text:
\section*{Introduction}

The algebraic structure of the spaces of polyvector fields and of polydifferential operators on a manifold play a crucial role in deformation quantization.
In fact, Kontsevich's famous formality theorem asserts that, for a smooth manifold $M$, the Hochschild--Kostant--Rosenberg map extends to an $L_\infty$ quasi-isomorphism from the dgla of polyvector fields on $M$ to the dgla of polydifferential operators on $M$ \cite{MR2062626,MR2699544,MR2102846}.

In this paper, we study ``polyvector fields'' and ``polydifferential operators'' on Lie pairs.
Throughout the paper, we use the symbol $\Bbbk$ to denote either of the fields $\RR$ and $\CC$.
A \emph{Lie algebroid} over $\Bbbk$ is a $\Bbbk$-vector bundle $L\to M$ together with a bundle map $\rho:L\to TM\otimes_\RR\Bbbk$ called \emph{anchor}
and a Lie bracket $[-,-]$ on the sections of $L$ such that $\rho:\sections{L}\to{\mathfrak{X}}(M)\otimes\Bbbk$ is a morphism of Lie algebras
and \[ [X,fY]=f[X,Y]+\big(\rho(X)f\big)Y ,\] for all $X,Y\in\Gamma(L)$ and $f\in C^\infty(M,\Bbbk)$.
By a \emph{Lie pair} $(L,A)$, we mean an inclusion $A\hookrightarrow L$ of Lie algebroids over a smooth manifold $M$.

Lie pairs arise naturally in a number of classical areas of mathematics such as Lie theory, complex geometry, foliation theory, and Poisson geometry.
A complex manifold $X$ determines a Lie pair over $\CC$: $L= T_X\otimes\CC$ and $A=T_X^{0,1}$.
A foliation on a smooth manifold $M$ determines a Lie pair over $\RR$: $L=TM$ and $A$ is the integrable distribution on $M$ tangent to the foliation.

Given a Lie pair $(L,A)$, the quotient $L/A$ is naturally an $A$-module.
When $L$ is the tangent bundle to a manifold $M$ and $A$ is an integrable distribution on $M$,
the infinitesimal $A$-action on $L/A$ is given by the Bott connection \cite{MR0362335}.

Given a Lie pair $(L,A)$, two cohomology groups serve as replacements for polyvector fields and polydifferential operators respectively.
These two groups $H^\bullet_{\CE}(A,\Xpoly{\bullet})$ and $H^\bullet_{\CE}(A,\Dpoly{\bullet})$ are defined as follows.
Denoting the algebra of smooth functions on the manifold $M$ by $R$,
we set $\Xpoly{k}=\sections{\Lambda^{k+1}(L/A)}$ for $k\geqslant 0$, $\Xpoly{-1}=R$,
and $\Xpoly{\bullet}=\bigoplus_{k=-1}^{\infty}\Xpoly{k}$.
The Bott $A$-connection on $L/A$ makes every $\Xpoly{k}$ an ${A}$-module.
The group $H^\bullet_{\CE}(A,\Xpoly{\bullet})$ is the Chevalley--Eilenberg hypercohomology of the complex of $A$-modules with trivial differential
\begin{equation*} \begin{tikzcd}
0 \arrow{r} & \Xpoly{-1} \arrow{r}{0} &
\Xpoly{0} \arrow{r}{0} & \Xpoly{1} \arrow{r}{0} &
\Xpoly{2} \arrow{r}{0} & \cdots
\end{tikzcd} \end{equation*}

Similarly, we set $\Dpoly{\bullet}=\bigoplus_{k=-1}^{\infty}\Dpoly{k}$,
where $\Dpoly{-1}=R$, $\Dpoly{0}=\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$, and
$\Dpoly{k}$ with $k\geqslant 1$ is the tensor product $\Dpoly{0}\otimes_R\cdots\otimes_R\Dpoly{0}$
of $k+1$ copies of the left $R$-module $\Dpoly{0}$.
Multiplication in $\enveloping{L}$ from the left by elements of $\sections{A}$ induces
an $A$-module structure on the quotient $\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$.
This action of $A$ on $\Dpoly{0}$ extends naturally to an action of $A$ on $\Dpoly{k}$ for each $k$.
In fact, $\Dpoly{0}$ is a cocommutative coassociative coalgebra over $R$
whose comultiplication $\Delta:\Dpoly{0}\to\Dpoly{0}\otimes_R\Dpoly{0}$
is a morphism of $A$-modules. Therefore, the Hochschild complex
\[ \begin{tikzcd}
0 \arrow{r} & \Dpoly{-1} \arrow{r}{d_{\mathscr{H}}} & \Dpoly{0} \arrow{r}{d_{\mathscr{H}}} &
\Dpoly{1} \arrow{r}{d_{\mathscr{H}}} & \Dpoly{2} \arrow{r}{d_{\mathscr{H}}} & \cdots
\end{tikzcd} \]
is a complex of $A$-modules.
The group $H^\bullet_{\CE}(A,\Dpoly{\bullet})$ is its Chevalley--Eilenberg hypercohomology.

For the Lie pair $L=T_X\otimes\CC$ and $A=T^{0,1}_X$ arising from a complex manifold $X$,
the Chevalley--Eilenberg hypercohomology group $H_{\CE}^{\bullet}(A,\Xpoly{\bullet})$ is isomorphic to the sheaf cohomology group
$H^\bullet(X,\wedge^\bullet T_X)$, while the Chevalley--Eilenberg hypercohomology group $H_{\CE}^{\bullet}(A,\Dpoly{\bullet})$
is isomorphic to the Hochschild cohomology group $HH^{\bullet}(X)$.

It is simple to see that both $H^\bullet_{\CE}(A,\Xpoly{\bullet})$
and $H^\bullet_{\CE}(A,\Dpoly{\bullet})$ are associative algebras ---
the multiplications in cohomology proceed from the wedge product in
$\Xpoly{\bullet}$ and the tensor product of left $R$-modules in
$\Dpoly{\bullet}$. However, it is in no way obvious that these two cohomology
groups carry Lie algebra structures. We are naturally led to the following
central question.

\begin{introques*}
Do the cohomology groups $H^\bullet_{\CE}(A,\Xpoly{\bullet})$ and $H^\bullet_{\CE}(A,\Dpoly{\bullet})$ associated with a Lie pair $(L,A)$ admit Gerstenhaber algebra structures?
\end{introques*}

To answer this question, we introduce the notion of Fedosov dg Lie algebroids --- they were constructed independently by Batakidis--Voglaire in the special case of matched pairs~\cite{1601.06254}.
Roughly speaking, a Fedosov dg Lie algebroid is a geometric object associated with a Lie pair $(L,A)$ that engenders cochain complexes
whose cohomology groups carry natural Gerstenhaber algebra structures.
These engendered cochain complexes happen to be quasi-isomorphic (in a style reminiscent of Dolgushev's Fedosov resolutions~\cite{MR2102846})
to the cochain complexes used to define $H^\bullet_{\CE}(A,\Xpoly{\bullet})$ and $H^\bullet_{\CE}(A,\Dpoly{\bullet})$.

Given a Lie pair $(L,A)$ and having chosen some additional geometric data,
one can endow the graded manifold $\mathcal{M}=L[1]\oplus L/A$ with a structure of dg manifold $(\mathcal{M},Q)$ quasi-isomorphic to $(A[1],d_A)$.
We will say that $(\mathcal{M},Q)$ is a Fedosov dg manifold associated with the Lie pair $(L,A)$.
A Fedosov dg Lie algebroid associated with the Lie pair $(L,A)$ is a dg Lie subalgebroid $\mathcal{A}\to\mathcal{M}$ of the tangent dg Lie algebroid $T_{\mathcal{M}}\to\mathcal{M}$ of a Fedosov dg manifold $(\mathcal{M},Q)$.
In other words, $\mathcal{A}$ is the dg Lie algebroid encoding a dg foliation of $(\mathcal{M},Q)$.

Since a Lie algebroid can be thought of as an extension of the tangent bundle of a manifold, the notions of polyvector fields and polydifferential operators
admit extensions to the context of a Lie algebroid and these each carry a natural Gerstenhaber algebra structure \cite{MR1675117,MR1815717}.
Likewise, the notions of polyvector fields and polydifferential operators can be extended in an appropriate sense to the context of a dg Lie algebroid.
The corresponding cohomology groups are naturally Gerstenhaber algebras.
The ``polyvector fields'' and ``polydifferential operators'' associated to a Fedosov dg Lie algebroid $\mathcal{A}\to\mathcal{M}$
may be viewed geometrically as polyvector fields and polydifferential operators tangent to the dg foliation on the Fedosov dg manifold $(\mathcal{M},Q)$.
In fact, one can identify the ``polyvector fields'' and ``polydifferential operators'' on $\mathcal{A}\to\mathcal{M}$ to a pair of cochain complexes
$(\sections{\Lambda^\bullet L^\vee}\otimes_R\verticalXpoly{\bullet}, Q)$
and $(\sections{\Lambda^\bullet L^\vee}\otimes_R\verticalDpoly{\bullet}, Q)$,
where $\verticalXpoly{\bullet}$ denotes the formal polyvector fields and $\verticalDpoly{\bullet}$
the formal polydifferential operators tangent to the fibers of the vector bundle $L/A\to M$.

Using these ideas, we prove the following

\begin{introthm}\label{Macau}
Given a Lie pair $(L, A)$, choosing a splitting $j:L/A\to L$ of the short exact sequence $0\to A \to L \to L/A \to 0$
and a torsion-free $L$-connection $\nabla$ on $L/A$ extending the Bott $A$-connection
determines Gerstenhaber algebra structures on $H^\bullet_{\CE}(A,\Xpoly{\bullet})$ and $H^\bullet_{\CE}(A,\Dpoly{\bullet})$.
\end{introthm}

When the short exact sequence $0\to A \to L \to L/A \to 0$ admits a splitting $j: L/A\to L$ whose image
$B:=j(L/A)$ is a Lie subalgebroid of $L$, i.e.\ when the Lie algebroid $L=A\oplus B$ results from a matched pair of Lie algebroids $A\bowtie B$,
the Gerstenhaber algebra structures on $H^\bullet_{\CE}(A, \Xpoly{\bullet})$ and
$H^\bullet_{\CE}(A,\Dpoly{\bullet})$ admit an explicit description, for, in the matched pair case exceptionally,
$(A[1]\oplus B, d_A^{\Bott})$ is a dg Lie algebroid over $(A[1], d_A)$.

\begin{introthm}
If the short exact sequence $0\to A \to L \to L/A \to 0$ admits a splitting $j:L/A\to L$ whose image $B:=j(L/A)$ is a Lie subalgebroid of $L$, then
the dg manifold $(A[1]\oplus B,d_A^{\Bott})$ is a dg Lie algebroid over the dg manifold $(A[1],d_A)$,
the cohomologies of the cochain complexes
$\big(\tot(\sections{\wedge^\bullet A^{\vee}\otimes\wedge^\bullet B}),d_A^{\Bott}+0\big)$ and
$\big(\tot(\sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}^{\otimes\bullet}),d_A^{\mathcal{U}}\pmd_{\mathscr{H}}\big)$
inherit canonical Gerstenhaber algebra structures,
and there are isomorphisms of Gerstenhaber algebras
$H^\bullet_{\CE}(A, \Xpoly{\bullet})\cong H^\bullet\big(\tot(\sections{\wedge^\bullet A^{\vee}\otimes\wedge^\bullet B}),d_A^{\Bott}+0\big)$
and $H^\bullet_{\CE}(A,\Dpoly{\bullet})\cong H^\bullet\big(\tot(\sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}^{\otimes\bullet}),d_A^{\mathcal{U}}\pmd_{\mathscr{H}}\big)$.
\end{introthm}

It is natural to ask whether, for nondescript Lie pairs, the Gerstenhaber algebra structures on
$H^\bullet_{\CE}(A, \Xpoly{\bullet})$ and $H^\bullet_{\CE}(A,\Dpoly{\bullet})$
depend on the choices of splittings and connections made.
This question will be addressed elsewhere; one can prove that the Gerstenhaber algebra structures are indeed independent of these choices and are therefore canonical.

In a forthcoming paper written in collaboration with Liao~\cite{LSX:16}, we establish a Kontsevich--Duflo theorem identifying the Gerstenhaber algebra structures on
$H^\bullet_{\CE}(A, \Xpoly{\bullet})$ and $H^\bullet_{\CE}(A,\Dpoly{\bullet})$ revealed in Theorem~\ref{Macau}.

The second main result of this paper is the proof of an Emmrich--Weinstein theorem for Lie pairs.
Given a smooth manifold $M$, Emmrich--Weinstein~\cite{MR1327535} constructed a dg manifold $(T_M[1]\oplus T_M,Q)$
using Fedosov's iteration method (see also~\cite{MR2102846}) and proved that this dg manifold is, in a certain sense,
equivalent to another one obtained by considering infinite jets of a geodesic exponential map.
The Emmrich--Weinstein theorem was recently extended to $\ZZ$-graded manifolds by Liao--Stiénon~\cite{1508.02780}.
In this paper, we present two equivalent constructions of the Fedosov dg manifold $(L[1]\oplus L/A,Q)$ associated with the Lie pair $(L,A)$.
One is based on the Fedosov iteration method while the other makes use of a Poincaré--Birkhoff--Witt map.

In~\cite{1408.2903}, together with Laurent-Gengoux, we showed that each choice of a splitting $j: L/A\to L$ of the short exact sequence of vector bundles
$0\to A \to L \to L/A \to 0$ and of an $L$-connection $\nabla$ on $L/A$ extending the Bott $A$-connection determines an exponential map
$\exp: L/A\to\groupoid{L}/\groupoid{A}$. Here $\groupoid{L}$ and $\groupoid{A}$ are local Lie groupoids corresponding to the Lie algebroids $L$ and $A$ respectively.
Considering infinite jets of this exponential map, we obtained an isomorphism of filtered $R$-coalgebras
$\pbw:\sections{S(L/A)}\to\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}}$,
which we called Poincaré--Birkhoff--Witt map.
Transfering the canonical infinitesimal $L$-action by coderivations on $\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}}$ through $\pbw$,
we discovered a \emph{flat} $L$-connection $\nabla^\lightning$ on $S(L/A)$:
\[ \nabla^\lightning_l(s)=\pbw^{-1}\big(l\cdot\pbw(s)\big) ,\]
for all $l\in\sections{L}$ and $s\in\sections{S (L/A)}$.
The covariant Chevalley--Eilenberg differential
\[ d_L^{\nabla^\lightning}: \sections{\wedge^\bullet L^\vee \otimes \hat{S} \big((L/A)^{\vee}\big)}\to \sections{\wedge^{\bullet+1} L^\vee \otimes \hat{S} \big((L/A)^{\vee}\big)} \]
of the induced flat $L$-connection on the dual bundle $\hat{S}\big((L/A)^{\vee}\big)$ is a derivation of degree $+1$ of the algebra
$\sections{\wedge^\bullet L^\vee \otimes \hat{S} (L/A)^{\vee}}$ of smooth functions on the graded manifold $L[1]\oplus L/A$.
As a consequence, $(L[1]\oplus L/A, d_L^{\nabla^\lightning})$ is a dg manifold.

In Section~\ref{Harbin}, we prove the following Emmrich--Weinstein theorem for Lie pairs, which could be of independent interest.

\begin{introthm}
Let $(L,A)$ be a Lie pair, let $j:L/A\to L$ be a splitting of the short exact sequence of vector bundles $0\to A\to L\to L/A\to 0$,
and let $\nabla$ be an $L$-connection on $L/A$ extending the Bott $A$-connection.
If $\nabla$ is torsion-free, then the cohomological vector field $Q$ constructed on the graded manifold $L[1]\oplus L/A$
using the Fedosov iteration method coincides with the cohomological vector field $d_L^{\nabla^\lightning}$
obtained using the Poincaré--Birkhoff--Witt map associated to $\nabla$ and $j$.
\end{introthm}

The above theorem reduces to the classical Emmrich--Weinstein theorem when $L=T_M$ and $A$ is trivial.

\subsection*{Terminology and notations}

\subsubsection*{Natural numbers} We use the symbol $\NN$ to denote the set of positive integers and the symbol $\NN_0$ for the set of nonnegative integers.

\subsubsection*{Field $\Bbbk$ and ring $R$} We use the symbol $\Bbbk$ to denote the field of either real or complex numbers.
The symbol $R$ always denotes the algebra of smooth functions on $M$ with values in $\Bbbk$.

\subsubsection*{Tensor products}
For any two $R$-modules $P$ and $Q$, we write $P\otimes_{R} Q$ to denote the tensor product of $P$ and $Q$ as $R$-modules and $P\otimes Q$ to denote the tensor product of $P$ and $Q$ regarded as $\Bbbk$-modules.

\subsubsection*{Completed symmetric algebra}
Given a module $M$ over a ring, the symbol $\hat{S}(M)$ denotes
the $\mathfrak{m}$-adic completion of the symmetric algebra $S(M)$,
where $\mathfrak{m}$ is the ideal of $S(M)$ generated by $M$.

\subsubsection*{Duality pairing}
For every vector bundle $E\to M$, we define a duality pairing
\[ \sections{\hat{S}(E^{\vee})}\times\sections{S(E)} \to R \]
by
\[ \duality{\nu_1\otimes\cdots\otimes\nu_p}{v_1\otimes\cdots\otimes v_q}
=\begin{cases}\sum_{\sigma\in S_p}\prod_{k=1}^p\duality{\nu_k}{v_{\sigma(k)}}
& \text{if } p=q , \\ 0 & \text{otherwise.} \end{cases} \]

\subsubsection*{Multi-indices}
Let $E\to M$ be a smooth vector bundle of finite rank $r$,
let $(\partial_i)_{i\in\{1,\dots,r\}}$ be a local frame of $E$
and let $(\chi_j)_{j\in\{1,\dots,r\}}$ be the dual local frame of $E^{\vee}$.
Thus, we have $\duality{\chi_i}{\partial_j}=\delta_{i,j}$.
Given a multi-index $I=(I_1,I_2,\cdots,I_r)\in\NO^r$,
we adopt the following multi-index notations:
\begin{gather*}
I! = I_1! \cdot I_2! \cdots I_r! \\
\abs{I} = I_1+I_2+\cdots+I_r \\
\partial^I=\underset{I_1 \text{ factors}}{\underbrace{\partial_1\odot\cdots\odot \partial_1}}
\odot\underset{I_2 \text{ factors}}{\underbrace{\partial_2\odot\cdots\odot\partial_2}}
\odot\cdots\odot\underset{I_r \text{ factors}}{\underbrace{\partial_r\odot\cdots\odot\partial_r}} \\
\chi^I=\underset{I_1 \text{ factors}}{\underbrace{\chi_1\odot\cdots\odot\chi_1}}
\odot\underset{I_2 \text{ factors}}{\underbrace{\chi_2\odot\cdots\odot\chi_2}}
\odot\cdots\odot\underset{I_r \text{ factors}}{\underbrace{\chi_r\odot\cdots\odot\chi_r}}
\end{gather*}
We use the symbol $e_k$ to denote the multi-index all of whose
components are equal to $0$ except for the $k$-th which is equal to $1$.
Thus $\chi^{e_k}=\chi_k$.

\subsubsection*{Shuffles}
A $(p,q)$-shuffle is a permutation $\sigma$ of the set $\{1,2,\cdots,p+q\}$
such that $\sigma(1)<\sigma(2)<\cdots<\sigma(p)$
and $\sigma(p+1)<\sigma(p+2)<\cdots<\sigma(p+q)$.
The symbol $\shuffle{p}{q}$ denotes the set of $(p,q)$-shuffles.

\subsubsection*{Graduation shift}
Given a graded vector space $V=\bigoplus_{k\in\ZZ}V^{(k)}$, $V[i]$ denotes the graded
vector space obtained by shifting the grading on $V$ according to the rule
$(V[i])^{(k)}=V^{(i+k)}$. Accordingly, if $E=\bigoplus_{k\in\ZZ}E^{(k)}$
is a graded vector bundle over $M$, $E[i]$ denotes
the graded vector bundle obtained by shifting the degree in the fibers
of $E$ according to the above rule.

\subsubsection*{Koszul sign}
The Koszul sign $\sgn(\sigma; v_1, \cdots, v_{n})$
of a permutation $\sigma$ of homogeneous vectors $v_1,v_2,\dots,v_n$
of a $\ZZ$-graded vector space $V=\bigoplus_{n\in\ZZ}V_n$ is determined by
the equality \[ v_{\sigma(1)}\odot v_{\sigma(2)}\odot\cdots\odot v_{\sigma(n)}
= \sgn(\sigma; v_1,\cdots,v_n) \cdot v_1\odot v_2\odot\cdots\odot v_n \]
in the graded commutative algebra $S(V)$.

\subsubsection*{$L_\infty$ algebra}
An $L_\infty[1]$-algebra~\cite{MR1235010,MR1183483,MR2440258}
is a $\ZZ$-graded vector space $V=\bigoplus_{n\in\ZZ}V_n$ endowed with
a sequence $(\lambda_k)_{k=1}^\infty$ of linear maps
$\lambda_k: S^k(V)\to V[1]$ satisfying the generalized Jacobi identities
\[ \sum_{p+q=n}\sum_{\sigma\in\shuffle{p}{q}}
\sgn(\sigma;v_1,\cdots,v_n)\
\lambda_{1+q}\big(\lambda_p(v_{\sigma(1)},\cdots,v_{\sigma(p)}),
v_{\sigma(p+1)},\cdots,v_{n}\big)=0 \]
for each $n\in\NN$ and for all homogeneous vectors $v_1,v_2,\dots,v_n\in V$.
\newline
A $\ZZ$-graded vector space $V$ is an $L_\infty$-algebra
if and only if $V[1]$ is an $L_\infty[1]$-algebra.

\subsection*{Acknowledgements}

We would like to thank Damien Broka, Camille Laurent-Gengoux, Hsuan-Yi Liao, Kirill Mackenzie, Dominique Manchon, Rajan Mehta, Michael Pevzner, Boris Shoikhet, and Dmitry Tamarkin
for fruitful discussions and useful comments. Stiénon is grateful to Université Paris~7 for its hospitality during his sabbatical leave in 2015--2016.

\section{Preliminaries}

\subsection{Lie algebroids and Lie pairs}

\subsubsection*{Lie algebroids}
We use the symbol $\Bbbk$ to denote either of the fields $\RR$ and $\CC$.
A \emph{Lie algebroid} over $\Bbbk$ is
a $\Bbbk$-vector bundle $L\to M$ together with
a bundle map $\rho:L\to TM\otimes_\RR \Bbbk$
called \emph{anchor} and a Lie bracket
$[-,-]$ on sections of $L$ such that
$\rho:\sections{L}\to{\mathfrak{X}}(M)\otimes \Bbbk$ is a morphism of Lie algebras
and \[ [X,fY]=f[X,Y]+\big(\rho(X)f\big)Y \]
for all $X,Y\in\Gamma(L)$ and $f\in C^\infty(M,\Bbbk)$.
In this paper `Lie algebroid' always means `Lie algebroid over $\Bbbk$' unless specified otherwise.

A $\Bbbk$-vector bundle $L\to M$ is a Lie algebroid if and only if $\sections{L}$ is a \emph{Lie--Rinehart algebra}~\cite{MR0154906} over the commutative ring $C^\infty(M, \Bbbk)$.

\subsubsection*{Lie pairs}
By a \emph{Lie pair} $(L,A)$, we mean an inclusion $A\hookrightarrow L$
of Lie algebroids over a smooth manifold $M$.

\begin{example}
If $\mathfrak{h}$ is a Lie subalgebra of a Lie algebra $\mathfrak{g}$,
then $(\mathfrak{g},\mathfrak{h})$ is a Lie pair over the one-point manifold $\{*\}$.
\end{example}

\begin{example}
If $X$ is a complex manifold, then $(T_X\otimes\CC,T^{0,1}_X)$ is a Lie pair over $X$.
\end{example}

\begin{example}
If $\mathcal{F}$ is a foliation of a smooth manifold $M$, then $(T_M,T_{\mathcal{F}})$ is a Lie pair over $M$.
\end{example}

\subsubsection*{Matched pairs}
A \emph{matched pair of Lie algebroids} $L=A\bowtie B$ is a Lie pair $(L,A)$ together with a splitting
$j:B\to L$ of the short exact sequence $0\to A\to L\to B\to 0$, whose image $j(B)$ happens to be a Lie subalgebroid of $L$ --- see \cite{MR1460632} for more details.

\begin{example}
If $X$ is a complex manifold, then $T_X\otimes\CC=T^{0,1}_X\bowtie T^{1,0}_X$ is a matched pair of complex Lie algebroids over $X$.
\end{example}

\begin{example}
Let $G$ be a Poisson Lie group and let $P$ be a Poisson $G$-space, i.e.\ a Poisson manifold $(P,\pi)$ endowed with a $G$-action
$G\times P\to P$ which happens to be a Poisson map. According to Lu~\cite{MR1430434}, the cotangent Lie algebroid $A=\big(T_P^\vee\big)_\pi$
and the transformation Lie algebroid $B=P\rtimes\mathfrak{g}$ form a matched pair of Lie algebroids over the manifold $P$.
\end{example}

\subsection{Chevalley--Eilenberg differential}

Let $L$ be a Lie algebroid over a smooth manifold $M$, and $R$ be the algebra of smooth functions on $M$ valued in $\Bbbk$.
The Chevalley--Eilenberg differential
\[ d_L:\sections{\Lambda^k L^{\vee}}\to\sections{\Lambda^{k+1} L^{\vee}} \]
defined by
\begin{multline*}
\big(d_L \omega\big)(v_0,v_1,\cdots,v_k)=
\sum_{i=0}^{n} \rho(v_i)\big(\omega(v_0,\cdots,\widehat{v_i},\cdots,v_k)\big) \\
+\sum_{i<j}\omega(\lie{v_i}{v_j},v_0,\cdots,\widehat{v_i},\cdots,\widehat{v_j},\cdots,v_k)
\end{multline*}
and the exterior product make $\bigoplus_{k\geqslant 0}\sections{\Lambda^k L^{\vee}}$
into a differential graded commutative $R$-algebra.

The following proposition is an immediate consequence of the definitions.
\begin{proposition}\label{Piacenza}
Let $L$ be a Lie algebroid and let $A$ and $B$ be two vector subbundles of $L$ such that $L=A\oplus B$. Set $\Omega^{u,v}=\sections{p\transpose(\Lambda^u A^{\vee})
\otimes q\transpose(\Lambda^v B^{\vee})}$ where $p:L\twoheadrightarrow A$ and $q:L\twoheadrightarrow B$ denote
the canonical projections.
\begin{enumerate}
\item If neither $A$ nor $B$ is a Lie subalgebroid of $L$, then
\[ d_L(\Omega^{u,v})\subset \Omega^{u+2,v-1}\oplus\Omega^{u+1,v}\oplus
\Omega^{u,v+1}\oplus\Omega^{u-1,v+2} .\]
\item If $A$ is a Lie subalgebroid of $L$, i.e.\ if $(L,A)$ is a Lie pair, then
\[ d_L(\Omega^{u,v})\subset \Omega^{u+1,v}\oplus
\Omega^{u,v+1}\oplus\Omega^{u-1,v+2} .\]
\item If both $A$ and $B$ are Lie subalgebroids of $L$, i.e.\ if $L=A\bowtie B$ is a matched pair, then \[ d_L(\Omega^{u,v})\subset \Omega^{u+1,v}\oplus\Omega^{u,v+1} .\]
\end{enumerate}
\end{proposition}

\subsection{Polydifferential operators for a Lie algebroid}
\label{universal}

Let $L$ be a Lie $\Bbbk$-algebroid over a smooth manifold $M$
and let $R$ denote the algebra of smooth functions on $M$ taking values in $\Bbbk$.
The vector space $\mathfrak{g}:=R\oplus\sections{L}$ admits a natural
Lie algebra structure given by the Lie bracket
\[ (f+X)\otimes(g+Y) \longmapsto X(g)-Y(f)+\lie{X}{Y} ,\]
where $f,g\in R$ and $X,Y\in\sections{L}$.
Its universal enveloping algebra $\enveloping{\mathfrak{g}}$ is the quotient
of the tensor algebra $T(\mathfrak{g})=\bigoplus_{k=0}^{\infty}
\big(\bigotimes_{\Bbbk}^k (R\oplus\sections{L})\big)$
by the ideal generated by the subset of all elements of the form
\[ (f+X)\otimes(g+Y)-(g+Y)\otimes(f+X)-\big(X(g)-Y(f)+\lie{X}{Y}\big) \]
with $f,g\in R$ and $X,Y\in\sections{L}$.
Let $i$ denote the natural inclusion of $\mathfrak{g}$ into $\enveloping{\mathfrak{g}}$
and let $\mathcal{V}(\mathfrak{g})$ denote the subalgebra of $\enveloping{\mathfrak{g}}$
generated by $i(\mathfrak{g})$.
The \emph{universal enveloping algebra $\enveloping{L}$ of the Lie algebroid $L$}
is the quotient of $\mathcal{V}(\mathfrak{g})$ by the two-sided ideal generated by the
elements of the form
\[ i(f)\otimes i(g+Y)-i(fg+fY) \]
with $f,g\in R$ and $Y\in\sections{L}$.
Note that we have implicitly used the left $R$-module structure of
 $\mathfrak{g}$.
The graduation of $T(\mathfrak{g})$ induces a natural ascending filtration
\begin{equation} \label{Watermael}
\cdots \hookrightarrow \mathcal{U}^{\leqslant n-1}(L) \hookrightarrow \mathcal{U}^{\leqslant n}(L) \hookrightarrow \mathcal{U}^{\leqslant n+1}(L) \hookrightarrow \cdots
\end{equation}
on $\enveloping{L}$.

\begin{lemma}\label{Pigalle}
For all $\sigma\in S_n$ and $X_1,\dots,X_n$ in $\sections{L}$, we have
\[ X_1\cdots X_n \equiv X_{\sigma(1)}\cdots X_{\sigma(n)}
\mod \mathcal{U}^{\leqslant n-1}(L) .\]
\end{lemma}

\begin{proof}
It suffices to prove that
\[ X_1\cdots X_n \equiv X_1\cdots X_{k-1} X_{k+1} X_k X_{k+2} \cdots X_n
\mod \mathcal{U}^{\leqslant n-1}(L) \]
for all $X_1,\dots,X_n\in\sections{L}$.
It follows from
$X_k\cdot X_{k+1}=X_{k+1}\cdot X_k+\lie{X_k}{X_{k+1}}$ that
\[ X_1\cdots X_n= X_1\cdots X_{k-1} X_{k+1} X_k X_{k+2} \cdots X_n
+ \underset{\in\ \mathcal{U}^{\leqslant n-1}(L)}{\underbrace{X_1\cdots X_{k-1}\lie{X_k}{X_{k+1}} X_{k+2} \cdots X_n}} .\qedhere\]
\end{proof}

When the base $M$ of the Lie algebroid $L$ is the one-point space so that the only fiber is a Lie algebra $\mathfrak{h}$,
the universal enveloping algebra of the Lie algebroid is the universal enveloping algebra of the Lie algebra $\mathfrak{h}$.
When the Lie algebroid $L$ is the tangent bundle $T_M\to M$,
its universal enveloping algebra $\enveloping{L}$ is the algebra of differential operators on $M$.
In general, when $L$ is a
Lie algebroid over $\RR$,
 its universal enveloping algebra $\enveloping{L}$
can canonically be identified with the associative algebra
of source-fiberwise differential operators on $C^\infty(\groupoid{L})$ invariant under left translations~\cite{MR1747916},
where $\groupoid{L}$ is a local Lie groupoid with Lie algebroid $L$.
Hence we can think of the elements of $\enveloping{L}$ as differential operators for the Lie algebroid $L$
and of the elements of $\bigotimes^{\bullet}_R\enveloping{L}$ as polydifferential operators for the Lie algebroid $L$.

The universal enveloping algebra $\enveloping{L}$ of the Lie algebroid $L\to M$ is a coalgebra over $R$~\cite{MR1815717}.
Its comultiplication
\[ \Delta:\enveloping{L}\to\enveloping{L}\otimes_R\enveloping{L} \]
is compatible with its filtration \eqref{Watermael} and characterized by the identities
\begin{gather*}
\Delta(1)=1\otimes 1; \\
\Delta(b)=1\otimes b+b\otimes 1, \quad \forall b\in \sections{L}; \\
\Delta(u\cdot v)=\Delta(u)\cdot\Delta(v), \quad \forall u,v\in\enveloping{L} ,
\end{gather*}
where $1\in R$ denotes the constant function on $M$ with value $1$ while the symbol $\cdot$ denotes the multiplication in $\enveloping{L}$. We refer the reader to~\cite{MR1815717} for the precise meaning of the last equation above.
Explicitly, we have
\begin{multline*}
\Delta(b_1\cdot b_2\cdot\cdots\cdot b_n)=
1\otimes(b_1\cdot b_2\cdot\cdots\cdot b_n)
+ \sum_{\substack{p+q=n \\ p,q\in\NN}}\sum_{\sigma\in\shuffle{p}{q}} (b_{\sigma(1)}\cdot\cdots\cdot b_{\sigma(p)}) \otimes
(b_{\sigma(p+1)}\cdot\cdots\cdot b_{\sigma(n)}) \\
+ (b_1\cdot b_2\cdot\cdots\cdot b_n)\otimes 1
,\end{multline*}
for all $b_1,\dots,b_n\in\sections{L}$.

\subsection{Connections and representations for Lie algebroids}
Let $M$ be a smooth manifold, let $L\to M$ be a Lie $\Bbbk$-algebroid with anchor map $\rho:L\to T_M\otimes_{\RR}\Bbbk$,
and let $E\xto{\varpi}M$ be a vector bundle over $\Bbbk$. The algebra of smooth functions on $M$ with values in $\Bbbk$ will be denoted $R$.

The traditional description of a (linear) $L$-connection on $E$ is in terms of a \emph{covariant derivative}
\[ \sections{L}\times\sections{E}\to\sections{E}: (l,e)\mapsto \nabla_l e \]
characterized by the following two properties:
\begin{gather}
\nabla_{f\cdot l} e=f\cdot \nabla_l e , \label{Molenbeek} \\
\nabla_l (f\cdot e)=\rho(l)f\cdot e+f\cdot\nabla_l e \label{Anderlecht}
,\end{gather}
for all $l\in\sections{L}$, $e\in\sections{E}$, and $f\in R$.

\begin{remark}
A covariant derivative
$\nabla:\sections{L}\times\sections{B}\to\sections{B}$
induces a covariant derivative
$\nabla:\sections{L}\times\sections{S(B)}\to\sections{S(B)}$
through the relation
\[ \nabla_l (b_1\odot\cdots\odot b_n) = \sum_{k=1}^n b_1\odot\cdots\odot\nabla_l b_k\odot\cdots\odot b_n ,\]
for all $l\in\sections{L}$ and $b_1,\dots,b_n\in\sections{B}$.
\end{remark}

\begin{remark}\label{Amalfi}
A covariant derivative
$\nabla:\sections{L}\times\sections{S(B)}\to\sections{S(B)}$
induces a covariant derivative
$\nabla:\sections{L}\times\sections{\hat{S}(B^{\vee})}\to\sections{\hat{S}(B^{\vee})}$
through the relation
\[ \rho(l)\duality{\sigma}{s}=\duality{\nabla_l\sigma}{s}+\duality{\sigma}{\nabla_l s} \]
for all $l\in\sections{L}$, $s\in\sections{B}$,
and $\sigma\in\sections{\hat{S}(B^{\vee})}$.
\end{remark}

A \emph{representation of a Lie algebroid} $A$ on a vector bundle $E\to M$ is
a flat $A$-connection $\nabla$ on $E$, i.e.\ a covariant derivative
$\nabla:\sections{A}\times\sections{E}\to\sections{E}$ satisfying
\begin{equation}\label{Jette}
\nabla_{a_1}\nabla_{a_2} e-\nabla_{a_2}\nabla_{a_1} e=\nabla_{\lie{a_1}{a_2}}e ,
\end{equation}
for all $a_1,a_2\in\sections{A}$ and $e\in\sections{E}$.
A vector bundle endowed with a representation of the Lie algebroid $A$ is called an \emph{$A$-module}.
More generally, given a left $R$-module $\mathcal{M}$, by an {\em infinitesimal action} of $A$ on
$\mathcal{M}$, we mean a $\Bbbk$-bilinear map $\nabla:\sections{A}\times\mathcal{M}\to\mathcal{M}$, $(a,e)\mapsto\nabla_a e$
satisfying Equations.~\eqref{Molenbeek}, \eqref{Anderlecht}, and~\eqref{Jette}.
In other words, $\nabla$ is a representation of the Lie--Rinehart algebra $(\sections{A},R)$~\cite{MR0154906}.

\begin{example}\label{Auderghem}
Let $(L,A)$ be a Lie pair, i.e.\ an inclusion $A\hookrightarrow L$ of Lie algebroids.
The \emph{Bott representation} of $A$ on the quotient $L/A$ is the flat connection defined by
\[ \nabla^{\Bott}_a q(l)=q\big(\lie{a}{l}\big), \quad\forall a\in\sections{A},l\in\sections{L} ,\]
where $q$ denotes the canonical projection $L\twoheadrightarrow L/A$.
Thus the quotient $L/A$ of a Lie pair $(L,A)$ is an $A$-module.
\end{example}

\subsection{Poincaré--Birkhoff--Witt isomorphisms}

Let $(L,A)$ be a pair of Lie algebroids over $\Bbbk$.

Writing $\enveloping{L}\sections{A}$ for the left ideal of $\enveloping{L}$ generated by $\sections{A}$,
the quotient $\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$ is automatically a filtered $R$-coalgebra since
\[ \Delta\big(\enveloping{L}\sections{A}\big)\subseteq \enveloping{L}\otimes_R \big(\enveloping{L}\sections{A}\big) + \big(\enveloping{L}\sections{A}\big) \otimes_R \enveloping{L} \]
and the filtration \eqref{Watermael} on $\enveloping{L}$ descends to a filtration
\begin{equation*}
\cdots \hookrightarrow \left(\frac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n-1} \hookrightarrow \left(\frac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n} \hookrightarrow \left(\frac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n+1} \hookrightarrow \cdots
\end{equation*}
of $\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$.

Likewise, deconcatenation defines an $R$-coalgebra structure on $\sections{S(L/A)}$.
The comultiplication
\[ \Delta:\sections{S(L/A)}\to\sections{S(L/A)}\otimes_R\sections{S(L/A)} \] is given by
\begin{multline*}
\Delta(b_1\odot b_2\odot\cdots\odot b_n)=
1\otimes(b_1\odot b_2\odot\cdots\odot b_n) \\
\shoveright{+ \sum_{\substack{p+q=n \\ p,q\in\NN}}\sum_{\sigma\in\shuffle{p}{q}} (b_{\sigma(1)}\odot\cdots\odot b_{\sigma(p)}) \otimes (b_{\sigma(p+1)}\odot\cdots\odot b_{\sigma(n)})} \\
+ (b_1\odot b_2\odot\cdots\odot b_n)\otimes 1
,\end{multline*}
for all $n\in\NN$ and $b_1,\dots,b_n\in\sections{L/A}$.
The symbol $\odot$ denotes the symmetric product in $\sections{S(L/A)}$.

The following theorem, which was obtained in \cite{1408.2903}, is an
extension of the classical Poincaré--Birkhoff--Witt isomorphism to Lie pairs.

\begin{theorem}[\cite{1408.2903}] \label{Nairobi}
Let $(L,A)$ be a Lie pair.
Given a splitting $j:L/A\to L$ of the short exact sequence $0\to A \to
L \to L/A \to 0$,
and a $L$-connection $\nabla$ on $L/A$ extending the Bott $A$-connection,
 there exists a unique isomorphism
 of filtered $R$-coalgebras
\[ \pbw^{\nabla,j}:\sections{S(L/A)}\to\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}} \]
satisfying
\begin{align}
\pbw^{\nabla,j}(f) &= f, \quad\forall f\in R; \label{pbwzero} \\
\pbw^{\nabla,j}(b) &= j(b), \quad\forall b\in\sections{L/A}; \label{pbwun} \\
\pbw^{\nabla,j}(b^{n+1}) &= j(b)\cdot\pbw^{\nabla,j}(b^n)-\pbw^{\nabla,j}\big(\nabla_{j(b)}(b^n)\big) \label{pbwpower}
\end{align}
for all $n\in \NN, b\in \sections{L/A}$.
\end{theorem}

\begin{remark}
Equation~\eqref{pbwpower} is equivalent to
\begin{multline}\label{Rome}
\pbw^{\nabla,j}(b_0 \odot \cdots \odot b_n) =\tfrac{1}{n+1}\sum_{i=0}^{n} \Big( j(b_i) \cdot
\pbw^{\nabla,j}(b_0 \odot \cdots \odot \widehat{b_i} \odot \cdots \odot b_n ) \\ -
\pbw^{\nabla,j}\big(\nabla_{j(b_i)} ( b_0 \odot \cdots \odot \widehat{b_i} \odot \cdots \odot b_n ) \big) \Big)
\end{multline}
for all $b_0,\dots,b_n \in \sections{L/A}$.
\end{remark}

It is immediate that Equations~\eqref{pbwzero}, \eqref{pbwun}, and~\eqref{Rome}
together define inductively a unique $R$-linear map $\pbw^{\nabla,j}$.

\begin{remark}
When $L=T_M$ and $A$ is the trivial Lie subalgebroid of $L$ of rank 0,
the $\pbw^{\nabla,j}$ map of Theorem~\ref{Nairobi} is the inverse of the so called ``complete symbol map,''
which is an isomorphism from the space $\enveloping{T_M}$ of differential operators on $M$
to the space $\sections{S(T_M)}$ of fiberwisely polynomial functions on $T^{\vee}_M$.
The complete symbol map was generalized to arbitrary Lie algebroids over $\RR$
by Nistor--Weinstein--Xu~\cite{MR1687747}.
It played an important role in quantization theory~\cite{MR706215,MR1687747}.
\end{remark}

\begin{lemma}\label{Stockholm}
For all $n\in\NN$ and $b_1,\dots,b_n\in\sections{L/A}$,
\[ \pbw^{\nabla,j}(b_1\odot\cdots\odot b_n)-\frac{1}{n!}\sum_{\sigma\in S_n}
j(b_{\sigma(1)})\cdot j(b_{\sigma(2)})\cdot \cdots \cdot j(b_{\sigma(n)}) \]
is an element of $\left(\frac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n-1}$.
\end{lemma}

\begin{proof}
It follows from the inductive relation \eqref{Rome} that
\begin{multline*} \pbw^{\nabla,j}(b_1\odot\cdots\odot b_n)-\frac{1}{n}\sum_{k=1}^n j(b_k)\cdot\pbw^{\nabla,j}(b_1\odot\cdots
\odot\widehat{b_k}\odot\cdots\odot b_n) \\
=-\frac{1}{n}\sum_{k=1}^n \pbw^{\nabla,j}\big(\nabla_{j(b_k)}(b_1\odot\cdots
\odot\widehat{b_k}\odot\cdots\odot b_n)\big) \end{multline*}
belongs to $\left(\frac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n-1}$
as \[ \nabla_{j(b_k)}(b_1\odot\cdots\odot\widehat{b_k}\odot\cdots\odot b_n)\in \sections{S^{n-1}(L/A)} \]
and $\pbw^{\nabla,j}$ respects the filtrations.
The result follows by induction on $n$.
\end{proof}

\begin{corollary}\label{Duroc}
For all homogeneous elements $b_1,\dots,b_n$ in $\sections{B}$, we have
\[ \pbw^{\nabla,j}(b_1\odot \cdots \odot b_n) \equiv j(b_1)\cdots j(b_n) \mod
\left(\frac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n-1} \]
\end{corollary}

\begin{proof}
It follows from Lemma~\ref{Stockholm} and Lemma~\ref{Pigalle} that
\[ \pbw^{\nabla,j}(b_1\odot \cdots \odot b_n)
\equiv \frac{1}{n!}\sum_{\sigma\in S_n} j(b_{\sigma(1)}) \cdots j(b_{\sigma(n)})
\equiv j(b_1) \cdots j(b_n) \mod
\left(\frac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n-1}
.\qedhere \]
\end{proof}

\subsection{Torsion-free connections extending the Bott representation}

Let $(L,A)$ be a pair of Lie algebroids over $\Bbbk$.
Consider the short exact sequence of vector bundles
\begin{equation}\label{Sydney}
\begin{tikzcd}
0 \arrow{r} & A \arrow{r}{i} & L \arrow{r}{q} & L/A \arrow{r} & 0
\end{tikzcd}
.\end{equation}

An $L$-connection $\nabla$ on $L/A$ is said to extend
the Bott $A$-connection on $L/A$ (see Example~\ref{Auderghem}) if
\[ \nabla_{i(a)} q(l)=\nabla^{\Bott}_a q(l)
=q\big(\lie{i(a)}{l}\big) ,\quad\forall a\in\sections{A},l\in\sections{L} .\]

Given an $L$-connection on $L/A$ extending the Bott $A$-connection on $L/A$,
the bundle map $T^\nabla:\Lambda^2 L\to L/A$ defined by
\begin{equation}\label{Tibet}
T^\nabla(l_1,l_2)=\nabla_{l_1}q(l_2)-\nabla_{l_2}q(l_1)-q\big(\lie{l_1}{l_2}\big),
\quad\forall l_1,l_2\in\sections{L}
\end{equation}
satisfies $T^\nabla(a,l)=0$ for all $a\in\sections{A}$ and $l\in\sections{L}$, so that there exists a unique bundle map
\begin{equation}\label{eq:dog}
\beta^\nabla:\Lambda^2(L/A)\to L/A,
\end{equation}
called the \emph{torsion} of $\nabla$, making the diagram
\[ \begin{tikzcd}
\Lambda^2 L \arrow{d}[swap]{q} \arrow{r}{T^\nabla} & L/A \\ \Lambda^2 (L/A) \arrow{ru} [swap]{\beta^\nabla} &
\end{tikzcd} \]
commute.

Torsion-free $L$-connections on $L/A$ extending
the Bott $A$-connection always exist:
\begin{lemma}
Given an $L$-connection $\nabla$ on $L/A$, setting
\[ \nabla'_{l_1}q(l_2)=\nabla_{l_1}q(l_2)-\tfrac{1}{2}T^\nabla(l_1,l_2) ,\quad\forall l_1,l_2\in\sections{L} \]
defines a torsion-free $L$-connection $\nabla'$ on $L/A$, i.e.\ $\beta^{\nabla'}=0$. \\
Moreover, if the $L$-connection $\nabla$ on $L/A$ extends the Bott $A$-connection, then so does
the torsion-free $L$-connection $\nabla'$.
\end{lemma}

\begin{proposition}\label{Glaciere}
Given a splitting $j:L/A\to L$ of the short exact sequence~\eqref{Sydney} and an $L$-connection $\nabla$ on $L/A$ extending the Bott $A$-connection
 on $L/A$, we have
\begin{equation*}
\pbw^{\nabla,j}(Y\odot Z) = j(Y)\cdot j(Z) -j\big(\nabla_{j(Y)} Z\big) +\tfrac{1}{2}\ j\circ\beta^\nabla(Y,Z)
\end{equation*}
for all elements $Y,Z$ of $\sections{L/A}$.
\end{proposition}

\section{Polydifferential operators and polyvector fields for Lie pairs}

\subsection{Chevalley--Eilenberg cohomology}

The Chevalley--Eilenberg covariant differential
associated to a representation $\nabla$ of a Lie algebroid $L\to M$
of rank $n$ on a vector bundle $B\to M$ is the operator
\[ d_L^{\nabla}: \sections{\Lambda^k L^{\vee}\otimes B}\to\sections{\Lambda^{k+1} L^{\vee}\otimes B} \]
that takes a section $\omega\otimes e$ of
$\Lambda^k L^{\vee}\otimes B$ to
\[ d_L^{\nabla}(\omega\otimes e)=(d_L\omega)\otimes e
+\sum_{j=1}^{n}(\nu_j\wedge\omega)\otimes \nabla_{v_j}e ,\]
where $v_1,v_2,\dots,v_n$ and $\nu_1,\nu_2,\dots,\nu_n$
are any pair of dual local frames for the vector bundles $L$ and $L^{\vee}$.
Because the connection $\nabla$ is flat, $d_L^{\nabla}$ is a coboundary operator:
$d_L^{\nabla}\circ d_L^{\nabla}=0$.

Let $A\to M$ be a Lie algebroid.
The symbol $\category{A}$ denotes the abelian category of left modules over $\enveloping{A}$.
Its bounded below derived category is denoted by $D^+(\category{A})$.
The Chevalley--Eilenberg cohomology in degree $k$
of a complex of $\enveloping{A}$-modules
\[ \begin{tikzcd}
0 \arrow{r} & \mathcal{E}^{-1} \arrow{r}{d} & \mathcal{E}^{0} \arrow{r}{d} & \mathcal{E}^{1} \arrow{r}{d} & \mathcal{E}^{2} \arrow{r}{d} & \cdots
\end{tikzcd} \]
is \[ H_{\CE}^k(A,\mathcal{E}^\bullet):=\Hom_{D^+ (\category{A})}(R,\mathcal{E}^\bullet[k]) ,\]
where $R$ is the algebra of functions on the base manifold $M$.
It is computed as the total cohomology in degree $k$ of the double complex
\[ \begin{tikzcd}[row sep=small]
\vdots & \vdots & \vdots & \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\mathcal{E}^{1} \arrow{u}{\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\mathcal{E}^{1} \arrow{u}{-\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\mathcal{E}^{1} \arrow{u}{\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} & \cdots \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\mathcal{E}^{0} \arrow{u}{\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\mathcal{E}^{0} \arrow{u}{-\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\mathcal{E}^{0} \arrow{u}{\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} & \cdots \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\mathcal{E}^{-1} \arrow{u}{\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\mathcal{E}^{-1} \arrow{u}{-\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\mathcal{E}^{-1} \arrow{u}{\id\otimes d} \arrow{r}{d_A^{\mathcal{E}}} & \cdots
\end{tikzcd} \]

\subsection{Polydifferential operators}

Given a Lie pair $(L,A)$, let $\Dpoly{-1}$ denote the algebra $R$ of smooth functions on the manifold $M$,
let $\Dpoly{0}$ denote the left $\enveloping{A}$-module
$\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$,
let $\Dpoly{k}$ denote the tensor product
$\Dpoly{0}\otimes_R\cdots\otimes_R\Dpoly{0}$ of $(k+1)$ copies of
the left $R$-module $\Dpoly{0}$, and set
$\Dpoly{\bullet}=\bigoplus_{k=-1}^{\infty}\Dpoly{k}$.
Since $\Dpoly{0}$ is a left $\enveloping{A}$-module
and $\enveloping{A}$, as a Hopf algebroid, is endowed with a
comultiplication, $\Dpoly{k}$ is also naturally a left $\enveloping{A}$-module
for each $k\geq -1$ \cite{MR1815717}.

\begin{lemma}
The $\enveloping{A}$-module $\Dpoly{0}$ is a cocommutative coassociative coalgebra over $R$
whose comultiplication $\Delta:\Dpoly{0}\to\Dpoly{0}\otimes_R\Dpoly{0}$ is
a morphism of $\enveloping{A}$-modules.
\end{lemma}

Since the comultiplication $\Delta$ is coassociative,
the Hochschild operator $d_{\mathscr{H}}:\Dpoly{k-1}\to\Dpoly{k}$ defined by
\begin{multline*}
d_{\mathscr{H}}(u_1\otimes\cdots\otimes u_k) = 1\otimes u_1\otimes\cdots\otimes u_k
+ \sum_{i=1}^{k} (-1)^i u_1\otimes\cdots\otimes\Delta(u_i)\otimes\cdots\otimes u_k \\
+ (-1)^{k+1} u_1\otimes\cdots\otimes u_k\otimes 1
\end{multline*}
is a coboundary operator, i.e.\ $d_{\mathscr{H}}^2=0$.

Moreover, $d_{\mathscr{H}}:\Dpoly{k-1}\to\Dpoly{k}$ is
 a morphism of $\enveloping{A}$-modules,
 since the comultiplication
$\Delta:\Dpoly{0}\to\Dpoly{0}\otimes_R\Dpoly{0}$ is
a morphism of $\enveloping{A}$-modules.
Therefore, the Hochschild complex
\begin{equation*}\begin{tikzcd}
0 \arrow{r} & \Dpoly{-1} \arrow{r}{d_{\mathscr{H}}} & \Dpoly{0} \arrow{r}{d_{\mathscr{H}}} &
\Dpoly{1} \arrow{r}{d_{\mathscr{H}}} & \Dpoly{2} \arrow{r}{d_{\mathscr{H}}} & \cdots
\end{tikzcd}\end{equation*}
is a complex of $\enveloping{A}$-modules.

The Chevalley--Eilenberg cohomology in degree $k$ of the Hochschild complex
of the pair $(L,A)$, which is defined as
\[ H^k_{\CE}(A,\Dpoly{\bullet}) := \Hom_{D^+ (\category{A})}(R,\Dpoly{\bullet}[k]) ,\]
can be computed as the degree $k$
 hypercohomology of the double complex
\[ \begin{tikzcd}[row sep=small]
\vdots & \vdots & \vdots & \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\Dpoly{1} \arrow{u}{\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\Dpoly{1} \arrow{u}{-\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\Dpoly{1} \arrow{u}{\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} & \cdots \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\Dpoly{0} \arrow{u}{\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\Dpoly{0} \arrow{u}{-\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\Dpoly{0} \arrow{u}{\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} & \cdots \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\Dpoly{-1} \arrow{u}{\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\Dpoly{-1} \arrow{u}{-\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\Dpoly{-1} \arrow{u}{\id\otimesd_{\mathscr{H}}} \arrow{r}{d_A^{\mathcal{U}}} & \cdots
\end{tikzcd} \]

The coboundary operator
$d_A^{\mathcal{U}}:\sections{\Lambda^p A^{\vee}}\otimes
\Dpoly{q}\to\sections{\Lambda^{p+1} A^{\vee}}\otimes
\Dpoly{q}$ is defined by
\[ \begin{split} d_A^{\mathcal{U}}(\omega\otimes u_0\otimes\cdots\otimes u_q)
=&\ (d_A\omega)\otimes u_0\otimes\cdots\otimes u_q \\
&\ +\sum_{j=1}^{\rk(A)}\sum_{k=0}^{q} (\alpha_j\wedge\omega)
\otimes u_0\otimes \cdots\otimes u_{k-1} \otimes
a_j\cdot u_k\otimes u_{k+1} \otimes \cdots\otimes u_q
,\end{split} \]
where $(a_i)_{i\in\{1,\dots,r\}}$ is any local frame of $A$
and $(\alpha_j)_{j\in\{1,\dots,r\}}$ is the dual local frame of $A^{\vee}$.

However, unlike the universal enveloping algebra $\enveloping{L}$
of the Lie algebroid $L$, $\Dpoly{0}$ is in general not a Hopf algebroid over $R$
(in fact, $\Dpoly{0}$ is not even an associative algebra).
Therefore, a priori, the Hochschild cohomology is only a vector space.
The following proposition is, however, quite obvious.

\begin{proposition}
For any Lie pair $(L, A)$, the Hochschild cohomology
$H^\bullet_{\CE}(A,\Dpoly{\bullet})$ is an associative
algebra whose multiplication stems from
the tensor product of left $R$-modules $\otimes_R$ in $\Dpoly{\bullet}$.
\end{proposition}

\subsection{Polyvector fields}

Likewise, given a Lie pair $(L,A)$, let $\Xpoly{-1}$ denote the algebra $R$ of smooth functions on the manifold $M$ and
set $\Xpoly{k}:=\sections{\Lambda^{k+1}(L/A)}$
for $k\geqslant 0$.
Consider $\Xpoly{\bullet}=\bigoplus_{k=-1}\Xpoly{k}$ as a complex of $\enveloping{A}$-modules with trivial differential:
\begin{equation*} \begin{tikzcd}
0 \arrow{r} & \Xpoly{-1} \arrow{r}{0} &
\Xpoly{0} \arrow{r}{0} & \Xpoly{1} \arrow{r}{0} &
\Xpoly{2} \arrow{r}{0} & \cdots
\end{tikzcd} \end{equation*}
Its Chevalley--Eilenberg cohomology in degree $k$
\[ H^k_{\CE}(A,\Xpoly{\bullet}) := \Hom_{D^+ (\category{A})}(R,\Xpoly{\bullet}[k]) ,\]
is computed as the degree $k$ hypercohomology of the double complex
\[ \begin{tikzcd}[row sep=small]
\vdots & \vdots & \vdots & \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\Xpoly{1} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\Xpoly{1} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\Xpoly{1} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} & \cdots \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\Xpoly{0} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\Xpoly{0} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\Xpoly{0} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} & \cdots \\
\sections{\Lambda^0 A^{\vee}}\otimes_R\Xpoly{-1} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^1 A^{\vee}}\otimes_R\Xpoly{-1} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^2 A^{\vee}}\otimes_R\Xpoly{-1} \arrow{u}{0} \arrow{r}{d_A^{\Bott}} & \cdots
\end{tikzcd} \]

The coboundary operator
$d_A^{\Bott}:\sections{\Lambda^p A^{\vee}}\otimes
\Xpoly{q}\to\sections{\Lambda^{p+1} A^{\vee}}\otimes
\Xpoly{q}$ is defined by
\[ \begin{split} d_A^{\Bott}(\omega\otimes b_0\wedge\cdots\wedge b_q)
=&\ (d_A\omega)\otimes b_0\wedge\cdots\wedge b_q \\
&\ +\sum_{j=1}^{\rk(A)}\sum_{k=0}^{q} (\alpha_j\wedge\omega)
\otimes b_0\wedge \cdots\wedge b_{k-1} \wedge
\nabla^{\Bott}_{a_j} b_k\wedge b_{k+1} \wedge\cdots\wedge b_q
,\end{split} \]
where $(a_i)_{i\in\{1,\dots,r\}}$ is any local frame of $A$
and $(\alpha_j)_{j\in\{1,\dots,r\}}$ is the dual local frame of $A^{\vee}$.

Again, a priori, $H^\bullet_{\CE}(A,\Xpoly{\bullet})$
is only a vector space. We have the following

\begin{proposition}
For any Lie pair $(L, A)$, the cohomology $H^\bullet_{\CE}(A,\Xpoly{\bullet})$
is an associative algebra whose multiplication stems from
the wedge product in $\Xpoly{\bullet}$.
\end{proposition}

\section{Fedosov dg manifolds and Emmrich--Weinstein theorem for Lie pairs}

\subsection{Homological perturbation}\label{Vilnius}

A cochain complex $(N,\delta)$ is said to \emph{contract} onto a cochain complex
$(M,d)$ if there exists two chain maps $\sigma:N\to M$ and $\tau:M\to N$
and an endomorphism $h:N\to N[1]$ of the graded module $N$ satisfying
the following five relations:
\begin{gather*} \sigma\tau=\id_M, \qquad \tau\sigma-\id_N=h\delta+\delta h, \\
\sigma h=0, \qquad h\tau=0, \qquad h^2=0 .\end{gather*}
If, furthermore, the cochain complexes $N$ and $M$ are filtered and the maps $\sigma$, $\tau$, and $h$ preserve the filtration, the contraction is said to be filtered~\cite[\S 12]{MR0056295}.

A \emph{perturbation} of the differential $\delta$ of a filtered chain complex
\[ \begin{tikzcd} \cdots \arrow{r} & N^{n-1} \arrow{r}{\delta} & N^{n} \arrow{r}{\delta} & N^{n+1} \arrow{r} & \cdots \end{tikzcd} \]
is an operator $\varrho:F^{k} N \to F^{k-1} N$ lowering the filtration and
satisfying \[ (\delta+\varrho)^2=0 \]
so that $\delta+\varrho$ is a new differential on $N$.

We refer the reader to~\cite[\S 1]{MR1109665} for
a brief history of the following lemma.

\begin{lemma}[Homological Perturbation~\cite{MR0220273}]
\label{Riga}
Let
\[ \begin{tikzcd}
\cdots \arrow{r} & N^{n-1} \arrow{d}{\sigma} \arrow{r}{\delta} &
N^{n} \arrow{d}{\sigma} \arrow{r}{\delta} \arrow{ddl}[near end, dashed]{h} &
N^{n+1} \arrow{d}{\sigma} \arrow{r} \arrow{ddl}[near end, dashed]{h}  & \cdots \\
\cdots \arrow{r} & M^{n-1} \arrow{d}[swap]{\tau} \arrow{r}[near start]{d}  &
M^{n} \arrow{d}[swap]{\tau} \arrow{r}[near start]{d}  &
M^{n+1} \arrow{d}[swap]{\tau} \arrow{r} & \cdots \\
\cdots \arrow{r} & N^{n-1} \arrow{r}{\delta} &
N^{n} \arrow{r}{\delta} &
N^{n+1} \arrow{r} & \cdots
\end{tikzcd} \]
be a filtered contraction.
Given a perturbation $\varrho$ of the differential $\delta$ on $N$, if the filtrations on $M$ and $N$ are complete,
then the series
\begin{gather*}
\vartheta:=\sum_{k=0}^{\infty} \sigma \varrho (h\varrho)^k\tau = \sum_{k=0}^{\infty}\sigma(\varrho h)^k\varrho\tau \\
\perturbed{\sigma}:=\sum_{k=0}^{\infty}\sigma(\varrho h)^k \\
\perturbed{\tau}:=\sum_{k=0}^{\infty}(h\varrho)^k\tau \\
\perturbed{h}:=\sum_{k=0}^{\infty}(h\varrho)^k h=\sum_{k=0}^{\infty} h(\varrho h)^k
\end{gather*}
converge, $\vartheta$ is a perturbation of the differential $d$ on $M$,
and
\[ \begin{tikzcd}[column sep=large]
\cdots \arrow{r} & N^{n-1} \arrow{d}{\perturbed{\sigma}} \arrow{r}{\delta+\varrho} &
N^{n} \arrow{d}{\perturbed{\sigma}} \arrow{r}{\delta+\varrho} \arrow{ddl}[near end, dashed]{\perturbed{h}}  &
N^{n+1} \arrow{d}{\perturbed{\sigma}} \arrow{r} \arrow{ddl}[near end, dashed]{\perturbed{h}} & \cdots \\
\cdots \arrow{r} & M^{n-1} \arrow{d}[swap]{\perturbed{\tau}} \arrow{r}[near start]{d+\vartheta} &
M^{n} \arrow{d}[swap]{\perturbed{\tau}} \arrow{r}[near start]{d+\vartheta} &
M^{n+1} \arrow{d}[swap]{\perturbed{\tau}} \arrow{r} & \cdots \\
\cdots \arrow{r} & N^{n-1} \arrow{r}{\delta+\varrho} &
N^{n} \arrow{r}{\delta+\varrho} &
N^{n+1} \arrow{r} & \cdots
\end{tikzcd} \]
constitutes a new filtered contraction.
\end{lemma}

\subsection{The coboundary operator $\delta$}
\label{section:pek}

Let $(L,A)$ be a Lie pair.
We use the symbol $B$ to denote the quotient vector bundle $L/A$ and the symbol $r$ to denote its rank.

Consider the endomorphism $\delta$ of the vector bundle
$\Lambda L^{\vee}\otimes\hat{S}B^{\vee}$ defined by
\[ \delta(\omega\otimes\chi^J)=\sum_{m=1}^r \big(q^\top(\chi_m)\wedge\omega\big)\otimes J_m\,\chi^{J-e_m} ,\]
for all $\omega\in\Lambda L^{\vee}$ and $J\in\NN^r$.
Here $\{\chi_k\}_{k=1}^r$ denotes an arbitrary local frame for the vector bundle $B^{\vee}$.

The operator $\delta$ is a derivation of degree $+1$ of the
graded commutative algebra $\sections{\Lambda^\bullet L^\vee
\otimes\hat{S} B^\vee}$ and satisfies $\delta^2=0$.
The resulting cochain complex
\[ \begin{tikzcd}[column sep=small]
\cdots \arrow{r}{} & \Lambda^{n-1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{\delta} &
\Lambda^{n} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{\delta} &
\Lambda^{n+1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{} & \cdots
\end{tikzcd} \]
deformation retracts onto the trivial complex
\[ \begin{tikzcd}
\cdots \arrow{r}{} & \Lambda^{n-1} A^{\vee} \arrow{r}{0} &
\Lambda^{n} A^{\vee} \arrow{r}{0} & \Lambda^{n+1} A^{\vee} \arrow{r}{} & \cdots
\end{tikzcd} \]
Indeed, for every choice of splitting $i\circ p+j\circ q=\id_L$ of the short exact sequence
\[ \begin{tikzcd}
0 \arrow{r} & A \arrow{r}{i} & L \arrow[bend left, dashed]{l}{p} \arrow{r}{q} & B \arrow{r} \arrow[bend left, dashed]{l}{j} & 0
\end{tikzcd} \]
and its dual
\[ \begin{tikzcd}
0 \arrow{r} & B^{\vee} \arrow{r}{q\transpose} & L^{\vee} \arrow[bend left, dashed]{l}{j\transpose} \arrow{r}{i\transpose}
& A^{\vee} \arrow{r} \arrow[bend left, dashed]{l}{p\transpose} & 0
\end{tikzcd} ,\]
the chain maps
\begin{gather*}
\sigma:\Lambda^\bullet L^{\vee} \otimes\hat{S} B^{\vee} \to\Lambda^\bullet A^{\vee}
\\ \intertext{and}
\tau:\Lambda^\bullet A^{\vee} \to\Lambda^\bullet L^{\vee} \otimes\hat{S} B^{\vee}
\end{gather*}
respectively defined by
\[ \sigma(\omega\otimes\chi^J)=\begin{cases} \omega\otimes\chi^J &
\text{if } v=0 \text{ and } \abs{J}=0 \\
0 & \text{otherwise,} \end{cases} \]
for all $\omega\in p^\top(\Lambda^u A^{\vee})\otimes q^\top(\Lambda^v B^{\vee})$ and all multi-indices $J\in\NO^r$,
and \[ \tau(\alpha)=p^\top(\alpha)\otimes 1 ,\] for all $\alpha\in\Lambda^\bullet(A^{\vee})$,
satisfy
\begin{equation*}
\sigma\tau=\id \qquad \text{and} \qquad \id-\tau\sigma= h\delta+\delta h
\end{equation*}
where the homotopy operator
\[ h:\Lambda^{\bullet} L^{\vee} \otimes\hat{S} B^{\vee} \to\Lambda^{\bullet-1} L^{\vee} \otimes\hat{S} B^{\vee} \]
is defined by
\[ h(\omega\otimes\chi^J)=\begin{cases}
\frac{1}{v+\abs{J}}\sum_{k=1}^r (\iota_{j(\partial_k)}\omega)\otimes\chi^{J+e_k} & \text{if } v\geqslant 1 \\
0 & \text{if } v=0 \end{cases} \]
for all $\omega\in p^\top(\Lambda^u A^{\vee})\otimes q^\top(\Lambda^v B^{\vee})$.
Here $\{\partial_k\}_{k=1}^r$ denotes the local frame for $B$ dual to $\{\chi_k\}_{k=1}^r$.

\begin{remark}
The operator $h$ is \emph{not} a derivation of the algebra $\sections{\Lambda L^\vee \otimes\hat{S} B^\vee }$.
\end{remark}

We note that $h\tau=0$, $\sigma h=0$, and $h^2 =0$.

Furthermore, the maps $\delta$, $\sigma$, $\tau$, and $h$ respect
the exhaustive, complete, descending filtrations
\begin{gather*}
\mathcal{F}^0\supset\mathcal{F}^1\supset\mathcal{F}^2\supset\mathcal{F}^3\supset\cdots \\ \intertext{and}
\mathscr{F}^0\supset\mathscr{F}^1\supset\mathscr{F}^2\supset\mathscr{F}^3\supset\cdots
\end{gather*}
on the complexes $\Lambda^\bullet A^{\vee} $ and $\Lambda^\bullet L^{\vee} \otimes\hat{S} B^{\vee} $
defined by \begin{gather*} \mathcal{F}^m=\bigoplus_{k\geqslant m}\Lambda^k A^{\vee} \\ \intertext{and}
\mathscr{F}^m=\prod_{k+p\geqslant m}\big( \Lambda^k L^{\vee} \otimes S^p B^{\vee} \big) .\end{gather*}

Hence, we have proved the following
\begin{proposition}\label{Geneva}
The diagram
\begin{dontbotheriftheequationisoverlong}
\begin{tikzcd}
\cdots \arrow{r} & \Lambda^{n-1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{d}{\sigma} \arrow{r}{-\delta} &
\Lambda^{n} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{d}{\sigma} \arrow{r}{-\delta} \arrow[dashed]{ddl}[near end]{h} &
\Lambda^{n+1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{d}{\sigma} \arrow{r} \arrow[dashed]{ddl}[near end]{h} & \cdots \\
\cdots \arrow{r} & \Lambda^{n-1} A^{\vee} \arrow{d}[swap]{\tau} \arrow{r}[near start]{0}  &
\Lambda^{n} A^{\vee} \arrow{d}[swap]{\tau} \arrow{r}[near start]{0} &
\Lambda^{n+1} A^{\vee} \arrow{d}[swap]{\tau} \arrow{r} & \cdots \\
\cdots \arrow{r} & \Lambda^{n-1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{-\delta} &
\Lambda^{n} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{-\delta} &
\Lambda^{n+1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r} & \cdots
\end{tikzcd} \end{dontbotheriftheequationisoverlong}
is a filtered contraction.\footnote{See Section~\ref{Vilnius}.}
\end{proposition}

\subsection{Fedosov dg manifold associated with a Lie pair}

\begin{lemma}\label{Brescia}
Let $(L,A)$ be a Lie pair, $\nabla$ an $L$-connection on $B$ extending
the Bott $A$-connection, and $\beta^\nabla$ its torsion as
in Equation~\eqref{eq:dog}.
Then $\beta^\nabla=0$ if and only if
$\delta d_L^{\nabla}+d_L^{\nabla}\delta=0$.
\end{lemma}

Consider the four maps $\etendu{\delta}$, $\etendu{\sigma}$, $\etendu{h}$, and $\etendu{\tau}$
\[ \sections{\Lambda^\bullet A^{\vee} \otimes B}
\overset{\etendu{\sigma}}{\underset{\etendu{\tau}}\leftrightarrows}
\sections{\Lambda^{\bullet} L^{\vee} \otimes\hat{S} B^{\vee} \otimes B}
\overset{\etendu{\delta}}{\underset{\etendu{h}}{\rightleftarrows}}
\sections{\Lambda^{\bullet+1} L^{\vee} \otimes\hat{S} B^{\vee} \otimes B} \]
defined by
\begin{align*}
\etendu{\delta}(\omega\otimes\sigma\otimes b)&=\delta(\omega\otimes\sigma)\otimes b
& \etendu{\sigma}(\omega\otimes\sigma\otimes b)&=\sigma(\omega\otimes\sigma)\otimes b
\\ \etendu{h}(\omega\otimes\sigma\otimes b)&=h(\omega\otimes\sigma)\otimes b
& \etendu{\tau}(\alpha\otimes b)&=\tau(\alpha)\otimes b
,\end{align*}
for all $\alpha\in\sections{\Lambda A^\vee}$,
$\omega\in\sections{\Lambda L^\vee}$, $\sigma\in\sections{\hat{S} B^{\vee} }$, and $b\in\sections{B}$. The maps
$\delta$, $\sigma$, $h$, and $\tau$ are defined in Section~\ref{section:pek}.

\begin{theorem}\label{strawberry}
Let $(L,A)$ be a Lie pair.
Given a splitting $j:L/A\to L$ of the short exact sequence
 $0\to A \to L \to L/A \to 0$
and a torsion-free $L$-connection $\nabla$ on $L/A$ extending the Bott
 $A$-connection, there exists a \emph{unique}
1-form valued in formal vertical vector fields of $L/A$:
\[ X^\nabla\in\sections{L^{\vee}\otimes\hat{S}^{\geqslant 2} B^{\vee} \otimes B} \] satisfying $\etendu{h}(X^\nabla)=0$
and such that the derivation
 $Q: \sections{\Lambda^\bullet L^\vee\otimes\hat{S} B^\vee}
\to \sections{\Lambda^{\bullet+1} L^\vee\otimes\hat{S} B^\vee}$ defined by
\begin{equation}\label{eq:Q} Q=-\delta+d_L^\nabla+X^\nabla \end{equation}
satisfies $Q^2=0$.
Here $X^\nabla$ acts on the algebra $\sections{\Lambda^\bullet L^\vee\otimes\hat{S} B^\vee }$
as a derivation in a natural fashion.
\newline As a consequence, $(L[1]\oplus B, Q)$ is a dg manifold.
\end{theorem}

\begin{proof}
Suppose there exists such a $X^\nabla$ and consider its decomposition $X^\nabla=\sum_{k=2}^{\infty} X_k$, where
$X_k\in\sections{L^{\vee}\otimes\hat{S}^{k}B^{\vee}\otimes B}$.
Then $Q=-\delta+d_L^\nabla+X_2+X_{\geqslant 3}$
with $X_{\geqslant 3}=\sum_{k=3}^{\infty} X_k$ and
\[ \begin{split}
Q^2 =&\ \delta^2
- \big(\delta d_L^\nabla+d_L^\nabla\delta\big)
+ \big\{d_L^\nabla d_L^\nabla-\deltaX_2-X_2\delta\big\} \\
&\ + \big\{d_L^\nablaX^\nabla+X^\nabla d_L^\nabla+{X^\nabla}^2
-\deltaX_{\geqslant 3}-X_{\geqslant 3}\delta\big\} \\
=&\ \delta^2 - \lie{\delta}{d_L^\nabla}
+ \big\{ R^\nabla-\lie{\delta}{X_2} \big\}
+ \big\{ \lie{d_L^\nabla+\tfrac{1}{2}X^\nabla}{X^\nabla}
- \lie{\delta}{X_{\geqslant 3}} \big\}
.
\end{split} \]

Let us write $\Lambda^p\otimes S^q$ for
$\sections{\Lambda^p L^\vee \otimes S^q B^\vee }$.

Since
\[ \begin{tikzcd}[column sep=huge,row sep=small]
& \Lambda^{p+1}\otimes S^{q-1} \\
\Lambda^{p}\otimes S^{q}
\arrow{ru}[near end]{-\delta}
\arrow{r}[near end]{d_L^\nabla}
\arrow{rd}[swap, near end]{X_2}
\arrow{rdd}[swap, near end]{X_{\geqslant 3}}
& \Lambda^{p+1}\otimes S^{q} \\
& \Lambda^{p+1}\otimes S^{q+1} \\
& \Lambda^{p+1}\otimes \hat{S}^{\geqslant q+2}
,\end{tikzcd} \]
we have
\[ \begin{tikzcd}[column sep=huge,row sep=small]
&& \Lambda^{p+2}\otimes S^{q-2} \\
&& \Lambda^{p+2}\otimes S^{q-1} \\
\Lambda^{p}\otimes S^{q}
\arrow{rruu}[near end]{\delta^2}
\arrow{rru}[swap, near end]{\lie{\delta}{d_L^\nabla}}
\arrow{rr}[swap, near end]{R^\nabla-\lie{\delta}{X_2}}
\arrow{rrd}[swap, near end]{\lie{d_L^\nabla+\frac{1}{2}X^\nabla}{X^\nabla}
-\lie{\delta}{X_{\geqslant 3}}}
&& \Lambda^{p+2}\otimes S^{q} \\
&& \Lambda^{p+2}\otimes \hat{S}^{\geqslant q+1}
.\end{tikzcd} \]

Since $\delta^2=0$ and $\lie{\delta}{d_L^\nabla}=0$ (by Lemma~\ref{Brescia}), we obtain the commutative diagram
\[ \begin{tikzcd}[row sep=small]
& \Lambda^{p}\otimes S^{q}
\arrow{dl}[swap]{R^\nabla-\lie{\delta}{X_2}}
\arrow{dd}{Q^2}
\arrow{dr}{\lie{d_L^\nabla+\frac{1}{2}X^\nabla}{X^\nabla}
-\lie{\delta}{X_{\geqslant 3}}}
& \\
\Lambda^{p+2}\otimes S^{q} &
& \Lambda^{p+2}\otimes\hat{S}^{\geqslant q+1} \\
& \Lambda^{p+2}\otimes\hat{S}^{\geqslant q}
\arrow[two heads]{lu}
\arrow[two heads]{ru}
&
\end{tikzcd} \]
The requirement $Q^2=0$ is thus equivalent to the pair of equations
\begin{align*}
\lie{\delta}{X_2}&=R^\nabla \\
\lie{\delta}{X_{\geqslant 3}}&=
\lie{d_L^\nabla+\tfrac{1}{2}X^\nabla}{X^\nabla}
\end{align*}

Note that $\etendu{\sigma}(X_2)=0$ and
$\etendu{\sigma}(X_{\geqslant 3})=0$ since
$X_2,X_{\geqslant 3}\in\sections{L^\vee\otimes\hat{S}^{\geqslant 2} B^\vee \otimes B}$ and also that $\etendu{h}(X_2)=0$ and $\etendu{h}(X_{\geqslant 3})=0$ as $\etendu{h}(X^\nabla)=0$.
Since $\etendu{\delta}\etendu{h}+\etendu{h}\etendu{\delta}=\id-\etendu{\tau}\etendu{\sigma}$,
we obtain $\etendu{h}\etendu{\delta}(X_2)=X_2$ and
$\etendu{h}\etendu{\delta}(X_{\geqslant 3})=X_{\geqslant 3}$.

It follows that \begin{align*}
X_2&=\etendu{h}\etendu{\delta}(X_2)=\etendu{h}(\lie{\delta}{X_2})=\etendu{h}(R^\nabla) \\
X_{\geqslant 3}&=\etendu{h}\etendu{\delta}(X_{\geqslant 3})=\etendu{h}(\lie{\delta}{X_{\geqslant 3}})=\etendu{h}
\lie{d_L^\nabla+\tfrac{1}{2}X^\nabla}{X^\nabla}
.\end{align*}
Projecting the second equation onto
$\sections{L^\vee\otimes\hat{S}^{k+1} B^\vee \otimes B}$,
we obtain
\begin{align}
X_2 &= \etendu{h}(R^\nabla) \label{deux}
\\ X_{k+1} &= \etendu{h}\Big(d_L^\nabla\circ X_k+X_k\circ d_L^\nabla+
\sum_{\substack{p+q=k+1 \\ 2\leqslant p,q\leqslant k-1}} X_p\circ X_q\Big), \qquad\text{for}\ k\geqslant 2
. \label{troisetplus}
\end{align}
The successive terms of $X^\nabla=\sum_{k=2}^{\infty} X_k$ can thus be computed sequentially starting from $X_2=\etendu{h}(R^\nabla)$.
Therefore, if it exists, the derivation $X^\nabla$ is uniquely determined
by the torsion-free connection $\nabla$ and the splitting $j:B\to L$.

Now, defining $X_k$ inductively by the relations \eqref{deux} and \eqref{troisetplus} and
setting $X^\nabla=\sum_{k=2}^{\infty} X_k$,
we have $\etendu{h}(X^\nabla)=\etendu{h}(X_2+X_{\geqslant 3})=\etendu{h}^2\big(R^\nabla+\etendu{\delta}(X_{\geqslant 3})\big)=0$
since $\etendu{h}^2=0$.
Moreover, we have $X_2=\etendu{h}(R^\nabla)\in\sections{L^{\vee}\otimes S^2(B^{\vee})\otimes B}$
as $R^\nabla\in\sections{\wedge^2 L^{\vee}\otimes B^{\vee}\otimes B}$.
Making use of Equation~\eqref{troisetplus}, one proves by induction on $k$ that
$X_k\in\sections{L^{\vee}\otimes S^k(B^{\vee})\otimes B}$.
This completes the proof of the existence of $X^\nabla$.
\end{proof}

\begin{remark}
When $L=TM$ and $A=0$, where $M$ is a smooth manifold,
Theorem~\ref{strawberry} reduces to a classical theorem of
Emmrich--Weinstein \cite{MR1327535} (see also \cite{MR2102846}). On the other hand,
when $L=TX\otimes \CC$ and $A=T^{0,1}X$, where $X$ is a complex
manifold, Theorem~\ref{strawberry} reduces to Theorem~5.9
in~\cite{MR2364075}.
\end{remark}

\subsection{Emmrich--Weinstein theorem for Lie pairs}

Alternatively, one can make use of the Poincaré--Birkhoff--Witt isomorphism $\pbw^{\nabla,j}$
in order to endow the graded manifold on $L[1]\oplus B$ with a homological vector field as in~\cite{1408.2903}.

