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 comorphism between graded-linear bundles satisfying natural axioms. We provide natural examples and discuss applications in geometric mechanics.


Introduction
The main goal of this paper is to introduce a concept of a (general) higher algebroid and study basic properties and applications of this notion. We work essentially within the framework of the theory of graded bundles and the theory of differential relations (Zakrzewski morphisms).
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 [Wei96] and, following seminal papers of Martínez [Mar01b,Mar01a], it developed into many different sub-branches (see a survey paper [CDLM + 06] 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 algebroids and variations is, perhaps, best described in two papers [GGU06,GG08], which themselves were inspired by much earlier works of Tulczyjew [Tul76a,Tul76b]. 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 algebroid bracket. In fact, putting aside the existence of real-life examples, it seems that the concept of a general algebroid [GU97,GU99] 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 higher algebroid, which is present whenever a variational problem involving higher velocities is considered. We have already tried to address this question from the perspective of the classical groupoid-algebroid reduction [JR15], yet now we propose a more abstract and a more general approach.
Towards a proper definition -objects In order one, the 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 an algebroid. Obviously, in higher-order variational calculus the role of TM is taken over by its higher analogue T k M , the bundle of k-velocities. Locally, T k M can be characterized by specifying the points, velocities, accelerations, and higher derivatives of curves on M . Thus, having the perspective of variational calculus in mind, it is natural to expect that a proper object hosting a higher algebroid structure would be a bundle of some sort with fiber coordinates sharing a similar derivative-like nature as coordinates on T k M . Graded bundles [GR11] meet these expectations. Shortly speaking these are fibre bundles with a typical fiber diffeomorphic with R n , however, in general, transition function 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). Intrinsic properties of graded bundles and their application in theoretical mechanics where studied in several papers ( [GR11,BGG15b,BGR16]) Towards a proper definition -structures The structure of an (order one) algebroid on σ : E → M is most commonly introduced as a bi-linear bracket operation on the space of sections of its host vector bundle σ. It is quite obvious that this definition has no direct generalization to the higher-order case as, in particular, the higher tangent bundle τ k M : T k M → M , which is an obvious candidate to host a higher algebroid structure, admits no natural bracket operation on its space of sections. Fortunately, the 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 Zakrzewski morphism) κ ⊂ TE × TE.
From the perspective of variational calculus, the latter characterization is the one most appealing. 1 This important role of κ can be easily explained. For the tangent algebroid structure on τ M : TM → M , κ is, in fact, the canonical isomorphism κ M : TTM → TTM interchanging 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 in fact constructed from the tangent lift of the curve of virtual displacements by means of κ M ). The situation in the higher-order case is no different and, perhaps, even easier to understand. Namely on a higher-tangent bundle τ k M : T k M → M , we have the canonical isomorphism κ k M : T k TM → TT k M which similarly as before allows to express a variation of a k th -tangent lift of a trajectory in M as a k thtangent lift of the related generator (a curve of virtual displacements) [JR14a]. 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 differentiable map but, in general, only a differential relation (but of a special kind). Again there is no essential difference between the first-and higher-order cases [JR15].
Summing up, geometrically relation κ ⊂ TE × TE is responsible for the construction of admissible variations of curves on an 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 algebroids can be, however, successfully developed in abstract terms (making the use of κ) without referring to the reduction procedure [GG08].
All the above suggests that the language of differential relations is the one most suitable to speak about higher algebroids (at least from the perspective of applications in variational calculus). The first order-case [GG08] and the known higher-order examples from [JR14b] advocate that these relations should have a special nature, namely be Zakrzewski morphisms (ZMs 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 Zakrzewski morphisms in the theory of Lie groupoids and Lie algebroids was recognized also in [CDW13] (here under the term comorphism).
Higher algebroids, examples and applications Summing up the above heuristic considerations, we postulate a higher algebroid to be a graded bundle σ k : E k → M equipped with a Zakrzewski morphism κ k ⊂ T k E 1 × TE k , which also should be naturally graded (Definition 4.1). Here E 1 is the natural reduction of E k to a first-order bundle, i.e., to a vector bundle.
For k = 1 we recover the standard notion of an algebroid. In fact, we devote entire Section 2 to carefully formulate the theory of (the first order) algebroids in the language of ZMs. This has a twofold purpose. First of all, the language of differential relations was so far hardly used in the theory of 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, natural examples of such structures are provided by higher tangent bundles T k M with isomorphisms κ k M mentioned above. Another important class of examples is given by the prolongations of almost-Lie algebroids introduced in [CdD11] and [JR15]. Prolongations naturally appear in higher-order variational calculus as reductions of higher tangent bundles of Lie groupoids. In our previous publications [JR14b,JR15] (see also [Mar15]) 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 algebroid introduced in this work by providing two particular applications. First of all, given a higher algebroid structure on σ k : E k → M we were able to construct a whole family of algebroid lifts of sections of E 1 to vector fields on E k . These generalize the well-know concepts of the vertical and the complete lifts of a section in the presence of an algebroid structure. Secondly, in Section 5 we show that the geometric formulation of higher-order variational calculus is possible within the framework of higher algebroids (from this perspective 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 [Col17] 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. In [Vor10] Voronov proposed that these should be understood as homological vector fields of weight one on a non-negatively graded supermanifold, per analogy to the description of Lie algebroids as Q-manifolds [Vai97]. Despite its elegance, this definition has limited applications in higher order variational calculus as higher tangent bundles are not examples of Voronov's higher algebroids.
A much more down-to-earth idea is to define higher algebroids as bundles of higher-jets of admissible curves on a standard Lie algebroid. In this paper we refer to this construction as a prolongation of an algebroids and briefly discuss it in Subsection 4.2. Such objects appeared for the first time in a seminar talk by Colombo and de Diego [CdD11], however without a full recognition of the relevant geometric structure. The latter was studied by us in [JR15] and successfully applied in higher-order variational problems on Lie algebroids and Lie groupoids [JR14b] (see also [Mar15]). Prolongations of algebroids in the above sense are a particular subclass of higher 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 [BGG16] they constructed a linearization functor, which, to a graded bundle σ k : E k → M of degree k, canonically associates a manifold l(E k ) equipped with a graded-linear structure of a bi-degree (k − 1, 1) over E 1 and E k−1 , respectively. Now the higher algebroid structure on σ k is defined as an algebroid structure on the vector bundle l(E k ) → E k−1 compatible with the second grading. The idea here is to mimic the canonical inclusion T k M ֒→ TT k−1 M , which makes the higher tangent bundle T k M a subobject of the tangent algebroid of T k−1 M . In this way the construction of [BGG16] admits higher tangent bundles and also prolongation of algebroids as examples of higher 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 degree 2. The construction of [BGG16] can be adapted to develop higher-order mechanics on graded bundles [BGG15b], equivalent to our results [JR14b] in comparable cases. So perhaps the difference here is of philosophical nature. In the classical case: should we treat k th -order variational calculus on a manifold M as the first-order theory on T k−1 M thus passing through unnecessary degrees of freedom (as would be in the spirit of [BGG16]) or do we prefer to work directly with T k M , as is in our formalism.
Organization of the paper In Section 2 we reformulate the theory of (general) algebroids in the language of Zakrzewski morphisms. Section 3 contains basic information concerning graded bundles 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 Subsection 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 natural higher algebroid structures inherited from higher tangent bundles of a Lie group G. In particular, we characterize all subalgebroids and quotients of T k G/G. In Section 7 we sketch a few perspectives of future research. Finally, in the Appendix A we have hidden most of the proofs of more technical results appearing in this work.

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 natural vector bundle structures are the tangent projection τ E : TE → E and Tτ E : TE → TM , the tangent lift of σ. The foundations of the theory of double vector bundles were laid by J. Pradines [Pra75] (see also Chapter 9 of [Mac05b] for basic definitions, examples and historical remarks). Due to the results of [GR09] 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 σ A : D → A, σ B : D → B such that for any t, s ∈ R and where · A and · B denote the multiplications by scalars in σ A and σ B , respectively. A morphism of DVBs (D, σ A , σ B ) and (D ′ , σ A ′ , σ B ′ ) is a smooth map φ : D → D ′ which is linear as a map from σ A to σ ′ A and simultaneously as a map from σ B to σ B ′ .
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 schematic diagram consisting of four vector bundle projections. All M , A, B can be seen as submanifolds of D via the zero section embeddings. In particular, the zero section 0 A : A → D defines a vector bundle structure on A as a substructure of the vector bundle structure on σ B : 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 σ A (the bundle projection σ A , the zero section 0 A , the scalar multiplication · A and the addition + A ) are vector bundle morphism with respect to the vector bundle structure of σ B . The notion of a double vector bundle naturally generalizes to the notion of a multi-graded bundle (see Section 3).

Algebroids as Zakrzewski morphisms
In this section we characterize algebroids by means of differential relations of a special kind (Zakrzewski morphisms, ZM in short). This will be the cornerstone of our definition of higher algebroids in Section 4.

The category of Zakrzewski morphisms
Following [JR15] we will recall a definition of a Zakrzewski morphism between vector bundles. Later we shall study basic properties of ZMs, i.e. the corresponding map on sections, morphisms between ZMs, and ZMs compatible with an additional linear structure.

