Geodesic
Factorial cluster algebras
Documenta mathematica, Tome 18 (2013), pp. 249-274
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We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.
DOI : 10.4171/dm/396
Classification : 13F15, 13F60, 17B37 
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     author = {Christof Gei{\ss} and Jan Schr\"oer and Bernard Leclerc},
     title = {Factorial cluster algebras},
     journal = {Documenta mathematica},
     pages = {249--274},
     year = {2013},
     volume = {18},
     doi = {10.4171/dm/396},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/396/}
}
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Christof Geiß; Jan Schröer; Bernard Leclerc. Factorial cluster algebras. Documenta mathematica, Tome 18 (2013), pp. 249-274. doi: 10.4171/dm/396

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