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Nonassociative analogs of Lie groupoids

We introduce nonassociative geometric objects generalising naturally Lie groupoids and called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids are canonically smooth loopoids again (it is nontrivial in the case of loopoids). We show also that this is not true if the cotangent bundles are concerned. After providing a few natural constructions, we show how the Lie-like functor associates with loopoids skew-algebroids and almost Lie algebroids and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.


Introduction
The rôle of the theory of Lie groups and Lie algebras in mathematics and theoretical physics cannot be overestimated. It is a fundamental tool for describing symmetries, starting from smooth geometry through the theory of smooth equations, and ending in field and quantum theories.
The classical Lie theory has been extended to Lie groupoids and Lie algebroids (for the survey and literature see the monograph [30] or [35]) which turned out to be even more useful concepts if the geometry is concerned. Especially, symplectic groupoids as 'noncommutative phase spaces', integrating Poisson structures and introduced by Weinstein, Karasev and Zakrzewski [47,49], give rise to numerous applications and fundamental geometric problems. This generalized Lie theory combined with symplectic geometry is one of the two main streams in geometry.
The recent development has shown the importance of nonassociative algebraic structures, such as quasigroups and loops. For instance, it is possible to say that nonassociativity is the algebraic equivalent of the smooth geometric concept of curvature. The nonassociative generalizations of Lie groups are smooth loops. It is clear what a smooth loop is, on a manifold G we have a smooth multiplication (no associativity is assumed) with unit e ∈ G such that the left and right translations are diffeomorphisms. In [16] we developed discrete mechanics on smooth loops, but there is still a need to consider nonassociative objects closer to these of Lie groupoids; we will call them smooth (quasi)loopoids. So, loopoids, or their weaker version quasiloopoids, will be related to Lie groupoids as smooth loops are related to Lie groups.
We show that a Lie-like functor can be defined for quasiloopoids and loopoids and it will lead to new applications. The theory of loopoids has to be developed from scratch.
The rough idea of a loopoid is clear, however the precise definition has to be an object of a deeper study. The definition should be based on important examples and determined by conceptual completeness. We shall use methods of differential geometry and algebra. Applications will be related to Lagrangian discrete mechanics (geometric integrators) and information geometry (statistical manifolds).
Note that here by a groupoid we understand a Brandt groupoid, i.e. a small category in which every morphism is an isomorphism, and not as an object which is called magma in algebra.
In the case of genuine groupoids, since the multiplication is only partially defined, the situation is more complicated. So the axioms of a loopoid must be essentially reformulated.
A convenient way of thinking about groupoids is by using group axioms with the difference that all maps are replaced by relations, like it has been done by Zakrzewski [49]. In particular, the unity is a relation ε : {e}−−✄ G , associating to a point e a subset M = ε(e) ⊂ G , the set of units. Shortly, a smooth loopoid is a smooth manifold G with surjective submersions α, β : G → M , on the set of units M , and a partially defined smooth multiplication such that the composition satisfies α(g)g = g , hβ(h) = h , α(gh) = α(g) , and β(gh) = β(h) (this reduces to the existence of an identity element for smooth loops). Note that no associativity or invertibility is assumed. Moreover, the canonical translations l g : F α (β(g)) → F α (α(g)) , x → gx , and r h : F β (α(h)) → F β (β(h)) , x → xh , where F α (x) = α −1 (x) and F β (x) = β −1 (x) denote the corresponding α and β-fibers, are supposed to be diffeomorphisms. Again, for smooth loops, the latter conditions just mean that the left and the right translations are bijections. In this sense, a smooth loopoid over one point is a smooth loop.
We prove that for a smooth loopoid α, β : G → M the tangent bundle Ìα, Ìβ : ÌG → ÌM is canonically a smooth loopoid with the multiplication Ìm . Contrary to the case of Lie groupoids this is a non-trivial theorem (we do not have inversion). Also the cotangent bundle to a Lie groupoid is canonically a Lie groupoid. We will show that this cannot be extend to the case smooth loopoids.
One of the aims of this paper is to extend the Lie functor to categories of nonassociative objects, like quasiloopoids and loopoids. Infinitesimal counterparts of Lie groupoids are Lie algebroids and the corresponding Lie theory is well established (cf. [30]). As the infinitesimal version of associativity is the Jacobi identity, the infinitesimal counterpart of loopoids will not satisfy the latter.
Note that in the literature there are already natural various generalizations of Lie algebroids, e.g. skew algebroids, almost Lie algebroids, or Dirac algebroids [8,9,14,17,18], where no Jacobi identity is assumed. For instance, the skew algebroid formalism is very useful in describing the geometry of nonholonomic systems [15].
As the Jacobi identity is an infinitesimal version of associativity, the 'integrated version' of these generalizations must be understood as 'nonassociative Lie groupoids'. The development of the corresponding theory is still a challenging task. This is a natural evolution of mathematical studies that not only we go deeper in our understanding of the known structures, but also we start to penetrate new lands of more general structures, hoping to extend the old results and discover new mathematical phenomena. Note that the nonassociativity is actually a form of noncommutativity: the left translations do not commute with the right translations.
We want to stress that our motivation comes from discrete mechanics, where Lie groupoids have been recently used for a geometric formulation of the Lagrangian formalism [5,23,24,33,34,32,44,46,48]. We expect that a natural application of smooth loopoids will be that in discrete Lagrangian mechanics. Another application can be found in information geometry. We have shown in [11] that a natural framework for information geometry is provided by Lie groupoids and Lie algebroids. We believe this can be, at least partially, generalized to a theory on smooth quasiloopoids and loopoids.
A nonassociative algebra, e.g. the theory of algebraic loops, is already quite well developed. On the other way, many authors limit their investigation to loops that satisfy various other structural conditions. Common examples of such notions are the left-and right-Bol loops, the Moufang loops (which are both left-Bol loops and right-Bol loops simultaneously), the generalized Bol loop, etc. Especially the theory of analytic Moufang loops and Mal'cev algebras [31] is quite close to the Lie theory with respect to a generalized Lie functor [28,36]. For general analytical loops the tangent algebra has a structure of an Akivis algebra [21].
The problem is that the axioms for smooth Moufang loops are quite restrictive and these algebraic concepts do not fit very well to the needs of Lie groupoid geometry. One wants to have an object similar to a Lie groupoid, so with a product which is only partially defined and nonassociative.
The question is, how far we can go with known constructions from the Lie theory in the nonassociative case. Investigating it will show what aspects are specific for the structures which are associative (or satisfying the Jacobi identity) and what we can get by assuming weaker axioms.
In the case of smooth loops, the theory resembles to some extent the theory of Lie groups and is pretty well-developed (see e.g. the book [37]). Of course, nonassociative generalizations of Lie groupoids provide much more problems. Some of them we discuss in this paper.
Contrary to the case of smooth loops (e.g. [6,31,28,38,39,43]), the literature in this subject, oriented on 'nonassociative Lie groupoids' is not very extensive. Besides some aspects contained in Sabinin's monograph [41], we can indicate our short introductory note [13] and the recent paper [1] (which appeared after the first version of our paper was prepared). In the latter paper, the notion of a quasigroupoid is introduced. There are obvious common points with our concept of a loopoid (e.g. the source and target maps), however, the authors consider only purely algebraic cases (no differential geometry) with the emphasis on the cases when the sets of units are finite.
Note that the term loopoid has already appeared in a paper by Kinyon [25] in a similar context. The motivating example, however, built as an object 'integrating' the Courant bracket on TM ⊕ M T * M , uses the group of diffeomorphisms of the manifold M as integrating the Lie algebra of vector fields on M , not the pair groupoid M × M as 'integrating' the Lie algebroid ÌM .

Quasigroups and loops
The remarkable development of smooth quasigroups and loops theory since the pioneering works of Mal'cev in 1955 (see [31]) was presented by Lev V. Sabinin in [41], where the large bibliography on the subject is given. We refer also to the books [4,40] and the survey articles [42,45] if terms and concepts from nonassociative algebra are concerned.
Let us recall that a quasigroup is an algebraic structure < G, · > with a binary operation (written usually as juxtaposition, a · b = ab) such that r g : x → xg (the right translation) and l g : x → gx (the left translation) are permutations of G, equivalently, in which the equations ya = b and ax = b are solvable uniquely for x and y respectively. A loop is a quasigroup with a two-sided identity element, e, ex = xe = x. A loop < G, ·, e > with identity e is called an inverse loop (or I.P. loop) if to each element a in G there corresponds an element a −1 in G such that for all b ∈ G . It can be then easily shown that in an inverse loop < G, ·, −1 , e > we have, for all a, b ∈ G , Indeed, the first two identities are trivial and Mimicking the Lie theory, one can define for any smooth loop, a 'Lie functor' associating with the loop a skew algebra, i.e. a real vector space with a bilinear skew operation [16,41,42].
Example 2.1. The octonions O are the noncommutative non-associative algebra which is one of the four division algebras that exist over the real numbers. Every octonion can be expressed in terms of a natural basis {e 0 , e 1 , ·, ·, ·, e 7 } where e 0 = 1 represents the identity element and the imaginary octonion units e i , {i = 1, ..., 7} satisfy the multiplication rule e i e j = −δ j i + f ijk e k , where δ j i is the Kronecker's delta and f ijk 's are completely antisymmetric structure constants which read as  The associator [g, h, k] = (gh)k − g(hk) of three octonions does not vanish in general but octonions satisfy a weak form of associativity known as alternativity, namely [g, h, g] = 0. The octonions are a generalization of the complex numbers, with seven imaginary units, so octonionic conjugation is given by reversing the sign of the imaginary basis units. Conjugation is an involution of O satisfying (gh) * = h * g * . The inner product on O is inherited from R 8 and can be rewritten and the norm of an octonion is just g 2 = gg * which satisfies the defining property of a normed division algebra, namely gh = g h . The scalar product is invariant with respect to the multiplication: ag, ah = g, h for a = 0 . Every nonzero octonion g ∈ O has an inverse g −1 = g * g 2 , such that which makes the set of invertible octonions into an inverse loop with respect to the octonion multiplication. We remark that the inverse in this loop is a genuine one, i.e. it satisfies the Inverse Property: which is stronger than the standard property (1) for non-associative algebra. Actually, the set O × of invertible octonions is a smooth Moufang loop under octonion multiplication.
Example 2.2. The set of all automorphisms of the algebra O , that is the set of invertible linear transformations A ∈ Aut(O) , forms a Lie group called G 2 which is the smallest of the exceptional Lie groups. We will show that the semidirect product O × ⋉ G 2 is an inverse loop under the multiplication with identity (1, Id) and inverse (g, . What needs to be checked is the following inverse property, Here we use the fact that Of course, because O × is not associative the above smooth loop is not a Lie group.
Some information about smooth loops and octonion bundles can be found in [19,20].
Loops that have only one-sided inverse are called left inverse loops (resp. right inverse loops). Left inverse loops appear naturally as algebraic structures on transversals or sections of a subgroup in a group. In this case the homogeneous structures are equipped with a binary operation. This observation, going back to R. Baer [3] (cf. also [7,26]), lies at the heart of much current research on loops, also in smooth geometry and analysis. Example 2.3. Let G be a group with the unit e , H be a subgroup, and S ⊂ G be a left transversal to H in G , i.e., S contains exactly one point from each coset gH in G/H. This means that any element g ∈ G has a unique decomposition g = sh , where s ∈ S and h ∈ H . This produces an identification G = S × H of sets. Let p S : G → S be the projection on S determined by this identification. If we assume that e ∈ S , then S with the multiplication s • s ′ = p S (ss ′ ) and e as the unit is a left inverse loop. Indeed, as e • s = s • e = p S (s) = s , e is the unit for this multiplication. For a, b ∈ S, there is h ∈ H such that p S (a −1 b) = a −1 bh . Hence, that shows that p S (a −1 b) is a solution of the equation a • x = b . If c, c ′ are two such solutions, then p S (ac) = p S (ac ′ ) , so there is h ∈ H such that ac = ac ′ h , so c = c ′ h and c = c ′ , since S is transversal to H.
We will be interested with smooth (or analytic) structures on (inverse) loops. In this case G is a smooth (analytic) manifold in which the multiplication (inverse) is a smooth (analytic) map.
Exactly like it is done for Lie groups we can define the corresponding local objects, smooth local loops, in an obvious way.
Example 2.4. Let G be a local smooth loop defined in a neighbourhood of 0 in R n with the product where [x, y] is a skew bilinear operation on R n . The unit is e = 0 and the translations are local diffeomorphisms: Every element x possesses −x such that x • (−x) = 0 but in general there is no inverse and no Inverse Property because If [x, [x, y]] = 0 , then we deal with a nilpotent Lie group.

Quasiloopoids and loopoids
Question 3.1. We will look for the 'right' concept of a smooth loopoid G such that: • it is in two ways a fibration over a submanifold M of 'units", α, β : G → M which determine the set of composable elements; • the translations are sufficiently regular; • a smooth loopoid over one point is a smooth loop; • it is as close as possible to the concept of a (Lie) groupoid; • it has sufficiently rich variety of natural examples.
Our concept of a structure satisfying the first three requirements is the following. We will consider weaker structures which we call (smooth) quasiloopoids and the corresponding linear objects: skew algebroids, with different versions of invertibility property. The first generalization of the concept of a Lie groupoid to nonassociative case is a smooth quasiloopoid.
is a smooth closed embedded submanifold of G × G . We also assume that gβ(g) = g and α(h)h = h, i.e., elements of M are units, and that the left and right translations maps l g : F α (β(g)) → G , l g (h) = gh , and r h : F β (α(h)) → G , r h (g) = gh , are injective immersions. By F α (a) = {h ∈ G | α(h) = a} and F β (a) = {g ∈ G | β(g) = a} we denote the target and source fibers (α-and β-fibers), respectively. Sometimes we denote the obvious embedding M ⊂ G as ε : M → G . Now, let us assume that a quasiloopoid G over M with a partial multiplication m and projections α, β : G → M , satisfies a very weak associativity condition, hereafter called unities associativity: (xy)z = x(yz) if one of x, y, z is a unit (i.e., belongs to M ) . (2) The above condition has to be understood as follows: if one side of equation (2) makes sense, the other makes sense and we have equality. The following proposition shows that the condition of unities associativity for a quasiloopoid over M is rather strong and implies that the anchor map (α, β) : G → M × M has nice properties, similar to these for Lie groupoids. The unities associativity assumption implies that the left and right translation maps can be viewed as injective immersions Proof. According to (2), for (g, h) ∈ G (2) we have so α(gh) = α(g). Analogously we can prove β(gh) = β(h) .
Definition 3.4. A quasiloopoid satisfying the unities associativity assumption and such that the maps (3) are diffeomorphisms will be called a loopoid. A (quasi)loopoid G over the manifold M will be denoted by G ⇒ M .
Remark 3.5. In a loop, the multiplication is globally defined, so the unity associativity is always satisfied by properties of the unity element. In this sense, smooth loops are smooth loopoids over one point. It is not true for quasigroups, and they are not quasiloopoids over a point. In fact the conditions gβ(g) = g and α(h)h = h require the existence of the two sided identity.
They can be considered as loopoids over a single point, because the above multiplications are clearly well defined when the source and target maps are the same (projections onto the unit). Note that when we speak about a general loopoid, the above multiplications are generally not well defined, so loopoids do not seem to be proper generalizations of Moufang loops except for smooth Moufang loops themselves.
The definition of a morphisms Φ : G 1 → G 2 between two (quasi)loopoids is completely clear: we impose that Φ(xy) = Φ(x)Φ(y). In particular, both sides have the meaning simultaneously. It follows that Φ induces a map of the sets of units φ : M 1 → M 2 and intertwines the target and source maps. In this way we obtain two categories for which, as we will see, one can construct a 'Lie functor' to the categories of skew-algebroids and almost-Lie algebroids. If Φ is inclusion of the submanifold, we call G 1 a (quasi)subloopoid of G 2 .
The elements of G (2) are referred to as composable (or admissible) pairs. Note that the full information about the loopoid is contained in the multiplication relation which is Of course, for loops the multiplication is globally defined and α, β : G → {e} , so that On a (quasi)loopoid we can consider inverse properties.
Definition 3.7. A smooth inverse in a quasiloopoid G ⇒ M is an inversion mapping (diffeomorphism) ι : G → G , satisfying the following properties (where we write g −1 for ι(g)) : We say also the G has the Inverse Property (I.P.).
Theorem 3.8. A quasiloopoid with the Inverse Property is a loopoid.
We will call this kind of loopoids inverse loopoids or I.P. loopoids. Like for the loops one can show that for I.P loopoids we have the identities We can also consider only left/right inverses and speak about left or right inverse quasiloopoids.  If G ′ is an inverse loop, then G is an I.P. loopoid with the inverse ι(x, s, t) = (x −1 , t, s) . In this example G ′ can be identified with the isotropy loop G u at each u ∈ M .
Remark 3.10. Let π : P → M be a fibration, that is a surjective submersion. The prolongation of a smooth quasiloopoid (loopoid) G ⇒ M over π , defined as the set is a smooth loopoid over P with structural maps given by . If G is an inverse loopoid, then P π G is an inverse loopoid with the inversion map ι(p, g, p ′ ) = (p ′ , g −1 , p) . Therefore, we can produce a loopoid out of a quasiloopid by the prolongation of that over a fibration.
Example 3.11. A more complicated example we can construct as follows. Consider the pair groupoid G = R 2 × R 2 with the standard target and source maps α(u It is a quasiloopoid, with the target and source maps inherited from G, and the partial multiplication Indeed, for fixed ((a 1 , b 1 ), (a 2 , b 2 )) ∈ R 2 × R 2 the left unit is (a 1 , b 1 ) and the right one is (a 2 , b 2 ) . For G we have α(gh) = α(g) , but generally, for non-linear ϕ , we have β(gh) = β(h) , so G is not an inverse loopoid. However, it is interesting that G has a left inverse Indeed, The last equality follows from the fact that (( All this implies that G is a left inverse quasiloopoid. Example 3.12. If the anchor map of a loopoid is diagonal (trivial), i.e., α = β, then the loopoid is actually a bundle of loops. This bundle may be locally trivial as a fiber bundle and not as a loop bundle, since a local trivialization (in the general sense of fiber bundles) is not necessarily a local trivialization in the sense of loop bundles. The loops for different fibers could be even non-isomorphic as loops. Conversely, any bundle of loops, i.e., a surjective submersion τ : G → M whose each fiber carries a loop structure, is canonically a loopoid over M . In this case M is embedded in G as the section of unit elements in each loop fiber, α = β = τ , and the partial multiplication k = g · h is defined if and only if τ (k) = τ (g) = τ (h) and is k = gh -the loop multiplication in each fiber [13].

Bisections and isotropy loops
The concept of global bisections for loopoids can be defined as for Lie groupoids. A submanifold B of loopoid G ⇒ M is called a bisection of G if the restricted maps of the source and the target maps α| B and β| B are both diffeomorphisms. It means that for any bisection B , there are corresponding α-section τ := (α| B ) −1 : M → B and β-section For each bisection B we can define a left translation l B : G → G and a right translation r B : G → G on the loopoid, given by Since the multiplication of two bisections Note that there is an equivalent definition of a bisection (or local bisection) [30], which takes a bisection (or local bisection) to be the pair of maps τ and σ, rather than submanifold B. We will use both equivalent definitions.
The following Lemma can be borrowed from the theory of Lie groupoids (see [ Proof. To show that the ranks of coincide at given points g, g ′ ∈ β −1 (m), let σ be a local bisection containing g ′ g −1 and let where u is the unique element in σ such that β(u) = α(h) . It is easy to see that Ψ σ (g) = g ′ and β • Ψ σ = β , so Ψ σ restricts to a diffeomorphism of each β-fiber. In the commutative diagram the horizontal maps are diffeomorphisms and the upper map takes g to g ′ . Hence the ranks of the vertical map at g and g ′ coincide.
Theorem 3.14. For every smooth inverse quasiloopoid G ⇒ M the intersection G m of F α (m) and F β (m) is an inverse loop ( isotropy loop) for all m ∈ M .
Proof. It is easy to see that G m is closed with respect to the multiplication and inverse inherited from G, so it is an inverse loop. It is a submanifold of G according to the above Lemma, so it is smooth.

Skew algebroids
Let τ : E → M be a vector bundle of rank n over an m-dimensional manifold M and let π : E * → M be its dual. Recall that the Grassmann algebra Gr(E) = ⊕ ∞ i=0 Sec(∧ i E) of multisections of E is a graded commutative associative algebra with respect to the wedge product. We use affine coordinates (x a , ξ i ) on E * and the dual coordinates (x a , y i ) on E , associated with dual local bases, (e i ) and (e i ) , of sections of E and E * , respectively. Definition 4.1. A skew algebroid structure on E is given by a linear bivector field Π on E * (linear skew Leibniz structure). In local coordinates, . If Π is a Poisson tensor, we speak about a Lie algebroid.
The bracket [·, ·] Π and the anchor ρ Π are related to the bracket {·, ·} Π according to the formulae: where we denoted with ι(X) the linear function on E * associated with the section X of E, i.e., ι(X)(e * p ) = X(p), e * p for each e * p ∈ E * p . The Lie algebroid bracket satisfies the Jacobi identity It follows that the anchor map is a Lie algebra homomorphism where the second bracket is the bracket of vector fields. If we assume (7) without the Jacobi identity, then we deal with an almost Lie algebroid.   for every X, Y, Z. One can see that Mal'cev algebras are skew algebroids over a point satisfying the above identity. We can reduce the prolongation of a Lie algebroid over a smooth map to a skew algebroid over a smooth map ( [22], [29]). In the following we will consider the process for the prolongation of a skew algebroid over a submersion. As we expect from the Remark 3.10, we will see that the prolongation of a skew algebroid over a fibration is an almost Lie algebroid.
where Ìπ : ÌP → ÌM is the tangent map to π , is equipped with an almost Lie algebroid structure.
Proof. The space P π (E) is a vector bundle over P with vector bundle projection τ π : P π E → P , τ π (X, V ) → τ P (V ) , where τ P : ÌP → P is the canonical projection. In fact, for p ∈ P every fiber where E π(p) is the fiber of E over the point π(p) ∈ M , is a finite-dimensional real vector and since π is a submersion, we have ρ(E π(p) ) + (Ì p π)(Ì p P ) = Ì π(p) M , for all p ∈ P , which implies that there exists c ∈ N such that dim ρ(E π(p) ) + (Ì p π)(Ì p P ) = c and therefore P π E has constant rank.
To know how the skew algebroid structure on P π E can be characterized, we need to describe the sections of the vector bundle τ π : P π E → P and for that we use the pullback vector bundle E over π. The pullback of the E over π π * E = {(p, X) ∈ P × E| π(p) = τ (X)} , is a vector bundle over P with vector bundle projection pr 1 : π * E → P , pr 1 (p, X) = p , and any section X of E over M induces a section of π * E over P , simply by defining We have to go one step further and take the pullback of in the category of vector bundles over P . Note that we denote the map P π E → E by ψ. Then, sections of the vector bundle τ π : P π E → P are of the form X ′ ⊕ σ, where X ′ ∈ Sec(ÌP ) and σ ∈ Sec(π * E) and dπ(X ′ ) = π * ρ(σ) , and so from (8), for X ∧ ∈ Sec(P π E) we have X ∧ (p) = (σ(p), X ′ (p)) with The vector bundle τ π : P π E → P is equipped with a skew algebroid structure with the anchor map ρ π (X ∧ ) = X ′ . The bracket of two sections ( One may prove that the bracket [[·, ·]] π on Sec(P π E) satisfies Jacobi identity if and only if the bracket [[·, ·]] on Sec(E) satisfies Jacobi identity. We see that   Exactly like for Lie algebroids we have the following.   [12]) If E i → M i , i = 1, 2 , are skew (almost Lie ) algebroids, then a skew (almost Lie) algebroid morphism between them is a vector bundle morphism φ : E 1 → E 2 such that its graph Example 4.11. In the Remark 4.6, the pair (pr 1 , π) is a morphism between the skew algebroids (P π E, [[·, ·]] π , ρ π ) and (E, [[·, ·]], ρ) , where pr 1 : P π E → E is the cononical projection on the first factor.

Generalized Lie functor
In the context of the Lie algebroid theory, the 'nonassociative' objects have already been studied by us and other authors [17,18,8,10,14]. The main objects are skew algebroids, i.e, vector bundles E with a skew-symmetric bracket operation on sections which does not satisfy the Jacobi identity but admits an anchor map a : E → TM . If the anchor is a bracket homomorphism, we speak about almost Lie algebroids. The homotopies of admissible paths in almost Lie algebroids are well defined [8,14]. The importance of such objects comes from the fact that they form a nice framework for generalizations of Lagrangian and Hamiltonian mechanics and nonholonomic systems [10,15]. We show that we can obtain skew and almost Lie algebroids as infinitesimal parts of smooth loopoids and that some of the standard geometric constructions for the tangent and cotangent groupoid for a given Lie groupoid can be extended to this category.
The infinitesimal part of a Lie groupoid G ⇒ M is the normal bundle AG = ν(G, M ) → M , ν(G, M ) = TG |M /TM equipped with the natural Lie algebroid bracket. The association G → AG is a functor from the category of Lie groupoids to the category of Lie algebroids; we will refer to it as the Lie functor. To any X ∈ Sec(AG) it corresponds the unique left-invariant vector field ← − X on G such that, for each m ∈ M the vector ← − X (m) represents X m in ν(G, M ) . Similarly we obtain the right-invariant vector fields − → X . Left invariant vector fields are closed with respect to the Lie bracket and define a Lie algebroid bracket [·, ·] l on AG such that Similarly, for the right-invariant vector fields In a Lie groupoid, left translations commute with right translations, i.e, We will try to follow these construction of the bracket on Sec(AG) for any (local) smooth quasiloopoid G ⇒ M .
Definition 5.1. For X ∈ Sec(AG) we define its left prolongation (left fundamental vector field on G) by Here,X α (a) ∈ TF α (a) represents X(a) in the normal bundle AG = ν(G, M ). This is a canonical isomorphism of the vector bundles AG → M and TF α |M → M . A similar construction we do for the right prolongation (right fundamental vector field on G) − → X (g) = (T α(g) r g )(X β (α(g)) , whereX β (a) ∈ TF β (a) represents X(a) in the normal bundle ν(G, M ) .
Theorem 5.2. For any X ∈ Sec(AG) the vector fields ← − X and − → X are smooth on G .
Proof. Let f be a smooth function on G. We will show that ← − X (f ) is smooth. The normal bundle AG and the tangent bundle of α-fibration along M are canonically isomorphic and we can take a smooth vector field Y on G which is tangent to α-fibers and extendsX α on M . The latter is a smooth section of the subbundle of tangent spaces to α-fibration along M . We have which clearly depends smoothly on g. Here (0, Y ) is a self-explaining notation for a vector field on G × G and m is the quasiloopoid multiplication. Note that the vector field (0, Y ) on G × G is really tangent to G (2) as Ìα(Y ) = 0 = Ìβ(0) . One can also follow the proof of Theorem 2.1 in [27].
The fact that the left and right translations are immersions implies the following. Proposition 5.3. For any smooth quasiloopoid G the maps ← − X (β(g)) → ← − X (g) and − → X (α(h)) → − → X (h) are linear isomorphisms of vector spaces. In other words, for any g, the vector spaces { ← − X (g) | X ∈ A β(g) G} and { − → X (h) | X ∈ A α(h) G} are canonically isomorphic with the fibers A β(g) G and A α(h) G , respectively.
In the case of quasiloopoids, the left (right) prolongations are no longer left (right) invariant and are not closed with respect to the Lie bracket of vector fields on G, as G is not associative. Also the left and right prolongations do not commute. However, we still can define two brackets, [·, ·] l and [·, ·] r , on sections of AG by putting We will denote AG equipped with these brackets by A l G and A r G, respectively. The fundamental fact is the following.
Proof. The brackets are clearly skew-symmetric and R-bilinear, as As which shows that ρ l is the anchor for the left bracket. Similarly for the right bracket. For X ∈ AG we put ρ(X) =X α −X β .
As the two anchors differ only by sign, we will call the left anchor just an anchor and denote it by ρ.
Proof. Consider a local loop multiplication in a neighborhood of 0 in R n with 0 as the neutral element with the Taylor expansion around 0 According to the fact that 0 is the unit, we have a As we see, the structure constants of the skew brackets are skew-symmetrization of c k ij with respect to lower indices. Moreover, Example 5.6. Consider the smooth inverse loop defined in the example 2.4. If we take {∂ x i } i=1,...,n as a basis for R n , then the left and right prolongations are where x j 's are the coefficients of the vector field x = x j ∂ x j ∈ ÌR n . Then, the skew bracket associated to the skew algebra is are structure constants of the skew symmetric bracket.
We could not find an example of a loopoid with entirely different left and right tangent algebras, so we propose the following conjecture.
Conjecture 5.7. The left and right skew algebroids of a smooth loopoid differ by sign, Example 5.8. Consider the left inverse quasiloopoid of the Example 3.11. We will characterize its infinitesimal counterpart. The source and the target maps are α ((a 1 , b 1 Since the projection α is constant on the first component, thenȧ 1 =ḃ 1 = 0 which implieṡ On the other hand the target and the source maps coincide on the base R 2 which means a 1 = a 2 and b 1 = b 2 on R 2 , so the vector bundle AG = Ker(Tα) TG over R 2 is AG = ((a 1 , b 1 ), (a 1 , b 1 )) , (0, 0), (φ ′ (0)ḃ 2 ,ḃ 2 ) : c(x, y) :=ḃ 2 .
Let {∂ x , ∂ y } be a basis of the tangent bundle of R 2 . The sections of AG are of the form which are one dimensional vector fields on R 2 . The vector field Y = φ ′ (0)∂ x + ∂ y is the generator of bundle AG and any other section of AG is of the form c(x, y)Y with the bracket Thus, the skew algebroid structure on AG , as a vector bundle of rank one, depends only on the anchor map. Therefore, which means, ρ(X) = φ ′ (0)∂ x + ∂ y . Note that In this one dimension case the anchor map is an algebra homomorphism, therefore AG is an almost-Lie algebroid.
Example 5.9. On R 2 consider a smooth local loop structure H defined by the multiplication x.y = (x 1 , x 2 ).(y 1 , y 2 ) = (x 1 + y 1 + x 1 y 2 , x 2 + y 2 + x 2 y 1 ) , and the identity (0, 0) . Taking X 1 = ∂ x 1 and X 2 = ∂ x 2 as a basis for Ì 0 H , we get Hence, so that the bracket for the corresponding skew algebra h ∼ = Ì 0 H is Using the loop structure of H and the pair groupoid R 2 × R 2 ⇒ R 2 , we define a smooth loopoid G := H × R 2 × R 2 over M = {(0, t, t)|t ∈ R 2 } with the source map α(x, t, s) = (0, t, t) , the target map β(x, t, s) = (0, s, s) , and the multiplication The vector bundle AG is the trivial vector bundle of rank four over M = R 2 . As a basis of its sections let us take the vector fields along the submanifold M . Here, [∂ x i ] denotes the class of the vector field ∂ x i in the normal bundle of M in G. Note that the representative ∂ x 1 , ∂ x 2 , ∂ x 5 , ∂ x 6 are tangent to α-fibers, and ∂ x 1 , ∂ x 2 , ∂ x 3 , ∂ x 4 are tangent to β-fibers. The left prolongations are The only non-zero bracket is Eventually, we get the brackets on sections of A l G : The rest of the brackets are zero. Now, identifying the vector bundle AG with ÌF β M , we get the right prolongations − → The only non-zero bracket is This induces the following bracket on sections of A r G: and the rest of the brackets equal zero.
Note that for Lie groupoids, the left and right algebroid structures differ just by sign. One of our main results is that this is also the case of smooth I.P. loopoids. In other words the conjecture 5.7 is true for I.P. loopoids. Adapting (2) in [34], proved for Lie groupoids, we obtain the following.
Lemma 5.10. For any smooth I.P. loopoid G with an inverse ι : G → G , for any Proof. Let X be an element of A a G . We take a curve γ : R → F α (a) such that its tangent vector v at a represents X , i.e,X α a = v =γ(0) . Let us observe that This proves (9), as Ì a β(v) is the anchor of v . Proof. According to Lemma 5.10, T a ι(X α a ) = −X β a . As ι is the inverse, we have In consequence, for a ∈ M , Remark 5.13. It is well known that the tangent algebra of a general analytical loop has a structure of an Akivis algebra [21]. Unfortunately, the same Akivis algebra might correspond to non-isomorphic local analytic loops, so a finer structure is required in order to determine the analytical loop. The situation is similar in our case: a skew algebroid can be the infinitesimal part of many smooth loopoids. Whether additionally we have a ternary operation that makes our skew algebroid into an 'Akivis algebroid' is not clear yet, but it is definitely an important question which is open up to now.

Functoriality
Lemma 5.14. If B ⇒ N is a smooth quasisubloopoid of the smooth quasiloopoid G ⇒ M , then the natural map of normal bundles, induced functorially by the map of pairs (B, N ) ֒→ (G, M ), defines a morphism of skew algebroids A l B ֒→ A l G with left brackets. The same is true for the right brackets.
Proof. It is immediate that AB → N is a subbundle of AG → M . Furthermore, X ∈ Sec(AG) restricts to a section of AB if and only if the vector field ← − X is tangent to B . If X, Y are two such sections, then [ ← − X , ← − Y ] again is tangent to B . That is, the space Sec(AG, AB) of sections AG that restrict to sections of AB , is a skew subalgebra with respect to the left bracket. Similarly, for the right bracket.
More generally, we have: Proof. F is a morphism of quasiloopoids if and only if graph(F ) ⊂ B × G is a quasi subgroupoid. But then In summary, we have constructed a generalized Lie functor from the category of smooth quasiloopoids (and their morphisms) to pairs of skew algebroids (and their morphisms) with opposite anchors. Note however that in the nonassociative case our Lie functor is not an equivalence of categories. In general, a morphism of skew algebroids do not induce a morphism of the quasiloopoids; one skew algebroid can be the infinitesimal part of many (local) smooth quasiloopoids.
This means that the anchor map  Proof. Let X ∈ A l G andX α ∈ T β(g) F α (β(g) be the corresponding vertical representant tangent to the α-fiber. LetX α be represented by a smooth curve γ : R → G, α(γ(t)) = β(g) . The curve is mapped by the anchor map (α, β) to the curve (β(g), β(γ(t))). It represents the tangent vector A l (α, β)(X) in TM viewed as A l (M × M ) . Proof. The anchor map ρ = A l (α, β) is a morphism of skew algebroids covering the identity on M , therefore ρ[X, Example 5.18. Let G = G ′ × N × N ⇒ M be the smooth loopoid in the Example 3.9.
The submanifold of units is diffeomorphic with N and the normal bundle AG = ν(G, M ) and (10) is satisfied.
Example 5.19. Let G ⇒ M be a quasiloopoid (or loopoid) and AG the corresponding skew algebroid. The prolongation of AG over a fibration π : P → M which is defined by is an almost Lie algebroid on P . Now, we denote by A(P π G) the almost Lie algebroid of the loopoid P π G ⇒ P defined in Remark 3.10. Then for p ∈ P and m = π(p) , it follows that One can see that there is a linear isomorphism which induces an isomorphism between the skew algebroids A(P π G) and P π (AG)) . For more details see [22].
6 Tangent and cotangent bundles of smooth loopoids 6

.1 Tangent bundle of a quasiloopoid
Like in the case of a Lie groupoid, the tangent bundle of a smooth quasiloopoid is a smooth quasiloopoid. The following is obvious.
, then X g and Y h can be represented by smooth curves . In this case In the following theorem we adapt the proof of Theorem A.1 in [34].
Theorem 6.2. Let G ⇒ M be a quasiloopoid with the multiplication m : G (2) → G . For the composable pairs (g, h) ∈ G (2) denote β(g) = α(h) = q . If τ and σ are local α-section and β-section defined in a neighborhood of the unit element q ∈ M such that g = σ(q) and h = τ (q) , then the tangent multiplication T (g,h) m : T (g,h) G (2) → T (g,h) G is given by where v q = T g β(v g ) = T h α(v h ) . Obviously, for loopoids over a point (loops) the multiplication will be Proof. Consider the tangent vector (v g , v h ) ∈ T (g,h) G (2) as the following decomposition and then apply the tangent map Ìm (g,h) to each term. For the first two terms the situation is rather obvious: The third term requires more attention. We will show that If we take a curve q(t) such that v q is tangent to it, then σ and τ maps this curve to curves g(t) and h(t) for which v g and v h are tangent to them, respectively. So , h(t)) = d dt t=0 m(σ(q(t)), τ (q(t))) .
Therefore, the lack of associativity is not an obstacle neither for definition of local bisections nor for the tangent multiplication in tangent bundle of quasiloopoids.
After differentiating we get that It remains to show that is a diffeomorphism (and similarly for r (Y h ) ) . First we show that l (Xg ) is regular (local diffeomorphism), i.e, the differential has trivial kernel. Take aẎ ∈ Ì (Y h ) F Tα (Ìβ(X g )) and suppose We will show thatẎ = 0 (Y h ) . We can representẎ with a curve Y h(t) (t) ∈ F Tα (Ìβ(X g )) , goes to vector Ì h (l g )(ḣ) represented by the curve gh(t) . In view of (14), Ì h (l g )(ḣ) = 0 gh , soḣ = 0 h because l g is a diffeomorphism. This means that we can reduce the curve Y h(t) (t) . We have then that the curve represents the 0-vector, soȲ h = 0 , as l g is a diffeomorphism. Hence,Ẏ = 0 (Y h ) and the map (13) is a local diffeomorphism. It remains to check global properties of l (Xg ) . First, we will show that it is injective.
Finally, it is easy to see that if ι : G → G is an inverse for (G ⇒ M, m), then Ìι : ÌG → ÌG is the inverse for (ÌG ⇒ ÌM, Ìm) .
Remark 6.4. In the case of smooth loops an analogous result can be found in [6]. The structure on ÌG is essentially the same as the one we propose here, however, the authors describe there the loop structure on ÌG by means of the canonical trivialization ÌG = G × Ì e G rather than by means of the general differential geometric tangent functor.

Cotangent bundle of a quasiloopoid
The structure of T * G is much more complicated. It is well known that for any Lie group G, the cotangent bundle T * G has a natural structure of a Lie groupoid having the dual Lie algebra g * of G as the unit submanifold. To go deeper in our understanding of structures of cotangent bundle of quasiloopoids, let us consider a quasiloopoid over a point which is actually a loop. Let < G, ·, e > be a loop with g as its corresponding skew algebra and g * as its dual skew algebra. In this case we can take the target and source maps α := r * h and β := l * g to be pullbacks of smooth left and right translations, defined by (l * g ξ)(X) = ξ(T e l g X) , (r * h ξ)(X) = ξ(T e r g X) , X ∈ g, ξ ∈ T * g G .
The possible multiplication µ g • ν h ∈ T * gh G is defined by the equation where X g • Y h is the multiplication in the tangent loop (11). Denote θ gh = µ g • ν h , then h θ gh and ν h = l * g θ gh and then from (15), we get l * g (r * h θ gh ) = r * h (l * g θ gh ) = σ , This is a sort of associativity which we do not have in our case.
One may see that this problem comes from the fact that the equation (16) defines properly µ g • ν h if and only if the left-hand side vanishes for (X g , Y h ) in the kernel of multiplication map Tm : ÌG (2) → G defined by (11). But (X g , Y h ) ∈ ker(Ìm) is equivalent If we insert the latter into (16), we get l * g (r * h ) −1 µ g = ν h , which in the lack of associativity is a contradiction with the composability condition (15). Eventually, we conclude that the standard way of defining the multiplication in T * G does not give a well-defined product and that is why there is not a canonical quasiloopoid(loopoid) structure on the cotangent bundle T * G of a smooth quasiloopoid(loopoid) G.
Although the cotangent bundles of quasiloopoid(loopoid) have no canonical structures of quasiloopoid(loopoid), we still have good candidates for the source and the target maps α andβ,α ,β : where A * G is the dual bundle of the normal bundle AG = ν(G, M ), defined by Proposition 6.5. For any smooth quasiloopoid G the mapsα,β : T * G → A * G are smooth surjective submersions.
Proof. This is a direct consequence of Proposition 5.3.
Proposition 6.6. For any quasiloopoid G ⇒ M , the cotangent bundle T * G is fibred over the dual vector bundle A * G of associated skew algebroid AG by the two fibrationsα andβ defined by (17). Thus T * G is a double fibred bundle.
Example 6.7. Consider a local smooth loop structure G on R with the multiplication x.y = x + y + x 2 y and the target and source map α, β : G → {0} . The identity element is zero and there is no inverse.

Discrete Lagrangian mechanics on loopoids
In this section, we introduce Lagrangian mechanics on smooth loopoids. Let us first recall the discrete Lagrangian mechanics on Lie groupoids. Let G ⇒ M be a Lie groupoid, α, β : G → M being its target and source maps, with a multiplication map m : ← − X (or right-invariant − → X ) vector fields on G associated with X ∈ Sec(AG) . Discrete mechanics on a groupoid G for a smooth, real-valued function L on G is defined as follows [48].
Let L (2) be the restriction to the set of composable pairs G (2) of the function (g, h) → L(g) + L(h) and Σ L ⊂ G (2) be the set of critical points of L (2) along the fibers of the multiplication map m; that is, the points in Σ L are stationary points of the function L(g) + L(h) when g and h are restricted to admissible pairs with the constraint that the product gh is fixed. Variations of the constraint are of the form (gu, u −1 h) ∈ G (2) .
A solution of the Hamilton principle for the Lagrangian function L is a sequence ..., g −2 , g −1 , g 0 , g 1 , g 2 , ... of elements of G , defined on some "interval" in Z , such that (g i , g i+1 ) ∈ Σ L for each i .
Discrete Euler-Lagrange equations on Lie groupoids can be derived from the variational principle. In [33], the discrete Euler-Lagrange equations take the form on the Lie groupoid G ⇒ M , for every section X of AG. Note that (g i , g i+1 ) ∈ G (2) and the left and right arrow denotes the right and left-invariant vector field on G associated with X ∈ Sec(AG) understood as a section on the normal bundle. In our previous work [16] on the category of smooth loops, we indicate that because of the lack of associativity there is not clear variations like what we have in Lie groups. But still we can define the discrete Euler-Lagrange equations using the smooth prolongation of vector fields. It can be extended to the definition of Euler-Lagrange equations on loopoids.
Definition 7.1. The discrete Euler-Lagrange equation for a discrete Lagrangian system on a smooth quasiloopoid G ⇒ M with Lagrangian L : G → R is given by for every X ∈ AG , where ← − X and − → X are the left and right prolongation, respectively. A sequence g 1 , g 2 , ... of elements G (2) is a solution of the Euler-Lagrange equations if g i and g i+1 satisfy (18) for i = 1, 2, . . . .
The discrete Euler-Lagrange operator where A * G is the dual of the skew algebroid AG, is given by A smooth mapping γ : G → G is called a discrete flow or discrete Lagrangian evolution operator for a discrete Lagrangian L if the graph graph(γ) = {(g, γ(g)) | g ∈ G} is a subset of G (2) and (g, γ(g)) is a solution of the discrete Euler-Lagrange equation for L for all g ∈ G . Equivalently, DL(g, γ(g)) = 0 for all g ∈ G .
We have the discrete Legendre transformation for a smooth loopoid G which is similar to what we have for Lie groupoids [34]. Given a Lagrangian L : G → R on a smooth loopoid G ⇒ M with the almost algebroid AG , we have two discrete Legendre transformations where dL : G → Ì * G, andα andβ are the two fibrations of the double fiberd bundle Ì * G defined by for X ∈ Sec(AG) . Directly from the definitions, we get the following. • A discrete Lagrangian L : G → R on smooth loopoid G ⇒ M is regular; • F − L is a local diffeomorphism; Moreover, L is hyperregular if and only if F − L is a global diffeomorphism. In this case the discrete Lagrangian evolution operator is a diffeomorphism.
The cotangent bundle Ì * G is equipped with a canonical symplectic structure but as we observed the lack of associativity is an obstacle for defining a natural loopoid structure on Ì * G analogous to the Lie groupoid. Nevertheless, it still is a double fibred bundle.
Setting aside the loopoid structure, for any function L : G → R on manifold G the submanifolds dL(G) ⊂ Ì * G is a Lagrangian submanifold of the cotangent bundle. The discrete Euler-Lagrange dynamics can be equivalently described as follows.
This means that the Legendre transformations are local diffeomorphism at the neighborhood of identity elements along the fibers, which implies the Lagrangian L is regular. It can be also computed by mathematical software such as Maple that as a discrete flow γ : G → G for the Lagrangian L can be taken the map γ(x, y, a, b, c, d) = (z, w, c, d, c, d) , where B = 1 + 2(x + y − yx + x 2 y + xy 2 − x 2 y 2 − x 3 − y 3 ) + 3(x 2 + y 2 ) + x 4 + y 4 .

Conclusions
We have introduced the categories of smooth quasiloopoids and loopoids as nonassociative analogs of Lie groupoids. We have shown that a big part of the Lie theory of Lie groupoids (e.g. Lie functor) can be generalized to this nonassociative case. In particular, infinitesimal parts of smooth loopoids have been recognized as skew algebroids. We have not obtained ternary structures on these skew algebroids producing 'Akivis algebroids'. This question we postpone to a forthcoming article. We have studied the properties of the tangent and cotangent bundles of smooth loopoids showing that the tangent bundles are smooth loopoids themselves. Smooth quasiloopoids and loopoids provide also a geometric framework for discrete Lagrangian mechanics. One can also use quasiloopoids and loopoids as a playground for information geometry (cf. [2,11]). The contrast (potential) function is in this picture a function F : G → R on the quasiloopoid/loopoid G ⇒ M with vanishing first jets along M . This defines a quasi-Riemannian metric g (an analog of the Fisher-Rao metric) on the corresponding skew algebroid AG , Since linear AG-connections and the Levi-Civita connection are well-defined even for skewalgebroids, we can then try to find the dualistic pair of such connections, the difference of which will define a symmetric covariant three-tensor on AG . This should lead to 'information geometry on loopoids'.