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\centerline {\bf Topological Quantum Groups, Star Products}
\psaut
\centerline {\bf and their relations}
\saut
\centerline {Mosh\'e Flato and Daniel Sternheimer}

\centerline {Physique Math\'ematique, Universit\'e de Bourgogne}

\centerline {F-21004 DIJON Cedex, FRANCE}

\centerline {\sevenrm (flato@satie.u-bourgogne.fr, daste@ccr.jussieu.fr)}
\gsaut
\saut
\centerline {\it Dedicated to our friend Ludwig Faddeev on his 60th birthday.}
\gsaut
\gsaut
\gsaut
\centerline {\bf Abstract.}
\saut
This short summary of recent developments in quantum compact groups and star
products is divided into 2 parts. In the first one we recast star products
in a more abstract form as deformations and review its recent developments.
The second part starts with a rapid presentation of standard quantum group theory
and its problems, then moves to their completion by introduction of suitable
Montel topologies well adapted to duality. Preferred deformations (by star products
and unchanged coproducts) of Hopf algebras of functions on compact groups and
their duals, are of special interest. Connection with the usual models of quantum
groups and the quantum double is then presented.
\vfill\eject
{\bf 0 - INTRODUCTION.}
\psaut
The idea that quantum theories are deformations of classical theories was presumably
in the back of the mind of many scientists, even before the mathematical notion
of deformation was formalized by Gerstenhaber [G] for algebraic structures. We
were even told by witnesses (many of whom contribute to this volume) that Ludwig
Faddeev mentioned that idea in his lectures on quantum mechanics in Leningrad
in the early 70's, around the time when the so-called geometric quantization was
developed.

However in all these approaches people were always considering that in the end
quantum theories have to be formulated in operator language, while an essential
point in our approach ([FS1], [B{\it ea}]) is that quantum theories can be developed
in an autonomous manner on the algebras of classical observables by deforming
the algebraic structures. The connection with operatorial formulation, whenever
possible, comes only afterwards and is optional. This applies both to quantum mechanics and
quantum field theories. Our approach is often referred to as star-products, or
deformation quantization.

Around the beginning of the 80's, when it became rather clear that constructive
quantum field theory (at least in 4 dimensions) was facing tremendous analytical
problems, the school of Faddeev tried a new approach to quantization of field
theories, first with 2-dimensional integrable models. Doing so they discovered
[KR] the beginning of what turned to be [FRT] a mathematical gold mine, to which both
mathematicians and theoretical physicists rushed (and the rush is still in full
speed): quantum groups.

In this short Note we shall present both theories in a context that makes the
relations between both quite natural. This presentation (especially its second
part) relies on a paper [BFGP]) now being published, where necessary details
can be found.
\saut
{\bf 1 - DEFORMATIONS, QUANTIZATIONS AND STAR PRODUCTS.}
\psaut
{\bf 1.1. The framework.} Let $A$ be an algebra. In the following it can be an
associative algebra (vector space with product and unit), a Lie algebra, a
bialgebra (associative algebra with coproduct), a Hopf algebra (bialgebra with
counit and antipode), etc., with the usual compatibility relations between
algebraic laws. For simplicity of notations we shall take the base field to be
$\Crm$ (the complex numbers). It can also be a topological algebra, i.e. any
of the above when the vector space is endowed with a topology such that all
algebraic laws are continuous mappings. We shall specify the kind of algebra
considered whenever needed.

An example of such an algebra is given by the Hopf algebra $\hskip.1cm \Crm [t]$ of complex
polynomials in one variable $t,$ with product $t^n \times t^p = \biggr(\matrix {
n+p \cr
p \cr }\biggl) \hskip.1cm t^{n+p},$ coproduct $\delta (t^n) = \displaystyle{\sum^n_{i=0}}
\hskip.1cm t^i \otimes t^{n-i},$ counit $\varepsilon (t^n) = \delta_{n 0}$
(Kronecker $\delta$) and antipode $S(t^n) = (-1)^n \hskip.1cm t^n.$

Its dual (in a sense we shall make precise in the following) is the bialgebra
of formal series $\Crm [[t]],$ with usual product and coproduct given by $\Delta f
(t,t') = f(t+t') \in \Crm [[t,t']]$ for $f \in \Crm [[t]].$

Now if we extend the base field to the ring $\hskip.1cm \Crm [[t]],$ we get from $A$ the
module $\tilde{A} =
A[[t]]$ of formal series in $t$ with coefficients in $A,$ on which
we can consider algebra structures.
\psaut
{\bf 1.2. Definition.} \it A deformation of an algebra $A$ is a (topologically
free in the case of topological algebras) $\hskip.1cm \Crm [[t]]$ algebra $\tilde{A}$ such
that the quotient of $\tilde{A}$ by the ideal $t \tilde{A}$ generated by $t$ is
isomorphic to $A.$ \rm

