Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

Higher-Order Analogs of Lie Algebroids
via Vector Bundle Comorphisms

Michał JÓŹWIKOWSKI † and Mikołaj ROTKIEWICZ ‡

M. Jóźwikowski and M. Rotkiewicz

† Institute of Mathematics, Polish Academy of Sciences,
† Śniadeckich 8, 00-656 Warszawa, Poland<EMAIL_ADDRESS>https://www.impan.pl/~mjoz/

‡ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,
‡ Banacha 2, 02-097 Warszawa, Poland<EMAIL_ADDRESS>https://www.mimuw.edu.pl/~mrotkiew/

Received January 10, 2018, in final form December 12, 2018; Published online
December 29, 2018

We introduce the concept of a higher algebroid, generalizing the notions of an
algebroid and a higher tangent bundle. Our ideas are based on a description of
(Lie) algebroids as vector bundle comorphisms – differential relations of a
special kind. In our approach higher algebroids are vector bundle comorphism
between graded-linear bundles satisfying natural axioms. We provide natural
examples and discuss applications in geometric mechanics.

higher algebroid; vector bundle comorphism; almost-Lie algebroid; graded
manifold; graded bundle; algebroid lift; variational principle

58A20; 58A50; 70G65; 58E30; 70H50

###### Contents

  1. 1 Introduction
    1. 1.1 Double vector bundles
  2. 2 (Lie) algebroids as vector bundle comorphisms
    1. 2.1 The category of vector bundle comorphisms
    2. 2.2 Description of (Lie) algebroids in terms of VBCs
  3. 3 Recollection of $\mathbb{N}$-graded manifolds
  4. 4 Higher (Lie) algebroids
    1. 4.1 On the Lie axiom
    2. 4.2 Prolongations of an AL algebroid
  5. 5 Variational calculus on (Lie) higher algebroids
    1. 5.1 The formalism of variational calculus
    2. 5.2 Examples
  6. 6 Further examples – substructures and quotients of $\mathrm{T}^{k}G/G$
  7. 7 Final remarks
  8. A Appendix – proofs of technical results
  9. References

##  1 Introduction

The main goal of this paper is to introduce a concept of a (general) _higher
algebroid_ 111In this paper under the name _algebroid_ we understand
generalizations of the notion of a Lie algebroid obtained by weakening its
axioms. In the geometric mechanics literature these generalizations are
commonly known as _almost-Lie_ (no Jacobi identity), _skew_ (no Jacobi and no
anchor-bracket compatibility), or _general_ (no Jacobi, no anchor-bracket
compatibility, no skew-symmetry of the bracket) algebroids (the latter are
often simply called _algebroids_). Throughout the paper we shall refer to
these objects as to _$($ Lie$)$ algebroids_ meaning “Lie algebroids and their
above-mentioned generalizations”. They should not be confused with _Courant
algebroids_ or other similarly named concepts. and study basic properties and
applications of this notion. We work essentially within the framework of the
theory of $\mathbb{N}$-graded manifolds and the theory of differential
relations (vector bundle comorphisms).

Why higher algebroids? Lie algebroids and their generalizations proved to be a
fruitful area of study in the last three decades, either on their own right,
as an offspring of Poisson geometry, and, particularly, in geometric
mechanics. The latter direction was originated by Weinstein [52] and,
following seminal papers of Martínez [36, 37], it developed into many
different sub-branches (see a survey paper [11] for a detailed discussion of
the available literature).

From our point of view one of the most spectacular achievements of the
algebroid-oriented study of mechanics is the recognition of the geometric
structures standing behind variational calculus. The relation between (Lie)
algebroids and variations is, perhaps, best described in two papers [13, 14],
which themselves were inspired by much earlier works of Tulczyjew [44, 45].
What makes the mentioned works particularly interesting is that the authors
were able to identify the true geometric essence of variational calculus,
which keeps working despite getting rid of several most natural assumptions,
such as the Jacobi identity and even the skew-symmetry of the Lie algebroid
bracket. In fact, putting aside the existence of real-life examples, it seems
that the concept of a general algebroid [18, 19] is the uttermost geometric
reality behind first-order variational calculus.

From the above perspective it is most natural to look for similar geometric
structures related with variational calculus of higher order. Thus we would
like to introduce a geometric object, a  _$($ Lie$)$ higher algebroid_, which
is present whenever a variational problem involving higher velocities is
considered. We have already tried to address this question from the
perspective of the classical groupoid–algebroid reduction [27], yet now we
propose a more abstract and a more general approach.

Towards a proper definition – objects. In order one, the (Lie) algebroid
structure “lives” on a vector bundle, which can be often interpreted as the
velocity bundle of some physical system, i.e., as a quotient (in a suitable
sense) of some tangent bundle $\mathrm{T}M$. The structure of the tangent
bundle is thus our most important example of a (Lie) algebroid. Obviously, in
higher-order variational calculus the role of $\mathrm{T}M$ is taken over by
its higher analogue $\mathrm{T}^{k}M$, the bundle of
$k$-velocities.222$\mathrm{T}^{k}M$ is the bundle of $k$-jets of curves in $M$
and it should not be mistaken with
$\mathrm{T}^{(k)}M=\mathrm{T}(\mathrm{T}(\ldots(\mathrm{T}M)\ldots))$ – the
iterated tangent bundle. Locally, $\mathrm{T}^{k}M$ can be characterized by
specifying the points, velocities, accelerations, and higher derivatives of
curves on $M$. Thus, having the perspective of variational calculus in mind,
it is reasonable to expect that a proper object hosting a higher algebroid
structure would be a bundle of some sort with fibre coordinates sharing a
similar derivative-like nature as coordinates on $\mathrm{T}^{k}M$. This
expectation is met by a class of fibre bundles with a typical fibre
diffeomorphic with $\mathbb{R}^{n}$, however, in general, transition functions
need not to be linear but are polynomials which are homogeneous with respect
to a prescribed gradation of coordinate functions (see Section 3 for a
detailed discussion). They are a special case of the concept of a _graded
manifold_ introduced by Voronov [47]. Namely, they are _purely even non-
negatively graded manifolds_ in the sense of Voronov, meaning that local
coordinates are graded by non-negative integers. We follow Voronov’s idea that
such a non-negatively graded manifold, which expands into a tower of
fibrations with polynomial transition functions (of a special form), should be
regarded as a generalization of a vector bundle [47], see also [48, 49, 50].
(For this, we use notation typical for (vector) bundles – see Section 3.)
Properties of such bundles and their application in theoretical mechanics
where also considered in several papers [4, 6, 17]. For brevity, we shall use
“$\mathbb{N}$-graded” for “non-negatively graded” (thinking that zero is
included into the natural numbers) or simply “graded” if no other gradings
will appear. Note that we consider only the even case, though the results are
expected to hold in the super setting too.

Towards a proper definition – structures. The structure of an order one (Lie)
algebroid on a vector bundle $\sigma\colon E\rightarrow M$ is most commonly
introduced as a bi-linear bracket operation on the space of sections of its
host bundle $\sigma$. It is quite obvious that this definition has no direct
generalization to the higher-order case as, in particular, the higher tangent
bundle $\tau^{k}_{M}\colon\mathrm{T}^{k}M\rightarrow M$, which is an obvious
candidate to host a higher (Lie) algebroid structure, admits no canonical
bracket operation on its space of sections. Fortunately, the (Lie) algebroid
structure on $\sigma$ has several other characterizations equivalent to the
standard definition, perhaps more suitable for a generalization. To mention
just a few, we may describe an algebroid on $\sigma$ as a certain homological
vector field on a graded supermanifold $E[1]$, a de Rham-like derivative in
the space of $E$-differential forms (i.e., sections of
$\bigwedge^{\bullet}E^{\ast}$), a certain linear 2-tensor field on the dual
bundle $\sigma^{\ast}$, a morphism
$\varepsilon\colon\mathrm{T}^{\ast}E\rightarrow\mathrm{T}E^{\ast}$ of double
vector bundles, or as a differential relation of a special kind (a vector
bundle comorphism) $\kappa\subset\mathrm{T}E\times\mathrm{T}E$.

From the perspective of variational calculus, the latter characterization is
the one most appealing to us.333Let us remark that, the double vector bundle
morphism $\varepsilon\colon\mathrm{T}^{\ast}E\rightarrow\mathrm{T}E^{\ast}$
favored in the so-called Tulczyjew approach to geometric mechanics [13, 14] is
just the dual of $\kappa$. This important role of $\kappa$ can be easily
explained. For the tangent algebroid structure on
$\tau_{M}\colon\mathrm{T}M\rightarrow M$, $\kappa$ is, in fact, the canonical
isomorphism
$\kappa_{M}\colon\mathrm{T}\mathrm{T}M\rightarrow\mathrm{T}\mathrm{T}M$
interchanging the two vector bundle structures on $\mathrm{T}\mathrm{T}M$ –
cf. Example 2.13. Its role in variational calculus is to present ‘a jet of
curves as a curve of jets’ (i.e., a variation of the tangent lift of a
trajectory in $M$ is constructed from the tangent lift of the curve of virtual
displacements by means of $\kappa_{M}$). The situation in the higher-order
case is no different and, perhaps, even easier to understand. Namely on a
higher-tangent bundle $\tau_{M}^{k}\colon\mathrm{T}^{k}M\rightarrow M$, we
have the canonical isomorphism
$\kappa^{k}_{M}\colon\mathrm{T}^{k}\mathrm{T}M\rightarrow\mathrm{T}\mathrm{T}^{k}M$
which, similarly as before, allows to express a variation of a
$k^{\mathrm{th}}$-tangent lift of a trajectory in $M$ as a
$k^{\mathrm{th}}$-tangent lift of the related generator (a curve of virtual
displacements) [26]. When performing the standard Lie groupoid – Lie algebroid
reduction the character (but not the role) of $\kappa$ changes – the reduced
object is no longer a differentiable map but, in general, only a differential
relation (but of a special kind). Again there is no essential difference
between the first- and higher-order cases [27].

Summing up, geometrically relation $\kappa\subset\mathrm{T}E\times\mathrm{T}E$
is responsible for the construction of admissible variations of curves on a
(Lie) algebroid on $E$. Its presence can be easily motivated by the groupoid –
algebroid reduction procedure: when performing a reduction of a variational
problem it is not enough to reduce the trajectories, but also the variations!
An this is how $\kappa$ appears. Variational calculus on (Lie) algebroids can
be, however, successfully developed in abstract terms (with a prominent role
of $\kappa$) without referring to the reduction procedure [13].

All the above suggests that the language of differential relations is suitable
to speak about higher (Lie) algebroids (at least from the perspective of
applications in variational calculus). The first order-case [13] and the known
higher-order examples from [25] advocate that these relations should have a
special nature, namely they should be _vector bundle comorphisms_ (_VBCs_ in
short; see Definition 2.1). This assumption, in principle, allows to uniquely
characterize admissible variations in terms of their generators (virtual
displacements). Let us remark that the important role of vector bundle
comorphisms in the theory of Lie groupoids and Lie algebroids was recognized
also in [7].

Higher (Lie) algebroids, examples and applications. Summing up the above
heuristic considerations, we postulate a _higher $($Lie$)$ algebroid_ to be a
$\mathbb{N}$-graded manifold $\sigma^{k}\colon E^{k}\rightarrow M$ equipped
with a vector bundle comorphism
$\kappa^{k}\subset\mathrm{T}^{k}E^{1}\times\mathrm{T}E^{k}$, which also should
be naturally graded (Definition 4.1). Here $E^{1}$ is the canonical reduction
of $E^{k}$ to a first-order bundle, i.e., to a vector bundle.

For $k=1$ we recover the standard notion of a (Lie) algebroid. In fact, we
devote entire Section 2 to carefully formulate the theory of (the first order)
Lie algebroids and their generalizations in the language of VBCs. This has a
two-fold purpose. First of all, the language of differential relations was so
far hardly used in the theory of (Lie) algebroids and most of the results of
this section are new (yet rather straightforward, so no true originality can
be claimed by us). Secondly, this first-order formulation has a direct
generalization to the higher order case, with most of the definitions being
completely analogous.

Apart from the first-order case, important examples of such structures are
provided by higher tangent bundles $\mathrm{T}^{k}M$ with the isomorphisms
$\kappa^{k}_{M}$ mentioned above. Another important class of examples is given
by the _prolongations of almost-Lie algebroids_ introduced in [9] and [27]
(see also [43]). Prolongations naturally appear in higher-order variational
calculus as reductions of higher tangent bundles of Lie groupoids. In our
previous publications [25, 27] (see also [39]) we study several concrete
cases. Additionally some examples related to Lie groups are discussed in
Section 6.

We further argue for the usefulness of the concept of a higher (Lie) algebroid
introduced in this work by providing two particular applications. First of
all, given a higher algebroid structure on $\sigma^{k}\colon E^{k}\rightarrow
M$ we were able to define a whole family of _algebroid lifts_ of sections of
$E^{1}$ to vector fields on $E^{k}$. These generalize the well-know
constructions of the vertical and the complete lifts of a section in the
presence of a (Lie) algebroid structure. Secondly, in Section 5 we show that
the geometric formulation of higher-order variational calculus is possible
within the framework of higher (Lie) algebroids (from this perspective the
algebroid lifts are directly related with conservation laws). In fact, we show
that the latter works fine also for a more general class of _higher pre-
algebroids_ , thus further generalization of our theory is possible. We refer
to a recent publication [8] and the references therein for numerous concrete
interesting examples of higher-order variational problems.

Alternative approaches. In the literature there have been a few attempts to
introduce higher analogs of (Lie) algebroids. The version of a higher analog
of Lie algebroids that we develop is quite distinct from the work of Mackenzie
[35] and Voronov [48, 50], though we make use of some of their concepts and
methods. Voronov [48] proposed that higher algebroids should be understood as
homological vector fields of weight one on a non-negatively graded
supermanifold (generalizing Vaintrob’s description of Lie algebroids as
$Q$-manifolds [46]). The language of $Q$-manifolds is central in the whole
bracket geometry. Connections between higher (Lie) algebroids in the sense of
this work and Voronov’s ones or $k$-fold Lie algebroids of Mackenzie are not
clear and need further studies.

A much more down-to-earth idea is to define higher (Lie) algebroids as bundles
of higher-jets of admissible curves on a standard Lie algebroid. In this paper
we refer to this construction as the _prolongation of an algebroid_ and
briefly discuss it in Section 4.2. Such objects appeared for the first time in
a seminar talk by Colombo and de Diego [9], however without a full recognition
of the relevant geometric structure. The latter was studied by us in [27] and
successfully applied in higher-order variational problems on Lie algebroids
and Lie groupoids [25] (see also [39]). Prolongations of algebroids in the
above sense constitute a particular subclass of higher (Lie) algebroids in the
understanding of this paper, perhaps the most important one as it contains
reductions of higher tangent bundles of Lie groupoids.

The most recent attempt is due to Bruce, Grabowska and Grabowski. In [5] they
constructed a _linearisation functor_ , which, to a $\mathbb{N}$-graded
manifold $\sigma^{k}\colon E^{k}\rightarrow M$ of order $k$, canonically
associates a manifold $l\big{(}E^{k}\big{)}$ equipped with a graded-linear
structure of a bi-order $(k-1,1)$ over $E^{1}$ and $E^{k-1}$, respectively.
Now the higher (Lie) algebroid structure on $\sigma^{k}$ is defined as a (Lie)
algebroid structure on the vector bundle $l\big{(}E^{k}\big{)}\rightarrow
E^{k-1}$ compatible with the second grading. The idea here is to mimic the
canonical inclusion
$\mathrm{T}^{k}M\hookrightarrow\mathrm{T}\mathrm{T}^{k-1}M$, which makes the
higher tangent bundle $\mathrm{T}^{k}M$ a sub-object of the tangent Lie
algebroid of $\mathrm{T}^{k-1}M$. In this way the construction of [5] admits
higher tangent bundles and also prolongations of algebroids as examples of
higher (Lie) algebroids, similarly to our approach. However, these two notions
of a higher algebroid are different, as can be easily seen from coordinate
calculations already in order 2. The construction of [5] can be adapted to
develop higher-order mechanics on $\mathbb{N}$-graded manifolds [4],
equivalent to our results [25] in comparable cases. So perhaps the difference
here is of a philosophical nature. In the classical case: should we treat
$k^{\mathrm{th}}$-order variational calculus on a manifold $M$ as the first-
order theory on $\mathrm{T}^{k-1}M$ thus passing through unnecessary degrees
of freedom (as would be in the spirit of [5]) or do we prefer to work directly
with $\mathrm{T}^{k}M$, as is in our formalism.

Organization of the paper. In Section 2 we reformulate the theory of (Lie)
algebroids in the language of vector bundle comorphisms. Section 3 contains
basic information concerning $\mathbb{N}$-graded manifolds and weighted
structures. In Section 4, using the result of the previous two sections, we
formulate the definition of a higher algebroid, provide a few natural examples
and study in detail higher algebroid lifts and the Lie axiom in Section 4.1.
In Section 5 we develop the framework of higher-order variational calculus on
higher algebroids, putting emphasis on admissible variations and their
relation with conservation laws. Section 6 contains a study of higher
algebroid structures inherited from higher tangent bundles of a Lie group $G$.
In particular, we characterize all higher subalgebroids and describe a class
of quotients of $\mathrm{T}^{k}G/G$. In Section 7 we sketch a few perspectives
of future research. Finally, in Appendix A we have hidden most of the proofs
of more technical results appearing in this work.

###  1.1 Double vector bundles

We shall frequently work with _double vector bundles_ , geometric objects
sharing two (compatible) vector bundle structures. A typical example is the
total space $\mathrm{T}E$ of the tangent bundle of a vector bundle
$\sigma\colon E\rightarrow M$. The two vector bundle structures are: the
tangent projection $\tau_{E}\colon\mathrm{T}E\rightarrow E$, and
$\mathrm{T}\sigma\colon\mathrm{T}E\rightarrow\mathrm{T}M$, the tangent lift of
$\sigma$. The foundations of the theory of double vector bundles were laid by
J. Pradines [41] (see also [34, Chapter 9] for basic definitions, examples and
historical remarks). Due to the results of [16] we may formulate a definition
of a double vector bundle in the following way.

######  Definition 1.1 (double vector bundle).

A structure of a _double vector bundle_ (_DVB_ , in short) on a manifold $D$
is a pair of vector bundles $\sigma_{A}\colon D\rightarrow A$,
$\sigma_{B}\colon D\rightarrow B$ such that for any $t,s\in\mathbb{R}$ and
$x\in D$

$\displaystyle t\cdot_{A}s\cdot_{B}x=s\cdot_{B}t\cdot_{A}x,$ (1.1)

where $\cdot_{A}$ and $\cdot_{B}$ denote the multiplications by scalars in
$\sigma_{A}$ and $\sigma_{B}$, respectively.

A _morphism of DVBs_ $(D,\sigma_{A},\sigma_{B})$ and
$(D^{\prime},\sigma_{A^{\prime}},\sigma_{B^{\prime}})$ is a smooth map
$\phi\colon D\rightarrow D^{\prime}$ which is linear as a map from
$\sigma_{A}$ to $\sigma_{A}^{\prime}$ and, simultaneously, as a map from
$\sigma_{B}$ to $\sigma_{B^{\prime}}$.

The bases $A$, $B$ of a double vector bundle as above carry induced vector
bundle structures over the common base $M$ giving rise to a diagram

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{A}}$$\scriptstyle{\sigma_{B}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{B}_{M}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{A}_{M}}$$\textstyle{M}$

consisting of four vector bundle projections. All $M$, $A$, $B$ can be seen as
submanifolds of $D$ via the zero section embeddings. In particular, the zero
section $0_{A}\colon A\rightarrow D$ defines a vector bundle structure on $A$
as a substructure of the vector bundle structure on $\sigma_{B}\colon
D\rightarrow B$. Double vector bundles are often said to be ‘vector bundles in
the category of vector bundles’ what can be understood as the condition that
all four structure maps of $\sigma_{A}$ (the bundle projection $\sigma_{A}$,
the zero section $0_{A}$, the scalar multiplication $\cdot_{A}$ and the
addition $+_{A}$) are vector bundle morphism with respect to the vector bundle
structure of $\sigma_{B}$. The notion of a double vector bundle naturally
generalizes to the notion of a multi-$\mathbb{N}$-graded manifold (see Section
3).

##  2 (Lie) algebroids as vector bundle comorphisms

In this section we characterize Lie algebroids and their relaxed versions
(such as generalized algebroids in the sense of Grabowski–Urbański [19]) by
means of differential relations of a special kind (vector bundle comorphisms,
VBC in short). This will be the cornerstone of our definition of higher
algebroids in Section 4.

###  2.1 The category of vector bundle comorphisms

Following [22, 34] we will recall a definition of a _comorphism_ between
vector bundles (VBC, in short). Later we shall study basic properties of VBCs,
i.e., the corresponding map on sections, morphisms between VBCs, and VBCs
compatible with an additional linear structure.

Vector bundle comorphisms.

######  Definition 2.1 (vector bundle comorphism, VBC).

A _vector bundle comorphism_ (_VBC_ , in short) from a vector bundle
$\sigma_{1}\colon E_{1}\rightarrow M_{1}$ to a vector bundle $\sigma_{2}\colon
E_{2}\rightarrow M_{2}$ is a relation $r\subset E_{1}\times E_{2}$ of a
special form. Namely, there exists an ordinary vector bundle morphism
$\phi\colon E_{2}^{\ast}\rightarrow E_{1}^{\ast}$ covering a smooth map
$\underline{\phi}\colon M_{2}\rightarrow M_{1}$ such that $r=\phi^{\ast}$,
i.e., $r$ is the union of graphs of linear maps
$r_{y}\colon(E_{1})_{\underline{\phi}(y)}\rightarrow(E_{2})_{y}$ such that
$\phi_{y}\colon(E_{2}^{\ast})_{y}\rightarrow(E_{1}^{\ast})_{\underline{\phi}(y)}$
is the dual of the linear map $r_{y}$, where $y$ varies in $M_{2}$. We also
say that the VBC $r$ _covers_ the base map $\underline{r}:=\underline{\phi}$.
Usually we shall denote a VBC as
$r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$ or as a
diagram

$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{r}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{2}}$$\textstyle{M_{1}}$$\textstyle{M_{2}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{r}}$

In particular, the VBC $r$ is a vector subbundle of
$\sigma_{1}\times\sigma_{2}$.

There is a general concept of comorphisms between fibre bundles, (Lie)
algebroid comorphisms and the corresponding (Lie) groupoid comorphisms. A full
presentation of these ideas can be found in [7] and [34] with details on the
history of these and related notions. The concept of a comorphism can be dated
back at least to Bourbaki [2] and has been appearing ever since, sometimes
under different names, see, e.g., [20, 21, 23]. We would like to acknowledge
that VBC is the simplest example of a groupoid morphism in the sense of
Zakrzewski [53, 54]. In the appendix of [27] we studied some properties of
VBCs (using the name _Zakrzewski morphism_ proposed in this context in [13]).

######  Remark 2.2 (VBCs and maps on sections).

An important property of a VBC is that it induces a mapping between the
corresponding spaces of sections. Namely, given a VBC
$r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$ as in
Definition 2.1 and a sections $s\in\operatorname{Sec}_{M_{1}}(E_{1})$ we
define $\hat{r}(s)\in\operatorname{Sec}_{M_{2}}(E_{2})$ by

$\displaystyle\hat{r}(s)(y):=r_{y}(s(\underline{r}(y))),\qquad\text{for every
$y\in M_{2}$.}$

In other words, the value of $\hat{r}(s)$ at a given point $y\in M_{2}$ is the
unique element of the fibre $(E_{2})_{y}$ which is $r$-related to the value of
$s$ at $\underline{r}(y)$. By the linearity of $r$, for every two sections
$s,s^{\prime}\in\operatorname{Sec}_{M_{1}}(E_{1})$ we have
$\hat{r}(s+s^{\prime})=\hat{r}(s)+\hat{r}(s^{\prime})$. Moreover, by
construction, for any $f\in\mathcal{C}^{\infty}(M_{1})$

$\displaystyle\hat{r}(f\cdot s)=\underline{r}^{\ast}(f)\cdot\hat{r}(s).$ (2.1)

Conversely, any ${\mathbb{R}}$-linear map
$\operatorname{Sec}_{M_{1}}(E_{1})\rightarrow\operatorname{Sec}_{M_{2}}(E_{2})$
satisfying (2.1) for some underlying map $\underline{r}\colon M_{2}\rightarrow
M_{1}$ gives rise to a VBC
$r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$. Checking this
property is left to the reader.

Morphisms between VBCs. A VBC can describe a single geometric object. As we
shall see shortly (Lemma 2.11 and Proposition 2.15), a (Lie) algebroid
structure on a bundle $\sigma\colon E\rightarrow M$ is fully encoded by a VBC
$\kappa$ from the tangent lift
$\mathrm{T}\sigma\colon\mathrm{T}E\rightarrow\mathrm{T}M$ to the tangent
bundle $\tau_{E}\colon\mathrm{T}E\rightarrow E$. This suggests that the
standard notion of a (Lie) algebroid morphism should be expressible by a,
properly defined, morphism between the corresponding vector bundle
comorphisms. Now we are going to introduce such a notion under the name
${\mathcal{VBC}}$-morphism, i.e., a morphism in the category ${\mathcal{VBC}}$
of vector bundle comorphisms.

######  Definition 2.3 (${\mathcal{VBC}}$-morphism).

Let $(\sigma_{i}\colon E_{i}\to M_{i})$ and $(\sigma_{i}^{\prime}\colon
E_{i}^{\prime}\to M_{i}^{\prime})$, $i=1,2$, be four VBs and let $r$ and
$r^{\prime}$ be the following vector bundle comorphisms:

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(2.2)

We say that a pair $(\phi_{1},\phi_{2})$ of VB morphisms $\phi_{i}\colon
E_{i}\to E_{i}^{\prime}$ covering smooth maps $\underline{\phi_{i}}\colon
M_{i}\to M_{i}^{\prime}$, $i=1,2$, is a _${\mathcal{VBC}}$ -morphism from $r$
to $r^{\prime}$_ (and write $(\phi_{1},\phi_{2})\colon r\Rightarrow
r^{\prime}$) if the following diagram

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(2.3)

is commutative in the following sense:

  1. (i)

The base maps commute, i.e.,
$\underline{\phi_{1}}\circ\underline{r}=\underline{r}^{\prime}\circ\underline{\phi_{2}}\colon
M_{2}\to M_{1}^{\prime}$.

  2. (ii)

At the level of fibres, for any $y\in M_{2}$ the following compositions of
linear maps

$\displaystyle
E_{1,x}\xrightarrow{r_{y}}E_{2,y}\xrightarrow{\phi_{2}}E_{2,y^{\prime}}^{\prime}\qquad\text{and}\qquad
E_{1,x}\xrightarrow{\phi_{1}}E_{1,x^{\prime}}^{\prime}\xrightarrow{r^{\prime}_{y^{\prime}}}E_{y^{\prime}}^{\prime}$

coincide. Here $x=\underline{r}(y)$, $y^{\prime}=\underline{\phi_{2}}(y)$,
$x^{\prime}=\underline{r}^{\prime}(y^{\prime})=\underline{\phi_{1}}(x)$ are
determined by $y\in M_{2}$.

It is easy to see that ${\mathcal{VBC}}$-morphisms can be naturally composed,
and that VBCs with ${\mathcal{VBC}}$-morphisms form a _category of vector
bundle comorphisms_ , denoted ${\mathcal{VBC}}$.

######  Remark 2.4 (dualization of VBCs).

By dualizing the diagram (2.3) we get a diagram of the same type:

$\textstyle{E_{1}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}^{\ast}}$$\textstyle{E_{2}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{\ast}}$$\scriptstyle{\sigma_{2}^{\ast}}$$\textstyle{E_{1}^{{}^{\prime}\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}^{\ast}}$$\scriptstyle{(\sigma_{1}^{\prime})^{\ast}}$$\textstyle{E_{2}^{{}^{\prime}\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{{}^{\prime}\ast}}$$\scriptstyle{\phi_{2}^{\ast}}$$\scriptstyle{(\sigma_{2}^{\prime})^{\ast}}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\phi_{1}}}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\phi_{2}}}$$\scriptstyle{\underline{r}}$$\textstyle{M_{1}^{\prime}}$$\textstyle{M_{2}^{\prime}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{r}^{\prime}}$

It is clear that $(\phi_{1},\phi_{2})\colon r\Rightarrow r^{\prime}$ is a
${\mathcal{VBC}}$-morphism if and only if
$(r^{\prime\ast},r^{\ast})\colon\phi_{2}^{\ast}\Rightarrow\phi_{1}^{\ast}$ is
so. Thus the notion of a morphism in the category ${\mathcal{VBC}}$ does not
reduce to the notion of a vector bundle morphism.

######  Example 2.5 (phase lift).

Let $\underline{\phi}\colon M\rightarrow M^{\prime}$ be a smooth map, and
consider the phase lift of $\underline{\phi}$ which is a VBC
$\mathrm{T}^{\ast}\underline{\phi}\colon\mathrm{T}^{\ast}M^{\prime}\rightarrow\\!\\!\vartriangleright\mathrm{T}^{\ast}M$
covering $\underline{\phi}$. Then the tangent lift of
$\mathrm{T}^{\ast}\underline{\phi}$ is a VBC as well and
$(\mathrm{T}\tau^{\ast}_{M^{\prime}},\tau_{\mathrm{T}^{\ast}M})\colon\mathrm{T}\mathrm{T}^{\ast}\underline{\phi}\Rightarrow\mathrm{T}^{\ast}\underline{\phi}$
is a morphism in the category ${\mathcal{VBC}}$.

Another useful formulation of the conditions presented in the definition of a
${\mathcal{VBC}}$-morphism is possible:

######  Proposition 2.6 (characterization of ${\mathcal{VBC}}$-morphisms).

Let $\phi_{i}\colon E_{i}\rightarrow E_{i}^{\prime}$ $(i=1,2)$ be vector
bundle morphisms and let $r,r^{\prime}$ be VBCs as in Definition 2.3. Then
$(\phi_{1},\phi_{2})\colon r\Rightarrow r^{\prime}$ is a
${\mathcal{VBC}}$-morphism if and only if for any $r$-related vectors $X\in
E_{1}$, $Y\in E_{2}$ their images $\phi_{1}(X)$ and $\phi_{2}(Y)$ are
$r^{\prime}$-related. Equivalently, $(\phi_{1}\times\phi_{2})(r)\subset
r^{\prime}$.

###### Proof.

Assume that $(\phi_{1},\phi_{2})\colon r\Rightarrow r^{\prime}$ is a
${\mathcal{VBC}}$-morphism and take $X$ and $Y$ such that $X\sim_{r}Y$. Denote
$x:=\sigma_{1}(X)$, $y:=\sigma_{2}(Y)$, $x^{\prime}:=\underline{\phi_{1}}(x)$
and $y^{\prime}:=\underline{\phi_{2}}(y)$. According to Definition 2.3
$r^{\prime}_{y^{\prime}}(\phi_{1}(X))=\phi_{2}(Y)$, in other words
$\phi_{1}(X)$ and $\phi_{2}(Y)$ are $r^{\prime}$-related. The proof the in
reverse direction is analogous and is left to the reader. ∎

Linear VBCs. Of our special interest will be _linear VBCs_ , i.e., VBCs
compatible with an additional linear structure. Later in Section 3 (Definition
3.6) we will generalize this notion to the concept of _weighted VBCs_ , which
are compatible with an additional graded structure. We shall follow a general
scheme already presented in the definitions of a _linear_ Poisson structure,
$\mathcal{VB}$-groupoids, $\mathcal{VB}$-algebroids, weighted algebroids etc.
[3, 34].

######  Definition 2.7 (linear VBC).

Let $r$ be a VBC from a vector bundle $\sigma\colon E\rightarrow M$ to a
vector bundle $\sigma^{\prime}\colon E^{\prime}\rightarrow M^{\prime}$. Assume
that the total spaces $E$ and $E^{\prime}$ carry linear structures $\tau\colon
E\rightarrow N$ and $\tau^{\prime}\colon E^{\prime}\rightarrow N^{\prime}$
compatible with $\sigma$ and $\sigma^{\prime}$, respectively (i.e., $E$ and
$E^{\prime}$ are DVBs in the sense of Definition 1.1). We say that $r$ is a
_linear VBC_ if $r\subset E\times E^{\prime}$ is a vector subbundle of
$\tau\times\tau^{\prime}$.

Note that a linear VBC $r\subset E\times E^{\prime}$ defined as above is a
vector subbundle of both $\sigma\times\sigma^{\prime}$ and
$\tau\times\tau^{\prime}$. It follows that $r$ projects to linear relations in
$M\times M^{\prime}$ and $N\times N^{\prime}$. In particular, the base map
$\underline{r}\colon M^{\prime}\rightarrow M$ is a VB morphism (cf. (i) and
(ii) in Lemma 2.9 below). Particular examples of linear VBCs (not all) are
provided by dualizing DVB morphisms.

Recall that if $\sigma\colon E\rightarrow M$ is a vector bundle, then the
tangent space $\mathrm{T}E$ carries two compatible vector bundle structures
$\mathrm{T}\sigma\colon\mathrm{T}E\rightarrow\mathrm{T}M$ and
$\tau_{E}\colon\mathrm{T}E\rightarrow E$. In the remaining part of this
paragraph we shall study linear VBCs intertwining these two VB structures. The
common kernel of $\mathrm{T}\sigma$ and $\tau_{E}$ (called the _core_
$\mathbf{C}(\mathrm{T}E)$ of DVB $(\mathrm{T}E,\mathrm{T}\sigma,\tau_{E})$) is
the subbundle $C=\mathbf{V}_{M}E\simeq E$ of the vertical bundle
$\mathbf{V}E=\ker\mathrm{T}\sigma$ consisting of vertical vectors based at
$M\subset E$. The two additive structures of $\mathrm{T}E$ coincide on the
core $C$. Note that the core $C$ acts on $\mathrm{T}E$ by an addition of
vertical vectors, i.e., for every $A\in\mathrm{T}_{a}E$ and every $e\in
C_{\sigma(a)}\simeq E_{\sigma(a)}$ we define

$\displaystyle
A\bm{\boldsymbol{+}}e:=A+_{\tau_{E}}\mathbf{V}_{a}e\in\mathrm{T}_{a}E,$

where
$\mathbf{V}_{a}\colon\mathbf{V}_{\sigma(a)}E\rightarrow\mathbf{V}_{a}E\subset\mathrm{T}_{a}E$
are canonical isomorphisms. Note that the above action does not affects the
two VB projections $\mathrm{T}\sigma$ and $\tau_{E}$ on $\mathrm{T}E$, i.e.,
$\mathrm{T}\sigma(A\bm{\boldsymbol{+}}e)=\mathrm{T}\sigma(A)$ and
$\tau_{E}(A\bm{\boldsymbol{+}}e)=\tau_{E}(A)$. Moreover, the following useful
identity holds

$\displaystyle\mathrm{T}(f\cdot
a)v=f\cdot_{\mathrm{T}\sigma}(\mathrm{T}a)v\bm{\boldsymbol{+}}v(f)\cdot a,$
(2.4)

where $f\in\mathcal{C}^{\infty}(M)$, $v\in\mathfrak{X}(M)$ and
$a\in\operatorname{Sec}(E)$.

######  Remark 2.8 (local form of a class of linear VBCs).

Let us write a general (local) form of a linear VBC $\kappa$ intertwining the
two compatible VB structures on $\mathrm{T}E$. For simplicity and further
applications we also assume that $\kappa$ projects onto
$\operatorname{graph}(\operatorname{id}_{M})$ under the bundle projection
$\tau_{M}\circ\mathrm{T}\sigma\times\sigma\circ\tau_{E}\colon\mathrm{T}E\times\mathrm{T}E\rightarrow
M\times M$, i.e., $\kappa$ relates only elements in the same fibre over $M$.
(Note that $\tau_{M}\circ\mathrm{T}\sigma=\sigma\circ\tau_{E}$.) Let
$(x^{a},y^{i})$ be local linear coordinates on $E$. We use the standard
notation $(x^{a},y^{i},\dot{x}^{b},\dot{y}^{j})$ for induced coordinates on
the tangent bundle $\mathrm{T}E$ and, to describe
$\kappa\subset\mathrm{T}E\times\mathrm{T}E$, we shall underline coordinates in
the second copy of $\mathrm{T}E$.

As $\kappa$ is a double vector subbundle of
$(\mathrm{T}E\times\mathrm{T}E,\mathrm{T}\sigma\times\tau_{E},\tau_{E}\times\mathrm{T}\sigma)$,
the graph of the base map $\rho_{L}:=\underline{\kappa}\colon
E\rightarrow\mathrm{T}M$ of $\kappa$ is a vector subbundle in
$E\times\mathrm{T}M$, i.e., it is a graph of a VB morphism. Thus it maps a
vector $(\underline{x}^{a},\underline{y}^{i})$ from $E$ to a vector of the
form $(x^{a}=\underline{x}^{a},\dot{x}^{b}=Q^{b}_{i}(x)\underline{y}^{i})$ in
$\mathrm{T}M$. Now, since for every fixed $a\in E$ relation
$\kappa_{a}\colon(\mathrm{T}E)_{\rho_{L}(a)}\rightarrow\mathrm{T}_{a}E$ is a
linear map, it turns out that $\kappa$ is given by a linear mapping of
coordinates $(y^{i},\dot{y}^{j})$ to coordinates
$(\underline{\dot{x}}^{b},\underline{\dot{y}}^{j})$ with coefficients
depending on coordinates $(\underline{x}^{a},\underline{y}^{i})$ in $E$.
Furthermore, we know that $\kappa$ is bi-homogeneous. Calculation on weights:
$w(y^{i})=(1,0)=w(\dot{\underline{x}}^{a})$,
$w(\dot{x}^{a})=(0,1)=w(\underline{y}^{i})$,
$w(\dot{y}^{i})=(1,1)=w(\dot{\underline{y}}^{i})$ ensures us that $\kappa$ is
determined by equations of the form

$\displaystyle\kappa\colon\ \begin{cases}x^{a}=\underline{x}^{a},\\\
\dot{x}^{b}=Q^{b}_{i}(x)\,\underline{y}^{i},\\\
\dot{\underline{x}}^{b}=\widetilde{Q}^{b}_{i}(x)y^{i},\\\
\dot{\underline{y}}^{k}=\alpha^{k}_{i}(x)\dot{y}^{i}+Q^{k}_{ij}(x)\underline{y}^{i}y^{j},\end{cases}$
(2.5)

where $x=(x^{a})$ and $Q^{b}_{i}(x)$, $\widetilde{Q}^{b}_{i}(x)$,
$\alpha^{j}_{i}(x)$ and $Q^{k}_{ij}(x)$ are some smooth functions defined
locally on $M$.

Note that if we additionally assume that $\kappa$ induces the identity on the
core bundle, then the coefficients $\alpha^{j}_{i}(x)$ are simply constants
$\delta^{j}_{i}$. We shall show later (see Lemma 2.11) that such linear VBCs
correspond to (Lie) algebroid structures. The anchor maps (left and right) are
locally given by functions $Q^{b}_{i}(x)$ and $\widetilde{Q}^{b}_{i}(x)$,
respectively, while functions $Q^{k}_{ij}(x)$ encode the bracket operation in
a given basis of sections of the bundle $\sigma$, dual to functions $(y^{i})$.

######  Lemma 2.9 (a class of linear VBCs).

Let $\sigma\colon E\rightarrow M$ be a vector bundle and let
$\kappa\colon\mathrm{T}\sigma\rightarrow\\!\\!\vartriangleright\tau_{E}$ be a
linear VBC. Assume additionally that $\kappa$ induces the identity on the core
bundles, i.e., $\kappa\cap(C\times
C)=\operatorname{graph}(\operatorname{id}_{C})$. Then:

  1. $(i)$

The base map $\rho_{L}:=\underline{\kappa}\colon E\rightarrow\mathrm{T}M$ is a
VB morphism covering the identity map $\operatorname{id}_{M}$.

  2. $(ii)$

Relation $\kappa^{\prime}:=(\tau_{E}\times\mathrm{T}\sigma)(\kappa)\subset
E\times\mathrm{T}M$, which is the $\tau_{E}\times\mathrm{T}\sigma$-projection
of $\kappa$, is a graph of a VB morphism $\rho_{R}\colon
E\rightarrow\mathrm{T}M$ covering the identity map $\operatorname{id}_{M}$.

  3. $(iii)$

Relation $\kappa$ respects the action of the core $C$, i.e., if
$(A,B)\in\kappa$ then
$(A\bm{\boldsymbol{+}}c,B\bm{\boldsymbol{+}}c)\in\kappa$, whenever the
addition of $c\in C$ makes sense.

  4. $(iv)$

The inverse relation $\kappa^{T}$ is a linear VBC from $\mathrm{T}\sigma$ to
$\tau_{E}$ over the base map $\rho_{R}\colon E\rightarrow\mathrm{T}M$ which
also induces the identity on the cores.

###### Proof.

The assertion has a rather straightforward geometric justification based
essentially on the bi-homogeneity of $\kappa$. For brevity, however, we prefer
the following local argument.

Since $\kappa$ is the identity on the cores, we conclude that
$\kappa\cap(M\times M)=\Delta_{M}\subset M\times M$ is a diagonal, and hence
$\kappa$ relates only the elements in the same fibres over $M$. Now we are
precisely in the situation described in Remark 2.8 and the assertion follows
easily from the local description (2.5). ∎

###  2.2 Description of (Lie) algebroids in terms of VBCs

In this part we shall recall the definition of a general algebroid, and latter
rephrase it in the language of vector bundle comorphisms introduced above. A
possibility of such a reformulation is of course well-recognized in the
literature since the very introduction of the concept of a general algebroid
[19] (see also [12, 38]). However, this topic was never systematically studied
for its own sake. In particular, we are not aware that the axioms of a (Lie)
algebroid were ever directly formulated in terms of the corresponding VBCs.
Despite this, we do not claim any originality in this area, as such a
formulation is straightforward and natural. Our goal is rather to show the
consistency and naturality of the approach to (Lie) algebroids based on
differential relations (the notions of a subalgebroid, a morphism between
algebroids, the Lie axiom, various specific types of algebroids, etc. are
intrinsically defined within the category of vector bundle comorphisms). In
consequence, we prepare the ground for a later definition of a higher
algebroid in Section 4.

General algebroids. General algebroids were introduced by Grabowski and
Urbański [18, 19] as double vector bundle morphisms of a special kind. Their
approach was motivated by the study of the geometry of mechanics and
variational calculus originated by Tulczyjew [44, 45]. Skew algebroids,
almost-Lie algebroids and Lie algebroids may be regarded as special subclasses
of this general notion.

######  Definition 2.10 (general algebroid).

A _general algebroid_ structure on a vector bundle $\sigma\colon
E\\!\rightarrow\\!M$ is given by a bilinear _bracket_ $[\cdot,\cdot]$ on the
space of smooth sections of $\sigma$, together with a pair of vector bundle
maps (_left_ and _right anchors_) $\rho_{L},\rho_{R}\colon
E\rightarrow\mathrm{T}M$ over the identity on $M$ such that

$\displaystyle[f\cdot a,g\cdot b]=f\rho_{L}(a)(g)\cdot b-g\rho_{R}(b)(f)\cdot
a+fg\cdot[a,b]$ (2.6)

for every sections $a,b\in\operatorname{Sec}_{M}(E)$ and every smooth
functions $f,g\in C^{\infty}(M)$.

In addition

  1. (i)

If the bracket is skew-symmetric, i.e., $[a,b]=-[b,a]$ (in particular, left
and right anchors coincide, i.e., $\rho_{L}=\rho_{R}=:\rho$ and we speak
simply about the _anchor_) we call $\sigma$ a _skew algebroid_.

  2. (ii)

If $\sigma$ is a skew algebroid and the anchor $\rho$ maps the algebroid
bracket $[\cdot,\cdot]$ to the Lie bracket of vector fields on $M$, i.e.,

$\displaystyle\rho[a,b]=[\rho(a),\rho(b)]_{\mathrm{T}M},$ (2.7)

we call $\sigma$ an _almost-Lie algebroid_ (AL algebroid, in short).

  3. (iii)

If $\sigma$ is an almost-Lie algebroid and the bracket $[\cdot,\cdot]$
satisfies the Jacobi identity, we call $\sigma$ a _Lie algebroid_.

We will refer to the objects from the above definition as to _$($ Lie$)$
algebroids_, i.e., writing that, say,
$(\sigma,\rho_{L},\rho_{R},[\cdot,\cdot])$ is a (Lie) algebroid, means that it
is either a Lie algebroid, an AL algebroid, a skew algebroid, or just a
general algebroid. By using the word “(Lie)” we emphasize that all the above
concepts were derived from the standard notion of the Lie algebroid by
relaxing its axioms. Moreover, this should prevent possible confusions with
Courant algebroid and other notions of an algebroid present in the literature.

From (Lie) algebroids to VBCs. We will now construct a canonical Zakrzewski
morphism related with a given (Lie) algebroid structure. Recall Lemma 2.9
describing the structure of a class of linear VBCs intertwining the two VB
structures on the total space of the tangent bundle $\mathrm{T}E$ of a vector
bundle $\sigma\colon E\rightarrow M$. It turns out that an algebroid structure
on $\sigma$ induces such a VBC.

######  Lemma 2.11 (from (Lie) algebroids to VBCs).

Let $(\sigma\colon E\rightarrow M,[\cdot,\cdot],\rho_{L},\rho_{R})$ be a
$($Lie$)$ algebroid. Formula

$\displaystyle\kappa_{a}\left[\mathrm{T}b(\rho_{L}(a))\right]:=\mathrm{T}a(\rho_{R}(b))\bm{\boldsymbol{+}}[a,b],$
(2.8)

where $a,b\in\operatorname{Sec}_{M}(E)$ are arbitrary sections, extends to the
unique relation $\kappa\subset\mathrm{T}E\times\mathrm{T}E$ such that

  1. $(i)$

$\kappa$ is a linear vector bundle comorphism from
$\mathrm{T}\sigma\colon\mathrm{T}E\rightarrow\mathrm{T}M$ to the tangent
bundle $\tau_{E}\colon\mathrm{T}E\rightarrow E$ covering the left anchor
$\rho_{L}\colon E\rightarrow\mathrm{T}M$,

  2. $(ii)$

$\kappa$ induces the identity on the core bundle, i.e., $\kappa\cap(C\times
C)=\operatorname{graph}(\operatorname{id}_{C})$.

Moreover, the $(\tau_{E}\times\mathrm{T}\sigma)$-projection of $\kappa$ is the
right anchor $\rho_{R}\colon E\rightarrow\mathrm{T}M$.

The proof is given in Appendix A.

######  Remark 2.12 (the inverse of $\kappa$).

Note that, according to Lemma 2.9(iv), the inverse relation $\kappa^{T}$ is
also a linear VBC (this time over the right anchor $\rho_{R}$) inducing the
identity on the cores. It clearly satisfies (we use the properties (i) and
(ii) of $\kappa$ to transform equality (2.8)) the condition

$\displaystyle\kappa_{b}^{T}\left[\mathrm{T}a(\rho_{R}(b))\right]=\mathrm{T}b(\rho_{L}(a))\bm{\boldsymbol{-}}[a,b],$

i.e., passing from $\kappa$ to $\kappa^{T}$ corresponds to changing the
bracket $[a,b]$ to $[a,b]^{T}:=-[b,a]$ (note that this operation interchanges
left and right anchors). Obviously $\kappa=\kappa^{T}$ if the bracket is skew-
symmetric.

######  Example 2.13 (the tangent algebroid).

The tangent bundle $\tau_{M}\colon\mathrm{T}M\rightarrow M$ of a manifold $M$
carries a canonical Lie algebroid structure. The left and right anchors are
the identity maps $\rho_{L}=\rho_{R}=\operatorname{id}_{\mathrm{T}M}$ and the
bracket $[\cdot,\cdot]_{\mathrm{T}M}$ is the commutator of vector fields.

In this case the corresponding VBC is, in fact, an isomorphism of DVB
$\kappa_{M}\colon\mathrm{T}\tau_{M}\rightarrow\tau_{\mathrm{T}M}$. If
$(x^{a})$ are local coordinates on $M$, $(x^{a},\dot{x}^{b})$ induced
coordinates on $\mathrm{T}M$ and $(x^{a},\dot{x}^{b},\delta
x^{c},\delta\dot{x}^{d})$ induced coordinates on $\mathrm{T}\mathrm{T}M$,
$\kappa_{M}$ is expressed as

$\displaystyle\kappa_{M}\colon\ \big{(}x^{a},\dot{x}^{b},\delta
x^{c},\delta\dot{x}^{d}\big{)}\longmapsto\big{(}x^{a},\delta
x^{b},\dot{x}^{c},\delta\dot{x}^{d}\big{)}.$

Note that $\kappa_{M}$ interchanges the two VB structures on
$\mathrm{T}\mathrm{T}M$, $\mathrm{T}\tau_{M}$ and $\tau_{\mathrm{T}M}$.

######  Example 2.14 (the tangent lift of a (Lie) algebroid).

If $(\sigma\colon E\rightarrow M,\rho_{L},\rho_{R},[\cdot,\cdot])$ is a (Lie)
algebroid structure, then
$\mathrm{T}\sigma\colon\mathrm{T}E\rightarrow\mathrm{T}M$ also carries a
canonical algebroid structure
$(\mathrm{T}\sigma,\mathrm{d}_{\mathrm{T}}\rho_{L}$,
$\mathrm{d}_{\mathrm{T}}\rho_{R},[\cdot,\cdot]_{\mathrm{d}_{\mathrm{T}}})$
called the _tangent lift of the algebroid_ structure on $\sigma$. This
structure is determined by conditions

$\displaystyle\mathrm{d}_{\mathrm{T}}\rho_{L}=\kappa_{M}\circ\mathrm{T}\rho_{L},\qquad\mathrm{d}_{\mathrm{T}}\rho_{R}=\kappa_{M}\circ\mathrm{T}\rho_{R}\qquad\text{and}\qquad[\mathrm{T}a,\mathrm{T}b]_{\mathrm{d}_{\mathrm{T}}}=\mathrm{T}[a,b]$

for any sections $a,b\in\operatorname{Sec}_{M}(E)$.

In this case (see [19]) the VBC $\mathrm{d}_{\mathrm{T}}\kappa$, corresponding
to the considered algebroid structure on $\mathrm{T}\sigma$, is the tangent
lift of the VBC $\kappa$ corresponding to the initial algebroid structure on
$\sigma$ composed with two canonical isomorphisms $\kappa_{E}$, i.e.,

$\displaystyle\mathrm{d}_{\mathrm{T}}\kappa=\kappa_{E}\circ\mathrm{T}\kappa\circ\kappa_{E}.$

This construction has a natural generalization to higher tangent lifts
$\mathrm{T}^{k}\sigma\colon\mathrm{T}^{k}E\rightarrow\mathrm{T}^{k}M$. It will
be discussed in detail in the second paragraph of Section 4.1.

From VBCs to algebroids. In fact, relation $\kappa$ introduced in Lemma 2.11
completely characterizes the (Lie) algebroid structure on $\sigma$.

######  Proposition 2.15 (from VBCs to (Lie) algebroids).

Let $\sigma\colon E\rightarrow M$ be a vector bundle. A linear vector bundle
comorphism $\kappa$ from
$\mathrm{T}\sigma\colon\mathrm{T}E\rightarrow\mathrm{T}M$ to the tangent
bundle $\tau_{E}\colon\mathrm{T}E\rightarrow E$, which induces the identity on
the core bundles, i.e., $\kappa\cap(C\times
C)=\operatorname{graph}(\operatorname{id}_{C})$, provides $\sigma$ with the
unique general algebroid structure.

The left anchor $\rho_{L}$ $($resp., the right anchor $\rho_{R})$ is given by
the base map of $\kappa$ $($resp., the base map of $\kappa^{T})$ and the
bracket is given by

$\displaystyle\mathbf{V}_{a}[a,b]:=\kappa_{a}\left[\mathrm{T}b(\rho_{L}(a))\right]-\mathrm{T}a(\rho_{R}(b)),$
(2.9)

for any sections $a,b\in\operatorname{Sec}_{M}(E)$.

The proof is given in Appendix A.

######  Remark 2.16 (alternative definition of a general algebroid).

By the results of Proposition 2.15 and Lemma 2.11 we can equivalently define a
general algebroid structure $(\sigma,\rho_{L},\rho_{R},[\cdot,\cdot])$ on
$\sigma\colon E\rightarrow M$ as a pair $(\sigma,\kappa)$, where
$\kappa\colon\mathrm{T}\sigma\rightarrow\\!\\!\vartriangleright\tau_{E}$ is a
linear VBC satisfying natural properties. In what follows we shall often refer
to this description.

######  Remark 2.17 (the dual of $\kappa$).

The fact that $\kappa$ induces the identity on the core implies that its dual
(which is a proper vector bundle morphism, and even a DVB morphism by the
linearity of $\kappa$)

$\textstyle{\mathrm{T}^{\ast}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{E}^{*}}$$\scriptstyle{\kappa^{\ast}}$$\textstyle{\mathrm{T}E^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{T}\sigma^{\ast}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{L}}$$\textstyle{\mathrm{T}M}$

covers the identity $\operatorname{id}_{E^{\ast}}$ under the projections
$\mathrm{T}^{\ast}E\rightarrow E^{\ast}$ and $\mathrm{T}E^{\ast}\rightarrow
E^{\ast}$ (the core of a DVB becomes a side bundle under dualization [29]). In
other words,
$\kappa^{\ast}\colon\mathrm{T}E^{\ast}\rightarrow\mathrm{T}^{\ast}E\simeq\mathrm{T}^{\ast}E^{\ast}$
corresponds to a linear bi-vector on $E^{\ast}$. This is an original point of
view of [18].

######  Remark 2.18 (a local form of $\kappa$).

By Remark 2.8, the local form of a linear VBC corresponding to a given general
algebroid structure is given by formulas (2.5) with
$\alpha^{k}_{i}(x)=\delta^{k}_{i}$. Within this description $\rho_{L}\colon
E\rightarrow\mathrm{T}M$, the base map of $\kappa$, is given by
$\rho_{L}\colon
E\ni(\underline{x}^{a},\underline{y}^{i})\longmapsto(x^{a}=\underline{x}^{a}$,
$\dot{x}^{b}=Q^{b}_{i}(x)\underline{y}^{i})\in\mathrm{T}M$. Similarly, the
right anchor reads as $\rho_{R}\colon
E\ni(x^{a},y^{i})\longmapsto(\underline{x}^{a}=x^{a}$,
$\dot{\underline{x}}^{b}=\widetilde{Q}^{b}_{i}(x)y^{i})\in\mathrm{T}M$.

Take now local sections $a(x)\sim(x^{a},y^{j}=a^{i}(x))$, and
$b(x)\sim(x^{a},y^{j}=b^{i}(x))$ of $E$. We can use formula (2.5) together
with (2.9) to calculate a local expression for an algebroid bracket of these
two sections. Simple calculations (which we omit here) lead to

$\displaystyle[a,b](x)\sim\left(x^{a},y^{k}=\frac{\partial b^{k}}{\partial
x^{a}}(x)Q^{a}_{i}(x)a^{i}(x)-\frac{\partial a^{k}}{\partial
x^{a}}(x)\widetilde{Q}^{a}_{j}(x)b^{j}(x)+Q^{k}_{ij}a^{i}(x)b^{j}(x)\right).$

(Lie) algebroid morphisms as $\bm{{\mathcal{VBC}}}$-morphisms. Next we shall
show that within the interpretation of (Lie) algebroids as vector bundle
comorphisms, the notion of an algebroid morphism corresponds to a
${\mathcal{VBC}}$-morphism between appropriate relations.

Intuitively, a morphism between two (Lie) algebroid structures on
$\sigma\colon E\rightarrow M$ and on $\sigma^{\prime}\colon
E^{\prime}\rightarrow M^{\prime}$ should be a vector bundle map $\phi\colon
E\rightarrow E^{\prime}$ over $\underline{\phi}\colon M\rightarrow M^{\prime}$
which intertwines the anchors and the algebroid bracket on sections. This
intuition, however, faces immediate problems as, in general, a VB morphism
$\phi$ does not map sections of $\sigma$ to sections of $\sigma^{\prime}$.
This problem is solved by passing to the pull-back bundles.

######  Definition 2.19 (morphism of (Lie) algebroids, [34]).

Let $(\sigma\colon E\rightarrow M,[\cdot,\cdot],\rho_{L},\rho_{R})$
and$(\sigma^{\prime}\colon E^{\prime}\rightarrow
M^{\prime},[\cdot,\cdot]^{\prime},\rho_{L}^{\prime},\rho_{R}^{\prime})$ be
(Lie) algebroids. A VB morphism $\phi\colon E\rightarrow E^{\prime}$ over
$\underline{\phi}\colon M\rightarrow M^{\prime}$ is a  _morphism_ between the
algebroid structures on $\sigma$ and $\sigma^{\prime}$ if and only if the VB
morphisms $\phi$ and $\mathrm{T}\underline{\phi}$ relate the left and the
right anchors of $\sigma$ and $\sigma^{\prime}$, i.e.,

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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36.00694pt\raise-39.36108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.00694pt\raise-39.36108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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(2.10)

and if for every sections $a,b\in\operatorname{Sec}_{M}(E)$ such that their
push-forwards
$\phi_{\ast}a,\phi_{\ast}b\in\operatorname{Sec}_{M}(\underline{\phi}^{\ast}E^{\prime})$
can be (locally) presented as finite sums
$\phi_{\ast}a=\sum_{i}f_{i}\cdot\underline{\phi}^{\ast}a_{i}$ and
$\phi_{\ast}b=\sum_{j}g_{j}\cdot\underline{\phi}^{\ast}b_{j}$ for some
functions $f_{i},g_{j}\in\mathcal{C}^{\infty}(M)$ and some sections
$a_{i},b_{j}\in\operatorname{Sec}_{M^{\prime}}(E^{\prime})$ we have

$\displaystyle\phi_{\ast}[a,b]=\sum_{i,j}\left(\rho_{L}(a)(g_{j})\cdot\underline{\phi}^{\ast}b_{j}-\rho_{R}(b)(f_{i})\cdot\underline{\phi}^{\ast}a_{i}+f_{i}g_{j}\cdot\underline{\phi}^{\ast}[a_{i},b_{j}]^{\prime}\right).$
(2.11)

######  Remark 2.20 (on notion of an algebroid morphism).

For any $x\in M$ there is a neighbourhood $U$ of $x$ in $M$ and a
neighbourhood $U^{\prime}$ of $\underline{\phi}(x)$ in $M^{\prime}$ such that
for any section $a$ of $\sigma$ we have
$\phi_{\ast}a=\sum_{i}f_{i}\underline{\phi}^{\ast}e_{i}^{\prime}$ where
$(e_{i}^{\prime})$ is a basis of sections of $\sigma^{\prime}$ over
$U^{\prime}$ and $f_{i}$ are some functions on $U$. This explains a local
character of the above definition. There are other equivalent and more elegant
formulation of the definition of an algebroid morphism – we discus some of
them in Proposition A.1 (see also [19, Definition 3], where general algebroids
are considered as a special type of Leibniz structures on the dual bundle and
[15] for a generalization of this notion to an algebroid relation).

It may seem unclear if condition (2.11) is well-posed, i.e., if it does not
depend on the presentation of $\phi_{\ast}a$ and $\phi_{\ast}b$ as finite sums
of sections with $\mathcal{C}^{\infty}(M)$ coefficients. To prove that this is
the case, we check that the right-hand side of (2.11) is tensorial with
respect to $a_{i}$ and $b_{j}$ (for this it is crucial that condition (2.10)
holds). The essential calculations (for a simpler case of a skew algebroid)
can be found in the classical book [34].

It turns out that the above definition has a very elegant (and much simpler)
interpretation in the language of VBCs naturally related with the (Lie)
algebroid structure.

######  Proposition 2.21 ((Lie) algebroid morphisms as
${\mathcal{VBC}}$-morphisms).

Let $(\sigma\colon E\\!\rightarrow\\!M,[\cdot,\cdot],\rho_{L},\rho_{R})$ and
$(\sigma^{\prime}\colon E^{\prime}\rightarrow
M^{\prime},[\cdot,\cdot]^{\prime},\rho_{L}^{\prime},\rho_{R}^{\prime})$ be
$($Lie$)$ algebroids with the corresponding vector bundle comorphisms
$\kappa\colon\mathrm{T}\sigma\rightarrow\\!\\!\vartriangleright\tau_{E}$ and
$\kappa^{\prime}\colon\mathrm{T}\sigma^{\prime}\rightarrow\\!\\!\vartriangleright\tau_{E^{\prime}}$,
respectively. A VB map $\phi\colon E\rightarrow E^{\prime}$ over
$\underline{\phi}\colon M\rightarrow M^{\prime}$ is a morphism between the
algebroid structures on $\sigma$ and $\sigma^{\prime}$ if and only if
$(\mathrm{T}\phi,\mathrm{T}\phi)$ is a ${\mathcal{VBC}}$-morphism from
$\kappa$ to $\kappa^{\prime}$.

The proof is given in Appendix A.

Subalgebroids and algebroidal relations. Let us now describe the notion of a
subalgebroid in terms of VBCs. Recall the following definition.

######  Definition 2.22 (subalgebroid, [15, 34]).

Let $(\sigma\colon E\rightarrow M,\rho_{L},\rho_{R},[\cdot,\cdot])$ be a (Lie)
algebroid. A vector subbundle $\sigma^{\prime}\colon E^{\prime}\rightarrow
M^{\prime}$ of $\sigma$ is called a _subalgebroid_ of $\sigma$ if the
following two conditions are satisfied

  1. (i)

The restrictions of the anchors $\rho_{L}|_{E^{\prime}}$ and
$\rho_{R}|_{E^{\prime}}$ map $E^{\prime}\subset E$ to
$\mathrm{T}M^{\prime}\subset\mathrm{T}M$.

  2. (ii)

If sections $a,b\in\operatorname{Sec}_{M}(E)$ are such that
$a|_{M^{\prime}},b|_{M^{\prime}}\in\operatorname{Sec}_{M^{\prime}}(E^{\prime})$,
then $[a,b]|_{M^{\prime}}\in\operatorname{Sec}_{M^{\prime}}(E^{\prime})$.

The first of the above conditions assures us that the bracket operation
$[\cdot,\cdot]_{E^{\prime}}$ defined on sections of $\sigma^{\prime}$ by
$[\widetilde{a},\widetilde{b}]_{E^{\prime}}:=[a,b]|_{M^{\prime}}$ does not
depend on the choice of the extensions $a,b\in\operatorname{Sec}_{M}(E)$ of
sections
$\widetilde{a},\widetilde{b}\in\operatorname{Sec}_{M^{\prime}}(E^{\prime})$.
The second condition guarantees that the section space
$\operatorname{Sec}_{M^{\prime}}(E^{\prime})$ is closed with respect to this
bracket. Clearly the subbundle $\sigma^{\prime}$ carries a general algebroid
structure inherited from $\sigma$.

In the face of the relationship between (Lie) algebroids and VBCs, the VBC
$\kappa\colon\mathrm{T}\sigma\\!\rightarrow\\!\\!\vartriangleright\\!\tau_{E}$
corresponding to the algebroid structure on $\sigma$ should induce some VBC
$\kappa^{\prime}\colon\mathrm{T}\sigma^{\prime}\rightarrow\\!\\!\vartriangleright\tau_{E^{\prime}}$
corresponding to the structure of a subalgebroid on $\sigma^{\prime}$
described above. We claim that such a $\kappa^{\prime}$ is a  _fine
restriction_ of $\kappa$ in the sense of the definition below.

######  Definition 2.23 (fine restriction).

Let, for $i=1,2$, $\sigma_{i}^{\prime}\colon E_{i}^{\prime}\rightarrow
M_{i}^{\prime}$ be a vector subbundle of $\sigma_{i}\colon E_{i}\rightarrow
M_{i}$, and let
$r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$ be a VBC over
a base map $\underline{r}\colon M_{2}\rightarrow M_{1}$. We say that $r$
_restricts fine to $\sigma_{1}^{\prime}\times\sigma_{2}^{\prime}$_ if
$\underline{r}(M_{2}^{\prime})\subset M_{1}^{\prime}$ and if for any $X\in
E_{1}$ and $Y\in E_{2}$ that are $r$-related and such that $X\in
E_{1}^{\prime}$ while $\sigma_{2}(Y)\in M_{2}^{\prime}$ we have $Y\in
E_{2}^{\prime}$. If this is the case, relation $r^{\prime}$, defined as the
intersection of $r$ with $E_{1}^{\prime}\times E_{2}^{\prime}$, defines a VBC
from $\sigma_{1}^{\prime}$ to $\sigma_{2}^{\prime}$.

The equivalence of the classical notion of a subalgebroid with the notion of a
fine restriction of the corresponding VBC can be easily proved.

######  Proposition 2.24 (on subalgebroids).

Let $(\sigma\colon\\!E\\!\rightarrow\\!M,\kappa)$ be a $($Lie$)$ algebroid and
let $\sigma^{\prime}\colon\\!E^{\prime}\\!\rightarrow\\!M^{\prime}$ be a
vector subbundle of $\sigma$. Then the following conditions are equivalent:

  1. $(i)$

The subbundle $\sigma^{\prime}$ is a subalgebroid of $\sigma$.

  2. $(ii)$

The inclusion map $\iota\colon\sigma^{\prime}\hookrightarrow\sigma$ is an
algebroid morphism.

  3. $(iii)$

The VBC $\kappa$ restricts fine to
$\mathrm{T}\sigma^{\prime}\times\tau_{E^{\prime}}\subset\mathrm{T}\sigma\times\tau_{E}$.

###### Proof.

It involves some elementary diagram-chasing to check that (iii) is equivalent
to $(\mathrm{T}\iota,\mathrm{T}\iota)$ being a ${\mathcal{VBC}}$-morphism
between $\kappa^{\prime}$ and $\kappa$. Due to Proposition 2.21 the latter
condition is equivalent to (ii). Finally the equivalence of conditions (i) and
(ii) is a standard fact in the theory of Lie algebroids (see [34, Chapter 4]).
∎

Following Grabowski [15], we recall the concept of an algebroidal relation,
which is a generalization of a morphism of algebroids. It is closely related
to the notion of a subalgebroid.

######  Definition 2.25 (algebroidal relation).

Let $(\sigma_{1}\colon E_{1}\rightarrow M_{1},\kappa_{1})$ and
$(\sigma_{2}\colon E_{2}\rightarrow M_{2},\kappa_{2})$ be (Lie) algebroids. A
relation $r\subset E_{1}\times E_{2}$ is called an _algebroidal relation_ if
the graph of $r$ is a subalgebroid of the product algebroid
$(\sigma_{1}\times\sigma_{2},\kappa_{1}\times\kappa_{2})$.

If $r$ is a vector bundle morphism, we recover the notion of an algebroid
morphism (see Proposition A.1). Algebroidal relations have an elegant
characterization in terms of the anchors and the algebroid brackets.

######  Proposition 2.26 (on algebroidal relations).

Let $(\sigma_{1}\colon E_{1}\rightarrow
M_{1},\rho_{1L},\rho_{1R},[\cdot,\cdot]_{1})\simeq(\sigma_{1},\kappa_{1})$ and
$(\sigma_{2}\colon E_{2}\rightarrow
M_{2},\rho_{2L},\rho_{2R},[\cdot,\cdot]_{2})\simeq(\sigma_{2},\kappa_{2})$ be
(Lie) algebroids and let
$r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$ be a VBC
covering $\underline{r}\colon M_{2}\rightarrow M_{1}$. The following are
equivalent:

  1. $(i)$

$r$ is an algebroidal relation,

  2. $(ii)$

    1. $(a)$

For any section $s_{1}\in\operatorname{Sec}_{M_{1}}(E_{1})$ and $I=L,R$, the
vector fields $\rho_{2I}(\hat{r}(s_{1}))$ and $\rho_{1I}(s_{1})$ are
$\underline{r}$-related and

    2. $(b)$

for any sections $s_{1},s_{1}^{\prime}\in\operatorname{Sec}_{M_{1}}(E_{1})$ we
have

$\displaystyle\hat{r}([s_{1},s_{1}^{\prime}]_{1})=[\hat{r}(s_{1}),\hat{r}(s_{1}^{\prime})]_{2}.$

The proof is given in Appendix A.

######  Remark 2.27.

We stress that although the anchor map is uniquely determined by the bracket
operation in any general algebroid, the above condition (iia) does not follow
from (iib). A simple counterexample is provided by the zero endomorphism in a
general algebroid.

The following result states that fine restrictions of algebroidal relations to
subalgebroids remain algebroidal relations.

######  Proposition 2.28 (restricting algebroidal relations).

Let, for $j=1,2$, $(\sigma_{j}^{\prime}\colon E_{j}^{\prime}\rightarrow
M^{\prime}_{j},\kappa_{j}^{\prime})$ be a subalgebroid of $(\sigma_{j}\colon
E_{j}\rightarrow M_{j},\kappa_{j})$. Let us assume that a VBC

$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{r}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{2}}$$\textstyle{M_{1}}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{r}}$

restricts fine to $r^{\prime}\subset E_{1}^{\prime}\times E_{2}^{\prime}$.
Then if $r$ is an algebroidal relation then so is $r^{\prime}$.

###### Proof.

We should check that $r^{\prime}$ is a subalgebroid of
$(\sigma_{1}^{\prime}\times\sigma_{2}^{\prime},\kappa_{1}^{\prime}\times\kappa_{2}^{\prime})$.
Since $r$ restricts fine to $E_{1}^{\prime}\times E_{2}^{\prime}$, the
relation $r^{\prime}$ is a vector subbundle of $E_{1}^{\prime}\times
E_{2}^{\prime}$. Let us take
$X,Y\in\mathrm{T}E_{1}^{\prime}\times\mathrm{T}E_{2}^{\prime}$ such that $X,Y$
are $\kappa_{1}^{\prime}\times\kappa_{2}^{\prime}$-related and
$X\in\mathrm{T}r^{\prime}$ while $Y$ lies over $y\in r^{\prime}\subset
E_{1}^{\prime}\times E_{2}^{\prime}$. We have to show that
$Y\in\mathrm{T}r^{\prime}$.

First note that $Y\in\mathrm{T}r$, because $r$ respects algebroid structures
of $E_{1}$ and $E_{2}$, and $X,Y$ are also
$\kappa_{1}\times\kappa_{2}$-related as $E_{1}^{\prime}\times E_{2}^{\prime}$
is a subalgebroid of $E_{1}\times E_{2}$, and $X\in\mathrm{T}r$ while $y\in
r^{\prime}\subset r$. Next, by [27, Theorem A.5], $\mathrm{T}r$ restricts fine
to $\mathrm{T}E_{1}^{\prime}\times\mathrm{T}E_{2}^{\prime}$. Therefore, since
$Y$ belongs also to $\mathrm{T}E_{1}^{\prime}\times\mathrm{T}E_{2}^{\prime}$,
we get
$Y\in\mathrm{T}r^{\prime}=\mathrm{T}r\cap(\mathrm{T}E_{1}^{\prime}\times\mathrm{T}E_{2}^{\prime})$
as we claimed. ∎

Characterization of algebroids with an additional structure. We shall now
express specific conditions (i)–(iii) from Definition 2.10 in terms of the
corresponding VBC $\kappa$.

######  Lemma 2.29 (characterization of various algebroids).

Let $(\sigma\colon E\rightarrow M,[\cdot,\cdot],\rho_{L},\rho_{R})$ be a
general algebroid structure on $\sigma$ and let
$\kappa\colon\mathrm{T}\sigma\rightarrow\\!\\!\vartriangleright\tau_{E}$ be
the corresponding vector bundle comorphism. Then $\sigma$ is

  1. $(i)$

A skew algebroid if and only if $\kappa$ is symmetric, i.e., if
$(X,Y)\in\kappa$ then $(Y,X)\in\kappa$ $($equivalently $\kappa=\kappa^{T})$.

  2. $(ii)$

An almost-Lie algebroid if and only if it is skew, and
$(\mathrm{T}\rho,\mathrm{T}\rho)\colon\kappa\Rightarrow\kappa_{M}$ is a
morphism of VBCs $($in the sense of Definition 2.3):

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35.99873pt\raise-31.64304pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\kappa_{M}}$}}}\kern
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(2.12)

Here $\rho=\rho_{L}=\rho_{R}$ is the anchor map.

  3. $(iii)$

A Lie algebroid if and only if it is almost-Lie and
$\kappa\subset\mathrm{T}E\times\mathrm{T}E$ is a subalgebroid of the product
algebroid
$(\mathrm{T}\sigma\times\tau_{E},\mathrm{d}_{\mathrm{T}}\kappa\times\kappa_{E})$.
In other words,
$\kappa\colon\mathrm{T}\sigma\rightarrow\\!\\!\vartriangleright\tau_{E}$ is an
algebroidal relation.

The proof is given in Appendix A.

An application – prolongations of AL algebroids. Throughout this section we
argued that VBCs provide a consistent language to describe (Lie) algebroids,
alternative to the standard treatment of the topic. Besides, some known
constructions in the theory of (Lie) algebroids have more evident definitions
in the language of vector bundle comorphism. To justify this claim we shall
now give a non-standard definition of a prolongation of an AL algebroid over a
fibration [11, 21], the crucial notion in the Lagrangian and Hamiltonian
formalisms for mechanics on Lie algebroids developed by Martínez in [36, 37].

######  Example 2.30 (prolongation of an AL algebroid over a fibration).

Let $\pi\colon P\rightarrow M$ be a fibration and let $(\sigma\colon
E\rightarrow M,\kappa)$ be an almost-Lie algebroid. Denote by $\rho\colon
E\rightarrow\mathrm{T}M$ the related anchor map. Then
$\mathcal{T}^{E}P:=E\times_{\mathrm{T}M}\mathrm{T}P=\\{(e,v)\in
E\times\mathrm{T}P\colon\rho(e)=\mathrm{T}\pi(v)\\}$ is a vector bundle over
$P$ which carries an almost-Lie algebroid structure defined by the restriction
of the relation
$\kappa\times\kappa_{P}\colon\mathrm{T}E\times\mathrm{T}\mathrm{T}P\rightarrow\\!\\!\vartriangleright\mathrm{T}E\times\mathrm{T}\mathrm{T}P$
to
$\mathrm{T}(\mathcal{T}^{E}P)\simeq\mathrm{T}E\times_{\mathrm{T}\mathrm{T}M}\mathrm{T}\mathrm{T}P$:

$\textstyle{\mathrm{T}E\times_{\mathrm{T}\mathrm{T}M}\mathrm{T}\mathrm{T}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa\times\kappa_{P}}$$\textstyle{\mathrm{T}E\times_{\mathrm{T}\mathrm{T}M}\mathrm{T}\mathrm{T}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{T}P}$$\textstyle{E\times_{\mathrm{T}M}\mathrm{T}P.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Indeed, the only thing to check is that $\mathcal{T}^{E}P$ is a subalgebroid
of the product algebroid $(\sigma\times\tau_{P}\colon
E\times\mathrm{T}P\rightarrow M\times P,\kappa\times\kappa_{P})$. This is
straightforward, given $X,X^{\prime}\in\mathrm{T}E$ and
$Y,Y^{\prime}\in\mathrm{T}\mathrm{T}P$ such that $(X,X^{\prime})\in\kappa$,
$Y^{\prime}=\kappa_{P}(Y)$, and $(X,Y)$ is tangent to $\mathcal{T}^{E}P$, we
need to prove that $(X^{\prime},Y^{\prime})$ is also tangent to
$\mathcal{T}^{E}P$, i.e., that
$\mathrm{T}\rho(X^{\prime})=\mathrm{T}\mathrm{T}\pi(Y^{\prime})$. But
$\mathrm{T}\rho(X^{\prime})=\kappa_{M}(\mathrm{T}\rho(X))$ while
$\mathrm{T}\mathrm{T}\pi(Y^{\prime})=\kappa_{M}(\mathrm{T}\mathrm{T}\pi(Y))$
(due to the assumption that $\kappa$ is almost-Lie) and the result follows.

Another non-standard characterization of the notion of the prolongation, also
emphasizing the role of the AL axiom (2.7), was recently provided by one of us
in [24, Proposition 3.1].

##  3 Recollection of $\mathbb{N}$-graded manifolds

$\mathbb{N}$-graded manifolds and homogeneity structures. _Higher order
algebroids_ which we shall introduce and study in the next section are modeled
on geometric objects which generalize the notion of a vector bundle. Such a
generalization (catching, in particular, the canonical graded structure of the
higher tangent bundle $\mathrm{T}^{k}M$ – a fundamental example from the point
of view of physical applications) was first proposed by Voronov in [47],
within the framework of supergeometry, as non-negatively graded
($\mathbb{N}$-graded, for short) manifolds. Voronov noticed that they expand
into a tower of fibrations and suggested to see a non-negative grading as a
generalization of a linear structure. In this paper we choose to work with
_purely even non-negatively graded manifolds_ , a particular subclass of
Voronov’s objects. In [17] they are referred to as _graded bundles_ (see also
the introduction to [6]). Here we shall recall basic properties and
constructions associated with these objects.

An important example is the _higher $(k^{\mathrm{th}}$-order$)$ tangent bundle
$\tau^{k}_{M}\colon\mathrm{T}^{k}M\rightarrow M$_ of a manifold $M$,
consisting of $k^{\mathrm{th}}$-order tangency classes (called _$k$
-velocities_) of curves in $M$. Bundle $\mathrm{T}^{1}M=\mathrm{T}M$ is just
the tangent bundle of $M$, however for $k>1$, $\tau^{k}_{M}$ is no longer a
vector bundle. We shall see this at the elementary level.

Given a smooth function $f$ on a manifold $M$ and an integer
$\alpha=0,1,\ldots,k$ one can construct a function $f^{(\alpha)}$ on
$\mathrm{T}^{k}M$, the so-called _$\alpha$ -lift of $f$_ (see [40]). It is
defined by

$\displaystyle
f^{(\alpha)}(\bm{\mathrm{t}}^{k}\gamma):=\left.\frac{\mathrm{d}^{\alpha}}{\mathrm{d}t^{\alpha}}\right|_{t=0}f(\gamma(t)),$

where $\bm{\mathrm{t}}^{k}\gamma$ denotes the $k$-jet of a curve
$\gamma\colon\mathbb{R}\rightarrow M$ at zero. We shall usually write
$\dot{f}$, $\ddot{f}$ instead of $f^{(1)}$, $f^{(2)}$, respectively. The
_adapted coordinates_ $(x^{a},\dot{x}^{a},\ddot{x}^{a},\ldots)$ for
$\mathrm{T}^{k}M$, induced by coordinates $(x^{a})$ on $M$, are obtained by
applying the above lifting procedure to coordinate functions $x^{a}$, and are
naturally graded. (We simply assign _weight_ $k$ to coordinates
$x^{a,(k)}:=(x^{a})^{(k)}$.) In particular, on $\mathrm{T}^{2}M$ they
transform as

$\displaystyle
x^{a^{\prime}}=x^{a^{\prime}}(x),\qquad\dot{x}^{a^{\prime}}=\frac{\partial
x^{a^{\prime}}}{\partial
x^{b}}\dot{x}^{b},\qquad\ddot{x}^{a^{\prime}}=\frac{\partial
x^{a^{\prime}}}{\partial
x^{b}}\ddot{x}^{b}+\frac{\partial^{2}x^{a^{\prime}}}{\partial x^{b}\partial
x^{c}}\dot{x}^{b}\dot{x}^{c}$ (3.1)

thus the weight of both left and right sides of the above equalities is the
same. From a geometric point of view, fibres of $\tau^{k}_{M}$ are equipped
with a special structure. Namely, we have a canonical action of the
multiplicative monoid of real numbers $(\mathbb{R},\cdot)$ on these fibres
defined by re-parametrizing curves representing the elements of
$\mathrm{T}^{k}M$:

$\displaystyle h\colon\
\mathbb{R}\times\mathrm{T}^{k}M\rightarrow\mathrm{T}^{k}M,\qquad\big{(}t,\bm{\mathrm{t}}^{k}\gamma\big{)}\mapsto\bm{\mathrm{t}}^{k}\gamma_{t},$

where $\gamma_{t}(s)=\gamma(ts)$. We clearly see, that the $\alpha$-lift of a
function $f$ on $M$ is a _homogeneous_ function on $\mathrm{T}^{k}M$ of
_weight_ $\alpha$, i.e.,

$\displaystyle f^{(\alpha)}(h_{t}(v))=t^{\alpha}f^{(\alpha)}(v),$

where $h_{t}(v):=h(t,v)$. This also explains why the adapted coordinates on
$\mathrm{T}^{k}M$ are graded.

Properties of the higher tangent bundle $\mathrm{T}^{k}M$ motivate the
concepts of a non-negatively graded manifold and a homogeneity structure. The
definition below is equivalent to the definition of a purely even positively-
graded manifold according to Voronov [47, Definition 4.1]. Here we reformulate
the condition of “cylindricity” of positive even coordinates from that
definition as the condition that fibers are diffeomorphic to $\mathbb{R}^{n}$.

######  Definition 3.1 ($\mathbb{N}$-graded manifold [47]).

A _$\mathbb{N}$ -graded manifold_ is a smooth fibration
$\sigma^{k}\colon\\!E^{k}{\rightarrow}M$ in which we are given a distinguished
class of fibre coordinates (called _graded coordinates_) with non-negative
integer _weights_ assigned. Moreover, it is assumed that these graded
coordinates identify the fibres with $\mathbb{R}^{n}$ (for some integer $n$),
and that the transition functions are multi-variable polynomials that preserve
the weights. The index $k$ in $E^{k}$ indicates that the weights on $E^{k}$
are less or equal $k$. We say that $E^{k}$ has _order_ $k$.

A _homogeneity structure_ (the idea of which dates back at least to [51]) is a
manifold $E$ equipped with a smooth action $h\colon\mathbb{R}\times
E\rightarrow E$ of the multiplicative monoid of real numbers
$(\mathbb{R},\cdot)$. Surprisingly, both concept coincide in the smooth
setting [17, Theorem 4.2], in particular, every $\mathbb{N}$-graded manifold
$\sigma^{k}\colon E^{k}\rightarrow M$ admits a canonical homogeneity structure
$h^{E^{k}}\colon\mathbb{R}\times E^{k}\rightarrow E^{k}$. For this reason we
shall use the terms $\mathbb{N}$-graded manifold and homogeneity structure
interchangeably.

Graded coordinates on a $\mathbb{N}$-graded manifold $\sigma^{k}\colon
E^{k}\rightarrow M$ can be denoted by $(x^{a},y^{i}_{w})$ where a
(superfluous) index $w=w(i)\in\mathbb{Z}_{>0}$ at $y^{i}$ indicates that a
fibre coordinate $y^{i}$ is homogeneous of weight $w$. The base coordinates
$(x^{a})$ are assumed to have weight zero. In such a notation the associated
homogeneity structure reads as

$\displaystyle h\colon\ \mathbb{R}\times E^{k}\rightarrow E^{k},\qquad
h\big{(}t,\big{(}x^{a},y^{i}_{w}\big{)}\big{)}=\big{(}x^{a},t^{w}y^{i}_{w}\big{)},\qquad
t\in\mathbb{R}.$

It is convenient and fruitful to encode the structure of a $\mathbb{N}$-graded
manifold by means of a canonical vector field on $E^{k}$ called the _weight
vector field_ which in graded coordinates is given by

$\displaystyle\Delta=\sum_{i}w\,y^{i}_{w}\frac{\partial}{\partial y^{i}_{w}}.$

For example, the canonical weight vector field on $\mathrm{T}^{k}M$ is
$\Delta_{M}^{k}=\sum_{\alpha=1}^{k}\sum_{a}\alpha
x^{a,(\alpha)}\partial_{x^{a,(\alpha)}}$. In fact, the weight vector field
provides an equivalent characterization of the structure of a
$\mathbb{N}$-graded manifold. For brevity, we often refer to a
$\mathbb{N}$-graded manifold as to a pair $(E^{k},\Delta)$.

A _morphism_ from a $\mathbb{N}$-graded manifold $\sigma_{E}\colon
E^{k}\rightarrow M$ to $\sigma_{F}\colon F^{k}\rightarrow M$ is a smooth map
$\phi\colon E^{k}\rightarrow F^{k}$ commuting with the respective homogeneity
structures, i.e., $\phi\circ h^{E}=h^{F}\circ\phi$. Equivalently $\phi$
relates the corresponding weight vector fields $\Delta_{E}$ and $\Delta_{F}$.
$\mathbb{N}$-graded manifolds of order $k$ form a _category_ denoted by
$\mathcal{GM}[k]$. In view of [16, Theorem 2.4], $\mathcal{GM}[1]$ is the
category of vector bundles.

Every $\mathbb{N}$-graded manifold has a _zero section_ $0_{M}^{k}\colon
M\rightarrow E^{k}$. The image of $M$ in $E^{k}$ is defined by _putting to
zero_ [6, Example 1.10] all graded coordinates of positive weight. More
generally, one can invariantly put to zero all _fibre_ coordinates of weights
less than a given number $j$. The obtained subset of $E^{k}$ is denoted by
$E^{k}[\Delta\geq j]$ and it is, actually, a graded subbundle of $E^{k}$.

Given a $\mathbb{N}$-graded manifold $\big{(}E^{k},\Delta\big{)}$ of order $k$
and an integer $0\leq j\leq k$ one can construct a canonical projection from
$E^{k}$ onto a $\mathbb{N}$-graded manifold of order $j$, denoted by
$E^{k}[\Delta\leq j]$, obtained by _removing all coordinates of weights
greater than_ $j$ [6, Definition 1.6]. As transformation rules for $E^{k}$ of
coordinates of weight $\leq j$ involve only coordinates of weights $\leq j$
the above construction is correct. It follows that a $\mathbb{N}$-graded
manifold $\sigma^{k}\colon E^{k}\to M$ induces a  _tower of affine fibrations_
[47]

$\displaystyle
E^{k}\xrightarrow{\sigma^{k}_{k-1}}E^{k-1}\to\cdots\xrightarrow{\sigma^{2}_{1}}E^{1}\xrightarrow{\sigma^{1}_{0}}M,$
(3.2)

where $E^{k}[\Delta\leq j]$ is denoted shortly by $E^{j}$.

######  Definition 3.2 (the top core of a $\mathbb{N}$-graded manifold).

Let $\sigma^{k}\colon E^{k}\rightarrow M$ be a $\mathbb{N}$-graded manifold of
order $k$. The _top core_ of $E^{k}$, denoted by $\widehat{E^{k}}$, is the set

$\displaystyle\widehat{E^{k}}=\big{\\{}v\in E^{k}\colon\sigma^{k}_{k-1}(v)\in
0_{M}^{k-1}(M)\subset E^{k-1}\big{\\}},$

where $\sigma^{k}_{k-1}\colon E^{k}\rightarrow E^{k-1}$ as in (3.2). Locally,
$\widehat{E^{k}}$ is defined by _putting to zero_ all fibre coordinates of
weights less than $k$:

$\displaystyle\widehat{E^{k}}=\big{\\{}\big{(}x^{A},y^{a}_{w}\big{)}\colon
y^{a}_{w}=0\text{ for any }y^{a}_{w}\text{ such that }1\leq
w<k\big{\\}}\subset E^{k},$

hence $\big{(}x^{A},y^{a}_{k}\big{)}$ are local coordinates on
$\widehat{E^{k}}$. The top core $\widehat{E^{k}}$ is naturally a vector bundle
over $M$ with the homotheties defined locally by

$\displaystyle t.\big{(}x^{A},y^{a}_{k}\big{)}=\big{(}x^{A},t\cdot
y^{a}_{k}\big{)}$

for $t\in\mathbb{R}$. Moreover, the top core $\widehat{\cdot}$ is a functor
from the category $\mathcal{GM}[k]$ to the category of vector bundles.

######  Example 3.3 (split $\mathbb{N}$-graded manifolds).

Given a sequence of vector bundles $E_{j}$, $j=1,\ldots,k$, over the same base
manifold $M$ we can turn the Whitney sum $E=\bigoplus_{j=1}^{k}E_{j}$ into a
$\mathbb{N}$-graded manifold of order $k$ by assuming that linear coordinates
on fibres of $E_{j}$ have weight $j$. The obtained $\mathbb{N}$-graded
manifold will be denoted by ${\sf X}_{j=1}^{k}E_{j}[j]$, where the notation
$V[j]$ means that we assign weight $j$ to linear coordinates on a vector space
$V$. In other words, we have a functor from the category of graded vector
bundles supported in degrees $-1,-2,\ldots,-k$ to the category
$\mathcal{GM}[k]$ of $\mathbb{N}$-graded manifolds of order $k$.

$\mathbb{N}$-graded manifolds equipped with a _splitting_ , i.e., a
$\mathbb{N}$-graded manifold isomorphism $p\colon E^{k}\rightarrow{\sf
X}_{j=1}^{k}E_{j}$ as above are called _split $\mathbb{N}$-graded manifolds_.
It is worth to remember that in the smooth real setting any
$\mathbb{N}$-graded manifold is (non-canonically) isomorphic to its _split
form_ which is obtained from the sequence of the top core bundles
$\widehat{E^{j}}$, $j=1,\ldots,k$: $E^{k}\simeq{\sf
X}_{j=1}^{k}\widehat{E^{j}}$.

Multi-$\mathbb{N}$-graded manifolds. Of particular interests are geometric
objects which admit several compatible $\mathbb{N}$-graded manifold
structures. Multi-grading appeared as a powerful tool in Voronov’s paper [50],
were it was used to study Mackenzie’s double Lie algebroids [32]. We define a
_$k$ -tuple $\mathbb{N}$-graded manifold_ $(E,\Delta_{1},\ldots,\Delta_{k})$
to be a manifold $E$ with $k$ pair-wise commuting weight vector fields, i.e.,
$[\Delta_{i},\Delta_{j}]=0$ for any $i,j=1,\ldots,k$. Equivalently, $E$ admits
$k$ pair-wise commuting homogeneity structures $h^{i}\colon\mathbb{R}\times
E\rightarrow E$ with $i=1,\ldots,k$. That is, $h^{i}_{t}\circ
h^{j}_{s}=h^{j}_{s}\circ h^{i}_{t}$ for any $i,j=1,2,\ldots,k$ and any
$t,s\in\mathbb{R}$. Interestingly, every $k$-tuple $\mathbb{N}$-graded
manifold admits local coordinates which are simultaneously graded with respect
to any of its graded structures [17, Theorem 5.2].

Double vector bundles, the tangent and cotangent bundles of a
$\mathbb{N}$-graded manifold $\sigma^{k}\colon\\!E^{k}{\rightarrow}M$ are
examples of double $\mathbb{N}$-graded manifolds. There are numerous other
constructions of multi-$\mathbb{N}$-graded manifolds: the iterated higher
tangent bundles $\mathrm{T}^{k_{1}}\cdots\mathrm{T}^{k_{r}}M$, the
substructures and quotients (e.g., the construction of the linearization of a
$\mathbb{N}$-graded manifold, the vertical subbundle
$\mathbf{V}E^{k}=\ker{\mathrm{T}\sigma^{k}}\subset TE^{k}$) and others (see
[6, Definitions 1.6 and 1.8] for “the removal of coordinates of weight greater
than a given number $l$” and “putting to zero coordinates of $X$-negative
weights” where $X$ is linear combination of weight vector fields
$\Delta_{i}$).

The notion of core of a DVB has a natural generalization for
multi-$\mathbb{N}$-graded manifolds.

######  Definition 3.4 (the core and the ultracore of a
multi-$\mathbb{N}$-graded manifold).

Let $(E,\Delta_{1},\ldots,\Delta_{k})$ be a $k$-tuple $\mathbb{N}$-graded
manifold and $k>1$. The _core $\mathbf{C}(E)$ of $E$_ is obtained by putting
to zero all fiber coordinates $y^{i}_{w}$ with multi-weight $w$ in which at
least one of $w_{1},\ldots,w_{k}$ is zero. Formally, using the construction
$E[X\geq 0]$ [6, Definition 1.8] of putting to zero all fiber coordinates with
negative weight with respect to a vector field $X$ being a linear combination
of weight vector fields $\Delta_{i}$, we may write the core of $E$ as the
intersection

$\displaystyle\mathbf{C}(E):=\bigcap_{j=1}^{k}E\bigg{[}N\Delta_{j}-\sum_{i\neq
j}\Delta_{i}\geq 0\bigg{]},$

for a sufficient large number $N$. Indeed, $E[N\Delta_{j}-\sum_{i\neq
j}\Delta_{i}\geq 0]$ is the subset of $E$ obtained by putting to zero all
_fiber_ coordinates $y^{i}_{w}$ of the total fibration $E\rightarrow
M=E[\Delta_{1}=\Delta_{2}=\cdots=\Delta_{k}=0]$ with multi-weight
$w=(w_{1},\ldots,w_{k})$ in which $w_{j}=0$. In general, $\mathbf{C}(E)$
remains a $k$-tuple $\mathbb{N}$-graded manifold.

By the _ultracore_ of $E$ we mean the top core $\widehat{E}$ of the
$\mathbb{N}$-graded manifold $(E,\Delta_{1}+\cdots+\Delta_{k})$.

Note that for a double vector bundle, the core in the above sense coincides
with the usual notion of the core of a DVB. What is more, our definition of
the ultracore coincides with the one introduced by Mackenzie [33].

Graded-linear manifolds. Let us now take a closer look at double
$\mathbb{N}$-graded manifolds with one of the homogeneity structures being
linear. If $\big{(}E,h^{1},h^{2}\big{)}$ is such a structure, we may treat it
as a vector bundle $\sigma\colon E\rightarrow M$ (say, that the VB structure
on $E$ corresponds to the homogeneity structure $h^{2}$) equipped with a
$\mathbb{N}$-graded manifold structure encoded in the homogeneity structure
$h^{1}\colon\mathbb{R}\times E\rightarrow E$ such that the structure maps of
the vector bundle $\sigma$ (vector bundle projection, zero section, addition,
and scalar multiplication) are _weighted_ or, in other words, homogeneous with
respect to $h^{1}$. In particular, for each $\lambda\in\mathbb{R}$, the
mapping $v\mapsto\lambda\cdot_{\sigma}v$, $v\in E$, should be a
$\mathbb{N}$-graded manifold morphism, or equivalently, it should commute with
the homotheties $h^{2}_{t}$ for any $t\in\mathbb{R}$. A double
$\mathbb{N}$-graded manifold $\big{(}E,h^{1},h^{2}\big{)}$ in which
$\big{(}E,h^{2}\big{)}$ is a vector bundle is called a  _graded-linear
manifold_ or, equivalently, a _weighted vector bundle_ , when we want to put
emphasis on the underlining linear structure. In the language of [3], we would
speak about _graded-linear bundles_. A graded-linear manifold $\sigma\colon
E\rightarrow M$ can be depicted in a diagram

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.20824pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.45908pt\raise 5.84583pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.55695pt\hbox{$\scriptstyle{\sigma^{k}_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 66.74722pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.00694pt\hbox{$\scriptstyle{\sigma^{\\{k\\}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-26.99998pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.94165pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 66.74722pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
75.12637pt\raise-19.38887pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.00694pt\hbox{$\scriptstyle{\sigma^{\\{0\\}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
75.12637pt\raise-27.13998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
30.95901pt\raise-31.59998pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.88889pt\hbox{$\scriptstyle{\underline{\sigma}^{k}_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
63.94165pt\raise-38.77774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.94165pt\raise-38.77774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
63.94165pt\raise-38.77774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M^{0}.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(3.3)

Here $E^{k}=E$, $\sigma^{\\{k\\}}=\sigma$ and the superscript $k$ at $E$
indicates the order of the homogeneity structure. The base $M=M^{k}$ of the
vector bundle $\sigma$ carries a $\mathbb{N}$-graded manifold structure
induced from $\sigma^{k}_{0}\colon E^{k}\rightarrow E^{0}$. The notation in
parentheses $\\{\ \\}$ is used in order to distinguish the vector bundle
projection $\sigma^{\\{k\\}}$ given by the homotheties $h^{2}$ from the
$\mathbb{N}$-graded manifold projection $\sigma^{k}_{0}$ given by the
homogeneity structure $h^{1}$.

An upcoming definition of a higher (Lie) algebroid involves two particular
examples of graded-linear manifolds: the tangent bundle of a
$\mathbb{N}$-graded manifold of order $k$, and the $k^{\mathrm{th}}$-order
tangent bundle of a vector bundle. Thus it will be important to understand the
core bundles of these examples in greater detail. For this we shall show now
that the core of any graded-linear manifold admits a canonical splitting into
a direct sum of vector bundles. In the special case of a double vector bundle
we recover the fact that the core is a vector bundle over the final base $M$.

######  Proposition 3.5 (the core of a graded-linear manifold).

The core of any order-$k$ graded-linear manifold $(D,\Delta_{1},\Delta_{2})$
is a graded vector bundle, i.e.,

$\displaystyle\mathbf{C}(D)=\bigoplus_{j=1}^{k}D_{j}[j]$

for some vector bundles $D_{j}$ canonically associated with $D$. In particular
cases,

$\displaystyle\mathbf{C}\big{(}\mathrm{T}^{k}E\big{)}\simeq E[1]\oplus
E[2]\oplus\cdots\oplus E[k],\qquad\text{and}$
$\displaystyle\mathbf{C}(\mathrm{T}E^{k})\simeq
E^{1}[1]\oplus\widehat{E^{2}}[2]\oplus\cdots\oplus\widehat{E^{k}}[k].$

###### Proof.

Bi-graded coordinates on $D$ have the form $\big{(}x^{A},y^{a}_{(i,j)}\big{)}$
where $i=0$ or $1$ and $0\leq j\leq k$, and $(i,j)\neq(0,0)$. As
$C:=\mathbf{C}(D)$ is given locally in $D$ by equations $y^{a}_{(1,0)}=0$ and
$y^{a}_{(0,j)}=0$, $1\leq j\leq k$, the transformation rules for the fibre
coordinates $y^{a}_{(1,j)}|_{C}$, $j=1,\ldots,k$, are of the form
$y^{a^{\prime}}_{(1,j)}|_{C}=Q^{a^{\prime}}_{j,b}\,y^{b}_{(1,j)}|_{C}$ for
some functions $Q^{a^{\prime}}_{j,b}$ on the final base $M$ of $D$. It follows
that the submanifold of $D$ defined locally by vanishing all fibre coordinates
except those of weight ${(1,j)}$, where $1\leq j\leq k$ is a fixed integer,
form a vector bundle $D_{j}$ over $M$ and $C$ is just the Whitney sum of these
vector bundles. The vector bundle $D_{j}$ can be recognized as the top core
bundle of $(D[\Delta_{2}\leq j],\Delta_{1}+\Delta_{2})$. The decomposition of
the core of $\mathrm{T}^{k}E$ and $\mathrm{T}E^{k}$ are due to the
isomorphisms $\widehat{\mathrm{T}^{k}E}\simeq E$ and
$\widehat{\mathrm{T}E^{k}}\simeq\widehat{E^{k}}$. ∎

Weighted structures and their reductions from ‘higher’ to ‘lower’ order. In a
similar manner we can introduce notion of other weighted geometrical objects
and structures, in particular VBCs.

######  Definition 3.6 (weighted VBC).

Let $\sigma_{i}\colon E_{i}\rightarrow M_{i}$, where $i=1,2$, be weighted VBs
of order $k$ with homogeneity structures $h_{i}$ on $E_{i}$. A VBC
$r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$ is called a
_weighted VBC_ if $r\subset E_{1}\times E_{2}$ is homogeneous with respect to
$h_{1}\times h_{2}$. In other words, $r$ is a graded submanifold of the
product of $\mathbb{N}$-graded manifolds
$(\sigma_{1})^{k}_{0}\times(\sigma_{2})^{k}_{0}\colon E_{1}\times
E_{2}\rightarrow E_{1}^{0}\times E_{2}^{0}$. (See diagram (3.3) for the
notation concerning weighted vector bundles.) In particular, the base map
$\underline{r}\colon M_{2}\rightarrow M_{1}$ is a $\mathbb{N}$-graded manifold
morphism. (We may assume that orders of both the homogeneity structures are
the same, as every $\mathbb{N}$-graded manifold of order $k$ is also of order
${k^{\prime}}$ for any $k^{\prime}>k$.)

######  Remark 3.7 (on weighted structures).

Weighted (Lie) algebroids and weighted (Lie) groupoids considered in [3] are
defined in the same spirit. One simply requests that the structure maps of a
(Lie) algebroid (resp. a (Lie) groupoid) are graded, in some sense. Particular
examples are $\mathcal{VB}$-algebroids and $\mathcal{VB}$-groupoids
extensively studied in the literature. (The graded structure is then of order
1.) We shall not use these notions in our paper.

For a weighted structure we may try to perform a _reduction from a ‘higher’ to
a ‘lower’ order_ in the tower of affine fibrations (3.2). If the starting
point is a graded-linear manifold $\big{(}E^{k},\Delta_{1},\Delta_{2}\big{)}$
of order $k$, then its $j$-order reduction $E^{j}=E^{k}[\Delta_{1}\leq j]$ is
still a graded-linear manifold and $\sigma^{k}_{j}\colon E^{k}\rightarrow
E^{j}$ is a morphism in the category of graded-linear manifolds. In
particular, given a weighted VBC
$r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$ between
weighted vector bundles $\sigma_{i}\colon E^{k}_{i}\rightarrow M_{i}^{k}$
($i=1,2$) of order $k$ we may consider $r$ as a $\mathbb{N}$-graded
submanifold in the product bundle $\sigma_{1}^{k}\times\sigma_{2}^{k}\colon
E^{k}_{1}\times E^{k}_{2}\rightarrow E^{0}_{1}\times E^{0}_{2}$ and then
project it to get its lower-order reduction $r^{j}\colon
E^{j}_{1}\rightarrow\\!\\!\vartriangleright E^{j}_{2}$, which is not only a
graded-linear submanifold, but also a weighted VBC of order $j$, for
$j=0,1,2,\ldots,k$. This and another useful observation in a similar spirit is
presented below.

######  Proposition 3.8 (reduction of weighted VBCs).

Let $r\colon\sigma_{1}\rightarrow\\!\\!\vartriangleright\sigma_{2}$ be a
weighted VBC where $\sigma_{i}\colon E_{i}^{k}\rightarrow M_{i}^{k}$ $(i=1,2)$
are weighted vector bundles of order $k$, and let $0\leq j\leq k$. Then

  1. $(i)$

the lower-order reductions
$r^{j}\colon\sigma^{\\{j\\}}_{1}\rightarrow\\!\\!\vartriangleright\sigma^{\\{j\\}}_{2}$
remain weighted VBCs.

  2. $(ii)$

If
$r^{\prime}\colon\sigma_{1}^{\prime}\rightarrow\\!\\!\vartriangleright\sigma_{2}^{\prime}$
is another weighted VBC and if $(\phi_{1},\phi_{2})\colon r\Rightarrow
r^{\prime}$, where $\phi_{i}\colon\sigma_{i}\rightarrow\sigma_{i}^{\prime}$
are morphism of graded-linear manifolds, is a ${\mathcal{VBC}}$-morphism, then
the lower-order reductions $\big{(}\phi_{1}^{j},\phi_{2}^{j}\big{)}\colon
r^{j}\Rightarrow{r^{\prime}}^{j}$ are also ${\mathcal{VBC}}$-morphisms.

The proof is given in Appendix A.

##  4 Higher (Lie) algebroids

Definitions and fundamental examples. As was already suggested in the
introduction, the higher tangent bundle
$\tau^{k}_{M}\colon\mathrm{T}^{k}M\rightarrow M$ (cf. the beginning of Section
3) together with the canonical isomorphism
$\kappa^{k}_{M}\colon\mathrm{T}^{k}\mathrm{T}M\rightarrow\mathrm{T}\mathrm{T}^{k}M$
will be our fundamental example of a higher (Lie) algebroid. According to our
considerations from the beginning of the previous section, a choice of local
coordinates $(x^{a})$ on $M$ induces canonical coordinates
$(x^{a},\dot{x}^{b})$ on $\mathrm{T}M$,
$\big{(}x^{a,(\alpha)}\big{)}_{\alpha=0,\ldots,k}$ on $\mathrm{T}^{k}M$,
$\big{(}x^{a},\dot{x}^{b},x^{a,(1)},\dot{x}^{b,(1)},\ldots,x^{a,(k)},\dot{x}^{b,(k)}\big{)}$
on $\mathrm{T}^{k}\mathrm{T}M$, and
$\big{(}x^{a,(\alpha)},\dot{x}^{b,(\beta)}\big{)}_{\alpha,\beta=0,\ldots,k}$
on $\mathrm{T}\mathrm{T}^{k}M$. In these coordinates $\kappa^{k}_{M}$ reads as

$\displaystyle\kappa^{k}_{M}\colon\
\big{(}x^{a},\dot{x}^{b},x^{a,(1)},\dot{x}^{b,(1)},\ldots,x^{a,(k)},\dot{x}^{b,(k)}\big{)}\longmapsto\big{(}x^{a},x^{a,(1)},\ldots,x^{a,(k)},\dot{x}^{b},\dot{x}^{b,(1)},\ldots,\dot{x}^{b,(k)}\big{)}.$

Clearly, for $k=1$ we get $\kappa^{1}_{M}=\kappa_{M}$ studied in Example 2.13.
It follows easily from the above coordinate formula that $\kappa^{k}_{M}$ is
an isomorphism in the category of graded-linear manifolds.

According to our motivating considerations from the introduction, we would now
like to generalize the above example by substituting the fibration
$\mathrm{T}^{k}M\rightarrow M$ with an arbitrary $\mathbb{N}$-graded manifold
$E^{k}\rightarrow M$ of order $k$. Instead of a graded-linear isomorphism
$\kappa^{k}_{M}$ we shall take a weighted vector bundle comorphism
$\kappa^{k}$ between $\mathrm{T}^{k}E^{1}$ and $\mathrm{T}E^{k}$. Such
postulates are motivated by the description of (Lie) algebroids (of order one)
in terms of a VBC $\kappa$ (cf. Proposition 2.15 and Lemma 2.11).

######  Definition 4.1 (general higher algebroid, HA).

A _general higher_ ($k^{\mathrm{th}}$-order) _algebroid_ (_HA_ , in short) is
a $\mathbb{N}$-graded manifold $\sigma^{k}\colon E^{k}\rightarrow M$ of order
$k$ together with a weighted VBC
$\kappa^{k}\subset\mathrm{T}^{k}E^{1}\times\mathrm{T}E^{k}$ from
$\mathrm{T}^{k}\sigma^{1}$ to $\tau_{E^{k}}$ (covering a (graded) mapping
$\rho^{k}\colon E^{k}\rightarrow\mathrm{T}^{k}M$) such that the relation
$\kappa^{1}\colon\mathrm{T}\sigma^{1}\rightarrow\\!\\!\vartriangleright\mathrm{T}\tau_{E^{1}}$
being the reduction to order one of $\kappa^{k}$ equips $\sigma^{1}\colon
E^{1}\rightarrow M$ with a general algebroid structure.

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(4.1)

In addition:

  1. (i)

If $\big{(}\sigma^{1},\kappa^{1}\big{)}$ is skew, we call
$\big{(}\sigma^{k},\kappa^{k}\big{)}$ a _skew_ HA.

  2. (ii)

If $\big{(}\sigma^{k},\kappa^{k}\big{)}$ is skew and
$\big{(}\mathrm{T}^{k}\rho^{1},\mathrm{T}\rho^{k}\big{)}\colon\kappa^{k}\Rightarrow\kappa^{k}_{M}$
is a morphism in the category of VBCs, where $\rho_{1}\colon
E^{1}\rightarrow\mathrm{T}M$ is the order-1 reduction of $\rho^{k}$, then we
call $\big{(}\sigma^{k},\kappa^{k}\big{)}$ an _almost-Lie $($AL$)$_ HA.

  3. (iii)

A skew higher algebroid $\big{(}\sigma^{k},\kappa^{k}\big{)}$ in which
$\kappa^{k}$ is a subalgebroid of the product of the algebroids
$\big{(}\mathrm{T}^{k}\sigma^{1},\mathrm{d}_{\mathrm{T}^{k}}\kappa^{1}\big{)}$
(the $k^{\mathrm{th}}$-tangent lift of $\big{(}\sigma^{1},\kappa^{1}\big{)}$ –
see Section 4.1 – and the tangent algebroid $(\tau_{E^{k}},\kappa_{E^{k}})$),
is called a _Lie_ HA. In fact a Lie HA must be an almost-Lie HA – see
Proposition 4.9. In the language of [7], $\big{(}\sigma^{k},\kappa^{k}\big{)}$
is Lie if $\kappa^{k}$ is a Lie algebroid comorphism.

  4. (iv)

A general HA in which $\kappa^{k}$ induces an isomorphism on the core bundles
$\mathbf{C}\big{(}\kappa^{k}\big{)}\colon\mathbf{C}\big{(}\mathrm{T}^{k}E\big{)}\rightarrow\mathbf{C}\big{(}\mathrm{T}E^{k}\big{)}$
is called a _strong_ HA.

Analogously to our convention from Section 2 we will colectively refer to the
above-defined objects as to _$($ Lie$)$ higher algebroids_.

We define a _morphism_ between (Lie) higher algebroids
$\big{(}\sigma^{k}_{E}\colon E^{k}\rightarrow M,\kappa^{k,E}\big{)}$ and
$\big{(}\sigma^{k}_{F}\colon F^{k}\rightarrow N,\kappa^{k,F}\big{)}$ to be a
$\mathbb{N}$-graded manifold morphism $\phi^{k}\colon E^{k}\rightarrow F^{k}$
such that
$\big{(}\mathrm{T}^{k}\phi^{1},\mathrm{T}\phi^{k}\big{)}\colon\kappa^{k,E}\Rightarrow\kappa^{k,F}$
is a ${\mathcal{VBC}}$-morphism. Obviously, general higher algebroids form a
_category_ which we denote by $\mathcal{HA}$, with Lie HA, AL HA, and skew HA
being its full subcategories.

Let us remark, that the symmetric role of the left and the right anchor
occurring in the first order case is no longer present for (Lie) higher
algebroids. The defining relation $\kappa^{k}$ covers a (graded) morphism
$\rho^{k}\colon E^{k}\rightarrow\mathrm{T}^{k}M$, and its reduction to order 0
induces a relation (in fact a linear map) $\kappa^{0}\colon
E^{1}\rightarrow\mathrm{T}M$. For $k=1$ these were the left and the right
anchors, respectively. As we see, these maps are of quite a different nature.
Note, however, that for skew higher algebroids $\kappa^{0}=\rho^{1}$.

Using Proposition 3.8 we easily see that reduction of a (Lie) HA
$\big{(}E^{k},\kappa^{k}\big{)}$ to a lower order $j=1,2,\ldots,k$ gives an
induced (Lie) higher algebroid structure on $E^{j}$ which is skew (resp. AL,
Lie, strong) if $\big{(}E^{k},\kappa^{k}\big{)}$ was so. Moreover, a (Lie) HA
morphism
$\phi^{k}\colon\big{(}E^{k},\kappa^{k,E}\big{)}\rightarrow\big{(}F^{k},\kappa^{k,F}\big{)}$
induces a (Lie) HA morphisms
$\phi^{j}\colon\big{(}E^{j},\kappa^{j,E}\big{)}\rightarrow\big{(}F^{j},\kappa^{j,F}\big{)}$.

Examples of (Lie) higher algebroids will be discussed later in Sections 4.2
and 6. For now let us take a closer look at general higher algebroids of order
2.

######  Remark 4.2 (a local form of a general higher algebroid of order 2).

Let $(x^{a},y^{i},z^{\mu})$ be graded coordinates on an order-$2$
$\mathbb{N}$-graded manifold $E^{2}\rightarrow M$. We shall find a general
form of an algebroid $\big{(}E^{2},\kappa^{2}\big{)}$ of order $2$. As in
Remark 2.8 we underline coordinates in $\mathrm{T}E^{2}$ when writing
equations for $\kappa^{2}\subset\mathrm{T}^{2}E^{1}\times\mathrm{T}E^{2}$.
Taking weights into account

weight | $(0,0)$ | $(0,1)$ | $(0,2)$ | $(1,0)$ | $(1,1)$ | $(1,2)$
---|---|---|---|---|---|---
$\mathrm{T}^{2}E^{1}$ | $x^{a}$ | $\dot{x}^{a}$ | $\ddot{x}^{a}$ | $y^{i}$ | $\dot{y}^{i}$ | $\ddot{y}^{i}$
$\mathrm{T}E^{2}$ | $x^{a}$ | $\underline{y}^{i}$ | $\underline{z}^{\mu}$ | $\dot{\underline{x}}^{a}$ | $\dot{\underline{y}}^{i}$ | $\dot{\underline{z}}^{\mu}$
(4.5)

we get