{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:FRA3DJQYZUVUP57YSEGM2DOX33","short_pith_number":"pith:FRA3DJQY","canonical_record":{"source":{"id":"1510.02645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-09T12:13:00Z","cross_cats_sorted":[],"title_canon_sha256":"8402fd77c8f86baeea2f853e1a76d3ff2f56ea86da417d19ba4ee686ca30334f","abstract_canon_sha256":"5c219b8a960828d45f17aff3d74ce90047d01742581465231611e9ec5a2a5e02"},"schema_version":"1.0"},"canonical_sha256":"2c41b1a618cd2b47f7f8910ccd0dd7dee8af4ae6b4ba4fee199aa33b47b96c37","source":{"kind":"arxiv","id":"1510.02645","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.02645","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"arxiv_version","alias_value":"1510.02645v1","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02645","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"pith_short_12","alias_value":"FRA3DJQYZUVU","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"FRA3DJQYZUVUP57Y","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"FRA3DJQY","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:FRA3DJQYZUVUP57YSEGM2DOX33","target":"record","payload":{"canonical_record":{"source":{"id":"1510.02645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-09T12:13:00Z","cross_cats_sorted":[],"title_canon_sha256":"8402fd77c8f86baeea2f853e1a76d3ff2f56ea86da417d19ba4ee686ca30334f","abstract_canon_sha256":"5c219b8a960828d45f17aff3d74ce90047d01742581465231611e9ec5a2a5e02"},"schema_version":"1.0"},"canonical_sha256":"2c41b1a618cd2b47f7f8910ccd0dd7dee8af4ae6b4ba4fee199aa33b47b96c37","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:41.841156Z","signature_b64":"RrKiPxmBhfLz9GcARg6A1nn91cKMDA1HKiODajR2MJzWEHTuVpmvuDD+7Vd+nOfiNdG8a6ePX6nBZNCtnUZwBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c41b1a618cd2b47f7f8910ccd0dd7dee8af4ae6b4ba4fee199aa33b47b96c37","last_reissued_at":"2026-05-18T01:30:41.840479Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:41.840479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.02645","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":"2026-05-18T01:30:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X+3mafoR6KfqsC7a1mseGfioOLLcmYNi7sw6GGEa1Dp+0Zh9BZfRTjotEAR0/uQSRQsxz5qDvTQGhfn+shVwAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T18:25:36.390249Z"},"content_sha256":"d1e1e271e520e401c21705d0b6c766c50ac38cc4ba12309580b619afe4433c5b","schema_version":"1.0","event_id":"sha256:d1e1e271e520e401c21705d0b6c766c50ac38cc4ba12309580b619afe4433c5b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:FRA3DJQYZUVUP57YSEGM2DOX33","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On semisimple Hopf algebras of dimension $2^{m}$, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Yevgenia Kashina","submitted_at":"2015-10-09T12:13:00Z","abstract_excerpt":"In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\\geq 2$ over an algebraically closed field of characteristic $0$ with the group of group-like elements isomorphic to $\\mathbb{Z}_{2^{n}}\\times \\mathbb{Z}_{2^{n}}$. Moreover we classify all such nonisomorphic Hopf algebras of dimension $32$ and show that they are not twist-equivalent to each other. More generally, given an abelian group of order $2^{m-1}$ we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension $2^{m}$ which have this group as t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02645","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":"2026-05-18T01:30:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Q13TZZb4v+E/lGRAz+9BP2pIb334DznOdg3lNbYeh3/vgEhGoL6tFAPz9mD1zPn0mORQ0MEhH5PhDFg4OOfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T18:25:36.390614Z"},"content_sha256":"12bf34787b578e0983d4edc0469536bf84e8780d3e013dda54a1bba77a77a7b2","schema_version":"1.0","event_id":"sha256:12bf34787b578e0983d4edc0469536bf84e8780d3e013dda54a1bba77a77a7b2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FRA3DJQYZUVUP57YSEGM2DOX33/bundle.json","state_url":"https://pith.science