For an associative algebra this means that on $\tilde{A}$ there is a new product,
denoted by $*$, such that for $a,b \in A$ :
$$a * b = \sum^\infty_{r=0} \hskip.1cm t^r \hskip.1cm C_r (a,b) \eqno{(1)}$$
where $C_0 (a,b) = ab$ (the product of $A$), and the cochains $C_r \in {\cal L}
(A \hat{\otimes} A, A),$ the space of linear (continuous) maps from the (completed,
for some adequate topology, in the topological case) tensor product $A \otimes A$
into $A.$ The associativity condition for $*$ gives as usual [G] conditions on
the cochains $C_r$ (e.g. $C_1$ is a cocycle for the Hochschild cohomology).

For a Lie algebra one has similar relations (with Chevalley cohomology), and for
bialgebras an adequate cohomology can be introduced [B1].

For a bialgebra, denoting by $\otimes_t$ the tensor product of $\hskip.1cm \Crm [[t]]$ modules,
one can identify $\tilde{A} \hat{\otimes}_t \tilde{A}$ with ($A \hat{\otimes} A)
[[t]]$ and therefore the deformed coproduct is defined by
$$\tilde{\Delta} (a) = \sum^\infty_{r=0} \hskip.1cm t^r D_r (a),
\hskip.1cm a \in A \eqno{(2)}$$
where $D_i \in {\cal L} (A, A \hat{\otimes} A)$ and $D_0$ is the coproduct $\Delta$
of $A.$

For a Hopf algebra, the deformed (Hopf) algebra has same unit and counit, but
in general not the same antipode.

As in the algebraic theory [G], two deformations are said {\it equivalent} if they are
isomorphic as $\Crm [[t]]$ (topological) algebras, the isomorphism being the
identity in degree $0$ (in $t$). And a deformation $\tilde{A}$ is said {\it trivial}
if it is equivalent to the deformation obtained by base field extensions from
the algebra $A.$
\psaut
{\bf 1.3. Example. Star products.} We take $A = C^\infty (W),$ with $W$ a
symplectic (or Poisson) manifold with $2$-form $\omega$. On $A$ we have a Poisson
bracket $(a,b) \mapsto P(a,b),$ which is a bidifferential operator of order (1,1).
We say that (1) defines a {\it star product} on the associative algebra $A$ (with pointwise
multiplication) if in addition:
$$C_1 (a,b) - C_1 (b,a) = 2P (a,b) \hskip.3cm a,b \in A. \eqno{(3)}$$
We do not assume here that the $C_r$ are bidifferential operators, nor n.c. (null
on constant functions, which implies that the function 1 is a unit for the deformed
algebra as well). If we do, then [B{\it ea}] it is coherent to restrict oneself to the
corresponding Hochschild cohomologies. But in star representations (see below)
one encounters often bipseudodifferential cochains $C_r.$

