Equivariant cohomology
1 Equivariant cohomology as a relative Lie algebra cohomology
To define the equivariant cohomology, we need to remember that 
is actually a differential Lie algebra.
It has a differentiation 
. In our definition of the cohomology, we have not yet used .
The way to use it is to consider a larger algebra 
, which is obtained from by
adding as an extra generator.
The Koszul dual of
has an additional generator which we call
. The differential on
(which comes from the non-homogeneity of
) acts as follows:
Notice that
is zero. We will simply put
. Then the differential formally coincides with the
given by Eq. (
3).
This is called
Weyl algebra (a commutative differential superalgebra).
Definition 1 We define the equivariant cohomology of
as the relative Lie algebra cohomology:
The ordinary (not relative) cohomology of
with coefficients in
is computed as follows:
with the differential acting as follows:
To define the
relative cohomology, we restrict to the subspace of cochains
which are:
|  -horizonthal:
|
| | | |
| and -invariant: |
| | | |
Now our cochains to not depend on
and are
-invariant; the differential becomes:
This is the Cartan model of equivariant cohomology.
2 Weyl model
To pass to the Weyl model, we act on cochains by
. This does not change the condition of invariance,
but changes the condition of horizonthality:
and
gets replaced with:
and 
.
on
, and the rest for the structure constants of
