coming from the dual of a Lie algebra we have $$\langle\frac{1}{h}CBH(h p_1,h p_2)-(p_1+p_2),x\rangle = \sum_{n\geq1}\frac{h^n}{n!}\sum_{\Gamma\in T_{n,2}}W_\Gamma \hat B_\Gamma(p_1,p_2,x)$$

This result is one of the main ingredient to prove that CBH-quantization is a deformation quantization in the case of the dual of a Lie algebra. It allows us to consider the generating function as a generalization of the CBH formula to the non-linear case.

By Proposition \[formalgroupoid\] we have that the deformed source and target maps are $$t_h(p,x) = \langle\frac{1}{h} \nabla_{p_1} CBH(0,h p),x\rangle = x +h \alpha^{ij}_k x^k p_j+\frac{h^2}{3} 
\alpha^{uv}_l\alpha^{ni}_v x^l p_u p_n+\dots$$ $$s_h(p,x) = \langle\frac{1}{h} \nabla_{p_2} CBH(h p,0),x\rangle = x -h \alpha^{ij}_k x^k p_j+\frac{h^2}{3} 
\alpha^{uv}_l\alpha^{ni}_v x^l p_u p_n+\dots$$

Perturbative form of the SGA equation {#Cayley}
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\[Cayleytree\]

The goal of this section is to formulate a perturbative version of the SGA equation. It is divided in two parts. First we introduce some tools and state the perturbative version of the SGA equation in Proposition \[pertSGA\]. The proof is then split into several lemmas.

Perturbative SGA and Cayley trees
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Let us recall that $B_n:= (\mathbb R^{*d})^n \times \mathbb R^d$. First suppose that we are looking for a generating function of the form $S_h = S_0+h S$ where $S_0(p_1,p_2,x) = (p_1+p_2)x$ 