# Poisson structure on the dual Lie algebroid

Let $E \to X$ be a Lie algebroid over the manifold $X$. Let $x_1,...x_n$ be local coordinates on $X$ and $e_1,...e_m$ be the basis of local sections of $E$. In terms of these coordinate functions Lie bracket and the anchor map $\rho$ are described like this: $$[e_i,e_j]_E = \sum\limits_k c_{ijk}e_k$$ $$\rho(e_i) = \sum\limits_j b_{ij}\frac{\partial}{\partial x_j}$$

Let $\xi_1,..\xi_m$ be the basis dual to $e_1,...e_m$. Now we define Poisson structure on $E^*$ by setting: $$\{x_i,x_j\} = 0$$ $$\{\xi_i,\xi_j\} = \sum\limits_k c_{ijk}\xi_k$$ $$\{\xi_i, x_j\} = -b_{ij}$$

This Poisson structure is independent of the choice of local coordinates and basis of local sections on $E$, so it's very possible that there exists a coordinate free expression of the Poisson structure like the one we have for dual Lie algebras. How does this expression look like?

Depending on you sign convention, this goes as follows. First you denote the bundle projection by $pr\colon E^* \longrightarrow X$. For a section $s \in \Gamma^\infty(E)$ you have a linear function $J(s) \in C^\infty(E^*)$ defined by pointwise evaluation, i.e. $J(s)(\alpha_p) = \alpha(s(p))$ where $\alpha_p \in E^*_p$ and $p \in X$. Then the linear Lie bracket on $E^*$ is uniquely determined by \begin{equation} \{pr^*f, pr^*g\} = 0, \quad \{pr^*f, J(s)\} = pr^*(\varrho(s)f), \quad \{J(s), J(t)\} = - J([s, t]), \end{equation} where $f, g \in C^\infty(X)$ are functions on the base and $s, t \in \Gamma^\infty(E)$ are sections of the Lie algebroid.
In fact, one can show that this Poisson structure is linear in the sense that for its Poisson tensor $\pi$ one has $L_\xi \pi = - \pi$ where $\xi$ is the Euler vector field on $E^*$. Conversely, every linaer Poisson structure on $E^*$ defines a Lie algebroid this way. It is a nice exercise the express anchor and Lie algebroid bracket in terms of the Poisson bracket for a linear Poisson tensor.