Unistructurality of cluster algebras of type A˜

Abstract

It is conjectured by Assem, Schiffler and Shramchenko in [ASS14a] that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of type Dynkin or rank 2 [ASS14a]. The aim of this paper is to prove it for algebras of type A. We use triangulations of annuli and algebraic independence of clusters to prove unistructurality for algebras arising from annuli, which are of type A. We also prove the automorphism conjecture [ASS14a] for algebras of type A as a direct consequence.

Unistructurality of cluster algebras

“…Conjecture 1.1 has been affirmed for cluster algebras of Dynkin type or rank 2 in [1], for cluster algebras of type à in [2]. Recently, Bazier-Matte and Plamondon have affirmed Conjecture 1.1 for cluster algebras from surfaces without punctures in [3].…”

Cluster automorphisms and quasi-automorphisms

“…For completeness, we also mention yet another notion of automorphisms introduced by Saleh in [20] as automorphisms of the ambient field that restrict to a permutation of the set of all cluster variables. The relation between Saleh's notion and that of cluster automorphisms above was studied in [3] and led to the open question of unistructurality of cluster algebras, see also [4].…”