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%\headline={ifodd\pageno\rightheadline \else\leftheadline\fi}
%\def\rightheadline{\ninerm\hfil MOSHE FLATO and DANIEL STERNHEIMER \hfil}
%\def\leftheadline{\ninerm\hfil STAR PRODUCTS, QUANTUM GROUPS... \hfil}
\font\ninerm=cmr9
\advance\vsize-14pt % -24pt for \makeheadline, + 10pt for \topskip
\font\bigtenrm=cmbx10 scaled \magstep2
\bigtenrm
\centerline {Star Products, Quantum Groups, Cyclic}
\saut
\centerline {Cohomology and Pseudodifferential Calculus.}
\tenrm
\gsaut
\saut
\centerline {MOSHE FLATO and DANIEL STERNHEIMER}
\gsaut
\gsaut
\gsaut
\font\sevenit=cmti7 \relax
{\sevenrm
ABSTRACT.
We start with a short historical overview of the developments of
deformation
(star) quantization on symplectic manifolds and of its relations with
quantum
groups. Then we briefly review the main points in the
deformation-quantization
approach, including the question of covariance (and related
star-representations)
and describe its relevance for a cohomological interpretation of
renormalization
in quantum field theory. We concentrate on the newly introduced notion
of closed
star product, for which a trace can be defined (by integration over the
manifold)
and is classified by cyclic (instead of Hochschild) cohomology ; this
allows to
define a character (the cohomology class of cocycle in the cyclic
cohomology bicomplex).
In particular we show that the star product of symbols of
pseudodifferential operators
on a compact Riemannian manifold is closed and that its character
coincides with
that given by the trace, thus is given by the Todd class, while in
general not
satisfying the integrality condition.

In the last section we discuss the relations between star products and
quantum
groups, showing in particular that "quantized universal enveloping
algebras" (QUEAs)
can be realized, essentially in a unique way (using a strong
star-invariance
condition), as star product algebras with a different quantization
parameter.
Finally we show (in the $\scriptstyle{sl(2)}$ case) that these QUEAs
are dense in a
model Fr\'echet-Hopf
algebra, stable under bialgebra deformations, containing all of them
(for different
parameter values) and that they have the same product and equivalent
coproducts with the original algebra.}
\gsaut
\gsaut
\gsaut
\noindent\underbar{\hbox{\hglue5cm}}

{\sevenrm 1991  Mathematics Subject Classification. Primary 81R50, 17B37, 81S10, 58G15, 58B30.

Lectures presented at the 10th annual Joint Summer Research Conference
(AMS - IMS - SIAM) on Conformal Field Theory, Topological Field Theory
and Quantum Groups (Mount Holyoke College, South Hadley, MA; June
13-19, 1992), delivered by D. Sternheimer.

In press in "Contemporary Mathematics" series of the AMS
(M. Flato, J.Lepowsky, N.Reshetikhin and P.Sally, eds.). }
\vglue2cm
\vfill\eject
\centerline {\bf 1. Introduction.}
\psaut
{\bf 1.1. Historical background.}
The mathematical theory of deformations is inherent to the development
of modern
physics. However this fact has essentially been recognized a posteriori
[1]. For
example Newtonian mechanics is invariant under the 10-parameter Galilei
group
of transformations of space-time, while relativistic mechanics brings in
its
deformation, the Poincar\'e group, obtained by attributing a non-zero
value to the
inverse of the velocity of light. If in addition one gives a (non-zero)
constant
curvature to space-time one gets the De Sitter groups and (since they
are simple)
the buck stops there - as far as Lie groups are concerned. Considering
the Lie
bialgebra structure permits, as we shall see below, to get "quantum
groups" by
deforming the coproduct - but then the "transformation group" point of
view is
lost, or at least needs to be seriously revised (which wasn't done yet).
Had the
theory of deformations of Lie groups been developed a century ago, one
could have
already then found  mathematically relativistic theories, before any
experimental fact.

Quantum mechanics can in a similar way be considered as a deformation of
classical
mechanics. However this fact was made explicit and rigorous only 15 years ago [2],
while again it might have been developed mathematically a century ago
(had the
mathematical tools been developed then). In 1927, H. Weyl [3] gave an
integral
formula (a Fourier transform with operatorial kernel) for a passage from
classical
to quantum observables, and E. Wigner gave in 1931 an inverse formula [4]. At the
end of the 40's, Moyal and Groenewold [5] used the latter to write the
symbols of
a bracket and a product (respectively) of quantum observables, in a form
from where
the structure of deformation could have been read off, had the theory of
deformation of algebras been developed and had they looked for its
physical interpretation
(instead of trying to interpret somehow the symbols as probability
densities, which
cannot be done). Many other works followed along similar lines including
some that
relied implicitely on the idea of deformation (e.g. the so-called
semi-classical
approximations), but none "hit the nail on the head" until the mid 70's.

Around 1970, coming from the representation theory of Lie groups, a
theory called
"geometric quantization" became quite popular [6]. The idea is to do for
more
general Lie groups (than the Heisenberg groups) and more general
symplectic manifolds
(than the flat ones) what the Weyl transform does, map functions on the
symplectic
manifold to operators on a Hilbert space of functions on "half" of the
variables.
While very interesting for representation theory, this approach proved
not very
effective physically; in particular polarizations (which in many cases
could be
found only in the complex domain) had to be introduced to get "half" of
the variables,
and relatively few classical observables could be thus quantized.

{\bf 1.2. Star-products and quantum groups.}
Our approach [2], sometimes called "deformation quantization", was
different - though
starting from similar premises. We did not look for polarizations or
operators,
but had the quantum theory "built in" the algebra of classical
observables by deforming
the composition laws, product and Poisson bracket of functions, to what
we called
star-products and the associated commutators. We could develop in an
autonomous
manner, even on general symplectic manifolds, a quantum theory that can
be mapped
into the usual one by a Weyl mapping (when there exists one). And this
has naturally
lead to a parallel development of representation theory of Lie groups,
without
operators (cf. a partial summary in [7]). Furthermore the same approach
can be developed for infinite-dimensional phase-spaces, giving rise to a
star-product approach to quantum field theories and in particular to a
cohomological interpretation of cancellation of infinities [8].

Around 1980, a new mathematical notion appeared in the quantization of
2-dimensional integrable models [9], and has gained tremendous popularity after Drinfeld [10]
coined the (somewhat misleading but very effective) term "quantum group" to qualify
it. The basic point is that, on a Lie group with a compatible Poisson
structure,
the functions have a natural Hopf algebra structure that can be deformed
by replacing
the usual product by a star-product; in the dual approach of Jimbo [11]
one
deforms the coproduct on a completion of the enveloping algebra of the
Lie algebra
(which has a bialgebra structure) and gets in particular the strange
commutation
relations obtained in [9]. A realization of these structures was given by Woronowicz [12]
using the basic representations of compact Lie groups while taking
coefficients in
$C^*$-algebras (with some relations); this was inspired by the
"non-commutative
geometry" of A. Connes [13].

{\bf 1.3. Main new points.}
In this paper we shall first briefly review the main points of the
star-product
theory, including the new developments of star-quantization in field
theory. Then
we shall develop the new notion of closed star-product [14], for which a
trace can
be defined by integration over phase-space, show its relation to cyclic
cohomology
and apply it to the pseudodifferential calculus on a compact Riemannian
manifold :
for the corresponding star-product a character can be defined (in the
cyclic cohomology)
that is given by the Todd class of the manifold (via the index theorem).

Then we shall concentrate on the relations between star-products and
quantum groups.
In particular we shall show, by the example of $sl(2),$ that "quantum
groups" are
neither quantum nor groups, but examples of star-products requiring
another star-product
in the background (with another parameter $\hbar$) in order to realize
the quantized
universal enveloping algebras (QUEAs) as star-product algebras and that
this realization is then essentially unique. Finally, we shall indicate
that
there exists a universal Fr\'echet-Hopf algebra (containing densely the
QUEAs), rigid as bialgebra, the dual of which contains the original
(simple compact) group as a "hidden group".
\saut
{\bf 2. Star-products, star quantization and closed star-products.}
\psaut
{\bf 2.1. The framework.}
{\it a. Phase-space.} The phase-space is a Poisson ma-

\noindent
nifold, i.e. a manifold where
a Poisson bracket can be defined. In the case of a finite-dimensional
manifold $W$ the
Poisson structure is given by a contravariant skew-symmetric 2-tensor
$\Lambda$
satisfying $[\Lambda, \Lambda] = 0$ in the sense of the supersymmetric
Schouten
brackets; the latter condition is equivalent [2] to the fact that the
Poisson
bracket :
$$P(u,v) = i(\Lambda) (du \wedge dv) \hskip.2cm; \hskip.2cm u,v \in
C^\infty (W) = N \leqno{(1)}$$
satisfies the Jacobi identity. When $\Lambda$ is everywhere non
degenerate, $W$
is a symplectic manifold, of even dimension $2l$ with a closed 2-form
$\omega.$
Poisson structures can however be defined also for infinite-dimensional
(e.g. Banach
or Fr\'echet) manifolds, for instance on the space of initial conditions
of a wave
equation such as Klein-Gordon ([15, 16]).

{\it b. Star-products, deformations of algebras and cohomologies.}
The relation between associative (resp. Lie) algebra deformations and
Hochschild
(resp. Chevalley) cohomology is well known [17], and can be made more
precise (at
each step) both for the existence question (determined by the
3-cohomology of the
algebra valued in itself) and the equivalence question (classified by
the 2-cohomology)
[2]. For the associative (resp. Lie) algebra $N$ it is consistent [2] to
restrict
oneself to differentiable cohomologies and to a formal series of
differential operators for the equivalence.

{\ninerm DEFINITION.}
\it A {\bf star-product} is an associative deformation of $N$ with a
complex parameter
$\nu$ :
$$u * v = \sum^\infty_{r=0} \nu^r \hskip.1cm C_r (u,v) \hskip.2cm;
\hskip.2cm
u,v \in N \leqno{(2)}$$
$$C_0 (u,v) = uv, \hskip.3cm C_1 (u,v) - C_1 (v,u) = 2P (u,v)$$
where the $C_r$ are bidifferential operators. \rm

We thus get a deformation of the Lie algebra $(N,P)$ by the
"star-commuta-

\noindent
tor" :
$${1 \over 2 \nu} (u * v - v * u) \equiv [u,v]_\nu = P(u,v) +
\sum^\infty_{r=1}
\nu^r \hskip.1cm C'_r (u,v) \leqno{(3)}$$
for which the relevant cohomologies are finite-dimensional (the
Chevalley 2-cohomology
has dimension $1 + b_2 (W),$ where $b_2 (W)$ is the second Betti
number of $W).$
Equivalence between two star-products $*$ and $*'$ (resp. brackets)
is given by a
formal series $T = I + \sum^\infty_{s=1} \nu^s \hskip.1cm T_s$ (where
the $T_s$
are necessarily differential operators [2]) such that $T (u *' v) = Tu
* Tv$ (and
similarly for the brackets). All this extends naturally from the algebra
$N$ of functions to ${\cal A} = N [[\nu]],$ the formal series in $\nu$
with coefficients in $N.$

For quantum groups one needs to consider Hopf algebras [10] where the
essential
ingredients are the product and coproduct, that together define a
bialgebra structure
for which a similar theory can be developed [18,19].

{\it c. Example.}
The typical example is $W = \Rrm^{2l}$ with the Moyal star-product
for which one takes
the $r^{th}$ powers of the bidifferential operator $P$ :
$$r ! C_r (u,v) = P^r (u,v) = \Lambda^{i_1j_1}... \Lambda^{i_rj_r}
(\partial_{i_1...i_r}
u) (\partial_{j_1...j_r} v) \leqno{(4)}$$
so that the star-product and bracket can be written :
$$u * v = \hbox{exp}(\nu P) (u,v) \leqno{(5)}$$
$$M(u,v) = \nu^{-1} \hbox{sinh}(\nu P) (u,v) \leqno{(6)}$$
The Weyl maps can be defined, for $u \in N,$ by
$$\Omega_w (u) = \int_{\Rrm^{2l}} {\cal F} (u) (\xi, \eta) w(\xi,
\eta) \hbox{ exp } ((\xi.P + \eta.Q) / 2 \nu) d^l \xi \hskip.1cm d^l
\eta \leqno{(7)}$$
where ${\cal F}$ is the inverse Fourier transform, $(P_\alpha, Q_\alpha,
I)$ are
generators of the Heisenberg Lie algebra ${\cal H}_l$ in the Von Neumann
representation
on $L^2 (\Rrm^l)$ satisfying $[P_\alpha, Q_\beta] = 2 \nu \delta_{\alpha
\beta} I$
$(\alpha, \beta = 1,...,l),$ and (in the Moyal case) the weight function
$w = 1$
and $2 \nu = i \hbar.$ Then $\Omega_1$ will map functions in $C^\infty
(\Rrm^{2l})$ into operators in $L^2 (\Rrm^l),$ star-products
into operator product, and the Moyal bracket $M$ into commutators. The
inverse map can be defined by a trace :
$$u = (2 \pi \hbar)^ {-1} Tr (\Omega_1 (u) e^{(\xi.P + \eta.Q) / i \hbar})
\leqno{(8)}$$
is one-to-one between $L^2 (\Rrm^{2l})$ and Hilbert-Schmidt operators on
$L^2 (\Rrm^l),$
and can be defined on larger spaces of functions (cf.e.g.[20]). One then
has,
for $\Omega_1 (u)$ trace-class :
$$Tr (\Omega_1 (u)) = (2 \pi \hbar)^{-l} \int_{\Rrm^{2l}} u \hskip.1cm
\omega^l
\equiv Tr_M (u) \leqno{(9)}$$
where $\omega = \sum^l_{\alpha = 1} dp_\alpha \wedge dq_\alpha$ is
the usual symplectic form on $\Rrm^{2l},$ so that
$$Tr_M (u * v) = Tr_M (v * u).$$

The latter property can be seen directly since (due to the skew-symmetry
of $\Lambda,$
by integration by parts), for all $r$ and $C_r$ defined by (4) :
$$\int_W C_r (u,v) \omega^l = \int_W C_r (v,u) \omega^l. \leqno{(10)}$$
For other orderings (other weight functions $w$ in (7)), formula (9) is
only
approximate, i.e. valid modulo higher powers of $\hbar.$ This is in
particular
true of the so-called standard ordering (all $q$'s on the left, as in
the usual
way of writing differential operators) for which [21] $w (\xi, \eta) =
exp ({1 \over 2}
i \xi \eta),$ that corresponds to the pseudodifferential calculus; if
we denote
by $\Omega_S$ the corresponding Weyl map, then [14], whenever defined :
$$Tr (\Omega_S (u)) = (2 \pi \hbar)^{-l} \int_{\Rrm^{2l}} u \omega^l +
O (\hbar^{1-l}). \leqno{(11)}$$
To define a trace on ${\cal A}$ with integration on $W$ one is thus lead
to look
at the coefficient of $\hbar^l$ in $u * v$ for $u,v \in {\cal A}.$

This will motivate our definition of closed star products.

{\bf 2.2. Closed star-products and cyclic cohomology.}

{\it a.} {\ninerm DEFINITION.} \it Let $W$ be a symplectic manifold of
 dimension $2l,$ and $*$
be a star-product on $N$ defined by (2). The star-product is said {\bf
closed} if
(10) holds for all $u,v \in N$ and $r \leq l,$ and {\bf strongly closed}
if (10)
holds for all $r.$ \rm

Equivalently, if $u = \sum^\infty_{k=0} \nu^k u_k \in {\cal A}$ and
$v = \sum^\infty_{j=0} \nu^j v_j \in {\cal A},$ then the coefficient
$a_l (u,v)$ of $\nu^l$ in $u * v$ is $\sum_{r+j+k=l} C_r (u_k, v_j)$ and
the condition
for closedness is
$$\int_W \hskip.1cm a_l (u,v) \hskip.1cm \omega^l = \int_W \hskip.1cm a_l (v,u)
\hskip.1cm \omega^l. \leqno{(12)}$$
Note that (10) is always true for $r=0$ and $1$ (thus all star-products
on
2-dimensional symplectic manifolds are closed). If the star-product is
closed, then
the map $\tau$ defined by
$${\cal A} \ni u = \sum^\infty_{k=0} \nu^k \hskip.1cm u_k \mapsto \tau
(u) = \int_W
\hskip.1cm u_l \hskip.1cm \omega^l \leqno{(13)}$$
has the properties of a trace for the algebra ${\cal A}$ with the product $*$ :
$$\tau (u * v) = \tau (v * u)$$
whenever both sides are defined.

{\it b. Cyclic cohomology} [13,14].
If $M$ is a $N$-module, the coboundary operation $b$ in the Hochschild
cohomology
can be defined on a $p$-cochain $C$ by :
$$b C (f_0,...,f_p) = f_0 \hskip.1cm C(f_1,...,f_p) -
C(f_0 f_1,...,f_p) + ...$$
$$+ (-1)^p \hskip.1cm C(f_0,...,f_{p-1} f_p) +
(-1)^{p+1} C(f_0,...,f_{p-1}) f_p. \leqno{(14)}$$
To every $f \in {\cal A}$ one can associate an element $\tilde{f}$
in the dual
${\cal A}^*$ by :
$$\tilde{f} : g \mapsto \int \hskip.1cm fg \hskip.1cm \omega^l.$$
The action of ${\cal A}$ on ${\cal A}^*$ is given by
$(x \varphi y) (a) = \varphi
(y ax)$ when $\varphi \in {\cal A}^*$ and $a,x,y \in {\cal A}.$ The
map $f \mapsto \tilde{f}$ gives then a map of $p$-cochains :
$$C^p ({\cal A}, {\cal A}) \fl C^p ({\cal A}, {\cal A}^*)$$
that is compatible with the coboundary operation : if $\tilde{C}_r$
denotes the
image in $C^p ({\cal A}, {\cal A}^*)$ of $C_r \in C^p ({\cal A},
{\cal A}),$ then
one has $\tilde{b C}_r = b \tilde{C}_r.$ If we define a bicomplex by
$\{ 0 \}$ for $n < m$ and, for $n \geq m$ :
$$C^{n,m} = C^{n-m} ({\cal A}, {\cal A}^*) \leqno{(15)}$$
the Hochschild coboundary $b$ is of degree $1.$ We can, in addition,
define another
operation $B$ of degree $-1$ that anticommutes with $b$ ($bB = -Bb,$
with $B^2 =
0 = b^2$) as $B = A_S B_0$ where $A_S$ is the cyclic antisymmetrization
and, for $\tilde{C} \in C^{n+1} ({\cal A}, {\cal A}^*)$ :
$$B_0 \tilde{C} (f_0,...,f_{n-1}) = \tilde{C} (1,
f_0,..., f_{n-1}) + (-1)^n \hskip.1cm \tilde{C} (f_0,...,f_{n-1}, 1).
\leqno{(16)}$$
Let us denote by $C^n_\lambda \subset C^n({\cal A},{\cal A}^*)$ the space of
cochains $\tilde{C}$ satisfying the cyclicity condition:
$$\tilde{C} (f_1,...,f_n) (f_0) = (-1)^n \hskip.1cm C(f_2,..., f_n, f_0)
(f_1). \leqno{(17)}$$

{\ninerm DEFINITION.} \it The {\bf cyclic cohomology} of ${\cal A},$
 denoted by
$HC^n ({\cal A}),$ is the cohomology of the complex $(C^n_\lambda, b).$
\rm

A fundamental (and non trivial) property of cyclic cohomology is :
\psaut
{\ninerm PROPOSITION.} \it At each level $n,$ one has \rm
$$HC^n ({\cal A}) = (ker b \cap ker B) \hskip.1cm / \hskip.1cm
b (ker B). \leqno{(18)}$$
\psaut
{\it c. Classification of closed star-products.}
For $r = 2,$ the closedness condition (10) can be written
$B \tilde{C}_2 = 0.$
(This condition is necessary; it is sufficient if $dim W = 4$). If we
start with a Hochschild $2$-cocycle $C_1$ (e.g. $C_1 = P$), standard
deformation
theory [2, 14, 17] gives a $3$-cocycle $E_2$ (determined by $C_1$) that
has to be
equal to $b C_2.$ Therefore $b \tilde{E}_2 = 0 = B \tilde{E}_2$ and
$b \tilde{C}_2 =
\tilde{E}_2 \in ker b \cap ker B.$ From (18) we therefore
get, since the same can be done [14] successively at each order of
deformation, and
in one degree less for the equivalence operators $T$ :
\saut
{\ninerm PROPOSITION.} \it At each order, the obstructions to the
 existence of closed
star-products are classified by $HC^3 ({\cal A}),$ and the obstructions
to equivalence
by $HC^2 ({\cal A}).$ \rm
\saut
If $C$ is a non-closed current on $W$ ($d C \neq 0$),
$\tilde{C}^2 (f_1,f_2) (f_0) =
< C, f_0 df_1 \wedge df_2 >$ will give $B \tilde{C}_2 \neq 0$ and
therefore we get in this way a non-closed star-product.

{\bf 2.3. Existence, uniqueness and rigidity of star-products.}
\saut
{\it a.} {\ninerm THEOREM} [22]. \it On any symplectic manifold $W$
 there exists a strongly closed star-product. \rm
\saut
The Moyal product is closed, and can be defined on any canonical chart of $W.$
The problem is to "glue" together these products, and to do it in a way
that preserves
the integrals of functions. The authors of [22] do both things together
by defining
globally on $W$ a locally trivial algebra bundle (called Weyl manifold)
giving a
(globally defined) formal completion of the Heisenberg enveloping
algebra; functions
in ${\cal A}$ are then extended to sections of this bundle (called Weyl
functions)
in an integration-preserving way.

Previous existence proofs of star-products and brackets were first done
by supposing
$b_3 (W) = 0,$ to control the multiple intersections of charts. Then,
after other
generalizations, Lecomte and De Wilde gave a very abstract existence
proof in the
general case, and later simplified a first existence proof by Omori,
Maeda and
Yoshioka by showing that Moyal products on canonical charts can be
transformed
by equivalences so as to coincide on intersections [23] - but these
equivalences
were not constructed in a way that obviously preserved closedness. All
these proofs
extend to regular Poisson manifolds (where the symplectic leaves have all the same
dimension).

{\it b. Uniqueness.} The Hochschild differentiable $p$-cohomology spaces
 $\tilde{H}^p
(N)$ for the associative algebra $N = C^\infty (W)$ are isomorphic [24]
to $\Lambda_p
(W),$ the space of skew-symmetric contravariant $p$-tensors on $W.$ More
precisely
[25], any differentiable $p$-cocycle can be written as such a tensor plus the
coboundary of a differentiable $(p-1)$ cochain. Therefore they are huge;
except for
$dim \hskip.1cm W = 2,$ the obstructions to existence (classified by
$\tilde{H}^3 (N))$ make the previous theorem highly non trivial.

However consideration of the Lie algebra $(N,P)$ reduces the choice. We
shall call
[2] {\bf Vey product} a star-product (2) for which the cochains $C_r$ are bidifferential
operators that vanish on constants (i.e. have no constants terms), have
the same
parity as $r$ (thus $C_1 = P$) and have the same principal part as $P^r$
in any
canonical chart (a star-product satisfying all these conditions except
the last
one can always be brought into the Vey form by equivalence [26]).

We then get associated brackets where only odd cochains $C_{2r+1}$ appear. The relevant
Chevalley cohomology spaces $H^p (N)$ are finite-dimensional. $H^2$ and
$H^3$ can
be explicitely computed [25], and in particular $dim \hskip.1cm H^2 (N) = 1 + b_2 (W).$
Explicit expressions for the cochains up to $C_4$ in a general Vey
product can also
be given ([25, 26]), in terms of a symplectic connection $\Gamma$ and
some tensors
(the general expressions for $C_5$ and above could not be written in
such terms; however the existence proof of Fedosov [23], which remained obscure until his MIT preprint of December 1992, gives a recurrence algorithm showing that a star-product can be constructed globally in terms of a symplectic connection and its covariant derivatives). In
particular one has, for a Vey product [26] :
$$C_2 = P^2_\Gamma + bH \hskip.2cm , \hskip.2cm C_3 = S^3_\Gamma + T +
3 \partial H \leqno{(19)}$$
where $H$ is a differential operator of maximal order $2,$ $T$ a
$2$-tensor
corresponding to a closed $2$-form, $\partial$ the Chevalley coboundary
operator, and $P^2_\Gamma$
and $S^3_\Gamma$ are given (in canonical coordinates) by expressions
similar to (4)
where, in $P^2,$ usual derivatives are replaced by covariant derivatives
and, in
$P^3,$ by the relevant components of the Lie derivative of $\Gamma$ in
the direction
of the vector field associated to the function ($u$ or $v$).

It is then possible [2] to work by steps of $2$ and to reduce the
possible choices
of Vey products from the infinity suggested by the Hochschild cohomology
to the
finite alternatives of Vey brackets. In particular :
\saut
{\ninerm PROPOSITION.} \it If $b_2 (W) = 0,$ the Moyal-Vey bracket is
 unique up to (mathematical)
equivalence. \rm
\saut
The more general definition of a star-product given here is required in
particular
because normal and standard orderings are star-products in this sense,
equivalent
to Moyal (on flat space) but not Vey products (the cochains $C_r$ are
not of the
same parity as $r$). Consideration of the associated Lie algebra is
however still
possible, and will give information on the skew-symmetric parts of the
cochains
$C_r.$ The analysis developed in [2] for Vey products can however be
carried over
to this more general definition.

{\it c. Rigidity.} A natural question then arises : is it the end of the
 story,
and in particular can the Moyal-Vey product be deformed ?

The answer to the latter is : essentially no. More precisely one can
show [27] that any (Vey)
star-product of the form
$$u *_{\nu \rho} v = \sum^\infty_{r,s=0} \nu^r \rho^s \hskip.1cm C_{rs}
(u,v),
\hskip.3cm C_{rs} (u,v) = (-1)^r C_r (u,v) \leqno{(20)}$$
can be transformed by equivalence to $*_\nu,$ by deforming the $2$-tensor $\Lambda$ with the parameter $\rho.$

However if the requirement that $C_1 = P$ is relaxed that result is no
longer true.
Examples can be given where an intertwinning operator will still exist,
but will
transform the Poisson into a Jacobi bracket (associated with a conformal
symplectic
structure) [28]: one writes
$$u *_{\nu \rho} v = u *_\nu f_{\nu \rho} *_\nu v \leqno{(21)}$$
where $f_{\nu \rho} = \sum^\infty_{r=0} \nu^{2r} \hskip.1cm f_{2r, \rho}
\in
(N[[\nu]]) [[\rho]]$ with $f_{0, \rho}$ invertible and $f_{0,0} = 1.$ In
particular
one can take for $f_{\nu \rho}$ the $*_\nu$ exponential of $\rho H,$ $H \in N$
(and $\rho$ proportional to the inverse temperature, in applications) :
the so-called
KMS conditions in statistical mechanics take then a simple expression
(the condition for a state to be KMS is similar to a trace).

Quantum groups are also, in a sense, deformations of the usual
star-products where
the covariance enveloping algebra is deformed. We shall now study more
closely that notion of covariance of a star-product.

{\bf 2.4. Invariance and covariance of star-products. Star
 representations.}
{\it a. Invariance.} The Poisson bracket $P$ is (by definition)
 invariant under
all symplectomorphisms. However its powers $P^r$ ($r \geq 2$) given
by (4) are
invariant only under the vector fields generated by the polynomials of
degree at
most $2.$ For general Vey products the cochains $C_2$ and $C_3$ given
by (19) are
invariant under a subgroup of the finite-dimensional Lie group of
symplectomorphisms
preserving the symplectic connection $\Gamma.$ That is also the case
in $\Rrm^{2l}$
for orderings other than Moyal (only the linear polynomials remain,
possibly supplemented
- e.g. for standard and normal orderings - by one second-degree
polynomial). We thus
have [2] a finite-dimensional Lie algebra ${\cal O}$ of "{\bf preferred
observables}",
under which the star-product will be {\bf invariant} :
$${\cal O} = \{ a \in N \hskip.1cm; \hskip.1cm [a,u]_\nu =
P(a,u) \hskip.3cm \forall u \in N \}. \leqno{(22)}$$

{\it b. Covariance.} A star-product will be said {\bf covariant} under
 a Lie
algebra ${\cal O}$ of functions if $[a,b]_\nu = P (a,b)$ for all
$a,b \in {\cal O}.$
It can then be shown [29] that $*$ is ${\cal O}$-covariant iff there
exists a
representation $\tau$ of the Lie group $G$ whose Lie algebra is
${\cal O}$ into $Aut \hskip.1cm ({\cal A}; *)$ such that
$$\tau_g u = (Id_N + \sum^\infty_{r=1} \nu^r \hskip.1cm \tau^r_g)
(g.u) \leqno{(23)}$$
where $g \in G, u \in N,$ $G$ acts on $N$ by the natural action induced
by the
vector fields associated with ${\cal O},$ $(g.u) (x) = u (g^{-1} x),$ and where the
$\tau^r_g$ are differential operators on $W.$ Invariance of course means
that the
geometric action preserves the star-product : $g.u * g.v = g (u * v).$

{\it c. Star representations.} Let $G$ be a Lie group (connected and
 simply connected),
acting by symplectomorphisms on a symplectic manifold $W$ (e.g. coadjoint orbits
in the dual of the Lie algebra ${\cal L}$ of $G$). The elements
$x,y \in {\cal L}$
will be supposed realized by functions $u_x, u_y$ in $N$ so that their
Lie bracket
$[x,y]_{\cal L}$ is realized by $P(u_x, u_y).$ Let us take a
$G$-covariant star-product
$*,$ so that $P (u_x, u_y) = [u_x, u_y]_\nu.$ We can now define the
{\bf star
exponential} :
$$E (e^x) = Exp (x) \equiv \sum^\infty_{n=0} (n!)^{-1} (u_x / 2 \nu)^{*n} \leqno{(24)}$$
where $x \in {\cal L}, \hskip.1cm e^x \in G$ and the power $*n$ denotes
the $n^{th}$
star-power of the corresponding function. By the Campbell-Hausdorff
formula one
can extend $E$ to a {\bf group homomorphism} :
$$E : G \fl (N [[\nu, \nu^{-1}]], *) \leqno{(25)}$$
where, in the formal series, $\nu$ and $\nu^{-1}$ are treated as
independent
parameters for the time being. Alternatively, the values of $E$ can be
taken in
the algebra $({\cal P} [[\nu^{-1}]], *),$ where ${\cal P}$ is the algebra generated by ${\cal L}$
with the $*$-product (a representation of the enveloping algebra).

We now call {\bf star representation} [2] of $G$ a distribution
${\cal E}$ (valued in
$Im \hskip.1cm E$) on $W$ defined by
$$D \ni f \mapsto {\cal E} (f) = \int_G f(g) \hskip.1cm E(g^{-1}) dg
\leqno{(26)}$$
where $D$ is some space of test-functions on $G.$ The corresponding
{\bf character}
$\chi$ is the (scalar-valued) distribution defined by
$$D \ni f \mapsto \chi (f) = \int_W \hskip.1cm {\cal E} (f) \hskip.1cm
d \mu \leqno{(27)}$$
where $d \mu$ is a quasi-invariant measure on $W.$ The character permit a comparison with usual representation theory.

This theory is now very developed, and parallels in many ways the usual
(operatorial)
representation theory. For a review of developments (and of star-product
theory)
until 1986, see e.g. [30] and references quoted therein. Among notable
results one may quote :

i) An exhaustive treatment of {\bf nilpotent} groups and of solvable
groups of exponential
type [31]. The coadjoint orbits are there symplectomorphic to
$\Rrm^{2l},$ and
one can lift the Moyal product to the orbits in a way that is adapted
to the
Plancherel formula. Polarizations are not required, and
"star-polarizations" can always be introduced to compare with usual
theory.

ii) For {\bf semi-simple} Lie groups an array of results is already
available, including
a complete treatment of the holomorphic discrete series [32] (that
includes the
case of compact Lie groups) and scattered results for specific examples.

iii) For semi-direct products, and in particular the Poincar\'e and
Euclidean groups,
an autonomous theory has also been developed (see e.g.[33]).

Comparison with the usual results of "operatorial" theory of Lie group
representations
can be performed in several ways, in particular by constructing an
invariant Weyl
transform generalizing (7), finding "star-polarizations" that always
exist, in
contradistinction with the geometric quantization approach (where at
best one can
find complex polarizations), study of spectra (of elements in the center
of the
enveloping algebra and of compact generators) in the sense of the next
subsection,
comparison of characters, etc. But our main insistence is that the
theory of
star-representations is an {\bf autonomous} one that can be formulated
completely within
this framework, based on coadjoint orbits (and some additional
ingredients when required).

{\bf 2.5. Star-quantization.}

{\it a.} {\ninerm DEFINITION.} \it Let $W$ be a symplectic (or Poisson)
 manifold and $N$ an
algebra of classical observables (functions, possibly including
distributions if
proper care is taken for the product). We shall call {\bf
star-quantization}
a star-product on $N$ invariant under some Lie algebra ${\cal O}$ of
"preferred observables". \rm

Invariance of the star-product ensures that the classical and quantum
evolutions
of observables under a Hamiltonian $H \in {\cal O}$ will coincide [2].
The typical
example is the case described in (2.1.c), the Moyal product on
$W = \Rrm^{2l}.$
Physicists often prefer to work with the so-called {\bf normal ordering}, for which
the Weyl map (7) is taken [21] with a weight $w(\xi, \eta) =
exp (- {1 \over 4} (\xi^2 + \eta^2)).$

{\it b. Spectrality.} Physicists want to get numbers that match
 experimental results,
e.g. for energy levels of a system. That is usually achieved by
describing the
spectrum of a given Hamiltonian $\hat{H}$ supposed to be a self-adjoint
operator so as
to get a real spectrum and so that the evolution operator (the
exponential of $it \hat{H}$)
is unitary (thus preserves probability). A similar spectral theory can
be done here,
in an autonomous manner. The most efficient way to achieve it is to
consider [2]
the star exponential (corresponding to the evolution operator)
$$Exp (Ht) \equiv \sum^\infty_{n=0} (n!)^{-1} \hskip.1cm (t / i \hbar)^n
\hskip.1cm
(H*)^n \leqno{(28)}$$
where $(H*)^n$ means the $n^{th}$ star power of $H \in N$ (or
${\cal A}$). Then
one writes its Fourier-Stieltjes transform $d \mu$ (in the distribution
sense),
formally
$$Exp (Ht) = \int \hskip.1cm e^{\lambda t / i \hbar} d \mu (\lambda).
\leqno{(29)}$$

{\ninerm DEFINITION.} \it The spectrum of $(H / \hbar)$ is the support
 $S$ of $d \mu.$ \rm

In the particular case that $H$ has discrete spectrum, the integral can
be written
as a sum : the distribution $d \mu$ is a sum of "delta functions"
supported at the
points of $S,$ multiplied by the symbols of the corresponding
eigenprojectors.

In different "orderings" given by (7) with various weight funtions $w$,
one gets
in general different operators for different classical observables $H,$
thus
different spectra. For $W = \Rrm^{2l}$ all those are mathematically
equivalent (to
Moyal under the Fourier transform $T_w$ of the weight function $w$ of
(7)). This
means that every observable $H$ will have the same spectrum under Moyal
ordering
as $T_w H$ under the equivalent ordering. But this does not imply
"physical equivalence",
i.e. the fact that $H$ will have the same spectrum under both orderings.
In fact,
the opposite is true :
\psaut
{\ninerm PROPOSITION} [34]. \it If two equivalent star-products are
 isospectral (give the
same spectrum for a "large family" of observables and all $\nu$) they
are identical. \rm
\saut
It is worth mentioning that our definition of spectrum permits to define
a spectrum
even for symbols of non-spectrable operators, such as the derivative on
a half-line
(that has different deficiency indices). That is one of the many
advantages of our autonomous approach to quantization.

{\it c. Applications.} i) In {\bf quantum mechanics} it is preferable to
 work (for
$W = \Rrm^{2l}$) with the star-product that has maximal symmetry, i.e.
$sp (\Rrm^{2l}).{\cal H}_l$
as algebra of preferred observables : the Moyal product (5). One indeed
finds
[2] that the star exponential of these observables (polynomials of order
$\leq 2$)
is proportional to the usual exponential. More precisely, if $X =
\alpha p^2 +
\beta pq + \gamma q^2 \in sl(2) \hskip.3cm (p,q \in \Rrm^l \hskip.1cm;
\hskip.1cm \alpha,
\beta, \gamma \in \Rrm),$ setting $d = \alpha \gamma - \beta^2$ and
$\delta = \vert d \vert^{1/2}$ one gets (as distributions)
$$(30) \hskip2.3cm Exp (Xt) = \left\{ \matrix {
(\hbox{cos }\delta t)^{-l} \hbox{exp} ((X / i \hbar \delta) \hskip.1cm
\hbox{tan}(\delta t)) \hskip.5cm \hbox{for } d > 0 \cr
\hbox{exp} (Xt / i \hbar) \hskip3.5cm \hbox{for } d = 0 \cr
(\hbox{cosh }\delta t)^{-l} \hbox{exp} ((X / i \hbar \delta)
\hskip.1cm \hbox{tanh}(\delta t)) \hskip.1cm \hbox{for } d < 0 \cr
}\right.$$
hence the Fourier decompositions
$$(31) \hskip3.8cm Exp (Xt) = \left\{ \matrix {
\sum^\infty_{n=0} \hskip.1cm \Pi^{(l)}_n \hskip.1cm e^{(n + {l \over 2})t}
\hskip1cm \hbox{for } d > 0 \cr
\int^\infty_{- \infty} \hskip.1cm e^{\lambda t / i \hbar} \Pi(\lambda, X) d \lambda
\hskip.5cm \hbox{for } d < 0. \cr
}\right.$$

We thus get the discrete spectrum $(n + {l \over 2}) \hbar$ of the
{\bf harmonic
oscillator} and the continuous spectrum $\Rrm$ for the dilation
generator $pq.$
The "eigenprojectors" $\Pi^{(l)}_n$ and $\Pi (\lambda, X)$ are given by
some special functions [2].

Other examples can be brought to this case by some functional
manipulations [2].
For instance the Casimir element $C$ of $so(\it l)$ representing
angular
momentum, which can be written $C = p^2 q^2 - (pq)^2 - l(l - 1)
{\hbar^2 \over 4},$
has $n (n + (l-2)) \hbar^2$ for spectrum. For the {\bf hydrogen atom,}
with Hamiltonian
$H = {1 \over 2} p^2 - \vert q \vert^{-1}$, the Moyal product on
$\Rrm^{2l+2}$ ($l=3$ in the
physical case) induces a star product on $W = T^* S^l$ and the energy
levels, solutions of $(H-E) * \phi = 0,$ are found to be (from (31), for
$l=3$) :

$E = {1 \over 2} (n+1)^{-2} \hbar^{-2}$ for the discrete spectrum

$E \in \Rrm_+$ for the continuous spectrum.

It is worth noting that the term ${l \over 2}$ in the harmonic oscillator spectrum,
obvious source of divergences in the infinite-dimensional case,
disappears if the
normal star-product is used instead of Moyal - which is one of the
reasons it is preferred in field theory.

ii) {\bf Path integrals} are intimately connected to star exponentials.
In fact,
in quantum mechanics the path integral of the action is nothing but the
partial
Fourier transform of the star exponential (28) with respect to the
momentum variables,
for $W = \Rrm^{2l}$ as phase space with the Moyal star-product [35]. For
compact
groups the star exponential $E$ given by (25) can be expressed in terms
of unitary
characters using a global coherent state formalism [36] based on the
Berezin dequantization
of compact group representation theory used in [32] (that gives
star-products
somewhat similar to normal ordering); the star exponential of any
Hamiltonian on
$G / T$ (where $T$ is a maximal torus in the compact group $G$) is then
equal to
the path integral for this Hamiltonian.

iii) For {\bf field theory} similar results hold. In particular [8] the
star
exponential of the Hamiltonian of the free scalar field (for the normal
star-product)
is equal to a path integral. That is only part of the results obtained
for field
theory [8]. It has indeed been found that other star-products "close to
normal"
enjoy similar properties and permit nonstandard quantizations of the
Klein-Gordon
equation (not necessarily leading to a free field theory). And for
interacting
fields one can show that by transforming the normal star-product by a
suitable
equivalence one can remove some of the divergences occurring in
$\lambda \phi^4_2$
theory, which indicates that {\bf the processus of renormalization is
cohomological
in essence} (removing an infinite coboundary to get a finite cocycle).

{\bf 2.6. Pseudodifferential calculus and character of a star-product.}

{\it a. Closedness of the star-product arising in pseudodifferential
 calculus.}

Let $g \in {\cal A}, \hskip.1cm g = \sum^\infty_{r=0} \nu^r
\hskip.1cm g_r \hskip.1cm
(g_r \in N, \nu = {1 \over 2} i \hbar).$ Let us define
$$\tau (g) = \int \hskip.1cm g_l \hskip.1cm \omega^l. \leqno{(32)}$$ From
(11) we see that, in the pseudodifferential calculus on
$\Rrm^{2l},$
$(2 \pi)^l \hskip.1cm Tr (\Omega_S (g))$ is equal to $\tau (g)$ modulo
multiples of $\hbar$ (and $\hbar^{-1}$).

Therefore $\tau$ enjoys the same properties as a trace with respect to
the standard
star product, which is thus closed. More precisely, for a
pseudodifferential
operator $D$ (with compact support) on $\Rrm^l$ one can define its symbol (on
$W = T^* \Rrm^l$) by [37]
$$\sigma_D (q_0, p) = D \hskip.1cm e^{i(q-q_0).p} \vert_{q=q_0}
\leqno{(33)}$$
so that one has for a product :
$$\sigma_{BD} = \sigma_B \hskip.1cm *_S \hskip.1cm \sigma_D =
\sum^\infty_{r=0}
\hskip.1cm {i^{-r} \over r!} \hskip.1cm {\partial^r \sigma_B \over
\partial p^r}
\hskip.1cm {\partial^r \sigma_D \over \partial q^r}. \leqno{(34)}$$
These formulas extend [37] to $W = T^* M, M$ Riemannian compact manifold, using
a globally defined function $L \in C^\infty (T^* M \times M)$ that can,
on canonical
charts, be written $L (q_0; p,q) = p.(q-q_0).$ As for standard ordering
one can
then show (by integration by parts e.g.) that the corresponding
star-product on $W = T^* M$ is closed.

{\it b. The character.} The deviation from algebra homomorphism of the
 identity
map on ${\cal A}$ between the algebras ${\cal A}_0$ (${\cal A}$ endowed
with commutative
product) and ${\cal A}_\nu$ (${\cal A}$ endowed with $*_\nu$ product) is
expressed by
$$\sigma (f,g) = f * g - fg \hskip.5cm (f,g \in {\cal A}). \leqno{(35)}$$
As for operator algebras [13], since the map $\tau$ behaves like a trace, one is
thus lead to define (for $f_i \in {\cal A})$ :
$$\varphi_{2k} (f_0, f_1,...,f_{2k}) = \tau (f_0 * \sigma (f_1, f_2)
*...* \sigma (f_{2k-1}, f_{2k})) \leqno{(36)}$$
and $\varphi_{2k} = 0$ except for $l \leq 2k \leq 2l.$ This makes sense
for a
general closed star-product.
\psaut
{\ninerm DEFINITION.} \it The {\it character} of a closed star-product
 $*$ is the cohomology
{\it class} of the cocycle $\varphi$ with components
$\varphi_{2k}$ defined by (38) in the cyclic cohomology bicomplex. \rm

The previous considerations then imply :
\psaut
{\ninerm PROPOSITION.} \it On $W = T^* M, \hskip.2cm M$ Riemannian
 compact, the character $\varphi$
defined by (36) (with standard ordering) coincides with the character
defined by
the trace on pseudodifferential operators. \rm
\psaut
{\it c. Examples.} The top component $\varphi_{2l}$ of $\varphi$ is given by [14]
$$\varphi_{2l} (f_0, f_1,...,f_{2l}) = \int \hskip.1cm f_0 P(f_1,f_2) ... P(f_{2l-1},
f_{2l}) \omega^l$$
$$\hskip0.9cm = \int f_0 \hskip.1cm df_1 \wedge ... \wedge df_{2l}.$$
If $dim \hskip.1cm W = 4,$ the computation of $\varphi$ can be carried
out easily and
one gets in addition $\varphi_2 = \tilde{C}_2$ (in the notations of
(2.2.b)) and thus
$b \varphi_2 = - {1 \over 2} B \varphi_4.$ Now, since $HC^2 (N) =
Z^2 (W, \Crm)
\oplus \Crm$ where $Z^2$ stands for the closed $2$-currents, a change
in $C_2$
will affect the class in $HC^4$ thus the class of $\varphi,$ which has
then no
reason to be an integer - as is the case in the traditional approaches
to quantization.
\psaut
{\it d. Consequences.} Combining the above proposition with the algebraic index
theory of A. Connes et al. [13] one gets [14] :
\saut
{\ninerm THEOREM.} \it The character $\varphi$ of the star product of
 the pseudodifferential
calculus belongs to $HC^{ev} (T^* M)$ and is given by the Todd class
$Td (T^* M)$ as
a current over $T^* M.$ \rm
\saut
Since the trace on compact operators is unique, whenever a star product
has an
associated Weyl map that includes them in the image, the cyclic cocycle
$\varphi$
will necessarily be proportional to an integral one. But we have seen
above that
the character is not necessarily integral. Therefore on one hand star
quantization
(with a closed star-product) can be applied in cases where there are
obstructions
to the traditional (Bohr-Sommerfeld) quantization; and on the other
hand it paves
the way for a {\bf generalized index theory where the index is no more
necessarily an integer.}

For finite-dimensional manifolds, because the trace formula (9) is exact
(not
modulo $\hbar$ terms) in the Moyal case, the corresponding "symmetric
pseudodifferential
calculus" should be of special interest. And in the infinite-dimensional
case since
orderings "close to normal" permit a cohomological interpretation of
cancellation
of infinities [8], the corresponding "complex pseudodifferential
calculus" appears to be an appropriate framework.
\saut
\centerline {\bf 3. Quantum groups and star-products.}
\psaut
{\bf 3.1. The background.} {\it a. Yang-Baxter equations and Hopf
 algebras.}

As mentioned in the introduction, "quantum groups" appeared around 1980
in the
quantization of $2$-dimensional integrable models [9]. A basic ingredient there
is an equation which the LOMI group called {\bf Yang-Baxter equation.}
If $R \in End (V \otimes V),$
where $V$ is a vector space, and if we define $R^{12} = R \otimes I \in
End (V \otimes
V \otimes V)$ and similarly for $R^{13}, R^{23},$ that equation writes
$$R^{12} \hskip.1cm R^{13} \hskip.1cm R^{23} = R^{23} \hskip.1cm R^{13}
\hskip.1cm
R^{12} \hbox{ with } R^{12} \hskip.1cm R^{21} = I. \leqno{(37)}$$
This is the quantum Yang-Baxter equation (QYBE). Writing
$$R = I + \sum^\infty_{j=1} \hskip.1cm \nu^j \hskip.1cm r_j \in End (V
\otimes V)
[[\nu]] \leqno{(38)}$$
one gets (at order 2) the classical Yang-Baxter equation (CYBE) :
$$[r^{12}_1, r^{13}_1] + [r^{12}_1, r^{23}_1] + [r^{13}_1, r^{23}_1] =
0 = r^{12}_1
+ r^{21}_1 \leqno{(39)}$$
Looking for solutions lead to a strange modification of the $sl(2)$
commutation
relations, which later was extended to a wide array of Lie groups and
became
systematized by Drinfeld in the framework of {\bf Hopf algebras} [10].
These are
{\bf bialgebras}, i.e. associative algebras ${\cal A}$ with a coproduct
 $\Delta :
{\cal A} \fl {\cal A} \otimes {\cal A},$ that posess in addition a counit $\varepsilon :
{\cal A} \fl \Crm$ and antipode map $S : {\cal A} \fl {\cal A},$
 satisfying the
expected (and well-known) relations and commutative diagrams.

{\it b. Poisson-Lie groups and Lie bialgebras.}
The typical example is the associative algebra $N$ of functions over a
Lie group
$G$ with pointwise multiplication, comultiplication $\Delta :
N \fl N \otimes N$
defined (by duality) from the group multiplication $G \times G \fl G,$
counit
defined by the value at the identity $e$ of $G$ and antipode by the value at the
inverse (in $G$). The group is said to be a {\bf Poisson-Lie group} when
it has
a Poisson structure $\Lambda$ for which the above $\Delta$ is a Poisson
morphism,
i.e.
$$\Delta P(u,v) = P (\Delta u, \Delta v) \leqno{(40)}$$
where $P$ denotes the Poisson bracket in $G$ and $G \times G,$ defined
by $\Lambda.$

The infinitesimal version (dual to it in the sense that the enveloping
algebra
${\cal U} (g)$ can be considered as a space of distributions on $G$ with
support at
the identity) is the notion of {\bf Lie bialgebras.} It is a Lie algebra
$\it g$
with a "compatible" bracket $\varphi^*$ on its dual ${\it g}^*$ (i.e.
$\varphi :
{\it g} \fl {\it g} \otimes {\it g}$ is a $1$-cocycle for the adjoint
 action of
$\it g$ on ${\it g} \otimes {\it g}).$

A case of special interest (triangular Poisson-Lie group [10]) is when
$\varphi$
is the coboundary of $r \in {\it g} \wedge {\it g}$ such that the
Poisson-Lie
structure $\Lambda$ on the corresponding group $G$ is the difference of
the left
and right invariant skew-symmetric $2$-tensors defined by $r$ : this $r$
is
solution of the CYBE (39).

{\it c. Their quantization.} It is now natural to try and "quantize" (40). This can
indeed be done. More precisely [38,39] on $N = C^\infty (G)$ where $G$ is a triangular
Poisson Lie group there exists an invariant star product defined by
invariant
bidifferential operators $F_k (x,y),$ i.e. by a formal series
$F = I + \sum^\infty_{k=1}
\nu^k \hskip.1cm F_k$ such that the associativity condition for the
star product
gives a solution $S (x,y) = F^{-1} (y,x) F(x,y)$ of the QYBE (37) on
${\cal U} ({\it g}) [[\nu]].$

Moreover, from that invariant star product one can build [38] an
equivalent one
($*'$, non invariant) such that (with the same coproduct $\Delta$ on
$N$) :
$$\Delta (u *' v) = \Delta u *' \Delta v. \leqno{(41)}$$
These results were more or less implicit in Drinfeld's work but precise
statements
and proofs can be found in [39], together with related results.

In the dual approach [11] it is the coproduct $\Delta,$ on a completion
${\cal U}_\nu ({\it g})$
of the enveloping algebra ${\cal U} ({\it g}),$ that is deformed to
$\Delta_\nu.$
The simplest example is when ${\it g} = {\it sl} (2)$ with generators $H$ and
$E^\pm$ (or more generally $H_\alpha$ and $E^\pm_\alpha$ for simple roots $\alpha$
on a simple Lie algebra ${\it g}$) where
$$\hskip.5cm \Delta_\nu (H) = H \otimes 1 + 1 \otimes H$$
$$\Delta_\nu (E^\pm) = E^\pm \otimes \hbox{exp} ({1 \over 2} \nu
H) + \hbox{exp} (-
{1 \over 2} \nu H) \otimes E^\pm$$
(see [40] for a very detailed exposition).

