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In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra $\\H(C_1(\\mathscr{P}))$ of $C_1(\\mathscr{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\\mathscr{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations in the generators in $\\H(C_1(\\mathscr{P}))$, and obtain quantum Serre relations"},"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":"1807.10892","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-07-28T05:17:28Z","cross_cats_sorted":["math.QA","math.RA"],"title_canon_sha256":"04bd4c65ddf616ab6b9ae422256e524f225361815b043dbf1902035b48b5940b","abstract_canon_sha256":"67aaff110b9ff71c63b958c2335917a35bb3ab0f820a27e28b73e8135bf47451"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:59.562789Z","signature_b64":"vUyCRQOzFGqGwPPt3++xi0tbQzjHuR4dktiH/jvL/hxScPhg4WWvQG58bxldzf2b/0aYQ5Xp9/Gt7bLE2op9DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b67955220f06bf5c25b68a54becd5fc953a6e9da33c3455e2d95559b2770ff05","last_reissued_at":"2026-05-18T00:08:59.562201Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:59.562201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal generators of Hall algebras of 1-cyclic perfect complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA","math.RA"],"primary_cat":"math.RT","authors_text":"Haicheng Zhang","submitted_at":"2018-07-28T05:17:28Z","abstract_excerpt":"Let $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\\mathscr{P})$ be the category of 1-cyclic complexes of projective $A$-modules. 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