{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:3DFNORAX2F6UEQMJV4AIDDWNMB","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":"57d2cf442534f592fe12b3e4def05460d2a7c09d8a3e8c5ea6fe9f1052a463ee","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2019-08-13T06:56:16Z","title_canon_sha256":"f793707244490ee7c130547fe6baee328f68bf3cd2a34ffee99afcfb32df03fe"},"schema_version":"1.0","source":{"id":"1908.04514","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.04514","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"arxiv_version","alias_value":"1908.04514v1","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.04514","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_12","alias_value":"3DFNORAX2F6U","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_16","alias_value":"3DFNORAX2F6UEQMJ","created_at":"2026-07-04T23:55:14Z"},{"alias_kind":"pith_short_8","alias_value":"3DFNORAX","created_at":"2026-07-04T23:55:14Z"}],"graph_snapshots":[{"event_id":"sha256:55435cad2eaa5c2ae699a13f3e6eedb832ac6b3f6afcdd50c26fb991d070e188","target":"graph","created_at":"2026-07-04T23:55:14Z","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/1908.04514/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate and review how Fourier transform is involved in the analysis of a twisted group algebra $L^1(G, \\sigma)$ for $G=\\widehat{\\Gamma}\\times \\Gamma$ and $\\sigma:G\\times G \\to \\mathbb{T}$ 2- cocycle where $\\Gamma$ is a locally compact abelian group and $\\widehat{\\Gamma}$ its Pontryagin dual. By weaving the Schr\\\"{o}dinger representation and Fourier transform, we construct the dual equivalence bimodule of the Heisenberg bimodule generated by the dual Schr\\\"{o}dinger representation and observe several relations between them including the application of noncommutative solitons.","authors_text":"Hyun Ho Lee","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2019-08-13T06:56:16Z","title":"Fourier transform, Schr\\\"odinger representation, and Heisenberg modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04514","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":"sha256:05b1e0fb2ebf160cba15d5db6e752703b3f74c7f932db1abb6e1ac4719caa1d7","target":"record","created_at":"2026-07-04T23:55:14Z","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":"57d2cf442534f592fe12b3e4def05460d2a7c09d8a3e8c5ea6fe9f1052a463ee","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2019-08-13T06:56:16Z","title_canon_sha256":"f793707244490ee7c130547fe6baee328f68bf3cd2a34ffee99afcfb32df03fe"},"schema_version":"1.0","source":{"id":"1908.04514","kind":"arxiv","version":1}},"canonical_sha256":"d8cad74417d17d424189af00818ecd607fc248a10d5d6f37d468f1056b83455e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8cad74417d17d424189af00818ecd607fc248a10d5d6f37d468f1056b83455e","first_computed_at":"2026-07-04T23:55:14.398330Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:55:14.398330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7KgOP8XgHe90n+TUi6SbZSxKmjpr6y/n4mtwZYqjCS+v6zn3LNpDbQhPhgmDS50MLZmNUunvyUtb9kt4mTGkDg==","signature_status":"signed_v1","signed_at":"2026-07-04T23:55:14.398780Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.04514","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:05b1e0fb2ebf160cba15d5db6e752703b3f74c7f932db1abb6e1ac4719caa1d7","sha256:55435cad2eaa5c2ae699a13f3e6eedb832ac6b3f6afcdd50c26fb991d070e188"],"state_sha256":"d6988ca3188aedfbbef124e28ee473e5020dff48a3cad673b7a22738d0e3ff18"}