Cluster Algebras of Finite Mutation Type Via Unfoldings

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 mutationfinite cluster algebras and triangulated marked bordered surfaces. ContentsResearch of Michael Shapiro was supported by grants DNS 0800671 and PHY 0555346.

Tubular cluster algebras II: Exponential growth

“…We would like to mention that Felikson, Shapiro and Tumarkin recently completed their above mentioned classification of mutation finite cluster algebras by covering also the skew symmetrizable cases [9]. Moreover, they determine in [10] (for the orbifold cases) and in [11] (with H. Thomas for the remaining exceptional cases) the growth rate for all mutation finite cluster algebras.…”

Cluster algebras and discrete integrability

“…However, it turns out that it is of finite mutation type, in the sense that there are only a finite number of exchange matrices produced under mutation from the initial B. Cluster algebras of finite mutation type have also been classified [16,17]: as well as those of finite type, they include cluster algebras associated with triangulated surfaces [19,20], cluster algebras of rank 2, plus a finite number of exceptional cases.…”

Strongly primitive species with potentials I: mutations

“…The general notion of unfolding for skew-symmetrizable matrices goes back at least to [11, Step 1, Section 2.4] and was then used by Dupont [9] and Demonet [5]. We present it in the form suggested by the second author several years ago (unpublished), and later reproduced in [10]. (1) the sum of entries in each column of each E i × E j block of C is equal to b i j ;…”