Equivariant cohomology

#### 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:
 (5)
 (6)
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:
 -horizonthal:
 and -invariant:
Now our cochains to not depend on and are -invariant; the differential becomes:
This is the Cartan model of equivariant cohomology.

#### 2Weyl 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:
 becomes
and gets replaced with: