An approach to non simply laced cluster algebras

TL;DR

This paper studies non simply laced cluster algebras via automorphisms of valued graphs, establishing their relation to quotient cluster algebras and proving mutation-finiteness for affine cases.

Contribution

It introduces a method to relate cluster algebras of valued graphs with automorphisms to quotient cluster algebras, extending understanding of their structure and finiteness properties.

Findings

Cluster algebra of the quotient graph is a subalgebra of the quotient of the original algebra.

For Dynkin diagrams, the quotient and the algebra of the automorphism group coincide.

Affine valued graphs are mutation-finite, providing an alternative proof to previous results.

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.