Recall that every choice of a splitting $j: B=L/A\to L$
of the short exact sequence of vector bundles~\eqref{Sydney}
and an $L$-connection $\nabla$ on $B$ extending the Bott $A$-connection
determines a Poincaré--Birkhoff--Witt map
\[ \pbw^{\nabla,j}:\sections{SB}\to\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}} ,\]
which is an isomorphism of filtered $R$-coalgebras according to Theorem~\ref{Nairobi}.
In what follows, we will use the simplified symbol $\pbw$ to denote the Poincaré--Birkhoff--Witt isomorphism $\pbw^{\nabla,j}$.

Being a quotient of the universal enveloping algebra $\enveloping{L}$ by
 a left ideal, the $R$-coalgebra
$\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}}$ is naturally a left $\enveloping{L}$-module. Hence
$\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}}$ is endowed with
a canonical infinitesimal $L$-action by coderivations.
Pulling back this infinitesimal action through $\pbw$, we obtain
 a \emph{flat} $L$-connection
$\nabla^\lightning$ on $SB$:
\begin{equation}\label{Vermont} \nabla^\lightning_l(s)=\pbw^{-1}\big(l\cdot\pbw(s)\big) ,\end{equation}
for all $l\in\sections{L}$ and $s\in\sections{S B}$.

The $L$-connection $\nabla^\lightning$ on $S(B)$ induces
an $L$-connections on the dual bundle $\hat{S}(B^{\vee})$ --- see Remark~\ref{Amalfi}.
By \[ d_L^{\nabla^\lightning}: \sections{\wedge^\bullet L^\vee \otimes \hat{S} B^{\vee}}
\to \sections{\wedge^{\bullet+1} L^\vee \otimes \hat{S} B^{\vee}} ,\]
we denote its corresponding Chevalley--Eilenberg differential.
It is known \cite{1408.2903} that the covariant derivative
 $\nabla^\lightning_l: \sections{SB}\to \sections{SB}$
 is a \emph{co}derivation of
 $\sections{SB}$ for all $l\in\sections{L}$
(However, $\nabla^\lightning_l$ need not be a derivation of
$\sections{SB}$ for any $l\in\sections{L}$).
Therefore, the covariant derivative
\begin{equation}
\nabla^\lightning_l:
\sections{\hat{S} B^{\vee}}\to \sections{\hat{S} B^{\vee}}
\end{equation}
 is a derivation of the symmetric algebra $\sections{\hat{S} B^{\vee}}$
and $d_L^{\nabla^\lightning}$ is a degree $+1$-derivation
on $\sections{\wedge^\bullet L^\vee \otimes \hat{S} B^{\vee}}$, i.e.\ a homological vector field on $L[1]\oplus B$.
As a consequence, $(L[1]\oplus B, d_L^{\nabla^\lightning})$
is indeed a dg-manifold.

The following theorem extends a classical theorem of
Emmrich--Weinstein \cite{MR1327535}.

\begin{theorem}
\label{thm:Lodz}
Let $(L,A)$ be a Lie pair,
$i\circ p+j\circ q=\id_L$ a splitting of the short exact sequence
\[ \begin{tikzcd}
0 \arrow{r} & A \arrow{r}{i} & L \arrow[bend left, dashed]{l}{p} \arrow{r}{q} & B \arrow{r} \arrow[bend left, dashed]{l}{j} & 0
\end{tikzcd} \]
and $\nabla$ an $L$-connection on $B$ extending the Bott $A$-connection.
If $\nabla$ is torsion-free, then the dg-manifold
$(L[1]\oplus B, d_L^{\nabla^\lightning})$
described above coincides with the Fedosov dg-manifold
$(L[1]\oplus B, Q)$ constructed by the Fedosov iteration as in
Theorem~\ref{strawberry}.
\end{theorem}

\subsection{Proof of Theorem~\ref{thm:Lodz}}
\label{Harbin}

The covariant derivative $\nabla^\lightning_l$ does not preserve the filtration
$\sections{S^{\leqslant p}(L/A)}$. Nevertheless, we have

\begin{proposition}
For all $l\in\sections{L}$, the covariant derivative $\nabla^\lightning_l$
maps $\sections{S^{\leqslant p}B}$
to $\sections{S^{\leqslant p+1}B }$.
Moreover, for all $a\in\sections{A}$, the covariant derivative $\nabla^\lightning_a$
maps $\sections{S^{\leqslant p}B}$
to $\sections{S^{\leqslant p}B}$.
\end{proposition}

\color{black}

At this stage, it is useful to introduce some new notations.
Let $\Theta$ be the bundle map
\[ \Theta: L\otimes SB\to SB \] defined by
\begin{equation}
\label{California}
\Theta(l;s)=\nabla^\lightning_l s -\nabla_l s-q(l)\odot s,
\quad\forall l\in\sections{L},s\in\sections{SB}
.\end{equation}

It follows immediately from Equations~\eqref{California} and~\eqref{Vermont} that
\begin{equation}\label{Oregon}
\Theta(l;1)=0 ,\quad\forall l\in\sections{L}
\end{equation}
since $\pbw(1)=1$ and $\pbw(b)=j(b)$ for all $b\in\sections{L/A}$.

Since $\pbw$ is invertible, Proposition~\ref{Glaciere} may be rewritten as
\[ \nabla^\lightning_l b=q(l)\odot b+\nabla_l b-\tfrac{1}{2}
\beta^\nabla(q(l),b) ,\quad\forall l\in\sections{L}, b\in\sections{B} .\]
Therefore,
\begin{equation}\label{Idaho}
\Theta(l;b)=-\tfrac{1}{2} \beta^\nabla(q(l),b)
,\quad\forall l\in\sections{L}, b\in\sections{B}
.\end{equation}

\begin{lemma}\label{Utah}
For all $l\in\sections{L}$,
the map $s\mapsto\Theta(l;s)$ is a coderivation of
the $R$-coalgebra $\sections{SB}$ and preserves the filtration
\begin{equation*}
\cdots \hookrightarrow S^{\leqslant n-1}B \hookrightarrow S^{\leqslant n}B \hookrightarrow S^{\leqslant n+1} B \hookrightarrow \cdots
.\end{equation*}
\end{lemma}

\begin{proof}
The verification that $\nabla^\lightning_l$, $\nabla_l$ and the map $s\mapsto q(l)\odot s$
are coderivations of $\sections{SB}$ for all $l\in\sections{L}$ is straightforward.
Hence $s\mapsto\Theta(l,s)$ is a coderivation of the $R$-coalgebra
 $\sections{SB}$
for all $l\in\sections{L}$ as well.
\newline
It follows from Equations~\eqref{California} and~\eqref{Vermont} together with Corollary~\ref{Duroc},
the fact that $\pbw$ respects the filtrations on $\Gamma\big(S(B)\big)$ and
$\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}}$, and Lemma~\ref{Pigalle} that
\[ \begin{split}
& \pbw\big(\Theta(l;b_1\odot\cdots\odot b_n)\big) \\
=\ & l\cdot\pbw(b_1\odot\cdots\odot b_n)
-\pbw\big(q(l)\odot b_1\odot\cdots\odot b_n-\nabla_l(b_1\odot\cdots\odot b_n)\big) \\
=\ & l\cdot j(b_1)\cdots j(b_n) - j\circ q(l)\cdot j(b_1)\cdots j(b_n) +u \\
=\ & p(l)\cdot j(b_1)\cdots j(b_n) +u \\
=\ & j(b_1)\cdots j(b_n)\cdot p(l) +v
\end{split} \]
where $u,v$ are elements of $\left(\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n}$.
Since $p(l)$ belongs to $\sections{A}$, we obtain
\[ \pbw\big(\Theta(l;b_1\odot\cdots\odot b_n)\big) = v
\in\left(\tfrac{\enveloping{L}}{\enveloping{L}\sections{A}}\right)^{\leqslant n} .\]
Therefore, we have $\Theta(l;b_1\odot\cdots\odot b_n)\in\Gamma\big(S^{\leqslant n}(B)\big)$
for all $l\in\sections{L}$ and $b_1,\dots,b_n\in\Gamma(B)$.
\end{proof}

\begin{proposition}\label{Mozambique}
For all $n\in\NN$ and all $b_0,b_1,\dots,b_n\in\sections{B}$, we have
\[ \sum_{k=0}^n \Theta\big(j(b_k);b_0\odot\cdots\odot
\widehat{b_k}\odot\cdot\odot b_n\big) =0 .\]
\end{proposition}

\begin{proof}
Set $b^{\{k\}}=b_0\odot\cdots\odot
\widehat{b_k}\odot\cdot\odot b_n$ and rewrite Equation~\eqref{Rome} as
\[ (n+1) \pbw(b_0\odot\cdots\odot b_n)
= \sum_{k=0}^n \Big\{j(b_k)\cdot\pbw(b^{\{k\}})
-\pbw\big(\nabla_{j(b_k)}(b^{\{k\}})
\big)\Big\} .\]
Applying $\pbw^{-1}$ to both sides, we obtain
\[ (n+1)\ b_0\odot\cdots\odot b_n =
\sum_{k=0}^n \Big\{
\nabla^\lightning_{j(b_k)}(b^{\{k\}}) - \nabla_{j(b_k)}(b^{\{k\}})
\Big\} ,\]
which is equivalent to
\[ \sum_{k=0}^n \Big\{
\nabla^\lightning_{j(b_k)}(b^{\{k\}}) - q\big(j(b_k)\big)\odot b^{\{k\}} - \nabla_{j(b_k)}(b^{\{k\}})
\Big\} =0 .\]
The result follows from Equation~\eqref{California}.
\end{proof}

Now consider the map
\[ \Xi^\nabla : \sections{S^\bullet B^{\vee} } \to
\sections{L^{\vee}\otimes\hat{S}^{\geqslant\bullet+1} B^{\vee} } \]
defined by
\begin{equation}\label{Rambuteau}
\duality{s}{\interior{l}\Xi^\nabla(\sigma)}
= \duality{\interior{l}\Theta^\nabla(l; s)}{\sigma}
,\end{equation}
for all homogeneous $l\in\sections{L}$, $s\in\sections{S B }$, and $\sigma\in\sections{\hat{S} B^{\vee} }$.

Here \[ \Gamma(S B )\otimes_{R}\Gamma(\hat{S} B^{\vee} )\xto{\duality{-}{-}} R \]
is the duality pairing defined by
\[ \duality{b_1\odot\cdots\odot b_p}
{\beta_1\odot\cdots\odot\beta_q} =
\begin{cases}
\sum_{\sigma\in S_p}
\interior{b_1}\beta_{\sigma(1)}\cdot
\interior{b_2}\beta_{\sigma(2)}\cdots
\interior{b_p}\beta_{\sigma(p)}
& \text{if } p=q \\
0 & \text{if } p\neq q
\end{cases} \]
for all $b_1,\dots,b_p\in\sections{B}$
and $\beta_1,\dots,\beta_q\in\sections{B^{\vee}}$.

A straightforward computation yields the following lemma.

\begin{lemma}\label{Bercy}
Let $(\partial_i)_{i\in\{1,\dots,r\}}$ be a local frame of $B$
and let $(\chi_j)_{j\in\{1,\dots,r\}}$ be the dual local frame of $B^{\vee}$.
We have
\[ \duality{\partial^I}{\chi^J}=I!\ \delta_{I,J}
,\quad\forall I,J\in\NO^n \]
and
\[ \sigma=\sum_{I\in\NN^N}\frac{1}{I!}
\duality{\partial^I}{\sigma}\, \chi^I
,\quad\forall \sigma\in\Gamma\big(\hat{S}(B^{\vee})\big) .\]
 \end{lemma}

\begin{lemma}\label{Goncourt}
For any $l\in \sections{L}$,
and all homogeneous $s\in\sections{S B }$,
and $\sigma\in\Gamma\big(\hat{S} B^{\vee} \big)$, we have
\[ \duality{s}{\interior{l}\delta(\sigma)}
=\duality{q(l)\odot s}{\sigma} .\]
\end{lemma}

\begin{proof}
It suffices to prove the relation for $s=\partial^I$ and $\sigma=\chi^J$.
We have
\begin{multline*}
\duality{\partial^I}{\interior{l}\delta(\chi^J)}
= \duality{\partial^I}{\sum_{k=1}^r \interior{q(l)}\chi_k \ J_k \chi^{J-e_k}}
= \sum_{k=1}^r \interior{q(l)}\chi_k \ \duality{\partial^I}{J_k \chi^{J-e_k}} \\
= \sum_{k=1}^r \interior{q(l)}\chi_k \ J_k\ I!\ \delta_{I,J-e_k}
= \sum_{k=1}^r \interior{q(l)}\chi_k \ J!\ \delta_{I+e_k,J}
= \sum_{k=1}^r \interior{q(l)}\chi_k \ \duality{\partial^{I+e_k}}{\chi^J} \\
= \duality{\sum_{k=1}^r \interior{q(l)}\chi_k \ \partial_k\odot\partial^I}{\chi^J}
= \duality{q(l)\odot\partial^I}{\chi^J}
.\qedhere\end{multline*}
\end{proof}

It follows that,
for all $\omega\in\sections{\Lambda L^{\vee}}$
and $\xi\in\sections{\hat{S}(B^{\vee})}$,
\[ \delta(\omega\otimes\xi)=\sum_{k=1}^{\rk(L)}
(\lambda_k\wedge\omega)\otimes
\duality{q(l_k)\odot-}{\xi}
=\sum_{k=1}^{r}
(q\transpose(\chi_k)\wedge\omega)\otimes
\duality{\partial_k\odot-}{\xi} ,\]
where $(l_k)_{k=1}^{\rk(L)}$ and
$(\lambda_k)_{k=1}^{\rk(L)}$ is any pair
of dual local frames for the vector bundles $L$ and $L^{\vee}$ and $(\partial_i)_{i\in\{1,\dots,r\}}$
and $(\chi_j)_{j\in\{1,\dots,r\}}$ is any pair
of dual local frames for the vector bundles $B$ and $B^{\vee}$.

\begin{proposition}
For every $l\in\sections{L}$,
the operator $\interior{l}\Xi^\nabla$ is a derivation of the algebra
$\Gamma\big(\hat{S} B^{\vee} \big)$.
\end{proposition}

\begin{proof}
The result follows immediately from Proposition~\ref{Utah}
since the algebra $\Gamma\big(\hat{S}(B^{\vee})\big)$ is dual to the coalgebra
$\Gamma(S(B))$ and $\interior{l}\Xi^\nabla$
is the transpose of
$\interior{l}\Theta^\nabla$ according to Equation~\eqref{Rambuteau}.
\end{proof}

Hence $\Xi^\nabla$ may be regarded as an element of
$\Gamma\big(L^{\vee}\otimes\hat{S}(B^{\vee})\otimes B\big)$.

\begin{proposition}\label{Oslo}
\begin{enumerate}
\item
If $\beta^\nabla=0$, then
$\Xi^\nabla\in\Gamma\big(L^{\vee}\otimes\hat{S}^{\geqslant 2} B^{\vee} \otimes B\big)$.
\item
If $\beta^\nabla\neq 0$, then
$\Xi^\nabla\in\Gamma\big(L^{\vee}\otimes\hat{S}^{\geqslant 1} B^{\vee} \otimes B\big)$.
\end{enumerate}
\end{proposition}

\begin{proof}
Let $(\partial_i)_{i\in\{1,\dots,r\}}$ be a local frame of $B$
and let $(\chi_j)_{j\in\{1,\dots,r\}}$ be the dual local frame of $B^{\vee}$. Fix any $l\in\sections{L}$.
Since $\interior{l}\Xi^\nabla$ is a derivation of the algebra
$\Gamma\big(\hat{S}(B^{\vee})\big)$, which is generated locally by $\chi_1,\dots,\chi_r$,
we have \[ \interior{l}\Xi^\nabla=\sum_{k=1}^r \interior{l}\Xi^\nabla(\chi_k) \partial_k ,\]
with
\begin{align*}
\interior{l}\Xi^\nabla(\chi_k) =& \sum_{I\in\NO^r} \frac{1}{I!}
\duality{\partial^I}{\interior{l}\Xi^\nabla(\chi_k)} \chi^I
&&\text{by Lemma~\ref{Bercy},} \\
=& \sum_{I\in\NO^r} \frac{1}{I!}
\duality{\interior{l}\Theta^\nabla(\partial^I)}{\chi_k} \chi^I
&&\text{by Equation~\eqref{Rambuteau}.}
\end{align*}
If $\beta^\nabla=0$, it follows from Equations~\eqref{Oregon} and~\eqref{Idaho} that
$\Theta^\nabla(\partial^I)=0$ for $\abs{I}\leqslant 1$ so that
$\interior{l}\Xi^\nabla\in\Gamma\big(\hat{S}^{\geqslant 2}(B^{\vee})\otimes B\big)$.
If $\beta^\nabla\neq 0$, it follows from Equations~\eqref{Oregon} that
$\Theta^\nabla(\partial^I)=0$ for $\abs{I}=0$ so that
$\interior{l}\Xi^\nabla\in\Gamma\big(\hat{S}^{\geqslant 1}(B^{\vee})\otimes B\big)$.
\end{proof}

We note that, for every pair of dual local frames $(\partial_i)_{i\in\{1,\dots,r\}}$ and
$(\chi_j)_{j\in\{1,\dots,r\}}$ for $B$ and $B^{\vee}$,
we have
\[ \Xi^\nabla=\sum_{k=1}^r \sum_{J\in\NO^r}
\frac{1}{J!} \duality{\partial^J}{\Xi^\nabla(\chi_k)}
\chi^J \partial_k \]
and hence
\[ \begin{split} \Xi^\nabla(\omega\otimes\chi^J)
=\ & \sum_{m=1}^{\rk(L)} (\lambda_m\wedge\omega)
\otimes \sum_{k=1}^r \sum_{J\in\NO^r}
\frac{1}{J!} \duality{\partial^J}{\interior{l_m}\Xi^\nabla(\chi_k)} \chi^J \partial_k \\
=\ & \sum_{m=1}^{\rk(L)} (\lambda_m\wedge\omega)
\otimes \sum_{k=1}^r \sum_{J\in\NO^r}
\frac{1}{J!} \duality{\Theta^\nabla(l_m;\partial^J)}{\chi_k} \chi^J \partial_k
.\end{split} \]

\begin{lemma}\label{Ragusa}
For all $\lambda\in\sections{L^\vee}$ and $J\in\NO^r$, we have \[ h(\lambda\otimes\chi^J)
= \frac{1}{1+\abs{J}}\sum_{k=1}^r
\interior{j(\partial_k)}\lambda\otimes\chi^{J+e_k} ,\]
where $(\partial_i)_{i\in\{1,\dots,r\}}$ is any local frame of $B$ and $(\chi_j)_{j\in\{1,\dots,r\}}$ is the dual local frame of $B^{\vee}$.
\end{lemma}

\begin{proof}
For $\lambda\in\sections{q^\top B^\vee}$, the result follows immediately from the very definition of $h$.
The results holds for $\lambda\in\sections{p^\top A^\vee}$ as well since, for all $\alpha\in\sections{A^\vee}$, we have $h(p^\top(\alpha)\otimes\chi^J) =0$ by the very definition of $h$ and $\interior{j(\partial_k)}p^\top(\alpha)=0$ as $p\circ j=0$.
\end{proof}

\begin{proposition}\label{Telegraphe}
$\etendu{h}(\Xi^\nabla)=0$
\end{proposition}

\begin{proof}
Let $(\partial_i)_{i\in\{1,\dots,r\}}$ be a local frame of $B$
and let $(\chi_j)_{j\in\{1,\dots,r\}}$ be the dual local frame of $B^{\vee}$.

From \[ \Xi^\nabla=\sum_{k=1}^r \sum_{J\in\NO^r} \frac{1}{J!}
\duality{\partial^J}{\Xi^\nabla(\chi_k)} \chi^J \partial_k ,\]
we obtain, using Lemma~\ref{Ragusa},
\begin{align*}
\etendu{h}(\Xi^\nabla) =&
\sum_{k=1}^r \sum_{J\in\NO^r}
h\bigg\{ \frac{1}{J!}\duality{\partial^J}{\Xi^\nabla(\chi_k)} \chi^J\bigg\} \partial_k
\\
=& \sum_{k=1}^r \sum_{J\in\NO^r}
\frac{1}{1+\abs{J}} \sum_{p=1}^r \frac{1}{J!}
\duality{\partial^J}{\interior{j(\partial_p)}\Xi^\nabla(\chi_k)} \chi_p \chi^J \partial_k \\
=& \sum_{k=1}^r \sum_{J\in\NO^r}
\frac{1}{1+\abs{J}} \sum_{p=1}^r \frac{1}{J!}
\duality{\interior{j(\partial_p)}\Theta^\nabla(\partial^J)}{\chi_k}
\chi^{J+e_p} \partial_k \\
=& \sum_{k=1}^r \sum_{M\in\NO^r}
\frac{1}{\abs{M}} \frac{1}{M!}
\duality{ \sum_{p=1}^r M_p\ \interior{j(\partial_p)}\Theta^\nabla(\partial^{M-e_p})}{\chi_k}
\chi^M \partial_k
.\end{align*}
It follows directly from Proposition~\ref{Mozambique} that
\[ \sum_{p=1}^r M_p \ \interior{j(\partial_p)}\Theta^\nabla(\partial^{M-e_p}) =0 \]
for every $M=(M_1,\dots,M_r)\in\NO^r$.
\end{proof}

The $L$-connections $\nabla$ and $\nabla^\lightning$ defined on $S B $ induce $L$-connections on the dual bundle $\hat{S} B^{\vee} $ --- see Remark~\ref{Amalfi}.
For all $l\in\sections{L}$, the covariant derivative $\nabla^\lightning_l$ is a derivation of the symmetric algebra
$\sections{\hat{S}B^{\vee}}$ which maps $\sections{\hat{S}^{\geqslant p}B^{\vee}}$
to $\sections{\hat{S}^{\geqslant p-1}B^{\vee}}$.

\begin{proposition}\label{Budapest}
$d_L^{\nabla^\lightning}=-\delta+d_L^{\nabla}-\Xi^\nabla$
\end{proposition}

\begin{proof}
From
\[ \duality{\nabla^\lightning_l s}{\sigma}+
\duality{s}{\nabla^\lightning_l \sigma} =
\rho(l) \duality{s}{\sigma} =
\duality{\nabla_l s}{\sigma}+
\duality{s}{\nabla_l \sigma} ,\]
we obtain
\begin{align*}
\duality{\nabla^\lightning_l s-\nabla_l s}{\sigma} &=
\duality{s}{\nabla_l \sigma-\nabla^\lightning_l \sigma} \\
\duality{q(l)\odot s+\interior{l}\Theta^\nabla(s)}{\sigma} &=
\duality{s}{\interior{l}\big(d_L^\nabla\sigma-d_L^{\nabla^\lightning}\sigma\big)} \\
\intertext{and, making use of Lemma~\ref{Goncourt}
and Equation~\eqref{Rambuteau},}
\duality{s}{\interior{l}\big(\delta\sigma+\Xi^\nabla\sigma\big)} &=
\duality{s}{\interior{l}\big(d_L^\nabla\sigma-d_L^{\nabla^\lightning}\sigma\big)}
\end{align*}
or, equivalently, \[ d_L^{\nabla^\lightning}=-\delta+d_L^\nabla-\Xi^\nabla .\qedhere\]
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:Lodz}]
Since $\Xi^\nabla\in\Gamma\big(L^{\vee}\otimes
\hat{S}^{\geqslant 2}(B^{\vee})\otimes B\big)$
provided $\beta^\nabla=0$ (Proposition~\ref{Oslo}),
$\etendu{h}(\Xi^\nabla)=0$ (Proposition~\ref{Telegraphe}),
and $d_L^{\nabla^\lightning}=-\delta+d_L^{\nabla}-\Xi^\nabla$
(Proposition~\ref{Budapest}) satisfies
$d_L^{\nabla^\lightning}\circ d_L^{\nabla^\lightning}=0$,
Theorem~\ref{strawberry} asserts that $X^\nabla=-\Xi^\nabla$ and $Q=d_L^{\nabla^\lightning}$.
\end{proof}

\section{Dolgushev--Fedosov quasi-isomorphisms}

\subsection{Contraction of the Fedosov dg manifold}

\begin{proposition}\label{Zurich}
The operator $\varrho$ defined as the difference
\[ \varrho=d_L^\nabla-\Xi^\nabla \]
is a perturbation\footnote{See Section~\ref{Vilnius}.}
of the cochain complex
\[ \begin{tikzcd}[column sep=scriptsize]
\cdots \arrow{r}{} & \Lambda^{n-1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{-\delta} &
\Lambda^{n} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{-\delta} &
\Lambda^{n+1} L^{\vee} \otimes\hat{S} B^{\vee} \arrow{r}{} & \cdots
\end{tikzcd} \]
compatible with its descending filtration
\[ \mathscr{F}^m=\bigoplus_{k=0}^{\rk(L)}
\Gamma \big(\Lambda^k L^{\vee} \otimes\hat{S}^{\geqslant m-k} B^{\vee} \big) .\]
\end{proposition}

\begin{proof}
We have $d_L^{\nabla^\lightning}=-\delta+(d_L^{\nabla}-\Xi^\nabla)$
and $(d_L^{\nabla^\lightning})^2 =0$
since the connection $\nabla^\lightning$ is flat.
Moreover, if $\beta^\nabla\neq 0$, we have
\[ \begin{tikzcd}[column sep=huge,row sep=small]
& \sections{\Lambda^{n+1}(L^{\vee})\otimes S^{p-1}(B^{\vee})} \\
\sections{\Lambda^n(L^{\vee})\otimes S^p(B^{\vee})}
\arrow[bend left]{ru}{-\delta} \arrow{r}{d_L^{\nabla}}
\arrow[bend right]{rd}[swap]{-\Xi^\nabla}
& \sections{\Lambda^{n+1}(L^{\vee})\otimes S^p(B^{\vee})}
\arrow[hook]{d} \\
& \sections{\Lambda^{n+1}(L^{\vee})\otimes\hat{S}^{\geqslant p}(B^{\vee})}
\end{tikzcd} ,\]

and if $\beta^\nabla=0$, then
\[ \begin{tikzcd}[column sep=huge,row sep=small]
& \sections{\Lambda^{n+1}(L^{\vee})\otimes S^{p-1}(B^{\vee})} \\
\sections{\Lambda^n(L^{\vee})\otimes S^p(B^{\vee})}
\arrow[bend left]{ru}{-\delta}
\arrow{r}{d_L^{\nabla}}
\arrow[bend right]{rd}[swap]{-\Xi^\nabla}
& \sections{\Lambda^{n+1}(L^{\vee})\otimes S^p(B^{\vee})} \\
& \sections{\Lambda^{n+1}(L^{\vee})\otimes\hat{S}^{\geqslant p+1}(B^{\vee})}
\end{tikzcd} \]
As a consequence, $d_L^\nabla-\Xi^\nabla$ maps $\mathscr{F}^{m}$ to $\mathscr{F}^{m+1}$.
This concludes the proof.
\end{proof}

\begin{proposition} \label{Ottignies}
The Chevalley--Eilenberg complex
$\big(\sections{\Lambda L^{\vee}\otimes\hat{S}B^{\vee}},d_{L}^{\nabla^\lightning}\big)$
contracts onto $\big(\sections{\Lambda A^{\vee}},d_{A}\big)$.
More precisely, we have the contraction
\begin{dontbotheriftheequationisoverlong} \begin{tikzcd}
\cdots \arrow{r} & \sections{\Lambda^{n-1} L^{\vee} \otimes\hat{S} B^{\vee} } \arrow{d}{\sigma} \arrow{r}{d_L^{\nabla^\lightning}} &
\sections{\Lambda^{n} L^{\vee} \otimes\hat{S} B^{\vee} } \arrow{d}{\sigma} \arrow{r}{d_L^{\nabla^\lightning}} \arrow[dashed]{ddl}[near end]{\perturbed{h}} &
\sections{\Lambda^{n+1} L^{\vee} \otimes\hat{S} B^{\vee}} \arrow{d}{\sigma} \arrow{r} \arrow[dashed]{ddl}[near end]{\perturbed{h}}  & \cdots \\
\cdots \arrow{r} & \sections{\Lambda^{n-1} A^{\vee} } \arrow{d}[swap]{\perturbed{\tau}} \arrow{r}[near start]{d_A} &
\sections{\Lambda^{n} A^{\vee}} \arrow{d}[swap]{\perturbed{\tau}} \arrow{r}[near start]{d_A} &
\sections{\Lambda^{n+1} A^{\vee}} \arrow{d}[swap]{\perturbed{\tau}} \arrow{r} & \cdots \\
\cdots \arrow{r} & \sections{\Lambda^{n-1} L^{\vee} \otimes\hat{S} B^{\vee} } \arrow{r}{d_L^{\nabla^\lightning}} &
\sections{\Lambda^{n} L^{\vee} \otimes\hat{S} B^{\vee}} \arrow{r}{d_L^{\nabla^\lightning}} &
\sections{\Lambda^{n+1} L^{\vee} \otimes\hat{S} B^{\vee}} \arrow{r} & \cdots
\end{tikzcd} \end{dontbotheriftheequationisoverlong}
where $\perturbed{\tau}=\sum_{k=0}^\infty(h\varrho)^k\tau$
and $\perturbed{h}=\sum_{k=0}^\infty(h\varrho)^k h$.
\end{proposition}

\begin{proof}
We proceed by homological perturbation (see Lemma~\ref{Riga}). Starting from the filtered contraction of Proposition~\ref{Geneva}, it suffices to perturb
the coboundary operator $-\delta$ by the operator $\varrho$ (see Proposition~\ref{Zurich}).
We have
$\sum_{k=1}^{\infty}\sigma(\varrho h)^k =0$
as, for all $n,p\in\NO$,
\[ \begin{tikzcd}[column sep=small]
\sections{\Lambda^n L^{\vee} \otimes S^p B^{\vee}}
\arrow{r}{h} &
\sections{\Lambda^{n-1} L^{\vee} \otimes S^{p+1} B^{\vee}} \arrow{r}{\varrho} &
\sections{\Lambda^n L^{\vee} \otimes\hat{S}^{\geqslant p+1} B^{\vee}} \arrow{r}{\sigma} & 0.
\end{tikzcd} \]
Therefore, we obtain
\[ \perturbed{\sigma}:=\sum_{k=0}^\infty\sigma(\varrho h)^k=\sigma \]
and
\[ \vartheta:=\sum_{k=0}^\infty\sigma(\varrho h)^k\varrho\tau=\sigma\varrho\tau
=\sigma(d_L^\nabla-\Xi^\nabla)(p^\top\otimes 1) =\sigma\big((d_L\circ p^\top)\otimes 1\big)=d_A .\]
The result follows immediately since
$-\delta+\varrho=d_L^{\nabla^\lightning}$ (Proposition~\ref{Budapest}).
\end{proof}

\begin{corollary}
The chain maps
\[ \big(\sections{\Lambda^\bullet L^{\vee} \otimes\hat{S} B^{\vee} },d_L^{\nabla^\lightning}\big)
\overset{\sigma}{\underset{\tau}{\rightleftarrows}}
\big(\sections{\Lambda^\bullet A^{\vee} },d_A\big) \]
are quasi-isomorphisms and homotopy inverses of each other.
\end{corollary}

Thus, we obtain the following
\begin{theorem}
Let $(L, A)$ be a Lie pair.
Given a splitting $j:L/A\to L$ of the short exact sequence
 $0\to A \to L \to L/A \to 0$
and a torsion-free $L$-connection $\nabla$ on $L/A$ extending the Bott
 $A$-connection, the natural inclusion
\[ (A[1], d_A)\xto{} (L[1]\oplus B, Q) \]
is a quasi-isomorphism of dg manifolds.
\end{theorem}

\begin{remark}\label{Fort-de-France}
We note that the homotopy operator $\perturbed{h}$ satisfies
$(\id-h\varrho)\perturbed{h}=h$ and the chain map $\perturbed{\tau}$ satisfies
$(\id-h\varrho)\perturbed{\tau}=\tau$.
\end{remark}

\begin{lemma}\label{Tremblay}
For every $y\in\sections{\Lambda^p L^{\vee}\otimes S^q(B^{\vee})}$,
there exists a unique $x\in\sections{\Lambda L^{\vee}\otimes \hat{S}(B^{\vee})}$
such that $(\id-h\varrho)x=y$.
\end{lemma}

\begin{proof}
First note that $y\in\sections{\Lambda^p L^{\vee}\otimes S^q(B^{\vee})}$ implies that
$(h\varrho)^k y\in\sections{\Lambda^p L^{\vee}\otimes \hat{S}^{\geqslant q+k}(B^{\vee})}$ as
\[ \sections{\Lambda^p L^{\vee}\otimes \hat{S}^{\geqslant q+1}(B^{\vee})} \xfrom{h} \sections{\Lambda^{p+1} L^{\vee}\otimes \hat{S}^{\geqslant q}(B^{\vee})} \xfrom{\varrho}
\sections{\Lambda^{p} L^{\vee}\otimes S^{q}(B^{\vee})} .\]
Therefore the series $\sum_{k=0}^{\infty} (h\varrho)^k y$
converges to an element $x$ in $\sections{\Lambda L^{\vee}\otimes \hat{S}(B^{\vee})}$.
Moreover, we have
\[ (\id-h\varrho)x = (\id-h\varrho) \sum_{k=0}^{\infty} (h\varrho)^k y = \sum_{k=0}^{\infty} (h\varrho)^k y -
\sum_{k=1}^{\infty} (h\varrho)^k y = y .\]

Suppose there exists another element $x'$ in
$\sections{\Lambda L^{\vee}\otimes \hat{S}(B^{\vee})}$
satisfying $(\id-h\varrho)x'=y$. It follows that
$(\id-h\varrho)(x'-x)=0$ and, consequently,
\[ x'-x=h\varrho(x'-x)=(h\varrho)^2(x'-x)=(h\varrho)^3(x'-x)
=\cdots \]
Since $x'-x\in\sections{\Lambda L^{\vee}\otimes \hat{S}^{\geqslant 0}(B^{\vee})}$, we have
\[ x'-x=(h\varrho)^k(x'-x) \in \sections{\Lambda L^{\vee}\otimes \hat{S}^{\geqslant k}(B^{\vee})} ,\qquad\forall k\in\NN .\]
Therefore,
\[ x'-x\in\bigcap_{k}
\sections{\Lambda L^{\vee}\otimes
\hat{S}^{\geqslant k}(B^{\vee})} = \{0\} \] so that $x=x'$.
\end{proof}

\subsection{First glimpse at matched pairs}

In this section, we establish explicit expressions for the quasi-isomorphism
\[ \perturbed{\tau}:\sections{\Lambda^{\bullet}A^{\vee}} \to \sections{\Lambda^{\bullet}L^{\vee}\otimes\hat{S}(B^{\vee})} \]
defined in Proposition~\ref{Ottignies} valid only in the special case of matched pairs.

Suppose the image of the chosen splitting $j:B\to L$ is a Lie subalgebroid of $L$
(i.e.\ $L=A\bowtie B$ is a matched pair).
Then $B$ is a Lie algebroid and composition of the morphism of associative algebras
$\enveloping{B}\to\enveloping{L}$ induced by $j$ with the canonical projection
$\enveloping{L}\twoheadrightarrow\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$ yields a canonical isomorphism of left $R$-coalgebras $\enveloping{B}\cong\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$.

Since $L=A\bowtie B$ is a matched pair, we have a Bott $B$-connection on $A$:
\[ \nabla^{\Bott}_b a = p\big(\lie{j(b)}{i(a)}\big) ,\quad\forall b\in\sections{B},a\in\sections{A} .\]
The dual $B$-connection on $A^{\vee}$ extends to the exterior algebra $\Lambda A^{\vee}$ by derivation:
\[ \nabla^{\Bott}_b \alpha = i\transpose\big( \liederivative{j(b)} (p\transpose\alpha) \big) ,\quad\forall b\in\sections{B},\alpha\in\sections{\Lambda A^{\vee}} .\]
One can show that, for every $b\in\sections{B}$, the diagram
\[ \begin{tikzcd}
\Lambda^\bullet A^{\vee} \arrow{r}{\nabla^{\Bott}_{b}} \arrow{d}{p\transpose} & \Lambda^\bullet A^{\vee} \arrow{d}{p\transpose} \\
\Lambda^\bullet L^{\vee} \arrow{r}[swap]{\liederivative{j(b)}} & \Lambda^\bullet L^{\vee}
\end{tikzcd} \]
commutes.
Since the Bott $B$-connection on $\Lambda A^{\vee}$ is flat, $\sections{\Lambda A^{\vee}}$ is a left $\enveloping{B}$-module: the action
\[ \enveloping{B}\times\sections{\Lambda A^{\vee}}\xto{\rtimes}\sections{\Lambda A^{\vee}} \] satisfies
\[ p\transpose\big( b_1 b_2\cdots b_n \rtimes\alpha \big) = \liederivative{j(b_1)} \liederivative{j(b_2)} \cdots \liederivative{j(b_n)}(p\transpose \alpha) ,\]
for all $b_1,b_2,\dots,b_n\in\sections{B}$ and $\alpha\in\sections{\Lambda A^{\vee}}$.

In the matched pair case, the chain map $\perturbed{\tau}:\sections{\Lambda^{\bullet}A^{\vee}}
\to \sections{\Lambda^{\bullet}L^{\vee}\otimes\hat{S}(B^{\vee})}$ defined in Proposition~\ref{Ottignies}
admits a simple description in terms of the splitting $j:B\hookrightarrow L$,
the associated left $\enveloping{B}$-module structure $\rtimes$ on $\sections{\Lambda A^{\vee}}$,
and the map $\pbw:\sections{S(B)}\to\enveloping{B}$ induced by $j$ and $\nabla$.

Consider the derivation $\mathscr{D}$ of
$C^{\infty}(A[1]\oplus B) = \sections{p\transpose(\Lambda A^{\vee})\otimes \hat{S}(B^{\vee})}$
defined by
\[ \mathscr{D}(p\transpose\alpha\otimes\chi^J)=
\sum_{k=1}^{r} \left\{ p\transpose(\nabla^{\Bott}_{\partial_k}\alpha)
\otimes\chi_k\cdot\chi^J +p\transpose\alpha\otimes\chi_k\cdot
\nabla_{j(\partial_k)}(\chi^J) \right\} ,\]
for all $\alpha\in\sections{\Lambda^{\bullet}A^{\vee}}$ and $J\in\NO^{r}$.

\begin{remark}
A analogue of the derivation $\mathscr{D}$ was introduced recently in \cite[Section~2.1 and Remark~2.3]{1512.07863}.
It would be interesting to understand the precise relation between these two derivations.
\end{remark}

The remainder of this section is devoted to the proof of the following theorem.
\begin{theorem}\label{Montmartre}
If the splitting
\[ \begin{tikzcd}
0 \arrow{r} & A \arrow{r}{i} & L \arrow[bend left, dashed]{l}{p} \arrow{r}{q} & B \arrow{r} \arrow[bend left, dashed]{l}{j} & 0
\end{tikzcd} \]
identifies $B$ with a Lie subalgebroid of $L$,
then $\sections{\Lambda A^{\vee}}$ is a left $\enveloping{B}$-module and
the chain map \[ \perturbed{\tau}:\sections{\Lambda^{\bullet}A^{\vee}}
\to \sections{\Lambda^{\bullet}L^{\vee}\otimes\hat{S}(B^{\vee})} \] defined in Proposition~\ref{Ottignies} satisfies
\[ \perturbed{\tau}=\exp(\mathscr{D})\circ\tau \]
and
\[ \perturbed{\tau}(\alpha)
= \sum_{J\in\NO^r} \frac{1}{J!} p\transpose\big( \pbw(\partial^J)\rtimes
\alpha\big)\otimes\chi^J ,\qquad\forall \alpha\in\sections{\Lambda A^{\vee}} .\]
\end{theorem}

The following Lemma is an analogue of Lemma~\ref{Ragusa},
which is proved \textit{mutans mutandis}.
\begin{lemma}\label{Ragusa2}
For all $\lambda\in\sections{p\transpose(\Lambda^u A^{\vee})\otimes
q\transpose(\Lambda^v B^{\vee})}\subset
\sections{\Lambda^{u+v}L^\vee}$ and $J\in\NO^r$, we have
\[ h(\lambda\otimes\chi^J) =
\begin{cases}
\frac{1}{v+\abs{J}}\sum_{k=1}^{r}
\interior{j(\partial_k)}\lambda\otimes\chi^{J+e_k} &\text{if } v\geqslant 1 \\
0 &\text{if } v=0, \end{cases} \]
where $(\partial_i)_{i\in\{1,\dots,r\}}$ denotes any local frame of $B$
and $(\chi_j)_{j\in\{1,\dots,r\}}$ denotes the dual local frame of $B^{\vee}$.
\end{lemma}

The following Lemma is an analogue of Lemma~\ref{Telegraphe},
which is proved \textit{mutans mutandis}.
\begin{lemma}\label{Telegraphe2}
For all $\alpha\in\sections{\Lambda^{\bullet}A^{\vee}}$ and $J\in\NO^{r}$,
we have $h\Xi(p\transpose\alpha\otimes\chi^J)=0$.
\end{lemma}

\begin{lemma}\label{Montrouge}
For all $\alpha\in\sections{\Lambda^{\bullet}A^{\vee}}$ and $J\in\NO^{\rk(B)}$,
we have
\begin{align*}
h \varrho(p\transpose\alpha\otimes\chi^J)
=& \frac{1}{1+\abs{J}}\sum_k\left\{
p\transpose(\nabla^{\Bott}_{\partial_k}\alpha) \otimes\chi_k\cdot\chi^J +p\transpose\alpha\otimes\chi_k\cdot \nabla_{j(\partial_k)}(\chi^J) \right\} \\
=& \frac{1}{1+\abs{J}} \mathscr{D}(p\transpose\alpha\otimes\chi^J)
.\end{align*}
\end{lemma}

\begin{proof}
Since $L=A\bowtie B$, Proposition~\ref{Piacenza} asserts that
\[ d_L(p\transpose\alpha)\in\Omega^{u+1,v}\oplus\Omega^{u,v+1} ,\]
where $\Omega^{u,v}=\sections{p\transpose(\Lambda^{u} A^{\vee})\otimes q\transpose(\Lambda^{v} B^{\vee})}$.
Therefore, we have
\begin{multline*} d_L^{\nabla}(p\transpose\alpha\otimes\chi^J)
= d_L(p\transpose\alpha)\otimes\chi^J +\sum_t\lambda_t\wedge(p\transpose\alpha)\otimes\nabla_{l_t}(\chi^J) \\
\in (\Omega^{u+1,v}\oplus\Omega^{u,v+1})\otimes_R\sections{S^{\abs{J}}(B^{\vee})} \end{multline*}
and it follows from Lemma~\ref{Ragusa2} that
\begin{align*} hd_L^{\nabla}(p\transpose\alpha\otimes\chi^J)
=& \frac{1}{1+\abs{J}} \sum_k \bigg\{ i_{j(\partial_k)}d_L(p\transpose\alpha)\otimes\chi^{J+e_k}
+\sum_t(i_{j(\partial_k)}\lambda_t)\cdot p\transpose\alpha\otimes\chi_k\cdot\nabla_{l_t}(\chi^J) \bigg\} \\
=& \frac{1}{1+\abs{J}} \sum_k \bigg\{ \liederivative{j(\partial_k)}(p\transpose\alpha)\otimes\chi^{J+e_k}
+p\transpose\alpha\otimes\chi_k\cdot\nabla_{\sum_t(i_{j(\partial_k)}\lambda_t) l_t}(\chi^J) \bigg\} \\
=& \frac{1}{1+\abs{J}} \sum_k \bigg\{ p\transpose(\nabla^{\Bott}_{\partial_k}\alpha)\otimes\chi^{J+e_k}
+p\transpose\alpha\otimes\chi_k\cdot\nabla_{j(\partial_k)}(\chi^J) \bigg\}
.\end{align*}
The desired result follows from Lemma~\ref{Telegraphe2}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{Montmartre}]
Reasoning by induction on $k$, one proves that
\[ \mathscr{D}^k\circ\tau(\alpha)\in\sections{p\transpose(\Lambda A^{\vee})\otimes
S^k(B^{\vee})} \] for all $\alpha\in\sections{\Lambda A^{\vee}}$ and $k\in\NN$.
Using Lemma~\ref{Montrouge} and reasoning by induction on $k$ once again,
one proves that \[ (h\varrho)^{k}\circ\tau=\frac{1}{k!}\mathscr{D}^k\circ\tau \]
for all $k\in\NN$. It follows that
\[ \breve{\tau} = \bigg(\sum_{k=0}^{\infty}(h\varrho)^k\bigg)\circ\tau
= \bigg(\sum_{k=0}^{\infty}\frac{1}{k!}\mathscr{D}^k\bigg)\circ\tau
= \exp(\mathscr{D})\circ\tau .\]

Set $\machin{\alpha}=\sum_{J\in\NO^r} \frac{1}{J!}p^\top\big( \pbw(\partial^J)\rtimes \alpha\big)\otimes\chi^J$
for all $\alpha\in\sections{\Lambda A^{\vee}}$.
We claim that $(\id-h\varrho)\machin{\alpha}=\tau(\alpha)$.
It then follows from Remark~\ref{Fort-de-France} and Lemma~\ref{Tremblay} that
$\breve{\tau}(\alpha)=\machin{\alpha}$ --- the desired result.

It remains to establish our claim.
From
\[ 0=\rho(j(\partial_k))\duality{\partial^K}{\chi^J}
=\duality{\nabla_{j(\partial_k)}(\partial^K)}{\chi^J}
+\duality{\partial^K}{\nabla_{j(\partial_k)}(\chi^J)} ,\]
we obtain
 \begin{equation}\label{Caracas} \nabla_{j(\partial_k)}(\chi^J)
 = \sum_K\frac{1}{K!}\duality{\partial^K}{\nabla_{j(\partial_k)}(\chi^J)}\chi^K
 = -\sum_K\frac{1}{K!}\duality{\nabla_{j(\partial_k)}(\partial^K)}{\chi^J}\chi^K
 .\end{equation}

From Lemma~\eqref{Telegraphe2}, Lemma~\eqref{Montrouge}, and Equation~\eqref{Caracas}, we obtain
\begin{multline*}
h\varrho\machin{\alpha} =
\sum_{J}\tfrac{1}{J!}\tfrac{1}{1+\abs{J}}\sum_k\bigg\{
p\transpose\Big(\nabla^{\Bott}_{\partial_k} \big(\pbw(\partial^J)\rtimes\alpha\big)\Big)\otimes
\chi_k\cdot\chi^J \\
-p\transpose\big(\pbw(\partial^J)\rtimes\alpha\big)\otimes\chi_k\cdot
\sum_K\tfrac{1}{K!}\duality{\nabla_{j(\partial_k)}(\partial^K)}{\chi^J}\chi^K\bigg\}
\end{multline*}
This can be rewritten as
\begin{multline*}
h\varrho\machin{\alpha} =
\sum_J\tfrac{1}{J!}\tfrac{1}{1+\abs{J}}\sum_k
p\transpose\big(j(\partial_k)\cdot\pbw(\partial^J)\rtimes\alpha\big)
\otimes\chi^{J+e_k} \\
-\sum_K\tfrac{1}{K!}\tfrac{1}{1+\abs{K}}\sum_k
p\transpose\Big(\pbw\big(\nabla_{j(\partial_k)}(\partial^K)\big)\rtimes\alpha\Big)
\otimes\chi^{K+e_k}
\end{multline*}
and then
\begin{multline*}
h\varrho\machin{\alpha} = \sum_{\substack{M\in\NO^r \\ \abs{M}\geqslant 1}}
\frac{1}{M!} p\transpose \bigg( \frac{1}{\abs{M}}\sum_k M_k
\Big( j(\partial_k)\cdot\pbw\big(\partial^{M-e_k}\big)
-\pbw\big(\nabla_{j(\partial_k)}\partial^{M-e_k}\big) \Big)
\rtimes\alpha \bigg)\otimes\chi^M
.\end{multline*}
Finally, it follows from Equation~\eqref{Rome} that
\[ h\varrho\machin{\alpha}
= \sum_{\substack{M\in\NO^r \\ \abs{M}\geqslant 1}}
\frac{1}{M!} p\transpose \big(\pbw(\partial^M)\rtimes\alpha\big)\otimes\chi^M
= \machin{\alpha}- p\transpose(\alpha)\otimes 1
= \machin{\alpha}-\tau(\alpha) .\qedhere\]
\end{proof}

\subsection{Fedosov dg Lie algebroids}

A $\ZZ$-graded manifold $\mathcal{M}$ with base manifold $M$ is
a sheaf of $\ZZ$-graded, graded-commutative algebras
$\{\mathcal{R}_U|U\subset M \ \text{open}\}$
over $M$, locally isomorphic to $C^\infty (U)\otimes
\hat{S}(V^\vee)$, where $U\subset M$ is an open submanifold,
$V$ is a $\ZZ$-graded vector space, and $\hat{S}(V^\vee)$ denotes
the graded algebra of formal polynomials on $V$.
By $C^\infty (\mathcal{M})$, we denote the $\ZZ$-graded, graded-commutative algebra
of global sections. By a dg manifold,
we mean a $\ZZ$-graded manifold endowed with a
homological vector field, i.e.\ a vector field $Q$ of degree $+1$ satisfying
$\lie{Q}{Q}=0$.

\begin{example}\label{example:Vauintrob dg vs Lie abd}
Let $A\to M$ be a Lie algebroid over $\Bbbk$. Then $A[1]$ is a dg manifold
with the Chevalley--Eilenberg differential
 $d_{\CE}$ as homological vector field.
According to Va\u{\i}ntrob~\cite{MR1480150},
there is a bijection between the Lie algebroid structures on the vector
 bundle $A\to M$
and the homological vector fields on the $\ZZ$-graded manifold $A[1]$.
\end{example}

\begin{example}
Let $\mathfrak g=\sum_{i\in \ZZ}\mathfrak g_i$ be a $\ZZ$-graded vector space of finite type, i.e.\ each $\mathfrak g_i$ is a finite-dimensional vector space. Then $\mathfrak g$ is said to be an $L_\infty$ algebra if $\mathfrak g[1]$ is a dg manifold.
\end{example}

Below we recall some basic notations regarding dg vector bundles.
For details, see ~\cite{MR2534186,MR3293862}.
A dg vector bundle is a vector bundle in
the category of dg manifolds.
Given a vector bundle $\mathcal{E}\stackrel{\pi}{\to} \mathcal{M}$ of graded
manifolds, its space of sections, denoted
$\sections{\mathcal{E}}$, is defined to be $\bigoplus_{j\in \ZZ}\Gamma (\mathcal{E})_j$,
where $\Gamma (\mathcal{E})_j$ consists of degree-$j$ sections, i.e.\ maps
$s\in \Hom (\mathcal{M}, \mathcal{E}[-j])$ such that $(\pi[-j])\circ s=\id_\mathcal{M}$.
Here $\pi[-j]: \mathcal{E}[-j]\to \mathcal{M}$ is the natural map induced from $\pi$ ---
see~\cite{MR2534186} for more details.
When $\mathcal{E}\to \mathcal{M}$ is a dg vector bundle, the homological
vector fields on $\mathcal{E}$ and $\mathcal{M}$ naturally induce
a degree $+1$ operator $\mathcal{Q}$ on $\sections{\mathcal{E}}$, making
$\sections{\mathcal{E}}$ a dg module over $C^\infty(\mathcal{M})$.
Since the space $\sections{\mathcal{E}^\vee}$ of linear functions on $\mathcal{E}$
together with the pull-back of $C^\infty (\mathcal{M})$ generates
$C^\infty(\mathcal{E})$, the converse is also true (see~\cite{MR3319134}).

In this case, the degree $+1$ operator
 $\mathcal{Q}$ on $\Gamma(\mathcal{E})$ gives rise to a cochain complex
\[ \cdots\to\Gamma (\mathcal{E})_i \xto{\mathcal{Q}}\Gamma (\mathcal{E})_{i+1}\to\cdots ,\]
whose cohomology group will be denoted
by $H^\bullet(\Gamma(\mathcal{E}),\mathcal{Q})$.

A dg Lie algebroid can be thought of as a Lie algebroid object
in the category of dg manifolds. For more details, we refer the reader
 to~\cite{MR2534186,MR2709144}, where dg Lie algebroids are called $Q$-algebroids.
It is simple to see that if $\mathcal{M}$ is a dg manifold, then $T_{\mathcal{M}}$ is
naturally a dg Lie algebroid.

One can make sense of ``polyvector fields'' and ``polydifferential operators''
on a dg Lie algebroid just as one does for ordinary Lie algebroids.
More precisely, a $k$-vector field on a dg Lie algebroid $\mathcal{L}\to\mathcal{M}$ is a section of the vector bundle $\wedge^k\mathcal{L}\to\mathcal{M}$
while a $k$-differential operator is an element of $\enveloping{\mathcal{L}}^{\otimes k}$, the
tensor product (as left $R$-modules) of $k$ copies of the universal enveloping algebra $\enveloping{\mathcal{L}}$.
The universal enveloping algebra $\enveloping{\mathcal{L}}$ of the dg Lie algebroid $\mathcal{L}\to\mathcal{M}$,
whose precise definition is a straightforward adaptation of the construction outlined
in Section~\ref{universal} for ordinary Lie algebroids,
is a dg Hopf algebroid over the dga $C^\infty( \mathcal{M} )$.
There exist natural Gerstenhaber algebra structures on the spaces of polyvector fields and polydifferential operators of a dg Lie algebroid.
For each $k\geq 0$, there is an induced differential $\mathcal{Q} : \UcL{k}\to \UcL{k}$ of degree $+1$.

