Extended Curvatures and Lie Algebroids

: The aim of the paper is to ﬁnd conditions in which the derived almost Lie vector bundle E ( 1 ) of an almost Lie vector bundle E is a Lie algebroid. The conditions are that some extended curvatures on E , considered in the paper, are vanishing. Two non-trivial examples are given. One example is when E 0 is a skew symmetric algebroid; the other one is when E 1 is not a skew symmetric algebroid.


Introduction
The Lie algebroids [1][2][3][4] are a good setting to consider some generalizations or extensions of classical constructions and notions from differential geometry, mechanics, physics, etc. Some less restrictive conditions in the definition of a Lie algebroid are studied under different names and using different forms (see [5] for an up-to-date review). Non-skew symmetric brackets can be involved, as in [6] or [7] and the references therein. In the present paper, we consider only skew symmetric brackets (as in [8][9][10]). Complex and/or Poisson geometry can be involved, as in [11]. An algebraic setting is studied in [12]. The Banach space setting is studied in [13][14][15][16].
Anchored vector bundles are considered in [17], where they are called relative tangent spaces; an associated skew symmetric bracket gives rise to an almost Lie vector bundle, called an almost Lie structure in [18][19][20]. In [19], it is proven that the anchor and bracket couple is in a one-to-one correspondence with a one-degree derivation of the exterior algebra of the given vector bundle. The Jacobiator of the bracket is linear only for skew symmetric algebroids, when the anchor compatibility of the brackets holds (see 3 of Proposition 1). The vanishing Jacobiator defines a Lie algebroid.
The first derived almost Lie vector bundle of a given almost Lie vector bundle is considered in [18], and it plays an important role in the present paper. A linear E-connection ∇ having a null torsion is used in the background. We define two extended curvatures of ∇ and we prove that if both these two extended curvatures vanish, then the derived almost Lie vector bundle E (1) is a Lie algebroid (the main Theorem 1).
In [10], we have considered an example of a skew symmetric algebroid E 0 whose anchor does not have a skew symmetric bracket with a vanishing Jacobiator, i.e., the anchor does not allow a Lie algebroid structure, but the derived almost Lie vector bundle E (1) 0 is a Lie algebroid. The calculations to settle this example are direct, but too long and only stated in the paper, without an effective proof. In the present paper, we give a different effective proof of this result, also extending the case n = 2 to an arbitrary n ≥ 2.
Two examples when Theorem 1 applies (involving the existence of two vanishing extended curvatures) are given. One example is when E 0 is a skew symmetric algebroid (the above mentioned example) and the other one is when E 1 is not a skew symmetric algebroid.
Here, E ∧ E denotes the exterior product bundle of E (i.e., the exterior product on each fiber). Now consider an almost Lie vector bundle E and let π A : A → M be a vector bundle over the same base M. A linear E-connection on A is a map ∇ : Γ(E) × Γ(A) → Γ(A) that verifies Koszul conditions: The curvature of ∇ is the map R : Γ(E) 2 × Γ(A) → Γ(A), given by the formula Proof. One can check that for a given s ∈ Γ(A), (X, Y) → R(X, Y)s is F (M)-linear in both arguments. However, for given X, Y ∈ Γ(E), the map s → R X∧Y s is additive, and for f ∈ F (M) we have This implies the conclusion.
In the particular case of an E-connection ∇ on E, we can consider its torsion given by the formula (∀)X, Y, Z ∈ Γ(E). The last notation above defines a linear E ∧ E-connection ∇ ∧ on E, according to the anchor ρ ∧ . The formula Notice that when the torsion vanishes, then If the E-connection ∇ is given, then the above Formula (1) defines the bracket [·, ·] E on E, such that ∇ has a null torsion.

The Derived Almost LIE Vector Bundle
Let E be an almost Lie vector bundle and ∇ be an E-connection on E that has a null torsion.

Proposition 4.
If the linear E-connection ∇ on the almost Lie vector bundle E has no torsion, then the following properties hold true: Proof. In order to prove 1, we have: In order to prove the second equality, we have:

Considering the analogous expressions for
, by summing, we obtain the second equality.
Besides the vanishing components, the Jacobiator on E (1) has other components that can be handled using two other curvatures involving ∇, which we define in the sequel.
Thus, let us consider the extended curvatures of ∇ (the first extended curvature and the second extended curvature, respectively). If f ∈ F (M) and X, Y, Z, W, U ∈ Γ(E), then we have and also Notice that, according to their properties, we can consider Using Proposition 2 for E (1) , we obtain the following true statement.

Proposition 6.
Let E be an almost Lie vector bundle. Then, the following conditions are equivalent: More exactly, Condition 2 can be read as follows.
Proof. Using 1 of Proposition 4, it follows that the condition D (1) = 0 reads D (1) The conclusion follows.
In the particular case when E is a skew symmetric algebroid, we obtain the following statement proven in [10] (Proposition 2.5). Proof. Using 2 of Proposition 6, it follows that D (1) = 0; then, using Formulas (4) and (5), the conclusion follows.
We can prove now the main result of the paper.

Theorem 1. Let E be an almost Lie vector bundle. If E (1) is a skew symmetric algebroid and both its extended curvatures vanish, then E (1) is a Lie algebroid.
Proof. Using Proposition 8, it follows that the extended curvatures are F (M)-linear in all arguments; thus, the vanishing conditions have sense. Moreover, using 2 of Proposition 4 and Proposition 5, we have J (1) = 0; thus, the conclusion follows.
Effectively, the conditions in the hypothesis of the above theorem are expressed by Relations (6) and (7), and

Some Examples
We consider below two relevant examples.
In the first example, we consider a skew-symmetric algebroid E 0 on R n that is not a Lie algebroid, and we prove that its derived almost Lie vector bundle E (1) 0 is a Lie algebroid. It is an extension of an example in [10], where the case n = 2 is considered and where it is proven that in this case, there is no algebroid Lie bracket associated with the anchor [10] (Theorem 2.3).
In the second example, we consider an almost Lie vector bundle E 1 that is not a skew symmetric algebroid, but its derived almost Lie vector bundle E 1 is a Lie algebroid. We proceed now with the first example. Let us consider the vector bundle E 0 = I R n × M n (I R) → I R n on the base manifold M = I R n , where M n (I R) is the set of square n-matrices with real entries. The anchor ρ on E 0 is defined as follows. In every point It is easy to see that the image by ρ of the sections of E 0 generates the whole tangent space Tx I R n forx =0 and {0} ⊂ T0 I R n forx =0 = (0, . . . , 0). A section on E 0 is in ker ρ if it is an F (I R n )-combination of sections X ijk = x j 2 X i k − x i 2 X j k , where 1 ≤ i < j, k ≤ n. We notice that these n 2 (n−1) 2 sections do not generate a (regular) vector sub-bundle of E 0 . Associated with the above anchor, we consider the bracket [·, ·] E 0 defined on generators by and the linear E 0 -connection ∇ on E 0 defined on generators by It is easy to see that ρ X i j , X k The curvature of ∇ is linear in all arguments and Since ρ ∇ X i j ∧X k l X u v = 0, then, using Corollary 1, it follows that the derived bundle is a skew symmetric algebroid as well. Using Proposition 8, it follows that the extended curvatures are F (R n )-linear in their arguments.
We proceed now with the next example, where Theorem 1 can also be used also in the case when E = E 1 is not a skew symmetric algebroid.
Consider M = R 2n+1 with coordinates x i , y i , z i=1,n and the vector fields Their Lie brackets are given by Let E 1 be the vector bundle generated by X i , Y j i,j=1,n . The anchor ρ 1 : is the natural inclusion. The corresponding bracket [·, ·] on E 1 extends the following values on generators: We consider also the linear E 1 -connection ∇ on E 1 , which extends the following values on generators: is not a skew symmetric algebroid and the curvature R is not F R 2n+1linear in all arguments.

Conclusions
A new construction providing Lie algebroids is considered in the paper. Some relaxed conditions in the Lie algebroid definition give rise to other kinds of structures. For a general almost Lie vector bundle, the Jacobiator can be non-null or nonlinear. In this paper, we consider the derived almost Lie vector bundle E (1) of a given almost Lie vector bundle E, and we define two extended curvatures of a given linear E-connection ∇ with null torsion. We prove that if these two extended curvatures vanish, then E (1) is a Lie algebroid. Two given examples show that the result can be applied not only when E is a skew-symmetric algebroid, but also when E is not.
Author Contributions: Both authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.