\documentclass{article}
\title{Existence of star products, a brief history}
\author{M\'elanie Bertelson}
\date{\today}
%\setlength{\voffset}{-1.5cm}
%\setlength{\textwidth}{6.6in}
%\setlength{\evensidemargin}{0in}
%\setlength{\oddsidemargin}{0in}
%\setlength{\textheight}{9in}
%\setlength{\topmargin}{0in}
%\setlength{\parskip}{1.5mm}
\def\R{{\cal R}}
\def\smoy{\ast_{\!\cal M}}
\begin{document}
\maketitle
This paper surveys work on existence of star products and deformations
of the Poisson bracket on a Poisson manifold.
\section{Introduction}
The mathematical framework of classical mechanics is a Poisson manifold, the
observables being smooth functions on the manifold. When formalizing
quantum mechanics, it is required that to each classical dynamical
system corresponds a quantum one, its {\em quantization}, and that the
latter yields the former when the Planck constant $\hbar$ is
neglected, or ``tends to zero''; this last requirement is called the
{\it correspondence principle}. The usual formalism, at least until
the 70's, made use of a Hilbert space $H$, the quantum observables
being operators on $H$, thus of a completely different nature from that
of the classical ones. This made the correspondence principle difficult
to carry out. In 1974, Berezin (\cite{Bz-q-74, Bz-gcq-75}),
followed in 1978 by Bayen, Flato, Fronsdal, Lichnerowicz and
Sternheimer (\cite{BFFLS-dtq-78}) proposed alternatives to this
formalism. In \cite{BFFLS-dtq-78}, the authors proposed a quantization
procedure based on the notion of (formal) deformation, in the sense of
Gerstenhaber (\cite{Gh-64}), of the structures of commutative
and of Lie algebra on the set of classical observables; this procedure
is referred to as {\it deformation quantization}. For a general account of
the subject, see \cite{FS-c-94} and \cite{W-dq-95}.
\section{Definitions}
Let $(M,\pi)$ be a Poisson manifold.
\begin{enumerate}
\item[1)] A formal deformation of the structure of algebra on $N=C^\infty(M)$,
or a {\it star product} is defined to be a map
$$
* : N\otimes N\to N[[\hbar]] : (u,v)\to \sum_{k=0}^{\infty}\, \hbar^k\, c_k(u,v)
$$
satisfying the following conditions
\begin{enumerate}
\item[(i)] formal associativity (when extended to $N[[\hbar]]\otimes N[[\hbar]]$), i.e.
for all $p\geq 0$
$$
\sum_{k+l=p} \left[c_k(c_l(u,v),w) - c_k(u,c_l(v,w)) \right] = 0
$$
\item[(ii)] $c_0(u,v) = uv$
\item[(iii)] $\frac{1}{2}\left(c_1(u,v)-c_1(v,u)\right) = \{u,v\}$ (Poisson
bracket)
\item[(iv)] It is required that each map $c_k : N\otimes N\to N$ be
a local, or a bidifferential operator, depending on the authors.
One sometimes asks for some of the additional requirements :
\item[(v)] $u*1=u=1*u$
\item[(vi)] $c_k(u,v)=(-1)^kc_k(v,u)$
\end{enumerate}
The necessity for something like (iii) was first observed by Dirac
(\cite{Di-pqm-30}).
\item[2)] A formal deformation of the Poisson bracket is a skewsymmetric map
$$
[,] : N\otimes N\to N[[\hbar]] : (u,v) \to \sum_{k=0}^{\infty}\, \hbar^k\, \Gamma_k(u,v)
$$
satisfying
\begin{enumerate}
\item[(a)] the formal Jacobi identity (when extended to $N[[\hbar]]\otimes N[[\hbar]]$), i.e.
for all $p \geq 0$
$$
\oint_{u,v,w}\sum_{k+l=p} \Gamma_k(
\Gamma_l(u,v),w) =0$$
($\oint_{u,v,w}$ means sum over the cyclic permutations of $\{u,v,w\}$)
\item[(b)] $\Gamma_0(u,v) = \{u,v\}$
\item[(c)] the analogue of (iv) for the $\Gamma_k$'s.
One sometimes requests :
\item[d)] $[1,u]=0$
\end{enumerate}
\end{enumerate}
\section{Existence on symplectic manifolds}
At the mathematical level, the first question encountered when studying
deformation quantization is the existence of star products, and of non trivial
deformations of the Poisson bracket. In the 70's,
the only example known was the so-called Moyal star product on a symplectic
vector space (\cite{M-qmst-49}), induced by the composition of operators
on $C^\infty(\R^n)$ by Weyl's correspondence (\cite{W-tgqm-31}). It is given
in coordinates by the formula
\begin{equation} \label{eq:moyal}
u \smoy v = \sum_{k=0}^{\infty}\, \left(\frac{-i \hbar}{2}\right)^k\, \frac{1}{k!} \,
\Lambda^{i_1j_1}\cdots\Lambda^{i_kj_k}\, \frac{\partial^k u}{\partial x^{i_1}\cdots\partial
x^{i_k}}\, \frac{\partial^k v}{\partial x^{j_1} \cdots\partial x^{j_k}}
\end{equation}
\noindent
where $\Lambda$ denotes the Poisson bivector associated to the given
symplectic structure.
Its skewsymmetric part
yields a deformation of the Poisson bracket. Formula (\ref{eq:moyal}) admits a
generalization to any Poisson manifold endowed with a flat torsionless
Poisson connection, obtained by replacing the usual derivatives in
(\ref{eq:moyal})
by covariant ones (flatness of the connection is necessary to obtain
associativity) (\cite{BFFLS-dtq-78});
or equivalently by defining
\begin{equation} \label{eq:moyal2}
u*v(x) =\left. \left[(exp^*_x u)(y) \smoy (exp^*_x v)(y) \right]\right|_{y=0}
\end{equation}
\noindent
where $exp_x : T_xM \to M$ denotes the exponential map induced by the given
connection.\\
An attempt to solve the existence problem in a more general context is
to proceed recursively, as described for deformations of general algebraic
structures by Gerstenhaber (\cite{Gh-64}); in this approach, one tries to
locate the obstruction to extending a star product or a deformation of
the Poisson bracket at order $k$
(at order $k$ means that i) or a) holds for $1\leq p\leq k$ only) to the next
order in some cohomology group, as small as possible. The
obstruction to extending a star product belongs to the third Hochschild
cohomology group of the commutative algebra N, while the obstruction
to extending a deformation of
the Poisson bracket belongs to the third Chevalley cohomology group of
the Lie algebra $(N,\{,\})$ (for the adjoint representation). (Given a
star product (resp. deformed bracket)
up to order $k$, it can be extended to the next order provided a certain
3-cocycle $F_k :N^3\to N$, constructed from the $c_i$'s (resp. $\Gamma_i$'s)
for $i=1,\ldots,k$, is Hochschild (resp. Chevalley)-exact; in fact,
the equation
$\delta c_{k+1}=F_k$ (resp. $\delta \Gamma_{k+1} = F_k$) has to be satisfied).
The next step is to understand the structure of the relevant cohomology groups;
this has been carried out by several authors (\cite{V-dcp-75, CG-lc-80, G-23-80, G-cC-84}).
