{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:AQHQTV5KECDBKBP5PQG26PH5X5","short_pith_number":"pith:AQHQTV5K","canonical_record":{"source":{"id":"1107.1775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-09T12:08:42Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"7e2e4d31cd306c6739758945b713b953c10dedeafab4a8f119178287a9ad7e24","abstract_canon_sha256":"9f179de0c9474f6b7f3da259b39490eca6f9e74084b2c32077552f55156723da"},"schema_version":"1.0"},"canonical_sha256":"040f09d7aa20861505fd7c0daf3cfdbf485e0e5636d089b1f12290bf94ac9cf7","source":{"kind":"arxiv","id":"1107.1775","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.1775","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"arxiv_version","alias_value":"1107.1775v1","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1775","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"pith_short_12","alias_value":"AQHQTV5KECDB","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AQHQTV5KECDBKBP5","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AQHQTV5K","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:AQHQTV5KECDBKBP5PQG26PH5X5","target":"record","payload":{"canonical_record":{"source":{"id":"1107.1775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-09T12:08:42Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"7e2e4d31cd306c6739758945b713b953c10dedeafab4a8f119178287a9ad7e24","abstract_canon_sha256":"9f179de0c9474f6b7f3da259b39490eca6f9e74084b2c32077552f55156723da"},"schema_version":"1.0"},"canonical_sha256":"040f09d7aa20861505fd7c0daf3cfdbf485e0e5636d089b1f12290bf94ac9cf7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:34.167651Z","signature_b64":"8tYiaOpWewgEmBVQgEFVG/eYGkfaecQkcHtx+0LiqVon5xoY6v8rxkdikAREpSaWWaHiSzdjDgHtHjG8oOiuCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"040f09d7aa20861505fd7c0daf3cfdbf485e0e5636d089b1f12290bf94ac9cf7","last_reissued_at":"2026-05-18T04:18:34.167027Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:34.167027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.1775","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":"2026-05-18T04:18:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dbK/HYlkdGbeY1hEIhb0LOgf5smzPKNGAJa4k11Z73FaOzYFeuWridfAtqN6W/LMGBu8+PeZQ8vqGH5HUr5HCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:03:38.619826Z"},"content_sha256":"c6427fbdac9cc20f43fc077c57f2c7801e499d33951c4e3ca263b342881e5377","schema_version":"1.0","event_id":"sha256:c6427fbdac9cc20f43fc077c57f2c7801e499d33951c4e3ca263b342881e5377"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:AQHQTV5KECDBKBP5PQG26PH5X5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Semidirect Product of Groupoids, Its Representations and Random Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Leszek Pysiak, Michael Heller, Micha{\\l} Eckstein, Wies{\\l}aw Sasin","submitted_at":"2011-07-09T12:08:42Z","abstract_excerpt":"One of pressing problems in mathematical physics is to find a generalized Poincar\\'e symmetry that could be applied to nonflat space-times. As a step in this direction we define the semidirect product of groupoids $\\Gamma_0 \\rtimes \\Gamma_1$ and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid $\\Gamma_1$ and prove that it is isomorphic to the convolutive algebra of the groupoid $\\Gamma_0 \\rtimes \\Gamma_1$. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1775","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":""},"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":"2026-05-18T04:18:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TK2ix0NU4vFvPGzKTwG3Fo7rhsjp2lhv5LRnAFIStZLmxJMneuii6AUGQTDmxnPiGcriL+ytTj5Q9OypiZAYAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:03:38.620230Z"},"content_sha256":"3f189a643b2a14e0f4a426b79e6205043fa54d13b274f56a4000c27405dd3639","schema_version":"1.0","event_id":"sha256:3f189a643b2a14e0f4a426b79e6205043fa54d13b274f56a4000c27405dd3639"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AQHQTV5KECDBKBP5PQG26PH5X5/bundle.json","state_url":"https://pith.