On Structure of cluster algebras of geometric type, II: Green's equivalences and paunched surfaces

TL;DR

This paper explores the structure of cluster algebras of geometric type, introducing partial seed homomorphisms and linking algebraic classifications to geometric surfaces, extending previous work on Green's equivalences.

Contribution

It introduces partial seed homomorphisms and relates sub-rooted cluster algebras to Green's equivalence classes and paunched surfaces, advancing the understanding of cluster algebra structures.

Findings

Isomorphism classes of sub-rooted cluster algebras correspond to regular D-classes of partial seed endomorphisms.

For cluster algebras from Riemann surfaces, these classes correspond to isomorphism classes of paunched surfaces.

The work extends the classification framework for cluster algebras of geometric type.

Abstract

Following our previous work [18], we introduce the notions of partial seed homomorphisms and partial ideal rooted cluster morphisms. Related to the theory of Green's equivalences, the isomorphism classes of sub-rooted cluster algebras of a rooted cluster algebra are corresponded one-by-one to the regular $\mathcal D$-classes of the semigroup consisting of partial seed endomorphisms of the initial seed. Moreover, for a rooted cluster algebra from a Riemannian surface, they are also corresponded to the isomorphism classes of the so-called paunched surfaces.