%_ ************************************************************************** %_ * 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. 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