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Duality

Take $X\in U(\gerg)$ . Then it defines a left invariant differential operator on

. Take $f\in F[G]$

$\begin{displaymath} (Xf)(e)=\scalar{X}{f} \end{displaymath}$

$\begin{displaymath} \scalar{\Dl X}{f_1\tensor f_2}=\scalar{X}{f_1f_2} \end{displaymath}$

It gives you a nondegenerate pairing of Hopf-*-algebras. In general it is a map

$\begin{displaymath} \scalar{-}{-}\: A\tensor B \to \bC \end{displaymath}$

such that
$\begin{align*} \scalar{a}{b}=0\;\forall\;a\in A &\implies b=0\\ \scalar{a}{b}=0... ...&= \scalar{a}{S(b)} \\ \scalar{a^*}{b} &= \Bar{\scalar{a}{S(b)^*}} \end{align*}$
So you have a pair of Hopf-*-algebras in nondegenerate duality. More structure when $(G, \Pi)$ is a Poisson-Lie group,

is a Poisson algebra such that multiplication $m\:G\times G\to G$ satisfies

$\begin{displaymath} \poiss{f_1\circ m}{f_2\circ m}_{G\times G}=\poiss{f_1}{f_2}_G\circ m \end{displaymath}$

$\begin{defn} \textbf{Poisson Hopf algebra} is defned by condition \begin{display... ...Dl f_1}{\Dl f_2}_{G\times G} = \Dl\poiss{f_1}{f_2}_G \end{displaymath}\end{defn}$
From our point of view it will be better to start with the infinitesimal description, i.e. universal enveloping algebra level. Let us first see what happens at the universal enveloping algebra of a Lie bialgebra.
$\begin{defn} A \textbf{coPoisson Hopf algebra} is a pair $(U, \Hat{\dl})$, where... ...nd the dual map $\dl^*\:U^*\tensor U^*\to U^*$\ is a Poisson bracket. \end{defn}$

$\begin{prop} Let $(\gerg, \dl)$\ be a Lie bialgebra and $U=U(\gerg)$\ its univer... ...ticular $U(\gerg)$\ has a canonical coPoisson Hopf algebra structure. \end{prop}$

$\begin{proof} The formula \begin{displaymath} \Hat{\dl}(ab)=(\Dl a) \Hat{\dl}(b)... ...aymath}[\Dl, a]=\ad_a \text{ on }\gerg\tensor\gerg. \end{displaymath}\end{proof}$

$\begin{defn} A topologically free Hopf algebra $H$\ over $\bC[[\hbar]]$\ is a \t... .../\hbar H\cong U(\gerg) \end{displaymath}for some Lie algebra $\gerg$. \end{defn}$

$\begin{prop} Let $H$\ be a quantized universal enveloping algebra. Then $\gerg$ ... ...d{displaymath}where $\Bar{X}$\ is any lifting of $X\in\gerg$\ to $H$. \end{prop}$

$\begin{proof} $\Dl \Bar{X}-\Dl^{\mathrm{op}}\Bar{X}\in \hbar H$\ because $U(\ger... ...ty. Cocycle condition follows from $\Dl$\ being an algebra morphism. \end{proof}$
So for us a quantum group will be the following set of data. A pair $F_{\hbar}[G]$ (quantum functions algebra), $U_{\hbar}(\gerg)$ (quantum universal enveloping algebra) of topological Hopf algebras over $\bC[[\hbar]]$ together with a nondegenerate Hopf pairing

$\begin{displaymath} \scalar{-}{-}\: F_{\hbar}[G] \times U_{\hbar}(\gerg) \to \bC[[\hbar]] \end{displaymath}$

The pairing gives you $U_{\hbar}(\gerg)$ as ''dual'' of $F_{\hbar}[G]$ and vice-versa. You can start with one of the two legs and construct the other. On the way you have some choices. Many technical problems containing remarkable details.

This ''pairing'' contains the kind of pairing, i.e. the interpretation of $U(\gerg)$ as differentiable distributions supported at . But it contains something completely different.

Next: Local, global, special quantizations Up: Quantization Previous: Introduction Contents

Pawel Witkowski 2006-06-26