Some generalizations of the notion of Lie algebra

We introduce the notion of left (and right) quasi-Loday algebroids and a"universal space"for them, called a left (right) omni-Loday algebroid, in such a way that Lie algebroids, omni-Lie algebras and omni-Loday algebroids are particular substructures.


Introduction
There are several ways in which Lie algebras can be generalized. Recall that if M is an R-module 1  ) is a Lie algebra structure over M . When R = R (resp. C) we speak about a real Lie algebra (resp. a complex Lie algebra). Let us briefly mention some of these generalizations: a) We can lift the restriction of antisymmetry. We then get the notion of Loday (or Leibniz 2 ) algebra [8,6]. More precisely, the pair (M, [ , ]) is a left Loday algebra if, instead of conditions (i), (ii) above, it satisfies the left Leibniz identity: [u, v], w] + [v, [u, w]]. 1 Here R is a commutative ring with unit element 1 R . 2 We prefer the denomination "Loday algebra", see [6] for an explanation.
Note that this condition can be expressed by saying that [u, ] is a derivation with respect to the product [ , ]. Analogously, we can define right Loday algebras over M , by imposing that [ , w] be a derivation with respect to [ , ]: (iii') [[u, v], w] = [ [u, w], v] + [u, [v, w]] Note that any one of (iii) or (iii') is equivalent to the Jacobi identity when [ , ] is antisymmetric.
b) Also, it is possible to consider a family of Lie algebras parametrized by points on a manifold M which, with some natural geometric assumptions, lead to the idea of Lie algebroid introduced by J. Pradines (see [5,9]). To be precise, a Lie algebroid over a manifold M (assume it real for simplicity) is given by a vector bundle π : E → M , an R-bilinear bracket [ , ] : ΓE×ΓE → ΓE defined on the C ∞ (M )-module of sections of E, and a mapping q E : ΓE → X(M ) (called the anchor map) such that, for all X, Y ∈ ΓE, f ∈ C ∞ (M ): (1) (ΓE, [ , ]) is a real Lie algebra, Note that, in the case when M reduces to a single point, a Lie algebroid over M = { * } is just a Lie algebra. A basic property of Lie algebroids is that the anchor map q E is a Lie algebra morphism when the bracket on X(M ) is taken as the Lie bracket of vector fields [7,2]. c) Finally, A. Weinstein introduced in [15] the concept of omni-Lie algebra, a structure that can be thought of as a kind of "universal space" for Lie algebras: take any natural number n ≥ 2 and consider the space product E n = gl n × R n endowed with the R-bilinear form { , } : E n × E n → E n given by is the n-dimensional omni-Lie algebra. The reason behind this denomination is that any n-dimensional real Lie algebra g is a closed maximal subspace of (E n , { , }).
Our goal is to define a structure for which the constructions mentioned in a),b),c) appear as particular cases. In an absolutely unimaginative way, we will call it a left omni-Loday algebroid (of course, there exists the corresponding "right" definition). As we will see, this also include as a particular case the notion of omni-Lie algebroid. Actually, the object we will construct will carry on a bracket that has already appeared in the literature, although under a different approach. In the paper [3], M. K. Kinyon and A. Weinstein attacked the problem of integrating (in the sense of S. Lie's "Third Theorem") a Loday algebra 3 , and they gave the following example: Take (h, [ , ]) a Lie algebra, and let V be an h−module with left action on V given by (ζ, x) → ζx. Then, we have the induced left action of h on h × V ζ(η, y) = ([ζ, η], ζy).
A binary operation · can be defined on It turns out that (E, ·) is a Loday algebra, and if h acts nontrivially on V , then (E, ·) is not a Lie algebra. Kinyon and Weinstein called E with this Loday algebra structure the hemisemidirect product of h with V . Our omni-Loday algebroid will be a particular case of this construction, taking gl(V ) as h (see Definition 4).
To achieve our goal, let us note that it is necessary to recast the definition of a Lie algebroid in a form more suitable to an algebraic treatment, as in a), c). This can be easily done, just note that C ∞ (M ) can be replaced by any Ralgebra A, with unit element 1 A and commutative, ΓE by a faithful A-module F , and X(M ) by the module of derivations Der R (A).
This idea was cleverly exploited by J. Grabowski who, in the paper [2], used it to prove the property of the anchor map of being a Lie algebra morphism. In the same paper, it is proved that there exist obstructions to the existence of Loday algebroid structures on vector bundles over a manifold M , stated in terms of the rank of these bundles (see Theorems 3, 4 below). As we will see, we can bypass these obstructions by considering left and right structures separately.
Acknowledgements The authors want to express their gratitude to Prof. M. K. Kinyon and Prof. Y. Sheng for their useful comments and for providing references to previous work on Loday algebras.

Quasi-derivations
The basic properties of a Lie algebroid are encoded in its anchor map, which in this context is a mapping ρ : F → Der R (A). We will assume that F is endowed with an R-bilinear bracket [[ , ]], then ρ is determined by two adjoints maps ad L A , ad R A : F → End R (F ), given respectively by ad L Under certain mild conditions, these mappings are quasi-derivations of F , a property which is basic in the study of ρ. For instance, the fact that ad L A , ad R A are quasi-derivations allows one to prove that ρ is a morphism of Lie algebras (when (F , [[ , ]]) is Lie and we take the commutator of endomorphisms as the bracket on End R (F )), see [2] (we refer the reader to that paper for the proof of the results stated in this section). We recall that an operator D ∈ End R (F ) is a quasi-derivation if for a given f ∈ A there exists g ∈ A such that is the commutator of endomorphisms of F , and µ h (X) = h.X, for any h ∈ A, X ∈ F . A quasi-derivation is called a tensor operator when Note that this is equivalent to D being A-linear (and not just R-linear). Some other straightforward properties of quasi-derivations are: (1) The set of all the quasi-derivations of F , QDer R (F ) is an R-module.
(2) The commutator of endomorphisms on Der R (F ) restricts to a closed bracket on is not just a Lie algebra, but also a Poisson algebra (with the product given by the composition of endomorphisms).
The following results will be crucial in the sequel.

Theorem 1.
There exists an R-linear mapping : The R-linear mapping extends to a Lie algebra morphism: Combining (4) above with Theorem 1, we also get:

Left Loday quasi-algebroids
The formula obtained in Corollary 2 looks very similar to condition (b2) in the definition of Lie algebroid. We can formalize this observation generalizing at once the definition, simply by replacing the Lie structure on ΓE (our F in the algebraic setting) by a Loday one. Thus, let (F , [[ , ]]) be a left Loday algebra. Given an X ∈ F , denote by ad L X , ad X R : F → F the endomorphisms This amount to the condition that, given X ∈ F , f ∈ A: and motivates the following definition.
this justifying the terminology with the "left" prefix.

Remark 1.
There is the corresponding notion of right Loday quasi-algebroid, when ad R X ∈ QDer R (F ). In this case, the formula reads ) is a morphism of left Loday R-algebras.

Proof. First, let us note that the condition of [[ , ]] being a Loday bracket on
As this is valid for all Z ∈ F , we get the stated equivalence. Now, Corollary 1 says that ∀ X, Y ∈ F : Remark 2. This result partly answers a question raised in Remark 3.3 (1) of [14].
The definitions just given can be particularized to the case of Lie algebras (i.e, [[ , ]] antisymmetric).
Remark 3. Note that in this case the distinction between the left and right cases is irrelevant: each left Lie quasi-algebroid with anchor q F is also a right Lie quasi-algebroid with anchor −q F .

Generation of Loday algebroids
As we have seen in the previous section, in order to get genuine examples of Loday quasi-algebroids, we must avoid that the two conditions ad L X ∈ QDer R (F ) and ad R X ∈ QDer R (F ) be satisfied simultaneously.
To get examples of this situation, it is useful to know how to generate Loday brackets from operators with certain features. First of all, note that given a left Loday bracket  then we can define: which satisfies the properties of R−bilinearity and the left Leibniz rule (so, it is a left Loday algebra).
Proof. Let us check first the R−bilinearity For the left Leibniz rule, we have: (c) A projector D, that is, D is an algebra morphism and D 2 = D. Then: Now, we can give a simple example of a left Loday quasi-algebroid which does not admit a right Loday quasi-algebroid structure.
Example 2. Consider F = Ω(R 6 ), which is an R-algebra with the exterior product ∧ and, moreover, a C ∞ (R 6 )-module (i.e R = R, A = C ∞ (M )). Define: is a left Loday algebra, as d is a square-zero operator (see (b) above). Now, we have, for any α, β ∈ Ω(R 6 ), f ∈ C ∞ (R 6 ): Thus, on the other hand, if f = x 6 , α = dx 1 ∧ dx 2 ∧ dx 3 ∧ dx 4 , β = dx 5 : Thus, (Ω(R 6 ), [[ , ]]) has a left Loday quasi-algebroid structure, with anchor q L Ω(R 6 ) ≡ 0, but it does not admit a right Loday quasi-algebroid structure. Note that q L Ω(R 6 ) is (trivially) tensorial. The following, less trivial, example was suggested to us by Y. Sheng. It shows that the kind of structures we are considering can appear in the more general context of higher order Courant algebroids (although here we just take n = 1 for simplicity) through the associated Dorfman bracket, see [11].
Example 3. Let M be a differential manifold. Consider the vector bundle T M ⊕ T * M whose sections are endowed with the Dorfman bracket: is a left Loday quasi-algebroid, with anchor map the projection onto the first factor: q L T M⊕T * M (X + α) = ad L X+α = X. Note that in this case the anchor is tensorial: If f, g ∈ C ∞ (M ) and X + α ∈ T M ⊕ T * M , then is indeed a left Loday algebroid. However, for ad R X+α we find: clearly spoils the possibility that ad R X+α be a quasi-derivation.

Left omni-Loday algebroids and omni-Lie algebroids
Having established the non-triviality of left Loday quasi-algebroids, we now turn to the question of whether an analogue of Weinstein's omni-Lie algebra exists for these structures. As before, let A be an associative algebra, commutative and with unit element 1 A over a ring R, commutative and with unit element 1 R . Also, let F a faithful A-module.
Definition 4. Consider the product space gl(F ) × F and define the bracket where [ , ] is he commutator of endomorphisms. Proof. F B is closed with respect to { , } if, and only if, for all X, Y ∈ F we have: that is, for all Z ∈ F : (that is, ad L X ∈ QDer R (F )). Proof. If (F , B) is a left Loday quasi-algebroid, it is also a left Loday algebra and then, by Proposition 2, F B is closed under { , }. On the other hand, the condition of being quasi-algebroid implies that for all X, Z ∈ F and for all For the second implication, consider F B closed with respect to { , }, so (F , B) is a left Loday algebra (see Proposition 2). Moreover, for all X, Z ∈ F and for all f ∈ A: Thus, (F , B) is a left quasialgebroid with anchor map ρ.
Let us try to get rid of the "quasi" prefix. Theorem 6. Let B : F × F −→ F be an R-bilinear form and ρ : F −→ Der R (A) a morphism of R-modules. Suposse that F B is closed with respect to B and that B is such that (that is, ad L X ∈ QDer R (F )). Then (F , B, ρ) is a left Loday algebroid.
Proof. We only need to prove that ρ is tensorial or, equivalenty, that for all X, Z ∈ F and for all g, f ∈ A the following holds: But the hypothesis in the statement guarantee that: Thus, (F , B, ρ) is a left Loday algebroid.
Remark 5. However, we can not say anything about the converse, as the Example 2 shows (there, we have a left Loday algebroid and the first condition (a) above is trivially satisfied while (b) is not).
We can avoid the "quasi" prefix if we add the condition of antisymmetry to B, thus entering into the realm of Lie structures. B(X, f.Z) = f.B(X, Z) + ρ(X)(f ).Z (that is, ad L X ∈ QDer R (A)). Proof. If (F , B, ρ) is a Lie algebroid, (F , B) is a Lie algebra, that is, (F , B) is a left (and right) Loday algebra and B is antisymmetric, so Proposition 2 tells us that F B is closed with respect to { , }. Now, let X, Z ∈ F , f ∈ A; then we have [ad L X , µ f ](Z) = µ ρ(X)(f ) (Z), which is the same as B(X, f.Z) = f.B(X, Z) + ρ(X)(f ).Z.
If now is F B closed with respect to { , }, Proposition 2 again tells us that (F , B) is a left Loday algebra, but as B is also antisymmetric, (F , B) is a Lie algebra. On the other hand, the hypothesis of Theorem 5 are satisfied, so we know that ad X is a quasi-derivation for all X ∈ F and (F , B) is a Lie quasi-algebroid with anchor map ρ. To finish, let us check (see Theorem 6) that for all X, Y, Z ∈ F , ∀ f ∈ A the following holds: The preceding results motivate the following definition.
Definition 5. Let A be an associative, commutative algebra with unit element 1 A over a commutative ring with unit element 1 R . Let F be an A-module. We call (gl(F ) × F , { , }) the left omni-Loday algebroid determined by F . Remark 7. In the case of Lie algebroids, we have the same situation as in the preceding remark: given an R-bilinear F -valued form B : F × F −→ F such that it is antisymmetric and satisfies ad L X ∈ QDer R (F ), by Theorem 7 there is a correspondence between Lie algebroids (F , B, ρ) and closed subspaces F B , but this time given by an "if and only if" statement. Thus, we could call (gl(F ) × F , { , }) an omni-Lie algebroid as well.
It is worth noting that a different definition for omni-Lie algebroids (based on the notion of Courant structures on the direct sum of the gauge Lie algebroid and the bundle of jets of a vector bundle E over a manifold M ), has been presented very recently in [1]. It would be interesting to know if this definition is equivalent to ours.