ON INVOLUTIVE CLUSTER AUTOMORPHISMS

We construct a special embedding of the translation quiver ZQ in the three-dimensional affine space ?(3) 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.