A quantum group can therefore be seen as a non commutative star-product
deformation
of the Hopf algebra of functions on a Poisson-Lie group, or as a non
cocommutative
deformation of its dual.

{\bf 3.2. Quantized Universal Enveloping Algebras as star-product
 algebras.}
{\it a. Example} [41]. First let us remark that the following functions
 on $\Rrm^2$ :
$$H = pq, \hskip.2cm J_+ = q, \hskip.2cm J^\eta_- = (1 - \hbox{ cos }\eta pq) /
\eta^2 q \leqno{(42)}$$
satisfy, with respect to Poisson brackets, the same commutation
relations as those
of the QUEA ${\cal U}_\eta (sl(2)).$ Since $H$ and $J_+$ are preferred
observables
this is also true for Moyal brackets and in particular
$$M (J_+, J^\eta_-) = - \eta^{-1} \hbox{sin}(\eta pq) \leqno{(43)}$$
Now from (30) one gets that a strong invariance condition
$$f (* t X / i \hbar) = \alpha (t) \hskip.1cm f(\beta (t) X / i \hbar)
\leqno{(44)}$$
holds when $f$ is an exponential or a trigonometric function (sine or
cosine),
where on the left-hand side we mean that we take (Moyal) star powers in
the power
series expansion of $f.$ Therefore (after a slight rescaling) (43) can
be rewritten
with (sin $*$) instead of sin, and the defining QUEA relations can
be expressed entirely in the $*$-product algebra generated by the Lie
generators.

{\it b. Generalizations.} The same can be done for all higher rank
 simple
algebras, e.g. the $A_n$ series [42]. The cubic Serre relations are also
expressed in the $*$-product algebra.

One can show [41] that {\bf for Moyal product, the only power series for
which
(44) holds, are sine, cosine and exponential} (with exception of these
series truncated at order two). Therefore if one requires that the
"quantum"
commutation relations be a deformation of the "classical" ones, the
strong invariance
condition (44) alone shows that the algebra found in the litterature is
unique
(except for the truncated sine $f(x) = a_1 x + a_3 x^3$).

For other star-products than Moyal there are even less functions $\it f$
satisfying
(44). For instance [42] for a family of star-products interpolating
between
standard and antistandard ordering, the only function is the exponential.

In conclusion, we have seen that the QUEAs can be realized essentially
uniquely,
as star-product algebras with a quantization  parameter $\hbar$ (in the
star-product)
different from the QUEA parameter. We have in fact a {\bf $2$-parameter
deformation}.

{\bf 3.3. A universal rigid model for quantum groups [43].}

{\it a. The model.} Let us start with $G = SU(2).$ Denote by
 $(\pi_n, V_n)$ its
$2n+1$-dimensional representation ($2n$ an integer) by matrices in
${\cal L}
(V_n).$ The direct product $A = \Pi^\infty_{n=0} \hskip.1cm {\cal L}
(V_n)$,
endowed with the product topology and algebra law, becomes a Fr\'echet
algebra
into which both $G$ and the enveloping algebra ${\cal U} ({\it g})$ can
be
imbedded by $x \mapsto (\pi_n (x)).$ These imbeddings are total because
$A$ is
the bicommutant of the direct sum representation $\pi = \sum \hskip.1cm
\pi_n$
on $V = \sum \hskip.1cm V_n,$ and every $\pi_n$ extends by continuity to
a representation of $A.$

{\it b. Universality.} We have :
\psaut
{\ninerm PROPOSITION.} {\it If $t \notin 2 \pi {\bf Q},$ {\it the QUE
 algebras}
${\cal U}_t ({\it g})$ {\it can be imbedded into a dense subalgebra
$A_t$ of
$A$; $\pi_n$ are still a complete set of representations of
$A_t,$ and the Hopf structures (coproduct $\Delta_t,$ counit
$\epsilon_t,$ and
antipode $S_t$) on ${\cal U}_t ({\it g})$ have unique extensions to
topological
Hopf structures (with equivalent coproducts) on $A,$ when $A \otimes A$
is endowed
with the completed projective tensor topology so that \rm
$$A \otimes A \approx \Pi_{n,p} \hskip.1cm {\cal L} (V_n \otimes V_p).$$

The classical limit is ${\cal U}_0 ({\it g}) \approx {\cal U} ({\it g})
\otimes \theta$ where $\theta^2 = 1$ (a parity), with coproduct
$\Delta_0$ equivalent to $\Delta_t$ : there exists $P(t) \in A \otimes A$ such that $\Delta_t = P(t) \hskip.1cm \Delta_0 \hskip.1cm P(t)^{-1}.$
The $R$-matrix is $R(t) = P(-t) \hskip.1cm P(t)^{-1}$ and can be chosen
so that $A_t$ is a quasitriangular Hopf algebra ($q = e^{it}$ in the
usual notations).

{\it c. The star-product.} The topological dual $A^*$ is isomorphic to
 $\oplus
{\cal L} (V_n) = {\it H},$ endowed with the inductive topology, and can
 thus be
viewed as polynomial functions on the complex extension $G_{\bf C}$ of
$G$; as
Hopf algebra,
$$H \approx {\bf C} [a,b,c,d] / (ad - bc = 1)$$
where the product is that of functions and the coproduct is defined by
the product
in $G.$ Each co-associative coproduct $\Delta_t$ induces an associative
star-product
on $H$ such that $(a+d)^{*k} = (a+d)^k, \hskip.1cm \forall k \in {\bf N},$ and
conversely if this is true for a $*$-product compatible with the usual
coproduct
$\Delta_0,$ this $*$ is induced by a coproduct $\Delta_t =
P(t) \Delta_0 P(t)^{-1}$
on $A$ : {\it we recover Manin's description of functions of
non-commutative
arguments.}

{\it d. Rigidity as bialgebra.} Let $B$ be a bialgebra, an associative
 algebra
$B$ with coproduct $\Delta : B \rightarrow B \otimes B.$ We then define
[18] in
a natural way the notion of isomorphism of bialgebras (associative
algebra
isomorphism with equivalence of coproducts), and of deformations of
such
structures, where we denote by $H (B, \Delta)$ the relevant 2-cohomology.

It can be shown [43] that if (for Hochschild cohomologies)
$$\tilde{H}^2 (B) = 0 = \tilde{H}^1 (B, B \otimes B)$$
then $H (B, \Delta) = 0.$ Whitehead's lemmas therefore prove that
$(A, \Delta_t)$
{\bf is rigid for all} $t \notin 2 \pi {\bf Q}.$

 {\ninerm REMARK.} These results can be extended to all simple compact
  groups. From this follows that (in the generic case) the QUEAs
  associated with such groups have a hidden group structure, that of
  the original group, included in a universal (reflexive) topological Hopf algebra completion.
\gsaut
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\gsaut
%\address
PHYSIQUE MATHEMATIQUE, UNIVERSITE DE BOURGOGNE

B.P. 138, F-21004 DIJON Cedex, FRANCE.

\par
%\endaddress
%\email{
e-mail addresses: flato@satie.u-bourgogne.fr, daste@ccr.jussieu.fr
%}\endemail
}}

\bye