manifold ${\boldsymbol{X}}$, and $E$ a finite rank complex vector bundle with a $B$-action as in [@MR1726784], then all cohomology groups ${\mathrm{H}^{\bullet}}(B,E)$ are finite dimensional.

It is therefore natural to ask when ${{A}^*}_H$ is elliptic. An easy calculation shows the following:

\[Vienna\] Let $a_*^H$ denote the anchor of ${{A}^*}_H$ and $\operatorname{C}:{{T^{0,1}{\boldsymbol{X}}}}\to{{T^{1,0}{\boldsymbol{X}}}}$ the complex conjugation. The bundle maps $\operatorname{Re}{\circ}a_*^H$ and $$\label{Paris}
{F}=(\operatorname{C}+\theta{^{\flat}})\oplus\pi{^{\sharp}}:{{T^{0,1}{\boldsymbol{X}}}}\oplus{{(T^{1,0}{\boldsymbol{X}})^*}}\to{{T^{1,0}{\boldsymbol{X}}}},$$ and the isomorphism of real vector bundles $\operatorname{Re}:{{T^{1,0}{\boldsymbol{X}}}}\to{T{\boldsymbol{X}}}$ fit into the commutative diagram $$\label{Diagram:alternateMapanchorRealCommute}
\xymatrix{ & {{T^{0,1}{\boldsymbol{X}}}}\oplus {{(T^{1,0}{\boldsymbol{X}})^*}}\ar[dl]_{{F}}\ar[dr]^{\operatorname{Re}{\circ}a_*^H} \\
{{T^{1,0}{\boldsymbol{X}}}}\ar[rr]^{\operatorname{Re}} & & {T{\boldsymbol{X}}}. }$$ As a consequence, ${{A}^*}_H$ is an elliptic Lie algebroid if and only if ${F}$ is surjective.

When $H=\pi$, or $\omega$, it is clear that ${{A}^*}_H$ is elliptic. On the other hand, if we consider the torus ${\boldsymbol{T}}$ endowed with the bivector field $\theta$ of Example \[Ex:realtorus\], the Lie algebroid ${{A}^*}_H$ is elliptic if and only if $f$ is not identically $1$.

Poisson cohomology
------------------

Given an extended Poisson manifold $({\boldsymbol{X}},H)$, the cohomology of the Lie algebroid ${{A}^*}_H$ is called the **Poisson cohomology** of the extended Poisson structure, and denoted ${\mathrm{H}^{\bullet}}
({\boldsymbol{X}},H)$. In other words, it is the cohomology of the cochain complex: $$\label{eq:H}
\cdots {\xrightarrow{{\bar{\partial}^H}}} {\Gamma(\wedge^{k}{A})} {\xrightarrow{{\bar{\partial}^H}}} {\Gamma(\wedge^{k+1}{A})} {\xrightarrow{{\bar{\partial}^H}}} \cdots
,$$ where ${\Gamma(\wedge^k{A})}=\oplus_{i+j=k}{\Omega^{0, j}}({T^{i,0}{\boldsymbol{X}}})$ and ${\bar{\partial}^H}={\bar{\partial}}+{[H,\cdot]}$.

