The Supergeometry of Loday Algebroids

A new concept of Loday algebroid (and its pure algebraic version - Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are interpreted as homological vector fields on a `supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given.


Introduction
The concept of Dirac structure, proposed by Dorfman [4] in the Hamiltonian framework of integrable evolution equations and defined in [3] as an isotropic subbundle of the Whitney sum T M = TM ⊕ M T * M of the tangent and the cotangent bundles and satisfying some additional conditions, provides a geometric setting for Dirac's theory of constrained mechanical systems. To formulate the integrability condition defining the Dirac structure, Courant [3] introduced a natural skew-symmetric bracket operation on sections of T M . The Courant bracket does not satisfy the Leibniz rule with respect to multiplication by functions nor the Jacobi identity. These defects disappear upon restriction to a Dirac subbundle because of the isotropy condition. Particular cases of Dirac structures are graphs of closed 2-forms and Poisson bivector fields on the manifold M .
The nature of the Courant bracket itself remained unclear until several years later when it was observed by Liu, Weinstein and Xu [32] that T M endowed with the Courant bracket plays the role of a 'double' object, in the sense of Drinfeld [5], for a pair of Lie algebroids (see [34]) over M . Let us recall that, in complete analogy with Drinfeld's Lie bialgebras, in the category of Lie algebroids there also exist 'bi-objects', Lie bialgebroids, introduced by Mackenzie and Xu [35] as linearizations of Poisson groupoids. On the other hand, every Lie bialgebra has a double which is a Lie algebra. This is not so for general Lie bialgebroids. Instead, Liu, Weinstein and Xu [32] showed that the double of a Lie bialgebroid is a more complicated structure they call a Courant algebroid, T M with the Courant bracket being a special case.
There is also another way of viewing Courant algebroids as a generalization of Lie algebroids. This requires a change in the definition of the Courant bracket and considering an analog of the non-antisymmetric Dorfman bracket [4], so that the traditional Courant bracket becomes the skew-symmetrization of the new one [43]. This change replaces one of the defects with another one: a version of the Jacobi identity is satisfied, while the bracket is no longer skewsymmetric. Such algebraic structures have been introduced by Loday [30] under the name Leibniz algebras, but they are nowadays also often called Loday algebras. Loday algebras, like their skew-symmetric counterparts -Lie algebras -determine certain cohomological complexes, defined on tensor algebras instead of Grassmann algebras.
The concepts of Loday algebroid we found in the literature do not seem to be exactly appropriate. The notion in [11], which assumes the existence of both anchor maps, is too strong and admits no real new examples, except for Lie algebroids and bundles of Loday algebras. The concept introduced in [45] requires a pseudo-Riemannian metric on the bundle, so it is too strong as well and does not reduce to a Loday algebra when we consider a bundle over a single point, while the other concepts [21,19,20,27,36,48], assuming only the existence of a left anchor, do not put any differentiability requirements for the first variable, so that they are not geometric and too weak (see Example 4.3). Only in [1] one considers some Leibniz algebroids with local brackets.
The aim of this work is to propose a modified concept of Loday algebroid in terms of an operation on sections of a vector bundle, as well as in terms of a homological vector field of a supercommutative manifold. We put some minimal requirements that a proper concept of Loday algebroid should satisfy. Namely, the definition of Loday algebroid, understood as a certain operation on sections of a vector bundle E, • should reduce to the definition of Loday algebra in the case when E is just a vector space; • should contain the Courant-Dorfman bracket as a particular example; • should be as close to the definition of Lie algebroid as possible.
We propose a definition satisfying all these requirements and including all main known examples of Loday brackets with geometric origins. Moreover, we can interpret our Loday algebroid structures as homological vector fields on a supercommutative manifold; this opens, like in the case of Lie algebroids, new horizons for a geometric understanding of these objects and of their possible 'higher generalizations'.
The paper is organized as follows. We first recall -in Section 2 -needed results on differential operators and derivative endomorphisms. In Section 3 we investigate, under the name of pseudoalgebras, algebraic counterparts of algebroids requiring varying differentiability properties for the two entries of the bracket. The results of Section 4 show that we should relax our traditional understanding of the right anchor map. A concept of Loday algebroid satisfying all the above requirements is proposed in Definition 4.7 and further detailed in Theorem 4.8. In Section 5 we describe a number of new Loday algebroids containing main canonical examples of Loday brackets on sections of a vector bundle. A natural reduction a Loday pseudoalgebra to a Lie pseudoalgebra is studied in Section 6. For the standard Courant bracket it corresponds that makes D(E) into a canonical example of a quantum Poisson algebra in the terminology of [16]. It was pointed out in [26] that the concept of derivative endomorphism can be traced back to N. Jacobson [22,23] as a special case of his pseudo-linear endomorphism. It has appeared also in [37] under the name module derivation and was used to define linear connections in the algebraic setting. In the geometric setting of Lie algebroids it has been studied in [34] under the name covariant differential operator. For more detailed history and recent development we refer to [26].
Algebraic operations in differential geometry have usually a local character in order to be treatable with geometric methods. On the pure algebraic level we should work with differential (or multidifferential) operations, as tells us the celebrated Peetre Theorem [39,40]. The algebraic concept of a multidifferential operator is obvious. For a K-multilinear operator D : E 1 ×· · ·×E p → E and each i = 1, . . . , p, we say that D is a differential operator of order ≤ k with respect to the ith variable, if, for all y j ∈ E j , j = i, is a differential operator of order ≤ k. In other words, where δ i (f )D(y 1 , . . . , y p ) = D(y 1 , . . . , f y i , . . . , y p ) − f D(y 1 , . . . , y p ) .
Note that the operations δ i (f ) and δ j (g) commute. We say that the operator D is a multidifferential operator of order ≤ n, if it is of order ≤ n with respect to each variable separately. This means that, fixing any p − 1 arguments, we get a differential operator of order ≤ n. A similar, but stronger, definition is the following Definition 2.4. We say that a multilinear operator D : Of course, a multidifferential operator of total order ≤ k is a multidifferential operator of order ≤ k. It is also easy to see that a p-linear differential operator of order ≤ k is a multidifferential operator of total order ≤ pk. In particular, the Lie bracket of vector fields (in fact, any Lie algebroid bracket) is a bilinear differential operator of total order ≤ 1.

