Let Ξ 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 A(Ξ/G) associated to the
valued quotient graph Ξ/G is a subalgebra of the quotient Ο(A(Ξ)) of the cluster algebra associated to Ξ by the action of G.
When Ξ is a Dynkin diagram, we prove that A(Ξ/G) and
Ο(A(Ξ)) coincide. As an example of application, we prove that
affine valued graphs are mutation-finite, giving an alternative proof to a
result of Seven.Comment: 36 pages. Minor correction