{"paper":{"title":"Using Catalan words and a $q$-shuffle algebra to describe a PBW basis for the positive part of $U_q({\\widehat {\\mathfrak{sl}}}_2)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Paul Terwilliger","submitted_at":"2018-06-28T23:15:46Z","abstract_excerpt":"The positive part $U^+_q$ of $U_q({\\widehat {\\mathfrak{sl}}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. In 1993 I. Damiani obtained a PBW basis for\n  $U^+_q$, consisting of some elements $\\lbrace E_{n\\delta+\\alpha_0}\\rbrace_{n=0}^\\infty$, $\\lbrace E_{n\\delta+\\alpha_1}\\rbrace_{n=0}^\\infty$, $\\lbrace E_{n\\delta}\\rbrace_{n=1}^\\infty$ that are defined recursively. Our goal is to describe these elements in closed form. We achieve this goal using Catalan words and a $q$-shuffle algebra."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11228","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"}