\documentclass[11pt]{article}
\usepackage{amssymb}
\usepackage{latexsym}
%\input amssym.def
%\input amssym
\renewcommand{\baselinestretch}{1.05}
\newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition}
\newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary}
\newtheorem{dfn}[thm]{Definition} \newtheorem{axiom}[thm]{Axiom}
\newcommand{\alg}{{\cal A}} \newcommand{\dive}{\mbox{div}}
\newcommand{\del}{\partial} \newcommand{\backl}{\mathbin{\vrule
width1.5ex height.4pt\vrule height1.5ex}} \newcommand{\bolde}{{\bf e}}
\newcommand{\reals}{{\Bbb R}} \title{{\bf The Modular Automorphism
Group of a Poisson Manifold}\\Dedicated to Professor A. Lichnerowicz} \author{Alan
Weinstein\thanks{Research partially supported by NSF Grant
DMS-93-09653.} \\Department of Mathematics\\ University of
California\\ Berkeley, CA 94720 USA\\
{\small(alanw@math.berkeley.edu)}} \begin{document} \maketitle
\section{Introduction} \label{sec-intro} The modular automorphism
group of a von Neumann algebra $A$ is a 1-parameter group of
automorphisms of $A$ which, modulo inner automorphisms, is canonically
associated to $A$. Since Poisson manifolds can be thought of as
``semiclassical limits'' of operator algebras, it is natural to ask
whether they, too, have modular automorphism groups. This paper will
show that they do. Thus, in the terms of Connes
\cite{co:noncommutative}, Poisson manifolds are, like von Neumann
algebras, intrinsically {\em dynamical} objects.
It appears that the study of the modular vector field will give some
geometric insight into the modular automorphisms of von Neumann
algebras, as well as being a useful tool in Poisson geometry.
The modular theory of von Neumann algebras has its
origins in the KMS theory in quantum mechanics and the closely related
Tomita-Takesaki theory. In these theories, a 1-parameter group of
automorphisms is related to a positive linear functional. The
independence modulo inner automorphisms of the 1-parameter group is
the content of Connes' noncommutative Radon-Nikodym theorem and was
extensively exploited by him for the classification of Type III
factors. We refer to \cite{co:noncommutative} for a discussion of
this material, with an extensive list of references.
I have been interested for many years in trying to understand the
classical limit of modular theory, stimulated by two sources. The
first is the similarities between KMS theory and symplectic geometry
appearing throughout Renault's book \cite{re:groupoid} (which was also
a principal stimulus for the development of the theory of symplectic
groupoids). The second is the work of Lichnerowicz and his
collaborators as reported in \cite{ba-fl-li-st:deformation}.
This interest was revived in recent discussions with Dimitri
Shlyakhtenko, who called to my attention the paper
\cite{ri-va:bounded} in the attempt to interpret some of its contents
in terms of symplectic linear algebra. By ``Poissonizing'' some of
the constructions in that paper, I arrived at a
definition of the modular automorphism group which is purely
geometric.
In fact, I soon discovered that the infinitesimal generator of the
modular automorphism group has
already appeared in Poisson geometry--it is the curl (``rotationnel''
in French) of the Poisson structure. Introduced without a name
by Koszul \cite{ko:crochet}, who already noted its modular nature in
the case of the dual of a Lie algebra, the curl was named in
\cite{du-ha:rotationnels}, where it was used in the classification of
quadratic Poisson structures (see also \cite{li-xu:quadratic}).
Finally, Jean-Luc Brylinski and Gregg Zuckerman have
recently studied the modular vector field in the context of complex
analytic Poisson geometry \cite{br-zu:outer}.
For background material on Poisson manifolds and operator algebras, we
refer the reader to \cite{va:lectures} and \cite{co:noncommutative}
respectively.
I would like to thank Dimitri Shlyakhtenko for reviving my interest in
modular theory and tutoring me in the theory of von Neumann algebras,
and Jean-Luc Brylinski, Alain Connes, Sam Evens, Viktor Ginzburg,
Jiang-hua Lu, Piotr Podles, Marc Rieffel, Ping Xu and Ilya Zakharevich
for helpful comments.
It is a pleasure to dedicate this paper to Professor Andr\'{e}
Lichnerowicz, for his personal support over many years, as well as for
the stimulation which his work has provided for much of my own,
including the particular research project reported upon here. I
regret only not having paid more attention at the time to the seminar talk
which he gave at Berkeley on the subject matter of
\cite{ba-fl-li-st:deformation}.
\section{Definition of the modular vector field}
\label{sec-definition}
Let $P$ be a Poisson manifold with Poisson tensor $\pi$, and choose a
positive smooth density $\mu$ on $P$. To this data we associate the operator
$\phi _{\mu }:f\mapsto \dive _{\mu }H_{f}$, where $H_{f}$ is the
hamiltonian vector field of $f$, and the
divergence $\dive _{\mu }\xi $ of a vector field $\xi $ is the
function $ {\cal L}_{\xi}\mu/\mu$. (${\cal L}_{\xi }$ is the Lie
derivative by $ \xi $.) Although $\phi
_{\mu }$ appears to be a second-order operator (a kind of
``laplacian''), a simple computation using the antisymmetry of the
Poisson tensor shows that $\phi _{\mu }$ is in fact a
derivation and hence a vector field; we call it the {\bf
modular vector field} of $(P,\pi)$ with respect to the density
$\mu$. A further calculation shows that ${\cal L}_{\phi _{\mu }}\mu $
and ${\cal L}_{\phi _{\mu }}\pi $ are both zero.\footnote{See Section
\ref{sec-regular}.}
The modular vector field $\phi _{\mu }$ is zero precisely when
$\mu $ is an invariant density for the flows of all hamiltonian
vector fields. In this case, simply refer to $\mu $ as an
{\bf invariant density} for the Poisson manifold, and
we call the Poisson manifold {\bf unimodular}.
If we replace $\mu$ by $a\mu$, where $a$ is a
positive function, the modular vector field becomes $\phi _{a\mu
}=\phi _{\mu }+ H_{-\log a}$. We conclude
that the modular vector field is well-defined modulo hamiltonian
vector fields. In other words, the set of
modular vector fields for all possible positive densities is an
element of
the first Poisson cohomology space of $P$ (Poisson
vector fields modulo hamiltonian vector fields). We call it the {\bf
modular class} of the Poisson manifold. It vanishes just for
unimodular Poisson manifolds.
In algebraic terms, the modular class is a derivation of the Poisson
algebra $C^{\infty}(P)$, modulo inner derivations. When we integrate
a particular modular vector field (assuming it to be complete),
the result is a 1-parameter subgroup in the group
$\mbox{Aut}(C^{\infty}(P))$ of automorphisms of the Poisson algebra
which is intrinsic modulo the subgroup $\mbox{Aut}_{0}(C^{\infty}(P))$
consisting of those automorphisms obtained by integrating
time-dependent hamiltonian vector fields. More precisely, the flows
of $\phi _{\mu }$ and $\phi _{a\mu }$ are related by a canonical
one-cocycle on $\reals $ with values in the group $\Gamma (P)$ of exact
lagrangian bi-sections of the symplectic groupoid of $P$. (See Section
\ref{sec-groupoids}, where we will also explain how the elements
$\mbox{Aut}_{0}(C^{\infty}(P))$ play the role of inner automorphisms.
A hint of this relation is already given in \cite{re:groupoid}: the
group $\Gamma (P)$ is in some sense the classical limit of the unitary group
in the multiplier algebra of the algebra whose classical limit is
$C^{\infty}(P)$.)
The modular vector field can also be viewed \cite{ko:crochet} as the
result of applying a differential operator to the Poisson tensor
itself. A smooth density $\mu $ on $P$ sets up an isomorphism $\alpha\mapsto
\alpha\backl \mu$ (defined
modulo a local choice of sign, which disappears in what follows) between
differential forms and multivector fields on $P$, so that the exterior
derivative becomes an operator on multivector fields. Applying this
to the bivector field $\pi $ yields its modular vector field.
When the modular vector field is zero, the form $\pi\backl \mu $ is
closed and therefore defines a deRham cohomology class, which is dual to
a homology class in $H_{2}(P,{\Bbb R})$.\footnote{Here we
must assume $P$ oriented or use twisted coefficients.} As $\mu$ runs
over all choices of invariant densities, we obtain a convex cone in
$H_{2}(P,{\Bbb R})$ whose elements may be called the {\bf fundamental
cycles} of the Poisson structure. If there exists an invariant
density of finite total volume, we can restrict attention to invariant
densities of total measure 1, in which case we obtain another convex
set--the {\bf normalized} fundamental cycles. We discuss these invariants in
further detail in Section \ref{sec-fundamental} below.
The modular vector field is also related to the canonical homology of
Koszul and Brylinski (see \cite{ko:crochet} and \cite{va:lectures}),
given by the complex in which the chains are differential forms and
the boundary operator is $\delta =i_{\pi }\circ d - d\circ i_{\pi }$.
Suppose that $P$ is oriented, so that we can identify
densities with differential forms of top degree. A density $\mu $ is
thus a top-dimensional chain for Poisson homology. Its boundary
$\delta \mu $ is equal to $-d(\pi \backl \mu )=-\pi \backl \phi _{\mu
}.$ Thus, the modular vector field corresponds to the ($d$ and $\delta
$ exact) $n-1$ form $\delta \mu =-d(\pi \backl \mu ).$ In the
unimodular situation, an invariant density $\mu $ is a
cocycle and thus defines a nonzero element of the top-degree Poisson
homology of $P$. Also note that the zeroth Poisson homology
$C^{\infty}(P)/\{C^{\infty}(P),C^{\infty}(P)\}$ is dual to the
``traces'' on $C^{\infty}(P).$ Such (smooth) traces are given
precisely by the invariant densities. (See Section \ref{sec-operator}
below.)
This relation between Poisson homology spaces in complementary
dimensions is evidently related to Poincar\'{e} duality. In fact,
such duality (or at least a pairing) has been studied recently by
Evens, Lu, and the author \cite{ev-lu-we:poincare}, as well as by
Brylinski and Xu.
\section{Relation to operator algebras}
\label{sec-operator}
For a von Neumann algebra $A$, the point of departure for the definition
of the modular automorphism group is the choice of a weight: a certain
type of positive linear functional on $A$. The modular automorphism
group measures the extent to which the weight fails to be a trace,
i.e. to vanish on commutator brackets. According to the
noncommutative Radon-Nikodym theorem, any other weight is related to
the first by the inner automorphisms associated to a cocycle, with
values in the unitary group of $A$.
Weights on the Poisson algebra of a Poisson manifold $P$ are by
definition positive (Borel) measures on $P$. To fix a measure class
(which corresponds to the selection of a particular enveloping von
Neumann algebra for an incomplete *-algebra), we choose the smooth
densities. (A discussion of more general densities, and the
corresponding modules obtained by a Poisson version of the GNS
construction, is planned for \cite{we:modules}.) Now recall that the
modular vector field measures the extent to which hamiltonian vector
fields are divergence free. Since, for compactly supported $g$, we
have by Stokes' theorem: $$ \int_{P}\{f,g\}\mu =\int_{P}g{\cal
L}_{H_{f}}\mu, $$ it follows that the modular vector field also
measures the extent to which integration with respect to $\mu $
vanishes on Poisson brackets (with at least one entry compactly
supported), i.e. when integration with respect to $\mu $ fails to be a
``Poisson trace''.
The relation
$$ \int_{P}\left(\{f,g\}-(\phi _{\mu f})g\right)\mu =0 $$
is called the {\bf infinitesimal KMS condition} relating the weight
$\mu $ to the vector field $\phi _{\mu }.$
\section{Some examples.}
\label{sec-examples}
The modular class of a symplectic manifold is zero. In fact, the
Liouville density associated to the symplectic structure is invariant
under all hamiltonian flows, so the corresponding modular vector field
is zero. If we take instead the density $a\mu $, where $\mu$
is the Liouville density, we obtain as modular vector field the hamiltonian
vector field $H_{-\log a}$. Writing $E$ for the hamiltonian $-\log
a$, we find a natural association between the hamiltonian flow of the
function $E$ and the density $e^{-E}\mu$. This association,
familiar in classical statistical mechanics, was the starting point
for the development of the KMS theory in quantum statistical
mechanics, which was subsequently shown to be essentially equivalent
to Tomita-Takesaki theory. (A geometric interpretation of the KMS theory in
terms of conformal deformations of Poisson brackets and their
quantizations was given in \cite{ba-fl-li-st:deformation}.)
If $P$ is the dual of a Lie algebra $\frak g$, with its Lie-Poisson
structure, the modular vector field with respect to any
translation-invariant density is the constant vector field with value
$\mbox{tr ad}$, the trace of the adjoint representation, or {\bf
modular character} of the Lie algebra. (This was already observed in
\cite{ko:crochet}.) In particular, the modular
class of ${\frak g}^{*}$ is zero just when ${\frak g}$ is
unimodular, since a hamiltonian vector field must vanish at the origin
in any Lie-Poisson structure. This fact motivates our use of the term
unimodular to describe Poisson structures with zero modular class.
(Such structures were called ``exact'' in
\cite{li-xu:quadratic}.)
At any singular point of a Poisson manifold, the projection of the
modular vector field into the normal space of the symplectic leaf is
equal to the modular character of the transverse Lie algebra.
Consequently, the transverse Poisson structure (and hence the Poisson
structure itself, at least locally) admits a 1-parameter group of
symmetries in this direction, even if the transverse structure is not
linearizable. (See \cite{we:poisson structures} and \cite{we:poisson
geometry} for discussion and examples of nonlinearizable structures.)
If $P$ is 2-dimensional, with Poisson
structure given in coordinates by $\{x,y\}=f(x,y)$, then the modular
vector field with respect to $\mu =|dx\wedge dy|$ is the same as the
hamiltonian vector field for $f$ {\em with respect to the canonical
bracket} $\{x,y\}=1$. In particular, the modular vector field is
tangent to the zero level of $f$, which is the singular set of the
Poisson structure, and the restriction of the modular vector field to
this singular set is invariantly attached to the Poisson structure.
For the Lie-Poisson structure on $\reals ^{2}$ with defining relation
$\{x,y\}=y$, the modular flow with respect to translation-invariant
measures is given by translations in the $x$-direction. The upper
half-plane $H^{+}$ in this space is a symplectic manifold. The smooth
measures on $H^{+}$ with smooth continuations to $\reals ^{2}$
could be thought of as a measure class on $H^{+}$ with
distinguished behavior ``at infinity'', giving rise to a nontrivial
modular flow at infinity.
The Poisson manifolds with boundary which are locally equivalent to
the product of the upper half plane with symplectic manifolds are exactly
the $b$-symplectic manifolds of Nest and Tsygan \cite{ne-ts:formal},
arising by generalization from the $b$-cotangent bundles of manifolds
with boundary studied by Melrose \cite{me:atiyah}. The modular vector
field on the boundary gives the obstruction to the existence of a
trace on algebras of $b$-pseudodifferential operators, or more
generally on quantized Poisson algebras of $b$-symplectic manifolds.
