Topics in Geometric Mechanics: Week 10 Tutorial

Say that $\bivector{P}$ is a Poisson tensor on the manifold $M$ if

• $\bivector{P}$ is a skew-symmetric $(2, 0)$ tensor;
• the bracket $\pb{}{}$ defined by \begin{align*} \pb{f}{g} &:= \bivectorwargs{P}{df}{dg}, && f, g \in \cinfty{M} \end{align*} defines a Poisson algebra structure on $\cinfty{M}$.

  Show that $\bivector{P}$ is a Poisson tensor iff \begin{align*} J^{ijk} = \sum_{\alpha} \bivector{P}^{i \alpha} \didi{\bivector{P}^{jk}}{x_{\alpha}} + \bivector{P}^{j \alpha} \didi{\bivector{P}^{ki}}{x_{\alpha}} + \bivector{P}^{k \alpha} \didi{\bivector{P}^{ij}}{x_{\alpha}} \end{align*} where $(x_i)$ is a coordinate system and $i, j, k = 1, \ldots, \dim M$.

Let $\liealg{g}$ be a Lie algebra. Let $a \in \liealg{g}^*$ be fixed. Define \begin{align*} \pb{f}{g}_a(x) = \ip{a}{\liebracket{df(x)}{dg(x)}}, && \forall f, g \in \cinfty{\liealg{g}^*}. \end{align*} Prove that $\pb{}{}_a$ defines a Poisson structure on $\liealg{g}^*$.