{"paper":{"title":"Noncommutative marked surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA","math.RT"],"primary_cat":"math.QA","authors_text":"Arkady Berenstein, Vladimir Retakh","submitted_at":"2015-10-09T10:56:20Z","abstract_excerpt":"The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface $\\Sigma$. This is a noncommutative algebra ${\\mathcal A}_\\Sigma$ generated by \"noncommutative geodesics\" between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Pl\\\"ucker relations. It turns out that the algebra ${\\mathcal A}_\\Sigma$ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of $\\Sigma$, which confirms its \"cluster nature\". As a surprising byproduct, we obtain a new topological invariant of $\\Sigma$, which is a free or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02628","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"}