Pseudoalgebras
Let us start this section with recalling that Loday, while studying relations between Hochschild and cyclic homology in the search for obstructions to the periodicity of algebraic K-theory, discovered that one can skip the skew-symmetry assumption in the definition of Lie algebra, still having a possibility to define an appropriate (co)homology (see [29,31] since we have no skew-symmetry. Loday called such structures Leibniz algebras, but to avoid collision with another concept of Leibniz brackets in the literature, we shall call them Loday algebras. This is in accordance with the terminology of [K-S], where analogous structures in the graded case are defined. Note that the identities (13) and (14) have an advantage over the identity (15) obtained by cyclic permutations, since they describe the algebraic facts that the left-regular (resp., right-regular) actions are left (resp., right) derivations. This was the reason to name the structure 'Leibniz algebra'. Of course, there is no particular reason not to define Loday algebras by means of (14) instead of (13) (and in fact, it was the original definition by Loday), but this is not a substantial difference, as both categories are equivalent via transposition of arguments. We will use the form (13) of the Jacobi identity.
Our aim is to find a proper generalization of the concept of Loday algebra in a way similar to that in which Lie algebroids generalize Lie algebras. If one thinks about a generalization of a concept of Lie algebroid as operations on sections of a vector bundle including operations (brackets) which are non-antisymmetric or which do not satisfy the Jacobi identity, and are not just A-bilinear, then it is reasonable, on one hand, to assume differentiability properties of the bracket as close to the corresponding properties of Lie algebroids as possible and, on the other hand, including all known natural examples of such brackets. This is not an easy task, since, as we will see soon, some natural possibilities provide only few new examples.
To present a list of these possibilities, we propose the following definitions serving in the pure algebraic setting.
Definition 3.1. Let E be a faithful module over an associative commutative algebra A over a field K of characteristic 0. A a K-bilinear bracket B = [·, ·] : E × E → E on the module E 1. is called a Kirillov pseudoalgebra bracket, if B is a bidifferential operator; 2. is called a weak pseudoalgebra bracket, if B is a bidifferential operator of degree ≤ 1; 3. is called a quasi pseudoalgebra bracket, if B is a bidifferential operator of total degree ≤ 1; 4. is called a pseudoalgebra bracket, if B is a bidifferential operator of total degree ≤ 1 and the adjoint map ad X = [X, ·] : E → E is a derivative endomorphism for each X ∈ E; 5. is called a QD-pseudoalgebra bracket, if the adjoint maps ad X , ad r X : E → E, associated with B are derivative endomorphisms (quasi-derivations); 6. is called a strong pseudoalgebra bracket, if B is a bidifferential operator of total degree ≤ 1 and the adjoint maps ad X , ad r X : E → E, are derivative endomorphisms.
We call the module E equipped with such a bracket, respectively, a Kirillov pseudoalgebra, weak pseudoalgebra etc. If the bracket is symmetric (skew-symmetric), we speak about Kirillov, weak, etc., symmetric (skew) pseudoalgebras. If the bracket satisfies the Jacobi identity (13), we speak about local, weak, etc., Loday pseudoalgebras, and if the bracket is a Lie algebra bracket, we speak about local, weak, etc., Lie pseudoalgebras. If E is the A = C ∞ (M ) module of sections of a vector bundle τ : E → M , we refer to the above pseudoalgebra structures as to algebroids.
is a pseudoalgebra bracket, then the map called the anchor map, is A-linear, ρ(f X) = f ρ(X), and for all X, Y ∈ E, f ∈ A. Moreover, if [·, ·] satisfies additionally the Jacobi identity, i.e., we deal with a Loday pseudoalgebra, then the anchor map is a homomorphism into the commutator bracket, Proof. Since the bracket B is a bidifferential operator of total degree ≤ 1, we have δ 1 (f )δ 2 (g)B = 0 for all f, g ∈ A. On the other hand, as easily seen, and the module is faithful, it follows ρ(f X) = f ρ(X). The identity (19) is a direct implication of the Jacobi identity combined with (18).
is a QD-pseudoalgebra bracket, then it is a weak pseudoalgebra bracket and admits two anchor maps ρ, ρ r : E → Der(A) , ρ(X) = ad X , ρ r = − ad r , for which we have for all X, Y ∈ E, f ∈ A. If the bracket is skew-symmetric, then both anchors coincide, and if the bracket is a strong QD-pseudoalgebra bracket, they are A-linear. Moreover, if [·, ·] satisfies additionally the Jacobi identity, i.e., we deal with a Loday QD-pseudoalgebra, then, for all X, Proof. Similarly as above, so B is a first-order differential operator with respect to the second argument. The same can be done for the first argument. Next, as for any QD-pseudoalgebra bracket B we have, analogously to (20), both anchor maps are A-linear if and only if D is of total order ≤ 1. The rest follows analogously to the previous theorem.
The next observation is that quasi pseudoalgebra structures on an A-module E have certain analogs of anchor maps, namely A-module For every X ∈ E we will view b(X) as an A-module homomorphism b(X) : where Ω 1 is the A-submodule of Hom A (Der(A); A) generated by dA = {df : f ∈ A} and df, D = D(f ). Elements of Der(A) ⊗ A End(E) act on elements of Ω 1 ⊗ A E in the obvious way: called generalized anchor maps, right and left, such that, for all X, Y ∈ E and all f ∈ A, The generalized anchor maps are actual anchor maps if they take values in Proof. Assume first that the bracket B is a bidifferential operator of total degree ≤ 1 and define a three-linear map of vector spaces A : It is easy to see that A is A-linear with respect to the first and the third argument, and a derivation with respect to the second. Indeed, as we get A-linearity with respect to the first argument. Similarly, from δ 2 (f )δ 2 (g)B = 0, we get the same conclusion for the third argument. We have also thus the derivation property. This implies that A is represented by an A-module homomorphism Analogous considerations give us the right generalized anchor map b r . Conversely, assume the existence of both generalized anchor maps. Then, the map A defined as above reads A similar reasoning for b r gives (δ 1 (f )δ 1 (g)B)(X, Y ) = 0, so the bracket is a bidifferential operator of total order ≤ 1.
In the case when we deal with a quasi algebroid, i.e., A = C ∞ (M ) and E = Sec(E) for a vector bundle τ : E → M , the generalized anchor maps (24) are associated with vector bundle maps that we denote (with some abuse of notations) also by b r , b l ,  Let us isolate and specify the most important particular cases of Definition 3.1.
1. A Kirillov-Loday algebroid (resp., Kirillov-Lie algebroid) on a vector bundle E over a base manifold M is a Loday bracket (resp., a Lie bracket) on the C ∞ (M )-module Sec(E) of smooth sections of E which is a bidifferential operator.

2.
A weak Loday algebroid (resp., weak Lie algebroid) on a vector bundle E over a base manifold M is a Loday bracket (resp., a Lie bracket) on the C ∞ (M )-module Sec(E) of smooth sections of E which is a bidifferential operator of degree ≤ 1 with respect to each variable separately.
3. A Loday quasi algebroid (resp., Lie quasi algebroid) on a vector bundle E over a base manifold M is a Loday bracket (resp., Lie bracket) on the C ∞ (M )-module Sec(E) of smooth sections of E which is a bidifferential operator of total degree ≤ 1.

4.
A QD-algebroid (resp., skew QD-algebroid, Loday QD-algebroid, Lie QD-algebroid) on a vector bundle E over a base manifold M is an R-bilinear bracket (resp., skew bracket, which the adjoint operators ad X and ad r X are derivative endomorphisms.

Remark 4.2.
Lie pseudoalgebras appeared first in the paper of Herz [18], but one can find similar concepts under more than a dozen of names in the literature (e.g. Lie modules, (R, A)-Lie algebras, Lie-Cartan pairs, Lie-Rinehart algebras, differential algebras, etc.). Lie algebroids were introduced by Pradines [41] as infinitesimal parts of differentiable groupoids. In the same year a book by Nelson was published where a general theory of Lie modules, together with a big part of the corresponding differential calculus, can be found. We also refer to a survey article by Mackenzie [33]. QD-algebroids, as well as Loday QD-algebroids and Lie QD-algebroids, have been introduced in [11]. In [17,9] Loday strong QD-algebroids have been called Loday algebroids and strong QD-algebroids have been called just algebroids. The latter served as geometric framework for generalized Lagrange and Hamilton formalisms.
In the case of line bundles, rk E = 1, Lie QD-algebroids are exactly local Lie algebras in the sense of Kirillov [24]. They are just Jacobi brackets, if the bundle is trivial, It is easy to see that this is a Loday bracket which admits the trivial left anchor, but the bracket is non-local and non-geometric as well. It is a bidifferential operator of order ≤ 1 and the total order ≤ 2. It is actually a Lie QD-algebroid bracket, as ad f and ad r f are, by definition, derivations (more generally, firstorder differential operators). Both anchor maps coincide and give the corresponding Hamiltonian vector fields, ρ(f )(g) = {f, g}. The map f → ρ(f ) is again a differential operator of order 1, so is not implemented by a vector bundle morphism ρ : M × R → TM. Therefore, this weak Lie algebroid is not a Lie algebroid.
Example 4.5. Various brackets are associated with a volume form ω on a manifold M of dimension n (see e.g. [28]). Denote with X k (M ) (resp., Ω k (M )) the spaces of k-vector fields (resp., k-forms) on M . As the contraction maps . A solution proposed in [28] depends on considering the algebra N of bivector fields modulo δ-exact bivector fields for which the Jacobi anomaly disappears, so that N is a Lie algebra.
Another option is to resign from skew-symmetry and define the corresponding Kirillov-Loday algebroid. In view of the duality between X 2 (M ) and Ω n−2 , it is possible to work with Ω n−2 (M ) instead. For γ ∈ Ω n−2 (M ) we define the vector field γ ∈ X (M ) from the formula i γ ω = dγ. The bracket in Ω n−2 (M ) is now defined by Therefore, so the Jacobi identity is satisfied and we deal with a Loday algebra. This is in fact a Kirillov-Loday algebroid structure on ∧ n−2 T * M with the left anchor ρ(γ) = γ. This bracket is a bidifferential operator which is first-order with respect to the second argument and second-order with respect to the first one.
Note that Lie QD-algebroids are automatically Lie algebroids, if the rank of the bundle E is > 1 [11,Theorem 3]. Also some other of the above concepts do not produce qualitatively new examples.
covering the identity on M , such that, for all X, Y ∈ Sec(E) and all f ∈ C ∞ (M ), If this is the case, the left anchor induces a homomorphism of the Loday bracket into the bracket [·, ·] vf of vector fields, Proof. This is a direct consequence of Theorem 3.5 and the fact that an algebroid bracket has the left anchor map. We just write the generalized right anchor map as b r = ρ ⊗ I − α.
To give a local form of a Loday algebroid bracket, let us recall that sections X of the vector bundle E can be identified with linear (along fibers) functions ι X on the dual bundle E * . Thus, fixing local coordinates (x a ) in M and a basis of local sections e i of E, we have a corresponding system (x a , ξ i = ι e i ) of affine coordinates in E * . As local sections of E are identified with linear functions σ = σ i (x)ξ i , the Loday bracket is represented by a bidifferential operator B of total order ≤ 1: Taking into account the existence of the left anchor, we have Since sections of End(E) can be written in the form of linear differential operators, we can rewrite (30) in the form Of course, there are additional relations between coefficients of B due to the fact that the Jacobi identity is satisfied.

Leibniz algebra
Of course, a finite-dimensional Leibniz algebra is a Leibniz algebroid over a point.

Courant-Dorfman bracket
The Courant bracket is defined on sections of T M = TM ⊕ M T * M as follows: This bracket is antisymmetric, but it does not satisfy the Jacobi identity; the Jacobiator is an exact 1-form. It is, as easily seen, given by a bidifferential operator of total order ≤ 1, so it is a skew quasi algebroid.
The Dorfman bracket is defined on the same module of sections. Its definition is the same as for Courant, except that the corrections and the exact part of the second Lie derivative disappear: This bracket is visibly non skew-symmetric, but it is a Loday bracket which is bidifferential of total order ≤ 1. Moreover, the Dorfman bracket admits the classical left anchor map which is the projection onto the first component. Indeed, For the right generalized anchor we have is a symmetric nondegenerate bilinear form on T M (while ·, · is the canonical pairing). We will refer to it, though it is not positively defined, as the scalar product in the bundle T M .
Note that α(Y + η) is really a section of TM ⊗ M End(TM ⊕ M T * M ) that in local coordinates Hence, the Dorfman bracket is a Loday algebroid bracket.
It is easily checked that the Courant bracket is the antisymmetrization of the Dorfman bracket, and that the Dorfman bracket is the Courant bracket plus d X + ω, Y + η +

Twisted Courant-Dorfman bracket
The Courant-Dorfman bracket can be twisted by adding a term associated with a 3-form Θ [44]: It turns out that this bracket is still a Loday bracket if the 3-form Θ is closed. As the added term is C ∞ (M )-linear with respect to X and Y , the anchors remain the same, thus we deal with a Loday algebroid. It was further observed [46,15] that the number of independent conditions can be reduced.

Courant algebroid
Note that (36) is equivalent to Similarly, (37) easily implies the invariance of the pairing (·, ·) with respect to the adjoint maps which in turn shows that ρ is the anchor map for the left multiplication: we get out of (38) that This, combined with (40), implies in turn so any Courant algebroid is a Loday algebroid.

Grassmann-Dorfman bracket
The Dorfman bracket (33)  k=0 Ω k (M ), is the Grassmann algebra of differential forms. The bracket, Grassmann-Dorfman bracket, is formally given by the same formula (33) and the proof that it is a Loday algebroid bracket is almost the same. The left anchor is the projection on the summand TM, and

Grassmann-Dorfman bracket for a Lie algebroid
All the above remains valid when we replace TM with a Lie algebroid (E, [·, ·] E , ρ E ), the de Rham differential d with the Lie algebroid cohomology operator d E on Sec(∧E * ), and the Lie derivative along vector fields with the Lie algebroid Lie derivative £ E . We define a bracket on sections of E ⊕ M ∧E * with formally the same formula This is a Loday algebroid bracket with the left anchor

Lie derivative bracket for a Lie algebroid
The above Loday bracket on sections of E ⊕ M ∧E * has a simpler version. Let us put simply This is again a Loday algebroid bracket with the same left anchor and and In particular, when reducing to 0-forms, we get a Leibniz algebroid structure on E × R, where the bracket is defined by [X + f, Y + g] = [X, Y ] E + ρ E (X)g, the left anchor by ρ(X, f ) = ρ E (X), and the generalized right anchor by In other words, Spaces equipped with skew-symmetric brackets satisfying the above identity have been introduced by Filippov [6] under the name n-Lie algebras.

Loday algebroids associated with a Nambu-Poisson structure
The concept of Leibniz (Loday) algebroid used in [21] is the usual one, without differentiability condition for the first argument. Actually, this example is a Loday algebroid in our sense as well. The bracket is defined for (n − 1)-forms by For the generalized right anchor we get Note that α is really a bundle map α : ∧ n−1 T * M → TM ⊗ M End(∧ n−1 T * M ), since it is obviously C ∞ (M )-linear in η and ω, as well as a derivation with respect to f.
In [19,20], another Leibniz algebroid associated with the Nambu-Poisson structure Λ is proposed. The vector bundle is the same, E = ∧ n−1 T * M , the left anchor map is the same as well, ρ(ω) = i ω Λ, but the Loday bracket reads Hence, so that for the generalized right anchor we get This Loday algebroid structure is clearly the one obtained from the Grassmann-Dorfman bracket on the graph of Λ, Actually, an n-vector field Λ is a Nambu-Poisson tensor if and only if its graph is closed with respect to the Grassmann-Dorfman bracket [2].

The Lie pseudoalgebra of a Loday algebroid
Let us fix a Loday pseudoalgebra bracket [·, ·] on an A-module E. Let ρ : E → Der(A) be the left anchor map, and let be the generalized right anchor map. For every X ∈ E we will view α(X) as a A-module It is a well-known fact that the subspace g 0 generated in a Loday algebra g by the symmetrized brackets X ⋄ Y = [X, Y ] + [Y, X] is a two-sided ideal and that g/g 0 is a Lie algebra.
we have then Indeed, symmetrized brackets are spanned by squares [X, X], so, due to the Jacobi identity, However, working with A-modules, we would like to have an A-module structure on E/E 0 . Unfortunately, E 0 is not a submodule in general. Let us consider therefore the A-submoduleĒ 0 of E generated by E 0 , i.e.,Ē 0 = A · E 0 . Lemma 6.1. For all f ∈ A and X, Y, Z ∈ E we have In particular, Proof. To prove (50) it suffices to combine the identity [X, Corollary 6.2. For all f ∈ A and X, Y ∈ E, and the left anchor vanishes onĒ 0 , Moreover,Ē 0 is a two-sided Loday ideal in E and the Loday bracket induces on the A-modulē E = E/Ē 0 a Lie pseudoalgebra structure with the anchor where [X] denotes the coset of X.
Proof. The first statement follows directly from (50). As [E 0 , E] = 0, the anchor vanishes on E 0 and thus onĒ 0 = A · E 0 . From we conclude thatĒ 0 is a two-sided ideal. AsĒ 0 contains all elements X ⋄ Y , The Loday bracket induces on E/Ē 0 a skew-symmetric bracket with the anchor (55) and satisfying the Jacobi identity, thus a Lie pseudoalgebra structure. Theorem 6.5. For any Loday pseudoalgebra structure on an A-module E there is a short exact sequence of morphisms of Loday pseudoalgebras over A, whereĒ 0 -the A-submodule in E generated by {[X, X] : X ∈ E} -is a Loday pseudoalgebra with the trivial left anchor andĒ = E/Ē 0 is a Lie pseudoalgebra.
Note that the Loday ideal E 0 is clearly commutative, while the modular idealĒ 0 is no longer commutative in general.
We first recall the definition of the Loday cochain complex associated to a bi-module over a Loday algebra [31].
Let K be a field of nonzero characteristic and V a K-vector space endowed with a (left) Loday bracket [·, ·]. A bimodule over a Loday algebra (V, [·, ·]) is a K-vector space W together with a left (resp., right) action µ l ∈ Hom(V ⊗ W, W ) (resp., µ r ∈ Hom(W ⊗ V, W )) that verify the following requirements for all x, y ∈ V.
The Loday cochain complex associated to the Loday algebra (V, [·, ·]) and the bimodule (W, µ l , µ r ), shortly -to B = ([·, ·], µ r , µ l ), is made up by the cochain space where we set Lin 0 (V, W ) = W , and the coboundary operator ∂ B defined, for any p-cochain c and any vectors x 1 , . . . , x p+1 ∈ V , by Let now ρ be a representation of the Loday algebra (V, [·, ·]) on a K-vector space W , i.e. a Loday algebra homomorphism ρ : V → End(W ). It is easily checked that µ l := ρ and µ r := −ρ endow W with a bimodule structure over V . Moreover, in this case of a bimodule induced by a representation, the Loday cohomology operator reads Note that the above operator ∂ B is well defined if only the map ρ : V → End(W ) and the bracket [·, ·] : V ⊗ V → V are given. We will refer to it as to the Loday operator associated with B = ([·, ·], ρ). The point is that ∂ 2 B = 0 if and only if [·, ·] is a Loday bracket and ρ is its representation. Indeed, the Loday algebra homomorphism property of ρ (resp., the Jacobi identity for [·, ·]) is encoded in ∂ 2 B = 0 on Lin 0 (V, W ) = W (resp., Lin 1 (V, W )), at least if Let now E be a vector bundle over a manifold M . In Lin • (Sec(E), C ∞ (M )) we can distinguish the subspace D • (Sec(E), C ∞ (M )) ⊂ Lin • (Sec(E), C ∞ (M )) of multidifferential operators.

Supercommutative geometric interpretation
Let E be a vector bundle over a manifold M .
Definition 8.1. For any ℓ ′ ∈ D p (Sec(E), C ∞ (M )) and ℓ ′′ ∈ D q (Sec(E), C ∞ (M )), p, q ∈ N, we define the shuffle product where the X i -s denote sections in Sec(E) and where sh(p, q) ⊂ S p+q is the subset of the symmetric group S p+q made up by all (p, q)-shuffles.
The next proposition is well-known.
Proposition 8.2. The space D • (Sec(E), C ∞ (M )) together with the shuffle multiplication ⋔ is a graded commutative associative unital R-algebra.
We refer to this algebra as the shuffle algebra of the vector bundle E → M , or simply, of E. Moreover, if no confusion is possible, we write D p (E) instead of D p (Sec(E), C ∞ (M )).
Let B = ([·, ·], ρ) be an anchored Kirillov algebroid structure on E and let ∂ B be the associated Loday operator in D • (E). Note that we would have ∂ 2 B = 0 if we had assumed that we deal with a Loday algebroid.
Denote now by D k (E) those k-linear multidifferential operators from D k (E) which are of degree 0 with respect to the last variable and of total degree ≤ k −1, and set D • (E) = ∞ k=0 D k (E). By convention, D 0 (E) = D 0 (E) = C ∞ (M ). Moreover, D 1 (E) = Sec(E * ). It is easy to see that D • (E) is stable for the shuffle multiplication. We will call the subalgebra (D • (E), ⋔), the reduced shuffle algebra, and refer to the corresponding graded ringed space as supercommutative manifold. Let us emphasize that this denomination is in the present text merely a terminological convention. The graded ringed spaces of the considered type are being investigated in a separate work.
Theorem 8.3. The coboundary operator ∂ B is a degree 1 graded derivation of the shuffle algebra of E, i.e.
for any ℓ ′ ∈ D p (E) and ℓ ′′ ∈ D q (E). Moreover, if [·, ·] is a pseudoalgebra bracket, i.e., if it is of total order ≤ 1 and ρ is the left anchor for [·, ·], then ∂ B leaves invariant the reduced shuffle The claim is easily checked on low degree examples. The general proof is as follows.
Proof. The value of the LHS of Equation (62) on sections X 1 , . . . , X p+q+1 ∈ Sec(E) is given by and In the sum S 2 , which is similar to S 1 , the index k runs through {p + 2, . . . , p + q + 1} (X τ k is then missing in ℓ ′′ ). The sum S 3 contains those shuffle permutations of 1 . . .k . . . p + q + 1 that send the argument [X k , X m ] with index m =: τ r into ℓ ′ , whereas S 4 is taken over the shuffle permutations that send [X k , X m ] into ℓ ′′ .
Let us stress that in S 3 and T 2 the bracket is in its natural position determined by the index τ r = m or σ j of its second argument, that, since sh(p, q) ≃ S p+q /(S p × S q ), the number of (p, q)-shuffles equals (p + q)!/(p! q!) , and that in S 1 the vector field ρ(X k ) acts on a product of functions according to the Leibniz rule, so that each term splits. It is now easily checked that after this splitting the number of different terms in ρ(X − ) (resp. [X − , X − ]) in the LHS and the RHS of Equation (62) is equal to 2(p + q + 1)!/(p! q!) (resp. (p + q)(p + q + 1)!/(2 p! q!)). To prove that both sides coincide, it therefore suffices to show that any term of the LHS can be found in the RHS.
We first check this for any split term of S 1 with vector field action on the value of ℓ ′ (the proof is similar if the field acts on the second function and also if we choose a split term in S 2 ), where k ∈ {1, . . . , p + 1} is fixed, as well as τ ∈ sh(p, q) -which permutes 1 . . .k . . . p + q + 1. This term exists also in T 1 . Indeed, the shuffle τ induces a unique shuffle σ ∈ sh(p + 1, q) and a unique i ∈ {1, . . . , p + 1} such that σ i = k. The corresponding term of T 1 then coincides with the chosen term in S 1 , since, as easily seen, sign σ (−1) i+1 = (−1) k+1 sign τ .
Consider now a term in S 3 (the proof is analogous for the terms of S 4 ), where k < m are fixed in {1, . . . , p + q + 1} and where τ ∈ sh(p, q) is a fixed permutation of 1 . . .k . . . p + q + 1 such that the section [X k , X m ] with index m =: τ r is an argument of ℓ ′ .
Finally this term is a term of T 2 , as it is again clear that (−1) k sign τ = sign σ (−1) i .
That D • (E) is invariant under ∂ B in the case of a pseudoalgebra bracket is obvious. This completes the proof.
Note that the derivations ∂ B of the reduced shuffle algebra (in the case of pseudoalgebra brackets on Sec(E)) are, due to formula (61), completely determined by their values on D 0 (E) ⊕ D 1 (E). More precisely, B = ([·, ·], ρ) can be easily reconstructed from ∂ B thanks to the formulae and where X, Y ∈ Sec(E), l ∈ Sec(E * ), and f ∈ C ∞ (M ).
The fact that ρ(X) is a derivation of C ∞ (M ) is a direct consequence of the shuffle algebra derivation property of ∂. Eventually, the map ρ is visibly associated with a bundle map ρ : E →

TM.
The bracket [·, ·] has ρ as left anchor. Indeed, since ∂l(X, Y ) is of order 0 with respect to Y , we get from (66) [ Similarly, as ∂l(X, Y ) is of order 1 with respect to X and of order 0 with respect to Y , the operator see (66), is C ∞ (M )-linear with respect to X and Y and a derivation with respect to f . The bracket [·, ·] is therefore of total order ≤ 1 with the generalized right anchor b r = ρ − α, where α is determined by the identity This corroborates that α is a bundle map from E to TM ⊗ M End(E). shuffle algebra that verify, for any c ∈ D 2 (E) and any X i ∈ Sec(E), i = 1, 2, 3, A homological vector field of the supercommutative manifold (M, D • (E)) is a square-zero derivation in Der 1 (D • (E), ⋔). Two homological vector fields of (M, D • (E)) are equivalent, if they coincide on C ∞ (M ) and on Sec(E * ).
Observe that Equation (68) implies that two equivalent homological fields also coincide on D 2 (E). We are now prepared to give the main theorem of this section.
Theorem 8.6. Let E be a vector bundle. There exists a 1-to-1 correspondence between equivalence classes of homological vector fields and Loday algebroid structures on E.
Remark 8.7. This theorem is a kind of a non-antisymmetric counterpart of the well-known similar correspondence between homological vector fields of split supermanifolds and Lie algebroids. Furthermore, it may be viewed as an analogue for Loday algebroids of the celebrated Ginzburg-Kapranov correspondence for quadratic Koszul operads [7]. According to the latter result, homotopy Loday structures on a graded vector space V correspond bijectively to degree 1 differentials of the Zinbiel algebra (⊗sV * , ⋆), where s is the suspension operator and wherē ⊗sV * denotes the reduced tensor module over sV * . However, in our geometric setting scalars, or better functions, must be incorporated (see the proof of Theorem 8.6), which turns out to be impossible without passing from the Zinbiel multiplication or half shuffle ⋆ to its symmetrization ⋔. Moreover, it is clear that the algebraic structure on the function sheaf should be associative.
Its left anchor is ρ = ρ ∂ and the generalized right anchor b r = ρ − α is determined by means of formula (67), where l runs through all sections of E * .
It is clear that the just detailed assignment of a Loday algebroid structure to any homological vector field can be viewed as a map on equivalence classes of homological vector fields.
Having a homological vector field ∂ associated with a Loday algebroid structure ([·, ·], ρ, α) on E, we can easily develop the corresponding Cartan calculus for the shuffle algebra D • (E).

The next proposition is obvious.
Proposition 8.9. The supercommutator L X := [∂, i X ] sc = ∂i X +i X ∂, X ∈ Sec(E), is a degree 0 graded derivation of the shuffle algebra. Explicitly, for any ℓ ∈ D p (E) and X 1 , . . . , X p ∈ Sec(E), (L X ℓ)(X 1 , . . . , X p ) = ρ(X) (ℓ(X 1 , . . . , X p )) − i ℓ(X 1 , . . . , We refer to the derivation L X as the Loday algebroid Lie derivative along X. If we define the Lie derivative on the tensor algebra T R (E) = ∞ p=0 Sec(E) ⊗ R p in the obvious way by and if we use the canonical pairing ℓ, X 1 ⊗ R . . . ⊗ R X p = ℓ(X 1 , . . . , X p ) between D • (E) and T R (E), we get The following theorem is analogous to the results in the standard case of a Lie algebroid E = TM and operations on the Grassmann algebra Ω(M ) ⊂ D • (TM ) of differential forms. Observe finally that Item (d) of the preceding theorem actually means that where the RHS is the restriction to interior products of the derived bracket on Der(D • (E), ⋔) defined by the graded Lie bracket [·, ·] sc and the interior Lie algebra derivation [∂, ·] sc of Der(D • (E), ⋔) induced by the homological vector field ∂.