Let us consider a BV algebra with the generator . Let be the Lie
superalgebra which is obtained from by forgetting the associative
algebra structure and flipping parity, and the corresponding
wavy Lie superalgebra.

We need to flip parity in order to turn into a Lie superalgebra operation.
If the parity of as an element of , is ,
then the parities of the corresponding elements of are:
and

Theorem 1
The following formulas define the representation of on :

We have to check that:

Indeed, we have:

2Form as an intertwiner

Let us consider the particular case when is the algebra of functions
on the odd symplectic manifold .

In this case, naturally acts on the differential forms on . Indeed, every
element determines the corresponding right-invariant vector
field on . Then would act as a Lie derivative along this vector field,
and acts as a contraction.

We can
consider as a linear map from to the space of differential forms on ;
for each , this map computes — the corresponding differential
form. Eqs. (12) and (13) can be interpreted as saying that is an
intertwiner: