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\title{\bf Poisson Geometry}
\author{Alan Weinstein\thanks{Research partially supported by NSF
Grant DMS-96-25122 and the Miller Institute for Basic Research in Science.}
\\Department of Mathematics\\
University of California\\
Berkeley, CA 94720 USA\\
{\small(alanw@math.berkeley.edu)}}
\begin{document}
\maketitle
\begin{abstract}
\noindent
This paper is a survey of Poisson geometry, with an emphasis on global
questions and the theory of Poisson Lie groups and groupoids.
\end{abstract}
\section{Introduction}
\label{sec-intro}
Poisson \cite{po:variation} invented his brackets as a tool for
classical dynamics.
Jacobi \cite{ja:vorlesungen}
realized the importance of these brackets and elucidated
their algebraic properties, and Lie
\cite{li:theorie} began the study of their geometry.
After a long dormancy, Poisson geometry has
become an active field of research during the past 30
years or so, stimulated by connections with a number of areas, including
harmonic analysis on Lie groups (Berezin \cite{be:remarks}),
infinite dimensional Lie algebras (Kirillov \cite{ki:local}),
mechanics of particles and continua
(Arnold \cite{ar:small3}, Lichnerowicz \cite{li:varietes},
Marsden--Weinstein \cite{ma-we:hamiltonian}), singularity theory
(Varchenko--Givental' \cite{va-gi:period}),
and completely integrable systems (Gel'fand--Dikii \cite{ge-di:family},
Kostant \cite{ko:solution}), to mention just a few examples.
A {\em Poisson algebra} is a commutative associative algebra $\cala$
over $\reals$ carrying a Lie
algebra bracket $\{~,~\}$ for which each adjoint operator
$X_h=\{~,h\}$ is a derivation of the associative algebra
structure.
There are also {\em noncommutative} Poisson algebras
\cite{bl-ge:quantization,fa-le:ring,
vo:poisson,xu:noncommutative}, but we will not treat them
in this paper. Of course, one can replace $\reals$ by another field.
For most of this paper,
$\cala$ will be the algebra $\cinf(M)$ of smooth
functions on a manifold $M$, in which case the bracket
is called a {\em Poisson structure} on $M$, and $(M,\{~,~\})$ is
called a {\em Poisson manifold}. The derivations $X_h$ are
represented in this ``spatial'' picture by vector fields, which are
called {\em hamiltonian} vector fields.
A {\em Poisson morphism} $\cala_1\rightarrow\cala_2$ is a map
between Poisson algebras which is a homomorphism for both the
associative and Lie algebra structures. When $\cala_1 = \cinf(M_1)$
and $\cala_2 = \cinf(M_2)$,
the algebra morphism
is the pullback operation for a smooth map
$M_1\leftarrow M_2$ which preserves Poisson brackets. We call these
maps {\em Poisson maps}. (See \cite{ab-ma-ra:manifolds} or
\cite{bk:ideaux} for a proof that
any associative algebra homomorphism between algebras of smooth
functions is the operation of pullback by some smooth map.)
Poisson manifolds occur as phase spaces for classical particles
(and, in the infinite dimensional case, fields), but Poisson
geometry is also
relevant to the algebras of observables in quantum mechanics, as well
as to more general noncommutative algebras.
In fact,
Kontsevich \cite{ko:deformation} has just shown that the
classification of formal deformations of the algebra $\cinf(M)$ for
any manifold $M$ is equivalent to the classification of formal
families of Poisson structures on $M$. Furthermore, the
derivations $X_h$ of a Poisson algebra behave rather like the inner
derivations of a noncommutative algebra, and consequently there are
strong analogies between Poisson geometry and
noncommutative algebra. We do not discuss these analogies here, but
refer the reader to \cite{ca-ha-we:lectures,sh:von neumann}.
Another category with close relations to Poisson geometry is that of
Lie algebroids, while the corresponding global objects, Lie groupoids,
can be used to encode the automorphisms of Poisson manifolds generated
by the hamiltonian vector fields $X_h$. It is no accident that
groupoids have also been a rich source of noncommutative
algebras \cite{co:noncommutative,we:groupoids}. We discuss Lie
algebroids and groupoids in Section \ref{sec-groupoids}.
There are now several
books available with
extensive discussions of Poisson geometry
\cite{bh-vi:poisson,ca-ha-we:lectures,
gu-st:techniques,ka-ma:nonlinear,li-ma:symplectic,va:lectures}. The
reader is referred
to
them for further details of much of what is merely sketched here.
This paper, then, is an idiosyncratic survey of
Poisson geometry, marked by my own interests, and
those of people with whom I have had close personal contact, including
several of my former students. I have tried to make the bibliography
representative of the literature, but it
is far from complete. I apologize for all
the relevant citations which have been omitted.
\section{Poisson manifolds and maps}
\label{sec-maps}
Since the bracket $\{ f,g\}$ of functions
on a Poisson manifold $M$ is a derivation
in each argument, it depends only on the first derivatives of $f$ and
$g$, and hence it can be written in the form
\[
\{ f,g\} = \pi (df,dg)\
\]
where $\pi\in\Gamma (\wedge^2TM)$ is a field of skew-symmetric
bilinear forms on $T^*M$,
i.e. a {\em bivector field}. We call $\pi$ the {\em
Poisson tensor}. The Jacobi identity for the bracket
implies that $\pi$ satisfies an integrability condition which is a
quadratic first-order
(semilinear) partial differential equation in local coordinates, and
has the invariant form $[\pi,\pi]=0$, where the bracket here is the
Schouten--Nijenhuis bracket on multivector fields (see \cite{va:lectures}).
In addition to the tensor $\pi$, we will occasionally refer to the
associated bundle map $\tilde{\pi}:T^*M\rightarrow TM$ defined by
$a(\tilde{\pi}(b))=\pi(b,a)$ for cotangent vectors $a$ and $b$.
Poisson maps can be characterized by
the following property of their graphs.
A submanifold $C$ of a Poisson manifold $M$ is called
{\em coisotropic} if the set of functions on $M$ which vanish on $C$
is closed under Poisson bracket. A differentiable map
$\phi$ between Poisson manifolds $M$ and $N$ is
a Poisson map if and only if its graph is a
coisotropic submanifold of $\overline{M}\times N$, where
$\overline{M}$ is $M$ with its Poisson structure multiplied by $-1$.
With this fact in mind, we call any coisotropic submanifold of
$\overline{M}\times N$ a {\em Poisson relation} from $M$ to $N$. The
{\em coisotropic calculus} \cite{we:coisotropic} in Poisson geometry is
based on the fact that the composition of two Poisson relations is
again a Poisson relation, as long as the composition satisfies a ``clean
intersection'' assumption which guarantees that this composition is a
smooth manifold. This calculus extends to Poisson geometry the
lagrangian calculus (composition of canonical relations) \cite{we:lectures}
for symplectic manifolds.
\section{Local structure of Poisson manifolds}
Local Poisson geometry begins with the following
theorem.
\begin{thm}[Splitting Theorem \cite{we:local}]
Centered at any point $O$ in a Poisson mani\-fold $M$, there are coordinates
$(\row qk$, $\row pk$, $\row y\ell )$ such that
\[
\pi =\sum_{i=1}^k\dbyd{}{q_i}\wedge\dbyd{}{p_i}+{1\over 2}
\sum_{i,j=1}^{\ell}\varphi_{ij}(y)\dbyd{}{y_i}\wedge\dbyd{}{y_j}
\ \ \ \mbox{ and } \ \ \ \varphi_{ij}(0)=0\ .
\]
\end{thm}
When $\ell=0$, the Poisson structure is called {\em symplectic}, and
the theorem is Darboux's theorem. Hence, a general Poisson manifold is
isomorphic near each point
to the product of an open subset of a standard symplectic
manifold $\reals^{2k}$ and a Poisson manifold for which the Poisson
tensor vanishes at the point in question. The local classification
question
can thus be reduced to that for structures of the second type, which
we call {\em totally degenerate}.
\subsection{Symplectic leaves}
The symplectic and totally degenerate factors are quite different when
they are
viewed intrinsically.
The symplectic factor is an open subset of a well-defined submanifold
$\calo_{O}$,
called the {\em symplectic leaf} through $O$. $M$ is
a disjoint union of these symplectic leaves, and the
Poisson bracket on $M$ ``assembles'' the canonical
Poisson brackets on these leaves. On an open dense subset
of $M$,
the {\em regular} part, the dimension of the leaves is locally
constant, and they form a foliation.
The totally degenerate factor,
on the other hand,
lives more naturally on a ``quotient'' of a neighborhood of $O$ by a
foliation (not intrinsically defined!) having $\calo_{O}$ as a leaf. It
can be shown that the degenerate factor is well-defined, but only up to
isomorphism; the isomorphism class of this {\em transverse structure} is
the same for all points of the symplectic leaf $\calo_O$.
\subsection{Totally degenerate structures and linearization}
Since our study will be local, we can assume that the underlying
manifold is a vector space $V$, and that the point of total degeneracy
is the origin. In linear coordinates, the coefficients of the
Poisson tensor can then be written as linear functions plus higher order
terms. If the higher order terms all vanish, the
dual space $V^*$ of linear functions on $V$ is a Lie algebra $\frakg$, so that
$V=\frakg^*$. The Poisson structure is completely determined by the Lie
algebra structure on $\frakg$: in linear coordinates
$(x_1,\ldots.x_n)$ on $\frakg^*$ (which can also be viewed as a basis of
$\frakg$), the Poisson tensor is
$$ \pi=\frac{1}{2}\sum_{i,j,k=1}^n
c_{ijk}x_k\dbyd{}{x_i}\wedge\dbyd{}{x_j},$$
where $c_{ijk}$ are the structure constants of the Lie algebra with
respect to this basis.
