{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:3V5KGPHBUGNIGTAHACKPEAHRQ4","short_pith_number":"pith:3V5KGPHB","canonical_record":{"source":{"id":"1408.0143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2014-08-01T12:03:45Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"43dda1c36e1cd915308f1473bb8836358337ab4a95dc0038c04cfc0014ae518d","abstract_canon_sha256":"ca7d63b097b1f06c914dbc66a277bd36220c6075bb490d455d93d275067ab1c3"},"schema_version":"1.0"},"canonical_sha256":"dd7aa33ce1a19a834c070094f200f1873cb88b7d8890696a2f0c7bdd3b306752","source":{"kind":"arxiv","id":"1408.0143","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.0143","created_at":"2026-05-18T02:46:03Z"},{"alias_kind":"arxiv_version","alias_value":"1408.0143v1","created_at":"2026-05-18T02:46:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.0143","created_at":"2026-05-18T02:46:03Z"},{"alias_kind":"pith_short_12","alias_value":"3V5KGPHBUGNI","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3V5KGPHBUGNIGTAH","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3V5KGPHB","created_at":"2026-05-18T12:28:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:3V5KGPHBUGNIGTAHACKPEAHRQ4","target":"record","payload":{"canonical_record":{"source":{"id":"1408.0143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2014-08-01T12:03:45Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"43dda1c36e1cd915308f1473bb8836358337ab4a95dc0038c04cfc0014ae518d","abstract_canon_sha256":"ca7d63b097b1f06c914dbc66a277bd36220c6075bb490d455d93d275067ab1c3"},"schema_version":"1.0"},"canonical_sha256":"dd7aa33ce1a19a834c070094f200f1873cb88b7d8890696a2f0c7bdd3b306752","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:03.762284Z","signature_b64":"dDXjxJhG+rmGsDf3O1lv8USXrL7AsO3hpJe+wakGjf/H1jBcn4RWTv1jBT1hxWIl6W1zRr/QKpzCKQMpZE0dDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd7aa33ce1a19a834c070094f200f1873cb88b7d8890696a2f0c7bdd3b306752","last_reissued_at":"2026-05-18T02:46:03.761821Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:03.761821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.0143","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:46:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h1vORy3eJZBisUyZeYSm65/wjmfKlC3P+ODtjaq6t5OL2+F2ZxrPpCuc1D/n4S2lpfGoNGEtI+rANPSlrn7uBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T09:47:57.520784Z"},"content_sha256":"6587c90ecf68c3630ba3b0637a6b8493be6dc924d5481149a02719a258ce5eb0","schema_version":"1.0","event_id":"sha256:6587c90ecf68c3630ba3b0637a6b8493be6dc924d5481149a02719a258ce5eb0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:3V5KGPHBUGNIGTAHACKPEAHRQ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Billiard Arrays and finite-dimensional irreducible $U_q(\\mathfrak{sl}_2)$-modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"Paul Terwilliger","submitted_at":"2014-08-01T12:03:45Z","abstract_excerpt":"We introduce the notion of a Billiard Array. This is an equilateral triangular array of one-dimensional subspaces of a vector space $V$, subject to several conditions that specify which sums are direct. We show that the Billiard Arrays on $V$ are in bijection with the 3-tuples of totally opposite flags on $V$. We classify the Billiard Arrays up to isomorphism. We use Billiard Arrays to describe the finite-dimensional irreducible modules for the quantum algebra $U_q(\\mathfrak{sl}_2)$ and the Lie algebra $\\mathfrak{sl}_2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0143","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:46:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ylXo/28Wpb7yCN905jTV6mlNEmKRVSv3TflorEslB5LzV7UeGX5DOkn1yRT5aN7eAZR92vdHVvFGPIX0/GW3Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T09:47:57.521441Z"},"content_sha256":"399ae5387bdc6e269894d9c6cc47f658ec1d524e5efce8ef641b0f2c24770ed2","schema_version":"1.0","event_id":"sha256:399ae5387bdc6e269894d9c6cc47f658ec1d524e5efce8ef641b0f2c24770ed2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3V5KGPHBUGNIGTAHACKPEAHRQ4/bundle.json","state_url