Cluster algebras and triangulated categories

Bernhard Keller

Université Paris 7

Cluster algebras were invented by S. Fomin and A. Zelevinsky in spring 2000 as a tool to approach Lusztig's theory of canonical bases in quantum groups and total positivity in algebraic groups. Since then, cluster algebras have become the center of a rapidly developing theory, which has turned out to be closely related to a large spectrum of other subjects, notably Lie theory, Poisson geometry, Teichmüller theory and quiver representations. Recent work by many authors has shown that this last link is best understood using the cluster category, which is a triangulated category associated with every Dynkin diagram. In this talk, I will report on these developments and present the cluster multiplication theorem, obtained in joint work with P. Caldero, which directly links the multiplication of the cluster algebra to the triangles in the cluster category.