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.