On the Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and   quantized enveloping algebras
  A Bernstein-Gelfand-Gelfand resolution for arbitrary Kac-Moody algebras and arbitrary subsets of the set of simple roots is proven. Moreover, quantum group analogs of the Bernstein-Gelfand-Gelfand resolution for symmetrizable Kac-Moody algebras are established. For quantized enveloping algebras with fixed complex nonzero deformation parameter q exactness is proven for all q which are not a root of unity. 
I. Heckenberger, S. Kolb