As was noted and exploited in \cite{du-ha:rotationnels} and
\cite{li-xu:quadratic}, the modular vector field of a quadratic
Poisson structure on a vector space (with respect to
translation-invariant measures) is a linear vector field (which is an
invariant of the modular class). For instance, the modular flow for
the structure $\{x,y\}=x^{2}+y^{2}$ consists of rotations around the
origin. Again, we could consider this as the modular flow at
``infinity'' (here represented by the origin) for the symplectic
structure on the punctured plane, with boundary conditions determined
by extendibility over the puncture. This Poisson structure is also
the local model for the singularity of the Bruhat-Poisson structure on
$S^{2}$. (See \cite{lu-we:classification}.)
The modular vector field has also been calculated in
\cite{ev-lu-we:poincare} for the Bruhat-Poisson structures on
higher-dimensional flag manifolds, as well as for the related
compact Poisson Lie groups. The nonvanishing of the modular class for
these manifolds seems to be related to the fact that the ``Haar
measure'' on the corresponding quantum groups is not a trace, but
rather satisfies a KMS-type condition in which the modular
automorphism is related to the square of the antipode. The recent
paper \cite{ma-na:neumann} contains an extensive discussion of quantum
groups from the von Neumann algebra point of view; a study of this
paper from the Poisson algebra point of view should give some new
insight both into quantum groups and their classical limit.
It would also be interesting to interpret the modular vector field on the
flag manifold in terms of the geometry at infinity of the ``big cell''
(an open dense symplectic leaf).
We close this section with a possible application of the modular
class. Tuynman \cite{tu:reduction} has pointed out a correction which should
be made to geometric quantization in order to make it compatible under
reduction by a nonunimodular group $G$ acting on a symplectic manifold
$P$. It seems that there should be an interpretation of his
results in terms of the nonunimodularity of the Poisson manifold
${\frak g}^*$ and that of $P/G$, which is Morita equivalent to ${\frak
g}^*$ when the $G$ action is free. In
fact, the first Poisson cohomology spaces of Morita equivalent Poisson
manifolds are isomorphic, according to Ginzburg and Lu
\cite{gi-lu:poisson}, and Ginzburg \cite{gi:private} has shown
that the modular classes
are compatible with this isomorphism.
\section{Regular Poisson manifolds}
\label{sec-regular}
Near any regular point $x$ on a Poisson manifold, we can introduce
canonical local coordinates and hence a measure which, near $x$, is
invariant under all hamiltonian flows. The modular vector field with
respect to this measure is therefore zero near $x$. Since $x$ is
arbitrary, the modular vector field with respect to {\em any} measure
is locally hamiltonian throughout the set of regular points of
$P$.\footnote{In \cite{br-zu:outer}, the modular vector field is
considered as a section of a sheaf of Poisson modulo hamiltonian
vector fields, so that it is actually supported on the set of singular
points. An advantage of this framework is that the modular vector
field is defined even when there is no global volume element, a
situation which often occurs in the holomorphic setting.} In
particular, any modular vector field is tangent to all the
regular symplectic leaves. \footnote{Since the regular points of any
Poisson manifold form a dense subset, this argument also gives a quick
proof that ${\cal L}_{\phi _{\mu }}\mu $ and ${\cal L}_{\phi _{\mu
}}\pi $ are zero on all of $P$.}
On the other hand, global conditions can cause the modular class of a
regular Poisson manifold to be nonzero. For example, we consider the
regular Poisson structures on $\reals ^{2}\times S^{1}$, with
coordinates $(x,y,\theta )$, of the form $\pi =\frac{\del }{\del
y}\wedge (\frac{\del }{\del \theta }+g(x)\frac{\del }{\del x})$, where
$g(x)=0$ just at $x=0$. The symplectic leaves for this structure
consist of the cylinder $C$ defined by $x=0$ and a family of planes
which spiral around this cylinder.
For $\mu =d\theta \wedge dx\wedge dy$, we
have $\pi \backl \mu = dx+g(x)d\theta $, $d(\pi \backl \mu
)=g'(x)dx\wedge d\theta $, and hence $\phi _{\mu }=-g'(x)\frac{\del }{\del y}$.
If $g'(0)\neq 0$, the restriction of $\phi _{\mu }$ to the cylinder
$C$ is a nonzero multiple of $\frac{\del }{\del y}$, which is not
hamiltonian, so the modular class of this Poisson structure is
nonzero. Perhaps more surprising is that, if $g'(0)=0$, $\phi_{\mu }$
is still not hamiltonian, even though its restriction to each
symplectic leaf is hamiltonian. (It is zero on $C$, and each of the
remaining leaves is simply connected.) To see this, we look at the
most general hamiltonian vector field
$$ H_{f}=\tilde{\pi }(df)=\frac{\del f}{\del y}(\frac{\del }{\del
\theta }+g(x)\frac{\del }{\del x})-(\frac{\del f}{\del \theta
}+g(x)\frac{\del f}{\del x})\frac{\del }{\del y}. $$ If $H_{f}$ is to
equal $\phi _{\mu }$, we must have
$$ \frac{\del f}{\del y}=0 ~~\mbox{and}~~\frac{\del f}{\del \theta
}+g(x)\frac{\del f}{\del x}=g'(x).$$
To see that this is impossible, we introduce the averaged hamiltonian function
$F(x)=\int_{0}^{2\pi }f(x,y,\theta )d\theta $ (independent of $y$ by
the first equation above), and we integrate the second equation above
with respect to $\theta$ to get
$$ g(x)F'(x)=g'(x). $$ For $x\neq 0$, we have $F(x)=\ln |g(x)|+C$,
where $C$ is constant on each semiaxis. As $x\rightarrow 0$,
$g(x)\rightarrow 0$ implies that $F(x)\rightarrow \infty$, which is
impossible if $f$ is continuous.
The modular vector field of a regular Poisson manifold is closely
related to an object which depends only on the foliation by symplectic
leaves. If ${\cal F}$ is a foliation on $P$ with tangent bundle
$F\subset TP$, we can define its modular class in the following way.
Choose a smooth transverse positive density $\nu $ for ${\cal F}$; i.e
a nowhere-vanishing section of
the highest exterior power of the normal bundle $TP/F$ (which for
simplicity we assume to be oriented--otherwise the usual twisting by
an orientation bundle is needed). If we denote by $\nabla $ the
Bott connection of the foliation (extended to densities), then $\nabla
\nu /\nu $ is a well-defined closed (because the Bott connection is
flat) 1-form along the leaves of ${\cal F}$, which we will denote by
$\psi _{\nu }$. The integral of $\psi _{\nu }$ around a loop in a
leaf of ${\cal F}$ is the logarithm of the determinant of the
linearized holonomy of the loop, and form $\psi _{\nu }$ is zero
exactly when $\nu $ defines an invariant transverse measure to the
foliation. If we multiply $\nu $ by a positive function $a$,
$\psi _{\nu }$ changes to $\psi _{\nu }+d_{\cal F}(\ln a)$, so that
$[\psi _{\nu }]$ is a well defined class in the tangential cohomology
of the foliation ${\cal F}$, which we call the {\bf modular class of
the foliation}.
On any regular Poisson manifold, multiplication by the canonical
symplectic density along the leaves gives a 1-1 correspondence between
the transverse densities to the symplectic leaf foliation and the
densities on the ambient manifold. Given a transverse density $\nu $,
we can apply the Poisson tensor to the closed form $\psi _{\nu }$
along the leaves to obtain a locally hamiltonian vector field tangent
to the leaves which is precisely the modular vector field $\phi _{\mu
}$ associated to the density $\mu $ corresponding to $\nu $.
When the symplectic leaf foliation is co-oriented of codimension 1, a
transverse density is given by a 1-form $\lambda $ which annihilates
the leaves. Integrability means that $d\lambda =\alpha \wedge \lambda
$ for a 1-form $\alpha $ defined up to a multiple of $\lambda $. The
restriction of $\alpha $ to the leaves depends only on $\lambda $ and
is in fact $\psi _{\lambda }$. When the Poisson structure is
unimodular, the form $\lambda $ can be chosen to be closed, so that
$\alpha \wedge d\alpha $, which represents the {\bf Godbillon-Vey
class} of the foliation, is zero. Hence a nonvanishing Godbillon-Vey
class implies nonunimodularity. The much stronger, but more
difficult to prove, theorem of Hurder-Katok
cited on p. 261 of \cite{co:noncommutative}, establishes that the
nonvanishing of the Godbillon-Vey class implies the nonexistence of any
invariant transverse measure, smooth or not.
\noindent
{\bf Example.} On the group $PSL(2,\reals)$ we have the
basis $(\bolde _{1},\bolde _{2},\bolde _{3})$ of left-invariant vector
fields, satisfying the commutation relations $$[\bolde _{1},\bolde
_{2}]=\bolde _{3}, ~[\bolde_{3},\bolde _{1}]=\bolde _{1},~[\bolde
_{3},\bolde _{2}]=-\bolde _{2}.$$ These pass to any quotient $PSL(2,\reals
)/\Gamma $ by a discrete subgroup $\Gamma $ acting from the left, as
does the dual basis $(\omega _{1},\omega _{2},\omega _{3})$. We
consider the Poisson structure $\pi =\bolde _{2}\wedge \bolde _{3}$
and the density $\mu =\omega _{1}\wedge \omega _{2}\wedge \omega
_{3}$, with respect to which the modular vector field is $-e_{2}$. If
$\Gamma $ is the fundamental group of a compact Riemann surface $M$,
then $PSL(2,\reals )$ is the unit tangent bundle of $M$, $\bolde _{3}$
is the geodesic flow for the Poincar\'{e} metric, and the foliation by
symplectic leaves of $\pi $ is the foliation by stable manifolds. The
symplectic form along the leaves is the area form for the natural
induced metric on the tangent bundle, and the modular vector field is
the generator of the horocycle flow! Since $e_{1}$ is the direction
of the unstable manifolds, if we travel around a closed geodesic, the
integral of the modular 1-form (which is $\omega _{3}$) must be
nonzero, so the modular class is nontrivial.
We can easily compute the Godbillon-Vey class in this example. Taking
$\omega _{1}$ as our transverse density $\lambda $, we have $d\omega
_{1}=\omega _{1}\wedge \omega _{3}$, so $\alpha =-\omega _{3}$, and
$\alpha \wedge d\alpha =\omega _{3}\wedge d\omega _{3}=-\omega
_{3}\wedge \omega _{1}\wedge \omega _{2},$ a volume form which gives a
nonzero Godbillon-Vey class.
It is interesting to compare our discussion with that on p. 58 of
\cite{co:noncommutative}. It is stated there that when the stable
foliation is ``suspended'' to the bundle of transverse densities, the
resulting foliation of type II has the same space of leaves as the
horocycle flow. We give a geometric explanation for this occurrence of the
horocycle flow in Section \ref{sec-weights}.
The Reeb foliation on $S^{3}$ provides another instructive example.
Its modular class is nonzero, since the contractive nature of the
holonomy around the central torus precludes the existence of a smooth
positive invariant transverse measure. However, the linearized
holonomy is zero around the torus, so the modular vector field can be
zero there. Since all the other leaves are planes, the modular
vector field is hamiltonian on every leaf separately, but it is not
globally hamiltonian. On the other hand, in this case, the
Godbillon-Vey class is zero. (See for example \cite{ta:topology}.)
The situation for regular Poisson manifolds suggests a point of view for an
arbitrary Poisson manifold $P$. A density on $P$ should be thought of
as a way of ``encoding'' a transverse measure to the foliation by
symplectic leaves, and the modular vector field measures the extent to
which this transverse measure fails to be ``invariant under
holonomy''. This point of view may shed some light on the
problem of defining the notion of transverse structure, holonomy, and
modular classes for singular foliations. (Compare
\cite{da:feuilletages} and \cite{su:holonomy}.)
Some further insight in this direction may be found in our discussion
of Lie algebroids in Section \ref{sec-groupoids}.
Given any foliation of a manifold $M$, the cotangent bundle $P$ along the
leaves is a Poisson manifold in a natural way. A transverse density
to the first foliation pulls back to a transverse density to the
symplectic foliation of $P$, and the modular 1-form pulls back
accordingly. Thus, the modular vector field of the cotangent bundle
to a foliation is generated by the pullback of the modular 1-form of
the foliation. In this way, the modular construction for Poisson
manifolds is a generalization, as well as a special case (!) of the
construction for foliations.
Finally, we note that Mikami \cite{mi:foliations} has also studied the
Godbillon-Vey class in the context of Poisson geometry.
\section{Intrinsic completeness}
\label{sec-completeness}
The modular flow of an operator algebra is always an action of the
real numbers. For a Poisson manifold, the modular vector field
defines an action of the real numbers only when it is complete. For a
given Poisson structure, this completeness may depend on which
representative of the modular class is chosen. (For instance, any
hamiltonian vector field on a symplectic manifold is a modular vector field.)
We will call a Poisson manifold {\bf intrinsically complete}
if {\em some} representative of the modular class is a complete vector
field.
Intrinsic completeness is clearly an invariant property of a Poisson
manifold. It would be nice to have a striking characterization of
this property, and some interpretation of its meaning (perhaps in
connection with some form of quantization), but we must content
ourselves here with some examples.
An open subset of $\reals ^{2}$ with the Poisson structure $\{x,y\}=y$
is intrinsically complete if and only if it contains either all or
none of the $x$-axis, since every modular vector field is constant and
nonzero on that axis. Of course, every symplectic manifold is
intrinsically complete. On the other hand, even a regular Poisson
manifold can fail to be intrinsically complete. For instance,
although the Poisson structures on $\reals ^{2}\times S^{1}$ described
in the previous section are intrinsically complete, the restriction to
the region $|y| < 1$ is not intrinsically complete when $g'(0)\neq 0$.
(It is not clear what happens when $g'(0)=0.)$ This follows from the
following lemma.
\begin{lemma}
\label{lemma-cylinder}
On the cylinder $(-1,1)\times \reals $ with symplectic structure
$dy\wedge d\theta ,$ every complete, locally hamiltonian vector field
is globally hamiltonian.
\end{lemma}
{\bf Proof.} Let $X$ be a locally hamiltonian vector field, $\alpha
=X\backl (dy\wedge d\theta) $. The integral of $\alpha $ around a loop
encircling the cylinder gives the flux of area through the
loop under the flow of $X$. Since the cylinder has finite area, this
flux must be zero, so $\alpha $ is exact, and $X$ is globally
hamiltonian.
