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:

1.2Definition of

Let us say that a subspace is admissible if:

is closed under . This implies that is closed under

For any exists such that:

(14)

is super-antisymmetric. This actually implies that satisfies the Jacobi identity of a super Lie algebra;
actually is a homomorphism of super Lie algebras:

is an injection;
in other words is isomorphic to

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

We will call the element of corresponding to , i.e. ,

1.3Abelian

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 .

2Proof of

The proof is based on the following algebraic interpretation of .

When satisfies the Master Equation, the RHS is our (usual, not equivariant).
But we can evaluate this map on arbitrary , not necessarily satisfying the Master Equation

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):