On involutive cluster automorphisms

TL;DR

This paper constructs a geometric embedding of a translation quiver to demonstrate the existence of an involutive anti-automorphism, leading to explicit characterization of cluster automorphisms for certain cluster algebras.

Contribution

It introduces a novel geometric embedding of translation quivers to explicitly describe involutive automorphisms in cluster algebra theory.

Findings

Existence of an involutive anti-automorphism of the translation quiver $ extbf{Z}Q$.

Explicit characterization of the group of cluster automorphisms for seed $(X,Q)$.

Identification of conditions where $Q$ and $Q^{op}$ are mutation equivalent.

Abstract

We construct a special embedding of the translation quiver $\mathbb{Z}Q$ in the three-dimensional affine space $\mathbb{R}^{3}$ where $Q$ is a finite connected acyclic quiver and $\mathbb{Z}Q$ 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 $\mathbb{Z}Q.$ 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.