{"paper":{"title":"Double Bruhat cells and symplectic groupoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jiang-Hua Lu, Victor Mouquin","submitted_at":"2016-07-02T16:01:44Z","abstract_excerpt":"Let $G$ be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure $\\pi_{{\\rm st}}$ determined by a pair of opposite Borel subgroups $(B, B_-)$. We prove that for each $v$ in the Weyl group $W$ of $G$, the double Bruhat cell $G^{v,v} = BvB \\cap B_-vB_-$ in $G$, together with the Poisson structure $\\pi_{{\\rm st}}$, is naturally a Poisson groupoid over the Bruhat cell $BvB/B$ in the flag variety $G/B$. Correspondingly, every symplectic leaf of $\\pi_{{\\rm st}}$ in $G^{v,v}$ is a symplectic groupoid over $BvB/B$. For $u, v \\in W$, we show that the double"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00527","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"}