Solving equation for a

Solving the equation for

Let us investigate the equation for

(15)

Let us solve it perturbatively as a series in

. We have:

where is:

(16)

(i.e. the definition of implies that should be -exact)

We can proceed similarly to higher orders. An immediate consistency check on Eq. (15) is that

-closed:

(17)

This should be checked order by order. And then we should prove order by order that

-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

-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”: