Felix Klein Lectures

Cluster algebras, cluster categories and periodicity

Professor Bernhard Keller (Paris)

Date: June 15 - July 18

Venue: Small lecture hall ("Kleiner Hörsaal"), Wegelerstraße 10

  • Wednesdays 10:00 – 12:00
  • Fridays 14:00 – 16:00

Abstract:

Cluster algebras were invented by S. Fomin and A. Zelevinsky at the beginning of this decade in a projet whose aim it was to develop a combinatorial framework for the study of total positivity in algebraic groups and of canonical bases in quantum groups. It was soon recognized that the combinatorics of cluster algebras also appeared in a large spectrum of other subjects, for example in Poisson geometry, higher Teichmuller theory, discrete dynamical systems, algebraic geometry, in combinatorics and notably the study of combinatorial polyhedra and, last not least, in the representation theory of quivers and finite-dimensional algebras.

In these lectures, we will give an introduction to cluster algebras and some of the many links to other subjects. We will then concentrate on the (additive) categorification of cluster algebras via cluster categories. These are certain triangulated Calabi-Yau categories obtained from categories of quiver representations. They can be constructed as orbit categories or as subquotients of derived categories of certain dg algebras, the Ginzburg algebras. Cluster categories allow one to give explicit solutions to the recurrence equations defining cluster algebras and thus to obtain conceptual proofs of some purely combinatorial conjectures about them. We will illustrate this by sketching a proof of the so-called perodicity conjecture, which originates in A. Zamolodchikov’s work in mathematical physics in the early nineties.

Prerequisites:

At the beginning of the course, only the contents of a basic algebra course will be assumed to be known. At an early stage, I will recall the definition and classification of finite root systems. Participants who have not seen them before might want to look at Serre's brief account of the subject beforehand. Later, it will be helpful to know some basic representation theory (modules over not necessarily commutative rings, semi-simple modules, Morita equivalence). Some basic topology (fundamental properties of Euler characteristics) will also be needed and later some basic algebraic geometry (as it can be found in the first chapter of Hartshorne's book). Basic homological algebra (resolutions, Ext-groups) will be assumed known. I will recall the definition of the derived category but only very briefly so it is good to have had some contact with it before. I will succinctly introduce more advanced notions as needed.