Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 3

On structure of cluster algebras of geometric type, II: Green’s equivalences and paunched surfaces

Pages: 451 – 490

DOI: https://dx.doi.org/10.4310/PAMQ.2015.v11.n3.a3

Authors

Min Huang (Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China)

Fang Li (Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China)

Abstract

Following our previous work, 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.

Keywords

seed homomorphism, rooted cluster morphism, subrooted cluster algebra, Green’s equivalence, paunched surface

2010 Mathematics Subject Classification

05E15, 13F60, 20M10

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Published 29 November 2016