On Involutive Cluster Automorphisms

Abstract

We construct a special embedding of the translation quiver ℤQ in the three-dimensional affine space 𝔸³ where Q is a finite connected acyclic quiver and ℤQ contains a local slice whose quiver is isomorphic to the opposite quiver Q ^op of Q. Via this embedding, we show that there exists an involutive anti-automorphism of the translation quiver ℤQ. As an immediate consequence, we characterize explicitly the group of cluster automorphisms of the cluster algebras of seed (X, Q), where Q and Q ^op are mutation equivalent.

Key Words:

• Cluster automorphisms
• Local slices
• Repetitive quivers

2010 Mathematics Subject Classification:

• 13F60
• 16G20

ACKNOWLEDGMENTS

This article is part of my Ph.D. thesis, under the supervision of Ibrahim Assem and Vasilisa Shramchenko. I would like to thank them deeply for their patience and availability.

Notes

Communicated by D. Zacharia.