{"paper":{"title":"Formulae of $\\imath$-divided powers in ${\\mathbf U}_q(\\mathfrak{sl}_2)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.QA"],"primary_cat":"math.RT","authors_text":"Collin Berman, Weiqiang Wang","submitted_at":"2017-03-02T03:33:11Z","abstract_excerpt":"The existence of the $\\imath$-canonical basis (also known as the $\\imath$-divided powers) for the coideal subalgebra of the quantum $\\mathfrak{sl}_2$ were established by Bao and Wang, with conjectural explicit formulae. In this paper we prove the conjectured formulae of these $\\imath$-divided powers. This is achieved by first establishing closed formulae of the $\\imath$-divided powers in basis for quantum $\\mathfrak{sl}_2$ and then formulae for the $\\imath$-canonical basis in terms of Lusztig's divided powers in each finite-dimensional simple module of quantum $\\mathfrak{sl}_2$. These formulae"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00602","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1703.00602/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"}