Cluster Automorphisms

Abstract

In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a $\ZZ$-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster automorphisms in detail for acyclic cluster algebras and cluster algebras from surfaces, and we compute this group explicitly for the Dynkin types and the Euclidean types.

35 pages, v2 final version, to appear in Proc. London Math. Soc

References in corpus (1)

• On triangulated orbit categories

Cited by in corpus (17)

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• Classification of Rank 2 Cluster Varieties
• Cluster algebras of infinite rank as colimits
• Cluster automorphism groups of cluster algebras with coefficients
• Lagrangian Fillings in A-type and their Kalman Loop Orbits
• A characterization of pseudo-Anosov mapping classes via tropical cluster transformations
• Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists
• Automorphisms of cluster algebras of rank 2
• On rooted cluster morphisms and cluster structures in $2$-Calabi-Yau triangulated categories
• Labelled seeds and the mutation group
• $q$-Painlevé equations on cluster Poisson varieties via toric geometry
• Unistructurality of cluster algebras of type $\tilde{\mathbb{A}}$
• Cluster automorphism groups and automorphism groups of exchange graphs