  Differential calculi of Poincare-Birkhoff-Witt type on universal enveloping
algebras of Lie algebras g are defined. This definition turns out to be
independent of the basis chosen in g. The role of automorphisms of g is
explained. It is proved that no differential calculus of Poincare-Birkhoff-Witt
type exists on semi-simple Lie algebras. Examples are given, namely gl_n,
Abelian Lie algebras, the Heisenberg algebra, the Witt and the Virasoro
algebra. Completely treated are the 2-dimensional solvable Lie algebra, and the
3-dimensional Heisenberg algebra.
