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Local, global, special quantizations

The discussion in the preceeding section was about local quantization. Their main advantage is that they are well suited to capture relations between the classical, semiclassical, and quantum properties (we will see some examples of these relations in more details later). However they miss part of the relevant information, or at least of the full geometry. For example local quantization does not allow to specialize the deformation parameter to complex values $\neq 0$ . Being $(\hbar)$ the only maximal ideal in the local ring $\bC[[\hbar]]$ they can describe only the limit $\hbar\to a$ . But we know of some relevant parts of the theory of quantum groups staying out of this range. This is the case, for example, of the theory of quantum groups at roots of unity, which links quantum groups to 3-manifold invariants and Lie algebras in characteristic

Let us denote with $\bC(q)$ the field of rational functions in the variable .
$\begin{defn} Let $A_q$\ be a $\bC(q)$-Hopf algebra. An \textbf{integer form} (re... ...=A_q \end{displaymath}(resp. $\sA\tensor_{\bQ[q, q^{-1}]}\bC(q)=A_q$) \end{defn}$

$\begin{defn} Given a $\bC(q)$-Hopf algebra $A_q$\ together with an integer form ... ... is taken with respect to $\vf\:\bZ[q, q^{-1}]\to \bC$, $\vf(q)=\la$. \end{defn}$
In this way starting from a $\bC(q)$ -Hopf algebra we obtain a $\bC$ -Hopf algebra.
$\begin{example} Let $\gerg$\ be a finitely dimensional complex simple Lie algebr... ...s\cdot (1-q^n) \end{displaymath}are the $q$-binomial coefficients. \end{example}$

$\begin{remark} \mbox{} \begin{itemize} \item If we have a relation $xy=qyx$, the... ...in between different groups with the same Lie algebra. \end{itemize}\end{remark}$

$\begin{defn} Let $F_q(\GL_n(\bC))$\ be the $\bC(q)$-algebra generated by $t_{ij}... ...)}t_{j\sg(1)}\hdots t_{i_{n-1}\sg(n-1)}t_{j\sg(n-1)} \end{displaymath}\end{defn}$
Here apparently there is no need to use the machinery of $\bC(q)$ -algebras and integer forms to specialize the parameter to complex values. This is why often in this context one does not mention integer forms. Still they are relevant in the duality between and $U_q(\gerg)$ .
$\begin{defn} Let $G$\ be an affine algebraic complex Poisson group. A \textbf{g... ... integer form $\sA$\ such that $A_{q=1}\cong F[G]$\ as Hopf algebras. \end{defn}$
Another good aspect of global quantization is that it provides you with genuine (non topological) Hopf algebras.

Next: Real structures Up: Quantization Previous: Duality Contents

Pawel Witkowski 2006-06-26