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Denote by $V(\\lambda)$ the extremal weight module of extremal weight $\\lambda$ with $v_\\lambda$ the extremal weight vector, and by $\\mathcal{B}(\\lambda)$ the crystal basis of $V(\\lambda)$ with $u_\\lambda$ the element corresponding to $v_\\lambda$. We prove that (i) $\\mathcal{B}(\\lambda)$ is connected, (ii) the subset $\\mathcal{B}(\\lambda)_{\\mu}$ of elements of weight $\\mu$ in $\\mathcal{B}(\\lambda)$ is a finite set for every "},"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":"1712.01009","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2017-12-04T11:15:29Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"8c146f63611e7e26e7f77e7157c8bf985d4822bdd53689b44eaccd425d8502a6","abstract_canon_sha256":"7fdce4eddf2255ae507f7d61ee0dd5b3ddc5cbe8a5bc0fa975746785e686b4fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:27.138231Z","signature_b64":"lnB2AA4ro4QJe+Z6oFmymQxwRjWC3zeoa5t8OMuyAtJNDf6yE9xzW1qraNAYBmVar2rai9iVxnSHA2Z0p2YKCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1dc699a78d120a702e7573fe8f0cbf3cf69c1a6d95a618bc8029c5a12cd29fdc","last_reissued_at":"2026-05-18T00:08:27.137801Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:27.137801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"Daisuke Sagaki, Dongxiao Yu","submitted_at":"2017-12-04T11:15:29Z","abstract_excerpt":"Let $\\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $\\lambda=\\Lambda_{1} - \\Lambda_{2}$, where $\\Lambda_{1}$, $\\Lambda_{2}$ are the fundamental weights. Denote by $V(\\lambda)$ the extremal weight module of extremal weight $\\lambda$ with $v_\\lambda$ the extremal weight vector, and by $\\mathcal{B}(\\lambda)$ the crystal basis of $V(\\lambda)$ with $u_\\lambda$ the element corresponding to $v_\\lambda$. We prove that (i) $\\mathcal{B}(\\lambda)$ is connected, (ii) the subset $\\mathcal{B}(\\lambda)_{\\mu}$ of elements of weight $\\mu$ in $\\mathcal{B}(\\lambda)$ is a finite set for every "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.01009","kind":"arxiv","version":2},"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":"1712.01009","created_at":"2026-05-18T00:08:27.137867+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.01009v2","created_at":"2026-05-18T00:08:27.137867+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.01009","created_at":"2026-05-18T00:08:27.137867+00:00"},{"alias_kind":"pith_short_12","alias_value":"DXDJTJ4NCIFH","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"DXDJTJ4NCIFHALTV","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"DXDJTJ4N","created_at":"2026-05-18T12:31:12.930513+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/DXDJTJ4NCIFHALTVOP7I6DF7HT","json":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT.json","graph_json":"https://pith.science/api/pith-number/DXDJTJ4NCIFHALTVOP7I6DF7HT/graph.json","events_json":"https://pith.science/api/pith-number/DXDJTJ4NCIFHALTVOP7I6DF7HT/events.json","paper":"https://pith.science/paper/DXDJTJ4N"},"agent_actions":{"view_html":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT","download_json":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT.json","view_paper":"https://pith.science/paper/DXDJTJ4N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.01009&json=true","fetch_graph":"https://pith.science/api/pith-number/DXDJTJ4NCIFHALTVOP7I6DF7HT/graph.json","fetch_events":"https://pith.science/api/pith-number/DXDJTJ4NCIFHALTVOP7I6DF7HT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT/action/storage_attestation","attest_author":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT/action/author_attestation","sign_citation":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT/action/citation_signature","submit_replication":"https://pith.science/pith/DXDJTJ4NCIFHALTVOP7I6DF7HT/action/replication_record"}},"created_at":"2026-05-18T00:08:27.137867+00:00","updated_at":"2026-05-18T00:08:27.137867+00:00"}