Department of Mathematics
Northeastern University
Cluster algebras of finite type: one more instance of the
Cartan-Killing classification
The famous
Cartan-Killing classification of simple Lie algebras is one of the most
important mathematical results of all time. The same combinatorial
objects that appear in this classification (Dynkin diagrams, finite
root systems, Cartan matrices of finite type) appear also in many other
important classification results: simple singularities, finite Coxeter
groups, finite subgroups of SL(2), quivers of finite representation
type, to name a few. I will present a new instance of this ubiquitous
classification: in a joint work with Sergey Fomin we show that cluster
algebras of finite type are also classified by Cartan-Killing types.
Cluster algebras introduced by Fomin and myself a few years ago are a
new class of commutative rings designed to provide an algebraic
framework for the study of canonical bases and total positivity in
semisimple Lie groups. The talk will be elementary and self-contained.