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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 is -closed:

(17)

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”:

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