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Cluster-Cyclic Quivers with three Vertices and the Markov Equation

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arxiv math/0612213 v2 pith:BGDISHSP submitted 2006-12-08 math.RA math.RT

Andre Beineke , Thomas Brüstle , Lutz Hille This is my paper · ORCID
classification math.RAmath.RT
keywords cluster-cyclicacyclicalgebraclusterdefinedequationmarkovquivers
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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.

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  1. Local and global patterns of rank 3 $G$-fans of totally-infinite type

    math.CO 2024-11 conditional novelty 6.0 of 10

    Infinite-type cluster algebras have G-fans that are never complete, and rank 3 local behavior falls into six types that correlate with global fan shapes.

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