Universal verma modules and W-resolvents over Kac-Moody algebras

We consider new aspects of external equations over symmetrizable Kac-Moody algebras. We develop nerw methode (reproducing classical finite-dimensional results) that can be applied to infinite-dimensional (affine) Lie algebras . We describe special extensions of universal enveloping algebras, investig ate the fine structure of W-resolvents, and use these methods to investigat e " extremal projectors " over Kac-Moody algebras.