We call such Poisson
structures on the dual spaces of Lie algebras {\em Lie--Poisson structures}.
A Poisson structure which becomes linear when expressed with respect
to suitable local coordinates at a point $O$ is said to be
{\em linearizable} at $O$. The candidate for the corresponding Lie
algebra structure is determined by the Poisson structure itself:
at a point $O$ where a Poisson structure on $M$ vanishes, there is a
well-defined bracket on the differentials at $O$ of functions on $M$,
defining the structure of a Lie algebra (the {\em tangent Lie
algebra})
on the cotangent space $T^*_O
M$ and hence a Lie--Poisson structure (the {\em tangent Poisson
structure}) on the tangent space $T_O M$. The structure on $M$ is
linearizable at $O$ iff it is locally isomorphic to its tangent Poisson
structure at that point.
There are Lie algebras $\frakg$ such that any totally degenerate
Poisson structure with tangent Lie algebra $\frakg$ is
linearizable. These algebras are analogous to nondegenerate quadratic
functions, from which additional higher order terms can be removed by a
coordinate transformation.
For this reason, we have
called such Lie algebras {\em nondegenerate} in \cite{we:local}.
However, the juxtaposition of ``degenerate'' and ``nondegenerate'' in
this context seems a bit confusing, so we will follow the terminology of
singularity theory and call these Lie algebras
{\em Poisson-determining}, with a modifier added to indicate
a particular class of Poisson structures and coordinate transformations.
The first result about linearization is due to Arnol'd (see Appendix
14 of \cite{ar:mathematical}), who showed that the 2-dimensional
nonabelian Lie algebra is Poisson-determining in any category. Next,
it was proved in \cite{we:local} that any semisimple Lie algebra is
formally Poisson-determining; i.e. a Poisson structure whose
coefficients are formal power series can be linearized by a formal
change of coordinates if the tangent Lie algebra is semisimple. An
example
in
the same paper showed that the semisimple Lie
algebra ${\mathfrak sl}(2,\reals)$ is not $C^{\infty}$
Poisson-determining. In \cite{we:poisson geometry}, it was shown that
no semisimple
algebra of noncompact type with real rank greater than 1 can be $C^{\infty}$
Poisson-determining.
Some examples of nonlinearizable transverse structures to symplectic
leaves in Lie--Poisson spaces are due to
Givental' (see the discussion in \cite{we:poisson structures}) and
Damianou \cite{da:transverse}.
(Mar\'i Beffa \cite{be:transverse} and Ovsienko--Khesin
\cite{ov-kh:symplectic}, by contrast, show linearizability of
transverse structure in some infinite-dimensional cases related to
integrable systems.)
Cahen--Gutt--Rawnsley \cite{ca-gu-ra:nonlinearizability} found
nonlinearizability at the identity element for many Poisson Lie
groups. (See Section \ref{subsec-groups and groupoids}.)
Positive results on Poisson-determinacy of semisimple algebras are due
to Conn. He showed first \cite{co:analytic} that any semisimple algebra is
analytically Poisson-determining, and then \cite{co:smooth} that any
semisimple algebra of compact type is $C^{\infty}$
Poisson-determining. These results were
extended by Molinier \cite{mo:these} to the sum of semisimple algebras
with $\reals$.
There is a striking similarity between these results and those for the
linearizability of actions of Lie algebras near a fixed point, so that
it is tempting to try to derive the former results using the latter,
whose proofs can be carried out quite simply by an averaging trick
in the compact case, together with analytic continuation to reduce the
noncompact analytic case to the compact one.
A global linearization theorem for the duals of compact semisimple Poisson Lie
groups (see Section \ref{subsec-groups and groupoids}) was obtained by
Ginzburg and the author in \cite{gi-we:lie-poisson}. Another proof of
this result was found by Alekseev \cite{al:poisson} as part of a
general theory which reduces actions of Poisson Lie groups to ordinary
symmetry actions.
When the tangent Lie algebra $\frakg$ is arbitrary, Wade
\cite{wa:normalisation} has shown that one can find a formal
coordinate system which linearizes all the components of the Poisson
tensor involving generators of the semisimple part of the Levi
decomposition of $\frakg$. Furthermore, Flato and Sternheimer
(private communication) have pointed out that the linearization theorem
in the semisimple case itself represents a Levi decomposition for the
infinite dimensional Poisson bracket Lie algebra of germs, or of formal power
series, since the functions vanishing at least quadratically at a
totally degenerate point form a ``topologically nilpotent'' ideal
with the tangent Lie algebra as quotient.
So far, though, no proof of an analytic or $\cinf$
Poisson linearization theorem using this idea has been found.
There are by now many examples of Poisson-determining Lie
algebras which are not semisimple. See, for example \cite{du:linearisation}.
\subsection{Quadratic Poisson structures and their perturbations}
After the linear Poisson structures, it is natural to look at
quadratic structures. It is perhaps surprising that these structures also
arise ``in nature,'' when Poisson structures on matrix Lie groups are
extended to the matrix algebras which contain them
\cite{ba-bu:quadratic,se:what}. Two basic questions
arise--classification and quadratization. A study of the
classification was begun
by Dufour--Haraki \cite{du-ha:rotationnels} and Liu--Xu
\cite{li-xu:quadratic}, and others. Quadratization (i.e. equivalence
to quadratic structures after a coordinate change) has been
established in some situations for structures with sufficiently nice
quadratic part by Dufour \cite{du:quadratisation} and Haraki
\cite{ha:quadratisation}.
\section{Global structure of regular Poisson manifolds}
The classification of regular Poisson structures on a given manifold $M$
can be subdivided as follows.
\begin{enumerate}
\item Classify the foliations on $M$.
\item Classify the Poisson structures having a given foliation
$\calf$ as its
symplectic leaf foliation.
\begin{enumerate}
\item Which foliations arise from some Poisson structure?
\item How many ``different'' Poisson structures can have the same
foliation by symplectic leaves?
\end{enumerate}
\end{enumerate}
Problem 1 is obviously beyond the scope of this paper, so we will
begin with a particular $\calf$ and
try to classify all the Poisson structures from which it can arise.
In its full generality, Problem 2 includes the classification of
symplectic manifolds, since
$M$ can always be foliated
by a single leaf. This classification is complete only for
2-dimensional manifolds (\cite{mo:volume} for the compact case,
\cite{gr-sh:diffeomorphisms}
for the noncompact case), although our knowledge in dimension 4 is
rapidly growing through the combination of flexible constructions
\cite{go:new} and the rigidity results arising from Seiberg-Witten theory
\cite{ta:from} (also see the extensive review of this paper
\cite{sa:review}).
For foliations of dimension 2 on
compact manifolds, Poisson structures which induce
a given orientation on the leaves certainly exist,
and they are classified by their classes in the second de Rham cohomology
along the leaves \cite{he-ma-sa:lemme}.
One is thus led to the computation of this
cohomology and the determination of the convex cone realized by Poisson
structures. Some examples are worked out in \cite{he-ma-sa:lemme}.
The higher dimensional problem is virtually untouched.
\section{Poisson cohomology and homology}
\label{sec-cohomology}
Lichnerowicz \cite{li:varietes} observed that the operation $[\pi,~]$ of
Schouten bracket with a Poisson tensor is a differential on
multivector fields, and he began the study of the resulting cohomology
theory for Poisson manifolds. In particular, he showed that the
map from differential forms to multivector fields determined by
$\tilde{\pi}:T^*M\rightarrow TM$ is a morphism from the de Rham
complex to the Poisson complex. In the symplectic case, this map is
an isomorphism, but the Poisson cohomology spaces $H_{\pi}^i(M)$ are
in general quite different from the de Rham cohomology.
As with most cohomology theories, the $H_{\pi}^i(M)$ have interesting
interpretations for the first few values of $i$.
The differential
in degree zero assigns to each function its
hamiltonian vector field, so $H_{\pi}^{0}(M)$ consists of the
functions which Poisson commute with everything, the so-called {\em
Casimir functions} on $M$. It is suggestive to think of them
as the ``smooth functions on the space of symplectic leaves.'' The
next differential maps each vector field $X$ to $-\call_X\pi$, so
$H_{\pi}^1(M)$ is the space of infinitesimal Poisson automorphisms
modulo hamiltonian vector fields, or, algebraically, the outer
derivations of the Poisson algebra. Ginzburg and Lu \cite{gi-lu:poisson}
make a case for thinking of $H_{\pi}^1(M)$ as the ``space of vector
fields on the space of symplectic leaves.'' Next, we can
interpret $H_{\pi}^2(M)$ as the space of infinitesimal deformations of
the Poisson structure modulo trivial deformations, while $H_{\pi}^3(M)$
receives the obstructions to extending infinitesimal deformations to
formal deformations of higher and higher order.
These cohomology spaces have also been known for some time to relate
to the deformation quantization of Poisson manifolds. For
instance, $H_{\pi}^2(M)$ is, at least in the symplectic case, a
parameter space for classifying deformation quantizations, while
$H_{\pi}^3(M)$ receives the possible
obstructions to constructing such quantizations.
Kontsevich \cite{ko:deformation} shows that these relations on the cohomology
level are simply the shadow of a much deeper relation between Poisson
geometry and deformations of $\cinf(M)$: the algebra of multivector
fields, with zero differential, is quasi-isomorphic to the
differential graded Lie algebra of multidifferential operators, with
its Gerstenhaber \cite{ge:cohomology} bracket, which controls the algebraic
deformations.
Computation of Poisson cohomology is generally quite difficult. For
regular Poisson manifolds, this cohomology reflects the topology of
the leaf space and the variation in the symplectic structure as
one passes from one leaf to another. Some references on the computation of
this cohomology are
\cite{gi:equivariant,gi-lu:poisson,va:lectures,vo-ka:poisson, xu:poisson}.
There is also a homology theory for Poisson manifolds, for which the
chains are differential forms. The boundary operator $\delta$ was
defined differential geometrically by Koszul \cite{ko:crochet} as
$i_{\pi}d-di_{\pi}$, where $i_{\pi}$ is the operator of contraction
with the Poisson tensor. Brylinski \cite{br:differential} found an
algebraic definition of the boundary operator
by taking the classical limit of the Hochschild boundary operator for a
quantized Poisson algebra. Brylinski also observed that $\delta$
becomes the ``Hodge dual'' of $d$ when the manifold is
symplectic, if one imitates Hodge
theory by using the symplectic structure in place of a riemannian
metric. Even when the manifold is not symplectic, the notion of
a Poisson-harmonic form turns out to be remarkably useful; see the end of
Section \ref{subsec-homogeneous}.
Huebschmann \cite{hu:poisson} studied both Poisson homology and
cohomology from the algebraic point of view and showed how they could be fit in
the standard homological framework of resolutions.
Some interesting identities relating the Poisson homology and
cohomology differentials were found by Bhaskara and Viswanath
\cite{bh-vi:poisson} (see also \cite{va:lectures}). More recently,
the duality between the two theories has been studied in depth by
Evens--Lu--Weinstein
\cite{ev-lu-we:poincare}, Huebschmann \cite{hu:duality}, and
Xu \cite{xu:gerstenhaber}. This work uses the fact that
Poisson cohomology is a special case of Lie
algebroid cohomology (see Section \ref{subsec-algebroids}),
while Poisson homology can be
identified with Lie algebroid cohomology with coefficients in the top
exterior power of the cotangent bundle.
This identification
establishes close relations between the homological theory and the modular
theory described in Section
\ref{sec-modular}. It is also related (see \cite{hu:duality})
to the use of ``dualizing
sheaves'' in algebraic geometry
\cite{ha:algebraic}.
\section{Completeness}
\label{sec-completeness}
There are several interesting notions of completeness in Poisson
geometry. Their quantum analogues should be related to notions of
self-adjointness.
\subsection{Complete functions}
A function $f:M\to \reals$ on a Poisson manifold $M$ will be called
complete if the hamiltonian vector field $X_f$ is a complete vector
field, i.e. if its flow is globally defined.
Any compactly supported
function is complete, and a riemannian
metric on a manifold $Q$ is complete iff the corresponding kinetic
energy function on $T^*Q$ is a complete function.
\subsection{Complete manifolds}
\label{subsec-complete manifolds}
A Poisson manifold $M$ will be called complete if every hamiltonian
vector field on $M$ is complete.
$M$ is complete if and only if every symplectic leaf is bounded in the
sense that its closure is compact. In fact, if every symplectic leaf
is bounded, every trajectory of every hamiltonian vector field is
contained in a compact set, so it can be continued for all time.
Conversely, in any
unbounded leaf $\calo$ we can find a sequence of points which leaves
every compact
subset of $M$. By connecting these points with embedded paths,
rounding corners, and removing intersections as necessary (a process
which is easy when the dimension of $\calo$ is more than 2 and harder
but possible in the 2-dimensional case), we can find a proper
embedding $\sigma:[0,1)\rightarrow M$ whose image is contained in
$\calo$. Now choose a 1-form $\alpha$ along the image of $\sigma$ so
that $
-\pitilde (\alpha (\sigma (t)))=\sigma '(t)$, and
construct a function $h$ on $M$ whose differential at each $\sigma(t)$
is $\alpha (\sigma(t))$. The trajectory of the hamiltonian vector
field $X_h$ which begins at $\sigma (0)$ then follows $\sigma$ but
``reaches infinity'' in a finite time, so $X_h$ is not complete, and
hence $M$ is not complete.
A Lie--Poisson space
$\frakg^*$ is complete if and only if $\frakg$ is the Lie algebra of a
compact group. The sufficiency of the compactness condition
is evident. For the necessity,
we note that
any group of transformations of a finite dimensional vector space with
all orbits bounded is contained in a set of operators which is
bounded, hence compact, so it admits an invariant,
positive definite, inner product. (I would like to thank Marc Rieffel
for providing me with this argument.) If $\frakg^*$ is complete, all
the coadjoint orbits are bounded, so the coadjoint representation
admits an invariant,
positive definite, inner product, and hence so does the adjoint
representation. It follows that $\frakg$ is the Lie algebra of a
compact group. (See for example Section 21 of \cite{mi:morse}.)
\subsection{Complete maps}
A Poisson map $\phi:M\to N$ will be called complete if the
pullback by $\phi$ of every complete function is complete.
It is a
nice exercise to show that a complete map to $\reals$
(with the zero Poisson structure, of course) is a complete function, and that
to check completeness it suffices to look at the pullbacks of
compactly supported functions. Also note that a Poisson manifold $M$
is complete if and only if every Poisson map with domain $M$ is
complete. The inclusion of each symplectic leaf into any Poisson
manifold is a
complete map.
The composition of complete Poisson maps is complete, and the Poisson
category becomes ``tamer'' when we restrict to complete maps. For
instance, the image of a complete Poisson map $M\to N$ is always a
union of symplectic leaves in $N$, so we do not have to deal with
inclusions of arbitrary open subsets. In this sense, complete maps
between Poisson manifolds are somewhat like proper maps between
locally compact topological spaces. (Note that any proper Poisson map
is complete.)
For the next two
examples, we refer to \cite{ca-ha-we:lectures} for further details.
If $N=\frakg^*$ is a Lie--Poisson space, then each Poisson
map $M\rightarrow N$ is the momentum map for a hamiltonian
action of the Lie algebra $\frakg$ on $M$. This action integrates to
a hamiltonian action of the corresponding (connected) simply-connected
Lie group $G$ if and only if the momentum map is complete.
A corollary of this result is that linear Poisson maps (which are just
the duals of Lie algebra homomorphisms) are always complete.
There is an analogous result for complete Poisson maps $\phi:M\to N$
when $N$ is a symplectic manifold. Any Poisson map from $M$ to a
symplectic manifold is a submersion, and the hamiltonian vector fields
on $M$ generated by functions pulled back from $N$ span an integrable
distribution on $M$. When the map $\phi$ is complete, integration of
these vector fields for compactly supported functions on $N$ can be
used to construct local trivializations, which shows that $\phi:M\to
N$ is a fibre bundle with a flat Ehresmann connection. The holonomy
of this connection is an action of the fundamental group of $N$ on the
typical fibre of $\phi$. Thus, there is a correspondence between
complete Poisson maps to symplectic $N$ and actions of the
fundamental group $\pi_1(N)$, rather analogous to that between
complete Poisson maps to $\frakg ^*$ and actions of $G$. It is
tempting to think of the symplectic manifold $N$ as the ``dual of the
Lie algebra of $\pi_1(N)$.''
The analogy above is not quite precise because one needs to choose a
basepoint in $N$ to define the fundamental group and to specify the
fibre on which it acts. It is actually better to think of the flat
connection on $\phi:M\to N$ as corresponding to an action of the
fundamental group{\em oid} of $N$ on $M$. (See \cite{gu-hu-je-we:group}
for a treatment of flat connections on vector bundles as
linear representations of fundamental groupoids.) The cases where $N$ is
Lie--Poisson or symplectic are then both subsumed under the theory of
moment maps for actions of symplectic groupoids
\cite{mi-we:moments}. In fact, the correct interpretation of
arbitrary complete maps $M\rightarrow N$ as momentum maps requires
the use of such groupoids, which we discuss in Section
\ref{subsec-symplectic}.
\subsection{Completeness of Poisson Lie groups}
\label{subsec-complete groups}
There is an independent notion of completeness for Poisson Lie
groups. We introduce these groups in
Section \ref{subsec-groups and groupoids}, along with the notion of
dressing transformation.
A Poisson Lie group $G$ is said to be complete if the
infinitesimal dressing transformations on the dual group $G^*$
corresponding to the elements of the Lie algebra $\frakg$ are complete
vector fields.
A theorem of Majid \cite{ma:matched} establishes that
$G$ is complete if and only if $G^*$ is complete; it would be nice to
have a Poisson-geometric proof of this result.
One can also ask when $G$ is complete just as a Poisson manifold. The
case $G=\frakg^*$ shows that this is rather different from
completeness as a Poisson Lie
group; a Lie--Poisson space is always complete as a
Poisson Lie group, since the dressing action on its dual is trivial,
but as we have seen earlier in this section, such a space may or not
be complete as a manifold.
There is more to be said about completeness and Poisson Lie groups,
but we defer this to Section \ref{subsec-actions}, after we have written
more about these groups, and about symplectic groupoids.
