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There is another Poisson algebra $\\{\\cdot,\\cdot\\}_{S2}$ on $\\mathcal{C}_{2,N}$ induced from the first Adler-Gelfand-Dickey Poissson algebra by Miura transformation. By observing that these two Poisson algebras are asymptotically related to the dual to the Virasoro algebra, finally, we prove that $\\{\\cdot,\\cdot\\}_{\\mathcal{C}_{2"},"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":"1412.4330","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-12-14T09:38:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"40f5225f5b2c87eff3f8f1bc155c31fd1b3abe7e33ec29bbccef2e98428d65f9","abstract_canon_sha256":"73238de816118f80ee1c7d7f1d5b86fab7e7609becf8fbca163bf4f2155faee8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:05.945922Z","signature_b64":"nb2PmhDncIwRNfIpExyw2FzhHdVMsYap+LtUDvjgsWsCqfTBVNuRsQxrsU0ZW1ekpTX3CBwLoo9qyjEBlXToDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2696bab74fbed01f22f52cacca76a92307fb15bf75fb47dc8bf71c39b0802f2c","last_reissued_at":"2026-05-18T01:04:05.945418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:05.945418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Swapping algebra, Virasoro algebra and discrete integrable system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DG","authors_text":"Zhe Sun","submitted_at":"2014-12-14T09:38:49Z","abstract_excerpt":"We induce a Poisson algebra $\\{\\cdot,\\cdot\\}_{\\mathcal{C}_{n,N}}$ on the configuration space $\\mathcal{C}_{n,N}$ of $N$ twisted polygons in $\\mathbb{RP}^{n-1}$ from the swapping algebra \\cite{L12}, which is found coincide with Faddeev-Takhtajan-Volkov algebra for $n=2$. 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By observing that these two Poisson algebras are asymptotically related to the dual to the Virasoro algebra, finally, we prove that $\\{\\cdot,\\cdot\\}_{\\mathcal{C}_{2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4330","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":"1412.4330","created_at":"2026-05-18T01:04:05.945492+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.4330v3","created_at":"2026-05-18T01:04:05.945492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4330","created_at":"2026-05-18T01:04:05.945492+00:00"},{"alias_kind":"pith_short_12","alias_value":"E2LLVN2PX3IB","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"E2LLVN2PX3IB6IXV","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"E2LLVN2P","created_at":"2026-05-18T12:28:25.294606+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/E2LLVN2PX3IB6IXVFSWMU5VJEM","json":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM.json","graph_json":"https://pith.science/api/pith-number/E2LLVN2PX3IB6IXVFSWMU5VJEM/graph.json","events_json":"https://pith.science/api/pith-number/E2LLVN2PX3IB6IXVFSWMU5VJEM/events.json","paper":"https://pith.science/paper/E2LLVN2P"},"agent_actions":{"view_html":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM","download_json":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM.json","view_paper":"https://pith.science/paper/E2LLVN2P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.4330&json=true","fetch_graph":"https://pith.science/api/pith-number/E2LLVN2PX3IB6IXVFSWMU5VJEM/graph.json","fetch_events":"https://pith.science/api/pith-number/E2LLVN2PX3IB6IXVFSWMU5VJEM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM/action/storage_attestation","attest_author":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM/action/author_attestation","sign_citation":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM/action/citation_signature","submit_replication":"https://pith.science/pith/E2LLVN2PX3IB6IXVFSWMU5VJEM/action/replication_record"}},"created_at":"2026-05-18T01:04:05.945492+00:00","updated_at":"2026-05-18T01:04:05.945492+00:00"}