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Dirac structures

$\begin{defn} Let $M$\ be a smooth manifold. A \textbf{Dirac structure} on $M$ is... ...]= ([X, Y], L_X \bt - L_Y \al + \half d(\al(Y)-\bt(X))) \end{equation}\end{defn}$

$\begin{remark} The Courant bracket is not a Lie bracket. However it turns out to be a Lie bracket on sections of a Dirac bundle. \end{remark}$

$\begin{prop} Let $\Pi\in\Ga(\La^2 TM)$\ be a bivector on $M$. Then $\graph(\Pi)$... ...re; $\Ga(\Pi)$ is a Dirac structure if and only if $\Pi$\ is Poisson. \end{prop}$

$\begin{remark} Not every Dirac structure comes from a Poisson bivector. \end{remark}$

$\begin{proof} For any $\Pi\in \Ga(\La^2 TM)$\ define \begin{displaymath} \Ga_{\P... ...\ is a Lie algebra map if and only if $[\Pi, \Pi]_{\textrm{SN}}=0$. \end{proof}$

$\begin{prop} Let $B$\ be a skewsymmetric bilinear form on $V$, $B\in \La^2 V^*$.... ...u+Bv) : (v, \mu)\in L\} \end{displaymath}is a linear Dirac structure. \end{prop}$

$\begin{proof} Dimension is obviously unchanged. Therefore it suffices to show is... ...e{B(v, w)+B(w,v)}_{=0})\\ &= ((v, \mu), (w, \eta)) = 0. \end{align*}\end{proof}$

$\begin{prop} Let $\Pi\in \La^2 V$\ and let $\Ga_{\Pi}$\ be the linear Dirac stru... ...\Pi(\xi, -) \end{displaymath}with the identifcation $V\simeq V^{**}$. \end{prop}$

$\begin{proof} \begin{displaymath} \sC_B(\Ga_{\Pi})=\Ga_{\Pi'} \Longleftrightarro... ...\flat_{B}\circ \char93 _{\Pi} \text{ is injective}. \end{displaymath}\end{proof}$

$\begin{prop} Let $L$\ be a Dirac structure on $M$\ and let $B\in \Om^2(M)$. Then... ...\sC_B(L)\text{ is Dirac } \Longleftrightarrow dB = 0 \end{displaymath}\end{prop}$

$\begin{proof} As we have already seen $\sC_B(L)$\ is pointwise a linear Dirac st... ...X, Y], Z) + B([X, Z], Y) - B([Y, Z], X). \end{displaymath}\end{proof}\end{proof}$

$\begin{defn} Two Poisson bivectors $\Pi_1, \Pi_2$\ on the manifold $M$\ are said... ...end{displaymath}such that $\Pi_0$\ and $\Pi_2$\ are gauge equivalent. \end{defn}$

$\begin{remark} Two symplectic structures on a given manifold are gauge equivalen... ...lent up to diffeomorphism if and only if they are symplectomorphic. \end{remark}$

Pawel Witkowski 2006-06-26