Cluster automorphisms and quasi-automorphisms

TL;DR

This paper explores the relationship between cluster automorphisms and quasi-automorphisms in cluster algebras, establishing conditions for their isomorphism and computing these groups for various algebra types.

Contribution

It proves that under certain conditions, the quasi-automorphism group is isomorphic to a subgroup of the automorphism group of a trivial coefficient algebra, and computes these groups for specific algebra types.

Findings

Quasi-automorphism group is isomorphic to a subgroup of the automorphism group of the trivial coefficient algebra.

The groups are isomorphic if the algebra has principal or universal coefficients.

Computed automorphism groups for all finite and skew-symmetric affine type cluster algebras.

Abstract

We study the relation between the cluster automorphisms and the quasi-automorphisms of a cluster algebra $\mathcal{A}$. We proof that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of $\mathcal{A}$ is isomorphic to a subgroup of the cluster automorphism group of $\mathcal{A}_{triv}$, and the two groups are isomorphic if $\mathcal{A}$ has principal or universal coefficients; here $\mathcal{A}_{triv}$ is the cluster algebra with trivial coefficients obtained from $\mathcal{A}$ by setting all frozen variables equal to the integer 1. We also compute the quasi-automorphism group of all finite type and all skew-symmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of $\mathcal{A}_{triv}$.