Using the comultiplication of $\enveloping{\mathcal{L}}$, one can define the Hochschild coboundary differential
\[ d_{\mathscr{H}}: {\enveloping{\mathcal{L}}}^{\otimes k}\to{\enveloping{\mathcal{L}}}^{\otimes (k+1)} \]
by the explicit algebraic expression
\begin{multline*}
d_{\mathscr{H}}(u_1\otimes\cdots\otimes u_k) = 1\otimes u_1\otimes\cdots\otimes u_k
+ \sum_{i=1}^{k} (-1)^i u_1\otimes\cdots\otimes\Delta(u_i)\otimes\cdots\otimes u_k \\
+ (-1)^{k+1} u_1\otimes\cdots\otimes u_k\otimes 1
.\end{multline*}

Similarly, the Gerstenhaber bracket of a $(u+1)$-differential operator
$\phi\in\enveloping{L}^{\otimes(u+1)}$ and a $(v+1)$-differential operator $\psi\in\enveloping{L}^{\otimes(v+1)}$
is the $(u+v+1)$-differential operator
\begin{equation*}
\gerstenhaber{\phi}{\psi} = \phi\star\psi - (-1)^{uv} \psi\star\phi \in \enveloping{L}^{\otimes(u+v+1)}
,\end{equation*}
where $\phi\star\psi\in \enveloping{L}^{\otimes(u+v+1)}$ is defined by
\[ \phi\star\psi = \sum_{k=0}^{u} (-1)^{kv}
d_0\otimes_R \cdots \otimes_R d_{k-1}\otimes_R (\Delta^v d_k)\cdot \psi \otimes_R d_{k+1}
\otimes_R \cdots \otimes_R d_u \]
if $\phi=d_0\cdot d_1 \cdots d_u$ for some $d_0,d_1,\dots,d_u\in\enveloping{L}$.

\begin{proposition}\label{pro:hongkong}
Let $\mathcal{L}\to \mathcal{M}$ be a dg Lie algebroid over $\Bbbk$.
\begin{enumerate}
\item When endowed with the differential $\mathcal{Q}$, the wedge product, and the Schouten bracket,
$\sections{\wedge^\bullet \mathcal{L}}$ is a differential Gerstenhaber algebra.
\item The cohomology group
 $H^\bullet \big(\sections{\wedge^\bullet \mathcal{L}}, \mathcal{Q})$
together with the wedge product and the Schouten bracket
is naturally a Gerstenhaber algebra.
\item
The Hochschild cohomology,
i.e.\ the total cohomology
$H^\bullet \big( \enveloping{\mathcal{L}}^{\otimes \bullet}, \mathcal{Q}+d_{\mathscr{H}} \big)$
together with the standard tensor product $\otimes_{C^{\infty}(\mathcal{M})}$, called the cup
product, and the Gerstenhaber bracket becomes a Gerstenhaber algebra.
\end{enumerate}
\end{proposition}

Let $(L,A)$ be a Lie pair.
Given a splitting $j:B\to L$ of the short exact sequence of vector bundles
$0\to A \to L \to B \to 0$ and a torsion-free $L$-connection $\nabla$ on $B$,
one constructs a Fedosov dg manifold $(L[1]\oplus B, Q)$ as in
Theorem~\ref{strawberry}.

Let $T_{\ver}B\to B$ denote the formal vertical tangent bundle of the
bundle $B\to M$, which consists of all
formal vertical tangent vectors of $B$. Then $\sections{B; T_{\ver}B}
\cong \sections{SB^\vee \otimes B}$. Indeed $T_{\ver}B$
is a double vector bundle \cite{math/9808081}, which
is isomorphic to $B\oplus B$. Consider the projection
$\pr: \mathcal{M}:= L[1] \oplus B \to B$. Denote by $\mathcal{A}$ the pull back bundle
$\pr^* T_{\ver}B$. Then $\mathcal{A}$ is a graded vector bundle over $\mathcal{M}$.
Note that as a graded manifold, $\mathcal{A}$ has support $M$ and
is defined by the graded vector bundle $L[1] \oplus B\oplus B\to M$.

\begin{proposition}
Let $(L[1]\oplus B, Q)$ be the Fedosov dg manifold as in Theorem~\ref{strawberry}.
Then $\mathcal{A}$ is a dg Lie algebroid over $\mathcal{M}$.
\end{proposition}

\begin{proof}
It is simple to see that $\mathcal{A}\to \mathcal{M}$ is a $\ZZ$-graded Lie algebroid.
It suffices to show that $\mathcal{A}$ admits a dg manifold structure compatible with
the $\ZZ$-graded Lie algebroid structure. According to
the observation above, it suffices to prove that
$\sections{\mathcal{M}; \mathcal{A}}$ is a dg module over $C^\infty(\mathcal{M})$.
It is clear that
 $\sections{\mathcal{M}; \mathcal{A}}\cong
\sections{\wedge^\bullet L^\vee \otimes \mathscr{X}_{\ver} (B)}$, where
$\mathscr{X}_{\ver} (B)$ denotes the space of formal vertical vector fields on $B$.
From Equation~\eqref{eq:Q}, it follows that
$\sections{\wedge^\bullet L^\vee \otimes \mathscr{X}_{\ver}(B)}$
is stable under the Lie derivative $L_Q$. Moreover,
we have
\[ \liederivative{Q}(\xi\cdot(\eta\otimes X))=Q(\xi)\cdot(\eta\otimes X)
+(-1)^{|\xi|}\xi\cdot \liederivative{Q}(\eta\otimes X) ,\]
for all $\xi\in\sections{\wedge^\bullet L^\vee\otimes\hat{S}B^\vee}$,
$\eta\in\sections{\wedge^\bullet L^\vee}$, and $X\in\mathscr{X}_{\ver}(B)$.
Therefore, $\sections{\mathcal{M};\mathcal{A}}$ is a dg module over $C^\infty(\mathcal{M})$.
\end{proof}

Such a dg Lie algebroid is called \emph{Fedosov dg Lie algebroid}
associated to the Lie pair $(L, A)$.

Set $\Xpoly{k}:=\sections{\wedge^{k+1} B}$
 and let $\verticalXpoly{k}(B)$ denote the space of formal
 vertical $(k+1)$-vector fields on $B$, i.e.\
\[ \verticalXpoly{k}(B) = \sections{\hat{S}(B^{\vee})}
\otimes_R \Xpoly{k} .\]

Then
\[ \sections{\Lambda^{\bullet}L^{\vee}}\otimes_R
\verticalXpoly{k}(B)
\cong \sections{\Lambda^{\bullet}L^{\vee}\otimes \hat{S}B^{\vee}}\otimes_R \Xpoly{k} \]

It is clear that
\[ \sections{\mathcal{M}; \wedge^k\mathcal{A}}\cong \sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalXpoly{k}(B) .\]

Let $\verticalDpoly{k}$ denote the space of formal vertical
 $(k+1)$-polydifferential operators on the vector bundle $B$,
 and $\verticalDpoly{\bullet}=
\bigoplus_{k=-1}^{\infty}\verticalDpoly{k}$.
Set $\mathscr{S}=\sections{\hat{S}(B^{\vee})}$.
There exists a canonical isomorphism
\[ \begin{tikzcd}
\sections{\hat{S}(B^{\vee})\otimes
\underset{k+1\text{ factors}}{\underbrace{S(B)\otimes\cdots\otimes S(B)}}}
\arrow{r}{\varphi}[swap]{\cong} &
\verticalDpoly{k}
\end{tikzcd} .\]

To $\chi^I\otimes\partial^{J_0}\otimes
\cdots\otimes\partial^{J_k}\in \sections{\hat{S}(B^{\vee})\otimes
\underset{k+1\text{ factors}}{\underbrace{S(B)\otimes\cdots\otimes S(B)}}}$,
the isomorphism $\varphi$ associates the poly\-differential operator
\[ \mathscr{S}^{\otimes k+1}\ni
\chi^{M_0}\otimes\cdots\otimes\chi^{M_k}\longmapsto
\chi^I\cdot\partial^{J_0}(\chi^{M_0})
\cdots\partial^{J_k}(\chi^{M_k})
\in\mathscr{S} .\]

The algebra of functions $C^\infty(L[1]\oplus B)$ is a
 module over its subalgebra
$\sections{\Lambda^\bullet L}\equiv\sections{\Lambda^\bullet L^{\vee}\otimes S^0(B^{\vee})}$.
The subspace of $D_{\poly}^{\bullet}(L[1]\oplus B)$ comprised of all
$\sections{\Lambda^\bullet L^{\vee}}$-multilinear polydifferential operators
is easily identified to
$\sections{\Lambda^\bullet L^{\vee}}\otimes_R\verticalDpoly{\bullet}$.
It is simple to see that the universal enveloping algebra
$\enveloping{\mathcal{A}}$ of $\mathcal{A}\to\mathcal{M}$, is naturally identified
with $\sections{\Lambda^\bullet L^{\vee}}\otimes_R\verticalDpoly{0}$,
which is a dg Hopf algebroid over $C^\infty (\mathcal{M})\cong
\sections{\Lambda^\bullet L^{\vee} \otimes\hat{S} B^{\vee} }$.
Moreover,
$\enveloping{\mathcal{A}}$ is a dg Hopf subalgebroid of $D_{\poly}^{0}(L[1]\oplus B)$.
Note that
\[ \enveloping{\mathcal{A}}^k\cong \sections{\Lambda^\bullet L^{\vee}}
\otimes_R\verticalDpoly{k-1} .\]

Thus, as a consequence of Proposition~\ref{pro:hongkong}, we have the following

\begin{proposition}
\label{pro:Metz}
\begin{enumerate}
\item The cochain complex $\Big(\tot\big(\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalXpoly{\bullet} (B)\big), \liederivative{Q}\Big)$
is a differential Gerstenhaber algebra.
\item The cohomology group $H^\bullet \Big(\tot\big(\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalXpoly{\bullet} (B)\big), \liederivative{Q}\Big)$
is a Gerstenhaber algebra.
\item The cohomology group $H^\bullet \Big(\tot\big(\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalDpoly{\bullet}\big), \gerstenhaber{Q\pm m}{-}\Big)$
is a Gerstenhaber algebra.
\end{enumerate}
\end{proposition}

\subsection{Dolgushev--Fedosov quasi-isomorphisms for $\Xpoly{\bullet}$}

\begin{lemma}
The subspace
$\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalXpoly{k}(B)$ of the space
$T_{\poly}^{k}(L[1]\oplus B)$
of $(k+1)$-vector fields on $L[1]\oplus B$
is stable under $\liederivative{\delta}$.
\end{lemma}

The following diagram commutes:
\[ \begin{tikzcd}
\sections{\Lambda^{i}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow{r}{\liederivative{\delta}}
\arrow[leftrightarrow]{d}{\equiv}
&
\sections{\Lambda^{i+1}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow[leftrightarrow]{d}{\equiv}
\\
\sections{\Lambda^{i}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R \Xpoly{k}
\arrow{r}[swap]{\delta\otimes\id}
&
\sections{\Lambda^{i+1}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R \Xpoly{k}
\end{tikzcd} \]

Since the vector field $\delta$ on $L[1]\oplus B$
is homological, we obtain the cochain complex
\[ \begin{tikzcd}
\cdots \arrow{r} &
\sections{\Lambda^{i}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow{r}{\liederivative{-\delta}} &
\sections{\Lambda^{i+1}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow{r} & \cdots
\end{tikzcd} \]
which admits the descending filtration
\[ \mathscr{F}^m=\bigoplus_{i=0}^{\rk(L)}
\sections{\Lambda^i L^{\vee}
\otimes\hat{S}^{\geqslant m-i} B^{\vee} }\otimes_R
\Xpoly{k} .\]

Adapting the proof of Proposition~\ref{Geneva}, we obtain
\begin{proposition}\label{Shiraz}
The cochain complex
$\big( \sections{\Lambda^{\bullet}L^{\vee}}
\otimes_R\verticalXpoly{k}, \liederivative{-\delta} \big)$
contracts onto
$\big( \sections{\Lambda^{\bullet}A^{\vee}}
\otimes_R\Xpoly{k}, 0 \big)$.
More precisely, we have the filtered contraction
\begin{dontbotheriftheequationisoverlong} \begin{tikzcd}
\cdots \arrow{r} & \sections{\Lambda^{n-1} L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow{d}{\etendu{\sigma}}
\arrow{r}{\liederivative{-\delta}} &
\sections{\Lambda^{n} L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow{d}{\etendu{\sigma}}
\arrow{r}{\liederivative{-\delta}}
\arrow[dashed]{ddl}[near end]{\etendu{h}}  &
\sections{\Lambda^{n+1} L^{\vee} }\otimes_R\verticalXpoly{k}
\arrow{d}{\etendu{\sigma}} \arrow{r}
\arrow[dashed]{ddl}[near end]{\etendu{h}} & \cdots \\
\cdots \arrow{r} &
\sections{\Lambda^{n-1} A^{\vee}}\otimes_R\Xpoly{k}
\arrow{d}[swap]{\etendu{\tau}} \arrow{r}[near start]{0} &
\sections{\Lambda^{n} A^{\vee}}\otimes_R\Xpoly{k}
\arrow{d}[swap]{\etendu{\tau}} \arrow{r}[near start]{0} &
\sections{\Lambda^{n+1} A^{\vee}}\otimes_R\Xpoly{k}
\arrow{d}[swap]{\etendu{\tau}} \arrow{r} & \cdots \\
\cdots \arrow{r} &
\sections{\Lambda^{n-1} L^{\vee} }\otimes_R\verticalXpoly{k}
\arrow{r}{\liederivative{-\delta}} &
\sections{\Lambda^{n} L^{\vee} }\otimes_R\verticalXpoly{k}
\arrow{r}{\liederivative{-\delta}} &
\sections{\Lambda^{n+1} L^{\vee} }\otimes_R\verticalXpoly{k}
\arrow{r} & \cdots
\end{tikzcd} \end{dontbotheriftheequationisoverlong}
where
$\etendu{\sigma}$, $\etendu{\tau}$ and $\etendu{h}$ are
the three maps making the following three diagrams commute:
\begin{equation} \begin{tikzcd}[row sep=tiny]
\sections{\Lambda^{i}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow{dr}{\etendu{\sigma}}
\arrow[leftrightarrow]{dd}{\equiv} & \\
& \sections{\Lambda^{i}A^{\vee}}\otimes_R\Xpoly{k} \\
\sections{\Lambda^{i}L^{\vee}\otimes\hat{S} B^{\vee} }
\otimes_R \Xpoly{k}
\arrow{ur}[swap]{\sigma\otimes\id} &
\end{tikzcd} \label{eq:etendusigma} \end{equation}

\[ \begin{tikzcd}[row sep=tiny]
& \sections{\Lambda^{i}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow[leftrightarrow]{dd}{\equiv} \\
\sections{\Lambda^{i}A^{\vee}}\otimes_R\Xpoly{k}
\arrow{ur}{\etendu{\tau}}
\arrow{dr}[swap]{\tau\otimes\id} & \\
& \sections{\Lambda^{i}L^{\vee}\otimes\hat{S} B^{\vee} }
\otimes_R \Xpoly{k}
\end{tikzcd} \]

\[ \begin{tikzcd}
\sections{\Lambda^{i}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow{r}{\etendu{h}}
\arrow[leftrightarrow]{d}{\equiv}
&
\sections{\Lambda^{i-1}L^{\vee}}\otimes_R\verticalXpoly{k}
\arrow[leftrightarrow]{d}{\equiv}
\\
\sections{\Lambda^{i}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R \Xpoly{k}
\arrow{r}[swap]{h\otimes\id}
&
\sections{\Lambda^{i-1}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R \Xpoly{k}
\end{tikzcd} \]
\end{proposition}

\begin{theorem}\label{Bagnolet}
The complex
$\big( \sections{\Lambda^{\bullet}L^{\vee}}
\otimes_R\verticalXpoly{k}, \liederivative{Q} \big)$
contracts onto
$\big( \sections{\Lambda^{\bullet}A^{\vee}}
\otimes_R\Xpoly{k}, d_A^{\Bott} \big)$.
More precisely, we have the (filtered) contraction
\begin{dontbotheriftheequationisoverlong} \begin{tikzcd}
\cdots \arrow{r} & \sections{\Lambda^{n-1}(L^{\vee})}\otimes_R\verticalXpoly{k}
\arrow{d}{\etendu{\sigma}}
\arrow{r}{\liederivative{Q}} &
\sections{\Lambda^{n}(L^{\vee})}\otimes_R\verticalXpoly{k}
\arrow{d}{\etendu{\sigma}}
\arrow{r}{\liederivative{Q}}
\arrow[dashed]{ddl}[near end]{\etendu{\perturbed{h}}} &
\sections{\Lambda^{n+1}(L^{\vee})}\otimes_R\verticalXpoly{k}
\arrow{d}{\etendu{\sigma}} \arrow{r}
\arrow[dashed]{ddl}[near end]{\etendu{\perturbed{h}}} & \cdots \\
\cdots \arrow{r} &
\sections{\Lambda^{n-1}(A^{\vee})}\otimes_R\Xpoly{k}
\arrow{d}[swap]{\etendu{\perturbed{\tau}}} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^{n}(A^{\vee})}\otimes_R\Xpoly{k}
\arrow{d}[swap]{\etendu{\perturbed{\tau}}} \arrow{r}{d_A^{\Bott}} &
\sections{\Lambda^{n+1}(A^{\vee})}\otimes_R\Xpoly{k}
\arrow{d}[swap]{\etendu{\perturbed{\tau}}} \arrow{r} & \cdots \\
\cdots \arrow{r} &
\sections{\Lambda^{n-1}(L^{\vee})}\otimes_R\verticalXpoly{k}
\arrow{r}{\liederivative{Q}} &
\sections{\Lambda^{n}(L^{\vee})}\otimes_R\verticalXpoly{k}
\arrow{r}{\liederivative{Q}} &
\sections{\Lambda^{n+1}(L^{\vee})}\otimes_R\verticalXpoly{k}
\arrow{r} & \cdots
\end{tikzcd} \end{dontbotheriftheequationisoverlong}
where $\etendu{\perturbed{h}}=\sum_{k=0}^\infty
(\etendu{h}\liederivative{\varrho})^k \etendu{h}$ and
\begin{equation}\label{eq:perturbedetendutau}
\etendu{\perturbed{\tau}}=\sum_{k=0}^\infty
(\etendu{h}\liederivative{\varrho})^k\etendu{\tau}
.\end{equation}
\end{theorem}

The proof requires the following technical results.

\begin{lemma}\label{Cergy}
Let $\pr_0$ denote the canonical projection
$\hat{S}(B^{\vee})\otimes B\twoheadrightarrow
S^0(B^{\vee})\otimes B$.
For all $a\in\sections{A}$ and $j\in\{1,\dots,r\}$, we have
\[ \pr_0\big(\schouten{\nabla^{\lightning}_a}{\partial_j}\big)
=\nabla^{\Bott}_a(\partial_j) .\]
\end{lemma}

\begin{proof}
We have seen that, for all $a\in\sections{A}$,
the operator $\nabla^{\lightning}_a$ is a derivation of
$\sections{\hat{S}(B^{\vee})}$, which stabilizes
the filtration $\sections{\hat{S}^{\geqslant n}(B^{\vee})}$.
Therefore, there exist local sections
$\theta_k^M$ of $A^{\vee}$ such that
\[ \nabla^{\lightning}_a \chi_k=
\sum_{\substack{M\in\NO^r \\ \abs{M}\geqslant 1}}
i_a\theta_k^M\cdot \chi^M .\]
It follows that $\nabla^{\lightning}_a$ may be regarded
as a section of $\hat{S}^{\geqslant 1}(B^{\vee})\otimes B$:
\[ \nabla^{\lightning}_a =
\sum_{k=1}^r \bigg(\sum_{\substack{M\in\NO^r \\ \abs{M}\geqslant 1}}
i_a\theta_k^M\cdot \chi^M\bigg) \partial_k .\]

On one hand, it follows from
\begin{multline*} \schouten{\nabla^{\lightning}_a}{\partial_j}
= \nabla^{\lightning}_a\circ\partial_j - \partial_j\circ
\nabla^{\lightning}_a =
\sum_{k=1}^r \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M\cdot\chi^M\partial_k\circ\partial_j
-\sum_{k=1}^r \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M\cdot
\partial_j\circ(\chi^M\partial_k) \\
=-\sum_{k=1}^r \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M\cdot M_j \chi^{M-e_j}\cdot\partial_k
\end{multline*}
that
\[ \pr_0\big(\schouten{\nabla^{\lightning}_a}{\partial_j}\big)
= -\sum_{k=1}^r i_a\theta_k^{e_j} \cdot \partial_k .\]

On the other hand, it follows from
\[ 0 = \rho(a)\underset{\delta_{k,j}}{\underbrace{\duality{\chi_k}{\partial_j}}} =
\duality{\nabla^{\lightning}_a\chi_k}{\partial_j}+
\duality{\chi_k}{\nabla^{\lightning}_a\partial_j} \]
and the fact that $\nabla^{\lightning}_a$ stabilizes the
subspace $\sections{S^1(B)}$ of $\sections{S(B)}$
that
\[ \begin{split}
\nabla^{\lightning}_a\big(\partial_j\big)
=&\ \sum_k \duality{\chi_k}{
\nabla^{\lightning}_a \partial_j}\partial_k \\
=&\ -\sum_k \duality{\nabla^{\lightning}_a \chi_k}{\partial_j} \partial_k \\
=&\ -\sum_k \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M\cdot\duality{\chi^M}{\partial_j}\cdot\partial_k \\
=&\ -\sum_k i_a\theta_k^{e_j}\cdot\partial_k
.\end{split} \]
Finally, for all $a\in\sections{A}$
and $B\in\sections{B}$, we have
$\nabla^{\lightning}_a(b)=\nabla^{\Bott}_a(b)$ as
\begin{multline*}
\pbw(\nabla^{\lightning}_a b-\nabla^{\Bott}_a b)
= a\cdot\pbw(b)-\pbw\big(q\lie{a}{j(b)}\big) \\
= a\cdot j(b)-j\circ q(\lie{a}{j(b)})
= j(b)\cdot a+ \underset{\in\sections{A}}
{\underbrace{p(\lie{a}{j(b)})}} =0
\end{multline*}
in $\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$.
The proof is complete.
\end{proof}

\begin{lemma}\label{Drancy}
$\etendu{\sigma}\liederivative{\varrho}\etendu{\tau}=d_A^{\Bott}$
\end{lemma}

\begin{proof}
Let $(l_k)_{k\in\{1,\dots,\rk{L}\}}$ denote any local frame of $L$ and let $(\lambda_k)_{k\in\{1,\dots,\rk{L}\}}$ denote the dual local frame of $L^{\vee}$.
Likewise let $(a_k)_{k\in\{1,\dots,\rk{A}\}}$ denote any local frame of $A$ and let $(\alpha_k)_{k\in\{1,\dots,\rk{A}\}}$ denote the dual local frame of $A^{\vee}$.
For all $\omega\in\sections{\Lambda^\bullet A^{\vee}}$, $n\in\NN$, and
$j_0,\dots,j_n\in\{1,\dots,r\}$, we have
\begin{align*}
&\ \etendu{\sigma}\Big(
\schouten{\varrho}
{\etendu{\tau}\big(\omega\otimes\partial_{j_0}\wedge
\cdots\partial_{j_n}\big)}\Big) \\
=&\ \etendu{\sigma}
\Big( \schouten{\varrho}
{p^\top\omega\otimes 1\otimes\partial_{j_0}\wedge
\cdots\wedge\partial_{j_n})} \Big) \\
=&\ \etendu{\sigma} \Big(
d_L(p^\top\omega)\otimes 1\otimes\partial_{j_0}\wedge
\cdots\wedge\partial_{j_n}) \\
&\qquad +\sum_k\lambda_k\wedge p^\top\omega\otimes
\schouten{\nabla_{l_k}-i_{l_k}\Xi}{1\otimes\partial_{j_0}\wedge\cdots\wedge\partial_{j_n}}
\Big) \\
=&\ \sigma\big(d_L(p^\top\omega)\otimes 1\big)
\otimes \partial_{j_0}\wedge\cdots
\wedge\partial_{j_n} \\
&\qquad +\sum_k \etendu{\sigma}
\Big( p^\top\alpha_k\wedge p^\top\omega
\otimes \schouten{\nabla^{\lightning}_{a_k}}
{1\otimes\partial_{j_0}\wedge\cdots\wedge
\partial_{j_n}} \Big) \\
=&\ d_A\omega\otimes \partial_{j_0}\wedge\cdots
\wedge\partial_{j_n} \\
&\qquad +\sum_k \etendu{\sigma}
\bigg(p^\top(\alpha_k\wedge\omega)\otimes
\bigg\{ \sum_{t=0}^n
1\otimes\partial_{j_0}\wedge\cdots\wedge
\schouten{\nabla^{\lightning}_{a_k}}{\partial_{j_t}}\wedge\cdots\wedge\partial_{j_n}
\bigg\} \bigg) \\
=&\ d_A\omega\otimes\partial_{j_0}\wedge\cdots
\wedge\partial_{j_n} \\
&\qquad +\sum_k\sum_{t=0}^n
\alpha_k\wedge\omega\otimes
\partial_{j_0}\wedge\cdots\wedge
\pr_0\schouten{\nabla^{\lightning}_{a_k}}
{\partial_{j_t}}\wedge\cdots\wedge\partial_{j_n}
.\end{align*}
It follows from Lemma~\ref{Cergy} that
\[ \pr_0\schouten{\nabla^{\lightning}_{a_k}}
{\partial_{j_t}} = \nabla^{\lightning}_{a_k}(\partial_{j_t})
= \nabla^{\Bott}_{a_k}(\partial_{j_t}) .\]
Hence, we conclude that $\etendu{\sigma}\circ\liederivative{\varrho}
\circ\etendu{\tau}=d_A^{\Bott}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{Bagnolet}]
We proceed by homological perturbation (see Lemma~\ref{Riga}). Starting from the filtered contraction of Proposition~\ref{Shiraz}, it suffices to perturb
the coboundary operator $\liederivative{-\delta}$ by the operator $\liederivative{\varrho}$.
One checks that $\etendu{\sigma}\liederivative{\varrho}\etendu{h}=0$.
It follows that
\[ \etendu{\perturbed{\sigma}}:=\sum_{k=0}^\infty
\etendu{\sigma}(\liederivative{\varrho}\etendu{h})^k
=\etendu{\sigma} \]
and, making use of Lemma~\ref{Drancy},
\[ \vartheta:=\sum_{k=0}^\infty
\etendu{\sigma}(\liederivative{\varrho} \etendu{h})^k\liederivative{\varrho}\etendu{\tau}
=\etendu{\sigma}\liederivative{\varrho}\etendu{\tau}=d_A^{\Bott} .\]
The result follows immediately since
$-\delta+\varrho=d_L^{\nabla^\lightning}=Q$.
\end{proof}

\begin{corollary}\label{Bari}
The chain maps
\[ \Big(\tot\big(\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalXpoly{\bullet}(B)\big), \liederivative{Q}\Big)
\overset{\etendu{\sigma}}{\underset{\etendu{\perturbed{\tau}}}{\rightleftarrows}}
\Big(\tot\big(\sections{\Lambda^\bullet A^{\vee}}\otimes_R \Xpoly{\bullet}\big)
,d_A^{\Bott}\Big) \]
are quasi-isomorphisms.
\end{corollary}

Combining Corollary~\ref{Bari} with Proposition~\ref{pro:Metz}, we are lead to the following
\begin{theorem}
Let $(L, A)$ be a Lie pair.
Choosing a splitting $j:L/A\to L$ of the short exact sequence
 $0\to A \to L \to L/A \to 0$
and a torsion-free $L$-connection $\nabla$ on $L/A$ extending the Bott
 $A$-connection determines a Gerstenhaber algebra structure
on $H^\bullet_{\CE}(A,\Xpoly{\bullet})$.
\end{theorem}

It is natural to ask whether the induced Gerstenhaber algebra structure on
$H^\bullet_{\CE}(A, \Xpoly{\bullet})$ is independent of these choices.
This question will be investigated somewhere else.

