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Dictionary

In the following we would like to set up a whole dictionary

classical	semiclassical	quantum
algebraic group	Poisson algebraic group	quantum group
compact group	Poisson compact group	compact quantum group
Lie algebra	Lie bialgebra	quantum universal enveloping agebra
	Poisson dual	quantum duality principle
	Poisson double	quantum double construction
point	0-leaf	character

It is known in examples that quantum groups have few characters (classical points). Why is it so ?
$\begin{prop} Let $A_{\hbar}$\ be a local quantization of $A_0=(F[M], \Pi)$. Ther... ...{\hbar}, A_{\hbar}])=0$) and 0-leaves of the Poisson bivector $\Pi$. \end{prop}$

$\begin{proof} Let $\eps$\ be the character of $A_{\hbar}$. Then $\eps$\ defines ... ...ch that $d_{x_0}f$\ generate $\Om_{x_0}^1(M)$\ we have $\Pi(x_0)=0$. \end{proof}$

$\begin{example} $F_q[\SU(2)]$\ (here *-characters - looking for real points) \be... ...displaymath} \eps(\al)=t,\quad \eps(\al^*)=t^{-1} \end{displaymath}\end{example}$
This is just an issue of a more general situation. In principle you would like to have a correspondence between primitive ideals of $F_{\hbar}[G]$ and symplectic foliation of $(G, \Pi)$ . For example if we take $U_q(\gerg)=F_q[\gerg^*]$ then by orbit method we obtain a homeomorphism between primitive ideals in $U_q(\gerg)$ and coadjoint orbits of

on $\gerg^*$ . It would be nice to have a ''quantum orbit method''. In fact it works for compact quantum groups.

Next: Quantum subgroups Up: Quantization Previous: Real structures Contents

Pawel Witkowski 2006-06-26