Chevalley-Eilenberg complex of a differential module
In this Section, there is no 
nor 
. We forget about them for now.
As a preparation for BV formalism, we will now discuss another formula similar to Eq. (11).
Consider a Lie superalgebra
, its universal enveloping algebra
, and its cone
, generated by
and
,
.
We consider the quadratic-linear dual coalgebra
.
The dual space
is the algebra of functions of the “ghost variables”
.
Following Section 3.4 of
LodayVallette, the Koszul twisting morphism is:
We will study the properties of the following operator:
Since
is quadratic-linear,
comes with the differential
. The dual differential on
is the
Chevalley-Eilenberg differential 
(the BRST operator):
Here
is the “internal” differential of
;
it comes from
being inhomogenous
(
i.e. quadratic-linear and not purely quadratic algebra).
The Chevalley-Eilenberg complex 
with coefficients in 
can be defined
for any -module . Consider the special case when is a -differential
module , i.e.
a representation of 
. We will denote
, 
and 
the elements of
representing elements ,
of 
and 
.
(Then is also a representation
of , where 
is represented by .)
Consider the Lie algebra cochain complex (= Chevalley-Eilenberg complex)
of
with coefficients in
. The differential is defined as follows:
All this can be defined for any
-module
. But when
is also an
-
differential
module (
i.e. a represenatation of
), then
and
are related:
where
. Notice that Eq. (
25) resembles Eq. (
11).
Indeed, both
of Eq. (
24) and
of Eq. (
4)
are maps from coalgebra to algebra, satisfying the Maurer-Cartan (MC) equation.
But the way MC equation is satisfied is different,
because
acts in the coalgebra (in
)
while
acts in the algebra (in
), see 〚
Are 
Faddev-Popov ghosts?〛.
Since
and
are both
-modules,
we can consider
a
-module, as a
of
two
-modules:
Proposition 1
In other words,
both
and
define on 
the structure of a differential -module,
and 
intertwines them.