Within this framework, Vey (\cite{V-dcp-75}) proved existence of non trivial
deformations of the
Poisson bracket of a symplectic manifold under the assumption that $H^3(M)=0$
(third de Rham cohomology group). Later, under the same assumption, Neroslavsky
and Vlassov proved existence of star products (\cite{NV-daf-81}), implying the
former result (actually when $H^3(M)=0$, the two results are equivalent
(\cite{L-da-82})). As shown by the examples in \cite{L-77} of star products
on some quotients of open subsets of $\R^{2n}$ by groups of affine symplectic
transformations, and of Poisson manifolds with a flat
torsionless Poisson connection, (to include torus), the condition
$H^3(M)=0$ is not necessary.\\
In 1982-1983, a series of existence proofs came out for larger and larger
classes of symplectic manifolds. Cahen and Gutt established the existence
of star products on the cotangent bundle of a parallelizable manifold
(\cite{CG-r*r-82}), (the proof consists in choosing astutely at each step
a 2-cochain $c_{k+1}$ among the ones satisfying $\delta c_{k+1} = F_k$, in
such a way as to force $F_{k+1}$ to be exact; this is possible
thanks to the extra structure assumed), and Gutt gave an explicit formula
in the case of a Lie group (\cite{G-e*p-83}). This result also yields existence
for the case of the Poisson manifold constituted by the dual of a (any) Lie
algebra ${\cal G}^*$ endowed with its natural (linear) Poisson structure,
the so-called Lie-Poisson structure; (the star product obtained is, when
restricted to the polynomial functions on ${\cal G}^*$, the one induced by the
associative product on the universal enveloping algebra
of ${\cal G}$ via the symmetrization map (\cite{Bz-sr-67})).
Shortly later, De Wilde and Lecomte gave proofs of existence of star products
and deformed brackets on
any cotangent bundle (\cite{DeWL-cb-83}); later, on any exact symplectic
manifold (\cite{DeWL-ee-85}); and eventually on arbitrary symplectic
manifolds (\cite{DeWL-ea-83} and \cite{DeWL-eed-88} for a revisited proof).
They show that there is no obstruction at all to the extension of a truncated
deformed bracket and an order $2k$ star product (the star products considered
there satisfy (vi)). They actually establish this for deformations
of Poisson brackets and show by a simple argument (using a characterization
of deformed Poisson brackets that are the skewsymmetric part of a star product)
that this implies the analogue for star products. The result
for Poisson brackets
is obtained by applying to each element of a contractible open covering
a refined version of the proof for exact symplectic manifolds, refined in such
a way as to allow a gluing.\\
After this first general proof (in the symplectic case), several
others, departing from the Gerstenhaber approach, appeared~:
\begin{enumerate}
\item[1)] Karasev and Maslov in 1984 (\cite{KM-po-84}, and \cite{KM-nl-93} for
more details).\\
As explained in \cite{W-dq-95}, the main objects here are local sheaves of
wave packets on the manifold $M$. On
each open subset $U$ of $M$, it is possible to construct such a sheaf,
(here denoted $S_U$);
Karasev and Maslov establish a correspondence between formal
power series with smooth functions on $U$ as coefficients,
and homomorphisms of the sheaf $S_U$.
It happens that, even though there might not exist a global sheaf of wave
packets on all of $M$ (indeed, the patching together of the local sheaves
is possible only when a certain non trivial ``quantization condition''
is satisfied), the composition of homomorphisms yields a globally
well-defined star product on $M$.
\item[2)] Fedosov in 1985 (\cite{F-fq-85}, \cite{F-it-89}, \cite{F-sg-94},
and \cite{F-dq-96}).\\
First published in Russia, this proof remained unknown in Western countries
until '93. It is algorithmic and geometric in nature. As in the proof of
Karasev and Maslov, the star product is obtained by transporting an associative
algebra structure to $N[[\hbar]]$ via a linear bijection, but here all objects
exist globally. The algebra in question is an algebra of flat (i.e.
parallel) sections of
a certain bundle for a certain flat connection on this bundle. The bundle,
called {\it Weyl bundle},
has as typical fiber the algebra obtained by completing
$(Pol(T_xM)[\hbar],\smoy)$ with respect to a suitable grading. To construct
the flat connection, or {\it Fedosov connection}, one starts from a symplectic
torsionless one, extends it to the Weyl bundle and then ``corrects'' it to make it
flat; the corrective term is obtained as the unique solution of a recursive
equation.
A unique flat section corresponding to a given element of $N[[\hbar]]$ appears
as solution to a similar equation.\\
Weinstein gave an interpretation of this proof in \cite{W-dq-95} as
being a generalization of the construction for Poisson manifolds
endowed with a flat torsionless Poisson connection; according to this
interpretation, the Fedosov
connection is a ``quantum'' version of the latter, and the
correspondence with the algebra of flat sections is a ``quantum
exponential''. The star product is then given by a formula of
type (\ref{eq:moyal2}) (see also \cite{EW-dg-94}, and \cite{X-F*p-96}).
