ClusterCyclic Quivers with three Vertices and the Markov Equation
Abstract
Acyclic cluster algebras have an interpretation in terms of tilting objects in a CalabiYau 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 $Q$ clusteracyclic in this case, otherwise clustercyclic. In this note we classify the clustercyclic quivers with three vertices using a Diophantine equation studied by Markov.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2006
 DOI:
 10.48550/arXiv.math/0612213
 arXiv:
 arXiv:math/0612213
 Bibcode:
 2006math.....12213B
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Representation Theory;
 16G20;
 11D25
 EPrint:
 11 pages, with an appendix by Otto Kerner