{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:V6QZODRV5LW2XITDQQXZDP2PWG","short_pith_number":"pith:V6QZODRV","schema_version":"1.0","canonical_sha256":"afa1970e35eaedaba263842f91bf4fb1b7687a6a6d743bc7c87b0ac5420094e6","source":{"kind":"arxiv","id":"1304.3611","version":3},"attestation_state":"computed","paper":{"title":"Green Rings of Finite Dimensional Pointed Rank One Hopf algebras of Nilpotent Type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Libin Li, Yinhuo Zhang, Zhihua Wang","submitted_at":"2013-04-12T12:03:17Z","abstract_excerpt":"Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and "},"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":"1304.3611","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-12T12:03:17Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"588804471f35ac9b6d9ff59777a4b84ec333967c2c267ab45e7e8cf3ab648aee","abstract_canon_sha256":"18ad03a529e5f432c496be14b4c946c499f872ebfc2a2b172b76268e912bba4c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:55.954352Z","signature_b64":"I9E0G4GtivL/r23FkVLuBaq2NkfamA7M+BTddeBteNgac5yycEJ7jP7b2meyvRC0zXh9mIO8dmFpu0ZuYoUyAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afa1970e35eaedaba263842f91bf4fb1b7687a6a6d743bc7c87b0ac5420094e6","last_reissued_at":"2026-05-18T03:13:55.953607Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:55.953607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Green Rings of Finite Dimensional Pointed Rank One Hopf algebras of Nilpotent Type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Libin Li, Yinhuo Zhang, Zhihua Wang","submitted_at":"2013-04-12T12:03:17Z","abstract_excerpt":"Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.3611","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":"1304.3611","created_at":"2026-05-18T03:13:55.953718+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.3611v3","created_at":"2026-05-18T03:13:55.953718+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.3611","created_at":"2026-05-18T03:13:55.953718+00:00"},{"alias_kind":"pith_short_12","alias_value":"V6QZODRV5LW2","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"V6QZODRV5LW2XITD","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"V6QZODRV","created_at":"2026-05-18T12:28:04.890932+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/V6QZODRV5LW2XITDQQXZDP2PWG","json":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG.json","graph_json":"https://pith.science/api/pith-number/V6QZODRV5LW2XITDQQXZDP2PWG/graph.json","events_json":"https://pith.science/api/pith-number/V6QZODRV5LW2XITDQQXZDP2PWG/events.json","paper":"https://pith.science/paper/V6QZODRV"},"agent_actions":{"view_html":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG","download_json":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG.json","view_paper":"https://pith.science/paper/V6QZODRV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.3611&json=true","fetch_graph":"https://pith.science/api/pith-number/V6QZODRV5LW2XITDQQXZDP2PWG/graph.json","fetch_events":"https://pith.science/api/pith-number/V6QZODRV5LW2XITDQQXZDP2PWG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG/action/storage_attestation","attest_author":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG/action/author_attestation","sign_citation":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG/action/citation_signature","submit_replication":"https://pith.science/pith/V6QZODRV5LW2XITDQQXZDP2PWG/action/replication_record"}},"created_at":"2026-05-18T03:13:55.953718+00:00","updated_at":"2026-05-18T03:13:55.953718+00:00"}