# Lie groups of Poisson diffeomorphisms

Wilmer<EMAIL_ADDRESS>
###### Abstract

While the symplectomorphism group of a symplectic manifold is well-understood
as an infinite-dimensional Lie group, little is known about the Poisson
diffeomorphism group of other classes of Poisson manifolds. In this paper we
develop an approach to obtain Lie group structures on (subgroups of) the
Poisson diffeomorphism groups of various classes of Poisson manifolds. As
applications, we show that the Poisson diffeomorphism groups of (normal-
crossing) log-symplectic, elliptic symplectic, scattering-symplectic and
cosymplectic manifolds are regular infinite-dimensional Lie groups.

###### Contents

  1. Introduction
  2. 1 Poisson geometry: Lagrangians and diffeomorphisms
    1. 1.1 Lagrangian submanifolds
    2. 1.2 Poisson Lie algebroids
    3. 1.3 Algebraic aspects of the Poisson diffeomorphisms group
    4. 1.4 Poisson structure of divisor type
    5. 1.5 Scattering-symplectic manifolds
  3. 2 Coisotropic bisections
    1. 2.1 Coisotropic bisections of a Poisson groupoid
    2. 2.2 The Lie group of (coisotropic) bisections
    3. 2.3 Lagrangian bisections of symplectic groupoids
    4. 2.4 Linearization of Poisson groupoids
  4. 3 Lie groups of Poisson diffeomorphisms
    1. 3.1 Almost injective Lie algebroids
    2. 3.2 Poisson structures of divisor type
    3. 3.3 Scattering-symplectic structures
    4. 3.4 Almost injective (co)symplectic Lie algebroids
    5. 3.5 Poisson structures of cosymplectic type

## Introduction

A Poisson structure on a manifold $M$ is a Lie bracket $\\{\cdot,\cdot\\}$ on
the space of smooth functions $C^{\infty}(M)$ subject to the Leibniz rule
$\\{fg,h\\}=f\\{g,h\\}+g\\{f,h\\}$ for $f,g,h\in C^{\infty}(M)$.
Alternatively, the Poisson structure can be described by a bivector field
$\pi\in\Gamma(\wedge^{2}TM)$ for which the Schouten-Nijenhuis bracket
$[\pi,\pi]$ vanishes. It induces the Lie bracket on $C^{\infty}(M)$ via
$\\{f,g\\}=\pi(df,dg)$.

A Poisson map $f:(M,\pi_{M})\to(N,\pi_{N})$ between Poisson manifolds
$(M,\pi_{M})$ and $(N,\pi_{N})$ is a smooth map $f$ for which $\pi_{M}$ is
$f$-related to $\pi_{N}$, meaning that
$\wedge^{2}T_{x}f(\pi_{M,x})=\pi_{N,f(x)}$ for all $x\in M$.

A Poisson manifold $(M,\pi)$ comes with a symplectic foliation, spanned by the
Hamiltonian vector fields

$X_{f}=\\{f,\cdot\\},\quad\mbox{for $f\in C^{\infty}(M)$}.$

A symplectic leaf $S$ comes with a symplectic stucture determined by
$\omega_{S}(X_{f},X_{g})=-\\{f,g\\}|_{S}$ for $f,g\in C^{\infty}(M)$.

A Poisson diffeomorphism of $(M,\pi)$ is a diffeomorphism of $M$ that is also
a Poisson map. This defines a subgroup $\operatorname{Diff}(M,\pi)$ of
$\operatorname{Diff}(M)$, called the Poisson diffeomorphism group. Note that a
diffeomorphism is Poisson if and only if it sends each symplectic leaf
symplectomorphically onto a (possibly different) symplectic leaf.

### The Poisson diffeomorphism group

The group of symplectomorphisms $\operatorname{Diff}(M,\omega)$ of a
symplectic manifold $(M,\omega)$ is well-understood as an infinite-dimensional
Lie group [7, 11]. Much less is known about $\operatorname{Diff}(M,\pi)$ for
general Poisson manifolds.

Let $(M,\pi)$ be a Poisson manifold. The group of Poisson diffeomorphisms
$\operatorname{Diff}(M,\pi)$ has many interesting subgroups:

  * •

Foliated Poisson diffeomorphisms $\operatorname{Fol}(M,\pi)$ are those that
send each leaf to itself. Correspondingly, one has the foliated Poisson vector
fields $\mathfrak{fol}(M,\pi)$, which are the Poisson v.f.’s that are tangent
to the symplectic foliation.

  * •

The group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,\pi)$ consists
of time-1 flows of Hamiltonian vector fields generated by a time-dependent
Hamiltonian function.

  * •

The locally Hamiltonian diffeomorphisms
$\operatorname{Ham}_{\mathrm{loc}}(M,\pi)$ are the of time-1 flows of locally
Hamiltonian vector fields generated by a time-dependent closed one-form.

######  Example 0.1 (Symplectic manifolds).

For a (compact) symplectic manifold $(M,\omega)$, the group of locally
Hamiltonian diffeomorphisms coincides with the identity component of the
symplectomorphism group. This follows from the fact that every symplectic
vector field is locally Hamiltonian. ∎

######  Example 0.2 (Zero Poisson manifolds).

For the zero Poisson manifold $(M,0)$, all of the above subgroups are trivial,
while $\operatorname{Diff}(M,\pi)=\operatorname{Diff}(M)$. ∎

######  Example 0.3 (Regular Poisson manifolds).

If $(M,\pi)$ is regular, with underlying symplectic foliation
$(\mathcal{F},\omega)$, then $\operatorname{Diff}(M,\pi)$ is generally larger
than $\operatorname{Fol}(M,\pi)$ (but not for instance in the case when all
the leaves are non-symplectomorphic). Also,
$\operatorname{Ham}_{\mathrm{loc}}(M,\pi)$ is generally smaller than (the
identity-component of) $\operatorname{Fol}(M,\pi)$, which, if they were Lie
groups, can be seen on the level of Lie algebras. The Lie algebra
$\mathfrak{fol}(M,\pi)$ is isomorphic via $\omega^{\flat}$ to the space
$\Omega^{1}_{\mathrm{cl}}(\mathcal{F})$ of closed foliated one-forms, and not
all closed foliated one-forms admit closed extensions in $\Omega^{1}(M)$. ∎

In this paper, we investigate the existence of Lie group structures on the
group of Poisson diffeomorphisms and on its subgroups for several classes of
Poisson manifolds.

### Coisotropic bisections of Poisson groupoids

When $(M,\omega)$ is a symplectic manifold, a manifold structure on
$\operatorname{Diff}(M,\omega)$ can be obtained in the following way. First,
symplectomorphisms on $(M,\omega)$ correspond to Lagrangian submanifolds of
$(M\times M,\overrightarrow{\omega}-\overleftarrow{\omega})$. Second, By means
of Weinstein’s Lagrangian neighbourhood theorem, the symplectic form
$\overrightarrow{\omega}-\overleftarrow{\omega}$ can be linearized around the
graph of a symplectomorphism. Finally, the linearization of $\omega$ around
the graph also linearizes the deformation space of the Lagrangian submanifold:
the Lagrangian submanifolds near the graph of a symplectomorphism form a
closed subspace of the sections of the normal bundle. This ‘models’ the
symplectomorphism group on an infinite-dimensional locally convex space,
namely $\Omega^{1}_{\mathrm{cl},c}(M)$, the space of compactly supported
closed one-forms on $M$.

For a general Poisson manifold $(M,\pi)$, things become much more complicated,
partly because of the intransitive nature on both the global and local level
of the Poisson diffeomorphism groups. For example, points on non-
symplectomorphic leaves can not be permuted, as any Poisson diffeomorphism
sends each symplectic leaf symplectomorphically onto another leaf.

A useful framework to capture intransitive symmetries in differential geometry
is provided by Lie groupoids and Lie algebroids. The group of bisections
$\operatorname{Bis}(\mathcal{G})$ of a Lie groupoid
$\mathcal{G}\rightrightarrows M$ naturally acts on $M$. Moreover, it is well-
known that $\operatorname{Bis}(\mathcal{G})$ comes with a natural Lie group
structure [19, 20].

When $(\mathcal{G},\Pi)\rightrightarrows(M,\pi)$ is a Poisson groupoid [23],
the group of coisotropic bisections $\operatorname{Bis}(\mathcal{G},\Pi)$ acts
by Poisson diffeomorphisms on $(M,\pi)$. This puts the object
$\operatorname{Diff}(M,\pi)$ in a more general perspective: coisotropic
bisections of the pair Poisson groupoid $(M\times
M,\overleftarrow{\pi}-\overrightarrow{\pi})\rightrightarrows(M,\pi)$ are in
one-to-one correspondence with Poisson diffeomorphisms of $(M,\pi)$.

In [14], Mărcut constructs an example of a compact regular codimension-2
Poisson manifold for which $\operatorname{Diff}(M,\pi)$ is not locally path-
connected in $\operatorname{Diff}(M)$. This is on itself not surprising or any
problem –it only tells that $\operatorname{Diff}(M,\pi)$ can not be embedded
in $\operatorname{Diff}(M)$– but it does indicate that the pair Poisson
groupoid might not be the right object to investigate, and we must broaden our
perspective to involve other Poisson groupoids. However, in general there is
no known or obvious Lie group structure on the group of coisotropic
bisections.

######  Example 0.4 (Lagrangian bisections).

A particularly important class of Poisson groupoids over a Poisson manifold
$(M,\pi)$ are, when integrable, the symplectic groupoids. Let
$(\mathcal{G},\Omega)\rightrightarrows(M,\pi)$ be a symplectic groupoid. Its
coisotropic bisection are better known as Lagrangian bisections, studied in
[18, 24]. As explained in more detail in Section 2.3, the group of Lagrangian
bisections $\operatorname{Bis}(\mathcal{G},\Omega)$ has interesting but
complicated interactions with the Poisson diffeomorphism group
$\operatorname{Diff}(M,\pi)$. For instance, the identity path component
$\operatorname{Bis}_{c,0}(\mathcal{G},\Omega)$ acts on $(M,\pi)$ by locally
Hamiltonian diffeomorphisms! In fact, any locally Hamiltonian diffeomorphism
comes from a Lagrangian bisection. The map
$\operatorname{Bis}_{c,0}(\mathcal{G},\Omega)\to\operatorname{Diff}(M,\pi)$
has a non-trivial kernel in general. ∎

### The Lie group of coisotropic bisections

Let $(\mathcal{G},\Pi)\rightrightarrows(M,\pi)$ be a Poisson groupoid. In
order to build charts for a manifolds structure on
$\operatorname{Bis}(\mathcal{G},\Pi)$, it is necessary to describe the
coisotropic deformations of the unit manifold $M\subset(\mathcal{G},\Pi)$. It
turns out $M$ is always Lagrangian in $(\mathcal{G},\Pi)$ (Proposition 2.2),
in which case it makes sense to ask whether the Poisson structure $\Pi$ is
linearizable around it. That is: does there exists a local Poisson
diffeomorphism $(\mathcal{G},\Pi)\dashrightarrow(NM,\Pi_{\mathrm{lin}})$, that
restricts to the identity on $M$? Here, the Poisson structure
$\Pi_{\mathrm{lin}}$ is the linearization of $\Pi$ on the normal bundle $NM$
of $M$ in $\mathcal{G}$, via any tubular neighbourhood (see section 1.1.1). If
the answer to this question is positive, we say that the Poisson groupoid
$(\mathcal{G},\Pi)$ is linearizable.

Under the additional assumption of linearizability of a Poisson groupoid, we
show that its coisotropic bisection group is a Lie group. The used framework
for infinite-dimensional manifolds is the convenient setting [11], in which
the space of smooth maps can be made into an infinite-dimensional manifold,
modelled on the space of compactly supported sections of a vector bundle
(denoted by $\Gamma_{c}(-)$), even when the domain is not compact. Our first
main result is the following.

######  Theorem 0.5 (Theorem 2.12).

Let $(\mathcal{G},\Pi)\rightrightarrows(M,\pi)$ be a Poisson groupoid with Lie
bialgebroid $(\mathcal{A},\mathcal{A}^{*})$. Assume that $(\mathcal{G},\Pi)$
is linearizable around $M$. Then $\operatorname{Bis}(\mathcal{G},\Pi)$ is a
regular embedded Lie subgroup of $\operatorname{Bis}(\mathcal{G})$ with Lie
algebra
$\Gamma_{c}(\mathcal{A},d_{\mathcal{A}^{*}})=\\{v\in\Gamma_{c}(\mathcal{A}):d_{\mathcal{A}^{*}}v=0\\}$,
where $d_{\mathcal{A}^{*}}$ is the differential associated to the Lie
algebroid $\mathcal{A}^{*}\Rightarrow M$.

This breaks up the search of Lie group structures on Poisson diffeomorphism
groups into two steps, each of which are very interesting on their own.

  * Step 1.

Find out which Poisson groupoids are linearizable.

  * Step 2.

Search for Poisson groupoids whose coisotropic bisections ‘compute’ relevant
groups of Poisson diffeomorphisms on the base.

In fact, to apply Theorem 0.5, one is often forced to look for Poisson
groupoids beyond the product, because of the following result from [21].

######  Theorem 0.6 ([21], Theorem 3.25).

Let $(M,\pi)$ be a Poisson manifold. If $(M\times
M,\overleftarrow{\pi}-\overrightarrow{\pi})$ is linearizable around the
diagonal, then $\pi$ must be regular.

The linearization problem of Poisson groupoids has been extensively addressed
in [21], where it is shown that integrations of triangular Lie algebroids of
so-called cosymplectic type, as well as dual integrations of arbitrary
triangular Lie bialgebroids are always linearizable. We recall these results
in more detail in section 2.4.

Building on this work, we show in this paper that several classes of Poisson
manifolds come with relevant Poisson groupoids that turn out to be
linearizable. This way, we obtain Lie group structures on their groups of
Poisson diffeomorphisms. The results are collected in the list of examples at
the end of the introduction.

######  Remark 0.7.

Up until now, there are no known examples of non-regular convenient Lie
groups. All the Lie group structures obtained in this paper are regular, as
they appear as a special case of theorem 0.5. To avoid repetition in the
introduction, the regularity statements for the Lie groups have been omitted.

### Lie groups of Poisson diffeomorphisms

When a Poisson structure $\pi$ on $M$ ‘lifts’ (in the sense of the diagram
below) to a Poisson structure
$\pi_{\mathcal{A}}\in\Gamma(\wedge^{2}\mathcal{A})$ on a Lie algebroid
$\mathcal{A}\Rightarrow M$, integrations of $\mathcal{A}$ are particularly
relevant to the group of Poisson diffeomorphisms.

${\mathcal{A}^{*}}$${\mathcal{A}}$${T^{*}M}$${TM}$$\scriptstyle{\pi_{\mathcal{A}}^{\sharp}}$$\scriptstyle{\pi^{\sharp}}$

We mainly focus on the case that $\mathcal{A}$ is an almost injective Lie
algebroid, in which case we obtain two result of a different kind. The first
is about inner (Poisson) automorphisms, and the second one involves the
($\mathcal{A}$-)locally Hamiltonian diffeomorphisms.

#### Inner automorphisms of almost injective Lie algebroids

When the Lie algebroid $\mathcal{A}\Rightarrow M$ is almost injective, the
bisections of the holonomy groupoid of $\mathcal{A}$ [1, 6] compute the inner
automorphisms of $\mathcal{A}$. Any automorphism of an almost injective Lie
algebroid is determined by its base map, so the inner automorphisms
$\operatorname{InnAut}(\mathcal{A})$ form a subgroup of
$\operatorname{Diff}(M)$. These observations combine to the following.

######  Theorem 0.8 (Corollary 3.4).

Let $\mathcal{A}\Rightarrow M$ be an almost injective Lie algebroid. Then the
group $\operatorname{InnAut}_{c}(\mathcal{A})$ is naturally a Lie group with
Lie algebra $\Gamma_{c}(\mathcal{A})$ for which the inclusion
$\operatorname{InnAut}_{c}(\mathcal{A})\to\operatorname{Diff}(M)$ is a smooth
immersion.

A k-cosymplectic structure on a Lie algebroid $\mathcal{A}\Rightarrow M$
consists of closed one-forms
$\alpha_{1},\dots,\alpha_{k}\in\Omega^{1}(\mathcal{A})$ and a closed 2-form
$\omega\in\Omega^{2}(\mathcal{A})$ such that
$\alpha_{1}\wedge\dots\wedge\alpha_{k}\wedge\omega\neq 0$ and
$\mathcal{A}=\mathcal{F}\oplus\ker\omega$, with
$\mathcal{F}=\cap_{i}\ker\alpha_{i}$. The subbundle
$\mathcal{F}\subset\mathcal{A}$ is involutive and thus can be regarded as a
subalgebroid. Note that $\omega|_{\mathcal{F}}$ is non-degenerate, and thus
induces an $\mathcal{A}$-Poisson structure $\pi_{\mathcal{A}}$ via

${\mathcal{F}^{*}}$${\mathcal{F}}$${\mathcal{A}^{*}}$${\mathcal{A}.}$$\scriptstyle{\left(\omega|_{\mathcal{F}}\right)^{\flat}}$$\scriptstyle{\pi_{\mathcal{A}}}$

We call an $\mathcal{A}$-Poisson structure $\pi_{\mathcal{A}}$ of
($k$-)cosymplectic type if it is induced by some ($k$-)cosymplectic structure
on $\mathcal{A}$.

######  Theorem 0.9 (Theorem 3.26).

Let $(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ be a Lie algebroid with an
$\mathcal{A}$-Poisson structure of cosymplectic type. Then
$\operatorname{InnAut}_{c}(\mathcal{A},\pi_{\mathcal{A}})$ is an embedded Lie
subgroup of $\operatorname{InnAut}_{c}(\mathcal{A})$ with Lie algebra
$\Gamma_{c}(\mathcal{A},\pi_{\mathcal{A}})=\left\\{v\in\Gamma_{c}(\mathcal{A}):[\pi_{\mathcal{A}},v]=0\right\\}$.

Under additional assumptions, the groups
$\operatorname{InnAut}_{c}(\mathcal{A})$ and
$\operatorname{InnAut}_{c}(\mathcal{A},\pi_{\mathcal{A}})$ are initial in
$\operatorname{Diff}(M)$, meaning that the smooth structure is determined by
$\operatorname{Diff}(M)$. This is important, because in some sense, it means
that the Lie group structure is the ‘correct’ one (i.e. it induces the natural
diffeology).

#### Locally Hamiltonian diffeomorphisms

Recall from example 0.4 that the group of Lagrangian bisections of a
symplectic groupoid surjects onto the group of locally Hamiltonian
diffeomorphisms of its underlying Poisson manifold. In case that $\pi$ is non-
degenerate almost everywhere, then the surjection has a discrete kernel. This
lays at the basis of the proof of the following result.

######  Theorem 0.10 (Theorem 3.14).

Let $\mathcal{A}\Rightarrow M$ be an almost injective Lie algebroid and let
$\pi_{\mathcal{A}}$ be a generically non-degenerate Poisson structure on
$\mathcal{A}$. Then
$\operatorname{Ham}_{\mathrm{loc,c}}(\mathcal{A},\pi_{\mathcal{A}})$ is a Lie
group with Lie algebra
$(\Omega^{1}_{\mathrm{cl},c}(\mathcal{A}),[\cdot,\cdot]_{\pi_{\mathcal{A}}})$.

The bracket $[\cdot,\cdot]_{\pi_{\mathcal{A}}}$ on
$\Gamma(\mathcal{A}^{*})=\Omega^{1}(\mathcal{A})$ is induced by the
$\mathcal{A}$-Poisson structure $\pi_{\mathcal{A}}$ (Remark 1.9), and the
group $\operatorname{Ham}_{\mathrm{loc,c}}(\mathcal{A},\pi_{\mathcal{A}})$
consists of $\mathcal{A}$-locally Hamiltonian Poisson automorphisms (see
Section 1.3).

Note that Theorem 0.10 applies in particular to the almost-regular Poisson
structures as studied by Androulidakis and Zambon in [1].

#### Applications

Concrete applications of our approach are collected in the following list of
examples.

######  Example 0.11 (Symplectic foliations).

Let $(M,\pi)$ be a regular Poisson manifold with associated symplectic
foliation $(\mathcal{F},\omega)$. Then $\operatorname{Fol}(M,\pi)$ is a
initial Lie subgroup of $\operatorname{Diff}(M)$, with Lie algebra
$\Omega^{1}_{\mathrm{cl},c}(\mathcal{F})$ of closed, compactly supported one-
forms on $\mathcal{F}$. Note that this is isomorphic via $\omega^{\flat}$ to
the Lie algebra of foliated vector fields $\mathfrak{fol}_{c}(M,\pi)$. This
Lie group structure coincides with the one in [17]. ∎

######  Example 0.12 (Cosymplectic manifolds).

Let $(M,\pi)$ be a Poisson manifold of $k$-cosymplectic type. Its Poisson
diffeomorphism group $\operatorname{Diff}(M,\pi)$ is an embedded Lie subgroup
of $\operatorname{Diff}(M)$, with Lie algebra $\mathfrak{X}_{c}(M,\pi)$
consisting of compactly supported Poisson vector fields. When all the
symplectic leaves are embedded in $M$, then $\operatorname{Fol}(M,\pi)$
becomes an embedded Lie subgroup of $\operatorname{Diff}(M,\pi)$. ∎

######  Example 0.13 (Log-symplectic manifolds).

Let $(M,\pi)$ be a normal-crossing log-symplectic manifold. All the groups
$\operatorname{Diff}(M,\pi)$, $\operatorname{Fol}(M,\pi)$ and
$\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$ are initial Lie subgroups of
$\operatorname{Diff}(M)$, whose Lie algebras are isomorphic to
$\mathfrak{X}_{c}(M,\pi)$, $\mathfrak{fol}_{c}(M,\pi)$ and
$\mathfrak{ham}_{\mathrm{loc},c}(M,\pi)$ respectively. In fact, the space
$\mathfrak{fol}_{c}(M,\pi)$ is isomorphic to
$\mathfrak{ham}_{\mathrm{loc},c}(M,\pi)$ and therefore
$\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$ coincides with the identity
component of $\operatorname{Fol}(M,\pi)$. ∎

######  Example 0.14 (Elliptic symplectic manifolds).

When $(M,\pi)$ is an elliptic symplectic manifold [5], then both
$\operatorname{Diff}(M,\pi)$ and $\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$
are initial Lie subgroups of $\operatorname{Diff}(M)$, with Lie algebras
$\mathfrak{X}_{c}(M,\pi)$ and $\mathfrak{ham}_{\mathrm{loc},c}(M,\pi)$,
respectively. ∎

######  Example 0.15 (Scattering-symplectic manifolds).

The class of scattering-symplectic Poisson structures [12] can also be treated
with our approach, with a small caveat: it requires specific linearization
results. More precisely, we prove a Lagrangian neighbourhood theorem for
scattering-symplectic manifolds.

######  Theorem 0.16 (Theorem 1.33).

Let $(M,\pi)$ be a scattering symplectic manifold, and $i:L\rightarrow M$ a
Lagrangian submanifold transverse to the degeneracy locus $Z\subset M$. Then
$\pi$ is linearizable around $Z$.

As a result, we show $\operatorname{Diff}(M,\pi)$ is an initial Lie subgroup
of $\operatorname{Diff}(M)$, integrating the Lie algebra
$\mathfrak{X}_{c}(M,\pi)$. ∎

### Acknowledgements

This article, combined with [21], is based on the author’s master’s thesis at
Utrecht University. I would like to thank my master’s thesis advisor Ioan
Mărcut for his guidance, many helpful conversations, and for introducing me to
the search for Lie group structures on groups of Poisson diffeomorphisms. My
thanks also goes to Aldo Witte for fruitful discussions. I also want to thank
Rui Loja Fernandes for useful comments on an early version of this paper.

##  1 Poisson geometry: Lagrangians and diffeomorphisms

### 1.1 Lagrangian submanifolds

First, we recall the definition of a coisotropic submanifold.

######  Definition 1.1.

Let $(M,\pi)$ be a Poisson manifold. A submanifold $C\subset M$ is coisotropic
when $\pi(\alpha_{1},\alpha_{2})=0$ for all
$\alpha_{1},\alpha_{2}\in(TC)^{\circ}$, where $(TC)^{\circ}\subset T^{*}M$ is
the annihilator of $TC$. Equivalently,
$(TC)^{\bot_{\pi}}:=\pi^{\sharp}\left((TC)^{\circ}\right)\subset TC$. ∎

Coisotropic submanifolds play a crucial role in this paper, because of the
following proposition, which is folklore.

######  Proposition 1.2.

Let $(M,\pi_{N})$ and $(N,\pi_{N})$ be Poisson manifolds. A map $f:M\to N$ is
a Poisson map if and only if the graph of $f$ is coisotropic in $(N\times
M,\pi_{N}\times(-\pi_{M}))$.

######  Example 1.3.

The graph of a Poisson diffeomorphism $\varphi:M\to M$ is a coisotropic
submanifold in $(M\times M,\pi\times(-\pi))$. There is more to it: the
intersection of the graph of $\varphi$ with the symplectic leaves of $(M\times
M,\pi\times(-\pi))$ is either empty or a Lagrangian submanifold of the leaf.
This hints to a more restrictive class of coisotropic submanifolds, suitable
for the study of Poisson diffeomorphisms. ∎

######  Definition 1.4 ([22], remark 7.8).

Let $(M,\pi)$ be a Poisson manifold. A submanifold $L\subset M$ is Lagrangian
when

$(TL)^{\bot_{\pi}}:=\pi^{\sharp}\left((TL)^{\circ}\right)=TL\cap\operatorname{im}\pi^{\sharp}.$

It is a clean Lagrangian when it intersects every leaf $(S,\omega_{S})$
cleanly in a Lagrangian submanifold of $(S,\omega_{S})$. ∎

The following lemma provides a practical characterization of Lagrangian
submanifolds.

######  Lemma 1.5.

Let $(M,\pi)$ be a Poisson manifold. A submanifold $L\subset M$ is Lagrangian
if and only if it is coisotropic and $\pi(\alpha_{1},\alpha_{2})=0$ for all
$\alpha_{1},\alpha_{2}\in(\pi^{\sharp})^{-1}(TL)$.

###### Proof.

A Lagrangian submanifold $L$ is clearly coisotropic. Moreover, for
$\alpha_{1},\alpha_{2}\in(\pi^{\sharp})^{-1}(TL)$ there exists
$\tilde{\alpha}_{1},\tilde{\alpha}_{2}\in(TL)^{\circ}$ with
$\pi^{\sharp}(\alpha_{1})=\pi^{\sharp}(\tilde{\alpha}_{1})$ and
$\pi^{\sharp}(\alpha_{2})=\pi^{\sharp}(\tilde{\alpha}_{2})$. In particular,
$\pi(\alpha_{1},\alpha_{2})=\pi(\tilde{\alpha}_{1},\tilde{\alpha}_{2})=0$.

To prove the converse, we have to find for $\alpha\in(\pi^{\sharp})^{-1}(TL)$
an element $\tilde{\alpha}\in(TL)^{\circ}$ such that
$\pi^{\sharp}(\alpha)=\pi^{\sharp}(\tilde{\alpha})$. Since
$\alpha|_{TL\cap\operatorname{im}\pi^{\sharp}}=0$, we can construct
$\tilde{\alpha}$ by extending $\alpha|_{\operatorname{im}\pi^{\sharp}}$ by 0
over $TL$. ∎

#### 1.1.1 Linear Poisson structures and the linearization problem for
Lagrangians

Let $E\to L$ be a vector bundle, with scalar multiplication $m_{t}:E\to E$. A
Poisson structure $\pi$ on the total space of $E$ is linear when
$tm_{t}^{*}\pi=\pi$ for all $t\in\mathbb{R}\setminus\\{0\\}$. Since a linear
Poisson structure $\pi$ on $E$ makes it into a Poisson groupoid over $L$, the
zero section must be a clean Lagrangian submanifold of $(E,\pi)$ by
Proposition 2.2. If $\pi$ is any Poisson structure on the total space of $E$
for which the zero section $L\subset E$ is a clean Lagrangian, we can call
$\pi$ linearizable around $L$ when there is a local Poisson diffeomorphism

$(E,\pi)\dashrightarrow(E,\pi_{\mathrm{lin}}),\quad\mbox{with
}\pi_{\mathrm{lin}}=\lim_{t\to 0}tm^{*}_{t}\pi,$

that restricts to the identity on $L$. More generally, if $(M,\pi)$ is any
Poisson manifold, and $L\subset(M,\pi)$ a clean Lagrangian, we can call
$(M,\pi)$ linearizable around $L$ if $(NL,\psi^{*}\pi)$ is linearizable around
$L$ for some (hence every) tubular neighbourhood $\psi:NL\to U\subset M$ of
$L$.

######  Question 1.6.

Let $L\subset(M,\pi)$ be a clean Lagrangian submanifold. When is $(M,\pi)$
linearizable around $M$?

We have addressed this question in more detail in [21], with emphasis on the
case that the ambient manifold is a Poisson groupoid, and the Lagrangian is
the unit section. As in [21], we adopt the following terminology.

######  Definition 1.7.

A Poisson groupoid $(\mathcal{G},\Pi)\rightrightarrows(M,\pi)$ is linearizable
when the Poisson structure $\Pi$ is linearizable around the unit space
$M\subset(\mathcal{G},\Pi)$. ∎

### 1.2 Poisson Lie algebroids

A Lie algebroid $\mathcal{A}\Rightarrow M$ over $M$ consists of a vector
bundle $\mathcal{A}\to M$ together with an anchor
$\rho_{\mathcal{A}}:\mathcal{A}\to TM$ and a Lie bracket
$[\cdot,\cdot]_{\mathcal{A}}$ on $\Gamma(\mathcal{A})$ subject to the Leibniz
rule:
$[v,fw]_{\mathcal{A}}=f[v,w]_{\mathcal{A}}+\mathcal{L}_{\rho_{\mathcal{A}}(v)}(f)w$.
Due to the presence of the bracket, a Lie algebroid behaves much like the
tangent bundle of a manifold $M$, and therefore it is often the case that
geometry on $TM$ can by ‘lifted’ to a Lie algebroid. In this section we
introduce Poisson Lie algebroids as the ‘lifts’ of Poisson structures.

Recall that the bracket of a Lie algebroid $\mathcal{A}\Rightarrow M$ can be
extended to a Schouten-Nijenhuis bracket on
$\Gamma(\wedge^{\bullet}\mathcal{A})$.

######  Definition 1.8.

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid. A Poisson structure on
$\mathcal{A}$, or an $\mathcal{A}$-Poisson structure, is a section
$\pi_{\mathcal{A}}\in\Gamma(\wedge^{2}\mathcal{A})$ satisfying
$[\pi_{\mathcal{A}},\pi_{\mathcal{A}}]=0$. The pair
$(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ is referred to as a Poisson Lie
algebroid. ∎

######  Remark 1.9.

A Lie bialgebroid $(\mathcal{A},\mathcal{A}^{*})$ consists of a a vector
bundle $\mathcal{A}$ with Lie algebroid structures on $\mathcal{A}$ and
$\mathcal{A}^{*}$ that are compatible in the following sense:

$d_{\mathcal{A}^{*}}[v,w]_{\mathcal{A}}=\left[d_{\mathcal{A}^{*}}v,w\right]+\left[v,d_{\mathcal{A}^{*}}w\right]$

for all $v,w\in\Gamma(\mathcal{A})$. Given a Poisson Lie algebroid
$(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$, the $\mathcal{A}$-Poisson
structure $\pi_{\mathcal{A}}$ anchors $\mathcal{A}^{*}$ to $\mathcal{A}$ via
the sharp map determined by
$\beta(\pi^{\sharp}_{\mathcal{A}}(\alpha))=\pi_{\mathcal{A}}(\alpha,\beta)$
for $\alpha,\beta\in\mathcal{A}^{*}$. Also, it induces a bracket on
$\Gamma(\mathcal{A}^{*})$ via the formula

$[\alpha,\beta]_{\pi_{\mathcal{A}}}=\mathcal{L}_{\pi_{\mathcal{A}}^{\sharp}(\alpha)}(\beta)-\mathcal{L}_{\pi_{\mathcal{A}}^{\sharp}(\beta)}(\alpha)-d_{\mathcal{A}}\left(\pi_{\mathcal{A}}(\alpha,\beta)\right).$

Together with the anchor map
$\rho_{\mathcal{A}^{*}}=\rho_{\mathcal{A}}\circ\pi_{\mathcal{A}}^{\sharp}$,
the vector bundle $\mathcal{A}^{*}$ becomes a Lie algebroid whose differential
on $\Gamma(\wedge^{\bullet}\mathcal{A})$ corresponds to
$[\pi_{\mathcal{A}},\cdot]$, from which easily follows that the pair
$(\mathcal{A},\mathcal{A}^{*})$ is a Lie bialgebroid. The data
$(\mathcal{A},\mathcal{A}^{*},\pi_{\mathcal{A}})$ constitutes a triangular Lie
bialgebroid, and the $\mathcal{A}$-Poisson structure $\pi_{\mathcal{A}}$ is
sometimes called the $r$-matrix (especially in the context of Lie bialgebras).

Let $(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ be a Poisson Lie algebroid.
It induces a Poisson structure $\pi$ on $TM$ via the following diagram.

${\mathcal{A}^{*}}$${\mathcal{A}}$${T^{*}M}$${TM}$$\scriptstyle{\pi_{\mathcal{A}}^{\sharp}}$$\scriptstyle{\rho_{\mathcal{A}}}$$\scriptstyle{\rho_{\mathcal{A}}^{*}}$$\scriptstyle{\pi^{\sharp}}$

Therefore, it is natural to refer to $\pi_{\mathcal{A}}$ as the lift of $\pi$
to $\mathcal{A}$.

######  Definition 1.10.

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid. A symplectic structure on
$\mathcal{A}$, or an $\mathcal{A}$-symplectic structure, is a closed, non-
degenerate two form $\omega_{\mathcal{A}}\in\Omega^{2}(\mathcal{A})$. In this
case, we call $(\mathcal{A},\omega_{\mathcal{A}})$ is a symplectic Lie
algebroid. ∎

If $\pi_{\mathcal{A}}$ is a non-degenerate Poisson structure on $\mathcal{A}$,
then the 2-from $\omega_{\mathcal{A}}\in\Omega^{2}(\mathcal{A})$ defined by

$\omega_{\mathcal{A}}^{\flat}=\left(\pi_{\mathcal{A}}^{\sharp}\right)^{-1},$

with $\omega_{\mathcal{A}}^{\flat}(v)(w)=\omega_{\mathcal{A}}(v,w)$ for
$v,w\in\mathcal{A}$, is closed, and therefore symplectic.

######  Lemma 1.11.

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid. Then there is a one-to-one
correspondence between non-degenerate $\mathcal{A}$-Poisson structures and
symplectic structures on $\mathcal{A}$, given by inversion.

### 1.3 Algebraic aspects of the Poisson diffeomorphisms group

Throughout this section, we fix a Poisson manifold $(M,\pi)$. A Poisson
diffeomorphism of $(M,\pi)$ is a diffeomorphism
$\varphi\in\operatorname{Diff}(M)$ that is also a Poisson map, i.e.
$\varphi_{*}(\pi)=\pi$. They form the group $\operatorname{Diff}(M,\pi)$ of
Poisson diffeomorphisms. A vector field $X\in\mathfrak{X}(M)$ is Poisson when
$\mathcal{L}_{X}(\pi)=0$. They form a subalgebra $\mathfrak{X}(M,\pi)$ of
$\mathfrak{X}(M)$ under the Lie bracket.

The Poisson manifold $(M,\pi)$ has an underlying symplectic foliation, leading
to the group of foliated Poisson diffeomorphisms $\operatorname{Fol}(M,\pi)$,
consisting of the maps that send each symplectic leaf symplectomorphically to
itself. Infinitesimally, the foliated Poisson vector fields, constituting a
subalgebra $\mathfrak{fol}(M,\pi)$ of $\mathfrak{X}(M,\pi)$, are those Poisson
vector fields that are tangent to the symplectic foliation.

The Lie algebra $\mathfrak{X}(M,\pi)$ has two more interesting subalgebras.
Let $\Omega^{1}_{\mathrm{cl}}(M)$ and $\Omega^{1}_{\mathrm{ex}}(M)$ be the
closed and exact one-forms on $M$, respectively. We introduce the Lie algebras
of Hamiltonian and locally Hamiltonian vector fields, respectively, as

$\mathfrak{ham}(M,\pi)=\pi^{\sharp}\left(\Omega^{1}_{\mathrm{ex}}(M)\right),\quad\mbox{and}\quad\mathfrak{ham}_{\mathrm{loc}}(M,\pi)=\pi^{\sharp}\left(\Omega^{1}_{\mathrm{cl}}(M)\right).$

A diffeomorphism is (locally) Hamiltonian if is the time-1 flow of a (locally)
Hamiltonian vector field generated by a time-dependent (closed) exact one-
form. These give rise to the group
$\operatorname{Ham}_{\mathrm{(loc)}}(M,\pi)$ of (locally) Hamiltonian
diffeomorphisms. Clearly, there are inclusions

${\mathfrak{ham}(M,\pi)}$${\mathfrak{ham}_{\mathrm{loc}}(M,\pi)}$${\mathfrak{fol}(M,\pi)}$${\mathfrak{X}(M,\pi),}$${\operatorname{Ham}(M,\pi)}$${\operatorname{Ham}_{\mathrm{loc}}(M,\pi)}$${\operatorname{Fol}(M,\pi)}$${\operatorname{Diff}(M,\pi).}$

that are generally not isomorphisms.

######  Proposition 1.12.

The groups $\operatorname{Fol}(M,\pi)$,
$\operatorname{Ham}_{\mathrm{loc}}(M,\pi)$ and $\operatorname{Ham}(M,\pi)$ are
normal subgroups of $\operatorname{Diff}(M,\pi)$.

###### Proof.

Establishing that $\operatorname{Ham}(M,\pi)$ and
$\operatorname{Ham}_{\mathrm{loc}}(M,\pi)$ are actually subgroups goes via the
usual argument: if $\varphi_{t}$ and $\psi_{t}$ are isotopies generated by
(locally) Hamiltonian vector fields $\pi^{\sharp}(\alpha_{t})$ and
$\pi^{\sharp}(\beta_{t})$, respectively, then $\varphi_{t}\circ\psi_{t}$ is
generated by

$\pi^{\sharp}(\alpha_{t})+(\varphi_{t})_{*}\pi^{\sharp}(\beta_{t})=\pi^{\sharp}(\alpha_{t}+(\varphi_{t})_{*}\beta_{t}),$

which is again (locally) Hamiltonian.

The foliated diffeomorphisms clearly form a normal subgroup of
$\operatorname{Diff}(M,\pi)$. Normality of $\operatorname{Ham}(M,\pi)$ and
$\operatorname{Ham}_{\mathrm{loc}}(M,\pi)$ also follows from the identity
$\varphi_{*}(\pi^{\sharp}(\alpha))=\pi^{\sharp}(\varphi_{*}(\alpha))$, which
holds for general $\varphi\in\operatorname{Diff}(M,\pi)$ and
$\alpha\in\Omega^{1}(M)$. ∎

######  Lemma 1.13.

Let $\varphi_{t}$ be an isotopy on $M$ generated by $X_{t}$. Then
$\varphi_{t}$ is a path in $\operatorname{Diff}(M,\pi)$ (resp. in
$\operatorname{Fol}(M,\pi)$) if and only if $X_{t}$ is in
$\mathfrak{X}(M,\pi)$ (resp. in $\mathfrak{fol}(M,\pi)$).

######  Remark 1.14.

A similar result for the (locally) Hamiltonian groups is hard. For symplectic
manifolds, the argument relies on a flux homomorphism, which is not available
for general Poisson manifolds. It is one of the interesting open questions
regarding the Poisson diffeomorphism group.

There are other open questions that will not be addressed in this paper. For
instance, if a time-dependent vector field $X_{t}$ is Hamiltonian for all $t$,
is it generated by a smooth time-dependent function?

######  Remark 1.15.

The infinitesimal algebraic structure of $\mathfrak{X}(M,\pi)$ is encoded in
the first Poisson cohomology group. For example,

$\frac{\mathfrak{X}(M,\pi)}{\mathfrak{ham}(M,\pi)}=H^{1}_{\pi}(M),\quad\frac{\mathfrak{ham}_{\mathrm{loc}}(M,\pi)}{\mathfrak{ham}(M,\pi)}=\operatorname{im}\left(\pi^{\sharp}:H^{1}(M)\to
H^{1}_{\pi}(M)\right).$

When $\pi$ is regular, and induced by the symplectic foliation
$(\mathcal{F},\omega)$, then

$\frac{\mathfrak{fol}(M,\pi)}{\mathfrak{ham}(M,\pi)}=H^{1}(\mathcal{F}),\quad\frac{\mathfrak{fol}(M,\pi)}{\mathfrak{ham}_{\mathrm{loc}}(M,\pi)}=\frac{H^{1}(\mathcal{F})}{\operatorname{im}\left(H^{1}(M)\to
H^{1}(\mathcal{F})\right)}.$

######  Remark 1.16.

There are many more interesting subgroups of $\operatorname{Diff}(M,\pi)$. For
instance, it is very natural to study leafwise Hamiltonian diffeomorphisms,
which are foliated Poisson diffeomorphisms that restrict to a Hamiltonian
diffeomorphism on each leaf. This merely illustrates that the algebraic
structure of the Poisson diffeomorphism group is very intricate indeed.

Finally, we define the $\mathcal{A}$-analogues of the geometric notions above.
Let $(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ be a Poisson Lie algebroid.
A Poisson section is a section $v\in\Gamma(\mathcal{A})$ such that
$\mathcal{L}_{v}(\pi_{\mathcal{A}})=[v,\pi_{\mathcal{A}}]=0$, defining a
subalgebra $\Gamma(\mathcal{A},\pi_{\mathcal{A}})$ of $\Gamma(\mathcal{A})$.
The closed $\mathcal{A}$-forms, $\Omega^{1}_{\mathrm{cl}}(\mathcal{A})$ form a
Lie subalgebra of $\Omega^{1}(\mathcal{A})$ under the bracket on
$\mathcal{A}^{*}$ (Remark 1.9). Therefore, we obtain a subalgebra of
$\Gamma(\mathcal{A},\pi_{\mathcal{A}})$, called the $\mathcal{A}$-locally
Hamiltonians, by

$\mathfrak{ham}_{\mathrm{loc}}(\mathcal{A},\pi_{\mathcal{A}})=\pi_{\mathcal{A}}^{\sharp}\left(\Omega^{1}_{\mathrm{cl}}(\mathcal{A})\right).$

An automorphism of $\mathcal{A}$ is $\mathcal{A}$-locally Hamiltonian when it
is the time-1 flow of a $\mathcal{A}$-locally Hamiltonian vector field
generated by a time-dependent closed $\mathcal{A}$-form, forming the group
$\operatorname{Ham}_{\mathrm{loc}}(\mathcal{A},\pi_{\mathcal{A}})$ regarded as
a subgroup of the group $\operatorname{InnAut}(\mathcal{A})$ of inner
automorphisms of $\mathcal{A}$.

### 1.4 Poisson structure of divisor type

To provide context to several central examples considered in this paper, we
give a brief summary of the Poisson structures of divisor type developed by
Klaasse in [10].

A (real) divisor on a smooth manifold $M$ is a pair $(L,\sigma)$ consisting of
a (real) line bundle $L\to M$ and a section $\sigma\in\Gamma(L)$ whose zero
locus $Z_{\sigma}$ is nowhere dense.

The section $\sigma$ of a divisor $(L,\sigma)$ defines an evaluation map
$\sigma:\Gamma(L^{*})\to C^{\infty}(M)$, and the ideal of $C^{\infty}(M)$
associated to $(L,\sigma)$ is its image $I_{\sigma}=\sigma(\Gamma(L^{*}))$.
Note that the vanishing set $Z_{I_{\sigma}}$ of $I_{\sigma}$ is exactly
$Z_{\sigma}$. The vanishing ideal $I_{Z_{\sigma}}$ of $Z_{\sigma}$ is
generally larger than $I_{\sigma}$. An ideal $I\subset C^{\infty}(M)$ that
comes from a divisor is a divisor ideal.

Every divisor ideal is locally principal, as it is generated by
$\langle\alpha,\sigma\rangle$, where $\alpha$ is a local trivialization of
$L^{*}$.

######  Definition 1.17 ([10], Definition 3.6).

A Lie algebroid $\mathcal{A}\Rightarrow M$ is of divisor type when

$\operatorname{div}(\mathcal{A})=\left(\det(\mathcal{A}^{*})\otimes\det(TM),\det(\rho_{\mathcal{A}})\right)$

is a divisor, or, equivalently, when the isomorphism locus of
$\rho_{\mathcal{A}}$ is dense in $M$. The divisor ideal associated to
$\operatorname{div}(\mathcal{A})$ is denoted $I_{\mathcal{A}}$. ∎

Let $I\subset C^{\infty}(M)$ be an ideal and $\mathcal{A}\Rightarrow M$ a Lie
algebroid. Following [10], we set

$\Gamma(\mathcal{A})_{I}=\left\\{v\in\Gamma(\mathcal{A}):\mathcal{L}_{v}(I)\subset
I\right\\},$

the set of sections of $\mathcal{A}$ that preserve $I$. This is a subalgebra
of $\Gamma(\mathcal{A})$ ([10], lemma 3.41).

When $\Gamma(\mathcal{A})_{I}$ is locally finitely generated and projective,
the Serre-Swan theorem gives a vector bundle $\mathcal{A}_{I}\to M$ such that
$\Gamma(\mathcal{A}_{I})=\Gamma(\mathcal{A})_{I}$. The inclusion of sections
determines a bundle map
$\rho_{\mathcal{A}_{I}}^{\mathcal{A}}:\mathcal{A}_{I}\to\mathcal{A}$. Together
with the Lie bracket on $\Gamma(\mathcal{A}_{I})$ inherited from
$\Gamma(\mathcal{A})$, and with anchor
$\rho_{\mathcal{A}_{I}}=\rho_{\mathcal{A}_{I}}^{\mathcal{A}}\circ\rho_{\mathcal{A}}$,
it becomes a Lie algebroid anchored to $\mathcal{A}$.

######  Definition 1.18.

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid and $I\subset C^{\infty}(M)$
an ideal. If it exists, the Lie algebroid $\mathcal{A}_{I}\Rightarrow M$ is
called the ideal Lie algebroid associated to $(\mathcal{A},I)$. ∎

This is often applied in the case that $\mathcal{A}=TM$, for which the
following terminology is introduced in [10].

######  Definition 1.19.

A divisor ideal is projective when the ideal Lie algebroid $T_{I}M:=TM_{I}$
exists. It is standard when it is projective and $I_{T_{I}M}=I$. ∎

By the Serre-Swan theorem, a divisor ideal $I$ is projective when
$\mathfrak{X}(M)_{I}$ is projective and locally finitely generated.

######  Definition 1.20.

A Poisson structure $(M^{2n},\pi)$ is of divisor type when
$(\wedge^{2n}TM,\wedge^{n}\pi)$ is a divisor. The associated divisor ideal is
denoted by $I_{\pi}$. ∎

We are interested in such Poisson structures, because of the following
proposition.

######  Proposition 1.21.

Let $(M,\pi)$ be a Poisson manifold of divisor type, whose divisor ideal
$I_{\pi}$ is projective. Then any Poisson vector field lifts to a section of
$T_{I_{\pi}}M$.

###### Proof.

Let $f\in I_{\pi}$. Then there exists a top-form $\omega\in\Omega^{2n}(M)$
such that $f=\langle\omega,\wedge^{n}\pi\rangle$. If $X\in\mathfrak{X}(M)$ is
a Poisson vector field, then

$\displaystyle\mathcal{L}_{X}(f)$
$\displaystyle=\langle\mathcal{L}_{X}(\omega),\wedge^{n}\pi\rangle+\langle\omega,\mathcal{L}_{X}(\wedge^{n}\pi)\rangle=\langle\mathcal{L}_{X}(\omega),\wedge^{n}\pi\rangle\in
I_{\pi}.$

It follows that $X$ is a section of $T_{I_{\pi}}M$. ∎

One of the main results in [10] (Theorem 4.35) is that whenever $I_{\pi}$ is
projective and $T^{*}_{I_{\pi}}M$ locally admits bases of closed sections,
then $\pi$ lifts to a Poisson structure on $T_{I_{\pi}}M$. If in addition
$I_{\pi}$ is standard, the lift is non-degenerate. This fact is used in the
examples below.

######  Example 1.22 (Log-symplectic manifolds).

A (normal-crossing) log-manifold222Also called a $c$-manifold in [15]. is a
pair $(M,Z)$ of a manifold $M$ and a (closed) hypersurface $Z\subset M$ that
is allowed to have self-intersections. More precisely, around each point $z\in
Z$ there are coordinates $(x_{1},\dots,x_{k},x_{k+1},\dots,x_{n})$ for which
$Z=\\{\prod_{i=1}^{k}x_{i}=0\\}$. Note that $k$ need not be fixed, and is
allowed to vary between different points in $Z$. We call such coordinates
adapted to $Z$. When $Z$ is an embedded hypersurface, we refer to the pair
$(M,Z)$ as a smooth log-manifold.

The vanishing ideal $I_{Z}$ is a standard divisor ideal, and a divisor
$(L,\sigma)$ with $I_{\sigma}=I_{Z}$ is called a normal-crossing log divisor.
The corresponding Lie algebroid $T_{Z}M:=T_{I_{Z}}M$ is called the log-tangent
bundle. The space of sections of $T_{Z}M$, called log-vector fields, is
denoted by $\mathfrak{X}(M,Z)$ and is locally generated by the vector fields
$x_{1}\partial_{x_{1}},\dots,x_{k}\partial_{x_{k}},\partial_{x_{k+1}},\dots,\partial_{x_{n}}$
in an adapted coordinate chart.

######  Definition 1.23.

A Poisson structure $\pi$ on $M^{2n}$ is log-symplectic when
$(\wedge^{2n}TM,\wedge^{n}\pi)$ is a normal-crossing log divisor. ∎

By theorem 4.35 in [10], a log-symplectic structure lifts to a symplectic
structure on the log-tangent bundle $T_{Z}M$. ∎

######  Example 1.24 (Elliptic symplectic manifolds).

A (real) divisor $|D|=(L,\sigma)$ is elliptic when $D:=Z_{\sigma}$ is a
codimension 2 submanifold and the normal Hessian
$\operatorname{Hess}(\sigma)\in\Gamma(\operatorname{Sym}^{2}N^{*}D\otimes L)$
is definite. Its associated ideal is usually denoted by $I_{|D|}$. An ideal
$I\subset C^{\infty}(M)$ is elliptic when it is the ideal of some elliptic
divisor.

An elliptic ideal $I_{|D|}$ is standard, and the associated Lie algebroid
$T_{|D|}M\Rightarrow M$ is called the elliptic tangent bundle. For any
elliptic divisor, there are coordinates $(x_{1},\dots,x_{n})$ such that
$I_{|D|}=\langle x_{1}^{2}+x_{2}^{2}\rangle$. Sections of $T_{|D|}M$ are
locally generated by
$r\partial_{r},\partial_{\theta},\partial_{x_{3}},\dots,\partial_{x_{n}}$.

######  Definition 1.25.

A Poisson structure $\pi$ on $M^{2n}$ is elliptic when
$(\wedge^{2n}TM,\wedge^{n}\pi)$ is an elliptic divisor. ∎

Elliptic symplectic structures appear naturally as the Poisson structure
associated to a stable generalized complex structure [5]. By theorem 4.35 in
[10], an elliptic symplectic structure $\pi$ on $M$ lifts to a symplectic
structure on $T_{|D|}M$, with $|D|=(\wedge^{2n}TM,\wedge^{n}\pi)$ (cf. [5],
lemma 3.4). ∎

### 1.5 Scattering-symplectic manifolds

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid and $\mathcal{B}\Rightarrow
Z$ a subalgebroid of $\mathcal{A}$ over a hypersurface $Z\subset M$. Consider
the module of sections of $\mathcal{A}$ that are "tangent" to $\mathcal{B}$:

$\Gamma(\mathcal{A},\mathcal{B})=\left\\{v\in\Gamma(\mathcal{A}):v|_{Z}\in\Gamma(\mathcal{B})\right\\}.$

This is a locally finitely generated subalgebra of $\Gamma(\mathcal{A})$ and
therefore by the Serre-Swan theorem corresponds to a vector bundle
$[\mathcal{A};\mathcal{B}]\rightarrow M$ that inherits a Lie algebroid
structure from $\mathcal{A}$ [8, 12]. The Lie algebroid
$[\mathcal{A};\mathcal{B}]\Rightarrow M$ is the rescaling of $\mathcal{A}$ by
$\mathcal{B}$.

######  Example 1.26.

Let $(M,Z)$ be a smooth log-manifold. Then $TZ\Rightarrow Z$ is a subalgebroid
of $TM$, and the rescaling $[TM;TZ]$ is exactly the log-tangent bundle
$T_{Z}M$ from Example 1.22. ∎

######  Example 1.27 (Zero-rescalings).

The zero algebroid $0_{Z}\Rightarrow Z$ is a subalgebroid of any Lie algebroid
$\mathcal{A}\Rightarrow M$. Then $[\mathcal{A};0_{Z}]$ is called the zero-
rescaling of $\mathcal{A}$ over $Z$. In the case that $\mathcal{A}=TM$, we
obtain the $0$-tangent bundle ${}^{0}T_{Z}M$. ∎

######  Example 1.28 (Scattering manifolds).

More interesting is the zero-rescaling of $T_{Z}M$ over $Z$. The resulting Lie
algebroid ${}^{sc}T_{Z}M=[T_{Z}M;0_{Z}]$ is the scattering-tangent bundle. A
Poisson structure $\pi$ in $M$ is scattering-symplectic when it lifts to a
symplectic structure $\omega=\pi^{-1}$ on ${}^{sc}T_{Z}M$. We refer to
$(M,Z,\omega)$ as a scattering-symplectic manifold.

In contrast with the zero-tangent bundle, scattering-symplectic structures are
ubiquitous. For example, every even dimensional sphere admists a scattering-
symplectic structure, whose hypersurface is the equator [12]. ∎

The following proposition is a consequence of the Serre-Swan theorem.

######  Proposition 1.29.

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid and $\mathcal{B}\Rightarrow
Z$ a subalgebroid over a hypersurface $Z\subset M$. Let
$\varphi:\mathcal{A}\to\mathcal{A}$ be a bundle map that restricts to a bundle
map $\varphi|_{\mathcal{B}}=\psi:\mathcal{B}\to\mathcal{B}$. Then $\varphi$
lifts to a (unique) map
$[\varphi;\psi]:[\mathcal{A};\mathcal{B}]\to[\mathcal{A};\mathcal{B}]$. If
$\varphi$ and $\psi$ are Lie algebroid morphisms, then so is $[\varphi;\psi]$.

As a consequence, every morphism of the log-tangent bundle $T_{Z}f:T_{Z}M\to
T_{Z}M$ induces a morphisms of the scattering-tangent bundle
${}^{sc}T_{Z}f:{}^{sc}T_{Z}M\to{}^{sc}T_{Z}M$.

#### 1.5.1 Linearization of scattering-symplectic structures

In this section we prove a specific linearization result of scattering
symplectic structures around Lagrangian submanifolds.

Let $(M,Z,\omega)$ be a scattering-symplectic manifold, with underlying
Poisson structure $\pi$, and $i:L\hookrightarrow Z$ a Lagrangian submanifold
transverse to $Z$. Certainly, the Poisson structure is linearizable around
$L\setminus Z$ in $M\setminus Z$, by Weinstein’s Lagrangian neighbourhood
theorem. Moreover, the scattering symplectic structure induces a contact
structure on $Z$ [12] which is linearizable around $\tilde{Z}=Z\cap L$ (which
is Legendrian in $Z$) by the Legendrian neighbourhood theorem in contact
geometry. This gives an intuition to why the Poisson structure is linearizable
around $L$.

The first step in the linearization is a Moser lemma adapted to scattering-
symplectic manifolds.

######  Lemma 1.30 ($sc$-Moser lemma).

Let $i:L\hookrightarrow(M,Z)$ be a submanifold transverse to $Z$. Let
$\omega_{0},\omega_{1}$ be two scattering symplectic forms such that
$\omega_{1}|_{L}=\omega_{0}|_{L}$. Then there exists neighbourhoods
$U_{0},U_{1}$ of $L$ in $M$ and a diffeomorphism $\varphi:(U_{0},U_{0}\cap
Z)\to(U_{1},U_{1}\cap Z)$ such that $\varphi^{*}\omega_{1}=\omega_{0}$ on
$U_{0}$ and $\varphi|_{L}=\operatorname{Id}_{L}$.

######  Remark 1.31.

For the scattering-symplectic forms $\omega_{0},\omega_{1}$, the condition
$\omega_{1}|_{L}=\omega_{1}|_{L}$ does _not_ imply that the cohomology classes
in the scattering-cohomology agree. In fact, often they do not (see [12]). The
scattering-morphism changes the cohomology class of the symplectic form.

###### Proof.

By the splitting theorem for Lie algebroids [3], applied to the log-tangent
bundle, we can assume that $M=E$ is a vector bundle over $L$ and the
hypersurface $Z$ is the bundle $E|_{\tilde{Z}}$ over the hypersurface
$\tilde{Z}=Z\cap L$ in $L$. We set
$\omega_{t}=\omega_{0}+t(\omega_{1}-\omega_{0})$.

On $M\setminus Z$, we can write $\omega_{1}-\omega_{0}=d\psi$ with

$\psi=\int_{0}^{1}\frac{1}{t}m^{*}_{t}(\iota_{\mathcal{E}}(\omega_{1}-\omega_{0}))dt.$

Here, $\mathcal{E}$ is the Euler vector field of $E$, and $m_{t}$ is scalar
multiplication by $t$. On $Z$, this is just a formal expression. We will show
that there is a family of log-vector fields $X_{t}\in\mathfrak{X}(M,Z)$ such
that

$\iota_{X_{t}}\omega_{t}=-\psi\quad\mbox{on $M\setminus Z$}.$ ($*$)

By non-degeneracy of $\omega_{t}$ away from $Z$, the family of vector fields
$X_{t}$ is uniquely determined on $M\setminus Z$, and satisfies
$X_{t}|_{L\setminus\tilde{Z}}=0$.

To show that this extends to a log-vector field over $Z$, we need to zoom in
on a neighbourhood of $Z$. Choose a defining function $\tilde{x}$ of
$\tilde{Z}$ in $L$. Then $x=p^{*}\tilde{x}$ is a defining function for $Z$,
with the nice property that $m^{*}_{t}(x)=x$ (and thus
$\iota_{\mathcal{E}}dx=0$). The key observation is that if $Y$ is a scattering
vector field, then $Y/x$ is a log-vector field.

######  Claim.

The form $x\cdot\iota_{\mathcal{E}}(\omega_{1}-\omega_{0})$ is a scattering
one-form.

###### Proof.

The Euler-vector field $\mathcal{E}$ is a log-vector field, and thus
$x\mathcal{E}$ is a scattering-vector field. Therefore,
$x\cdot\iota_{\mathcal{E}}(\omega_{1}-\omega_{0})=\iota_{x\mathcal{E}}(\omega_{1}-\omega_{0})$
is a scattering one-form. ∎

From the claim, it follows that

$x\cdot\psi=x\int_{0}^{1}\frac{1}{t}m^{*}_{t}(\iota_{\mathcal{E}}(\omega_{1}-\omega_{0}))dt=\int_{0}^{1}\frac{1}{t}m^{*}_{t}(x\cdot\iota_{\mathcal{E}}(\omega_{1}-\omega_{0}))dt$

is a scattering form ($m_{t}$ is a log-map and thus a scattering algebroid
morphism by Proposition 1.29). Hence, we can find scattering vector fields
$\tilde{X}_{t}$ near $Z$ such that
$\iota_{\tilde{X}_{t}}\omega_{t}=-x\cdot\psi$ in a neighbourhood of $Z$. The
log-vector fields $X_{t}=\tilde{X}_{t}/x$ extend the solution to ($*$ ‣ 1.5.1)
over $Z$.

Let $\varphi$ be the time-1 flow of $X_{t}$, which is defined in a
neighbourhood $U$ of $L$ because $X_{t}|_{L}=0$. Then $\varphi$ is a $b$-map
that satisfies $\varphi^{*}\omega_{1}=\omega_{0}$ outside $Z$ by the standard
Moser argument. But since both are scattering forms, they must agree also on
$Z$. ∎

#### 1.5.2 Scattering Lagrangian neighbourhood theorem

Let $i:(L,\tilde{Z})\hookrightarrow(M,Z,\omega)$ be a Lagrangian submanifold
transverse to $Z$. By the splitting theorem, we can assume that
$(M,Z)=(NL,NL|_{\tilde{Z}})$. We can already linearize $\omega$ on the
scattering algebroid level.

######  Lemma 1.32.

The limit

$\omega_{\mathrm{lin}}=\lim_{t\to 0}\frac{m^{*}_{t}\omega}{t}$

defines a linear scattering 2-form on $(NL,NL|_{\tilde{Z}})$ that satisfies
$\omega_{\mathrm{lin}}|_{L}=\omega|_{L}$.

###### Proof.

For every $t\in\mathbb{R}$, the map $m_{t}:NL\to NL$ is a log-morphism and
thus a scattering morphism by Proposition 1.29. Therefore, $m^{*}_{t}\omega$
is scattering 2-form for all $t\in\mathbb{R}$. Because $L$ is Lagrangian,
$m_{0}^{*}\omega=0$. Since $m_{t}^{*}\omega$ is smooth in $t$, the limit
$\lim_{t\to 0}m^{*}_{t}\omega/t$ exists and the resulting form is again
smooth. Clearly,
$m^{*}_{\lambda}\omega_{\mathrm{lin}}=\lambda\omega_{\mathrm{lin}}$, so the
resulting form is indeed linear. ∎

######  Theorem 1.33 ($sc$-Lagrangian neighbourhood theorem).

Let $(M,Z,\omega)$ be a scattering symplectic manifold with Poisson structure
$\pi$ and $i:L\hookrightarrow(M,Z)$ a Lagrangian submanifold transverse to
$Z$. Then there are neighbourhoods $U$ of $L$ in $M$ and $V$ of $L$ in $NL$
and a Poisson diffeomorphism on $\varphi:(U,\pi)\to(V,\pi_{\mathrm{lin}})$
that restricts to the identity on $L$.

###### Proof.

The theorem follows from Lemma 1.30 and Lemma 1.32. ∎

##  2 Coisotropic bisections

In this section, we establish general properties of coisotropic bisections of
Poisson groupoids and their connection to Poisson diffeomorphisms. After that,
we show that they form a Lie group whenever the Poisson groupoid is
linearizable.

### 2.1 Coisotropic bisections of a Poisson groupoid

Let $\mathcal{G}\rightrightarrows M$ be a groupoid. A bisection
$\sigma\in\operatorname{Bis}(\mathcal{G})$ is a section of the source map
$s:\mathcal{G}\to M$ with the property that $l_{\sigma}:=t\circ\sigma:M\to M$
is a diffeomorphism. Any bisection corresponds to and is determined by a left
translation map

$L_{\sigma}:\mathcal{G}\to\mathcal{G},\quad g\mapsto m(\sigma(t(g)),g).$

If $X\in\Gamma(\mathcal{A})$ is a section of the Lie algebroid of
$\mathcal{A}$, then the flow $\varphi_{t}$ of its right-invariant vector field
$\overrightarrow{X}$ gives rise to a family of bisections
$\sigma_{t}=\varphi_{t}|_{M}$, and in fact, it holds that
$\varphi_{t}=L_{\sigma_{t}}$.

Bisections of a Lie groupoid form a group, and can be multiplied: if
$\sigma,\tau\in\operatorname{Bis}(\mathcal{G})$, we define $\sigma\cdot\tau$
as

$\sigma\cdot\tau(x)=m(\sigma(t\circ\tau(x)),\tau(x)),\quad\mbox{ for $x\in
M$,}$

or, equivalently, as $L_{\sigma\cdot\tau}=L_{\sigma}\circ L_{\tau}$. Several
authors already described $\operatorname{Bis}(\mathcal{G})$ as a Lie group,
e.g. in [19, 20].

######  Definition 2.1.

Let $(\mathcal{G},\Pi)\rightrightarrows(M,\pi)$ be a Poisson groupoid. A
bisection $\sigma\in\operatorname{Bis}(\mathcal{G})$ is coisotropic when
$\operatorname{im}\sigma$ is a coisotropic submanifold of $(\mathcal{G},\Pi)$.
The collection of coisotropic bisections is denoted by
$\operatorname{Bis}(\mathcal{G},\Pi)$. ∎

We establish some properties of coisotropic bisections.

######  Proposition 2.2.

Let $(\mathcal{G},\Pi)\rightrightarrows(M,\pi)$ be a Poisson groupoid.

  * (i).

The coisotropic bisections form a subgroup
$\operatorname{Bis}(\mathcal{G},\Pi)$ of $\operatorname{Bis}(\mathcal{G})$.

  * (ii).

A bisection $\sigma\in\operatorname{Bis}(\mathcal{G})$ is coisotropic if and
only if $L_{\sigma}$ is a Poisson map.

  * (iii).

For any $\sigma\in\operatorname{Bis}(\mathcal{G},\Pi)$, the induced
diffeomorphism on the base $l_{\sigma}:M\to M$ is a Poisson map.

  * (iv).

The image of a coisotropic bisection is a clean Lagrangian submanifold.

######  Remark 2.3.

Although every coisotropic bisection is Lagrangian for any Poisson groupoid,
we reserve the term "Lagrangian bisections" for symplectic groupoids only.

###### Proof.

The first assertion follows from the second, while (ii) and (iii) can be
proved by means of the coisotropic calculus developed in [23].

As for (iv), we only have to show that the unit section $M\subset\mathcal{G}$
is a clean Lagrangian, because of (ii). First we show that it is Lagrangian by
applying Lemma 1.5. For $\alpha_{1},\alpha_{2}\in(\Pi^{\sharp})^{-1}(TM)$ we
can write

$\alpha_{1}=\beta_{1}+s^{*}\gamma_{1},\quad\alpha_{2}=\beta_{2}+t^{*}\gamma_{2},$

for $\beta_{1},\beta_{2}\in(TM)^{\circ}$ and $\gamma_{1},\gamma_{2}\in
T^{*}M$. It follows that

$\Pi(\alpha_{1},\alpha_{2})=\Pi(s^{*}\gamma_{1},t^{*}\gamma_{2})+\Pi(\alpha_{1},\beta_{2})+\Pi(\beta_{1},\alpha_{2})-\Pi(\beta_{1},\beta_{2}).$

The first term vanishes because $(\ker Ts)^{\circ}$ and $(\ker Tt)^{\circ}$
are Poisson orthogonal for Poisson groupoids ([23], proposition 4.2.6), the
second and third therm are zero because $\beta_{i}\in(TM)^{\circ}$ and
$\Pi^{\sharp}(\alpha_{j})\in TM$, and the last term vanishes because $M$ is
coisotropic in $(\mathcal{G},\Pi)$. Thus, $M$ is Lagrangian.

Next we show that the intersection with the symplectic leaves if $\Pi$ is
clean. Let $(\mathcal{A},\mathcal{A}^{*})$ be the Lie bialgebroid of
$(\mathcal{G},\Pi)$. The Poisson structure $\Pi$ induces a Lie groupoid
morphism

${T^{*}\mathcal{G}}$${T\mathcal{G}}$${\mathcal{A}^{*}}$${TM.}$$\scriptstyle{\Pi^{\sharp}}$$\scriptstyle{\rho_{\mathcal{A}^{*}}}$

######  Claim.

When $S$ is an orbit of $T^{*}\mathcal{G}$, the path-components of $M\cap S$
are orbits of $\mathcal{A}^{*}$.

Proof of claim. Let $s,t:\mathcal{G}\to M$ be the source- and target maps, and
$u:M\to\mathcal{G}$ the unit section. Decorate the structure maps of
$T^{*}\mathcal{G}$ with a tilde (e.g.
$\tilde{s},\tilde{t}:T^{*}\mathcal{G}\to\mathcal{A}^{*}$ for the source- and
target). A path $\gamma$ in $M\cap S$ is tangent to $M$ as well as
$\Pi^{\sharp}(T^{*}\mathcal{G})$, hence
$\dot{\gamma}(t)=Tu(v)=\Pi^{\sharp}(\alpha)$ for some $v\in TM$ and $\alpha\in
T^{*}\mathcal{G}$. But then

$v=Ts\circ
Tu(v)=Ts\circ\Pi^{\sharp}(\alpha)=\rho_{\mathcal{A}^{*}}\circ\tilde{s}(\alpha),$

so $v$ is tangent to $\mathcal{A}^{*}$.

Conversely, if $\gamma$ is a path tangent to $\mathcal{A}^{*}$, then
$\tilde{u}\circ\gamma$ is a path tangent to $\operatorname{Im}\Pi^{\sharp}$,
and therefore also to $S$.

The claim equips the (path-components of) $M\cap S$ with the structure of a
manifold. The intersection is clean, since for $x\in M\cap S$

$\displaystyle T_{x}(M\cap S)$
$\displaystyle=\operatorname{Im}\rho_{\mathcal{A}^{*},x}=\Pi^{\sharp}_{x}\left(\left(T_{x}M\right)^{\circ}\right)=T_{x}M\cap\operatorname{Im}\Pi^{\sharp}_{x}=T_{x}M\cap
T_{x}S,$

where in the third equality we used that $M$ is already Lagrangian in
$(\mathcal{G},\Pi)$. ∎

######  Proposition 2.4 (Curves of coisotropic bisections).

Let $(\mathcal{G},\Pi)\rightrightarrows M$ be a Poisson groupoid with Lie
bialgebroid $(\mathcal{A},\mathcal{A}^{*})$. The following Lie algebras are
isomorphic.

  * (i).

The Lie algebra $\mathfrak{X}^{R}(\mathcal{G},\Pi)$ of right-invariant Poisson
vector fields, equipped with the Lie bracket.

  * (ii).

The Lie algebra $\Gamma(\mathcal{A},d_{\mathcal{A}^{*}})$ of
$\mathcal{A}^{*}$-closed sections of $\mathcal{A}$, equipped with the bracket
on $\Gamma(\mathcal{A})$.

The isomorphism is given by restricting
$\overrightarrow{v}\in\mathfrak{X}^{R}(\mathcal{G},\Pi)$ along $M$. Moreover,
if an isotopy $\sigma_{t}$ of bisections on $\mathcal{G}$ is generated by
$v_{t}\in\Gamma(\mathcal{A})$, then
$\sigma_{t}\in\operatorname{Bis}(\mathcal{G},\Pi)$ for all $t$ if and only if
$v_{t}\in\Gamma(\mathcal{A},d_{\mathcal{A}^{*}})$.

###### Proof.

The differential $d_{\mathcal{A}^{*}}$ is determined by
$\overrightarrow{d_{\mathcal{A}^{*}}v}=[\Pi,\overrightarrow{v}]$ for
$v\in\Gamma(\mathcal{A})$ (Corollary 11.4.9 in [13]), which proves that the
correspondence is an isomorphism. The second statement now follows directly
from combining Proposition 2.2, (ii) with Lemma 1.13. ∎

######  Example 2.5 (Linear Poisson structures).

It is well-known that there is a duality between linear Poisson structures and
Lie algebroids ([13], Chapter 9). Therefore, we always think of a vector
bundle $\mathcal{A}^{*}\to M$ with a linear Poisson structure
$\pi_{\mathrm{lin}}$ on its total space as the dual of a Lie algebroid
$\mathcal{A}\Rightarrow M$. Since $(\mathcal{A}^{*},\pi_{\mathrm{lin}})$ can
be interpreted as a Poisson groupoid, a section $\alpha:M\to\mathcal{A}^{*}$
with coisotropic image is always a clean Lagrangian submanifold by Proposition
2.2. The coisotropic bisections of $(\mathcal{A}^{*},\pi_{\mathrm{lin}})$ form
a vector space by Proposition 2.6 below, and will play a crucial role in the
proof of Theorem 2.12. ∎

######  Proposition 2.6 ([21], Corollary 2.10).

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid. The image of a section
$\alpha\in\Gamma(\mathcal{A}^{*})$ is coisotropic in
$(\mathcal{A}^{*},\pi_{\mathrm{lin}})$ if and only if
$d_{\mathcal{A}}\alpha=0$, in which case it is a clean Lagrangian submanifold.

### 2.2 The Lie group of (coisotropic) bisections

The convenient setting of Kriegl and Michor [11] is a framework for infinite-
dimensional manifolds that is based on smooth curves. At it’s core lies
Boman’s theorem, which states that a map between finite dimensional manifolds
is smooth if and only if it sends smooth curves to smooth curves.

Likewise, smooth maps between convenient manifolds –topological spaces
modelled on convenient topological vector spaces– are characterized by the
same property, making smoothness easy to check, relative to the setting of
locally convex manifolds (e.g. [16]). Nevertheless, up to Fréchet manifolds,
the convenient setting entirely coincides with the locally convex setting.

A (convenient) Lie group $G$ is a convenient manifold together with a group
structure for which the multiplication $m:G\times G\to G$ and inversion
$\iota:G\to G$ are smooth. The tangent space $\mathfrak{g}=T_{e}G$ at the
identity inherits a Lie bracket from $G$.

The (right-)logarithmic derivative
$\delta^{r}\gamma:\mathbb{R}\to\mathfrak{g}$ of a smooth curve
$\gamma:\mathbb{R}\to G$ starting at $e\in G$ is defined by

$(\delta^{r}\gamma)(t)=\frac{d}{d\epsilon}\big{|}_{\epsilon=0}\gamma(t+\epsilon)\gamma(t)^{-1}.$

A Lie group is regular when the inverse of $\delta^{r}$,

$\operatorname{Evol}^{r}_{G}:C^{\infty}(\mathbb{R},\mathfrak{g})\to
C^{\infty}((\mathbb{R},0),(G,e))$

exists and is bijective, and the time-1 evolution map

$\operatorname{evol}^{r}_{G}:C^{\infty}(\mathbb{R},\mathfrak{g})\to
G,\quad\operatorname{evol}^{r}_{G}(\gamma)=\operatorname{Evol}^{r}_{G}(\gamma)(1)$

is smooth. When restricted to constant curves in $\mathfrak{g}$, the evolution
map coincides with the usual exponential map for Lie groups. All the Lie
groups in this paper are regular. In fact, to the author’s knowledge, up until
now all known convenient Lie groups are regular.

Let $G$ be a Lie group. An immersed Lie subgroup is a Lie group $H$ together
with an injective smooth group homomorphism $i:H\to G$ for which
$T_{h}i:T_{h}H\to T_{i(h)}G$ is injective for all $h\in H$. An initial Lie
subgroup is an immersed Lie subgroup $i:H\to G$ for which $i$ is an initial
map, meaning that a curve $\gamma:\mathbb{R}\to H$ is smooth if and only if
$i\circ\gamma:\mathbb{R}\to G$ is. An initial Lie subgroup $i:H\to G$ is
embedded when $H$ is an embedded submanifold of $G$, that is, $G$ admits
charts $(\mathcal{V},\Phi)$ modelled on a convenient vector space $V$ covering
$H$ for which $\Phi(H\cap\mathcal{V})=W\cap\Phi(\mathcal{V})$, where $W\subset
V$ is a closed subspace.

The following lemma will be of use later.

######  Lemma 2.7 ([11], 38.7).

Let $G$ be a regular Lie group and $i:K\hookrightarrow G$ an initial subgroup
with the following property: there exists an open neighbourhood $U$ of $e$ in
$G$ and a smooth mapping $p:U\to E$ into a convenient vector space $E$ such
that $p^{-1}(0)=K\cap U$ and $p$ is constant on the left cosets $Kg\cap U$.
Then $K$ is regular.

#### 2.2.1 The Lie group of bisections

Interestingly, the group of bisections of a Lie groupoid can always be given
the structure of a Lie group.

######  Theorem 2.8 ([19], Theorem 2.2).

Let $\mathcal{G}\rightrightarrows M$ be a (not necessarily Hausdorff) Lie
groupoid with Lie algebroid $\mathcal{A}\Rightarrow M$ over a Hausdorff
manifold $M$. Then $\operatorname{Bis}(\mathcal{G})$ is a regular (Hausdorff)
Lie group integrating the Lie algebra $\Gamma_{c}(\mathcal{A})$ of compactly
supported sections of $\mathcal{A}$.

######  Remark 2.9.

In the locally convex setting, the Lie group of bisections has been
extensively studied by Schmeding and Wockel [20]. However, since we are
working in the convenient setting, and our groupoids are not Hausdorff, we
preferred to cite the paper by Rybicki [19]. Our result on the coisotropic
bisections (Theorem 2.12) is equally valid in the locally setting. See also
Remark 2.11.

To proceed to the group of coisotropic bisections, a description of the smooth
structure on $\operatorname{Bis}(\mathcal{G})$ is necessary.

#### Charts

A chart of $\operatorname{Bis}(\mathcal{G})$ around a bisection $\sigma$ can
be obtained from any tubular neighbourhood as follows. Identify
$N\operatorname{im}\sigma\cong\sigma^{*}\ker Ts=:\mathcal{A}_{\sigma}$. Let
$\varphi:\mathcal{A}_{\sigma}\supset U\to V\subset\mathcal{G}$ be a tubular
neighbourhood around $\operatorname{im}\sigma$. Define the $C^{1}$-opens

$\displaystyle\mathcal{U}$
$\displaystyle=\left\\{v\in\Gamma_{c}(\mathcal{A}_{\sigma}):\begin{subarray}{c}\mbox{
$\operatorname{im}v\subset U$}\\\ \mbox{ $s\circ\varphi\circ v$ and
$t\circ\varphi\circ v$ are diffeo's}\end{subarray}\right\\},$
$\displaystyle\mathcal{V}$
$\displaystyle=\left\\{\tau\in\operatorname{Bis}_{c}(\mathcal{G}):\begin{subarray}{c}\mbox{$\operatorname{im}\tau\subset
V$}\\\ \mbox{ $p_{\mathcal{A}_{\sigma}}\circ\varphi^{-1}\circ\tau$ is a
diffeo}\end{subarray}\right\\},$

and the map $\Phi:\mathcal{V}\to\mathcal{U}$ by

$\Phi(\tau)=\varphi^{-1}\circ\tau\circ\left(p_{\mathcal{A}_{\sigma}}\circ\varphi^{-1}\circ\tau\right)^{-1}\in\mathcal{U}.$

Then $(\mathcal{V},\Phi)$ is a chart modelled on
$\Gamma_{c}(\mathcal{A}_{\sigma})$ around the identity. The crux is that any
tubular neighbourhood around $\operatorname{im}\sigma$ defines a chart of
$\operatorname{Bis}(\mathcal{G})$ around $\sigma$ this way.

#### Topological and smooth structure

Let $E\to M$ be a vector bundle and $K\subset M$ a compact subset. Denote by
$\Gamma_{K}(E)$ the space of sections of $E$ supported in $K$. Equipped with
the $C^{\infty}$-topology, it becomes a Fréchet space.

The space of compactly supported sections $\Gamma_{c}(E)$ is endowed with the
direct limit locally convex topology inherited from $\Gamma_{K}(E)$, ranging
over compact subsets of $M$.

A curve $\gamma:\mathbb{R}\to\Gamma_{c}(E)$ is smooth if and only if the
associated map $\gamma^{\wedge}:\mathbb{R}\times M\to E$ is smooth and is
timely proper in the following sense: for every bounded interval
$[a,b]\subset\mathbb{R}$ there exists a compact $K\subset M$ for which
$\gamma(t)$ is constant on $M\setminus K$ for all $t\in[a,b]$.

As the group of bisections of a Lie groupoid $\mathcal{G}\rightrightarrows M$
is modelled on $\Gamma_{c}(\mathcal{A})$, we have the following
characterization of smooth curves in $\operatorname{Bis}(\mathcal{G})$.

######  Lemma 2.10 ([11], 42.5).

A curve $\gamma:\mathbb{R}\to\operatorname{Bis}(\mathcal{G})$ is smooth if and
only its associated map $\gamma^{\wedge}:\mathbb{R}\times M\to\mathcal{G}$ is
smooth and $\gamma$ is timely proper: for every bounded interval
$[a,b]\subset\mathbb{R}$ there exists a compact $K\subset M$ such that
$\gamma(t)$ is constant on $M\setminus K$ for all $t\in[a,b]$.

######  Remark 2.11 (Topologies).

In the convenient setting, manifolds are endowed with the
$c^{\infty}$-topology (not to be confused with the Whitney
$C^{\infty}$-topology): the initial topology with respect to all smooth
curves. When $V$ is a Fréchet space, the $c^{\infty}$-topology coincides with
the locally convex topology, but is finer for arbitrary LCS’s. If $E$ is a
bornological LCS, then the $c^{\infty}$-topology and the locally convex
topology have the same convex open sets. Since $\Gamma_{c}(E)$ is a
bornological space (it is an LF-space when $M$ is second countable), the
$c^{\infty}$-topology on $\operatorname{Bis}(\mathcal{G})$ is strictly finer,
but only slightly so, than $\operatorname{Bis}(\mathcal{G})$ as a locally
convex manifold in the sense of [16, 20]. The smooth curves are the same in
both settings. When $M$ is compact, then $\Gamma_{c}(E)$ is a Fréchet space,
and the two settings coincide.

#### 2.2.2 The Lie group of coisotropic bisections

Under the assumption of linearizability of a Poisson groupoid
$(\mathcal{G},\Pi)$ around $M$, the coisotropic bisections form a Lie group.

######  Theorem 2.12.

Let $(\mathcal{G},\Pi)\rightrightarrows(M,\pi)$ be a Poisson groupoid with
bialgebroid $(\mathcal{A},\mathcal{A}^{*})$. Assume that $(\mathcal{G},\Pi)$
is linearizable around $M$. Then $\operatorname{Bis}(\mathcal{G},\Pi)$ is a
regular embedded Lie subgroup of $\operatorname{Bis}(\mathcal{G})$, whose Lie
algebra corresponds to
$\Gamma_{c}(\mathcal{A},d_{\mathcal{A}^{*}})=\\{v\in\Gamma_{c}(\mathcal{A}):d_{\mathcal{A}^{*}}v=0\\}$,
the space of closed compactly supported one-forms of $\mathcal{A}^{*}$.

###### Proof.

We will construct charts of $\operatorname{Bis}(\mathcal{G})$ adapted to
$\operatorname{Bis}(\mathcal{G},\Pi)$. Choose a linearization
$\varphi:(\mathcal{A},\pi_{\mathrm{lin}})\supset U\to
V\subset(\mathcal{G},\Pi)$ around $M$. Let
$\sigma\in\operatorname{Bis}(\mathcal{G},\Pi)$. As $L_{\sigma}$ is a Poisson
map, $L_{\sigma}\circ\varphi$ linearizes $\Pi$ around
$\operatorname{im}\sigma$. Additionally, let $(\mathcal{V},\Phi)$ be the chart
obtained from $\varphi$ as described above. Then, according to Proposition
2.6, $\tau\in\mathcal{V}$ is coisotropic if and only if $\Phi(\tau)$ is
coisotropic in $\mathcal{A}$, which happens if and only if
$d_{\mathcal{A}^{*}}\Phi(\tau)=0$. It follows that
$\Phi(\operatorname{Bis}(\mathcal{G},\Pi)\cap\mathcal{V})=\Gamma_{c}(\mathcal{A},d_{\mathcal{A}^{*}})\cap\Phi(\mathcal{V})$.
Hence, $\operatorname{Bis}(\mathcal{G},\Pi)$ is an embedded $C^{1}$-closed
submanifold of $\operatorname{Bis}(\mathcal{G})$, modelled on
$\Gamma_{c}(\mathcal{A},d_{\mathcal{A}^{*}})$. This shows that
$\operatorname{Bis}(\mathcal{G},\Pi)$ is an embedded Lie subgroup.

By Proposition 2.4, the Lie algebra of $\operatorname{Bis}(\mathcal{G},\Pi)$
with this Lie group structure is exactly
$\Gamma_{c}(\mathcal{A},d_{\mathcal{A}^{*}})$.

By considering the map
$p:\operatorname{Bis}(\mathcal{G})\to\mathfrak{X}^{2}(\mathcal{G})$,
$\sigma\mapsto L_{\sigma}^{*}(\Pi)$, regularity of
$\operatorname{Bis}(\mathcal{G},\Pi)$ follows from Proposition 2.2, (ii), and
Lemma 2.7. ∎

### 2.3 Lagrangian bisections of symplectic groupoids

When $(\mathcal{G},\Omega)\rightrightarrows(M,\pi)$ is a symplectic groupoid,
the Poisson structure is always linearizable around $M$ by Weinstein’s
Lagrangian neighbourhood theorem. As a special case of Theorem 2.12, we
recover the result by Rybicki [18]: the group of Lagrangian bisections
$\operatorname{Bis}(\mathcal{G},\Omega)$ of a symplectic groupoid is a regular
Lie group, whose Lie algebra is given by closed, compactly supported one-
forms. Already this group has many interesting connections with the Poisson
diffeomorphism group on the base.

Analogous to Hamiltonian diffeomorphisms on symplectic manifolds, one has the
notion of (compactly supported) exact Lagrangian bisections
$\operatorname{Bis}_{\mathrm{ex},c}(\mathcal{G},\Omega)$. These are the time-
one flows of (compactly supported) exact time-dependent one-forms
$\Omega^{1}_{\mathrm{ex},c}(M)$. By means of a flux homomorphism for
Lagrangian bisections [18, 24], the exact sequence of Lie algebras

$0\rightarrow\Omega^{1}_{\mathrm{ex},c}(M)\rightarrow\Omega^{1}_{\mathrm{cl},c}(M)\rightarrow
H^{1}_{c}(M)\rightarrow 0$

integrates, whenever a certain flux group $\Gamma\subset H^{1}_{c}(M)$ is
discrete, to an exact sequence of Lie groups

$1\rightarrow\operatorname{Bis}_{\mathrm{ex},c}(\mathcal{G},\Omega)\rightarrow\operatorname{Bis}_{c,0}(\mathcal{G},\Omega)\to
H^{1}_{c}(M)/\Gamma\to 1,$

where $\operatorname{Bis}_{c,0}(\mathcal{G},\Omega)$ is the identity component
of $\operatorname{Bis}(\mathcal{G},\Omega)$, consisting of time-1 flows of
(compactly supported, time dependent) closed one-forms. In future work we hope
to investigate how the flux homomorphism for Lagrangian bisections as
described in [18, 24] extends to the setting of this paper.

In light of Proposition 2.2, the Lagrangian bisections act on $(M,\pi)$ by
Poisson diffeomorphisms. The map $\pi^{\sharp}$ induces surjective Lie algebra
homomorphisms

$\Omega^{1}_{\mathrm{cl},c}(M)\to\mathfrak{ham}_{\mathrm{loc},c}(M,\pi),\quad\Omega^{1}_{\mathrm{ex},c}(M)\to\mathfrak{ham}_{c}(M,\pi).$

so that it covers the exact sequence

${0}$${\Omega^{1}_{\mathrm{ex},c}(M)}$${\Omega^{1}_{\mathrm{cl},c}(M)}$${H^{1}_{c}(M)}$${0}$${0}$${\mathfrak{ham}_{c}(M,\pi)}$${\mathfrak{ham}_{\mathrm{loc},c}(M,\pi)}$${\pi^{\sharp}\left(H^{1}_{c}(M)\right)}$${0.}$

The Lie algebra homomorphisms induced by $\pi^{\sharp}$ integrate to
surjective group homomorphisms

$\operatorname{Bis}_{c,0}(\mathcal{G},\Omega)\to\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi),\quad\operatorname{Bis}_{\mathrm{ex},c}(\mathcal{G},\Omega)\to\operatorname{Ham}_{c}(M,\pi).$

In future work, we hope to investigate the existence of a flux homomorphism on
the locally Hamiltonian group for arbitrary Poisson manifolds.

The Picard group [2, 4], that we will not define here, is also strongly
related to the Poisson diffeomorphisms. By Proposition 5.1 in [4]: the kernel
of the map $\operatorname{Diff}_{c,0}(M,\pi)\to\operatorname{Pic}(M,\pi)$ is
exactly $\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$.

### 2.4 Linearization of Poisson groupoids

Before we proceed to applications of our framework, the main results in [21]
the linearization of Poisson groupoids are stated for later use. Both theorems
linearize integrations of a triangular bialgebroid
$(\mathcal{A},\mathcal{A}^{*},\pi_{\mathcal{A}})$ (Remark 1.9). The
fundamental difference is that the first one is about integrations of
$\mathcal{A}^{*}$, while the second one concerns integrations of
$\mathcal{A}$. The latter does require extra conditions on the
$\mathcal{A}$-Poisson structure.

######  Theorem 2.13 ([21], Theorem 3.16).

Let $(\mathcal{A},\mathcal{A}^{*},\pi_{\mathcal{A}})$ be a triangular Lie
bialgebroid over $M$. Any (local) Poisson groupoid
$(\mathcal{G}^{*},\Pi)\rightrightarrows(M,\pi)$ integrating
$(\mathcal{A}^{*},\mathcal{A})$ is linearizable around $M$.

######  Theorem 2.14 ([21], Theorem 3.23).

Let $(\mathcal{A},\mathcal{A}^{*},\pi_{\mathcal{A}})$ be a triangular Lie
bialgebroid, and assume that $\pi_{\mathcal{A}}$ is of $k$-cosymplectic type.
Then for any (local) integration $\mathcal{G}\rightrightarrows M$ of
$\mathcal{A}$, the Poisson groupoid
$(\mathcal{G},\overleftarrow{\pi_{\mathcal{A}}}-\overrightarrow{\pi_{\mathcal{A}}})$
is linearizable.

We refer to the introduction or Definition 3.24 for details on
$\mathcal{A}$-Poisson structures of cosymplectic type.

#### 2.4.1 Poisson groupoids over scattering-symplectic manifolds

To place scattering-symplectic manifolds in our approach, we must prove
specific linearization results that are not included in [21], as will be done
in the current section.

Let $(M,Z)$ be a smooth log-manifold, with $p_{Z}:T_{Z}M\to M$ and
$p_{sc}:{}^{sc}T_{Z}M\to M$ the bundle projections. The scattering Lie
algebroid ${}^{sc}T_{Z}M$ is naturally anchored to the log-tangent bundle
$T_{Z}M$, and therefore, according to [21], if $\mathcal{G}\rightrightarrows
M$ is an integration of $T_{Z}M$, the following squares are a double Lie
algebroid and an LA-groupoid, respectively.

${p_{sc}^{!}T_{Z}M}$${T_{Z}M}$${{}^{sc}T_{Z}M}$${M,}$
${p_{sc}^{!}\mathcal{G}}$${\mathcal{G}}$${{}^{sc}T_{Z}M}$${M.}$

######  Proposition 2.15.

The pair $(T_{Z}M,T_{Z}M|_{Z})$ is a log-manifold. Furthermore:

  * (i).

the prolongation algebroid $p^{!}_{Z}(T_{Z}M)\to T_{Z}M$ is isomorphic to the
log-tangent bundle $T_{(T_{Z}M|_{Z})}(T_{Z}M)$,

  * (ii).

the prolongation algebroid $p^{!}_{sc}(T_{Z}M)\to T_{Z}M$ is isomorphic to the
scattering algebroid ${}^{sc}T_{(T_{Z}M|_{Z})}(T_{Z}M)$.

If $\mathcal{G}\rightrightarrows M$ is an integration of $T_{Z}M$, then, with
$\mathcal{H}=s^{-1}(Z)=t^{-1}(Z)$ as a critical hypersurface, the following
hold:

  * (iii).

the Lie algebroid $p^{!}_{Z}\mathcal{G}\Rightarrow\mathcal{G}$ is isomorphic
to the log-tangent bundle $T_{\mathcal{H}}\mathcal{G}$.

  * (iv).

the Lie algebroid $p^{!}_{sc}\mathcal{G}\Rightarrow\mathcal{G}$ is isomorphic
to the scattering-tangent bundle ${}^{sc}T_{\mathcal{H}}\mathcal{G}$.

###### Proof.

We follow the notation in [21], Section 2.

Recall that $p^{!}_{Z}(T_{Z}M)$ has side bundles and core isomorphic to
$T_{Z}M$. The same is true for $T_{(T_{Z}M|_{Z})}(T_{Z}M)$.

The sections of $p^{!}_{Z}(T_{Z}M)\to T_{Z}M$ are generated by the complete
lifts $\widetilde{v}$ and vertical lifts $v^{\uparrow}$ for
$v\in\Gamma(T_{Z}M)$. Since the $\varphi_{t}$ of a section
$v\in\Gamma(T_{Z}M)$ preserves $T_{Z}M|_{Z}$, it follows that $\widetilde{v}$
is tangent to $T_{Z}M|_{Z}$, and thus a section of
$T_{(T_{Z}M|_{Z})}(T_{Z}M)$. This shows that the anchor of
$p^{!}_{Z}(T_{Z}M)\to T_{Z}M$ factors through $T_{(T_{Z}M|_{Z})}(T_{Z}M)$. The
map $p^{!}_{Z}(T_{Z}M)\to T_{(T_{Z}M|_{Z})}(T_{Z}M)$ is a map of double vector
bundles, because it sends linear sections to linear sections and core sections
to core sections. Now it follows that the map is an isomorphism as it is the
identity on the side bundles and the core.

For the scattering bundles, the argument is similar. The sections of
$p_{sc}^{!}(T_{Z}M)\to T_{Z}M$ are generated by $T_{Z}v$ and the core lifts
$\widehat{v}$ for $v\in\Gamma({}^{sc}T_{Z}M)$. The anchor sends $T_{Z}v$ to
the vector field $\widetilde{v}$, which vanishes (as a log-vector field) over
$T_{Z}M|_{Z}$ (indeed, the flow of scattering vector fields induce the
identity on $T_{Z}M|_{Z}$), and is therefore a scattering-vector field. The
anchor sends $\widehat{v}$ to $v^{\uparrow}$, which too vanishes as a log-
vector field over $T_{Z}M|_{Z}$. It follows that there is an induced map of
double vector bundles $p_{sc}^{!}(T_{Z}M)\to{}^{sc}T_{(T_{Z}M|_{Z})}(T_{Z}M)$
that is an isomorphism because it is the identity on the side bundles and the
core.

The proofs of statements (iii) and (iv) are similar. ∎

Let $(M,Z,\omega)$ be a scattering-symplectic manifold, with underlying
Poisson structure $\pi$. If
$\tilde{t},\tilde{s}:p_{sc}^{!}\mathcal{G}\to{}^{sc}T_{Z}M$ are the source-
and target morphisms, then $\tilde{s}^{*}\omega-\tilde{t}^{*}\omega$ is a
multiplicative symplectic structure on the LA-groupoid
$p_{sc}^{!}\mathcal{G}\Rightarrow\mathcal{G}$. By Proposition 2.15,
$(\mathcal{G},\mathcal{H},\tilde{s}^{*}\omega-\tilde{t}^{*}\omega)$ is a
scattering-symplectic manifold with underlying Poisson structure
$\overleftarrow{\pi_{Z}}-\overrightarrow{\pi_{Z}}$ on $\mathcal{G}$, where
$\pi_{Z}\in\Gamma(\wedge^{2}T_{Z}M)$ is the Poisson structure on the log-
tangent bundle induced by $\tilde{t}^{*}\omega-\tilde{s}^{*}\omega$. Since
$M\subset\mathcal{G}$ is transverse to the hypersurface $\mathcal{H}$, the
Poisson groupoid $(\mathcal{G},\Pi)$ is linearizable around $M$ by the
scattering-Lagrangian neighbourhood Theorem 1.33. We have proven the following
result.

######  Proposition 2.16.

Let $(M,Z,\omega)$ be a scattering-symplectic manifold, with underlying
$T_{Z}M$-Poisson structure $\pi_{Z}$. Let $\mathcal{G}\rightrightarrows M$ be
an integration of the log-tangent bundle $T_{Z}M$. Then the Poisson groupoid
$(\mathcal{G},\overleftarrow{\pi_{Z}}-\overrightarrow{\pi_{Z}})$ is
linearizable.

######  Remark 2.17.

If $(M,Z,\omega)$ is a scattering-symplectic manifold, an integration of the
scattering-tangent bundle $\mathcal{G}_{sc}\rightrightarrows M$ always carries
a linearizable Poisson structure by Theorem 2.14. This groupoid, however, is
less interesting from the perspective of Poisson diffeomorphisms.

##  3 Lie groups of Poisson diffeomorphisms

### 3.1 Almost injective Lie algebroids

Many interesting Lie algebroids are determined by the image of the anchor on
sections.

######  Definition 3.1.

A Lie algebroid $\mathcal{A}\Rightarrow M$ is almost injective if the anchor
is injective on sections. ∎

Isomorphisms of almost injective Lie algebroids are uniquely determined by the
base map.

######  Lemma 3.2.

Let $\mathcal{A}\Rightarrow M$ be an almost injective Lie algebroid and
$f\in\operatorname{Diff}(M)$. Then $Tf$ extends to a (necessarily unique) Lie
algebroid isomorphism $(\varphi,f):\mathcal{A}\to\mathcal{A}$ if and only if
$f_{*}(\rho(\Gamma_{c}(\mathcal{A})))=\rho(\Gamma_{c}(\mathcal{A}))$.

Almost injective Lie algebroids are always integrable. Moreover, the holonomy
groupoid (the final object in the category of integrations of
$\mathcal{A}\Rightarrow M$) can be recovered from any integration (see also
[6]).

######  Proposition 3.3 (Proposition 1.9, [1]).

Let $\mathcal{G}\rightrightarrows M$ be any integration of an almost-injective
Lie algebroid $\mathcal{A}\Rightarrow M$. Then, as a set,
$\operatorname{Hol}(\mathcal{A})\rightrightarrows M$ is given by
$\left(\mathcal{G}/\sim\right)\rightrightarrows M$, where

$g\sim h\iff\begin{subarray}{c}\mbox{there exists local bisections
$\sigma_{g}$ and $\sigma_{h}$}\\\ \mbox{ through $g$ and $h$ such that
$t\circ\sigma_{g}=t\circ\sigma_{h}$.}\end{subarray}$

The holonomy groupoid $\operatorname{Hol}(\mathcal{A})\rightrightarrows M$
carries the unique structure of a Lie groupoid for which the map
$\mathcal{G}\to\operatorname{Hol}(\mathcal{A})$ is a local diffeomorphism.

It follows that the groupoid anchor $\chi:\operatorname{Hol}(\mathcal{A})\to
M\times M$ induces an injective Lie group homomorphism
$\chi_{*}:\operatorname{Bis}(\operatorname{Hol}(\mathcal{A}))\to\operatorname{Diff}(M)$
which an immersion because it differentiates at the identity to the injective
map $\rho:\Gamma_{c}(\mathcal{A})\to\mathfrak{X}_{c}(M)$.

Recall that the inner automorphisms $\operatorname{InnAut}(\mathcal{A})$ of a
Lie algebroid $\mathcal{A}\Rightarrow M$ are given by the time-1 flows of
(time-dependent) Lie algebroid sections. If $M$ is not compact, we also want
to consider $\operatorname{InnAut}_{c}(\mathcal{A})$, the inner automorphisms
of $\mathcal{A}$ with compact support. For an almost injective Lie algebroid,
an automorphism $(\varphi,f):\mathcal{A}\to\mathcal{A}$ is uniquely determined
by $f$, and can thus be regarded as an element of $\operatorname{Diff}(M)$.

######  Corollary 3.4.

Let $\mathcal{A}\Rightarrow M$ be an almost injective Lie algebroid. Then the
group $\operatorname{InnAut}_{c}(\mathcal{A})$ is a regular immersed Lie
subgroup of $\operatorname{Diff}(M)$.

###### Proof.

The group $\operatorname{InnAut}_{c}(\mathcal{A})$ corresponds to
$\operatorname{Bis}_{c,0}(\operatorname{Hol}(\mathcal{A}))$, the identity
component of the group of compactly supported bisections of
$\operatorname{Hol}(\mathcal{A})$, which is a regular Lie group. The tangent
map at the identity of the inclusion
$\operatorname{InnAut}_{c}(\mathcal{A})\to\operatorname{Diff}(M)$ is
$\rho:\Gamma_{c}(\mathcal{A})\to\mathfrak{X}_{c}(M)$, which is injective, so
it must be an immersion. ∎

We expect that the smooth structure on
$\operatorname{InnAut}_{c}(\mathcal{A})$ is determined by the inclusion into
$\operatorname{Diff}_{c}(M)$. That is, it should be an initial map. However,
we only prove it under additional assumptions, as we will now explain.

######  Definition 3.5.

Let $\mathcal{A}\Rightarrow M$ be an almost injective Lie algebroid. We call
$\mathcal{A}$ isotopically closed when $\Gamma_{c}(\mathcal{A})$ generates all
of its isotopies of inner automorphisms. That is, if an isotopy $\varphi_{t}$
in $\operatorname{Diff}(M)$ generated by $X_{t}$ is such that
$\varphi_{t}\in\operatorname{InnAut}_{c}(\mathcal{A})$ for all $t$, then
$X_{t}\in\rho(\Gamma_{c}(\mathcal{A}))$ for all $t$. ∎

######  Example 3.6.

A regular foliation $\mathcal{F}\subset TM$ is always isotopically closed.
More generally, if $\mathcal{A}\Rightarrow M$ is an almost injective Lie
algebroid such that $X\in\rho(\Gamma_{c}(\mathcal{A}))$ if and only if $X$ is
tangent to the leaves of $\mathcal{A}$, then $\mathcal{A}$ is isotopically
closed. In particular, the log-tangent bundle of a log-manifold is
isotopically closed. ∎

######  Example 3.7.

If $I$ be a projective divisor ideal (Definition 1.19), then $T_{I}M$ is
isotopically closed. Indeed, if $\varphi$ is an inner automorphism, then
$\varphi$ is the time-1 flow of a (time-dependent) vector field that preserves
$I$, and so $\varphi^{*}I=I$. If $\varphi_{t}$ is an isotopy of inner
automorphisms, smooth in $\operatorname{Diff}(M)$ and generated by $X_{t}$,
then

$\frac{d}{dt}\varphi^{*}_{t}(I)=\varphi_{t}^{*}\left(\mathcal{L}_{X_{t}}(I)\right)\subset
I,$

hence $X_{t}$ is in $\rho(\Gamma(T_{I}M))$.

In particular, the (normal-crossing) log-tangent bundle (Example 1.22) and the
elliptic tangent bundle (Example 1.24) are isotopically closed. ∎

######  Example 3.8.

If $\mathcal{A}\Rightarrow M$ is an isotopically closed almost injective Lie
algebroid, and $\mathcal{B}\Rightarrow Z$ an (almost injective) isotopically
closed subalgebroid over a hypersurface $Z\subset M$, then the rescaling
$[\mathcal{A};\mathcal{B}]$ is also isotopically closed. This applies to the
scattering-tangent bundle (Example 1.28). ∎

######  Remark 3.9.

In all of the above examples, the anchor also provides an embedding
$\rho:\Gamma_{c}(\mathcal{A})\to\mathfrak{X}_{c}(M)$ of topological vector
spaces, and thus it is an initial map. In general, when
$\rho(\Gamma_{c}(\mathcal{A}))$ is closed in $\mathfrak{X}_{c}(M)$, the map
$\rho$ is an embedding by the open mapping theorem for Fréchet spaces
(although $\Gamma_{c}(\mathcal{A})$ is not a Fréchet space,
$\Gamma_{K}(\mathcal{A})$ is for every compact $K\subset M$).

We expect that every almost injective Lie algebroid is isotopically closed, at
least for those for which the anchor
$\rho:\Gamma_{c}(\mathcal{A})\to\mathfrak{X}_{c}(M)$ is an embedding. However,
we were not able to prove it in this paper. Therefore, it is added as a
separate assumption in the proposition below.

######  Proposition 3.10.

Let $\mathcal{A}\Rightarrow M$ be an _isotopically closed_ almost injective
Lie algebroid for which $\rho:\Gamma_{c}(\mathcal{A})\to\mathfrak{X}_{c}(M)$
is an embedding of topological vector spaces. Then
$\operatorname{InnAut}_{c}(\mathcal{A})$ is a regular initial Lie subgroup of
$\operatorname{Diff}_{c,0}(M)$.

###### Proof.

Let $\varphi_{t}$ be an isotopy in $\operatorname{InnAut}_{c}(\mathcal{A})$
that is smooth in $\operatorname{Diff}_{c}(M)$, and generated by
$X_{\epsilon}\in\mathfrak{X}_{c}(M)$. Since $\mathcal{A}$ is isotopically
closed, it follows that $X_{\epsilon}=\rho(v_{\epsilon})$ for some
$v_{\epsilon}\in\Gamma_{c}(\mathcal{A})$. Because $\rho$ is an embedding, it
is initial, and therefore $v_{\epsilon}$ is a smooth curve in
$\operatorname{InnAut}_{c}(\mathcal{A})=\operatorname{Bis}_{c,0}(\operatorname{Hol}(\mathcal{A}))$.
By regularity of $\operatorname{Bis}(\operatorname{Hol}(\mathcal{A}))$, it
integrates to a smooth curve $\sigma_{\epsilon}$ in
$\operatorname{Bis}_{c,0}(\operatorname{Hol}(\mathcal{A}))$. Now, since
$\mathcal{A}$ is almost-injective, the curve $t\circ\sigma_{\epsilon}$ is
generated by $X_{t}$, and therefore must agree with $\varphi_{t}$. It follows
that $\varphi_{t}$ is smooth in $\operatorname{InnAut}_{c}(\mathcal{A})$. ∎

######  Remark 3.11.

Let $\mathcal{G}\rightrightarrows M$ be any integration of an almost injective
Lie algebroid $\mathcal{A}\Rightarrow M$. Since
$\mathcal{G}\to\operatorname{Hol}(\mathcal{A})$ is an isomorphism in a
neighbourhood of $M$, the map
$\operatorname{Bis}_{c,0}(\mathcal{G})\to\operatorname{Bis}_{c,0}(\operatorname{Hol}(\mathcal{A}))$
is surjective with discrete kernel

$K=\ker\left(\operatorname{Bis}_{c,0}(\mathcal{G})\to\operatorname{Bis}_{c,0}(\operatorname{Hol}(\mathcal{A})\right)=\ker\left((\chi_{G})_{*}:\operatorname{Bis}_{c,0}(\mathcal{G})\to\operatorname{Diff}_{c,0}(M)\right)$

Thus the quotient $\operatorname{Bis}_{c,0}(\mathcal{G})/K$ is again a regular
Lie group. It follows that the maps below are Lie group isomorphisms:

$\operatorname{Bis}_{c,0}(\mathcal{G})/K\to\operatorname{Bis}_{c,0}(\operatorname{Hol}(\mathcal{A}))\to\operatorname{InnAut}_{c}(\mathcal{A}).$

######  Example 3.12.

When a Lie algebroid $\mathcal{F}\Rightarrow M$ is almost-injective and
regular, it corresponds to a regular foliation. In this case, there is an
identification
$\operatorname{InnAut}_{c}(\mathcal{F})=\operatorname{Fol}_{c,0}(\mathcal{F})$,
where $\operatorname{Fol}(\mathcal{F})$ is the group of diffeomorphisms
sending each leaf to itself, that equips $\operatorname{Fol}(\mathcal{F})$
with the structure of a regular initial Lie subgroup of
$\operatorname{Diff}(M)$. It coincides with the one described in [17]. ∎

######  Remark 3.13.

Even for a regular foliation $\mathcal{F}\Rightarrow M$, the subgroup
$\operatorname{InnAut}(\mathcal{F})=\operatorname{Fol}(\mathcal{F})$ may not
be embedded in $\operatorname{Diff}(M)$. Consider for instance a foliation on
the torus $T^{2}$ with coordinates $(\theta_{1},\theta_{2})$ whose leaves are
integral curves of a constant vector field
$X=\partial_{\theta_{1}}+a\partial_{\theta_{2}}$, where $a\in\mathbb{R}$ is
irrational. For each $t\in\mathbb{R}$, the map
$\varphi_{t}(\theta_{1},\theta_{2})=(\theta_{1}+t,\theta_{2})$ preserves the
foliation, and for a dense subset of $\mathbb{R}$, the maps $\varphi_{t}$ are
foliated. Hence, there is a sequence
$\varphi_{t_{n}}\in\operatorname{Fol}(\mathcal{F})$ such that
$\varphi_{t_{n}}$ converges to the identity in $\operatorname{Diff}(M)$, but
does not converge in $\operatorname{Fol}(\mathcal{F})$.

Another interesting application arises when we consider almost everywhere non-
degenerate Poisson structures, combined with the Lagrangian bisections of its
integrating symplectic groupoid.

######  Theorem 3.14.

Let $(M,\pi)$ be a Poisson manifold whose non-degeneracy locus is open and
dense. Then the group $\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$ is a
regular immersed Lie subgroup of $\operatorname{Diff}(M,\pi)$, that is initial
whenever $T^{*}M$ is isotopically closed and
$\pi^{\sharp}:\Gamma_{c}(T^{*}M)\to\mathfrak{X}_{c}(M)$ is an embedding.

More generally, if $(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ is an almost
injective Poisson Lie algebroid and $\pi_{\mathcal{A}}$ is generically non-
degenerate, then
$\operatorname{Ham}_{\mathrm{loc,c}}(\mathcal{A},\pi_{\mathcal{A}})$ is a
regular immersed Lie subgroup of $\operatorname{Diff}(M,\pi)$ that is initial
whenever $\mathcal{A}^{*}$ is isotopically closed and
$\rho_{\mathcal{A}}\circ\pi_{\mathcal{A}}^{\sharp}:\Omega^{1}_{c}(\mathcal{A})\to\mathfrak{X}_{c}(M)$
is an embedding.

###### Proof.

We prove the general case. First, the dual $\mathcal{A}^{*}$ becomes an
almost-injective Lie algebroid as in Remark 1.9. Let
$(\mathcal{G}^{*},\Pi)\rightrightarrows(M,\pi)$ be a Poisson groupoid
integrating the bialgebroid $(\mathcal{A}^{*},\mathcal{A})$. As in Remark
3.11, we set

$K=\ker\left((\chi_{\mathcal{G}^{*}})_{*}:\operatorname{Bis}_{c,0}(\mathcal{G}^{*})\to\operatorname{Diff}(M)\right),$

which is a discrete subgroup of $\operatorname{Bis}_{c,0}(\mathcal{G}^{*})$.
By Theorem 2.13, the Poisson groupoid $(\mathcal{G}^{*},\Pi)$ is linearizable,
and therefore the identity component
$\operatorname{Bis}_{c,0}(\mathcal{G}^{*},\Pi)$ of the group of Lagrangian
bisections is an embedded submanifold of $\operatorname{Bis}(\mathcal{G}^{*})$
(Theorem 2.12), it follows that

$K_{\Pi}=K\cap\operatorname{Bis}_{c,0}(\mathcal{G}^{*},\Pi)$

is discrete in $\operatorname{Bis}_{c,0}(\mathcal{G}^{*})$. Because
$\rho_{\mathcal{A}}\circ\pi_{\mathcal{A}}^{\sharp}:\mathcal{A}^{*}\to TM$ is
almost-injective, the bijection

$\operatorname{Bis}_{c,0}(\mathcal{G}^{*},\Pi)/K_{\Pi}\cong\operatorname{Ham}_{\mathrm{loc,c}}(\mathcal{A},\pi_{\mathcal{A}})$

equips $\operatorname{Ham}_{\mathrm{loc,c}}(\mathcal{A},\pi_{\mathcal{A}})$
with the structure of a regular immersed Lie subgroup of
$\operatorname{Diff}_{c}(M)$.

Under the additional assumptions, it becomes an initial Lie subgroup, because
$\operatorname{InnAut}_{c}(\mathcal{A}^{*})$ is. ∎

### 3.2 Poisson structures of divisor type

Our approach is particularly effective for a class of generically non-
degenerate Poisson structures, namely those of divisor type considered [10]
(see Section 1.4). Let $(M,\pi)$ be a Poisson structure of divisor type with
projective divisor ideal $I_{\pi}$. The algebroid $T_{I_{\pi}}M$ is almost
injective, and by Proposition 1.21, every Poisson vector field is a section of
$T_{I_{\pi}}M$. When $T_{I_{\pi}}^{*}M$ locally admits bases of closed
sections, the Poisson structure $\pi$ lifts to a Poisson structure $\pi_{I}$
on $T_{I_{\pi}}M$ by Theorem 4.35 in [10]. If in addition $I_{\pi}$ is a
standard ideal, the lift $\pi_{I}$ is non-degenerate.

######  Theorem 3.15.

Let $(M,\pi)$ be a Poisson structure of standard divisor type. Suppose that
$T_{I_{\pi}}M$ locally admits bases of closed sections. Then
$\operatorname{Diff}(M,\pi)$ is a regular initial Lie subgroup of
$\operatorname{Diff}(M)$.

###### Proof.

We will equip the path component $\operatorname{Diff}_{c,0}(M,\pi)$ of
$\operatorname{Diff}(M,\pi)$ with the structure of a regular initial Lie
subgroup of $\operatorname{Diff}_{c,0}(M)$. By Proposition 1.21, every Poisson
diffeomorphism in $\operatorname{Diff}_{c,0}(M,\pi)$ is an inner automorphism
of $T_{I_{\pi}}M$, to which $\pi$ lifts to a non-degenerate Poisson structure
$\pi_{I}$. By Example 3.7, the Lie algebroid $T_{I_{\pi}}M$ is isotopically
closed, and therefore $\operatorname{InnAut}_{c}(T_{I_{\pi}}M,\pi_{I})$ is a
regular initial Lie subgroup of $\operatorname{Diff}_{c,0}(M)$ by Theorem
3.26. ∎

#### 3.2.1 Log-symplectic structures

Let $(M,Z)$ be a (normal-crossing) log-manifold (Example 1.22). Analogous to
the boundary of a manifold with corners, the hypersurface $Z$ comes with a
natural stratification $Z=\cup_{j=0}^{2n-1}Z_{j}$ (the $Z_{j}$’s are (unions
of) the $j$-dimensional orbits of $T_{Z}M$). Note that by the local form, each
$Z_{j}$ is embedded (but not closed!). Because
$(M\setminus\left(\cup_{i=1}^{2n-2}Z_{i}\right),Z_{2n-1})$ is a smooth log-
manifold, the log-symplectic structure induces a cosymplectic structure on
$i:Z_{2n-1}\hookrightarrow M$, whose one-form is given by
$\alpha=i^{*}(\iota_{\mathcal{E}}\omega)$, where $\mathcal{E}$ is the Euler
vector field obtained via any tubular neighbourhood of $Z_{2n-1}$ ([9],
Proposition 10).

######  Lemma 3.16.

Let $(M,Z,\omega)$ be a log-symplectic manifold with underlying Poisson
structure $\pi$. A vector field $X\in\mathfrak{X}(M)$ is tangent to the
symplectic foliation of $\pi$ if and only if there is a one-form
$\eta\in\Omega^{1}(M)$ such that $\pi^{\sharp}(\eta)=X$.

###### Proof.

Any vector field of the form $\pi^{\sharp}(\eta)$ it tangent to the symplectic
foliation. Conversely, if $X$ is tangent to the symplectic foliation, it is
certainly tangent to $Z$ and thus can be regarded as a section of $T_{Z}M$,
and thus we can set $\eta=\omega^{\flat}(X)$. Since $X$ is tangent to the
symplectic foliation, it must be in the kernel of
$\alpha=i^{*}(\iota_{\mathcal{E}}\omega)$ when restricted to $Z_{2n-1}$. In
local coordinates $(x_{i})$ adapted to $Z$, we have
$Z_{2n-1}=\\{\prod_{i=1}^{k}x_{i}=0:\mbox{no two $x_{i}$'s are zero}\\}$.
Therefore, on $Z_{2n-1}\cap\\{x_{i}=0\\}$, the Euler vector field is given by
$x_{i}\partial_{x_{i}}$. We find that

$0=\iota_{X}\alpha=\omega\left(x_{i}\frac{\partial}{\partial{x_{i}}},X\right)\big{|}_{Z_{2n-1}\cap\\{x_{i}=0\\}}=-\eta\left(x_{i}\frac{\partial}{\partial{x_{i}}}\right)\big{|}_{Z_{2n-1}\cap\\{x_{i}=0\\}}.$

By continuity, $\iota_{x_{i}\partial_{x_{i}}}\eta=0$ along $\\{x_{i}=0\\}$.
Since this is true for all $i=1,\dots,k$, it follows that
$\eta\in\Omega^{1}(M)$. ∎

Consequently, for log-symplectic manifolds, the locally Hamiltonian and
foliated vector fields coincide! This implies that
$\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$ is the identity component of
$\operatorname{Fol}_{c}(M,\pi)$.

######  Theorem 3.17.

Let $(M,Z)$ be a log-manifold. Then

$\operatorname{Diff}(M,Z)=\\{\varphi\in\operatorname{Diff}(M):\varphi(Z)=Z\\}$

is a regular initial Lie subgroup of $\operatorname{Diff}(M)$ with Lie algebra
$\mathfrak{X}_{c}(M,Z)$.

Let $(M,Z,\omega)$ be a log-symplectic manifold with underlying Poisson
structure $\pi$. Then $\operatorname{Diff}(M,\pi)$ is a regular embedded Lie
subgroup of $\operatorname{Diff}(M,Z)$, with Lie algebra
$\mathfrak{X}_{c}(M,\pi)$.

Finally, $\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$ coincides with the
identity component of $\operatorname{Fol}(M,\pi)$, and comes with the
structure of a regular Lie group for which it is initial in
$\operatorname{Diff}(M,\pi)$, with Lie algebra
$\mathfrak{ham}_{\mathrm{loc},c}(M,\pi)=\mathfrak{fol}_{c}(M,\pi)$.

######  Remark 3.18.

If $Z$ is a smooth hypersurface, Ebin and Marsden proved in [7] that
$\operatorname{Diff}(M,Z)$ is actually embedded in $\operatorname{Diff}(M)$ by
means of a metric for which $Z$ is geodesically closed. This is not immediate
from our approach.

###### Proof.

Let $\mathcal{G}\rightrightarrows M$ be the holonomy groupoid of $T_{Z}M$. The
identity path-component of $\operatorname{Diff}(M,Z)$ identifies with the
(compactly supported) inner automorphisms of $T_{Z}M$, which is an initial
subgroup of $\operatorname{Diff}(M)$ by Proposition 3.10 after identifying
$\operatorname{Diff}_{c,0}(M,Z)=\operatorname{Bis}_{c,0}(\mathcal{G})$. The
second statement is a special case of Theorem 3.15. Finally, due to Lemma
3.16, the identity component of $\operatorname{Fol}(M,\pi)$ coincides with
$\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$. The cotangent algebroid $T^{*}M$
is almost injective an isotopically closed by Lemma 1.13, and therefore
$\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$ obtains the structure of a
regular initial Lie subgroup of $\operatorname{Diff}(M)$ via Theorem 3.14. ∎

######  Remark 3.19.

When $Z\subset M$ is a smooth hypersurface, then explicit integrations of the
log-tangent bundle $T_{Z}M$ have been constructed in [8]. One of its
integrations, that we call the log-pair groupoid $\operatorname{Pair}_{Z}(M)$,
can be obtained by blowing up $Z\times Z$ in $M\times M$. After removing a
certain subset (namely the arrows that point into and out of $Z$),
$\operatorname{Pair}_{Z}(M)\rightrightarrows M$ becomes a groupoid integrating
$T_{Z}M$. If $\omega$ is a log-symplectic structure on $T_{Z}M$, then
$\Pi^{-1}=s^{*}\omega-t^{*}\omega$ is a log-symplectic structure on
$p_{Z}^{!}\operatorname{Pair}_{Z}(M)=T_{\mathcal{H}}\operatorname{Pair}_{Z}(M)$
(see Proposition 2.15). For the resulting Poisson groupoid
$(\operatorname{Pair}_{Z}(M),\Pi)$, the anchor to $M\times M$ induces
isomorphisms
$\operatorname{Bis}(\operatorname{Pair}_{Z}(M))\cong\operatorname{Diff}(M,Z)$
and
$\operatorname{Bis}(\operatorname{Pair}_{Z}(M),\Pi)\cong\operatorname{Diff}(M,\pi)$.

#### 3.2.2 Elliptic symplectic structures

Let $|D|$ be an elliptic divisor on $M$. If $X\in\mathfrak{X}(M)$ is a vector
field with the property that $[X,Y]$ is elliptic for all
$Y\in\mathfrak{X}(M,|D|)$, then $X\in\mathfrak{X}(M,|D|)$. This means that the
automorphisms of $T_{|D|}M$ are generated by its inner automorphisms, so
$\operatorname{Diff}_{c,0}(M,|D|):=\operatorname{Aut}_{c,0}(T_{|D|}M)\cong\operatorname{InnAut}_{c}(T_{|D|}M)$.

Let $(M,\pi)$ be an elliptic symplectic manifold with elliptic divisor $|D|$.
Since $T_{|D|}^{*}M$ locally admits bases of closed sections, the following is
a special case of Proposition 3.10 and Theorem 3.15.

######  Theorem 3.20.

Let $|D|$ be an elliptic divisor. Then $\operatorname{Diff}(M,|D|)$ is a
regular initial Lie subgroup of $\operatorname{Diff}(M)$ with Lie algebra
$\mathfrak{X}(M,|D|)$.

Let $(M,\pi)$ be an elliptic symplectic manifold. Then
$\operatorname{Diff}(M,\pi)$ is a regular initial Lie subgroup of
$\operatorname{Diff}(M)$ with Lie algebra $\mathfrak{X}_{c}(M,\pi)$.

The symplectic foliation of an elliptic symplectic manifold $(M,\pi)$ depends
on the elliptic residue of the elliptic two-form
$\omega\in\Omega^{2}(T_{|D|}M)$, defined as follows. Near the degeneracy locus
$i:D\hookrightarrow M$, we can write

$\omega=\lambda d\log r\wedge d\theta+d\log
r\wedge\alpha_{1}+d\theta\wedge\alpha_{2}+\beta$

where $\lambda\in\Omega^{0}(D;\mathfrak{k}^{*})$, with
$\mathfrak{k}=\det\left(\ker\rho_{T_{|D|}M}|_{D}\right)$, is locally constant.

If $\lambda\neq 0$, then the symplectic foliation of $(M,\pi)$ consists of
(the connected components of) $M\setminus D$ and $D$. It follows that
$\operatorname{Fol}(M,\pi)$ is an open subgroup of
$\operatorname{Diff}(M,\pi)$.

When $\lambda=0$, the symplectic foliation is given by connected components of
$M\setminus D$ and the foliation induced by the kernels of the closed one-
forms $i^{*}\alpha_{1},i^{*}\alpha_{2}\in\Omega^{1}(D)$. At the moment, we do
not have a description of a Lie group structure on $\operatorname{Fol}(M,\pi)$
in this case.

Finally, note that $\operatorname{Ham}_{\mathrm{loc,c}}(M,\pi)$ carries the
structure of an initial Lie subgroup of $\operatorname{Diff}(M,\pi)$ by
Theorem 3.14. In contrast to the log-symplectic case, not every foliated
vector field is locally Hamiltonian.

### 3.3 Scattering-symplectic structures

The following is a consequence of the linearization from Proposition 2.16 for
scattering-symplectic manifolds.

######  Theorem 3.21.

Let $(M,Z,\omega)$ be a scattering-symplectic manifold with underlying Poisson
structure $\pi$. Then $\operatorname{Diff}(M,\pi)$ is a regular initial Lie
subgroup of $\operatorname{Diff}(M)$, with Lie algebra
$\mathfrak{X}_{c}(M,\pi)$.

###### Proof.

We equip the path component $\operatorname{Diff}_{c,0}(M,\pi)$ with a Lie
group structure as follows. Any Poisson vector field $X$ must be tangent to
$Z$, hence it follows
$\operatorname{Diff}_{c,0}(M,\pi)\subset\operatorname{Diff}_{c,0}(M,Z)$.
Recall from the proof of Theorem 3.17 that $\operatorname{Diff}_{c,0}(M,Z)$
obtains its Lie group structure by identifying it with
$\operatorname{Bis}_{c,0}(\mathcal{G})$, with $\mathcal{G}\rightrightarrows M$
the holonomy groupoid of $T_{Z}M$. By Proposition 2.16, $(\mathcal{G},\Pi)$ is
linearizable around $M$, Therefore, Theorem 2.12 says that
$\operatorname{Bis}_{c,0}(\mathcal{G},\Pi)$ is a regular embedded Lie subgroup
of $\operatorname{Bis}_{c,0}(\mathcal{G})$. Clearly,
$\operatorname{Bis}_{c,0}(\mathcal{G},\Pi)$ identifies with
$\operatorname{Diff}_{c,0}(M,\pi)$. ∎

### 3.4 Almost injective (co)symplectic Lie algebroids

An inner $\mathcal{A}$-Poisson automorphism of a Poisson algebroid
$(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ is an inner automorphism
$\varphi\in\operatorname{InnAut}(\mathcal{A})$ that preserves the bivector
$\pi_{\mathcal{A}}$. This gives rise to the subgroup
$\operatorname{InnAut}(\mathcal{A},\pi_{\mathcal{A}})$ of
$\operatorname{InnAut}(\mathcal{A})$. Its infinitesimal generators correspond
precisely to Poisson sections, which are the sections of $\mathcal{A}$ closed
with respect to the Lie algebroid structure on $\mathcal{A}^{*}$.

######  Proposition 3.22.

Let $(\varphi_{t},\Phi_{t}):\mathcal{A}\to\mathcal{A}$ be the flow of a time-
dependent section $v_{t}\in\Gamma(\mathcal{A})$. Then $\varphi_{t}$ is a path
in $\operatorname{InnAut}(\mathcal{A},\pi_{\mathcal{A}})$ if and only if
$[\pi_{\mathcal{A}},v_{t}]=0$ for all $t$.

In this section we consider a class of regular $\mathcal{A}$-Poisson
structures with which admit a lot of transverse Poisson sections.

######  Definition 3.23.

Let $\mathcal{A}\Rightarrow M$ be a Lie algebroid. A cosymplectic structure of
type $k$ (or $k$-cosymplectic structure) consists of closed one-forms
$\alpha_{1},\dots,\alpha_{k}\in\Omega^{1}(\mathcal{A})$ and a closed 2-form
$\omega\in\Omega^{2}(\mathcal{A})$ of constant rank $2n$ such that
$\omega^{n}\wedge\alpha_{1}\wedge\dots\wedge\alpha_{k}$ is a nowhere vanishing
top-form on $\mathcal{A}$. ∎

A cosymplectic structure $(\alpha_{1},\dots,\alpha_{k},\omega)$ on
$\mathcal{A}\Rightarrow M$ induces an isomorphism

$\flat:\mathcal{A}\to\mathcal{A}^{*},\quad
v\mapsto\iota_{v}\omega+\sum_{i=1}^{k}\alpha_{i}(v)\alpha_{i},$

called the flat map. The Reeb sections of the cosymplectic structures are
given by $R_{i}:=\flat^{-1}(\alpha_{i})$

The 2-form $\omega$ restricts to an $\mathcal{F}$-symplectic structure
$\omega_{\mathcal{F}}=\omega|_{\mathcal{F}}$ on the $\mathcal{A}$-foliation
$\mathcal{F}=\cap_{i=1}^{k}\ker\alpha_{i}$, and thus determines an
$\mathcal{A}$-Poisson structure $\pi_{\mathcal{A}}$ via the diagram

${\mathcal{F}^{*}}$${\mathcal{F}}$${\mathcal{A}^{*}}$${\mathcal{A}.}$$\scriptstyle{\omega_{\mathcal{F}}^{\flat}}$$\scriptstyle{\pi^{\sharp}_{\mathcal{A}}}$

######  Definition 3.24.

Let $(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ be a Poisson algebroid. We
call $\pi_{\mathcal{A}}$ of $\mathcal{A}$-cosymplectic type if there exists a
cosymplectic structure on $\mathcal{A}$ inducing $\pi_{\mathcal{A}}$. ∎

######  Remark 3.25.

The Reeb sections $R_{i}$ of a cosymplectic structure inducing
$\pi_{\mathcal{A}}$ are ‘Poisson sections’ meaning that
$[\pi_{\mathcal{A}},R_{i}]=0$.

We can now prove the following general result.

######  Theorem 3.26.

Let $(\mathcal{A},\pi_{\mathcal{A}})\Rightarrow M$ be an almost-injective
Poisson algebroid of cosymplectic type. The group
$\operatorname{InnAut}_{c}(\mathcal{A},\pi_{\mathcal{A}})$ is a regular
embedded Lie subgroup of $\operatorname{InnAut}_{c}(\mathcal{A})$ with Lie
algebra $\Gamma_{c}(\mathcal{A},\pi_{\mathcal{A}})$. This way it becomes an
immersed Lie subgroup of $\operatorname{Diff}(M)$.

If in addition $\mathcal{A}$ is isotopically closed and
$\rho:\Gamma_{c}(\mathcal{A})\to\mathfrak{X}_{c}(M)$ is initial, then
$\operatorname{InnAut}_{c}(\mathcal{A},\pi_{\mathcal{A}})$ is an initial Lie
subgroup of $\operatorname{Diff}(M)$.

###### Proof.

The holonomy groupoid $\operatorname{Hol}(\mathcal{A})$ integrates
$\mathcal{A}$, and the Poisson structure
$\overleftarrow{\pi_{\mathcal{A}}}-\overrightarrow{\pi_{\mathcal{A}}}$ on
$\operatorname{Hol}(\mathcal{A})$ is linearizable by Theorem 2.14.

If $\sigma\in\operatorname{Bis}(\operatorname{Hol}(\mathcal{A}))$, then its
image in $(M\times M,\pi\times(-\pi))$ is coisotropic precisely when $\sigma$
is coisotropic in
$(\operatorname{Hol}(\mathcal{A}),\overleftarrow{\pi_{\mathcal{A}}}-\overrightarrow{\pi_{\mathcal{A}}})$,
because $\mathcal{A}$ is almost-injective. This implies that $t\circ\sigma$
induces an $\mathcal{A}$-Poisson isomorphism.

Under the additional assumptions, it becomes initial by Proposition 3.10. ∎

### 3.5 Poisson structures of cosymplectic type

In this section we consider the symmetry group of Poisson stuctures of
cosymplectic type on the tangent bundle of a manifold.

######  Lemma 3.27.

Let $(M,\pi)$ be a Poisson manifold of cosymplectic type, with underlying
foliation $\mathcal{F}$. Then $\mathcal{F}$ has trivial holonomy.

###### Proof.

Choose a cosymplectic structure $(\alpha_{1},\dots,\alpha_{k},\omega)$
inducing $\pi$, The Reeb vector fields $R_{1},\dots,R_{k}$ are pairwise
commuting and transverse to $\mathcal{F}$, while also preserving
$\mathcal{F}$. Let $L$ be a leaf of $\mathcal{F}$ and $x\in L$ and
$\gamma:[0,1]\to L$ a loop based at $x$. The flows $\varphi^{i}_{\epsilon}$ of
$R_{i}$ induce an embedding
$(\epsilon_{1},\dots,\epsilon_{k})\mapsto\varphi^{1}_{\epsilon_{1}}\circ\dots\circ\varphi^{k}_{\epsilon_{k}}(x)$
of a small disc through $x$ transverse to $\mathcal{F}$. Since the image of
$\gamma$ is compact, for $\epsilon_{i}$ small enough,
$\varphi_{\epsilon_{1}}^{1}\circ\dots\circ\varphi^{k}_{\epsilon_{k}}\circ\gamma$
is a loop tangent to $\mathcal{F}$ starting at
$\varphi_{\epsilon_{1}}^{1}\circ\dots\circ\varphi^{k}_{\epsilon_{k}}(x)$.
Clearly, at $t=1$, it has returned to its starting point. ∎

It follows that $\operatorname{Hol}(\mathcal{F})$ coincides with the relation
groupoid

$\operatorname{Rel}(\mathcal{F})=\\{(x,y)\in M\times M:\mbox{$x$ and $y$
belong to the same leaf}\\}.$

We aim to strengthen Theorem 3.26 in the case that $\mathcal{A}=TM$.

######  Theorem 3.28.

Let $(M,\pi)$ be a Poisson manifold of cosymplectic type with underlying
foliation $\mathcal{F}$. Then $\operatorname{Diff}(M,\pi)$ is an embedded
regular Lie subgroup of $\operatorname{Diff}(M)$. Furthermore,
$\operatorname{Fol}(M,\pi)$ is an initial Lie subgroup of
$\operatorname{Diff}(M,\pi)$.

If the relation groupoid $\operatorname{Rel}(\mathcal{F})$ is embedded in
$M\times M$, then $\operatorname{Fol}(M,\pi)$ is embedded in
$\operatorname{Diff}(M,\pi)$.

Note that the relation groupoid $\operatorname{Rel}(\mathcal{F})$ is embedded
in $M\times M$ when every symplectic leaf of $(M,\pi)$ is embedded.

Let $(M,\pi)$ be a Poisson manifold of cosymplectic type, and choose a
cosymplectic structure $(\alpha_{1},\dots,\alpha_{k},\omega)$ inducing $\pi$.
To obtain charts of $\operatorname{Diff}(M,\pi)$ adapted to
$\operatorname{Fol}(M,\pi)$, it is crucial to keep track of the one-forms that
are part of the cosymplectic structure in the linearization of the pair
Poisson groupoid $(M\times M,\overleftarrow{\pi}-\overrightarrow{\pi})$ (which
is not part of Theorem 2.14), because of the following observation.

######  Lemma 3.29.

Suppose that $M$ is connected. Let $\alpha_{1},\dots,\alpha_{k}$ be everywhere
independent closed one-forms on $M$ and set
$\mathcal{F}=\cap_{i}\ker\alpha_{i}$. The relation groupoid
$\operatorname{Rel}(\mathcal{F})$ is the leaf of the foliation
$\cap_{i}\ker(\overrightarrow{\alpha_{i}}-\overleftarrow{\alpha_{i}})$ on
$M\times M$ through the diagonal.

To keep track of the cosymplectic structure, we identify
$N\Delta=\\{(-v,v):v\in TM\\}\subset TM\times TM$ and choose a tubular
neighbourhood $\psi:TM\to U\subset M\times M$ of $\Delta$ that respects the
splittings along $M$, meaning that $T\psi$ sends $(v,w)\in T_{x}(TM)\cong
T_{x}M\oplus T_{x}M$ to $(v-w,v+w)\in T_{(x,x)}(M\times M)$. Also, we let
$p:TM\to M$ and $p_{*}:T^{*}M\to M$ be the bundle projections.

######  Lemma 3.30.

Let $\flat:TM\to T^{*}M$ be the flat map induced by the cosymplectic structure
$(\alpha_{1},\dots,\alpha_{k},\omega)$, with inverse $\sharp=\flat^{-1}$. Then
the following equalities hold along $M$:

$\displaystyle p_{*}^{*}\alpha_{i}|_{M}$
$\displaystyle=\sharp^{*}\psi^{*}\left(\frac{1}{2}\left(\overrightarrow{\alpha_{i}}+\overleftarrow{\alpha_{i}}\right)\right)\big{|}_{M},$
$\displaystyle d\ell_{R_{i}}\big{|}_{M}$
$\displaystyle=\sharp^{*}\psi^{*}\left(\frac{1}{2}\left(\overrightarrow{\alpha_{i}}-\overleftarrow{\alpha_{i}}\right)\right)\big{|}_{M},$
$\displaystyle\left(\omega_{\mathrm{can}}-\sum_{i=1}^{k}d(\ell_{R_{i}}p_{*}^{*}\alpha_{i})\right)\big{|}_{M}$
$\displaystyle=\sharp^{*}\psi^{*}\left(\frac{1}{2}\left(\overrightarrow{\omega}-\overleftarrow{\omega}\right)\right)\big{|}_{M}.$

###### Proof.

Since the map $\sharp$ is a vector bundle map, its derivative along the zero
section is given in terms of the natural splittings by
$T\sharp(v,w)=(v,\sharp(w))$.

For the first two equalities, we have on one hand, for $(v,\beta)\in
T_{x}(T^{*}M)$:

$p^{*}_{*}\alpha_{i}(v,\beta)=\alpha_{i}(v),\quad
d\ell_{R_{i}}(v,\beta)=\beta(R_{i}),$

while the right side evaluates to:

$\displaystyle\sharp^{*}\psi^{*}\left(\frac{1}{2}\left(\overrightarrow{\alpha_{i}}+\overleftarrow{\alpha_{i}}\right)\right)(v,\beta)$
$\displaystyle=\frac{1}{2}\left(\overrightarrow{\alpha_{i}}+\overleftarrow{\alpha_{i}}\right)(v-\sharp(\beta),v+\sharp(\beta))=\alpha_{i}(v),$

and the second equality goes similar. For the third one, let
$(v,\beta),(w,\gamma)\in T_{x}(T^{*}M)$. Then,

$\displaystyle\sharp^{*}\psi^{*}\left(\frac{1}{2}\left(\overrightarrow{\omega}-\overleftarrow{\omega}\right)\right)\left((v,\beta),(w,\gamma)\right)$
$\displaystyle=\frac{1}{2}\left(\overrightarrow{\omega}-\overleftarrow{\omega}\right)\left(T\psi(v,\sharp(\beta)),T\psi(w,\sharp(\gamma))\right)$
$\displaystyle=\omega(\sharp(\beta),w)+\omega(v,\sharp(\gamma))$
$\displaystyle=\beta(w)-\gamma(v)-\left(\sum_{i}\beta(R_{i})\alpha_{i}(w)-\gamma(R_{i})\alpha_{i}(v)\right),$

where we used that
$\iota_{\sharp(\beta)}\omega=\beta-\sum_{i}\beta(R_{i})\alpha_{i}$. On the
other hand,

$\displaystyle\omega_{\mathrm{can}}((v,\beta),(w,\gamma))$
$\displaystyle=\beta(w)-\gamma(v)$ $\displaystyle d\ell_{R_{i}}\wedge
p^{*}_{*}\alpha_{i}\left((v,\beta),(w,\gamma)\right)$
$\displaystyle=\beta(R_{i})\alpha_{i}(w)-\gamma(R_{i})\alpha_{i}(v).$

The third equality now follows easily. ∎

######  Remark 3.31.

The cosymplectic structure $(p^{*}_{*}\alpha_{1},d\ell_{R_{1}},\dots
p^{*}_{*}\alpha_{k},d\ell_{R_{k}},\omega_{\mathrm{can}}-\sum_{i}d(\ell_{R_{i}}p^{*}_{*}\alpha_{i}))$
induces a linear Poisson structure on $T^{*}M$, which is Poisson diffeomorphic
via the flat map to the linear Poisson structure on $TM$ dual to the cotangent
algebroid. Using the cosymplectic structure to write
$T^{*}M=\mathcal{F}^{*}\times\mathbb{R}^{k}$, we see that a section
$(\eta,f)\in\Gamma(\mathcal{F}^{*}\times\mathbb{R}^{k})$ has a coisotropic
image in $T^{*}M$ if and only if $d_{\mathcal{F}}f=0$ and
$d_{\mathcal{F}}\eta=0$. This is equivalent to $\sharp(\eta,f)$ being a
Poisson vector field.

###### Proof of Theorem 3.28.

The first part is just Theorem 3.26 in the case that $\mathcal{A}=TM$.
Alternatively, the Poisson pair groupoid is linearizable by Theorem 2.14, and
so by Theorem 2.12 its subgroup of coisotropic bisections is embedded. Its
group of foliated diffeomorphisms is an initial Lie subgroup by Theorem 3.26,
taking $\mathcal{A}=\mathcal{F}$.

Suppose now that $\operatorname{Rel}(\mathcal{F})$ is embedded in $M\times M$.
According to Lemma 3.30 and the Moser lemma for cosymplectic Lie algebroids
([21], Lemma 3.1), we can find

  * •

open neighbourhoods $U\subset T^{*}M$ of $M$ and $V\subset M\times M$ of the
diagonal;

  * •

a Poisson diffeomorphism $\varphi:U\to V$ such that
$\varphi^{*}\left(\frac{1}{2}\left(\overrightarrow{\alpha_{i}}+\overleftarrow{\alpha_{i}}\right)\right)=p^{*}_{*}\alpha_{i}$
and
$\varphi^{*}\left(\frac{1}{2}\left(\overrightarrow{\alpha_{i}}-\overleftarrow{\alpha_{i}}\right)\right)=d\ell_{R_{i}}$.

Identify $T^{*}M=\mathcal{F}^{*}\times\mathbb{R}^{k}$ as in Remark 3.31. Since
$\operatorname{Rel}(\mathcal{F})$ is embedded, and
$\varphi^{*}\left(\frac{1}{2}\left(\overrightarrow{\alpha_{i}}-\overleftarrow{\alpha_{i}}\right)\right)=d\ell_{R_{i}}$,
we can assume by Lemma 3.29, after possibly shrinking $U$, that
$\varphi(\eta,f)\in\operatorname{Rel}(\mathcal{F})$ if and only if $f=0$. In
this tubular neighbourhood, the $C^{1}$-open

$\mathcal{U}=\left\\{(\eta,f)\in\Gamma_{c}(\mathcal{F}^{*}\times\mathbb{R}^{k}):\begin{subarray}{c}\mbox{$d_{\mathcal{F}}\eta=0$,
$d_{\mathcal{F}}f=0$, $\mathrm{im}(\eta,f)\subset U$, }\\\ \mbox{
$s\circ\varphi\circ(\eta,f)$ and $t\circ\varphi\circ(\eta,f)$ are
diffeo's}\end{subarray}\right\\}$

is the image of some chart $(\mathcal{V},\Phi)$ if
$\operatorname{Diff}(M,\pi)$ as described in Section 2.2. Now, a section
$(\eta,f)$ has image in $\operatorname{Rel}(\mathcal{F})$ if and only if
$f=0$. It follows that $\Phi$ restricts to $\operatorname{Fol}(M,\pi)$ as

$\operatorname{Fol}(M,\pi)\cap\mathcal{V}\xrightarrow{\Phi}\Omega^{1}_{\mathrm{cl},c}(\mathcal{F})\cap\mathcal{U}.$

Hence, the chart $(\mathcal{V},\Phi)$ is adapted to
$\operatorname{Fol}(M,\pi)$. ∎

######  Remark 3.32.

Without the presence of the 2-form in a cosymplectic structure, our approach
remains valid. Indeed, let $\alpha\in\Omega^{1}(M)$ be a closed nowhere
vanishing one-form, and set $\mathcal{F}=\ker\alpha$. A bisection $\sigma:M\to
M\times M$ corresponds to a one-form preserving diffeomorphism if and only if
$\sigma^{*}(\overrightarrow{\alpha})=\sigma^{*}(\overleftarrow{\alpha})=\alpha$.
Furthermore, it corresponds to a foliation preserving diffeomorphism if and
only if it is a foliated map $\sigma:(M,\mathcal{F})\to(M\times
M,\mathcal{F}\times\mathcal{F})$. The Moser lemma ([21], Lemma 3.1) can be
rephrased without the two-form. Using these observations, we can argue similar
as in the proof of Theorem 3.28 to obtain charts of $\operatorname{Diff}(M)$
adapted to $\operatorname{Diff}(M,\alpha)$ and
$\operatorname{Diff}(M,\mathcal{F})$. The result is the following theorem.

######  Theorem 3.33.

Let $\alpha_{1},\dots,\alpha_{k}\in\Omega^{1}(M)$ be closed, everywhere
independent one-forms and set $\mathcal{F}=\cap_{i=1}^{k}\ker\alpha_{i}$. Then
$\operatorname{Diff}(M,\mathcal{F})$ and
$\operatorname{Diff}(M,\alpha_{1},\dots,\alpha_{k})$ are regular embedded Lie
subgroups of $\operatorname{Diff}(M)$, with Lie algebras

$\displaystyle\mathfrak{X}_{c}(M,\mathcal{F})$
$\displaystyle=\\{X\in\mathfrak{X}_{c}(M):[X,\Gamma(\mathcal{F})]\subset\Gamma(\mathcal{F})\\}\mbox{
and }$ $\displaystyle\mathfrak{X}_{c}(M,\alpha_{1},\dots,\alpha_{k})$
$\displaystyle=\\{X\in\mathfrak{X}_{c}(M):\mathcal{L}_{X}(\alpha_{1})=\dots=\mathcal{L}_{X}(\alpha_{k})=0\\},$

respectively.

## References

  * [1] I. Androulidakis and M. Zambon.  Almost regular Poisson manifolds and their holonomy groupoids.  Selecta Mathematica, 23(3):2291–2330, 2017.
  * [2] H. Bursztyn and R. L. Fernandes.  Picard groups of Poisson manifolds.  Journal of Differential Geometry, 109(1):1–38, 2018.
  * [3] H. Bursztyn, H. Lima, and E. Meinrenken.  Splitting theorems for Poisson and related structures.  Journal für die reine und angewandte Mathematik (Crelles Journal), 2019:281 – 312, 2016.
  * [4] H. Bursztyn and A. Weinstein.  Picard groups in Poisson geometry.  Moscow Mathematical Journal, 4(1):39–66, 2004.
  * [5] G. R. Cavalcanti and M. Gualtieri.  Stable generalized complex structures.  Proceedings of the London Mathematical Society, 116(5):1075–1111, 2017.
  * [6] C. Debord.  Holonomy groupoids of singular foliations.  Journal of Differential Geometry, 58(3):467 – 500, 2001.
  * [7] D. G. Ebin and J. Marsden.  Groups of diffeomorphisms and the motion of an incompressible fluid.  Annals of Mathematics, 92(1):102–163, 1970.
  * [8] M. Gualtieri and S. Li.  Symplectic groupoids of log symplectic manifolds.  International Mathematics Research Notices, 2014(11):3022–3074, 2014.
  * [9] V. Guillemin, E. Miranda, and A. R. Pires.  Symplectic and Poisson geometry on b-manifolds.  Advances in Mathematics, 264:864 – 896, 2014.
  * [10] R. Klaasse.  Poisson structures of divisor type, arXiv:1811.04226, 2018.
  * [11] A. Kriegl and P. Michor.  The convenient setting of global analysis.  Mathematical Surveys and Monographs, 53, 1997.
  * [12] M. Lanius.  Symplectic, Poisson, and contact geometry on scattering manifolds.  Pacific Journal of Mathematics, 310(1):213–256, 2021.
  * [13] K. Mackenzie.  General Theory of Lie Groupoids and Lie Algebroids.  London Mathematical Society Lecture Note Series. Cambridge University Press, 2005.
  * [14] I. Mărcut.  Poisson structures whose Poisson diffeomorphism group is not locally path-connected, arXiv:2001.00644, 2020.
  * [15] E. Miranda and G. Scott.  The geometry of E-manifolds.  Revista Matemática Iberoamericana, 37(3):1207–1224, 2020.
  * [16] K.-H. Neeb.  Towards a Lie theory for locally convex groups, arXiv:1501.06269, 2015.
  * [17] T. Rybicki.  On foliated, Poisson and Hamiltonian diffeomorphisms.  Differential Geometry and its Applications, 15(1):33 – 46, 2001\.
  * [18] T. Rybicki.  On the group of Lagrangian bisections of a symplectic groupoid.  Banach Center Publications, 54:235–247, 2001.
  * [19] T. Rybicki.  A Lie group structure on strict groups.  Publicationes Mathematicae, 61, 2002.
  * [20] A. Schmeding and C. Wockel.  The Lie group of bisections of a Lie groupoid.  Annals of Global Analysis and Geometry, 48:87–123, 2015.
  * [21] W. Smilde.  Linearization of Poisson groupoids.  2021\.
  * [22] I. Vaisman.  Lectures on the Geometry of Poisson Manifolds.  Birkhäuser Basel, 1994.
  * [23] A. Weinstein.  Coisotropic calculus and Poisson groupoids.  Journal of the Mathematical Society of Japan, 40(4):705–727, 1988\.
  * [24] P. Xu.  Flux homomorphism on symplectic groupoids.  Mathematische Zeitschrift, 226(4):575–597, 1997.