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When $R=\\Z$, we prove that this map extends to a group homomorphism $\\rho_{\\lambda,\\Z}: G(\\Z) \\to G^{\\lambda}(\\Z).$ We prove that the kernel $K^{\\lambda}$ of the map $\\rho_{\\lam,\\Z}: G(\\Z)\\to G^{\\lam}(\\Z)$ lies in $H(\\C)$ and if the group homomorphism $\\v"},"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":"1512.04623","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-12-15T01:59:17Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"3cdee8084555326bc7a3440973e158bc67fa95785e20ecf28539881d73e664dc","abstract_canon_sha256":"54a145e55f17d30edb998868f15225a821979babce37dfe13f32ac5ca7f5bba7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:11.056566Z","signature_b64":"7h/RMfs8K9Iq2NOxFc7lfLX/HMBeBCd5fyzhJkY1p6rU9AzeV7qCCznFu6Solbh8va+Z/1L941kkjhcFIt6JBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8cca0c8788f4059307ad9cea5360ea57d092e4f70b8d1d4aaae665fb8f2bdc6c","last_reissued_at":"2026-05-18T01:21:11.056012Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:11.056012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness of representation--theoretic hyperbolic Kac--Moody groups over $\\Z$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Frank Wagner, Lisa Carbone","submitted_at":"2015-12-15T01:59:17Z","abstract_excerpt":"For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and Carbone to a representation--theoretic construction $G^{\\lambda}(R)$ corresponding to an integrable representation $V^{\\lambda}$ with dominant integral weight $\\lambda$. 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