%_ **************************************************************************
%_ * The TeX source for AMS journal articles is the publisher's TeX code    *
%_ * which may contain special commands defined for the AMS production      *
%_ * environment.  Therefore, it may not be possible to process these files *
%_ * through TeX without errors.  To display a typeset version of a journal *
%_ * article easily, we suggest that you either view the HTML version or    *
%_ * retrieve the article in DVI, PostScript, or PDF format.                *
%_ **************************************************************************
% Author Package file for use with AMS-LaTeX 1.2%
\controldates{29-OCT-1997,29-OCT-1997,29-OCT-1997,29-OCT-1997}
 
\documentstyle[newlfont]{era-l}
\issueinfo{3}{19}{January}{1997}
\dateposted{November 4, 1997}
\pagespan{119}{120}
\PII{S 1079-6762(97)00034-6}
\def\copyrightyear{1997}

%\issueinfo{3}{1}{September}{1997}

\copyrightinfo{1997}{American Mathematical Society}

%\def
\renewcommand \Bbb{\mathbf}
 %\def
\newcommand \R{{\mathbf R}}
%\def
%\newcommand \P{{\Bbb P}}
%\def
%\newcommand \S{{\Bbb S}}
%\def
\newcommand \A{{\cal A}}
%\def
%\newcommand \Vect{{\rm Vect}}
%\def
\newcommand \V{{\cal V}}
%\def
\newcommand \Vp{\V_P}
%\def
\newcommand \Vq{\V_Q}
%\def
\newcommand \D{{\cal D}}
%\def
\renewcommand\d{\mathcal{D}}
%\def
\newcommand \bDp{\overline{\D_P}}
%\def
\newcommand \bDq{\overline{\D_Q}}
%\def
\newcommand \Dp{\D_P}
%\def
\newcommand  \Dq{\D_Q}
%\def
\newcommand \dX{\delta_X}
%\def
\newcommand \E{{\cal E}}
%\def
\newcommand \ep{\E_{P}}
%\def
\newcommand \eq{\E_{Q}}
%\def
%\newcommand \Sp{\S_{P,Q}}
\newcommand \Spq{\mathbf{S}_{P,Q}}
%\def
\newcommand \C{{\cal C}}
%\def
\newcommand \Cp{\C_P}
%\def
\newcommand \Cq{\C_Q}
%\def
\renewcommand \mp{m_{P,Q}}
%\def
\newcommand\arrow{\to}
%\def
\newcommand\semi{\stackrel{.}{\times}}
%\def
\newcommand\follows{\to}
%\def
\newcommand\Vr{\V_r}
%\def
\newcommand\F{{\cal F}}
%\def
%\newcommand\Fun{{\rm Fun}}
%\def
%\newcommand \ad{{\rm ad}}
%\def
\newcommand \be{\begin{equation}}
%\def
\newcommand \ee{\end{equation}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\textheight=20truecm
%\textwidth=13truecm
 
%\def
\newcommand \schmendric{\sim} 


\newtheorem{Theorem}{Theorem}
 \newtheorem{Prop}{Proposition}
 \newtheorem{Proposition}{Proposition}
 \newtheorem{Lemma}{Lemma}
 \newtheorem{lemma}{Lemma}
 \newtheorem{Corollary}{Corollary} 

\begin{document}

\title{Quantization of Poisson structures on $\R^2$}

\author{Dmitry Tamarkin}

\address{Department of Mathematics, Pennsylvania State University, 
218 McAllister Building, University Park, PA 16802}

\email{tamarkin@@math.psu.edu}

\date{September 2, 1997}

\commby{Alexandre Kirillov}

\subjclass{Primary 81Sxx}

% \maketitle
\begin{abstract}
An `isomorphism' between the `moduli space' of 
star products on $\R^2$ and the `moduli space' of all 
formal Poisson structures on $\R^2$ is established.
\end{abstract}

\maketitle

The problem of quantization of Poisson structures has been posed  in \cite{FFF}.
It is well known that any Poisson structure on a two-dimensional manifold is quantizable. In this paper 
we establish an `isomorphism' between the `moduli space' of 
star products on $\R^2$ and the `moduli space' of all 
formal Poisson structures on $\R^2$ by construction of a 
map from Poisson structures to star products. Certainly, this isomorphism 
follows from the Kontsevich formality conjecture \cite{Kon}. Most likely,
our map can be used as a first step in constructing 
an $L_{\infty}$-quasiisomorphism in the formality conjecture for $\R^2$.
The author would like to thank Boris Tsygan and Paul Bressler for the attention
and helpful suggestions.
   
The set of all star-products $ %\S
\Bbb S$ is acted upon by the group
$\d\semi{\operatorname{Diffeo}}\R^2$, where $\d$ is the group of operators of the form
$1+hD_1+h^2D_2+\cdots$ with $D_k$ to be arbitrary differential operators.
The set of all formal Poisson structures $%\P
\Bbb P$ consists of
formal series in $h$ with bivector fields as the coefficients. Formal Poisson
structures are acted upon by the group
${\operatorname{Diffeo}\R^2}\semi{\operatorname{exp}}(h{\operatorname{Vect}}[[h]])$, 
where ${\operatorname{Vect}}$ is the Lie algebra of vector fields on $\R^2$.
These actions define  equivalence relations. We want to have a pair of maps
$f_1: %\S
\Bbb S\arrow%\P
\Bbb P$ and $f_2:%\P
\Bbb P\arrow %\S
\Bbb S$ such that 
\[ %$$
f_1\circ f_2(x)\schmendric x,\quad
f_2\circ f_1(x)\schmendric x, 
\] %$$
%\be
\begin{equation}
x\schmendric y\follows f_{1,2}x\schmendric f_{1,2}y.\label{equiv}
\end{equation}
%\ee
 By a map from $ %\S
\Bbb S$ we mean a differential expression in terms of the coefficients of the bidifferential operators
corresponding to the star products. Maps from $%\P
\Bbb P$ are defined similarly.

We can replace $ %\S
\Bbb S$ by a subspace. Let $P,Q$ be a nondegenerate pair of
(real) polarizations of $\R^2$. Define a subset $ %\S
\Bbb{S}_{P,Q}$ of $ %\S
\Bbb S$ in the following
way: 
$m\in  %\Sp
\Spq$ iff $m(f,g)=fg$ if $f$ is constant along $P$ or $g$ is constant along
$Q$.
\begin{Prop} 
Let $x,y$ be a nondegenerate coordinate system on $\R^2$ such that $x$
is constant along $Q$ and $y$ is constant along $P$. Then there exists a unique
map $
 %\S
\Bbb{S}\arrow\d :m\mapsto U(m)=1+hV(m)$ such that
\begin{equation}\label{polar}
\begin{split}
1)\quad&  \mp(m)=U^{-1}(m(Uf,Ug))\in %\Sp
\Spq,\\ %$$ 
2)\quad& Ux=x,\ Uy=y,\ U1=1.
\end{split}
\end{equation}
%\ee
$U$ is uniquely defined by the condition
$U(x^{*m}*y^{*n})=x^my^n$ (where star denotes the star product $m$).
\end{Prop}
We denote by $\mp: %\S
\Bbb S\to %\Sp
\Spq$ the map which sends $m$ to $\mp(m)$. 
Further, $x,y$ will mean the same as in Proposition 1.
Thus, it is enough to find maps $p_1: %\Sp
\Spq\arrow%\P
\Bbb P$ and $p_2:%\P
\Bbb P\arrow %\Sp
\Spq$
 with the same properties as  $f_1,f_2$ have. Indeed, put
\begin{equation}
f_2  =  i\circ p_2, \qquad
f_1  =  p_1\circ \mp \label{masya} 
\end{equation}
(here $i:  %\Sp
\Spq\arrow  %\S
\Bbb S$ is
the inclusion).
 
The following theorem gives an explicit construction for $p_2$ which appears
to be a bijective map so that we can put $p_1=p_2^{-1}$. Denote by 
$\Cp$
(resp. $\Cq$) the space of functions, constant along $Q$ (resp. $P$). Denote by $\Vp$
(resp. $\Vq$) the space of
vector fields preserving the polarizations and tangent to $P$ (resp. $Q$).
Denote by $\Dp$ the subalgebra of the algebra of differential operators
consisting of operators $D$ such that $D(\Cq)\subset \Cq$ and
$D(fg)=fD(g)$ if $f\in \C_P$.  Denote by $\Dq$ the same algebra where
$P$ and $Q$ are interchanged. In the coordinates $x,y$
we have $\Cp=\{f(x)\}$, $\Vp=\{f(x)\partial_x\}$, 
$\Dp=\sum f_i(x)\partial_x^i$ 
and the same things with $P$ replaced by $Q$ and $x$ replaced by $y$. 
Denote by $\bDp$ (resp. $\bDq$) the subring of $\Dp$
(resp. $\Dq$) consisting of operators which annihilate
constant functions.

Note that the space of bivector fields is isomorphic to
$\Vp\otimes_{\R}\Vq$. Let $\D_{P,k}$ be the space of maps
$Vp^{\otimes k}\arrow \bDp$ (which are differential operators
in terms of the coefficients).

