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
Adresse :IHP


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