/pith/FRA3DJQYZUVUP57YSEGM2DOX33/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FRA3DJQYZUVUP57YSEGM2DOX33/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":"2026-06-04T18:25:36Z","links":{"resolver":"https://pith.science/pith/FRA3DJQYZUVUP57YSEGM2DOX33","bundle":"https://pith.science/pith/FRA3DJQYZUVUP57YSEGM2DOX33/bundle.json","state":"https://pith.science/pith/FRA3DJQYZUVUP57YSEGM2DOX33/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FRA3DJQYZUVUP57YSEGM2DOX33/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FRA3DJQYZUVUP57YSEGM2DOX33","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":"5c219b8a960828d45f17aff3d74ce90047d01742581465231611e9ec5a2a5e02","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-09T12:13:00Z","title_canon_sha256":"8402fd77c8f86baeea2f853e1a76d3ff2f56ea86da417d19ba4ee686ca30334f"},"schema_version":"1.0","source":{"id":"1510.02645","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.02645","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"arxiv_version","alias_value":"1510.02645v1","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02645","created_at":"2026-05-18T01:30:41Z"},{"alias_kind":"pith_short_12","alias_value":"FRA3DJQYZUVU","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"FRA3DJQYZUVUP57Y","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"FRA3DJQY","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:12bf34787b578e0983d4edc0469536bf84e8780d3e013dda54a1bba77a77a7b2","target":"graph","created_at":"2026-05-18T01:30:41Z","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":"In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\\geq 2$ over an algebraically closed field of characteristic $0$ with the group of group-like elements isomorphic to $\\mathbb{Z}_{2^{n}}\\times \\mathbb{Z}_{2^{n}}$. Moreover we classify all such nonisomorphic Hopf algebras of dimension $32$ and show that they are not twist-equivalent to each other. More generally, given an abelian group of order $2^{m-1}$ we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension $2^{m}$ which have this group as t","authors_text":"Yevgenia Kashina","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-09T12:13:00Z","title":"On semisimple Hopf algebras of dimension $2^{m}$, II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02645","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:d1e1e271e520e401c21705d0b6c766c50ac38cc4ba12309580b619afe4433c5b","target":"record","created_at":"2026-05-18T01:30:41Z","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":"5c219b8a960828d45f17aff3d74ce90047d01742581465231611e9ec5a2a5e02","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-09T12:13:00Z","title_canon_sha256":"8402fd77c8f86baeea2f853e1a76d3ff2f56ea86da417d19ba4ee686ca30334f"},"schema_version":"1.0","source":{"id":"1510.02645","kind":"arxiv","version":1}},"canonical_sha256":"2c41b1a618cd2b47f7f8910ccd0dd7dee8af4ae6b4ba4fee199aa33b47b96c37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c41b1a618cd2b47f7f8910ccd0dd7dee8af4ae6b4ba4fee199aa33b47b96c37","first_computed_at":"2026-05-18T01:30:41.840479Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:41.840479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RrKiPxmBhfLz9GcARg6A1nn91cKMDA1HKiODajR2MJzWEHTuVpmvuDD+7Vd+nOfiNdG8a6ePX6nBZNCtnUZwBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:41.841156Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.02645","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d1e1e271e520e401c21705d0b6c766c50ac38cc4ba12309580b619afe4433c5b","sha256:12bf34787b578e0983d4edc0469536bf84e8780d3e013dda54a1bba77a77a7b2"],"state_sha256":"3a53b2505b72381ad1e80864fb70f8ea58b5510f9f63e42c2926ebe08ce59eed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HwoflbaZ+cmaZiOROoZ0lhyLlzni5k93fR21GRjcZqtaIrLHLILzFs9+0JwwC3LrPPI562jQfIKZibdz4FA/Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T18:25:36.392604Z","bundle_sha256":"8980a82513f92a5a6fb0fdb77cefc77f49c858d0212a95e850a1a64f70a9efe1"}}