# Skew-symmetric cluster algebras of finite mutation type

Anna Felikson; Michael Shapiro; Pavel Tumarkin

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 4, page 1135-1180
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topFelikson, Anna, Shapiro, Michael, and Tumarkin, Pavel. "Skew-symmetric cluster algebras of finite mutation type." Journal of the European Mathematical Society 014.4 (2012): 1135-1180. <http://eudml.org/doc/277265>.

@article{Felikson2012,

abstract = {In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems $E_6,E_7,E_8,\tilde\{E\}_6,\tilde\{E\}_7,\tilde\{E\}_8, E_6^\{(1,1)\}, E_7^\{(1,1)\}, E_8^\{(1,1)\}$; two remaining were found by Derksen and Owen in [DO]. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.},

author = {Felikson, Anna, Shapiro, Michael, Tumarkin, Pavel},

journal = {Journal of the European Mathematical Society},

keywords = {cluster algebra; finite mutation type; triangulated surface; growth rate; Quiver mutations; Cluster algebras; Quiver mutations},

language = {eng},

number = {4},

pages = {1135-1180},

publisher = {European Mathematical Society Publishing House},

title = {Skew-symmetric cluster algebras of finite mutation type},

url = {http://eudml.org/doc/277265},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Felikson, Anna

AU - Shapiro, Michael

AU - Tumarkin, Pavel

TI - Skew-symmetric cluster algebras of finite mutation type

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 4

SP - 1135

EP - 1180

AB - In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems $E_6,E_7,E_8,\tilde{E}_6,\tilde{E}_7,\tilde{E}_8, E_6^{(1,1)}, E_7^{(1,1)}, E_8^{(1,1)}$; two remaining were found by Derksen and Owen in [DO]. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.

LA - eng

KW - cluster algebra; finite mutation type; triangulated surface; growth rate; Quiver mutations; Cluster algebras; Quiver mutations

UR - http://eudml.org/doc/277265

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.