Michael Shapiro: Mutationally finite cluster algebras
Talk from the conference "Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)," given at Northeastern University on April 24th, 2013. The other talks can be found in this playlist: http://www.youtube.com/playlist?list=... Abstract: In 2003, 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). Ia a series of papers with A.Felikson, P.Tumarkin and H.Thomas we classified all mutationally finite cluster algebras. Except finitely many cases, almost all mutationally finite cluster algebras are associated with triangulations of 2-dimensional surfaces (generally speaking, surfaces with orbifold points). All mutationally finite non skew-symmetric cases are obtained from skew-symmetric cases by construction of folding (notion due to A.Zelevinsky). Based on the mutational finite classification we described growth rate of cluster algebras.
Talk from the conference "Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)," given at Northeastern University on April 24th, 2013. The other talks can be found in this playlist: http://www.youtube.com/playlist?list=... Abstract: In 2003, 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). Ia a series of papers with A.Felikson, P.Tumarkin and H.Thomas we classified all mutationally finite cluster algebras. Except finitely many cases, almost all mutationally finite cluster algebras are associated with triangulations of 2-dimensional surfaces (generally speaking, surfaces with orbifold points). All mutationally finite non skew-symmetric cases are obtained from skew-symmetric cases by construction of folding (notion due to A.Zelevinsky). Based on the mutational finite classification we described growth rate of cluster algebras.