\subsection{Completeness in Poisson cohomology classes}
Motivated by the example of the modular class (see Section
\ref{sec-modular} below), it is interesting to look at the
completeness of Poisson vector fields which are not hamiltonian,
looking at Poisson cohomology classes in
$H^1_{\pi}(M)$ one at a time.
The zero class, i.e. the hamiltonian vector fields, always contains
{\em some} complete vector fields, e.g. the zero field,
but this may not be true
for nonzero classes. For instance, on the ``truncated'' cylinder $(-1,1)\times
\torus $ with symplectic structure $dy\wedge d\theta ,$ every
complete, locally hamiltonian vector field is globally hamiltonian
(Lemma 6.1 in \cite{we:modular}), so the nonzero classes in
$H^1_{\pi}(M)=\reals$
do not contain any complete vector fields at all. On the other hand, on the
full cylinder $\reals \times \torus$, one can find both complete and
incomplete vector fields in each cohomology class.
On the product $\torus ^2 \times \reals$ with coordinates
$(\theta_1,\theta_2,t)$, we may compare the Poisson structures
$$\dbyd{}{\theta_1}\wedge \dbyd{}{\theta_2} \mbox{~and~}
(1+t^2)\dbyd{}{\theta_1}\wedge \dbyd{}{\theta_2}.$$ In each case,
there is a natural map $\tau$ from $H^1_{\pi}$ onto the vector fields on
the $t$-axis, and completeness of a Poisson vector field depends only
on the image under
$\tau$ of its cohomology class. For the first structure, $\tau$ is
surjective since the leaves are all isomorphic as symplectic
manifolds; as a result, there are many cohomology classes which
contain no complete vector fields. For the second structure, $\tau$
is zero because the symplectic volume varies from leaf to leaf, so
every Poisson vector field is complete.
It would be interesting to find a manifold on which Poisson vector
fields are forced to be complete for reasons less trivial than in the previous
example. Perhaps some ``dynamical'' property of the leaves could
prevent trajectories from escaping to infinity in a finite time.
\section{Lie algebroids and Lie groupoids}
\label{sec-groupoids}
Up to now, when we have thought of a Poisson algebra $\cala$ as an object with
two structures, the multiplicative structure has been
the primary one, and the Lie algebra structure secondary. But it is
possible to take the opposite point of view, thinking of $\cala$ first
of all as a Lie algebra. This Lie algebra can in many cases
be integrated to a Lie group
$\calg$ (of infinite dimension),
and we may then ask what the associative algebra
structure on $\cala$ implies for $\calg$.
We will limit our attention to the case of Poisson manifolds,
referring to \cite{hu:poisson} for general Poisson algebras.
As background references for the material on Lie algebroids and Lie
groupoids in this section, we suggest Mackenzie's book \cite{ma:lie}
and the notes \cite{ca-ha-we:lectures}.
\subsection{Lie algebroids}
\label{subsec-algebroids}
Instead of working directly with the Lie algebra
$\cala=C^{\infty}(M)$,
we introduce
a Lie algebra structure over $\reals$ on the space $\cale$ of 1-forms
on a Poisson manifold $M$ by the formula
$$[a,b]=\call_{\tilde{\pi}a}b-\call_{\tilde{\pi}b}a -d\pi(a,b),$$
where $\pitilde$ is the bundle map defined in Section \ref{sec-maps}.
This bracket of
1-forms has the property $[df,dg]=d\{f,g\}$, so
the Lie algebra $\cala/\reals$ appears as a subalgebra of $\cale.$
The bracket on $\cale$ is related to multiplication
by functions through the Leibniz-type identity
\begin{equation}
\label{eq-anchor}
[a,fb]=f[a,b]+(\rho (a)\cdot f)b,
\end{equation} where $\rho=\tilde{\pi}$.
For any commutative associative algebra $\cala$ over $\reals$,
an $(\reals,\cala)$ {\em Lie algebra} is a Lie algebra
$\cale$ over $\reals$ carrying the additional structure of an
$\cala$-module, together with a map $\rho$ from $\cale$ to the
derivations of $\cala$ which is a homomorphism for both the
$\cala$-module and Lie algebra structures, and which satisfies
identity (\ref{eq-anchor}) above. When $\cala=\cinf(M)$
and $\cale$ is the $\cinf(M)$-module
of sections of a vector bundle $E$, the map $\rho$ is realized by a
bundle map from $E$ to $TM$, which we will also denote by $\rho$.
The bracket on sections of $E$ together with the map $\rho$,
called the {\em anchor}, is called a {\em Lie algebroid} structure on
$E$.
Thus the cotangent bundle $T^*M$ of a Poisson manifold $M$ is a Lie
algebroid in a natural way.
The bracket on 1-forms on $M$ was discovered
independently by many people, beginning apparently with Fuchssteiner
\cite{fu:lie}. The
Lie algebroid structure on $T^*M$ was first used in
\cite{co-da-we:groupoides,we:symplectic groupoids}.
There is also a connection in the reverse direction between Poisson
manifolds and Lie algebroids: the dual vector bundle $E^*$ of any Lie
algebroid carries a natural Poisson structure
\cite{co-da-we:groupoides,co:dirac}.
Like the Lie--Poisson
structure on the dual of a Lie algebra, it is determined by the
brackets among a small class of functions--in this case, the fibrewise
affine functions on $E^*$, which are identified with sections of $E$
plus
functions on $M$.
Besides the cotangent bundles of Poisson manifolds, there are many
other interesting examples of Lie algebroids. The tangent bundle $TM$
of any manifold is a basic example, isomorphic as a Lie
algebroid to $T^*M$ when $M$ is a nondegenerate (i.e. symplectic)
Poisson manifold. Other examples are integrable subbundles of $TM$,
Lie algebras $\frakg$ (which are Lie algebroids over a point), and product
bundles $M\times \frakg$ for manifolds $M$ carrying actions of
$\frakg$, with a bracket on $\cinf(M,\frakg)$ induced from the
action.
It can be useful to think of a Lie algebroid $E$ over $M$ as a ``new
tangent bundle'' for $M$. The algebra $\Omega_E(M)$ of
multilinear alternating forms on $E$
then plays the role of differential forms for $M$ with this new
structure. In fact, the Lie algebroid properties enable one to define
a differential $d_E$ on the algebra $\Omega_E(M)$ which makes it into
a complex whose cohomology is known as {\em Lie algebroid
cohomology}. Special cases of this cohomology include de Rham
cohomology ($E=TM$), Poisson cohomology ($E=T^*M$), Chevalley
cohomology of Lie algebras ($M$ is a point), and leafwise de Rham
cohomology of foliations ($E$ is the tangent bundle along the leaves).
In fact, the Lie algebroid structure on $E$ is essentially equivalent to
the differential $d_E$ on $\Omega_E(M)$. In this way, a Lie algebroid
can be viewed as a supermanifold (the space on which $\Omega_E(M)$ is
the functions) carrying an odd vector field with square zero
(the derivation $d_E$)
\cite{al-sc-za-ko:geometry,va:lie}.
\subsection{Lie groupoids}
Since an $(\reals,\cala)$ Lie algebra $\cale$ acts by
derivations on the algebra $\cala$,
a Lie group $\calg$ whose Lie algebra is $\cale$
should act on $\cala$ by automorphisms, at least when
some completeness condition is satisfied. In
addition, the $\cala$ module structure on $\cale$ should be reflected
in some further structure on $\calg$.
We shall limit ourselves here to the geometric situation where $\cale$
is the space of sections of a Lie algebroid $E$ over a manifold $M$.
The $\cinf(M)$ module structure on $E$ is
``integrated'' to the fact that $\calg$ is itself a space of sections
of a ``bundle'' $\beta:G\rightarrow M$,
and the action of $\calg$ on $\cinf(M)$ becomes an action by
diffeomorphisms of $M$, which is encoded geometrically by a second
map $\alpha:G\rightarrow M$. The structure on the manifold $G$ is
that of a {\em Lie groupoid}.
To be precise, we recall first that a {\em small category}
$G$ over a {\em base} $M$ is a set $G$ equipped with {\em
source} and {\em target} maps $\beta$ and $\alpha$ from $G$ to
$M$, a {\em unit section} $\epsilon:M\rightarrow G$, and a {\em
multiplication} operation $(x,y)\mapsto xy$ defined
on the set
$G*G=\{(x,y)\in G\times G|\beta(x)=\alpha(y)\}$ of {\em composable
pairs}. These operations satisfy the conditions that
$\alpha(xy)=\alpha(x)$,
$\beta(xy)=\beta(y)$, $(xy)z=x(yz)$ when either side is defined, and
the elements of $\epsilon(M)$ in $G$ act as identities for
multiplication. If all the elements of $G$ have inverses with
respect to these identities, $G$ is called a {\em groupoid}. If $G$
and $M$ are manifolds, and the structural maps are smooth (one
requires $\alpha$ and $\beta$ to be submersions to insure that the
domain of multiplication is a manifold), then $G$ is called a {\em Lie
groupoid}.
The Lie algebra of vector fields on a Lie groupoid $G$ contains a
distinguished subalgebra $\cale$ of fields which are left-invariant in
a certain
sense; these are the sections of a vector bundle $E$ which can be
identified with the normal bundle to $\epsilon(M)$ in $G$, and then with
the kernel of $T\alpha$.
$T\beta$ is then an anchor map $E\rightarrow TM$ for a Lie algebroid
on $E$. We call this {\em the Lie algebroid of the Lie
groupoid} $G$.
As one knows already from the case of groups (groupoids for which $M$
has one element), there may be nonisomorphic groupoids having
isomorphic Lie algebroids--the groupoids are distinguished by the
zeroth and first homotopy groups of their $\alpha$-fibres. But unlike
(finite dimensional) Lie algebras, not all Lie algebroids can be
integrated to Lie groupoids \cite{al-mo:suites}. They can be integrated
\cite{pr:troisieme} to
{\em local Lie groupoids}, the natural candidates for a
neighborhood of $\epsilon(M)$ in a Lie groupoid.
In any [local] Lie groupoid $G$ over $M$, the set $S(G)$ of submanifolds which
project diffeomorphically onto $M$ under both $\alpha$ and
$\beta$ (they are called {\em bi-sections})
inherits a [local] group structure from the multiplication on
$G$. The Lie algebra of this group (defined in terms of its
1-parameter subgroups) can be identified with the sections of the Lie
algebroid of $G$.
\section{Poisson groupoids and their actions}
\subsection{Symplectic groupoids}
\label{subsec-symplectic}
We have seen that the cotangent bundle $T^*M$ of a Poisson manifold
has a natural Lie algebroid structure derived from the Poisson bracket
of functions. If there is a Lie groupoid $G$ whose Lie algebroid is
isomorphic to $T^*M$, we say that $M$ is an {\em integrable} Poisson
manifold.
When the fibres of $\alpha$ are connected and simply connected (and
sometimes even when they are not),
the canonical symplectic structure on $T^*M$ induces
on $G$ a symplectic structure for which the
multiplication is symplectic in the sense that its graph
$\{(z,x,y)|z=xy\}$ is a lagrangian submanifold of the product
$G\times\overline{G}\times\overline{G}$.
This property of multiplication implies
that the set $LS(G)$ consisting of the lagrangian bi-sections is a
subgroup of $S(G)$, as is its subgroup $HLS(G)$ consisting of the
lagrangian bi-sections obtained from $\epsilon(M)$ by hamiltonian
deformations. The Lie algebras of these two groups
consist of the closed and exact 1-forms respectively. In particular,
$HLS(G)$ is a group whose Lie algebra is $\cinf(M)/\reals$. There is
a natural action of $HLS(G)$ on $M$ by Poisson automorphisms which can
be considered as ``inner,'' since they are generated by hamiltonian
vector fields.
We refer to \cite{xu:flux} for further
discussion of these groups.
The target map $\alpha:G\rightarrow M$ of a [local] symplectic
groupoid is always a Poisson map. This shows that any Poisson manifold
can be realized as the quotient of a symplectic manifold by a
foliation compatible with the symplectic Poisson brackets. Such
``symplectic realizations'' were first studied by Lie
\cite{li:theorie}, who used them to prove his ``third theorem'': the
existence of a local Lie group corresponding to any finite dimensional
Lie algebra. Global realizations for arbitrary Poisson manifolds were
first found by Karasev \cite{ka:analogues} and the author
\cite{we:symplectic groupoids}.
\subsection{Groupoid actions}
A groupoid $G$ over a set $M$ can act on a space $Q$
equipped with a map $J:Q\rightarrow M$, which we call the {\em moment
map} of the action. By definition, the action is a map $(x,q)\mapsto
xq$ from $G*Q=\{(x,q)|\beta(x)=J(q)\}$ to $Q$ satisfying the condition
$\alpha(xq)=\alpha (x)$ and the usual laws for an action. For
instance, $G$ acts on itself by left multiplication with $J=\alpha$
and on $M$ with $J$ the identity. A groupoid action of $G$ on $Q$
induces a group action of $S(G)$ on $Q$.
If $G$ is a symplectic groupoid for the Poisson manifold $M$, a $G$
action on a Poisson manifold $Q$ is a {\em Poisson action} if its graph
$\{(r,x,q)|r=xq\}$ is a coisotropic submanifold of
$Q\times\overline{G}\times \overline{Q}$. The
moment map $J:Q\rightarrow M$ is then a Poisson map. This
statement is also true for ``local actions.'' Conversely, we have
the following theorem of Dazord
\cite{da:groupoides} and Xu
\cite{xu:morita poisson}.
\begin{thm}
\label{thm-action}
Let $G$ be a local
symplectic groupoid for $M$. Every Poisson map
$J:Q\rightarrow M$ is the moment map for a unique local
action of $G$ on $Q$. If $M$ is integrable and $G$ is its
$\alpha$-connected and
$\alpha$-simply connected (global) symplectic groupoid, then $J$ is
the moment map for an action of
$G$ on $Q$ if and only if it is complete. The action is unique.
\end{thm}
This theorem unifies the two examples of complete Poisson maps which
were classified in Section \ref{sec-completeness}. When $M$ is a
Lie-Poisson space $\frakg^*$, $G$ is $T^*G_0$, where $G_0$ is
the connected and simply connected Lie group whose Lie algebra is
$\frakg$. Poisson actions of $G$ then correspond to hamiltonian actions of
$G_0$ together with their momentum maps. When $M$ is symplectic,
$G$ is $\pi(M)$, the {\em fundamental groupoid} of $M$ consisting of
homotopy classes of paths with fixed endpoints. An action of $G$ is
just a flat connection on a fibre bundle over $M$; it is a Poisson
action when the fibre is a Poisson manifold and the parallel
translations are Poisson maps.
To close this section, we note that Zakrzewski \cite{za:quantum} has
introduced a notion of morphism between symplectic groupoids such that
each morphism induces a complete Poisson map between the underlying
Poisson manifolds. The morphisms are not in general functors or even
mappings between the groupoids considered as categories, but are
rather Poisson relations which are lagrangian submanifolds
compatible with the groupoid structures.
\subsection{Poisson Lie groups and groupoids}
\label{subsec-groups and groupoids}
Symplectic groupoids appeared in the 1980's with the
independent work of Karasev \cite{ka:analogues}, Zakrzewski
\cite{za:quantum}, and the
author \cite{we:symplectic groupoids},
motivated by quantization problems. Meanwhile, a
theory of Poisson Lie groups had been
developing through the work of Drinfel'd
\cite{dr:hamiltonian} and Semenov-Tian-Shansky \cite{se:what,se:dressing}
on completely integrable systems and quantum
groups. It was therefore natural to unify these two theories with a
notion of Poisson groupoid \cite{we:coisotropic}.
A {\em Poisson groupoid} (for consistency, we should
probably say ``Poisson Lie groupoid'')
is a Lie
groupoid $G$ with a Poisson structure for which the graph of
multiplication is a coisotropic submanifold of
$G\times\overline{G}\times\overline{G}$. If $G$ happens to be
symplectic, then the graph of
multiplication is forced to be lagrangian, and we have a symplectic
groupoid as defined above.
The base $M$ of a Poisson groupoid $G$ has
a unique Poisson structure making the target and source maps
Poisson and anti-Poisson respectively, but this structure no longer
determines the local groupoid as it did in the symplectic case.
For example, if $M$ is a point, $G$ can be any
Poisson Lie group.
The infinitesimal object which encodes the local structure of a Poisson
groupoid is a {\em Lie bialgebroid}. This is a Lie algebroid $E$ for which
the dual vector bundle $E^*$ also carries a Lie algebroid structure
which is compatible in a certain way with that on $E$. The
compatibility condition was first determined by Mackenzie and Xu
\cite{ma-xu:lie} and then reinterpreted by Kosmann-Schwarzbach
\cite{ko:exact} in the language of Gerstenhaber algebras (a form of
graded Poisson algebras).
Mackenzie and Xu \cite{ma-xu:integration} proved
that, if $E$ is the Lie algebroid of a groupoid $G$ with connected and
simply connected $\alpha$-fibres, then a compatible Lie algebroid
structure on $E^*$ always comes from a Poisson groupoid structure on $G$.
All of this structure was discovered much earlier
\cite{dr:hamiltonian} in the case of a Poisson Lie group $G$.
Multiplication in such a group is a Poisson map from $G\times G$ to
$G$. The Poisson tensor vanishes at the identity element, so that
$\frakg^*$ has the tangent Lie algebra structure. The corresponding
Lie group, called the {\em dual Poisson Lie group} to $G$ and denoted
by $G^*$, is uniquely determined only locally; this ambiguity is
frequently ignored in the literature.
The compatibility condition between
Lie algebra structures on $\frakg$ and $\frakg^*$ in a {\em Lie bialgebra}
is that the dual of the second
structure is a Lie algebra 1-cocycle for $\frakg$ with values in the
second exterior power of the
adjoint representation. This condition has another very useful
formulation: the direct sum $\frakd=\frakg \oplus \frakg^*$ carries a
unique Lie algebra structure for which the natural symmetric bilinear form
(in which the summands are isotropic) is invariant under the adjoint
representation and the summands are subalgebras
with the given structures. The algebra $\frakd$ is
called the {\em double} of the Lie bialgebra, and the corresponding
(connected, simply connected) Lie group $D$ is the double of the
Poisson Lie group $G$. Any Lie algebra $\frakd$ provided with
an invariant symmetric bilinear form and a complementary pair of
isotropic subalgebras is called a {\em Manin triple}; these objects
are equivalent to Lie bialgebras.
