Cluster algebras of finite mutation type via unfoldings

TL;DR

This paper completes the classification of mutation-finite cluster algebras by extending unfoldings to skew-symmetrizable matrices, linking them to triangulated surfaces and skew-symmetric cases.

Contribution

It introduces a method to embed skew-symmetrizable mutation-finite matrices into skew-symmetric ones via unfoldings, expanding classification techniques.

Findings

Every mutation-finite skew-symmetrizable matrix admits an unfolding.

Unfoldings embed their mutation class into that of a skew-symmetric matrix.

Establishes a correspondence with triangulated marked bordered surfaces.

Abstract

We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram characterizing the matrix admits an unfolding which embeds its mutation class to the mutation class of some mutation-finite skew-symmetric matrix. In particular, this establishes a correspondence between a large class of skew-symmetrizable mutation-finite cluster algebras and triangulated marked bordered surfaces.