Graded Bundles in the Category of Lie Groupoids

Graded Bundles in the Category of Lie Groupoids\star\star$\star$This paper is
a contribution to the Special Issue on Poisson Geometry in Mathematics and
Physics. The full collection is available at
http://www.emis.de/journals/SIGMA/Poisson2014.html

Andrew James BRUCE †, Katarzyna GRABOWSKA ‡ and Janusz GRABOWSKI †

A.J. Bruce, K. Grabowska and J. Grabowski

† Institute of Mathematics, Polish Academy of Sciences, Poland
<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ Faculty of Physics, University of Warsaw, Poland<EMAIL_ADDRESS>
Received February 25, 2015, in final form November 05, 2015; Published online
November 11, 2015

We define and make initial study of Lie groupoids equipped with a compatible
homogeneity (or graded bundle) structure, such objects we will refer to as
_weighted Lie groupoids_. One can think of weighted Lie groupoids as graded
manifolds in the category of Lie groupoids. This is a very rich geometrical
theory with numerous natural examples. Note that $\mathcal{VB}$-groupoids,
extensively studied in the recent literature, form just the particular case of
weighted Lie groupoids of degree one. We examine the Lie theory related to
weighted groupoids and _weighted Lie algebroids_ , objects defined in a
previous publication of the authors, which are graded manifolds in the
category of Lie algebroids, showing that they are naturally related via
differentiation and integration. In this work we also make an initial study of
_weighted Poisson–Lie groupoids_ and _weighted Lie bi-algebroids_ , as well as
_weighted Courant algebroids_.

graded manifolds; homogeneity structures; Lie groupoids; Lie algebroids

22A22; 55R10; 58E40; 58H05

##  1 Introduction

Lie groupoids and Lie algebroids are ubiquitous throughout differential
geometry and are playing an ever increasing rôle in mathematical physics. Lie
groupoids provide a unifying framework to discuss diverse topics in modern
geometry including the theory of group actions, foliations, Poisson geometry,
orbifolds, principal bundles, connection theory and so on. The infinitesimal
counterpart to Lie groupoids are Lie algebroids. The Lie theory here is very
rich and not as simple as the corresponding Lie theory for Lie groups and Lie
algebras. In particular, while it is true that every Lie groupoid can be
differentiated to obtain a Lie algebroid, the reverse procedure of global
integration has an obstruction [7, 26]. Not all Lie algebroids can be globally
integrated to obtain a Lie groupoid, although one can always integrate Lie
algebroids to ‘local’ Lie groupoids.

Another important notion in modern geometry is that of a graded manifold. Such
notions have their conception in the ‘super’ context via the BV–BRST formalism
of gauge theories in physics. We remark that Lie algebroids appear in physics
as symmetries of field theories that do not arise from Lie groups or Lie
algebras. Moreover, these symmetries cannot always be directly separated from
the space of fields. Such symmetries are naturally accommodated in the BV–BRST
formalism. The concept of a graded manifold has been put to good use in
describing Lie algebroids, Lie bi-algebroids, Courant algebroids and related
notions [38, 39, 44, 46]. In the purely commutative setting, Grabowski and
Rotkiewicz [15] define what they referred to as _graded bundles_. Loosely, a
graded bundle is a natural generalisation of the concept of a vector bundle.
We will discuss graded bundles in a little more detail shortly.

There has been some recent interest in $\mathcal{VB}$-groupoids and
$\mathcal{VB}$-algebroids, see for example [1, 5, 16, 17]. From a categorical
point of view $\mathcal{VB}$-groupoids are vector bundles in the category of
Lie groupoids and similarly $\mathcal{VB}$-algebroids are vector bundles in
the category of Lie algebroids. It is known, via [1, 5], that the Lie functor
restricts to the category of $\mathcal{VB}$-groupoids and
$\mathcal{VB}$-algebroids; that is we can differentiate
$\mathcal{VB}$-groupoids to get $\mathcal{VB}$-algebroids and
$\mathcal{VB}$-algebroids integrate to $\mathcal{VB}$-groupoids. Let us
mention that $\mathcal{VB}$-groupoids and $\mathcal{VB}$-algebroids have been
studied in relation to Poisson geometry and representations. Indeed,
representations of Lie groupoids on vector bundles gives rise to
$\mathcal{VB}$-groupoids.

As graded bundles are generalisations of vector bundles, it seems natural that
one should study graded bundles in the category of Lie groupoids and Lie
algebroids, including their Lie theory. The first work in this direction was
by the authors of this paper, where various descriptions of so-called
_weighted Lie algebroids_ were given in [3]. The motivation for that work was
to uncover a practical notion of a “higher Lie algebroid” suitable for higher-
order geometric mechanics in the spirit of Tulczyjew. We presented details of
the Lagrangian and Hamiltonian formalisms on graded bundles using weighted Lie
algebroids in a separate publication [2]. As an application, the higher-order
Euler–Lagrange equations on a Lie algebroid were derived completely
geometrically, and in full agreement with the independent approach of Martínez
using variational calculus [32]. This and other results have convinced us of
the potential for further applications of weighted Lie algebroids.

The natural question of integrating weighted Lie algebroids was not posed at
all in [3] or [2]. We address this question in this paper. We define _weighted
Lie groupoids_ as Lie groupoids that carry a compatible homogeneity structure,
i.e., an action of the monoid ${\mathbb{R}}$ of multiplicative reals; we will
make this precise in due course. We show that these structures, which are a
generalisation of $\mathcal{VB}$-groupoids, are the objects that integrate
weighted Lie algebroids. Just as the tangent bundle of a Lie groupoid is a
canonical example of a $\mathcal{VB}$-groupoid, the $k$-th order higher
tangent bundle of a Lie groupoid is a canonical example of a weighted Lie
groupoid.

The results found in this paper can be viewed as the ‘higher order’
generalisations of the results found in [5]. In particular, Bursztyn et al.
[5] treat $\mathcal{VB}$-groupoids/algebroids as Lie groupoids/algebroids
equipped with a compatible regular homogeneity structure. The idea of
simplifying $\mathcal{VB}$-‘objects’ by describing then in terms of regular
homogeneity structures goes back to [14], where symplectic structures were
studied in this context. By passing to graded bundles, we do not really
further simplify the notion of $\mathcal{VB}$-‘objects’, but by understanding
them as part of a larger class of structures we simplify working with them.

Following our intuition, we also examine the notion of _weighted Poisson–Lie
groupoids_ as Poisson–Lie groupoids equipped with a compatible homogeneity
structure. The infinitesimal versions of weighted Poisson–Lie groupoids are
_weighted Lie bi-algebroids_ , a notion we carefully define in this paper.
Associated with any Lie bi-algebroid is a Courant algebroid. We show that this
notion very naturally passes over to the weighted case and motivates a more
general notion of a _weighted Courant algebroid_ , which is a natural
generalisation of a $\mathcal{VB}$-Courant algebroid. The use of a compatible
homogeneity structure provokes the question of replacing the monoid
${\mathbb{R}}$ by its subgroup ${\mathbb{R}}^{\times}$ of multiplicative reals
$\neq 0$. We claim that this is a natural way of defining contact and Jacobi
groupoids, see [4].

Summing up, one can say that in this paper we generalise the notion of
$\mathcal{VB}$-‘objects’ by passing to the category of graded bundles and
categorical objects therein. On the other hand, these concepts and their
applications are far from being fully exploited. This work is only the first
step in this direction.

Our use of supermanifolds. We recognise the power and elegance of the
framework of supermanifolds in the context of algebroid-like objects and we
will exploit this formalism. In particular we will make use of the so called
$Q$-manifolds, that is supermanifolds equipped with a homological vector
field. Although we will not dwell on fundamental issues from the theory of
supermanifolds, we will technically follow the “Russian school” and understand
supermanifolds in terms of locally superringed spaces. However, for the most
part the intuitive and correct understanding of a supermanifold as a
‘manifold’ with both commuting and anticommuting coordinates will suffice.
When necessary we will denote the Grassmann parity of an object by ‘tilde’.

Arrangement of paper. In Section 2 we briefly recall the necessary parts of
theory of graded bundles, weighted Lie algebroids and Lie groupoids as needed
in the rest of this paper. In Section 3 we introduce the main objects of
study, that is weighted Lie groupoids. Weighted Lie theory is the subject of
Section 4. Finally in Section 5 we apply some of the ideas developed earlier
in this paper to weighted Poisson–Lie groupoids, weighted Lie bi-algebroids
and weighted Courant algebroids. We end with some remarks on contact and
Jacobi groupoids.

##  2 Preliminaries

In this section we briefly recall parts of the theory of graded bundles,
$n$-tuple graded bundles and weighted algebroids as needed in later sections
of this paper. Everything in this paper will be in the smooth category. The
interested reader should consult the original literature [3, 12, 14, 15] for
details, such as proofs of the statements made in this section. We will also
set some notation regarding Lie groupoids and recall the groupoid/algebroid
version of Lie’s second theorem, which was first proved by Mackenzie and Xu
[30]. For an overview of the general theory of Lie groupoids and Lie
algebroids the reader can consult Mackenzie [28]. We will also very briefly
recall the notion of a $Q$-manifold and a $QS$-manifold as needed throughout
this paper.

###  2.1 Graded and $\boldsymbol{n}$-tuple graded bundles

Manifolds and supermanifolds that carry various extra gradings on their
structure sheaf are now an established part of modern geometry and
mathematical physics. The general theory of graded manifolds in our
understanding was formulated by Voronov in [44]; we must also mention the
works of Kontsevich [21], Mehta [33], Roytenberg [38] and Ševera [41] where
various notions of a $\mathbb{Z}$-graded supermanifold appear. The graded
structure on such (super)manifolds is conveniently encoded in the _weight
vector field_ whose action via the Lie derivative counts the degree of tensor
and tensor-like objects on the (super)manifold.

In this introductory section we will concentrate our attention on just genuine
manifold and only sketch the theory for supermanifolds, as the extension of
the results here to supermanifolds is almost straightforward. In later
sections we will make use of supermanifolds that carry additional gradings
coming from a _homogeneity structure_ which will be defined in a moment.

An important class of graded manifolds are those that carry non-negative
grading. It will be convenient to denote homogeneous local coordinates in the
form $(y^{a}_{w})$ (or $(y^{w}_{a})$), where $w=0,1,\dots,k$ is the weight of
$y^{w}_{a}$. A canonical example of such a structure is the bundle
${\mathtt{T}}^{k}M=J^{k}_{0}({\mathbb{R}},M)$ of $k$-velocities, i.e., $k$th-
jets (at $0$) of curves $\gamma\colon{\mathbb{R}}\to M$. We will furthermore
require that (like in the case of ${\mathtt{T}}^{k}M$) this grading is
associated with a smooth action $h\colon{\mathbb{R}}_{\geq 0}\times F\to F$ of
the monoid $({\mathbb{R}}_{\geq 0},\cdot)$ of multiplicative non-negative
reals on a manifold $F$. Let us recall in this context that a function $f\in
C^{\infty}(F)$ we call _homogeneous of degree $k\in{\mathbb{R}}$_ if

$\displaystyle h_{t}^{*}(f):=f(h(t,\cdot))=t^{k}f$ (2.1)

for all $t>0$. We call $k$ the _weight_ or _degree_ of $f$.

Such actions are known as _homogeneity structures_ in the terminology of
Grabowski and Rotkiewicz [15] who proved that only _non-negative integer
weights_ are allowed, so the algebra $\mathcal{A}(F)\subset C^{\infty}(F)$
spanned by homogeneous function has a canonical $\mathbb{N}$-grading,
$\mathcal{A}(F)=\bigoplus_{i\in\mathbb{N}}\mathcal{A}^{i}(F)$. This algebra is
referred to as the _algebra of polynomial functions_ on $F$. This action
reduced to ${\mathbb{R}}_{>0}$ is the one-parameter group of diffeomorphism
integrating the weight vector field, thus the weight vector field is in this
case _${\rm h}$ -complete_ [12]. Note also that the homogeneity structure
always has a canonical extension to the action $h\colon{\mathbb{R}}\times F\to
F$ of the monoid $({\mathbb{R}},\cdot)$ such that any homogeneous function $f$
of degree $k\in\mathbb{N}$ satisfies (2.1) this time for all
$t\in{\mathbb{R}}$; it will be convenient to speak about a homogeneity
structure as this extended action.

Importantly, we have that for $t\neq 0$ the action $h_{t}$ is a diffeomorphism
of $F$ and when $t=0$ it is a smooth surjection $\tau=h_{0}$ onto a
submanifold $F_{0}=M$, with the fibres being diffeomorphic to $\mathbb{R}^{N}$
(cf. [15]). Thus, the objects obtained are particular kinds of _polynomial
bundles_ $\tau\colon F\to M$, i.e., fibrations which locally look like
$U\times{\mathbb{R}}^{N}$ and the change of coordinates (for a certain choice
of an atlas) are polynomial in ${\mathbb{R}}^{N}$. For this reason graded
manifolds with non-negative weights and h-complete weight vector fields are
also known as _graded bundles_ [15]. Furthermore, the h-completeness condition
implies that graded bundles are determined by the algebra of homogeneous
functions on them. Canonical examples of graded bundles are higher tangent
bundles ${\mathtt{T}}^{k}M$. The canonical coordinates
$(x^{a},\dot{x}^{b},\ddot{x}^{c},\dots)$ on ${\mathtt{T}}^{k}M$, associated
with local coordinates $(x^{a})$ on $M$, have degrees, respectively,
$0,1,2,\dots$, so the homogeneity structure reads

$\displaystyle
h^{*}_{t}\big{(}x^{a},\dot{x}^{b},\ddot{x}^{c},\dots\big{)}=\big{(}x^{a},t\dot{x}^{b},t^{2}\ddot{x}^{c},\dots\big{)}.$

A fundamental result is that any smooth action of the multiplicative monoid
$({\mathbb{R}},\cdot)$ on a manifold leads to a $\mathbb{N}$-gradation of that
manifold. A little more carefully, the category of graded bundles is
equivalent to the category of $(\mathbb{R},\cdot)$-manifolds and equivariant
maps. A canonical construction of this correspondence goes as follows. Take
$p\in F$ and consider $t\mapsto h_{t}(p)$ as a smooth curve $\gamma_{h}^{p}$
in $F$. This curve meets $M$ for $t=0$ and is constant for $p\in M$.

######  Theorem 2.1 ([15]).

For any homogeneity structure $h\colon{\mathbb{R}}\times F\to F$ there is
$k\in\mathbb{N}$ such that the map

$\displaystyle\Phi_{h}^{k}\colon\
F\to{\mathtt{T}}^{k}F_{|M},\qquad\Phi^{k}_{h}(p)={\mathtt{j}}^{k}_{0}(\gamma_{h}^{p}),$
(2.2)

is an equivariant $($with respect to the monoid actions of
$({\mathbb{R}},\cdot)$ on $F$ and ${\mathtt{T}}^{k}F)$ embedding of $F$ into a
graded submanifold of the graded bundle ${\mathtt{T}}^{k}F_{|M}$. In
particular, there is an atlas on $F$ consisting of homogeneous function.

Any $k$ described by the above theorem we call a _degree_ of the homogeneity
structure $h$. We stress that a graded bundle is not just a manifold with
consistently defined homogeneous local coordinates. The additional condition
is that the weight vector field encoding the graded structure is h-complete;
that is the associated one-parameter group of diffeomorphisms can be extended
to an action of the monoid $(\mathbb{R},\cdot)$.

######  Remark 2.2.

The above theorem has a counterpart for graded supermanifolds, the proof of
which we will quickly sketch here. Following Voronov [44], we do not require
that weights induce parity. The compatibility condition just means that
$\mathbb{N}$-weights commute with $\mathbb{Z}_{2}$-grading. In other words, a
homogeneity structure on a supermanifold ${\mathcal{M}}$ is a smooth action
$h\colon{\mathbb{R}}\times{\mathcal{M}}\to{\mathcal{M}}$ of the monoid
$({\mathbb{R}},\cdot)$ such that $h_{t}\colon{\mathcal{M}}\to{\mathcal{M}}$
are morphisms, i.e., respect the parity. Equivalently, we have an
$({\mathbb{R}},\cdot)$-action on the super-algebra
${\mathcal{O}}_{\mathcal{M}}=C^{\infty}({\mathcal{M}})$ by homomorphisms.

The method of the proof of [20, Theorem 1.13] can be easily adapted to the
supermanifold case to show that ${\mathcal{M}}_{0}=h_{0}({\mathcal{M}})$ is a
submanifold of ${\mathcal{M}}$. We can also adapt the methods of the proofs of
Lemma 4.1 and Theorem 4.1 in [15]. First, define
$H(t)\colon{\mathtt{T}}{\mathcal{M}}_{|{\mathcal{M}}_{0}}\to{\mathtt{T}}{\mathcal{M}}_{|{\mathcal{M}}_{0}}$
as the derivative ${\mathtt{T}}{h_{t}}$ restricted to
${\mathtt{T}}{\mathcal{M}}_{|{\mathcal{M}}_{0}}$. We have $H(t)H(s)=H(ts)$, so
$Q_{0}=H(0)$ is a projector. Since both, $Q_{0}$ and $\mathbbmss{1}-Q_{0}$ are
projectors whose rank cannot fall locally, the rank of $Q_{0}$ is constant
along ${\mathcal{M}}_{0}$ (assuming ${\mathcal{M}}$ is connected) and
$E_{0}=Q_{0}({\mathtt{T}}{\mathcal{M}}_{|{\mathcal{M}}_{0}})$ is a vector
subbundle of ${\mathtt{T}}{\mathcal{M}}_{|{\mathcal{M}}_{0}}$. One can show
inductively that
$Q_{r}=\frac{\operatorname{d}^{r}H}{\operatorname{d}t^{r}}(0)$ is a projector
operator defined on the kernel of $Q_{r-1}$ and we put $E_{r}$ to be its
range. According to the argument similar to that of [15, Proposition 4.2],
$E_{k+1}={\mathtt{T}}{\mathcal{M}}_{0}$ for some $k$. We have therefore a
vector bundle decomposition

$\displaystyle{\mathtt{T}}{\mathcal{M}}_{|{\mathcal{M}}_{0}}=E_{0}\oplus_{{\mathcal{M}}_{0}}\dots\oplus_{{\mathcal{M}}_{0}}E_{k+1}$

such that, roughly speaking,

$\displaystyle\frac{\partial}{\partial x}\frac{\partial^{i}h}{\partial
t^{i}}(0,x),$

viewed as a liner map on ${\mathtt{T}}{\mathcal{M}}_{|{\mathcal{M}}_{0}}$,
vanish on $E_{r}$ for $r>i$ and are identical on $E_{i}$. This implies that
the analog of the map (2.2),
$\Phi^{k}_{h}\colon{\mathcal{M}}\to{\mathtt{T}}^{k}{\mathcal{M}}_{|{\mathcal{M}}_{0}}$.
whose coordinate form is

$\displaystyle\Phi^{k}_{h}(x)=\left(h(0,x),\frac{\partial h}{\partial
t}(0,x),\dots,\frac{\partial^{k}h}{\partial t^{k}}(0,x)\right),$

is of maximal rank at ${\mathcal{M}}_{0}$. Since it intertwines the actions of
$({\mathbb{R}},\cdot)$ (cf. [15, Lemma 4.1]), it is actually an embedding into
a graded submanifold of ${\mathtt{T}}^{k}{\mathcal{M}}_{|{\mathcal{M}}_{0}}$.
Indeed, the maximal rank argument is sufficient for odd coordinates, while for
even ones we can refer to [15].

Morphisms between graded bundles are necessarily polynomial in the non-zero
weight coordinates and respect the weight. Such morphisms can be characterised
by the fact that they relate the respective weight vector fields, or
equivalently the respective homogeneity structures [15].

######  Remark 2.3.

It is possible to consider manifolds with gradations that do not lead to
h-complete weight vector fields. From the point of view of this paper such
manifolds are less rigid in their structure and will exhibit pathological
behavior. We remark that the definition given by Voronov [44] of a non-
negatively graded supermanifold states that the Grassmann even coordinates are
‘cylindrical’. Similarly, Roytenberg [38] also insists on this ‘cylindrical’
condition in his definition of an $N$-manifold. This together with the fact
that functions of Grassmann odd coordinates are necessarily polynomial, means
that the weight vector fields on non-negatively graded supermanifolds and the
closely related $N$-manifolds are h-complete. One should also note that Ševera
[41] takes as his _definition_ of an $N$-manifold a supermanifold equipped
with an action of $(\mathbb{R},\cdot)$ such that $-1\in\mathbb{R}$ acts as the
parity operator (it flips sign of any Grassmann odd coordinate). Note however,
that Ševera does not offer a proof that homogeneous local coordinates can
always be found and so it is not immediately clear if his notion of an
$N$-manifold exactly coincides with that of Roytenberg. This equivalence
follows from minor modification of the arguments made in the previous remark.

A graded bundle of degree $k$ admits a sequence of polynomial fibrations,
where a point of $F_{l}$ is a class of the points of $F$ described in an
affine coordinate system by the coordinates of weight $\leq l$, with the
obvious tower of surjections

$\displaystyle
F=F_{k}\stackrel{{\scriptstyle\tau^{k}}}{{\longrightarrow}}F_{k-1}\stackrel{{\scriptstyle\tau^{k-1}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\tau^{3}}}{{\longrightarrow}}F_{2}\stackrel{{\scriptstyle\tau^{2}}}{{\longrightarrow}}F_{1}\stackrel{{\scriptstyle\tau^{1}}}{{\longrightarrow}}F_{0}=M,$

where the coordinates on $M$ have zero weight. Note that $F_{1}\rightarrow M$
is a linear fibration and the other fibrations $F_{l}\rightarrow F_{l-1}$ are
affine fibrations in the sense that the changes of local coordinates for the
fibres are linear plus and additional additive terms of appropriate weight.
The model fibres here are $\mathbb{R}^{N}$ (cf. [15]). We will also use on
occasion $\tau:=\tau^{k}_{0}\colon F_{k}\rightarrow M$.

Canonical examples of graded bundles are, for instance, vector bundles,
$n$-tuple vector bundles, higher tangent bundles ${\mathtt{T}}^{k}M$, and
multivector bundles $\wedge^{n}{\mathtt{T}}E$ of vector bundles $\tau\colon
E\to M$ with respect to the projection
$\wedge^{n}{\mathtt{T}}\tau\colon\wedge^{n}{\mathtt{T}}E\to\wedge^{n}{\mathtt{T}}M$
(see [13]). If the weight is constrained to be either zero or one, then the
weight vector field is precisely a vector bundle structure on $F$ and will be
generally referred to as an _Euler vector field_.

The notion of a double vector bundle [37] (or a higher $n$-tuple vector
bundle) is conceptually clear in the graded language in terms of mutually
commuting homogeneity structures, or equivalently weight vector fields; see
[14, 15]. This leads to the higher analogues known as _$n$ -tuple graded
bundles_, which are manifolds for which the structure sheaf carries an
$\mathbb{N}^{n}$-grading such that all the weight vector fields are h-complete
and pairwise commuting. In particular a _double graded bundle_ consists of a
manifold and a pair of mutually commuting and h-complete weight vector fields.
If all the weights are either zero or one, then we speak of an _$n$ -tuple
vector bundle_.

###  2.2 $\boldsymbol{Q}$-manifolds and related structures

Let us briefly define some structures on supermanifolds that we will employ
throughout this paper. The main purpose is to set some nomenclature and
notation. The graded versions will be fundamental throughout this paper.

######  Definition 2.4.

An odd vector field $Q\in\mathtt{Vect}(\mathcal{M})$ on a supermanifold
$\mathcal{M}$ is said to be a  _homological vector field_ if and only if
$2Q^{2}=[Q,Q]=0$. Note that, as we have an odd vector field, this condition is
generally non-trivial. A pair $(\mathcal{M},Q)$, where $\mathcal{M}$ is a
supermanifold and $Q\in\mathtt{Vect}(\mathcal{M})$ is called a  _$Q$
-manifold_. A  _morphism of $Q$-manifolds_ is a morphism of supermanifolds
that relates the respective homological vector fields.

The nomenclature ‘$Q$-manifold’ is due to Schwarz [40]. As is well known,
$Q$-manifolds, equipped with an additional grading, give a very economical
description of Lie algebroids as first discovered by
Va$\breve{\textrm{{\i}}}$ntrob [43]. Recall the standard notion of a Lie
algebroid as a vector bundle $E\rightarrow M$ equipped with a Lie bracket on
the sections
$[\bullet,\bullet]\colon\mathtt{Sec}(E)\times\mathtt{Sec}(E)\rightarrow\mathtt{Sec}(E)$
together with an anchor $\rho\colon\mathtt{Sec}(E)\rightarrow\mathtt{Vect}(M)$
that satisfy the Leibniz rule

$\displaystyle[u,fv]=\rho(u)[f]v+f[u,v],$

for all $u,v\in\mathtt{Sec}(E)$ and $f\in C^{\infty}(M)$. The Leibniz rule
implies that the anchor is actually a Lie algebra morphism:
$\rho\left([u,v]\right)=[\rho(u),\rho(v)]$ (see, e.g., [11]). If we pick some
local basis for the sections $(e_{a})$, then the structure functions of a Lie
algebroid are defined by

$\displaystyle[e_{a},e_{b}]=C_{ab}^{c}(x)e_{c},\qquad\rho(e_{a})=\rho_{a}^{A}(x)\frac{\partial}{\partial
x^{A}},$

where we have local coordinates $(x^{A})$ on $M$. These structure functions
satisfy some compatibility conditions which can be neatly encoded in a
homological vector field of weight one on $\Pi E$. Let us employ local
coordinates

$\displaystyle\big{\\{}\underbrace{x^{A}}_{0},\,\underbrace{\xi^{a}}_{1}\big{\\}},$

on $\Pi E$. Here the weight of coordinates corresponds to the natural weight
induced by the homogeneity structure associated with the vector bundle
structure [14] and this also corresponds to the Grassmann parity. We just
briefly remark that one can also define Lie algebroids in the category of
supermanifolds where it natural to consider the weight and Grassmann parity as
being independent. The homological vector field encoding the Lie algebroid is

$\displaystyle Q=\xi^{a}\rho_{a}^{A}(x)\frac{\partial}{\partial
x^{A}}+\frac{1}{2}\xi^{a}\xi^{b}C_{ba}^{c}(x)\frac{\partial}{\partial\xi^{c}}.$

The axioms of a Lie algebroid are then equivalent to $2Q^{2}=[Q,Q]=0$. We
will, by minor abuse of nomenclature, also refer to the graded $Q$-manifold
$(\Pi E,Q)$ as a Lie algebroid.

A morphism of Lie algebroid is then understood as a morphism of graded
$Q$-manifolds. That is, we have a morphism of super graded bundles that
relates the respective homological vector fields. Note that the definition of
a Lie algebroid morphism in terms of $Q$-manifold morphisms is equivalent to
the less obvious notion of a morphism of Lie algebroids as described in [28].

######  Example 2.5.

Any Lie algebra $(\mathfrak{g},[\,,\,])$ can be encoded in a homological
vector field on the linear supermanifold $\Pi\mathfrak{g}$. Let us employ
local coordinates $(\xi^{a})$ on $\Pi\mathfrak{g}$; then we have

$\displaystyle
Q=\frac{1}{2}\xi^{a}\xi^{b}C_{ba}^{c}\frac{\partial}{\partial\xi^{c}},$

where $C_{ba}^{c}$ is the structure constant of the Lie algebra in question.
In this case the base manifold is just a point and the anchor map is trivial.

######  Example 2.6.

In the other extreme, the tangent bundle of a manifold ${\mathtt{T}}M$ is
naturally a Lie algebroid for which the anchor is the identity map. The
homological vector on $\Pi{\mathtt{T}}M$ that encodes the Lie algebroid
structure is nothing other than the de Rham differential.

######  Definition 2.7.

An odd Hamiltionain $S\in C^{\infty}({\mathtt{T}}^{*}\mathcal{M})$ that is
quadratic in momenta (i.e., fibre coordinates) is said to be a _Schouten
structure_ if and only if $\\{S,S\\}=0$, where the bracket is the canonical
Poisson bracket on the cotangent bundle. A pair $(\mathcal{M},S)$, where
$\mathcal{M}$ is a supermanifold and $S\in
C^{\infty}({\mathtt{T}}^{*}{\mathcal{M}})$ is a Schouten structure, is called
a _$S$ -manifold_. The associated _Schouten bracket_ on $C^{\infty}(M)$ is
given as a derived bracket

$\displaystyle[[f,g]]_{S}:=\\{\\{f,S\\},g\\}.$

We must remark here that non-trivial Schouten structures only exist on
supermanifolds. A Schouten bracket is also known as an odd Poisson bracket and
satisfies the appropriate graded versions of the Jacobi identity and Leibniz
rule.

######  Example 2.8.

A Lie algebroid structure on a vector bundle $E\rightarrow M$ can be encoded
in a weight one Schouten structure on the supermanifold ${\mathtt{T}}^{*}\Pi
E^{*}$. Let us employ natural local coordinates on ${\mathtt{T}}^{*}\Pi E^{*}$
(with indicated bi-degrees)

$\displaystyle\big{(}\underbrace{x^{A}}_{(0,0)},\,\underbrace{\chi_{a}}_{(0,1)},\,\underbrace{p_{B}}_{(1,1)},\,\underbrace{\theta^{b}}_{(1,0)}\big{)},$

Then, the Schouten structure encoding a Lie algebroid is given by

$\displaystyle
S=\theta^{a}\rho_{a}^{A}(x)p_{A}+\frac{1}{2}\theta^{a}\theta^{b}C_{ba}^{c}(x)\chi_{c}$

and the axioms of a Lie algebroid are equivalent to $\\{S,S\\}=0$, where the
bracket here is the canonical Poisson bracket on the cotangent bundle.

######  Definition 2.9.

A supermanifold $M$ is said to be a _$QS$ -manifold_ if it is simultaneously a
$Q$-manifold and a $S$-manifold such that $L_{Q}S:=\\{\mathcal{Q},S\\}=0$,
where $\mathcal{Q}\in C^{\infty}({\mathtt{T}}^{*}M)$ is the symbol of the
homological vector field $Q$.

As the symbol map sends the Lie bracket of vector fields to the Poisson
bracket, a $QS$-manifold can be considered as a supermanifold equipped with
odd Hamiltonians, one linear and one quadratic in momenta, that satisfy
$\\{\mathcal{Q},\mathcal{Q}\\}=0$, $\\{S,S\\}=0$ and $\\{\mathcal{Q},S\\}=0$.
The definition of a $QS$-manifold goes back to Voronov [44].

Graded $QS$-manifolds give us a convenient way to understand Lie bi-
algebroids, which is due to Voronov [44], but also see Kosmann-Schwarzbach
[22] who modified the original definition of Mackenzie and Xu [29]. The
original definition involved the differential associated with the dual Lie
algebroid and Lie bracket on sections of the Lie algebroid, and not the
associated Schouten bracket on ‘multivector fields’. Following Kosmann-
Schwarzbach, a Lie bi-algebroid is a pair of Lie algebroids $(E,E^{*})$ such
that

$\displaystyle
Q_{E}[[X,Y]]_{E^{*}}=[[Q_{E}X,Y]]_{E^{*}}+(-1)^{\widetilde{X}}[[X,Q_{E}Y]]_{E^{*}},$

for all ‘multivector fields’ $X$ and $Y\in C^{\infty}(\Pi E)$. That is the
homological vector field encoding the Lie algebroid structure on $E$ must be a
derivation with respect to the Schouten bracket that encodes the Lie algebroid
structure on the dual vector bundle $E^{*}$. It is not hard to see that this
definition is equivalent to the compatibility condition

$\displaystyle\mathcal{L}_{Q_{E}}S_{E^{*}}=\\{\mathcal{Q}_{E},S_{E^{*}}\\}=0,$

and so we have a $QS$-manifold. Here we use the canonical isomorphism
${\mathtt{T}}^{*}\Pi E\simeq{\mathtt{T}}^{*}\Pi E^{*}$. In natural local
coordinates the symbol of the homological vector field and the Schouten
structure are given by

$\displaystyle\mathcal{Q}_{E}=\theta^{a}\rho_{a}^{A}(x)p_{A}+\frac{1}{2}\theta^{a}\theta^{b}C_{ba}^{c}(x)\chi_{c},\qquad
S_{E^{*}}=\chi_{a}\rho^{aA}(x)p_{A}+\frac{1}{2}\chi_{a}\chi_{b}C^{ba}_{c}(x)\theta^{c},$

which are clearly of bi-weight $(2,1)$ and $(1,2)$ respectively. We can then
_define_ a Lie bi-algebroid as the graded $QS$-manifold $({\mathtt{T}}^{*}\Pi
E,Q_{E},S_{E^{*}})$.

The above definition of a Lie bi-algebroid is not manifestly symmetric in $E$
and $E^{*}$, however, due to the isomorphism

$\displaystyle({\mathtt{T}}^{*}\Pi
E,Q_{E},S_{E^{*}})\simeq({\mathtt{T}}^{*}\Pi E^{*},S_{E},Q_{E^{*}}),$

it is clear that if $(E,E^{*})$ is a Lie bi-algebroid, then so is $(E^{*},E)$.

###  2.3 Weighted Lie algebroids

One can think of a _weighted Lie algebroid_ as a Lie algebroid in the category
of graded bundles or, equivalently, as a graded bundle in the category of Lie
algebroids [3]. Thus one should think of weighted Lie algebroids as Lie
algebroids that carry an additional compatible grading. For the purposes of
this paper, we will define weighted Lie algebroids here using the notion of
homogeneity structures as this will turn out to be a useful point of view when
dealing with the associated Lie theory. Let us recall the definition of a
graded-linear bundle, which is fundamental in the notion of a weighted Lie
algebroid.

######  Definition 2.10.

A manifold $D_{(k-1,1)}$ equipped with a pair of homogeneity structures
$(\widehat{\textnormal{h}},\widehat{\textnormal{l}})$ of degree $k-1$ and $1$
respectively is called a _graded-linear bundle_ of degree $k$, which we will
abbreviate as $\mathcal{GL}$-bundle, if and only if the respective actions
commute.

The above definition is equivalent to the definition given in [3] in terms of
mutually commuting h-complete weight vector fields of degree $k-1$ and $1$. In
all, by passing to total weight, a $\mathcal{GL}$-bundle is a graded bundle of
degree $k$. We will denote the base defined by the vector bundle structure as
$B_{k-1}\rightarrow M$. We will generally employ the shorthand notation
$D_{k}$ for a $\mathcal{GL}$-bundle $D_{(k-1,1)}$ of degree $k$ in this paper.

######  Example 2.11 ([3]).

Let $F_{k-1}$ be a graded bundle of degree $k-1$, then ${\mathtt{T}}F_{k-1}$
is canonically a $\mathcal{GL}$-bundle of degree $k$. The degree $k-1$
homogeneity structure is simply the tangent lift of the homogeneity structure
on the initial graded bundle, while the degree one homogeneity structure is
that associated with the natural vector bundle structure of the tangent
bundle. That is, if we equip $F_{k-1}$ with local coordinates
$(x^{\alpha}_{u})$, here $0\leq u<k$, then ${\mathtt{T}}F_{k-1}$ can be
naturally equipped with coordinates

$\displaystyle\big{(}\underbrace{x^{\alpha}_{u}}_{(u,0)},\,\underbrace{\delta
x_{u+1}^{\beta}}_{(u,1)}\big{)}.$

######  Example 2.12 ([3]).

Similarly to the above example, if $F_{k-1}$ be a graded bundle of degree
$k-1$, then ${\mathtt{T}}^{*}F_{k-1}$ is canonically a $\mathcal{GL}$-bundle
of degree $k$. Note there we have to employ the _phase lift_ of the
homogeneity structure on $F_{k-1}$ to ensure that we do not leave the category
of graded bundles [12]; that is we do not want negative weight coordinates.
This amounts to an appropriate shift in the weight and allows us to employ
homogeneous local coordinates of the form

$\displaystyle\big{(}\underbrace{x^{\alpha}_{u}}_{(u,0)},\,\underbrace{p_{\beta}^{k-u}}_{(k-1-u,1)}\big{)}.$

Note that as we have a linear structure on $\mathcal{GL}$-bundles, that is we
have a homogeneity structure of weight one, or equivalently an Euler vector
field, applying the parity reversion functor makes sense. Thus, we can
consider the $\mathcal{GL}$-antibundle $(\Pi
D_{k},\Pi\,\widehat{\textnormal{h}},\Pi\,\widehat{\textnormal{l}})$, where we
define viz

$\displaystyle\Pi\widehat{\textnormal{h}}_{t}\colon\ \Pi D_{k}\rightarrow\Pi
D_{k},$

and similarly for $\Pi\widehat{\textnormal{l}}$. We can now define a weighted
Lie algebroid as follows;

######  Definition 2.13.

A _weighted Lie algebroid_ of degree $k$ is the quadruple

$\displaystyle\big{(}\Pi
D_{k},\Pi\widehat{\textnormal{h}},\Pi\widehat{\textnormal{l}},Q\big{)},$

where $(\Pi D_{k},Q)$ is a $Q$-manifold and $(\Pi
D_{k},\Pi\widehat{\textnormal{h}},\Pi\widehat{\textnormal{l}})$ is a
$\mathcal{GL}$-antibundle such that

$\displaystyle
Q\circ\big{(}\Pi\widehat{\textnormal{h}}_{t}\big{)}^{*}=\big{(}\Pi\,\widehat{\textnormal{h}}_{t}\big{)}^{*}\circ
Q,$ (2.3) $\displaystyle
sQ\circ\big{(}\Pi\widehat{\textnormal{l}}\big{)}_{s})^{*}=\big{(}\Pi\widehat{\textnormal{l}}_{s}\big{)}^{*}\circ
Q,$

for all $t$ and all $s\in\mathbb{R}$.

The above definition is really just the statement that a weighted Lie
algebroid can be defined as a $\mathcal{GL}$-antibundle equipped with a
homological vector of bi-weight $(0,1)$. By passing to total weight, weighted
Lie algebroids produce examples of higher Lie algebroids, cf. Voronov [45].

######  Example 2.14.

It is not hard to see that a weighted Lie algebroid of degree $1$ is just a
standard Lie algebroid; the additional homogeneity structure is trivial and so
we have a graded super bundle of degree one and a weight one homological
vector field.

######  Example 2.15.

Similarly to the previous example, weighted Lie algebroids of degree $2$
(i.e., of bi-degree $(1,1)$) are $\mathcal{VB}$-algebroids; we now have a
double super vector bundle structure and a weight $(0,1)$ homological vector
field [16].

Further natural examples of weighted Lie algebroids include the tangent bundle
of a graded bundle and the higher order tangent bundles of a Lie algebroid,
see [3].

The above definition allows for a further more economical definition via the
following proposition, which follows immediately from (2.3).

######  Proposition 2.16.

A weighted Lie algebroid of degree $k$ can equivalently be defined as a Lie
algebroid $(\Pi E,Q)$ equipped with a homogeneity structure of degree $k-1$
such that

$\displaystyle\Pi\widehat{\textnormal{h}}_{t}\colon\ \Pi E\rightarrow\Pi E,$

is a Lie algebroid morphism for all $t\in\mathbb{R}$.

######  Remark 2.17.

Bursztyn, Cabrera and del Hoyo [5, Theorem 3.4.3] establish that
$\mathcal{VB}$-algebroids can be defined as Lie algebroids equipped with a
compatible regular homogeneity structure. Note that they make explicit use of
the Poisson structure associated with a Lie algebroid, where we prefer the
(equivalent) description in terms of $Q$-manifolds. Their motivation for
describing $\mathcal{VB}$-algebroids in terms of compatible homogeneity
structures is, much like our motivation, to tackle the associated Lie theory.

To get a more traditional description of a weighted Lie algebroid, let us
employ homogeneous local coordinates

$\displaystyle\big{(}\underbrace{x^{\alpha}_{u}}_{(u,0)},\,\underbrace{\theta^{I}_{u+1}}_{(u,1)}\big{)},$

on $\Pi D_{k}$, where $0\leq u\leq k-1$ and this label refers to the total
weight. Note that the second component of the bi-weight encodes the Grassmann
parity of the coordinates; that is the $\theta$’s are anticommuting
coordinates. In these homogeneous local coordinates the homological vector
field encoding the structure of a weighted Lie algebroid is

$\displaystyle
Q=\theta^{I}_{u-u^{\prime}+1}Q_{I}^{\alpha}[u^{\prime}](x)\frac{\partial}{\partial
x^{\alpha}_{u}}+\frac{1}{2}\theta^{I}_{u^{\prime\prime}-u^{\prime}+1}\theta^{J}_{u-u^{\prime\prime}+1}Q_{JI}^{K}[u^{\prime}](x)\frac{\partial}{\partial\theta_{u+1}^{K}},$

where $Q_{I}^{\alpha}[u^{\prime}]$ and $Q_{IJ}^{K}[u^{\prime}]$ are the
homogenous parts of the structure functions of degree $u^{\prime}$. In the
notation employed here, any $\theta$ with seemingly zero or negative total
weight are set to zero.

In [3] homogeneous linear sections of $D_{k}\rightarrow B_{k-1}$ of weight $r$
were defined functions on the $\mathcal{GL}$-bundle $D_{k}^{*}$ of bi-weight
$(r-1,1)$. Let us equip $D_{k}^{*}$ with homogeneous local coordinates

$\displaystyle\big{(}\underbrace{x^{\alpha}_{u}}_{(u,0)},\,\underbrace{\pi_{I}^{u+1}}_{(u,1)}\big{)}.$

In these local coordinates a linear section of $D_{k}$ is given by

$\displaystyle s_{r}=s_{r-u-1}^{I}(x)\pi_{I}^{u+1}.$

Such linear sections can also be understood as vertical vector fields constant
along the fibres of $\Pi D_{k}$. In the graded language we consider vector
fields that are of weight $-1$ with respect to the second component of the bi-
weight on $\Pi D_{k}$. Note that we have a shift in the bi-weight and also the
Grassmann parity. The reason we use $\Pi D_{k}$ rather than $D_{k}$ will
become clear in a moment when we consider how to construct the Lie algebroid
bracket and the anchor. By employing the homogeneous local coordinates on $\Pi
D_{k}$ introduced perviously we have the identification

$\displaystyle s_{r}\leftrightsquigarrow
i_{s_{r}}:=s^{I}_{r-u-1}(x)\frac{\partial}{\partial\theta_{k-u}^{I}},$

which is really no more than the ‘interior product’ generalised to sections of
a vector bundle. We do not require any extra structure here for this
identification, just the vector bundle structure is required. Note the shift
in the bi-weight $(r-1,1)\rightarrow(r-k,-1)$, or in other words we have a
shift of $(-(k-1),-2)$. This shift will be very important.

We can now employ the derived bracket formalism [23] to construct the Lie
algebroid bracket;

$\displaystyle i_{[s_{1},s_{2}]}=[[Q,i_{s_{1}}],i_{s_{2}}],$

up to a possible overall minus sign by convention. By inspection we see that
$i_{[s_{1},s_{2}]}$ is of bi-weight $(r_{1}+r_{2}-2k,-1)$, where $s_{1}$ is a
linear section of degree $r_{1}$ and similarly for $s_{2}$. This means that
the Lie algebroid bracket $[s_{1},s_{2}]$ is of bi-weight
$(r_{1}+r_{2}-1-k,1)$; that is the Lie algebroid bracket carries weight $-k$.

######  Definition 2.18.

A _morphism of weighted Lie algebroids_ is a morphism of the underlying Lie
algebroids (understood as a morphism of $Q$-manifolds) that intertwines the
respective additional homogeneity structures.

In particular, there is a forgetful functor from the category of weighted Lie
algebroids to Lie algebroids given by forgetting the homogeneity structure.
The image of this forgetful functor is not a full subcategory of the category
of Lie algebroids. We will denote the category of Lie algebroids as Algd and
the subcategory of weighted Lie algebroids as wAlgd.

######  Proposition 2.19.

If $(\Pi D_{k},Q)$ is weighted Lie algebroid of degree $k$, then each $\Pi
D_{j}$ for $0\leq j<k$ comes with the structure of a weighted Lie algebroid of
degree $j$. In particular, $\Pi E=\Pi D_{1}$ is a ‘genuine’ Lie algebroid.
Moreover, the projection form any ‘higher level’ to a ‘lower level’ is a
morphism of weighted Lie algebroids.

###### Proof.

Because of the condition that the weight of $Q$ is $(0,1)$ and that there are
no negatively graded coordinates, we know that $Q$ is projectable to any of
the ‘levels’ of the affine fibrations

$\displaystyle\Pi D_{(k-1,1)}\rightarrow\Pi
D_{(k-2,1)}\rightarrow\cdots\rightarrow\Pi D_{(1,1)}\rightarrow\Pi D_{(0,1)}.$
(2.4)

This can be directly checked using the local expression for the homological
vector field. Moreover, the projection of $Q$ to any level is a homological
vector field of weight $(0,1)$ and hence we have the structure of a weighted
Lie algebroid. The fact that the projections give rise to weighted Lie
algebroid morphisms follows directly. ∎

###  2.4 Lie groupoids

Here we shall set some notation and recall some well-known results. Let
$\mathcal{G}\rightrightarrows M$ be an arbitrary Lie groupoid with _source
map_ $\underline{s}\colon\mathcal{G}\rightarrow M$ and _target map_
$\underline{t}\colon\mathcal{G}\rightarrow M$. There is also the inclusion map
$\iota\colon M\rightarrow\mathcal{G}$ and a _partial multiplication_
$(g,h)\mapsto g\cdot h$ which is defined on
$\mathcal{G}^{(2)}=\\{(g,h)\in\mathcal{G}\times\mathcal{G}\colon\underline{s}(g)=\underline{t}(h)\\}$.
Moreover, the manifold $\mathcal{G}$ is foliated by $\underline{s}$-fibres
$\mathcal{G}_{x}=\\{g\in\mathcal{G}\,|\,\underline{s}(g)=x\\}$, where $x\in
M$. As by definition the source and target maps are submersions, the
$\underline{s}$-fibres are themselves smooth manifolds. Geometric objects
associated with this foliation will be given the superscript $\underline{s}$.
In particular, the distribution tangent to the leaves of the foliation will be
denoted by ${\mathtt{T}}^{\underline{s}}\mathcal{G}$. To ensure no
misunderstanding with the notion of a Lie groupoid morphism we recall the
definition we will be using.

######  Definition 2.20.

Let $\mathcal{G}_{i}\rightrightarrows M_{i}$ $(i=1,2)$ be a pair of Lie
groupoids. Then a _Lie groupoid morphisms_ is a pair of maps $(\Phi,\phi)$
such that the following diagram is commutative

$\textstyle{\mathcal{G}_{1}}$$\textstyle{\mathcal{G}_{2}}$$\textstyle{M_{1}}$$\textstyle{M_{2}}$$\textstyle{\Phi}$$\textstyle{s_{1}}$$\textstyle{t_{1}}$$\textstyle{s_{2}}$$\textstyle{t_{2}}$$\textstyle{\phi}$

in the sense that $\underline{s}_{2}\circ\Phi=\phi\circ\underline{s}_{1}$ and
$\underline{t}_{2}\circ\Phi=\phi\circ\underline{t}_{1}$, subject to the
further condition that $\Phi$ respect the (partial) multiplication; if
$g,h\in\mathcal{G}_{1}$ are composable, then $\Phi(g\cdot
h)=\Phi(g)\cdot\Phi(h)$. It then follows that for $x\in M_{1}$ we have
$\Phi(\mathbbmss{1}_{x})=\mathbbmss{1}_{\phi(x)}$ and
$\Phi(g^{-1})=\Phi(g)^{-1}$.

We will denote the category of Lie groupoids as Grpd. It is well known that
via a differentiation procedure one can construct the _Lie functor_

$\displaystyle\textnormal{{Grpd}}\stackrel{{\scriptstyle\textnormal{Lie}}}{{\xrightarrow{\hskip
30.0pt}}}\textnormal{{Algd}},$

that sends a Lie groupoid to its Lie algebroid, and sends morphisms of Lie
groupoids to morphisms of the corresponding Lie algebroids. However, as is
also well known, we do not have an equivalence of categories as not all Lie
algebroids arise as the infinitesimal versions of Lie groupoids. There is no
direct generalisation of Lie III, apart from the local case. The obstruction
to the integrability of Lie algebroids, the so called _monodromy groups_ ,
were first uncovered by Crainic and Fernandes [7]. For the transitive case the
topological obstruction was uncovered by Mackenzie [26]. The interested reader
can consult the original literature or the very accessible lecture notes also
by Crainic and Fernandes [8]. To set some notation and nomenclature, given a
Lie groupoid $\mathcal{G}$, we say that
$\textnormal{Lie}(\mathcal{G})=\textnormal{A}(\mathcal{G})$ _integrates_
$\textnormal{A}(\mathcal{G})$. Moreover, if
$\Phi\colon\mathcal{G}\rightarrow\mathcal{H}$ is a morphism of Lie groupoids,
then we will write
$\Phi^{\prime}=\textnormal{Lie}(\Phi)\colon\textnormal{A}(\mathcal{G})\rightarrow\textnormal{A}(\mathcal{H})$
for the corresponding Lie algebroid morphism, which actually comes from the
differential
${\mathtt{T}}\Phi\colon{\mathtt{T}}\mathcal{G}\rightarrow{\mathtt{T}}\mathcal{H}$
restricted to $\underline{s}$-fibres at the submanifold $M$.

Let us just recall Lie II theorem, as we will need it later on.

######  Theorem 2.21 (Lie II).

Let $\mathcal{G}\rightrightarrows M$ and $\mathcal{H}\rightrightarrows N$ be
Lie groupoids. Suppose that $\mathcal{G}$ is source simply-connected and that
$\phi\colon\textnormal{A}(\mathcal{G})\rightarrow\textnormal{A}(\mathcal{H})$
is a Lie algebroid morphism between the associated Lie algebroids. Then,
$\phi$ integrates to a unique Lie groupoid morphisms
$\Phi\colon\mathcal{G}\rightarrow\mathcal{H}$ such that $\Phi^{\prime}=\phi$.

This generalisation of Lie II to the groupoid case was first proved by
Mackenzie ans Xu [30]. A simplified proof was obtained shortly after by
Moerdijk and Mrčun [36]. Note that the assumption that the Lie groupoid
$\mathcal{G}$ must be source simply-connected is essential.

##  3 Weighted Lie groupoids

To avoid any possible confusion with the overused adjective ‘graded’,
following the nomenclature in [3] we will refer to weighted rather than graded
Lie groupoids. The term ‘graded groupoid’ appears in the work of Mehta [33] as
groupoids in the category of $\mathbb{Z}$-graded manifolds; we will comment on
this further. In this work we will content ourselves with working in the
strictly commutative (in opposition to graded-commutative) setting. The basic
idea is to take the definition of a Lie groupoid and replace ‘smooth manifold’
everywhere with ‘graded bundle’. To do this economically, we will make use of
the description of graded bundles in terms of smooth manifolds equipped with a
homogeneity structure of degree $k$, so that morphisms of graded bundles are
just smooth maps intertwining the respective homogeneity structures. In
essence, we require that the Lie groupoid structure functions respect the
graded structure which is encoded in the homogeneity structure.

###  3.1 The definition of weighted Lie groupoids via homogeneity structures

Let us formalise the comments in the beginning of this section with the
following definition;

######  Definition 3.1.

A _weighted Lie groupoid_ of degree $k$ is a Lie groupoid
$\Gamma_{k}\rightrightarrows B_{k}$, together with a homogeneity structure
$\underline{\textnormal{h}}\colon\mathbb{R}\times\Gamma_{k}\rightarrow\Gamma_{k}$
of degree $k$, such that $\underline{\textnormal{h}}_{t}$ is a Lie groupoid
morphism for all $t\in\mathbb{R}$. Such homogeneity structures will be
referred to as _multiplicative homogeneity structures_.

######  Remark 3.2.

Equivalently one could think of a weighted Lie groupoid in terms of
multiplicative weight vector fields. This follows from the definition of a
weighted Lie groupoid and the fact that weight vector fields associated with
homogeneity structures are complete. However, it will be convenient to use the
homogeneity structures rather than weight vector fields from the perspective
of Lie theory.

Let us unravel some of the structure here. First, as both the source and
target maps are submersions and by assumption $\Gamma_{k}$ is a graded bundle
of degree $k$, $B_{k}$ is also a graded bundle of degree $k$ (we will consider
lower degree graded bundles to be included as higher degree graded bundles if
necessary). Secondly, it will be convenient to consider the homogeneity
structure $\underline{\textnormal{h}}$ as a pair of structures:
$\underline{\textnormal{h}}_{t}=(\textnormal{h}_{t},\textnormal{g}_{t})$.
Then, consider the following commutative diagram

$\textstyle{\Gamma_{k}}$$\textstyle{\Gamma_{k}}$$\textstyle{B_{k}}$$\textstyle{B_{k}}$$\textstyle{\textnormal{h}_{t}}$$\textstyle{\underline{s}}$$\textstyle{\underline{t}}$$\textstyle{\underline{s}}$$\textstyle{\underline{t}}$$\textstyle{\textnormal{g}_{t}}$

which means that
$\underline{s}\circ\textnormal{h}_{t}=\textnormal{g}_{t}\circ\underline{s}$,
and
$\underline{t}\circ\textnormal{h}_{t}=\textnormal{g}_{t}\circ\underline{t}$.
That is, the source and target maps intertwine the respective homogeneity
structure. Furthermore, we have $\textnormal{h}_{t}(g\cdot
h)=\textnormal{h}_{t}(g)\cdot\textnormal{h}_{t}(h)$, meaning that the
homogeneity structure respects the (partial) multiplication, or indeed vice
versa. From the definition of a Lie groupoid morphism, compatibility with the
identities and inverses follows. In short, the graded structure is fully
compatible with the Lie groupoid structure. Or, in other words, we have a Lie
groupoid object in the category of graded bundles or equivalently vice-versa.

######  Definition 3.3.

A _morphism of weighted Lie groupoids_ is a Lie groupoid morphism that
intertwines the respective homogeneity structures.

In other words, a morphism of weighted Lie groupoids must in addition to being
a morphism of Lie groupoids be a morphism of graded bundles. Clearly we have a
forgetful functor from the category of weighted Lie groupoids to the category
of Lie groupoids which forgets the homogeneity structure. Similar to the case
of algebroids, the image of this forgetful functor is not a full subcategory
of the category of Lie groupoids. We will denote the category of weighted Lie
groupoids as wGrpd.

######  Remark 3.4.

Graded groupoids in the sense of Mehta [33, 34] are understood as groupoid
objects in the category of $\mathbb{Z}$-graded manifolds. The
$\mathbb{Z}_{2}$-grading is a consequence of the $\mathbb{Z}$-grading and so
$\mathbb{Z}$-manifolds are not quite as general as Voronov’s graded
supermanifolds (cf. [44]). Without details, a graded groupoid is a pair of
$\mathbb{Z}$-graded manifolds $(\mathcal{G},M)$, equipped with surjective
submersions $s,t\colon\mathcal{G}\rightrightarrows M$ (the source and target),
together with a partial multiplication map, an identity map and an inverse map
which all satisfy the appropriate commutative diagrams to define a groupoid
structure. To be very clear, all the structural maps are morphisms in the
category of $\mathbb{Z}$-graded manifolds. One should of course be reminded of
the definition of Lie supergroups as group objects in the category of
supermanifolds and indeed Mehta’s constructions cover the case of Lie
supergroupoids.

From the _definition_ of a weighted Lie groupoid one can check directly that
all the structural maps are morphisms in the category of graded bundles; all
the maps relate the respective weight vector fields (or equivalently the
actions of the respective homogeneity structures). Thus, one could start with
a definition of a weighted Lie groupoid in terms closer to that used by Mehta
to define his graded groupoids. However, in light of Lie theory it will turn
out to be very convenient to describe weighted Lie groupoids as Lie groupoids
with a compatible homogeneity structure. Mehta by working in the category of
$\mathbb{Z}$-graded manifolds did not have anything like a homogeneity
structure at his disposal. While it is straight forward to show that
differentiating graded groupoids leads to graded algebroids, the question of
integration seems less tractable.

###  3.2 Relation with $\boldsymbol{\mathcal{VB}}$-groupoids

Bursztyn, Cabrera and del Hoyo (cf. [5, Theorem 3.2.3]) establish a one-to-one
correspondence between $\mathcal{VB}$-groupoids, which have several equivalent
definitions (see for example [17]), and Lie groupoids that have a regular
action of $(\mathbb{R},\cdot)$ as Lie groupoid morphisms. In essence, they
show the equivalence between $\mathcal{VB}$-groupoids and weighted Lie
groupoids of degree one. We take the attitude that the ‘correct definition’ of
a $\mathcal{VB}$-groupoid is implied by this correspondence. We consider the
double structure, that is the pair of Lie groupoids and the pair of vector
bundles together with a collection of not-so obvious compatibility conditions,
as being derived rather than fundamental. This helps motivate our definition
of weighted Lie groupoids using homogeneity structures.

######  Proposition 3.5.

For any weighted Lie groupoid $\Gamma_{k}\rightrightarrows B_{k}$ there is an
underlying ‘genuine’ or ‘ungraded’ Lie groupoid $\mathcal{G}\rightrightarrows
M$ described by the following surjective groupoid morphism

$\textstyle{\Gamma_{k}}$$\textstyle{\mathcal{G}}$$\textstyle{B_{k}}$$\textstyle{B_{0}}$$\textstyle{\textnormal{h}_{0}}$$\textstyle{\underline{s}}$$\textstyle{\underline{t}}$$\textstyle{\underline{\sigma}}$$\textstyle{\underline{\tau}}$$\textstyle{\textnormal{g}_{0}}$

Moreover, if $\Gamma_{k}$ is source simply-connected, then so is
$\mathcal{G}$.

###### Proof.

By construction we have projections
$\textnormal{h}_{0}\colon\Gamma_{k}\rightarrow\Gamma_{0}:=\mathcal{G}$ and
$\textnormal{g}_{0}\colon B_{k}\rightarrow B_{0}:=M$. Thus
$\mathcal{G}\subset\Gamma_{k}$ and $M\subset\mathcal{G}$ are embedded
manifolds. Clearly $\mathcal{G}$ and $M$ define a (set theoretical)
subgroupoid of $\Gamma_{k}\rightrightarrows B_{k}$. As
$(\textnormal{h}_{t},\textnormal{g}_{t})$ is a morphism of Lie groupoids for
all $t\in\mathbb{R}$, including zero, it follows that the source map
$\underline{\sigma}\colon\mathcal{G}\rightarrow M$ is a submersion. Thus we
have the structure of a Lie groupoid. Now consider any point
$p\in\mathcal{G}$. As $\textnormal{h}_{0}(p)=p$ the we have an induced
retraction map $\underline{s}^{-1}(p)\rightarrow\underline{\sigma}^{-1}(p)$
and thus we have a homomorphisms between the respective fundamental groups.
Thus if $\Gamma_{k}$ is source simply-connected, then so is $\mathcal{G}$. ∎

Actually, slightly modifying the above proof, we can derive the groupoid
version of the tower of Lie algebroid structures (2.4).

######  Proposition 3.6.

If $\Gamma_{k}\rightrightarrows B_{k}$ is a weighted Lie groupoid of degree
$k$, then we have the following tower of weighted groupoid structures of lower
order and their morphisms:

$\textstyle{\Gamma_{k}}$$\textstyle{\Gamma_{k-1}}$$\textstyle{\cdots}$$\textstyle{\Gamma_{1}}$$\textstyle{\mathcal{G}}$$\textstyle{B_{k}}$$\textstyle{B_{k-1}}$$\textstyle{\cdots}$$\textstyle{B_{1}}$$\textstyle{M}$$\textstyle{\tau^{k}}$$\textstyle{\tau^{k-1}}$$\textstyle{\tau^{2}}$$\textstyle{\tau^{1}}$$\textstyle{\underline{s}_{k}}$$\textstyle{\underline{t}_{k}}$$\textstyle{\underline{s}_{k-1}}$$\textstyle{\underline{t}_{k-1}}$$\textstyle{\underline{s}_{1}}$$\textstyle{\underline{t}_{1}}$$\textstyle{\underline{\sigma}}$$\textstyle{\underline{\tau}}$$\textstyle{\pi^{k}}$$\textstyle{\pi^{k-1}}$$\textstyle{\pi^{2}}$$\textstyle{\pi^{1}}$

In particular, $\Gamma_{1}\rightrightarrows B_{1}$ is a
$\mathcal{VB}$-groupoid.

###  3.3 Examples of weighted Lie groupoids

In this subsection we sketch some simple examples of weighted Lie groupoids
following adaptation of some of the standard examples of Lie groupoids and
graded bundles. We expect more intricate examples to follow, some of which
maybe more useful in applications.

######  Example 3.7.

All graded bundles can be considered as weighted Lie groupoids over themselves
by taking all the structure maps to be the identity. This leads to the notion
of a _weighted unit Lie groupoid_.

######  Example 3.8.

Just as Lie groups are examples of Lie groupoids (over a single point), we can
consider _weighted Lie groups_ of degree $k$ which are Lie groups $G_{k}$
equipped with a compatible homogeneity structure of degree $k$. The
compatibility condition is simply $\textnormal{h}_{t}(g\cdot
h)=\textnormal{h}_{t}(g)\cdot\textnormal{h}_{t}(h)$ for all elements $g,h\in
G_{k}$.

######  Example 3.9.

If $F_{k}\rightarrow M$ is a graded bundle of degree $k$, then we can
construct the _weighted pair groupoid_ in the obvious way; $F_{k}\times
F_{k}\rightrightarrows F_{k}$. Underlying this is the ‘genuine’ pair groupoid
$M\times M\rightrightarrows M$. Moreover, as we have a series of affine
fibrations associated with any graded bundle the weighted pair groupoid can be
‘fibred’ as $F_{k}\times_{F_{q}}F_{k}\rightrightarrows F_{k}$ in the obvious
way.

######  Example 3.10.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Then associated with
this is the _higher tangent Lie groupoid_
${\mathtt{T}}^{k}\mathcal{G}\rightrightarrows{\mathtt{T}}^{k}M$, which is
naturally a weighted Lie groupoid and is formed by applying the higher tangent
functor of order $k$ to the structure maps of $\mathcal{G}\rightrightarrows
M$. Clearly the initial Lie groupoid is the ‘genuine’ Lie groupoid underlying
the higher tangent Lie groupoid.

######  Example 3.11.

Similar to the previous example, one can consider iterating the tangent
functor $k$-times to construct the _iterated tangent Lie groupoid_
${\mathtt{T}}^{(k)}\mathcal{G}\rightrightarrows{\mathtt{T}}^{(k)}M$. By
passing to total weight we have a weighted Lie groupoid of degree $k$.

######  Example 3.12.

As already commented on, $\mathcal{VB}$-groupoids are weighted Lie groupoids
of degree $1$. Standard examples of $\mathcal{VB}$-groupoids include
${\mathtt{T}}\mathcal{G}\rightrightarrows{\mathtt{T}}M$ and
${\mathtt{T}}^{*}\mathcal{G}\rightrightarrows\textnormal{A}^{*}(\mathcal{G})$
for any Lie groupoid $\mathcal{G}\rightrightarrows M$.

######  Example 3.13.

Combining Examples 3.10 and 3.12, we can consider the _higher Pontryagin Lie
groupoid_

$\displaystyle{\mathtt{T}}^{k}\mathcal{G}\times_{\mathcal{G}}{\mathtt{T}}^{*}\mathcal{G}\rightrightarrows{\mathtt{T}}^{k}M\times_{M}\textnormal{A}^{*}(\mathcal{G}),$

as a weighted Lie groupoid by passing from the natural bi-weight to total
weight.

######  Example 3.14.

Let $(G_{l},\textnormal{h})$ be a weighted Lie group and
$(F_{k},\textnormal{g})$ be a graded bundle. We assume that $G_{l}$ acts on
$F_{k}$. Then we can construct the _weighted action Lie groupoid_ as follows;
$G_{l}\times F_{k}\rightrightarrows F_{k}$, where the source and target maps
are given by

$\displaystyle\underline{s}(g,x)=x,\qquad\underline{t}(g,x)=gx,$

and the (partial) multiplication being

$\displaystyle(h,y)\cdot(g,x)=(hg,x).$

We define $\underline{\textnormal{h}}:=(\textnormal{h},\textnormal{g})$ as the
homogeneity structure on the Lie groupoid. Compatibility with the source map
and partial multiplication is automatic from the definition of a homogeneity
structure, but compatibility with the target map requires the natural
condition $\textnormal{g}_{t}(gx)=(\textnormal{h}_{t}g)(\textnormal{g}_{t}x)$.
Thus we will define a weighted action Lie groupoid via the construction of an
action Lie groupoid, with the addition of compatible homogeneity structures as
described above.

######  Example 3.15.

Any $\mathcal{GL}$-bundle
$(D_{k},\widehat{\textnormal{h}},\widehat{\textnormal{l}})$ can be considered
as a weighed Lie groupoid in the following way. The base is defined as
$B_{k-1}:=\widehat{\textnormal{l}}_{0}(D_{k})$ and the source and target maps
are set to the projection
$\underline{s}=\underline{t}=\widehat{\textnormal{l}}_{0}$. Thus, the
$\underline{s}$-fibres have the structure of graded spaces, cf. [15] (i.e.,
graded bundles over a point). The partial multiplication is just the addition
in the $\underline{s}$-fibres. The compatibility of the Lie groupoid and
graded bundle structures follows directly from the definition of a
$\mathcal{GL}$-bundle.

######  Example 3.16.

The fibred product

$\displaystyle{\mathtt{T}}\mathcal{G}\times_{\mathcal{G}}{\mathtt{T}}\mathcal{G}\times_{\mathcal{G}}\cdots\times_{\mathcal{G}}{\mathtt{T}}\mathcal{G}\rightrightarrows{\mathtt{T}}M\times_{M}{\mathtt{T}}M\times_{M}\cdots\times_{M}{\mathtt{T}}M,$

where $\mathcal{G}\rightrightarrows M$ is an arbitrary Lie groupoid, can be
considered as a weighted Lie groupoid in the following way. The natural
homogeneity structure associated with the $i$-th factor
${\mathtt{T}}\mathcal{G}\rightrightarrows{\mathtt{T}}M$ can be shifted by
$i\in\mathbb{N}$ and then, by passing to total weight, we have a weighted Lie
groupoid of degree $k$, assuming we the fibred product consists of $k$
factors.

##  4 Lie theory for weighted groupoids and algebroids

We now turn our attention to the Lie theory relating weighted Lie groupoids
and weighted Lie algebroids. The differentiation of a weighted Lie groupoid
produces rather naturally a weighted Lie algebroid. One can use the standard
construction here, just taking a little care with the weights. The integration
of a weighted Lie algebroid to a weighted Lie groupoid also follows via the
standard constructions, again taking care with the graded structure. We will
not address the full question of global integrability, rather we will assume
integrability as a Lie algebroid and show that the weighted structure is
naturally inherited.

###  4.1 Differentiation of weighted Lie groupoids

It is clear that as a Lie groupoid a weighted Lie groupoid always admits a Lie
algebroid associated with it following the classical constructions. The
question is _if the associated Lie algebroid is in fact a weighted Lie
algebroid_?

######  Theorem 4.1.

If $\Gamma_{k}\rightrightarrows B_{k}$ is a weighted Lie groupoid of degree
$k$ with respect to a homogeneity structure $h$ on $\Gamma_{k}$, then
$\textnormal{A}(\Gamma_{k})\rightarrow B_{k}$ is a weighted Lie algebroid of
degree $k+1$ with respect to the homogeneity structure $\widehat{h}$ defined
by

$\displaystyle\widehat{h}_{t}=(h_{t})^{\prime}=\textnormal{Lie}(h_{t}).$ (4.1)

###### Proof.

We need only follow the classical constructions taking care of the weight as
we proceed. In particular, we need only show that $\textnormal{A}(\Gamma_{k})$
has the structure of a $\mathcal{GL}$-bundle and that the corresponding Lie
algebroid brackets are of weight $-(k+1)$.

1\. As
$\textnormal{A}(\Gamma_{k}):=\ker({\mathtt{T}}\underline{s})|_{\iota(B_{k})}$,
it is clear that as a subbundle of ${\mathtt{T}}\Gamma_{k}$ that we have the
structure of a $\mathcal{GL}$-bundle. The degree $k$ homogeneity structure
$\widehat{h}$ is inherited from the tangent lift of the homogeneity structure
on $\Gamma_{k}$ and the degree $1$ homogeneity structure is inherited from the
natural one related to the vector bundle structure of tangent bundles. In
other words, $\widehat{h}_{t}=(h_{t})^{\prime}$ is the ‘differential’ of $h$.

2\. Sections of weight $r$ are by our definition functions on
$\textnormal{A}(\Gamma_{k})^{*}$ of bi-weight $(r-1,1)$, see Section 2. Thus,
associating sections with right invariant vector fields requires a shift in
the weights of $(-k,-2)$, which comes from associating momenta with
derivatives, i.e., use weighted principle symbols. Thus, the prolongation of a
section of weight $r$ to a left invariant vector field is a vector field of
bi-weight $(r-1-k,-1)$. Thus if we take the Lie bracket of two sections of
weight $r_{1}$ and $r_{2}$ as right invariant vector fields the resulting will
be a right invariant vector field of weight $(r_{1}+r_{2}-2-2k,-1)$. Now upon
restriction and ‘shifting back’ to functions on
$\textnormal{A}(\Gamma_{k})^{*}$, the resulting section is of bi-weight
$(r_{1}+r_{2}-1-(k+1),1)$. Thus the Lie algebroid bracket carries weight
$-(k+1)$. ∎

Theorem 4.1 tells us that weighted groupoids are the objects that integrate
weighted algebroids, but of course it does not tell us that we can actually
(globally) integrate weighted Lie algebroids. As a ‘corollary’ we see that
$\mathcal{VB}$-groupoids are the objects that integrate
$\mathcal{VB}$-algebroids.

######  Example 4.2.

The weighted Lie algebroid associated with the weighted pair Lie groupoid
$F_{k}\times F_{k}\rightrightarrows F_{k}$ is the tangent bundle
${\mathtt{T}}F_{k}$.

######  Example 4.3.

The $k$-th order tangent bundle of a Lie groupoid
${\mathtt{T}}^{k}\mathcal{G}$ naturally comes with the structure of a weighted
Lie groupoid over ${\mathtt{T}}^{k}M$. The associated Lie algebroid is
$\textnormal{A}({\mathtt{T}}^{k}\mathcal{G})\simeq{\mathtt{T}}^{k}\textnormal{A}(\mathcal{G})$,
see for example [24] for details. From [3] we know that that
${\mathtt{T}}^{k}\textnormal{A}(\mathcal{G})$ is a weighted Lie algebroid.

######  Proposition 4.4.

Via the appropriate forgetful functors, the Lie functor restricts to the
subcategories of weighted Lie groupoids and weighted Lie algebroids, i.e.,

$\displaystyle\textnormal{{wGrpd}}\stackrel{{\scriptstyle\textnormal{Lie}}}{{\xrightarrow{\hskip
30.0pt}}}\textnormal{{wAlgd}}.$

###### Proof.

This follows from Theorem 4.1 and the fact that both weighted Lie groupoid and
weighted Lie algebroid morphisms intertwine the respective actions of the
homogeneity structure. ∎

###  4.2 Integration of weighted Lie algebroids

We will not address the full problem of integration of Lie algebroids here,
one should consult Crainic and Fernandes [7] for details of the obstruction to
(global) integrability. For a details of the obstruction to the integration of
$\mathcal{VB}$-algebroids see [1]. We remark though, that passing to the world
of differentiable stacks and the notion of a Weinstein groupoid all Lie
algebroids are integrable [42], though in this work we will strictly stay in
the world of Lie groupoids. The question we address is not one of the
integrability of weighted Lie algebroids as Lie algebroids, but rather if they
integrate to weighted Lie groupoids. That is, _does the homogeneity structure
on the weighted Lie algebroid integrate to a homogeneity structure on the
associated source simply-connected Lie groupoid such that we have the
structure of a weighted Lie groupoid?_

######  Theorem 4.5.

Let $D_{k+1}\rightarrow B_{k}$ be a weighted Lie algebroid of degree $k+1$
with respect to a homogeneity structure $\widehat{\rm h}$ and $\Gamma_{k}$ its
source simply-connected integration groupoid. Then $\Gamma_{k}$ is a weighted
Lie groupoid of degree $k$ with respect to the homogeneity structure ${\rm h}$
uniquely determined by (4.1).

###### Proof.

A weighted Lie algebroid is a Lie algebroid together with a homogeneity
structure $\widehat{\textnormal{h}}_{t}\colon D_{k+1}\rightarrow D_{k+1}$ such
that for any $t\neq 0$ the homogeneity structure acts as a Lie algebroid
automorphism, and for $t=0$ the homogeneity structure is a Lie algebroid
morphism to the underlying non-graded Lie algebroid. Thus we can use Lie II to
uniquely integrate $\widehat{\textnormal{h}}_{t}$ (for a given $t$) to a Lie
groupoid morphism

$\displaystyle\textnormal{h}_{t}\colon\ \Gamma\rightarrow\Gamma,$

such that ($\textnormal{h}_{t})^{\prime}=\widehat{\textnormal{h}}_{t}$. The
corresponding action $\textnormal{h}\colon{\mathbb{R}}\times\Gamma\to\Gamma$
is smooth. Indeed, the smooth action $\widehat{\textnormal{h}}$ on the Lie
algebroid $D_{k+1}$ induces clearly a smooth action on the Banach manifold of
$C^{1}$-admissible paths in $D_{k+1}$ whose projection on $\Gamma$ gives h.
The uniqueness of the integration of Lie algebroid morphisms together with the
functorial properties of differentiating Lie groupoids, namely
$(\textnormal{h}_{t}\circ\textnormal{h}_{s})^{\prime}=\textnormal{h}^{\prime}_{t}\circ\textnormal{h}^{\prime}_{s}$,
implies that $\textnormal{h}\colon\mathbb{R}\times\Gamma\rightarrow\Gamma$ is
a homogeneity structure. Thus $\Gamma\rightrightarrows B$ is a weighted Lie
groupoid.

The only remaining question is that of the degree of the homogeneity structure
h. This in fact follows directly from the Lie functor. The degree of the
homogeneity structure is a local question and so we can examine everything in
local coordinates. In particular, on $\Gamma\rightrightarrows B$ we can employ
local coordinates adapted to the source fibration

$\displaystyle\big{(}x^{\alpha}_{u},Y_{u}^{I}\big{)},$

for $0\leq u\leq k^{\prime}$, for some $k^{\prime}$ to be determined. As the
homogeneity structure commutes with the source, for simplicity we can assume
that the weight of both coordinates run over the same range. Following the
construction of the associated Lie algebroid, we equip ${\mathtt{T}}\Gamma$
with homogeneous coordinates

$\displaystyle\big{(}\underbrace{x^{\alpha}_{u}}_{(u,0)},\underbrace{Y_{u}^{I}}_{(u,0)},\underbrace{\dot{x}^{\beta}_{u+1}}_{(u,1)},\underbrace{\dot{Y}_{u+1}^{J}}_{(u,1)}\big{)},$

where we have applied the tangent lift of the homogeneity structure to define
the weight of the coordinates on the tangent bundle. The second component is
simply the natural weight induced by the vector bundle structure. Locally we
obtain $D_{k+1}\rightarrow B_{k}$ by setting $\dot{x}=0$ and $Y=0$.

Note that as $D_{k+1}$ is a graded bundle of degree $k$ (forget the linear
structure) and that $(x,\dot{Y})$ give homogeneous coordinates on the weighted
Lie algebroid $D_{k+1}\subset{\mathtt{T}}\Gamma$. The tangent bundle of a
graded bundle of degree $k^{\prime}$ is also a graded bundle of degree
$k^{\prime}$ (again forgetting the linear structure). Meaning that the weight
of the coordinates $\dot{Y}$ on the algebroid are directly inherited from the
coordinates $Y$ on the groupoid. Thus we conclude that $k^{\prime}=k$. Upon
setting $\Gamma=\Gamma_{k}$ we have established the theorem. ∎

######  Remark 4.6.

The results of Bursztyn et al. [5] on integration of $\mathcal{VB}$-algebroids
rely on some technical issues regarding regular homogeneity structures. In
particular it is some work to show that regularity is preserved under
integration. By considering the larger category of graded bundles and not just
vector bundles we bypass this technicality. The fact that integration of a
weighted Lie algebroid produces a weighted Lie groupoid is quite clear; the
degree of the integrated homogeneity structure then follows from the graded
bundle structure of tangent bundle of the integration Lie groupoid rather than
any difficult technical arguments. The philosophy is that graded bundles give
us a ‘little more room to manoeuvre’ as compared with just vector bundles.

Because
$\textnormal{A}(\textnormal{h}_{0}(\Gamma_{k}))=\widehat{\textnormal{h}}_{0}(D_{k+1})$,
it is clear that the integrability of $D_{k+1}\rightarrow B_{k}$ as a Lie
algebroid implies that $A\rightarrow M$, where $A=\widehat{\rm
h}_{0}(D_{k+1})$ is also integrable as a Lie algebroid. The converse is not
true and examples of $\mathcal{VB}$-algebroids with integrable base Lie
algebroids for which the total Lie algebroid is not integrable exist, see [1,
5].

##  5 Weighted Poisson–Lie groupoids and their Lie theory

Following our general ethos that we can construct weighted versions of our
favorite geometric structures by simply including a compatible homogeneity
structure, we now turn to the notion of weighted Poisson–Lie groups and
weighted Lie bi-algebroids. As the propositions in this section follow by mild
adaptation of standard proofs by including weights into considerations, we
will not generally present details of the proofs.

###  5.1 Weighted Poisson–Lie groupoids

######  Definition 5.1.

A _weighted Poisson–Lie groupoid_ $(\Gamma_{k},\Lambda)$ of degree $k$ is a
Poisson–Lie groupoid equipped with a multiplicative homogeneity structure
$\underline{\textnormal{h}}\colon\mathbb{R}\times\Gamma_{k}\rightarrow\Gamma_{k}$
of degree $k$, such that the Poisson structure is of weight ${-}k$.

Recall that a Poisson–Lie groupoid is a Lie groupoid equipped with a
multiplicative Poisson structure; this means that

$\displaystyle\textnormal{graph}(\textnormal{mult})=\\{(g,h,g\circ
h)\,|\,\underline{s}(g)=\underline{t}(h)\\}$

is coisotropic inside
$(\Gamma_{k}\times\Gamma_{k},\times\Gamma_{k},\Lambda\oplus\Lambda\oplus{-}\Lambda)$.
The associated Poisson bracket on $C^{\infty}(\Gamma_{k})$ is of weight
${-}k$. Weighted Poisson–Lie groupoids should be thought of as the natural
generalisation on $\mathcal{PVB}$-groupoids as first defined by Mackenzie
[27]. A $\mathcal{PVB}$-groupoid is a $\mathcal{VB}$-groupoid equipped with a
compatible Poisson structure.

An alternative, yet equivalent and illuminating way to view weighted
Poisson–Lie groupoids is as follows. Let $(G,\Lambda)$ be a Lie groupoid
equipped with a Poisson structure, we will temporarily forget the graded
structure and place no conditions on the Poisson structure. Associated with
the Poisson structure is the fibre-wise map

$\displaystyle\Lambda^{\\#}\colon\ {\mathtt{T}}^{*}G\rightarrow{\mathtt{T}}G.$

This fibre-wise map is of course completely independent of the groupoid
structure on $G$. Note, however, that if $G$ is a Lie groupoid, then both
${\mathtt{T}}^{*}G\rightrightarrows\textnormal{A}^{*}(G)$ and
${\mathtt{T}}G\rightrightarrows{\mathtt{T}}M$ are naturally Lie groupoids
[28]. Then one can then define a Poisson–Lie groupoid as a Lie groupoid
equipped with a Poisson structure such that the induced map $\Lambda^{\\#}$ is
a morphism of Lie groupoids.

The only extra requirement for the case of a weighted Poisson–Lie groupoid is
that $\Lambda^{\\#}\colon{\mathtt{T}}^{*}\Gamma_{k}$
$\rightarrow{\mathtt{T}}\Gamma_{k}$ must be a morphisms of double graded
bundles, and this forces the Poisson structure to be of weight ${-}k$.
Furthermore note that ${\mathtt{T}}^{*}\Gamma_{k}$ and
${\mathtt{T}}\Gamma_{k}$ are both $\mathcal{GL}$-bundles, cf. Section 2.

######  Example 5.2.

Rather trivially, any weighted Lie groupoid of degree $k$ is weighted
Poisson–Lie groupoid of degree $k$ when equipped with the zero Poisson
structure.

######  Example 5.3.

Any Poisson–Lie groupoid can be considered as a weighted Poisson–Lie groupoid
of degree $0$ by equipping it with the trivial homogeneity structure. Lie
groups equipped with multiplicative Poisson structures are specific examples.

######  Example 5.4.

Consider a graded bundle of degree $k$ equipped with a Poisson structure of
weight $-k$, $(F_{k},\Lambda)$. The pair groupoid $F_{k}\times
F_{k}\rightrightarrows F_{k}$ is a weighted Poisson–Lie groupoid of degree $k$
if we equip $F_{k}\times F_{k}$ with the ‘difference’ Poisson structure.

######  Example 5.5.

Consider a Poisson–Lie groupoid $(\mathcal{G}\rightrightarrows M,\pi)$, which
we can consider as a trivial weighted Poisson–Lie groupoid. Recall that
${\mathtt{T}}^{k}\mathcal{G}\rightrightarrows{\mathtt{T}}^{k}M$ is a weighted
Lie groupoid of degree $k$. Furthermore, in [24] it was shown that the higher
tangent lift of a multiplicative Poisson structure is a multiplicative Poisson
structure itself. By inspection we see that Poisson structure lifted to the
$k$-th order tangent bundle $\Lambda:=\textnormal{d}_{{\mathtt{T}}}^{k}\pi$ is
of weight $-k$, with respect to the natural homogeneity structure on the
higher tangent bundle. Thus
$({\mathtt{T}}^{k}\mathcal{G}\rightrightarrows{\mathtt{T}}^{k}M,\textnormal{d}_{{\mathtt{T}}}^{k}\pi)$
is a weighted Poisson–Lie groupoid of degree $k$. This example we consider as
the archetypal weighted Poisson–Lie groupoid.

###  5.2 Weighted Lie bi-algebroids

Follow Roytenberg [38], Voronov [44] and Kosmann-Schwarzbach [22] we are lead
to the following definition.

######  Definition 5.6.

A pair of weighted Lie algebroids $(D_{k},D_{k}^{*})$ both of degree $k$ is a
_weighted Lie bi-algebroid_ if and only if their exists odd Hamiltonians
$\mathcal{Q}_{D_{k}}$ and $S_{D_{k}^{*}}$ on ${\mathtt{T}}^{*}(\Pi D_{k})$ of
weight $(k-1,2,1)$ and $(k-1,1,2)$, respectively, that Poisson commute.

Let us employ homogeneous local coordinates

$\displaystyle\big{(}\underbrace{x_{u}^{\alpha}}_{(u,0,0)},\underbrace{\theta_{u+1}^{I}}_{(u,1,0)},\underbrace{p_{\beta}^{u+2}}_{(u,1,1)},\underbrace{\chi_{J}^{u+1}}_{(u,0,1)}\big{)},$

on ${\mathtt{T}}^{*}(\Pi D_{k})$. In these homogeneous coordinates we have

$\displaystyle\mathcal{Q}=\theta_{u-u^{\prime}+1}^{I}Q_{I}^{\alpha}[u^{\prime}](x)p_{\alpha}^{k+1-u}+\frac{1}{2}\theta_{u^{\prime\prime}-u^{\prime}+1}^{I}\theta_{u-u^{\prime\prime}+1}^{J}Q_{JI}^{K}[u^{\prime}](x)\chi_{K}^{k-u},$
$\displaystyle
S=\chi_{I}^{u-u^{\prime}+1}Q^{I\alpha}[u^{\prime}](x)p_{\alpha}^{k+1-u}+\frac{1}{2}\chi_{I}^{u^{\prime\prime}-u^{\prime}+1}\chi_{J}^{u-u^{\prime\prime}+1}Q_{K}^{JI}[u^{\prime}](x)\theta_{k-u}^{K}.$

Since we have a Schouten structure here of tri-weight $(k-1,1,2)$, we have a
graded fibre-wise map

$\displaystyle S^{\\#}\colon\ {\mathtt{T}}^{*}(\Pi
D_{k})\longrightarrow\Pi{\mathtt{T}}(\Pi D_{k}),$

given in local coordinates by

$\displaystyle(S^{\\#})^{*}\textnormal{d}x^{\alpha}_{u+1}=\frac{\partial
S}{\partial
p_{\alpha}^{k+1-u}},\qquad(S^{\\#})^{*}\textnormal{d}\theta_{u+2}^{I}=\frac{\partial
S}{\partial\chi_{\alpha}^{k-u}},$

where we have employed homogeneous local coordinates

$\displaystyle\big{(}\underbrace{x_{u}^{\alpha}}_{(u,0,0)},\underbrace{\theta_{u+1}^{I}}_{(u,1,0)},\underbrace{\textnormal{d}x^{\beta}_{u+1}}_{(u,0,1)},\underbrace{\textnormal{d}\theta^{J}_{u+2}}_{(u,1,1)}\big{)},$

on $\Pi{\mathtt{T}}(\Pi D_{k})$.

######  Proposition 5.7.

Let $(D_{k},D_{k}^{*})$ be a weighted Lie bi-algebroid of degree $k$. The
fibre-wise map $S^{\\#}$ is a graded morphisms of $Q$-manifolds

$\displaystyle\big{(}{\mathtt{T}}^{*}(\Pi
D_{k}),Q_{0}:=\\{\mathcal{Q},\bullet\\}\big{)}\longrightarrow\big{(}\Pi{\mathtt{T}}(\Pi
D_{k}),\mathcal{L}_{Q}=[\textnormal{d},i_{Q}]\big{)}.$

###### Proof.

It follows via direct computation and so we omit details. ∎

######  Remark 5.8.

The above proposition holds for general $QS$-manifolds and the additional
gradings play no critical role. In fact, we do not need a Schouten structure,
but just an odd quadratic Hamiltonian such that
$\mathcal{L}_{Q}S:=\\{\mathcal{Q},S\\}=0$.

######  Example 5.9.

If $(\textnormal{A},\textnormal{A}^{*})$ is a Lie bi-algebroid, then
$({\mathtt{T}}^{k}\textnormal{A},{\mathtt{T}}^{k}\textnormal{A}^{*})$ is a Lie
bi-algebroid of degree $k+1$. See for details [24], although the authors do
not make mention of the graded structure.

######  Example 5.10.

Consider a weighted Lie algebroid $D_{k}$ of degree $k$, together with an even
function $\mathcal{P}\in C^{\infty}(\Pi D_{k}^{*})$ of bi-weight $(k-1,2)$
such that $[[\mathcal{P},\mathcal{P}]]=0$, where the bracket is the Schouten
bracket encoding the Lie algebroid structure. Such functions we consider as
Poisson structures on weighted Lie algebroids. Then, following well-known
classical results, $(D_{k},D_{k}^{*})$ is a weighted Lie bi-algebroid of
degree $k$. This example is the weighted version of a triangular Lie bi-
algebroid. As a canonical example,
$({\mathtt{T}}({\mathtt{T}}^{k-1}M),{\mathtt{T}}^{*}({\mathtt{T}}^{k-1}M))$ is
a weighted Lie bi-algebroid of degree $k$. The homological vector field is
supplied by the de Rham differential on $\Pi{\mathtt{T}}({\mathtt{T}}^{k-1}M)$
and the Schouten structure is that associated with the canonical Schouten
bracket on $\Pi{\mathtt{T}}^{*}({\mathtt{T}}^{k-1}M)$.

###  5.3 The Lie theory of weighted Poisson–Lie groupoids
and weighted Lie bi-algebroids

It is well known that the infinitesimal object associated with a Poisson–Lie
groupoid is a Lie bi-algebroid. Let us quickly sketch the association of a Lie
bi-algebroid with a Poisson–Lie groupoid. The construction of the underlying
Lie algebroid is as standard, we only need to show how the appropriate
Schouten structure is generated and that it carries the correct weight. With
this in mind, let us outline the classical construction.

It can be shown that there are two natural isomorphisms

$\displaystyle\Pi\vartheta\colon\
\Pi\textnormal{A}({\mathtt{T}}^{*}G)\longrightarrow{\mathtt{T}}^{*}(\Pi\textnormal{A}(G)),\qquad\Pi
j\colon\
\Pi{\mathtt{T}}(\Pi\textnormal{A}(G))\longrightarrow\Pi\textnormal{A}({\mathtt{T}}G),$

which follow from application of the parity reversion functor to the standard
non-super isomorphisms (see [28, 30] and [24], where the higher tangent
versions are also discussed). Let $(G,\Lambda)$ be a Poisson–Lie groupoid,
then as we have
$\Lambda^{\\#}\colon{\mathtt{T}}^{*}G\rightarrow{\mathtt{T}}G$, we also have
the ‘superised’ version:

$\displaystyle\Pi(\Lambda^{\\#})^{\prime}\colon\
\Pi\textnormal{A}({\mathtt{T}}^{*}G)\rightarrow\Pi\textnormal{A}({\mathtt{T}}G).$

Furthermore, the above gives rise to a Schouten structure on A viz

$\displaystyle\Pi(\Lambda^{\\#})^{\prime}:=\Pi j\circ
S^{\\#}\circ\Pi\vartheta,$

where
$S^{\\#}\colon{\mathtt{T}}^{*}(\Pi\textnormal{A}(G))\longrightarrow\Pi{\mathtt{T}}(\Pi\textnormal{A}(G))$.
One now only has to check the weight when dealing with the weighted versions.

In the other direction, it is also well known an integration procedure for
passing from Lie bi-algebroids to Poisson–Lie groupoids. Using Lie’s second
theorem one can show that if $(A,A^{*})$ is a Lie bi-algebroid and
$G\rightrightarrows M$ is the source simply-connected Lie groupoid integrating
$A\rightarrow M$, then there exists a unique Poisson structure on $G$ such
that $(G,\Lambda)$ is a Poisson–Lie groupoid integrating $(A,A^{*})$. Again,
it is straightforward to check the weights when dealing with the weighted
versions. We are lead to the following proposition.

######  Proposition 5.11.

There is a canonical correspondence between weighted Poisson–Lie groupoids of
degree $k$ and $($integrable$)$ weighted Lie bi-algebroids of degree $k+1$.

###  5.4 The Courant algebroid associated with weighted Lie bi-algebroids

Following our general ethos, a weighted Courant algebroid is a Courant
algebroid (see [6, 25]) with an additional compatible homogeneity structure.

######  Definition 5.12.

A _weighted Courant algebroid of degree $k$_ consists of the following data:

  1. 1.

A double graded supermanifold $({\mathcal{M}},\textnormal{h},\textnormal{l})$
of bi-degree $(k-1,2)$ such that $({\mathcal{M}},\textnormal{l})$ is an
$N$-manifold (the Grassmann parity of homogeneous coordinates is determined by
l).

  2. 2.

An even symplectic form $\omega$ on $\mathcal{M}$ of bi-degree $(k-1,2)$.

  3. 3.

An odd function $\Theta\in C^{\infty}(\mathcal{M})$ of bi-degree $(k-1,3)$,
such that $\\{\Theta,\Theta\\}_{\omega}=0$, where the bracket is the Poisson
bracket induced by $\omega$.

The corresponding derived bracket

$\displaystyle[[\sigma^{1},\sigma^{2}]]_{\Theta}:=\big{\\{}\big{\\{}\sigma^{1},\Theta\big{\\}}_{\omega},\sigma^{2}\big{\\}}_{\omega}$

is closed on functions of bi-degree $(k-1,1)$ (so functions of total degree
$k$) and defines the higher analog of the Courant–Dorfman bracket.

Our motivating example is associated with an arbitrary weighted Lie bi-
algebroid $(D_{k},D_{k}^{*})$. Let us ‘collapse’ the tri-weight on
${\mathtt{T}}^{*}(\Pi D_{k})$ to a bi-weight by taking the sum of the last two
weight vector fields. Then, the Hamiltonian vector field

$\displaystyle Q_{\lambda}:=\\{\mathcal{Q}+\lambda
S,\bullet\\}\in\mathtt{Vect}({\mathtt{T}}^{*}(\Pi D_{k}))$

is of bi-weight $(0,1)$. Here we take $\lambda\in\mathbb{R}$ to be some free
parameter (carrying no weight) and so we have a pencil of such vector fields.
Clearly $Q_{\lambda}$ is a homological vector field. Moreover, as
$Q_{\lambda}$ is a Hamiltonian vector field its Lie derivative annihilates the
canonical symplectic structure on ${\mathtt{T}}^{*}(\Pi D_{k})$. Note that the
canonical symplectic form has natural bi-weight $(k-1,2)$. Thus, following
Roytenberg [38], we have objects that deserve to be called _weighted Courant
algebroids_ of degree $k$.

To uncover the natural pairing, bracket and anchor structure, first note that
$\Pi D_{k}\times_{B_{k-1}}\Pi D_{k}^{*}$ is a quotient supermanifold of
${\mathtt{T}}^{*}\Pi D_{k}$ defined (locally) by projecting out the
coordinates $p^{u+1}_{\alpha}$. Thus we will naturally identify functions on
${\mathtt{T}}^{*}(\Pi D_{k})$ of bi-weight $(r-1,1)$ as homogeneous sections
of the vector bundle $D_{k}\oplus D^{*}_{k}=D_{k}\oplus_{B_{k-1}}D_{k}^{*}$ of
degree $r$, up to a shift in the Grassmann parity. In the homogeneous
coordinates introduced above, any homogeneous section of degree $r$ is of the
form

$\displaystyle\sigma_{r}=s_{r-u-1}^{I}(x)\chi_{I}^{u+1}+\theta_{u+1}^{I}s_{I}^{r-u-1}(x).$

The _natural pairing_ between sections, that is a fibre-wise pseudo-Riemannian
structure on $D_{k}\oplus D_{k}^{*}$, comes from the Poisson bracket on
${\mathtt{T}}^{*}(\Pi D_{k})$,

$\displaystyle\langle\sigma^{1},\sigma^{2}\rangle=\big{\\{}\sigma^{1},\sigma^{2}\big{\\}}\in
C^{\infty}(B_{k-1}),$

and thus the pairing carries bi-weight $(-k+1,-2)$, or we can think of the
total weight as $-k-1$. The fact that we have identified homogeneous sections
with particular odd functions on ${\mathtt{T}}^{*}(\Pi D_{k})$ does not effect
the identification of the metric structure and the Poisson structure;
everything is linear and so the parity reversion does not add any real
complications here. Let us simplify notation slightly and define
$\Theta_{\lambda}:=\mathcal{Q}+\lambda S$. Then the derived bracket, given by

$\displaystyle[[\sigma^{1},\sigma^{2}]]_{\lambda}:=\big{\\{}\big{\\{}\sigma^{1},\Theta_{\lambda}\big{\\}},\sigma^{2}\big{\\}},$

carries bi-weight $(-k+1,-1)$ (so the total weight $-k$), similarly to the
case of a weighted Lie algebroid. It is closed on functions of bi-degree
$(k-1,1)$, so sections of the vector bundle $D_{k}\oplus D^{*}_{k}$ of degree
$k$. The latter is the ‘higher’ analog of the _Courant–Dorfman bracket_ ,
which is not a Lie bracket, but rather a Loday (Leibniz) bracket. The anchor
is then defined as

$\displaystyle\rho_{\lambda}\colon\ \mathtt{Sec}(D_{k}\oplus
D_{k}^{*})\rightarrow\mathtt{Vect}(B_{k-1}),\qquad\rho_{\lambda}(\sigma)[f]:=\\{\\{\sigma,\Theta_{\lambda}\\},f\\},$

for all $f\in C^{\infty}(B_{k-1})$.

######  Theorem 5.13.

Via the above construction, if $(D_{k},D_{k}^{*})$ is a weighted Lie bi-
algebroid of degree $k$, then ${\mathtt{T}}^{*}(\Pi D_{k})$ naturally has the
structure of a weighted Courant algebroid of degree $k$ $($setting $\lambda=1$
for simplicity$)$.

######  Example 5.14.

The $k=1$ case is just that of the Courant structure associated with a
standard Lie bi-algebroid. The case of $k=2$ gives an example of a
$\mathcal{VB}$-Courant algebroid.

######  Example 5.15.

As any weighted Lie algebroid $D_{k}$ can be considered as a weighted Lie bi-
algebroid with the trivial structure on the dual bundle, ${\mathtt{T}}^{*}(\Pi
D_{k})$ can be considered as a weighted Courant algebroid with the obvious
homological vector field.

######  Example 5.16.

If we take $D_{k}={\mathtt{T}}({\mathtt{T}}^{k-1}M)$ then we have the natural
structure of a weighted Courant algebroid given in local coordinates as

$\displaystyle
Q=\big{\\{}\theta_{u+1}^{\alpha}p_{\alpha}^{k+1-u},\bullet\big{\\}}=\theta_{u+1}^{\alpha}\frac{\partial}{\partial
x^{\alpha}_{u}}+p_{\alpha}^{u+1}\frac{\partial}{\partial\chi_{\alpha}^{u+1}}\in\mathtt{Vect}\big{(}{\mathtt{T}}^{*}\big{(}\Pi{\mathtt{T}}\big{(}{\mathtt{T}}^{k-1}M\big{)}\big{)}\big{)}.$

This example should be considered as an natural generalisation of the
canonical Courant algebroid on the generalised tangent bundle
${\mathtt{T}}M\oplus{\mathtt{T}}^{*}M$, which is a substructure of
${\mathtt{T}}({\mathtt{T}}^{*}M)$.

###  5.5 Remarks on contact and Jacobi groupoids

The notion of a _weighted symplectic groupoid_ is clear: it is just a weighted
Poisson groupoid with an invertible Poisson, thus symplectic, structure. By
replacing the homogeneity structure, i.e., an action of the monoid
${\mathbb{R}}$ of multiplicative reals, with a smooth action of its subgroup
$\mathbb{R}^{\times}={\mathbb{R}}{\setminus}\\{0\\}$, one obtains a principal
$\mathbb{R}^{\times}$-bundle in the category of symplectic (in general
Poisson) groupoids. We get in this way the ‘proper’, in our belief, definition
of a _contact_ (resp.  _Jacobi_) _groupoid_.

This belief is based on the general and well-known _credo_ , presented, e.g.,
in [12], that the geometry of contact (more generally, Jacobi) structures on a
manifold $M$ is nothing but the geometry of ‘homogeneous symplectic’ (resp.
‘homogeneous Poisson’) structures on an ${\mathbb{R}}^{\times}$-principal
bundle $P\to M$. The word ‘homogeneous’ refers to the fact that the symplectic
form (resp., Poisson tensor) is homogeneous with respect to the
${\mathbb{R}}^{\times}$-action. Of course, it is pretty well known that any
contact (resp. Jacobi) structure admits a symplectization (resp.
poissonization) and these facts are frequently used in proving theorems on
contact and Jacobi structures (see, e.g., [35]). However, in [12] the
symplectization/poissonization was taken seriously as a genuine definition of
a contact (Jacobi) structure. The general _credo_ then says that any object
related to a contact (Jacobi) structure ought to be considered as the
corresponding object in the symplectic (Poisson) category, equipped
additionally with an ${\mathbb{R}}^{\times}$-principal bundle structure,
compatible with other structures defining the object. The only problem is what
the compatibility means in various cases.

For Lie groupoids (and also Lie algebroids) is natural to expect that the
condition of an ${\mathbb{R}}^{\times}$-action to be compatible with the
groupoid (algebroid) structure is the same as for the action of multiplicative
${\mathbb{R}}$, i.e., it consists of groupoid (algebroid) isomorphisms. This
is the reason why we comment on contact and Jacobi groupoids here. We present
the detailed study of contact/Jacobi groupoids in a separate paper [4].

At this point, note only that our definition of a contact groupoid turns out
to be equivalent to the definition of Dazord [10]. The first and frequently
used definition, presented in [18], is less general and involves an arbitrary
multiplicative function, that is due to the fact that in this approach contact
structures are forced to be trivial, i.e., to admit a global contact 1-form.
Note that very similar idea of using ‘homogeneous symplectic’ manifolds in the
context of contact groupoids appears already in [9] in a slightly less general
framework of ${\mathbb{R}}_{+}$-actions.

To finish, we want to stress that our use of the group ${\mathbb{R}}^{\times}$
rather than ${\mathbb{R}}_{+}$ is motivated by the need of including non-
trivial line bundles into the picture; there is a nice one-to-one
correspondence between principal ${\mathbb{R}}^{\times}$-bundles and line
bundles. In this understanding, the contact structure does not need to admit a
global contact form and the Jacobi bracket is defined on sections of a line
bundle rather than on functions, so it is actually a _local Lie algebra_ in
the sense of Kirillov [19] or a _Jacobi bundle_ in the sense of Marle [31].
This, in turn, comes from the observation (cf. [12, Remark 2.4]) that the
Jacobi bracket is _par excellence_ a Lie bracket related to a module
structure, even if the regular module action of an associative commutative
algebra on itself looks formally as the multiplication in the algebra.
Consequently, allowing nontrivial modules (line bundles) is necessary for the
full and correct picture.

### Acknowledgements

The authors are indebted to the anonymous referees whose comments have served
to improve the content and presentation of this paper. The research of K.
Grabowska and J. Grabowski was funded by the Polish National Science Centre
grant under the contract number DEC-2012/06/A/ST1/00256. A.J. Bruce graciously
acknowledges the financial support of the Warsaw Centre for Mathematics and
Computer Science in the form of a postdoctoral fellowship.

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