{\bf QED}
\section{Modular classes of Lie algebroids and groupoids}
\label{sec-groupoids}
The foliation and Lie-Poisson examples are special cases of a more
general construction. Given a Lie algebroid ${\cal A}$, the dual
bundle carries a natural Poisson structure. The modular vector field
of this structure with respect to a suitably chosen density is tangent
to the fibres of ${\cal A}^{*}$ and translation invariant along each
fibre. Thus this vector field can be identified with a section of
${\cal A}^{*}$. As such, it is a 1-cochain for the Lie algebroid
cohomology \cite{ma:lie} of ${\cal A}$; it turns out to be a cocycle
whose cohomology class is well defined. We have therefore attached to
each Lie algebroid a {\bf modular class} in its first Lie algebroid
cohomology.
In fact, this class can be defined directly in terms of the Lie
algebroid itself. We summarize here the detailed discussion which
can be found in \cite{ev-lu-we:poincare}.
Given a Lie algebroid $\alg$ over $P$, we introduce the bundle
$Q_{\alg }=\wedge^{top}\alg \otimes \wedge ^{top}T^{*}P$. When $\alg
$ is an integrable subbundle of $TP$, this is the bundle whose
sections are (not necessarily invariant or positive) transverse smooth
measures to the corresponding foliation, so we should think of
sections of $Q_{\alg }$ in general as being ``transverse measures to $\alg $.''
It turns out that the ``difference'' between Lie derivative
operators on $\wedge ^{top}\alg ^{*}$ and $\wedge ^{top}T^{*}P$
defines a representation of $\alg $ on the line bundle $Q_{\alg }$.
Any nowhere vanishing section of this line bundle (or its square, in
case the ``transverse space is nonorientable'') has a ``divergence''
which gives an invariantly defined class $\theta _{\alg }$, the
{\bf modular class} of $A$, in the Lie algebroid cohomology of $\alg $ (with
coefficients in the trivial line bundle). This construction
reproduces the class defined above using the Poisson structure on
$\alg ^{*}$.
Since the cotangent bundle of a Poisson manifold is a Lie algebroid,
we can consider the modular class of this Lie algebroid. But the Lie
algebroid cohomology of $T^*P$ is just the Poisson cohomology of $P$,
and in fact we arrive back at the modular class of the Poisson
manifold $P$ itself, if to each density $\mu $ on $P$ we construct a section
$\rho _{\mu }$ of $Q_{TP}$ by first taking the ``square'' of $\mu $ as a section of
$Q_{TP}$ and then taking its square root in the same line bundle.
We turn now to Lie groupoids. Once again, we give here a brief
description of results to be presented in more detail in
\cite{ev-lu-we:poincare}. The modular class of a Lie algebroid is
in fact the infinitesimalization of the modular class of a
corresponding (local) Lie groupoid, which may be defined as a ratio of
right- and left-invariant ``measures,'' as in
\cite{re:groupoid}, Chapter 1, Section 3. For a Lie groupoid with Lie
algebroid $\alg $ we can
give an description of the modular class which does not require the
separate choice of a Haar system and a measure on the base. Instead,
we choose a section of $Q_{\alg }$, which plays the role of these two
objects simultaneously.
In fact, it can be shown that $G$ acts in a
natural way on the line bundle $Q_{\alg }$, so that a trivialization
of $Q_{\alg }$ turns this action into a 1-cocycle on $G$, with values
in the multiplicative group $\reals ^{\times }$ of the real numbers;
this cocycle is called the {\bf modular function} of the groupoid with
respect to the given section.\footnote{Multiplication by powers of
this function gives the modular automorphism group of the von Neumann
algebra of the groupoid; see \cite{re:groupoid}, p. 115.} A change of
trivialization corresponds to a zero cocycle, whose coboundary gives
the change in the modular function, so that the {\bf modular class} is
a well defined element of $H^{1}(G;\reals ^{\times }).$
Differentiation of this modular class (more precisely, of its
representative cocycles) along the identities of $G$ gives the modular
class of the Lie algebroid $\alg $.
When $\alg $ is the cotangent bundle of a Poisson manifold $P$, then
the groupoid $G$ can be taken to be a symplectic groupoid for $P$, if
it exists.\footnote{See \cite{co-da-we:groupoides}
or\cite{va:lectures} for a discussion of symplectic groupoids, and
\cite{we-xu:extensions} for the lifting of Poisson automorphisms to
groupoid automorphisms.} This gives the following picture. Given a
density $\mu $ on $P$, its modular flow\footnote{All flows in this
paragraph will be local, if necessary.} lifts in a natural way to a
flow on the symplectic groupoid $G$ by symplectic groupoid
automorphisms. This lifted flow is hamiltonian and is generated by
the logarithm of the modular function $f$ on $G$. When $f$
is the coboundary of a function $h$ on $P$, the logarithm of
$h$ is a hamiltonian for the modular flow on $P$. In this situation,
we can now give a precise sense in which the modular flow is
``inner''. (Compare \cite{re:groupoid}, p. 111.) Namely, the
pullback of $f$ to $G$ by the source map of this groupoid itself
generates a hamiltonian flow on $G$ which moves the identity section
into a 1-parameter subgroup of the group of $\Gamma (P)$ of exact
lagrangian bi-sections in $G$. The action of this 1-parameter
subgroup by conjugation on $G$ is exactly the lifted modular flow, and
its restricted action on the identity section is the modular flow
itself. But the group $\Gamma (P)$ should be thought of as being
essentially a classical limit of the unitary elements in the
(multiplier algebra of) the ``quantized'' algebra whose classical
limit is the functions on $P$. (The Lie algebra of $\Gamma (P)$
consists of the real-valued functions on $P$, which is the classical
limit of the self-adjoint elements of the quantized algebra.)
We note here that an $\reals ^{\times }$-valued 1-cocycle $\phi $ on
the symplectic groupoid $G$ gives rise to a 1-parameter group of
automorphisms of the groupoid algebra of $G$ in two different ways:
first by multiplication by the powers $\phi ^{it}$ (see
\cite{re:groupoid}), and second by the hamiltonian flow of $\log |\phi
|$. The relation between these two groups of automorphisms is not
completely clear to us, but it must somehow involve the passage, via
quantization as in \cite{we-xu:extensions}, from the groupoid algebra
of $G$ to the quantized algebra of functions on the underlying Poisson
manifold $P$.
We also remark that the inversion map in the symplectic groupoid $G$
should be the canonical transformation underlying the *-operator in
the quantized algebra of functions on $P$, but there may also need to
be an ``amplitude'' as in the theory of Fourier integral operators.
We suspect that this amplitude is related to the modular function via
the operator-algebraic construction of the latter, as described for instance in
\cite{ri-va:bounded}. We also suspect a relation with the square of
the antipode in the ``hopfoid algebra'' of functions on the groupoid,
which should lead us back to the ideas about quantum groups mentioned
at the end of Section \ref{sec-examples}.
\section{The flow of weights}
\label{sec-weights}
In the theory of von Neumann algebras, the modular flow leads to two
ways of passing between algebras of type II and type III. In this
section, we discuss the Poisson analogues of those constructions.
Given a Poisson manifold $P$ with positive density $\mu $ and
corresponding modular vector field $\phi _{\mu }$, we can construct
the Poisson semidirect product of $P$ by this modular flow (even
if it is not complete). This Poisson analogue of the crossed product of
an algebra by a 1-parameter group of automorphisms is defined as
follows.
