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Cluster algebras and triangulated orbifolds (2012)
Journal Article
Felikson, A., Shapiro, M., & Tumarkin, P. (2012). Cluster algebras and triangulated orbifolds. Advances in Mathematics, 231(5), 2953-3002. https://doi.org/10.1016/j.aim.2012.07.032

We construct geometric realizations for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston [10] to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hy... Read More about Cluster algebras and triangulated orbifolds.

Skew-symmetric cluster algebras of finite mutation type (2012)
Journal Article
Felikson, A., Shapiro, M., & Tumarkin, P. (2012). Skew-symmetric cluster algebras of finite mutation type. Journal of the European Mathematical Society, 14(4), 1135-1180. https://doi.org/10.4171/jems/329

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... Read More about Skew-symmetric cluster algebras of finite mutation type.

Cluster algebras of finite mutation type via unfoldings (2012)
Journal Article
Felikson, A., Shapiro, M., & Tumarkin, P. (2012). Cluster algebras of finite mutation type via unfoldings. International Mathematics Research Notices, 2012(8), 1768-1804. https://doi.org/10.1093/imrn/rnr072

We complete the 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 characterizin... Read More about Cluster algebras of finite mutation type via unfoldings.