Introduction

The purpose will be here to give a definition of quantization and estabilish a vocabulary given us the link between two languages: Poisson geometry and noncommutative algebras. Something like

classical	semiclassical	quantum
manifold	Poisson manifold	noncommutative algebra
group	Poisson Lie group	noncommutative Hopf algebra
point	0-leaf	character

Of course to state all this correctly we need to be very precise on the setting in which we will work. Apart from some preliminaries we will content ourselves to deal with the group case where, for a number of reasons and still with a high degree of attention on details, such dictionary behaves particularly well (i.e. is a functor).

Let us start with a general definition of quantization. On the formal level that will first require from us some definitions. We will work over the field $k=\bC$. Basically all what follows work on any field of characteristic 0 and a not so trivial part still holds in characteristic $p$.

Let us denote with $\bC[[\hbar]]$ the ring of formal power series in an indeterminant $\hbar$ with coefficients in $\bC$. The algebraic structure here is obvious:

$$\sum_{n\geq 0}a_n\hbar^n + \sum_{n\geq 0}b_n\hbar^n = \sum_{n\geq 0}\sum_{n\geq 0}(a_n+b_n)\hbar^n$$

$$\left(\sum_{n\geq 0}a_n\hbar^n \right)\cdot \left(\sum_{n\ge... ...\right) =\sum_{n\geq 0}\left(\sum_{p+q=n}a_pb_q\right)\hbar^n$$

This is a ring with unit $1$. Invertible elements are exactly those power series with $a_0\neq 0$ (check this as an exercise).

Let now $M$ be a $\bC[[\hbar]]$-module. For every $x\in M$ define

$$\kappa(x):=\max \{k : x\in \hbar^kM\}$$

Define for every $x, y\in M$

$$d(x, y):= 2^{-k(x-y)}$$

This metric induces a topology on $M$ which is called the $\hbar$-adic topology. A $\bC[[\hbar]]$-module is called torsion free if the multiplication by $\hbar$ is an injective map.



A module $M$ over $\bC[[\hbar]]$ of this form is called a topologically free module.

Take $A$ to be a topologically free $\bC[[\hbar]]$-algebra (completed tensor product). Then being $\hbar A$ an ideal $A/\hbar A$ is an algebra over $\bC$.





In fact, when you have a Lie group, then you have two algebraic objects to describe with: $F[G]$ and $U(\gerg)$. What is their relation ?

$U(\gerg)$ is a Hopf algebra (cocommutative). The ''right'' choice of $F[G]$ is a Hopf algebra:

• $G$ affine algebraic group and $\bC[G]$ algebra of regular functions (sheaf of Hopf algebras wnen you do not have affine)
• $K$ compact group and $R[K]$ algebra of representative functions (matrix elements of irreducible representations)
• $G$ Lie group and $\bC_f[G]$ algebra of formal functions

If you consider everything as real objects you have a Hopf-*-algebras. $(H, m, \Dl, \eps, S)$ is a Hopf-*-algebra if $*\: A\to A$ is an involution, i.e.
$$\begin{align*} (ab)^* &= b^*a^*\\ (\la a)^* & =\bar{\la}a^*\\ \text{and}\\ \Dl(a^*) &= (\Dl a)^* \\ (a\tensor b)^* &= a^*\tensor b^* \end{align*}$$
(this implies $(*\circ S)^2=\id$). Then $U(\gerg)$ and $F[G]$ can be seen as Hopf-*-algebras.