\begin{Theorem} 
a) There exists a unique sequence $c_k\in \D_{P,k}\otimes
\D_{Q,k}, k=0,1,2,\ldots,$ $c_k=\sum_i a^i_k\otimes b^i_k$, 
$c_0(X,Y)=1\otimes 1$, such that for any 
bivector field $\Psi=\sum_i X_i\wedge Y_i$, $X_i\in \Vp,Y_i\in \Vq$, the formula
\begin{equation}\label{posya} 
\begin{split}
m(\Psi,P,Q,f,g)&=fg+\sum_{k,i_1,\ldots, 
i_{k+1}}h^{k+1}L_{X_{i_1}}\{a^n_k(X_{i_2},X_{i_3},\ldots,X_{i_{k+1}})f\}\\ 
&\qquad\qquad\qquad\qquad\times L_{Y_{i_1}}\{b^n_k(Y_{i_2},Y_{i_3},\ldots,Y_{i_{k+1}})g\}\\
 &=\sum_k m_k(f,g).
\end{split}   
\end{equation} 
gives a star-product.

b) Put $p_2:%\P
\Bbb P\to %\Sp
\Spq: \Psi\to m(\Psi,P,Q,\cdot,\cdot)$.
Put $p_1=p_2^{-1}$. Then $p_1$ and $p_2$ provide an isomorphism of $%\P
\Bbb P$
and $S$ by \eqref{masya}.
\end{Theorem}


\begin{thebibliography}{123}
  
\bibitem{FFF}{F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz,
and D. Sternheimer, Deformation theory and quantization, I, II, 
Ann. Phys. 11 (1978), 61--151.} 
\MR{58:14737a}, \MR{58:14737b}
\bibitem{Kon}{M. Kontsevich, Formality conjecture, preprint, to appear in
Proc. of Summer School on Deformation Quantization in Ascona.}
%\bibitem{Ber}{F. Berezin, Secondary Quantization.}

\end{thebibliography}

\end{document}
\endinput
20-Oct-97 08:09:23-EST,311083;000000000000
Return-path: 
Received: from AXP14.AMS.ORG by AXP14.AMS.ORG (PMDF V5.1-8 #1)
 id <01IP0TTW0F4G000L5I@AXP14.AMS.ORG>; Mon, 20 Oct 1997 08:09:21 EST
Date: Mon, 20 Oct 1997 08:09:19 -0400 (EDT)
From: "pub-submit@ams.org " 
Subject: ERA/9700
To: pub-jour@MATH.AMS.ORG
Reply-to: pub-submit@MATH.AMS.ORG
Message-id: <01IP0TYEFEN6000L5I@AXP14.AMS.ORG>
MIME-version: 1.0
Content-type: TEXT/PLAIN; CHARSET=US-ASCII


Return-path: 
Received: from gate1.ams.org by AXP14.AMS.ORG (PMDF V5.1-8 #1)
 with SMTP id <01IOX1GDHUN4000BHN@AXP14.AMS.ORG>; Fri, 17 Oct 1997 15:01:11 EST
Received: from leibniz.math.psu.edu ([146.186.130.2])
 by gate1.ams.org via smtpd (for axp14.ams.org [130.44.1.14]) with SMTP; Fri,
 17 Oct 1997 19:00:40 +0000 (UT)
Received: from weber.math.psu.edu (aom@weber.math.psu.edu [146.186.130.202])
 by math.psu.edu (8.8.5/8.7.3) with ESMTP id PAA25843 for
 ; Fri, 17 Oct 1997 15:00:39 -0400 (EDT)
Received: (aom@localhost) by weber.math.psu.edu (8.8.5/8.6.9)
 id PAA21702 for pub-submit@MATH.AMS.ORG; Fri, 17 Oct 1997 15:00:37 -0400 (EDT)
Date: Fri, 17 Oct 1997 15:00:37 -0400 (EDT)
From: Alexander O Morgoulis 
Subject: *accepted to ERA-AMS, Volume 3, Number 1, 1997*
To: pub-submit@MATH.AMS.ORG
Message-id: <199710171900.PAA21702@weber.math.psu.edu>
%% Modified October 15, 1997 by A. Morgoulis
%% Modified September 29, 1997 by A. Morgoulis



% Date: 20-OCT-1997
%   \pagebreak: 0   \newpage: 0   \displaybreak: 0
%   \eject: 0   \break: 0   \allowbreak: 0
%   \clearpage: 0   \allowdisplaybreaks: 0
%   $$: 4   {eqnarray}: 2
%   \linebreak: 0   \newline: 0
%   \mag: 0   \mbox: 0   \input: 0
%   \baselineskip: 0   \rm: 8   \bf: 1   \it: 0
%   \hsize: 0   \vsize: 0
%   \hoffset: 0   \voffset: 0
%   \parindent: 0   \parskip: 0
%   \vfil: 0   \vfill: 0   \vskip: 0
%   \smallskip: 0   \medskip: 0   \bigskip: 0
%   \sl: 0   \def: 34   \let: 0   \renewcommand: 0
%   \tolerance: 0   \pretolerance: 0
%   \font: 0   \noindent: 0
%   ASCII 13 (Control-M Carriage return): 0
%   ASCII 10 (Control-J Linefeed): 0
%   ASCII 12 (Control-L Formfeed): 0
%   ASCII 0 (Control-@): 0