\subsection{Dolgushev--Fedosov quasi-isomorphisms for $\Tpoly{\bullet,\bullet}$}

Set $\Tpoly{r,s}:=\sections{(B^{\vee})^{\otimes r}\otimes B^{\otimes s}}$ and let $\verticalTpoly{r,s}(B)$ denote the space of formal vertical tensors of type $(r,s)$ on $B$, i.e.\
\[ \verticalTpoly{r,s}(B) = \sections{\hat{S}(B^{\vee})}
\otimes_R \Tpoly{r,s} .\]

Hence $\sections{\wedge^\bullet L^{\vee}}\otimes_R \verticalTpoly{r,s}(B)
\cong \sections{\wedge^\bullet L^{\vee}\otimes \hat{S}B^{\vee}}\otimes_R
\Tpoly{r,s}$.
 It is simple to see that
\[ \sections{\mathcal{M}; (\mathcal{A}^{\vee})^{\otimes r}\otimes \mathcal{A}^{\otimes s}}
\cong \sections{\wedge^\bullet L^{\vee}\otimes \hat{S}B^{\vee}}\otimes_R
\Tpoly{r,s} \]

Since $Q:=d_L^{\nabla^\lightning}$ is a homological vector field on the graded manifold $L[1]\oplus B$, the Lie derivative $\liederivative{Q}$
is a coboundary operator on the space
$T^{r,s}(L[1]\oplus B)$ of tensors of type $(r,s)$.
Since $\mathcal{A}$ is a dg Lie subalgebroid of $T_{L[1]\oplus B}$,
 thus we have the following

\begin{lemma}
The subspace
$\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalTpoly{r,s}(B)$ of $T^{r,s}(L[1]\oplus B)$
is stable under $\liederivative{Q}$.
Moreover, the diagram
\[ \begin{tikzcd}
\sections{\Lambda^k L^{\vee}}\otimes_R
\verticalTpoly{r,s}(B) \arrow{r}{\cong}
\arrow{d}[swap]{\liederivative{Q}} &
\sections{\Lambda^k L^{\vee}\otimes\hat{S}(B^{\vee})
\otimes(B^{\vee})^{\otimes r}\otimes B^{\otimes s}}
\arrow{d}{d_L^{\nabla^\lightning}} \\
\sections{\Lambda^{k+1} L^{\vee}}\otimes_R
\verticalTpoly{r,s}(B) \arrow{r}[swap]{\cong} &
\sections{\Lambda^{k+1} L^{\vee}\otimes\hat{S}(B^{\vee})
\otimes(B^{\vee})^{\otimes r}\otimes B^{\otimes s}}
\end{tikzcd} \]
commutes.
\end{lemma}

By $\etendu{\sigma}$ and $\etendu{\perturbed{\tau}}$, we denote the maps
\[ \sections{\wedge^\bullet L^{\vee}}\otimes_R\verticalTpoly{r,s}(B)=
\sections{\wedge^\bullet L^{\vee}\otimes \hat{S}B^{\vee}}\otimes_R
\Tpoly{r,s}\xto{\sigma\otimes\id} \sections{\Lambda^\bullet A^{\vee}}\otimes_R\Tpoly{r,s} \]
and
\[ \sections{\Lambda^\bullet A^{\vee}}\otimes_R\Tpoly{r,s}
\xto{\perturbed{\tau}\otimes\id}
\sections{\wedge^\bullet L^{\vee}\otimes \hat{S}B^{\vee}}\otimes_R \Tpoly{r,s}
=\sections{\wedge^\bullet L^{\vee}}\otimes_R\verticalTpoly{r,s}(B) \]
respectively.

The following proposition can be proved using a similar argument as in Theorem~\ref{Bagnolet} and Corollary~\ref{Bari}.

\begin{proposition}
The chain maps
\[ \Big(\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalTpoly{r,s}(B), \liederivative{Q}\Big)
\overset{\etendu{\sigma}}{\underset{\etendu{\perturbed{\tau}}}{\rightleftarrows}}
\Big(\sections{\Lambda^\bullet A^{\vee}}\otimes_R
\Tpoly{r,s},d_A^{\nabla^{\Bott}}\Big) \]
are quasi-isomorphisms.
\end{proposition}

\subsection{Dolgushev--Fedosov quasi-isomorphisms for $\Dpoly{\bullet}$}

Let $C^n:=\Hom_{R}(\mathscr{C}^{\otimes n+1},\mathscr{C})$
denote the space of Hochschild $(n+1)$-cochains
of the algebra $\mathscr{C}:=C^\infty(L[1]\oplus B)$.
The Gerstenhaber bracket of two
cochains $\phi\in C^u$ and $\psi\in C^v$
is the cochain
\begin{equation*}
\gerstenhaber{\phi}{\psi} = \phi\star\psi - (-1)^{uv} \psi\star\phi \in C^{u+v}
\end{equation*}
where $\phi\star\psi\in C^{u+v}$ is defined by
\[ \phi\star\psi (a_0\otimes a_1\otimes \cdots\otimes a_{u+v}) = \sum_{k=0}^{u} (-1)^{kv}
\phi\big(a_0\otimes\cdots\otimes a_{k-1}\otimes
\psi(a_k\otimes\cdots\otimes a_{k+v})\otimes a_{k+1+v}
\otimes \cdots\otimes a_{u+v} \big) .\]
The Gerstenhaber bracket satisfies the graded Jacobi identity.
Since the multiplication $m$ in $C^\infty(L[1]\oplus B)$
is associative, $\gerstenhaber{m}{m}=0$ and the standard Hochschild coboundary operator $\gerstenhaber{m}{-}$ turns $C^\bullet$ into a cochain complex.

The space $D_{\poly}^{\bullet}(L[1]\oplus B)$ of polydifferential operators on $L[1]\oplus B$ is a subspace of
$C^\bullet$ containing $m$ and closed under the Gerstenhaber bracket.
The Hochschild cohomology of the Fedosov dg manifold $(L[1]\oplus B,Q)$ is the cohomology of the cochain complex
$\big(D_{\poly}^{\bullet}(L[1]\oplus B),\gerstenhaber{Q\pm m}{-}\big)$.
The algebra of functions $C^\infty(L[1]\oplus B)$ is a module over its
subalgebra $\sections{\Lambda^\bullet L}\equiv\sections{\Lambda^\bullet L^{\vee}\otimes S^0(B^{\vee})}$.
The subspace of
$D_{\poly}^{\bullet}(L[1]\oplus B)$ comprised of all
$\sections{\Lambda^\bullet L^{\vee}}$-multilinear polydifferential operators is easily identified to
$\sections{\Lambda^\bullet L^{\vee}}\otimes_R\verticalDpoly{\bullet}$.

Since $\mathcal{A}\to\mathcal{M}$ is a dg Lie subalgebroid of the tangent bundle $T_{\mathcal{M}}\to\mathcal{M}$
of the Fedosov dg manifold $\mathcal{M}=(L[1]\oplus B,Q)$, it follows that
the subspace $\sections{\Lambda^\bullet L^{\vee}}\otimes_R\verticalDpoly{\bullet}$ of
$D_{\poly}^{\bullet}(L[1]\oplus B)$ is stable under the Hochschild coboundary
 operator ${\gerstenhaber{Q\pm m}{-}}$ of the Fedosov dg manifold
$(L[1]\oplus B, Q)$.

We also have the following
\begin{lemma}
The subspace $\sections{\Lambda^\bullet L^{\vee}}\otimes_R\verticalDpoly{\bullet}$
of $D_{\poly}^{\bullet}(L[1]\oplus B)$ is stable under $\gerstenhaber{-\delta}{-}$.
\end{lemma}

\begin{lemma}
The diagram
\[ \begin{tikzcd}
\sections{\Lambda^p L^{\vee}}\otimes_R\verticalDpoly{v}
\arrow{r}{\gerstenhaber{\delta}{-}} &
\sections{\Lambda^{p+1} L^{\vee}}\otimes_R\verticalDpoly{v} \\
\sections{\Lambda^p L^{\vee}}\otimes_R\verticalDpoly{v-1}
\arrow{u}{\gerstenhaber{m}{-}}
\arrow{r}{\gerstenhaber{\delta}{-}} &
\sections{\Lambda^{p+1} L^{\vee}}\otimes_R\verticalDpoly{v-1}
\arrow{u}{\gerstenhaber{m}{-}}
\end{tikzcd} \]
commutes.
\end{lemma}

\begin{proof}
It suffices to verify that the diagrams
\[ \begin{tikzcd}[column sep=huge]
\sections{\Lambda^p L^{\vee}}\otimes_R\verticalDpoly{v}
\arrow{r}{\gerstenhaber{\delta}{-}} &
\sections{\Lambda^{p+1} L^{\vee}}\otimes_R\verticalDpoly{v} \\
\sections{\Lambda^p L^{\vee}\otimes\hat{S}(B^{\vee})}\otimes_R\sections{(SB)^{\otimes v+1}}
\arrow{r}{\delta\otimes\id}
\arrow{u}{\id\otimes\varphi}[swap]{\equiv} &
\sections{\Lambda^{p+1} L^{\vee}\otimes\hat{S}(B^{\vee})}\otimes_R\sections{(SB)^{\otimes v+1}}
\arrow{u}{\id\otimes\varphi}[swap]{\equiv}
\end{tikzcd} \]
and
\[ \begin{tikzcd}[column sep=huge]
\sections{\Lambda^p L^{\vee}}\otimes_R\verticalDpoly{v-1}
\arrow{r}{\gerstenhaber{m}{-}} &
\sections{\Lambda^{p} L^{\vee}}\otimes_R\verticalDpoly{v} \\
\sections{\Lambda^p L^{\vee}\otimes\hat{S}(B^{\vee})}\otimes_R\sections{(SB)^{\otimes v}}
\arrow{r}{\id\otimes(-1)^{v-1}d_{\mathscr{H}}}
\arrow{u}{\id\otimes\varphi}[swap]{\equiv} &
\sections{\Lambda^p L^{\vee}\otimes\hat{S}(B^{\vee})}\otimes_R\sections{(SB)^{\otimes v+1}}
\arrow{u}{\id\otimes\varphi}[swap]{\equiv}
\end{tikzcd} \]
commute.
\end{proof}

It follows that

\begin{proposition}\label{Liege}
The diagram
\[ \begin{tikzcd}[column sep=large,row sep=small]
\vdots & \vdots & \vdots & \\
\sections{\Lambda^0 L^{\vee}}\otimes_R\verticalDpoly{1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} &
\sections{\Lambda^1 L^{\vee}}\otimes_R\verticalDpoly{1} \arrow{u}{-\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} &
\sections{\Lambda^2 L^{\vee}}\otimes_R\verticalDpoly{1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} & \cdots \\
\sections{\Lambda^0 L^{\vee}}\otimes_R\verticalDpoly{0} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} &
\sections{\Lambda^1 L^{\vee}}\otimes_R\verticalDpoly{0} \arrow{u}{-\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} &
\sections{\Lambda^2 L^{\vee}}\otimes_R\verticalDpoly{0} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} & \cdots \\
\sections{\Lambda^0 L^{\vee}}\otimes_R\verticalDpoly{-1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} &
\sections{\Lambda^1 L^{\vee}}\otimes_R\verticalDpoly{-1} \arrow{u}{-\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} &
\sections{\Lambda^2 L^{\vee}}\otimes_R\verticalDpoly{-1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta}{-}} & \cdots
\end{tikzcd} \]
is a double complex.
\end{proposition}

Its total complex
\[ \cdots\to\tot^{n}\Big(\sections{\Lambda^{\bullet} L^{\vee}}\otimes_R\verticalDpoly{\bullet}\Big) \xto{\gerstenhaber{-\delta\pm m}{-}} \tot^{n+1}\Big(\sections{\Lambda^{\bullet} L^{\vee}}\otimes_R\verticalDpoly{\bullet}\Big)\to\cdots \]
admits the descending filtration
$\mathscr{F}^{0}\supset \mathscr{F}^{1}\supset \mathscr{F}^{2}\supset\mathscr{F}^{3}\supset\cdots$
defined by
\[ \mathscr{F}^m=\bigoplus_{k=0}^{\rk(L)}
\sections{\Lambda^k(L^{\vee})}\otimes_R\varphi
\Big(\bigoplus_{q=-1}^{\infty}
\Gamma\big(\hat{S}^{\geqslant m-k}(B^{\vee})\otimes
\underset{q+1\text{ factors}}{\underbrace{S(B)\otimes\cdots\otimes S(B)}}\big) \Big) .\]

We note that, since $\pbw:\sections{S(B)}\to\Dpoly{0}$
is an isomorphism of $R$-coalgebras,
\begin{equation*} \begin{tikzcd}
0 \arrow{r} & \sections{S^0(B)}
\arrow{d}{\id} \arrow{r}{d_{\mathscr{H}}}
& \sections{S(B)} \arrow{r}{d_{\mathscr{H}}}
\arrow{d}{\pbw} &
\sections{S(B)}^{\otimes 2} \arrow{r}{d_{\mathscr{H}}}
\arrow{d}{\pbw^{\otimes 2}}
& \sections{S(B)}^{\otimes 3} \arrow{r}{d_{\mathscr{H}}}
\arrow{d}{\pbw^{\otimes 3}}
& \cdots
\\
0 \arrow{r} & \Dpoly{-1} \arrow{r}{d_{\mathscr{H}}} & \Dpoly{0} \arrow{r}{d_{\mathscr{H}}} &
\Dpoly{1} \arrow{r}{d_{\mathscr{H}}} & \Dpoly{2} \arrow{r}{d_{\mathscr{H}}} & \cdots
\end{tikzcd} \end{equation*}
is an isomorphism of cochain complexes of $R$-modules.

Adapting once again the proof of Proposition~\ref{Geneva}, we obtain
\begin{proposition}\label{Damas}
The cochain complex
$\big(\tot\big( \sections{\Lambda^{\bullet}L^{\vee}}
\otimes_R\verticalDpoly{\bullet}\big),
\gerstenhaber{-\delta\pm m}{-}\big)$
contracts onto
$\big(\tot\big(\sections{\Lambda^{\bullet}A^{\vee}}
\otimes_R\Dpoly{\bullet}\big),\id\otimes\pmd_{\mathscr{H}}\big)$.
More precisely, we have the filtered contraction
\begin{dontbotheriftheequationisoverlong} \begin{tikzcd}[column sep=huge]
\cdots \arrow{r} &
\tot^{n}\Big(\sections{\Lambda^{\bullet} L^{\vee} }\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{d}{\etendu{\sigma}}
\arrow{r}{\gerstenhaber{-\delta\pm m}{-}} &
\tot^{n+1}\Big(\sections{\Lambda^{\bullet} L^{\vee} }\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{d}{\etendu{\sigma}} \arrow{r}
\arrow[dashed]{ddl}[near end]{\etendu{h}} & \cdots \\
\cdots \arrow{r} &
\tot^{n}\Big(\sections{\Lambda^{\bullet} A^{\vee} }\otimes_R\Dpoly{\bullet}\Big)
\arrow{d}[swap]{\etendu{\tau}} \arrow{r}[near start]{\id\otimes\pmd_{\mathscr{H}}} &
\tot^{n+1}\Big(\sections{\Lambda^{\bullet} A^{\vee} }\otimes_R\Dpoly{\bullet}\Big)
\arrow{d}[swap]{\etendu{\tau}} \arrow{r} & \cdots \\
\cdots \arrow{r} &
\tot^{n}\Big(\sections{\Lambda^{\bullet} L^{\vee} }\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{r}{\gerstenhaber{-\delta\pm m}{-}} &
\tot^{n+1}\Big(\sections{\Lambda^{\bullet} L^{\vee} }\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{r} & \cdots
\end{tikzcd} \end{dontbotheriftheequationisoverlong}
where
$\etendu{\sigma}$, $\etendu{\tau}$ and $\etendu{h}$ are the three maps making the following three diagrams commute:
\[ \begin{tikzcd}[row sep=tiny]
\sections{\Lambda^{u}L^{\vee}}\otimes_R\verticalDpoly{v}
\arrow{dr}{\etendu{\sigma}} & \\
& \sections{\Lambda^{u}A^{\vee}}\otimes_R\Dpoly{v} \\
\sections{\Lambda^{u}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R\sections{(SB)^{\otimes v+1}}
\arrow{uu}{\id\otimes\varphi}[swap]{\equiv}
\arrow{ur}[swap]{\sigma\otimes\pbw^{\otimes v+1}} &
\end{tikzcd} \]

\[ \begin{tikzcd}[row sep=tiny]
& \sections{\Lambda^{u}L^{\vee}}\otimes_R\verticalDpoly{v} \\
\sections{\Lambda^{u}A^{\vee}}\otimes_R\Dpoly{v}
\arrow{ur}{\etendu{\tau}}
\arrow{dr}[swap]{\tau\otimes(\pbw^{-1})^{\otimes v+1}} & \\
& \sections{\Lambda^{u}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R\sections{(SB)^{\otimes v+1}}
\arrow{uu}{\id\otimes\varphi}[swap]{\equiv}
\end{tikzcd} \]

\[ \begin{tikzcd}
\sections{\Lambda^{u}L^{\vee}}\otimes_R\verticalDpoly{v}
\arrow{r}{\etendu{h}}
&
\sections{\Lambda^{u-1}L^{\vee}}\otimes_R\verticalDpoly{v}
\\
\sections{\Lambda^{u}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R \sections{(SB)^{\otimes v+1}}
\arrow{r}[swap]{h\otimes\id}
\arrow{u}{\id\otimes\varphi}[swap]{\equiv}
&
\sections{\Lambda^{u-1}L^{\vee}\otimes\hat{S}(B^{\vee})}
\otimes_R \sections{(SB)^{\otimes v+1}}
\arrow{u}{\id\otimes\varphi}[swap]{\equiv}
\end{tikzcd} \]
\end{proposition}

\begin{lemma}\label{Namur}
The diagram
\[ \begin{tikzcd}
\sections{\Lambda^p L^{\vee}}\otimes_R\verticalDpoly{v+1}
\arrow{r}{\gerstenhaber{\varrho}{-}} &
\sections{\Lambda^{p+1} L^{\vee}}\otimes_R\verticalDpoly{v+1} \\
\sections{\Lambda^p L^{\vee}}\otimes_R\verticalDpoly{v}
\arrow{u}{\gerstenhaber{m}{-}}
\arrow{r}{\gerstenhaber{\varrho}{-}} &
\sections{\Lambda^{p+1} L^{\vee}}\otimes_R\verticalDpoly{v}
\arrow{u}{\gerstenhaber{m}{-}}
\end{tikzcd} \]
commutes.
\end{lemma}

\begin{proof}[Sketch of proof]
We have $\gerstenhaber{\varrho}{m}=0$ because,
for every $l\in\sections{L}$, the operator $i_l\varrho$
is a derivation for the multiplication $m$ on $C^\infty(L[1]\oplus B)$.
\end{proof}

It follows from Proposition~\ref{Liege} and Lemma~\ref{Namur} that
\[ \begin{tikzcd}[column sep=large,row sep=small]
\vdots & \vdots & \vdots & \\
\sections{\Lambda^0 L^{\vee}}\otimes_R\verticalDpoly{1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} &
\sections{\Lambda^1 L^{\vee}}\otimes_R\verticalDpoly{1} \arrow{u}{-\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} &
\sections{\Lambda^2 L^{\vee}}\otimes_R\verticalDpoly{1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} & \cdots \\
\sections{\Lambda^0 L^{\vee}}\otimes_R\verticalDpoly{0} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} &
\sections{\Lambda^1 L^{\vee}}\otimes_R\verticalDpoly{0} \arrow{u}{-\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} &
\sections{\Lambda^2 L^{\vee}}\otimes_R\verticalDpoly{0} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} & \cdots \\
\sections{\Lambda^0 L^{\vee}}\otimes_R\verticalDpoly{-1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} &
\sections{\Lambda^1 L^{\vee}}\otimes_R\verticalDpoly{-1} \arrow{u}{-\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} &
\sections{\Lambda^2 L^{\vee}}\otimes_R\verticalDpoly{-1} \arrow{u}{\gerstenhaber{m}{-}} \arrow{r}{\gerstenhaber{-\delta+\varrho}{-}} & \cdots
\end{tikzcd} \]
is a double complex.

Indeed, the operator $\gerstenhaber{\varrho}{-}$
is a perturbation of the filtered complex
\[ \cdots\to\tot^{n}\Big(\sections{\Lambda^{\bullet} L^{\vee}}\otimes_R\verticalDpoly{\bullet}\Big) \xto{\gerstenhaber{-\delta\pm m}{-}} \tot^{n+1}\Big(\sections{\Lambda^{\bullet} L^{\vee}}\otimes_R\verticalDpoly{\bullet}\Big)\to\cdots \]

\begin{proposition}\label{Dhaka}
The cochain complex
$\big(\tot\big( \sections{\Lambda^{\bullet}L^{\vee}}
\otimes_R\verticalDpoly{\bullet}\big),
\gerstenhaber{Q\pm m}{-}\big)$
contracts onto
$\big(\tot\big(\sections{\Lambda^{\bullet}A^{\vee}}
\otimes_R\Dpoly{\bullet}\big), d_A^{\mathcal{U}}\otimes\pmd_{\mathscr{H}}\big)$.
More precisely, we have the (filtered) contraction
\begin{dontbotheriftheequationisoverlong} \begin{tikzcd}[column sep=large]
\cdots \arrow{r} &
\tot^{n}\Big(\sections{\Lambda^{\bullet}(L^{\vee})}\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{d}{\etendu{\sigma}}
\arrow{r}{\gerstenhaber{Q\pm m}{-}} &
\tot^{n+1}\Big(\sections{\Lambda^{\bullet}(L^{\vee})}\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{d}{\etendu{\sigma}} \arrow{r}
\arrow[dashed]{ddl}[near end]{\etendu{\perturbed{h}}} & \cdots \\
\cdots \arrow{r} &
\tot^{n}\Big(\sections{\Lambda^{\bullet}(A^{\vee})}\otimes_R\Dpoly{\bullet}\Big)
\arrow{d}[swap]{\etendu{\perturbed{\tau}}} \arrow{r}[near start]{d_A^{\mathcal{U}}\otimes\pmd_{\mathscr{H}}} &
\tot^{n+1}\Big(\sections{\Lambda^{\bullet}(A^{\vee})}\otimes_R\Dpoly{\bullet}\Big)
\arrow{d}[swap]{\etendu{\perturbed{\tau}}} \arrow{r} & \cdots \\
\cdots \arrow{r} &
\tot^{n}\Big(\sections{\Lambda^{\bullet}(L^{\vee})}\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{r}{\gerstenhaber{Q\pm m}{-}} &
\tot^{n+1}\Big(\sections{\Lambda^{\bullet}(L^{\vee})}\otimes_R\verticalDpoly{\bullet}\Big)
\arrow{r} & \cdots
\end{tikzcd} \end{dontbotheriftheequationisoverlong}
where $\etendu{\perturbed{\tau}}=\sum_{k=0}^\infty
(\etendu{h}\circ\gerstenhaber{\varrho}{-})^k\circ\etendu{\tau}$
and $\etendu{\perturbed{h}}=\sum_{k=0}^\infty
(\etendu{h}\circ\gerstenhaber{\varrho}{-})^k\circ\etendu{h}$.
\end{proposition}

The proof requires the following technical results.

\begin{lemma}\label{Vincennes}
Let $\pr_0$ denote the canonical projection
$\hat{S}(B^{\vee})\otimes S(B)\twoheadrightarrow
S^0(B^{\vee})\otimes S(B)$.
For all $a\in\sections{A}$ and $J\in\NO^r$, we have
\[ \pr_0\big(\gerstenhaber{\nabla^{\lightning}_a}{\partial^J}\big)
=\nabla^{\Bott}_a(\partial^J) .\]
\end{lemma}

\begin{proof}
We have seen that, for all $a\in\sections{A}$,
the operator $\nabla^{\lightning}_a$ is a derivation of
$\sections{\hat{S}(B^{\vee})}$, which stabilizes
the filtration $\sections{\hat{S}^{\geqslant n}(B^{\vee})}$.
Therefore, there exist local sections
$\theta_k^M$ of $L^{\vee}$ such that
\[ \nabla^{\lightning}_a \chi_k=
\sum_{\substack{M\in\NO^r \\ \abs{M}\geqslant 1}}
i_a\theta_k^M\cdot \chi^M .\]
It follows that $\nabla^{\lightning}_a$ may be regarded
as a section of $\hat{S}^{\geqslant 1}(B^{\vee})\otimes B$:
\[ \nabla^{\lightning}_a =
\sum_{k=1}^r \bigg(\sum_{\substack{M\in\NO^r \\ \abs{M}\geqslant 1}}
i_a\theta_k^M\cdot \chi^M\bigg) \partial_k .\]

On one hand, it follows from
\[ \gerstenhaber{\nabla^{\lightning}_a}{\partial^J}
= \nabla^{\lightning}_a\star\partial^J - \partial^J\star
\nabla^{\lightning}_a =
\sum_{k=1}^r \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M\cdot\chi^M\partial^{J+e_k}
-\sum_{k=1}^r \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M\cdot\big(\partial^J\star\chi^M\partial_k\big) ,\]
that
\[ \begin{split}
\pr_0\big(\gerstenhaber{\nabla^{\lightning}_a}{\partial^J}\big)
=&\ -\sum_{k=1}^r \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M \frac{J!}{M!(J-M)!}\partial^M(\chi^M)
\cdot\partial^{J-M+e_k} \\
=&\ -\sum_{k=1}^r \sum_{\abs{M}\geqslant 1}
i_a\theta_k^M \frac{J!}{(J-M)!}\cdot\partial^{J-M+e_k}
.\end{split} \]

