Cluster Algebras and Lie Theory
Abstract.
Cluster algebras, invented in 2001 by Fomin and Zelevinsky,
have become a ubiquitous combinatorial device in representation theory,
integrable systems, Poisson geometry, Teichmüller theory, etc.
After an introduction to the basic definitions and structural properties
of cluster algebras, the lectures will present two families of examples
occuring in Lie theory : (1) coordinate rings of algebraic varieties
arising in the representation theory of semisimple algebraic groups
(Graßmannians, flag varieties, Bruhat cells, ...); (2) Grothendieck
rings of categories of representations of quantum loop algebras.