Classification of graded cluster algebras generated by rank 3 quivers

Abstract

We consider gradings on cluster algebras generated by [Formula: see text] skew-symmetric matrices. We show that, except in one particular case, mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have only finitely many associated cluster variables. For mutation-acyclic matrices, we prove that all occurring degrees are associated with infinitely many variables. We also give a direct proof that the gradings are balanced in this case (i.e. that there is a bijection between the cluster variables of degree [Formula: see text] and [Formula: see text] for each occurring degree [Formula: see text]).