dc.contributor.author Berenstein, Arkady dc.contributor.author Retakh, Vladimir dc.date.accessioned 2015-11-18T12:00:01Z dc.date.accessioned 2016-10-05T14:14:03Z dc.date.available 2015-11-18T12:00:01Z dc.date.available 2016-10-05T14:14:03Z dc.date.issued 2015-11-18 dc.identifier.uri http://publications.mfo.de/handle/mfo/1104 dc.description Research in Pairs 2014 en_US dc.description.abstract 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-Plücker 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 a 1-relator group easily computable in terms of any triangulation of $\Sigma$. Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems. en_US dc.language.iso en en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2015,16 dc.title Noncommutative Marked Surfaces en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2015-16 local.scientificprogram Research in Pairs 2014 local.series.id OWP-2015-16
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