Differential Calculi of Poincare-Birkhoff-Witt type on Universal Enveloping Algebras

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.