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Furthermore, using Dirichlet's prime number theorem we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if $d = {\\rm gcd}(m,\\nu(n))$ and $\\frac{\\nu(n)}{d} = p_1^{\\alpha_1} \\cdots p_r^{\\alpha_r}$ is the prime decomposition of $\\frac{\\nu(n)}{d}$ then the number of t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1611.05674","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.RA","submitted_at":"2016-11-17T13:30:53Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"e39fa6862d8bec7026054213ef141f04bbf44b587104e2394cc93ba010864a9e","abstract_canon_sha256":"f8b357e1c19f395f1839ffd375b0198fbe8669f1fba3f19b29e83c9ad9877671"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:17.605780Z","signature_b64":"uYkwGmDWPwBFiDJXaNz7L/KifbYdUR7p1Vzv/yyMHvKSdzyV/UHbua5xsjaKmVwfjfF6hNESwAjoXBdlxF3uCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62a37b2a14ed851c7e5a65fd223e7e0a7ddfc1a47d45a8d419bdf8f174a01f30","last_reissued_at":"2026-05-18T00:20:17.605135Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:17.605135Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hopf Algebras which Factorize through the Taft Algebra $T_{m^{2}}(q)$ and the Group Hopf Algebra $K[C_{n}]$","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.RA","authors_text":"Ana-Loredana Agore","submitted_at":"2016-11-17T13:30:53Z","abstract_excerpt":"We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra $T_{m^{2}}(q)$ and the group Hopf algebra $K[C_{n}]$: they are $nm^{2}$-dimensional quantum groups $T_{nm^{2}}^ {\\omega}(q)$ associated to an $n$-th root of unity $\\omega$. Furthermore, using Dirichlet's prime number theorem we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if $d = {\\rm gcd}(m,\\nu(n))$ and $\\frac{\\nu(n)}{d} = p_1^{\\alpha_1} \\cdots p_r^{\\alpha_r}$ is the prime decomposition of $\\frac{\\nu(n)}{d}$ then the number of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05674","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1611.05674","created_at":"2026-05-18T00:20:17.605219+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.05674v3","created_at":"2026-05-18T00:20:17.605219+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05674","created_at":"2026-05-18T00:20:17.605219+00:00"},{"alias_kind":"pith_short_12","alias_value":"MKRXWKQU5WCR","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MKRXWKQU5WCRY7S2","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MKRXWKQU","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ","json":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ.json","graph_json":"https://pith.science/api/pith-number/MKRXWKQU5WCRY7S2MX6SEPT6BJ/graph.json","events_json":"https://pith.science/api/pith-number/MKRXWKQU5WCRY7S2MX6SEPT6BJ/events.json","paper":"https://pith.science/paper/MKRXWKQU"},"agent_actions":{"view_html":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ","download_json":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ.json","view_paper":"https://pith.science/paper/MKRXWKQU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.05674&json=true","fetch_graph":"https://pith.science/api/pith-number/MKRXWKQU5WCRY7S2MX6SEPT6BJ/graph.json","fetch_events":"https://pith.science/api/pith-number/MKRXWKQU5WCRY7S2MX6SEPT6BJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ/action/storage_attestation","attest_author":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ/action/author_attestation","sign_citation":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ/action/citation_signature","submit_replication":"https://pith.science/pith/MKRXWKQU5WCRY7S2MX6SEPT6BJ/action/replication_record"}},"created_at":"2026-05-18T00:20:17.605219+00:00","updated_at":"2026-05-18T00:20:17.605219+00:00"}