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Let $U_\\A(\\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \\in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $\\Phi: U_\\A^-(\\g) \\to K_0(R)$ and that $\\Phi$ is an isomorphism if $a_{ii}\\ne 0$ for all $i\\in"},"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":"1102.5165","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-02-25T05:56:44Z","cross_cats_sorted":[],"title_canon_sha256":"ed7c9549a766af1e246711c7231a103922b4cfdd9526cb99fbd9c5b8af2a8581","abstract_canon_sha256":"53b0e4d992ae2ee48d1ad4e97ce12a5d9455c9fed6793cbedacd4c1cc9d9e58b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:34.346446Z","signature_b64":"VcifxALeqHYYW2nXWxhtLwr1s7e/vW7/y/xqBqYXMhKg2TSTaR5fB66zbmdtfVMdWqBsS5uZNUSxdV/MhHRADA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"509f7d151b737fbeb5cefdc5bc2e78208fee42356f30321ada56e68d66d33a56","last_reissued_at":"2026-05-18T03:48:34.346001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:34.346001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Euiyong Park, Se-Jin Oh, Seok-Jin Kang","submitted_at":"2011-02-25T05:56:44Z","abstract_excerpt":"We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^\\lambda$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\\A(\\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \\in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $\\Phi: U_\\A^-(\\g) \\to K_0(R)$ and that $\\Phi$ is an isomorphism if $a_{ii}\\ne 0$ for all $i\\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5165","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":"1102.5165","created_at":"2026-05-18T03:48:34.346063+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.5165v3","created_at":"2026-05-18T03:48:34.346063+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.5165","created_at":"2026-05-18T03:48:34.346063+00:00"},{"alias_kind":"pith_short_12","alias_value":"KCPX2FI3ON73","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"KCPX2FI3ON735NOO","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"KCPX2FI3","created_at":"2026-05-18T12:26:32.869790+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/KCPX2FI3ON735NOO7XC3YLTYEC","json":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC.json","graph_json":"https://pith.science/api/pith-number/KCPX2FI3ON735NOO7XC3YLTYEC/graph.json","events_json":"https://pith.science/api/pith-number/KCPX2FI3ON735NOO7XC3YLTYEC/events.json","paper":"https://pith.science/paper/KCPX2FI3"},"agent_actions":{"view_html":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC","download_json":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC.json","view_paper":"https://pith.science/paper/KCPX2FI3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.5165&json=true","fetch_graph":"https://pith.science/api/pith-number/KCPX2FI3ON735NOO7XC3YLTYEC/graph.json","fetch_events":"https://pith.science/api/pith-number/KCPX2FI3ON735NOO7XC3YLTYEC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC/action/storage_attestation","attest_author":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC/action/author_attestation","sign_citation":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC/action/citation_signature","submit_replication":"https://pith.science/pith/KCPX2FI3ON735NOO7XC3YLTYEC/action/replication_record"}},"created_at":"2026-05-18T03:48:34.346063+00:00","updated_at":"2026-05-18T03:48:34.346063+00:00"}