Cluster-Cyclic Quivers with three Vertices and the Markov Equation

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.

11 pages, with an appendix by Otto Kerner

References in corpus (1)

• Mutation classes of skew-symmetric 3x3-matrices

Cited by in corpus (1)

• An approach to non simply laced cluster algebras