Proof of equivariance
1 Restriction of symmetries
1.1 New operation 
Symmetries are generated by Hamiltonians of the form:
for arbitrary
.
We want to restrict the possible to belong to some subspace 
Let us introduce on functions on
the operation
as follows:
1.2 Definition of 
Let us say that a subspace
is
admissible if:
The requirement 3 almost follows from 1 and 2.
The requirement 4 is probably technical, but it is very convenient:
To any
corresponds a symmetry generated by
,
and this correspondence is one-to-one
1.3 Abelian
The simplest case is when
is abelian,
i.e. when
for any
and
.
In particular, this is the case for theories
coming from BRST formalism.
In this case it is very easy to solve Eq. (
14):
| | (for abelian ) |
| | |
However,
not all abelian are admissible; we also have to check closedness under
.
2 Proof of 
The proof is based on the following algebraic interpretation of 
.
Consider the linear map:
| [space of 1/2-densities on  ]  |
| | | |
| | | | |
We observe that this map is an intertwiner between:
action of  on half-densities in | and | action of on PDFs on  |
| | |
This is true even when
. In particular, applying this intertwiner to the product
in Eq. (
7):
is closed under 
exists 
such that:
is a homomorphism of super Lie algebras: 
is an injection;
in other words 
is isomorphic to 
the element of
, i.e.
,
satisfies the Master Equation, the RHS is our
(usual, not equivariant). But we can evaluate this map on arbitrary
