Algèbres de Calabi-Yau et algèbres N-Koszul

Du 22 au 26 octobre 2007 au CIRM

Organisateurs: Roland Berger et Bernard Leclerc

1. Scientific Presentation

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.

2. Mini-courses

Four three-hour long mini-courses will be taught by: Victor Ginzburg (Chicago, USA), Idun Reiten (Trondheim, Norway),  James J. Zhang (Seattle, USA), Roland Berger (Saint-Etienne, France).

3. Talks

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).

4. Scientific Committee

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.