science/pith/AQHQTV5KECDBKBP5PQG26PH5X5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AQHQTV5KECDBKBP5PQG26PH5X5/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":"2026-05-28T22:03:38Z","links":{"resolver":"https://pith.science/pith/AQHQTV5KECDBKBP5PQG26PH5X5","bundle":"https://pith.science/pith/AQHQTV5KECDBKBP5PQG26PH5X5/bundle.json","state":"https://pith.science/pith/AQHQTV5KECDBKBP5PQG26PH5X5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AQHQTV5KECDBKBP5PQG26PH5X5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:AQHQTV5KECDBKBP5PQG26PH5X5","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":"9f179de0c9474f6b7f3da259b39490eca6f9e74084b2c32077552f55156723da","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-09T12:08:42Z","title_canon_sha256":"7e2e4d31cd306c6739758945b713b953c10dedeafab4a8f119178287a9ad7e24"},"schema_version":"1.0","source":{"id":"1107.1775","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.1775","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"arxiv_version","alias_value":"1107.1775v1","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1775","created_at":"2026-05-18T04:18:34Z"},{"alias_kind":"pith_short_12","alias_value":"AQHQTV5KECDB","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AQHQTV5KECDBKBP5","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AQHQTV5K","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:3f189a643b2a14e0f4a426b79e6205043fa54d13b274f56a4000c27405dd3639","target":"graph","created_at":"2026-05-18T04:18:34Z","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":"One of pressing problems in mathematical physics is to find a generalized Poincar\\'e symmetry that could be applied to nonflat space-times. As a step in this direction we define the semidirect product of groupoids $\\Gamma_0 \\rtimes \\Gamma_1$ and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid $\\Gamma_1$ and prove that it is isomorphic to the convolutive algebra of the groupoid $\\Gamma_0 \\rtimes \\Gamma_1$. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. A","authors_text":"Leszek Pysiak, Michael Heller, Micha{\\l} Eckstein, Wies{\\l}aw Sasin","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-09T12:08:42Z","title":"Semidirect Product of Groupoids, Its Representations and Random Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1775","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:c6427fbdac9cc20f43fc077c57f2c7801e499d33951c4e3ca263b342881e5377","target":"record","created_at":"2026-05-18T04:18:34Z","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":"9f179de0c9474f6b7f3da259b39490eca6f9e74084b2c32077552f55156723da","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-07-09T12:08:42Z","title_canon_sha256":"7e2e4d31cd306c6739758945b713b953c10dedeafab4a8f119178287a9ad7e24"},"schema_version":"1.0","source":{"id":"1107.1775","kind":"arxiv","version":1}},"canonical_sha256":"040f09d7aa20861505fd7c0daf3cfdbf485e0e5636d089b1f12290bf94ac9cf7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"040f09d7aa20861505fd7c0daf3cfdbf485e0e5636d089b1f12290bf94ac9cf7","first_computed_at":"2026-05-18T04:18:34.167027Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:34.167027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8tYiaOpWewgEmBVQgEFVG/eYGkfaecQkcHtx+0LiqVon5xoY6v8rxkdikAREpSaWWaHiSzdjDgHtHjG8oOiuCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:34.167651Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.1775","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6427fbdac9cc20f43fc077c57f2c7801e499d33951c4e3ca263b342881e5377","sha256:3f189a643b2a14e0f4a426b79e6205043fa54d13b274f56a4000c27405dd3639"],"state_sha256":"d143d9c28e7b781c252d5b6dc527a638ce86439390b6d17da44d99222d431dcb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sLdhdLWcO7Fg7mzYDVAYm9irJZw/AZQnIVPcvwlT9U5ddx1GkW//So6JCmtueEWcaMBXjKv85wZQTU53jU20Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T22:03:38.623490Z","bundle_sha256":"88050ddb10bc936453a019141a55f1b95836bf3ef454bab653dab1a694a99a6d"}}