  Dirac cohomology is a new tool to study unitary and admissible
representations of semisimple Lie groups. It was introduced by Vogan and
further studied by Kostant and ourselves \cite{V2}, \cite{HP1}, \cite{Kdircoh}.
The aim of this paper is to study the Dirac cohomology for the Kostant cubic
Dirac operator and its relation to Lie algebra cohomology. We show that the
Dirac cohomology coincides with the corresponding nilpotent Lie algebra
cohomology in many cases, while in general it has better algebraic behavior and
it is more accessible for calculation.
