Cluster-Cyclic Quivers with three Vertices and the Markov Equation

Details Cluster-Cyclic Quivers with three Vertices and the Markov Equation Cluster-Cyclic Quivers with three Vertices and the Markov Equation

2006-12-08

arxiv.org/abs/math/0612213v2

Mathematics

Rings and Algebras

11 pages, with an appendix by Otto Kerner

Scientific paper

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.

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