On LA-Courant algebroids and Poisson Lie 2-algebroids

This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs of 2-representations. We use the known geometric examples of LA-Courant algebroids in order to provide new examples of Poisson Lie 2-algebroids, and we explain in this general context Roytenberg's equivalence of Courant algebroids with symplectic Lie 2-algebroids. We study further the core of an LA-Courant algebroid and we prove that it carries an induced degenerate Courant algebroid structure. In the nondegenerate case, this gives a new construction of a Courant algebroid from the corresponding symplectic Lie 2-algebroid. Finally we completely characterise VB-Dirac and LA-Dirac structures via simpler objects, that we compare to Li-Bland's pseudo-Dirac structures.

We prove that a split Poisson Lie 2-algebroid is equivalent to the matched pair of a self-dual 2-representation with a split Lie 2-algebroid. Here, the self-dual 2-representation is the one that is equivalent to the Poisson structure of degree −2. Given a Poisson Lie 2-algebroid, a splitting of the underlying [2]-manifold is equivalent to a decomposition of the corresponding metric double vector bundle [7]. The induced split Lie 2-algebroid is then equivalent to a VB-Courant algebroid structure in this decomposition [8] and the induced split Poisson structure of degree −2 is equivalent to the decomposition of a (metric) VB-algebroid structure on the other side of the metric double vector bundle [7]. The compatibility of the graded Poisson structure and the Lie 2-algebroid structure is equivalent to the VB-Courant algebroid and the metric VB-algebroid defining together an LA-Courant algebroid (short for Lie algebroid Courant algebroid ) [10]. Hence, we find that a metric double vector bundle with a VB-Courant algebroid structure and a metric VB-algebroid structure define together an LA-Courant algebroid if and only if, in any decomposition, the induced split Lie 2-algebroid and the induced self-dual 2representation build a matched pair.
The original definition of an LA-Courant algebroid [10] is inspired from and as technical as Mackenzie's first definition of a double Lie algebroid [15]. In short, a VB-Courant algebroid with a linear Lie algebroid on its other side is an LA-Courant algebroid if a relation defined by the Lie algebroid structure in the tangent prolongation of the VB-Courant algebroid -the underlying geometric structure is a triple vector bundle -is a Dirac structure in this tangent prolongation. In an earlier version of this work [5], we deduced from this definition our equations in the definition of the matched pair of a split Lie 2-algebroid with a self-dual 2representation. Then we found easily that those equations were also equivalent to the homological vector field defining the Lie 2-algebroid to be a Poisson vector field.
However, working out the equations directly from Li-Bland's LA-Courant algebroid condition is very long and technical (see the Appendix B of [5]). In this paper, we prefer therefore using Li-Bland's equivalence [10] in order to find the characterisation of an LA-Courant algebroid via the matched pair. We provide so a better (more handy) definition of LA-Courant algebroids. We explain again along the way the parallels between the theory of Lie algebroids, double Lie algebroids and 2-representations on the one hand, and Courant algebroids, LA-Courant algebroids and Lie 2-algebroids on the other hand. We find in particular that a matched pair of 2-representations [3] not only defines a split Lie 2-algebroid [8], but also a split Poisson Lie 2-algebroid -this is a different construction.
We prove further that the core of an LA-Courant algebroid has an induced degenerate Courant algebroid structure -just like the core of a double Lie algebroid has an induced Lie algebroid structure [15,3]. This allows us to explain in a constructive manner the equivalence of symplectic Lie 2-algebroids with Courant algebroids [18].
Finally, we study VB-and LA-Dirac structures in VB-and LA-Courant algebroids, and in particular, we prove that LA-Dirac structures are double Lie algebroids. More precisely, any double Lie algebroid can be understood as an LA-Dirac structure in an appropriate LA-Courant algebroid; just like any Lie algebroid can be seen as a Dirac structure in the induced Courant algebroid. This shows that the 7 equations defining a matched pair of 2-representations [3] can in fact be deduced from the 5 equations defining the matched pair of a split Lie 2-algebroid with a self-dual 2-representation. For completeness, we explain as well how Li-Bland's pseudo-Dirac structures [11] fit in our description of LA-Dirac structures in the tangent prolongation of a Courant algebroid.
Outline of the paper. In Section 2, we recall some general notions and conventions around dull brackets, Dorfman connections, Courant algebroids, graded manifolds and double vector bundles. In Section 2.5 we recall the equivalence of metric double vector bundles with [2]-manifolds, of metric VB-algebroids with Poisson [2]-manifolds and with self-dual 2-representations when decomposed. We recall also the equivalence of decompositions of VB-Courant algebroids with split Lie 2algebroids. In Section 3 we give the definition of the matched pair of a split Lie 2-algebroid with a self-dual 2-representation, and we prove our main theorem. In Section 4 we study the core of an LA-Courant algebroid and in Section 5 we describe VB-and LA-Dirac structures in decompositions. The text is illustrated with several examples that were prepared in [6,8].

