Structure of the obstacle
Some map 
always exists, the question is, can
we pick a 
-invariant one?
Let us choose some 
. Any map 
can be thought of as
a family of maps:
parametrized by 
satisfying:
For each element
there is a corresponding vector field
on
. Let us
consider
. It is a vector field on
:
Because of Eq. (
2), the scaling degree of
correlates with the power of
.
In particular,
is a
linear vector field on the linear space
. This is the
action of
on
. (The
is a quadratic vector field:
,
and
is qubic,
etc..)
Can we choose
so that
?
We would like to reformulate this question as a cohomological problem.
For two elements
and
of
, we have:
Let us introduce the “ghosts” 
and the “BRST operator”:
This defines the differential in the Lia algebra
cohomology complex of with values in the space of vector fields on having zero
of at least second order at the point 
. (The action of the second term, 
,
is by the commutator of vector fields.)
Let us define
as the sum of nonlinear terms in
(
cf. Eq. (
4)):
It satisfies the Maurer-Cartan equation:
Ambiguity in the choice of 
translates to gauge transformations
of :
Obstacle to the existence of
-invariant
is a solution of the MC
Eq. (
7) modulo gauge transformations defined in Eq. (
8)
In particular, if the first non-removable term is
, then the corresponding obstacle is:
