Séminaires : Séminaire d'Algèbre

Equipe(s) : gr,
Responsables :J. Alev, D. Hernandez, B. Keller, Th. Levasseur, et S. Morier-Genoud.
Email des responsables : Jacques Alev <jacques.alev@univ-reims.fr>, David Hernandez <david.hernandez@imj-prg.fr>, Bernhard Keller <bernhard.keller@imj-prg.fr>, Thierry Levasseur <Thierry.Levasseur@univ-brest.fr>, Sophie Morier-Genoud <sophie.morier-genoud@imj-prg.fr>
Salle : à distance / remote
Adresse :IHP

Depuis le 23 mars 2020, le séminaire se tient à distance. Pour les liens et mots de passe, merci de contacter l'un des organisateurs ou de souscrire à la liste de diffusion https://listes.math.cnrs.fr/wws/info/paris-algebra-seminar. L'information nécessaire sera envoyée par courrier électronique peu avant chaque exposé. Les notes et transparents sont disponibles ici.


Since March 23, 2020, the seminar has been taking place remotely. For the links and passwords, please contact one of the organizers or

subscribe to the mailing list at https://listes.math.cnrs.fr/wws/info/paris-algebra-seminar. The connexion information will be emailed shortly before each talk. Slides and notes are available here.


Orateur(s) Christof GEISS - ,
Titre Crystal graphs and semicanonical functions for symmetrizable Cartan matrices.
Horaire14:00 à 15:00
RésumeIn joint work with B. Leclerc and J. Schröer we propose a 1-Gorenstein algebra $H$, defined over an arbitrary field $K$, associated to the datum of a symmetrizable Cartan Matrix $C$, a symmetrizer $D$ of $C$ and an orientation $\Omega$. The $H$-modules of finite projective dimension behave in many aspects like the modules over a hereditary algebra, and we can associate to $H$ a generalized preprojective algebra $\Pi$. If we look, for $K$ algebraically closed, at the varieties of representations of $\Pi$ which admit a filtration by generalized simples, we find that the components of maximal dimension provide a realization of the crystal $B_C(-\infty)$. For K being the complex numbers we can construct, following ideas of Lusztig, an algebra of constructible functions which contains a family of ``semicanonical functions'', which are naturally parametrized by the above mentioned components of maximal dimensions. Modulo a conjecture about the support of the functions in the ``Serre ideal'' those functions would yield a semicanonical basis of the enveloping algebra $U(n)$ of the positive part of the Kac-Moody Lie algebra $g(C)$.
Salleà distance / remote