Zakrzewski morphisms
Definition 2.1 (Zakrzewski morphism, ZM). A Zakrzewski morphism (ZM, in short) from a vector bundle σ 1 : E 1 → M 1 to a vector bundle σ 2 : E 2 → M 2 is a relation r ⊂ E 1 × E 2 of a very special form. Namely, there exists an ordinary vector bundle morphism φ : E * 2 → E * 1 covering a smooth map φ : M 2 → M 1 such that r = φ * , i.e., r is the union of graphs of linear maps r y : (E 1 ) φ(y) → (E 2 ) y such that φ y : (E * 2 ) y → (E * 1 ) φ(y) is the dual of the linear map r y , where y varies in M 2 . We also say that the ZM r covers the base map r := φ. Usually we shall denote a ZM as r : In particular, a ZM r is a vector subbundle of σ 1 × σ 2 . Zakrzewski morphisms are sometimes called morphisms of the second kind or comorphisms [Mac05b]. The concept of a ZM should be attributed to Zakrzewski [Zak90a,Zak90b] -see the Appendix of [JR15] for properties of ZMs and the justification for the name we use.
Remark 2.2 (ZMs and maps on sections). An important property of a ZM is that it induces a mapping between the corresponding spaces of sections. Namely, given a ZM r : σ 1 → ⊲ σ 2 as in Definition 2.1 and a sections s ∈ Sec M 1 (E 1 ) we definer(s) ∈ Sec M 2 (E 2 ) bŷ r(s)(y) := r y (s(r(y))), for every y ∈ M 2 .
In other words, the value ofr(s) at a given point y ∈ M 2 is the unique element of the fiber (E 2 ) y which is r-related to the value of s at r(y). By the linearity of r, for every two sections s, s ′ ∈ Sec M 1 (E 1 ) we haver(s + s ′ ) =r(s) +r(s ′ ). Moreover, by construction, for any f ∈ C ∞ (M 1 ) Conversely, any linear map Sec M 1 (E 1 ) → Sec M 2 (E 2 ) satisfying (2.1) for some underlying map r : M 2 → M 1 gives rise to a ZM r : σ 1 → ⊲ σ 2 . Checking this property is left to the reader.
Morphisms between ZMs A ZM can describe a single geometric object. As we shall see shortly (Lemma 2.11 and 2.15), a (Lie) algebroid structure on a bundle σ : E → M is fully encoded by a ZM κ from the tangent lift Tσ : TE → TM to the tangent bundle τ E : TE → E. This suggests that the standard notion of a (Lie) algebroid morphism should be expressible by a, properly defined, morphism between the corresponding Zakrzewski morphisms. Now we are going to introduce such a notion.
be four VBs and let r and r ′ be the following Zakrzewski morphisms: , is a ZM-morphism from r to r ′ (and write (φ 1 , φ 2 ) : r ⇒ r ′ ) if the following diagram is commutative in the following sense: (ii) At the level of fibers, for any y ∈ M 2 the following compositions of linear maps It is easy to see that ZM-morphisms can be naturally composed, and that ZMs with ZM-morphisms form a category of Zakrzewski morphisms, denoted ZM.
Remark 2.4 (Dualization of ZMs). By dualizing the diagram (2.3) we get a diagram of the same type: Thus the notion of a morphism in the category ZM does not reduce to the notion of a vector bundle morphism. Another useful formulation of the conditions presented in the definition of a ZM-morphism is possible: , 2) be vector bundle morphisms and let r, r ′ be ZMs as in Definition 2.3. Then (φ 1 , φ 2 ) : r ⇒ r ′ is a ZM-morphism if and only if for any r-related vectors X ∈ E 1 , Y ∈ E 2 their images φ 1 (X) and φ 2 (Y ) are r ′ -related. Equivalently, (φ 1 × φ 2 )(r) ⊂ r ′ .
Linear ZMs Of our special interest will be linear ZMs, i.e. ZMs compatible with an additional linear structure. Later in Section 3 (Definition 3.6) we will generalize this notion to the concept of weighted ZMs, which are compatible with an additional graded structure. We shall follow a general scheme already presented in the definitions of linear Poisson structure, VB-groupoids, VBalgebroids, weighted algebroids etc. [Mac05b,BGG15a].
Definition 2.7 (Linear ZM). Let r be a ZM 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 ZM if r ⊂ E × E ′ is a vector subbundle of τ × τ ′ .
Note that a linear ZM 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. Lemma 2.9 (i) and (ii) below). Particular examples of linear ZMs (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 τ E : TE → E. In the remaining part of this paragraph we shall study linear ZMs intertwining these two VB structures. The common kernel (the core) of Tσ and τ E is C = C(TE) = V M E ≃ E. The two additive structures of TE coincide on the core C.
Note that the core C naturally acts on TE by an addition of a vertical vector, i.e. for every A ∈ T a E and every e ∈ C σ(a) ≃ E σ(a) we define Note that the above action does not affects the two VB projections Tσ and τ E on TE, i.e. Tσ(A+ + +e) = Tσ(A) and τ E (A+ + +e) = τ E (A). Moreover, the following useful identity holds where f ∈ C ∞ (M ), v ∈ X(M ) and a ∈ Sec(E). Let (x a , y i ) be local linear coordinates on E. We use the standard notation (x a , y i ,ẋ b ,ẏ j ) for induced coordinates on the tangent bundle TE and, to describe κ ⊂ TE × TE, we shall underline coordinates in the second copy of TE.
The graph of the base map ρ L := 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 (x a , y i ) from E to a vector of the form (x a = x a ,ẋ b = Q b i (x)y i ) in TM . Now, since for every fixed a ∈ E relation κ a : (TE) ρ L (a) → T a E is a linear map, it turns out that κ is given by a linear mapping of coordinates (y i ,ẏ j ) to coordinates (ẋ b ,ẏ j ) with coefficients depending on coordinates (x a , y i ) in E. Furthermore, we know that κ is bi-homogeneous. Calculation on weights: w(y i ) = (1, 0) = w(ẋ a ), w(ẋ a ) = (0, 1) = w(y i ), w(ẏ i ) = (1, 1) = w(ẏ i ) ensures us that κ is determined by equations of the form (2.5) κ : and Q k ij (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 α j i (x) are simply constants δ j i . We shall show later (see Lemma 2.11) that the such linear ZMs correspond to general algebroid structures. The anchor maps (left and right) are locally given by functions Q b i (x) and Q b i (x), respectively, while functions Q k ij (x) encode the bracket operation (in a given basis of sections of the bundle σ, dual to (y i )).
Lemma 2.9 (A class of linear ZMs). Let σ : E → M be a vector bundle and let κ : Tσ → ⊲ τ E be a linear ZM. Assume additionally that κ induces the identity on the core bundles, i.e. κ ∩ (C × C) = graph(id C ). Then: (i) The base map ρ L := κ : E → TM is a VB morphism covering the identity map id M .
(iv) The inverse relation κ T is a linear ZM from Tσ to τ E over the base map ρ R : E → TM which also induces the identity on the cores.
Proof. The assertion has a rather straightforward geometric justification based essentially on the bihomogeneity 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 fiber 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).

Description of algebroids in terms of ZMs
In this part we shall recall the definition of a (general) algebroid, and latter rephrase it in the language of Zakrzewski morphisms introduced above. A possibility of such a reformulation is of course wellrecognized in the literature since the very introduction of the concept of a general algebroid [GU99] (see also [LMM05,Mar08]). However, this topic was never systematically studied for its own sake. In particular, we are not aware that the axioms of an algebroid were ever directly formulated in terms of the corresponding ZMs. 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 algebroids based on differential relations (the notions of a subalgebroid, a morphism between algebroids, the Lie axiom, various types of algebroids, etc. are intrinsically defined within the category of Zakrzewski morphisms). 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 [GU97,GU99] as double vector bundle morphism of a special kind. Their approach was motivated by the study of the geometry of mechanics and variational calculus originated by Tulczyjew [Tul76a,Tul76b]. 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) ρ L , ρ R : E → TM over the identity on M such that for every sections a, b ∈ Sec M (E) and every smooth functions f, g ∈ C ∞ (M ). (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.
(iii) If σ is an almost-Lie algebroid and the bracket [·, ·] satisfies the Jacobi identity, we call σ a Lie algebroid.
From algebroids to ZMs We will now construct a canonical Zakrzewski morphism related with a given algebroid structure. Recall Lemma 2.9 describing the structure of a class of linear ZMs 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 ZM.
The proof is given in Appendix A.
Remark 2.12 (Inverse of relation κ). Note that, according to Lemma 2.9 (iv), the inverse relation κ T is also a linear ZM (this time the over the right anchor ρ R ) 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 i.e. passing form κ to κ T corresponds to changing the bracket Note that κ M interchanges the two VB structures on TTM , Tτ M and τ TM .
Example 2.14 (The tangent lift of an algebroid). If (σ : E → M, ρ L , ρ R , [·, ·]) is an algebroid structure, then Tσ : TE → TM also carries a natural algebroid structure (Tσ, called the tangent lift of the algebroid structure on σ. This structure is determined by conditions for any sections a, b ∈ Sec M (E).
In this case (see [GU99]) the ZM d T κ corresponding to the considered algebroid structure on Tσ is the tangent lift of the ZM κ corresponding to the initial algebroid structure on σ composed with two canonical isomorphisms κ E , i.e.
This construction has a natural generalization to higher tangent lifts T k σ : T k E → T k M . It will be discussed in the second paragraph of Subsection 4.1 in detail.
From ZMs to algebroids In fact, relation κ introduced in Lemma 2.11 completely characterizes the algebroid structure on σ.
Proposition 2.15 (From ZMs to algebroids). Let σ : E → M be a vector bundle. A linear Zakrzewski morphisms κ from Tσ : TE → TM to the tangent bundle τ E : TE → E, which induces the identity on the core bundles, i.e. κ ∩ (C × C) = graph(id C ), provides σ with the unique algebroid structure.
The left anchor ρ L (the right anchor ρ R ) is given by the base map of κ (base map of κ T ) and the bracket is given by The proof is given in Appendix A.
Remark 2.16 (Alternative definition of an algebroid). By the results of Proposition 2.15 and Lemma 2.11 we can equivalently define an algebroid structure (σ, In what follows we shall often refer to this characterization. Remark 2.17 (The dual of relation κ). The fact that κ induces the identity on the core implies that its dual (which is a proper vector bundle morphism, in fact a DVB morphism by the linearity of κ) covers the identity id E * under the projections T * E → E * and TE * → E * (the core of a DVB becomes a side bundle under dualization [KU99]). 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 [GU97].
Remark 2.18 (Local form of κ). By Remark 2.8, the local form of a linear ZM corresponding to a given general algebroid structure is given by formulas (2.5) with α k i (x) = δ k i . Within this description ρ L : E → TM , the base map of κ, is given by Similarly, the right anchor reads as ρ R : 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 Algebroid morphisms as ZM-morphisms Next we shall show that within the interpretation of algebroids as Zakrzewski morphisms, the notion of an algebroid morphism corresponds to a ZMmorphism between appropriate relations. Intuitively, a morphism between two 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. (2.10) Remark 2.20 (On notion of an algebroid morphism). For any is a basis of sections of σ ′ over U ′ and f i are some functions on U . This explains a local character of the above definition. There are other equivalent and even more natural formulation of the definition of an algebroid morphism -we discus some of them in Proposition A.1 (see also Definition 3 in [GU99], where general algebroids are considered as a special type of Leibniz structures on the dual bundle and [Gra12] for a generalization of this notion to an algebroid relation).
It may seem unclear if condition (2.11) is well-posted, 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 justify this fact we check that the right-hand side of (2.11) is tensorial with respect to a i and b j (for this it is crucial that condition (2.10) holds). The essential calculations (for a simpler case of a skew algebroid) can be found in the classical book [Mac05b].
It turns out that the above definition has a very elegant (and much simpler) interpretation in the language of ZMs naturally related with the algebroid structure.
Subalgebroids and algebroidal relations Let us now describe the notion of a subalgebroid in terms of ZMs. Recall the following definition.
The first of the above conditions assures us that the bracket operation The second condition guarantees that the section space Sec M ′ (E ′ ) is closed with respect to this bracket. Clearly the subbundle σ ′ carries an algebroid structure inherited from σ.
In the face of the relationship between algebroids and ZMs, the ZM κ : Tσ → ⊲ τ E corresponding to the algebroid structure on σ should induce some ZM κ ′ : Tσ ′ → ⊲ τ E ′ corresponding 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 σ i : E i → M i , and let r : σ 1 → ⊲ σ 2 be a ZM over a base map r : M 2 → M 1 . We say that r restricts and if for any X ∈ E 1 and Y ∈ E 2 that are r-related and such that If this is the case the restriction r ′ defined as the intersection of r with The equivalence of the classical notion of a subalgebroid with the notion of a fine restriction of the corresponding ZM can be easily proved.
Proposition 2.24 (On subalgebroids). Let (σ : E → M, κ) be an algebroid and let σ ′ : E ′ → M ′ be a vector subbundle of σ. Then the following conditions are equivalent: Proof. It involves some elementary diagram-chasing to check that (iii) is equivalent to (Tι, Tι) being a ZM-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 algebroids (see Chapter 4 in [Mac05b]).
A concept of an algebroidal relation, introduced by Grabowski in [Gra12], is a generalization of a morphism of algebroids. It is closely related to the notion of a subalgebroid.
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 bracket.
Remark 2.27. We stress that although the anchor map is uniquely determined by the bracket operation in any general algebroid, condition (iia) does not follows 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.
Proof. We should check that r ′ is a subalgebroid of (σ First note that Y ∈ Tr, because r respects algebroid structures of E 1 and E 2 , and X, Y are also , and X ∈ Tr while y ∈ r ′ ⊂ r. Next, by Theorem A.5 of [JR15], Tr restricts fine to TE ′ 1 × TE ′ 2 . Therefore, since Y belongs also to Characterization of algebroids with an additional structure We shall now express specific conditions (i)-(iii) from Definition 2.10 in terms of the corresponding ZM κ.
(iii) a Lie algebroid if and only if it is almost-Lie and κ ⊂ TE × TE is a subalgebroid of the product algebroid (Tσ × τ E , d T κ × κ E ). In other words, κ : Tσ → ⊲ τ E is an algebroidal relation.
The proof is given in Appendix A.
An application -prolongations of AL algebroids Throughout this section we argued that ZMs provide a consistent language to describe algebroids, alternative to the standard treatment of the topic. Besides, some known constructions in the theory of algebroids have more evident definitions in the language of Zakrzewski morphism. To justify this claim we shall give now a non-standard definition of a prolongation of an algebroid over a fibration [HM90, CDLM + 06], the crucial notion in the Lagrangian and Hamiltonian formalisms for mechanics on algebroids developed by Martínez in [Mar01a,Mar01b]. In a forthcoming section we shall present of a natural notion of (higher) tangent lift of an algebroid.
Example 2.30 (Prolongation of an 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 } is a vector bundle over P which carries an almost Lie algebroid structure defined by the restriction of the relation κ × κ P : Indeed, the only thing to check is that T E P is a subalgebroid of the product algebroid (σ × τ P : ) (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 Prop. 3.1 in [Jóź17].

Graded bundles
Graded bundles and homogeneity structures Higher order algebroids which we shall introduce and study in the next section are modelled on graded bundles [GR11], geometric objects which generalize the notion of a vector bundle. Here we shall recall basic properties and constructions associated with graded bundles.
An important example is a higher (k th -order) tangent bundle τ k M : T k M → M of a manifold M , consisting of k th -order tangency classes (called k-velocities) of curves in M . Bundle T 1 M = TM is just the tangent bundle of M , however for k > 1, τ k M is no longer a vector bundle. We shall see this at the elementary level.
Given a smooth function f on a manifold M and an integer α = 0, 1, . . . , k one can construct function f (α) on T k M , the so-called α-lift of f (see [Mor70]). It is defined by where t k γ denotes the k-velocity of a curve γ : R → M at zero. We shall usually writeḟ ,f instead of f (1) , f (2) , respectively. The adapted coordinates (x a ,ẋ a ,ẍ a , . . .) for T k M induced by coordinates (x a ) on M are obtained by applying the above lifting procedure to coordinate functions x a , and are naturally graded. (We simply assign weight k to coordinates x a,(k) := (x a ) (k) .) In particular, on T 2 M they transform as thus the weight of both left and right sides of the above equalities is the same. From a geometric point of view, fibers of τ k M are equipped with a special structure. Namely, we have a natural action of the multiplicative monoid of real numbers (R, ·) on these fibers defined by re-parametrizing curves representing the elements of T k M : where γ t (s) = γ(ts). We clearly see, that the α-lift of a function f on M is a homogeneous function . This also explains why the adapted coordinates on T k M are graded. Properties of the higher tangent bundle T k M motivate the concepts of a graded bundle and a homogeneity structure.
Definition 3.1 (Graded bundle). A graded bundle is a smooth fibration σ k : E k → M in which we are given a distinguished class of fiber coordinates (called graded coordinates) with non-negative integer weights assigned. Moreover, it is assumed that these graded coordinates identify the fibers with R n (for some integer n), and that the transition functions are multi-variable polynomials that preserve the weights. The index k in E k indicates that the weights on E k are less or equal k. We say that E k has degree k.
A homogeneity structure is a manifold E equipped with a smooth action h : R × E → E of the multiplicative monoid of real numbers (R, ·). Surprisingly, both concept coincide in the smooth setting ( [GR11], Theorem 4.2), in particular, every graded bundle σ k : E k → M admits a unique homogeneity structure h E k : R × E k → E k . For this reason we shall use the terms graded bundle and homogeneity structure interchangeably.
Graded coordinates on a graded bundle σ k : E k → M can be denoted by (x a , y i w ) where a (superfluous) index w = w(i) ∈ Z >0 at y i indicates that a fiber coordinate y i is homogeneous of weight w. The base coordinates (x a ) are assumed to have weight zero. In such a notation the associated homogeneity structure reads as It is convenient and fruitful to encode the structure of a graded bundle by means of a canonical vector field on E k called the weight vector field which in graded coordinates is given by For example, the canonical weight vector field on T k M is ∆ k M = k α=1 a αx a,(α) ∂ x a,(α) . In fact, the weight vector field provides an equivalent characterization of the structure of a graded bundle.
A morphism from a graded bundle σ E : E k → M to σ F : F k → M is a smooth map φ : E k → F k between the total spaces commuting with the respective homogeneity structures, i.e., Given a graded bundle (E k , ∆) of degree k and an integer 0 ≤ j ≤ k one can construct a canonical projection from E k onto a graded bundle of degree j, denoted by E k [∆ ≤ j], obtained by removing all coordinates of weights greater than j ([BGR16], Definition 1.6). As transformation rules for E k of coordinates of weight ≤ j involve only coordinates of weights ≤ j the above construction is correct. It follows that a graded bundle σ k : E k → M induces a tower of affine fibrations Definition 3.2 (The top core of a graded bundle). Let σ k : E k → M be a graded bundle of degree k.
The top core of E k , denoted by E k , is the set Locally, E k is defined by putting to zero all fiber coordinates of weights less than k: : y a w = 0 for any y a w such that 1 ≤ w < k} ⊂ E k , hence (x A , y a k ) are local coordinates on E k . The top core E k is naturally a vector bundle over M with the homotheties defined locally by t.(x A , y a k ) = (x A , t · y a k ). for t ∈ R. Moreover, the top core · is a functor from the category GB[k] to the category of vector bundles.
Example 3.3 (Split graded bundles). Given a sequence of vector bundles E j , j = 1, . . . , k, over the same base manifold M we can turn the Whitney sum E = k j=1 E j into a graded bundle of degree k by assuming that linear fiber coordinates on E j have weight j. The obtained graded bundle will be denoted by X k j=1 E j [j], where the notation V [j] means that we assign weight j to linear coordinates on a vector space V . In other words, we have a functor from the category of graded vector spaces supported in degrees −1, −2, . . . , −k to the category of graded bundles of degree k.
Graded bundles equipped with a splitting, i.e., a graded bundle isomorphism p : E k → X k j=1 E j as above are called split graded bundles. It is worth to remember that in the smooth real setting any graded bundle is (non-canonically) isomorphic to its split form which is obtained from the sequence of the top core bundles E j , j = 1, . . . , k: Multi-graded bundles Of particular interests are geometric objects which admit several compatible graded bundle structures. We define a k-tuple graded bundle (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 h i : Definition 3.4 (The core and the ultracore of a multi-graded bundle). Let (E, ∆ 1 , . . . , ∆ k ) be a ktuple graded bundle. The core of E is the intersection where E[−∆ j ≥ 0] denotes the subset of E obtained by putting to zero all fiber coordinates y i w of the total fibration E → M with multi-weight w = (w 1 , . . . , w k ) in which w j = 0. Thus C(E) is obtained by putting to zero all fiber coordinates y i w with multi-weight w in which at least one of w 1 , . . . , w k is zero. In general, C(E) is still a k-tuple graded bundle with trivial side bundles. Note that the core C(E) coincides with the intersection of of k 2 kernels of vector bundle morphisms, namely C(E) = i<j ker(p i , p ij ), where By the ultracore of E we mean the top core E of the graded bundle (E, ∆ 1 + . . . + ∆ k ).
Note that for a double vector bundle, the core in 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 considered by Mackenzie [Mac05a].
Graded-linear bundles Let us now take a closer look at double graded bundles with one of the homogeneity structures linear. If (E, h 1 , h 2 ) 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 h 1 ) equipped with a graded bundle structure encoded in a homogeneity structure h 2 : 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 with respect to h 2 . In particular, for each λ ∈ R, the mapping v → λ · σ v, v ∈ E, should be a graded bundle morphism, or equivalently, it should commute with h 2 t for any t ∈ R. A double graded bundle (E, h 1 , h 2 ) in which (E, h 1 ) is a vector bundle is called a linear-graded bundle while its flip (E, h 2 , h 1 ) a graded-linear bundle [BGG15a]. Thus weighted vector bundles are nothing more but graded-linear or linear-graded bundles.
A coming definition of higher algebroids involves two particular examples of graded-linear bundles, the tangent bundle of a order-k graded bundle and k th -order tangent bundle of a vector bundle. Thus it will be important to understand the core bundles of both weighted bundles in more details. For this we shall show now that the core of any graded-linear bundle 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 bundle). The core of any degree k graded-linear bundle D is a graded vector bundle, i.e., for some vector bundles D j canonically associated with D. In particular cases, Proof. Bi-graded coordinates on D have the form (x A , y a (i,j) ) where i = 0 or 1 and 0 ≤ j ≤ k, and (i, j) = (0, 0). As C := C(D) is given locally in D by equations y a (1,0) = 0 and y a (0,j) = 0, 1 ≤ j ≤ k, the transformation rules for the fiber coordinates y a (1,j) | C , j = 1, . . . , k, are of the form y a ′ (1,j) | C = Q a ′ j,b y b (1,j) | C for some functions Q a ′ j,b on the final base M of D. It follows that the submanifold of D defined locally by vanishing all fiber coordinates except those of weight (1, j), where 1 ≤ j ≤ k is a fixed integer, form a vector bundle D j over M and C is just the Whitney sum of these vector bundles.
Weighted structures and their reductions from "higher" to "lower" In similar manner we can introduce notion of other weighted geometrical objects and structures, in particular ZMs.
Definition 3.6 (Weighted ZM). Let σ : E → M and σ ′ : E ′ → M ′ be two weighted VBs, with homogeneity structures h, h ′ on E and E ′ , respectively. A ZM r : σ → ⊲ σ ′ is called a weighted ZM if r ⊂ E × E ′ is homogeneous with respect to h × h ′ . In other words, r is a linear-graded subbundle of the product of linear-graded bundles E × E ′ → M × M ′ . In particular, the base map r : M ′ → M is a graded bundle morphism.
Remark 3.7 (On weighted structures). Weighted algebroids and weighted groupoids considered in [BGG15a] are defined in the same spirit. We simply request that the structure maps of an algebroid (resp. a groupoid) is graded, in some sense. A particular examples are VB-algebroids and VBgroupoids extensively studied in the literature. (The graded structure is then of degree 1.) We shall not need these notions in our paper.
For a weighted structure we may try to perform a reduction from a "higher" to a "lower" level in the tower of affine fibrations (3.2). If the starting point is a graded-linear bundle (E k , ∆ 1 , ∆ 2 ) of degree k, then its j-level reduction E j = E k [∆ 1 ≤ j] is still a graded-linear bundle and σ k j : E k → E j is a morphism in the category graded-linear bundles. In particular, given a weighted ZM r : σ 1 → ⊲ σ 2 between graded-linear bundles σ i : E k i → M i (i = 1, 2) of degree k we may consider r as a graded-linear subbundle in the product bundle σ 1 × σ 2 and then get its lower-level reductions r j : E j 1 → ⊲ E j 2 which are not only graded-linear subbundles but also weighted ZMs of degree j, where j = 0, 1, 2, . . . , k.
This and another useful observation in a similar spirit is presented below. The proof is left to the reader.

Higher algebroids
Definitions and fundamental examples As was already suggested in the introduction, the higher tangent bundle τ k M : T k M → M (cf. the beginning of Section 3) together with the canonical isomorphism κ k M : T k TM → TT k M will be our fundamental example of a higher algebroid. According to our considerations from the beginning of the previous section, a choice of local coordinates (x a ) on M induces canonical coordinates (x a ,ẋ b ) on TM , (x a,(α) ) α=0,...,k on T k M , (x a ,ẋ b , x a,(1) ,ẋ b,(1) , . . . , x a,(k) ,ẋ b,(k) ) on T k TM , and (x a,(α) ,ẋ b,(β) ) α,β=0,...,k on TT k M . In these coordinates κ k M reads as , x a,(1) , . . . , x a,(k) ,ẋ b ,ẋ b,(1) , . . . ,ẋ b,(k) ) . Clearly, for k = 1 we get κ 1 M = κ M studied in Example 2.13. It follows easily from the above coordinate formula that κ k M is an isomorphism in the category of graded-linear bundles. According to our motivating considerations from the introduction, we would now like to generalize the above example by substituting the fibration T k M → M with an arbitrary graded bundle E k → M of degree k. Instead of a graded-linear isomorphism κ k M we shall take a weighted Zakrzewski morphism κ k between T k E 1 and TE k . Such postulates are naturally motivated by the description of general algebroids (of order one) in terms of a ZM κ (cf. Proposition 2.15 and Lemma 2.11).
Definition 4.1 (Higher algebroid, HA). A (general) higher (k th -order) algebroid (HA, in short) is a graded bundle σ k : E k → M of degree k together with a weighted ZM κ k ⊂ T k E 1 ×TE k from T k σ 1 to τ E k (covering a (graded) mapping ρ k : E k → T k M ) such that the relation κ 1 : Tσ 1 → ⊲ Tτ E 1 being the reduction to level one of κ k equips σ 1 : E 1 → M with an algebroid structure. (4.1) In addition: (i) If (σ 1 , κ 1 ) is skew, we call (σ k , κ k ) a skew HA.
(ii) If (σ k , κ k ) is skew and (T k ρ 1 , Tρ k ) : κ k ⇒ κ k M is a morphism in the category of ZMs, where ρ 1 : E 1 → TM is the level 1 reduction of ρ k , then we call (σ k , κ k ) an almost Lie (AL) HA.
(iii) A skew higher algebroid (σ k , κ k ) in which κ k is a subalgebroid of the product of the algebroids (T k σ 1 , d T k κ 1 ) (the k th -tangent lift of (σ 1 , κ 1 ) -see Subsection 4.1 -and the tangent algebroid (τ E k , κ E k )), then (σ k , κ k ) is called a Lie HA. In fact a Lie HA must be an almost Lie HA -see Proposition 4.9. In the language of [CDW13], (σ k , κ k ) is Lie if κ k is a Lie algebroid comorphism.
(iv) A general HA in which κ k induces an isomorphism on the core bundles C(κ k ) : C(T k E) → C(TE k ) is called a strong HA.
We define a morphism between higher algebroids (σ k E : E k → M, κ k,E ) and (σ k F : F k → N, κ k,F ) to be a graded bundle morphism φ k : E k → F k such that (T k φ 1 , Tφ k ) : κ k,E ⇒ κ k,F is a ZM-morphism. Obviously, higher algebroids form a category which we denote by HA.
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 higher algebroids. The defining relation κ k covers a (graded) morphism ρ k : E k → T k M , and its reduction to level 0 induces a relation (in fact a linear map) κ 0 : E 1 → TM . For k = 1 they were the left and the right anchor, respectively. As we see, these maps are of quite a different nature. Note, however, that for higher skew algebroids κ 0 = ρ 1 .
Using Proposition 3.8 we easily see that reduction of a HA (E k , κ k ) to a lower level j = 1, 2, . . . , k gives an induced higher algebroid structure on E j which is skew (resp. AL, Lie, strong) if (E k , κ k ) was so. Moreover, a HA morphism φ k : (E k , κ k,E ) → (F k , κ k,F ) induces a HA morphisms φ j : (E j , κ j,E ) → (F j , κ j,F ).
Examples of higher algebroids will be discussed later in Subsection 4.2 and Section 6. For now let us take a closer look at higher algebroids of order 2.
Remark 4.2 (A local form of a higher algebroid of order 2). Let (x a , y i , z µ ) be graded coordinates on a degree 2 graded bundle E 2 → M . We shall find a general form of an algebroid (E 2 , κ 2 ) of order 2. As in Remark 2.8 we underline coordinates in TE 2 when writing equations for κ 2 ⊂ T 2 E 1 × TE 2 . Taking weights into account for some structure functions Q ··· ··· . For a skew algebroid (E 2 , κ 2 ) we have Q ′ a i = Q a i and Q i jk = −Q i kj . An algebroid is strong if the matrix Q µ i 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 natural notion of higher subalgebroid. Definition 4.3 (Subalgebroid of a HA). Let (σ k : E k → M, κ k ) be a higher algebroid and let σ ′ k : E ′ k → M ′ be a graded subbundle of σ k . We call σ ′ k a higher subalgebroid of (σ k , κ k ) if the ZM κ k restricts fine to T k σ ′ 1 × τ E ′k ⊂ T k σ 1 × τ E k .

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) 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 ∈ Sec M (E) its section. The assignment Sec M (E) ∋ s → T k s ∈ Sec T k M (T k E) is injective, and R-linear, yet it cannot be onto, since rank(T k σ) = (k + 1) · rank(σ) > rank(σ). It is, however, possible to characterize all sections of T k σ 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 T k R × T k E → T k E of T k R, the latter understood as an R-algebra with addition and multiplication defined as T k -lifts of the standard addition and multiplication in R. In fact, as an algebra T k R is canonically isomorphic to the truncated polynomial algebra R[ε]/ ε k+1 (this algebra is often denoted D k and called the algebra of k th -order numbers). From the point of view of the theory of natural bundles, T k is a product-preserving bundle functor associated with the Weil algebra D k -see [KMS93]. In conclusion, on T k E we have an operation of multiplication by the elements of D k , which is in fact determined by the action of the generator ε ∈ D k . In coordinates (x a,(α) , y i,(α) ) on T k σ induced from linear coordinates (x a , y i ) on σ it reads as ε · (x a,(α) ; y i,(0) , y i,(1) , . . . , y i,(k) ) = (x a,(α) ; 0, y i,(0) , 2y i,(1) , . . . , ky i,(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 T k σ defined as In particular s (0) = 1 k! ε k · T k s is called the vertical and s (k) = T k s the total lift of s, in agreement with the standard notions for k = 1.
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 Sec T k M (T k E). Recall the notion of an α-lift of a smooth function introduced a the beginning of Section 3. For any function f ∈ C ∞ (M ) the operation of the k-lift has the following property We recognize the standard formula for the iterated derivative. An analogous formula for (f · s) (k−α) (with binomial coefficients) can be easily derived from the latter.
Higher tangent lifts of algebroids It is well-known that if a VB σ : E → M carries an algebroid structure, then its k th -tangent lift T k σ : T k E → T k M has the so-called lifted algebroid structure (e.g. [KWN13]). It can be elegantly described in terms of the ZM κ related with the initial algebroid structure on σ.
We shall now check that d T k κ indeed defines and algebroid structure and describe it in the classical terms of the anchor and the bracket operations.
Proposition 4.6 (Properties of the lifted algebroid). The relation d T k κ := κ k,E •T k κ•κ −1 k,E presented in (4.4) defines an algebroid structure on T k σ. Moreover, (i) The (left and right) anchor maps on T k σ are given by for any integers α, β = 0, 1, . . . , k such that α + β ≤ k and any sections s 1 , s 2 ∈ Sec M (E), and [s This property fully determines the bracket [·, ·] d T k on the space of sections of T k σ.
The proof is given in Appendix A.
Remark 4.7 (Characterization of lifts by natural pairings). The natural pairing ·, · σ : E * × M E → R lifts to the pairing T k ·, · σ : T k E * × T k M T k E → R which enables us to identify sections of the vector bundle T k σ with linear functions on T k E * . As T k E * is, in particular, a graded bundle of degree k the space of sections of T k σ 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 T k E * ) coincides with the (k − α)-lifts (k−α) of the linear functions : E * → R canonically associated with the section s. Taking into account the graded bundle structure of T k E * , 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, assuming an 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 T k σ form a graded Lie algebra concentrated in degrees −k, −k +1, . . .. It has a Lie subalgebra Sec T k M,≤0 (T k E) 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 ZM between two vector bundles induces a natural 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 higher algebroid structure, we can naturally combine these two constructions to arrive at Definition 4.8 (Algebroid lift of a section). Let (σ k : E k → M, κ k : T k σ 1 → ⊲ τ E k ) be a higher algebroid. Given a section s ∈ Sec M (E 1 ) and an integer α = 0, 1, . . . , k we define the (k − α)algebroid lift of s as 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 [GU99].
In [GU97] Grabowski and Urbański observed that the Jacobi identity can be elegantly reformulated in terms of the algebroid lifts. Their results easily generalize to the setting of higher Lie algebroids.
Proposition 4.9 (Characterization of the Lie axiom). The map κ k : Sec T k M (T k E) → X(E k ) preserve gradation, i.e. for a homogeneous section s of T k σ the vector field κ k (s) has the same weight as s. Moreover, a higher algebroid (σ k : E k → M, κ k : T k σ 1 → ⊲ τ E k ) is Lie if and only if it is almost Lie and if the associated algebroid lift satisfies for any sections s 1 , s 2 ∈ Sec M (E 1 ) and any integers α, β = 0, 1, . . . , k. Above [·, ·] E k denotes the standard Lie bracket of vector fields on E k , and [·, ·] σ 1 the algebroid bracket on the reduced (Lie) algebroid (σ 1 , κ 1 ).
Equivalently, an AL algebroid (E k , κ k ) is Lie if and only if is a graded Lie algebra morphism, where Sec T k M,≤0 (T k E) (resp. X ≤0 (E k )) are Lie algebras generated by the homogeneous sections of T k σ (resp. vector fields on E k ) of non-positive degree.
Proof. Let s ∈ Sec T k M (T k E) be a homogeneous section, denote X := κ k (s) and consider the corresponding functionss ∈ C ∞ (T k E * ), andX ∈ C ∞ (T * E k ) on the dual vector bundles (cf. Remark 4.7). They are related byX =s • ε k , where ε k : T * E k → T k E * is a (weighted) vector bundle morphism dual to κ k . Since κ k is bi-homogeneous, the same is ε k , hence weights ofX ands are the same, and our first assertion follows. By definition, (σ k , κ k ) is a Lie HA if and only if it is skew and if κ k is an algebroidal relation between (T k σ 1 , d T k κ 1 ) and (τ E k , κ E k ). Proposition 2.26 provides a useful characterization of algebroidal relations. It follows that (σ k , κ k ) is a Lie HA if and only if for any sections s, s ′ ∈ Sec T k M (T k E 1 ) we have (note that both (T k σ 1 , d T k κ 1 ) and (τ E k , κ E k ) are skew algebroids and their anchor maps are κ k M • T k ρ 1 and id TE k , respectively) The first of these conditions is precisely the commutativity of the diagram i.e. the axiom of an AL algebroid. The second condition for lifts s = s , with s 1 , s 2 ∈ Sec M (E 1 ), gives equality (4.6). Since all (k − α)-lifts of sections from Sec M (E 1 ) span the whole space Sec T k M (T k E 1 ) the assertion follows.
Remark 4.10. Note that Sec T k M,≤0 (T k E) and X ≤0 (E k ) 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 s 1 = e i , s 2 = e j where (e i ) is a local basis of sections of σ 1 .

Subalgebroids of Lie higher algebroids
We shall end our considerations concerning the Lie axiom for HA by showing that it is preserved when passing to a subalgebroid. First we will need the following property of higher tangent lifts. (i) Let r : σ 1 → ⊲ σ 2 be a ZM that restricts fine to a ZM r ′ : σ ′ 1 → ⊲ σ ′ 2 . Then T k r : T k σ 1 → ⊲ T k σ 2 is a ZM which restricts fine to a ZM T k r ′ : T k σ ′ 1 → ⊲ T k σ ′ 2 . (ii) Let (σ, κ) be an algebroid and (σ ′ , κ ′ ) its subalgebroid. Then the k th -tangent lift algebroid (iii) Let (σ 1 , κ 1 ) and (σ 2 , κ 2 ) be algebroids and let r : σ 1 → ⊲ σ 2 be an algebroidal relation. Then T k r : T k σ 1 → ⊲ T k σ 2 is also an algebroidal relation between the k th -tangent lift algebroids The proof is given in Appendix A.
Proof. Assume that (σ k F : F k → N, κ k,F ) is a higher subalgebroid of a higher AL algebroid (σ k E : E k → M, κ k,E ). To prove that (σ k F , κ k,F ) is also AL we must verify that elements X ∈ T k F 1 and Y ∈ TF k are κ k,F -related if and only if their images T k ρ 1 F (X) and Tρ k F (Y ) are κ k M -related. This follows from a trivial observations that the anchor map ρ k F : F k → T k N is the restriction ρ k E | F k and that κ k,E restricts fine to κ k,F , hence for two elements X and Y as above, the condition (X, Y ) ∈ κ k,E implies (X, Y ) ∈ κ k,F . Now assume that (σ E , κ k,E ) is Lie. This means that κ k,E is an algebroid relation between T k σ 1 E and τ E k . Note that T k σ 1 F is a subalgebroid of T k σ 1 E (by Proposition 2.26 (ii)) and that τ F k is a subalgebroid of τ E k . Moreover, by definition, κ k,F is a fine restriction of κ k,E . Thus, by the results of Proposition 2.28, κ k,F is an algebroidal relation, i.e. (σ k F : F k → N, κ k,F ) is a Lie higher algebroid.

Prolongations of a general algebroid
In our previous publication [JR15] we studied a class of objects crafted to play the role of prototypes of 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 higher algebroids in the sense of Definition 4.1. Let us now recall their construction. See also [CdD11] for an equivalent definition (under the name of higher order Lie algebroid) based on tangent lifts of admissible curves in a Lie algebroid.
The total spaces of bundles σ [k] are defined inductively: where ρ : E → TM is the anchor map of (σ, κ  [JR15], Proposition 4.6): Since κ is a ZM, it is easy to see that so is κ k−1 E • T k−1 κ. We stress that the fact that the latter restricts fine to a ZM from T k E to TE [k] is a non-trivial result which relies strongly on the fact that σ has an AL algebroid structure.
In particular, if we start from the tangent algebroid (σ, κ) = (τ M , κ M ), then σ [k] is just the higher tangent bundle τ k M : T k M → M and κ [k] is the canonical isomorphism κ k M . In general, is a graded bundle of rank (r, r, . . . , r) where r is the rank of E (see [JR15], 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 k th -prolongation can be interpreted as a reduction of the higher tangent bundle T k G by the action of G. Thus such structures, appear naturally in variational problems reduced by symmetries [JR14b,JR15]. We refer to these papers of several concrete examples.
Properties of the prolongations Prolongations of AL algebroids provide an example of strong higher AL algebroids. Moreover, prolongations of a Lie algebroids are Lie higher algebroids. The proof is given in Appendix A.

Variational calculus on higher algebroids
In this section we shall discuss a geometric formalism of the k th -order variational calculus based on the notion of a higher algebroid (σ k : E k → M, κ k ). Actually the geometry here is exactly the same as in our previous study on prolongations of algebroids [JR14b]. The idea is that the relation κ k 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 κ k we may represent the variation of an action functional as a pairing of a certain curve of covectors with the k th -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 [JR14a].
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 this section we provide a rather trival 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-Poincare and Hummel equations within our formalism can be found in [JR14b].

Pre-algebroids
Definition 5.1 (Pre-algebroid). By a pre-algebroid of order k we understand a triple (σ k , τ, κ k ) consisting of a graded bundle σ k : E k → M of degree k, a vector bundle τ : F → M (over the same base), and a weighted ZM κ k : T k τ → ⊲ τ E k .
The reduction of the relation κ k to order zero defines a vector bundle morphism ρ F = κ 0 : F → TM . We call a pre-algebroid (σ k , τ, κ k ) almost Lie if (T k ρ F , Tρ k ) : κ k ⇒ κ k M is a ZM-morphism.
The difference with Definition 4.1 is that we allow the weighted ZM κ k to relate the tangent bundle τ E k : TE k → E k with the lifted bundle T k τ : T k F → T k M of an arbitrary VB τ , a priori not related with σ k . By taking τ = σ 1 we recover the definition of a 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 ∈ Sec M (F ) of τ and a number α = 0, 1, . . . , k we assign a vector field s [k−α] := κ k (s (k−α) ) ∈ X(E k ).
Remark 5.2. We believe that examples of pre-algebroids may appear naturally in the theory of reductions. Let G be a Lie group (or more generally a Lie groupoid). In the standard reduction of a k th -order variational problem on G one divides the higher tangent bundle T k G by the the natural action of G. In fact, here G acts on curves in G by, say, left multiplication and the action on T k G is the natural differential consequence of this action on k th -jets. However, one can consider more general actions by a subgroup H < T k G which properly contains G, and thus which would not be induced by the action of G on curves. In proper circumstances, the quotient T k G/H will have a structure of a graded bundle (space) E k . However, in general the knowledge of E 1 may not be enough to define admissible variations, if for example E 1 is trivial but the entire E k is not. Thus a reduction of κ k G (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 section.
Admissible paths and admissible variations Consider now a pre-algebroid (σ k , τ, κ k ). Let γ : R → E k be a curve over γ := σ k (γ) : R → M . Choose another curve a : R → F and consider its k th -order tangent lift t k a : R → T k F . Along γ we would like to construct a vector field δ a γ by the formula δ a γ(t) := (κ k ) γ(t) (t k a(t)) ∈ T γ(t) E k .
Note that, by the properties of κ k , this definition is correct if and only if γ and a share the same base path, i.e. σ k (γ) = γ = τ (a) and if t k γ = ρ k (γ). The latter equation is known as the (left) admissibility of γ. We call δ a γ 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(E k ), while the set of all admissible variations by VA(E k ). 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 E k -valued path γ is a vector field along γ, i.e., δγ : t → T γ(t) E k . Hence a variation δγ can be represented by a family of paths γ s : R → E k , where γ 0 = γ and δγ(t) = t 1 (s → γ s (t)). In a very general, but intuitive sense, a variation δγ is tangent to a subset A of paths in E k if it can be represented by a family γ s lying in A. Results of [Mar08] and Theorem 3 in [GG08] show that for an algebroid of order one the subspace Adm(E 1 ) of admissible curves is a Banach submanifold in the space of all C 1 -paths in E 1 . Moreover, the condition T Adm(E 1 ) = VA(E 1 ) 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.
Lemma 5.3 (Tangent space for admissible variations). Let δγ ∈ T γ E k be a vector field along an admissible path γ in a pre-algebroid (σ k : E k → M, τ : F → M, κ k ). If δγ is tangent to Adm(E k ) then where t k δγ denotes the k th -tangent lift of the curve δγ = Tσ k (δγ).
Proof. Assume that δγ is a variation of an admissible path γ ∈ Adm(E k ) tangent to Adm(E k ), i.e. δγ(t) = t 1 [s → γ(t, s)] and paths t → γ(t, s) are admissible for each s. Then Due to the definition of κ k M , formula (5.2) follows.
Remark 5.4 (Problems with the Banach-manifold structure). According to [Mar08], for k = 1 the converse of above lemma is also true, i.e., equation (5.2) describes the space T Adm(E 1 ). For k > 1 some additional conditions are necessary to assure that the subspace of admissible curves in E k is a Banach submanifold. Such conditions may follow directly from the geometry of the considered problem. For example in our preprint [JR14b] 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 (σ k : E k → M, τ : F → M, κ k ) is almost Lie, γ is an admissible path, and a is a curve in F such that τ (a) = γ, then the admissible variation δ a γ satisfies equation (5.2).
Proof. Consider a diagram which is commutative as our pre-algebroid(σ k , τ, κ k ) is almost Lie. As (t k a, δ a γ) ∈ κ k , we get The last equality follows from the commutativity of the diagram Variational calculus We understand Lagrangian mechanics on a k th -order pre-algebroid (σ k , τ, κ k ) as the study how a functional, obtained by integrating the Lagrangian function L : E k → R along an admissible curve γ : R → E k , behaves under movement in the direction of admissible variations. By construction where (κ k ) * : T * E k → T k F * is a morphism of graded-linear bundles dual to κ k . We have thus arrived at the pairing of a T k F * -valued curve with the k th -tangent lift t k a : R → T k F of the generator a of δ a γ. Using our earlier results on the k th -order geometric integration by parts -see [JR14a] -we may present this pairing as a sum of a paring ·, · τ : F * × F → R evaluated on the generator a with a complete derivative Here EL k (·) denotes the Euler-Lagrange operator and P k (·) the momentum operator associated with the Lagrangian. An admissible curve γ satisfies Euler-Lagrange equations EL k (t k γ) = 0 (i.e. is a trajectory of the considered Lagrangian system) if and only if for every virtual displacement a we have or in a more familiar integral version The presented formalism generalizes the construction of geometric Lagrangian mechanics on a general algebroid from [GG08].
Symmetries and conservation laws As explained above, Euler-Lagrange equations on higher prealgebroids are derived by studying changes of the Lagrangian L(γ) in the direction of every admissible variation δ a γ. By contrast, conservation laws are related with special properties of a particular admissible variation. Given an admissible curve γ : R → E k over γ : R → M we basically look for a generator a : R → F of an admissible variation δ a γ such that dL(γ(t)), δ a γ(t) = d dt f (t) for some smooth function f : R → R. Now if γ is a solution of the Euler-Lagrange equations, by (5.5) we have d 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 EL k (t k γ) = 0. Instead it is better to look for a generator universal for every γ by considering a = s| γ , where s ∈ Sec M (F ) is some section of τ . Note that in this case s [k] | γ = δ a γ, i.e. the admissible variation δ a γ is the restriction of the k th -order algebroid lift (note that df may be regarded as a map df : T k M ⊂ TT k−1 M → R), we call section s ∈ Sec M (F ) a generator of the symmetry of L. If this is the case then i.e., dL(γ), δ a γ is a total derivative regardless of the choice of an admissible curve γ, as intended.
If follows that f (t k−1 γ) − P k (t k−1 γ), ρ k−1 (γ)(s) T k−1 σ is constant along the trajectory γ, i.e. we derived a conservation law related with the symmetry of L.
A simple example 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 variational problem on the Euclidean plane R 2 , constituted by a Lagrangian function L : T 2 R 2 → R. In this case the HA in question is simply the second tangent bundle τ 2 M : T 2 R 2 → R 2 equipped with the standard higher algebroid structure κ 2 R 2 : T 2 TR 2 ≃ −→ TT 2 R 2 . Formula (5.4) leads to the standard Euler-Lagrange equations: where (x, y) are standard coordinates on R 2 and (x, y,ẋ,ẏ,ẍ,ÿ) the adapted coordinates on T 2 R 2 . Consider now the following action of (a, b, c) ∈ R 3 on (the germs at t = 0 of) curves in R 2 : (a, b, c) • (x(t), y(t)) = (x(t) + a +ẋ(0)bt, y(t) + c) .
If L is invariant under this action, then (5.6) reduces to 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 of T 2 R 2 by the action of R 3 is naturally a graded space (i.e., a graded bundle over a point) σ 2 : On E 2 we can introduce natural graded coordinates (y 1 , x 2 , y 2 ) induced byẏ,ẍ andÿ, respectively.
It is clear that the invariant Lagrangian L induces a function l : E 2 → R such that L = p • l for p : T 2 R 2 → E 2 = T 2 R 2 /R 3 being the quotient map.
We can further equip σ 2 with a pre-algebroid structure For this structure the admissibility condition is empty, since τ : R 2 → {pt} has a trivial base. However, an additional condition (5.8)ẏ 1 = y 2 should be imposed on admissible curves if we want to maintain the correspondence of admissible curves in E 2 with the admissible curves in T 2 R 2 under the reduction procedure (cf. Remark 5.4).
It is now an easy exercise to show that equations (5.7) are the Euler-Lagrange equations for the variational problem on (σ 2 , τ, κ 2 ) constituted by function l : E 2 → R and the admissibility condition (5.8).
The above example suggests that pre-algebroids may naturally appear in reductions of standard variational problems by actions which are non-trivial on higher jets. (Note that in the standard Lie groupoid-Lie algebroid reduction [Mac05b] and its generalization to higher jets [JR15] the action of the groupoid on higher jets is merely a consequence of its action on points -see Remark 5.2.) 6 Further examples -substructures and quotients of T k G/G Another interesting class of examples of higher algebroids is obtained from the reduction T k G/G of the k th -tangent bundle of a Lie group G. The resulting space E k := T k e G, consisting of k-velocities in G based at the identity element e ∈ G, is a graded bundle over a single point e ∈ G (i.e., a graded space -see [GR11]). It can be equipped with the canonical HA structure κ k g which, in fact, can be identified with the k th -prolongation (in the sense of Subsection 4.2) of (g, [·, ·]) -the Lie algebra of the group G.
Throughout this part we shall describe all higher subalgebroids of T k G/G, as well as characterize HA quotients of T k G/G by which we understand higher algebroids (F k , κ k ) obtained from T k G/G by means of surjective HA morphism onto F k .
A higher algebroid structure on T k e G We shall begin by describing the higher algebroid structure on T k e G. First of all, T k e G 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 T k 0 exp : T k 0 g ≃ T k e G of graded spaces. (Note that in general, for k ≥ 2, the k th -tangent space of a manifold at a given point has no canonical vector space structure.) Next, T k 0 g (and so T k e G) has a canonical identification with T k−1 g 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, T k e G is naturally 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 T k e G = T k−1 g = g ⊗ R[t]/ t k . Now we would like to recall a ZM κ k g : T k g → ⊲ TT k−1 g constituting a HA structure on T k e G ≃ T k−1 g (see Section 6 of [JR15] for details). An element X ∈ T k g represented by a curve t → k j=0 X j t j ∈ g can be identified with a (k + 1)-tuple (X 0 , X 1 , . . . , X k ) of elements of g. In a similar manner a vector Y ∈ TT k−1 g ≃ T k−1 g × T k−1 g (note that TT k−1 g is a tangent bundle of a vector space), can be identified with a 2k-tuple (Y 0 , . . . , Y k−1 ,Ẏ 0 , . . . ,Ẏ k−1 ) of elements of g. Now (X, Y ) ∈ κ k g if and only if , for every l = 0, 1, . . . , k − 1.
The details of this relation can be found in Proposition 6.3 in [JR15].
We recall that homogeneous functions of weight j + 1 on T k e G correspond to linear functions on the summand t j g. In particular, V 0 should be a Lie subalgebra of g and additionally each subspace V i should contain V 0 and be preserved by the action of the latter subspace.
Quotients of T k e G General quotients of Lie algebroids and the concept of an ideal in a Lie algebroid are quite involved subjects (see [Mac05b]). For this reason in the following definition of a quotient higher algebroid structure we shall restrict our attention only to "quotient" maps covering the identity on the base. We call a graded bundle morphism φ k : E k → F k surjective if for each 1 ≤ j ≤ k the induced top core vector bundle morphisms φ j : E j → F j are fiber-wise surjective linear maps.
In particular, such a morphisms is a surjective mapping E k → F k but not vice versa: a graded bundle morphism which is a surjective mapping need not induce surjective mapping between the core bundles.
Definition 6.2 (A quotient of a HA). We shall say that a k th -order algebroid (σ k F : F k → M, κ k,F ) is a quotient of a higher algebroid (σ k E : E k → M, κ k,E ) if there is a surjective graded bundle morphism φ k : E k → F k covering the identity on the base manifolds such that (T k φ 1 , Tφ k ) is a ZM morphism κ k,E ⇒ κ k,F . In other words, elements X ′ ∈ T k F 1 and Y ′ ∈ TF k are κ k,F -related if and only if they are images under projections T k φ 1 and Tφ k , respectively, of some κ k,E -related elements X ∈ T k E 1 and Y ∈ TE k : Proposition 6.3 (Quotients of HA). Let φ k : T k−1 g → F k be a surjective graded space morphism. Then F k inherits the quotient higher algebroid structure from (T k−1 g, κ k g ) if and only if (up to a noncanonical HA isomorphism) φ k is the quotient map T k−1 g → (T k−1 g)/J where J = k−1 j=0 J j is a graded ideal in the Lie algebra T k−1 g of a special form. Namely, J 0 ⊂ J j for each j = 0, 1, . . . , k−1.
In particular, F k shares a graded Lie algebra structure.
The proof is given in Appendix A.
Remark 6.4. It follows that any reduction of the HA (T k−1 g, κ g k ) can be decomposed into two steps. The first step is the natural projection T k−1 g → T k−1 h where h = g/I 0 is the quotient Lie algebra. In the second step we have a reduction of the form T k−1 h → (T k−1 h)/J, where J = k−1 j=0 J j t j + t k is a graded Lie ideal of T k−1 h such that J 0 = {0}.
Example 6.5 (A class of examples). A slight generalization of the example (T k−1 g, κ k g ) is possible. Let E k = k−1 i=0 g i be a graded Lie algebra. In particular, g 0 has a Lie algebra structure and so T k g 0 is another graded Lie algebra. Let α : T k−1 g 0 → E k be a graded Lie algebra homomorphism such that α 0 = id g 0 , where α 0 is the restriction of α to the subalgebra g 0 being the component in degree 0 of the graded Lie algebra T k−1 g 0 . For X ∈ T k g 0 let (X0, X1) ∈ TT k−1 g 0 ≃ T k−1 g 0 × T k−1 g 0 be the image of X under the canonical embedding T k g 0 ⊂ TT k−1 g 0 . We consider E k as a (split) graded space defined by the weight vector field on E k given by ∆ = k−1 i=0 (i + 1)∆ g i , where ∆ g i is the Euler vector field on g i . Then the formula where y ∈ E k , turns E k 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 → [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.)

Final remarks
The main idea of this paper was to use the framework of Zakrzewski morphisms in order to provide a proper language to describe higher analogs of general 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 ZMs κ : Tσ → ⊲ τ E which induce the identity on the core. By modifying the latter condition we can define natural generalization of the concept of an algebroid. For example if κ is a linear ZM which induces the identity on the base κ ∩ (M × M ) = graph(id M ) ⊂ M × M then the left and right anchors ρ L , ρ R : E → TM are still well-defined. Moreover, such a κ induces a VB endomorphism on the cores φ : σ → σ over id M . 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) 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 ZM κ : Tσ → ⊲ τ E which induces an endomorphism φ on the cores. For example, a linear connection on the tangent bundle TM is an R-bi-linear operator ∇ : Pre-algebroids and reduction As we have seen in Section 5 for many interesting applications, 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 naturally appear as reductions of higher tangent bundles, while considering (pseudo) group actions on the space of smooth curves on manifolds in the spirit of [KL16] (see a natural example of such a situation in the last paragraph of Section 5). Thus it would be interesting to initiate 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 ρ E : E → TM and ρ F : F → TM a VB morphism φ : F → E and a bi-linear bracket operation [·, ·] : Sec(E) × Sec(F ) × Sec(E) satisfy, for every sections a ∈ Sec(E), b ∈ Sec(F ) and functions f, g ∈ C ∞ (M ), a Leibniz-like rule In particular, left-twisted algebroids provide examples of pre-algebroids (with τ = σ).
Structure of higher algebroids One of us initiated a study of the internal structure of higher algebroids. It turns out that, at least in degree 2, 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 [Rot17].
is valid for every sections a, b, c ∈ Sec M (E). We shall show that this generalized formula defines a differential relation κ satisfying conditions (i) and (ii). This will end the proof.
Denote F (a, b) := Tb(ρ L (a)) and G(a, b) := Ta(ρ R (b))+ + +[a, b]. Leaving aside (for a moment) the problem of correctness of the definition of κ, note that formula (A.1) relates an element F (a, b)+ + +c, whose Tσand τ E -projections are, respectively, ρ L (a) and b, with an element G(a, b)+ + +c, for which these projections are ρ R (b) and a, respectively. We clearly see that ZM κ (if correctly defined) covers the left anchor ρ L and projects to the graph of the right anchor ρ R under τ E × Tσ.
We want to prove that κ is a ZM over ρ L . Note that every element B of the Tσ-fibre (TE) ρ L (a) for any a ∈ E x can be represented as B = Tb| x (ρ L (a))+ + +c| x for some sections b, c ∈ Sec M (E), and thus (A.1) gives us the value of κ a on every element of (TE) ρ L (a) . (Note that if ρ L (a) = 0, then also Tb| x (ρ L (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 κ a for every possible element of (TE) ρ L (a) . This justifies the need of using the extended formula (A.1).) It is enough to show that the value κ a (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 κ a (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 κ a (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 ′′ ∈ Sec M (E). By our assumption, and we can conclude that ρ L (a)(f ) · b ′′ | x = (c − c ′ )| x . Now we can use this fact to get It follows that κ a (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 · τ E A) ∈ κ and (f · τ E B, f · Tσ A) ∈ κ. This can be easily done using the following properties od F and G: 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 ZM 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 ∈ Sec M (E) the right-hand side of (2.9) is a difference of two elements of TE with the same τ E -projection a and the same Tσ-projection ρ R (b), thus a vertical vector. That is, [a, b] ∈ Sec M (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 → f · a and b → 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 ZM-morphism (Tφ, Tφ) : κ ⇒ κ ′ , then (see diagram (2.3)) the commutativity at the level of base maps (the left anchors) gives us precisely the commutativity of the first diagram in (2.10). The commutativity of the second diagram follows from an observation that (Tφ, Tφ) is also a ZM-morphism between κ T 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 ZM-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 ∈ Sec M (E) and let φ * a = i f i · φ * a i and φ * b = j g i · φ * b j as in Definition 2.19. In the following calculations we understand a section s ∈ Sec M (φ * E ′ ) as a map s : Similarly, and we conclude that Applying the tangent map Tφ to formula (2.9) gives us (ii) The graph of φ is a subalgebroid of the product algebroid on σ × σ ′ .
Proof. Recall ( [GU99]) that a Leibniz manifold (M, Λ) is a manifold M equipped with a contravariant 2-tensor Λ and that, what is more, linear contravariant 2-tensors on the dual E * of a vector bundle σ : E → M are in one-to-one correspondence with general algebroid structures on σ. The construction is completely parallel to that between linear Poisson tensors and Lie algebroid structures. The equivalence between (ii) and (iii) is proved in [Gra12], Theorem 5.1 for skew algebroids, however, the proof can be directly rewritten for general algerboids. 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 (ρ L , ρ ′ L ) and (ρ R , ρ ′ R ) map F to TN ⊂ T(M × M ′ ) we have commuting diagrams in (2.10). To prove (2.11) write φ * a = i f i φ * a i and similarly, φ * b = j g j φ * b j and notice that (x, φ(x)) → (a(x), φ * a(x)) is a section of F whose extension to a section of σ × σ ′ can be written as followsã (x, y) = (a(x), i f i (x)a i (y)), and similarly for section b. We clearly have hence [ã,b](x, φ(x)) ∈ F implies (2.11). This reasoning can be inverted proving the equivalence of conditions (i) and (ii).
First observe that for sections s 1 ∈ Sec M 1 (E 1 ) and s 2 ∈ Sec M 2 (E 2 ) the restriction (s 1 , s 2 )| M ′ is a section of the vector subbundle σ ′ ⊂ σ 1 × σ 2 if and only if s 2 =r(s 1 ) and, moreover, every section of σ ′ can be presented as such a restriction. Thus condition (iia) is equivalent to the fact that σ I = (σ 1I , σ 2I ) maps E ′ ⊂ E 1 × E 2 to TM ′ = T graph(r) ⊂ TM 1 × TM 2 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 σ 1 × σ 2 is closed with respect to sections of the form (s 1 , r(s 1 )), where s 1 ∈ Sec M 1 (E 1 ). As we already observed such sections are σ 1 × σ 2 -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 ZM κ is given by formula (2.8). As has been observed in Remark 2.12, κ T corresponds to another 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 = κ T is equivalent to the antisymmetry of the bracket [a, b] = −[b, a]. This proves (i).
Assume now that the algebroid structure on σ is skew. To prove (ii) apply Tρ to (2.9) to get which ends the proof of point (i).
To show (ii) observe that by our hypothesis, κ restricts fine to TE ′ × TE ′ , hence by (i) T k κ : T k Tσ → ⊲ T k τ E restricts fine to T k TE ′ × T k TE ′ . Since d T k κ is obtained by composing T k κ with the vector bundle isomorphisms κ k E and its inverse, the relation d T k κ also restricts fine to TT k E ′ × TT k E ′ , hence T k σ ′ is a subalgebroid of T k σ, as was claimed.
A proof of Proposition 4.13 Assume first that (σ, κ) is an AL algebroid. From the results of Theorem 4.5(ix) in [JR15] we already know that κ [k] is a ZM. We shall prove first that it is a weighted ZM. For this it is enough to check that κ [k] is a graded subbundle of τ k E × Tτ [k] : T k E × TE [k] → E×TM . A general idea is to consider the latter as a graded subbundle of T k−1 TE×TT k−1 E → E× TM and notice that although κ interchanges the two homogeneity structures on TE, the homogeneity structure in our concern (which is defined by the sum of weight vector fields of degree 1 and k − 1) is respected by both T k−1 κ and κ k−1 E , thus also by κ [k] (which is a fine restriction of the composition of these relations). In detail, TT k−1 E is a triple-graded bundle of 3-degree (1, k − 1, 1) with legs TT k−1 τ : TT k−1 E → TT k−1 M , Tτ k−1 E : TT k−1 E → TE, and τ T k−1 E : TT k−1 E → T k−1 E, respectively. The multi-graded bundle structure on TT k−1 E is defined by the weight vector fields TT k−1 ∆ E , T∆ k−1 E , and ∆ 1 T k−1 E . 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 T k−1 κ is homogeneous with respect to Similarly, κ k−1 E is homogeneous with respect to In particular, T k−1 κ is homogeneous with respect to (T k−1 ∆ 1 E + ∆ k−1 TE , T k−1 T∆ E + ∆ k−1 TE ) while κ k−1 E is homogeneous with respect to (T k−1 T∆ E +∆ k−1 TE , TT k−1 ∆ E +T∆ k−1 E ) thus the composition κ k−1 E • T k−1 κ is homogeneous with respect to (T k−1 ∆ 1 E + ∆ k−1 TE , TT k−1 ∆ E + T∆ k−1 E ). The result follows because the degree k graded bundle structures on T k E → E and TE [k] → TM are encoded by T k−1 ∆ 1 E + ∆ k−1 TE and TT k−1 ∆ E + T∆ k−1 E , respectively (see Theorem 4.5(i) in [JR15]). The commutativity of the diagram Take any y ∈ F k , X = (X 0 , . . . , X k ) ∈ T k g, and any Y0 ∈ T k−1 g such that φ k (Y0) = y. Since κ k,F is a ZM, the image of the affine subspace (κ k g ) Y0 (X + T k I 0 ) ⊂ T Y0 T k−1 g under the linear mapping T Y0 φ k should be a single vector of T y F k . But (κ k g ) Y0 : T k g → T Y0 T k−1 g is a linear map, hence T Y0 φ k should annihilate the subspace (κ k g ) Y0 (T k I 0 ), and thus φ k should be constant on affine subspaces of T k−1 g consisting of vectors of the form Y0 + Y1 ∈ T k−1 g for which (Y0, Y1) is κ k g -related with a vector from the vector subspace T k I 0 ⊂ T k g. Equations (6.1) for κ k g easily imply that (Y0, T k−1 I 0 ) ⊂ (κ k g ) Y0 (T k I 0 ) hence φ k is constant on any affine subspace Y0 + T k−1 I 0 , and hence it factors to a map ψ k : T k−1 (g/I 0 ) → F k which is also a graded bundle morphism. Moreover, looking only at the first order reduction φ 1 : g → F 1 , we easily find that I 0 is a Lie ideal of g.
Step 2. We may now assume that F k is a HA reduction of T k−1 h (h is a Lie algebra) by means of a graded bundle morphism ψ k : T k−1 h → F k such that ker ψ 1 = {0}. We shall show that the corresponding kernels J i−1 := ker ψ i of the top core morphisms form an ideal {0} ⊕ J 1 ⊕ . . . ⊕ J k−1 of the Lie algebra T k−1 h.
We know that Tψ k should take the same value at points (κ k h ) Y0 (X) while Y0 varies in the preimage (ψ k ) −1 (y) for fixed points y ∈ F k and X ∈ T k h. Replace Y0 with Y ′ 0 = Y0+ + +z, where z ∈ J k−1 ⊂ h is considered as an element of the top core vector subspace of T k−1 h. Obviously, ψ k maps both Y0 and Y ′ 0 on the same element y ∈ F k . Denote, (Y ′ 0 , Y ′ 1 ) := (κ k h ) Y ′ 0 (X). The corresponding change Y ′ 1 − Y1 is in the top core space of T k−1 h, and from equations (6.1) we read that the condition that Tψ k is equal on both vectors (Y ′ 0 , Y ′ 1 ) and (Y0, Y1) implies that the top kernel bundle J k−1 is a Lie ideal of h. More generally, by putting z = (z 0 , . . . , z k−1 ) ∈ k−1 i=0 J i we find that Tψ k (Y0, Y1) = Tψ k (Y ′ 0 , Y ′ 1 ) implies that [h, J i ] ⊂ J k−1 . Applying the same argument to φ j we see that the ideals J i should satisfy [h, J i ] ⊂ J j if i ≤ j. It follows that J = {0} ⊕ J 1 ⊕ . . . ⊕ J k−1 is a Lie ideal of T k−1 h.
Step 3. Assume k−1 j=0 J j is an ideal in the Lie algebra T k−1 h and J 0 = {0}. Then the natural map π k J : T k−1 h → (T k−1 h)/J gives a HA reduction (T k−1 h, κ k h ) → ((T k−1 h)/J, κ k J ). This is immediately seen from the equation (6.1) as X1 + [X0, Y0 + J] T k−1 h has a well defined value in T k−1 h/J.