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Poisson homogeneous spaces


\begin{defn} A \textbf{Poisson homogeneous space} is a Poisson manifold $(M, \Pi_M)$\ together with a transitive Poisson action of a Poisson Lie group. \end{defn}

\begin{remark} The covariance condition is \begin{displaymath} \Pi_M(g\cdot x) =... ...e. $\Pi_M$\ is uniquely determined by its value at one fixed point. \end{remark}
Homogeneous $G$-spaces are of the form $G/H$ for a closed Lie subgroup $H$. We will show how, and why, such description does not work any more at the Poisson level. First we need to describe properties of subgroups of Poisson Lie group.
\begin{defn} A Lie subgroup $H$\ of a Poisson Lie group $G$\ is called a \textb... ...d a \textbf{coisotropic subgroup} if it is a coisotropic submanifold. \end{defn}

\begin{remark} Let $H\leq G$\ be a Poisson (coisotropic) Lie subgroup and $g\in ... ..._g(H)=gH g^{-1}$\ may be Poisson, coisotropic or none of the above. \end{remark}

\begin{prop} Let $H$\ be a connected Lie subgroup of a Poisson Lie group $(G, \P... ...if and only if $\gh^{\perp}$ is a Lie subalgebra. \end{enumerate}\par \end{prop}

\begin{proof} $H$\ is a Poisson submanifold if and only if \begin{displaymath} I... ...rp}$\ is still spanned by $d_e f$, $f\in I_H$, therefore the thesis. \end{proof}
Poisson homogeneous spaces $\phi\: G\times M\to M$ contain a number of special cases.
  1. Invariant Poisson structures ($\Pi_G=0$)
  2. Affine Poisson structures ($M=G$)
  3. Non symplectic covariant (i.e. $\Pi_G\neq 0$) Poisson structures, which include
    1. ''Highly singular'' covariant Poisson structures ( $\exists\; x_0\; \Pi_M(x_0)=0$)
    2. Quotients by coisotropic subgroups
    3. Quotients by Poisson Lie subgroups
    Furthermore $(a) = (b) \supset (c)$.
Some relevant examples of Poisson Lie groups:
\begin{exer} Classify Poisson Lie subgroups of $\SU(2)$. \end{exer}
Hint: Compute the dual Lie bialgebra. Classify ideals in this 3-dimensional Lie algebra, distinguishing between 2-dimensional ideals and 1-dimensional ideals. Check which of them is the $\perp$ of a Lie algebra, and you have that the only pair $(\gh,\gh^{\perp})$ such that $\gh$ is a Lie subalgebra of $\gsu(2)$ and $\gh^{\perp}$ is a Lie ideal in $\gsu(2)^*$ is when $\gh=\langle H\rangle$, $H$ being the Cartan diagonal element. Therefore the only connected Poisson-Lie subgroup is $\bS^1$ diagonally embedded in $\SU(2)$ and the disconnected ones are its discrete subgroups.


\begin{exer} Classify Poisson Lie subgroups of $\SL(n, \bC)$\ with respect to the standard structure. \end{exer}
It requires some work. A good start is to look at the first pages of [69].

Coisotropy condition is much weaker. For example let $H\leq G$ be a Lie subgroup of codimension 1. Then $H$ is coisotropic. In fact $\dim \gh^{\perp}=1$ and therefore $\gh^{\perp}$ is a Lie algebra, $[X, X]=0$.

Let $M$ be a Poisson homogeneous space. Fix $x\in M$

\begin{displaymath} T_xM\simeq \gerg/\gh_x,\quad \gh_x\text{ - stabilizer of }x \end{displaymath}


\begin{prop} For any $v\in \La^2\gerg/\gh_x$ \begin{displaymath} L_x:=\{ X+\xi :... ...v)=X+\gh_x\} \end{displaymath}is a Lagrangian subspace of the double. \end{prop}

\begin{proof} \begin{displaymath} \scalar{X+\xi}{Y+\eta}=(\xi\tensor\eta + \eta\... ...follows from surjectivity of $X+\xi\to X$, which implies maximality. \end{proof}


\begin{thm} For any $x\in M$\ let $L_x$\ be the Lagrangian subspace in $D\gerg$.... ...bras such that if $x\in M$\ then $L_x\cap \gerg=\gh_x$. \end{enumerate}\end{thm}

\begin{remark} % latex2html id marker 7567Let $D\gerg = \gerg\oplus \gerg^*$\ ... ... orbits are ''models'' for Poisson homogeneous spaces (\cite{elu}). \end{remark}


\begin{prop} Let $M$\ be a Poisson homogeneous space of $(G, \Pi_G)$. For $x_0\i... ... G : gx_0=x_0\}$) is coisotropic; $M\simeq G/H_{x_0}$ \end{enumerate}\end{prop}

\begin{proof} % latex2html id marker 7584$(1)\implies (2)$\ Take the same $x_0... ...et \gh\land\gerg$, that is precisely when $H_{x_0}$\ is coisotropic. \end{proof}

\begin{prop} Let $G$\ be a Poisson Lie group. Let $K$\ be a Poisson Lie subgrouo... ...th} i\: K/ K\cap H' \to G/H' \end{displaymath}is a Poisson embedding. \end{prop}

\begin{proof} \begin{displaymath} I_K:= \{f\in\Coo(G) : f\vert _K=0\} \end{displ... ...r] \ar[d] & G \ar[d]\\ K/ H'\cap K \ar[r] & G/H' } \end{displaymath}\end{proof}
The following example is carried out in all details in [20].
\begin{example} Take $\SU(n)$\ with standard Poisson Lie structure \begin{align*... ...h}where odd spheres have the standard Poisson structure. \end{prop}\end{example}
Recall that we have defined a Lie bracket on $\Om^1(M)$ (where $M$ is Poisson)

\begin{displaymath}[\al, \bt]=L_{\char93 _{\Pi}(\al)}\bt - L_{\char93 _{\Pi}(\bt)}\al - d(\Pi(\al, \bt)) \end{displaymath} (2)

What happens to this bracket when $M=G$ is a Poisson-Lie group ?
\begin{thm} % latex2html id marker 7761 [Dazord-Karasev-Weinstein] The left (res... ...ermore this induces a Lie bracket on $\gerg^*$\ isomorphic to $^t\dl$. \end{thm}

\begin{proof} % latex2html id marker 7764Let $\al, \bt$\ be left invariant 1-f... ...df_2](e),\text{ where }\xi_1=d_ef_1,\;\xi_2=d_ef_2. \end{displaymath}\end{proof}


\begin{exer} Consider the standard Poisson Lie group structure on $\SU(2)$. Then... ... that $\bS^2_{\la}\ncong \bS^2_{\la'}$ for $\la\neq\la'$\ in $[0,1]$. \end{exer}

next up previous contents
Next: Dressing actions Up: Poisson actions Previous: Poisson actions   Contents
Pawel Witkowski 2006-06-26