Algebras and Representation Theory

, Volume 14, Issue 1, pp 97–112

Cluster-Cyclic Quivers with Three Vertices and the Markov Equation

With an appendix by Otto Kerner


    • Fakultät für MathematikUniversität Bielefeld
  • Thomas Brüstle
    • Département de MathématiquesUniversité de Sherbrooke
    • Department of MathematicsBishop’s University
  • Lutz Hille
    • Mathematisches InstitutFachbereich Mathematik und Informatik der Universität Münster

DOI: 10.1007/s10468-009-9179-9

Cite this article as:
Beineke, A., Brüstle, T. & Hille, L. Algebr Represent Theor (2011) 14: 97. doi:10.1007/s10468-009-9179-9


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 algebrasMutationsBraid group

Mathematics Subject Classification (2000)


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© Springer Science+Business Media B.V. 2009