An approach to non simply laced cluster algebras

Abstract

Let $Δ$ be an oriented valued graph equipped with a group of admissible automorphisms satisfying a certain stability condition. We prove that the (coefficient-free) cluster algebra $\mathcal A(Δ/G)$ associated to the valued quotient graph $Δ/G$ is a subalgebra of the quotient $π(\mathcal A(Δ))$ of the cluster algebra associated to $Δ$ by the action of $G$. When $Δ$ is a Dynkin diagram, we prove that $\mathcal A(Δ/G)$ and $π(\mathcal A(Δ))$ coincide. As an example of application, we prove that affine valued graphs are mutation-finite, giving an alternative proof to a result of Seven.