Solving the equation for
Let us investigate the equation for :
Let us solve it perturbatively as a series in . We have:
We can proceed similarly to higher orders. An immediate consistency check on Eq. (15) is that is -closed:
This should be checked order by order. And then we should prove order by order that is -exact.
The validity of Eq. (17) can be proven by induction order by order as follows:
The first term is zero by Jacobi identity. That the second term is zero can be proven using an additional assumption that is -invariant; this should be checked order by order.
Then we should check that is -exact.
Suppose that all these assumptions are true. Then we have a “formal solution”: