An Approach to Omni-Lie Algebroids Using Quasi-Derivations

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. MSC 2010: 17A32, 53D17, 58H99

Let us briefly mention some of these generalizations.
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 parameterized by points on a manifold M which, with some natural geometric assumptions, leads 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 [ , ] : Γ E × Γ E → Γ E defined on the C ∞ (M )-module of sections of E, and a mapping q E : Γ E → X(M ) (called the anchor map) such that, for all X, Y ∈ Γ E, f ∈ C ∞ (M ): (1) (Γ E, [ , ]) is a real Lie algebra, (2) 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 [2,7]. (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 En = gl n × R n endowed with the R-bilinear form { , } : En × En → En given by The reason behind this denomination is that any n-dimensional real Lie algebra g is a closed maximal subspace of (En, { , }).
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 includes 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 [4], 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 : 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 17).
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 R-algebra A, with unit element 1 A , and commutative Γ E 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 [2] who used it to prove that the property of the anchor map is 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 11, 12 below). As we will see, we can bypass these obstructions by considering left and right structures separately.

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 adjoint 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 us 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 where [ , ] 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 as follows: (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 Defining, for any f ∈ A and D ∈ QDer R (F ), 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 :

Corollary 2.
The R-linear mapping extends to a Lie algebra morphism: Combining (4) above with Theorem 1, we also get the following corollary.

Left Loday quasi-algebroids
The formula obtained in Corollary 3 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 Γ E (our F in the algebraic setting) by a Loday one. Thus, let (F , [[ , ]]) be a left Loday algebra. Given an X ∈ F, denote by Note that if [[ , ]] is antisymmetric, then ad L is called a left Loday quasi-algebroid if ad L X ∈ QDer R (F ), for all X ∈ F. This amounts to the condition that, given X ∈ F, f ∈ A, and motivates the following definition.

The condition in Definition 4 now reads
with this justifying the terminology with the "left" prefix.

Remark 6.
There is the corresponding notion of right Loday quasi-algebroid, when ad R X ∈ QDer R (F ). In this case, the formula reads Proof. First, let us note that the condition of [[ , ]] being a Loday bracket on F means that To check this, let Z ∈ F and compute As this is valid for all Z ∈ F, we get the stated equivalence. Now, Corollary 2 says that for all X, Y ∈ F, The definitions just given can be particularized to the case of Lie algebras (i.e. [[ , ]] antisymmetric).
Remark 10. Note that in this case the distinction between the left and right cases is irrelevant: each left Lie quasialgebroid with anchor q F is also a right Lie quasi-algebroid with anchor −q F . How different are left (and right) Loday quasi-algebroids, Lie quasi-algebroids and Loday algebroids? In some cases, there is no such distinction: if we take R = R, A = C ∞ (M ), F = Γ E, with π : E → M a vector bundle over a manifold M , Grabowski calls a QD-Loday (resp. Lie) algebroid a left Loday (resp. Lie) quasi-algebroid (i.e. ad L X ∈ QDer R (F ), for all X ∈ F) such that ad R X ∈ QDer R (F ), for all X ∈ F; then, he proves the following. Theorem 11. Every QD-Loday algebroid (resp. QD-Lie) with rank ≥ 1 is a Loday algebroid (resp. Lie).

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 ) are satisfied simultaneously.  If (A, ·) is an associative R-algebra 4 and, moreover, is endowed with an R-linear mapping D : 4 That is, A is an R-module endowed with an associative product · : A × A → A.
Journal of Generalized Lie Theory and Applications 5 then one 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 quasialgebroid structure.

Thus, on the other hand, if
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 16. Let M be a differential manifold. Consider the vector bundle T M ⊕T * M whose sections are endowed with the Dorfman bracket: Then we have a left Loday algebra, as [[ , ]] is clearly R-bilinear and is a left Loday quasi-algebroid, with anchor map the projection onto the first factor: 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 is 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 that is commutative and with unit element 1 R . Also, let F be a faithful A-module.
Definition 17. Consider the product space gl(F ) × F and define the bracket where [ , ] is the commutator of endomorphisms.
Remark 18. It is straightforward to check that { , } is R-bilinear. However, it does not satisfy Jacobi's identity (here denotes cyclic sum), as we have As stated in the introduction, the bracket { , } satisfies instead of the left Leibniz identity the following: Now, let B : F × F → F be an R-bilinear form. Define the "graph" of B as which is the left Leibniz identity, that is, (F , B) is a left Loday algebra.
For left Loday quasi-algebroids, we have the following.
Proof. If (F , B) is a left Loday quasi-algebroid, it is also a left Loday algebra and then, by Proposition 19, 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 f ∈ A, we have that is, For the second implication, consider F B closed with respect to { , }, so (F , B) is a left Loday algebra (see Proposition 19). Moreover, for all X, Z ∈ F and for all f ∈ A, that is, ad L X is a quasi-derivation for all X ∈ F. Thus, (F , B) is a left quasi-algebroid with anchor map ρ.
Let us try to get rid of the "quasi" prefix.
Proof. On one hand, applying (a) then (b), and on the other hand, first applying (b), Let {Y j } j∈I be a basis of F . Taking X = Y i and Z = Y j for some distinct i, j ∈ I in (5.1), we have for all f, g ∈ A. Now, let X = j∈I k j · Y j and f ∈ A. By the R-linearity of ρ and (5.2), that is, ρ is tensorial. Thus, (F , B, ρ) is a left Loday algebroid.
Remark 22. However, we can not say anything about the converse, as Example 15 shows (there, we have a left Loday algebroid and the first condition (a) above is trivially satisfied while (b) is not).
We also can avoid the "quasi" prefix if we add the condition of antisymmetry to B, thus entering into the realm of Lie structures.
Theorem 23. Let F be a free module of rank k > 1, B : F × F → F an R-bilinear form, and ρ : F → Der R (A) a morphism of R-modules. Then, (F , B, ρ) is a Lie algebroid if and only if F B is closed with respect to { , }, B is antisymmetric and, for all X, Z ∈ F and f ∈ A, the following holds: 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 19 tells us that F B is closed with respect to { , }. Now, let X, Z ∈ F, f ∈ A; then we have which is the same as If now is F B closed with respect to { , }, Proposition 19 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 20 is 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 21) that for all X, Y, Z ∈ F and for all f ∈ A the following holds: So (F , B, ρ) is a Lie algebroid.
The preceding results motivate the following definition.
Definition 24. Let A be an associative, commutative algebra with unit element 1 A over a commutative ring with unit element 1 R . Let F be a free A-module of rank k > 1. We call (gl(F ) × F, { , }) the left omni-Loday algebroid determined by F . (F , B, ρ) is a left Loday algebroid then, in particular, it is a left Loday quasi-algebroid and thus F B ⊂ gl(F ) × F is closed with respect to { , }, as by Theorem 20: every left Loday algebroid can be seen as a closed subspace of left omni-Loday algebroid.

Remark 25. Note that if
Remark 26. In the case of Lie algebroids, we have the same situation as in the preceding remark: given an Rbilinear F -valued form B : F × F → F such that it is antisymmetric and satisfies ad L X ∈ QDer R (F ), by Theorem 23 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.