KTH/SU Mathematics Colloquium

Andrei ZELEVINSKY, Northeastern University

Cartan-Killing classification: old and new

The famous Cartan-Killing classification of semisimple Lie algebras by Dynkin diagrams is one of the most important mathematical results of all time. Dynkin diagrams appear also in many other important classification results: simple singularities, finite crystallographic reflection groups, finite subgroups of SL(2), quivers of finite representation type, etc. I will present a new instance of this ubiquitous classification: as shown jointly with S.Fomin, cluster algebras of finite type are also classified by Dynkin diagrams. However the underlying combinatorics of the new classification is very different from that of the old ones: it is based on skew-symmetrizable analogues of Cartan matrices. We discuss an interplay between symmetrizable and skew-symmetrizable matrices, leading to a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type (this is a joint work with M.Barot and C.Geiss). The talk will be elementary and self-contained.