Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms
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...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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2018
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| Цитувати: | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms / M. Jóźwikowski, M. Rotkiewicz // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 54 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859538514535776256 |
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| author | Jóźwikowski M. Rotkiewicz M. |
| author_facet | Jóźwikowski M. Rotkiewicz M. |
| citation_txt | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms / M. Jóźwikowski, M. Rotkiewicz // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 54 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 comorphisms between graded-linear bundles satisfying natural axioms. We provide natural examples and discuss applications in geometric mechanics.
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| first_indexed | 2025-12-04T15:12:48Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 135, 46 pages
Higher-Order Analogs of Lie Algebroids
via Vector Bundle Comorphisms
Micha l JÓŹWIKOWSKI † and Miko laj ROTKIEWICZ ‡
† Institute of Mathematics, Polish Academy of Sciences,
Śniadeckich 8, 00-656 Warszawa, Poland
E-mail: m.jozwikowski@mimuw.edu.pl
URL: https://www.impan.pl/~mjoz/
‡ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,
Banacha 2, 02-097 Warszawa, Poland
E-mail: mrotkiew@mimuw.edu.pl
URL: https://www.mimuw.edu.pl/~mrotkiew/
Received January 10, 2018, in final form December 12, 2018; Published online December 29, 2018
https://doi.org/10.3842/SIGMA.2018.135
Abstract. 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 geomet-
ric mechanics.
Key words: higher algebroid; vector bundle comorphism; almost-Lie algebroid; graded mani-
fold; graded bundle; algebroid lift; variational principle
2010 Mathematics Subject Classification: 58A20; 58A50; 70G65; 58E30; 70H50
Contents
1 Introduction 2
1.1 Double vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 (Lie) algebroids as vector bundle comorphisms 6
2.1 The category of vector bundle comorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Description of (Lie) algebroids in terms of VBCs . . . . . . . . . . . . . . . . . . . . . . . 10
3 Recollection of N-graded manifolds 16
4 Higher (Lie) algebroids 21
4.1 On the Lie axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Prolongations of an AL algebroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Variational calculus on (Lie) higher algebroids 29
5.1 The formalism of variational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Further examples – substructures and quotients of TkG/G 35
7 Final remarks 37
A Appendix – proofs of technical results 38
References 44
mailto:m.jozwikowski@mimuw.edu.pl
https://www.impan.pl/~mjoz/
mailto:mrotkiew@mimuw.edu.pl
https://www.mimuw.edu.pl/~mrotkiew/
https://doi.org/10.3842/SIGMA.2018.135
2 M. Jóźwikowski and M. Rotkiewicz
1 Introduction
The main goal of this paper is to introduce a concept of a (general) higher algebroid1 and study
basic properties and applications of this notion. We work essentially within the framework of
the theory of N-graded manifolds and the theory of differential relations (vector bundle comor-
phisms).
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 Mart́ınez [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 ve-
locities 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 TM . 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 TM is taken over by its higher analogue TkM , the
bundle of k-velocities.2 Locally, TkM 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 TkM . This expectation is met by a class of fibre bundles with
a typical fibre diffeomorphic with Rn, 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
1In 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.
2TkM is the bundle of k-jets of curves in M and it should not be mistaken with T(k)M = T(T(. . . (TM) . . .)) –
the iterated tangent bundle.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 3
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 “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) al-
gebroid on a vector bundle σ : E → M is most commonly introduced as a bi-linear bracket
operation on the space of sections of its host bundle σ. It is quite obvious that this definition
has no direct generalization to the higher-order case as, in particular, the higher tangent bun-
dle τkM : TkM →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 σ 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 σ
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
∧•E∗), a certain linear 2-tensor field on
the dual bundle σ∗, a morphism ε : T∗E → TE∗ of double vector bundles, or as a differential
relation of a special kind (a vector bundle comorphism) κ ⊂ TE × TE.
From the perspective of variational calculus, the latter characterization is the one most ap-
pealing to us.3 This important role of κ can be easily explained. For the tangent algebroid
structure on τM : TM →M , κ is, in fact, the canonical isomorphism κM : TTM → TTM inter-
changing the two vector bundle structures on TTM – 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 κM ). The situation in the higher-order case is no different and, perhaps, even easier
to understand. Namely on a higher-tangent bundle τkM : TkM → M , we have the canonical
isomorphism κkM : TkTM → TTkM which, similarly as before, allows to express a variation of
a kth-tangent lift of a trajectory in M as a kth-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 κ changes – the reduced object is no longer a dif-
ferentiable 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 κ ⊂ TE × TE 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 κ
appears. Variational calculus on (Lie) algebroids can be, however, successfully developed in
abstract terms (with a prominent role of κ) 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].
3Let us remark that, the double vector bundle morphism ε : T∗E → TE∗ favored in the so-called Tulczyjew
approach to geometric mechanics [13, 14] is just the dual of κ.
4 M. Jóźwikowski and M. Rotkiewicz
Higher (Lie) algebroids, examples and applications. Summing up the above heuristic
considerations, we postulate a higher (Lie) algebroid to be a N-graded manifold σk : Ek → M
equipped with a vector bundle comorphism κk ⊂ TkE1 × TEk, which also should be naturally
graded (Definition 4.1). Here E1 is the canonical reduction of Ek 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 gener-
alizations 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 TkM with the isomorphisms κkM 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 σk : Ek → M we were able to define a whole family of algebroid lifts of sections
of E1 to vector fields on Ek. 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 N-graded manifold σk : Ek → M of order k, canonically
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 5
associates a manifold l
(
Ek
)
equipped with a graded-linear structure of a bi-order (k − 1, 1)
over E1 and Ek−1, respectively. Now the higher (Lie) algebroid structure on σk is defined as
a (Lie) algebroid structure on the vector bundle l
(
Ek
)
→ Ek−1 compatible with the second
grading. The idea here is to mimic the canonical inclusion TkM ↪→ TTk−1M , which makes the
higher tangent bundle TkM a sub-object of the tangent Lie algebroid of Tk−1M . 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 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 kth-order variational calculus
on a manifold M as the first-order theory on Tk−1M thus passing through unnecessary degrees
of freedom (as would be in the spirit of [5]) or do we prefer to work directly with TkM , 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
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 TkG/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 TE of the tangent bundle of
a vector bundle σ : E → M . The two vector bundle structures are: the tangent projection
τE : TE → E, and Tσ : TE → TM , the tangent lift of σ. 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 σA : D → A, σB : D → B such that for any t, s ∈ R
and x ∈ D
t ·A s ·B x = s ·B t ·A x, (1.1)
where ·A and ·B denote the multiplications by scalars in σA and σB, respectively.
A morphism of DVBs (D,σA, σB) and (D′, σA′ , σB′) is a smooth map φ : D → D′ which is
linear as a map from σA to σ′A and, simultaneously, as a map from σB to σB′ .
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
D
σA
��
σB // B
σBM
��
A
σAM //M
6 M. Jóźwikowski and M. Rotkiewicz
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 0A : A → D defines a vector
bundle structure on A as a substructure of the vector bundle structure on σB : D → 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 σA (the bundle projection σA, the
zero section 0A, the scalar multiplication ·A and the addition +A) are vector bundle morphism
with respect to the vector bundle structure of σB. The notion of a double vector bundle naturally
generalizes to the notion of a multi-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 σ1 : E1 → M1 to a vector bundle σ2 : E2 → M2 is a relation
r ⊂ E1 × E2 of a special form. Namely, there exists an ordinary vector bundle morphism
φ : E∗2 → E∗1 covering a smooth map φ : M2 →M1 such that r = φ∗, i.e., r is the union of graphs
of linear maps ry : (E1)φ(y) → (E2)y such that φy : (E∗2)y → (E∗1)φ(y) is the dual of the linear
map ry, where y varies in M2. We also say that the VBC r covers the base map r := φ. Usually
we shall denote a VBC as r : σ1 →B σ2 or as a diagram
E1
σ1
��
r � ,2E2
σ2
��
M1 M2.
roo
In particular, the VBC r is a vector subbundle of σ1 × σ2.
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 : σ1 →B σ2 as
in Definition 2.1 and a sections s ∈ SecM1(E1) we define r̂(s) ∈ SecM2(E2) by
r̂(s)(y) := ry(s(r(y))), for every y ∈M2.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 7
In other words, the value of r̂(s) at a given point y ∈M2 is the unique element of the fibre (E2)y
which is r-related to the value of s at r(y). By the linearity of r, for every two sections s, s′ ∈
SecM1(E1) we have r̂(s+ s′) = r̂(s) + r̂(s′). Moreover, by construction, for any f ∈ C∞(M1)
r̂(f · s) = r∗(f) · r̂(s). (2.1)
Conversely, any R-linear map SecM1(E1) → SecM2(E2) satisfying (2.1) for some underlying
map r : M2 →M1 gives rise to a VBC r : σ1 →B σ2. 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
σ : E → M is fully encoded by a VBC κ from the tangent lift Tσ : TE → TM to the tangent
bundle τE : TE → 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 VBC-morphism,
i.e., a morphism in the category VBC of vector bundle comorphisms.
Definition 2.3 (VBC-morphism). Let (σi : Ei →Mi) and (σ′i : E
′
i →M ′i), i = 1, 2, be four VBs
and let r and r′ be the following vector bundle comorphisms:
E1
σ1
��
r � ,2E2
σ2
��
M1 M2,
roo
E′1
σ′1
��
r′ � ,2E′2
σ′2
��
M ′1 M ′2.
r′oo
(2.2)
We say that a pair (φ1, φ2) of VB morphisms φi : Ei → E′i covering smooth maps φi : Mi →M ′i ,
i = 1, 2, is a VBC-morphism from r to r′ (and write (φ1, φ2) : r ⇒ r′) if the following diagram
E1
φ1
}}{{{{{{{{
σ1
��
r � ,2E2
φ2
}}{{{{{{{{
σ2
��
E′1
σ′1
��
r′ � ,2E′2
σ′2
��
M1
φ1
}}{{{{{{{{
M2
φ2
}}{{{{{{{{
roo
M ′1 M ′2
r′oo
(2.3)
is commutative in the following sense:
(i) The base maps commute, i.e., φ1 ◦ r = r′ ◦ φ2 : M2 →M ′1.
(ii) At the level of fibres, for any y ∈M2 the following compositions of linear maps
E1,x
ry−→ E2,y
φ2−→ E′2,y′ and E1,x
φ1−→ E′1,x′
r′
y′−−→ E′y′
coincide. Here x = r(y), y′ = φ2(y), x′ = r′(y′) = φ1(x) are determined by y ∈M2.
It is easy to see that VBC-morphisms can be naturally composed, and that VBCs with VBC-
morphisms form a category of vector bundle comorphisms, denoted VBC.
8 M. Jóźwikowski and M. Rotkiewicz
Remark 2.4 (dualization of VBCs). By dualizing the diagram (2.3) we get a diagram of the
same type:
E∗1
σ∗1
��
E∗2
r∗oo
σ∗2
��
E
′∗
1
φ∗1
< 9C
|||||||
(σ′1)∗
��
E
′∗
2
r
′∗
oo
φ∗2
< 9C|||||||
(σ′2)∗
��
M1
φ1
}}{{{{{{{{
M2
φ2
}}zzzzzzzz
roo
M ′1 M ′2.
r′oo
It is clear that (φ1, φ2) : r ⇒ r′ is a VBC-morphism if and only if (r′∗, r∗) : φ∗2 ⇒ φ∗1 is so. Thus
the notion of a morphism in the category VBC does not reduce to the notion of a vector bundle
morphism.
Example 2.5 (phase lift). Let φ : M →M ′ be a smooth map, and consider the phase lift of φ
which is a VBC T∗φ : T∗M ′ →B T∗M covering φ. Then the tangent lift of T∗φ is a VBC as well
and (Tτ∗M ′ , τT∗M ) : TT∗φ⇒ T∗φ is a morphism in the category VBC.
Another useful formulation of the conditions presented in the definition of a VBC-morphism
is possible:
Proposition 2.6 (characterization of VBC-morphisms). Let φi : Ei → E′i (i = 1, 2) be vector
bundle morphisms and let r, r′ be VBCs as in Definition 2.3. Then (φ1, φ2) : r ⇒ r′ is a VBC-
morphism if and only if for any r-related vectors X ∈ E1, Y ∈ E2 their images φ1(X) and φ2(Y )
are r′-related. Equivalently, (φ1 × φ2)(r) ⊂ r′.
Proof. Assume that (φ1, φ2) : r ⇒ r′ is a VBC-morphism and take X and Y such that X ∼r Y .
Denote x := σ1(X), y := σ2(Y ), x′ := φ1(x) and y′ := φ2(y). According to Definition 2.3
r′y′(φ1(X)) = φ2(Y ), in other words φ1(X) and φ2(Y ) are r′-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,
VB-groupoids, VB-algebroids, weighted algebroids etc. [3, 34].
Definition 2.7 (linear VBC). Let r be a VBC from a vector bundle σ : E → M to a vector
bundle σ′ : E′ →M ′. Assume that the total spaces E and E′ carry linear structures τ : E → N
and τ ′ : E′ → N ′ compatible with σ and σ′, respectively (i.e., E and E′ are DVBs in the sense
of Definition 1.1). We say that r is a linear VBC if r ⊂ E ×E′ is a vector subbundle of τ × τ ′.
Note that a linear VBC r ⊂ E × E′ defined as above is a vector subbundle of both σ × σ′
and τ × τ ′. It follows that r projects to linear relations in M ×M ′ and N ×N ′. In particular,
the base map r : M ′ → 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 σ : E →M is a vector bundle, then the tangent space TE carries two compatible
vector bundle structures Tσ : TE → TM and τE : TE → E. In the remaining part of this
paragraph we shall study linear VBCs intertwining these two VB structures. The common kernel
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 9
of Tσ and τE (called the core C(TE) of DVB (TE,Tσ, τE)) is the subbundle C = VME ' E
of the vertical bundle VE = ker Tσ consisting of vertical vectors based at M ⊂ E. The two
additive structures of TE coincide on the core C. Note that the core C acts on TE by an
addition of vertical vectors, i.e., for every A ∈ TaE and every e ∈ Cσ(a) ' Eσ(a) we define
A+++e := A+τE Vae ∈ TaE,
where Va : Vσ(a)E → VaE ⊂ TaE are canonical isomorphisms. Note that the above action does
not affects the two VB projections Tσ and τE on TE, i.e., Tσ(A+++e) = Tσ(A) and τE(A+++e) =
τE(A). Moreover, the following useful identity holds
T(f · a)v = f ·Tσ (Ta)v+++v(f) · a, (2.4)
where f ∈ C∞(M), v ∈ X(M) and a ∈ Sec(E).
Remark 2.8 (local form of a class of linear VBCs). Let us write a general (local) form of
a linear VBC κ intertwining the two compatible VB structures on TE. For simplicity and
further applications we also assume that κ projects onto graph(idM ) under the bundle projection
τM ◦ Tσ × σ ◦ τE : TE × TE →M ×M , i.e., κ relates only elements in the same fibre over M .
(Note that τM ◦ Tσ = σ ◦ τE .) Let (xa, yi) be local linear coordinates on E. We use the
standard notation (xa, yi, ẋb, ẏj) for induced coordinates on the tangent bundle TE and, to
describe κ ⊂ TE × TE, we shall underline coordinates in the second copy of TE.
As κ is a double vector subbundle of (TE × TE,Tσ × τE , τE × Tσ), the graph of the base
map ρL := κ : E → TM of κ is a vector subbundle in E × TM , i.e., it is a graph of a VB
morphism. Thus it maps a vector (xa, yi) from E to a vector of the form (xa = xa, ẋb =
Qbi(x)yi) in TM . Now, since for every fixed a ∈ E relation κa : (TE)ρL(a) → TaE is a linear
map, it turns out that κ is given by a linear mapping of coordinates (yi, ẏj) to coordinates
(ẋb, ẏj) with coefficients depending on coordinates (xa, yi) in E. Furthermore, we know that κ
is bi-homogeneous. Calculation on weights: w(yi) = (1, 0) = w(ẋa), w(ẋa) = (0, 1) = w(yi),
w(ẏi) = (1, 1) = w(ẏi) ensures us that κ is determined by equations of the form
κ :
xa = xa,
ẋb = Qbi(x) yi,
ẋb = Q̃bi(x)yi,
ẏk = αki (x)ẏi +Qkij(x)yiyj ,
(2.5)
where x = (xa) and Qbi(x), Q̃bi(x), αji (x) and Qkij(x) are some smooth functions defined locally
on M .
Note that if we additionally assume that κ induces the identity on the core bundle, then the
coefficients αji (x) are simply constants δji . 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 Qbi(x) and Q̃bi(x), respectively, while functions Qkij(x) encode the bracket
operation in a given basis of sections of the bundle σ, dual to functions (yi).
Lemma 2.9 (a class of linear VBCs). Let σ : E → M be a vector bundle and let κ : Tσ →B τE
be a linear VBC. Assume additionally that κ induces the identity on the core bundles, i.e.,
κ ∩ (C × C) = graph(idC). Then:
(i) The base map ρL := κ : E → TM is a VB morphism covering the identity map idM .
(ii) Relation κ′ := (τE × Tσ)(κ) ⊂ E × TM , which is the τE × Tσ-projection of κ, is a graph
of a VB morphism ρR : E → TM covering the identity map idM .
10 M. Jóźwikowski and M. Rotkiewicz
(iii) Relation κ respects the action of the core C, i.e., if (A,B) ∈ κ then (A+++c,B+++c) ∈ κ,
whenever the addition of c ∈ C makes sense.
(iv) The inverse relation κT is a linear VBC from Tσ to τE over the base map ρR : E → TM
which also induces the identity on the cores.
Proof. The assertion has a rather straightforward geometric justification based essentially on
the bi-homogeneity of κ. For brevity, however, we prefer the following local argument.
Since κ is the identity on the cores, we conclude that κ ∩ (M ×M) = ∆M ⊂ M ×M is
a diagonal, and hence κ 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 σ : E→M
is given by a bilinear bracket [·, ·] on the space of smooth sections of σ, together with a pair of
vector bundle maps (left and right anchors) ρL, ρR : E → TM over the identity on M such that
[f · a, g · b] = fρL(a)(g) · b− gρR(b)(f) · a+ fg · [a, b] (2.6)
for every sections a, b ∈ SecM (E) and every smooth functions f, g ∈ C∞(M).
In addition
(i) If the bracket is skew-symmetric, i.e., [a, b] = −[b, a] (in particular, left and right anchors
coincide, i.e., ρL = ρR =: ρ and we speak simply about the anchor) we call σ a skew
algebroid.
(ii) If σ is a skew algebroid and the anchor ρ maps the algebroid bracket [·, ·] to the Lie bracket
of vector fields on M , i.e.,
ρ[a, b] = [ρ(a), ρ(b)]TM , (2.7)
we call σ an almost-Lie algebroid (AL algebroid, in short).
(iii) If σ is an almost-Lie algebroid and the bracket [·, ·] satisfies the Jacobi identity, we call σ
a Lie algebroid.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 11
We will refer to the objects from the above definition as to (Lie) algebroids, i.e., writing that,
say, (σ, ρL, ρR, [·, ·]) 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 TE of a vector bundle σ : E →M . It turns out that an algebroid structure on σ induces
such a VBC.
Lemma 2.11 (from (Lie) algebroids to VBCs). Let (σ : E → M, [·, ·], ρL, ρR) be a (Lie) alge-
broid. Formula
κa [Tb(ρL(a))] := Ta(ρR(b))+++[a, b], (2.8)
where a, b ∈ SecM (E) are arbitrary sections, extends to the unique relation κ ⊂ TE × TE such
that
(i) κ is a linear vector bundle comorphism from Tσ : TE → TM to the tangent bundle
τE : TE → E covering the left anchor ρL : E → TM ,
(ii) κ induces the identity on the core bundle, i.e., κ ∩ (C × C) = graph(idC).
Moreover, the (τE × Tσ)-projection of κ is the right anchor ρR : E → TM .
The proof is given in Appendix A.
Remark 2.12 (the inverse of κ). Note that, according to Lemma 2.9(iv), the inverse relation κT
is also a linear VBC (this time over the right anchor ρR) inducing the identity on the cores. It
clearly satisfies (we use the properties (i) and (ii) of κ to transform equality (2.8)) the condition
κTb [Ta(ρR(b))] = Tb(ρL(a))−−−[a, b],
i.e., passing from κ to κT corresponds to changing the bracket [a, b] to [a, b]T := −[b, a] (note
that this operation interchanges left and right anchors). Obviously κ = κT if the bracket is
skew-symmetric.
Example 2.13 (the tangent algebroid). The tangent bundle τM : TM → M of a manifold M
carries a canonical Lie algebroid structure. The left and right anchors are the identity maps
ρL = ρR = idTM and the bracket [·, ·]TM is the commutator of vector fields.
In this case the corresponding VBC is, in fact, an isomorphism of DVB κM : TτM → τTM .
If (xa) are local coordinates on M , (xa, ẋb) induced coordinates on TM and (xa, ẋb, δxc, δẋd)
induced coordinates on TTM , κM is expressed as
κM :
(
xa, ẋb, δxc, δẋd
)
7−→
(
xa, δxb, ẋc, δẋd
)
.
Note that κM interchanges the two VB structures on TTM , TτM and τTM .
Example 2.14 (the tangent lift of a (Lie) algebroid). If (σ : E → M,ρL, ρR, [·, ·]) is a (Lie)
algebroid structure, then Tσ : TE → TM also carries a canonical algebroid structure (Tσ, dTρL,
dTρR, [·, ·]dT
) called the tangent lift of the algebroid structure on σ. This structure is determined
by conditions
dTρL = κM ◦ TρL, dTρR = κM ◦ TρR and [Ta,Tb]dT
= T[a, b]
for any sections a, b ∈ SecM (E).
12 M. Jóźwikowski and M. Rotkiewicz
In this case (see [19]) the VBC dTκ, corresponding to the considered algebroid structure
on Tσ, is the tangent lift of the VBC κ corresponding to the initial algebroid structure on σ
composed with two canonical isomorphisms κE , i.e.,
dTκ = κE ◦ Tκ ◦ κE .
This construction has a natural generalization to higher tangent lifts Tkσ : TkE → TkM . It will
be discussed in detail in the second paragraph of Section 4.1.
From VBCs to algebroids. In fact, relation κ introduced in Lemma 2.11 completely
characterizes the (Lie) algebroid structure on σ.
Proposition 2.15 (from VBCs to (Lie) algebroids). Let σ : E →M be a vector bundle. A linear
vector bundle comorphism κ from Tσ : TE → TM to the tangent bundle τE : TE → E, which
induces the identity on the core bundles, i.e., κ ∩ (C × C) = graph(idC), provides σ with the
unique general algebroid structure.
The left anchor ρL (resp., the right anchor ρR) is given by the base map of κ (resp., the base
map of κT ) and the bracket is given by
Va[a, b] := κa [Tb(ρL(a))]− Ta(ρR(b)), (2.9)
for any sections a, b ∈ SecM (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 (σ, ρL, ρR, [·, ·]) on
σ : E → M as a pair (σ, κ), where κ : Tσ →B τE is a linear VBC satisfying natural properties.
In what follows we shall often refer to this description.
Remark 2.17 (the dual of κ). The fact that κ 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 κ)
T∗E
τ∗E
��
κ∗ // TE∗
Tσ∗
��
E
ρL // TM
covers the identity idE∗ under the projections T∗E → E∗ and TE∗ → E∗ (the core of a DVB
becomes a side bundle under dualization [29]). In other words, κ∗ : TE∗ → T∗E ' T∗E∗
corresponds to a linear bi-vector on E∗. This is an original point of view of [18].
Remark 2.18 (a local form of κ). 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 αki (x) = δki . Within this
description ρL : E → TM , the base map of κ, is given by ρL : E 3 (xa, yi) 7−→ (xa = xa,
ẋb = Qbi(x)yi) ∈ TM . Similarly, the right anchor reads as ρR : E 3 (xa, yi) 7−→ (xa = xa,
ẋb = Q̃bi(x)yi) ∈ TM .
Take now local sections a(x) ∼ (xa, yj = ai(x)), and b(x) ∼ (xa, yj = bi(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
[a, b](x) ∼
(
xa, yk =
∂bk
∂xa
(x)Qai (x)ai(x)− ∂ak
∂xa
(x)Q̃aj (x)bj(x) +Qkija
i(x)bj(x)
)
.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 13
(Lie) algebroid morphisms as 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 VBC-morphism between appropriate relations.
Intuitively, a morphism between two (Lie) algebroid structures on σ : E → M and on
σ′ : E′ →M ′ should be a vector bundle map φ : E → E′ over φ : M → M ′ which intertwines
the anchors and the algebroid bracket on sections. This intuition, however, faces immediate
problems as, in general, a VB morphism φ does not map sections of σ to sections of σ′. This
problem is solved by passing to the pull-back bundles.
Definition 2.19 (morphism of (Lie) algebroids, [34]). Let (σ : E → M, [·, ·], ρL, ρR) and
(σ′ : E′ → M ′, [·, ·]′, ρ′L, ρ′R) be (Lie) algebroids. A VB morphism φ : E → E′ over φ : M → M ′
is a morphism between the algebroid structures on σ and σ′ if and only if the VB morphisms φ
and Tφ relate the left and the right anchors of σ and σ′, i.e.,
E
φ //
ρL
��
E′
ρ′L
��
TM
Tφ
// TM ′
and
E
φ //
ρR
��
E′
ρ′R
��
TM
Tφ
// TM ′,
(2.10)
and if for every sections a, b ∈ SecM (E) such that their push-forwards φ∗a, φ∗b ∈ SecM (φ∗E′)
can be (locally) presented as finite sums φ∗a =
∑
i fi · φ
∗ai and φ∗b =
∑
j gj · φ
∗bj for some
functions fi, gj ∈ C∞(M) and some sections ai, bj ∈ SecM ′(E
′) we have
φ∗[a, b] =
∑
i,j
(
ρL(a)(gj) · φ∗bj − ρR(b)(fi) · φ∗ai + figj · φ∗[ai, bj ]′
)
. (2.11)
Remark 2.20 (on notion of an algebroid morphism). For any x ∈M there is a neighbourhood U
of x in M and a neighbourhood U ′ of φ(x) in M ′ such that for any section a of σ we have
φ∗a =
∑
i fiφ
∗e′i where (e′i) is a basis of sections of σ′ over U ′ and fi 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 φ∗a and φ∗b as finite sums of sections with C∞(M) coefficients. To prove that
this is the case, we check that the right-hand side of (2.11) is tensorial with respect to ai and bj
(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 VBC-morphisms). Let (σ : E→M, [·, ·], ρL, ρR)
and (σ′ : E′ →M ′, [·, ·]′, ρ′L, ρ′R) be (Lie) algebroids with the corresponding vector bundle comor-
phisms κ : Tσ →B τE and κ′ : Tσ′ →B τE′, respectively. A VB map φ : E → E′ over φ : M →M ′
is a morphism between the algebroid structures on σ and σ′ if and only if (Tφ,Tφ) is a VBC-
morphism from κ to κ′.
The proof is given in Appendix A.
Subalgebroids and algebroidal relations. Let us now describe the notion of a subalge-
broid in terms of VBCs. Recall the following definition.
14 M. Jóźwikowski and M. Rotkiewicz
Definition 2.22 (subalgebroid, [15, 34]). Let (σ : E → M,ρL, ρR, [·, ·]) be a (Lie) algebroid.
A vector subbundle σ′ : E′ →M ′ of σ is called a subalgebroid of σ if the following two conditions
are satisfied
(i) The restrictions of the anchors ρL|E′ and ρR|E′ map E′ ⊂ E to TM ′ ⊂ TM .
(ii) If sections a, b ∈ SecM (E) are such that a|M ′ , b|M ′ ∈ SecM ′(E
′), then [a, b]|M ′ ∈ SecM ′(E
′).
The first of the above conditions assures us that the bracket operation [·, ·]E′ defined on
sections of σ′ by [ã, b̃]E′ := [a, b]|M ′ does not depend on the choice of the extensions a, b ∈
SecM (E) of sections ã, b̃ ∈ SecM ′(E
′). The second condition guarantees that the section space
SecM ′(E
′) is closed with respect to this bracket. Clearly the subbundle σ′ carries a general
algebroid structure inherited from σ.
In the face of the relationship between (Lie) algebroids and VBCs, the VBC κ : Tσ→BτE
corresponding to the algebroid structure on σ should induce some VBC κ′ : Tσ′ →B τE′ corre-
sponding to the structure of a subalgebroid on σ′ described above. We claim that such a κ′ is
a fine restriction of κ in the sense of the definition below.
Definition 2.23 (fine restriction). Let, for i = 1, 2, σ′i : E
′
i → M ′i be a vector subbundle of
σi : Ei → Mi, and let r : σ1 →B σ2 be a VBC over a base map r : M2 → M1. We say that r
restricts fine to σ′1×σ′2 if r(M ′2) ⊂M ′1 and if for any X ∈ E1 and Y ∈ E2 that are r-related and
such that X ∈ E′1 while σ2(Y ) ∈ M ′2 we have Y ∈ E′2. If this is the case, relation r′, defined as
the intersection of r with E′1 × E′2, defines a VBC from σ′1 to σ′2.
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 (σ :E→M,κ) be a (Lie) algebroid and let σ′ :E′→M ′
be a vector subbundle of σ. Then the following conditions are equivalent:
(i) The subbundle σ′ is a subalgebroid of σ.
(ii) The inclusion map ι : σ′ ↪→ σ is an algebroid morphism.
(iii) The VBC κ restricts fine to Tσ′ × τE′ ⊂ Tσ × τE.
Proof. It involves some elementary diagram-chasing to check that (iii) is equivalent to (Tι,Tι)
being a VBC-morphism between κ′ and κ. 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 gener-
alization of a morphism of algebroids. It is closely related to the notion of a subalgebroid.
Definition 2.25 (algebroidal relation). Let (σ1 : E1 → M1, κ1) and (σ2 : E2 → M2, κ2) be
(Lie) algebroids. A relation r ⊂ E1 × E2 is called an algebroidal relation if the graph of r is
a subalgebroid of the product algebroid (σ1 × σ2, κ1 × κ2).
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 (σ1 : E1 →M1, ρ1L, ρ1R, [·, ·]1) ' (σ1, κ1) and
(σ2 : E2 → M2, ρ2L, ρ2R, [·, ·]2) ' (σ2, κ2) be (Lie) algebroids and let r : σ1 →B σ2 be a VBC
covering r : M2 →M1. The following are equivalent:
(i) r is an algebroidal relation,
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 15
(ii) (a) For any section s1 ∈ SecM1(E1) and I = L,R, the vector fields ρ2I(r̂(s1)) and ρ1I(s1)
are r-related and
(b) for any sections s1, s
′
1 ∈ SecM1(E1) we have
r̂([s1, s
′
1]1) = [r̂(s1), r̂(s′1)]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, (σ′j : E′j → M ′j , κ
′
j) be
a subalgebroid of (σj : Ej →Mj , κj). Let us assume that a VBC
E1
σ1
��
r � ,2E2
σ2
��
M1 M2
roo
restricts fine to r′ ⊂ E′1 × E′2. Then if r is an algebroidal relation then so is r′.
Proof. We should check that r′ is a subalgebroid of (σ′1 × σ′2, κ′1 × κ′2). Since r restricts fine to
E′1 × E′2, the relation r′ is a vector subbundle of E′1 × E′2. Let us take X,Y ∈ TE′1 × TE′2 such
that X,Y are κ′1×κ′2-related and X ∈ Tr′ while Y lies over y ∈ r′ ⊂ E′1×E′2. We have to show
that Y ∈ Tr′.
First note that Y ∈ Tr, because r respects algebroid structures of E1 and E2, and X,Y are
also κ1 × κ2-related as E′1 × E′2 is a subalgebroid of E1 × E2, and X ∈ Tr while y ∈ r′ ⊂ r.
Next, by [27, Theorem A.5], Tr restricts fine to TE′1 ×TE′2. Therefore, since Y belongs also to
TE′1 × TE′2, we get Y ∈ Tr′ = Tr ∩ (TE′1 × TE′2) 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 κ.
Lemma 2.29 (characterization of various algebroids). Let (σ : E →M, [·, ·], ρL, ρR) be a general
algebroid structure on σ and let κ : Tσ →B τE be the corresponding vector bundle comorphism.
Then σ is
(i) A skew algebroid if and only if κ is symmetric, i.e., if (X,Y ) ∈ κ then (Y,X) ∈ κ
(equivalently κ = κT ).
(ii) An almost-Lie algebroid if and only if it is skew, and (Tρ,Tρ) : κ⇒ κM is a morphism of
VBCs (in the sense of Definition 2.3):
TE
Tρ
��
κ � ,2TE
Tρ
��
TTM
κM // TTM.
(2.12)
Here ρ = ρL = ρR is the anchor map.
16 M. Jóźwikowski and M. Rotkiewicz
(iii) A Lie algebroid if and only if it is almost-Lie and κ ⊂ TE × TE is a subalgebroid of the
product algebroid (Tσ × τE , dTκ × κE). In other words, κ : Tσ →B τE 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 stan-
dard 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 fibra-
tion [11, 21], the crucial notion in the Lagrangian and Hamiltonian formalisms for mechanics on
Lie algebroids developed by Mart́ınez in [36, 37].
Example 2.30 (prolongation of an AL algebroid over a fibration). Let π : P →M be a fibration
and let (σ : E → M,κ) be an almost-Lie algebroid. Denote by ρ : E → TM the related anchor
map. Then T EP := E ×TM TP = {(e, v) ∈ E × TP : ρ(e) = Tπ(v)} is a vector bundle
over P which carries an almost-Lie algebroid structure defined by the restriction of the relation
κ× κP : TE × TTP →B TE × TTP to T(T EP ) ' TE ×TTM TTP :
TE ×TTM TTP
��
κ×κP � ,2TE ×TTM TTP
��
TP E ×TM TP.oo
Indeed, the only thing to check is that T EP is a subalgebroid of the product algebroid
(σ×τP : E×TP →M×P, κ×κP ). This is straightforward, given X,X ′ ∈ TE and Y, Y ′ ∈ TTP
such that (X,X ′) ∈ κ, Y ′ = κP (Y ), and (X,Y ) is tangent to T EP , we need to prove that
(X ′, Y ′) is also tangent to T EP , i.e., that Tρ(X ′) = TTπ(Y ′). But Tρ(X ′) = κM (Tρ(X)) while
TTπ(Y ′) = κM (TTπ(Y )) (due to the assumption that κ 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 N-graded manifolds
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 TkM – 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 (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 (kth-order) tangent bundle τkM : TkM → M of a mani-
fold M , consisting of kth-order tangency classes (called k-velocities) of curves in M . Bundle
T1M = TM is just the tangent bundle of M , however for k > 1, τkM is no longer a vector
bundle. We shall see this at the elementary level.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 17
Given a smooth function f on a manifold M and an integer α = 0, 1, . . . , k one can construct
a function f (α) on TkM , the so-called α-lift of f (see [40]). It is defined by
f (α)(tkγ) :=
dα
dtα
∣∣∣∣
t=0
f(γ(t)),
where tkγ denotes the k-jet of a curve γ : R → M at zero. We shall usually write ḟ , f̈ in-
stead of f (1), f (2), respectively. The adapted coordinates (xa, ẋa, ẍa, . . .) for TkM , induced by
coordinates (xa) on M , are obtained by applying the above lifting procedure to coordinate func-
tions xa, and are naturally graded. (We simply assign weight k to coordinates xa,(k) := (xa)(k).)
In particular, on T2M they transform as
xa
′
= xa
′
(x), ẋa
′
=
∂xa
′
∂xb
ẋb, ẍa
′
=
∂xa
′
∂xb
ẍb +
∂2xa
′
∂xb∂xc
ẋbẋc (3.1)
thus the weight of both left and right sides of the above equalities is the same. From a geo-
metric point of view, fibres of τkM are equipped with a special structure. Namely, we have
a canonical action of the multiplicative monoid of real numbers (R, ·) on these fibres defined by
re-parametrizing curves representing the elements of TkM :
h : R× TkM → TkM,
(
t, tkγ
)
7→ tkγt,
where γt(s) = γ(ts). We clearly see, that the α-lift of a function f on M is a homogeneous
function on TkM of weight α, i.e.,
f (α)(ht(v)) = tαf (α)(v),
where ht(v) := h(t, v). This also explains why the adapted coordinates on TkM are graded.
Properties of the higher tangent bundle TkM 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 Rn.
Definition 3.1 (N-graded manifold [47]). A N-graded manifold is a smooth fibration σk :Ek→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 Rn (for some integer n), and that the transition functions are multi-
variable polynomials that preserve the weights. The index k in Ek indicates that the weights
on Ek are less or equal k. We say that Ek 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 : R × E → E of the multiplicative monoid of real num-
bers (R, ·). Surprisingly, both concept coincide in the smooth setting [17, Theorem 4.2], in
particular, every N-graded manifold σk : Ek → M admits a canonical homogeneity structure
hE
k
: R× Ek → Ek. For this reason we shall use the terms N-graded manifold and homogeneity
structure interchangeably.
Graded coordinates on a N-graded manifold σk : Ek →M can be denoted by (xa, yiw) where
a (superfluous) index w = w(i) ∈ Z>0 at yi indicates that a fibre coordinate yi is homogeneous
of weight w. The base coordinates (xa) are assumed to have weight zero. In such a notation the
associated homogeneity structure reads as
h : R× Ek → Ek, h
(
t,
(
xa, yiw
))
=
(
xa, twyiw
)
, t ∈ R.
18 M. Jóźwikowski and M. Rotkiewicz
It is convenient and fruitful to encode the structure of a N-graded manifold by means of a canon-
ical vector field on Ek called the weight vector field which in graded coordinates is given by
∆ =
∑
i
w yiw
∂
∂yiw
.
For example, the canonical weight vector field on TkM is ∆k
M =
∑k
α=1
∑
a αx
a,(α)∂xa,(α) . In fact,
the weight vector field provides an equivalent characterization of the structure of a N-graded
manifold. For brevity, we often refer to a N-graded manifold as to a pair (Ek,∆).
A morphism from a N-graded manifold σE : Ek → M to σF : F k → M is a smooth map
φ : Ek → F k commuting with the respective homogeneity structures, i.e., φ ◦ hE = hF ◦ φ.
Equivalently φ relates the corresponding weight vector fields ∆E and ∆F . N-graded manifolds
of order k form a category denoted by GM[k]. In view of [16, Theorem 2.4], GM[1] is the
category of vector bundles.
Every N-graded manifold has a zero section 0kM : M → Ek. The image of M in Ek 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 Ek is denoted by Ek[∆ ≥ j] and it is, actually, a graded subbundle of Ek.
Given a N-graded manifold
(
Ek,∆
)
of order k and an integer 0 ≤ j ≤ k one can construct
a canonical projection from Ek onto a N-graded manifold of order j, denoted by Ek[∆ ≤ j],
obtained by removing all coordinates of weights greater than j [6, Definition 1.6]. As transfor-
mation rules for Ek of coordinates of weight ≤ j involve only coordinates of weights ≤ j the
above construction is correct. It follows that a N-graded manifold σk : Ek →M induces a tower
of affine fibrations [47]
Ek
σkk−1−−−→ Ek−1 → · · ·
σ2
1−→ E1 σ1
0−→M, (3.2)
where Ek[∆ ≤ j] is denoted shortly by Ej .
Definition 3.2 (the top core of a N-graded manifold). Let σk : Ek →M be a N-graded manifold
of order k. The top core of Ek, denoted by Êk, is the set
Êk =
{
v ∈ Ek : σkk−1(v) ∈ 0k−1
M (M) ⊂ Ek−1
}
,
where σkk−1 : Ek → Ek−1 as in (3.2). Locally, Êk is defined by putting to zero all fibre coordinates
of weights less than k:
Êk =
{(
xA, yaw
)
: yaw = 0 for any yaw such that 1 ≤ w < k
}
⊂ Ek,
hence
(
xA, yak
)
are local coordinates on Êk. The top core Êk is naturally a vector bundle over M
with the homotheties defined locally by
t.
(
xA, yak
)
=
(
xA, t · yak
)
for t ∈ R. Moreover, the top core ·̂ is a functor from the category GM[k] to the category of
vector bundles.
Example 3.3 (split N-graded manifolds). Given a sequence of vector bundles Ej , j = 1, . . . , k,
over the same base manifold M we can turn the Whitney sum E =
⊕k
j=1Ej into a N-graded
manifold of order k by assuming that linear coordinates on fibres of Ej have weight j. The
obtained N-graded manifold will be denoted by Xkj=1Ej [j], where the notation V [j] means that
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 19
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, . . . ,−k to the cate-
gory GM[k] of N-graded manifolds of order k.
N-graded manifolds equipped with a splitting, i.e., a N-graded manifold isomorphism p : Ek →
Xkj=1Ej as above are called split N-graded manifolds. It is worth to remember that in the smooth
real setting any N-graded manifold is (non-canonically) isomorphic to its split form which is
obtained from the sequence of the top core bundles Êj , j = 1, . . . , k: Ek ' Xkj=1Ê
j .
Multi-N-graded manifolds. Of particular interests are geometric objects which admit
several compatible 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 N-graded manifold (E,∆1, . . . ,∆k) to be a manifold E with k pair-wise commuting
weight vector fields, i.e., [∆i,∆j ] = 0 for any i, j = 1, . . . , k. Equivalently, E admits k pair-wise
commuting homogeneity structures hi : R× E → E with i = 1, . . . , k. That is, hit ◦ h
j
s = hjs ◦ hit
for any i, j = 1, 2, . . . , k and any t, s ∈ R. Interestingly, every k-tuple 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 N-graded manifold σk :Ek→M
are examples of double N-graded manifolds. There are numerous other constructions of multi-
N-graded manifolds: the iterated higher tangent bundles Tk1 · · ·TkrM , the substructures and
quotients (e.g., the construction of the linearization of a N-graded manifold, the vertical sub-
bundle VEk = ker Tσk ⊂ TEk) 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 ∆i).
The notion of core of a DVB has a natural generalization for multi-N-graded manifolds.
Definition 3.4 (the core and the ultracore of a multi-N-graded manifold). Let (E,∆1, . . . ,∆k)
be a k-tuple N-graded manifold and k > 1. The core C(E) of E is obtained by putting to zero
all fiber coordinates yiw with multi-weight w in which at least one of w1, . . . , wk is zero. Formally,
using the construction E[X ≥ 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 ∆i, we may write the core of E as the intersection
C(E) :=
k⋂
j=1
E
[
N∆j −
∑
i 6=j
∆i ≥ 0
]
,
for a sufficient large number N . Indeed, E[N∆j −
∑
i 6=j ∆i ≥ 0] is the subset of E obtained
by putting to zero all fiber coordinates yiw of the total fibration E → M = E[∆1 = ∆2 =
· · · = ∆k = 0] with multi-weight w = (w1, . . . , wk) in which wj = 0. In general, C(E) remains
a k-tuple N-graded manifold.
By the ultracore of E we mean the top core Ê of the N-graded manifold (E,∆1 + · · ·+ ∆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 N-graded manifolds with
one of the homogeneity structures being linear. If
(
E, h1, h2
)
is such a structure, we may treat it
as a vector bundle σ : E →M (say, that the VB structure on E corresponds to the homogeneity
structure h2) equipped with a N-graded manifold structure encoded in the homogeneity structure
h1 : R×E → E such that the structure maps of the vector bundle σ (vector bundle projection,
zero section, addition, and scalar multiplication) are weighted or, in other words, homogeneous
20 M. Jóźwikowski and M. Rotkiewicz
with respect to h1. In particular, for each λ ∈ R, the mapping v 7→ λ ·σ v, v ∈ E, should be
a N-graded manifold morphism, or equivalently, it should commute with the homotheties h2
t
for any t ∈ R. A double N-graded manifold
(
E, h1, h2
)
in which
(
E, h2
)
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 σ : E →M can be depicted in a diagram
Ek
σk0 //
σ{k}
��
E0
σ{0}
��
Mk
σk0 //M0.
(3.3)
Here Ek = E, σ{k} = σ and the superscript k at E indicates the order of the homogeneity
structure. The base M = Mk of the vector bundle σ carries a N-graded manifold structure
induced from σk0 : Ek → E0. The notation in parentheses { } is used in order to distinguish
the vector bundle projection σ{k} given by the homotheties h2 from the N-graded manifold
projection σk0 given by the homogeneity structure h1.
An upcoming definition of a higher (Lie) algebroid involves two particular examples of graded-
linear manifolds: the tangent bundle of a N-graded manifold of order k, and the kth-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,∆1,∆2) is a graded vector bundle, i.e.,
C(D) =
k⊕
j=1
Dj [j]
for some vector bundles Dj canonically associated with D. In particular cases,
C
(
TkE
)
' E[1]⊕ E[2]⊕ · · · ⊕ E[k], and
C(TEk) ' E1[1]⊕ Ê2[2]⊕ · · · ⊕ Êk[k].
Proof. Bi-graded coordinates on D have the form
(
xA, ya(i,j)
)
where i = 0 or 1 and 0 ≤ j ≤ k,
and (i, j) 6= (0, 0). As C := C(D) is given locally in D by equations ya(1,0) = 0 and ya(0,j) = 0,
1 ≤ j ≤ k, the transformation rules for the fibre coordinates ya(1,j)|C , j = 1, . . . , k, are of the
form ya
′
(1,j)|C = Qa
′
j,b y
b
(1,j)|C for some functions Qa
′
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 ≤ j ≤ k is a fixed integer, form a vector bundle Dj over M and C is just the
Whitney sum of these vector bundles. The vector bundle Dj can be recognized as the top core
bundle of (D[∆2 ≤ j],∆1 + ∆2). The decomposition of the core of TkE and TEk are due to the
isomorphisms T̂kE ' E and T̂Ek ' Êk. �
Weighted structures and their reductions from ‘higher’ to ‘lower’ order. In a sim-
ilar manner we can introduce notion of other weighted geometrical objects and structures, in
particular VBCs.
Definition 3.6 (weighted VBC). Let σi : Ei →Mi, where i = 1, 2, be weighted VBs of order k
with homogeneity structures hi on Ei. A VBC r : σ1 →B σ2 is called a weighted VBC if r ⊂
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 21
E1 × E2 is homogeneous with respect to h1 × h2. In other words, r is a graded submanifold of
the product of N-graded manifolds (σ1)k0×(σ2)k0 : E1×E2 → E0
1×E0
2 . (See diagram (3.3) for the
notation concerning weighted vector bundles.) In particular, the base map r : M2 →M1 is a N-
graded manifold morphism. (We may assume that orders of both the homogeneity structures
are the same, as every N-graded manifold of order k is also of order k′ for any k′ > 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
VB-algebroids and 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(
Ek,∆1,∆2
)
of order k, then its j-order reduction Ej = Ek[∆1 ≤ j] is still a graded-linear
manifold and σkj : Ek → Ej is a morphism in the category of graded-linear manifolds. In
particular, given a weighted VBC r : σ1 →B σ2 between weighted vector bundles σi : E
k
i → Mk
i
(i = 1, 2) of order k we may consider r as a N-graded submanifold in the product bundle
σk1 ×σk2 : Ek1 ×Ek2 → E0
1 ×E0
2 and then project it to get its lower-order reduction rj : Ej1 →B E
j
2,
which is not only a graded-linear submanifold, but also a weighted VBC of order j, for j =
0, 1, 2, . . . , k. This and another useful observation in a similar spirit is presented below.
Proposition 3.8 (reduction of weighted VBCs). Let r : σ1 →B σ2 be a weighted VBC where
σi : E
k
i →Mk
i (i = 1, 2) are weighted vector bundles of order k, and let 0 ≤ j ≤ k. Then
(i) the lower-order reductions rj : σ
{j}
1 →B σ{j}2 remain weighted VBCs.
(ii) If r′ : σ′1 →B σ′2 is another weighted VBC and if (φ1, φ2) : r ⇒ r′, where φi : σi → σ′i are
morphism of graded-linear manifolds, is a VBC-morphism, then the lower-order reductions(
φj1, φ
j
2
)
: rj ⇒ r′j are also 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 τkM : TkM → M (cf. the beginning of Section 3) together with the
canonical isomorphism κkM : TkTM → TTkM 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 (xa) on M induces canonical coordinates (xa, ẋb) on TM ,
(
xa,(α)
)
α=0,...,k
on TkM ,
(
xa, ẋb, xa,(1), ẋb,(1), . . . , xa,(k), ẋb,(k)
)
on TkTM , and
(
xa,(α), ẋb,(β)
)
α,β=0,...,k
on TTkM .
In these coordinates κkM reads as
κkM :
(
xa, ẋb, xa,(1), ẋb,(1), . . . , xa,(k), ẋb,(k)
)
7−→
(
xa, xa,(1), . . . , xa,(k), ẋb, ẋb,(1), . . . , ẋb,(k)
)
.
Clearly, for k = 1 we get κ1
M = κM studied in Example 2.13. It follows easily from the above
coordinate formula that κkM 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 TkM → M with an arbitrary N-
graded manifold Ek →M of order k. Instead of a graded-linear isomorphism κkM we shall take
a weighted vector bundle comorphism κk between TkE1 and TEk. Such postulates are motivated
by the description of (Lie) algebroids (of order one) in terms of a VBC κ (cf. Proposition 2.15
and Lemma 2.11).
22 M. Jóźwikowski and M. Rotkiewicz
Definition 4.1 (general higher algebroid, HA). A general higher (kth-order) algebroid (HA, in
short) is a N-graded manifold σk : Ek → M of order k together with a weighted VBC κk ⊂
TkE1 × TEk from Tkσ1 to τEk (covering a (graded) mapping ρk : Ek → TkM) such that the
relation κ1 : Tσ1 →B TτE1 being the reduction to order one of κk equips σ1 : E1 → M with
a general algebroid structure.
TkE1
Tkσ1
��
κk � ,2TEk
τ
Ek
��
TkM Ek.
ρkoo
(4.1)
In addition:
(i) If
(
σ1, κ1
)
is skew, we call
(
σk, κk
)
a skew HA.
(ii) If
(
σk, κk
)
is skew and
(
Tkρ1,Tρk
)
: κk ⇒ κkM is a morphism in the category of VBCs,
where ρ1 : E1 → TM is the order-1 reduction of ρk, then we call
(
σk, κk
)
an almost-Lie
(AL) HA.
(iii) A skew higher algebroid
(
σk, κk
)
in which κk is a subalgebroid of the product of the
algebroids
(
Tkσ1,dTkκ
1
)
(the kth-tangent lift of
(
σ1, κ1
)
– see Section 4.1 – and the tangent
algebroid (τEk , κEk)), 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],
(
σk, κk
)
is Lie if κk is a Lie algebroid comorphism.
(iv) A general HA in which κk induces an isomorphism on the core bundles C
(
κk
)
: C
(
TkE
)
→
C
(
TEk
)
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
(
σkE : Ek →M,κk,E
)
and
(
σkF : F k →
N,κk,F
)
to be a N-graded manifold morphism φk : Ek → F k such that
(
Tkφ1,Tφk
)
: κk,E ⇒ κk,F
is a VBC-morphism. Obviously, general higher algebroids form a category which we denote by
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 κk covers
a (graded) morphism ρk : Ek → TkM , and its reduction to order 0 induces a relation (in fact
a linear map) κ0 : E1 → TM . 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 κ0 = ρ1.
Using Proposition 3.8 we easily see that reduction of a (Lie) HA
(
Ek, κk
)
to a lower order
j = 1, 2, . . . , k gives an induced (Lie) higher algebroid structure on Ej which is skew (resp. AL,
Lie, strong) if
(
Ek, κk
)
was so. Moreover, a (Lie) HA morphism φk :
(
Ek, κk,E
)
→
(
F k, κk,F
)
induces a (Lie) HA morphisms φj :
(
Ej , κj,E
)
→
(
F j , κj,F
)
.
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 (xa, yi, zµ) be graded
coordinates on an order-2 N-graded manifold E2 → M . We shall find a general form of an
algebroid
(
E2, κ2
)
of order 2. As in Remark 2.8 we underline coordinates in TE2 when writing
equations for κ2 ⊂ T2E1 × TE2. Taking weights into account
weight (0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2)
T2E1 xa ẋa ẍa yi ẏi ÿi
TE2 xa yi zµ ẋa ẏi żµ
(4.2)
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 23
we get
κ2 :
ẋa = Qai y
i,
ẍa = 1
2Q
a
ij y
iyj +Qaµ z
µ, where Qaij = Qaji,
ẋa = Q
′a
i y
i,
ẏi = ẏi +Qijk y
jyk,
żµ = Qµi ÿ
i +Qµijy
iẏj +Qµνiz
νyi + 1
2Q
µ
ij,ky
iyjyk, where Qµij,k = Qµji,k,
(4.3)
for some structure functions Q······ of base coordinates xa. For a skew algebroid
(
E2, κ2
)
we have
Q
′a
i = Qai and Qijk = −Qikj . An algebroid is strong if the matrix Qµi is invertible. The conditions
of being an almost-Lie or a Lie algebroid result in more complicated systems of equations for
the structure functions.
Our considerations from Section 2 lead to a notion of a higher subalgebroid.
Definition 4.3 (subalgebroid of a (Lie) HA). Let
(
σk : Ek →M,κk
)
be a (Lie) higher algebroid
and let σ′k : E′k → M ′ be a N-graded submanifold of σk. We call σ′k a higher subalgebroid of(
σk, κk
)
if the VBC κk restricts fine to Tkσ′1 × τE′k ⊂ Tkσ1 × τEk .
4.1 On the Lie axiom
In order to discuss the notion of a Lie higher algebroid, we shall first take a look at higher tangent
lifts of a first order (Lie) algebroid structure. We will begin our considerations by studying the
lifts of sections of a vector bundle.
Higher lifts of sections of a vector bundle. Let σ : E → M be a vector bundle and
s ∈ SecM (E) its section. The assignment SecM (E) 3 s 7→ Tks ∈ SecTkM
(
TkE
)
is injective, and
R-linear, yet it cannot be onto, since rank
(
Tkσ
)
= (k + 1) · rank(σ) > rank(σ). It is, however,
possible to characterize all sections of Tkσ in terms of sections of σ by means of the following
procedure.
First, note that the multiplicative action of the reals R × E → E lifts to the multiplicative
action TkR × TkE → TkE of TkR, the latter understood as an R-algebra with addition and
multiplication defined as Tk-lifts of the standard addition and multiplication in R. In fact, as an
algebra TkR is canonically isomorphic to the truncated polynomial algebra R[ε]/〈εk+1〉. (This
algebra is often denoted Dk and called the algebra of kth-order numbers.) From the point of
view of the theory of natural bundles [28], Tk is a product-preserving bundle functor associated
with the Weil algebra Dk. In conclusion, on TkE we have an operation of multiplication by the
elements of Dk, which is in fact determined by the action of the generator ε ∈ Dk. In coordinates(
xa,(α), yi,(α)
)
on Tkσ induced from linear coordinates (xa, yi) on σ it reads as
ε ·
(
xa,(α); yi,(0), yi,(1), . . . , yi,(k)
)
=
(
xa,(α); 0, yi,(0), 2yi,(1), . . . , kyi,(k−1)
)
.
This construction leads us to
Definition 4.4 (lifts of a section). Let s be a section of σ and α = 0, 1, . . . , k an integer. By
the (k − α)-lift of s we understand the section of Tkσ defined as
s(k−α) :=
(k − α)!
k!
εα · Tks.
In particular s(0) = 1
k!ε
k ·Tks is called the vertical and s(k) = Tks the total lift of s, in agreement
with the standard notions for k = 1.
24 M. Jóźwikowski and M. Rotkiewicz
Using local coordinates it is easy to check that sections of the form s(k−α) with α = 0, 1, . . . , k
span locally the full space of sections SecTkM
(
TkE
)
. Recall the notion of an (α)-lift of a smooth
function introduced at the beginning of Section 3. For any function f ∈ C∞(M) the operation
of the (k)-lift has the following property
(f · s)(k) = Tk(f · s) = Tkf · Tks
=
(
k∑
α=0
f (α) ε
α
α!
)
· s(k) =
k∑
α=0
f (α)
α!
k!
(k − α)!
s(k−α) =
k∑
α=0
(
k
α
)
f (α)s(k−α).
We recognize the standard formula for the iterated derivative. An analogous formula for
(f · s)(k−α) can be easily derived from the latter.
Higher tangent lifts of (Lie) algebroids. It is well-known that if a VB σ : E →M carries
a (Lie) algebroid structure, then its kth-tangent lift Tkσ : TkE → TkM has the so-called lifted
(Lie) algebroid structure (e.g., [30]). It can be elegantly described in terms of the VBC κ related
with the initial algebroid structure on σ.
Definition 4.5 (higher tangent lift of a (Lie) algebroid). Let (σ : E→M,ρL, ρR : E→TM, [·, ·])
' (σ, κ : Tσ →B τE) be a (Lie) algebroid. A VBC dTkκ : TTkσ →B τTkE over an anchor map
dTkρL : TkE → TTkM obtained, up to natural identifications, by applying functor Tk to κ:
TTkE
TTkσ
��
'κ−1
k,E // TkTE
Tkκ � ,2
TkTσ
��
TkTE
TkτE
��
'κk,E // TTkE
τ
TkE
��
TTkM TkTM
'κkMoo TkE
TkρLoo TkE,
=oo
(4.4)
i.e., dTkκ = κ−1
k,E ◦Tkκ ◦ κk,E , and dTkρL := TkρL ◦ κMk , equips Tkσ : TkE → TkM with a (Lie)
algebroid structure. We call it the kth-tangent lift of the algebroid (σ, κ).
We shall now check that dTkκ indeed defines a (Lie) algebroid structure and then describe it
in the classical terms of the anchors and the bracket operation.
Proposition 4.6 (properties of the lifted (Lie) algebroid). The relation dTkκ := κk,E◦Tkκ◦κ−1
k,E
defines a (Lie) algebroid structure on Tkσ. Moreover,
(i) The (left and right) anchor maps on Tkσ are given by dTkρL := κkM ◦TkρL and dTkρR :=
κkM ◦ TkρR, respectively.
(ii) The algebroid bracket [·, ·]d
Tk
on Tkσ satisfies[
k!
(k − α)!
s
(k−α)
1 ,
k!
(k − β)!
s
(k−β)
2
]
d
Tk
=
k!
(k − α− β)!
([s1, s2]σ)(k−α−β), (4.5)
for any integers α, β = 0, 1, . . . , k such that α+ β ≤ k and any sections s1, s2 ∈ SecM (E),
and
[
s
(k−α)
1 , s
(k−β)
2
]
d
Tk
= 0 if α + β > k. These properties fully determine the bracket
[·, ·]d
Tk
on the space of sections of Tkσ.
(iii) If (σ, κ) is a skew/AL/Lie algebroid, then so is
(
Tkσ, dTkκ
)
.
The proof is given in Appendix A.
Remark 4.7 (characterization of lifts by the canonical pairings). The pairing 〈·, ·〉σ : E∗×ME →
R lifts to the pairing Tk 〈·, ·〉σ : TkE∗ ×TkM TkE → R which enables us to identify sections of
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 25
the vector bundle Tkσ with linear functions on TkE∗. As TkE∗ is, in particular, a N-graded
manifold of order k, the space of sections of Tkσ is naturally graded.
Consider a section s of σ. It is easy to see that its (k − α)-lift s(k−α) (considered as a linear
function on TkE∗) coincides with the (k − α)-lift s̃(k−α) of the linear function s̃ : E∗ → R
canonically associated with the section s. Taking into account the N-graded manifold structure
of TkE∗, a section of the form f (α)s(k−β), where f ∈ C∞(M) and s ∈ Sec(E), would have weight
α+k−β. However, in a presence of a (Lie) algebroid structure on σ, it is reasonable to shift this
gradation by −k, so that the section f (α)s(k−β) has weight α−β. Then formula (4.5) shows that
homogeneous sections of Tkσ form a graded Lie algebra concentrated in degrees −k,−k+ 1, . . ..
It has a Lie subalgebra SecTkM,≤0
(
TkE
)
consisting of sections of non-positive weights which is
of finite rank (over C∞(M)).
Algebroid lifts of sections and the Lie axiom. By Remark 2.2 a VBC between two vector
bundles induces a canonical map between the associated spaces of sections of these bundles. On
the other hand, in Definition 4.4 we introduced natural lifts of sections of a vector bundle to
sections of its higher tangent lift. Given a (Lie) higher algebroid structure, we can combine
these two constructions to arrive at
Definition 4.8 (algebroid lift of a section). Let
(
σk : Ek → M,κk : Tkσ1 →B τEk
)
be a (Lie)
higher algebroid. Given a section s ∈ SecM
(
E1
)
and an integer α = 0, 1, . . . , k we define the
(k − α)-algebroid lift of s as
s[k−α] := κ̂k
(
s(k−α)
)
∈ X
(
Ek
)
.
The above concept naturally generalizes the notion of a vertical lift and the algebroid lift (for
(k, α) = (1, 0) and (k, α) = (1, 1), respectively) known in the literature [19].
In [18] Grabowski and Urbański observed that the Jacobi identity can be elegantly reformu-
lated in terms of the algebroid lifts. Their results easily generalize to the setting of Lie higher
algebroids.
Proposition 4.9 (characterization of the Lie axiom). The map κ̂k : SecTkM
(
TkE
)
→ X
(
Ek
)
preserve gradation, i.e., for a homogeneous section s of Tkσ the vector field κ̂k(s) has the same
weight as s. Moreover, a higher algebroid
(
σk : Ek →M,κk : Tkσ1 →B τEk
)
is Lie if and only if
it is almost-Lie and if the associated algebroid lift satisfies[
k!
(k − α)!
s
[k−α]
1 ,
k!
(k − β)!
s
[k−β]
2
]
Ek
=
k!
(k − α− β)!
([s1, s2]σ1)[k−α−β] (4.6)
for any sections s1, s2 ∈ SecM
(
E1
)
and any integers α, β = 0, 1, . . . , k. Above [·, ·]Ek denotes
the standard Lie bracket of vector fields on Ek, and [·, ·]σ1 the algebroid bracket on the reduced
(Lie) algebroid
(
σ1, κ1
)
.
Equivalently, an AL higher algebroid
(
Ek, κk
)
is Lie if and only if
κ̂k : SecTkM,≤0
(
TkE
)
→ X≤0
(
Ek
)
(4.7)
is a graded Lie algebra morphism, where SecTkM,≤0
(
TkE
)
(resp. X≤0
(
Ek
)
) are Lie algebras
generated by the homogeneous sections of Tkσ (resp. vector fields on Ek) of non-positive weight.
Proof. Let s ∈ SecTkM
(
TkE
)
be a homogeneous section, denote X := κ̂k(s) and consider the
corresponding functions s̃ ∈ C∞
(
TkE∗
)
, and X̃ ∈ C∞
(
T∗Ek
)
on the dual vector bundles (cf.
Remark 4.7). They are related by X̃ = s̃ ◦ εk, where εk : T∗Ek → TkE∗ is a (weighted) vector
26 M. Jóźwikowski and M. Rotkiewicz
bundle morphism dual to κk. Since κk is bi-homogeneous, so is εk, hence weights of X̃ and s̃
are the same, and our first assertion follows.
By definition,
(
σk, κk
)
is a Lie HA if and only if it is skew and if κk is an algebroidal relation
between
(
Tkσ1, dTkκ
1
)
and (τEk , κEk). Proposition 2.26 provides a useful characterization of
algebroidal relations. It follows that
(
σk, κk
)
is a Lie HA if and only if for any sections s, s′ ∈
SecTkM
(
TkE1
)
we have (note that both
(
Tkσ1,dTkκ
1
)
and (τEk , κEk) are skew algebroids and
their anchor maps are κkM ◦ Tkρ1 and idTEk , respectively):
Tρk
(
κ̂k(s)
)
= κkM
(
Tkρ1(s)
)
= dTkρ(s) (4.8)
and
κ̂k([s, s′]d
Tk
) =
[
κ̂k(s), κ̂k(s′)
]
Ek
. (4.9)
The first of these conditions means that for any section s ∈ Sec
(
Tkσ
)
the vector fields κ̂k(s) ∈
X
(
Ek
)
and κkM ◦ Tkρ1(s) = dTkρ(s) ∈ X
(
TkM
)
are ρk-related and it is equivalent to the
commutativity of the diagram
TkE1 κk � ,2
Tkρ1
��
TEk
Tρk
��
TkTM
'κkM // TTkM,
i.e., the axiom of an AL algebroid. The second condition for lifts s = s
(k−α)
1 and s′ = s
(k−β)
2 ,
with s1, s2 ∈ SecM
(
E1
)
, gives equality (4.6).
Assume now that the equation (4.6) holds and
(
Ek, κk
)
is an almost-Lie higher algebroid.
The latter condition implies (4.8), and from (4.6) it follows that equation (4.9) holds for sections
of the form s = s
(k−α)
1 and s′ = s
(k−β)
2 . To prove that
(
Ek, κk
)
is Lie, we need to show that (4.9)
holds for any sections s, s′ ∈ Sec
(
Tkσ
)
.
Assuming that (4.9) holds for given sections s and s′, we shall show that it also holds for
sections s := φs and s′ for any φ ∈ C∞
(
TkM
)
. Indeed, we have
κ̂k([φs, s′]d
Tk
)
(2.6)
= κ̂k
[
φ[s, s′]d
Tk
− dTkρ(s′)(φ)s)
]
(2.1)
=
(
ρk
)∗
(φ)κ̂k([s, s′]d
Tk
)−
(
ρk
)∗
(dTkρ(s′)(φ))κ̂k(s),
while [
κ̂k(φs), κ̂k(s′)
]
Ek
(2.1)
=
[(
ρk
)∗
(φ)κ̂k(s), κ̂k(s′)
]
Ek
(2.6)
=
(
ρk
)∗
(φ)
[
κ̂k(s), κ̂k(s′)
]
Ek
− κ̂k(s′)
((
ρk
)∗
(φ)
)
κ̂k(s)
=
(
ρk
)∗
(φ)
[
κ̂k(s), κ̂k(s′)
]
Ek
−
(
ρk
)∗(
κ̂k(s′)(φ)
)
κ̂k(s)
(4.8)
=
(
ρk
)∗
(φ)
[
κ̂k(s), κ̂k(s′)
]
Ek
−
(
ρk
)∗
(dTkρ(s′)(φ))κ̂k(s).
Comparison of the above equalities proves our claim. Now, since all (k − α)-lifts (with α ≥ 0)
of sections from SecM
(
E1
)
span the whole space SecTkM
(
TkE1
)
as the C∞
(
TkM
)
-module,
equation (4.9) holds for every sections s, s′ ∈ SecTkM
(
TkE1
)
. This ends the proof. �
Remark 4.10. Note that SecTkM,≤0
(
TkE
)
and X≤0
(
Ek
)
are C∞(M)-modules of finite rank.
It follows that the Lie axiom for HA can be reduced to a finite number of equations of the
form (4.6) with s1 = ei, s2 = ej where (ei) is a local basis of sections of σ1.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 27
Subalgebroids of Lie higher algebroids. We shall end our considerations concerning
the Lie axiom for HA by showing that it is inherited by subalgebroids. First we will need the
following properties of higher tangent lifts.
Proposition 4.11 (properties of the lifted algebroid). Let k be a positive integer. Consider
vector bundles σ : E →M , σ1 : E1 →M1 and σ2 : E2 →M2 and their subbundles σ′ : E′ →M ′,
σ′1 : E′1 →M ′1 and σ′2 : E′2 →M ′2, respectively.
(i) Let r : σ1 →B σ2 be a VBC that restricts fine to a VBC r′ : σ′1 →B σ′2. Then Tkr : Tkσ1→B
Tkσ2 is a VBC which restricts fine to a VBC Tkr′ : Tkσ′1 →B Tkσ′2.
(ii) Let (σ, κ) be a (Lie) algebroid and (σ′, κ′) its subalgebroid. Then the kth-tangent lift alge-
broid
(
Tkσ′,dTkκ
′) is a subalgebroid of
(
Tkσ, dTkκ
)
.
(iii) Let (σ1, κ1) and (σ2, κ2) be (Lie) algebroids and let r : σ1 →B σ2 be an algebroidal rela-
tion. Then Tkr : Tkσ1 →B Tkσ2 is also an algebroidal relation between the kth-tangent lift
algebroids
(
Tkσ1,dTkκ1
)
and
(
Tkσ2,dTkκ2
)
.
The proof is given in Appendix A.
Proposition 4.12 (the Lie axiom and subalgebroids). A higher subalgebroid of a Lie (resp.
almost-Lie) higher algebroid is Lie (resp. almost-Lie).
Proof. Assume that
(
σkF : F k → N,κk,F
)
is a higher subalgebroid of an AL higher algebroid(
σkE : Ek → M,κk,E
)
. To prove that
(
σkF , κ
k,F
)
is also AL we must verify that if two elements
X ∈ TkF 1 and Y ∈ TF k are κk,F -related then their images Tkρ1
F (X) and TρkF (Y ) are κkN -
related. Now as κk,F is a fine restriction of κk,E , elements X and Y as above are κk,E-related,
and thus, since
(
σkE : Ek → M,κk,E
)
was almost-Lie, elements Tkρ1
E(X) and TρkE(Y ) are κkM -
related. The assertion follows from a simple observation that the anchor maps ρkF : F k → TkN
and ρ1 : F 1 → TM are the restrictions ρkF = ρkE |Fk and ρ1
F = ρ1
E |F 1 , respectively.
Now assume that
(
σkE , κ
k,E
)
is Lie. This means that κk,E is an algebroidal relation between
Tkσ1
E and τEk . Moreover, by definition, κk,F is a fine restriction of κk,E . In particular,
(
σ1
E , κ
1,F
)
is a subalgebroid of
(
σ1
E , κ
1,E
)
, hence by Proposition 4.11(ii), Tkσ1
F is a subalgebroid of Tkσ1
E .
Note also that τFk is a subalgebroid of τEk . Thus, by Proposition 4.11(iii), κk,F is also an
algebroidal relation, i.e.,
(
σkF : F k → N,κk,F
)
is a Lie higher algebroid. �
4.2 Prolongations of an AL algebroid
In our previous publication [27] we studied a class of objects crafted to play the role of prototypes
of (Lie) higher algebroids. Our construction was motivated by the procedure of reduction of
a higher tangent bundle of a Lie groupoid. In fact, as we shall see shortly, these objects provide
natural examples of (Lie) higher algebroids in the sense of Definition 4.1. Let us now recall their
construction.
Construction of the prolongations. Let (σ : E → M,κ) be an AL algebroid. For each
k = 1, 2, . . ., we will construct bundles σ[k] : E[k] → M and relations κ[k] ⊂ TkE × TE[k]. We
shall call the pair
(
σ[k], κ[k]
)
the kth-prolongation of the algebroid (σ, κ). The total spaces of
bundles σ[k] are defined inductively:
E[1] := E, E[2] := {A ∈ TE : Tσ(A) = ρ ◦ τE(A)},
E[k+1] := TE[k] ∩ TkE, for k ≥ 2,
where ρ : E → TM is the anchor map of (σ, κ). By the inductive hypothesis, E[k] is considered,
as a subset of Tk−1E, hence both TE[k] and TkE can be understood here as subsets of TTk−1E,
28 M. Jóźwikowski and M. Rotkiewicz
and σ[k] is simply the projection E[k] ⊂ TTk−1E → E
σ→M . (Recall that for any manifold M ,
TkM can be consider a subset of TTk−1M in a natural way.)
The bundle σ[k] : E[k] →M has a very interesting algebraic structure encoded in a relation κ[k]
which can be defined directly as κ[1] = κ, κ[k] =
(
κk−1
E ◦ Tk−1κ
)
∩
(
TkE × TE[k]
)
(see [27,
Proposition 4.6]):
Tk−1TE
Tk−1κ � ,2Tk−1TE
κk−1
E // TTk−1E
TkE
� ?
OO
κ[k] � ,2_________________ TE[k].
� ?
OO
(4.10)
Since κ is a VBC, it is easy to see that so is κk−1
E ◦Tk−1κ. We stress that the fact that the latter
restricts fine to a VBC from TkE to TE[k] is a non-trivial result which strongly depends on the
fact that σ has an AL algebroid structure.
In particular, if we start from the tangent algebroid (σ, κ) = (τM , κM ), then σ[k] is just the
higher tangent bundle τkM : TkM → M and κ[k] is the canonical isomorphism κkM . In general,
σ[k] : E[k] → M is a N-graded manifold of rank (r, r, . . . , r) where r is the rank of E (see [27,
Theorem 4.5]). If (σ, κ) is the Lie algebroid of a Lie groupoid G (i.e., it is a reduction of the
tangent algebroid TG by the action of G), then its kth-prolongation Ak(G) can be interpreted
as a reduction of the higher tangent bundle TkG by the action of G. More precisely, Ak(G) =
TkGα
∣∣
M
, where M is the base of G and α : G → M denotes the source map [27]. That is,
bundle Ak(G) consists of k-jets of curves in G which are tangent to the α-fibres at the base.
To our best knowledge the kth-order prolongations of Lie algebroids were for the first time
studied by Saunders [43], who introduced bundles Ak(G) discussed above. He also proposed
an abstract construction of the second prolongation of a Lie algebroid which is not necessarily
integrable. Later Colombo and De Diego [9] generalized his ideas defining bundle E[k+1] (under
the name of higher order Lie algebroid) as the space of kth-tangent lifts of admissible curves in
a Lie algebroid E. Their construction was later adopted by Martinez [39] to develop higher-
order variational calculus on Lie algebroids. Later in Remark 5.6 we briefly discuss his and other
applications of the notion of a prolongation in variational calculus.
Let us remark that in all the above-cited sources prolongations are constructed out of Lie
algebroids, whereas our results from [27] recognize the almost-Lie class as fundamental to define
a prolongation. Moreover, in all these sources prolongations are understood only as bundles
σ[k] : E[k] →M , and the prominent role of the VBC κ[k], being the generalization of the algebroid
bracket operation, remains unrecognized.
Properties of the prolongations. Prolongations of AL algebroids provide an example of
strong AL higher algebroids. Moreover, prolongations of Lie algebroids are Lie higher algebroids.
Proposition 4.13 (properties of the prolongations). Let (σ : E →M,κ) be a Lie (resp. almost-
Lie) algebroid. Then
(
σ[k] : E[k] → M,κ[k]
)
is an example of a strong Lie (resp. almost-Lie)
higher algebroid.
The proof is given in Appendix A.
Example 4.14 (a local form of a second order prolongation). Let σ : E →M be a vector bun-
dle with local coordinates (xa, yi) and consider an AL algebroid (σ, κ) given locally by structure
function Qbi(x) = Q̃bi(x) and Qijk(x) (cf. Remark 2.8). On T2E consider induced coordinates
(xa, yi, ẋb, ẏj , ẍc, ÿk), and coordinates
(
xa, yi, yj,(1), ẋb, ẏk, ẏl,(1)
)
on TE[2]. The inductive for-
mula (4.10) allows to calculate the expression for κ[2] : T2E →B TE[2] (recall a general local
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 29
form of a second-order algebroid given in Remark 4.2) which reads as
κ[2] :
ẋa = Qai (x)yi,
ẍa =
∂Qai
∂xb
(x)Qbj(x)yiyj +Qai (x)yi,(1),
ẋa = Qai (x)yi,
ẏi = ẏi +Qijk(x)yjyk,
ẏi,(1) = ÿi +Qijk(x)yj ẏk +Qijk(x)ẏjyk +
∂Qijk
∂xa
(x)Qal (x)ylyjyk.
(4.11)
Observe that the equation for ẍi can be obtained from the formula ẋa = Qai (x)yi by a formal
derivation, assuming that (yi)̇ = yi,(1). In a similar manner, formula ẏi,(1) can be derived from
the equation ẏi = ẏi + Qijk(x)yjyk. Thus it is instructive to think about κ[2] as of first-order
differential consequences of κ = κ[1]. We will use this property later in Section 5.2.
5 Variational calculus on (Lie) higher algebroids
In this section we shall discuss a geometric formalism of the kth-order variational calculus based
on the notion of a (Lie) higher algebroid
(
σk : Ek → M,κk
)
. Actually the geometry here is
exactly the same as in our previous study on prolongations of AL algebroids [25]. The idea
is that the relation κk is responsible for the construction of admissible variations of potential
extremals of the system (the role of κ is to change a ‘jet of curves’ into a ‘curve of jets’ –
cf. our discussion in Section 1). By dualizing κk we may represent the variation of an action
functional as a pairing of a certain curve of covectors with the kth-tangent lift of the curve of
virtual displacements. Then all that is left to do is to perform an integration by parts according
to the recipe from [26].
Actually, it is straightforward to observe that the geometric formalism described above works
well for even more general structures which we shall call pre-algebroids. We believe that they
may have some potential applications in the theory of reductions. In fact, in the last paragraph of
Section 5.1 we provide a rather trivial example of that sort. From the point of view of complexity
it is not interesting, yet still it is an example of a non-standard reduction – cf. Remark 5.2. On
the other hand, more complex examples including the derivation of higher-order Euler–Poincaré
and Hummel equations within our formalism can be found in [25].
5.1 The formalism of variational calculus
Pre-algebroids.
Definition 5.1 (pre-algebroid). By a pre-algebroid of order k we understand a triple
(
σk, τ, κk
)
consisting of a N-graded manifold σk : Ek →M of order k, a vector bundle τ : F →M (over the
same base), and a weighted VBC κk : Tkτ →B τEk inducing the identity on M :
TkF
Tkτ
��
κk � ,2TEk
τ
Ek
��
TkM Ek.
ρkoo
(5.1)
The reduction of the relation κk to order zero defines a vector bundle morphism ρF = κ0 : F →
TM . We call a pre-algebroid
(
σk, τ, κk
)
almost-Lie if
(
TkρF ,Tρk
)
: κk ⇒ κkM is a VBC-
morphism.
30 M. Jóźwikowski and M. Rotkiewicz
The difference with Definition 4.1 is that we allow the weighted VBC κk to match the tangent
bundle τEk : TEk → Ek with the lifted bundle Tkτ : TkF → TkM of an arbitrary VB τ : F →M ,
a priori not related with σk. By taking τ = σ1 and assuming that κ1 gives the identity on the core
bundles we recover the definition of a general HA. Pre-algebroids of order one have an elegant
description in terms of a certain bracket operation. We discuss it briefly in the concluding
Section 7.
It is easy to see that the construction of the algebroid lift can be straightforwardly extended
to pre-algebroids. This time to a section s ∈ SecM (F ) of τ and a number α = 0, 1, . . . , k we
assign a vector field s[k−α] := κ̂k
(
s(k−α)
)
∈ X
(
Ek
)
.
Remark 5.2. We believe that examples of pre-algebroids may appear in the theory of reduc-
tions. Let G be a Lie group (or more generally a Lie groupoid). In the standard reduction of
a kth-order variational problem on G one divides the higher tangent bundle TkG by the natural
action of G. In fact, here G acts on curves in G by, say, left multiplication and the action
on TkG is the differential consequence of this action on kth-jets. However, one can consider
more general actions by a subgroup H < TkG which properly contains G, and thus which would
not be induced by the action of G on curves. In proper circumstances, the quotient TkG/H will
have a structure of a N-graded manifold (space) Ek. However, in general the knowledge of E1
may not be enough to define admissible variations, if for example E1 is trivial but the entire Ek
is not. Thus a reduction of κkG (if it can be properly defined) may lead to a natural example of
a pre-algebroid.
A very simple example in this direction is provided in the last paragraph of this subsection.
Admissible paths and admissible variations. Consider now a pre-algebroid
(
σk, τ, κk
)
.
Let γ : R → Ek be a curve over γ := σk(γ) : R → M . Choose another curve a : R → F and
consider its kth-order tangent lift tka : R → TkF . Along γ we would like to construct a vector
field δaγ by the formula
δaγ(t) :=
(
κk
)
γ(t)
(
tka(t)
)
∈ Tγ(t)E
k. (5.2)
Note that, by the properties of κk, this definition is correct if and only if γ and a share the
same base path, i.e., σk(γ) = γ = τ(a) and if tkγ = ρk(γ). The latter equation is known as the
(left) admissibility of γ. If these conditions hold, we call δaγ an admissible variation along γ
and a its generator or a virtual displacement. The set of all admissible curves on the considered
pre-algebroid will be denoted by Adm
(
Ek
)
, while the set of all admissible variations by VA
(
Ek
)
.
Let us explain a relation between the notions of admissible variations and variations as
understood in the standard variational calculus. In a general setting, a variation δγ of an Ek-
valued path γ is a vector field along γ, i.e., δγ : t 7→ Tγ(t)E
k. Hence a variation δγ can be
represented by a family of paths γs : R→ Ek, where γ0 = γ and δγ(t) = t1(s 7→ γs(t)). In a very
general, but intuitive sense, a variation δγ is tangent to a subset A of paths in Ek if it can be
represented by a family γs lying in A. Results of [38] and [13, Theorem 3] show that for an
algebroid of order one the subspace Adm
(
E1
)
of admissible curves is a Banach submanifold in
the space of all C1-paths in E1. Moreover, the condition T Adm
(
E1
)
= VA
(
E1
)
is equivalent
to the AL axiom, i.e., for AL algebroids admissible variations are precisely variations tangent
to the space of admissible curves.
In the higher-order case we can prove the following two Lemmas in this direction. Actually,
the presented reasonings simplify the proofs available in the literature for the order-one case.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 31
Lemma 5.3 (tangent space for admissible variations). Let δγ ∈ TγE
k be a vector field along
an admissible path γ in a pre-algebroid
(
σk : Ek → M, τ : F → M,κk
)
. If δγ is tangent to
Adm
(
Ek
)
then(
Tρk
)
(δγ) = κkM
(
tkδγ
)
, (5.3)
where tkδγ denotes the kth-tangent lift of the curve δγ = Tσk(δγ).
Proof. Assume that δγ is a variation of an admissible path γ ∈ Adm
(
Ek
)
tangent to Adm
(
Ek
)
,
i.e., δγ(t) = t1[s 7→ γ(t, s)] and paths t 7→ γ(t, s) are admissible for each s. Then
Tρk(δγ)(t) = t1
[
s 7→ ρk(γ(t, s))
]
= t1[s 7→ tk[t 7→ γ(t, s)]] ∈ TTkM,
while tkδγ = tk
[
t 7→ t1[s 7→ γ(t, s)]
]
. Due to the definition of κkM , formula (5.3) follows. �
Remark 5.4 (problems with the Banach-manifold structure). According to [38], for k = 1
the converse of above lemma is also true, i.e., equation (5.3) describes the space T Adm
(
E1
)
.
For k > 1 some additional conditions are necessary to assure that the subspace of admissible
curves in Ek is a Banach submanifold. Such conditions may follow directly from the geometry
of the considered problem. For example in our preprint [25] additional assumptions imposed on
admissible paths were motivated by the reduction procedure.
Lemma 5.5 (a property of AL pre-algebroids). If a pre-algebroid
(
σk : Ek →M, τ : F →M,κk
)
is almost-Lie, γ is an admissible path, and a is a curve in F such that τ(a) = γ, then the
admissible variation δaγ satisfies equation (5.3).
Proof. Consider a diagram
TkF
TkρF
��
κk � ,2TEk
Tρk
��
TkTM
κkM // TTkM,
which is commutative as our pre-algebroid
(
σk, τ, κk
)
is almost-Lie. As
(
tka, δaγ
)
∈ κk, we get
Tρk(δaγ) = κkM
(
TkρF
(
tka
))
= κkM
(
tkρF (a)
)
= κkM
(
tkδaγ
)
. The last equality follows from the
commutativity of the diagram
TkF
τkF
��
κk � ,2TEk
Tσk
��
F
ρF // TM,
from which we get ρF (a) = δaγ as tka ∼κk δaγ. �
Variational calculus. We understand Lagrangian mechanics on a kth-order pre-algebroid(
σk, τ, κk
)
as the study how a functional, obtained by integrating the Lagrangian function
L : Ek → R along an admissible curve γ : R→ Ek, behaves under movement in the direction of
admissible variations. By construction
〈dL(γ), δaγ〉τ
Ek
=
〈
dL(γ),
(
κk
)
γ
(
tka
)〉
τ
Ek
=
〈(
κk
)∗
(dL(γ)), tka
〉
Tkτ
, (5.4)
where
(
κk
)∗
: T∗Ek → TkF ∗ is a morphism of graded-linear bundles dual to κk. We have
thus arrived at the pairing of a TkF ∗-valued curve with the kth-tangent lift tka : R → TkF
32 M. Jóźwikowski and M. Rotkiewicz
of the generator a of δaγ. Using our earlier results on the kth-order geometric integration by
parts – see [26] – we may present this pairing as a sum of a complete derivative, and a paring
〈·, ·〉τ : F ∗ × F → R evaluated on the generator a
〈(
κk
)∗
(dL(γ)), tka
〉
Tkτ
=
〈
ELk
(
tkγ
)
, a
〉
τ
+
d
dt
〈
Pk
(
tk−1γ
)
, tk−1a
〉
Tk−1τ
. (5.5)
Here ELk(·) denotes the Euler–Lagrange operator and Pk(·) the momentum operator associated
with the Lagrangian (see [26] for the precise definitions). An admissible curve γ satisfies Euler–
Lagrange equations ELk
(
tkγ
)
= 0 (i.e., is a trajectory of the considered Lagrangian system) if
and only if for every virtual displacement a we have
〈dL(γ), δaγ〉τ
Ek
=
d
dt
〈
Pk
(
tk−1γ
)
, tk−1a
〉
Tk−1τ
, (5.6)
or in a more familiar integral version∫ T
0
〈dL(γ), δaγ〉τ
Ek
dt =
〈
Pk
(
tk−1γ
)
, tk−1a
〉
Tk−1τ
∣∣T
0
.
The presented formalism generalizes the construction of geometric Lagrangian mechanics on
a general algebroid from [13].
Remark 5.6. In the literature there have been a few attempts to develop higher-order mechanics
on generalizations of Lie algebroids. With the sole exception of [4], all trials known to us used the
structure of a prolongation of a Lie algebroid
(
E[k], κ[k]
)
(see Section 4.2), naturally appearing
in reduction of higher tangent bundles of Lie groupoids.
As we already pointed out when discussing the notion of an AL algebroid prolongation for
the first time, the role of the VBC κ[k] in the structure of an algebroid prolongation remained
unrecognised in all the primary sources [9, 39, 43]. On the other hand, in the theory presented
above, κ[k] is responsible for the construction of admissible variations. Mart̀ınez in [39] con-
structed the kth-algebroid lift s[k] of a section s ∈ SecM (E) directly form the first-order data
(E, κ) (which should not be surprising as the prolongation is completely determined by its base
object). Then he defined an admissible variation generated by s as a restriction of s[k] to an
admissible trajectory γ. Apparently, his approach is equivalent to ours as s[k]|γ = δaγ for a = s|γ
(see the discussion of symmetries and conservation laws below). Summing up, on the prolon-
gation of a Lie algebroid the geometry of Mart̀ınez’s formalism is the same as ours, despite the
presence of κ[k] not being directly observed.
The case of k = 2 prolongations was intensively studied by Colombo and his collaborators
(see [1, 8] and the references therein). In his formalism the second prolongation E[2] is treated
as a subset in the prolongation of E along the projection τ : E → M (see Example 2.30), i.e.,
E[2] ⊂ T EE. Now the initial problem on E[2] can be treated as a constrained problem on T EE,
the latter being a first-order algebroid. Equations of motion are derived by using the first-order
formalism of [36] applied to T EE. This method seems to be quite complicated (prolongations
of prolongations are required) in comparison to our approach, the presence of κ[2] is hidden
somewhere inside the inclusion E[2] ⊂ T EE, and the direct relation with variational calculus is
hard to observe.
The approach of [4] is conceptually similar to the one of Colombo, but much more general.
Given a N-graded manifold F k, the authors consider a first-order algebroid structure on its
linearisation D
(
F k
)
. Again we have a canonical inclusion F k ⊂ D
(
F k
)
, and the kth-order
dynamics on F k is derived as the first-order constrained dynamics on the algebroid D
(
F k
)
. The
relation of the methods of [4] to the ones of this work is so far unclear, although in the case of
prolongation of algebroids both approaches give consistent results.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 33
Symmetries and conservation laws. As explained above, Euler–Lagrange equations on
higher pre-algebroids are derived by studying changes of the Lagrangian L(γ) in the direction
of every admissible variation δaγ. By contrast, conservation laws are related with special prop-
erties of a particular admissible variation. Below we will sketch basic concepts standing behind
conservation laws in higher-order variational calculus.
Let γ : R → Ek be an admissible curve over γ : R → M . The idea is to look for a generator
a : R→ F of an admissible variation δaγ such that 〈dL(γ(t)), δaγ(t)〉 = d
dtf(t) for some smooth
function f : R → R. Now if γ is a solution of the Euler–Lagrange equations, by (5.6) we
have d
dtf = d
dt
〈
Pk
(
tk−1γ
)
, tk−1a
〉
Tk−1τ
, and hence the difference f −
〈
Pk
(
tk−1γ
)
, tk−1a
〉
Tk−1τ
is
constant along γ.
In practice such a method of finding constants of motion is not very effective, as a is defined
only along γ which is unknown until we actually solve (at least partially) the Euler–Lagrange
equations ELk
(
tkγ
)
= 0. Instead it is better to look for a generator universal for every pos-
sible γ by considering a = s|γ , where s ∈ SecM (F ) is some section of τ . Note that in this
case s[k]|γ = δaγ, i.e., the admissible variation δaγ is the restriction of the kth-order algebroid
lift s[k] of s. Note that
〈
dL, s[k]
〉
τ
Ek
is a smooth function on Ek. If
〈
dL, s[k]
〉
τ
Ek
= df ◦ ρk for
some smooth function f ∈ C∞(Tk−1M) (note that df may be regarded as a map df : TkM ⊂
TTk−1M → R), we call the section s ∈ SecM (F ) a generator of the symmetry of L. If this is
the case then
〈dL(γ), δaγ〉 =
〈
dL|γ , s[k]|γ
〉
τ
Ek
= df
(
ρk(γ)
)
= df
(
tkγ
)
=
d
dt
f
(
tk−1γ
)
,
i.e., 〈dL(γ), δaγ〉 is a total derivative regardless of the choice of an admissible curve γ, as in-
tended. If follows that f
(
tk−1γ
)
−
〈
Pk
(
tk−1γ
)
,Tk−1s ◦ ρk−1(γ)
〉
Tk−1τ
is constant, i.e., we derived
a conservation law related with the symmetry of L.
A simple example of a prealgebroid. Let us end this part with a very simple example of
a variational problem on a pre-algebroid. Our starting point is a standard second-order varia-
tional problem on the Euclidean plane R2, constituted by a Lagrangian function L : T2R2 → R.
In this case the HA in question is simply the second tangent bundle τ2
M : T2R2 → R2 equipped
with the standard higher algebroid structure κ2
R2 : T2TR2 '−→ TT2R2. Formula (5.5) leads to
the standard Euler–Lagrange equations:
d2
dt2
(
∂L
∂ẍ
)
− d
dt
(
∂L
∂ẋ
)
+
∂L
∂x
= 0 and
d2
dt2
(
∂L
∂ÿ
)
− d
dt
(
∂L
∂ẏ
)
+
∂L
∂y
= 0, (5.7)
where (x, y) are standard coordinates on R2 and (x, y, ẋ, ẏ, ẍ, ÿ) the adapted coordinates on T2R2.
Consider now the following action of (a, b, c) ∈ R3 on (the germs at t = 0 of) curves in R2:
(a, b, c) ◦ (x(t), y(t)) = (x(t) + a+ ẋ(0)bt, y(t) + c).
Obviously this action reduces to the R3-action on T2R2 given by
(a, b, c) ◦ (x, y, ẋ, ẏ, ẍ, ÿ) = (x+ a, y + c, ẋ(1 + b), ẏ, ẍ, ÿ).
If L is invariant under this action, then (5.7) reduces to
d2
dt2
(
∂L
∂ẍ
)
= 0 and
d2
dt2
(
∂L
∂ÿ
)
− d
dt
(
∂L
∂ẏ
)
= 0. (5.8)
On the other hand, the latter equations can be obtained within the framework of variational
calculus on pre-algebroids by an easy reduction procedure. Namely, note that the quotient
34 M. Jóźwikowski and M. Rotkiewicz
of T2R2 by the action of R3 is naturally a graded space (i.e., a N-graded manifold over a point)
σ2 : E2 = R[1]⊕ R2[2]→ {pt}. On E2 we can introduce graded coordinates (y1, x2, y2) induced
by ẏ, ẍ and ÿ, respectively.
It is clear that an invariant Lagrangian L induces a function l : E2 → R such that L = p ◦ l
for p : T2R2 → E2 = T2R2/R3 being the quotient map.
We can further equip σ2 with a pre-algebroid structure
T2R2
T2τ
��
κ2 � ,2TE2
τE2
��
{pt} E2oo
as follows(
κ2
)
(y1,x2,y2)
: (a, b, ȧ, ḃ, ä, b̈) 7→ (y1, x2, y2, ẏ1 = ḃ, ẋ2 = ä, ẏ2 = b̈).
For this structure the admissibility condition is empty, since τ : R2 → {pt} has a trivial base.
However, an additional condition
ẏ1 = y2 (5.9)
should be imposed on admissible curves if we want to maintain the correspondence of admissible
curves in E2 with the admissible curves in T2R2 under the reduction procedure (cf. Remark 5.4).
It is now an easy exercise to show that equations (5.8) are the Euler–Lagrange equations for
the variational problem on
(
σ2, τ, κ2
)
constituted by function l : E2 → R and the admissibility
condition (5.9).
The above example suggests that pre-algebroids may naturally appear in reductions of stan-
dard variational problems by actions which are non-trivial on higher jets. (Note that in the
standard Lie groupoid–Lie algebroid reduction [34] and its generalization to higher jets [27]
the action of the groupoid on higher jets is merely a consequence of its action on points – see
Remark 5.2.)
5.2 Examples
Now we shall provide some basic examples of higher-order Euler–Lagrange equations obtained
using the formalism introduced in Section 5.2. For simplicity we will reduce our attention to
the case k = 2.
Variational problem on a prolongation of an AL algebroid. In Example 4.14 we
derived the local form of the second prolongation
(
E[2], κ[2]
)
of an AL algebroid structure (E, κ).
Now we can use it to obtain a coordinate formula of the second-order Euler–Lagrange equations.
Consider an admissible curve γ(t) =
(
xa(t), yi(t), yj,(1)(t)
)
∈ E[2]. From (4.11) the admissibility
conditions for γ(t) are ẋa = Qai (x)yi and ẍa =
∂Qai
∂xb
(x)Qbj(x)yiyj+Qai (x)yi,(1). We can deduce the
latter equation from the former under an additional assumption that yi,(1)(t) = ẏi(t), which can
be easily justified by the reduction procedure (see [25]). A generator a(t) = (xa(t), ξi(t)) ∈ Eγ(t)
produces the following admissible variation
δaγ(t) :
δxa = Qai (x)ξi(t),
δyi = ξ̇i +Qijk(x) yjξk,
δyi,(1) = ξ̈i +Qijk(x)yj ξ̇k +Qijk(x)ẏjξk +
∂Qijk
∂xa
(x)Qal (x)ylyjξk =
(
δyi
)·
.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 35
Now, for the Lagrangian L : E[2] → R, the left hand-side of (5.4) reds as
〈dL(γ), δaγ〉 =
∂L
∂xa
(
x, y, y(1)
)
δxa +
∂L
∂yi
(
x, y, y(1)
)
δyi +
∂L
∂yi,(1)
(
x, y, y(1)
)
δyi,(1).
Expressing it in terms of ξi’s and integrating by parts leads us to
〈dL(γ), δaγ〉 =
〈(
δki
d
dt
−Qkij(x)yj
)(
d
dt
∂L
∂yk,(1)
− ∂L
∂yk
)
+Qai (x)
∂L
∂xa
, ξi
〉
+
d
dt
(〈
∂L
∂yi,(1)
, ξ̇i
〉
−
〈(
δki
d
dt
−Qkij(x)yj
)
d
dt
∂L
∂yk,(1)
− ∂L
∂yi
, ξi
〉)
.
And hence (cf. (5.5)) the second-order Euler–Lagrange equations become(
δki
d
dt
−Qkij(x)yj
)(
d
dt
∂L
∂yk,(1)
− ∂L
∂yk
)
+Qai (x)
∂L
∂xa
= 0 and ẋa = Qai (x) yi.
Euler–Poincaré equations. Let (g, [·, ·]) ' (g, κg) be a Lie algebra, and let l : Tg '
g× g→ R be a Lagrangian function its the second prolongation
(
g[2] = Tg, κ
[2]
g
)
. By setting in
the previous example Qai (x) = 0 and Qijk(x) = cijk, where cijk denote the structural constants
of g in a given basis {ej}, we easily arrive at the second-order Euler–Poincaré equations for
a curve (a(t), ȧ(t)) ∈ Tg:(
d
dt
− ad∗a(t)
)(
d
dt
∂l
∂ȧ
− ∂l
∂a
)
(a(t), ȧ(t)) = 0. (5.10)
An alternative, more geometric derivation is possible directly from (5.2) and (5.4) using for-
mula (6.1) – see [25] for the treatment of the general case. The obtained results are consistent
with these of Colombo and De Diego [10].
6 Further examples – substructures and quotients of TkG/G
Another interesting class of examples of (Lie) higher algebroids is obtained from the reduction
TkG/G of the kth-tangent bundle of a Lie group G. The resulting space Ek := Tk
eG, consisting of
k-velocities in G based at the identity element e ∈ G, is a N-graded manifold over a single point
e ∈ G (i.e., a graded space – see [17]). It can be equipped with the canonical HA structure κkg
which, in fact, can be identified with the kth-prolongation (in the sense of Section 4.2) of (g, [·, ·]) –
the Lie algebra of the group G.
Throughout this part we shall describe all higher subalgebroids of TkG/G, and give examples
of HA quotients of TkG/G by which we understand higher algebroids (F k, κk) obtained from
TkG/G by means of surjective HA morphism onto F k.
A (Lie) higher algebroid structure on Tk
eG. We shall begin by describing the higher
algebroid structure on Tk
eG. First of all, Tk
eG is a split graded space concentrated in weights
1, 2, . . . , k in which each homogeneous component is identified with g, the Lie algebra of G. This
canonical identification is possible thanks to the group structure on G and is obtained by means
of the local diffeomorphism exp: g→ G inducing an isomorphism Tk
0 exp: Tk
0g
'−→ Tk
eG of graded
spaces. (Note that in general, for k ≥ 2, the kth-tangent space of a manifold at a given point
has no canonical vector space structure.) Next, Tk
0g (and so Tk
eG) has a canonical identification
with Tk−1g which is, on the other hand, a graded Lie algebra (see Proposition 4.6), yet the
latter gradation is shifted by −1 with respect to the former. Summing up, Tk
eG is canonically
equipped with two structures: of a graded space (which is split) and of a graded Lie algebra. Both
structures are clearly recognized after canonical identifications Tk
eG = Tk−1g = g⊗ R[t]/〈tk〉.
36 M. Jóźwikowski and M. Rotkiewicz
Now we would like to recall a VBC κkg : Tkg →B TTk−1g constituting a HA structure on
Tk
eG ' Tk−1g (see [27, Section 6] for details). An element X ∈ Tkg represented by a curve
t 7→
∑k
j=0Xjt
j ∈ g can be identified with a (k + 1)-tuple (X0, X1, . . . , Xk) of elements of g. In
a similar manner a vector Y ∈ TTk−1g ' Tk−1g×Tk−1g (note that TTk−1g is a tangent bundle
of a vector space), can be identified with a 2k-tuple (Y0, . . . , Yk−1, Ẏ0, . . . , Ẏk−1) of elements of g.
Now (X,Y ) ∈ κkg if and only if
Ẏl = (l + 1)Xl+1 −
∑
i+j=l
[Xi, Yj ], for every l = 0, 1, . . . , k − 1. (6.1)
The details of this relation can be found in [27, Proposition 6.3].
Higher subalgebroids of Tk
eG. Let F k be a split graded subspace of Tk
eG ' g[t]/〈tk〉, so
there are linear subspaces Vj ⊂ g (0 ≤ j ≤ k − 1) such that F k =
{∑k−1
j=0 vjt
j + 〈tk〉 : vj ∈ Vj
}
.
We recall that homogeneous functions of weight j + 1 on Tk
eG correspond to linear functions on
the summand tjg.
Proposition 6.1 (subalgebroids of Tk
eG). A split graded subbundle F k ⊂ Tk
eG is a higher
subalgebroid if and only if
Tk−1V0 ⊂ F k and [V0, Vi] ⊂ Vj for each i = 0, 1, . . . , k − 1 and j ≥ i.(6.2)
In particular, V0 should be a Lie subalgebra of g and additionally each subspace Vi should con-
tain V0 and be preserved by the action of the latter subspace.
Proof. We should basically check if κkg restricts fine to a VBC from TkV0 to TF k. Take any Y =
(Y0, . . . , Yk−1) ∈ F k ⊂ Tk−1g and X = (X0, . . . , Xk) ∈ TkV0 ⊂ Tkg. So all the components Xj
are in V0 and Yi ∈ Vi for i = 0, . . . , k − 1. Let Y := (Y , Ẏ ) =
(
κkg
)
Y
(X) be the element in
TY Tk−1g which is κkg -related to X. The components of Ẏ = (Ẏ0, . . . , Ẏk−1) are determined
by (6.1). We should examine when Ẏ ∈ F k, i.e., when Ẏi ∈ Vi for each i = 0, 1, . . . , k − 1. This
happens if and only if
V0 +
j∑
i=0
[V0, Vi] ⊂ Vj
for each j = 0, 1, . . . , k − 1. This gives (6.2). �
Quotients of Tk
eG. General quotients of Lie algebroids and the concept of an ideal in a Lie
algebroid are quite involved subjects (see [34]). For this reason in the following definition of
a quotient (Lie) higher algebroid structure we shall restrict our attention only to “quotient”
maps covering the identity on the base. We call a N-graded manifold morphism φk : Ek → F k
surjective if for each 1 ≤ j ≤ k the induced top core vector bundle morphisms φ̂j : Êj → F̂ j are
fibre-wise surjective linear maps. In particular, such a morphisms always is a surjective mapping
Ek → F k but not vice versa: a N-graded manifold morphism which is a surjective mapping need
not induce surjective mapping between the core bundles.
Definition 6.2 (a quotient of a (Lie) HA). We shall say that a kth-order algebroid
(
σkF : F k →
M,κk,F
)
is a quotient of a (Lie) higher algebroid
(
σkE : Ek → M,κk,E
)
if there is a surjective
N-graded manifold morphism φk : Ek → F k covering the identity on the base manifolds such
that
(
Tkφ1,Tφk
)
is a VBC-morphism κk,E ⇒ κk,F . In other words, elements X ′ ∈ TkF 1 and
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 37
Y ′ ∈ TF k are κk,F -related if and only if they are images under projections Tkφ1 and Tφk,
respectively, of some κk,E-related elements X ∈ TkE1 and Y ∈ TEk:
TkE1
Tkφ1
��
κk,E � ,2TEk
Tφk
��
TkF 1 κk,F � ,2______ TF k.
(6.3)
Example 6.3 (a class of examples). A slight generalization of the example
(
Tk−1g, κkg
)
is
possible. Let Ek =
⊕k−1
i=0 gi be a graded Lie algebra. In particular, g0 has a Lie algebra structure
and, consequently, Tkg0 also is graded Lie algebra. Let α : Tk−1g0 → Ek be a graded Lie algebra
homomorphism such that α0 = idg0 , where α0 is the restriction of α to the subalgebra g0 being
the component in degree 0 of the graded Lie algebra Tk−1g0. For X ∈ Tkg0 let (X0̄, X1̄) ∈
TTk−1g0 ' Tk−1g0×Tk−1g0 be the image of X under the canonical embedding Tkg0 ⊂ TTk−1g0.
We consider Ek as a (split) graded space defined by the weight vector field on Ek given by
∆ =
∑k−1
i=0 (i+ 1)∆gi , where ∆gi is the Euler vector field on gi. Then the formula(
κk
)
y
(X) = α(X1̄) + [y, α(X0̄)]Ek , (6.4)
where y ∈ Ek, turns Ek into a higher Lie algebroid. The proof is left to the reader. (One should
notice that vector fields F(a,b) ∈ X(g), F(a,b) : y 7→ [y, a]g + b, where g is an arbitrary Lie algebra
and y, a, b ∈ g, form a Lie subalgebra of X(g) isomorphic with g ⊕ εg.) We shall show that(
Ek, κk
)
is a quotient of
(
Tk−1g0, κ
k
g0
)
if α is surjective. Let J := kerα = ⊕k−1
i=0 Ji, so gi = g0/Ji.
Consider the diagram (6.3) in our case:
Tkg0
Tkα0=id
Tkg0 ��
κkg0 � ,2TTk−1g0
Tα
��
Tkg0
κk � ,2_______ TEk.
Take y ∈ Ek and X ∈ Tkg0 as above. Let ỹ ∈ y + J be any pre-image of y under the
projection α. Using (6.1) we find that the vector
(
κkg0
)
ỹ
X ∈ TỹT
k−1g0 is represented by the
curve t 7→ ỹ+ t(X1̄ + [ỹ, X0̄]) ∈ Tk−1g0. Therefore Tα
((
κkg0
)
ỹ
X
)
∈ TEk does not depend on the
choice of ỹ and is given by the formula (6.4) as α(ỹ) = y and α([ỹ, X0̄]) = [y, α(X0̄)]Ek .
7 Final remarks
The main idea of this paper was to use the framework of vector bundle comorphisms in order to
provide a proper language to describe higher analogs of (Lie) algebroids, having in mind potential
applications in variational calculus and geometric mechanics. Our studies suggest a few (in our
opinion interesting) directions of future research, which we discuss below.
Left-twisted algebroids. In Lemma 2.9 we studied linear VBCs κ : Tσ →B τE which induce
the identity on the core. By modifying the latter condition we can define generalization of the
concept of an algebroid. For example if κ is a linear VBC which induces the identity on the
base κ ∩ (M ×M) = graph(idM ) ⊂ M ×M then the left and right anchors ρL, ρR : E → TM
are still well-defined. Moreover, such a κ induces a VB endomorphism on the cores φ : σ → σ
over idM . By an analogous argument to the one used in the proof of Proposition 2.15 such a κ
defines a left-φ-twisted algebroid structure which satisfies a modified equation (2.6)
[f · a, g · b] = fρL(a)(g) · φ(b)− gρR(b)(f) · a+ fg · [a, b].
38 M. Jóźwikowski and M. Rotkiewicz
The converse implication is also true (the argument used in the proof of Lemma 2.11 works
without any change): a left-φ-twisted algebroid structure on σ uniquely determines a linear
VBC κ : Tσ →B τE which induces an endomorphism φ on the cores.
For example, a linear connection on the tangent bundle TM is an R-bi-linear operator
∇ : X(M)×X(M)→ X(M) satisfying ∇X(f · Y ) = f · ∇XY +X(f) · Y and ∇f ·XY = f · ∇XY
for any vector fields X,Y ∈ X(M) and any smooth function f ∈ C∞(M). Now [X,Y ] :=
−∇YX is a left-φ-twisted algebroid structure with both anchors being the identities idTM , and
φ : TM → TM being the null map. The related VBC κ maps a given element A ∈ TTM to the
∇-horizontal lift of τTM (A) at TτM (A).
Pre-algebroids and reduction. As we have seen in Section 5 for many interesting ap-
plications, we do not need the full structure of a (higher) algebroid, but it is enough to have
a less rigid structure of a (higher) pre-algebroid. It seems to us that such objects can na-
turally appear as reductions of higher tangent bundles, while considering (pseudo) group ac-
tions on the space of smooth curves on manifolds in the spirit of [31] (see an example of
such a situation in the last paragraph of Section 5). Thus it would be interesting to ini-
tiate a systematic study of such objects. For example a structure of a pre-algebroid of order
one (σ : E → M, τ : F → M,κ) can be equivalently characterized as a pair of anchor maps
ρE : E → TM and ρF : F → TM a VB morphism φ : F → E over the identity on M and
a bi-linear bracket operation [·, ·] : Sec(E) × Sec(F ) → Sec(E) satisfying, for every sections
a ∈ Sec(E), b ∈ Sec(F ) and functions f, g ∈ C∞(M), a Leibniz-like rule
[f · a, g · b] = fρE(a)(g) · φ(b)− gρF (b)(f) · a+ fg · [a, b].
In particular, left-twisted algebroids provide examples of pre-algebroids (with τ = σ).
Applications to variational problems. In Section 5 we only sketched some possibilities
of applications of the theory of higher (pre)algebroids in geometric mechanics. We postpone
a more detailed study to a separate publication. Let us only mention here the problem of de-
veloping a constrained higher-order Lagrangian dynamics in this setting. Following the elegant
treatment of the first-order case in [13], for a Lagrangian system constituted by a function
L : Ek → R on a higher algebroid κk : TkE1 →B TEk, there are basically two different concepts
of constraints. Vakonomic constraints should be understood as a graded subbundle Dk ⊂ F k
and the related dynamics is generated by these admissible variations which are valued in TDk.
On the other hand, nonholonomic constraints are constituted by restricting the set of genera-
tors to a subbundle D ⊂ E1. That is, admissible variations are obtained from the restriction
κk|TkD : TkD →B TEk.
Structure of (Lie) higher algebroids. One of us initiated a study of the internal struc-
ture of (Lie) higher algebroids. It turns out that, at least in order 2, (Lie) higher algebroids
have a geometric description in terms of a collection of bundle maps and differential operators
which should satisfy certain compatibility conditions. The details can be find in a forthcoming
publication [42].
A Appendix – proofs of technical results
A proof of Lemma 2.11. Note that, according to Lemma 2.9(iii), every relation satisfying (i)
and (ii) must respect the natural action of the core C ' E on TE. Thus if the assertion holds,
the following generalization of formula (2.8)
κa [Tb(ρL(a))+++c] := Ta(ρR(b))+++[a, b]+++c (A.1)
is valid for every sections a, b, c ∈ SecM (E). We shall show that this generalized formula defines
a differential relation κ satisfying conditions (i) and (ii). This will end the proof.
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 39
Denote F (a, b) := Tb(ρL(a)) and G(a, b) := Ta(ρR(b))+++[a, b]. Leaving aside (for a mo-
ment) the problem of correctness of the definition of κ, note that formula (A.1) relates an
element F (a, b)+++c, whose Tσ- and τE-projections are, respectively, ρL(a) and b, with an ele-
ment G(a, b)+++c, for which these projections are ρR(b) and a, respectively. We clearly see that
the VBC κ (if correctly defined) covers the left anchor ρL and projects to the graph of the right
anchor ρR under τE × Tσ.
We want to prove that κ is a VBC over ρL. Note that every element B of the Tσ-fibre
(TE)ρL(a) for any a ∈ Ex can be represented as B = Tb|x(ρL(a))+++c|x for some sections b, c ∈
SecM (E), and thus (A.1) gives us the value of κa on every element of (TE)ρL(a). (Note that if
ρL(a) = 0, then also Tb|x(ρL(a)) is a null vector regardless of the choice of section b and hence
the initial formula (2.8) is not sufficient to determine the values of κa for every possible element
of (TE)ρL(a). This justifies the need of using the extended formula (A.1).) It is enough to show
that the value κa(B) does not depend on the chosen presentation B = F (a, b)|x+++c|x, and that
the resulting differential relation is bi-linear.
It follows directly from (2.6) that for any smooth function f ∈ C∞(M) we have G(f · a, b) =
f ·Tσ G(a, b), i.e., G(a, b) is tensorial with respect to a, and hence the value of κa(B) given
by (A.1) (for B as above) does not depend on the particular choice of the section a, but only
on the value a(x). Now assume that we present B = F (a, b)|x+++c|x in a different way as B =
F (a, b′)|x+++c′|x. We shall show that formula (A.1) gives the same value of κa(B) for both
presentations. Clearly, b|x = b′|x and hence we may present b′ − b = f · b′′ for some smooth
function f vanishing at x and some (local) section b′′ ∈ SecM (E). By our assumption,
B = F (a, b)|x+++c|x = Tb|x(ρL(a))+++c|x = Tb′|x(ρL(a))+++c′|x = T(b+ f · b′′)|x(ρL(a))+++c′|x
= F (a, b)|x + f |x · F (a, b′′)|x+++ρL(a)(f) · b′′|x+++c′|x = F (a, b)|x+++ρL(a)(f) · b′′|x+++c′|x,
and we conclude that ρL(a)(f) · b′′|x = (c− c′)|x. Now we can use this fact to get
G(a, b)|x+++c|x = Ta|x(ρR(b))+++[a, b]|x+++c|x = Ta|x(ρR(b′))+++[a, b− f · b′′]|x+++c|x
= G(a, b′)−−−f |x · [a, b′′]|x−−−ρL(a)(f) · b′′|x+++c|x
= G(a, b′)−−−ρL(a)(f) · b′′|x+++c|x = G(a, b′)|x+++c′|x.
It follows that κa(B) is indeed well-defined.
Finally to show that κ defined by formula (A.1) is bi-linear we need to check that if (B,A) ∈ κ
then also (f ·Tσ B, f ·τE A) ∈ κ and (f ·τE B, f ·Tσ A) ∈ κ. This can be easily done using the
following properties od F and G:
G(f · a, b) = f ·Tσ G(a, b), F (f · a, b) = f ·τE F (a, b),
G(a, f · b) = f ·τE G(a, b)+++ρL(a)(f)b, F (a, f · b) = f ·Tσ F (a, b)+++ρL(a)(f)b, (A.2)
which follow directly from (2.6).
A proof of Proposition 2.15. From the results of Lemma 2.9 we already know that every
linear VBC inducing the identity on the cores gives rise to a well-defined left and right anchor
maps. Thus we need only to check if formula (2.9) defines an algebroid bracket compatible
with these anchors. First note that given any sections a, b ∈ SecM (E) the right-hand side
of (2.9) is a difference of two elements of TE with the same τE-projection a and the same
Tσ-projection ρR(b), thus a vertical vector. That is, [a, b] ∈ SecM (E) is well-defined. To check
that this bracket satisfies the Leibniz rule (2.6) we have to study the behaviour of formula (2.9)
under rescaling a 7→ f · a and b 7→ g · b. The calculations are basically the same as in the proof
of Lemma 2.11 – see formulas (A.2).
A proof of Proposition 2.21. Note first that if we have a VBC-morphism (Tφ,Tφ) : κ⇒ κ′,
then (see diagram (2.3)) the commutativity at the level of base maps (the left anchors) gives
40 M. Jóźwikowski and M. Rotkiewicz
us precisely the commutativity of the first diagram in (2.10). The commutativity of the se-
cond diagram follows from an observation that (Tφ,Tφ) is also a VBC-morphism between κT
and κ′T, whose base maps are the right anchors (cf. Lemma 2.9 and Remark 2.12). This is an
immediate consequence of Proposition 2.6 where the characterisation of being a VBC-morphism
does not depend on the direction of the considered relations, in contrast to the formulation of
Definition 2.3. From now on we can thus assume that (2.10) holds.
Choose any two sections a, b ∈ SecM (E) and let φ∗a =
∑
i fi · φ
∗ai and φ∗b =
∑
j gi · φ
∗bj
as in Definition 2.19. In the following calculations we understand a section s ∈ SecM (φ∗E′) as
a map s : M → E′ such that s(x) ∈ E′φ(x) and thus Vss
′, where s, s′ ∈ SecM (φ∗E′), is a map
M → TE′, x 7→ Vss
′(x), where Vss
′(x) is represented by the curve t 7→ s(x) + ts′(x) ∈ E′φ(x).
We have
Tφ[Ta(ρR(b))] =
∑
i
[ρR(b)(fi) ·Vφ∗aφ
∗ai + fi ·Tσ′ TaiTφ(ρR(b))]
(2.10)
=
∑
i
[ρR(b)(fi) ·Vφ∗aφ
∗ai + fi ·Tσ′ Tai ρ′R(φ∗b)]
=
∑
i
ρR(b)(fi) ·Vφ∗aφ
∗ai +
∑
i,j
fi ·Tσ′ gj ·τE′ Taiρ
′
R(φ∗bj).
Similarly,
Tφ[Tb(ρL(a))] =
∑
j
ρL(a)(gj) ·Vφ∗bφ
∗bj +
∑
i,j
gj ·Tσ′ fi ·τE′ Tbjρ
′
L(φ∗ai)
and we conclude that
κ′φ(a)[Tφ[Tb(ρL(a))]]− Tφ[Ta(ρR(b))]
= Vφ∗a
[∑
j
ρL(a)(gj) · φ∗bj −
∑
i
ρR(b)(fi) · φ∗ai +
∑
i,j
figj · φ∗[ai, bj ]′
]
.
Applying the tangent map Tφ to formula (2.9) leaves us with
Vφ∗aφ∗[a, b] := Tφ[κa[Tb(ρL(a))]]− Tφ[Ta(ρR(b))].
The comparison of the latter two equations shows that (2.11) holds if and only if Tφ[κa[Tb(ρL(a))]]
= κ′φ(a)[Tφ[Tb(ρL(a))]]. The latter equality (for every a and b) is equivalent to (Tφ,Tφ) being
a VBC-morphism between κ and κ′.
Equivalent characterizations of algebroid morphisms.
Proposition A.1. Let σ : E →M and σ′ : E′ →M ′ be vector bundles carrying (Lie) algebroid
structures as in Definition 2.19. Let φ : E → E′ over φ : M → M ′ be a VB morphism. The
following are equivalent:
(i) φ is an algebroid morphism (in the sense of Definition 2.19).
(ii) The graph of φ is a subalgebroid of the product (Lie) algebroid on σ × σ′.
(iii) The graph of φ∗ is a coisotropic submanifold of the Leibniz manifold [19] (E∗×E′∗,−ΛE +
ΛE′).
Proof. Recall [19] that a Leibniz manifold (M,Λ) is a manifoldM equipped with a contravariant
2-tensor Λ, and that linear contravariant 2-tensors on the dual σ∗ : E∗ →M of a vector bundle
σ : E → M are in one-to-one correspondence with general algebroid structures on σ. This
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 41
correspondence is completely parallel to that between linear Poisson tensors and Lie algebroid
structures.
The equivalence between (ii) and (iii) is proved in [15, Theorem 5.1] for skew algebroids,
however, the proof can be directly rewritten for general algebroids. Thus we need only to show
the equivalence between (i) and (ii).
Let F ⊂ E × E′, N ⊂ M ×M ′ be the graphs of φ and φ, respectively, and assume that
σ×σ′|F : F → N is a subalgebroid of σ×σ′. As φ is a vector bundle morphism, the base map φ
is smooth. Because (ρL, ρ
′
L) and (ρR, ρ
′
R) map F to TN ⊂ T(M ×M ′) we have commuting
diagrams in (2.10). To prove (2.11) take sections a, b ∈ SecM (E), write φ ◦ a =
∑
i fiφ
∗ai and
similarly, φ ◦ b =
∑
j gjφ
∗bj and notice that (x, φ(x)) 7→ (a(x), φ ◦ a(x)) is a section of F whose
extension to a section of σ × σ′ can be written as follows
ã(x, y) =
(
a(x),
∑
i
fi(x)ai(y)
)
,
and similarly for section b. We clearly have
[ã, b̃]σ×σ′(x, y)
=
(
[a, b](x),
∑
i,j
fi(x)gj(x)[ai, bj ]
′(y) + ρL(a)(gj)(x)bj(y)− ρR(b)(fi)(x)ai(y)
)
,
hence [ã, b̃](x, φ(x)) ∈ F implies (2.11). This reasoning can be inverted proving the equivalence
of conditions (i) and (ii). �
A proof of Proposition 2.26. By definition, condition (i) is equivalent to κ being a subal-
gebroid of the product (Lie) algebroid structure on σ1×σ2. Note that for this structure the left
and right anchors are, respectively, ρL := (ρ1L, ρ2L) and ρR := (ρ1R, ρ2R), whereas the algebroid
bracket reads as
[(s1, s2), (s′1, s
′
2)] := ([s1, s
′
1]1, [s2, s
′
2]2)
for any sections s1, s
′
1 ∈ SecM1(E1) and s2, s
′
2 ∈ SecM2(E2). It is thus enough to prove that
conditions (iia) and (iib) are equivalent to conditions (i) and (ii) of Definition 2.22 taken for
σ′ := σ1×σ2|r : E1×E2 ⊃ r =: E′ →M ′ := graph(r) ⊂M1×M2, σ := σ1×σ2 and κ := κ1×κ2.
First observe that for sections s1 ∈ SecM1(E1) and s2 ∈ SecM2(E2) the restriction (s1, s2)|M ′
is a section of the vector subbundle σ′ ⊂ σ1 × σ2 if and only if s2 = r̂(s1) and, moreover, every
section of σ′ can be presented as such a restriction. Thus condition (iia) is equivalent to the
fact that σI = (σ1I , σ2I) maps E′ ⊂ E1 ×E2 to TM ′ = T graph(r) ⊂ TM1 ×TM2 for I = L,R.
That is precisely condition (i) of Definition 2.22 for σ′, σ and κ as above.
Next note that condition (iib) means that the algebroid bracket on σ1 × σ2 is closed with
respect to sections of the form (s1, r̂(s1)), where s1 ∈ SecM1(E1). As we already observed such
sections are σ1 × σ2-extensions of all possible sections of σ′. Since, by the remark following
Definition 2.22, condition (i) of this definition guarantees that the induced bracket defined on σ′
does not depend on the choice of extensions of sections of σ′, we conclude that if (iia) holds
then (iib) is equivalent to condition (ii) of Definition 2.22 for σ′, σ and κ as above. This ends
the proof.
A proof of Lemma 2.29. The relation between the algebroid bracket and VBC κ is given
by formula (2.8). As has been observed in Remark 2.12, κT corresponds to another general
algebroid structure with the bracket [a, b]T := −[b, a] (note that this passage is possible due
to the fact that κ is bi-linear and that it induces the identity on the core). Consequently the
symmetry condition κ = κT is equivalent to the antisymmetry of the bracket [a, b] = −[b, a].
This proves (i).
42 M. Jóźwikowski and M. Rotkiewicz
Assume now that the algebroid structure on σ is skew. To prove (ii) apply Tρ to (2.9) to get
Vρ(a)ρ([a, b]) = Tρ(κa[Tb(ρ(a))])− Tρ(a)(ρ(b)).
On the other hand, formula (2.9) for the commutator of vector fields ρ(a), ρ(b) ∈ SecM (TM)
gives us
Vρ(a)[ρ(a), ρ(b)]TM := (κTM )ρ(a)[Tρ(b)(ρ(a))]− Tρ(a)(ρ(b)).
By comparing the above two formulas it is clear that (2.7) is equivalent to (2.12).
Point (iii) is equivalent to the fact that the dual map κ∗ : T∗E → TE∗ (cf. Remark 2.17)
is a linear Poisson map, which is a well-known equivalent characterization of the Jacobi iden-
tity [19]. By the results of [15] the latter condition is equivalent to κ being the algebroidal
relation.
A proof of Proposition 3.8.
(i) Let
(
xaw, X
A
w
)
,
(
yiw, Y
I
w
)
be graded coordinates on Ek1 and Ek2 , respectively, where the
capital letters indicate linear coordinates, i.e., XA
w and Y I
w are of weight (1, w). A VBC r
is given locally in Ek1 × Ek2 by a system of equations of the form
r :
{
xaw = faw(y),
Y I
w =
∑
AX
A
w′g
I
A,w−w′(y),
(A.3)
where faw and gIA,w are (local) functions on Mk
2 of weight w. Equations for rj ⊂ Ej1 × E
j
2
are obtained from (A.3) by removing all equations in which at least one coordinate has
weight greater than j. Clearly, such a system defines a VBC as rj is a vector subspace of
Ej1 × E
j
2 → M0
1 ×M0
2 with the property: given (yiw) ∈ M j
2 and
(
xaw, X
A
w
)
∈ Ej1 such that
xaw = faw(y) there is a unique
(
yiw, Y
I
w
)
∈ Ej2 (namely, Y I
w =
∑
AX
A
w′ g
I
A,w−w′(y)) satisfying
the equations for rj .
(ii) In view of Proposition 2.6, we should check whether
(
Aj , Bj
)
∈ rj implies
(
φj1
(
Aj
)
, φj2
(
Bj
))
∈ (r′)j . This is straightforward: there exist
(
Ak, Bk
)
∈ rk which project to
(
Aj , Bj
)
. We
have
(
φ1
(
Ak
)
, φ2
(
Bk
))
∈ (r′)k as (φ1, φ2) : r ⇒ r′ is a VBC-morphism. But the latter
element projects under the bundle reduction map (σ1)kj × (σ2)kj to
(
φj1
(
Aj
)
, φj2
(
Bj
))
which
is therefore an element of rj .
A proof of Proposition 4.6. By Proposition 2.15, to prove that dTkκ defines a general
algebroid structure on Tkσ it is enough to show that it is a linear VBC and that it induces
the identity on the cores. By the results of [27, Theorem A.5], the higher tangent lift Tkκ has
these two desired properties. So do κkE and (κkE)−1, which are not only VBCs but, in fact, even
isomorphisms of graded-linear bundles. We conclude that dTkκ, which is a composition of these
three VBCs, also shares these properties, i.e., dTkκ indeed defines a graded algebroid structure.
In fact, all the considered VBCs are also TkR-linear, with respect to the natural TkR-action on
the Tk-lift of a vector bundle introduced in at the beginning of Section 4.1. It follows that dTkκ
is also TkR-linear. We will use this property shortly.
The formula for anchor maps follows immediately from the definition of dTkκ – we simply
need to calculate the relevant projections of this VBC to the legs of the considered DVBs.
Consider now any two sections s1, s2 ∈ SecM (E). By applying the functor Tk to (2.8), we get
TkκTks1
[
TkTs2
(
TkρL
(
Tks1
))]
:= TkTs1
(
TkρR
(
Tks2
))
+++Tk[s1, s2].
Note that in the above formula element Tk[s1, s2] belongs to the core. Now let us compose this
equality with maps κkE and
(
κkE
)−1
, which are core-identities, and use the definition of dTkκ to get
(dTkκ)Tks1
[
TTks2
(
dTkρL
(
Tks1
))]
:= TTks1
(
dTkρR
(
Tks2
))
+++Tk[s1, s2].
Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms 43
By comparing the above with formula (2.8) for the algebroid structure
(
Tkσ, dTkκ
)
, we may
conclude that the algebroid bracket [·, ·]d
Tk
on Tkσ satisfies[
Tks1,T
ks2
]
d
Tk
= Tk[s1, s2], i.e.,
[
s
(k)
1 , s
(k)
2
]
d
Tk
= [s1, s2](k)
for any sections s1, s2 ∈ SecM (E). What is more, since, as we have observed before, dTkκ is
TkR-linear, the resulting bracket operation is TkR-bilinear, and hence (4.5) holds:[
k!
(k − α)!
s
(k−α)
1 ,
k!
(k − β)!
s
(k−β)
2
]
d
Tk
=
[
εα · s(k)
1 , εβ · s(k)
2
]
d
Tk
= εα+β · [s1, s2](k) =
k!
(k − α− β)!
[s1, s2](k−α−β).
This formula completely determines the algebroid bracket on Tkσ since (k − α)-lifts of sections
of σ for all possible α’s span the module of sections of Tkσ (cf. the remark following Defini-
tion 4.4).
Finally, the fact that the lifted algebroid structure preserves the properties of being
skew/AL/Lie follows directly from the anchor and the bracket formulas.
A proof of Proposition 4.11. To prove (i) note first that, by [27, Theorem A.5], the
relations Tkr and Tkr′ are VBCs. We have to prove that Tkr restricts fine to a subbundle
TkE′1 × TkE′2. Take a k-jet vk2 ∈ TkM ′2 which is represented by a curve γ
2
(t) ∈M ′2 and denote
γ
1
(t) := r(γ
2
(t)). Since r(M ′2) ⊂ M ′1, the curve γ
1
(t) lies in M ′1 . Let us take any k-jet
vk1 ∈ TkE′1 which lies over vk1 := tkγ
1
∈ TkM ′1. We can choose a representative γ1(t) ∈ E1 of vk1
such that γ1(t) projects to γ
1
(t). Then we may write(
Tkr
)
vk2
(
vk1
)
= tkrγ
2
(t)(γ1(t)) ∈ TkE′2,
which ends the proof of point (i).
To show (ii) observe that by our hypothesis, κ restricts fine to TE′ × TE′, hence by (i)
Tkκ : TkTσ →B TkτE restricts fine to TkTE′×TkTE′. Since dTkκ is obtained by composing Tkκ
with the vector bundle isomorphisms κkE and its inverse, the relation dTkκ also restricts fine to
TTkE′ × TTkE′, hence
(
Tkσ′, dTkκ
′) is a subalgebroid of
(
Tkσ, dTkκ
′), as was claimed.
Finally, point (iii) follows easily from (ii). Since r is a subalgebroid of (σ1 × σ2, κ1 × κ2),
by (ii) Tkr is a subalgebroid of
(
Tk(σ1 × σ2), dTk(κ1 × κ2)
)
'
(
Tkσ1 × Tkσ2,dTkκ1 × dTkκ2
)
,
hence Tkr is an algebroidal relation.
A proof of Proposition 4.13. Assume first that (σ, κ) is an AL algebroid. From the
results of [27, Theorem 4.5(ix)] we already know that κ[k] is a VBC. We shall prove first
that it is a weighted VBC. For this it is enough to check that κ[k] is a N-graded submani-
fold of τkE × Tτ [k] : TkE × TE[k] → E × TM . A general idea is to consider the latter as
a N-graded submanifold of Tk−1TE × TTk−1E → E × TM and notice that although κ in-
terchanges the two homogeneity structures on TE, the homogeneity structure in our concern
(which is defined by the sum of weight vector fields of degrees 1 and k − 1) is respected by
both Tk−1κ and κk−1
E , thus also by κ[k] (which is a fine restriction of the composition of these
relations). In detail, TTk−1E is a triple-N-graded manifold of 3-order (1, k − 1, 1) with legs
TTk−1τ : TTk−1E → TTk−1M , Tτk−1
E : TTk−1E → TE, and τTk−1E : TTk−1E → Tk−1E, re-
spectively. The multi-N-graded manifold structure on TTk−1E is defined by the weight vector
fields TTk−1∆E , T∆k−1
E , and ∆1
Tk−1E
. Since κ is homogeneous with respect to
(
∆1
E ,T∆E
)
∈
X(TE × TE) and
(
T∆E ,∆
1
E
)
, its (k − 1)th-tangent lift Tk−1κ is homogeneous with respect to(
Tk−1∆1
E ,T
k−1T∆E
)
,
(
Tk−1T∆E ,T
k−1∆1
E
)
,
(
∆k−1
TE ,∆
k−1
TE
)
∈ X
(
Tk−1TE × Tk−1TE
)
.
44 M. Jóźwikowski and M. Rotkiewicz
Similarly, κk−1
E is homogeneous with respect to(
Tk−1T∆E ,TTk−1∆E
)
,
(
∆k−1
TE ,T∆k−1
E
)
,
(
Tk−1∆1
E ,∆
1
Tk−1E
)
∈ X
(
Tk−1TE × TTk−1E
)
.
In particular, Tk−1κ is homogeneous with respect to
(
Tk−1∆1
E +∆k−1
TE ,T
k−1T∆E +∆k−1
TE
)
while
κk−1
E is homogeneous with respect to
(
Tk−1T∆E + ∆k−1
TE ,TTk−1∆E + T∆k−1
E
)
thus the compo-
sition κk−1
E ◦Tk−1κ is homogeneous with respect to
(
Tk−1∆1
E +∆k−1
TE ,TTk−1∆E +T∆k−1
E
)
. The
result follows because the order-k N-graded manifold structures on TkE → E and TE[k] → TM
are encoded by Tk−1∆1
E +∆k−1
TE and TTk−1∆E +T∆k−1
E , respectively (see [27, Theorem 4.5(i)]).
The commutativity of the diagram
TkE
Tkρ
zzuuuuuuuuu
Tkτ
��
κ[k] � ,2TE[k]
Tρ[k]
zzttttttttt
τ
E[k]
��
TkTM
TkτM
��
κkM � ,2TTkM
τ
TkM
��
TkM
=
zzuuuuuuuuu
E[k]
ρ[k]
zzttttttttt
ρ[k]oo
TkM TkM
=oo
(A.4)
follows immediately from [27, Theorem 4.5(xi)], while the reduction of κ[k] from order k to
order 1 coincides obviously with the relation κ defining AL algebroid structure on E.
Now if (σ, κ) is Lie, κ is an algebroidal relation between Tσ : TE → TM and τE : TE → E.
If follows from Proposition 4.11(iii) that Tk−1κ is an algebroidal relation between the (k− 1)st-
tangent lift algebroids on Tk−1Tσ and Tk−1τE . What is more, κk−1
E is an algebroid isomorphism
between Tk−1τE and τTk−1E , and thus the composition κk−1
E ◦ Tk−1κ is an algebroidal relation
between Tk−1Tσ and τTk−1E . By [27, Proposition 4.6], κk−1
E ◦ Tk−1κ ⊂ Tk−1TE × TTk−1E
restricts fine to κ[k] ⊂ TkE×TE[k], and so, in light of Proposition 2.28, κ[k] is also an algebroidal
relation, i.e.,
(
σ[k], κ[k]
)
is Lie.
It remains to prove that
(
σ[k], κ[k]
)
is strong. We shall proceed by induction on k and assume
that κ[k−1] is the identity on the core bundle. To show the same for κ[k] we observe that the
reduction from order k to k − 1 of κ[k] is κ[k−1], hence it is enough to show that κ[k] is the
identity on the ultracore bundle. But the inclusions TkE ⊂ Tk−1TE and TE[k] ⊂ TTk−1E
of graded-linear bundles give identity on the ultracores. The same is true for relations Tk−1κ
and κk−1
E what completes our proof.
Acknowledgments
The authors are grateful to the anonymous referees who put a lot of effort in improving the
quality and clarity of the manuscript. This research was supported by the Polish National
Science Center under the grant DEC-2012/06/A/ST1/00256.
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https://arxiv.org/abs/math.DG/0105237
https://doi.org/10.1063/1.3527417
https://arxiv.org/abs/1010.2503
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https://doi.org/10.1007/BF02097706
https://doi.org/10.1007/BF02097706
https://doi.org/10.1007/BF02097707
https://doi.org/10.1007/BF02097707
1 Introduction
1.1 Double vector bundles
2 (Lie) algebroids as vector bundle comorphisms
2.1 The category of vector bundle comorphisms
2.2 Description of (Lie) algebroids in terms of VBCs
3 Recollection of N-graded manifolds
4 Higher (Lie) algebroids
4.1 On the Lie axiom
4.2 Prolongations of an AL algebroid
5 Variational calculus on (Lie) higher algebroids
5.1 The formalism of variational calculus
5.2 Examples
6 Further examples – substructures and quotients of Tk G/G
7 Final remarks
A Appendix – proofs of technical results
References
|
| id | nasplib_isofts_kiev_ua-123456789-209869 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-04T15:12:48Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Jóźwikowski M. Rotkiewicz M. 2025-11-28T09:34:27Z 2018 Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms / M. Jóźwikowski, M. Rotkiewicz // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 54 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58A20; 58A50; 70G65; 58E30; 70H50 arXiv: 1708.03174 https://nasplib.isofts.kiev.ua/handle/123456789/209869 https://doi.org/10.3842/SIGMA.2018.135 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 comorphisms between graded-linear bundles satisfying natural axioms. We provide natural examples and discuss applications in geometric mechanics. The authors are grateful to the anonymous referees who put a lot of effort into improving the quality and clarity of the manuscript. This research was supported by the Polish National Science Center under the grant DEC-2012/06/A/ST1/00256. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms Article published earlier |
| spellingShingle | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms Jóźwikowski M. Rotkiewicz M. |
| title | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms |
| title_full | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms |
| title_fullStr | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms |
| title_full_unstemmed | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms |
| title_short | Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms |
| title_sort | higher-order analogs of lie algebroids via vector bundle comorphisms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209869 |
| work_keys_str_mv | AT jozwikowskim higherorderanalogsofliealgebroidsviavectorbundlecomorphisms AT rotkiewiczm higherorderanalogsofliealgebroidsviavectorbundlecomorphisms |