A triangulated category C
is said to be d-Calabi-Yau if the dth power of
its suspension functor is a Serre
functor. This terminology comes from the following
example : if X is a smooth projective variety of dimension d, the bounded
derived category of coherent sheaves on X is d-Calabi-Yau
if and only if X is a Calabi-Yau variety. Often an algebra is said to be d-Calabi-Yau if a naturally associated triangulated category (depending
on the authors, it could be its derived category or its stable category ...) is
d-Calabi-Yau. Some important examples of d-Calabi-Yau algebras are polynomial algebras twisted by group
algebras of finite subgroups of SL_d. When d=2, these
algebras were largely studied in relation with the MacKay correspondence. Thus
the study of d-Calabi-Yau algebras provides
interesting examples in non-commutative algebraic geometry. Recently, the study
of tilting modules of a d-Calabi-Yau algebra for d=2,3
has been related to the quiver mutation introduced by Fomin
and Zelevinsky in the framework of cluster algebras.

N-Koszul
algebras are defined as natural extensions of usual Koszul
algebras. N-Koszul algebras appear in many different
areas such as theoretical physics, N-complexes, quivers, A-infinity algebras, symplectic reflection algebras, non-commutative algebraic
geometry. A duality theory exists for N-Koszul
algebras (He and Lu) which is based on A-infinity algebra duality due to Lu, Palmieri, Wu and Zhang. The N-Koszulity
property is an essential ingredient for the N-generalisation of the Poincaré-Birkhoff-Witt Theorem (Fløystad
and Vatne, Berger and Ginzburg).
The latter was used to get an N-generalisation of symplectic
reflection algebras.

It turns out that graded 3-Calabi-Yau
algebras are N-Koszul. The following are significant
examples of graded 3-Calabi-Yau algebras: Yang-Mills algebras (Connes and Dubois-Violette), Artin-Schelter regular algebras of dimension 3, some
special N-symmetric algebras. Thus for these examples, the N-Koszulity property implies the existence of a duality
theory and of Poincaré-Birkhoff-Witt deformations.

The aim of the conference
is to bring together some of the best experts on the subject, and moreover, to
make the subject accessible to young researchers and to any mathematician
interested in these topics.

Four three-hour long mini-courses
will be taught by: Victor Ginzburg (

The list of the other
speakers is the following: Ralf Bocklandt (Anvers, Belgique), Joe Chuang (Bristol, UK), Claude Cibils
(Montpellier, France), Michel Dubois-Violette (Orsay, France), Christof Geiss (UNAM, Mexique), Iain
Gordon (Edimbourg, UK), Osamu Iyama
(Nagoya, Japon), Bernhard Keller (Jussieu,
France), Jean-Michel Oudom (Montpellier, France), Raphaël Rouquier (Oxford, UK), Nicole
Snashall (Leicester, UK), Rachel Taillefer
(Saint-Etienne, France), Michel Van den Bergh (Hasselt,
Belgique).

Jacques Alev (Reims, France), Roland Berger (Saint-Etienne, France), Bernhard Keller (Paris, France), Bernard Leclerc (Caen, France), Thierry Levasseur (Brest, France).

Ce colloque est organisé par le GDR 2432 et soutenu par le réseau de formation par la recherche Européen Liegrits.

Retour à la page du GDR

29 août 2007.