Cohomology of 
vs Lie algebra cohomology
1 Definition of the Lie algebra cohomology 
The space
is a representation of
; the action of
is slightly easier
to write down at the level of the corresponding action of the Lie group
;
acts on
as follows:
Therefore:
where
is defined as in Eq. (
2). We define
to coincide with
of Eq. (
6):
To follow the Faddeev-Popov notations, we introduce
| the Faddeev-Popov ghost:  | |
|
| | | |
| then  |
|  (cp. Eq. (2))
|
| | |
| | | | |
Eqs. (
14) and (
15) are equivalent to saying:
where
is the structure constants of
:
It is straightforward to verify using Eqs. (
12) and (
13) that:
The subgroup
preserves
and therefore
:
This implies:
2 Proof that 
is the same as 
This is similar to the statement that for any Lie group
, the de Rham subcomplex of right-invariant forms on
is the same as the Lie cohomology complex of
with coefficients in the trivial representation:
Our case is a variation on this theme:
As
we explained,
consists of functions of
and
. To obtain the corresponding element
of
, we replace
with
with
(as in Eq. (
15)), and
with
.
Under this identification
becomes
.
with coefficients in
.
is not just the Faddeev-Popov ghost; it is the product of the Faddeev-Popov ghost
with
.
