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The first Pontryagin class of a quadratic Lie 2-algebroid

In this paper, first we give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid TM to a strict Lie 3-algebroid constructed from derivations. Then we introduce the notion of a quadratic Lie 2-algebroid and define its first Pontryagin class, which is a cohomology class in H^5(M). Associated to a CLWX 2-algebroid, there is a quadratic Lie 2-algebroid naturally. Conversely, we show that the first Pontryagin class of a quadratic Lie 2-algebroid is the obstruction class of the existence of a CLWX-extension. Finally we construct a quadratic Lie 2-algebroid from a trivial principle 2-bundle with a Gamma-connection and show that its first Pontryagin class is trivial.


Introduction
The notion of a Lie algebroid was introduced by Pradines in 1967, which is a generalization of Lie algebras and tangent bundles. Just as Lie algebras are the infinitesimal objects of Lie groups, Lie algebroids are the infinitesimal objects of Lie groupoids. See [16] for general theory about Lie algebroids. A transitive Lie algebroid (i.e. ρ is surjective) (A, [·, ·] A , ρ) is called a quadratic Lie algebroid if there is a nondegenerate symmetric invariant bilinear form S on ker(ρ). A connection of a quadratic Lie algebroid is a splitting σ : T M −→ A of the short exact sequence The curvature R ∈ Hom(∧ 2 T M, ker(ρ)) is defined by Then the first Pontryagin class of a quadratic Lie algebroid is defined to be the cohomology class [S(R, R)] in H 4 (M ).
The notion of a Courant algebroid was introduced in [14] in the study of the double of a Lie bialgebroid. An alternative definition was given in [20]. See the review article [12] for more information. Roughly speaking, a Courant algebroid is a vector bundle E, whose section space is a Leibniz algebra, together with an anchor map ρ : E −→ T M and a nondegenerate symmetric bilinear form S, such that some compatibility conditions are satisfied. For a transitive Courant algebroid (i.e. ρ is surjective), E/ρ * (T * M ) is a quadratic Lie algebroid. Conversely, it is proved in [8,9] that a quadratic Lie algebroid admits a Courant-extension if and only if its first Pontryagin class is trivial.
Recently, people have paid more attention to higher categorical structures by reasons in both mathematics and physics. A Lie 2-algebra is the categorification of a Lie algebra [3]. The Jacobi identity is replaced by a natural transformation, called the Jacobiator, which also satisfies some coherence laws of its own. If a skew-symmetric bracket is used in the definition of a Courant algebroid, the authors showed that the underlying algebraic structure of a Courant algebroid is a Lie 2-algebra [21]. Along this approach, the author defined split Lie n-algebroids in [26] using the language of graded vector bundles. Usually an NQ-manifold of degree n is considered as a Lie n-algebroid [27]. The equivalence between the category of split Lie n-algebroids and the category of NQ-manifolds of degree n was given in [6]. CLWX 2-algebroids (named after Courant-Liu-Weinstein-Xu) were introduced in [13], which can be viewed as the categorification of Courant algebroids, and one-to-one correspond to symplectic NQ-manifolds of degree 3. See [11] for more applications in 4D topological field theory. Similar as the case of Courant algebroids, we can obtain a quadratic Lie 2-algebroid (called the ample Lie 2-algebroid) from an exact CLWX 2-algebroid by modulo ρ * (T * M ). Then it is natural to ask the following question: • Whether every quadratic Lie 2-algebroid admits a CLWX-extension? If not, what is the obstruction?
Motivated by the above question, first we study the structure of a transitive Lie 2-algebroid (A −1 ⊕ A 0 ; ρ, l 1 , l 2 , l 3 ). It turns out that a transitive Lie 2-algebroid is a nonabelian extension of the tangent Lie algebroid T M by a graded bundle of Lie 2-algebra A −1 ⊕ker(ρ). By choosing a splitting (a connection), we give precise formulas of a transitive Lie 2-algebroid. One important property that we need to stress here is that the corresponding curvatures include a ker(ρ)-valued 2-form and an A −1 -valued 3-form. Furthermore, we construct a strict Lie 3-algebroid using derivations 2 Preliminaries

Leibniz 2-algebras
As a model for "Leibniz algebras that satisfy Jacobi identity up to all higher homotopies", the notion of a strongly homotopy Leibniz algebra, or a Lod ∞ -algebra was given in [15] by Livernet, which was further studied by Ammar and Poncin in [1]. In [25], the authors introduced the notion of a Leibniz 2-algebra, which is the categorification of a Leibniz algebra, and proved that the category of Leibniz 2-algebras and the category of 2-term Lod ∞ -algebras are equivalent. Definition 2.1. A Leibniz 2-algebra V consists of the following data: • a complex of vector spaces V : • bilinear maps l 2 : • a trilinear map l 3 such that for all w, x, y, z ∈ V 0 and m, n ∈ V −1 , the following equalities are satisfied: (f) the Jacobiator identity: We usually denote a Leibniz 2-algebra by (V −1 ⊕ V 0 ; d, l 2 , l 3 ), or simply by V. In particular, if both l 2 and l 3 are skew-symmetric, we obtain the notion of a Lie 2-algebra [3]. and

Lie n-algebroids and CLWX 2-algebroids
The notion of a split Lie n-algebroid was introduced in [26].
A split Lie n-algebroid is said to be strict if l i = 0 for all i > 2. In this paper, we will only use split Lie 2-algebroids and strict split Lie 3-algebroids. See [6] for more details about the category of Lie n-algebroids. Lemma 2.4. Let (A; ρ, l 1 , l 2 , · · · , l n+1 ) be a Lie n-algebroid. Then we have When n = 1, a Lie 1-algebroid is exactly a Lie algebroid. Associated to any vector bundle E, the covariant differential operator bundle D(E) is a Lie algebroid naturally. Let (A; a, l 2 ) be a Lie algebroid and ∇ : A −→ D(E) a bundle map. Then there is a differential operator for all X 1 , · · · , X k+1 ∈ Γ(A). d 2 ∇ = 0 if and only if ∇ is a Lie algebroid morphism, i.e. a representation of A on E. See [16] for more details.

Transitive Lie 2-algebroids
In this section, we study the structure of a transitive Lie 2-algebroid. In Subsection 3.1, by choosing a splitting, we obtain some structure maps and we give the conditions that these structure maps satisfy. Then in Subsection 3.2, we characterize these conditions by a morphism from the tangent Lie algebroid T M to a strict Lie 3-algebroid constructed from derivations of a graded bundle of Lie 2-algebras.

General description of a transitive Lie 2-algebroid
and a bundle map γ : Here denotes the inclusion map.
After choosing a splitting, we have A 0 ∼ = T M ⊕ G and ρ is simply the projection pr T M . Define Define the bundle map R : Define the bundle map I : Define totally skewsymmetric bundle map J : Define totally skewsymmetric bundle map K : Transfer the transitive Lie 2-algebroid structure from , for which we use the same notation (ρ, l 1 , l 2 , l 3 ), we have where Remark 3.2. R γ and I γ can be viewed as curvatures.

Transitive Lie 2-algebroids and Lie 3-algebroid morphisms
In this subsection, we give a conceptual explanation of equations listed in Theorem 3.3. We show that they give rise to a morphism from the Lie algebroid T M to a strict Lie 3-algebroid constructed from derivations of the graded bundle of Lie 2-algebras ( First we give the definition of a morphism from a Lie algebroid to a strict Lie 3-algebroid. See [6, Section 4.1] for details for general formulas of a morphism between Lie n-algebroids. such that for all X, Y, Z, X i ∈ Γ(A), i = 1, 2, 3, 4, we have Then we construct a strict Lie 3-algebroid using derivations of a graded bundle of Lie 2-algebras G = (G −1 ⊕ G 0 ; l 1 , l 2 , l 3 ). Denote by D(G 0 ) and D(G −1 ) the covariant differential operator bundle associated to G 0 and G −1 respectively. We have the following exact sequences: It is obvious that the set of degree 0 derivations is the section space of a vector bundle, which we denote by Der 0 (G). Furthermore, we define Der −1 (G) = Hom(G 0 , G −1 ).
On the graded bundle Der(G) = Der 0 (G) ⊕ Der −1 (G), we define δ : where It is straightforward to deduce that We go on constructing a strict Lie 3-algebroid. On the 3-term graded vector bundles is a strict Lie 3-algebroid.
Proof. The proof is a straightforward verification. We leave it to readers.

The first Pontryagin class of a quadratic Lie 2-algebroid
In this section, we give the notion of a quadratic Lie 2-algebroid and define its first Pontryagin class, which is a cohomology class in H 5 (M ).
Let (A −1 ⊕ A 0 ; ρ, l 1 , l 2 , l 3 ) be a quadratic Lie 2-algebroid. As before, denote by G = ker(ρ). By the nondegeneracy of S, we deduce that A −1 ∼ = G * and we will write A −1 = G * directly. Then the pairing S is simply given by By (30)-(32), we deduce that for the graded bundle of Lie 2-algebras (G * ⊕ G; l 1 , l 2 , l 3 ), there holds: Proof. It follows from that the invariant conditions in the definition of a quadratic Lie 2-algebroid is equivalent to (34)-(39).

Proof. First we have
Then by (37), we can deduce that (41) holds after a careful computation. We omit details.
Thus, S(R γ , I γ ) is a closed 5-form. Furthermore, by Lemma 4.4 and (19), we have which implies that the cohomology class does not depend on the choices of γ.
If we choose another section σ ′ : We have and Therefore, by the following Lemma 4.5-Lemma 4.9, we have which implies that S(R ′ γ , I ′ γ ) and S(R γ , I γ ) are in the same cohomology class.

Lemma 4.5. With the above notations, we have
Proof. By Lemma 4.4, (21) and (38), we have The proof is finished.

Lemma 4.8. With the above notations, we have
The proof is finished.

Lemma 4.9. With the above notations, we have
Proof. It follows from (35), (36) and the Jacobiator identity for l 3 .

Skeletal exact CLWX 2-algebroids
In this subsection, we associate a skeletal exact CLWX 2-algebroid a cohomology class in H 4 (M ), can call it the higher analogue of the Ševera class.

Definition 5.3. An isomorphism from a CLWX 2-algebroid (E
We have seen that different skeletal exact CLWX 2-algebroids are differed by closed 4-forms. Now we show that if two closed 4-forms H and H ′ are in the same cohomology class, the corresponding skeletal exact CLWX 2-algebroids are isomorphic. Thus, skeletal exact CLWX 2-algebroids are classified by H 4 (M ). This is a higher analogous result of that exact Courant algebroids are classified by H 3 (M ).
It is known that exact Courant algebroids are classified by the Ševera class [24]. Thus, for a skeletal exact
By (37), we have which implies that Axiom (ii) in Definition 2.5 holds. Finally, by Proposition 4.2, we can deduce that Axioms (iii)-(iv) in Definition 2.5 hold directly. The proof is finished. 6 The first Pontryagin class of a trivial principle 2-bundle with a Γ-connection

Strict Lie 2-groups and strict Lie 2-algebras
A group is a monoid where every element has an inverse. A 2-group is a monoidal category where every object has a weak inverse and every morphism has an inverse. Denote the category of smooth manifolds and smooth maps by Diff, a (semistrict) Lie 2-group is a 2-group in DiffCat, where DiffCat is the 2-category consisting of categories, functors, and natural transformations in Diff. For more details, see [3,4]. Here we only give the definition of a strict Lie 2-group. In the following we will denote the composition • in the Lie groupoid structure by · v : C 1 × C0 C 1 −→ C 1 and call it the vertical multiplication. Denote the Lie 2-group multiplication by · h : C × C −→ C and call it the horizontal multiplication.
It is well known that strict Lie 2-groups can be described by crossed modules of Lie groups.
and Ψ satisfies the so called Perffer identity: Theorem 6.3. There is a one-to-one correspondence between crossed modules of Lie groups and strict Lie 2-groups.
Roughly speaking, given a crossed module (H 1 , H 0 , Ψ, Φ) of Lie groups, there is a strict Lie 2-group for which C 0 = H 0 and C 1 = H 0 ⋉ H 1 , the semidirect product of H 0 and H 1 . In this strict Lie 2-group, the source and target maps s, t : C 1 −→ C 0 are given by the vertical multiplication · v is given by: the horizontal multiplication · h is given by Example 6.5. For any Lie algebra k, the adjoint action ad is a Lie algebra morphism from k to Der(k). Then (k, Der(k), ad, id) is a crossed module of Lie algebras. Theorem 6.6. There is a one-to-one correspondence between 2-term DGLAs (strict Lie 2-algebras) and crossed modules of Lie algebras.
In short, the formula for the correspondence can be given as follows: A 2-term DGLA (V −1 ⊕ V 0 ; l 1 , l 2 ) gives rise to a Lie algebra crossed module with h 1 = V −1 and h 0 = V 0 , where the Lie brackets are given by: and ψ = l 1 , φ : h 0 −→ Der(h 1 ) is given by φ u (m) = l 2 (u, m). The DGLA structure gives the Jacobi identity for [·, ·] h1 and [·, ·] h0 , and various other conditions for crossed modules.

The transitive Lie 2-algebroid associated to a trivial principle 2bundle with a Γ-connection
First we review the notion of a principle 2-bundle for a strict Lie 2-group Γ on the basis of [28,30]. Let Γ be a strict Lie 2-group corresponding to the Lie group crossed module (H 1 , H 0 , Ψ, Φ). Let (h 1 , h 0 , ψ, φ) be the crossed module of Lie algebras corresponding to (H 1 , H 0 , Ψ, Φ), and (h 1 ⊕ h 0 ; l 1 , l 2 ) the associated strict Lie 2-algebra. Definition 6.7. A principle Γ-2-bundle over a differential manifold M is a Lie groupoid P, a surjective submersion functor π : P −→ M dis , and a smooth right action R of Γ on P that preserves π, such that the smooth functor Remark 6.9. As proposed by the referee, it is natural to consider the case of nontrivial principle 2bundle. It is known that principle 2-bundles are classified by the nonabelian differential cohomology. See [19,29,30] for more details. A nonabelian cocycle is represented by an open cover {U i } of M , together with a collection of smooth maps g ij : U ij −→ H 0 and a ijk : U ijk −→ H 1 such that g ik = Ψ(a ijk )g ij g jk and a ikl Φ g kl a ijk = a ijl a jkl . One can see that g ij does not satisfy the cocycle condition. Thus, the naive idea of gluing the local standard model given in Theorem 6.8 to obtain a global object does not work. We will study this interesting question in the future.

The first Pontryagin class of the quadratic Lie 2-algebroid associated to a Γ-connection
In this subsection, we show that the first Pontryagin class of the quadratic Lie 2-algebroid associated to an Γ-connection is trivial. First we give an interesting example of quadratic strict Lie 2-algebras.
Example 6.10. Let (g, [·, ·] g , K) be a quadratic Lie algebra. That is, K is a symmetric nondegenerate bilinear form on g. Denote by K ♯ the induced map from g to g * , i.e.