an anti-holomorphic section of the holomorphic vector bundle ${{(T^{0,1}{\boldsymbol{X}})^*}}$) is also zero. Thus $$[{{\tfrac{\partial}{\partial{z^{i}}}}},{{\tfrac{\partial}{\partial{z^{j}}}}}]=0, \qquad [d{\overline{z_i}},d{\overline{z_j}}]=0, \quad \text{and} \quad [{{\tfrac{\partial}{\partial{z^{i}}}}},d{\overline{z_j}}]=0 .$$ Together with the Leibniz rule, the above three rules completely determine the bracket of any two arbitrary sections of $A$. Similarly, one endows ${{A}^*}$ with a complex Lie algebroid structure as well. It is simple to see that $({A},{{A}^*})$ constitutes a Lie bialgebroid. Indeed $A$ and $A^*$ are transversal Dirac structures of the Courant algebroid ${T_{{\mathbb{C}}}{\boldsymbol{X}}}\oplus{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}$, for they are the eigenbundles of the generalized complex structure on ${\boldsymbol{X}}$ induced by its complex manifold structure [@MR2013140; @GualtieriThesis]. In the sequel we will use the symbols $${{T^{1,0}{\boldsymbol{X}}}}\bowtie{{(T^{0,1}{\boldsymbol{X}})^*}}\quad \text{and} \quad {{T^{0,1}{\boldsymbol{X}}}}\bowtie{{(T^{1,0}{\boldsymbol{X}})^*}}$$ to refer to ${A}$ and ${{A}^*}$ when seen as Lie algebroids [@MR2439547].

Moreover, one has $$\begin{gathered}
\wedge^k {A}\cong \bigoplus_{i+j=k} {T^{i,0}{\boldsymbol{X}}} \otimes {(T^{0,j}{\boldsymbol{X}})^*} ,\\
\wedge^k {{A}^*}\cong \bigoplus_{i+j=k} {T^{0,i}{\boldsymbol{X}}}\otimes {(T^{j,0}{\boldsymbol{X}})^*} .\end{gathered}$$

The Lie algebroid differentials associated to the Lie algebroid structures on ${{A}^*}$ and ${A}$ are the usual ${\bar{\partial}}$- and ${\partial}$-operators, respectively: $$\begin{gathered}
{{d}_*}={\bar{\partial}}:~ {\Omega^{0,j}} ({T^{i,0}{\boldsymbol{X}}}) \to {\Omega^{0,j+1}} ({T^{i,0}{\boldsymbol{X}}}) ,\\
{d}={\partial}:~ {\Omega^{j,0}} ({T^{0, i}{\boldsymbol{X}}}) \to {\Omega^{j+1,0}} ({T^{0, i}{\boldsymbol{X}}}) .\end{gathered}$$

Extended Poisson structures
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An **extended Poisson manifold** $({\boldsymbol{X}},H)$ is a complex manifold ${\boldsymbol{X}}$ equipped with an $H\in{\Gamma(\wedge^2{A})}$ which is an Hamiltonian operator with 