1 Restriction of symmetries 1.1 New operation 1.2 Definition of 1.3 Abelian 2 Proof of

Proof of equivariance

#### 1Restriction of symmetries

##### 1.1New 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:

##### 1.2Definition of

Let us say that a subspace is admissible if:
1. is closed under . This implies that is closed under

2. For any exists such that:
 (14)

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

4. 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 .

Consider the linear map:

 [space of 1/2-densities on ]

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):