":"https://pith.science/pith/3V5KGPHBUGNIGTAHACKPEAHRQ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3V5KGPHBUGNIGTAHACKPEAHRQ4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T09:47:57Z","links":{"resolver":"https://pith.science/pith/3V5KGPHBUGNIGTAHACKPEAHRQ4","bundle":"https://pith.science/pith/3V5KGPHBUGNIGTAHACKPEAHRQ4/bundle.json","state":"https://pith.science/pith/3V5KGPHBUGNIGTAHACKPEAHRQ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3V5KGPHBUGNIGTAHACKPEAHRQ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:3V5KGPHBUGNIGTAHACKPEAHRQ4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ca7d63b097b1f06c914dbc66a277bd36220c6075bb490d455d93d275067ab1c3","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2014-08-01T12:03:45Z","title_canon_sha256":"43dda1c36e1cd915308f1473bb8836358337ab4a95dc0038c04cfc0014ae518d"},"schema_version":"1.0","source":{"id":"1408.0143","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.0143","created_at":"2026-05-18T02:46:03Z"},{"alias_kind":"arxiv_version","alias_value":"1408.0143v1","created_at":"2026-05-18T02:46:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.0143","created_at":"2026-05-18T02:46:03Z"},{"alias_kind":"pith_short_12","alias_value":"3V5KGPHBUGNI","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3V5KGPHBUGNIGTAH","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3V5KGPHB","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:399ae5387bdc6e269894d9c6cc47f658ec1d524e5efce8ef641b0f2c24770ed2","target":"graph","created_at":"2026-05-18T02:46:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce the notion of a Billiard Array. This is an equilateral triangular array of one-dimensional subspaces of a vector space $V$, subject to several conditions that specify which sums are direct. We show that the Billiard Arrays on $V$ are in bijection with the 3-tuples of totally opposite flags on $V$. We classify the Billiard Arrays up to isomorphism. We use Billiard Arrays to describe the finite-dimensional irreducible modules for the quantum algebra $U_q(\\mathfrak{sl}_2)$ and the Lie algebra $\\mathfrak{sl}_2$.","authors_text":"Paul Terwilliger","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2014-08-01T12:03:45Z","title":"Billiard Arrays and finite-dimensional irreducible $U_q(\\mathfrak{sl}_2)$-modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.0143","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6587c90ecf68c3630ba3b0637a6b8493be6dc924d5481149a02719a258ce5eb0","target":"record","created_at":"2026-05-18T02:46:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ca7d63b097b1f06c914dbc66a277bd36220c6075bb490d455d93d275067ab1c3","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2014-08-01T12:03:45Z","title_canon_sha256":"43dda1c36e1cd915308f1473bb8836358337ab4a95dc0038c04cfc0014ae518d"},"schema_version":"1.0","source":{"id":"1408.0143","kind":"arxiv","version":1}},"canonical_sha256":"dd7aa33ce1a19a834c070094f200f1873cb88b7d8890696a2f0c7bdd3b306752","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd7aa33ce1a19a834c070094f200f1873cb88b7d8890696a2f0c7bdd3b306752","first_computed_at":"2026-05-18T02:46:03.761821Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:46:03.761821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dDXjxJhG+rmGsDf3O1lv8USXrL7AsO3hpJe+wakGjf/H1jBcn4RWTv1jBT1hxWIl6W1zRr/QKpzCKQMpZE0dDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:46:03.762284Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.0143","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6587c90ecf68c3630ba3b0637a6b8493be6dc924d5481149a02719a258ce5eb0","sha256:399ae5387bdc6e269894d9c6cc47f658ec1d524e5efce8ef641b0f2c24770ed2"],"state_sha256":"906dafbaf682b49f929096e3eeeb8adc43c7cf5febe5c6fd6bfa34301307d06f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1eVPfc/H0FXviqeHukhvclmutlrRWt2Vk/lxR3bSSyEX8ammlyfPro1uINJ6WF1L8j60KZJXRtpbfiVImnV6Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T09:47:57.524545Z","bundle_sha256":"0ebf37508059cd316d2af7e7843985eaa34f90fc132835b09649175d033fef0a"}}