>From (3) follows that the star product defines a deformation of the Lie algebra
$(A,P)$ by :
$$[a,b]_* \equiv {1 \over 2t} (a*b-b*a) = P(a,b) + \sum^\infty_{r=2} \hskip.1cm
t^{r-1} (C_r (a,b) - C_r (b,a)). \eqno{(4)}$$
This allows (in the differentiable case) to use instead of the infinite-dimensional
Hochschild cohomologies, the finite-dimensional Chevalley cohomology spaces. E.g.
the dimension of Chevalley $2$-cohomology is (in the n.c. case) $1+b_2 (W)$ where $b_2 (W)$ is the second Betti
number of $W$ which permits (as in [B{\it ea}]) to show that at
each level there are only $1+b_2 (W)$ choices.
\psaut
{\bf 1.4. Typical example : Moyal on $\Rrm^{2n}$.} In 1927, H. Weyl [W] gave
 a rule for passing from a classical observable $a \in A = C^\infty (\Rrm^{2l})$
to an operator in $L^2 (\Rrm^l)$ which represents a quantization of this
observable. It can be written
$$A \ni a \mapsto \Omega_W (a) = \int \tilde{a} (\xi, \eta) \hskip.1cm exp
(i(P \xi + Q \eta) / \hbar) \hskip.1cm w (\xi, \eta) \hskip.1cm d \xi \hskip.1cm
d^l \eta \eqno{(5)}$$
where $\tilde{a}$ is the inverse Fourier transform of $a,$ $P$ and $Q$ satisfy
the canonical commutation relations $[P_\alpha, Q_\beta] = i \hbar \hskip.1cm
\delta_{\alpha \beta} \hskip.1cm (\alpha, \beta = 1,...,l),$ $w$ is a weight function
($=1$ in the case of Weyl) and the integral is taken in the weak operator topology.
An inverse formula was given a few years later by E. Wigner [Wi], and numerous
variants exist.
Whenever either side is defined, the trace can be given by:
$$Tr (\Omega_1 (a)) = (2 \pi \hbar)^{-l} \hskip.1cm \int_{\Rrm^{2l}} \hskip.1cm a
\hskip.1cm \omega^l \eqno{(6)}$$
In the end of the 40's, starting from a point of view different from ours, Moyal
[M] and Groenewold [Gr] found that the commutator and product (resp.) of quantum
observables correspond, in the Weyl rule, to sine and exponential of the Poisson
bracket (resp.), with the parameter $t = {1 \over 2} \hskip.1cm i \hbar.$ Thus
$\Omega_1 (a) \hskip.1cm \Omega_1 (b) = \Omega_1 (a *_M b)$ where $*_M$ is given
by (1) with (for $r \geq 1$) $r ! C_r (a,b) = P^r (a,b),$ the $r^{th}$ power of the
bidifferential operator $P.$
\psaut
{\bf 1.5. Quantizations.} In 1975, inspired by our earlier works [FLS] on $1$-differentiable
deformations of the Lie algebras $(A,P),$ J. Vey [V] obtained what turned to be
the Moyal bracket as an example of differentiable deformation, and showed its
existence on any symplectic $W$ with $b_3 (W) = 0.$ We then not only made the
connection with quantization but also showed, with examples, that quantization
should in fact be considered as a deformation of a classical theory, with the
same algebra of observables and a star-product [B{\it ea}]. Around the same time and
independently, Berezin [B] had shown that the normal ordering of physicists (weight
$w(\xi, \eta) = exp (- {1 \over 4} (\xi^2 + \eta^2))$ in (5)) can be
defined for
more general manifolds than $\Rrm^{2l}.$ That ordering is the analogue
(for complex
coordinates $\xi \pm i \eta$) of the standard ordering (weight $w(\xi, \eta) =
exp ( - {1 \over 2} i \xi \eta))$ which mathematicians are using in pseudodifferential
operator theory, and is preferred for field theory quantization.

In our approach, we have an autonomous definition of the spectrum of an observable.
To that effect we consider the star exponential (the analogue of the evolution
operator)
$$Exp (sa) = \sum^\infty_{n=0} \hskip.1cm {1 \over n!} \hskip.1cm s^n (i \hbar)^{-n}
\hskip.1cm (a*)^n \eqno{(7)}$$
(the sums involved being taken in the distribution sense) and define the spectrum
of the observable $a$ to be that (in the sense of L. Schwartz) of the star exponential
distribution, i.e. the support of its Fourier-Stieltjes transform (in $s$). For
the harmonic oscillator for instance, one gets $(n + {1 \over 2} l) \hbar$ with
Moyal ordering $(n \in \Nrm)$ and $n \hbar$ with normal ordering (which explains
why it is favored when $l \fl \infty$). But many other examples can be treated,
e.g. the hydrogen atom with $W = T^\forall \hskip.1cm S^3$ for manifold.

Star products can also be defined when dim $W = \infty,$ and there one
can e.g. find some cohomological cancellations of infinities [Di] by taking
orderings "in the neighbourhood of normal ordering" : this amounts to substracting
an infinite coboundary from an infinite cocycle to get a finite ("renormalized")
cocycle.
\psaut
{\bf 1.6. Closed star products.} Whenever there is a (generalized) Weyl mapping
between $A = C^\infty (W)$ (plus possibly some distributions, or part of it
only) and operators on a Hilbert space (typically a space of square integrable
functions in "half" of the variables, via some polarization), some of these operators
will have a trace. Therefore it is natural to ask whether a functional with the
properties of a trace can be defined on the algebra $(A, *).$

For Moyal ordering one has (6). For other orderings on $\Rrm^{2l}$ that formula
is valid modulo higher powers of $\hbar.$ Therefore [CFS] a natural requirement
is to look at the coefficient of $\hbar^l$ in $a*b,$ where $a,b \in A[[\hbar]],$
and require that its integral over $W$ is the same as that of $b*a.$ Or equivalently :
$$\int_W \hskip.1cm C_r (a,b) \hskip.1cm \omega^l = \int_W \hskip.1cm C_r (b,a)
\hskip.1cm \omega^l \eqno{(8)}$$
whenever defined for $a,b \in A$ and $1 \leq r \leq l.$ A star-product (1) satisfying
(8) is called {\it closed}. If (8) is true for all $r$ we call it {\it strongly
closed}. Note that, in view of (3), (8) is always true for $r=1$ - so that all
star products on $2$-dimensional manifolds are closed. It has been pointed out
to us by Pierre Lecomte that, in view of (4) and Prop. 2.1. (iii) of [DW], all
differentiable n.c. star products are closed. (There exist however non closed
star products, that are not e.g. null on constants).

An interesting feature of closed star products [CFS] is that they are classified
by {\it cyclic} cohomology [C], instead of only Hochschild cohomology. This suggests
to define, in parallel to the similar notion for operator algebras [C], the {\it character}
of a closed star-product as a cocycle $\varphi$ in the cyclic cohomology bicomplex
with components (non zero only for $l \leq 2k \leq 2l$) :
$$\varphi_{2k} (a_0, a_1,..., a_{2k}) = \int_W \hskip.1cm a_0 * \tau (a_1,a_2)
*...* \tau (a_{2k-1}, a_{2k}) \hskip.1cm \omega^l \eqno{(9)}$$
where $\tau (a,b) = a*b - ab$ measures the noncommutativity of the star product.
It can be shown [CFS] that for $W = T^* M, M$ compact Riemannian manifold, and
for the star product of standard ordering (composition of symbols of pseudodifferential
operators), the character coincides with that given by the trace on pseudodifferential
operators. Therefore, using the algebraic index theorem of [CM], it is given by the
Todd class Td$(T^*M)$ as a current over $T^*M.$
\psaut
{\bf 1.7. Existence.} Jacques Vey [V] had obtained the existence of star brackets
for all symplectic manifolds with $b_3 = 0,$ and this was extended ([NV], [L]) to
star products (under the same hypothesis). The underlying idea is to "glue" Moyal
products on Darboux charts, and the condition $b_3 = 0$ is needed to control
multiple intersections of charts. But we knew from the beginning [B{\it ea}] that this
condition is not necessary. Then M. Cahen and S. Gutt showed existence for $W = T^*M,$
$M$ parallelisable, and soon afterwards [LDW1] existence was shown for any $W$
symplectic (or regular Poisson) manifold.

In 1985-86 (in obscure form, made more clear only recently) B. Fedosov [F] gave
a geometrical and algorithmic construction of star products on any $W$ by viewing
$A[[t]]$ as a space of flat sections in the bundle of (formal) Weyl algebras on
$W$ (and pulling back the multiplication of sections; a flat connection on that
bundle is algorithmically constructed starting with any symplectic connection on
$W$). The geometric background of Fedosov's construction has been recently explicited
further by several authors ([Gu], [EW]).

Using also Weyl algebras, but here essentially [LDW2] to build compatible local
equivalences that allow to "glue together" Moyal products a Darboux charts, it has
been possible [OMY] to give another and more concrete proof of existence of
star products on any $W,$ and even to do it in a way that proves directly also
existence of closed star products.
\psaut
{\bf 1.8. Star representations.} When $a$ is a generator of a Lie algebra ${\cal G}$
of functions (e.g. on a coadjoint orbit of a Lie group $G$), the star exponential
(7) gives the corresponding one-parameter group. And if the star commutator (4)
coincides, for $a,b \in {\cal G},$ with $P(a,b),$ the Poisson bracket (which is
the Lie bracket in this case), one can (by the Campbell-Hausdorff-Dynkin formula)
generate a realization of $\tilde{G}$ (the connected and simply connected Lie
group with Lie algebra ${\cal G}$) by the star exponentials (7) and their star products.
Such a star product is said {\it covariant}.

It is said {\it invariant} if $[a,b] = P(a,b) \hskip.1cm \forall a \in {\cal G}$
and $b \in A$ (this is the geometric invariance of the star product under the
action of $G$). There do not always exist invariant star products (e.g. for
nilpotent groups of length $>2$), but covariant ones always exist. For covariant star products,
the geometric action of $G$ is modified by a $t$-dependent multiplier.

We call {\it star representation} the distribution on $G$ defined by the star
exponential associated with a covariant star product. Such representations have
been built for all compact and all solvable Lie groups, some series of representations
of semi-simple groups (including some of those with unipotent orbits), and other
examples. The cochains $C_r$ obtained here are in general pseudodifferential.
\saut
{\bf 2 - TOPOLOGICAL QUANTUM GROUPS.}
\psaut
{\bf 2.1. The setting.} Let $G$ be a Poisson-Lie group, i.e. a Lie group with
Poisson structure, such that for the usual coproduct $\Delta$ on the Hopf algebra
$H = C^\infty (G),$ (i.e. $\Delta a(g,g') = a(gg') \hskip.1cm ; \hskip.1cm g,g' \in
G$), the Poisson bracket $P$ (on $G$ or $G \times G$) satisfies
$$\Delta \hskip.1cm P(a,b) = P(\Delta a, \hskip.1cm \Delta b) \hskip.3cm a,b \in
H \eqno{(10)}$$

Equivalently we can consider the Lie bialgebra ${\cal G}$;
the dual ${\cal G}^*$
has a bracket $\varphi^* : {\cal G}^* \wedge {\cal G}^* \fl {\cal G}^*$ such that
its dual $\varphi$ is a $1$-cocycle for the adjoint action. When $\varphi$ is
the coboundary of some $r \in {\cal G} \wedge {\cal G}$ (solution of the classical
Yang-Baxter equation) it is said that the Poisson-Lie group is triangular. In
that case there exists a $G$-invariant differentiable star product on $H,$ and
the associativity condition for that star product gives a solution to the quantum
Yang-Baxter equation : the deformed algebra $H$ is the realization of a quantum
groups [D]. Furthermore [T] there exists a (non-invariant) equivalent star product
$*'$ on $H$ such that (for the same $\Delta$ as above)
$$\Delta (a*'b) = \Delta a *' \Delta b \eqno{(11)}$$
and the same for the commutator, which is clearly a quantization of (10).

In the "dual" approach of Jimbo [J], one deforms $\Delta$ to some $\Delta_t$ on
some completion ${\cal U}_t ({\cal G})$ of the enveloping algebra ${\cal U} ({\cal G}).$
It is this deformation  that was first discovered [KR], for ${\cal G} = sl(2)$ :
the commutation relations which define ${\cal U}_t$ have a deformed form (one of
them becoming a sine instead of a linear function).

In line with our philosophy, it is thus natural to ask whether the deformed
algebra ${\cal U}_t$ can be realized (instead of an operatorial realization)
by classical functions and some star product giving the deformed commutators. It turns
out that this is possible [FS], with a star-product using a new parameter $\hbar$
unrelated to $t.$ In fact, since there is some duality between $H$ and ${\cal U}_t$
(we shall make this more precise later), the two parameters $t$ and $\hbar$ are
in a way dual one to the other : the deformed algebra $H[[t]]$ (with star product)
gives a deformed coproduct on ${\cal U}_t$ that induces deformed commutation
relations expressible with another star product (with a new parameter $\hbar$).
Moreover the latter expression is essentially unique [FS2] due to a strong
invariance property that essentially characterizes the star-sine for the Moyal
star product. These star realizations (with $\hbar$) can be given ([Lu], [FLuS])
for various series of classical Lie algebras.

We have just seen that duality plays an important r\^ole in the Hopf algebraic
formulation of quantum groups. But there is a fundamental difficulty, that until
recently was quietly avoided : the algebraic dual of an infinite-dimensional Hopf
algebra $A$ is not Hopf and the bidual is strictly larger than $A.$ So (unless
$G$ is a finite group !) one has to be extremely careful in dualizing - or topologize
in a suitable fashion.
\psaut
{\bf 2.2. Topological quantum groups : the classical case [BFGP].}
\psaut
{\bf a. Definition.} A topological algebra (resp. bialgebra, Hopf algebra) $A$
is said {\it well behaved} if the underlying (complete) topological vector
space is nuclear and either Fr\'echet (F) or dual of Fr\'echet (DF) [Tr].

The topological dual $A^*$ is then also well-behaved, and the bidual $A^{**} = A.$
This is the case when $A$ has countable dimension, with the strict inductive
limit of finite-dimensional subspaces as topology. For example, $A = \Crm [t]$ (the
polynomials) is well-behaved, and so is $A^* = \Crm [[t]].$

{\bf b. The models.} Let $G$ be a compact connected Lie group. Then $H(G) = C^\infty (G)$
and its dual $A(G) = {\cal D}' (G)$ (the distributions) are well-behaved topological
Hopf algebras.

Now $G$ can be imbedded in ${\cal D}' (G)$ as Dirac distributions at points of
$G,$ and its linear span is dense in ${\cal D}' (G).$ The product on ${\cal D}' (G)$
is the convolution of (compactly supported) distributions, and the coproduct is
defined by $\Delta (x) = x \otimes x$ for $x \in G$ (considered as a Dirac distribution).

