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Poisson Lie groups

Recall the two presentations of a Poisson manifold:

such that
- $\poiss{-}{-}$ is a Lie bracket (antisymmetric + Jacobi identity)
- $\poiss{f}{gh} = \poiss{f}{g}h + g\poiss{f}{h}$ (Leibniz rule)
$(M, \Pi)$ , $\Pi\in \Ga(\La^2 TM)$ such that $[\Pi, \Pi]=0$

connected by the equality

$\begin{displaymath} \poiss{f}{g}(x)=\scalar{\Pi(x)}{d_xf\ox d_xg} \end{displaymath}$

Recall that a smooth map $\phi\: M\to N$ between Poisson manifolds is a map that preserves Poisson brackets

$\begin{displaymath} \poiss{f_1}{f_2}_M\circ \phi = \poiss{f_1\circ \phi}{f_2\circ\phi}_N \end{displaymath}$

or equivalently

$\begin{displaymath} \phi^{\ox 2}_{*, x}\Pi_M(x)=\Pi_N(\phi(x)) \end{displaymath}$

Recall also that if

are Poisson manifolds the structure of product Poisson manifold on $M\times N$ is the one given by

$\begin{displaymath} \poiss{f_1}{f_2}_{M\times N}(x, y)=\poiss{f_1(-, y)}{f_2(-,y)}_M(x) +\poiss{f_1(x, -)}{f_2(x,-)}_N(y) \end{displaymath}$

or equivalently

$\begin{displaymath} \Pi_{M\times N}=\Pi_M\oplus \Pi_N\in \Ga(\La^2 T(M\times N))= \Ga(\La^2 TM\oplus \La^2 TN) \end{displaymath}$

$\begin{prop} Let $G$\ be a Lie group, $\Pi$\ Poisson tensor on $G$. Then the fol... ...splaymath}and $L_{g, *}$, $R_{g, *}$\ are derivatives. \end{enumerate}\end{prop}$

$\begin{proof} Let $m\: G\times G\to G$\ be a Poisson map, that is \begin{display... ...(g_1)}{d_{g_1}f_1\ox d_{g_1} f_2} \end{displaymath}hence the thesis. \end{proof}$

$\begin{defn} % latex2html id marker 6521When one of the conditions of proposit... ...oup}) is verified $(G, \Pi)$\ is called a \textbf{Poisson Lie group}. \end{defn}$

$\begin{remarks} \mbox{} \begin{itemize} \item For a Poisson Lie group $(G, \Pi)$... ...s called an \textbf{affine Poisson structure on} $G$. \end{itemize}\end{remarks}$
Let us move on to the infinitesimal description of the Poisson Lie groups. Consider

$\begin{displaymath} \eta\: G\to \La^2\gerg \end{displaymath}$

given by right translating the Poisson tensor

$\begin{displaymath} \eta(g)= R_{g^{-1}, *}\Pi(g) \end{displaymath}$

(obviously $\eta(e)=0$ ). Now
$\begin{align*} \eta(g_1g_2) &= R_{(g_1 g_2)^{-1}, *} \Pi(g_1 g_2) \\ &= R_{g_1^... ..._{g_1}R_{g_2^{-1}, *}\Pi(g_2) \\ &= \eta(g_1) + \Ad_{g_1}\eta(g_2) \end{align*}$
i.e. $\Pi$ multiplicative $\implies$ $\eta$ is a cocycle of

with values in $\La^2\gerg$ .

Define now

$\begin{displaymath} \dl\:\gerg\to \La^2\gerg \end{displaymath}$

to be its derivative at

, i.e.

$\begin{displaymath} \dl(X) := \ddt{t}\eta(e^{tX})\vert _{t=0} \end{displaymath}$

What are the properties of $\dl$ coming from the fact that $\Pi$ is Poisson and multiplicative ?
$\begin{prop} \mbox{} \begin{enumerate} \item $\Pi$\ multiplicative $\implies$ \b... ... \to \gerg \end{displaymath}satisfies Jacobi identity. \end{enumerate}\end{prop}$

$\begin{proof} \begin{enumerate} \item \begin{displaymath} \eta(e^{tX} e^{tY}) = ... ..._e\Jac_{\poiss{-}{-}}(f_1, f_2, f_3) \end{displaymath}\end{enumerate}\end{proof}$

Next: Lie bialgebras Up: Poisson Lie groups Previous: Poisson Lie groups Contents

Pawel Witkowski 2006-06-26