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Lie-Poisson bracket

Let $(M, \Pi)$ be a Poisson manifold.
$\begin{thm} There exists a unique $\bR$-linear skewsymmetric bracket $[-,-]_{\Pi... ...math}[\smap(\al), \smap(\bt)]=\smap([\al,\bt]_{\Pi}). \end{displaymath}\end{thm}$

$\begin{proof} Such $[-,-]_{\Pi}$\ should be local i.e. if $\bt_1\vert _U = \bt_2... ...gn*}The Jacobi identity is proved locally using $adf$, $bdg$, $cdh$. \end{proof}$

$\begin{defn} Let $M$\ be a manifold, $E\to M$\ vector bundle. Then $E$\ is calle... ...]+(\rho(v)f)w$\ for all $v,w\in\Ga(E)$, $f\in\Coo(M)$. \end{enumerate}\end{defn}$
Remarks:

Given any Lie algebroid, the image of the anchor is always a generalized integrable distribution; its maximal integrable submanifolds are called orbits of the Lie algebroid.
The tangent bundle to a manifold is always a Lie algebroid with the trivial anchor map $\rho=\id$ .
Theorem proves that for any Poisson manifold , its cotangent bundle is a Lie algebroid with anchor the sharp map. The orbits of this algebroid are the symplectic leaves of

Pawel Witkowski 2006-06-26