of any two sections $u,v\in{\Gamma({K})}$, the restriction to ${\boldsymbol{Y}}$ of ${[\widetilde{u},\widetilde{v}]^H_*}$ is a section of ${K}$ which does not depend on the choice of extensions.

Poisson relations
-----------------

Following Weinstein [@MR959095], we introduce the following

Let $({\boldsymbol{X}}_1,H_1)$ and $({\boldsymbol{X}}_2,H_2)$ be extended Poisson manifolds. A Poisson relation from $({\boldsymbol{X}}_2,H_2)$ to $({\boldsymbol{X}}_1,H_1)$ is a coisotropic submanifold of the product manifold ${\boldsymbol{X}}_1\times{\boldsymbol{X}}_2^{\vee}$ (i.e. ${\boldsymbol{X}}_1\times{\boldsymbol{X}}_2$ endowed with the extended Poisson structure $(H_1, H_2^{\vee})$, see Corollary \[Cor:Hminus\]).

We call a holomorphic map $f:{\boldsymbol{X}}_2\to{\boldsymbol{X}}_1$ between extended Poisson manifolds $({\boldsymbol{X}}_1,H_1)$ and $({\boldsymbol{X}}_2,H_2)$ an **extended Poisson map** if its graph $$G_f={\left\{(f(x),x) {\;\text{s.t.}\;}x\in{\boldsymbol{X}}_2\right\}}\subset{\boldsymbol{X}}_1\times{\boldsymbol{X}}_2^{\vee}$$ is a Poisson relation.

Let $({\boldsymbol{X}}_1,H_1)$ and $({\boldsymbol{X}}_2,H_2)$ be extended Poisson manifolds, where the extended Poisson structures decompose as $H_i=\pi_i+\theta_i+\omega_i$ ($i=1,2$). Then a holomorphic map $f:{\boldsymbol{X}}_2\to{\boldsymbol{X}}_1$ is an extended Poisson map if and only if $f_*\pi_2=\pi_1$; $f^*\omega_1=\omega_2$; and $f_*\circ\theta{^{\flat}}_2=\theta{^{\flat}}_1\circ f_*$.

The proof is a direct verification and is left to the reader. As a consequence, we have

The composition of two extended Poisson maps is again an extended Poisson map.

Koszul-Brylinski Poisson homology
=================================

In this section we will introduce homology groups for extended Poisson manifolds based on the Evens-Lu-Weinstein module of a Lie algebroid.

Koszul-Brylinski cochain complex
--------------------------------

First we recall the notion of Clifford 