Algebras and Representation Theory

, Volume 14, Issue 1, pp 97-112

First online:

Cluster-Cyclic Quivers with Three Vertices and the Markov Equation

With an appendix by Otto Kerner
  • Andre BeinekeAffiliated withFakultät für Mathematik, Universität Bielefeld Email author 
  • , Thomas BrüstleAffiliated withDépartement de Mathématiques, Université de SherbrookeDepartment of Mathematics, Bishop’s University
  • , Lutz HilleAffiliated withMathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster

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Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver Q it is thus desirable to decide if the cluster algebra defined by Q is acyclic. We call Q cluster-acyclic in this case, otherwise cluster-cyclic. In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov.


Cluster algebras Mutations Braid group

Mathematics Subject Classification (2000)