We know that the enveloping algebra ${\cal U} ({\cal G})$
can be identified with differential operators on $G,$ i.e. all distributions with
support at the identity. Its "completion" ${\cal U}_t$ will involve some entire
functions of Lie algebra generators, i.e. an infinite sum Dirac $\delta$'s and
derivatives, and thus take us outside ${\cal D}'.$ In order to include this model
as well one will therefore have to restrict oneself to a subalgebra of $H.$ The
natural choice is the space ${\cal H} (G)$ of $G$-finite vectors of the regular
representation, which is generated by the coefficients (matrix elements) of the
irreducible (unitary) representations. Thus ${\cal H} (G) = \displaystyle{\sum_{\rho \in \hat{G}}}
\hskip.1cm {\cal L} (V_\rho),$ where $V_\rho$ is the space on which the representation
$\rho \in \hat{G}$ is realized. Its dual is then
$${\cal H}^* (G) = {\cal A} (G) = \prod_{\rho \in \hat{G}} \hskip.1cm {\cal L}
(V_\rho) \supset {\cal D}' (G). \eqno{(11)}$$
The imbedding ${\cal U} ({\cal G}) \ni u \mapsto i(u) = (\rho (u)) \in {\cal A}
(G)$ has a dense image for the topology of ${\cal A}$ (the image is of course
in ${\cal D}' (G),$ but is {\it not} dense for the ${\cal D}'$ topology).
\psaut
{\bf 2.3. Topological quantum groups : the deformations.}
\psaut
We shall restrict ourselves here to a summary of the main notions and results
of the theory in the framework explained before, referring to [BFGP] and references
quoted therein for more details.

Duality and deformations work very well together in our setting. More precisely :
\psaut
{\bf Proposition 1.} \it Let $\tilde{A}$ be a bialgebra (resp. Hopf) deformation of a
well-behaved topological bialgebra (resp. Hopf algebra) $A.$ Then the $\hskip.1cm \Crm [[t]]$
dual $\tilde{A}^*_t$ is a deformation of the topological Hopf algebra $A^*.$  Two
deformations $\tilde{A}$ and $\tilde{A}'$ of $A$ are equivalent iff $\tilde{A}^*_t$
and $\tilde{A}'^*_t$ are equivalent deformations of $A^*.$ \rm

The known models of quantum groups lead us to select a special type of deformations:

{\bf Definition} (see also [GS]). \it A deformation of the bialgebra ${\cal H} (G)$
(resp. $C^\infty (G)$) with unchanged coproduct is called a preferred deformation. \rm

This definition is motivated by the following :

{\bf Proposition 2.} \it Let $({\cal H} [[t]], * , \tilde{\delta})$ be a
coassociative deformation of the bialgebra ${\cal H}.$ Then, up to equivalence,
one can assume that $\tilde{\delta} = \delta$ (the coproduct in ${\cal H}$) ;
the product is quasi-commutative and quasi-associative, the counit unchanged, and
if the product is associative then ${\cal H} [[t]]$ is a $\Crm [[t]]$ Hopf algebra
with same unit and counit as ${\cal H}.$ The same holds for $H.$ \rm

(By quasi-associativity, etc., we means as usual that the associativity, etc.,
condition is satisfied up to a factor). That result is proved by using duality
from the following results for the duals ${\cal A} (G) = {\cal H} (G)^*$ and
$A (G) = H (G)^* = {\cal D}' (G)$ :

{\bf Theorem 1.} \it Let $A$ be either ${\cal A} (G)$ or $A(G).$ Then any
associative algebra deformation of $A$ is trivial, and $A$ is rigid
in the category of bialgebras; any associative bialgebra
deformation of $A$ is quasi-cocommutative and quasi-coassociative. \rm

   More specifically $H^n (A,A) = 0 \hskip.2cm \forall n \geq 1 $ and
$ H^1 (A, A \hat{\otimes} A) = 0 $ (for algebraic and continuous Hochschild
cohomologies), which shows the rigidity of $A$ as bialgebra in the
sense of [B1].
Moreover, if $(A [[t]], \tilde{\Delta})$ is an associative bialgebra
deformation of $A$ with unchanged product, then there exists $\tilde{P} \in
(A \hat{\otimes} A) [[t]]$ such that $\tilde{\Delta} = \tilde{P} \hskip.1cm \Delta_0
\hskip.1cm P^{-1}$ (where $\Delta_0$ is the coproduct in $A$), the counit is
unchanged, and there exists an antipode $\tilde{S}$ for $A[[t]]$ that is given
by $\tilde{S} = \tilde{a} \hskip.1cm S_0 \hskip.1cm \tilde{a}^{-1}$ where $S_0$ is
the antipode of $A$ and $\tilde{a}$ is some element in $A[[t]].$
Our topological notion of duality also gives us, automatically, that
the deformed product $*$ on the topological dual $H$ (either ${\cal H} (G)$
or $C^\infty (G)$) of $A$ is a {\it star product} (starting with the Poisson
bracket) in the sense of part 1, for all $G$ compact.

In addition, the restriction of a Hopf deformation of $H(G)$ defines a Hopf
deformation of ${\cal H} (G).$ If $\Gamma$ is a normal subgroup of $G,$ any
preferred deformation of ${\cal H} (G)$ gives a preferred deformation of ${\cal H}
(G/\Gamma)$ (and the same with $H(G)$): we can define {\it quotient deformations},
a useful notion e.g. to pass from $SU(2)$ to $SO(3)$, etc.
\psaut
{\bf 2.4. Topological quantum groups : the models.}
\psaut
We shall now explain how the known models of quantum groups relate to the general
framework presented in the previous section.

{\bf a. Generators of ${\cal H} (G).$} The algebra ${\cal H} (G),$ $G$ compact,
is a finitely generated domain. We say that a set $\{ \pi_1,...,\pi_2 \} \subset \hat{G}$
of irreducible representations (irrep.) is {\it complete} if its coefficients
generate ${\cal H} (G).$ For $SU(n), SO(n)$ and $Sp(n),$ the standard representation
is in itself a complete set. For $Spin (n),$ we take the irreducible spin representation(s)
(one for $n$ odd, $2$ for $n$ even).
For $E_6$ (resp. $E_7$) there exist(s) two (resp. 1) irrep. that form a complete
set.
For all other exceptional (simply connected compact) groups, any irrep. is a
complete set.

Define $\pi_0 = \displaystyle{\oplus^r_{i=1}} \hskip.1cm \pi_i,$
and call $\{ C_{ij} \}$ the coefficients of $\pi_0$ in a given fixed basis : they
form a topological generator system for the preferred Hopf deformation $({\cal H} [[t]],
  * )$ of ${\cal H}.$ The quasi-commutativity of that deformation can then
be expressed as follows: if $T$ is the matrix $[C_{ij}], T_1 = T \otimes Id,
\hskip.1cm T_2 = Id \otimes T,$ there exists an invertible $R$ in ${\cal L}$
($V_{\pi_0} \otimes V_{\pi_0}) [[t]]$ such that
$R \hskip.1cm (T_1 \hskip.1cm * T_2) =
 (T_1 \hskip.1cm * \hskip.1cm T_2) \hskip.1cm R.$

{\bf b. The Drinfeld models} [D1]. Let ${\cal U} = {\cal U} ({\cal G})$ be the
enveloping algebra. Drinfeld has shown [D2] that it is rigid (as algebra), and
there exists a Hopf deformation ${\cal U}_t$ of ${\cal U}$ (endowed with its natural
topology) that is a topologically free complete $\Crm [[t]]$-module: there is
an isomorphism $\tilde{\varphi}: {\cal U}_t \simeq {\cal U} [[t]]$ as $\Crm [[t]]$-modules,
and also as algebras; we call such a $\tilde{\varphi}$ a {\it Drinfeld isomorphism}.
The coproduct $\tilde{\Delta}$ of ${\cal U}_t$ is obtained from the original
coproduct by a twist : $\tilde{\Delta} = \tilde{P} \hskip.1cm \Delta_0 \hskip.1cm
P^{-1}$ for some $\tilde{P} \in {\cal U}_t \hat{\otimes}_t \hskip.1cm {\cal U}_t.$

Using the fact that ${\cal U} ({\cal G}) \subset A (G) \subset {\cal A} (G)$
we can extend the Hopf deformation ${\cal U}_t$ to a Hopf deformation of $A(G)$
or ${\cal A} (G)$ with unchanged product, unit and counit. By $\Crm [[t]]$
duality this gives a preferred deformation of $H(G)$ or ${\cal H} (G)$ (resp.).

All this construction depends on the choice of a Drinfeld isomorphism $\tilde{\varphi},$
but in an inessential way : two Drinfeld isomorphisms $\tilde{\varphi}$ and
$\tilde{\psi}$ give equivalent  preferred deformations of ${\cal H} (G).$
Note that the above $ R$-matrix can be specified to be a solution of the Yang-Baxter
equation.

