An approach to non simply laced cluster algebras

Abstract

Let $\Delta$ 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 $\mathcal A(\Delta/G)$ associated to the valued quotient graph $\Delta/G$ is a subalgebra of the quotient $\pi(\mathcal A(\Delta))$ of the cluster algebra associated to $\Delta$ by the action of $G$. When $\Delta$ is a Dynkin diagram, we prove that $\mathcal A(\Delta/G)$ and $\pi(\mathcal A(\Delta))$ coincide. As an example of application, we prove that affine valued graphs are mutation-finite, giving an alternative proof to a result of Seven.