On the other hand, it follows from
\[ 0 = \rho(a)\underset{K!\cdot\delta_{K,J}}{\underbrace{\duality{\chi^K}{\partial^J}}} =
\duality{\nabla^{\lightning}_a\chi^K}{\partial^J}+
\duality{\chi^K}{\nabla^{\lightning}_a\partial^J} \]
that
\[ \begin{split}
\nabla^{\lightning}_a\big(\partial^J\big)
=&\ \sum_K \frac{1}{K!} \duality{\chi^K}{
\nabla^{\lightning}_a \partial^J}\partial^K \\
=&\ -\sum_K \frac{1}{K!}
\duality{\nabla^{\lightning}_a \chi^K}{\partial^J} \partial^K \\
=&\ -\sum_K \frac{1}{K!}
\duality{\sum_k K_k \chi^{K-e_k} \nabla^{\lightning}_a
\chi_k}{\partial^J} \partial^K \\
=&\ -\sum_K \frac{1}{K!} \sum_k K_k
\sum_{\abs{M}\geqslant 1}
i_a\theta_k^M \underset{J!\cdot \delta_{K-e_k+M,J}}{\underbrace{\duality{\chi^{K-e_k+M}}{\partial^J}}}
\partial^K \\
=&\ -\sum_k \sum_{\abs{M}\geqslant 1}
\frac{J!}{(J-M+e_k)!} (J_k-M_k+1)
 i_a\theta_k^M \partial^{J-M+e_k}
\\ =&\ -\sum_k \sum_{\abs{M}\geqslant 1} \frac{J!}{(J-M)!}
i_a\theta_k^M \partial^{J-M+e_k}
.\end{split} \]
The proof is complete.
\end{proof}

\begin{lemma}\label{Pantin}
$\etendu{\sigma}\circ\gerstenhaber{\varrho}{-}\circ\etendu{\tau}=d_A^{\mathcal{U}}$
\end{lemma}

\begin{proof}
Let $(l_k)_{k\in\{1,\dots,\rk{L}\}}$ denote any local frame of $L$ and let $(\lambda_k)_{k\in\{1,\dots,\rk{L}\}}$ denote the dual local frame of $L^{\vee}$.
Likewise let $(a_k)_{k\in\{1,\dots,\rk{A}\}}$ denote any local frame of $A$ and let $(\alpha_k)_{k\in\{1,\dots,\rk{A}\}}$ denote the dual local frame of $A^{\vee}$.
For all $\omega\in\sections{\Lambda^\bullet A^{\vee}}$, $n\in\NN$, and
$J_0,\dots,J_n\in\NO^{r}$, we have
\begin{align*}
&\ \etendu{\sigma}\Big(
\gerstenhaber{\varrho}
{\etendu{\tau}\big(\omega\otimes
\pbw(\partial^{J_0})\otimes\cdots\otimes\pbw(\partial^{J_n})
\big)}\Big) \\
=&\ \etendu{\sigma}\Big( \gerstenhaber{\varrho}
{p^\top\omega\otimes\varphi(
1\otimes\partial^{J_0}\otimes\cdots\otimes\partial^{J_n}
)} \Big) \\
=&\ \etendu{\sigma} \Big(
d_L(p^\top\omega)\otimes\varphi(
1\otimes\partial^{J_0}\otimes\cdots\otimes\partial^{J_n}
) \\
&\qquad +\sum_k\lambda_k\wedge p^\top\omega\otimes
\gerstenhaber{\nabla_{l_k}-i_{l_k}\Xi}{\varphi(
1\otimes\partial^{J_0}\otimes\cdots\otimes\partial^{J_n})}
\Big) \\
=&\ \sigma\big(d_L(p^\top\omega)\otimes 1\big)
\otimes \pbw(\partial^{J_0})\otimes\cdots
\otimes\pbw(\partial^{J_n}) \\
&\qquad +\sum_k \etendu{\sigma}
\Big( p^\top\alpha_k\wedge p^\top\omega
\otimes \gerstenhaber{\nabla^{\lightning}_{a_k}}
{\varphi(1\otimes\partial^{J_0}\otimes\cdots\otimes
\partial^{J_n})} \Big) \\
=&\ d_A\omega\otimes \pbw(\partial^{J_0})\otimes\cdots
\otimes\pbw(\partial^{J_n}) \\
&\qquad +\sum_k \etendu{\sigma}
\bigg(p^\top(\alpha_k\wedge\omega)\otimes\varphi
\bigg\{ \sum_{t=0}^n
1\otimes\partial^{J_0}\otimes\cdots\otimes
\gerstenhaber{\nabla^{\lightning}_{a_k}}{\partial^{J_t}}\otimes\cdots\otimes\partial^{J_n}
\bigg\} \bigg) \\
=&\ d_A\omega\otimes \pbw(\partial^{J_0})\otimes\cdots
\otimes\pbw(\partial^{J_n}) \\
&\qquad + \sum_k \sum_{t=0}^n
\alpha_k\wedge\omega\otimes
\pbw(\partial^{J_0})\otimes\cdots\otimes
\pbw\big(\pr_0\gerstenhaber{\nabla^{\lightning}_{a_k}}
{\partial^{J_t}}\big)\otimes\cdots\otimes\pbw(\partial^{J_n})
.\end{align*}
It follows from Lemma~\ref{Vincennes} that
\[ \pbw\big(\pr_0\gerstenhaber{\nabla^{\lightning}_{a_k}}
{\partial^{J_t}}\big) = \pbw\big(\nabla^{\lightning}_{a_k}(\partial^{J_t})\big) = a_k\cdot\pbw(\partial^{J_t}) .\]
Hence, we conclude that $\etendu{\sigma}\circ\gerstenhaber{\varrho}{-}\circ\etendu{\tau}=d_A^{\mathcal{U}}$.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{Dhaka}]
We proceed by homological perturbation (see Lemma~\ref{Riga}). Starting from the filtered contraction of Proposition~\ref{Damas}, it suffices to perturb
the coboundary operator $\gerstenhaber{-\delta\pm m}{-}$ by the operator $\gerstenhaber{\varrho}{-}$.
One checks that
$\etendu{\sigma}\circ\gerstenhaber{\varrho}{-}\circ\etendu{h}=0$.
Therefore, we obtain
\[ \etendu{\perturbed{\sigma}}:=\sum_{k=0}^\infty
\etendu{\sigma}\circ(\gerstenhaber{\varrho}{-}\circ\etendu{h})^k
=\etendu{\sigma} \]
and, making use of Lemma~\ref{Pantin},
\[ \vartheta:=\sum_{k=0}^\infty
\etendu{\sigma}\circ(\gerstenhaber{\varrho}{-}\circ\etendu{h})^k\circ
\gerstenhaber{\varrho}{-}\circ\etendu{\tau}=\etendu{\sigma}\circ
\gerstenhaber{\varrho}{-}\circ\etendu{\tau}=d_A^{\mathcal{U}} .\]
The result follows immediately since
$-\delta+\varrho=d_L^{\nabla^\lightning}=Q$.
\color{black}
\end{proof}

\begin{corollary}\label{Verona}
The chain maps
\[ \Big(\tot\big(\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalDpoly{\bullet}\big), \gerstenhaber{Q\pm m}{-}\Big)
\overset{\etendu{\sigma}}{\underset{\etendu{\perturbed{\tau}}}{\rightleftarrows}}
\Big(\tot\big(\sections{\Lambda^\bullet A^{\vee}}\otimes_R
\Dpoly{\bullet}\big),d_A^{\mathcal{U}}\pmd_{\mathscr{H}} \Big) \]
are quasi-isomorphisms.
\end{corollary}

As a consequence of Corollary~\ref{Verona} and Proposition~\ref{pro:Metz},
we see that $H^\bullet_{\CE}(A,\Dpoly{\bullet})$ carries
a Gerstenhaber algebra structure.
 In summary, we have

\begin{theorem}
\label{thm:Dpoly}
Let $(L, A)$ be a Lie pair.
Choosing a splitting $j:L/A\to L$ of the short exact sequence
 $0\to A \to L \to L/A \to 0$
and a torsion-free $L$-connection $\nabla$ on $L/A$ extending the Bott
 $A$-connection determines a Gerstenhaber algebra structure
on $H^\bullet_{\CE}(A,\Dpoly{\bullet})$.
\end{theorem}

As in the case of polyvector fields,
it is natural to ask whether the induced Gerstenhaber algebra structure on
$H^\bullet_{\CE}(A, \Dpoly{\bullet})$ is independent of these choices.
This question will be investigated somewhere else.

\subsection{Matched pair case}

Let $A\bowtie B$ be a matched pair of Lie algebroids.
It is standard that $A\oplus B \to A$ is a
Lie algebroid \cite{MR2831518}, whose structure
maps can be described explicitly as follows.
The space of sections of the bundle $A\oplus B \to A$
can be identified with $C^\infty (A)\otimes_{C^\infty (M)}\sections{B}$.
The anchor map is essentially determined by
the map $\rho_v : \sections{B}\to {\mathfrak{X}} (A)$ given by
\[ \rho_v (b)(l_\xi)=\nabla^{\Bott}_b \xi, \qquad
\rho_v (b) (\pi^* f)= \pi^* (\rho(b)(f)) ,\]
for all $b\in \sections{B}$, $\xi\in\sections{A^{\vee}}$, and $f\in C^\infty(M)$,
where $\pi: A\to M$ is the projection.
The bracket between two sections in
$C^\infty (A)\otimes_{C^\infty (M)}\sections{B}$
is the extension of the Lie bracket of $\sections{B}$
by Leibniz rule.

Similarly, $A\oplus B$ is also a Lie algebroid over $B$. The following
classical result is due to Mackenzie \cite{MR2831518}.

\begin{lemma}
If $A\bowtie B$ is a matched pair of Lie algebroids, then
\[ \begin{tikzcd}[column sep=small,row sep=small]
A\oplus B \arrow{r} \arrow{d} & B \arrow{d} \\
A \arrow{r} & M
\end{tikzcd} \]
is a double Lie algebroid.
\end{lemma}

According to Gracia-Saz and Mehta~\cite{MR2581370}, any double Lie algebroid induces
a pair of dg Lie algebroids. As an immediate consequence, we have
the following

\begin{corollary}
If $A\bowtie B$ is a matched pair of Lie algebroids, then
$(A[1]\oplus B, d^{\Bott}_A )$ is a dg Lie algebroid over
$(A[1], d_A)$.
\end{corollary}

Here the dg manifold structures on $(A[1]\oplus B, d_A^{\Bott})$
and $(A[1], d_A)$ are induced, respectively, from
the Lie algebroid structures on $A\oplus B\to B$ and
$A\to M$ as in Example~\ref{example:Vauintrob dg vs Lie abd}.
In what follows, denote by $\mathcal{B}$ the dg manifold $A[1]\oplus B$.

Again the space of sections of $\mathcal{B}\to A[1]$
can be naturally identified with $\sections{\wedge^\bullet A^{\vee} \otimes B}$.
The bracket on $\sections{\wedge^\bullet A^{\vee} \otimes B}$ is given by
\begin{equation}
\label{eq:Brussels1}
[\xi_1\otimes b_1, \xi_2\otimes b_2]
=\xi_1\wedge \xi_2 \otimes [b_1, b_2]
+\xi_1\wedge \nabla^{\Bott}_{b_1} \xi_2 \otimes b_2
- \nabla^{\Bott}_{b_2} \xi_1\wedge \xi_2 \otimes b_1
\end{equation}
for all $\xi_1,\xi_2\in\sections{\wedge^\bullet A^{\vee}}$ and
$b_1,b_2\in\sections{B}$. And the anchor map is given by
\begin{equation}
\label{eq:Brussels2}
\sections{\wedge^\bullet A^{\vee} \otimes B}\xto{\bar\rho}\Der
(\wedge^\bullet A^{\vee}), \qquad \bar{\rho}(\xi \otimes b)(\eta)=
\xi \wedge \nabla^{\Bott}_b \eta
\end{equation}
for all $\xi,\eta\in\sections{\wedge^\bullet A^{\vee}}$ and
$b\in \sections{B}$.

Finally, the induced differential on the space of sections in
$\mathcal{B}\to A[1]$ is simply the the coboundary
differential corresponding to the Bott representation
$d_A^{\Bott}: \sections{\wedge^\bullet A^{\vee} \otimes B}
\to \sections{\wedge^{\bullet+1} A^{\vee} \otimes B}$.

According to Proposition~\ref{pro:hongkong}, the dg Lie algebroid
$\mathcal{B}\to A[1]$ induces a differential Gerstenhaber algebra
structure on $\sections{\wedge^\bullet \mathcal{B}}\cong
 \sections{\wedge^\bullet A^{\vee} \otimes \wedge^\bullet B}$,
where the differential is the coboundary
differential corresponding to the Bott representation
\begin{equation}
\label{eq:Vienna}
d_A^{\Bott}: \sections{\wedge^\bullet A^{\vee} \otimes \wedge^\bullet B}
\to \sections{\wedge^{\bullet+1} A^{\vee} \otimes \wedge^\bullet B},
\end{equation}
and the Gerstenhaber bracket is the Schouten bracket of the
dg Lie algebroid $\mathcal{B}\to A[1]$, which
is essentially the extension of Equations~\eqref{eq:Brussels1} and~\eqref{eq:Brussels2} by graded Leibniz rule.
Note that the cochain complex \eqref{eq:Vienna} is
 exactly the same one on the r.h.s.\ of Corollary~\ref{Bari}.
 Indeed we have the following

\begin{theorem} \label{thm:Bonn}
Let $(L,A)$ be a Lie pair and let $j:B=L/A\to L$ be a splitting
of the short exact sequence $0\to A \to L \to B \to 0$ such that $j(B)$ is a Lie subalgebroid of L.
If we identify $B$ with $j(B)$, then $L=A\bowtie B$ is a matched pair of
Lie algebroids.
Choose a torsion-free $L$-connection $\nabla$ on $B$ extending the Bott
 $A$-connection. Let $(L[1]\oplus B, Q)$ be the corresponding
 Fedosov dg-manifold.
Then the chain map $\etendu{\sigma}$ as in Theorem~\ref{Bari} (1):
\[ \begin{tikzcd}[column sep=large]
\Big(\tot\big(\sections{\Lambda^\bullet L^{\vee}}\otimes_R
\verticalXpoly{\bullet}(B)\big), \liederivative{Q}\Big)
\arrow{r}{{\etendu{\sigma}}}
& \Big(\tot\big(\sections{\Lambda^\bullet A^{\vee}\otimes \Lambda^\bullet B}
\big),d_A^{\Bott}\Big)
\end{tikzcd} \]
is a quasi-isomorphism of differential Gerstenhaber algebras.
Therefore the Gerstenhaber algebra $H^\bullet_{\CE}(A,\Xpoly{\bullet})$
is isomorphic to the Gerstenhaber algebra
$H^\bullet\Big( \tot\big(\sections{\Lambda^\bullet A^{\vee}\otimes\Lambda^\bullet B}, d_A^{\Bott}\big)$
induced from the dg Lie algebroid $\mathcal{B}\to A[1]$.
\end{theorem}

Next, consider the universal enveloping algebra
$\enveloping{\mathcal{B}}$ of the dg Lie algebroid $\mathcal{B}\to A[1]$.

\begin{proposition}
As dg Hopf algebroids over $(\sections{\wedge^\bullet A^{\vee}}, d_A)$,
the universal enveloping algebra $\enveloping{\mathcal{B}}$ is isomorphic to
$\sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}$,
whose structure maps are described below.
\begin{enumerate}
\item the product is generated by the following rule: both
$\sections{\wedge^\bullet A^{\vee}} \otimes 1$ and $1\otimes\enveloping{B}$
are subalgebras, and
\[ (1\otimes d)(\xi\otimes 1)=(\xi\otimes 1)(1\otimes d) \pm d\cdot \xi ,\]
$\forall d \in\enveloping{B}$, $\xi \in \sections{\wedge^\bullet A^{\vee}}$,
and $d\cdot \xi=\nabla^{\Bott}_{b_1}\cdots \nabla^{\Bott}_{b_k} \xi$
if $d=b_1\cdots b_k$;

\item both source and target maps $\alpha, \beta: \sections{\wedge^\bullet A^{\vee}}\to \sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}$ are inclusion;

\item the differential
$D: \sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}
\to\sections{\wedge^{\bullet+1} A^{\vee}}\otimes_R\enveloping{B}$
is the coboundary differential with $\enveloping{B}$ being considered
as a natural $A$-module by identifying
$\enveloping{B}\cong\frac{\enveloping{L}}{\enveloping{L}\sections{A}}$,
on which the Lie algebroid $A$ acts by mutiplication from the left.

\item the comultiplication $\Delta$ is defined by the following commutative diagram:
\[ \begin{tikzcd}[row sep=small]
& \big( \sections{\wedge^\bullet A^{\vee}}\otimes_R \enveloping{B}\big)
\otimes_{\sections{\wedge^\bullet A^{\vee}}}
\big( \sections{\wedge^\bullet A^{\vee}}\otimes_R \enveloping{B}\big)
\arrow{dd}{\cong} \\
\sections{\wedge^\bullet A^{\vee}}\otimes_R \enveloping{B}
\arrow{ru}[near start]{\Delta} \arrow{rd}[swap]{\id\otimes \Delta_{\enveloping{B}}} & \\
& \sections{\wedge^\bullet A^{\vee}}\otimes_R \enveloping{B}\otimes_R \enveloping{B}
, \end{tikzcd} \]
where $\Delta_{\enveloping{B}}: \enveloping{B}\to\enveloping{B}\otimes_R\enveloping{B}$
denotes the comultiplication of the Hopf algebroid $\enveloping{B}$.
\end{enumerate}
\end{proposition}

As a consequence, we have

\begin{corollary}
\label{cor:Tiajiang}
For a matched pair of Lie algebroids, the Hochschild cochain complex
$\big(\sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}^{\otimes \bullet}, D\pmd_{\mathscr{H}}\big)$
of the dg Lie algebroid $\mathcal{B}\to A[1]$ coincides with
$\big(\tot\big(\sections{\Lambda^\bullet A^{\vee}}\otimes_R
\Dpoly{\bullet}\big),d_A^{\mathcal{U}}\pmd_{\mathscr{H}}\big)$,
the r.h.s.\ one in Corollary~\ref{Verona} (1).
\end{corollary}

On the other hand, since $\mathcal{B}\to A[1]$ is a dg Lie algebroid,
according to Proposition~\ref{pro:hongkong},
$H^\bullet \big(\sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}^{\otimes \bullet},
 D\pmd_{\mathscr{H}} \big)$ is a Gerstenhaber algebra.
Hence, there exists an induced Gerstenhaber algebra structure on $H^\bullet_{\CE}(A,\Dpoly{\bullet})$.
In fact, we have the following
\begin{theorem}
\label{thm:Berlin}
Under the same hypothesis as in Theorem~\ref{thm:Bonn},
the Gerstenhaber algebra $H^\bullet_{\CE}(A,\Dpoly{\bullet})$
as in Theorem~\ref{thm:Dpoly} is isomorphic to the Gerstenhaber algebra
$H^\bullet \big(\sections{\wedge^\bullet A^{\vee}}\otimes_R\enveloping{B}^{\otimes \bullet},
 D\pmd_{\mathscr{H}}\big)$
induced from the dg Lie algebroid $\mathcal{B}\to A[1]$ as in Corollary~\ref{cor:Tiajiang}.
\end{theorem}

Proposition~\ref{thm:Bonn} and Proposition~\ref{thm:Berlin}
essentially follow from the following

\begin{proposition}
\label{pro:Geneva}
Under the same hypothesis as in Theorem~\ref{thm:Bonn}, the map
\begin{equation}
\big((\sections{\wedge^\bullet A^{\vee}\otimes B}, d_A^{\Bott}),
 (\sections{\wedge^\bullet A^{\vee}}, d_A )\big)
\xto{\perturbed{\tau}}
\big(
(\sections{\wedge^\bullet L^\vee \otimes \mathscr{X}_{\ver} (B)},\liederivative{Q}),
(\sections{\wedge^\bullet L^\vee \otimes \hat{S} B^{\vee}}, Q)
\big)
\end{equation}
is a quasi-isomorphism of dg Lie--Rinehart algebras.
\end{proposition}

In what follows, we identify $L$ with $A\oplus B$, and therefore
we have the decomposition
\[ \sections{\wedge^\bullet L^\vee }\cong
\bigoplus \sections{\wedge^\bullet A^{\vee}
\otimes \wedge^\bullet B^{\vee}} .\]

The following lemma can be easily verified.

\begin{lemma}
\label{lem:Suzhou}
We have
\[ \perturbed{\tau}: \sections{\wedge^\bullet A^{\vee}\otimes B}\xto{}
\sections{\wedge^\bullet A^\vee \otimes \mathscr{X}_{\ver} (B)} .\]
\end{lemma}

Let $X^\nabla\in\sections{L^{\vee}\otimes\hat{S}^{\geqslant 2} B^{\vee} \otimes B}$
be the 1-form valued in formal vertical vector fields of $B$ as in Theorem~\ref{strawberry}.
Write \[ X^\nabla= X^{1, 0}+X^{0, 1} ,\]
where $X^{0, 1}\in\sections{A^{\vee}\otimes\hat{S}^{\geqslant 2} B^{\vee}\otimes B}$
and $X^{1, 0}\in\sections{B^{\vee}\otimes\hat{S}^{\geqslant 2} B^{\vee} \otimes B}$.
Note that a torsion-free $L$-connection $\nabla$ on $B$ extending the Bott
$A$-connection corresponds to a torsion-free $B$-connection $\nabla^B$ on $B$.
Hence $d_L^\nabla: \sections{\Lambda^\bullet L^\vee\otimes\hat{S} B^\vee }
\to \sections{\Lambda^{\bullet+1} L^\vee\otimes\hat{S} B^\vee }$
is decomposed as $d_L^\nabla=d_A^{\Bott}+d_B^{\nabla^B}$, where
\[ d_A^{\Bott}: \sections{\Lambda^\bullet A^\vee \otimes \Lambda^\bullet B^\vee \otimes\hat{S} B^\vee}
\to \sections{\Lambda^{\bullet+1} A^\vee \otimes \Lambda^\bullet B^\vee
\otimes\hat{S} B^\vee } \]
is the coboundary differential of the Lie algebroid $A$ with respect to
the induced Bott representation on $\Lambda^\bullet B^\vee \otimes\hat{S} B^\vee$, while
\[ d_B^{\nabla^B}: \sections{\Lambda^\bullet A^\vee \otimes \Lambda^\bullet B^\vee \otimes\hat{S} B^\vee}
\to \sections{\Lambda^{\bullet} A^\vee \otimes \Lambda^{\bullet+1} B^\vee\otimes\hat{S} B^\vee} \]
is the covariant differential of the Lie algebroid $B$ with respect
to the $B$-connection $\nabla^{\Bott}\otimes 1+1\otimes \nabla^B$
on ${\Lambda^\bullet A^\vee \otimes\hat{S} B^\vee}$.

Set \[ D^{1, 0}=-\delta+d_B^{\nabla^B}+X^{1, 0} \qquad \text{and} \qquad D^{0, 1}=d_A^{\Bott}+X^{0,1} .\]

We have
\begin{lemma}
Let $D$ be the Fedosov differential as in Theorem~\ref{strawberry} corresponding to the $L$-connection
on $B$ induced by $ \nabla^B$. Then $D=D^{0, 1}+D^{1, 0}$ and
$(D^{0, 1})^2=(D^{1, 0})^2=D^{0, 1}D^{1, 0}+D^{1, 0}D^{0, 1}=0$.
\end{lemma}

Since $\liederivative{D}\circ\perturbed{\tau}=\perturbed{\tau}\circ d_A^{\Bott}$,
from Lemma~\ref{lem:Suzhou},
it follows that
\begin{gather*} \liederivative{D^{1, 0}} \circ\perturbed{\tau} = 0 \\
\liederivative{D^{0, 1}}\circ\perturbed{\tau} = \perturbed{\tau}\circd_A^{\Bott}
.\end{gather*}

We also need the following technical lemma, which can be proved by the Fedosov iteration method.
We leave it for the reader.

\begin{lemma} \label{lem:Dijon}
If $\alpha\in\sections{\wedge^\bullet A^\vee \otimes \mathscr{X}_{\ver} (B)}$
satisfies $\liederivative{D^{1, 0}}\alpha =0$ and $\etendu{\sigma} (\alpha )=0$, then $\alpha=0$.
\end{lemma}

We are now ready to prove Proposition~\ref{pro:Geneva}, which in fact is
a direct corollary of the following

\begin{lemma}
For any $b, c\in \sections{B}$, and $\xi\in \sections{\wedge^\bullet A^{\vee}}$,
we have the following identities
\begin{align}
[\perturbed{\tau}b, \perturbed{\tau}c] &= \perturbed{\tau}[b, c]; \label{eq:wuxi1}\\
\perturbed{\tau} (\xi\otimes b) &= \perturbed{\tau} (\xi)\cdot \perturbed{\tau} b; \label{eq:wuxi2}\\
[\perturbed{\tau}b, \perturbed{\tau}\xi] &= \perturbed{\tau}(\nabla_b^{\Bott}\xi); \label{eq:wuxi3}
\end{align}
\end{lemma}

\begin{proof}
Let $\alpha=\perturbed{\tau}[b, c]-[\perturbed{\tau}b, \perturbed{\tau}c]$.
From Lemma~\ref{lem:Suzhou}, it is clear
that $\alpha\in\sections{\wedge^\bullet A^\vee\otimes\mathscr{X}_{\ver}(B)}$.
Moreover,
\[ \liederivative{D^{1, 0}}\alpha=\liederivative{D^{1, 0}}\perturbed{\tau}[b, c]-\liederivative{D^{1, 0}}[\perturbed{\tau}b, \perturbed{\tau}c]
= \liederivative{D^{1, 0}}\perturbed{\tau}[b, c]-[\liederivative{D^{1, 0}}\perturbed{\tau} b, \perturbed{\tau} c]
-[\perturbed{\tau}b, \liederivative{D^{1, 0}} \perturbed{\tau}c]=0 \]
Finally, one can check directly using Equations~\ref{eq:etendusigma} and~\eqref{eq:perturbedetendutau} that
\[ \etendu{\sigma} [\perturbed{\tau}b, \perturbed{\tau}c]=[b, c] .\]
Therefore we have
\[ \etendu{\sigma}\alpha=\etendu{\sigma}(\perturbed{\tau}[b, c])-\etendu{\sigma} [\perturbed{\tau}b, \perturbed{\tau}c]=0 .\]
From Lemma~\ref{lem:Dijon}, it follows that $\alpha=0$. Equation~\eqref{eq:wuxi1} is thus proved.
Equations~\eqref{eq:wuxi2} and~\eqref{eq:wuxi3} can be proved in a similar fashion.
\end{proof}

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