and comments, including Camille Laurent-Gengoux, Giovanni Felder, Jiang-Hua Lu, Pierre Schapira and Alan Weinstein.

Preliminaries
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Lie bialgebroids
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A complex Lie algebroid [@arXiv:math/0601752] consists of a complex vector bundle ${A}\to M$, a bundle map ${a}:{A}\to T_{\mathbb{C}}M$ called anchor, and a Lie algebra bracket ${[\cdot,\cdot]}$ on the space of sections ${\Gamma({A})}$ such that ${a}$ induces a Lie algebra homomorphism from ${\Gamma({A})}$ to ${\mathfrak{X}}_{\mathbb{C}}(M)$ and the Leibniz rule $$\label{1} {[u,fv]}=\big({a}(u)f\big)v+f{[u,v]}$$ is satisfied for all $f\in{C^{\infty}(M,{\mathbb{C}})}$ and $u,v\in{\Gamma({A})}$.

It is well-known that a Lie algebroid $({A},{[\cdot,\cdot]},{a})$ is equivalent to a Gerstenhaber algebra $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{A})},\wedge,{[\cdot,\cdot]})$ [@MR1675117]. On the other hand, for a Lie algebroid structure on a vector bundle ${A}$, there is also a degree 1 derivation ${d}$ of the graded commutative algebra $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{{A}^*})},\wedge)$ such that ${d}^2=0$. The differential ${d}$ is given by $$\begin{gathered}
\label{m2} ({d}\alpha)(u_0,u_1,\cdots,u_n)=\sum_{i=0}^n (-1)^i
{a}(u_i) \alpha(u_0,\cdots,\widehat{u_i},\cdots,u_n) \\
+ \sum_{i<j} (-1)^{i+j}
\alpha({[u_i,u_j]},u_0,\cdots,\widehat{u_i},\cdots,\widehat{u_j},\cdots,u_n).\end{gathered}$$ Indeed, a Lie algebroid structure on ${A}$ is also equivalent to a differential graded algebra $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{{A}^*})},\wedge,{d})$.

Let ${A}\to M$ be a complex vector bundle. Assume that ${A}$ and its dual ${{A}^*}$ both carry Lie algebroid structures with anchor maps ${a}:{A}\to
T_{\mathbb{C}}M$ and ${{a}_*}:{{A}^*}\to T_{\mathbb{C}}M$, brackets on sections ${\Gamma({A})}\otimes_{{\mathbb{C}}}{\Gamma({A})}\to{\Gamma({A})}:u\otimes
v\mapsto{[u,v]}$ and ${\Gamma({{A}^*})}\otimes_{{\mathbb{C}}}{\Gamma({{A}^*})}\to{\Gamma({{A}^*})}:\alpha\otimes
\beta\mapsto{[\alpha,\beta]_*}$, and differentials ${d}:{\Gamma(\wedge^{\bullet}{{A}^*})}\to{\Gamma(\wedge^{\bullet+1}{{A}^*})}$ and ${{d}_*}:{\Gamma(\wedge^{\bullet}{A})}\to{\Gamma(\wedge^{\bullet+1}{A})}$.

This pair of 