For a comparison between De Wilde, Lecomte and Fedosov, see
\cite{Dg-c-95}.
\item[3)] Omori, Maeda and Yoshioka in 1991 (\cite{OMY-Wm-91} and
\cite{OMY-c*p-92}).\\
Independently from Fedosov, the authors use Weyl bundles, or more
precisely local
Weyl bundles. They introduce the notion of Weyl function associated
to an element of $N[[\hbar]]$, these are sections of the local Weyl
bundles very similar to Fedosov's flat sections in the case where
the initial Poisson connection is already flat. Here the problem is
to patch together the local Weyl functions associated to a given
global power series in $N[[\hbar]]$. To do this, they need suitable
isomorphisms between the local Weyl bundles preserving the class of
Weyl functions; (equivalently, they need a complete set of
trivializations of the global Weyl bundle compatible with the Weyl
functions, only defined locally).
\item[4)] De Wilde and Lecomte in 1990 (\cite{DeWL-r-90}).\\
They show how to carry out the gluing of Moyal star products on open
chart domains using suitable equivalences on the intersections.
A simplification of the process used in \cite{OMY-Wm-91} to patch together
the local Weyl bundles provides them with the equivalences.
\end{enumerate}
Let us mention some papers concerned with the study of star products
specialized to K\"{a}hler manifolds : \cite{Bz-Km-75, CG-Km-95, K-Km-96}
and references therein.
\section{Existence on Poisson manifolds}
So far, it is not known whether every Poisson manifold admits a star product. There
are classes of examples~:
\begin{enumerate}
\item Regular Poisson manifolds.\\
Fedosov's construction extends to that case (\cite{F-sg-94}), but there is no
hope to extend it further because of the key role that a Poisson connection
plays in the construction (such a connection does not exist on a non regular
Poisson manifold). Other proofs have been given by several authors ~:
\cite{M-td-92, Dz-ce-97, NT-fd-97}.
\item The dual of a Lie algebra with its canonical Lie-Poisson structure
(\cite{G-e*p-83}).
\item Poisson Lie groups (\cite{EK-q-96}).
\item Omori, Maeda, Yoshioka in \cite{OMY-Pa-94} develop Gerstenhaber's theory
of deformation in the general framework of Poisson algebras. In the
following example of Poisson manifolds, the obstruction to extending an order
$k$ star product disappears :
\begin{enumerate}
\item[a)]Any Poisson manifold whose third Poisson cohomology group vanishes.
\item[b)]A vector space endowed with a quadratic Poisson structure.
\end{enumerate}
\item Poisson structures induced by symplectic Lie algebroids
(\cite{W-NI-94, NT-fd-97}). (A star product on a symplectic
Lie algebroid can be constructed by a generalization of
Fedosov's method, and it in turn induces a star product
on the associated Poisson manifold.)
\end{enumerate}
Penkava and Vanhaecke (\cite{PV-dq-96}) have proven that locally,
a star product
of order three exists on
every general non-linear Poisson algebra.
The hope now lies principally in Kontsevich's conjecture (\cite{K-IHES-95},
\cite{K-Ber-95}, and \cite{K-a-97}) which, if true, would imply
existence of a star product on
an arbitrary Poisson manifold (see also \cite{V-qP}, and the paper by Yakimov
in the present volume).
\bibliographystyle{alpha}
\newcommand{\etalchar}[1]{$^{#1}$}
\begin{thebibliography}{BFF{\etalchar{+}}78}
\bibitem[Ber67]{Bz-sr-67}
F.~A. Berezin.
\newblock Some remarks about the associated envelope of a {Lie} algebra.
\newblock {\em Funct. Anal. Appl.}, 1:91--102, 1967.
\bibitem[Ber74]{Bz-q-74}
F.~A. Berezin.
\newblock Quantization.
\newblock {\em Math. USSR Izv.}, 8:1109--1165, 1974.
\bibitem[Ber75a]{Bz-gcq-75}
F.~A. Berezin.
\newblock General concept of quantization.
\newblock {\em Commun. Math. Phys.}, 40:153--174, 1975.
\bibitem[Ber75b]{Bz-Km-75}
F.~A. Berezin.
\newblock Quantization of {K\"{a}hler} manifolds.
\newblock {\em Comment. Math. Phys.}, 40:153, 1975.
\bibitem[BFF{\etalchar{+}}78]{BFFLS-dtq-78}
F.~Bayen, M.~Flato, C.~Fronsdal, A.~Lichnerowicz, and D.~Sternheimer.
\newblock Deformation theory and quantization {I} and {II}.
\newblock {\em Ann. Phys.}, 111:61--151, 1978.
\bibitem[CG80]{CG-lc-80}
M.~Cahen and S.~Gutt.
\newblock Local cohomology of the algebra of ${C}^{\infty}$ functions on a
compact manifold.
\newblock {\em Lett. Math. Phys.}, 4:157--167, 1980.
\bibitem[CG82]{CG-r*r-82}
M.~Cahen and S.~Gutt.
\newblock Regular $*$-representations of {Lie} algebras.
\newblock {\em Lett. Math. Phys.}, 6:395--404, 1982.
\bibitem[CG95]{CG-Km-95}
M.~Cahen and S.~Gutt.
\newblock Quantization of {K\"{a}hler} manifolds. {IV}.
\newblock {\em Lett. Math. Phys.}, 34:159--168, 1995.
\bibitem[Daz97]{Dz-ce-97}
P.~Dazord.
\newblock Construction explicite de produits \'etoiles sur les vari\'et\'es de
{Poisson} r\'eguli\`eres.
\newblock {\em Pr\'epublications de L'Institut G. Desargues}, 1, 1997.
\bibitem[Del95]{Dg-c-95}
P.~Deligne.
\newblock D\'eformations de l'alg\`ebre des fonctions d'une vari\'et\'e
symplectique : {Comparaison} entre {Fedosov} et {De Wilde}, {Lecomte}.
\newblock {\em {Selecta} {Mathematica}, {New} {Series}}, 1 no. 4, 1995.
\bibitem[Dir30]{Di-pqm-30}
P.~A.~M. Dirac.
\newblock {\em The principles of quantum mechanics}.
\newblock The international series of monographs on Physics. Clarendon {Press},
Oxford, 1930.
\bibitem[DL83a]{DeWL-ea-83}
M.~{De Wilde} and P.~Lecomte.
\newblock Existence of star-products and of formal deformations of the
{Poisson} {Lie} algebra of arbitrary symplectic manifolds.
\newblock {\em Lett. Math. Phys.}, 7:427--496, 1983.
\bibitem[DL83b]{DeWL-cb-83}
M.~{De Wilde} and P.~Lecomte.
\newblock Star-products on cotangent bundles.
\newblock {\em Lett. Math. Phys.}, 7, no. 3:235--241, 1983.
\bibitem[DL85]{DeWL-ee-85}
M.~{De Wilde} and P.~Lecomte.
\newblock Existence of star-products on exact symplectic manifolds.
\newblock {\em Ann. Inst. Fourier, Grenoble}, 35,2:117--143, 1985.
\bibitem[DL88]{DeWL-eed-88}
M.~{De Wilde} and P.~Lecomte.
\newblock Formal deformations of the {Poisson} {Lie} algebra of a symplectic
manifold and star-products. {Existence}, equivalence, derivations.
\newblock In {\em Deformation theory of algebras and structures and
applications}, pages 897--960, Dordrecht, 1988. Kluwer Acad. Pub.
\bibitem[DL90]{DeWL-r-90}
M.~{De Wilde} and P.~Lecomte.
\newblock Existence of star-products revisited.
\newblock {\em Note di Matematica}, 10, Suppl. n. 1:205--216, 1990.
\bibitem[EK96]{EK-q-96}
P.~Etingof and D.~Kazhdan.
