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Quantum homogeneous spaces

Let $B$ be a unital *-algebra and let $A$ be a Hopf-*-algebra.
\begin{defn} A *-algebra homomorphism $\dl\: B\to b\tensor A$\ is a \textbf{rig... ...\id \end{displaymath}$B$\ is called \textbf{$A$-right quantum space}. \end{defn}
Which right coactions correspond to homogeneous actions ? Here we mean $A=F[G]$, $B=F[X]$, $\dl$ dual of action $\phi\: G\times X\to X$.
\begin{defn} Two right quantum spaces $(B, \dl)$, $(B', \dl')$\ are \textbf{equi... ...[d] & B'\ar[d] \\ B\tensor A \ar[r] & B'\tensor A } \end{displaymath}\end{defn}
Modifying the following definition replacing the identity in ([*]) by a *-algebra morphism $\Psi\: A\to A'$

\begin{displaymath} \dl'\circ\Phi=(\Phi\tensor \Psi)\circ\dl \end{displaymath}

gives the definition of equivariant map of quantum spaces on different Hopf algebras.
\begin{prop} Let $(B, \dl)$\ be $A$-right quantum space. There is a 1:1 correspo... ...laymath}\begin{displaymath} \Tilde{\eps}=\eps\circ i \end{displaymath}\end{prop}

\begin{proof} Say $\Tilde{\eps}\: B\to \bC$\ is given. Define \begin{displaymath... ... \id)\circ \dl = (\eps\tensor\id)\circ\Dl\circ i =i \end{displaymath}\end{proof}
Thus for any $A$-right quantum space $(B,\dl)$ such that $B$ has a character there exists an equivariant map between $(B,\dl)$ and a subalgebra $(i_{\eps}(B), \Dl\vert _{i_{\eps}(B)})$ of $A$.

What is $i_{\eps}(B)$ in usual language ? Take a $G$-space $X$. Fix $x_0\in X$. Then consider

\begin{displaymath} F[X]\to F[G],\quad f\mapsto \Tilde{f_{x_0}} \end{displaymath}

where $\Tilde{f_{x_0}}(g):=f(gx_0)$. When $X$ is a classical homogeneous space we have that this map is injective.
\begin{defn} An \textbf{embeddable} quantum homogeneous space is an $A$-right qu... ...$\Tilde{\eps}\: B\to \bC$ such that $i_{\Tilde{\eps}}$\ is injective. \end{defn}
Identifying $(B,\dl)$ with $(i_{\Tilde{\eps}}(B), \Dl\vert _{i_{\Tilde{\eps}}(B)})$ we can equivalently declare an embeddable quantum homogeneous space to be a *-subalgebra and right coideal of $F_q[G]$.
\begin{remark} This is not the most general fdefinition of quantum homogeneous s... ...ter, which is in noncommutative algebras something not so trivial. \end{remark}
Let us understand this from the point of view of semiclassical limit. Everything above can be rephrased on $\bC[\hbar]$-Hopf-*-algebras. Now we have

\begin{displaymath} \xymatrix{ {\begin{array}{l} \text{Having one}\\ \text{char... ...F_{\hbar[X]}\hookrightarrow F_{\hbar}[G] \end{array}} \ar[u] } \end{displaymath}

But we have seen already this at the semiclassical level

\begin{displaymath} \xymatrix{ \text{SEMICLASSICAL} & \text{QUANTUM} \\ {\begin... ...E WANT}\\ \text{TO FILL THIS !} \end{array}} \ar[l] \ar[u] } \end{displaymath}

Before going into this we want to understatnd the relation betwwen quantum subgroups and embeddable quantum homogeneous spaces.
\begin{prop} Let $F_q[G]=A$\ be a quantum group and let $F_q[H]$\ be a quantum ... ...ore $B_H$ is $S^2$-invariant and $p_H(b)=\eps(b)1$\ for all $b\in B$. \end{prop}

\begin{proof} Remark that \begin{displaymath} y\in B_H\tensor A \quad\Longleftri... ... $(p_H\tensor \id)\circ\Dl b=1\tensor b$ to prove $p_H(b)=\eps(b)1$. \end{proof}
We would like to check whether all quantum homogeneous spaces are of this form. We have a necessary condition, $S^2$-invariance. Is it always verified ?
\begin{example} Consider on the standard $F_q[E(2)]$ \begin{displaymath} z=\la v... ...s $z, S^2(z)\in B$, so $n\in B$, which is not true if $\la\neq 0$. \end{example}

\begin{example} Similarly consider $F_q[\SU(2)]$. Take \begin{displaymath} K := ... ...uotient by a quantum subgroup if and only if $s=1$. \end{enumerate}\end{example}
We are looking for a quantum analogue of a coisotropic subgroup.

next up previous contents
Next: Coisotropic creed Up: Quantization Previous: Quantum subgroups   Contents
Pawel Witkowski 2006-06-26