Given a Poisson vector field $\phi $ on a Poisson manifold $P$, we
define the {\bf semidirect product}\footnote{This construction may be
found in Section 2 of \cite{kr-ma:hamiltonian}, or the Appendix of
\cite{we:poisson geometry}. The name comes from the fact that, when
$P$ is the dual of the Lie algebra on which a group acts by
automorphisms, the construction produces the dual of the semidirect
product Lie algebra.} to be the quotient of $P\times T^{*}\reals $ by
the diagonal action of $\reals$, which acts on its own cotangent
bundle by left translations. Although this construction appears to
depend on the integration of $\phi$ to a flow, we can identify the
quotient with $P\times \reals ^{*}$, on which the induced Poisson
structure is given purely in terms of the Lie algebra action by the
requirement that the projections on $P$ and $\reals ^{*}$ be Poisson
maps, and that $\{f,\tau \}=\phi \cdot f$, where $f$ is any function on
$P$ and $\tau $ is the standard coordinate on $\reals ^{*}$.
(Our notation has been chosen to suggest how this
construction can be extended to the case where $\reals $ is replaced
by an arbitrary Lie algebra acting on $P$ by Poisson vector fields.)
We denote $P\times \reals^{*} $ with the resulting Poisson structure
by $P\times_{\phi } \reals ^{*}$. If the vector field $\phi$ is
globally hamiltonian, then the choice of a hamiltonian function
produces an isomorphism between $P\times_{\phi } \reals^{*} $ and the
direct product $P\times \reals ^{*}$ (see \cite{kr-ma:hamiltonian}).
More generally, the Poisson isomorphism class of the semidirect product depends
on only the equivalence class of $\phi $ modulo globally hamiltonian vector
fields.
Applying the semidirect product construction to the modular flow, we
obtain a Poisson manifold $P\times _{\phi _{\mu }}\reals ^{*}$ which
is determined up to isomorphism by the Poisson manifold $P$ itself.
(This is the Poisson counterpart of a result in
\cite{ta:automorphisms}, where it is established that the cross
product by the modular flow is weight-independent.) It turns out that
$P\times _{\phi _{\mu }}\reals ^{*}$ is a unimodular Poisson manifold;
in fact, it is simple to check that $\mu \wedge e^{\tau }d\tau $ is an
invariant density.
The {\bf flow of weights}, denoted by ${\bf mod}(P)$, is defined to be
the flow of the vector field $\frac{\del }{\del \tau }$ on the space
of leaves of $P\times _{\phi _{\mu }}\reals ^{*}$.
In case $P$ is a regular Poisson manifold, the objects introduced
above have a rather simple description. The symplectic leaves of
$P\times _{\phi _{\mu }}\reals ^{*}$ are coverings of the symplectic
leaves of $P$; in fact, they are the parallel (multivalued) sections
for a flat connection along the symplectic leaves of $P$ on the bundle
$P\times \reals^{*} $ Upon identification of $P\times \reals^{*} $
with the space of transverse densities to the symplectic leaf
foliation (via exponentiation and multiplication by the transverse
density associated with $\mu $), this connection is just the usual
Bott connection. This gives a completely intrinsic description
(corresponding to the ``functorial construction'' on p. 496 of
\cite{co:noncommutative}) of the semidirect product manifold $P\times
_{\phi _{\mu }}\reals ^{*}$ and hence of the flow of weights: the flow
is given by the action of $\reals^{+} $ by multiplication on the
``space of multivalued invariant transverse densities'' to the
symplectic leaf foliation.
Note that the flow of weights, unlike the modular flow itself, depends
on only the symplectic leaf foliation. In fact, it corresponds
precisely to the flow of weights for a foliation algebra as described
in Proposition 9c on p. 58 of \cite{co:noncommutative}. We also
remark that any group of Poisson automorphisms of $P$ acts naturally
on the flow of weights.
The second way to pass from a general Poisson manifold to a unimodular
one is to divide by the modular flow. Of course, the resulting
quotient, which we denote by $P/\phi _{\mu }$, is a manifold, even
locally, only if the modular vector field is either identically zero
or nowhere zero. In the latter case, it can be shown at least
formally (i.e. without worrying about the global structure of quotient
spaces) that the semidirect product $P\times _{\phi _{\mu }}\reals
^{*}$ is Morita equivalent as a Poisson manifold \cite{xu:morita poisson} to
the quotient space. (Locally, the semidirect product is the product of the quotient space
by a 2-dimensional symplectic manifold.)
We can show directly that the leaf spaces of $P\times _{\phi _{\mu
}}\reals ^{*}$ and $P/\phi_{\mu }$ are isomorphic when $P$ is regular
and $\phi _{\mu }$ is nowhere zero. To do this, we restrict attention
to one symplectic leaf ${\cal O}$ of $P$ at a time.
Let $F\subset T{\cal O}$ be the symplectic orthogonal space to $\phi
_{\mu }$; it is also the null bundle of the closed 1-form $\psi _{\mu
}$ corresponding to $\phi _{\mu }$ via the symplectic structure on
${\cal O}$. Choose a vector field $X$ on ${\cal O}$ such that $\psi
_{\mu }(X)=1$, and let $\tilde{X}$ be its horizontal lift to ${\cal
O}\times \reals ^{*}$ for the ``Bott connection''. $\tilde{X}$ is a
complete vector field, and $d\tau (\tilde{X})=1$ , so the coordinate function
$\tau $ is a fibration to $\reals^{*} $ on each leaf of ${\cal O}\times
_{\phi _{\mu }}\reals ^{*}$. Hence, the intersection of each such
leaf with $\tau ^{-1}(0)$ is connected and must be a leaf of the
foliation ${\cal F}$ determined by $F$. Thus, the symplectic leaf
space of ${\cal O}\times _{\phi _{\mu }}\reals ^{*}$ is isomorphic to
the leaf space of ${\cal F}$. On the other hand, it is a basic fact
about Poisson reduction that the symplectic leaf
space of ${\cal O}/\phi _{\mu }$ is the same as
the leaf space of ${\cal F}$.
In particular, the symplectic leaf space for $P\times _{\phi _{\mu
}}\reals ^{*}$ in the example of $PSL(2,\reals )/\Gamma $ in Section
\ref{sec-regular} is the same as the space of leaves of the horocycle
foliation. (When the symplectic leaves of $P$ are 2-dimensional, the
bundle $F$ is the same as the span of $\phi _{\mu }$.) This
corresponds to a similar statement about the horocycle foliation on p.
58 of \cite{co:noncommutative}.
To close this section, we propose the study of the Poisson analogue of the
invariants $R$ and $S$ of von Neumann algebras discussed in
\cite{co:noncommutative}.
\section{The fundamental class of a unimodular Poisson manifold}
\label{sec-fundamental}
Let $(P,\pi )$ be a unimodular Poisson manifold of dimension $n$. For
simplicity, we will assume that $P$ is oriented.\footnote{Everything
we will do also works in the nonorientable case if we use forms,
homology, and cohomology with twisted coefficients. When $P$ is
noncompact, we use homology with locally finite chains to insure
Poincar\'{e} duality.} Beginning with
an invariant positive density $\mu $ on $P$ we obtain in succession
the closed $n-2$-form $\pi \backl \mu $, the de Rham cohomology class
$[\pi \backl \mu ] \in H^{n-2}(P;\reals )$, and the dual homology
class $\sigma _{\pi }(\mu )\in H_{2}(P;\reals )$. When $P$ is
noncompact, we use homology with locally finite chains to insure
Poincar\'{e} duality.
Let $I(P,\pi)$ be the convex cone of invariant positive densities, and
$I_{1}(P,\pi )$ its convex subset of normalized elements (i.e.
satisfying the condition
$\int_{P}\mu =1.$) We say that $(P,\pi )$ has {\bf finite type} when
$I_{1}(P,\pi )$ is nonempty. The images of $I(P,\pi )$ and
$I_{1}(P,\pi )$ under $\sigma _{\pi }$ are a convex cone ${\cal
C}(P,\pi )$and a convex
set ${\cal C}_{1}(P,\pi )$ respectively in $H_{2}(P;\reals )$ whose
elements we call the {\bf
fundamental cycles} and {\bf normalized fundamental cycles} of the
unimodular Poisson manifold.
