Double Lie algebroids and representations up to homotopy

We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.


Introduction
Double Lie algebroids first arose as the infinitesimal form of double Lie groupoids [11,14]. In the same way as the Lie theory of Lie groupoids and Lie algebroids expresses many of the basic infinitesimalization and integration results of differential geometry, the process of taking the double Lie algebroid of a double Lie groupoid captures two-stage differentiation processes, such as the iterated tangent bundle of a smooth manifold, and the relations between a Poisson Lie group, its Lie bialgebra and its symplectic double groupoid.
The transition from a double Lie groupoid to its double Lie algebroid is straightforward. To define an abstract concept of double Lie algebroid, however, is much more difficult, since there is no meaningful way in which a Lie algebroid bracket can be said to be a morphism with respect to another Lie algebroid structure. The solution ultimately found was to extend the duality between Lie algebroids and Lie-Poisson structures to the double context, using the duality properties of double vector bundles [16]. This definition was given a simple and elegant reformulation in terms of super geometry by Th. Voronov [26]. In terms of super geometry, an ordinary Lie algebroid structure on a vector bundle corresponds to a homological vector field Q of weight 1 on the parity-reversed bundle. A double vector bundle D with Lie algebroid structures on both bundle structures on D therefore involves two homological vector fields and if these are of suitable weights with respect to the grading from the double vector bundle structure, the main compatibility condition of [26] is that they commute.
In the present paper we give a third formulation of double Lie algebroids, in terms of representations up to homotopy as defined in [1,4]; this differs from the concept of the same name introduced in [3].
In fact the representations up to homotopy which are relevant are concentrated in degrees 0 and 1 and we refer to them as 2-representations for brevity (see Definition 2.8). A 2-representation is a special form of Quillen's notion of superconnection; see below following 2.7. We could paraphrase the content of the paper as showing that double Lie algebroids can be regarded as matched pairs of flat Lie algebroid superconnections.
We now give a more detailed description of the background to the paper. Consider first a double vector bundle D with Lie algebroid structures on two parallel sides, which are compatible with the vector bundle structures on the other sides; these are the LA-vector bundles of [16,Definition 3.3] and, in an equivalent reformulation, the VB-algebroids of [4]. Further suppose that D is 'decomposed'; that is, as a manifold it is the fibre product A × M B × M C of three vector bundles A, B, C on the same base M, and the vector bundle structures on D are the pullbacks of B ⊕ C to A and of A ⊕ C to B. Then [4] showed that LA-vector bundle structures on D are in bijective correspondence with 2-representations defined in terms of A, B and C.
Decompositions of double vector bundles may themselves be regarded as extensions of the notion of connection. One definition of a connection in a vector bundle (E, q, M) is a map C : E × M T M → T E which is linear both as a map from the pullback of T M over the projction E → M and as a map from the pullback of E over the projection T M → M, and which is right-inverse to both projections. This formulation, which can be found in Dieudonné, is precisely a decomposition of T E in our sense. However for a decomposition of a double vector bundle to possess connection-like properties it is necessary for there to be bracket structures on at least one pair of parallel sides; that is, for it to have a LA-vector bundle structure.
We assume, as part of the definition of a double vector bundle, that a decomposition exists; this property is preserved by the various prolongation and dualization processes studied here and in the references. Decompositions may be regarded as trivializations of D at the double level; in this paper we do not need to trivialize A, B and C. For a formulation in coordinate terms, see [26]. Now consider an arbitrary double Lie algebroid D. The Lie algebroid structures on D may be considered as a pair of LA-vector bundle structures and accordingly a decomposition of D expresses the double Lie algebroid structure as a pair of 2-representations. Our main result (Theorem 3.6) determines the compatibility conditions between these and, conversely, proves that a suitable pair of 2-representations defines a double Lie algebroid structure on D.
Our formulation is significantly different from both the original formulation and that of [26]. Our treatment resembles the coordinate treatment in [26] inasmuch as the three intrinsic conditions of [16] are replaced by a greater number of conditions which are dependent on auxiliary data, but which are easier to work with. On the other hand, our methods are entirely 'classical' rather than supergeometric, and rely on a global decomposition rather than a local trivialization.
Our formulation may also be regarded as a considerable generalization of the description of a vacant double Lie algebroid in terms of a matched pair of representations [16, §6]-that is, of representations of Lie algebroids in the strict sense, without curvature. For this reason we regard the conditions (M1) to (M7) in Definition 3.1 as defining a matched pair of 2-representations.
In turn, [7] shows that the bicrossproduct of a matched pair of 2-representations is a split Lie 2-algebroid, in the same way that the bicrossproduct of a matched pair of representations of Lie algebroids is a Lie algebroid [10,18]. In a different direction, [8] will apply our main result to show that double Lie algebroids which are transitive in a sense appropriate to the double structure are determined by a simple diagram of morphisms of ordinary Lie algebroids.
We now describe the contents of the paper. In Sect. 2 we recall the basic notions needed throughout the paper. We begin with double vector bundles, the special classes of sections with which it is easiest to work, and the nonstandard pairing between their duals. In Sect. 2.2 we recall LA-vector bundles and double Lie algebroids, and in Sect. 2.3 we finally define 2-representations.
In Sect. 3 we state our main result and apply it to the double Lie algebroids which arise from the tangent and cotangent prolongations of a Lie algebroid. The main work of the proof of Theorem 3.6 is given in Sect. 4.
We have included definitions of the key concepts required; in particular it is not necessary to have a detailed knowledge of [1,4,16] or [26].

Double vector bundles, decompositions and dualization
We briefly recall the definitions of double vector bundles, of their linear and core sections, and of their linear splittings and lifts. We refer to [4,15,20] for more detailed treatments.
satisfying the following four conditions: 1. all four sides are vector bundles; 2. π B is a vector bundle morphism over q A ; 3.
together with a fifth condition (5) below.
The corresponding statements for the operations in the bundle D → A follow. Given a double vector bundle (D; A, B; M), the vector bundles A and B are called the side bundles. The core C of a double vector bundle is the intersection of the kernels of π A and π B . It has a natural vector bundle structure over M, the restriction of either structure on D, the projection of which we call q C : C → M. The inclusion C → D is usually denoted by Given any three vector bundles A, B, C on the same base manifold M, the fibre product D := A × M B × M C has a natural structure of double vector bundle with side bundles A and B and core C; the vector bundle structure on D → A is the pullback of B ⊕ C to A and likewise the vector bundle structure on D → B is the pullback of A ⊕ C to B. We can now state the fifth condition of Definition 2.1: where C is the core of D, which is the identity on A, B and C and is an isomorphism of vector bundles D → D over A and B.
A morphism of double vector bundles from (D; Given a double vector bundle (D; A, B; M), the space of sections B (D) is generated as a C ∞ (B)-module by two distinguished classes of sections (see [16]), the linear and the core sections which we now describe.

Definition 2.2 For a section
We denote the corresponding core section A → D by c † also, relying on the argument to distinguish between them.
The space of core sections of D over B will be written c B (D) and the space of linear sections B (D). Given ψ ∈ (B * ⊗ C), there is a linear section ψ : B → D over the zero section 0 A : M → A given by We call ψ a core-linear section.
and similarly for c † : In particular, the fibered product A × M B is a double vector bundle with side bundles A and B and core M × 0.

Decompositions and lifts
as in Definition 2.1; that is, I is an isomorphism of double vector bundles over the identity maps on A, B and C.
Given an injective morphism of double vector bundles : A × M B → D over the identity on the sides A and B, define a decomposition of D by Decompositions of D are in bijective correspondence with such morphisms . We will often refer to as a decomposition of D.
Decompositions of D are also equivalent to splittings σ A of the short exact sequence of C ∞ (M)-modules where the third map is the map that sends a linear section (ξ, a) to its base section a ∈ (A). The splitting σ A will be called a lift. Given a decomposition, the lift In the case of the tangent double of a vector bundle E → M, the lift from vector fields on M to vector fields on E (see Sect. 2.1.2) would be the horizontal lift corresponding to a connection. We avoid the word 'horizontal' here since 'horizontal' and 'vertical' refer to the two structures on D.
By the symmetry of a decomposition, this implies that a lift Note finally that two decompositions of D differ by a section φ 12 of A * ⊗ B * ⊗C Hom(A, B * ⊗ C) Hom(B, A * ⊗ C) in the following sense. For each a ∈ (A) the difference σ 1 A (a) − B σ 2 A (a) of lifts is the core-linear section defined by φ 12 (a) ∈ (B * ⊗ C). By symmetry,

The tangent double vector bundle of a vector bundle
Let q E : E → M be a vector bundle. Then the tangent bundle T E has two vector bundle structures; one as the tangent bundle of the manifold E, and the second as a vector bundle over T M. The structure maps of T E → T M are the derivatives of the structure maps of E → M. The space T E is a double vector bundle with core bundle E → M.
The core vector field corresponding to e ∈ (E) is the vertical lift e ↑ : E → T E, i.e. the vector field with flow φ : E × R → E, φ t (e m ) = e m + te(m). An element of E (T E) = X (E) is called a linear vector field. It is well-known (see e.g. [15]) that a linear vector field ξ ∈ X l (E) covering X ∈ X(M) corresponds to a derivation D * : (E * ) → (E * ) over X ∈ X(M), and hence to a derivation D : (E) → (E) over X ∈ X(M) (the dual derivation). The precise correspondence is given by 1 for all ε ∈ (E * ) and f ∈ C ∞ (M). Here ε is the linear function E → R corresponding to ε. We will write D for the linear vector field corresponding in this manner to a derivation D of (E). The choice of a decomposition for (T E; T M, E; M) is equivalent to the choice of a connection on E: Since a decomposition gives us a linear vector field σ T M (X ) ∈ X l (E) for each X , we can define ∇ : and a decomposition of E. We recall as well the relation between the connection and the Lie bracket of vector fields on E. Given ∇, it is easy to see using the equalities in (3) that, writing σ for σ ∇ T M : for all X, Y ∈ X(M) and e, e ∈ (E). That is, the Lie bracket of vector fields on M and the connection encode completely the Lie bracket of vector fields on E. One result of this paper is an extension of this description to general double Lie algebroids. Now let us have a quick look at the other structure on the double vector bundle T E.

Dualization and lifts
Recall that double vector bundles can be dualized in two distinct ways. We denote the dual of D as a vector bundle over A by D A and likewise for D B. The dual D A is itself a double vector bundle, with side bundles A and C * and core B * [13,16]. Note also that by dualizing again D B over C * , we get with core B * . In the same manner, we get a double vector bundle D A C * with sides B and C * and core A * .
Recall first of all that the vector bundles D B → C * and D A → C * are, up to a sign, naturally in duality to each other [13]. The pairing is defined as follows: for ∈ D A and ∈ D B projecting to the same element , d B does not depend on the choice of d and we set , This implies in particular that D A is canonically (up to a sign) isomorphic to D B C * and D B is isomorphic to D A C * . We will use this below.
Each linear section ξ ∈ B (D) over a ∈ (A) induces a linear section ξ ∈ C * (D B C * ) over a. Namely ξ induces a function ξ : D B → R which is fibrewise-linear over B and, using the definition of the addition in D B → C * [16, Equation (7)], it follows that ξ is also linear over C * . The corresponding section of D B C * → C * is denoted ξ [16]. Thus for ∈ D B such that π B ( ) = b and π C * ( ) = γ .
for all a ∈ (A). We now use the (canonical up to a sign) isomorphism of D A with D B C * to construct a canonical decomposition of D A given a decomposition of D. We identify D A with D B C * using − · , · . Thus we define the lift σ A : for all a ∈ (A). Note that by (6), this implies that for all γ ∈ C * and b ∈ B. The choice of sign in (7) is necessary for σ A (a) to be a linear section of D A over a. To be more explicit, check or recall from [15,Equation (28), p. 352] that σ A (a), α † = −q * C * α, a for all α ∈ (A * ) (and α † the corresponding core section of D B over C * ). But σ ,B A (a), α † C * = q * C * a, α by definition of the pairing of D B C * with D B. Hence, without the choice of sign that we make, σ A (a) would be linear over −a, hence not a lift.
By (skew-)symmetry, given the lift (This time, we do not need the minus sign.) As a summary, we have the equations:

LA-vector bundles and double Lie algebroids
The vector bundle A → M is then also a Lie algebroid, with anchor ρ A and bracket defined as follows: if Equivalently, D is a double Lie algebroid if the pair (D A, D B) with the induced Lie algebroid structures on base C * and the pairing ·, · , is a Lie bialgebroid. One aim of this paper is to reformulate this definition in terms of specific classes of sections, so as to allow the user to bypass frequent use of the duality of doubles; see Theorem 3.6.

Representations up to homotopy and LA-vector bundles
Let A → M be a Lie algebroid and consider an A-connection ∇ on a vector bundle E → M. Then the space • (A, E) of E-valued Lie algebroid forms has an induced operator d ∇ given by the É. Cartan formula: for all ω ∈ k (A, E) and a 1 , . . . , a k+1 ∈ (A).
Let now E = k∈Z E k be a graded vector bundle. Consider the space • (A, E) with grading given by

Definition 2.7 [1] A representation up to homotopy of A on E is a map
with total degree 1 and such that D 2 = 0 and Note that Gracia-Saz and Mehta [4] defined this concept independently and called it a "superrepresentation"; it is related to Quillen's notion of superconnection [22] in the same way that representations of Lie algebroids are related to the general notion of A-connection.
The representations up to homotopy which we will consider are always on graded vector bundles E = E 0 ⊕ E 1 concentrated on degrees 0 and 1; so-called 2-term graded vector bundles. In this case the data of [1] can be reformulated as follows (see [1,4]).

Definition 2.8 Let
A be a Lie algebroid and let E = E 0 ⊕ E 1 be a 2-term graded vector bundle on the same base. Then a 2-term representation up to homotopy or, for brevity, a 2-representation of A on E, consists of (i) a map ∂ :

Then D extends to a representation up to homotopy of A on E. This defines a bijective correspondence between 2-representations of A on E and representations up to homotopy of A on E.
Let (D → B; A → M) be an LA-vector bundle. Recall that since the anchor B is linear, it sends a core section c † , c ∈ (C) to a vertical vector field on B. This defines the core-anchor ∂ B : C → B given by, B (c † ) = (∂ B c) ↑ for all c ∈ (C) (see [11]).
Choose further a decomposition : A × M B → D. Since the anchor of a linear section is linear, for each a ∈ (A) the vector field B (σ A (a)) defines a derivation of (B) with symbol ρ(a) (see Sect. 2.1.2). This defines a linear connection for all a ∈ (A). Since the bracket of a linear section with a core section is again a core section, we find a linear connection ∇ AC : ] is a core-linear section for all a 1 , a 2 ∈ (A). This defines a vector valued Lie algebroid form R ∈ 2 (A, Hom(B, C)) such that for all a 1 , a 2 ∈ (A). See [4] for more details on these constructions.
The following theorem is proved in [4]. The following formulas for the brackets of linear and core sections with core-linear sections will be very useful in the proof of our main theorem. In the situation of the previous theorem, we have and for all a ∈ (A), φ ∈ (Hom(B, C)) and c ∈ (C). To see this, write φ as j and one can use the formulas in Theorem 2.10 and the Leibniz rule to compute the brackets with σ A (a) and c † .
for all a, a 1 , a 2 ∈ (A). the 2-representation of the Lie algebroid A on ∂ B : C → B. We have seen above that (D A; A, C * ; M) has an induced LA-vector bundle structure, and we have shown that the decomposition induces a natural decomposition

Dualization and 2-representations
This is easy to verify, and proved in the appendix 2 of [2]. One only needs to recall for the proof that, by construction, σ A (a) equals σ A (a) as a function on D B.

The tangent of a Lie algebroid
Given a T M-connection on A, and so a decomposition ∇ of T A as in Section where the connections are defined by and ∇ bas : with R bas ∇ ∈ 2 (A, Hom(T M, A)) given by for all X ∈ X(M), a, a 1 , a 2 ∈ (A).

Main theorem and examples
We define in this section the notion of matched pair of 2-representations. We then state our main result, Theorem 3.6: a double vector bundle endowed with two LA-vector bundle structures (one horizontal, one vertical) is a double Lie algebroid if and only if, for each decomposition, the two induced 2-representations form a matched pair.
In the second part of this section, we work out the example of the structures on the tangent double vector bundle of a Lie algebroid.

Matched pairs of 2-representations and main result
where R B is seen as an element of 1 (A, ∧ 2 B * ⊗ C) and R A as an element of 1 (B, ∧ 2 A * ⊗ C).

Remark 3.2 The conditions in Definition 3.1 imply that
Specifically, if A has nonzero rank, then (12) can be obtained from (M2) by replacing a with f a for f ∈ C ∞ (M). If B has nonzero rank, then (12) can similarly be obtained from (M3). If both A and B have rank zero, then it is trivially satisfied.

Remark 3.3 The conditions in Definition 3.1 also imply that
for all a ∈ (A) and b ∈ (B). Specifically, if A has nonzero rank, then (M0) can be obtained from (M5) by replacing a 2 with f a 2 for f ∈ C ∞ (M). If B has nonzero rank, then (M0) can be similarly obtained from (M6). If both A and B have rank zero, then it is trivially satisfied.  The proof of the Jacobi identity is not completely straightforward; it follows from (M2), (M3) and (M4). A detailed proof of a more general result, but with the same type of computation, is given in [7,Theorem 7.7]. Note that (M2) together with The Lie algebroid structure on C was defined intrinsically in [16]. Referring to Definition 2.6, the Lie bialgebroid (D A C * , D B C * ) induces a Poisson structure (natural up to sign) on its base C * ; this Poisson structure is linear [16, §4] and so induces a Lie algebroid structure on C. As with the specific formula for the bracket above, the sign of the Poisson structure is determined by the requirement that ∂ A and ∂ B be morphisms of Lie algebroids.

Comparison with the equations of Voronov
In [26] Th. Voronov extended the notion of double Lie algebroid to supergeometry and thereby gave an exceptionally elegant reformulation of the original definition. A vector bundle A may be characterized as a Lie algebroid if its parity-reversion A has a vector field Q of degree +1 which is homological in the sense that Q 2 = 0 [24,25].
Consider now a supermanifold D with a double vector structure (D; A, B Most of the calculations in [26] are carried out in terms of coordinates. Equations (47) and (48) of [26] give coordinate descriptions of the homological vector fields that are equivalent to the two Lie algebroid structures D A → C * and D B → C * .
The main work of [26] is to establish an equivalence between the Lie bialgebroid condition in the definition of a double Lie algebroid (Condition III in [26]) and the commutativity condition which we refer to briefly as In what follows, we refer to equations from [26] with an initial 'V'. The coefficients in (V 47) can be formulated in our terms as where R, ∇ and ∂ refer to the 2-representations induced by the local decomposition. We believe that the term ξ i η α Q j iα in (V 47) should have a minus sign for consistency with (V 26) and (V 28).
Similarly the coefficients in (V 48) are We now describe briefly how Voronov's nine equations (V 50) to (V 58) correspond to our seven equations (M1)-(M7) together with (12) and (M0). We treat two equations in detail and state the remaining correspondences, leaving details to the reader.

is satisfied for all μ and a if and only if
for all μ and a. But this is and we have ∂ * B η α , z μ η α = η α , ∂ B z μ η α = ∂ B (z μ ) and ∂ * A ξ i , z μ ξ i = ξ i , ∂ A z μ ξ i = ∂ A (z μ ) because of the choices of the dual basis {ξ i } and {ξ i }, and {η α } and {η α }. Hence, Equation (V 50) now reads ρ A (∂ A (z μ ))(x a ) = ρ B (∂ B (z μ ))(x a ) for all μ, a. Since x a are coordinates on the base manifold and {z μ } is a basis for C, for all μ, ν, λ. As before, this is which is exactly (M1). As the reader can see, the proof of the equivalences requires writing out Voronov's equations in terms of the 2-representations (see (13) and (14)), and using the duality of the coordinates. The proof of the remaining equivalences can be done in the same manner. We leave the details to the reader.

Equation (V 56):
, providing the last term in (V 56) is changed to a positive sign.

Equation (V 58):
is (M6), providing the third term in (V 58) is changed to a negative sign.

Remark 3.9
Regarding differences in signs, it is worth observing that, given a Lie bialgebroid (E, E * ), taking the opposite structure on either E or E * , or reversing their order, still results in a Lie bialgebroid. Thus varying the Lie bialgebroid chosen in the definition of double Lie algebroid has no important consequence, though some choices are easier to work with than others. We now verify equations (M1) to (M7) on a basic example.

The tangent double vector bundle of a Lie algebroid
Let A → M be a Lie algebroid with anchor ρ. We have seen in Sect. 2.3.2 that We check that these two 2-representations form a matched pair. This will provide a new proof of the fact that the tangent double of a Lie algebroid is a double Lie algebroid [16]. In the following, X, X 1 , X 2 will be arbitrary vector fields on M and a, a 1 , a 2 arbitrary sections of A. (M4) The left-hand side of (M4) is The second and sixth terms add up to R(X, ρ(a 2 ))a 1 + ∇ [X,ρ(a 2 )] a 1 and the first, third, and fifth terms to −R bas The definition of ∇ bas : (A) × X(M) → X(M) yields then immediately the right hand side of (M4), namely (M6) This equation is easily verified: To get the second equality, we use the Jacobi identity for the Lie bracket of vector fields. The four remaining terms cancel pairwise.
(M7) As one would expect, checking (M7) is a long, but straightforward computation. We carry this out in detail here, but will omit similar calculations in later cases. We begin by computing This expands out to Twelve terms of this equation cancel pairwise as shown, and a reordering of the remaining terms yields

By (M6), this equals
This completes the verification that the 2-representations associated with the tangent double of a Lie algebroid form a matched pair.
Remark 3.10 Infinitesimal ideal systems [5,9] may also be understood in terms of 2-representations. It is proved in [2] that infinitesimal ideal systems are in bijective correspondence with double subbundles of (T

The cotangent double of a Lie bialgebroid
Let (A, ρ, [· , ·]) be a Lie algebroid over a smooth manifold M, and assume that A * also has a Lie algebroid structure, with anchor denoted by ρ and with bracket [· , ·] .
Since A * is a Lie algebroid, A has a linear Poisson structure, and so (T * A → A, A * → M) has a LA-vector bundle structure. The Lie algebroid bracket is given by for all α, α 1 , α 2 ∈ (A * ) and θ, θ 1 , θ 2 ∈ 1 (M). The anchor is given by (see Sect. 2.1.2 for the notation). Likewise the Lie algebroid structure on A induces a LA-vector bundle structure on (T * (A * ) → A * , A → M), satisfying corresponding equations. Using the natural diffeomorphism T * (A * ) → T * A of [17], this LA-vector bundle structure may be transferred to (T * A → A * , A → M) and equips the double vector bundle with two LA-vector bundle structures. If (A, A * ) is a Lie bialgebroid then (18) is a double Lie algebroid with core T * M ( [16]; see also [23]). We now establish an equivalent result in terms of 2-representations. A linear connection ∇ : X(M) × (A) → (A), is equivalent to a decomposition ∇ of T A and so to a decomposition ∇ of (18). Using (8), one can check that ∇ : A× A * → T * A sends (a m , α m ) to d a m α −(T a m q) * ∇ · α, a m for any α ∈ (A * ) such that α(m) = α m (see also [6]).
A computation shows that the representation up to homotopy defined by (T * A → A, A * → M) and the decomposition is given by the morphism ρ * : T * M → A, by the connections (19) and by R bas (Recall from the previous example that since we have an ordinary connection ∇ * -the dual connection to ∇-on A * , we can define the basic A * -connections on A * and T M. The two connections above are their duals.) Next the LA-vector bundle Recall that the decomposition ∇ of T A and this LA-vector bundle structure define the 2-representation

Proof of the main theorem
We will prove the theorem by checking the Lie bialgebroid condition only on particular families of sections; the linear sections and the core sections. The main difficulty is to understand the additional conditions which have to be verified by the families of sections for the proof to be complete. This is done in Sect. 4.1. In Sect. 4.2, we will show how the equations found in Sect. 4.1 imply (M1)-(M7) and vice-versa.

Families of sections of Lie bialgebroids
We recall the definition of a Lie bialgebroid [17]; see also [15,Chapter 12]. We will then show how the equation defining a Lie bialgebroid (A, A * ) need be verified only on families of spanning sections of A and A * .
The brackets on the RHS are extensions to 2-vectors by standard Schouten calculus. It is often very convenient to check this condition only on the elements of a given set of sections S ⊆ (A) which spans (A) as a C ∞ (M)-module. We will formalize this technique shortly. We first need to recall some consequences of the definition.
The proof of the following proposition is a straightforward computation.

Proposition 4.2 Let A and A * be dual vector bundles with Lie algebroid structures.
For a 1 , a 2 ∈ (A), α 1 , α 2 ∈ (A * ) and f ∈ C ∞ (M), we have Now assume that (A, A * ) is a Lie bialgebroid. Take any a 1 ∈ A and any nonvanishing local section α 1 ∈ (A * ). Choose a (local) nonvanishing a 2 ∈ (A) and an α 2 ∈ (A * ) such that a 2 , α 1 = 0 and a 2 , α 2 = 1. (If A has rank 1 then (25) below is vacuously true.) Equation (24) now reduces to for all a 1 ∈ (A), f ∈ C ∞ (M), and local nonvanishing α 1 ∈ (A * ). A straightworward computation shows that the left-hand side of (25) is tensorial in the term α 1 . Hence, (25) holds for all α 1 ∈ (A * ). (For another proof, see [15, 12.1.8].) On the other hand, the left-hand side of (25) is not tensorial in the term a 1 . We multiply a 1 by a function g ∈ C ∞ (M) in this equation, expand out, and subtract We get that Again, since a 1 and α 1 were arbitrary, we have found which is easily seen to be equivalent to see also [17], [15, §12.1]. The map ρ * • ρ * : T * M → T M defines a Poisson structure on M, which we take to be the Poisson structure on M induced by the Lie bialgebroid structure.
These considerations lead to the following result.
Proof We proved above that these three conditions hold when (A, A * ) is a Lie bialgebroid. For the converse, a quick computation using (B1) and the considerations before the proposition shows that for all a 1 , a 2 ∈ S, α 1 , α 2 ∈ (A * ) and f, g ∈ C ∞ (M). This vanishes by (B1) and (B2). Since the Lie bialgebroid condition is additive and (A) is spanned as a C ∞ (M)-module by S, we are done.

The Lie bialgebroid conditions on lifts and on core sections
We write here A : D A → T C * for the anchor of D A → C * and B : D B → T C * for the anchor of D B → C * . We set and B ( (B)).

Proposition 4.5 Condition (B3) on S and R is equivalent to ρ
Proof Since A • * B and B • * A are vector bundle maps T * C * → T C * , it is sufficient to check (B3) on dF for F ∈ C ∞ (C * ). In fact, it is even sufficient to check and for any α ∈ (A * ): * This shows that * We get consequently In the same manner, We continue with linear functions. Choose c ∈ (C). Then for any section b ∈ (B), we get * This shows * where ∇ · c, · is seen as an element of (Hom(C * , B * )). This leads to for all c ∈ (C) and As a corollary of this proof, we find the following result. Recall that the map B • * A : T * C * → T C defines a Poisson structure on C * (see (26) and the considerations following it).

Corollary 4.6
The Poisson structure on C * induced by the Lie bialgebroid structure is the linear Poisson structure dual to the Lie algebroid structure on C as in Remark 3.5. More explicitly, it is given by Remark 4.7 Note that the apparent asymmetry between the structures over A and B arises from unavoidable choices in the identifications between the various duals. The Poisson structure on C * is nonetheless determined by requiring ∂ A and ∂ B to be morphisms of Lie algebroids.
For the study of (B1) and (B2), we will need the following lemma. Recall that for a Lie algebroid A, the Lie derivative £ : for all a, a ∈ (A) and α ∈ (A * ).
for all a ∈ (A), b ∈ (B), α ∈ (A * ) and β ∈ (B * ). The Lie derivative £ : Note that in these equations, R(a, ·)b is seen as a section of Hom(C * , A * ) and R(b, ·)a is seen as a section of Hom(C * , B * ).
Then we compute is a section with values in the core, and for arbitrary a 1 , a 2 ∈ (A) and α ∈ (A * ). This proves the equality £ σ A (a 1 ) α † = £ a 1 α † . The identity shows that £ σ A (a 1 ) σ B (b) is the sum of σ B (∇ a 1 b) with a section with values in the core. To find out this core term, we finally compute The formulas describing the Lie derivative £ : can be verified in the same manner. Proof The idea of this proof is to check (B2) on linear and core sections in S and R, and on linear and q C * -pullback functions on C * . We start with core sections. Choose α ∈ (A * ) and β ∈ (B * ). We have By Lemma 4.8 and with β † , α † = 0, we find that (B2) is trivially satisfied on α † , β † and any element of C ∞ (C * ). Now choose a ∈ (A), α ∈ (A * ). Using Lemma 4.8 we find for all F ∈ C ∞ (C * ) In particular, for . This shows that (B2) is in this case equivalent to (M2). In the same manner, (B2) on β † ∈ S for β ∈ (B * ), by Lemma 4.8. This is (M0). Finally we compute (B2) on σ A (a), σ B (b) and c , for c ∈ (C). This is by Lemma 4.8. We find hence that (B2) on σ A (a), σ B (b) and c is equivalent to (M4).
On the other hand, we can check as above that [d  R(a 1 , a 2 )b + a 1 , ∇ * ∇ a 2 b α − a 2 , ∇ * ∇ a 1 b α . Now using (10) and (11) we finally get We hence find that α(a 1 , a 2  By replacing in this equation a 2 by f a 2 with f ∈ C ∞ (M), we find (M0) since a 1 , a 2 , b and α were arbitrary. Then, using (M0) twice, we obtain (M5).
This completes the proof of Theorem 3.6.