Prerequisites, notation and conventions
We recall in this section some necessarily background, and we set our notation convention.
2.1. General conventions. We write p M : T M → M , q E : E → M for vector bundle maps. For a vector bundle Q → M we often identify without further mentioning the vector bundle (Q * ) * with Q via the canonical isomorphism. We write · , · for the canonical pairing of a vector bundle with its dual; i.e. q m , τ m = τ m (q m ) for q m ∈ Q and τ m ∈ Q * . We use several different pairings; in general, which pairing is used is clear from its arguments. Given a section ε of E * , we always write ℓ ε : E → R for the linear function associated to it, i.e. the function defined by e m → ε(m), e m for all e m ∈ E.
Let M be a smooth manifold. We denote by X(M ) and Ω 1 (M ) the sheaves of local smooth sections of the tangent and the cotangent bundle, respectively. For an arbitrary vector bundle E → M , the sheaf of local sections of E will be written Γ(E).
2.3. N-Manifolds of degree 2. An N-manifold M of degree 2 and dimension (p; r 1 , r 2 ) is a smooth manifold M of dimension p together a sheaf C ∞ (M) of Ngraded, graded commutative, associative, unital C ∞ (M )-algebras over M , that is locally freely generated by r 1 + r 2 elements ξ 1 1 , . . . , ξ r1 1 , ξ 1 2 , . . . , ξ r2 2 with ξ j i of degree i for i = 1, 2 and j ∈ {1, . . . , r i }. We write "[2]-manifold" for "N-manifold of degree 2". A morphism of N-manifolds µ : N → M over a smooth map µ 0 : N → M of the underlying smooth manifolds is a morphism µ ⋆ : C ∞ (M) → C ∞ (N ) of sheaves of graded algebras over µ * 0 : C ∞ (M ) → C ∞ (N ). Let E 1 and E 2 be smooth vector bundles of finite ranks r 1 , r 2 over M and assign the degree i to the fiber coordinates of E i , for each i = 1, . . . , n. The direct sum E = E 1 ⊕ E 2 is a graded vector bundle with grading concentrated in degrees 1 and 2. The has the elements of local frames of E * i as local generators of degree i, for i = 1, 2, and so dimension (p; [2]-manifolds over the bases M and N , respectively, consists of a smooth map µ 0 : N → M , three vector bundle morphisms µ 1 : Double Lie algebroids and matched pairs of 2-representations. We refer to Section 2.2 of [7] for the definition of a double vector bundle, and for the necessary background on their linear and core sections, and on their linear splittings and dualisations. Sections 2.3-2.5 of [7] recall the definition of a VB-algebroid, and also the equivalence of 2-term representations up to homotopy (called here "2representations" for short) with linear decompositions of VB-algebroids [4]. The notation that we use here is the same as in [7]. Of course, we also refer to [16,14,4] for more details on double vector bundles and VB-algebroids.
In this section we only recall the correspondence of decompositions of double Lie algebroids with matched pairs of 2-representations.
If (D, A, B, M ) is a VB-algebroid with Lie algebroid structures on D → B and A → M , then the dual vector bundle D * B → B has a Lie-Poisson structure (a linear Poisson structure), and the structure on D * B is also Lie-Poisson with respect to D * B → C * [15, 3.4]. Dualising this bundle gives a Lie algebroid structure on (D * B ) * C * → C * . This equips the double vector bundle ((D * B ) * C * ; C * , A; M ) with a VB-algebroid structure. Using the isomorphism defined by the non-degenerate pairing − ·, · : D * A × C * D * B → R, (see [14] and [7, §2.2.4] for a summary and our sign convention), the double vector bundle (D * A → C * ; A → M ) is also a VBalgebroid. In the same manner, if (D → A, B → M ) is a VB-algebroid then we use the non-degenerate pairing ·, · : be a linear splitting of D and denote by (∇ B , ∇ C , R A ) the induced 2-representation of the Lie algebroid A on ∂ B : C → B (see [4]; this is also recalled in Section 2.5 of [7]). The linear splitting Σ induces a linear splitting The 2-representation of A that is associated to this splitting is then (∇ C * , ∇ B * , −R * A ) on the complex ∂ * B : B * → C * . This is proved in the appendix of [2].
A double Lie algebroid [15] is a double vector bundle (D, A, B, M ) with core C, and with Lie algebroid structures on each of A → M , B → M , D → A and D → B such that each pair of parallel Lie algebroids gives D the structure of a VB-algebroid, and such that the pair (D * A , D * B ) with the induced Lie algebroid structures on base C * and the pairing ·, · , is a Lie bialgebroid.
Consider a double vector bundle (D; A, B; M ) with core C and a VB-Lie algebroid structure on each of its sides. After the choice of a splitting Σ : A × M B → D, the Lie algebroid structures on the two sides of D are described by two 2representations [4]. We prove in [3] that (D * A , D * B ) is a Lie bialgebroid over C * if and only if, for any splitting of D, the two induced 2-representations form a matched pair as in the following definition [3].
be two Lie algebroids and assume that A acts on ∂ B : C → B up to homotopy via (∇ B , ∇ C , R A ) and B acts on ∂ A : C → A up to homotopy via (∇ A , ∇ C , R B ) 2 . Then we say that the two representations up to homotopy form a matched pair if where R B is seen as an element of Ω 1 (A, ∧ 2 B * ⊗C) and R A as an element of Ω 1 (B, ∧ 2 A * ⊗ C).

The equivalence of [2]-manifolds with metric double vector bundles.
We begin by summarising the correspondence found in [7] between double vector bundles endowed with a linear metric, and N-manifolds of degree 2.
A linear splitting Σ : Q × M B → E is said to be Lagrangian if its image is maximal isotropic in E → B. The corresponding horizontal lifts σ Q : Γ(Q) → Γ l B (E) and σ B : Γ(B) → Γ l Q (E) are then also said to be Lagrangian. By definition, a horizontal lift σ Q : Γ(Q) → Γ l B (E) is Lagrangian if and only if σ Q (q 1 ), σ Q (q 2 ) = 0 for all q 1 , q 2 ∈ Γ(Q). Showing the existence of a Lagrangian splitting of E is relatively easy [7]: Note that a general linear decomposition Σ of a metric double vector bundle defines as follows a section Λ of S 2 (Q) ⊗ B * : 2 For the sake of simplicity, we write in this definition ∇ for all the four connections. It will always be clear from the indexes which connection is meant. We write ∇ A for the A-connection induced by ∇ AB and ∇ AC on ∧ 2 B * ⊗ C and ∇ B for the B-connection induced on ∧ 2 A * ⊗ C.
for all q 1 , q 2 ∈ Γ(Q). In particular, Λ(q, ·) : Q → B * is a morphism of vector bundles for each q ∈ Γ(Q). Define a new horizontal lift σ ′ Q : Γ(Q) → Γ l B (E) by for all q ∈ Γ(Q). It is easy to check that the corresponding linear decomposition Σ ′ is Lagrangian. Further, if Σ 1 and Σ 2 : Q × M B → E are Lagrangian, then the change of splitting We define 3 a morphism Ω : F → E of metric double vector bundles as a pair of maps ω ⋆ : C(E) → C(F), ω ⋆ P : Γ(Q * ) → Γ(P * ) together with a smooth map ω 0 : N → M such that all morphisms of vector bundles over a smooth map ω 0 : . We write MDVB for the obtained category of metric double vector bundles. In [7] we established an equivalence between the category of involutive double vector bundles and the category of [2]-manifolds. We also proved there that there is a 3 A metric double vector bundle (E, B, Q, M ) is dual (over Q) to an involutive double vector bundle [7]. A morphism Ω : F → E of metric double vector bundles is defined as a relation Ω ⊆ F × E that is the dual of a morphism of involutive double vector bundles ω : F * P → E * Q [7]. The characterisation given here is proved in [7].
(contravariant) dualisation equivalence of the categry of involutive double vector bundles with MDVB. This yields the following theorem. We quickly describe the functors between the two categories. To construct the geometrisation functor G : [2]−Man → MDVB, take a [2]-manifold and considers its local trivialisations. Changes of local trivialisation define a set of cocycle conditions, that correspond exactly to cocycle conditions for a double vector bundle atlas. The local trivialisations can hence be collated to a double vector bundle, which naturally inherits an involution. See [7] for more details, and remark that this construction is as simple as the construction of a vector bundle from a locally free and finitely generated sheaf of C ∞ (M )-modules. Conversely, the algebraisation functor A sends a metric double vector bundle E to the [2]-manifold defined as follows: the functions of degree 1 are the sections of Γ c Q (E) ≃ Γ(Q * ), and the functions of degree 2 are the elements of C(E). The multiplication of two core sections τ 1 , τ 2 ∈ Γ(Q * ) is the core-linear section τ 1 ∧ τ 2 ∈ C(E). [2]-manifolds. The correspondence above of split 2-manifolds with decomposed metric double vector bundles induces a correspondence of split Poisson [2]-manifolds with decomposed metric VB-algebroids. This bijection extends to an equivalence of Poisson [2]-manifolds with metric VBalgebroids [7], but we only need the split objects here.

Metric VB-algebroids and Poisson
Consider a metric double vector bundle (E, B, Q, M ) with a linear Lie algebroid on E → Q over a Lie algebroid structure on B → M . Then E is a metric VBalgebroid [7] if the bijection and only if the 2-representation is dual to itself [7]: The bracket is graded skew-symmetric; {ξ, η} = −(−1) |ξ| |η| {η, ξ} and satisfies the graded Leibniz and Jacobi identities Via the identification of the underlying metric double vector bundle [7]): This identification is compatible with changes of splittings of the [2]-manifolds and changes of decomposition of metric VB-algebroids: The category of Poisson [2]manifolds is equivalent to the category of metric VB-algebroids [7].

Example 2.4. Consider a metric vector bundle E → M and a metric connection
The metric VB-algebroid structure on T E → E is just the standard Lie algebroid structure on the tangent bundle of E. See [7] for more details.

VB-Courant algebroids and Lie 2-algebroids. A VB-Courant algebroid
[10] is a metric double vector bundle (E → B, · , · ) with side Q and core Q * , together with a linear anchor Θ : E → T B and a linear Courant algebroid bracket on sections of E → B.
Example 2.5. [8] We consider here a Courant algebroid (E → M, ρ, · , · , · , · ). We use the pairing to identify E with E * . After the choice of a metric connection on E and so of a Lagrangian decomposition ∆ bas e e ′ = e, e ′ + ∇ ρ(e ′ ) e, which we call the basic Dorfman connection associated to ∇. The dual dull bracket is given by (14) e, e ′ ∆ bas = e, e ′ − ρ * ∇ · e, e ′ for all e, e ′ ∈ Γ(E). The linear connection is ∇ bas : for all e 1 , e 2 ∈ Γ(E) and X ∈ X(M ).

a split Lie 2-algebroid
The category of Lie 2-algebroid is equivalent via the correspondence described above to the category of VB-Courant algebroids [10,8]. Note that a morphism µ : We refer to §3.5 of [8] for the characterisation of a morphism of split Lie 2-algebroids in terms of its components (∂ * B : B * → Q, ρ Q , · , · , ∇, ω).

LA-Courant algebroids vs Poisson Lie 2-algebroids
In this section, we prove that a split Poisson Lie 2-algebroid is equivalent to the matched pair of a split Lie 2-algebroid with a self-dual 2-representation.
Take a double vector bundle (E, B, Q, M ) with core Q * , with a VB-Lie algebroid structure on (E → Q, B → M ) and a VB-Courant algebroid structure on (E → B, Q → M ). In this section we show that the double vector bundle is an LA-Courant algebroid [10] if and only if the VB-algebroid is metric and the self-dual 2representation defined by any Lagrangian decomposition of E and the VB-algebroid side forms a matched pair with the split Lie 2-algebroid describing the Courant algebroid side.
We begin with the following definition.
and from (M2) if B has positive rank. If both Q and B have rank zero, The equation (20) [ follows from (M3) if B has positive rank, and from (M4) if Q has positive rank. If both Q and B have rank zero, then it is trivially satisfied.
3.1. Poisson Lie 2-algebroids via matched pairs. We begin this subsection with the definition of a Poisson Lie 2-algebroid.  Proof. The idea of this proof is very simple, but requires rather long computations. We will leave some of the detailed verifications to the reader. We check (23) in coordinates, by using the formulae found in (16), (17), (18) and (10) for Q and {· , ·}, respectively.
g} by the graded skew-symmetry and {τ, f } = 0 for τ ∈ Γ(Q * ). Then we have for τ ∈ Γ(Q * ): In a similar manner, we have Q{b, . By comparing the Γ(B) and the Ω 2 (Q)-terms, we find that (20) holds, then (24) is (M4). Next we study the equation Finally, we choose b 1 , b 2 ∈ Γ(B) and we study the equation Note that in the expression d Q (R(b 1 , b 2 )), the object R(b 1 , b 2 ) is understood as an element of Ω 2 (Q), and in the expression The right-hand side of this equation is easily calculated to be the pairing of (d for all e 1 , e 2 ∈ Γ(E). Later we will need the following lemma in Section 5. We leave the proof to the reader.

Consider a Lie algebroid (q
for all a, b, c ∈ Γ(A). (Note that in [10], the relation is defined in a different manner. Checking that this alternative definition is correct is rather long. Details can be obtained in the appendix of [5]). The relation Π E defined as above by the Lie algebroid structure on E over Q is then a relation Π E of the triple vector bundles [10] T [10] is the following. We have the following theorem. The proof of this theorem is very long and technical (see the appendix of [5]), showing that Li-Bland's definition of an LA-Courant algebroid is hard to handle. Hence our result provides a new definition of LA-Courant algebroids, that is much simpler to express and probably also easier to use.
Further, we now explain how this theorem shows that LA-Courant algebroids are equivalent to Poisson Lie 2-algebroids. This has already been found by Li-Bland in [10]. First, morphisms of LA-Courant algebroids are morphisms of metric double vector bundles that preserve the Courant algebroid structure and the Lie algebroid structure [10]. Hence, the category of LA-Courant algebroids is a full subcategory of the intersection of the category of metric VB-algebroids and the category of VB-Courant algebroids.
On the other hand, Definition 3.3 shows that the category of Poisson Lie 2algebroids is a full subcategory of the intersection of the categories of Poisson . A straightforward computation resembling the one in [3, Section 3.2] for the tangent double of a Lie algebroid shows that this 2-representation and this split Lie 2-algebroid are matched, and so that T E is an LA-Courant algebroid (see also [10]).
The Poisson structure on the [2]-manifold corresponding to T E is, via the equivalence of [2]-manifolds with metric double vector bundles, just the Poisson structure that is dual to the Lie algebroid T E → E. Hence, it is symplectic (see [7], in particular §4.5.1).
Hence, the class of LA-Courant algebroids that is equivalent to the symplectic Lie 2-algebroids is just the class of tangent prolongations of Courant algebroids.

The standard Courant algebroid over a Lie algebroid. Let
A be a Lie algebroid. Then T A ⊕ T * A is a double vector bundle with sides A and T M ⊕ A * and with core A⊕T * M . It has a linear Courant algebroid structure on T A⊕ A T * A → A (see [8]) and a metric VB-algebroid structure (T A ⊕ A T * A → T M ⊕ A * , A → M ) (see [7]). Set The objects l, · , · , ∇ * , ω define a split Lie 2-algebroid; the standard split Lie 2-algebroid defined by the dull bracket (or equivalently by the dual Dorfman connection).
The direct sum D ⊕ B D * B over B has also a VB-algebroid structure (D ⊕ B D * B → A⊕C * , B → M ) with core C ⊕A * . The linear decompositionΣ : , see [7], §4.5.2.
A straightforward computation shows that the matched pair conditions for the 2-representations describing the sides of D imply that the 2-representation (28)   In the case of the double Lie algebroid T A, for A → M a Lie algebroid, the two LA-Courant algebroids obtained in this manner are T A⊕ A T * A described in Section 3.3.2, and the tangent prolongation as in Section 3.3.1 of the Courant algebroid A ⊕ A * → M ; (a 1 , α 1 ), (a 2 , α 2 ) = ([a 1 , a 2 ], £ a1 α 2 − i a2 dα 1 ) for a 1 , a 2 ∈ Γ(A) and α 1 , α 2 ∈ Γ(A * ).

The core of an LA-Courant algebroid
We prove in this section that the core of an LA-Courant algebroid inherits a natural structure of degenerate Courant algebroid. We discuss some examples and we deduce a new way of describing the equivalence between Courant algebroids and symplectic Lie 2-algebroids.
. This structure does not depend on the choice of the Lagrangian splitting, and the map ∂ B : Q * → B is compatible with the brackets and the anchors: for all τ 1 , τ 2 ∈ Γ(Q * ).

by the definition of a 2-representation, and (22), this is
We write β := ∇ · ∂ Q τ 1 , τ 2 ∈ Γ(B * ). Since ρ Q • ∂ Q = ρ B • ∂ B and ∇ • ∂ Q = ∂ Q • ∇, we find β = ∂ Q ∇ * · τ 1 , τ 2 = ∇ * · τ 1 , ∂ Q τ 2 ∈ Γ(B * ). To see that (31), which is a section of Q * , vanishes, we evaluate it on an arbitrary q ∈ Γ(Q). We use (12) and the definition of a 2-representation and we get Since the Dorfman connection ∆ is dual to the skew-symmetric dull bracket · , · ∆ , this is ρ Q (∂ * B β) q, τ 3 . Because ρ Q • ∂ * B = 0 by (11), we can conclude. We finally prove that the degenerate Courant algebroid structure does not depend on the choice of the Lagrangian splitting. Clearly the pairing and anchor are independent of the splitting, so we only need to check that the bracket remains the same if we choose a different Lagrangian splitting. Assume that Σ 1 , We have hence proved that the Courant algebroid associated to a symplectic Lie 2-algebroid can be defined directly from any of the splittings of the Lie 2-algebroid, and so does not need to be obtained as a derived bracket.  (a 1 , θ 1 ), (a 2 , θ 2 ) A⊕T * M = (a 1 , θ 1 ), (ρ A , ρ * A )(a 2 , θ 2 ) and the bracket defined by for all a, a 1 , a 2 ∈ Γ(A) and θ, θ 1 , θ 2 ∈ Ω 1 (M ). To see this, use Lemma 5.16 in [6] or the next example; this degenerate Courant algebroid plays a crucial role in the infinitesimal description of Dirac groupoids [9], i.e. in the definition of Dirac bialgebroids.
for all α 1 , α 2 ∈ Γ(A * ) and c 1 , c 2 ∈ Γ(C), and the bracket by Note that the restriction to Γ(C) of the Courant bracket is the Lie algebroid bracket induced on C by the matched pair, see §2.4.
The second degenerate Courant algebroid is C ⊕ B * → M with the anchor for all β 1 , β 2 ∈ Γ(B * ) and c 1 , c 2 ∈ Γ(C), and the bracket Here again, by (m1), the restriction to Γ(C) of the Courant bracket is the Lie algebroid bracket induced on C by the matched pair, as in §2.4.

VB-Dirac structures, LA-Dirac structures and pseudo Dirac structures
In this section, we study isotropic subalgebroids of metric VB-algebroids and Dirac structures in VB-and LA-Courant algebroids. While we paid attention in the preceding sections to bridge [2]-geometric objects to geometric structures on metric double vector bundles, we are here more interested in classifications of VB-Dirac structures via the simple geometric descriptions that we found before for VB-Courant algebroids and LA-Courant algebroids.

VB-Dirac structures.
Let (E, B, Q, M ) be a VB-Courant algebroid with core Q * and anchor Θ : E → T B. Let D be a double vector subbundle structure over B ′ ⊆ B and U ⊆ Q and with core K. Choose a linear splitting Σ : B × M Q → E that is adapted 5 to D, i.e. such that Σ(B ′ × M U ) ⊆ D. Then D is spanned as a vector bundle over B ′ by the sections σ Q (u)| B ′ for all u ∈ Γ(U ) and τ † | B ′ for all τ ∈ Γ(K).
We get immediately the following proposition.  Proof. As before, let U ⊆ Q and B ′ ⊆ B be the sides of D. Then by Proposition 5.2 the core of D is the vector bundle U • ⊆ Q * . Choose a linear splitting Σ : Q× M B → E that is adapted to D. Then D is spanned as a vector bundle over B ′ by the sections σ Q (u)| B ′ for all u ∈ Γ(U ) and τ † | B ′ for all τ ∈ Γ(U • ). As in (6), transform Σ into a new Lagrangian linear splitting Σ ′ . We need to show that σ ′ Q (u)| B ′ − σ Q (u)| B ′ is equivalent to a section of B ′ * ⊗ U • for all u ∈ Γ(U ). But σ ′ Q (u) − σ Q (u) = 1 2 Λ(u, ·) by construction and, since D is isotropic, we have Λ(u, u ′ )| B ′ = 0 for all u, u ′ ∈ Γ(U ). The existence of Lagrangian splittings of E adapted to maximal isotropic double subbundles D will now be used to study the involutivity of D. Note that in a very early version of this work, we studied VB-Courant algebroids via general (not necessarily Lagrangian) linear splittings. We found some more general objects than split Lie 2-algebroids; involving also Λ ∈ S 2 (Q, B * ) defined in (5). The study of the involutivity of general (not necessarily isotropic) double subbundles D of E is therefore also possible in this more general framework, and yields very similar results.
A Dirac double subbundle D of a VB-Courant algebroid E as in the proposition is called a VB-Dirac structure.
If φ ∈ Γ(Q * ∧ Q * ⊗ B * ) is the tensor defined as in the proof of Theorem 4.1 by a change of Lagrangian splitting adapted to D, then, by Proposition 4.7 in [8], for all u, u ′ . But since both splittings Σ 1 , Σ 2 : B × M Q → E are adapted to D, we know that σ 1 Q (u) and σ 2 Q (u) have values in D, and their difference σ 1 Q (u) − σ 2 Q (u) = φ(u) is a core-linear section of D → B. Hence it must takes values in U • , and φ(u, u ′ ) must so vanish for all u, u ′ ∈ Γ(U ). As a consequence, u, u ′ 1 = u, u ′ 2 .
The following two corollaries are now easy to prove. The first one was already given in [10].
In a splitting Q[−1] ⊕ B * [−2] of M, the homological vector field Q is given by (16)- (18). Choose an open subset V of M with a local frame (u 1 , . . . , u r , q r+1 , . . . , q l ) of Q over V such that (u 1 , . . . , u r ) is a local frame for U over V . Let (τ 1 , . . . , τ l ) be the dual smooth frame for Q * over V . Then we have This shows that the bracket on U must be the restriction to Γ(U ) of the dull bracket on Γ(Q). Finally 0 = Q U (µ ⋆ b) = µ ⋆ (Q(b)) = − i<j<k<r ω(u i , u j , u k )(b)τ iτjτk for all b ∈ Γ(B) shows that ω(u 1 , u 2 , u 3 ) must be zero for all u 1 , u 2 , u 3 ∈ Γ(U ). This is equivalent to (3) in Proposition 5.5 (with B ′ = B). Note that since B ′ = B, (1) and (2) in Proposition 5.5 are trivially satisfied. Hence we can conclude.
The Lie algebroid structure on U is the base Lie algebroid from the VB-algebroid D → B in the following corollary. The proof is immediate.
Hence, we also have the following result. In a local splitting, we find easily that this implies ∂ Q (U • ) ⊆ U , ∇ * b τ ∈ Γ(U • ) for all b ∈ Γ(B) and τ ∈ Γ(U • ), and the restriction to U of R(b 1 , b 2 ) has image in U • . By Proposition 5.9, we can conclude. The second claim follows with Corollary 5.7.
As a corollary of Theorem 3.7, Proposition 5.5 and Proposition 5.9, we get the following theorem. By Proposition 5.5 and Corollary 5.6, the restriction to Γ(U ) of the dull bracket on Γ(Q) that is dual to ∆ defines a Lie algebroid structure on U , ω| U⊗U⊗Q can be seen as an element of Ω 2 (U, Hom(B, U • )) and since ∆ u τ ∈ Γ(U • ) for all u ∈ Γ(U ) and τ ∈ Γ(U • ), the Dorfman connection ∆ restricts to a map ∆ D : , we find that this restriction is in fact an ordinary connection.
Note finally that with a different approach as the one adopted in this paper, we could deduce the main result in [3]

Pseudo-Dirac structures.
We explain here the notion of pseudo-Dirac structures that was introduced in [10,11] and we compare it with our approach to VBand LA-Dirac structures in the tangent of a Courant algebroid. Consider a VB-Courant algebroid E with core Q * , and a double vector subbundle in E with core K, as in the following diagrams.
. This is a double vector bundle with sides B and U and with core Q * . The total quotient of E| U by D is the map q from After the choice of a linear splitting of E that is adapted to D, we know that each element of E| U can be written Conversely it is easy to see that D can be recovered from q. Recall that if In particular, e 1 , e 2 = π Q (e 1 ), q(e 2 ) + π Q (e 2 ), q(e 1 ) for all e 1 , e 2 ∈ E| U if and only if Λ| U⊗U vanishes and K = U • , i.e. if and only if D is maximal isotropic (Proposition 5.2). Now we recall Li-Bland's definition of a pseudo-Dirac structure [11]. and Consider the tangent double (T E, T M, E, M ) where E is a Courant algebroid over M . Choose a linear (wide) Dirac structure D in T E, over the side U ⊆ E and a metric connection ∇ : X(M ) × Γ(E) → Γ(E) that is adapted to D. Li-Bland defines the pseudo-Dirac structure associated to D [11] as the map ∇ p : Γ(U ) → Ω 1 (M, U * ) that is defined by ∇ p u = q • T u for all u ∈ Γ(U ). By definition of σ ∇ E , we have T u = σ ∇ E (u)+ ∇ · u and we find that ∇ p u(v m ) = ∇ vm u = [∇] vm u. The pseudo-Dirac structure is nothing else than the invariant part of the metric connection that is adapted to D (Remark 5.4). Condition (2) in Definition 5.13 is then for all u 1 , u 2 ∈ Γ(U ) and Condition (1) is The bracket · , · p is then the bracket defined in (14). Finally, a straightforward computation shows that the left-hand side of (33) equals R bas ∆ (u 1 , u 2 ) * u 3 ∈ Γ(B * ), which is zero by Proposition 5.5. Li-Bland proves that the bracket · , · p defines a Lie algebroid structure on U . More explicitly, he finds that the left-hand side Ψ(u 1 , u 2 , u 3 ) of (33) defines a tensor Ψ ∈ Ω 3 (U, T * M ) that is related as follows to the Jacobiator of · , · p : Jac · ,· p = (β −1 • ρ * E )Ψ. He proves so that (wide) linear Dirac structures in T E are in bijection with pseudo-Dirac structures on E. Hence, our result in Proposition 5.5 is a generalisation of Li-Bland's result to linear Dirac structures in general VB-Courant algebroids.
Further, our Theorem 5.9 can be formulated as follows in Li-Bland's setting.
Theorem 5.14. In the correspondence of linear Dirac structures with pseudo-Dirac connections in [11], LA-Dirac structures correspond to pseudo-Dirac connections We propose to call these pseudo-Dirac connections quadratic pseudo-Dirac connections. Note that ∇ p equals∇ : X(M ) × Γ(U/U ⊥ ) → Γ(U/U ⊥ )∇ Xū = ∇ X u, u ∈ Γ(U ) and X ∈ X(M ), for any metric connection ∇ : X(M )×Γ(U ) → Γ(U ) such that [∇] = ∇ p . Such a connection must preserve U by Condition (2) in Proposition 5.9, and so also U ⊥ since it is metric. The condition R ∇ (X 1 , X 2 )u ∈ Γ(U ⊥ ) for all X 1 , X 2 ∈ X(M ) and u ∈ Γ(U ) in Proposition 5.9 is then equivalent to R∇ = 0. 5.4. The Manin pair associated to an LA-Dirac structure. Consider as before an LA-Courant algebroid E with sides B and Q and with core Q * , and an LA- This vector bundle is anchored by the map Note that this map is well-defined because for all τ ∈ U • . We will show that there is a symmetric non-degenerate pairing · , · B on B × M B and a bracket · , · B on Γ(B) such that is a Courant-algebroid. We define the pairing on B by It is easy to check that this pairing is well-defined and non-degenerate and that the induced map D B : C ∞ (M ) → Γ(B) given by can alternatively be defined by D B f = 0 ⊕ ρ * Q df . Choose as before a Lagrangian splitting of E that is adapted to D, and recall that the linear Courant algebroid structure and the linear Lie algebroid structure on E are then encoded by a split Lie 2-algebroid and by a self-dual 2-representation, respectively, both denoted as usual. We define the bracket on Γ(B) by with the anchor ρ B , the pairing · , · B and the bracket · , · B , is a Courant algebroid. Further, U is a Dirac structure in B, via the inclusion U ֒→ B, u → u ⊕ 0.
In the case of an LA-Courant algebroid (T A⊕ A T * A, T M ⊕A * , A, M ) as in §3.3.2, for a Lie algebroid A, we could show in [9] that Manin pairs as in Corollary 5.16 are in bijection with LA-Dirac structures on A. That is, given a Manin pair (B, U ) with an inclusion U ֒→ T M ⊕ A * and a degenerate Courant algebroid morphism A ⊕ T * M → B satisfying (1)-(4), then via (37), there exists a Lagrangian splitting of T A ⊕ A T * A such that the Courant bracket on B is given by (36).
We have used (11) and (2) in the first line, as well as for the first equality. To conclude, we have used ∂ B • ∆ q ′ = ∇ q ′ • ∂ B by (12).