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
Article

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

Abstract

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.

Keywords

Cluster algebrasMutationsBraid group

Mathematics Subject Classification (2000)

16G20

Copyright information

© 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