MODELS for HIGHER ALGEBROIDS Micha L Jуzwikowski Miko Laj Rotkiewicz

JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2015.7.317 c American Institute of Mathematical Sciences Volume 7, Number 3, September 2015 pp. 317–359

MODELS FOR HIGHER ALGEBROIDS

Micha lJo´´zwikowski∗ Institute of Mathematics Polish Academy of Sciences Sniadeckich´ 8, 00-656 Warszawa, Poland MikolajRotkiewicz Faculty of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2, 02-097 Warszawa, Poland and Institute of Mathematics Polish Academy of Sciences Sniadeckich´ 8, 00-656 Warszawa, Poland

(Communicated by Manuel de Le´on)

Abstract. Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a diﬀerential relation of special kind (a Zakrzewski morphism). In the paper we investigate basic properties of higher algebroids and present some examples.

Contents 1. Introduction 318 2. Preliminaries 323 2.1. Notation, basic constructions 323 2.2. Almost Lie algebroids and Zakrzewski morphisms 325 3. Reductions of Lie groupoids 328 4. Higher algebroids 333 5. Integrable higher algebroids 344 6. Examples 348 7. Conclusions and outlook 351 Appendix A. On Zakrzewski morphisms between vector bundles 352 Acknowledgments 357 REFERENCES 357

2010 Mathematics Subject Classiﬁcation. Primary: 58A20, 58A50; Secondary: 70G65, 70H50. Key words and phrases. Higher algebroids, almost Lie algebroids, graded bundles, reduction of Lie groupoids. This research was supported by the Polish National Science Center grant under the contract number DEC-2012/06/A/ST1/00256. ∗ Corresponding author: Micha lJ´o´zwikowski.

317 318 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

1. Introduction. Motivations. In this paper we undertake preliminary studies on the concept of a higher algebroid. In our intuitive understanding a higher algebroid (of order k) should be a geometric structure generalizing the notion of a (kth-order) tangent bundle in the same way as an algebroid generalizes the notion of the tangent bundle. Schematically this idea can be presented as the proportion

higher algebroid algebroid = . higher tangent bundle tangent bundle Recall that the most common way [30] to motivate the concept of a Lie algebroid is studying the reduction of the tangent bundle TG of a Lie groupoid α, β : G ⇒ M. As a geometric object a Lie algebroid of G is the bundle τ : A(G) → M consisting of those velocities in TG which are attached at the base points in M ⊂ G and are tangent to the natural source fibration in G (the fibration by α fibers), i.e. A(G) = α TM G . An additional geometric structure associated with τ, being inherited from the groupoid multiplication, is the Lie bracket on the space of sections of τ induced by the natural Lie bracket of right invariant vector fields on G. Taking this as the starting point it is a natural to construct a kth-order higher algebroid of G as the reduction of the kth-order tangent bundle TkG of G. A natural geometric object obtained from such a reduction is the bundle τ k : Ak(G) = k α TM G → M consisting of those k-velocities on G which are attached at the base points M ⊂ G and are tangent to the source fibration of the groupoid. In other k k words, elements of A (G) are k-jets of the form tt=0γ(t) where γ(t) is a curve in G such that the source map of G is constant along the image of γ and such that γ(0) ∈ M (note that τ k has a natural graded bundle structure [15] inherited from TkG). It is, however, unclear what natural geometric structure associated with such an Ak(G) plays the role of the Lie bracket on A(G). It seems that TkG does not admit any natural bracket operation. Thus the fundamental question arises

What structure on Ak(G) is inherited from the groupoid multiplication? (A)

In this paper we propose an answer to this problem. To root Problem (A) in the mathematical mainstream, note that its study for k = 1 and G being a Lie group can be seen as the foundations of the theory of Lie algebras. As indicated above for a general groupoid G and k = 1 the solution is also well known: it is the Lie algebroid structure on A(G). As far as we are concerned, Problem (A) has never been studied even for k = 2 and G being a Lie group, although the bundle Ak(G) where introduced already in [37] as ‘higher-order analogue of the usual Lie algebroid A(G), but without the bracket operation . . . ’.

Our results. In Sec.3 we study the reduction of the higher tangent bundle k k k k τG :T G → G of a Lie groupoid G to the bundle τ : A (G) → M providing an answer to Problem (A). The fundamental idea behind this solution is a reformulation of the definition of an algebroid (τ : E → M, ρ, [·, ·]) in terms of a relation κ canonically associated with an algebroid structure [18]. A simple definition of κ which, however, does not reveal its geometric importance, states that κ is the relation dual to a vector bundle morphism ε :T∗E → TE∗ over ρ : E → TM. The latter is defined as the composition ε = Λ˜ ◦ R, where Λ:T˜ ∗E∗ → TE∗ is associated with the MODELS FOR HIGHER ALGEBROIDS 319 canonical Poisson tensor [18] on E∗ and R :T∗E → T∗E∗ is the canonical anti- symplectomorphism. The details are provided in Subsection 2.2. The graph of κ is T not only a vector subbundle of Tτ × τE supported by the graph of ρ :TM →B E −1 −1 but, moreover, it is a union of graphs of linear mappings Tτ (ρ(y)) → τE (y) as y varies in E, the property which characterize Zakrzewski morphisms (ZMs in short, AppendixA). We present such a κ in a form of a diagram

κ TE ,2TE

Tτ τE ρ TME.o The geometric meaning of κ is concealed in the description of the homotopy relation of admissible curves on a Lie algebroid in the integration problem of Lie algebroids (see [20] for details). In fact κ, even if not commonly recognized as equivalent to the algebroid structure, is well-known in particular special cases. For a tangent algebroid E = TM, the relation κ is the canonical ﬂip κM : TTM → TTM, while if E = g is a Lie algebra then (x, X) ∈ Txg ≈ g and (y, Y ) ∈ Tyg ≈ g are κ-related if and only if X − Y = [x, y]. Here we identify the tangent space of a vector space at a given point with the vector space itself. k It turned out that for A (G) there exists an analogous linear relation κk between kth tangent prolongation of τ denoted by Tkτ (which is a vector bundle projection) and the tangent bundle of Ak(G)

κ TkA(G) k ,2TAk(G)

τ Tkτ Ak(G)

ρk TkM o Ak(G).

k We deﬁne κk as the natural reduction of the canonical isomorphism κk,G :T TG → k TT G (Ex. 1.1 shows the construction of κk in the concrete case when G is the Atiyah groupoid). We prove that κ1 coincides with κ deﬁned above and that κk is a Zakrzewski morphism, thus the dual of κk

ε T∗Ak(G) k / TkA∗(G)

∗ k ∗ τAk(G) T τ

ρk Ak(G) / TkM

k is a vector bundle morphism. We call the pair (A (G), κk) a higher algebroid of a Lie groupoid G. We found it more natural and handy to work with κk rather than with εk. k A higher algebroid (A (G), κk) as constructed above is completely determined by the Lie groupoid structure on G and the natural properties of (higher) tangent functors Tk. In light of the famous result on the integrability of Lie brackets [6], we know that the Lie groupoid structure on G is (locally) determined by (A(G), κ)– k the Lie algebroid of G. This suggests that (A (G), κk) can be constructed directly from (A(G), κ) without referring to a quite complicated reduction procedure. We provide such a construction in Theorem 3.5. 320 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

The latter result is the cornerstone of our considerations in Section4. By mimick- ing the construction from Theorem 3.5 we propose a definition of a higher algebroid k k (E , κk) associated with an algebroid (E, κ). Here E is the total space of a graded k k k k bundle τ : E → M associated with E and κk :T E →B TE is a linear relation, k namely a vector subbundle of T τ × τEk . For (E, κ) = (A(G), κ) we recover the k k notion of a higher algebroid (A (G), κk) of G. Actually the construction of (E , κk) does not requires the initial algebroid (E, κ) to be a Lie algebroid. In Section4 we restrict our attention to the case when (E, κ) is an almost Lie algebroid, i.e. it satisfies all the axioms of a Lie algebroid apart from the Jacobi identity. The choice of this class of objects is justified by the results of Theorem 4.5 where we k list the fundamental properties of higher algebroids (E , κk). The most important of these states that whenever (E, κ) is an almost Lie algebroid, then the relation k k κk :T E →B TE is of special kind, namely it is a Zakrzewski morphism. This fact ∗ k k ∗ can be equivalently stated as follows: εk :T E → T E – the dual of κk – is a true k k vector bundle morphism (not just a relation!) over the anchor map ρk : E → T M. It has fundamental meaning in applications of the discussed structures in Geometric Mechanics and Variational Calculus (see the next paragraph). The remaining properties listed in Theorem 4.5 concern mostly the natural graded bundle structure on k E and the interaction of this gradation with the relation κk.

Contexts and applications. Let us explicitly state some important aspects related with of our concept of a higher algebroid:

• The notion of a higher algebroid consists of two parts: a geometric object (a k graded bundle E ) and a geometric structure (the canonical relation κk). The object on its own is not a higher algebroid. Analogously, the tangent space at the unit of a Lie group is not a Lie algebra without deﬁning the Lie bracket on it. • The language of linear relations, ZMs and its tangent lifts seems to be unavoid- able. The reason of this is, basically, the fact that a reduction of a map is, in general, a relation not a map. Being aware that this subject is not commonly known in the community, we provided a concise discussion of the concept of ZM (based on the original papers of Zakrzewski [43, 44]) in AppendixA. • The substitution of the notion of a Lie algebroid by an almost Lie algebroid in our considerations is not just a vain generalization. It can be understood as the study of the limits of the theory, i.e. to what extend we can relax the requirements input on the studied objects without loosing their most important features. It turns out that our theory does not require the Jacobi identity (but the compatibility of the anchor and the bracket is crucial). It is worth to mention that the structure of an almost Lie algebroid seems to be promising in the context of nonholonomic mechanics [14]. k k • Bundles E are naturally graded. This situates higher algebroids (E , κk) in the context of recently developing theory of graded bundles [1,2, 15, 16]. In fact our work should be treated rather as a study of a particular, yet interesting from the point of view of applications, and special, but fundamental, case of a general higher algebroid (that is why we used the word model in the title). Such an object should be a graded bundle equipped with a ZM with properties based on the properties of κk enlisted in Theorem 4.5. A small insight into MODELS FOR HIGHER ALGEBROIDS 321

such a theory is provided in Section7. We postpone the detailed discussion to our next publication [24]. • It is worth to mention that Lie algebroids were quite early postulated as the proper language for Geometric Mechanics by Weinstein [42]. Starting from the seminal papers of Mart´ınez[31, 32], this task was undertaken by many mathematicians and many solutions were proposed (see for example [10, 11] or the survey papers [5, 34] and the references therein). The significance of the notion of a Lie algebroid in Geometric Mechanics lies basically in the fact that tangent bundles reduced by symmetries provide natural examples of such structures. Consequently, Lie algebroids provide a universal framework for Geometric Mechanics and Variational Calculus suitable for both standard and reduced systems. This perspective justifies the need of studying geometric structures related with the reductions of higher tangent bundles. Actually one of our key motivations when initiating this research was the hope to construct a universal geometric framework for higher-order Geometric Mechanics and Variational Calculus. With the help of geometric tools introduced in our previous publication [22] we were able to achieve this goal. Due to the limitations of space we provide only a flavor of our results in this direction in Section7. The preliminary version of our theory can be found in our preprint [23] and is planned to be published as a separate paper [25]. In this context it is worth to mention recent results [2, 35].

Similar results. The topic of higher algebroids is almost untouched in literature with just a few exceptions known to us. In [39] Voronov introduced the concept of a higher algebroid in the frames of Graded Geometry. In his approach a higher algebroid is an N-manifold equipped with a homological vector field of weight 1, per analogy to the characterization of a Lie algebroid as a homological vector field of weight 1 on an N-manifold associated with a vector bundle. Despite the similarity of names Voronov’s ideas point in a different direction then ours. We postpone the detailed comparison of his and our approaches to a separate publication [24]. We are also aware that some aspects of the theory of higher algebroids were know to Colombo and de Diego. In [4] they introduced bundles Ek (as objects) under the name higher-order algebroids using a definition equivalent to ours. These authors, k however, did not recognize the canonical geometric structure κk on E , nor gave any examples. Therefore, no idea how to solve Problem (A) is present there. After submitting the preliminary version of this paper Bruce, Grabowska and Grabowski [1] studied a concept of a weighted algebroid which is a generalization of the notion of a VB-algebroid [19]. Weighted algebroids are simply usual algebroids equipped with an additional (and compatible) graded bundle structure, thus the total space of a weighted algebroid is a double-graded bundle of degree (k, 1) (so called graded-linear bundle). The canonical example of a higher algebroid TkM in the sense of this paper, by the authors of [1] is regarded as a subbundle of TTk−1M equipped with the canonical Lie algebroid structure (such an understanding of higher order structures is motivated by the work of Grabowska and Vitagliano on higher order mechanics [12]). The authors of [1] discovered that actually any 322 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ graded bundle τ k : Ek → M of degree k can be canonically regarded as a subbundle of a canonically associated graded-linear bundle D(Ek) called the linearization of Ek and propose to say that “Ek carries the structure of a weighted algebroid” if D(Ek) is a weighted algebroid. The differences between our approach and the concepts of [1] are outlined in Section 5 of the latter work. To close the discussion it is worth to mention the appearance of (usually unrec- ognized) structure of a higher algebroid in some works in Geometric Mechanics. In [8] variational problems on a higher tangent bundle of a Lie group are studied and the associated EL equations (higher-order Euler-Poincaréequations) are derived. In a recent paper of Mart´ınez[35] the equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found using the methods of his previous work [33]. Of course, they are equivalent to (7.1). In fact, the author of [33] constructs k explicitly bundles E yet pays no attention to the canonical relation κk. In his paper this relation is used implicitly ([35] Prop. 4.1) to construct variational vector fields (i.e., higher-order variations). Let us end this introductory section with a motivating example. Example 1.1 (kth-order Atiyah algebroid). Consider a right principal G-bundle p : P → M = P/G. The associated right G-action rP,0 : P × G → P induces the k k k canonical right G-action rP,k :T P × G → T P , defined as (rP,k)g = T (rP,0)g for each g ∈ G. This action is still free and proper and thus TkP/G is a smooth manifold (in fact, it is a graded bundle over M as we shall see in Section4). Bundle TkP/G can be equipped with the natural structure of a higher alge- th k k broid (called the k -order Atiyah algebroid) κk :T (TP/G) →B T T P/G . The relation κk can be defined as a reduction of the canonical flip κk,P :

κk,P TkTP / TTkP

k k k κk k k T (TP/G) T TP/T G ______,2TT P/TG T T P/G , that is, two elements in the bottom row are κk-related if and only if they are images of two κk,P -related elements in the top row. Here the left and right down- k pointing arrows are projections by T rP,1 and TrP,k-action, respectively. Later in k th Proposition 3.6 we shall show that (T P/G, κk) is a k -order algebroid associated with the canonical groupoid structure on ΓP = (P × P )/G (the so-called Atiyah groupoid). The reduced bundle TkP/G appears naturally in variational calculus. Consider, namely, a standard variational problem on P of minimizing the action associated with a kth-order smooth Lagrangian Le :TkP → R. Let us assume that Le is invariant with respect to the action rP,k. Therefore, it factorizes through a smooth function L :TkP/G → R: k Le T P /7 R o o ∃!L o o o o o TkP/G. It is worth to mention that in the language of Mart´ınez[35] Lagrangians L and L0 are related by the natural Lie algebroid morphism TP → TP/G. MODELS FOR HIGHER ALGEBROIDS 323

Not so obviously, also the canonical relation κk is important in variational calculus [23, 25]. To see the appearance of κk is the standard higher-order variational calculus on a manifold M, the Reader is directed toward [22].