{\bf c. The Faddeev-Reshetikhin-Takhtajan models.} These [FRT] models are recovered
by a good choice of the Drinfeld isomorphism : if $\tilde{\rho}$ is a representation
of ${\cal U}_t$ and $\pi = \rho_0 \in \hat{G}$ is its classical limit, then there
is a Drinfeld isomorphism $\tilde{\varphi}$ such that $\tilde{\rho} = \pi \circ
\tilde{\varphi}.$

When we apply this to $G = SU(n), SO(n)$ or $Sp(n)$ we recover the [FRT] quantizations
of these groups as preferred Hopf deformations of ${\cal H} (G)$ that extend
to preferred Hopf deformations of $C^\infty (G).$

{\bf d. The Jimbo models} [J]. These models are somewhat special, because we get here
nontrivial deformations. We shall explain it here for the case ${\cal G} = sl(2).$
The general case is similar, the main difference being that there ${\cal U} ({\cal G})$
is extended by Rank$({\cal G})$ parities.

Consider the quantum algebra $A_t$ generated by 4 generators $\{ F,F',S,C \}$ with
relations :
$$[F,F'] = 2 SC, \hskip.1cm FS = (S \hskip.1cm {\rm cost} - C) \hskip.1cm F, \hskip.1cm
FC = (C \hskip.1cm {\rm cost} + S \hskip.1cm {\rm sin}^2 t) \hskip.1cm F \eqno{(12a)}$$
$$F'S = (S {\rm cost} + C) \hskip.1cm F', \hskip.1cm F'C = ({\rm cost} - S \hskip.1cm {\rm sin}^2 t)
F', \hskip.1cm C^2 + S^2 {\rm sin}^2 t = 1, \hskip.1cm [S,C] = 0. \eqno{(12b)}$$
The more familiar form is obtained by setting $q = e^{it}$ $(t \notin 2 \pi {\bf Q})$
and $S = {K-K^{-1} \over q-q^{-1}}, \hskip.1cm C = {1 \over 2} (K+K^{-1})$ for
some new generators $K$ and $K^{-1}.$ But we prefer (12) because it is not singular
at $t=0,$ and we can thus define $\tilde{A}_t$ as the $\Crm [[t]]$ algebra $A_t$
when $t$ is a formal parameter. The usual commutation rules of $sl(2)$ are obtained
with $SC, FC$ and $F'C$; therefore $A_0 \simeq {\cal U} (sl(2)) \otimes P$ where
$P \simeq \Crm [x] / (x^2 - 1)$ is generated by a parity $C$ $(C^2=1$ when $t=0).$

The formal algebra $\tilde{A}_t$ is thus a deformation of $A_0.$ But it is a
domain, while $A_0$ is not and therefore the $\hskip.1cm \Crm [[t]]$ algebras $\tilde{A}_t$
and $A_0 [[t]]$ cannot be isomorphic: the deformation is {\it nontrivial}.

Similarly $A_t$ and $A_0$ cannot be isomorphic for $t \notin 2 \pi {\bf Q}.$
Furthermore, $\tilde{A}_{t_0 + t}$ is a non trivial deformation of $A_{t_0}$ because
the Casimir element $Q_t = F'F+SC+S^2 {\rm cos}t$ takes different values
in $A_{t_0}$ and $A_{t_0+t}$ : in the $(2k+1)$-dimensional representation its value
is sin$(k t )$sin $(k+1) t / {\rm sin}^2 t.$ Therefore, in contradistinction
with the other models, the Jimbo models are not rigid.

{\bf e. Topological quantum double.} Now that we have good models with a nice
duality between them, it is possible to have a good formulation of the quantum
double. To this effect we shall consider ${\cal H}_t (G) \bar{\otimes} {\cal A}_t (G)$
(with inductive tensor product topology) ; its dual is ${\cal A}_t (G) \hat{\otimes}
{\cal H}_t (G)$ (with the projective tensor product topology). Similarly we can
consider $C^\infty_t (G) \bar{\otimes} {\cal D}'_t (G).$ The following is true
[B2].

{\bf Theorem 2.} \it Let $A$ denote ${\cal A} (G)$ or ${\cal D}' (G)$ or their
deformed versions, and $H$ denote ${\cal H} (G)$ or $C^\infty (G)$ or their
deformed versions. Then the double is $D(A) = A^* \bar{\otimes} A = H \bar{\otimes} A,$
and its dual is $D(A)^* = A \hat{\otimes} A^* = A \hat{\otimes} H.$ We have
$D(A)^{**} = D(A),$ and these algebras are rigid. \rm
\saut
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\end