08be4"},"schema_version":"1.0","source":{"id":"2407.01450","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01450","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01450v2","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01450","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"pith_short_12","alias_value":"6JF3S6BARKUQ","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"pith_short_16","alias_value":"6JF3S6BARKUQHBRH","created_at":"2026-07-05T11:45:53Z"},{"alias_kind":"pith_short_8","alias_value":"6JF3S6BA","created_at":"2026-07-05T11:45:53Z"}],"graph_snapshots":[{"event_id":"sha256:a15198d987c803391e901e5b37dce398dec804431770493fdc1f829737e0bda8","target":"graph","created_at":"2026-07-05T11:45:53Z","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/2407.01450/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\\mathfrak{g})$ for classical $\\mathfrak{g}$, both through the decomposition of $V\\otimes V$ into irreducibles $U_{r,s}(\\mathfrak{g})$-submodules as well as by evaluating the universal $R$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $U^{\\pm}_{r,s}(\\mathfrak{g})$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affi","authors_text":"Alexander Tsymbaliuk, Ian Martin","cross_cats":["hep-th","math-ph","math.MP","math.QA","nlin.SI"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2024-07-01T16:43:50Z","title":"Two-Parameter Quantum Groups and $R$-Matrices: Classical Types"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01450","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":"sha256:54cf1b328ff7cb534c744001d300681719a641bb3d16924580492a298c792a94","target":"record","created_at":"2026-07-05T11:45:53Z","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":"f1f97a458301b3fa85530c6629bd5b47fa1fe808351bd506f3207a14c4bf86a3","cross_cats_sorted":["hep-th","math-ph","math.MP","math.QA","nlin.SI"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RT","submitted_at":"2024-07-01T16:43:50Z","title_canon_sha256":"72bb9e11474db7f81e6ff675b426099fd756724b06dafd75ce28001cd5f08be4"},"schema_version":"1.0","source":{"id":"2407.01450","kind":"arxiv","version":2}},"canonical_sha256":"f24bb978208aa9038627628a93e6789677352c9737c4fcdbb511210a395e59c3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f24bb978208aa9038627628a93e6789677352c9737c4fcdbb511210a395e59c3","first_computed_at":"2026-07-05T11:45:53.057702Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:45:53.057702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ru69OQ9omdCd+FSDr2zR/qSPaZpC4HI9xOr7ExIJTRhG9Iu5dBbiseEgVePOjW1CEHhGy++k0y0chG+A+R1yDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:45:53.058224Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.01450","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:54cf1b328ff7cb534c744001d300681719a641bb3d16924580492a298c792a94","sha256:a15198d987c803391e901e5b37dce398dec804431770493fdc1f829737e0bda8"],"state_sha256":"d9a0095ffef1f0e46c19068360d1c416da502fda90dddfb1e3c5d4edd47aa67c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GzXz9wxgNlb+VspKgodOJfieS7rdd5oLkx67rUs/mPbPl6JEFfP6+V320pLi4qY5iwHoVHlJdzpN0Vg8styLAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T07:10:15.988293Z","bundle_sha256":"358f33a4a6fa80776a0bab1cc94ef7d5f7c41889c4e0d41f2104ba4496c7e5aa"}}