{"paper":{"title":"Cluster Nature of Quantum Groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.QA"],"primary_cat":"math.RT","authors_text":"Linhui Shen","submitted_at":"2022-09-13T18:35:29Z","abstract_excerpt":"We present a rigid cluster model to realize the quantum group ${\\bf U}_q(\\mathfrak{g})$ for $\\mathfrak{g}$ of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group ${\\bf U}_q(\\mathfrak{g})$ to a quotient algebra of the Weyl group invariants of the Fock-Goncharov quantum cluster algebra $\\mathcal{O}_q(\\mathscr{P}_{{\\rm G},\\odot})$. By applying the quantum duality of cluster algebras, we show that ${\\bf U}_q(\\mathfrak{g})$ admits a natural basis $\\bar{\\bf \\Theta}$ whose structural coefficients are in $\\mathbb{N}[q^{\\frac{1}{2}}, q^{-\\frac{1}{2}}]$. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.06258","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2209.06258/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}