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 kind of 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 $B(-\infty)$

corresponding to C. For K beeing the complex numbers we can construct,

following ideas of Lusztig, an algebra of constructible functions which

contains a family of "semicanonical functions". Those are naturally

parametrized by the above mentioned components of maximal dimension.

Modulo a conjecture about the support of the functions in the "Serre ideal"

the semicanonical functions yield a basis of the enveloping algebra U(n) of

the positive part of the Kac-Moody Lie algebra g(C).

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