Algebras and Representation Theory

, Volume 14, Issue 1, pp 97–112 | Cite as

Cluster-Cyclic Quivers with Three Vertices and the Markov Equation

With an appendix by Otto Kerner


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)



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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Département de MathématiquesUniversité de SherbrookeSherbrookeCanada
  3. 3.Department of MathematicsBishop’s UniversitySherbrookeCanada
  4. 4.Mathematisches InstitutFachbereich Mathematik und Informatik der Universität MünsterMünsterGermany

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