It turns out \cite{se:dressing}
that an open dense subset of $D$ (equal to all of $D$ in some
cases) carries a natural
symplectic structure, and that there is a Poisson projection from this
open set to $G$. This projection resembles the target map of a
symplectic groupoid, and in fact Lu and Weinstein \cite{lu-we:groupoides} have
shown that a slight modification of $D$ produces a symplectic manifold
$\tilde{D}$ which carries a pair of ``commuting'' symplectic groupoid
structures with bases $G$ and $G^*$. This {\em symplectic double
groupoid} is in a sense a geometric model for a Hopf algebra, or
``quantum group,'' of which the bialgebra $(\frakg,\frakg^*)$ represents the
classical limit. (Zakrzewski \cite{za:quantum} replaces
one of the the groupoid products by a
``coproduct'' to model more faithfully the Hopf algebra structure.)
In \cite{we-xu:classical},
the double groupoid is used in a geometric analog
of part of the construction of knot invariants from Hopf algebras; in
particular, many ``set theoretic solutions of the quantum Yang-Baxter
equation'' are found, answering a question of Drinfel'd \cite{dr:unsolved}.
The doubles of general Lie bialgebroids are discussed in Section
\ref{subsec-brackets}.
\subsection{Poisson group actions}
\label{subsec-actions}
If $G$ is a Poisson groupoid over the Poisson manifold $M$, a $G$
action on a Poisson manifold $Q$ is a {\em Poisson action} if its graph
$\{(r,x,q)|r=xq\}$ is a coisotropic submanifold of
$Q\times\overline{G}\times \overline{Q}$. As in the symplectic case,
the moment map $J:Q\rightarrow M$ is necessarily a Poisson map. We
have already examined such actions when $G$ is symplectic; now we will
look at the case where $G$ is a group.
An action on $Q$ of a Poisson Lie group $G$ is a Poisson action when
the ``action map'' $G\times Q\rightarrow Q$ is a Poisson map. Each
element $x$ of $G$ acts on $Q$ as a diffeomorphism, but this
diffeomorphism is not necessarily a Poisson map.
Recall that, when $G$ has the zero Poisson structure (so that a
Poisson action is just an action by Poisson automorphisms), certain Poisson
actions of $G$ are distinguished as ``hamiltonian.'' These are
generated by momentum maps to $\frakg^*$ and induce actions of the
symplectic groupoid $T^*G$ of $\frakg^*$. This theory of momentum maps
has been generalized by Lu \cite{lu:momentum} to the case of general Poisson
Lie groups. Her momentum maps take values in the dual group $G^*$ and
exist for all Poisson actions on simply connected symplectic
manifolds, and for many other actions as well, which we than call
``hamiltonian'' as before. One can lift these
actions to the symplectic
groupoid $\tilde{D}$ of $G^*$. Conversely, if $G$ is connected and
simply connected, by Theorem
\ref{thm-action} above, one can
integrate an arbitrary Poisson map $J:Q\rightarrow G^*$ to a local
action of $\tilde{D}$ on $Q$, which is global if $J$ is complete.
Now, if $G$ is complete as a Poisson group (see Section
\ref{subsec-complete groups}),
$\tilde{D}$ contains a copy of $G$ naturally embedded in its group
of bi-sections, so there is an induced action of $G$ on $Q$. These
bi-sections are not lagrangian, so the action is not by Poisson maps;
rather it is a Poisson action--in fact, it is the action whose momentum
map is $J$.
The identity map on $G^*$ is an important special case. The
local action of $G$ on $G^*$ generated by this map is
called the {\em dressing action}. (The
term originated in the theory of completely integrable differential
equations (see \cite{se:dressing}),
in which ``dressing transformations'' were used to build
complicated solutions from simpler ones.) This
action is global if and only if $G^*$ is a complete Poisson Lie group
in the sense of \ref{subsec-complete groups}. This shows that a Poisson
map $J:Q\rightarrow G^*$ is guaranteed to be the momentum map of a
global Poisson action of $G$ only if $J$ is a complete map {\em and} $G$ is a
complete Poisson Lie group.
\subsection{Poisson homogeneous spaces}
\label{subsec-homogeneous}
Transitive Poisson actions of a Poisson Lie group $G$ were identified
by Drinfel'd \cite{dr:poisson} with maximal isotropic subalgebras of the double
$\frakd$ of the Lie bialgebra $(\frakg,\frakg^*)$. This result has
been applied by Karolinsky \cite{ka:homog-compact}
to the classification of Poisson homogeneous spaces of compact
semi-simple Lie groups with the standard Poisson Lie group structure.
Lu \cite{lu:classical} has
obtained a similar classification and, in addition, has shown that
these homogeneous spaces correspond to solutions of the so-called
{\em classical dynamical Yang-Baxter equation} \cite{fe:conformal}.
It is quite
intriguing that these solutions have also been shown by
Etingof--Varchenko \cite{et-va:geometry}
to produce examples of Poisson groupoids.
while Lu \cite{lu:homogeneous} has found a more conceptual approach to
Drinfeld's identification by connecting Poisson homogeneous spaces
directly to
Lie algebroids. A very beautiful theory should appear when the
connections among all these results are sorted out.
The homogeneous spaces of compact semisimple groups known as {\em flag
manifolds} carry very interesting homogeneous Poisson structures for
which the symplectic leaves are the cells of the Bruhat (or Schubert)
decomposition \cite{lu-we:poisson}. These Poisson structures have
been analyzed in great detail by Evens and Lu \cite{ev-lu:poisson,
lu:coordinates}, who use them to explain geometrically
some hitherto mysterious calculations by Kostant \cite{ko:lie} of
cohomology in Lie algebras and flag manifolds. In particular, they
show that Kostant's ``harmonic forms'' are
precisely the forms which are Poisson-harmonic in the sense of Section
\ref{sec-cohomology}. Their work suggests the possibility of using
Poisson structures as a tool for studying the topology of singular
spaces. (I owe this remark to Sam Evens and J.-H. Lu.)
\subsection{Moment maps vs. momentum maps}
The terms ``moment map'' and ``momentum map'' are usually
used interchangeably in the literature, with different authors
preferring each of these two translations of Souriau's
\cite{so:structure}
French term, ``moment.'' By contrast, in this paper, we have used the terms
in different ways. For us, a ``momentum map'' is a Poisson map
$J:Q\rightarrow G^*$ to the dual of a Poisson Lie group $G$,
generating a hamiltonian action of $G$ on $Q$. On the other hand, a
``moment map'' is a map to the base of a groupoid which is acting on
$Q$; for a Poisson groupoid action, the moment map is a Poisson
map.
These two objects are related in the following way. The momentum map
$J:Q\rightarrow G^*$ for a hamiltonian action of a Poisson group $G$
is also the moment map for the corresponding Poisson groupoid action
of the symplectic groupoid action of $\tilde{D}$ on $Q$. But some
pieces of the puzzle are missing here. Among the Poisson actions of a
Poisson groupoid $G$ on $Q$, there should be some which are
distinguished as ``hamiltonian,'' having momentum maps to the dual
Poisson groupoid $G^*$ (defined as in the case of groups by
integrating the dual of the Lie algebroid, when this is possible).
These momentum maps should then generate actions on $Q$ by a
symplectic groupoid $\tilde{D}$ of $G^*$. A first look at the problem
suggests that hamiltonian actions may be rare for groupoids which are
not groups.
\section{Modular theory}
\label{sec-modular}
The {\em modular automorphism group} of a von Neumann algebra $\cala$
is a one-parameter group of automorphisms of $\cala$ which is
generated by a weight of $\cala$. It is the trivial group whenever
the weight is a trace. This group was first constructed
by Tomita, who was motivated by the
special case of the convolution algebra of a locally compact group, on
which the modular function measures the discrepancy between left- and
right-invariant measures. An exposition by Takesaki rendered
Tomita's work much more
accessible, and the modular
theory is now known as {\em Tomita--Takesaki theory}. We refer to
Section V.3 of Connes' book \cite{co:noncommutative} for further
references to this theory and its connections with the KMS
(Kubo--Martin--Schwinger) equilibrium conditions in statistical
mechanics.
The modular theory was a basic ingredient in Connes' classification
theory for type III von Neumann algebras. In particular, Connes
sharpened the theory by proving a ``Radon--Nikodym theorem'' to the
effect that the modular automorphism groups generated by two different
weights are equal modulo inner automorphisms. As a
result, each von Neumann algebra $\cala$ carries a well-defined
one-parameter subgroup of its outer automorphism group. This group is
trivial if and only if $\cala$ admits a trace.
When the algebra $\cala$ is commutative, every weight is a trace, and
so the modular automorphism group is always trivial. Nevertheless,
Gallavotti and Pulvirenti \cite{ga-pu:classical} observed that the
modular ideas could be applied to classical statistical mechanics if a
subalgebra $\cala$ of the complex-valued continuous functions $C(M)$
on a topological space $M$ is equipped with a skew-symmetric bracket,
with values in $C(M)$, satisfying some axioms resembling those of a
Poisson bracket (not including the Jacobi identity!). They define a
{\em state} in this context to be a probability measure $\mu$ on $M$
such that $\cala$ is contained and dense in $L^2(M,\mu)$. The measure
$\mu$ is the classical analog of a weight, and the analog of the trace
condition is that it should vanish when applied (via integration) to
any Poisson bracket $\{f,g\}.$
Gallavotti and Pulvirenti then impose some further conditions so
that the skew-hermitian bilinear form $\mu\{\overline{f},g\}$ on
$\cala$ is defined and is realized by a densely defined operator
$X_{\mu}$ for which $iX_{\mu}$ is essentially self-adjoint. $X_{\mu}$ is
a derivation (generally unbounded); the main theorem in \cite{ga-pu:classical}
states that this derivation generates a
one-parameter group of automorphisms of $L^\infty(M,\mu)$ coming from
a one-parameter group of automorphisms of the measure space
$(M,\mu)$.
The analysis just described becomes very simple when
$M$ is a Poisson manifold and $\cala$ is the
algebra of smooth functions. $X_{\mu}$ is a
smooth Poisson vector field when $\mu$ is a smooth
measure. The construction of this {\em modular vector field} is
local, so there is no reason to require
$\mu$ to be of total measure 1, or even finite.
This vector field $X_{\mu}$ was constructed by Koszul \cite{ko:crochet} for
reasons totally unconnected with operator algebras or statistical
mechanics (though he did observe that, for $M$ a Lie--Poisson space
$\frakg ^*$ and $\mu$ a translation-invariant measure, $X_\mu$ is the
infinitesimal modular character defined as the trace of the adjoint
representation). It was used as a tool for the classification of
Poisson structures by Dufour--Haraki \cite{du-ha:rotationnels} and others
\cite{gr-ma-pe:poisson,li-xu:quadratic}.
Stimulated by the modular theory in operator algebras, Weinstein
\cite{we:modular} rediscovered the vector field $X_{\mu}$ by defining
it as the operator which assigns to each function $f$ the divergence
with respect to $\mu$ of its hamiltonian vector field $X_f$. He observed
that a change in $\mu$ simply adds to $X_{\mu}$ a
hamiltonian vector field, the analogue in Poisson terms of an inner
derivation. Thus, the class of $X_{\mu}$ modulo hamiltonian vector
vector fields is a well-defined element of the Poisson cohomology
space $H^1_{\pi}(M)$ which may be called the {\em modular class} of
$M$.
The existence of the modular automorphism {\em group} of a Poisson
manifold $M$ just depends on whether some representative of the modular
class is a complete vector field. Some examples where this may or may
not be the case are given in \cite{we:modular}.
When $M$ is a symplectic manifold, the modular vector field is zero
when $\mu$ is the symplectic measure. Thus, all modular vector fields
are hamiltonian, and when the correspondence ``measure
$\rightarrow$ hamiltonian function'' is reversed, one obtains the
Gibbs measure associated to each hamiltonian system. This aspect of
classical KMS theory was investigated by Basart, Flato, Lichnerowicz,
and Sternheimer \cite{ba-fl-li-st:deformation}
in the setting of conformal symplectic geometry and
deformation theory.
The modular class for Poisson manifolds is
a special case of a construction for Lie algebroids and groupoids
\cite{ev-lu-we:poincare} and even for $(\reals,\cala)$ Lie algebras
\cite{hu:duality}, where it plays a key role in Poincar\'e duality theory.
\section{Self-similarities and Liouville vector fields}
\label{sec-self}
A {\em dilation} of a Poisson manifold $M$ is a pair $(\phi,\lambda)$,
where $\lambda$ is a nonzero real number and
$\phi:M\rightarrow M$ is a diffeomorphism such that
$\{\phi^*f,\phi^*g\}=\lambda\phi^*\{f,g\}$ for all $f$ and $g$ in
$\cinf(M)$. Equivalently, $\phi^*\pi=\lambda^{-1}\pi.$
The dilations of $M$ form a group, and the image $F(M)$ of this
group under the projection to the multiplicative group
$\reals^{\times}$ of nonzero real numbers will be called the {\em
Murray--von Neumann fundamental group}
of $M$, by analogy with a similar invariant for von Neumann
algebras (it is the group denoted by $F(N)$ on p. 457 of
\cite{co:noncommutative}), which is sometimes referred to in that
subject simply as the
``fundamental group'' of a von Neumann algebra. This
similarity was called to
our attention by D. Shlyakhtenko \cite{sh:von neumann}.
$F(M)$ contains all the positive real numbers when $M$ admits a
complete {\em Liouville vector field}, i.e. a vector field $X$ for
which $[X,\pi] = -\pi$. This definition extends the one where $\pi$
comes from a symplectic form $\omega$, in which case $X$ is required
to satisfy $\call_X \omega = \omega$. The existence of such $X$ in
the symplectic case is equivalent to the exactness of the form
$\omega$, and in the general Poisson case to the exactness of $\pi$ as
a cocycle for Poisson cohomology. We therefore call a Poisson
manifold admitting a Liouville vector field {\em exact}.
Although a compact symplectic manifold cannot be exact, examples
in \cite{he-ma-sa:lemme,we-xu:hochschild}
show that compact Poisson manifolds can
be exact. The meaning of this condition in deformation
quantization is discussed in \cite{we-xu:hochschild}, the last section
of which also contains some examples of manifolds for which $F(M)$ is
countable. For a translation-invariant Poisson structure on a torus,
the fundamental group turns out to be closely related to the group of
units in the integers of an algebraic number field
\cite{mi-we:scaling}.
There is a related notion for
Poisson manifolds $M$ which have vanishing modular class, and admitting only
constant Casimir functions. Such a manifold admits a unique ray of
smooth invariant measures. For every Poisson automorphism $\phi$ of $M$,
there is a unique positive real number $\nu$ such that, for any smooth
invariant measure $\mu$ on $M$, $\mu(\phi^*f) = \nu \mu(f)$ for all
$f$ in $\cinf(M)$. The set of all such $\nu$ as $\phi$ ranges over
the automorphism group is the {\em trace scaling group} of $M$.
Finally we mention another connection between the fundamental group and the
modular automorphism group. If $(\phi,\lambda)$ is a dilation of $M$
and $\mu$ is a smooth measure on $M$, then $\phi^* X_{\mu}
=\lambda^{-1}X_{\phi^* \mu}$, so that $\phi^*$ multiplies the modular
class by $\lambda^{-1}$. An infinitesimal version of this argument
shows that, if $X$ is a Liouville vector field, then
$[X,X_{\mu}]+X_{\mu}$ is a hamiltonian vector field. These facts can
be used to show the nonexistence of nontrivial dilations or
Liouville vector fields under certain conditions, such as in the
following example.
On the Poisson Lie group $SU(2)$ with the
Bruhat--Poisson structure, the modular vector field for any measure
is nonzero and tangent to the circle of diagonal matrices. The period
of this vector field is an invariant under diffeomorphisms; hence
$-1$ is the only possible nontrivial element of the fundamental group (it is
realized by the group inversion map, as is the case on any
Poisson--Lie group), and there are no Liouville vector fields. A
similar argument applies to the reduced Poisson structure on the flag
manifold $S^2$. This structure has the form $\{x,y\}=x^2+y^2$ in
local coordinates near one point; the linearized modular vector
field there is a rotation with period $2 \pi$. J.-H. Lu (private
communication)
has shown that
these arguments can be generalized to all the compact groups with
Bruhat--Poisson structure. She proves that, if the modular class is
non-zero, and one of its representatives
acts semi-simply on the
multivector fields on $M$, then a Liouville vector field cannot
exist. (A lemma for this proof is that a modular vector field always
acts trivially on Poisson cohomology.) She then
uses an explicit description \cite{ev-lu-we:poincare} of the modular
vector fields of Bruhat--Poisson structures to show that they act
semi-simply.
It is interesting to contrast the results above with the result of
Sheu (private communication)
that the Bruhat--Poisson Lie groups
$SU(n)$, all admit {\em continuous} Poisson dilations with all possible
scaling factors. (These maps are homeomorphisms which
are symplectic dilations on all symplectic leaves.) It is not known
whether such dilations exist for the Bruhat--Poisson structures on the
other series of classical groups.
\section{Generalizations}
\label{sec-generalizations}
There are two ways in which the notion of Poisson manifold can be
usefully generalized. The first is to keep the Poisson algebra idea,
but to allow the underlying space to be something other than a
differentiable manifold. The second is to alter the axioms for the
Poisson bracket itself.
\subsection{Poisson spaces}
The derivation property of the operations $\{~,h\}$ on a Poisson
algebra $\cala$ suggests that, when $\cala$ is an algebra of functions
on a space, this space should have some kind of ``differential
structure.'' Even on a Poisson manifold, though, it is not really necessary
to differentiate except along the symplectic leaves. This suggests
that there may be useful notions of ``Poisson topological space,''
or even ``Poisson measurable space,'' specified by Poisson algebras
which are dense in the
continuous
or measurable functions on a topological or measure space $M$. We
have already mentioned (see Section \ref{sec-modular})
work in this direction by Gallavotti and Pulvirenti
\cite{ga-pu:classical}.
Poisson topological spaces seem to occur naturally as classical limits
of $C^*$-algebras. For instance, Sheu
\cite{sh:quantization} constructs an algebra of functions on a quantum $SU(2)$
by deforming a Poisson algebra of continuous functions
on the classical $SU(2)$. Landsman \cite{la:concepts} has introduced
a notion of Poisson space designed to serve the purposes of {\em both}
classical and quantum mechanics.
A guide to the measurable case, which should provide classical models
for some von Neumann algebras, may be the notion of ``foliated space''
used by Moore and Schochet \cite{mo-sc:global} in their study of index
theory for foliations. For example, the union $M_0$ of the bounded
symplectic leaves on a Poisson manifold (see Section
\ref{subsec-complete manifolds}), i.e. the ``complete'' part of
$M$, is generally neither open nor closed, but it is measurable. More
generally, there should be a decomposition of any Poisson manifold
into ``ergodic components'' of its symplectic leaf foliation. These
components will generally be measurable rather than smooth spaces.
