Résume | In 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 $g(C)$. |