If $(P,\pi )$ is a connected symplectic manifold of dimension $n=2m$
with symplectic form $\omega$, oriented by $\omega ^{m}$, then
$I(P,\pi )=\{c\omega ^{m}|c>0\}$. If $P$ has finite symplectic
volume, then it is of finite type, and $I_{1}(P,\pi )=\{\omega
^{m}/v(M)\}$, where $v(P)=\int_{P}\omega ^{m}$, which is $n!$ times
the symplectic volume. Since $\pi \backl \omega ^{m}=\omega ^{m-1}$, if
we denote by $u_{\omega }$ the homology class dual to $\omega ^{m-1}$,
then ${\cal C}(P,\pi )$ is the ray through $u_{\omega }$, and ${\cal
C}_{1}(P,\pi )$ consists of the single class $u_{\omega
}/v(P).$ Note that the pairing $\langle [\omega
],u_{\omega }\rangle $ is by definition $\int_{P}\omega \wedge \omega
^{m-1}=v(P)$, so the homology class in ${\cal
C}_{1}(P,\pi )$ is normalized so that its pairing with the symplectic
class $[\omega ]$ equals 1.
Next suppose that $(P,\omega )$ is a bundle of $2k$-dimensional
connected symplectic manifolds; i.e. its symplectic leaves are the
fibres of a smooth fibration $\gamma :P\rightarrow M$. Let $\omega $
be a 2-form on $P$ (not necessarily closed) which restricts to the
given symplectic structure $\omega (y)$ on each leaf $P(y)=\gamma
^{-1}(y)$. The invariant densities on $P$ are then of the form
$\omega ^{k}\wedge \gamma ^{*}\nu $, where $\nu $ runs over the volume
elements on $M$ consistent with the orientation of $M$ corresponding
to a given orientation of $P$. (This follows from the corresponding
fact about the densities on an exact sequence of vector spaces.) The
corresponding closed form $\pi \backl (\omega ^{k}\wedge \gamma
^{*}\nu ) $ equals $\omega ^{k-1}\wedge \gamma ^{*}\nu$.
In terms of the homology maps
$i(y):H_{2}(P(y);\reals)\rightarrow H_{2}(P;\reals)$ induced by the
inclusions, the homology class dual to $\pi
\backl (\omega ^{k}\wedge \gamma ^{*}\nu ) $ is then
$\int_{P}i(y)u_{\omega (y)}\nu $, so that ${\cal C}(P,\pi )$ consists
of the superpositions (with strictly positive weights) of
fundamental cycles of the symplectic leaves, inserted into
$H_{2}(P;\reals )$ by the inclusion maps. (These inserted homology
classes are defined because the inclusions of the fibres are proper
maps.)
{\bf Remark.} Suppose that $\pi =0$. Then any $\mu $ is invariant,
but $\pi \backl \mu $ is always zero, so that ${\cal C}(P,0 )=\{0\}$.
This is consistent with the fact that each symplectic leaf has
$H_{2}=\{0\}$.
To determine the normalized fundamental cycles on a bundle of
symplectic manifolds, we first note that $\int_{P}\omega ^{k}\wedge
\gamma ^{*}\nu =\int_{M}v(P(y))\nu ,$ so that $(P,\pi )$ has finite
type when the symplectic volume function of the leaves is
locally $L^{1}$ on $M$. In particular, $v(P(y))$ must be finite
for almost all $y$. (Note that the class of locally $L^{1}$
functions is determined by the smooth structure of $M$. It is independent of
$\nu $, which can always be chosen to make a given locally $L^{1}$
function have integral equal to 1.)
%***IS THERE AN OPERATOR ALGEBRA VERSION OF THIS? This would require
%each of the fibre algebras (a factor?) to have a canonical real
%``dimension'' which blows up as we reach a singular point. One could
%imagine doing this with a ``step function of matrix algebras'', since
%these have canonical traces which have a natural normalization. For
%type $II_{1}$ algebras, the situation seems to be different: there is
%no natural ``size''. A related question is whether the algebra
%obtained by quantizing a ball of finite volume has a finite ``size''.
%In fact, it is not even clear what the quantum algebra is in this
%case, beyond a formal quantization.***
Now for $\nu$ such that $\omega ^{k}\wedge \gamma ^{*}\nu $ is
normalized we have $$ \int_{M} i(y)u_{\omega (y)}\nu =
\int_{M}i(y)\frac{u_{\omega }(y)}{v(P(y))}v(P(y))\nu ,$$ which is the
integral over $M$ of the images in $H_{2}(P;\reals )$ of the normalized
fundamental cycles of the symplectic leaves, with respect to the normalized
measure $v(P(y))\nu $. In other words, the set of
normalized fundamental cycles of $P$ is the ``open convex hull''
of the inserted normalized cycles of the leaves.
If $(P,\omega )$ is a finite union of connected
submanifolds, possibly of different dimensions, then the fundamental
cycles are again given by taking convex combinations of the
fundamental cycles of the components.
The examples above suggest that the fundamental cycles for any
unimodular Poisson manifold should be thought of as some kind of
superpositions of fundamental cycles of the symplectic leaves. The
next example will give some meaning to this viewpoint in the case
where $P$ is compact, but the symplectic leaves are not of finite type.
Let $P$ be a 3-torus with a translation invariant Poisson structure
$\pi $ whose symplectic leaves form a foliation by planes, each of
which is dense in the torus. In terms of coordinates
$(x_{1},x_{2},x_{3})$ (defined modulo ${\Bbb Z}$), we can write $\pi =
(\frac{\del }{\del x_{1}}+ a_{1} \frac{\del }{\del x_{3}})\wedge
(\frac{\del }{\del x_{2}} +a_{2}\frac{\del }{\del x_{3}})$, where
$a_{2}$ and $a_{3}$ are irrational and have irrational ratio. The
density $\mu =b dx_{1}\wedge dx_{2} \wedge dx_{3}$ is invariant only
when $b$ is constant and normalized when $b=1$; in fact, $\pi \backl
\mu = b(dx_{3}-a_{1}dx_{1}-a_{2}dx_{2})$, which is closed just when
$b$ is constant along the leaves of the foliation. If we denote by
$c_{ij}$ the fundamental homology class of the oriented product of the
$i$th and $j$th coordinate circles, the the homology class dual to
$(dx_{3}-a_{1}dx_{1}-a_{2}dx_{2})$ is
$c_{12}-a_{1}c_{23}-a_{2}c_{13}$. This is the unique normalized
fundamental cycle, which generates the ray of fundamental cycles. It
should be thought of as an inserted fundamental class of any leaf.
The most general Poisson structure having the same symplectic leaf
foliation is of the form $c\pi $ for some nowhere vanishing function
$c$. The invariant densities now have the form $b|c^{-1}|\mu $ for
positive b, and the ray of fundamental cycles remains the same (up to
the sign of $c$), while the normalized fundamental cycle is multiplied
by $(\int_{P}c^{-1}\mu )^{-1}$. This homology class is therefore an
invariant of the Poisson structure; we refer to \cite{he-ma-sa:lemme}
for a proof that this is essentially the only invariant when the
symplectic leaf foliation is translation invariant.
We end this section with a question and a remark. When does the set
of normalized fundamental cycles have a compact closure? It seems
that these cycles might be related to Ruelle-Sullivan currents for
foliations \cite{ru-su:currents} and related currents for group
actions studied by Brylinski \cite{br:noncommutative}.
\begin{thebibliography}{99}
\bibitem{ba-fl-li-st:deformation}
Basart, H., Flato, M., Lichnerowicz A., and Sternheimer, D.,
Deformation theory applied to quantization and statistical
mechanics, {\em Lett. Math. Phys.} {\bf 8} (1984), 483-494.
\bibitem{br:noncommutative}
Brylinski, J.L., Noncommutative Ruelle-Sullivan type currents.
{\em The Grothendieck Festschrift, Vol. I}, Birkh\"{a}user,
Boston (1990) pp. 477-498.
\bibitem{br-zu:outer}
Brylinski, J.-L., and Zuckerman, G., The outer derivation of a complex
Poisson manifold, preprint 1996.
\bibitem{co:noncommutative}
Connes, A., {\em Noncommutative Geometry}, Academic Press, San Diego, 1994.
\bibitem{co-da-we:groupoides}
Coste, A., Dazord, P., et Weinstein, A., Groupo\"{i}des symplectiques,
{\em Publications du D\'{e}partement de Math\'{e}matiques,
Universit\'{e} Claude Bernard-Lyon I} {\bf 2A} (1987), 1-62.
\bibitem{da:feuilletages}
Dazord, P., Feuilletages \`{a} singularit\'{e}s, {\em Nederl. Akad. Wetensch.
Indag. Math.} {\bf 47} (1985), 21-39.
\bibitem{du-ha:rotationnels}
Dufour, J.-P., and Haraki, A., Rotationnels et structures de Poisson
quadratiques, {\em C.R.A.S. Paris} {\bf 312} (1991), 137-140.
\bibitem{ev-lu-we:poincare}
Evens, S., Lu., J-H., and Weinstein, A., Poincar\'{e} duality for
Lie algebroid cohomology, preprint, University of Arizona, Tucson (1996).
\bibitem{gi:private}
Ginzburg, V.L., Paper in preparation.
\bibitem{gi-lu:poisson}
Ginzburg, V.L., and Lu, J.-H., Poisson cohomology of Morita-equiva\-lent
Poisson manifolds, {\em Duke Math. J.} {\bf 68} (1992), A199-A205.
\bibitem{he-ma-sa:lemme}
Hector, G., Mac\'{i}as, E., and Saralegi, M., Lemme de Moser
feuillet\'{e} et classification des vari\'{e}t\'{e}s de Poisson
r\'{e}guli\`{e}res, {\em Publicacions Matem\`{a}tiques} {\bf 33}
(1989), 423-430.
\bibitem{ko:crochet}
Koszul, J. L., Crochet de Schouten-Nijenhuis et cohomolo\-gie,
{\em As\-t\'{e}risque}, hors serie, (1985), 257 - 271.
\bibitem{kr-ma:hamiltonian}
Krishnaprasad, P. S., and Marsden, J. E.,
Hamiltonian structures and stability for rigid bodies with flexible
attachments, {\it Arch. Rational Mech. Anal.} {\bf 98} (1987), 71-93.
\bibitem{li-xu:quadratic}
Liu, X.-J., and Xu, P., On quadratic Poisson structures
{\em Lett. Math. Phys.} {\bf 26} (1992), 33-42.
\bibitem{lu-we:poisson}
Lu, J.-H., and Weinstein, A., Poisson Lie groups, dressing transformations,
and the Bruhat decomposition, {\em J. Diff. Geom.} {\bf 31} (1990), 501-526.
\bibitem{lu-we:classification}
Lu, J.-H., and Weinstein, A., Classification of $SU(2)$-covariant
Poisson structures on $S^{2}$, (Appendix to a paper of Albert J.-L.
Sheu), {\em Comm. Math. Phys.} {\bf 135} (1991), 229-232.
\bibitem{ma:lie}
Mackenzie, K., {\em Lie Groupoids and Lie Algebroids in Differential
Geometry}, LMS Lecture Notes Series, {\bf 124}, Cambridge Univ. Press, 1987.
\bibitem{ma-na:neumann} Masuda, T., and Nakagami, Y., A von Neumann
algebra framework for the duality of the quantum groups,
{\em Publ. RIMS Kyoto Univ.} {\bf 30} (1994), 799.850.
\bibitem{me:atiyah}
Melrose, R.B., {\em The Atiyah-Patodi-Singer Index Theorem}, A.K.
Peters, Wellesley, 1993.
\bibitem{mi:foliations}
Mikami, K., Foliations of Poisson structures and their Godbillon-Vey
classes, preprint, Akita Univ., 1995.
\bibitem{mi-we:moments}
Mikami, K., and Weinstein, A., Moments and reduction for symplectic
groupoid actions, {\em Publ. RIMS Kyoto Univ.} {\bf 24}
(1988),121-140.
\bibitem{ne-ts:formal}
Nest, R., and Tsygan, B., Formal deformation quantization of
symplectic manifolds with boundary, preprint.
\bibitem{re:groupoid}
Renault, J., A groupoid approach to $C^{*}$ algebras, {\em Lecture
Notes in Math.} {\bf 793} (1980).
\bibitem{ri-va:bounded}
Rieffel, M.A., and van Daele, A., A bounded operator approach to
Tomita-Takesaki theory.
{\em Pacific J. Math.} {\bf 69} (1977), 187-221.
\bibitem{ru-su:currents}
Ruelle, D., and Sullivan, D., Currents, flows and diffeomorphisms,
{\em Topology} {\bf 14} (1975), 319-327.
\bibitem{su:holonomy}
Suzuki, H., Holonomy groupoids of generalized foliations. II.
Transverse measures and modular classes,
{\em Symplectic geometry, groupoids,
and integrable systems, S\'{e}minaire sud-Rhodanien
de g\'{e}om\'{e}trie \`{a} Berkeley (1989)}, P. Dazord and A.
Weinstein, eds., Springer-MSRI Series (1991), 267-279.
\bibitem{ta:automorphisms}
Takesaki, M., Automorphisms and von Neumann algebras of type ${\rm
III}$, Operator algebras and applications, Part 2 (Kingston,
Ont., 1980), {\em Proc. Sympos. Pure Math.} {\bf 38},
(1982), 111-135.
\bibitem{ta:topology}
Tamura, I., {\em Topology of Foliations: an Introduction}, Amer. Math.
Soc., Providence, 1992.
\bibitem{tu:reduction}
Tuynman, G.M., Reduction, quantization, and nonunimodular groups,
{\em J. Math. Phys.} {\bf 31} (1990), 83-90.
\bibitem{va:lectures}
Vaisman, I., {\em Lectures on the geometry of Poisson manifolds},
Birkh\"{a}user, Basel, 1994.
\bibitem{we:poisson structures}
Weinstein, A., Poisson structures and Lie algebras, {\em
Ast\'{e}risque}, hors s\'{e}rie (1985), 421--434.
\bibitem{we:poisson geometry}
Weinstein, A., Poisson geometry of the principal series and
nonlinearizable structures, {\em J. Diff. Geom.} {\bf 25} (1987), 55--73.
\bibitem{we:modules}
Weinstein, A., Poisson modules (in preparation).
\bibitem{we-xu:extensions}
Weinstein, A., and Xu, P., Extensions of symplectic groupoids and
quantization, {\em J. Reine Angew. Math.} {\bf 417} (1991), 159-189.
\bibitem{xu:morita poisson}
Xu, P. Morita equivalence of Poisson manifolds,
{\em Comm. Math. Phys.} {\bf 142} (1991), 493-509.
\end{thebibliography}
\end{document}