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Nikulin","submitted_at":"1995-03-10T12:35:58Z","abstract_excerpt":"This is a continuation of our \"Lecture on Kac--Moody Lie algebras of the arithmetic type\" \\cite{25}.\n  We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\\times M \\to {\\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\\subset W(S)$, fundamental polyhedron $\\Cal M$ of $W$ and an acceptable (corresponding to twisting coefficients) set $P({\\Cal M})\\subset M$ of vectors orthogonal to faces of $\\Cal M$ (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra ${\\goth g}={\\goth g}^{\\prime\\prime}(A(S,W,P({\\Cal M})))$ which is graded b"},"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":"alg-geom/9503003","kind":"arxiv","version":6},"metadata":{"license":"","primary_cat":"alg-geom","submitted_at":"1995-03-10T12:35:58Z","cross_cats_sorted":["hep-th","math.AG","math.QA","q-alg"],"title_canon_sha256":"b7222da0108e16d95496c8e5c6b6b75de1a95d5cffde17501e4ef3c72fd6caff","abstract_canon_sha256":"881bc1f0ef62b5d550e9be701a58da17b752377379467e4b0eb1bf98d002cfc2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:40:48.563866Z","signature_b64":"FvHlUDFbKIaD77Q7PTtxgIes132UENHupBdbjlRYgJca+0e5fFiZbmkPH6+42iNS+8plPQOMLxbcO/72X6OAAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2428bb6a8d5c351d8e78ea4efc667d8c70c5f3507fbd0e8954b649204542f128","last_reissued_at":"2026-05-18T01:40:48.563159Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:40:48.563159Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras","license":"","headline":"","cross_cats":["hep-th","math.AG","math.QA","q-alg"],"primary_cat":"alg-geom","authors_text":"Viacheslav V. 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One can construct the corresponding Lorentzian Kac--Moody Lie algebra ${\\goth g}={\\goth g}^{\\prime\\prime}(A(S,W,P({\\Cal M})))$ which is graded b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"alg-geom/9503003","kind":"arxiv","version":6},"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":"alg-geom/9503003","created_at":"2026-05-18T01:40:48.563290+00:00"},{"alias_kind":"arxiv_version","alias_value":"alg-geom/9503003v6","created_at":"2026-05-18T01:40:48.563290+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.alg-geom/9503003","created_at":"2026-05-18T01:40:48.563290+00:00"},{"alias_kind":"pith_short_12","alias_value":"EQULW2UNLQ2R","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_16","alias_value":"EQULW2UNLQ2R3DTY","created_at":"2026-05-18T12:25:47.700082+00:00"},{"alias_kind":"pith_short_8","alias_value":"EQULW2UN","created_at":"2026-05-18T12:25:47.700082+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/EQULW2UNLQ2R3DTY5JHPYZT5RR","json":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR.json","graph_json":"https://pith.science/api/pith-number/EQULW2UNLQ2R3DTY5JHPYZT5RR/graph.json","events_json":"https://pith.science/api/pith-number/EQULW2UNLQ2R3DTY5JHPYZT5RR/events.json","paper":"https://pith.science/paper/EQULW2UN"},"agent_actions":{"view_html":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR","download_json":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR.json","view_paper":"https://pith.science/paper/EQULW2UN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=alg-geom/9503003&json=true","fetch_graph":"https://pith.science/api/pith-number/EQULW2UNLQ2R3DTY5JHPYZT5RR/graph.json","fetch_events":"https://pith.science/api/pith-number/EQULW2UNLQ2R3DTY5JHPYZT5RR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR/action/storage_attestation","attest_author":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR/action/author_attestation","sign_citation":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR/action/citation_signature","submit_replication":"https://pith.science/pith/EQULW2UNLQ2R3DTY5JHPYZT5RR/action/replication_record"}},"created_at":"2026-05-18T01:40:48.563290+00:00","updated_at":"2026-05-18T01:40:48.563290+00:00"}