\newblock Quantization of {Lie} bialgebras, {I}.
\newblock {\em Selecta Mathematica, New Series}, 2 no. 1:1--41, 1996.
\bibitem[EW94]{EW-dg-94}
C.~Emmrich and A.~Weinstein.
\newblock The differential geometry of {Fedosov's} quantization.
\newblock In {\em Lie theory and geometry, in honnor of {B}. {Kostant},
Progress in Mathematics}, pages 214--239, New York, 1994. {Birkh\"{a}user}.
\bibitem[Fed85]{F-fq-85}
B.~V. Fedosov.
\newblock Formal quantization.
\newblock {\em Some topics of Modern Mathematics and their applications to
problems of Mathematical Physics (in Russian), Moscow}, pages 129--136, 1985.
\bibitem[Fed89]{F-it-89}
B.~V. Fedosov.
\newblock Index theorem in the algebra of quantum observables.
\newblock {\em Sov. Phys. Dokl.}, 34:318--321, 1989.
\bibitem[Fed94]{F-sg-94}
B.~V. Fedosov.
\newblock A simple geometrical construction of deformation quantization.
\newblock {\em J. Diff. Geom.}, 40:213--238, 1994.
\bibitem[Fed96]{F-dq-96}
B.~V. Fedosov.
\newblock {\em Deformation Quantization and Index Theory}.
\newblock Akademie Verlag, Berlin, 1996.
\bibitem[FS94]{FS-c-94}
M.~Flato and D.~Sternheimer.
\newblock Closedness of star products and cohomologies.
\newblock In {\em Lie Theory and Geometry, in Honor of B. Kostant}, Progress in
Mathematics, pages 241--259, New York, 1994. {Birkh\"{a}user}.
\bibitem[GDL84]{G-cC-84}
S.~Gutt, M.~{De Wilde}, and P.~B.~A. Lecomte.
\newblock A propos des 2\`eme et 3\`eme groupe de cohomologie de {Chevalley}
d'une vari\'et\'e symplectique.
\newblock {\em Ann. Inst. H. Poincar\'e}, 40:77--83, 1984.
\bibitem[Ger64]{Gh-64}
M.~Gerstenhaber.
\newblock On the deformation of rings and algebras.
\newblock {\em Ann. Math.}, 79:59--90, 1964.
\bibitem[Gut80]{G-23-80}
S.~Gutt.
\newblock Second et troisi\`eme espaces de cohomologie diff\'erentiable de
l'alg\`ebre de {Lie} de {Poisson} d'une vari\'et\'e symplectique.
\newblock {\em Ann. Inst. H. Poincar\'e}, 33:1--31, 1980.
\bibitem[Gut83]{G-e*p-83}
S.~Gutt.
\newblock An explicit $*$-product on the cotangent bundle of a {Lie} group.
\newblock {\em Lett. Math. Phys.}, 7, no. 3:249--258, 1983.
\bibitem[Kar96]{K-Km-96}
A.~V. Karabegov.
\newblock On the deformation quantization, on a {K\"{a}hler} manifold,
associated with a {Berezin} quantization.
\newblock {\em Funct. Anal. Appl.}, 30, No. 2:142--144, 1996.
\bibitem[KM84]{KM-po-84}
M.~V. Karasev and V.~P. Maslov.
\newblock Pseudodifferential operators and a canonical operator in general
symplectic manifolds.
\newblock {\em Math. USSR Izv.}, 23:277--305, 1984.
\bibitem[KM93]{KM-nl-93}
M.~V. Karasev and V.~P. Maslov.
\newblock Nonlinear {Poisson} brackets : geometry and quantization.
\newblock {\em Translation of mathematical monographs, Amer. Math. Soc.,
Providence}, 119, 1993.
\bibitem[Kon95a]{K-IHES-95}
M.~Kontsevich.
\newblock Lectures at {IHES}.
\newblock Fall 1995.
\bibitem[Kon95b]{K-Ber-95}
M.~Kontsevich.
\newblock Lectures on deformation theory, {University} of {California},
{Berkeley}.
\newblock Spring 1995.
