From the introduction: Cluster algebras were introduced by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497-529 (2002; Zbl 1021.16017)] in the context of canonical bases and total positivity. These are \(\mathbb Z\)-algebras whose generators are grouped into sets called clusters, and one passes from one cluster to another using an operation called mutation.
We are interested in the question whether one can study cluster algebras as a category. This means understanding the morphisms which preserve their very particular structure, that is, that keep invariant the grouping of generators into clusters and are compatible with mutations. As a first step in this direction, we study here what we call cluster automorphisms. We define a cluster automorphism of a given cluster algebra as an automorphism of \(\mathbb Z\)-algebras sending a cluster to another and commuting with mutations. Thus, in this paper, we study the symmetries of a given cluster algebra and compute the cluster automorphism group for the best known classes of cluster algebras, those arising from an acyclic quiver and those arising from a surface. Observe that, in [Invent. Math. 154, No. 1, 63-121 (2003; Zbl 1054.17024)], S. Fomin and A. Zelevinsky have considered a related notion of strong isomorphisms, by which they mean an isomorphism of the cluster algebras that maps every seed to an isomorphic seed. As will follow from our results, a strong automorphism of a cluster algebra is what we call here a direct cluster automorphism.
Let \(\mathcal A=\mathcal A(\mathbf x,Q)\) be a cluster algebra. Among the most interesting properties of the automorphism group \(\operatorname{Aut}\mathcal A\) of \(\mathcal A\) is the fact that an element of this group sends the quiver \(Q\) either to itself or to the opposite quiver \(Q^{\text{op}}\). This allows to define a subgroup \(\operatorname{Aut}^+\mathcal A\) of \(\operatorname{Aut}\mathcal A\) consisting of those automorphisms sending \(Q\) to itself. We prove that the index of \(\operatorname{Aut}^+\mathcal A\) in \(\operatorname{Aut}\mathcal A\) is two if and only if \(Q\) is mutation equivalent to \(Q^{\text{op}}\) and otherwise \(\operatorname{Aut}\mathcal A=\operatorname{Aut}^+\mathcal A\). We first compute these groups in the context of acyclic cluster algebras. In this case, the combinatorics of the cluster algebra is nicely encoded in the cluster category introduced in [A. B. Buan et al., Adv. Math. 204, No. 2, 572-618 (2006; Zbl 1127.16011)] and, for type \(\mathbb A\), also in [P. Caldero et al., Trans. Am. Math. Soc. 358, No. 3, 1347-1364 (2006; Zbl 1137.16020)]. In particular, we recall that the Auslander-Reiten quiver of the cluster category of an acyclic cluster algebra has a particular connected component, called the transjective component.
Theorem 1.1. Let \(\mathcal A\) be an acyclic cluster algebra and \(\Gamma_{\text{tr}}\) the transjective component of the Auslander-Reiten quiver of the associated cluster category. Then \(\operatorname{Aut}^+\mathcal A\) is the quotient of the group \(\operatorname{Aut}(\Gamma_{\text{tr}})\) of the quiver automorphisms of \(\Gamma_{\text{tr}}\), modulo the stabiliser \(\text{Stab}(\Gamma_{\text{tr}})_0\) of the points of this component. Moreover, if \(\Gamma_{\text{tr}}\cong\mathbb Z\Delta\), where \(\Delta\) is a tree or of type \(\widetilde{\mathbb A}\) then \(\operatorname{Aut}\mathcal A=\operatorname{Aut}\mathcal A\rtimes\mathbb Z_2\) and this semidirect product is not direct.
As an easy consequence, we compute the automorphism groups of the cluster algebras of Dynkin and Euclidean types.
The paper is organised as follows. In Section 2, we define our notion of cluster automorphism and prove some of its elementary properties; Section 3 is devoted to the case of acyclic cluster algebras and Section 4 to that of cluster algebras arising from surfaces. Finally, in Section 5, we consider the finiteness of the automorphism group.