Cluster-Cyclic Quivers with Three Vertices and the Markov Equation

By A. Beineke, T. Brüstle, L. Hille. 2006

62 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7467470a902690e476bc38534b24475175adc8a8

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

A survey on maximal green sequences (2019, background): “bers a, b, c ě 2 admits a maximal green sequence On the other hand, the quiver Q2,2,3 is mutation-acyclic (the quiver Qa,b,c is mutationacyclic iﬀ minpa,b,cq ă 2 or a2 ` b2 ` c2 ´ abc ą 4 as shown in [7]). Thus, the existence of a maximal green sequence is not preserved under mutation [91]. 4.9. Exi”
Positivity for cluster algebras (2013, unknown): “her case. Note that if ℓ = 1 the condition c[r] ℓ+1ξ − c [r] ℓ ω > 0 becomes ξ > 0, and the condition c[r] ℓ−1ω − c [r] ℓ ξ < 0 becomes ω > 0. Proof. This is generalization of a result in [3], where the ﬁrst case is considered. If c[r] ℓ+1ξ − c[r] ℓ ω > 0, for 1 ≤ ℓ ≤ n, we proceed by”
R-Twisting and 4d/2d Correspondences (2010, background): “The classification for mutation equivalent quivers with ≤ 3 vertices [45, 108] corresponds to the classification of 2d susy models with at most 3 vacua (see §§.”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “iver 1 r //2 t 3 s ^=^= = , CLUSTER ALGEBRAS, REPRESENTATIONS, TRIANGULATED CATEGORIES 11 where r, s and t are non negative integers, is mutation-equivalent to a quiver without a 3-cycle: As shown in [15], this is the case iﬀ the ‘Markoﬀ inequality’ r2 +s2 +t2 −rst > 4 holds or one among r, s and”