MTH-696A: Topics in Geometric Mechanics Assignment 9 (by Dr. Leo Butler)

$ %% not provided in align* environment \def\newsavebox#1{} \def\intertext#1{\text{#1}} \def\let#1#2{} \def\makeatletter{} \def\makeatother{} \newenvironment{theorem}{\textbf{Theorem.}\rm}{} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: %% sets \def\R{\mathbf{R}} \def\Z{\mathbf{Z}} \def\C{\mathbf{C}} \def\CP{\C{}P} \def\Q{\mathbf{Q}} \def\sphere#1{\mathbf{S}^{#1}} \def\rp#1{\R P^{#1}} \def\set#1{\left\{ #1 \right\}} \def\st{\,\mathrm{s.t.}\,} \def\extalg#1#2{\Lambda^{#1}(#2)} \def\symalg#1#2{\mathsf{S}^{#1}(#2)} \def\tenalg#1#2{\mathsf{T}^{#1}(#2)} \def\hom#1#2{\mathrm{Hom}(#1;#2)} \def\extproduct{\wedge} \def\symmetricproduct{\cdot} \def\tensorproduct{\otimes} \def\image#1{\mathrm{Im}\,#1} \def\kernel#1{\mathrm{Ker}\,#1} \def\Trace[#1]#2{\mathrm{Tr}\ifx#1\empty\else{}_{#1}\fi\ifx#2\empty\else\,(#2)\fi} \def\Trce#1{\mathrm{Tr}(#1)} \def\orbit#1#2{{#1}\cdot{#2}} \def\stab#1#2{\mathrm{stab}_{#1}(#2)} \def\Span{\mathrm{span}} \def\cinfty#1{C^{\infty}(#1)} %% Lie groups \def\GL#1{\mathrm{GL}(#1)} \def\SL#1{\mathrm{SL}(#1)} \def\Symp#1{\mathrm{Sp}(#1)} \def\Orth#1{\mathrm{O}(#1)} \def\SOrth#1{\mathrm{SO}(#1)} \def\Unitary#1{\mathrm{U}(#1)} \def\SUnitary#1{\mathrm{SU}(#1)} \def\Torus#1{\mathbf{T}^{#1}} \def\H#1{\mathbf{H}^{#1}} \def\Diagonal#1{\mathbf{Diag}(#1)} \def\diag#1{\mathbf{diag}\left(#1\right)} \def\Liegp#1{\mathrm{#1}} %% Lie algebras \def\Mat#1#2{\mathrm{Mat}_{{#1} \times {#1}}(#2)} \def\Matrect#1#2#3{\mathrm{Mat}_{{#1} \times {#2}}(#3)} \def\gl#1{\mathfrak{gl}(#1)} \def\spl#1{\mathfrak{sl}(#1)} \def\symp#1{\mathfrak{sp}(#1)} \def\orth#1{\mathfrak{o}(#1)} \def\sorth#1{\mathfrak{so}(#1)} \def\unitary#1{\mathfrak{u}(#1)} \def\sunitary#1{\mathfrak{su}(#1)} \def\torus#1{\mathfrak{t}^{#1}} \def\liealg#1{\mathfrak{#1}} \def\liebracket#1#2{[#1,#2]} \def\Ad#1#2{\mathrm{Ad}_{#1}{#2}} \def\ad#1#2{\mathrm{ad}_{#1}{#2}} \def\coAd#1#2{\mathrm{Ad}^*_{#1}{#2}} \def\coad#1#2{\mathrm{ad}^*_{#1}{#2}]} \def\pb#1#2{\left\{{#1},{#2}\right\}} %% operators \def\D#1{\,\mathrm{d}{#1}\,} \def\Dat#1#2{\,\mathrm{d}_{#1}{#2}\,} \def\lieder#1#2{{\sf{L}}_{#1}{#2}} \def\covder#1#2{\,\frac{D #2}{d{#1}}} \def\liebracket#1#2{\left[#1,#2\right]} \def\crossproduct{{\boldsymbol{\times}}} \def\ip#1#2{\langle #1, #2 \rangle} \def\tlangle{\langle\hspace{-1.1mm}\langle} \def\trangle{\rangle\hspace{-1.1mm}\rangle} \def\dotp#1#2{\tlangle #1, #2 \trangle} \def\interiorproduc#1#2{\iota_{#1}{#2}} \def\norm#1{|#1|} \def\lt{<} \def\gt{>} \def\ddt#1#2{\frac{d{#2}}{d{#1}}} \def\dndtn#1#2#3{\frac{d^{#3}{#2}}{d{#1}^{#3}}} \def\wint#1#2{\int\limits_{#1}^{#2}} \def\didi#1#2{\frac{\partial{#1}}{\partial{#2}}} \def\dindin#1#2#3{\frac{\partial^{#3}{#1}}{\partial{#2}^{#3}}} \def\eulerlagrange#1#2#3#4{\ddt{#1}{}\didi{#4}{#3} - \didi{#4}{#2}} \def\imagpart#1{\mathrm{Im}(#1)} \def\realpart#1{\mathrm{Re}(#1)} \def\equivclass#1{\left[#1\right]} \def\bivector#1{{\mathcal #1}} \def\bivectorwargs#1#2#3{\bivector{#1}(#2, #3)} %% labels, etc. \def\labelenumi{{\bf\Alph{enumi}.}} \def\solutioncolor{blue} \def\solutiontopsep{3mm} \def\solutionbotsep{0mm} \newenvironment{solution}{\vspace{\solutiontopsep}\noindent\textbf{Solution}. \textcolor{\solutioncolor}\bgroup}{\egroup\vspace{\solutionbotsep}} %% redo quote environment \def\quote{} \def\endquote{} %% redo enumerate environment \makeatletter \let\oldenumerate\enumerate \let\oldendenumerate\endenumerate \def\enumerate{\bgroup\oldenumerate\setlength{\itemsep}{3mm}\setlength{\topsep}{3mm}} \def\endenumerate{\oldendenumerate\egroup} \makeatother \def\museincludegraphicsoptions{% width=0.25\textwidth% } \def\museincludegraphics{% \begingroup \catcode`\|=0 \catcode`\\=12 \catcode`\#=12 \expandafter\includegraphics\expandafter[\museincludegraphicsoptions] } \def\musefigurewidth{!} \def\musefigureheight{!} \newsavebox{\mypicboc}% \def\includefigure#1#2#3#4{% \edef\fileending{#4}% \def\svgend{svg}% \def\pdftend{pdf_t}% \def\pdftexend{pdf_tex}% \def\bang{!}% \begin{lrbox}{\mypicboc}% \ifx\fileending\pdftend% \input{#3.#4}% \else% \ifx\fileending\pdftexend \input{#3.#4}% \else% \ifx\fileending\svgend% \input{#3.#4}% \fi\fi% \includegraphics{#3.#4}% \fi% \end{lrbox}% \edef\thiswidth{#1}\edef\thisheight{#2}% \ifx\thiswidth\bang% \ifx\thisheight\bang% \resizebox{!}{!}{\usebox{\mypicboc}}% \else% \resizebox{!}{\thisheight}{\usebox{\mypicboc}}% \fi% \else% \ifx\thisheight\bang% \resizebox{\thiswidth}{!}{\usebox{\mypicboc}}% \else% \resizebox{\thiswidth}{\thisheight}{\usebox{\mypicboc}}% \fi\fi} $

We will say that $(M,\pb{}{})$ is a Poisson manifold if, for all smooth function $f$ and $g$, $\pb{f}{g} = \bivectorwargs{P}{df}{dg}$ for some $(2, 0)$ skew-symmetric tensor field $\bivector{P}$, and $\pb{}{}$ satisfies the Jacobi identity. We call $\bivector{P}$ the associated Poisson tensor.

Let $(M,\pb{}{})$ be a Poisson manifold. Show that $\bivector{P}$ is preserved by all Hamiltonian vector fields: $\lieder{X}{P} = 0$ for all hamiltonian vector fields $X$. (Hint: show that the Jacobi identity implies this.)
Let $\liealg{g}$ be a Lie algebra with a non-trivial centre. Prove that the centre of Poisson algebra $(\cinfty{\liealg{g}^*},\cdot,\pb{}{})$ with the canonical Poisson structure is non-trivial.
Let $(M,\pb{}{})$ be a Poisson manifold with associated Poisson tensor $\bivector{P}$.
1. Show that, for each $x \in M$, the subspace $H_x = \set{ X_f(x) \mid f \in \cinfty{M} }$ equals the subspace $\image{\bivector{P}_x}$ where $\bivector{P}_x : T^*_x M \to T_x M$.
2. Prove that $H_x$ is even dimensional and that $\bivector{P}_x$ induces a non-degenerate skew-symmetric $2$-form on $H_x$.