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):