2. Preliminaries.

2.1. Notation, basic constructions. Higher tangent bundles. Throughout the paper we will work with higher tangent bundles. We will use standard notation TkM for the kth tangent bundle of a manifold M. Points in the total space of this bundle will be called k-velocities. An k element represented by a curve t 7→ γ(t) ∈ M at t0 will be denoted by t γ(t0) or tk γ(t), so tkγ is the kth tangent lift of the curve γ. t=t0 The k + 1st tangent bundle is canonically included in the tangent space of the kth tangent bundle (see, e.g., [38]): 1,k k+1 k k+1 1 k ι :T M ⊂ TT M, tt=0 γ(t) 7−→ tt=0ts=0γ(t + s). k The composition of this injection with the canonical projection τTkM : TT M → TkM defines the structure of the tower of higher tangent bundles TkM −→ Tk−1M −→ Tk−2M −→ ... −→ TM −→ M. The canonical projections from higher- to lower-order tangent bundles will be de- k k s k k noted by τs,M :T M → T M (for k ≥ s). Instead of τ0,M :T M → M we will k 1 1 write simply τM and instead of τ0,M = τM :TM → M we will use the standard symbol τM . The cotangent fibration will be denoted with the standard symbol ∗ ∗ τM :T M → M. Another important constructions are iterated tangent bundles T(k)M := T ... TM and iterated higher tangent bundles Tn1 ... Tnr M. These bundles admit natural projections to lower-order jet bundles. (Iterated higher) tangent bundles are subject 1,k k+1 k to a number of natural inclusions such as already mentioned ιM :T M ⊂ TT M, l,k k+l l k ιM :T M ⊂ T T M, etc. We will use these extensively. Given a smooth function f on a manifold M one can construct functions f (α) on TkM, the so called (α)-lifts of f (see [36]), where 0 ≤ α ≤ k. These are defined by α (α) k d f (t γ(0)) := α f(γ(t)). dt t=0 By iterating this construction we obtain functions f (α,β) := (f (β))(α) on TkTlM for 0 ≤ α ≤ k, 0 ≤ β ≤ l. A coordinate system (xa) on M gives rise to the so-called a,(α) k a,(α,β) adapted coordinate systems (x )0≤α≤k on T M and (x )0≤α≤k,0≤β≤l on TkTlM where xa,(α) and xa,(α,β) are as above. Let us remark that in the definition of f (α) we follow the convention of [8, 38]. The original convention of [36] is slightly 1 different, namely it contains a normalizing factor α! in front of the derivative.

Graded bundles and homogeneity structures. The bundle TkM is the fundamental example of a graded bundle [15], a generalization of the notion of a vector bundle. A graded bundle has an atlas in which one can assign weights (non-negative integers) to the local coordinates, in such a way that coordinate transformations preserve the gradation deﬁned by this assignment. For example for TkM, one as- signs weight α to coordinates xa,(α). Since the number of coordinates of each weight is m = dim M and the highest weight assigned is k, the rank of TkM is said to be 324 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

(m, . . . , m) and the degree is said to be k.A graded space is a graded bundle over a point, thus a generalization of a vector space. A total space of a graded bundle τ : E → M is equipped with the canonical smooth action h : R × E → E by homotheties of the monoid (R, ·), thus a ho- k k mogeneity structure [15]. For the higher tangent bundle τM :T M → M this action is defined by a reparametrization of a curve representing the k-velocity: k k h(u, t0 γ) = t0 γ˜ whereγ ˜(t) = γ(ut). The crucial fact is that the converse holds (see [15]): any smooth action h : R × E → E of (R, ·) on a manifold E induces a graded bundle structure on τ := h(0, ·): E → M := h({0} × E). Moreover, if f : E1 → E2 is a smooth map between graded bundles τj : Ej → Mj, j = 1, 2, such that f commutes with the homotheties h1(t, ·), h2(t, ·) in E1 and E2, for any t ∈ R then, automatically, f is a graded bundle morphism. In other words, the categories of graded bundles and homogeneity structures are isomorphic. This fact has many applications even in the case of vector bundles (see [16]). A graded bundle τ k : Ek → M of degree k (k ≥ 1) is fibrated by submanifolds defined invariantly by fixing values of all coordinates of weight less or equal j (0 ≤ j ≤ k) in a given graded coordinate system on (τ k)−1(U), U ⊂ M. The quotient space is a graded bundle of degree j equipped with an atlas inherited from the atlas of Ek in an obvious way. The obtained bundle will be denoted by τ j : Ek[j] → M. Note that Ek[j][j0] ' Ek[j0], where j ≥ j0 ≥ 0. The following sequence of graded bundle projections

Ek → Ek[k − 1] → Ek[k − 2] → ... → Ek[1] → M. (2.1) will be called the tower of graded bundles associated with Ek. For Ek = TkM we have Ek[j] = TjM. If no confusions would arise, we shall simply write Ej for Ek[j]. Iterated higher tangent bundles are examples of r-tuple graded bundles, since they r admit an atlas with coordinates with weights in N0. Any r-tuple graded bundle can be considered as a graded bundle by taking the total weight into account.

Tangent lifts of vector bundles and canonical pairings. Of our special inter- ests will be (iterated) higher tangent bundles of vector bundles. Let τ : E → M be a vector bundle. It is clear that τ may be lifted to vector bundles Tkτ :TkE → TkM, TkTlτ :TkTlE → TkTlM, etc. Throughout the paper we denote by (xa) coordinates on the base of a vector i bundle τ : E → M, by (y ) linear coordinates on ﬁbers of this bundle and by (ξi) linear coordinates on ﬁbers of the dual bundle τ ∗ : E∗ → M. Natural weighted coordinates on (iterated higher) tangent lifts of τ and τ ∗ are constructed from xa, i y , ξj by the mentioned lifting procedure. They are denoted by adding a proper weight to a coordinate symbol. ∗ Recall (e.g. [25]) that the natural pairing h·, ·iτ : E ×M E → R can be lifted to a non-degenerate pairing denoted by

k ∗ k h·, ·iTkτ :T E ×TkM T E −→ R, D a,(α) (α) a,(α) i,(α) E X X k (α) i,(k−α) (x , ξi ), (x , y ) = ξi y . (2.2) Tkτ α i 0≤α≤k

The canonical ﬂip κk,M . It is well known that the iterated tangent bundle TTM MODELS FOR HIGHER ALGEBROIDS 325 admits an involutive double vector bundle isomorphism (called the canonical ﬂip)

κM : TTM −→ TTM, which intertwines projections τTM : TTM → TM and TτM : TTM → TM. This object can be generalized to a family of isomorphisms (also known as canonical ﬂips) k k k 1 1 k κk,M :T TM −→ TT M, tt tsγ(t, s) 7−→ tstt γ(t, s), k k k k k which map projection T τM :T TM → T M to τTkM : TT M → T M over k k k k idTkM and τTM :T TM → TM to TτM : TT M → TM over idTM . Morphisms κk,M can be also deﬁned inductively as follows: κ1,M := κM and κk+1,M := Tκk,M ◦

κTkM Tk+1TM , i.e.,

Tκk,M κ k TTkTM / TTTkM T M / TTTkM (2.3) O O

? κk+1,M ? Tk+1TM ______/ TTk+1M. a,(α,) k Local description of κk,M is very simple. If x are natural coordinates on T TM and xa,(,α) are natural coordinates on TTkM (here α = 0, 1, . . . , k and ∈ {0, 1}), a,(α,) a,(,α) then κk,M changes x to x . It is worth to mention that the dual of κk, which is a canonical isomorphisms between T∗TkM and TkT∗M was studied in the literature [3].

2.2. Almost Lie algebroids and Zakrzewski morphisms. Deﬁnition 2.1 ([13]). A skew-symmetric algebroid is a vector bundle τ : E → M equipped with a vector bundle morphism ρ : E → TM, called the anchor map, and a skew-symmetric bracket [·, ·] : Sec(E) × Sec(E) → Sec(E) on the space of sections of τ such that [X, fY ] = f[X,Y ] + ρ(X)(f)Y (the Leibniz rule) (a) for f ∈ C∞(M) and X,Y ∈ Sec(E). If, in addition,

ρ([X,Y ]) = [ρ(X), ρ(Y )]TM (compatibility of the anchor and the bracket) (b) then E is called an almost Lie algebroid (AL algebroid, in short). Thus, an almost Lie algebroid is a structure which satisﬁes all axioms of a Lie algebroid except for the Jacobi identity. Most of the results of this section are valid also for skew-symmetric algebroids. Let us comment that we do not assume the Jacobi identity as it is not necessary in the construction of the tower of higher k algebroids (E , κk) performed in Section4. However, the axiom of the compatibility of the anchor and the bracket is essential in this construction, as well as in the variational calculus on algebroids ([21] Lem. 2.5). For these reasons we shall work with AL algebroids. Although we do not provide any example of using higher-order algebroids associated with an almost Lie (but not Lie) algebroid, there is a possible ﬁeld of such applications in nonholonomic mechanics [14]. In standard local coordinates (xa, yi) on a vector bundle τ : E → M, an AL a k algebroid can be described in terms of local functions ρi (x), cij(x) on M given by k a [ei, ej] = cij(x) ek, ρ(ei) = ρi (x) ∂xa , 326 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

i i where (ei) is a basis of local sections of τ such that y (ej) = δj. The skew-symmetry k of [·, ·] results in the skew-symmetry of cij(x) in lower indices, while the axiom (b) of an AL algebroid reads as ∂ρa(x) ∂ρa(x) k ρb(x) − j ρb (x) = ρa(x)ci (x). (2.4) ∂xb j ∂xb k i jk A basic example is the tangent bundle TM with the standard Lie bracket of vector fields and ρ = idTM . The canonical flip κM : TTM → TTM has its algebroid version κ :TE →B TE, which is no longer a bundle map but a Zakrzewski morphism (ZM in short; see AppendixA). For understanding this paper it is generally enough to know that a ZM is a relation which is dual to a true vector bundle morphism (see Theorem A.3). ZMs will be often presented in a form of diagram (A.2). The Zakrzewski morphism κ turned out to be very useful in defining geometric objects in the context of mechanics on algebroids ([10, 11, 28]). It will play a crucial role also in our paper. Let us recall the geometric construction of κ. See also [28, 33] where a closely related map called the canonical involution associated with a Lie algebroid is introduced by means of an exterior differential. A skew-symmetric bracket [·, ·] on Sec(E) satisfying the Leibniz rule gives rise to a linear bi-vector field Λ on the total space E∗ of the dual bundle to τ. This correspondence is completely analogous to one between a Lie algebroid structure on τ : E → M and a linear Poisson tensor on τ ∗ : E∗ → M. In local coordinates a ∗ a i (x , ξi) on E , dual to (x , y ) on E, we have ([17, 18])

1 k a Λ(x, ξ) = c (x) ξ ∂ ∧ ∂ + ρ (x) ∂ ∧ ∂ a . 2 ij k ξi ξj i ξi x Let us interpret Λ as a map Λ:T˜ ∗E∗ → TE∗. Let R :T∗E → T∗E∗ denote the canonical anti-symplectomorphism ([7, 11, 18, 29]) called Tulczyjew-Dufour or Mackenzie-Xu transformation [40]. We deﬁne ε :T∗E → TE∗ as the composition ε := Λ˜ ◦ R. It is a vector bundle morphism over ρ : E → TM: ε T∗E / TE∗ .

∗ ∗ τE Tτ ρ E / TM a i ∗ a a ˙ ∗ In local coordinates (x , y , pb, πj) on T E and (x , ξi, x˙ , ξi) on TE the morphism ε reads as a i a a a i ˙ k i a ε(x , y , pb, πj) = (x , ξi = πi, x˙ = ρi (x)y , ξj = cij(x)y πk + ρj (x)pa). The relation κ is a Zakrzewski morphism deﬁned as dual to ε: κ TE lr TE .

τE T τ ρ E / TM

That is, vectors X,Y ∈ TE are κ-related if and only if ρ(τE(Y )) = (Tτ)X and hε(ω),Xi = hω, Y i (2.5) Tτ τE for every ω ∈ T∗E such that τ ∗ (ω) = τ (Y ). Here h·, ·i is the canonical pairing E E τE between the tangent and cotangent bundles of E and h·, ·iTτ is the tangent lift of the MODELS FOR HIGHER ALGEBROIDS 327 canonical pairing between E and E∗ (see Eq. (2.2)). The linear map whose graph −1 −1 −1 −1 is κ ∩ (Tτ) ({ρ(y)}) × τE ({y}) is denoted by κy : (Tτ) ({ρ(y)}) → τE ({y}) for y ∈ E (cf. AppendixA). Formula (2.5) enables us to calculate the local form of κ. Consider namely a i a i a i a i a i X ∼ (x , y , x˙ , y˙ ), Y ∼ (x , y , x˙ , y˙ ) and ω ∼ (x , y , pb, πj) as above. According to (2.2), hω, Y i =x ˙ ap +y ˙jπ , while τE a j hε(ω),Xi = ξ y˙j + ξ˙ yj = π y˙j + ck (x)yiπ yj + ρa(x)p yj. T τM j j j ij k j a By (2.5), we ﬁnd that Y ∈ κ(X) if and only if a a a a j a a j k k k i j x = x , x˙ = ρj (x) y , x˙ = ρj (x) y , y˙ =y ˙ + cij(x) y y . (2.6) a We observe that κ fully encodes the AL-algebroid structure functions ρi (x) and k cij(x). Hence it makes sense to think about an AL algebroid as a ZM κ satisfying some properties. We will explore this point of view in the forthcoming publication [24]. For this moment let us only state the following simple, yet not commonly- known, relation between the ZM κ and the algebroid bracket [·, ·]. Proposition 2.2. Given two sections a, b : M → E of τ their bracket at x ∈ M is equal to the vector κa(x)(B) − A, where A = Ta(ρ(b(x))) ∈ Ta(x)E; B = Tb(ρ(a(x))) ∈ Tb(x)E: [a, b](x) = κa(x)(B) − A. (2.7)

Note that both κa(x)(B),A ∈ Ta(x)E project to the same vector ρ(b(x)) ∈ TxM, hence κa(x)(B) − A is vertical, and thus it can be considered as an element of E.) The above characterization can be checked directly in local coordinates. For the proof in the special case when E is the tangent bundle algebroid the reader may consult [27]. For brevity, we shall call a pair (E, κ) consisting of a vector bundle τ : E → M and a ZM κ :TE →B TE over a map ρ : E → TM a (skew, AL or Lie) algebroid if the derived bracket on sections of τ and the anchor map ρ satisfy appropriate axioms. To end this part let us state some properties of the ZM κ which will be important in our considerations. Proposition 2.3. ZM κ described above has the following properties: (a) κ is symmetric: Y ∈ κ(X) if and only if X ∈ κ(Y ), i.e., κT = κ. (b) Axiom (b) in Deﬁnition 2.1 is equivalent to the commutativity of the following diagram: κ TE ,2TE . (2.8)

T ρ Tρ

κM TTM / TTM (c) The graph of κ is a vector subbundle of (TE × TE, τE × Tτ, E × TM) as well, hence κ is a linear relation with respect to both vector bundle structures on TE × TE. It follows that κ TE ,2TE (2.9)

τE T τ ρ E / TM 328 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

is commutative, i.e., Tτ ◦ κ = ρ ◦ τE as the composition of relations. (d) An element A ∈ TE is κ-invariant (i.e., (A, A) ∈ κ ⊂ TE × TE) if and only if

Tτ(A) = ρ ◦ τE(A). (2.10) Proof. Properties (a), (c) and (d) can be easily checked in local coordinates using formula (2.6). Property (b) means that κM (Tρ(X)) = Tρ(Y ) whenever Y ∈ κ(X). For proof consider X ∼ (xa, yi, x˙ b, y˙j) and Y ∼ (xa, yi, x˙ b, y˙j) such that Y ∈ κ(X). Then a i a a i a κM (Tρ(X)) = Tρ(Y ) means that ρi (x)y =x ˙ , ρi (x)y =x ˙ and ∂ρa(x) ∂ρa(x) i yix˙ b + ρi (x)y ˙i = i yix˙ b + ρi (x)y ˙i, ∂xb a ∂xb a what simpliﬁes to (2.4) after substituting forx ˙ a,x ˙ a andy ˙i respective formulas from (2.6).

3. Reductions of Lie groupoids. In this section we study reductions of higher tangent bundles TkG of a Lie groupoid G. A straightforward generalization of the canonical reduction of the tangent bundle TG of a Lie groupoid G to the total bundle of the associated Lie algebroid A(G), leads to the construction (from the higher tangent bundle TkG) of a graded bundle Ak(G). Since, as it will follow from Lemma 3.4, the canonical relation κ :TA(G) →B TA(G) constituting the Lie algebroid structure on the reduced bundle A(G) (cf. Proposition 2.2) can be obtained as the reduction of the canonical ﬂip κG : TTG → TTG (cf. also [13]), it k k is natural to consider reductions of its higher-order analog κk,G :T TG → TT G. k 1 k In this way we obtain a canonical relation κk :T A (G) →B TA (G). The pair k th (A (G), κk) constitutes the k -order Lie algebroid of the Lie groupoid G.

The Lie algebroid of a Lie groupoid [20, 30]. Consider a Lie groupoid G with source and target maps α, β : G → M, identity section M ⊂ G and partial multiplication (h, g) 7→ hg; G ∗ G = {(h, g): α(h) = β(g)} → G. Manifold G is foliated by α-fibers Gx = {g ∈ G : α(g) = x} where x ∈ M. We will refer to objects associated with this foliation by adding a superscript α to a proper object. α In particular, g(t) ∈ G will denote a curve g(t) lying in a single α-leaf Gx ⊂ G. The distribution tangent to the leaves of this foliation will be denoted by TGα. α Note that TG has a vector bundle structure over G as a subbundle of τG :TG → G consisting of those elements of TG which belong to TGx for some α-leaf Gx. In a similar manner we can consider subbundles TkGα ⊂ TkG,T(k)Gα ⊂ T(k)G, etc. (in general, TAGα ⊂ TAG for a Weil functor TA), which consist of all higher (iterated) k α velocities tangent to some α-leaf Gx ⊂ G. Note the difference between TT G – the k k α union of all spaces TT Gx for all α-leaves Gx – and T T G – the tangent space to TkGα. Clearly TTkGα ⊂ T(TkGα), TkTGα ⊂ Tk(TGα) but, in general, there is no equality (calculate the dimensions of the corresponding bundles). Consider now the right action of G on itself Rg : h 7→ hg. This action maps α α α-fibers to α-fibers, and hence TRg preserves TG . A vector field X ∈ Sec(TG ) is said to be a right-invariant vector field (RIVF) on G if TRg(X(h)) = X(hg). Note that every RIVF on G is uniquely determined by its values along the identity section M ⊂ G, i.e., X(g) = Rg(X(β(g))). Therefore it is natural to consider MODELS FOR HIGHER ALGEBROIDS 329

α S α A(G) = TM G = x∈M TxG , which has a structure of a vector bundle over M as a α subbundle of τG TGα :TG → G. We will denote the projection τG A(G) : A(G) → M by τ. Sections of τ can be canonically identiﬁed with RIVFs on G. The bracket of RIVFs induces a bracket on Sec(τ), which provides A(G) with the structure of a Lie algebroid. It is commonly known as the Lie algebroid of the Lie groupoid G. α α Observe that maps TRg−1 :TgG → Tβ(g)G , considered point-wise for all g ∈ G, give rise to a vector bundle map (the reduction map)

R1 TGα / A(G) (3.1)

τG τ TGα β G / M, which is a ﬁber-wise isomorphism. It plays a role analogous to the role of the map −1 TG → g := TeG given byg ˙ 7→ gg˙ in the theory of Lie groups and Lie algebras.

Reduction of TkGα. In a similar manner the higher tangent lift of the action k k α k α k α T Rg preserves T G and hence the collection of maps T Rg−1 :TgG → Tβ(g)G , considered point-wise for all g ∈ G, give rise to the map of graded bundles (the kth reduction map)

Rk TkGα / Ak(G) (3.2)

τ k τ k G TkGα β G / M, which is also a ﬁber-wise isomorphism. Here k k α A (G) := TM G k k k with τ := τG Ak(G) : A (G) → M has a natural structure of a graded bundle as a k k α 1 1 subbundle of τG TkGα :T G → G. Clearly τ : A (G) → M is just τ : A(G) → M deﬁned before. Remark 3.1. It is worth mentioning that the construction of graded bundles Ak(G) and related reduction maps Rk can be generalized in the frame of the theory of Weil functors [26, 27, 41]. Namely, for a Weil functor F = TA one can consider F A α A α A A −1 maps Rg : Tg G → Tβ(g)G given by t (γ) 7→ t (γ · g ) for any representative k α A A α F γ : R → G of an element t (γ) ∈ T G such that γ(0) = g ∈ G. Maps Rg considered point-wise for all g ∈ G give rise to a bundle map

F α R α F F G / FM G := A (G)

τ F F G TF Gα τ β G / M,

k k which is also a ﬁber-wise isomorphism. In particular, AT (G) = Ak(G) and RT = Rk.

Higher algebroids of a Lie groupoid. It turns out that bundles Ak(G) are 330 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ naturally equipped with an additional geometric structure hidden in a relation κk deﬁned in a natural way as follows.

k k Definition 3.2. Define a relation κk :T A(G) →B TA (G) as the reduction of the canonical flip κk,G:

κk,G TkTGα TkTGα / TTkGα (3.3)

TkR1 TRk TkTGα TTkGα κ TkA(G) k ______,2TAk(G). The dashed line in this diagram should be understood in the following sense: two k k elements in T A(G) and TA (G) are κk-related if and only if they are projections of k α k α k α some κk,G-related elements in T TG and TT G . Note that κk,G maps T TG ⊂ Tk(TGα) to TTkGα ⊂ T(TkGα). Note that the vertical arrows need not to be projections (think of G = M × G, where G is a Lie group). Deﬁnition 3.3. For a Lie groupoid G, bundle τ k : Ak(G) → M, together with the k k th relation κk :T A(G) →B TA (G), will be called a higher (k -order) Lie algebroid of the Lie groupoid G.

1 Note that the bundle τ = τ1 : A(G) = A (G) → M is naturally equipped with two, a priori different, algebroid structures. On the one hand we have the relation 1 1 κ1 :TA (G) →B TA (G) defined as the reduction of the form (3.3). On the other hand, the natural Lie algebroid structure (τ, [·, ·], ρ) induced by the Lie algebra of RIVFs on G can be encoded by a differential relation κ :TA(G) → TA(G) related with the Lie bracket [·, ·] by Proposition 2.2. It turns out that both structures coincide, i.e. κ1 = κ, and therefore it is natural to use the simplified notation 1 (A (G), κ1) =: (A(G), κ). st 1 Lemma 3.4. The 1 -order algebroid (A (G), κ1) of G coincides with the Lie algebroid (A(G), κ) induced by the Lie algebra of RIVFs on G. Proof. We need to show that two vectors in TA(G) are κ-related (in the sense of Proposition 2.2, where [·, ·] is the Lie bracket induced by RIVFs on G) if and only if they are κ1-related, that is if they are projections of some κG-related elements in TTGα (cf. Definition 3.2). For the “only if” part, take any κ-related vectors A ∈ TaA(G) and B ∈ TbA(G) at x ∈ M. Then Tτ(A) = ρ(b) and Tτ(B) = ρ(a), hence we can extend a and b to local sections of A(G) in such a way that A = Ta(ρ(b)) and B = Tb(ρ(a)). By Proposition 2.2 the algebroid bracket of sections a and b vanishes at x: [a, b](x) = 0. By the construction of τ : A(G) → M, local sections a and b can be lifted to local α −1 right-invariant vector fields ea,eb ∈ Sec(TG ) around some g ∈ β (x) such that 1 1 1 a = R (ea), b = R (eb) and [a, b] = R ([ea,eb]). 1 Since R is a fiber-wise isomorphism, [ea,eb](g) = 0. Consequently, vectors Ae := Tea(eb(g)) and Teb(ea(g)) are κG-related. This follows from Proposition 2.2 for the 1 standard tangent Lie algebroid structure on τG :TG → G. Clearly A = TR (Ae) 1 and B = TR (Be) and thus A and B are κ1-related. MODELS FOR HIGHER ALGEBROIDS 331

The “if” part is easier. We take any κ -related vectors A ∈ T TGα and B ∈ G e ea e T TGα at g ∈ G and extend a and b to local right-invariant vector fields on G eb e e (commuting at g, by Proposition 2.2 for the standard tangent Lie algebroid structure 1 on τG :TG → G). Since R is a fiber-wise isomorphism which maps the Lie bracket 1 of right-invariant vector fields on G to the algebroid bracket on A(G), sections R (ea) and R1(eb) commute at β(g) and hence (again by Proposition 2.2 this time for the Lie algebroid structure (A(G), κ) induced by the RIVFs on G) vectors TR1(Ae) and TR1(Be) are κ-related.

At this point we suggest the Reader to verify that if G is a Lie group then g κ1 :Tg →B Tg coincides with (2.6) or κ described in (2.7) where g is the Lie algebra of G. We point that to be consistent with the deﬁnitions of the Lie bracket on g inherited from RIVFs and the Lie bracket deﬁned by the adjoint action one d should use formula [X,Y ] = − dt |t=0 Ad(Exp tX)(Y ) as in [30] Proposition 3.7.1. k Higher Lie algebroids (A (G), κk) have many interesting properties: bundles Ak(G) are naturally graded, and the natural projection Ak(G) → Ak−1(G) relates κk with κk−1 ; the relation κk is naturally a Zakrzewski morphism; one can con- k k struct natural generalization of the algebroid anchor map ρk : A (G) → T M, etc. We postpone the discussion of these structures and their properties until the next Section4, where they will be considered in a more general abstract context (cf. Theorem 4.5).

The inductive deﬁnition of a higher algebroid of a Lie groupoid. By its k construction, the whole information about the higher Lie algebroid (A (G), κk) is contained in the structure of the Lie groupoid G and in natural properties of higher k k tangent functors T and their iterations. In fact, to reconstruct (A (G), κk) the explicit knowledge of G is not necessary. Actually, all we need is just the 1st-order Lie algebroid (A(G), κ). This observation should be clear in the context the well-known results about the integrability of Lie algebroids [6]: the Lie algebroid (A(G), κ) integrates to the associated Lie groupoid G (actually to its α-simply connected k covering Ge), and thus knowing (A(G), κ) we know G and can build (A (G), κk). It is, however, natural to look for a simpler construction, which does not refer to the quite complicated integrability theory. Below we state such a result, describing th k (in an inductive manner) a direct construction of the k -order algebroid (A (G), κk) from (A(G), κ). Since the justiﬁcation of this result is not essential at this point, we postpone the proof until Section5.

Theorem 3.5. For k = 1 the pair (A(G), κ) is the standard Lie algebroid of the k Lie groupoid G. For higher k’s the pairs (A (G), κk) can be inductively constructed as follows. • For k = 2 we may characterize the total space of the bundle A2(G) as the space of κ-invariant elements in TA(G). • For k ≥ 3 the total space of the bundle Ak(G) can be deﬁned as

Ak(G) := TAk−1(G) ∩ Tk−1A(G), (3.4)

where the spaces on the right hand-side are regarded as subsets of TTk−2A(G). • For k ≥ 2, κk can be deﬁned inductively as k+1 k+1 κk+1 := κAk(G) ◦ Tκk ∩ T A(G) × TA (G) : 332 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

Tκ κAk(G) TTkA(G) k ,2TTAk(G) / TTAk(G) (3.5) O O

? κk+1 ? Tk+1A(G) ______,2TAk+1(G).

Motivating Example 1.1 revisited – higher Atiyah algebroids. Below we discuss natural examples of such objects for G being an Atiyah groupoid. Further examples, including a Lie group and the action groupoid, are discussed in Section6. k Now we shall interpret pairs (T P/G, κk) introduced in Example 1.1 as higher Lie algebroids of a Lie groupoid in the sense of Deﬁnition 3.3. Recall the notation used in Example 1.1. With the G-bundle p : P → M one can canonically associate a Lie groupoid ΓP = (P × P )/G with base M called the Atiyah groupoid of p. The source and target maps are deﬁned by α([(y, x)]) = [x], β([(y, x)]) = [y] and multiplication is simply [(z, y)] · [(y, x)] = [(z, x)], where [(y, x)] denotes the G-orbit of (y, x) ∈ P × P and [x] the G-orbit of x ∈ P . k Our goal is to describe higher algebroids (A (ΓP ), κk) associated with ΓP . We will call these objects higher Atiyah algebroids.

Observe that a curve in a fixed α-fiber (ΓP )[x] is of the form [(γ(t), x)] for some curve γ(t) ∈ P . Curves starting at a base point [(x, x)] ≈ [x] satisfy ad- k k α ditionally γ(0) = x. Hence every element of A (ΓP ) = T ΓP M is of the form tk[(γ(t), x)], where γ(t) ∈ P is a curve starting at x. One easily checks that the k k map tt=0[(γ(t), x)] 7→ [t γ] is well-defined, depends on the class [x] ∈ M, and defines the canonical isomorphism

k k α k A (ΓP ) = T ΓP M ' T P/G. (3.6)

k k Hence we may identify A (ΓP ) ' T P/G. k k k k k The anchor map on T P/G is simply ρk :T P/G → T M;[t γ(t)] 7→ t [γ(t)]. k k k To describe the relation κk :T A(ΓP ) →B TA (ΓP ) observe first that T A(ΓP ) ≈ k k k k T TP/T G and TA (ΓP ) ≈ TT P/TG, where the quotients are taken with respect k k k k k to natural Lie groups actions T rP,1 :T TP × T G → T TP and TrP,k : TT P × TG → TTkP (recall that for any Weil algebra A, the structure of a group on TAG is obtained by applying the Weil functor TA to the structure maps of the group G (see [26])). We claim that the relation κk is defined by means of the following diagram (cf. Definition A.7):

κk,P TkTP / TTkP (3.7)

k k k κk k k T (TP/G) T TP/T G ______,2TT P/TG T T P/G , where the down-pointing arrows are just natural reduction maps. To see this, A A α A observe that for any Weil algebra A, the reduction map R :T ΓP → T P/G = A A A (ΓP ) (cf. Rem. 3.1) factorizes through T P after extending the domain by P MODELS FOR HIGHER ALGEBROIDS 333 over M, i.e.,

A A α A ∃Re A T ΓP ×M P / / T P ______/ T P/G , ffff3 A ffff R fffff Π1 ffff fffff fffff A α ffff T ΓP where the above maps on representatives read

A A 0 A 0 Re A A 0 t [(γ(t), x)], x / t γ (t) / [t γ(t)] = [t γ (t)] , 2 _ A eeee R eeeeee Π1 eee eeeeee eeeeee % eeeeee tA[(γ(t), x)] e for x, x0 ∈ P lying in the same G-orbit (i.e., [x] = [x0]) and where γ0(t) is the unique 0 0 curve in P such that [(γ(t), x)] = [(γ (t), x )] in ΓP . Hence diagram (3.3) deﬁning κk can be factorized as:

κ ×id k α k,ΓP P k α T TΓP ×M P / TT ΓP ×M P p NNN ppp NN Π1 ppp NNN pp Π1 NN xpp κ N' k α k k,P k k α T TΓP T TP / TT P TT ΓP . NN k 1 p NN T R pp NN TkR1 TRk pp NN e e p k NN pppTR N& κ wpp TkTP/TkG k ______,2TTkP/TG The commutativity of the upper square of the above diagram can be checked directly on representatives. Thus we have proved the following

th Proposition 3.6. The k -order Lie algebroid of the Atiyah groupoid ΓP is k (T P/G, κk), where κk is given by (3.7).

Note that the relation κk can be interpreted as a relation of the coincidence of k k k orbits. Namely classes [X] ∈ T TP/T G and [Y ] ∈ TT P/TG are κk-related if the k k k T G-orbit of X ∈ T TP and the TG-orbit of κk,P (Y ) ∈ T TP have a non-empty k k k intersection. Here T TP is equipped with a natural T TG action T TrP,0, whereas TkG and TG act as subgroups of TkTG.

4. Higher algebroids. In the previous Section3 we constructed higher Lie al- k gebroids (A (G), κk) associated with a given Lie groupoid G. We also made an k important observation (see Theorem 3.5) that the structure of (A (G), κk) is fully st 1 determined by the 1 -order algebroid (A (G), κ1). Thus it is natural to repeat the construction described in Theorem 3.5 taking any algebroid (E, κ) (not necessarily 1 (A (G), κ1)) as the starting point. In this way we will construct higher algebroids k (E , κk) associated with (E, κ). Later in Theorem 4.5 we study the fundamental properties of these objects.

Higher algebroids associated with an almost Lie algebroid. Throughout this section (τ : E → M, κ) will be an almost Lie algebroid. (Recall that κ fully 334 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ encodes the algebroid structure on E – cf. the remark before Proposition 2.2.) By ρ : E → TM we will denote the associated anchor map. Definitions 4.1– 4.3 given below are motivated directly by Theorem 3.5. Definition 4.1. Let (E, κ) be an almost Lie algebroid. From (E, κ) we define inductively the sequence of bundles τ k : Ek → M: • We take E1 := E and τ 1 = τ : E → M. • We define the total space E2 as the set of all κ-invariant elements in TE. Note that, by Proposition 2.3 (d), E2 can be characterized as 2 E := {A ∈ TE :Tτ(A) = ρ ◦ τE(A)}. (4.1) • For k ≥ 2 the total space of the bundle Ek+1 is defined by the following formula Ek+1 := TEk ∩ TkE, (4.2) where Ek is considered, by the inductive assumption, as a subset of Tk−1E, hence both TEk and TkE are understood here as subsets of TTk−1E. By the above construction, for each k greater than 1 we have two canonical inclusions ιk−1,1 : Ek ⊂ Tk−1E and ι1,k−1 : Ek ⊂ TEk−1. Using the latter we obtain τ k 1,k−1 k k−1 Ek−1 k−1 the natural projection τk−1 := τEk−1 ◦ ι : E ⊂ TE → E and, k k−1 k k consequently, we can define τ := τ ◦ τk−1 : E → M. As shall be proved later in Theorem 4.5, τ k has a natural structure of a graded bundle. Note that the inclusion ιk−1,1 : Ek ⊂ Tk−1E and ι1,k−1 : Ek ⊂ TEk−1 k−1,1 k k−1 1,k−1 k generalize canonical inclusions ιM :T M ⊂ T TM and ι :T M ⊂ TTk−1M. Usually we will write just Ek ⊂ Tk−1E and Ek ⊂ TEk−1. Similarly the k k projections τk−1 are natural generalizations of the canonical projections τk−1,M : TkM → Tk−1M. Definition 4.2. Starting with an almost Lie algebroid (E, κ), for each k we define k k inductively relations κk :T E →B TE as follows: • We take κ1 := κ. k+1 k+1 • For κ ≥ 1 we define κk+1 = (κEk ◦ Tκk) ∩ T E × TE , i.e., κk+1 is the k k restriction of the relation κek+1 := κEk ◦ Tκk ⊂ TT E × TTE to the subset Tk+1E × TEk+1:

Tκ κ k TTkE k ,2TTEk E / TTEk (4.3) O O

? κk+1 ? Tk+1E ______,2TEk+1.

Relations κk are, in fact, ZMs as will be proved later in Theorem 4.5. It is essential that we work with AL algebroids, not general skew-symmetric algebroids. Actually the constructions from Definitions 4.1 and 4.2 can be stated for an arbitrary algebroid (E, κ), yet κk will be a ZM only if we start with an almost Lie algebroid. In light of Theorem 3.5 it is natural to propose the following Definition 4.3. Let (τ : E → M, κ) be an almost Lie algebroid. The bundle Ek, k k together with the canonical relation κk :T E →B TE defined above, will be called a higher (kth-order) algebroid associated with E. Let us stress again that a the structure of a higher algebroid is not only the k k bundle τ : E → M itself, but also the canonical relation κk. MODELS FOR HIGHER ALGEBROIDS 335

Example 4.4. At this moment it is instructive to write down the formula for κ2 in coordinates. This calculation will show explicitly that to guarantee that κ2 is a ZM, we need the assumption that the initial algebroid (E, κ) is almost Lie. a k Let ρi (x), cij(x) be local structure functions of an AL algebroid E as in Subsec- tion 2.2. It follows immediately from (4.1) that

2 a i a i a i E = {(x , y , x˙ , y˙ ) | x˙ = ρa(x)yi} ⊂ TE in standard coordinates (xa, yi, x˙ a, y˙i) on TE, so (xa, yi, y˙i) can be taken as coor- 2 dinates on E . Let us see that the ZM r := κE ◦ Tκ restricts fine (cf. Remark A.6) 2 2 2 2 2 2 to a ZM from T τ :T E → T M to τE2 :TE → E . Take b ∈ E ⊂ TE, a i i a a i 2 b ∼ (x , y , y˙ ), sox ˙ (b) = ρi (x)y . Let a = Tρ(b) ∈ TTM. Obviously, a ∈ T M, ∂ρa(x) ∂ρa(x) a ∼ (xa = xa, x˙ a = ρa(x)yi, xä = i x˙ byi + ρa(x), y˙i = i ρb(x)yiyj + ρa(x)y ˙i) i ∂xb i ∂xb j i (4.4) Take any A ∈ T2E such that (T2ρ)A = a, A ∼ (xa = xa(a), x˙ a =x ˙ a(a), xä = a i i i 0 0 x¨ (a), y , y˙ , y¨ ). Let B := rb(A) and B = κE(B), so (A, B ) ∈ Tκ. To calculate B we differentiate equations (2.6) for κ and get

a a 0 a i dx (B) =x ˙ (B ) = ρi (x)y (A), (4.5) k k 0 k k i j dy (B) =y ˙ (B ) =y ˙ + cij(x)y y , (4.6) ∂ρa(x) ∂ρa(x) dx˙ a(B) = dx˙ a(B0) = i dxb yi + ρa(x)dyi = i ρbyj yi + ρa(x)y ˙i, ∂xb i ∂xb j i ∂ck (x) dy˙k(B) = dy˙k(B0) = dy˙k + ij yiyjdxa + ck (x)(yjdyi + yidyj) ∂xa ij ∂ck (x) =y ¨k + ij yiyjx˙ a + ck (x)(yjy˙i + yiy˙j). (4.7) ∂xa ij The equations on the right hold due to the fact that A ∈ T2E, so for example k k i i y¨ (A) = dy˙ (A), andy ˙ (A) = dy (A). We should check that B ∈ TE2 TE is tangent to E2 ⊂ TE at b, i.e. that B satisfies the following equation (obtained by differentiating the equation describing E2): ∂ρa(x) dx˙ a(B) = i dxb(B)yi + ρa(x)dyi(B). (4.8) ∂xb i We substitute dx˙ a(B), dxb(B) and dyi(B) with the expressions given above and find that (4.8) simplifies to (2.4) by comparing coefficients at yiyj, i.e., to the axiom of an AL algebroid. Thus κ2 is a ZM. It seems to be hardly possible to prove that κk is a ZM for any k by a direct calculation. Instead, we shall prove it in Theorem 4.5 using some geometric reasoning. Summarizing, for A ∈ T2E, B ∈ TE2, if A ∼ (xa, x˙ a, xä, yi, y˙i, yï) and B ∼ a i i a i i (x , y , y˙ , dx , dy , dy˙ ) then (A, B) ∈ κ2 if and only if the equations (4.4), (4.5), (4.6) and (4.7) are satisfied. Note that κ2 is bi-graded if we assign the following weights (00, 10, 20, 01, 11, 21) to (xa, x˙ a, xä, yi, y˙i, yï) and (xa, yi, y˙i, dxa, dyi, dy˙i).

Fundamental properties of higher algebroids. Below we explore some basic properties of higher algebroids from Deﬁnition 4.3. Most of them regard the natural graded bundle structure on τ k : Ek → M and the commutativity of various diagrams naturally associated with this bundle. In general, these results mimic 336 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

analogous properties of higher tangent bundles TkM and their proofs are more or less direct. The most important and, at the same time, most difficult to prove is the property that κk is a Zakrzewski morphism. In the higher tangent bundle case the relation k k κk is just the canonical flip κk,M :T TM → TT M which is a map, not a relation, so there is nothing to check. Let us stress that the proof is truly not-trivial as, in general, a restriction of a ZM is not necessarily a ZM. This property is, however, crucial from the point of view of potential applications. Namely, it allows to define ∗ k k ∗ a map (in fact even a vector bundle morphism) εk :T E → T E which is k k in a natural way dual to the relation κk :T E →B TE . The map εk plays a fundamental role in defining variational calculus on higher algebroids. Some details are provided in Section7.

k Theorem 4.5 (Properties of higher algebroids). Higher algebroids (E , κk) have the following properties for each k ≥ 1: (i) Ek is a graded subbundle [15] of Tk−1E → M of degree k and rank (r, . . . , r) where r = rank E. In particular, Ek is a submanifold of Tk−1E. (ii) Linear coordinates (xa, yi) on E induce canonical graded coordinates (xa, yi, yi,(1), . . . , yi,(k−1)) on Ek inherited, under Ek ⊂ Tk−1E, from the canoni- a,(α) i,(β) k−1 a cal coordinates x , y α,β=0,1,...,k−1 on T E. Coordinates x are of weight 0, whereas yi,(α) of weight α + 1. k k k−1 k k (iii) τk−1 : E → E coincides with the canonical projection E → E [k − 1] in the tower of graded bundles (2.1) associated with τ k : Ek → M. k+1 k k k−1 k+1 k (iv) τk is the restriction of τk−1,E :T E → T E to E ⊂ T E :

k τk−1,E TkE / Tk−1E O O

? τ k+1 ? Ek+1 k / Ek.

k+1 k k k−1 k+1 k (v) τk is the restriction of Tτk−1 :TE → TE to E ⊂ TE :

k Tτk−1 TEk / TEk−1 O O

? τ k+1 ? Ek+1 k / Ek.

(vi) Elements of Ek+1 can be characterized as those elements A ∈ TkE for which

k k−1 k T τ(A) = T ρ ◦ τk−1,E(A). (4.9)

(vii) Elements of Ek+1 can be characterized as those elements A ∈ TEk for which

k τEk (A) = Tτk−1(A). (4.10)

(viii) Tk−1ρ :Tk−1E → Tk−1TM maps Ek into TkM respecting the canonical gradings and, moreover, (T k−1ρ)−1(T kM) = Ek. MODELS FOR HIGHER ALGEBROIDS 337

k−1 k k (ix) The relation κk is a Zakrzewski morphism over ρk := T ρ|Ek : E → T M:

κ TkE k ,2TEk (4.11)

k τ T τ Ek

ρk TkMEo k.

k k k−1 k (x) For k ≥ 2 vector bundle morphisms τk−1,E :T E → T E and Tτk−1 : k k−1 TE → TE relate ZMs κk and κk−1 (in the sense of explained in Deﬁni- tion A.7), i.e., κ TkE k ,2TEk (4.12)

k k τk−1,E Tτk−1

κk−1 Tk−1E ,2TEk−1. k k k (xi) For every k ≥ 1 vector bundle morphisms T ρ :T E → T TM and Tρk : k k TE → TT M relate ZMs κk with the canonical ﬂip κk,M , i.e.,

κ TkE k ,2TEk (4.13)

k T ρ Tρk

κk,M TkTM / TTkM.

Proof. Let us ﬁrst prove properties (iv), (v) and (vii). Observe that τTk−1E TkE = k k k−1 τk−1,E and since E ⊂ T E, we have τTk−1E TEk = τEk :

τ k Ek k TE _ / E _

τ k−1 TTk−1E T E / Tk−1E O

k ? τk−1,E TkE / Tk−1E.

k+1 k k+1 df k+1 Since E ⊂ TE , we have τk = τEk Ek+1 = τTk−1E Ek+1 . Now since E ⊂ k k T E, the later equals τk−1,E Ek+1 . That gives (iv) k+1 k τk = τk−1,E Ek+1 . (4.14) To prove (v) consider the following commutative diagram

k−1 Tτk−2,E TTk−1E / TTk−2E O O

k ? τk−1,E ? Ek+1 ⊂ TkE / Tk−1E. It follows that

k (4.14) k−1 k (4.14) k+1 Tτk−1 Ek+1 = Tτk−2,E Ek+1 = τk−1,E Ek+1 = τk . 338 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

To check (vii) recall that elements of TkE can be characterized as those elements k−1 k−1 A ∈ TT E for which τTk−1E(A) = Tτk−2,E(A):

k−1 Tτk−2,E TkE ⊂ TTk−1E / TTk−2E. j j j τ j j Tk−1E j j j j j j j j Tk−1E j j We conclude that A ∈ TEk lies in Ek+1 = TkE ∩ TEk if and only if k−1 τTk−1E TEk (A) = Tτk−2,E TEk (A).

k−1 (4.14) k Yet, τTk−1E TEk = τEk and Tτk−2,E TEk = Tτk−1, which proves (4.10). Now we will prove inductively properties (i), (ii)(iii), (vi) and (viii). Case k = 1 2 is very simple. Indeed, property (vi)k=1 is just the description of E given by (4.1). 1 Properties (i)k=1 and (ii)k=1 hold since E = E has a canonical graded structure of degree 1 as a vector bundle over M. Finally (iii)k=1 and (viii)k=1 are trivial. Assume now that the considered properties holds for a given k. We shall prove them for k + 1. The scheme of the proof of the above-mentioned properties is presented below (an arrow represents a logical dependency).

(iii)k (ii)k (i)k (viii)k (vi)k t t tt tt tt tt tt ttt ztt ytt (iii)k+1 o (ii)k+1 o (i)k+1 (viii)k+1 (vi)k+1 Bundle TkE is a double graded bundle of degrees 1 and k over bases TkM and k E, respectively. The associated homotheties are hT τ : R × TkE → TkE – the kth tangent lift of the canonical homogeneity structure ([15]) on a vector bundle τ; and hk,E : R × TkE → TkE – the canonical homogeneity structure associated with a higher tangent bundle structure on TkE → E. Both actions commute, i.e.,

k,E Tkτ Tkτ k,E ht ◦ hs = hs ◦ ht for every t, s ∈ R, k where ht stands for h(t, ·). It follows that T E is a graded bundle of degree k + 1 (the total degree) with the homogeneity structure

k+1 k,E Tkτ Ht := ht ◦ ht . Let us now consider a map k k k k−1 k Πk := (τk−1,E, T τ):T E −→ T E ×Tk−1M T M.

Map Πk is a smooth ﬁbration with typical ﬁber Ex (r = rank E-dimensional) as can be easily seen from the coordinate description

a,(α) i,(β) a,(α) i,(β) Πk : x , y α=0,1,...,k 7−→ x , y α=0,1,...,k, , β=0,1,...,k β=0,1,...,k−1 a i a,(α) i,(β) where (x , y ) are linear coordinates on E and x , y α,β=0,...,k canonical induced coordinates on TkE. k−1 k Observe that if Πk(A) = (a, b) ∈ T E ×Tk−1M T M, then

k,E Tkτ k−1,E Tk−1τ k,M Πk ht ◦ hs (A) = ht ◦ hs (a), ht (b) MODELS FOR HIGHER ALGEBROIDS 339 and consequently k+1 k k,M Πk Ht (A) = Ht (a), ht (b) . (4.15) k−1 k k k+1 Due to (viii)k,T ρ maps E into T M and hence (4.9)k says that E is a k−1 k pullback of bundle Πk with respect to the inclusion graph T ρ Ek ⊂ E ×Tk−1M k k−1 k k+1 k T M ⊂ T E ×Tk−1M T M. We conclude that E is a subbundle of T E, so k+1 k+1 to prove (i)k+1 we need to check whether E is preserved by Ht . Consider (4.9) k+1 k k k−1 k A ∈ E ⊂ T E, i.e., A such that Πk(A) = (a, T ρ(a)) for some a ∈ E . Now (viii) k+1 (4.15) k k,M k−1 k k k−1 k Πk Ht (A) = Ht (a), ht (T ρ(a)) = Ht (a), T ρ(Ht (a)) .

k k−1 k k By (i)k, E is a graded subbundle of T E, so Ht (a) ∈ E and hence, by (4.9)k, k+1 k+1 Ht (A) ∈ E which proves (i)k+1. The coordinate description (ii)k+1 follows k+1 easily from the above characterization of E as a pullback bundle of Πk. k+1 k+1 k k k By (iv)k, we may consider τk : E → E as the restriction of τk−1,E :T E → k−1 k+1 k k+1 T E to E . The image of the projection τk−1,E of a point of E [k], thus a subset of Ek+1 ⊂ TkE, is a single point in Ek ⊂ Tk−1E. Hence we have deﬁned a canonical map Ek+1[k] → Ek intertwining the projections from Ek+1 which is an isomorphism of graded bundles what is easily seen in coordinates inherited from k+1 E and given in (ii)k+1: Ek+1 u DD uu DD uuu DD uu DD zuu D" Ek+1[k] / Ek

k+1 k+1 k Thus we have proved (iii)k+1. In particular τk : E → E is a graded bundle morphism. k+1 k To check (vi)k+1 observe that, due to (4.9)k,TE is characterized in TT E as k k k−1 k {X ∈ TT E : TT τ(X) = TT ρ ◦ Tτk−1,E(X)}. (4.16) (The tangent lift of a pullback f ∗E of a bundle σ : E → M with respect to a map f : M 0 → M is the pullback (Tf)∗(TE).) If such an X belongs to Tk+1E (and k+1 k+1 k+2 k k+1 k hence to T E ∩TE = E ), then TT τ(X) = T τ(X) and Tτk−1,E(X) = k+1 k k+1 k k+1 τk,E (X) ∈ T E, so in this case (4.16) simpliﬁes to T τ(X) = T ρ ◦ τk,E (X). This gives (vi)k+1. Finally, assuming (viii)k we get Tkρ : Ek+1 = TEk ∩ TkE → TTkM ∩ TkTM = Tk+1M. Since Tkρ is a morphism of double graded bundles

Tkρ TkE / TkTM k { k v τE { τTM v { k vv k {{ T τ v T τM {{ vv }{{ {vv E TkM TM TkM, we get k k k k+1 k k,E T τ k,TM T τM k T ρ Ht (A) = T ρ ht ◦ ht (A) = ht ◦ ht T ρ(A) . 340 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

k k,TM T τM k+1,M k+1 k Since ht ◦ ht = ht on T M ⊂ T TM and we already know that k k+1 k+1,M k T ρ(A) belongs to T M; the r.h.s. of the above equality is ht T ρ(A) , which proves that Tkρ is a morphism of graded bundles. Take now any X ∈ TkE such that Y := Tkρ(X) ∈ Tk+1M. Let us consider the following diagram:

TkM 3 Y 00 f3 6 k fff mm T τ fffff mm fffff mmm ffff mmTkτ ffk f mm M fffffT ρ m X ∈ TkE / TkTM 3 Y

k k τk−1,E τk−1,TM Tk−1ρ X ∈ Tk−1E / Tk−1TM 3 Y 0,

k 0 k 00 k 0 00 where X = τk−1,E(X), Y = τk−1,TM (Y ) and Y = T τM (Y ). Note that Y = Y k+1 k−1 k k since Y ∈ T M, hence we have T ρ ◦ τk−1,E(X) = T τ(X). Therefore, by k+1 (vi)k, X ∈ E , which ends the proof of (viii)k+1. Now we shall prove the remaining properties (ix)–(xi) making use of already proved properties (i)–(viii). Again we shall use the inductive approach. Property

(ix)k=1 holds since κ1 = κ which is a ZM over ρ1 = ρ. Similarly, (xi)k=1 is just a diagram (2.8) of an almost Lie algebroid. Property (x)k=2 (k = 2 is the lowest value in this case) shall be checked in a moment. th Assume now that on the k level the relation κk is a well-deﬁned ZM satisfying (4.12) and (4.13). We shall prove that κk+1 is a well-deﬁned ZM, checking k k (ix)k+1. Observe that κek+1 := κEk ◦ Tκk is a ZM over Tρk :TE → TT M as the composition of a ZM Tκk with a vector bundle isomorphism:

Tκ κ k TTkE k ,2TTEk E / TTEk

k τ TT τ TτEk TEk

Tρk TTkM o TEk TEk.

We are going to show that κk+1, the restriction ofκ ˜k+1 to the linear subbundle k+1 k k+1 k T τ × τEk+1 of TT τ × τTEk is a ZM. Let us ﬁx A ∈ T E and B ∈ TTE such that B ∈ κEk ◦ Tκk(A). It amounts to show (cf. Rem. A.6) that if B lies over k+1 k k+1 b := τTEk (B) ∈ E ⊂ TE then B is tangent to E . We will consider cases k + 1 = 2 and k + 1 > 2, separately. For k + 1 = 2, since E2 is described by (4.1), we get

2 TE = {X ∈ TTE : TTτ(X) = Tρ ◦ TτE(X)}. (4.17) MODELS FOR HIGHER ALGEBROIDS 341

Consider now the following diagram of the ZM κE ◦ Tκ:

Tρ κ a0 ∈ TE / TTM M / TTM 3 b0 q A ©D p8 @ q ££ © Tρ p ¡¡ q ££ © p ¡ q £ ©© p p ¡¡ q ££ κ © = p ¡¡ a ∈ TE q £ ,2TE ©© / TE 3 b ¡ O ££ O © O ¡¡TTτ £ TτE ©©TTτ ¡¡ τTE £ τTE TτE ¡ ££ ©© ¡ £ © ¡¡ £ Tκ © κE ¡ A ∈ TTE ,2TTE / TTE 3 B

TTτ TτE τTE

Tρ = TTM o TE / TE 3 b. It is the tangent lift of (2.9) in the upper left parallelogram. The other “squares” are obvious. It is a commutative diagram of relations if we take only “solid” arrows into 2 account. We have already ﬁxed A ∈ T E and B ∈ TTE such that B ∈ κE(Tκ(A)). 0 0 Let us denote: a = τTE(A), a = TτE(A), b = TτE(B), b = TTτ(B). We know that a = a0 as A ∈ T2E. Due to (4.17), it is enough to show that Tρ(b) = b0. (4.18) From the commutativity of the diagram we read that 0 0 b ∈ κ(a), b = κM (Tρ)(a ). (4.19) Now let us consider the commutative diagram (2.8):

κ a = a0 ∈ TE ,2TE 3 b Tρ Tρ κM TTM / TTM 3 b0,

(4.19) 0 0 (4.19) which proves (4.18) as {T ρ(b)} ⊂ Tρ(κ(a)) = Tρ(κ(a )) = {κM (Tρ(a ))} = {b0}. (Observe that we didn’t use explicitly the fact that b ∈ E2. By property (viii) this is, however, a necessary condition for TTτ(A) ∈ T2M ⊂ TTM.)

Having proved (ix)k=2 we can conclude (x)k=2. Indeed, consider the following commutative diagram:

Tκ κE T2E / TTE ,2TTE / TTE o ? _TE2 HH w HH ww H τTE τTE TτE w 2 HH ww 2 τ1,E H w Tτ1 H# κ {ww TE ,2TE TE.

By (ix)k=2, we know that the composition κE ◦ Tκ restricts to κ2, which gives (4.12)k=2. For the case k + 1 > 2 the reasoning is similar. Consider A ∈ Tk+1E, and k k+1 k B ∈ (κek+1)(A) ⊂ TTE lying over b = τTEk (B) ∈ E ⊂ TE . Let us denote: 0 k 0 k a = τTkE(A), a = Tτk−1,E(A), b = TτEk (B), b = TTτk−1(B). We know that a = a0 as A ∈ Tk+1E. By (4.10), vector B ∈ TTEk belongs to TEk+1 if and only if k TτEk (B) = TTτk−1(B), so using the notation above we have to check if b = b0. (4.20) 342 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

From the following diagram (which combines the tangent prolongation of κk with κEk ) Tκ κ k A ∈ TTkE k ,2TTEk E / TTEk 3 B

τ τ TkE TEk TτEk κ a ∈ TkE k ,2TEk TEk 3 b we see that b ∈ κk(a); i.e., b is an element κk-related to a which lies over b := τEk (b) = τEk ◦ TτEk (B) = τEk ◦ τTEk (B) = τEk (b):

τ k B ∈ TTEk TE / TEk 3 b

τ TτEk Ek

τ k b ∈ TEk E / Ek 3 b.

Similarly, taking the tangent lift of (4.12)k composed with κEk we get

Tκ κ k A ∈ TTkE k ,2TTEk E / TTEk 3 B

k k k Tτk−1,E TTτk−1 TTτk−1

Tκk−1 κ k−1 a0 ∈ TTk−1E ,2TTEk−1 E / TTEk−1 3 b0, which implies that 0 0 b ∈ κek(a ) = κek(a). 0 k (4.10) k Using property (vii) we get that b lies over Tτk−1(b) = τEk (b) = b ∈ E :

τ k B ∈ TTEk TE / TEk ⊃ Ek+1 3 b

k k TTτk−1 Tτk−1

τ k−1 b0 ∈ TTEk−1 TE / TEk−1 3 b.

0 k k In other words, b is an element κek-related to a ∈ T E lying over b ∈ E . By our inductive assumption κ = κ , so we have ek TkE×TEk k 0 b ∈ κk(a). 0 k We see that both b and b are κk-related to a and lie over the same element b ∈ E . k Since by our inductive assumption κk is a ZM, in the ﬁber TbE there can be at most one element κk-related to a, which implies (4.20). To prove properties (x)k+1 and (xi)k+1 one needs to apply the tangent functor k+1 to (4.12)k and (4.13)k, compose it with κEk and observe that restrictions to T E and TEk+1 project to analogous restrictions on lover levels. For example, for (x) consider the following diagram (the last square on the right is due to property (v)):

κ Tκk Ek Tk+1E / TTkE ,2TTEk / TTEk o ? _TEk+1 τ k+1 Tτ k TTτ k TTτ k Tτ k+1 k,E k−1,E k−1 k−1 k κ Tκk−1 Ek−1 TkE / TTk−1E ,2TEk−1 / TEk−1 o ? _TEk, MODELS FOR HIGHER ALGEBROIDS 343 which gives

κk+1 Tk+1E ,2TEk+1

k+1 τk,E Tτk+1k

κk+1 TkE ,2TEk. Property (xi) can be proved analogously with help of (2.3).

Another description of κk. Due to Theorem 4.5 we know that the canonical relation κk introduced in Deﬁnition 4.2 is a ZM. This deﬁnition is, however, not very useful from the practical point of view as the construction of κk requires a consecutive construction of all κl’s for l < k. Therefore it is desirable to simplify the description of κk and construct it more directly.

Proposition 4.6. The ZM κk can be equivalently deﬁned as: κ1 = κ; κk+1 = k k+1 k+1 κk,E ◦ T κ ∩ T E × TE :

Tkκ κk,E TkTE ,2TkTE / TTkE O O

? κk+1 ? Tk+1E ______,2TEk+1.

k k+1 Proof. We shall proceed by induction. Denote by κek :T E →B TE the relation deﬁned in the assertion. By deﬁnition, κe1 = κ1 = κ. Let us assume that κek = κk for k+1 k+1 a given k. Consequently κk+1 = (κEk ◦ Tκk) ∩ T E × TE is a restriction k−1 of the relation κEk ◦ Tκek ⊂ κTk−1E ◦ T κk−1,E ◦ T κ , i.e.,

k−1 TT κ Tκk−1,E κTk−1E TTk−1TE ,2TTk−1TE / TTTk−1E / TTTk−1E O O O

? Tκ =Tκ ? κ k ? TTkE k ek ,2TTEk E / TTEk O O

? κk+1 ? Tk+1E ______,2TEk+1.

On the other hand, since for every manifold M the canonical ﬂip κk,M is a restriction k k of the relation κTk−1M ◦ Tκk−1,M to T TM × TT M (see (2.3)), also the relation k−1 κek+1 can be obtained as a restriction of the relation κTk−1E ◦ Tκk−1,E ◦ TT κ:

k−1 TT κ Tκk−1,E κTk−1E TTk−1TE ,2TTk−1TE / TTTk−1E / TTTk−1E O O O

? Tkκ ? κk,E ? TkTE ,2TkTE / TTkE O O

? κk+1 ? Tk+1E e ______,2TEk+1. 344 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

To prove that κk+1 = κek+1 we need just to check that both restrictions coincide. That is, we need to check that the two diagrams below commute:

TTk−1TE o ? _TkTE and TTTk−1E o ? _TTkE O O O O

? ? ? ? TTkE o ? _Tk+1E TTEk o ? _TEk+1. The ﬁrst diagram is easy to check. The second is just the tangent lift of

TTk−1E o ? _TkE O O

? ? TTk−1E ∩ TTEk−1 = TEk o ? _Ek+1 = TkE ∩ TEk, where all inclusions are natural. Obviously, two possible passages from bottom-right to top-left in this diagram are just two equivalent ways of expressing the canonical inclusion TkE ⊂ TTk−1E restricted to Ek+1.

5. Integrable higher algebroids. In this section we want to relate the results of Sections3 and4. That is, our main goal will be to show that the construction k of a higher algebroid of a Lie groupoid (A (G), κk) obtained as a natural reduction k k of T G coincides with the inductive construction of (E , κk) from the standard algebroid (E, κ) provided that we put (E, κ) = (A(G), κ). In light of Deﬁnitions 4.1 and 4.3, this fact is an immediate consequence of the assertion of Theorem k 3.5. Since for (E , κk) the assertion of Theorem 4.5 holds, we deduce that also k higher algebroids (A (G), κk) are subject to Theorem 4.5. In particular, the relation k k κk :T A(G) →B TA (G) is a ZM. The remaining part of this section will be devoted to giving the proof of Theorem 3.5. We shall need the following technical results. Lemma 5.1. The kth tangent lift of the reduction map (3.1) as well as its restriction to TkTGα ⊂ Tk(TGα),

k 1 T R |TkTGα TkTGα / TkA(G) (5.1)

k k T τG |TkTGα T τ k T β|TkGα TkGα / TkM, are ﬁber-wise vector bundle isomorphisms. More generally, for a Weil algebra A k A,k A k the reduction maps R give rise to morphisms R := T R |TATkGα of graded bundles of order k RA,k TATkGα / TAAk(G) (5.2)

A k A k T τG |TATkGα T τ A T β|TAGα TAGα / TAM which are ﬁber-wise isomorphisms. Moreover, if A ⊂ B is an inclusion of Weil algebras then A,k B,k R = R |TATkGα . (5.3) MODELS FOR HIGHER ALGEBROIDS 345

Proof. First of all, the kth tangent lift of a fiber-wise vector bundle isomorphism is a fiber-wise vector bundle isomorphism. A particular case is TkR1 :Tk(TGα) → TkA(G) covering Tkβ :TkG → TkM. Clearly, the submanifold TkTGα is invariant k k α k with respect to the homotheties of the vector bundle T τG :T (TG ) → T G, k k α k k α hence T τG|TkTGα is a vector bundle projection onto T G = T τG(T TG ), in k 1 light of Theorem 2.3 [16]. Therefore, the restriction T R |TkTGα is at least fiber- wise injective, but the dimensions of fibers of the corresponding bundles are the same and equal (k + 1) times the rank of τ : A(G) → M. Hence, the assertion on (5.1) follows. A similar reasoning works for RA,k. We use [15] (Thm 4.2 and results of Sec. 5) to prove that TA is a functor in the category of graded bundles. An easy calculation A k shows that the dimension of a fiber of the graded subbundle T τG |TATkGα is dim A· rank τ k and thus coincides with the dimension of the fibers of TAτ k. Therefore, RA,k has to be a fiber-wise graded bundle isomorphism. The equality (5.3) is clear, since for any smooth map f : M → N we have B A T f|TAM = T f.

Lemma 5.2. For k ≥ 2 the bundles Ak(G) are subject to natural inclusions

1,k−1 k k−1 1,k−1 k 1 k−1 −1 ι : A (G) ,→ TA (G), ι (t0 γ) := tt=0ts=0 γ(t + s)γ(t) , (5.4) and

k−1,1 k k−1 k−1,1 k k−1 1 −1 ι : A (G) ,→ T A(G), ι (t0 γ) := tt=0 ts=0γ(t + s)γ(t) (5.5) where γ(t) ∈ Gα represents an element of Ak(G) ⊂ TkGα. Moreover, the following diagram involving ι1,k−1 and ιk−1,1 is commutative

ι1,k−1 Ak(G) / TAk−1(G) (5.6)

ιk−1,1 Tιk−2,1 1,k−2 ιA(G) Tk−1A(G) / TTk−2A(G) .

Proof. We can deﬁne ι1,k−1 and ιk−1,1, respectively, as the following composition of maps

k−1 Ak(G) ⊂ TkGα ⊂ TTk−1Gα ⊂ T(Tk−1Gα) −−−−→TR TAk−1(G) and

k−1 1 Ak(G) ⊂ TkGα ⊂ Tk−1TGα ⊂ Tk−1(TGα) −−−−−→T R Tk−1A(G) .

Obviously, ι1,k−1 and ιk−1,1 are well-deﬁned maps, yet it is not clear that they are indeed injective. We will prove this fact inductively starting from ι1,1 : A2(G) → TA(G). Consider the following diagram, in which the top row presents ι1,1 and the bottom row is the identity A1(G) = A(G):

TR1 A2(G) / T2Gα / TTGα / T(TGα) / TA(G)

2 2 τ α τ α τA(G) τ1 τ1,G TG TG = = R1 A1(G) / T1Gα / TGα / TGα / A(G) , 346 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

Take now any X,Y ∈ A2(G) such that ι1,1(X) = ι1,1(Y ). Note that the projections α α α 2 α α TτG : T(TG ) → TG and τTGα : T(TG ) → TG coincide on T G ⊂ TTG ⊂ T(TGα), hence we may write 1,1 1,1 TτG(X) = τTGα (X) = τA(G)(ι (X)) = τA(G)(ι (Y )) = τTGα (Y ) = TτG(Y ) We conclude that X and Y lie in the same ﬁber of T(TGα) → TG and that their images under TR1 are equal. Thus X = Y as, by Lemma 5.1, (TR1, Tβ) is a ﬁber-wise isomorphism. Consequently ι1,1 is injective. Assume now that ιk−1,1 is an inclusion. We would like to show that so is ιk,1. The general scheme of the proof will be very similar to the one provided for the injectivity of ι1,1. Namely, consider the following diagram in which the top and bottom rows presents ιk,1 and ιk−1,1, respectively:

TkR1 Ak+1(G) / Tk+1Gα / TkTGα / Tk(TGα) / TkA(G)

k+1 τ k+1 τ k τ k τ k τk k,G k−1,TGα k−1,TGα k−1,A(G)

Tk−1R1 Ak(G) / TkGα / Tk−1TGα / Tk−1(TGα) / Tk−1A(G) , k+1 k,1 k,1 k Take now X,Y ∈ A (G) such that ι (X) = ι (Y ). Since the projections T τG : k α k k k α k−1 α k+1 α T (TG ) → T G and τk−1,TGα :T (TG ) → T (TG ) coincide on T G ⊂ TkTGα ⊂ Tk(TGα), the elements X,Y must belong to the same ﬁber of Tk(TGα) → TkG. Thus, by Lemma 5.1, they must be equal. We conclude that ιk,1 is an inclusion. An analogous reasoning shows that ι1,k−1 is injective. It is based on a presentation of ι1,k−1 as a composition

k−1 Ak(G) ⊂ TkGα ⊂ TTk−1Gα ⊂ T(Tk−1Gα) −−−−→TR TAk−1(G) The details are left to the Reader. Finally, the commutativity of (5.6) can be checked directly on representatives: k−2,1 1,k−1 k 1 k−2,1 k−1 −1 Tι ◦ ι (t0 γ) = tt=0ι (ts=0 γ(t + s)γ(t) ) | {z } δ(t,s) 1 k−2 1 −1 = tt=0ts=0 tu=0δ(t, s + u)δ(t, s) , 1,k−2 k−1,1 k 1,k−2 k−1 1 1 k−2 1 ιA(G) ◦ ι (t0 γ) = ιA(G) tt=0 tu=0δ(t, u) = tt=0ts=0 tu=0δ(t + s, u), and δ(t, s+u)δ(t, s)−1 = γ(t+s+u)γ(t)−1(γ(t+s)γ(t)−1)−1 = γ(t+s+u)γ(t+s)−1 = δ(t + s, u). Now we are ﬁnally ready to prove Theorem 3.5.

k Proof of Theorem 3.5. Our goal is to show that higher algebroids (A (G), κk) defined as the reductions of TkG (Definitions 3.2 and 3.3) coincide with the ones defined inductively as in the assertion of our theorem. To distinguish these two, a k priori different, objects, denote by (A (G), κk) higher algebroids defined inductively k by (3.4) and (3.5). By the results of Section4,( A (G), κk) is subject to Theorem 4.5. k k We would like to show that (A (G), κk) = (A (G), κk) for each k. Our reasoning will 1 1 be inductive. Note that the first step, i.e. the equality (A (G), κ1) = (A (G), κ1), is guaranteed by the results of Lemma 3.4. k At first we shall concentrate our attention on proving that Ak(G) = A (G). k Observe that both Ak(G) and A (G) are naturally graded bundles of the same MODELS FOR HIGHER ALGEBROIDS 347 degree k and the same rank (r, . . . , r) where r is the rank of A(G). Indeed the graded structure on Ak(G) is induced from the natural structure on TkGα, while k the existence of the graded structure on A (G) is guaranteed by Theorem 4.5. In k fact it will be enough to show that there is an inclusion Ak(G) ⊂ A (G) covering the identity on the base M and respecting he graded structures. Indeed, the equality of ranks of these two graded bundles implies that the inclusion must be an isomorphism on each fiber, and thus, since the bases are respected, the isomorphism of graded bundles. For k = 2 consider any element X ∈ A2(G). By definition it is an R2-projection 2 α 2 α of some element Xe ∈ T G . Now T G can be characterized as the set of κG- α α invariant elements in TTG . Thus Xe regarded as an element of TTG is κG- 2 1 invariant. By (3.3)k=1 we conclude that X = R (Xe) = TR (Xe) ∈ TA(G) (note that R2 is a natural restriction of TR1, and that we use here the natural inclusion 2 2 A (G) ⊂ TA(G)) is κ1-related to itself, i.e., it belongs to A (G) ⊂ TA(G). We 2 conclude that A2(G) ⊂ A (G). k Assuming that Ak(G) = A (G) for some k ≥ 2, by the results of Lemma 5.2 we have k k+1 Ak+1(G) ⊂ TAk(G) ∩ TkA(G) = TA (G) ∩ TkA(G) = A (G) . Note that the above inclusion respects the natural graded structures on TAk(G) k and TkA(G), and that the graded structure on A (G) is induced by the graded k k+1 structures on TA (G) and TkA(G). Thus Ak+1(G) ⊂ A (G) is an inclusion of graded bundles. By our previous considerations this implies the equality Ak+1(G) = k+1 A (G). k Knowing already that Ak(G) = A (G) allows us now to prove the equality of the relations κk and κk. Again our reasoning will be inductive: assuming that κk = κk we will prove that κk+1 = κk+1 (recall that the case initial case k = 1 is already proved). k+1 k+1 Take any (X,Y ) ∈ κk+1 ⊂ T A(G) × TA (G). By (3.3)k+1 there exist k+1 α k+1 α Xe ∈ T TG and Ye ∈ TT G which project to X and Y and are κk+1,G- related. Now using (2.3) with M = G, we know that Xe and Ye (regarded as elements k α k α of TT TG and TTT G , respectively) are related by the composition κTkG ◦Tκk,G. Consider the following diagram

Tκk,G κTkG TTkTGα / TTTkGα / TTTkGα (5.7)

TTkR1 TTRk TTRk

Tκ =Tκ κAk(G) TTkA(G) k k ______,2TTAk(G) / TTAk(G) as the composition of the tangent lift of (3.3) (actually, we take only Tκk,G restricted to the subbundle TTkTGα ⊂ T(TkTGα)) with an obviously commuting diagram. Due to Proposition A.8, any Tκk,G-related elements project to Tκk-related ones. Since the image of Xe ∈ Tk+1TGα ⊂ TTkTGα under TTkR1 is precisely X (we use the inclusion Tk+1A(G) ⊂ TTkA(G) and the fact that Tk+1R1 is a restriction of TTkR1) and that the image of Ye ∈ TTk+1Gα ⊂ TTTkGα under TTRk is Y (we use the inclusion TAk+1(G) ⊂ TTAk(G) and the fact that TRk+1 is a restriction k of TTR ), we conclude that X and Y are related by the restriction of κAk(G) ◦ Tκk 348 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

k+1 k+1 to the product T A(G) × TA (G). In other words, (X,Y ) ∈ κk+1. Thus κk+1 ⊂ κk+1. k+1 k+1 Conversely, take any pair (X,Y ) ∈ κk+1 ⊂ T A(G) × TA (G). By (3.5) elements X and Y (regarded as elements TTkA(G) and TTAk(G), respectively) are k α κAk(G) ◦Tκk = κAk(G) ◦Tκk-related. By Proposition A.8 there exist Xe ∈ T(T TG ) k α and Ye ∈ T(TT G ) which are κTkG ◦ Tκk,G-related. It is easy to check that k α k α k α k α κTkG ◦ Tκk,G ∩ T(T TG ) × T(TT G ) ⊂ TT TG × TTT G , hence Xe ∈ TTkTGα and Ye ∈ TTTkGα. We shall now apply Lemma 5.1 for the inclusion of Weil functors Tk+1 ⊂ TTk and associated lifts of reduction maps: TTkTGα o ? _Tk+1TGα fiber-wise isomorphisms Tk+1A(G) / TTkA(G) (Tk+1R1, Tk+1β) and / (TTkR1, TTkβ) TTkGα o ? _Tk+1Gα Tk+1M / TTkM Since X was in fact an element of Tk+1A(G) ⊂ TTkA(G) and the reduction maps in the diagram above are fiber-wise isomorphisms, we conclude that necessarily Xe k+1 α k α A is an element of T TG ⊂ TT TG . Similarly, we consider (5.2)k for T = TT, and find that Ye must be an element of TTk+1Gα ⊂ TTTkGα. By (2.3) for M = G, elements Xe and Ye are κk+1,G-related. Since X and Y are projections of the κk+1,G- related elements Xe and Ye, in light of (3.3)k+1, we have (X,Y ) ∈ κk+1. Thus κk+1 ⊂ κk+1 and since we have already proved the opposite inclusion, we get κk+1 = κk+1. This ends the reasoning.

6. Examples. Reductions of higher tangent bundles of Lie groupoids provide a wide spectrum of natural examples of higher algebroids. Perhaps the simplest class of groupoid-born examples are the usual higher tangent bundles Ek = TkM with k k the canonical relation κk = κk,M :T TM → TT M and trivial anchor map ρk = idTkM . In this case the corresponding groupoid is simply the pair groupoid G = M × M. Another important, and more general, class are higher Atiyah algebroids k k (A (ΓP ) = T P/G, κk) considered in Proposition 3.6. They originate from the Atiyah groupoid ΓP Clearly they are higher algebroids associated with the Atiyah 1 algebroid (A (G) = TP/G, κ1) In this section we would like to discuss in detail the higher algebroid structures related with a Lie group G with a natural G-action on the right, being an extreme example of a principal G-bundle. The corresponding Lie groupoid is simply the Lie group G itself in this case. It is well known that TkG is a Lie group whose Lie algebra is Tkg with structure maps lifted from G and g by means of the kth tangent functor. On the other hand, we consider the reduction Rk of TkG, thus a higher k k−1 Lie algebra Te G which can be canonically identiﬁed with T g as a graded space. We shall prove in Proposition 6.3 that the Lie algebra structure on Tkg and the k higher Lie algebra structures of Te G are closely related.

Higher Lie algebras. The Lie algebroid associated with a Lie group G regarded as a Lie groupoid is A(G) = TG/G =: g – with the structure of a Lie algebra of the Lie group G. By Proposition 3.6 higher algebroids associated with G are k k g A (G) = T G/G with the canonical relations κk =: κk described by means of MODELS FOR HIGHER ALGEBROIDS 349

k g (3.7) with P = G. We will refer to higher algebroids (T G/G, κk) as to higher Lie algebras. Now we are going to give a more concrete description of these objects. th k In general the k tangent space TmM to a manifold M at a point m ∈ M is only a graded space (not a vector space if k > 1) of rank (r, . . . , r) where r = dim M (see Subsection 2.1) admitting no natural splitting. The group structure of G, however, k induces the canonical decomposition of Te G into the direct sum of graded spaces of degrees 1, 2, . . . , k, as is explained in the following Proposition 6.1. The kth tangent bundle to a Lie group G has the canonical decomposition k k ⊕k T G ' G × Te G ' G × g . (6.1) Proof. Clearly, the reduction map Rk deﬁnes a trivialization of the graded bundle k k k τ :T G ' G×Te G. As the exponential map exp : g → G is a local diﬀeomorphism k k k around 0 ∈ g, the map T exp | k :T g → T G is an isomorphism of graded T0 g 0 e k spaces. Obviously, T0 g splits canonically into direct sum of copies of g as for any k k k k+1 vector space V ,T V ' V ⊗ D where D ' R[ν]/ ν is the polynomial algebra associated with the functor Tk.

g Let us recall the canonical relation κ =: κ :Tg →B Tg encoding the structure of a Lie algebra on g. Due to (3.7) it can be described as the reduction: κ TTG G / TTG

κg Tg = TTG/TG ,2TTG/TG = Tg, where the quotients on both sides are taken with respect to the same action TrG,1 : TTG×TG → TTG (we denote by rG,0 : G×G → G the standard G-action on itself and by rG,1 :TG × G → TG the induced action on TG). Note that rG,1 coincides with the action of G as a subgroup G ≈ OG(G) of TG (symbol OM denotes the inclusion of M ⊂ TM as the zero section). Therefore TrG,1 coincides with the action of the subgroup TG ≈ TOG(TG) of the group TTG. g A direct description of κ in terms of the Lie bracket [·, ·]g is the following. g Elements X = (X0,X1) and Y = (Y0,Y1) in Tg ≈ g × g are κ -related if and only if (cf. [10] and Proposition 2.2):

[X0,Y0]g = X1 − Y1. (6.2) Our next step will be the description of Tkg equipped with the structure of the Lie algebra of the group TkG. It is well-known (cf. [27]) that, for any Weil algebra A, the Lie algebra of TAG is TAg ' g ⊗ A with the Lie bracket given by

[X ⊗ a, Y ⊗ b]TAg = [X,Y ]g ⊗ ab, (6.3) for X,Y ∈ g and a, b ∈ A. The functor Tk corresponds, of course, to the polynomial k k+1 algebra D ' R[ν]/ ν . Tkg k k On the other hand, the relation κ : TT g →B TT g is deﬁned as the reduction

κ k TTTkG T G / TTTkG

k κT g TTkg = TTTkG/TTkG ,2TTTkG/TTkG = TTkg, 350 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ where the quotients are taken with respect to the action TTTkG×TTkG → TTTkG k k k of the subgroup TT G ≈ TOTkG(TT G) of the group TTT G. It turns out that k the relation κT g can be nicely described in terms of the relation Tkκg.

k Lemma 6.2. The relations κT g and Tkκg are related by the following commutative diagram:

Tkκg TkTg ,2TkTg (6.4)

κk,g κk,g

k κT g TTkg ,2TTkg.

k g k k k Proof. Note that T κ is deﬁned by the reduction of T κG :T TTG → T TTG k k k k by the action of the subgroup T TG ≈ T TOG(T TG) ⊂ T TTG. The following commutative diagram:

TkTO Tkκ TkTG G / TkTTG G / TkTTG

κk,G κk,TG κk,TG

TTkO TTkG G / TTkTG TTkTG

Tκk,G Tκk,G

TO k κ k TTkG T G / TTTkG T G / TTTkG

k k k shows that the subgroup T TG ≈ T TOG(T TG) is transformed by means of k k k Tκk,G ◦ κk,TG to the subgroup TT G ≈ TOTkG(TT G) ⊂ TTT G. Since κk,g is Tkg the reduction of Tκk,G ◦ κk,TG by the action of the mentioned subgroups and κ k k is the reduction of κTkG by the action of the subgroup TT G ≈ TOTkG(TT G), the assertion follows.

As a simple consequence we obtain the following characterization of the higher- g k k k−1 k algebroid relation κk :T A(G) = T g →B TT g = TA (G):

g Tk−1g k Proposition 6.3. The relation κk is the restriction of the relation κ to T g ⊂ TTk−1g:

TTk−1g (6.5) O RR k−1 RRRκT g RRR RRR ? κg R $, Tkg k ,2TTk−1g,

k k−1 k−1 k−1 i.e., vectors X = (X0,X1) ∈ T g ⊂ TT g ≈ T g × T g and Y = (Y0,Y1) ∈ k−1 k−1 k−1 g TT g ≈ T g × T g are κk-related if and only if

[X0,Y0]Tk−1g = X1 − Y1, (6.6)

k−1 where [·, ·]Tk−1g is the Lie bracket on T g. MODELS FOR HIGHER ALGEBROIDS 351

Proof. From Proposition 4.6 we know that κk is the following restriction of κk−1,g ◦ Tk−1κg:

Tk−1κg κk−1,g Tk−1Tg ,2Tk−1Tg / TTk−1g O

? κg Tkg k ,2TTk−1g.

Tk−1g By Lemma 6.2 the upper row in this diagram equals κ ◦ κk−1,g, and since

κk−1,g Tkg = idTkg we get (6.5). This, in turn, due to (6.2) considered for the Lie algebra Tk−1g, implies (6.6).

7. Conclusions and outlook. To close the discussion of the construction of higher algebroids presented in this paper let us point some possible applications and ex- tensions of this notion.

Geometric mechanics on higher algebroids. In fact, our initial motivation to study higher algebroids lies in Geometric Mechanics. We wanted to obtain a natural geometric structure suitable for constructing higher-order Euler-Lagrange equations for both standard system and systems reduced by inner symmetries (that is why we began our considerations from studying a groupoid reduction in Section3). Due to the limitations of space we postpone the detailed discussion of geometric mechanics (and variational calculus) on higher algebroids to a separate publication [25], yet we will give a flavor if our approach below. The interested Reader may consult the previous version of this work available as a preprint [23] or consult alternative approaches of Bruce et al. [2] or Mart´ınez[35]. In our approach the construction of geometric mechanics on a higher algebroid k (E , κk) requires the following two key ingredients: k k • The fact that κk :T E →B TE is a ZM allows us to introduce its dual ∗ k k ∗ k k εk :T E → T E being a vector bundle map over ρk : E → T M. Let us stress that εk is a true map, even though κk was just a differential relation (cf. Theorem A.3). • In our previous publication [22] we introduced a natural vector bundle map k,k ∗ ∗ k,k ∗ Υk,τ ∗ : Te E → E defined on a certain natural subbundle Te E ⊂ TkTkE∗. This map is naturally associated with the kth-order integration by parts operation on bundle E. It can be constructed regardless of the (higher) algebroid structure on E or Ek. Now given a kth-order Lagrangian, i.e. a smooth function L : Ek → R, the force k k k along a trajectory a :[t0, t1] → E (a should be admissible in a natural sense) reads as k Fak (t) := Υk,τ ∗ t ΛL(t) ,

k where ΛL(t) := εk(d L(a (t))) uses the notion of the dual map and the diﬀerential of the Lagrangian along the trajectory. Higher-order Euler-Lagrange equations describe the free movement, and hence they read as

k Υk,τ ∗ t ΛL(t) = 0 . (7.1) 352 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

Geometrically the above construction can be seen as follows

ker Υ ∗ k,τ m6 _ ∃? m m m m m k k εk m t ΛL(a ) T∗Ek / TkE∗ ______/ Tk,kE∗ H 7 e ∗ ΛL τ k ∗ Υ ∗ d L Ek T τ k,τ k a k ρk k ∗ [t0, t1] / E / T ME . To end this short explication let us just mention that Eq. (7.1) is not just a geometric construction but it has a natural variational interpretation. In fact on a k higher algebroid (E , κk) one can introduce the full framework of variational calculus, including external forces, generalized momenta and various types of boundary conditions. It covers many examples of practical importance, among the others: higher-order mechanical systems on manifolds and their reductions by inner symmetries. For details the Reader is directed toward our preprint [23].

k Abstract higher algebroids. Many basic properties of higher algebroids (E , κk) collected in Theorem 4.5 are expressible in terms of the natural graded structure on Ek. Note, however, that this graded structure is rather special – it closely mimics the natural graded structure on higher tangent bundles (e.g., the rank of Ek as a graded bundle is of a very special form (r, r . . . , r)). It is therefore natural to consider similar structures hosted on more general graded bundles. This can be a promising area of studies as the recently introduced theory of graded bundles seems to be gaining more interest [1, 15]. For example, a natural notion of a 2nd-order abstract algebroid should consist 2 2 2 1 2 of a degree-2 graded bundle σ : F → M and a bi-graded ZM κ2 :T F → TF (here F 1 is the vector bundle in the tower of graded bundles F 2 → F 1 → M 2 associated with F ). We insist that κ2 is a ZM since, as the first paragraph of 2 this section suggests, it opens possible applications in mechanics. The pair (F , κ2) should satisfy some additional axioms, e.g. the one presented in Theorem 4.5 in 1 1 diagram (4.13) such that the induced ZM κ1 :TF →B TF constitutes an algebroid structure. Locally such a ZM has a form  a a i x˙ = Qi y  xä = 1 Qa yiyj + Qa zµ  2 ij µ a 0a i κ2 : x˙ = Qi y y˙ i = Qi y˙j + Qi yjyk  j jk  µ µ i µ i j µ ν i 1 µ i j k z˙ = Qi y¨ + Qij y y˙ + Qνi z y + 2 Qij,ky y y where (xa, x˙ a, xä, yi, y˙i, yï) and (xa, yi, zµ, x˙ a, y˙ i, z˙ µ) are coordinate systems on T2F 1 and TF 2, respectively, of degrees (00, 10, 20, 01, 11, 21), respectively. The structure · functions Q··· should satisfy some equations reflecting the axioms of an abstract higher algebroid. We shall explore this subject in a forthcoming publication [24].

Appendix A. On Zakrzewski morphisms between vector bundles. In [43] S. Zakrzewski observed that if in the usual axioms for a group one replaces maps by relations, in particular if the group multiplication is replaced by a partial multiplication, what one gets is the standard notion of a groupoid. However, in this case the MODELS FOR HIGHER ALGEBROIDS 353 notion of a morphism is not the usual one. Denote by r : X →B Y a relation, i.e. a subset of X ×Y called graph of r with a specified domain X and a codomain Y . Re- lations with compatible domains and codomains can be composed. Relations form a category with respect to this composition. Therefore, it makes sense to speak about commutative diagrams in which arrows represent relations, not necessary mappings. The composition r2 ◦ r1 of relations r1 : X →B Y , r2 : Y →B Z is called simple, what we indicate by writing r2 • r1, if for any (x, z) ∈ r2 ◦ r1 there is only one T y ∈ Y such that (x, y) ∈ r1,(y, z) ∈ r2. The transposed relation r : Y →B X and product of relations are defined in an obvious way. For x ∈ X and r as above we write r(x) for {y ∈ Y :(x, y) ∈ r}. A relation r : X →B Y is smooth (in this case X and Y√are smooth manifolds) if its graph is a submanifold of X × Y . An example x 7→ 3 x, x ∈ R, shows that it can happen that a non-smooth map is smooth when considered as a relation. Definition A.1. A groupoid (in the sense of Zakrzewski, called in [43] a U ∗- algebra) is a quadruple (Γ, m, e, s), where set Γ, partial multiplication m :Γ×Γ→B Γ, unit e : {pt} →B Γ and inverse s :Γ →B Γ are such that m ◦ (m × id) = m ◦ (id ×m), m ◦ (e × id) = m ◦ (id ×e) = id, s ◦ s = id, (A.1) s ◦ m = m ◦ (s × s) ◦ ex, where ex(x, y) = (y, x) for x, y ∈ Γ, ∅= 6 m(s(x), x) ⊂ e(pt), for any x ∈ Γ.

A Zakrzewski morphism (ZM, in short) from a groupoid (Γ1, m1, e1, s1) to a groupoid (Γ2, m2, e2, s2) is a relation f :Γ1 →B Γ2 that commutes with the structure relations, i.e.,

f ◦ m1 = m2 ◦ (f × f), f ◦ s1 = s2 ◦ f, f ◦ e1 = e2. Note that the relation s has to be a mapping. Indeed, the image s(x) cannot be empty, and hence #s(x) ≤ 1, since otherwise #s ◦ s(x) > 1. The image M := e(pt) ⊂ Γ is called the base of Γ. The source and target maps α, β :Γ → M are uniquely defined by properties ([43], Lem. 2.2) m(x, α(x)) 6= ∅, m(β(x), x) 6= ∅. −1 −1 Let us denote Γx = α (x) (resp. xΓ = β (x)). It turns out that the ZM f :Γ1 →B Γ2 induces a mapping f : M2 → M1 on the bases of groupoids Γ1, Γ2 ([43], Lem. 2.5). Namely, the graph of f is the transpose of graph(f) ∩ (M1 × M2). Moreover, for any y ∈ M2 the restrictions graph(f) ∩ (Γ1,x × Γ2,y), where x = f(y), are mappings as well ([43], Lem. 3.3). We denote them by fy :Γ1,f(y) → Γ2,y. Similarly, we obtain mappings between β-fibers. We shall refer to the mapping f by saying that a ZM f covers the map f. Differential (i.e., Lie) groupoids in the standard sense can be defined using Za- krzewski’s approach. (They are called regular D∗-algebras in [44].) Lie groupoids are groupoids in the sense of Definition A.1 whose structure relations m, e and s are smooth and additionally satisfy some transversality conditions ([44], p. 375). The notion of a smooth Zakrzewski morphism ([44], pp. 375-376) requires the notion of transversality of relations. Recall that the tangent lift of a smooth relation r : X →B Y is a smooth relation Tr :TX →B TY whose graph is T graph(r). The ∗ ∗ ∗ phase lift of r is, in turn, a smooth relation T r :T X →B T Y whose graph consists 354 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

∗ ∗ of pairs (ξ, η) ∈ TxX ×TyY such that hξ, vi = hη, ui for any (v, u) ∈ T(x,y) graph(r). We say that relations r1 : X →B Y and r2 : Y →B Z have transverse composition, which is denoted by r2 t r1, if the graph of r2 ◦r1 is a smooth submanifold of X ×Z and both tangent and phase lifts of r1 and r2 have simple composition: Tr2 • Tr1, ∗ ∗ T r2 • T r1.

Deﬁnition A.2. A Zakrzewski morphism f :Γ1 →B Γ2 of Lie groupoids (Γi, mi.ei, si), i = 1, 2, is smooth if the graph of f is a smooth submanifold of Γ2 × Γ1 and, moreover,

(i) f t e1, (ii) m2 t (f × f). It turns out that by applying the tangent lift or the phase lift to the structure relations of a Lie groupoid Γ one obtains again a Lie groupoid ([44], Prop. 3.4). Moreover, Lie groupoids and smooth ZMs form a category closed with respect to taking tangent and phase lifts ([44], Prop. 4.2). A map g : R → R of the groupoid Γ = R = M given by f(x) = x3 for x ∈ R, gives a simple example of a ZM (and a smooth relation) whose tangent lift Tg is not a ZM. Indeed, by Lemma 5.2 in [43] for every (not necessary smooth) ZM f :Γ1 →B Γ2 the compositions f ◦ e1 and m2 ◦ (f × f) are always simple. One can check that Tg fails to satisfy this property at x = 0. From what was said above f is not a smooth ZM (although it is a ZM and a smooth relation). A simple but important class of groupoids is provided by vector bundles (all vector bundles we consider will be of ﬁnite rank). Any vector bundle τ : E → M gives rise to a groupoid (E, +, 0M , −), where 0M : M → E is the zero section. In case of groupoids associated with vector bundles, the notion of smooth ZM simpliﬁes considerably.

Theorem A.3. Let f : E1 →B E2 be a Zakrzewski morphism of groupoids associated with vector bundles τi : Ei → Mi, i = 1, 2 (no smoothness condition is assumed). Then the following conditions are equivalent: (a) f is a smooth ZM, ∗ (b) The map g : f (E1) → E2 deﬁned by g(y, v) = fy(v), for v ∈ E1, y ∈ M2

such that f(y) = τ1(v), is a vector bundle morphism covering idM2 , ∗ ∗ ∗ (c) The dual f : E2 → E1 is a vector bundle morphism. Proof. (a)⇒(b): By ([44], Lem. 4.1) the graph of f is a graph of a smooth section g of the projection τ × id : E × E → M × E over f ∗(E ) ⊂ M × E . We e 2 E1 2 1 2 1 1 2 1 have g = pr ◦ g, hence g is a smooth mapping. A smooth map between vector E2 e spaces respecting addition group structures is automatically linear. Therefore g is a vector bundle morphism. ∗ ∗ ∗ ∗ ∗ (b)⇔(c): Given g we ﬁnd that f is the composition of g : E2 → (f (E1)) = ∗ ∗ ∗ ∗ ∗ f (E1 ) with the canonical vector bundle morphism f (E1 ) → E1 . Conversely, dual ∗ ∗ ∗ ∗ to the canonical morphism E2 → f (E1 ) associated with f and covering idM2 is g, hence g is a vector bundle morphism. ∗ ∗ ∗ (c) ⇒(a): Clearly a vector bundle morphism h : F2 → F1 gives a ZM h : F1 →B F2 . We have to check that for h = f ∗, h∗ = f is a smooth ZM. Obviously, the graph of f ◦ e1 is M2, so a smooth submanifold of E2. Similarly, the graph of m2 ◦ (f × f) is canonically identiﬁed with the graph of m2 ◦ (g × g), which is a smooth submanifold ∗ ∗ of E2 ×M2 f (E1) ×M2 f (E1) ⊂ E2 × E1 × E1. MODELS FOR HIGHER ALGEBROIDS 355

What is left to check is the fact that the compositions Tf ◦ Te1,Tm2 ◦ T(f × f), ∗ ∗ ∗ ∗ ∗ T f ◦ T e1 and T m2 ◦ T (f × f) are simple. It is so indeed, because (Tf , Tf): ∗ ∗ TE2 → TE1 is a vector bundle morphism covering Tf :TM2 → TM1, hence its dual, which is Tf :TE1 →B TE2, is a ZM. Therefore the compositions Tf ◦ Te1, Tm2 ◦ (Tf × Tf) are simple, like for any ZM of groupoids. Similarly for the phase ∗ ∗ ∗ ∗ ∗ lift: T f :T E1 →B T E2 is a ZM dual to a vector bundle morphism (Tf , f ): ∗ ∗ ∗ ∗ ∗ TE2 → TE1 but now covering f : E2 → E1 . In what follows we always assume that a ZM is smooth. A Zakrzewski morphism between vector bundles will be presented in a form

f E1 ,2E2 (A.2)

τ1 τ2 f M1 o M2. The ZM f as above is a particular example of a linear relation, that is a linear subbundle of the product vector bundle E1 × E2. Corollary A.4. The contravariant functor E → E∗ is an equivalence of categories between the category of vector bundles with vector bundle morphisms and opposite to the category of vector bundles with Zakrzewski morphisms. In literature (e.g. [10]) ZMs between vector bundles are sometimes called vector bundle morphisms of the second kind. In [27] category of star bundles FM∗ is considered. Its objects are fibered manifolds and morphism are of the form similar τ1 τ2 as in Theorem A.3, i.e., a morphism φ :(Y1 −→ M1) → (Y2 −→ M2) between fibered manifolds is a pair (φ0, φ1) where φ0 : M1 → M2 is a smooth map and ∗ φ1 :(φ0) Y2 → Y1 is a fibered morphism over idM . By iterated application of the tangent functor to a relation r : X →B Y we get (k) (k) (k) relations T r :T X →B T Y . In a similar way we get the higher-order tangent k k k (r) r ∗ prolongations of r, i.e., relations T r :T X →B T Y . By applying T ,T or T to the structure relation of a (Lie) groupoid Γ we obtain again a (Lie) groupoid. The following theorem follows from a remark under Lemma 4.1 in [44], but can be also easily derived directly from Theorem A.3.

Theorem A.5. If f : E1 →B E2 is a Zakrzewski morphism of vector bundles then so are

(r) r ∗ (r) T f (r) r T f r ∗ T f ∗ T E1 ,2T E2 T E1 ,2T E2 and T E1 ,2T E2

(r) (r) Tr τ Tr τ T τ1 T τ2 1 2 r T(r)f T f f ∗ (r) (r) r r ∗ ∗ T M1 o T M2, T M1 o T M2 E1 o E2 . The associated vector bundle morphisms of dual bundles are

(r) ∗ r ∗ ∗ (r) ∗ T f (r) ∗ r ∗ T f r ∗ ∗ Tf ∗ T E2 / T E1 T E2 / T E1 and TE2 / TE1

r T(r)f T f f ∗ (r) (r) r r ∗ ∗ T M2 / T M1, T M2 / T M1 E2 / E1 . 356 MICHAL JO´ ZWIKOWSKI´ AND MIKOLAJ ROTKIEWICZ

Let us end with two simple remarks about relations. The ﬁrst one concerns the problem of restricting a relation to a subset.

Remark A.6 (Reductions of ZMs). Let f : E1 →B E2 be a ZM as in (A.2). Let 0 0 0 0 τ1 : E1 → M1 be a subbundle of τ1 : E1 → M1 and let M2 be a submanifold of M2 0 0 such that f(M2) ⊆ M1. Then, by restricting f ﬁber-wise we get a ZM denoted here by f 0 |E1

f|E0 0 1 0 E1 ,2E2

0 0 τ1 τ2 f 0 |M2 M1 o M2, 0 0 where τ2 is the pullback of τ2 with respect to the inclusion M2 ⊂ M2. It may 0 00 00 0 happen that f(E1) is contained in a subbundle τ2 : E2 → M2. This way we get a 0 0 00 ZM denoted here by f : E → E covering f 0 and say that the ZM f restricts 1 B 2 0|M2 0 00 0 0 00 ﬁne to a ZM E1 →B E2 . We can write shortly f = f ∩ (E1 × E2 ). Such situation can be presented on the following diagram:

f 0 E0 ,2f(E0 ) ___ / E00 1 p 1 Mm 2 CC w C ι1 ww CC ww CC ww ι2 C! f {ww E1 ,2E2

0 00 τ1 τ1 τ2 τ2 f M1 o M2 {= cGG {{ GG {{ GG . {{ GG1 Q 0 { 0 M1 o M2. 0 ∗ 00 ∗ 0 ∗ It is interesting to see how the dual morphism (f ) :(E2 ) → (E1) behaves in ∗ 0 ∗ ∗ ∗ this situation. We claim that f factorizes to (f ) through ι1 and ι2:

(f 0)∗ 0 ∗ 00 ∗ (E1) o (E2 ) (A.3) _LR _LR ∗ ∗ ι1 ι2 ∗ ∗ f ∗ E1 o E2 . 0 00 ∗ 0 0 ∗ 0 0 ∗ 0 Indeed, take any ψ ∈ (E2 )y and consider φ = (f ) ψ ∈ (E1)f(y). By deﬁnition φ 0 00 is the unique element such that for every V ∈ (E1)f(y) and W ∈ (E2 )y which are f 0-related we have hφ0,V i = hψ0,W i . ∗ 0 ∗ ∗ ∗ Now take any ψ ∈ E2 such that ψ = ι2ψ and let us calculate ι1f ψ. For V,W as above we have ∗ ∗ ∗ hι1f ψ, V i = hf ψ, ι1(V )i . 0 0 00 Now ι1(V ) and ι2(W ) are f-related as f = f ∩ (E1 × E2 ). Hence the later equals ∗ 0 hψ, ι2(W )i = hι2ψ, W i = hψ ,W i , ∗ ∗ 0 and hence ι1f ψ = φ . This ends the reasoning. MODELS FOR HIGHER ALGEBROIDS 357

The second remark concerns a problem of deﬁning new relations from old ones. Let us start with the following natural deﬁnition.

Deﬁnition A.7. By saying that a relation r : A →B B is deﬁned by a diagram f M1 ,2M2 (A.4)

p1 p2

r N1 ___ ,2N2 we mean that elements xi ∈ Ni are r-related (x1, x2) ∈ r if and only if there exist yj ∈ Mj, j = 1, 2 such that pj(yj) = xj and (y1, y2) ∈ f. The maps pj need not to be projections. We also say that p1, p2 relate the relations f and r. Note the following simple observation:

Proposition A.8. Let us assume that a smooth relation r : N1 →B N2 is deﬁned by a diagram (A.4) in which f is a smooth relation and p1, p2 are smooth maps. Then the tangent lift of r is deﬁned by the following diagram

T f TM1 ,2TM2

T p1 T p2 T r TN1 ___ ,2TN2.

Proof. Let us assume first that X1 ∈ TN1, X2 ∈ TN2 are projections of some elements Y1 ∈ TM1, Y2 ∈ TM2 such that (Y1,Y2) ∈ Tf. We may assume that Yj, j = 1, 2, are represented by curves yj(t) ∈ Mj such that (y1(t), y2(t)) ∈ f. Then X1, X2 are represented by curves p1(x1(t)), p2(x2(t)), respectively, which are r-related, and so (X1,X2) ∈ T r. Conversely, given Tr-related elements X1 ∈ TN1, X2 ∈ TN2 we may assume that they are represented by some curves xj : R → Nj such that (x1(t), x2(t)) ∈ r for any t ∈ R. Due to our hypothesis on r, we can find yj(t) ∈ Mj, j = 1, 2, such that pj(yj(t)) = xj(t) and (y1(t), y2(t)) ∈ f. Therefore, X1, X2 are projections of 1 1 Tf-related elements t0y1, t0y2 as it was claimed. Acknowledgments. This research was supported by the Polish National Science Center grant under the contract number DEC-2012/06/A/ST1/00256. The authors are grateful to professors PawelUrbański and Janusz Grabowski for reference suggestions. We wish to thank especially the second of them for reading the manuscript and giving helpful remarks.

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