Double Lie algebroids and representations up to homotopy

We show that double Lie algebroids, together with a chosen linear splitting, are equivalent to pairs 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 tangent of a Lie algebroid.


Introduction
Double Lie algebroids first arose as the infinitesimal form of double Lie groupoids [9,12]. 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 [14]. This definition was immediately given a simple and elegant reformulation in terms of super geometry and coordinates by Th. Voronov [20]. In terms of super geometry, a 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, of suitable weights, and the main compatibility condition of [20] 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 [5] and [1]; this differs from the concept of [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. 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 variously called LA-groupoids or VBgroupoids, and are the 'preliminary case' of double Lie algebroids in [14]. 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 [5] showed that VB-groupoid structures on D are in bijective correspondence with 2-representations defined in terms of A, B and C.
It is always possible to 'decompose' a double vector bundle; that is, any double vector bundle is isomorphic to a decomposed double vector bundle. 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 [20]. Now consider an arbitrary double Lie algebroid D. The Lie algebroid structures on D may be considered as a pair of VB-structures and accordingly a decomposition of D expresses the double Lie algebroid structure as a pair of 2-representations. Our main result (Theorem 3.4) 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 in some respects midway between the original formulation and that of [20]. Our treatment resembles that of Voronov inasmuch as the three intrinsic conditions of [14] are replaced by a greater number of conditions which are dependent on auxiliary data, but 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 local coordinates.
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 [14, §6] that is, of representations of Lie algebroids in the strict sense, without curvature. For this reason we regard the conditions (M1) to (M9) in Definition 3.1 as defining a matched pair of 2-representations.
In turn, [6] will show 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 [16,8]. In a different direction, [7] 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 §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 §2.2 we recall VB-algebroids and double Lie algebroids, and in §2.3 we finally define 2-representations.
In §3 we state our main result and the main work of the proof is given in §4.
We have included definitions of the key concepts required; in particular it is not necessary to have a detailed knowledge of [1], [5], [14] or [20].

Background and definitions
2.1. 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 [18,13,5] 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 ; is the addition map for the vector bundle D → B, and (4) the scalar multiplication R × D → D in the bundle D → B is a vector bundle morphism over the scalar multiplication R × A → A.
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 : . Given a double vector bundle (D; A, B; M ), the space of sections Γ B (D) is generated as a C ∞ (B)-module by two distinguished classes of sections (see [14]), the linear and the core sections which we now describe.
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 Γ c B (D) and the space of linear sections Γ ℓ B (D). Given ψ ∈ Γ(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 over the sides A and B and has core M × 0.
2.1.1. Linear splittings and lifts. A linear splitting 1 of (D; A, B; M ) is an injective morphism of double vector bundles Σ : A × M B ֒→ D over the identity on the sides A and B. That every double vector bundle admits local linear splittings was proved by [4]. Local linear splittings are equivalent to double vector bundle charts. Pradines originally defined double vector bundles as topological spaces with an atlas of double vector bundle charts [17]. Using a partition of unity, he proved that (provided the double base is a smooth manifold) this implies the existence of a global double splitting [18]. Hence, any double vector bundle in the sense of Definition 2.1 admits a (global) linear splitting.
A linear splitting Σ of D is equivalent to a splitting σ A of the short exact sequence of where the third map is the map that sends a linear section (ξ, a) to its base section a ∈ Γ(A). The splitting σ A will be called a lift. Given Σ, the lift σ A : 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 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 linear splitting, this implies that a lift The core vector field corresponding to e ∈ Γ(E) is the vertical lift e ↑ : E → T E, i.e. the vector field with flow φ : is called a linear vector field. It is well-known (see e.g. [13]) that a linear vector field ξ ∈ X l (E) covering X ∈ X(M ) corresponds to a derivation D * : Γ(E * ) → Γ(E * ) over X ∈ 1 Note that a linear splitting of D is equivalent to a decomposition of D, i.e. an isomorphism I : A × M B × M C → D of double vector bundles over the identities on the sides and core. Given a linear splitting Σ, the corresponding decomposition I is given by I(am, bm, cm) = Σ(am, bm) + B (0 bm + A cm). Given a decomposition I, the corresponding linear splitting Σ is given by Σ(am, bm) = I(am, bm, 0 C m ).
X(M ), and hence to a derivation D : Γ(E) → Γ(E) over X ∈ X(M ) (the dual derivation). The precise correspondence is given by 2 Here ℓ 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 Γ(E). The choice of a linear splitting Σ for (T E; T M, E; M ) is equivalent to the choice of a connection on E: Since a linear splitting gives us a linear vector field σ T M (X) ∈ X l (E) for each X, we can define ∇ : 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 ′ ∈ Γ(E). That is, the Lie bracket of vector fields on M and the connection encode completely the Lie bracket of vector fields on E.
Now let us have a quick look at the other structure on the double vector bundle T E. The , v ∈ T M, e ∈ Γ(E). 2.1.3. 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 * [11,14].
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 Φ ∈ D A and Ψ ∈ D B projecting to the same element γ m in 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 ∈ Γ B (D) over a ∈ Γ(A) induces a linear section ξ ⊓ ∈ Γ ℓ C * (D B C * ) over a. Namely ξ induces a function ℓ ξ : D B → R which is fibrewise-linear over B and, using the definition of the addition in D B → C * ([14, Equation (7)], it follows that ℓ ξ is also linear over C * . The corresponding section of D B C * → C * is denoted ξ ⊓ [14]. Thus Given a linear splitting Σ : We now use the (canonical up to a sign) isomorphism of D A with D B C * to construct a canonical linear splitting of D A given a linear splitting of D. We identify D A with 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 [13,Equation (28) 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: 2.2. VB-algebroids and double Lie algebroids. What we are here calling VB-algebroids were defined in [10,14] and called LA-vector bundles. 3 Let (D; A, B; M ) be a double vector bundle 3 The terminology 'LA-vector bundle' followed that of LA-groupoids, which were defined in [9, §4], on the model of Pradines' [19] concept of VB-groupoid. In [10] and [14, 3.3] LA-vector bundles were seen as a special case of LA-groupoids. The terminology 'VB-algebroid' of [5] distingushes the equivalent formulation in terms of bracket conditions on the linear and core sections.
Let now E = k∈Z E k [k] be a graded vector bundle. Consider the space Ω(A, E) with grading given by with total degree 1 and such that D 2 = 0 and is the Lie algebroid differential.
Note that Gracia-Saz and Mehta [5] defined this concept independently and called them "superrepresentations".
Let A be a Lie algebroid. 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. These representations are equivalent to the following data (see [1,5]): (1) a map ∂ : two A-connections, ∇ 0 and ∇ 1 on E 0 and E 1 , respectively, such that ∂ where ∇ Hom is the connection induced on Hom(E 1 , E 0 ) by ∇ 0 and ∇ 1 . We will call such a 2-term representation up to homotopy a 2-representation for brevity.
Let (D → B; A → M ) be a VB-algebroid. Then since the anchor ρ D is linear, it sends a core section c † , c ∈ Γ(C) to a vertical vector field on B. This defines the core-anchor [9]).
Choose further a linear splitting Σ : Since the anchor of a linear section is linear, for each a ∈ Γ(A) the vector field ρ D (σ A (a)) defines a derivation of Γ(B) with symbol ρ(a) (see §2.1.2). This defines a linear connection ∇ AB : for all a ∈ Γ(A). Since the bracket of a linear section with a core section is again linear, we find a linear connection ∇ AC : ] is a corelinear section for all a 1 , a 2 ∈ Γ(A). This defines a vector valued Lie algebroid form R ∈ Ω 2 (A, Hom(B, C)) such that for all a 1 , a 2 ∈ Γ(A). See [5] for more details on these constructions.
The following theorem is proved in [5]. 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 j and one can use the formulas in Theorem 2.8 and the Leibniz rule to compute the brackets with σ A (a) and c † .
Note that (9) and (10) can also be proved by diagrammatic methods.

Main theorem and examples
We define in this section the notion of matched pair of representations up to homotopyas above, we consider only representations up to homotopy which are concentrated in degrees 0 and 1; that is, 2-representations. We then state our main result: a double vector bundle endowed with two VB-algebroid structures (on each of its sides) is a double Lie algebroid if and only if, for each linear splitting, the two induced representations up to homotopy form a matched pair.
In the second part of this section, we work out the example of the tangent double of a Lie algebroid.
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).   It is easy to see using Remark 2.10 that the induced Lie algebroid structure on the core C of the double Lie algebroid does not depend on the choice of splitting.  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. (M6) The left-hand side of (M6) is The second and sixth term add up to R(X, ρ(a 2 ))a 1 + ∇ [X,ρ(a2)] a 1 and the first, third, and fifth term to −R bas ∇ (a 1 , a 2 )X+∇ ∇ bas a 2 X a 1 −∇ ∇ bas a 1 X a 2 . The definition of ∇ bas : Γ(A)×X(M ) → X(M ) yields then immediately the right hand side of (M6), namely R(X, ρ(a 2 ))a 1 − R bas ∇ (a 1 , a 2 )X.

(M8) 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).
(M9) As one would expect, checking (M9) is a long, but straightforward computation. We carry this out in detail here, because we will not give all the details of the proof of Theorem 3.4. 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

Proof of the 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 Subsection 4.1. In Subsection 4.2, we will show how the equations found in Subsection 4.1 imply (M1)-(M9) and vice-versa.

Families of sections of Lie bialgebroids.
We recall the definition of a Lie bialgebroid [15]; see also [13,Chapter 12]. We will then show how the equation defining a Lie bialgebroid (A, A * ) can 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 ⊆ Γ(A) which spans Γ(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.
These considerations lead to the following result.
for all a ∈ S, α ∈ Γ(A * ) and f ∈ C ∞ (M ), and (B3) −ρ • ρ * * = ρ * • ρ * . 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 ∈ Γ(A * ) and f, g ∈ C ∞ (M ). This vanishes by (B1) and (B2). Since the Lie bialgebroid condition is additive and Γ(A) is spanned as a C ∞ (M )-module by S, we are done. Proof. Since Θ A • Θ * B , Θ B • Θ * A : T * C * → T C * . are vector bundle maps, it is sufficient to check (B3) on dF for F ∈ C ∞ (C * ). In fact, it is even sufficient to check (B3) on d(q * C * f ) for f ∈ C ∞ (M ) and dℓ c for c ∈ Γ(C). Choose first f ∈ C ∞ (M ) and consider q * C * f ∈ C ∞ (C * ). We have for any section b ∈ Γ(B): In the same manner, we find We continue with linear functions. Choose c ∈ Γ(C). Then for any section b ∈ Γ(B), we get and for any α ∈ Γ(A * ): where ∇ · c, · is seen as an element of Γ(Hom(C * , B * )). This leads to As a corollary of this proof, we find the following result. Recall that the map Θ B • Θ * A : T * C * → T C defines a Poisson structure on C * (see (14) 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 3.3. 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 ′ ∈ Γ(A) and α ∈ Γ(A * ).
is given by the following identities: for all a ∈ Γ(A), b ∈ Γ(B), α ∈ Γ(A * ) and β ∈ Γ(B * ). The Lie derivative £ : ·)a. 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 * ).

Proof. We have
Then we compute is a section with values in the core, and . We also find α, a 2 for arbitrary a 1 , a 2 ∈ Γ(A) and α ∈ Γ(A * ). This proves the equality with a section with values in the core. To find out this core term, we finally compute

This shows that
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 ∈ Γ(A * ) and β ∈ Γ(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 ∈ Γ(A), α ∈ Γ(A * ). Using Lemma 4.8 we find for all F ∈ C ∞ (C * ) (15) and for F = ℓ c , c ∈ Γ(C), this is (16). But this equals q * C * ( α, ∂ A (∇ a c) − ∇ ∂B c a − [a, ∂ A c] ). This shows that (B2) is in this case equivalent to (M3). In the same manner, (B2) on β † ∈ S for β ∈ Γ(B * ), σ ⋆ B (b) ∈ R for b ∈ Γ(B) and F ∈ C ∞ (C * ) is equivalent to (M4).
We conclude the proof of Theorem 3.4 with the study of (B1) on linear and core sections. In the proof of this proposition, we will use the following formulas. Let A and A * be a pair of Lie algebroids in duality. Then, for all a ∈ Γ(A) and α 1 , α 2 ∈ Γ(A * ): For all a 1 , a 2 ∈ Γ(A) and α 1 , α 2 ∈ Γ(A * ), we have For β 1 , β 2 ∈ Γ(B * ) and a 1 , a 2 ∈ Γ(A), we find using Lemma 4.8 1 , a 2 )) .
On the other hand, we can check as above that [d Recall (9) and (10). Using this, we finally get σ ⋆ We hence find that for all a 1 , a 2 ∈ Γ(A * ). This is Hence, using (M5) twice, we obtain (M7) since α was arbitrary.
We conclude the proof of the theorem with the most technical formula. Choose b 1 , b 2 ∈ Γ(B). We want to study the equation 2 ) * † and we find easily that both sides of (18) vanish on β † 1 , β † 2 , for β 1 , β 2 ∈ Γ(B * ). We have for a ∈ Γ(A) and β ∈ Γ(B * ): Thus, we find that the two sides of (18) are equal on (σ ⋆ A (a), β † ) if and only if (M8) is satisfied.
Hence, we find that the two sides of (18) coincides on (σ ⋆ A (a 1 ), σ ⋆ A (a 2 )) if and only if (M9) is satisfied.