Universal verma modules and w-resolvents over kač-moody algebras

We consider new aspects of extremal equations over symmetrizable Kač-Moody algebras. We develop new methods (reproducing classical finite-dimensional results) that can be applied to infinite-dimensional (affine) Lie algebras. We describe special extensions of universal enveloping algebras, investigate the line structure of W-resolvents, and use these methods to investigate "extremal projectors" over Kač Moody algebras.