Cluster-Cyclic Quivers with three Vertices and the Markov Equation

2006/12/08 by Andre Beineke, Thomas Brüstle, Beineke, Andre +3 · 3 citations
Mathematics · Physics and Astronomy · #11D25 #16G20 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.math/0612213

openalex publication_date 2006/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Orderings of k-Markov Numbers
• Sign-coherence and tropical sign pattern for rank 3 real cluster-cyclic exchange matrices
• On q-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges
• An approach to non simply laced cluster algebras

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