There are also singular spaces arising from compact group actions.
For example,
the notion of ``stratified symplectic space,'' developed by Lerman and
Sjamaar \cite{sj-le:stratified} plays an essential role in the theory
of reduction by compact group actions (see \cite{ki:moment}).
An important application of
this theory is to moduli spaces of connections (see the paper of
Huebschmann
\cite{hu:certain}).
Egilsson \cite{eg:embedding} has looked at the quotient spaces of
$\reals^{2n}$ by symplectic circle actions and shown that, in certain
cases, these singular Poisson
varieties cannot be embedded into Poisson manifolds.
A general study of Poisson algebraic geometry was begun by Berger
\cite{be:geometrie}, and Saint-Germain \cite{sa:these} has applied
the algebraic point of view to Lie--Poisson geometry for
nilpotent algebras, while Polishchuk \cite{po:algebraic} has
considerably developed the theory in the complex case, where the usual
rigidity associated with compact complex manifolds leads to very
strong classification results.
Finally, Farkas and Letzer (see {fa-le:ring} and several
other recent papers by Farkas) study Poisson structures on commutative
and noncommutative rings.
\subsection{Other brackets}
\label{subsec-brackets}
In the same paper \cite{ki:local} where he characterized Poisson
structures, Kirillov also introduced more general brackets on
sections of line bundles. These structures, dubbed {\em Jacobi
structures} by Lichnerowicz \cite{li:jacobi}, include as special cases
Poisson structures, contact structures, conformal symplectic
structures, and foliations with leaves of all these types. They
naturally arise from Poisson structures as quotients by the flows of
Liouville vector fields (see Section \ref{sec-self} above). Their
local structure was analyzed in \cite{da-li-ma:structure}.
In the geometric study of completely integrable systems, an important
role has been played by pairs of Poisson structures
which are compatible in the sense that their sum is again a Poisson
structure. The theory of so-called {\em bi-hamiltonian structures}
has been extensively developed. We refer to the papers of
Kosmann-Schwarzbach and Magri \cite{ko-ma:nijenhuis}
and Vaisman \cite{va:lecture} for discussion of the related {\em
Poisson--Nijenhuis strucures} from a viewpoint close to that of this
survey. This papers also include references to earlier work on
bi-hamiltonian structures.
Although quotients of Poisson manifolds by groups of Poisson
automorphisms (and even by Poisson group actions) inherit Poisson
structures, passing to submanifolds takes us to a larger category, that
of Dirac structures. Roughly speaking, a Dirac structure is a
singular foliation together with a Poisson structure on the leaf
space. More precisely, a Dirac structure is defined by a subbundle
$L$ of $TM\oplus T^*M$ which is maximal isotropic for the natural
symmetric bilinear form, and which satisfies an integrability
condition discovered by Courant \cite{co:dirac}. When $L$ is the
graph of a map from $T^*M$ to $TM$, the map is $-\tilde{\pi}$ for a
Poisson structure $\pi$. When $L$ is the graph of a map from $TM$ to
$T^*M$, the map comes from a closed 2-form. In general, the
projection of $L$ to $T^*M$ defines a (singular) characteristic
foliation, with a Poisson bracket defined on the leaf space. The
projection of $L$ to $TM$ defines another singular foliation whose
leaves carry closed (but possibly degenerate) 2-forms.
Courant's integrability condition is that the sections of $L$ should be
closed under a natural bracket which he constructed on the sections of
$TM\oplus T^*M$. This suggests that the
theory of Dirac structures should be a special case of a Lie algebroid version
of Drinfel'd's theory of Poisson homogeneous spaces (see Section
\ref{subsec-homogeneous}). However, although the pair $(TM,T^*M)$,
with the usual bracket on vector fields and the zero bracket on 1-forms,
does form a Lie bialgebroid, Courant's bracket satisfies the Jacobi
identity only modulo a ``coboundary term.'' Thus, the double of a Lie
bialgebroid is not a Lie algebroid, but rather a new kind of object
called a {\em Courant algebroid}. The theory of these objects,
resulting in a Lie algebroid version of Manin triples and Poisson
homogeneous spaces (of Poisson groupoids) was developed by Liu,
Weinstein, and Xu \cite{li-we-xu:manin,li-we-xu:dirac}.
In addition, Roytenberg and Weinstein \cite{ro-we:lie} have
interpreted Courant algebroid brackets as strongly homotopy Lie
algebra structures \cite{la-ma:strongly},
a general class of bracket operations on complexes,
which satisfy the Jacobi identity modulo coboundary terms.
Poisson brackets which do not quite satisfy the Jacobi identity are
also quite relevant to the study of mechanical systems with
non-holonomic constraints. See, for example, the work of Koon and Marsden
\cite{ko-ma:poisson}.
Another interesting variation on the notion of Poisson algebra is that
of graded Poisson algebra, with various sign conditions for the
commutativity of the product and anti-commutativity of the bracket.
This leads both to Gerstenhaber algebras
\cite{hu:lie-rinehart,
ko:exact,xu:gerstenhaber} and to other types of super-Poisson
algebras and manifolds \cite{ca-ib:introduction}.
Finally, there is the idea of Nambu \cite{na:generalized} of
describing physical systems by an antisymmetric bracket on more than
two variables in the algebra of observables. Both classical and
quantum aspects of this {\em Nambu
mechanics} have been pursued vigorously in recent years; see the
survey by Flato, Dito, and Sternheimer \cite{fl-di-st:nambu}.
\section{Odds and ends}
This section collects some miscellaneous examples and
questions about Poisson geometry.
In answer to a question raised in an early version of this paper,
Zakrzewski \cite{za:poisson} used ideas from \cite{za:phase} to
construct examples of Poisson structures on any $\reals^{2n}$ having
just two symplectic leaves: the origin and its complement. It turns
out that a simple way to get some of Zakrzewski's examples is to begin
with the standard symplectic structure on $\reals^{2n}$ and then
perform an inversion through the unit sphere. The resulting Poisson
structure on the complement of the origin turns out to have
coefficients which are quartic polynomials and which therefore extend
smoothly over the whole space. Using the inversion again, one can
easily extend these structures to give Poisson structures on the
even-dimensional spheres with just two symplectic leaves: a point and
its complement. It should be interesting to study these examples
further, finding their Poisson cohomology, quantizations, etc.
The examples above are special members of the
class of Poisson manifolds
for which the decomposition into symplectic leaves is locally finite.
The Bruhat--Poisson structures on flag manifolds (see Section
\ref{subsec-homogeneous} above) are examples of such structures, and others can
be constructed on toric varieties, by pushing forward a nondegenerate bivector
field on the Lie algebra of a complex torus. Are these examples ``typical'' in
any sense? Is there an interesting generic class of such Poisson
structures? How are they related to the stratified symplectic
manifolds of Sjamaar--Lerman \cite{sj-le:stratified}? Sam Evens and J.-H. Lu
(private communication) have suggested that Poisson structures with
locally finite symplectic leaf decomposition may be a useful geometric
tool for studying the topology of the spaces on which they are defined,
with \cite{ev-lu:poisson} being only a model example.
There seems to be a close relation between the
Atiyah--Guillemin--Stern\-berg \cite{at:convexity,gu-st:convexity}
theorem on the convexity properties of momentum maps and the
Duistermaat--Heckman theorem \cite{du-he:variation} on the variation
of the symplectic structure of reduced spaces with the value of the
momentum map. (See Kirwan's survey \cite{ki:moment} for
generalizations to actions of nonabelian groups.) Both theorems refer
to a hamiltonian action of a torus $T$ on a symplectic manifold
$M$. The first concerns the momentum map $M\rightarrow {\mathfrak
t}^*$, while the second concerns the ``dual'' map $M\rightarrow M/T$.
Since variation of the cohomology classes of the symplectic forms on
leaves of a Poisson manifold is related to Poisson cohomology, it may
be that ideas of Morita equivalence \cite{gi-lu:poisson,xu:morita
poisson} could be used to establish some duality between these two
results.
In the paper \cite{te:convexity}, Terng proved a convexity theorem for
projections of isoparametric submanifolds of euclidean spaces on their
normal spaces. This theorem in riemannian geometry includes (and was
suggested by) the convexity theorem for projections of coadjoint
orbits of compact Lie groups onto to the dual of a Cartan subalgebra.
Is there some Poisson geometry behind Terng's theorem? Is there a
theorem which encompasses both Terng's results and the general
convexity theorems for momentum maps?
In \cite{ch-me-sc:existence}, a
modification of the Lie--Poisson structure is used to construct connections
with interesting holonomy. What are the Poisson-geometric
implications of this construction?
\section{Acknowledgments}
I would
like to thank Mark Gotay for soliciting this paper and thereby forcing
me to review this broad subject, and for his comments on the manuscript.
I have learned about Poisson geometry from too many people to list
here, but I would like to thank a few people for their timely
help in the of preparation of this paper and for their
comments on the manuscript:
David Ben-Zvi, Johannes Huebschmann, Francis Kirwan,
Yvette Kosmann-Schwarzbach,
Jiang-Hua Lu, Kirill
Mackenzie, Jerry Marsden, Tudor Ratiu, Marc Rieffel, Izu
Vaisman, Ping
Xu. and the referee.
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\end{document}