On Involutive Cluster Automorphisms

We construct a special embedding of the translation quiver ?Q in the three-dimensional affine space 𝔸³ where Q is a finite connected acyclic quiver and ?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 ?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.