{"paper":{"title":"Two descriptions of the quantum affine algebra $U_v(\\hat{\\mathfrak{sl}}_2)$ via Hall algebra approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.RT","authors_text":"Igor Burban, Olivier Schiffmann","submitted_at":"2009-03-27T15:33:04Z","abstract_excerpt":"We compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver $\\overr{Q}$ and of the category of coherent sheaves on ${\\mathbb P}^1$. Using this approach, we show that the Drinfeld--Beck isomorphism for the quantized enveloping algebra $U_v(\\hat{\\mathfrak{sl}}_2)$ is a corollary of an equivalence between the derived categories $D^b\\bigl(\\Rep(\\overr{Q})\\bigr)$ and $D^b\\bigl(\\Coh({\\mathbb P}^1)\\bigr)$. This technique also allows to reprove several technical results on the integral form of $U_v(\\hat{\\mathfrak{sl}}_2)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4828","kind":"arxiv","version":4},"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"}