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Compatible Poisson tensors

Say that $\Pi_1$, $\Pi_2$ are Poisson bivectors on $M$.

Question: is $\Pi_1 + \Pi_2$ a Poisson bivector on $M$ ?

If this is the case we will say that they are compatible Poisson tensors.
\begin{prop} $\Pi_1$\ and $\Pi_2$\ are compatible if and only if $[\Pi_1, \Pi_2]=0$. \end{prop}
In this case $a\Pi_1 + b\Pi_2$ is Poisson for all $a,b\in\bR$ and $\{a\Pi_1 +b\Pi_2 : a,b\in\bR\}$ is called a Poisson pencil.
\begin{proof} \begin{displaymath}[\Pi_1 + \Pi_2, \Pi_1 + \Pi_2]= \underbrace{[\... ...}[a\Pi_1+b\Pi_2, a\Pi_1+b\Pi_2]=2ab [\Pi_1, \Pi_2]. \end{displaymath}\end{proof}


Pawel Witkowski 2006-06-26