\bibitem[Kon97]{K-a-97}
M.~Kontsevich.
\newblock to appear in Proceedings of {Ascona} conference (June 1996), 1997.
\bibitem[Lic77]{L-77}
A.~Lichnerowicz.
\newblock Construction of twisted products for cotangent bundles of classical
groups and {Stiefel} manifolds.
\newblock {\em Lett. Math. Phys.}, 2:133--143, 1977.
\bibitem[Lic82]{L-da-82}
A.~Lichnerowicz.
\newblock D\'eformations d'alg\`ebres associ\'ees \`a une vari\'et\'e
symplectique (les $*_\nu$-produits).
\newblock {\em Ann. Inst. Fourier}, 32-1:157--209, 1982.
\bibitem[Mas92]{M-td-92}
A.~Masmoudi.
\newblock Tangential deformations of the {Poisson} bracket and tangential
$*$-products on a regular {Poisson} manifold.
\newblock {\em J. of Geom. Phys.}, 9:156--177, 1992.
\bibitem[Moy49]{M-qmst-49}
J.~Moyal.
\newblock Quantum mechanics as a statistical theory.
\newblock {\em Proc. Camb. Phil. Soc.}, 45:99--124, 1949.
\bibitem[NT97]{NT-fd-97}
R.~Nest and B.~Tsygan.
\newblock Formal deformations of symplectic {Lie} algebroids.
\newblock preprint, 1997.
\bibitem[NV81]{NV-daf-81}
O.~M. Neroslavsky and T.~Vlasov.
\newblock Sur les d\'eformations de l'alg\`ebre des fonctions d'une vari\'et\'e
symplectique.
\newblock {\em CRAS Paris}, 292:71--73, 1981.
\bibitem[OMY91]{OMY-Wm-91}
H.~Omori, Y.~Maeda, and A.~Yoshioka.
\newblock Weyl manifolds and deformation quantization.
\newblock {\em Advances in Math.}, 85:224--255, 1991.
\bibitem[OMY92]{OMY-c*p-92}
H.~Omori, Y.~Maeda, and A.~Yoshioka.
\newblock Existence of closed star products.
\newblock {\em Lett. Math. Phys.}, 26:285--294, 1992.
\bibitem[OMY94]{OMY-Pa-94}
H.~Omori, Y.~Maeda, and A.~Yoshioka.
\newblock Deformation quantization of {Poisson} algebras.
\newblock In {\em Symplectic geometry and quantization (Sanda and Yokohama,
1993)}, volume 179 of {\em Contemp. Math.}, pages 213--240, Providence, {RI},
1994. Amer. Math. Soc.
\bibitem[PV96]{PV-dq-96}
M.~Penkava and P.~Vanhaecke.
\newblock Deformation quantization of non-linear {Poisson} brackets.
\newblock preprint, 1996.
\bibitem[Vey75]{V-dcp-75}
J.~Vey.
\newblock D\'eformation du crochet de {Poisson} sur une vari\'et\'e
symplectique.
\newblock {\em Comment. Math. Helvetici}, 50:421--454, 1975.
\bibitem[Vor97]{V-qP}
A.~A. Voronov.
\newblock Quantizing {Poisson} manifolds.
\newblock preprint, 1997.
\bibitem[Wei94]{W-NI-94}
A.~Weinstein.
\newblock Lectures at the {Newton} {Institute}.
\newblock July 1994.
\bibitem[Wei95]{W-dq-95}
A.~Weinstein.
\newblock {\em Deformation quantization}, volume no.227, Exp. No. 789, 5 of
{\em S\'eminaire Bourbaki, Vol. 1993/94}.
\newblock Ast\'erisque, 1995.
\bibitem[Wey31]{W-tgqm-31}
H.~Weyl.
\newblock {\em The theory of groups and quantum mechanics}.
\newblock Dover, New-York, 1931.
\bibitem[Xu96]{X-F*p-96}
P.~Xu.
\newblock Fedosov $*$-products and quantum momentum maps.
\newblock preprint, 1996.
\end{thebibliography}
\end{document}