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Coisotropic creed

When $A_{\hbar}$ is a quantization of $(M, \Pi)$ then one-sided ideals in $A_{\hbar}$ should correspond to coisotropic submanifolds. The motivation for this comes from characterization

$\begin{displaymath} \xymatrix{ {\begin{array}{l} \text{Poisson submanifold } N \... ...NGER THAN} \\ \text{BEING A SUBALGEBRA} \end{array}} \ar[l] } \end{displaymath}$

$\begin{prop} Let $A_{\hbar}$\ is a quantization of $M$. Take $I$\ to be a right ... ..._0=I/\hbar I$\ is an ideal in $A_0$\ and a Poisson subalgebra in $M$. \end{prop}$

$\begin{proof} Let $i\in I$, $f\in A_{\hbar}$ \begin{displaymath} f*i = fi + \hba... ...ss{i}{j}\in I_0 \end{displaymath}so $I_0$\ is a Poisson subalgebra. \end{proof}$
We will stick to this creed and declare the following
$\begin{defn} Let $A$\ be (*)-Hopf algebra. A \textbf{right (real) coisotropic qu... ...ed with an involution $\sg$ such that $p\circ(*\circ S)=\sg\circ p$). \end{defn}$

$\begin{prop} $C$\ is a right (real) coisotropic quantum subgroup if and only if ... ... such that \begin{displaymath} p\: A\to A/I_C\cong C \end{displaymath}\end{prop}$

$\begin{remark} All Poisson subgroups can be quantized in a context of functorial... ...in such context whether all coisotropic subgroups can be quantized. \end{remark}$

$\begin{prop} \mbox{} \begin{enumerate} \item Let $C$\ be a coisotropic quantum s... ...laymath}is a right ideal and two sided coideal of $A$. \end{enumerate}\end{prop}$
Is this

$\begin{displaymath} \xymatrix{ {\begin{array}{l} \text{coisotropic quantum}\\ \... ...e quantum}\\ \text{homogeneous spaces} \ar[l] \end{array}} } \end{displaymath}$

a bijective correspondence ? Is it true that quotient by quantum subgroups are characterized by

-invariance ? Almost.

Let be a right coideal subalgebra. Take

$\begin{displaymath} AB^+:= B\cap \ker\eps = \{ b-\eps(b)1 : b\in B\} \end{displaymath}$

In general $B\subseteq A^{\mathrm{co} A/AB^+}$ but not necessarily equal. If the antipode is bijective and we restrict to left faithfully flat right coideal subalgebras and left faithfully coflat coisotropic quantum subgroups, then in that case

-invariance corresponds to quotient by a coisotropic quantum subgroup.

Pawel Witkowski 2006-06-26