Cluster-Cyclic Quivers with Three Vertices and the Markov Equation
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- Beineke, A., Brüstle, T. & Hille, L. Algebr Represent Theor (2011) 14: 97. doi:10.1007/s10468-009-9179-9
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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.