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Poisson homology

$\begin{defn} \textbf{Canonical} (or \textbf{Brylinski}) \textbf{operator} \begin... ...\Pi}:=i_{\Pi} d - d i_{\Pi}\: \Om^k M\to \Om^{k+1} M \end{displaymath}\end{defn}$

$\begin{prop} The following identities are verified \begin{enumerate} \item $d\de... ...\Pi} - i_{\Pi}\del_{\Pi}=0$. \item $\del_{\Pi}^2 = 0$. \end{enumerate}\end{prop}$

$\begin{proof}\mbox{} \begin{enumerate} \item $d\del_{\Pi} = d i_{\Pi} d = -\del_... ... &= 2i_{\Pi}di_{\Pi}d-di_{\Pi}i_{\Pi}d=0. \end{align*}\end{enumerate}\end{proof}$

$\begin{defn} The homology of the complex $(\Om^{\bullet}, \del_{\Pi})$ is called... ...{canonical}) \textbf{homology} and it is denoted by $\rH^{\Pi}_k(M)$. \end{defn}$

Pawel Witkowski 2006-06-26