1Equivariant 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:
where and and .
This is the usual BRST-like operator defining Lie algebra cohomology.
We have visually separated the terms into two groups: the first three terms account for the action of on ,
and the rest for the structure constants of .
To define the relative cohomology, we restrict to the subspace of cochains which are:
Now our cochains to not depend on and are -invariant; the differential becomes:
This is the Cartan model of equivariant cohomology.
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: