Double Lie algebroids and representations up to homotopy

Abstract

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.

1 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 $A\times _M B\times _M C$ of three vector bundles A, B, C on the same base M, and the vector bundle structures on D are the pullbacks of $B\oplus C$ to A and of $A\oplus C$ 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 $\mathsf {C}:E\times _M TM\rightarrow TE$ which is linear both as a map from the pullback of TM over the projction $E\rightarrow M$ and as a map from the pullback of E over the projection $TM\rightarrow M$, and which is right-inverse to both projections. This formulation, which can be found in Dieudonné, is precisely a decomposition of TE 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].

2 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 [4, 15, 20] for more detailed treatments.

Definition 2.1

A double vector bundle is a commutative square

satisfying the following four conditions:

• 1. all four sides are vector bundles;

• 2. $\pi _B$ is a vector bundle morphism over $q_A$;

• 3. $+_B: D\times _B D \rightarrow D$ is a vector bundle morphism over $+: A\times _M A \rightarrow A$, where $+_B$ is the addition map for the vector bundle $D\rightarrow B$;

• 4. the scalar multiplication $\mathbb {R}\times D\rightarrow D$ in the bundle $D\rightarrow B$ is a vector bundle morphism over the scalar multiplication $\mathbb {R}\times A\rightarrow A$,

together with a fifth condition (5) below.

The corresponding statements for the operations in the bundle $D\rightarrow A$ 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 $\pi _A$ and $\pi _B$. It has a natural vector bundle structure over M, the restriction of either structure on D, the projection of which we call $q_C: C \rightarrow M$. The inclusion $C \hookrightarrow D$ is usually denoted by

$$\begin{aligned} C_m \ni c \longmapsto \overline{c} \in \pi _A^{-1}(0^A_m) \cap \pi _B^{-1}(0^B_m). \end{aligned}$$

Given any three vector bundles A, B, C on the same base manifold M, the fibre product $\overline{D}:=A\times _M B\times _M C$ has a natural structure of double vector bundle with side bundles A and B and core C; the vector bundle structure on $\overline{D}\rightarrow A$ is the pullback of $B\oplus C$ to A and likewise the vector bundle structure on $\overline{D}\rightarrow B$ is the pullback of $A\oplus C$ to B. We can now state the fifth condition of Definition 2.1:

• 5. there is a diffeomorphism from D to the double vector bundle $\overline{D} = A\times _M B\times _M C$, where C is the core of D, which is the identity on A, B and C and is an isomorphism of vector bundles $D\rightarrow \overline{D}$ over A and B.

A morphism of double vector bundles from (D; A, B; M) to $(D';A',B';M')$ consists of smooth maps $\phi :D\rightarrow D'$, $\phi _A:A\rightarrow A'$, $\phi _B:B\rightarrow B'$ and $\phi _M:M\rightarrow M'$ such that each of $(\phi , \phi _A)$, $(\phi , \phi _B)$, $(\phi _A, \phi _M)$ and $(\phi _B, \phi _M)$ is a vector bundle morphism.

Given a double vector bundle (D; A, B; M), the space of sections $\Gamma _B(D)$ is generated as a $C^{\infty }(B)$-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 $c:M \rightarrow C$, the corresponding core section $c^\dagger :B \rightarrow D$ is defined as

$$\begin{aligned} c^\dagger (b_m) = \tilde{0}_{{b_m}} +_A \overline{c(m)}, \,\, m \in M, \, b_m \in B_m. \end{aligned}$$

(1)

We denote the corresponding core section $A\rightarrow D$ by $c^\dagger$ also, relying on the argument to distinguish between them.

Definition 2.3

A section $\xi \in \Gamma _B(D)$ is called linear if $\xi :B \rightarrow D$ is a bundle morphism from $B \rightarrow M$ to $D \rightarrow A$ over a section $a\in \Gamma (A)$.

The space of core sections of D over B will be written $\Gamma _B^c(D)$ and the space of linear sections $\Gamma ^\ell _B(D)$. Given $\psi \in \Gamma (B^*\otimes C)$, there is a linear section $\widetilde{\psi }:B\rightarrow D$ over the zero section $0^A:M\rightarrow A$ given by

$$\begin{aligned} \widetilde{\psi }(b_m) = \widetilde{0}_{b_m}+_A \overline{\psi (b_m)}. \end{aligned}$$

(2)

We call $\widetilde{\psi }$ a core-linear section.

Example 2.4

Let $A, \, B, \, C$ be vector bundles over M and consider $\overline{D}=A\times _M B \times _M C$ with the vector bundle structures described in Definition 2.1. We call a double vector bundle of this type decomposed or a trivial double vector bundle with core C. The core sections are given by

$$\begin{aligned} c^\dagger :b_m \mapsto (0^A_m, b_m, c(m)),\quad \text { where } m \in M, \quad b_m \in B_m,\quad c \in \Gamma (C), \end{aligned}$$

and similarly for $c^\dagger :A\rightarrow D$. The space of linear sections $\Gamma ^\ell _B(D)$ is naturally identified with $\Gamma (A)\oplus \Gamma (B^*\otimes C)$ via

$$\begin{aligned} (a, \psi ): b_m \mapsto (a(m), b_m, \psi (b_m)), \quad \text { where } \psi \in \Gamma (B^*\otimes C), \quad a\in \Gamma (A). \end{aligned}$$

In particular, the fibered product $A\times _M B$ is a double vector bundle with side bundles A and B and core $M\times 0$.

2.1.1 Decompositions and lifts

A decomposition of (D; A, B; M) is a diffeomorphism $\mathbb {I}:D\rightarrow \overline{D} = A\times _M B\times _M C$, as in Definition 2.1; that is, $\mathbb {I}$ is an isomorphism of double vector bundles over the identity maps on A, B and C.

Given an injective morphism of double vector bundles $\Sigma :A\times _M B\hookrightarrow D$ over the identity on the sides A and B, define a decomposition of D by $\mathbb {I}(a_m,b_m,c_m)=\Sigma (a_m,b_m)+_B(\tilde{0}_{{b_m}} +_A \overline{c_m})$. Decompositions of D are in bijective correspondence with such morphisms $\Sigma$. We will often refer to $\Sigma$ as a decomposition of D.

Decompositions of D are also equivalent to splittings $\sigma _A$ of the short exact sequence of $C^\infty (M)$-modules

$$\begin{aligned} 0 \longrightarrow \Gamma (B^*\otimes C) \hookrightarrow \Gamma ^\ell _B(D) \longrightarrow \Gamma (A) \longrightarrow 0, \end{aligned}$$

where the third map is the map that sends a linear section $(\xi ,a)$ to its base section $a\in \Gamma (A)$. The splitting $\sigma _A$ will be called a lift. Given a decomposition, the lift $\sigma _A:\Gamma (A)\rightarrow \Gamma _B^\ell (D)$ is given by $\sigma _A(a)(b_m)=\Sigma (a(m), b_m) = {\mathbb {I}}^{-1}(a(m), b_m,0_m)$ for all $a\in \Gamma (A)$ and $b_m\in B$.

In the case of the tangent double of a vector bundle $E\rightarrow M$, 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 $\sigma _A:\Gamma (A)\rightarrow \Gamma _B^\ell (D)$ is equivalent to a lift $\sigma _B:\Gamma (B)\rightarrow \Gamma _A^\ell (D)$. Given a lift $\sigma _A:\Gamma (A)\rightarrow \Gamma ^\ell _B(D)$, the corresponding lift $\sigma _B:\Gamma (B)\rightarrow \Gamma ^\ell _A(D)$ is given by $\sigma _B(b)(a(m))=\sigma _A(a)(b(m))$ for all $a\in \Gamma (A)$, $b\in \Gamma (B)$.

Note finally that two decompositions of D differ by a section $\phi _{12}$ of $A^*\otimes B^*\otimes C\simeq {\text {Hom}}(A,B^*\otimes C)\simeq {\text {Hom}}(B,A^*\otimes C)$ in the following sense. For each $a\in \Gamma (A)$ the difference $\sigma _A^1(a)-_B\sigma _A^2(a)$ of lifts is the core-linear section defined by $\phi _{12}(a)\in \Gamma (B^*\otimes C)$. By symmetry, $\sigma _B^1(b)-_A\sigma _B^2(b)=\widetilde{\phi _{12}(b)}$ for each $b\in \Gamma (B)$.

2.1.2 The tangent double vector bundle of a vector bundle

Let $q_E:E\rightarrow M$ be a vector bundle. Then the tangent bundle TE has two vector bundle structures; one as the tangent bundle of the manifold E, and the second as a vector bundle over TM. The structure maps of $TE\rightarrow TM$ are the derivatives of the structure maps of $E\rightarrow M$. The space TE is a double vector bundle with core bundle $E \rightarrow M$.

The core vector field corresponding to $e \in \Gamma (E)$ is the vertical lift $e^{\uparrow }: E \rightarrow TE$, i.e. the vector field with flow $\phi :E\times \mathbb {R}\rightarrow E$, $\phi _t(e'_m)=e'_m+te(m)$. An element of $\Gamma ^\ell _E(TE)=\mathfrak {X}^\ell (E)$ is called a linear vector field. It is well-known (see e.g. [15]) that a linear vector field $\xi \in \mathfrak {X}^l(E)$ covering $X\in \mathfrak {X}(M)$ corresponds to a derivation $D^*:\Gamma (E^*) \rightarrow \Gamma (E^*)$ over $X\in \mathfrak {X}(M)$, and hence to a derivation $D:\Gamma (E)\rightarrow \Gamma (E)$ over $X\in \mathfrak {X}(M)$ (the dual derivation). The precise correspondence is given by¹

$$\begin{aligned} \xi (\ell _{\varepsilon }) = \ell _{D^*(\varepsilon )} \quad \text { and } \quad \xi (q_E^*f)= q_E^*(X(f)) \end{aligned}$$

(3)

for all $\varepsilon \in \Gamma (E^*)$ and $f\in C^\infty (M)$. Here $\ell _\varepsilon$ is the linear function $E\rightarrow \mathbb {R}$ corresponding to $\varepsilon$. We will write $\widehat{D}$ for the linear vector field corresponding in this manner to a derivation D of $\Gamma (E)$. The choice of a decomposition for (TE; TM, E; M) is equivalent to the choice of a connection on E: Since a decomposition gives us a linear vector field $\sigma _{TM}(X)\in \mathfrak {X}^l(E)$ for each X, we can define $\nabla :\mathfrak {X}(M)\times \Gamma (E)\rightarrow \Gamma (E)$ by $\sigma _{TM}(X)=\widehat{\nabla _X}$ for all $X\in \mathfrak {X}(M)$. Conversely, a connection $\nabla :\mathfrak {X}(M)\times \Gamma (E)\rightarrow \Gamma (E)$ defines a lift $\sigma _{TM}^\nabla$ for (TE; TM, E; M) and a decomposition of E.

We recall as well the relation between the connection and the Lie bracket of vector fields on E. Given $\nabla$, it is easy to see using the equalities in (3) that, writing $\sigma$ for $\sigma _{TM}^\nabla$:

$$\begin{aligned} \left[ \sigma (X), \sigma (Y)\right] =\sigma [X,Y]-R_\nabla (X,Y)^\uparrow ,\quad \left[ \sigma (X), e^\uparrow \right] =(\nabla _Xe)^\uparrow ,\quad \left[ e^\uparrow ,e'^\uparrow \right] =0, \end{aligned}$$

(4)

for all $X,Y\in \mathfrak {X}(M)$ and $e,e'\in \Gamma (E)$. 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 TE. The lift $\sigma _{E}^\nabla :\Gamma (E)\rightarrow \Gamma _{TM}^\ell (TE)$ is given by

$$\begin{aligned} \sigma _{E}^\nabla (e)(v) = Te(v) +_{TM} (T0^E(v) -_E \overline{\nabla _v e}), \quad v \in TM, \quad e \in \Gamma (E). \end{aligned}$$

(5)

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 and likewise for . The dual is itself a double vector bundle, with side bundles A and $C^*$ and core $B^*$ [13, 16].

Note also that by dualizing again over $C^*$, we get

with core $B^*$. In the same manner, we get a double vector bundle with sides B and $C^*$ and core $A^*$.

Recall first of all that the vector bundles and are, up to a sign, naturally in duality to each other [13]. The pairing

is defined as follows: for and projecting to the same element $\gamma _m$ in $C^*$, choose $d\in D$ with $\pi _A(d)=\pi _A(\Phi )$ and $\pi _B(d)=\pi _B(\Psi )$. Then $\langle \Phi , d\rangle _A-\langle \Psi ,d\rangle _B$ does not depend on the choice of d and we set .

This implies in particular that is canonically (up to a sign) isomorphic to and is isomorphic to . We will use this below.

Each linear section $\xi \in \Gamma _B(D)$ over $a\in \Gamma (A)$ induces a linear section over a. Namely $\xi$ induces a function which is fibrewise-linear over B and, using the definition of the addition in [16, Equation (7)], it follows that $\ell _\xi$ is also linear over $C^*$. The corresponding section of is denoted $\xi ^\sqcap$ [16]. Thus

$$\begin{aligned} \langle \xi ^\sqcap (\gamma ), \Psi \rangle _{C^*} = \ell _\xi (\Psi ) = \langle \Psi ,\, \xi (b)\rangle _B \end{aligned}$$

(6)

for such that $\pi _B(\Psi )=b$ and $\pi _{C^*}(\Psi )=\gamma$.

Given a decomposition $\Sigma :A\times _M B\rightarrow D$ of D, we get hence a decomposition

, defined by the corresponding lift :

$$\begin{aligned} \sigma _{A}^{\star ,B}(a)=(\sigma _A(a))^\sqcap \end{aligned}$$

for all $a\in \Gamma (A)$.

We now use the (canonical up to a sign) isomorphism of with to construct a canonical decomposition of given a decomposition of D. We identify with using . Thus we define the lift by

(7)

for all $a\in \Gamma (A)$. Note that by (6), this implies that

$$\begin{aligned} \langle \sigma _A^\star (a)(\gamma ), \sigma _A(a)(b)\rangle _A=0 \end{aligned}$$

(8)

for all $\gamma \in C^*$ and $b\in B$. The choice of sign in (7) is necessary for $\sigma _A^\star (a)$ to be a linear section of over a. To be more explicit, check or recall from [15, Equation (28), p. 352] that for all $\alpha \in \Gamma (A^*)$ (and $\alpha ^\dagger$ the corresponding core section of over $C^*$). But $\langle \sigma _A^{\star ,B}(a), \alpha ^\dagger \rangle _{C^*}=q_{C^*}^*\langle a, \alpha \rangle$ by definition of the pairing of with . Hence, without the choice of sign that we make, $\sigma _A^\star (a)$ would be linear over $-a$, hence not a lift.

By (skew-)symmetry, given the lift $\sigma _B:\Gamma (B)\rightarrow \Gamma ^\ell _A(D)$, we identify with using and define the lift by for all $b\in \Gamma (B)$. (This time, we do not need the minus sign.) As a summary, we have the equations:

(9)

for all $a\in \Gamma (A)$, $b\in \Gamma (B)$, $\alpha \in \Gamma (A^*)$ and $\beta \in \Gamma (B^*)$. See also [15, §9.2].

2.2 LA–vector bundles and double Lie algebroids

Let (D; A, B; M) be a double vector bundle

with core C. Then (D; A, B; M) is a LA–vector bundle [12] if there are Lie algebroid structures on $D\rightarrow B$ and $A\rightarrow M$, such that the vector bundle operations in $D\rightarrow A$ are Lie algebroid morphisms over the corresponding operations in $B\rightarrow M$. As a consequence, the anchor $\Theta _B:D\rightarrow TB$ is a morphism of double vector bundles.

An equivalent concept was introduced in [4] with the name VB-algebroid: $(D \rightarrow B; A \rightarrow M)$ is a VB-algebroid if $D \rightarrow B$ is a Lie algebroid, the anchor $\Theta _D:D \rightarrow TB$ is a bundle morphism over $\rho _A: A \rightarrow TM$ and the Lie bracket is linear:

$$\begin{aligned}{}[\Gamma ^\ell _B(D), \Gamma ^\ell _B(D)] \subset \Gamma ^\ell _B(D), \quad [\Gamma ^\ell _B(D), \Gamma ^c_B(D)] \subset \Gamma ^c_B(D), \quad [\Gamma ^c_B(D), \Gamma ^c_B(D)]= 0. \end{aligned}$$

The vector bundle $A\rightarrow M$ is then also a Lie algebroid, with anchor $\rho _A$ and bracket defined as follows: if $\xi _1,\xi _2\in \Gamma ^\ell _B(D)$ are linear over $a_1,a_2\in \Gamma (A)$, then the bracket $[\xi _1,\xi _2]$ is linear over $[a_1,a_2]$.

Example 2.5

For a vector bundle E on M, the tangent double vector bundle (TE; E, TM; M) has an LA–vector bundle structure $(TE\rightarrow E, TM\rightarrow M)$ with respect to the standard Lie algebroid structures on $TE\rightarrow E$ and $TM\rightarrow M$.

If D is an LA–vector bundle with Lie algebroid structures on $D\rightarrow B$ and $A\rightarrow M$ the dual vector bundle has a Lie-Poisson structure (a linear Poisson structure), and the structure on is also Lie-Poisson with respect to [16, 3.4]. Dualizing this bundle gives a Lie algebroid structure on . This equips the double vector bundle with a LA–vector bundle structure. Using the isomorphism defined by , the double vector bundle also has a LA–vector bundle structure.

Definition 2.6

[16]  A double Lie algebroid is a double vector bundle (D; A, B; M) with core denoted C, and with Lie algebroid structures on each of $A\rightarrow M$, $B\rightarrow M$, $D\rightarrow A$ and $D\rightarrow B$ such that each pair of parallel Lie algebroids gives D the structure of an LA–vector bundle, and such that with the induced Lie algebroid structures on base $C^*$ as defined above, is a Lie bialgebroid.

Equivalently, D is a double Lie algebroid if the pair 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.

2.3 Representations up to homotopy and LA–vector bundles

Let $A\rightarrow M$ be a Lie algebroid and consider an A-connection $\nabla$ on a vector bundle $E\rightarrow M$. Then the space $\Omega ^\bullet (A,E)$ of E-valued Lie algebroid forms has an induced operator $\mathbf {d}_\nabla$ given by the É. Cartan formula:

$$\begin{aligned} \mathbf {d}_\nabla \omega (a_1,\ldots ,a_{k+1})&=\sum _{i<j}(-1)^{i+j}\omega ([a_i,a_j],a_1,\ldots ,\hat{a}_i,\ldots ,\hat{a}_j,\ldots , a_{k+1})\\&\quad +\sum _i(-1)^{i+1}\nabla _{a_i}(\omega (a_1,\ldots ,\hat{a}_i,\ldots ,a_{k+1})) \end{aligned}$$

for all $\omega \in \Omega ^k(A,E)$ and $a_1,\ldots ,a_{k+1}\in \Gamma (A)$.

Let now ${\mathcal E}= \bigoplus _{k\in \mathbb {Z}} E_k$ be a graded vector bundle. Consider the space $\Omega ^\bullet (A,{\mathcal E})$ with grading given by

$$\begin{aligned} \Omega ^\bullet (A,{\mathcal E})^k = \bigoplus _{i+j=k}\Omega ^i(A, E_j). \end{aligned}$$

Definition 2.7

[1] A representation up to homotopy of A on ${\mathcal E}$ is a map ${\mathcal D}:\Omega ^\bullet (A, {\mathcal E}) \rightarrow \Omega ^\bullet (A,{\mathcal E})$ with total degree 1 and such that ${\mathcal D}^2=0$ and

$$\begin{aligned} {\mathcal D}(\alpha \wedge \omega ) = \mathbf {d}_A\alpha \wedge \omega + (-1)^{|\alpha |} \alpha \wedge {\mathcal D}(\omega ),\quad \text { for } \alpha \in \Gamma (\wedge ^\bullet A^*),\quad \omega \in \Omega ^\bullet (A,{\mathcal E}), \end{aligned}$$

where $\mathbf {d}_A:\Gamma (\wedge ^\bullet A^*) \rightarrow \Gamma (\wedge ^\bullet A^*)$ is the Lie algebroid differential.

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 ${\mathcal E}=E_0\oplus E_1$ 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 ${\mathcal E}= E_0\oplus E_1$ 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 ${\mathcal E}$, consists of

• (i) a map $\partial :E_0\rightarrow E_1$,

• (ii) two A-connections, $\nabla ^0$ and $\nabla ^1$ on $E_0$ and $E_1$, respectively, such that $\partial \circ \nabla ^0 = \nabla ^1 \circ \partial$,

• (iii) an element $R \in \Omega ^2(A, {{\mathrm{Hom}}}(E_1, E_0))$ such that $R_{\nabla ^0} = R\circ \partial$, $R_{\nabla ^1}=\partial \circ R$ and $\mathbf {d}_{\nabla ^{{{\mathrm{Hom}}}}}R=0$, where $\nabla ^{{{\mathrm{Hom}}}}$ is the connection induced on ${{\mathrm{Hom}}}(E_1,E_0)$ by $\nabla ^0$ and $\nabla ^1$.

Proposition 2.9

Let A be a Lie algebroid and let ${\mathcal E}= E_0\oplus E_1$ be a 2-term graded vector bundle on the same base. Given a 2-representation of A on ${\mathcal E}$, define ${{\mathcal D}}:\Gamma E^0 \rightarrow \Omega ^1(A,E^0) \oplus \Gamma E^1$ by ${{\mathcal D}}(e^0) = \nabla ^0(e^0) \oplus \partial (e^0)$ and define ${{\mathcal D}}:\Omega ^1(A,E^0) \oplus \Gamma E^1 \rightarrow \Omega ^2(A,E^0) \oplus \Omega ^1(A,E^1)$ by

$$\begin{aligned} {{\mathcal D}}(\omega \oplus e^1) = (-R(e^1) - d_{\nabla ^0}(\omega ))\oplus (\partial (\omega ) - \nabla ^1(e^1)). \end{aligned}$$

Then ${{\mathcal D}}$ extends to a representation up to homotopy of A on ${\mathcal E}$. This defines a bijective correspondence between 2-representations of A on ${\mathcal E}$ and representations up to homotopy of A on ${\mathcal E}$.

Let $(D \rightarrow B; A \rightarrow M)$ be an LA–vector bundle. Recall that since the anchor $\Theta _B$ is linear, it sends a core section $c^\dagger$, $c\in \Gamma (C)$ to a vertical vector field on B. This defines the core-anchor $\partial _B:C\rightarrow B$ given by, $\Theta _B(c^\dagger )=(\partial _Bc)^\uparrow$ for all $c\in \Gamma (C)$ (see [11]).

Choose further a decomposition $\Sigma :A\times _MB\rightarrow D$. Since the anchor of a linear section is linear, for each $a\in \Gamma (A)$ the vector field $\Theta _B(\sigma _A(a))$ defines a derivation of $\Gamma (B)$ with symbol $\rho (a)$ (see Sect. 2.1.2). This defines a linear connection $\nabla ^{AB}:\Gamma (A)\times \Gamma (B)\rightarrow \Gamma (B)$:

$$\begin{aligned} \Theta _B(\sigma _A(a))=\widehat{\nabla _a^{AB}} \end{aligned}$$

for all $a\in \Gamma (A)$. Since the bracket of a linear section with a core section is again a core section, we find a linear connection $\nabla ^{AC}:\Gamma (A)\times \Gamma (C)\rightarrow \Gamma (C)$ such that

$$\begin{aligned}{}[\sigma _A(a),c^\dagger ]=(\nabla _a^{AC}c)^\dagger \end{aligned}$$

for all $c\in \Gamma (C)$ and $a\in \Gamma (A)$. The difference $\sigma _A[a_1,a_2]-[\sigma _A(a_1), \sigma _A(a_2)]$ is a core-linear section for all $a_1,a_2\in \Gamma (A)$. This defines a vector valued Lie algebroid form $R\in \Omega ^2(A,{\text {Hom}}(B,C))$ such that

$$\begin{aligned}{}[\sigma _A(a_1), \sigma _A(a_2)]=\sigma _A[a_1,a_2]-\widetilde{R(a_1,a_2)}, \end{aligned}$$

for all $a_1,a_2\in \Gamma (A)$. See [4] for more details on these constructions.

The following theorem is proved in [4].

Theorem 2.10

Let $(D \rightarrow B; A \rightarrow M)$ be a LA–vector bundle and choose a decomposition $\Sigma :A\times _MB\rightarrow D$. The triple $(\nabla ^{AB},\nabla ^{AC},R)$ defined as above is a 2-representation of A on the complex $\partial _B:C\rightarrow B$.

Conversely, let (D; A, B; M) be a double vector bundle such that A has a Lie algebroid structure and choose a decomposition $\Sigma :A\times _MB\rightarrow D$. Then if $(\nabla ^{AB},\nabla ^{AC},R)$ is a 2-representation of A on a complex $\partial _B:C\rightarrow B$, then the four equations above define a LA–vector bundle structure on $(D\rightarrow B; A\rightarrow M)$.

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

$$\begin{aligned} \left[ \sigma _A(a),\widetilde{\phi }\right] =\widetilde{\nabla _a^\mathrm{Hom}\phi } \end{aligned}$$

(10)

and

$$\begin{aligned} \left[ c^\dagger ,\widetilde{\phi }\right] =(\phi (\partial _Bc))^\dagger \end{aligned}$$

(11)

for all $a\in \Gamma (A)$, $\phi \in \Gamma ({\text {Hom}}(B,C))$ and $c\in \Gamma (C)$. To see this, write $\phi$ as $\sum f_{ij}\cdot \beta _i\cdot c_j$ with $f_{ij}\in C^\infty (M)$, $\beta _i\in \Gamma (B^*)$ and $c_j\in \Gamma (C)$. Then $\widetilde{\phi }=\sum q_B^*f_{ij}\cdot \ell _{\beta _i}\cdot c_j^\dagger$ and one can use the formulas in Theorem 2.10 and the Leibniz rule to compute the brackets with $\sigma _A(a)$ and $c^\dagger$.

Note that (10) and (11) can also be proved by diagrammatic methods.

Example 2.11

Choose a linear connection $\nabla :\mathfrak {X}(M)\times \Gamma (E)\rightarrow \Gamma (E)$ and consider the corresponding decomposition $\Sigma ^\nabla$ of TE as in Sect. 2.1.2. The description of the Lie bracket of vector fields in (4) shows that the 2-representation induced by $\Sigma ^\nabla$ is the 2-representation of TM on ${{\mathrm{Id}}}_E:E\rightarrow E$ given by $(\nabla ,\nabla ,R_\nabla )$.

Remark 2.12

Let $\Sigma _1,\Sigma _2:A\times _M B\rightarrow D$ be two decompositions of a LA–vector bundle $(D\rightarrow B, A\rightarrow M)$, and write $\phi _{12}$ for the section of $A^*\otimes B^*\otimes C$ defined in Sect. 2.1.1. Regarding $\phi _{12}$ as an element of ${{\mathrm{Hom}}}(A, B^* \otimes C)$ and as an element of ${{\mathrm{Hom}}}(B, A^* \otimes C)$, the two corresponding 2-representations are related by the following identities [4].

$$\begin{aligned} \nabla ^{B,2}_a=\nabla ^{B,1}_a+\partial _B\circ \phi _{12}(a), \quad \nabla ^{C,2}_a=\nabla ^{C,1}_a+\phi _{12}(a)\circ \partial _B \end{aligned}$$

and

$$\begin{aligned}&R^2(a_1,a_2)=R^1(a_1,a_2)+(\mathbf {d}_{\nabla ^{{\text {Hom}}(B,C)}}\phi _{12})(a_1,a_2) +\phi _{12}(a_1)\partial _B\phi _{12}(a_2)\\&\quad -\phi _{12}(a_2)\partial _B\phi _{12}(a_1) \end{aligned}$$

for all $a,a_1,a_2\in \Gamma (A)$.

2.3.1 Dualization and 2-representations

Let (D; A, B; M) be a LA–vector bundle with Lie algebroid structures on $D\rightarrow B$ and $A\rightarrow M$. Let $\Sigma :A\times _MB\rightarrow D$ be a decomposition of D and denote by $(\nabla ^B,\nabla ^C,R)$ the 2-representation of the Lie algebroid A on $\partial _B:C\rightarrow B$. We have seen above that has an induced LA–vector bundle structure, and we have shown that the decomposition $\Sigma$ induces a natural decomposition of . The 2-representation of A that is associated to this decomposition is then $({\nabla ^C}^*,{\nabla ^B}^*,-R^*)$ on the complex $\partial _B^*:B^*\rightarrow C^*$. This is easy to verify, and proved in the appendix² of [2]. One only needs to recall for the proof that, by construction, $\ell _{\sigma _A^\star (a)}$ equals $\ell _{\sigma _A(a)}$ as a function on .

2.3.2 The tangent of a Lie algebroid

Let $(A\rightarrow M,\rho ,[\cdot \,,\cdot ])$ be a Lie algebroid. Then the tangent $TA\rightarrow TM$ has a Lie algebroid structure with bracket defined by $[Ta_1, Ta_2]=T[a_1,a_2]$, $[Ta_1, a_2^\dagger ]=[a_1,a_2]^\dagger$ and $[a_1^\dagger , a_2^\dagger ]=0$ for all $a_1,a_2\in \Gamma (A)$. The anchor of Ta is $\widehat{[\rho (a),\cdot ]}\in \mathfrak {X}(TM)$ and the anchor of $a^\dagger$ is $\rho (a)^\uparrow$ for all $a\in \Gamma (A)$. This defines a LA–vector bundle structure $(TA\rightarrow TM; A\rightarrow M)$ on (TA; TM, A; M).

Given a TM-connection on A, and so a decomposition $\Sigma ^\nabla$ of TA as in Section 2.1.2, the 2-representation of A on $\rho :A\rightarrow TM$ encoding the LA–vector bundle is $(\nabla ^\mathrm{bas},\nabla ^\mathrm{bas}, R_\nabla ^\mathrm{bas})$, where the connections are defined by

$$\begin{aligned} \nabla ^\mathrm{bas} :\Gamma (A)\times \mathfrak {X}(M)\rightarrow \mathfrak {X}(M), \quad \nabla ^\mathrm{bas}_aX =[\rho (a), X]+\rho (\nabla _Xa), \end{aligned}$$

and

$$\begin{aligned} \nabla ^\mathrm{bas} :\Gamma (A)\times \Gamma (A)\rightarrow \Gamma (A), \quad \nabla ^\mathrm{bas}_{a_1}a_2 =[a_1,a_2]+\nabla _{\rho (a_2)}a_1, \end{aligned}$$

with $R_\nabla ^\mathrm{bas}\in \Omega ^2(A,{\text {Hom}}(TM,A))$ given by

$$\begin{aligned} R_\nabla ^\mathrm{bas}(a_1,a_2)X=-\nabla _X[a_1,a_2]+[\nabla _Xa_1,a_2]+[a_1,\nabla _Xa_2]+\nabla _{\nabla ^\mathrm{bas}_{a_2}X}a_1-\nabla _{\nabla ^\mathrm{bas}_{a_1}X}a_2 \end{aligned}$$

for all $X\in \mathfrak {X}(M)$, $a, a_1,a_2\in \Gamma (A)$.

3 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.

3.1 Matched pairs of 2-representations and main result

Definition 3.1

Let $(A\rightarrow M, \rho _A, [\cdot \,,\cdot ])$ and and $(B\rightarrow M, \rho _B, [\cdot \,,\cdot ])$ be two Lie algebroids and assume that A acts on $\partial _B:C\rightarrow B$ up to homotopy via $(\nabla ^{AB},\nabla ^{AC}, R_{A})$ and B acts on $\partial _A:C\rightarrow A$ up to homotopy via $(\nabla ^{BA},\nabla ^{BC}, R_{B})$. Then we say that the two 2-representations form a matched pair if the following hold:³

• (M1) $\nabla _{\partial _Ac_1}c_2-\nabla _{\partial _Bc_2}c_1=-\nabla _{\partial _Ac_2}c_1+\nabla _{\partial _Bc_1}c_2$,

• (M2) $[a,\partial _Ac]=\partial _A(\nabla _ac)-\nabla _{\partial _Bc}a$,

• (M3) $[b,\partial _Bc]=\partial _B(\nabla _bc)-\nabla _{\partial _Ac}b$,

• (M4) $\nabla _b\nabla _ac-\nabla _a\nabla _bc-\nabla _{\nabla _ba}c+\nabla _{\nabla _ab}c= R_{B}(b,\partial _Bc)a-R_{A}(a,\partial _Ac)b$,

• (M5) $\partial _A(R_{A}(a_1,a_2)b)=-\nabla _b[a_1,a_2]+[\nabla _ba_1,a_2]+[a_1,\nabla _ba_2]+\nabla _{\nabla _{a_2}b}a_1-\nabla _{\nabla _{a_1}b}a_2$,

• (M6) $\partial _B(R_{B}(b_1,b_2)a)=-\nabla _a[b_1,b_2]+[\nabla _ab_1,b_2]+[b_1,\nabla _ab_2]+\nabla _{\nabla _{b_2}a}b_1-\nabla _{\nabla _{b_1}a}b_2$,

for all $a,a_1,a_2\in \Gamma (A)$, $b,b_1,b_2\in \Gamma (B)$ and $c,c_1,c_2\in \Gamma (C)$, and

• (M7) $\mathbf {d}_{\nabla ^A}R_{B}=\mathbf {d}_{\nabla ^B}R_{A}\in \Omega ^2(A, \wedge ^2B^*\otimes C)=\Omega ^2(B,\wedge ^2 A^*\otimes C)$, where $R_{B}$ is seen as an element of $\Omega ^1(A, \wedge ^2B^*\otimes C)$ and $R_{A}$ as an element of $\Omega ^1(B, \wedge ^2A^*\otimes C)$.

Remark 3.2

The conditions in Definition 3.1 imply that

$$\begin{aligned} \rho _A\circ \partial _A=\rho _B\circ \partial _B. \end{aligned}$$

(12)

Specifically, if A has nonzero rank, then (12) can be obtained from (M2) by replacing a with fa for $f\in C^\infty (M)$. 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 $a \in \Gamma (A)$ and $b \in \Gamma (B)$. Specifically, if A has nonzero rank, then (M0) can be obtained from (M5) by replacing $a_2$ with $fa_2$ for $f\in C^\infty (M)$. 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.

Remark 3.4

Note that if C is trivial, then $\partial _A,\partial _B$, $R_A, R_B$ and $\nabla ^{AC}, \nabla ^{BC}$ are trivial. In that case, equations (M1)–(M4), (M7) and the left hand sides of (M5) and (M6) vanish. Bearing in mind that (M5)–(M6) imply (M0), we find the definition of a matched pair of representations of Lie algebroids [10, 18]. In particular we find that (M0) is in fact redundant in the definition of a matched pair of representations of Lie algebroids.

Remark 3.5

The vector bundle C inherits a Lie algebroid structure with anchor $\rho _C:=\rho _A\circ \partial _A=\rho _B\circ \partial _B$ and with bracket given by $[c_1, c_2]=\nabla _{\partial _Ac_1}c_2-\nabla _{\partial _Bc_2}c_1$ for all $c_1,c_2\in \Gamma (C)$. Using Remark 2.12 one can see that the Lie algebroid structure on C does not depend on the choice of splitting.

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 $\partial _A \circ \nabla ^{BC} = \nabla ^{BA} \circ \partial _A$ (Equation (ii) in Definition 2.8) shows that $\partial _A :C \rightarrow A$ is a Lie algebroid morphism. In the same manner, (M3) together with $\partial _B \circ \nabla ^{AC} = \nabla ^{AB} \circ \partial _B$ shows that $\partial _B :C \rightarrow B$ is a Lie algebroid morphism.

The Lie algebroid structure on C was defined intrinsically in [16]. Referring to Definition 2.6, the Lie bialgebroid 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 $\partial _A$ and $\partial _B$ be morphisms of Lie algebroids.

Theorem 3.6 is our main result. The proof is in Sect. 4.

Theorem 3.6

Let (D; A, B; M) be a double vector bundle with LA–vector bundle structures on both $(D\rightarrow A, B\rightarrow M)$ and $(D\rightarrow B, A\rightarrow M)$. Choose a decomposition $\Sigma$ of D and let ${\mathcal D}_A$ and ${\mathcal D}_B$ be the two 2-representations defined by the lifts $\sigma _A$ and $\sigma _B$. Then (D; A, B; M) is a double Lie algebroid if and only if the two 2-representations form a matched pair.

Remark 3.7

An immediate consequence of Theorem 3.6 is that, if ${\mathcal D}_A$ and ${\mathcal D}_B$ are 2-representations that form a matched pair, and ${\mathcal D}'_A$ and ${\mathcal D}'_B$ are the 2-representations obtained by the transformation in Remark 2.12, then ${\mathcal D}'_A$ and ${\mathcal D}'_B$ will also form a matched pair. In other words, equations (M1) to (M7) are invariant under the transformation in Remark 2.12,

Remark 3.8

Given a matched pair of representations of Lie algebroids A and B on the same base M, the direct sum vector bundle $A\oplus B$ has a Lie algebroid structure, the bicrossproduct Lie algebroid, denoted $A\bowtie B$ [10, 18]. The matched pair structure also induces on the decomposed double vector bundle $A\times _M B$ a double Lie algebroid structure and, conversely, any vacant double Lie algebroid (that is, a double Lie algebroid for which the core is zero) arises from a matched pair of Lie algebroids in this way [16, §6].

3.2 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 $\Pi A$ 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; M). Write $\Pi _AD$ and $\Pi _BD$ for the parity reversions and let $Q_A$ and $Q_B$ be homological vector fields on $\Pi _AD$ and $\Pi _BD$ which have degrees (1, 0) and (0, 1). It follows that $Q_A$ and $Q_B$ induce Lie algebroid structures on $\Pi A$ and $\Pi B$. Then [26] defines D to be a (super) double Lie algebroid if $[Q_A,Q_B] = 0$.

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 and .

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 $[Q_A,Q_B] = 0$. (In fact $Q_A$ and $Q_B$ are defined on different bundles, but admit“parity reversions” to $\Pi ^2 D = \Pi _A \Pi _B D \cong \Pi _B \Pi _A D$ and it is these parity reversions which must commute.)

This is expressed in terms of eight coordinate equations (50)–(57). We now relate these to our equations (M1) to (M7) and (M0).

A choice of coordinates as in [26] constitutes a local choice of decomposition of D, and yields local decompositions of the duals and . By Remark 3.7, it is in fact enough to check (M1) to (M7) in local decompositions. As in [26] the letters $\xi _i$ denote basis sections for A and $\xi ^i$ denote the dual basis sections for $A^*$. The letters $\eta _\alpha$ denote basis sections for B and $\eta ^\alpha$ denote the dual basis sections for $B^*$. The letters $x_a$ denote coordinates on the base manifold M. Finally the letters $z_\mu$ are basis sections for C and $z^\mu$ are the dual basis sections for $C^*$.

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

$$\begin{aligned} Q_\alpha ^a= & {} \rho _B(\eta _\alpha )(x_a),\quad Q_{i\alpha }^j=-\langle \xi ^j,\nabla _{\eta _\alpha }\xi _i\rangle , \quad Q_\mu ^j=\langle \partial _A^*\xi ^j,z_\mu \rangle ,\nonumber \\ Q_{i\beta \alpha }^\lambda= & {} \langle R(\eta _\alpha ,\eta _\beta )\xi _i,z^\lambda \rangle ,\quad Q_{\beta \alpha }^\gamma =\langle [\eta _\beta ,\eta _\alpha ]_B,\eta ^\gamma \rangle , \quad Q_{\mu \alpha }^\lambda =-\langle z^\lambda ,\nabla _{\eta _\alpha }z_\mu \rangle , \end{aligned}$$

(13)

where R, $\nabla$ and $\partial$ refer to the 2-representations induced by the local decomposition. We believe that the term $\xi _i\eta ^\alpha Q_{i\alpha }^j$ in (V 47) should have a minus sign for consistency with (V 26) and (V 28).

Similarly the coefficients in (V 48) are

$$\begin{aligned} Q_i^a&=\rho _A(\xi _i)(x_a), \quad Q_{\alpha i}^\beta =-\langle \eta ^\beta ,\nabla _{\xi _i}\eta _\alpha \rangle , \quad Q_\mu ^\beta =\langle \partial _B^*\eta ^\beta ,z_\mu \rangle ,\nonumber \\ Q_{\alpha ji}^\lambda&= \langle R(\xi _i,\xi _j)\eta _\alpha ,z^\lambda \rangle , \quad Q_{ji}^k=\langle [\xi _j,\xi _i]_A,\xi ^k\rangle , \quad Q_{\mu i}^\lambda =-\langle z^\lambda ,\nabla _{\xi _i}z_\mu \rangle . \end{aligned}$$

(14)

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.

Equation (V 50): $Q_\alpha ^aQ_\mu ^\alpha -Q_\mu ^iQ_i^a=0$ is satisfied for all $\mu$ and a if and only if

$$\begin{aligned} \rho _B(\eta _\alpha )(x_a)\langle \partial _B^*\eta ^\alpha ,z_\mu \rangle = \rho _A(\xi _i)(x_a)\langle \partial _A^*\xi ^i,z_\mu \rangle \end{aligned}$$

for all $\mu$ and a. But this is

$$\begin{aligned} \rho _B\left( \langle \partial _B^*\eta ^\alpha ,z_\mu \rangle \eta _\alpha \right) (x_a)= \rho _A\left( \langle \partial _A^*\xi ^i,z_\mu \rangle \xi _i\right) (x_a) \end{aligned}$$

and we have $\langle \partial _B^*\eta ^\alpha ,z_\mu \rangle \eta _\alpha =\langle \eta ^\alpha ,\partial _Bz_\mu \rangle \eta _\alpha =\partial _B(z_\mu )$ and $\langle \partial _A^*\xi ^i,z_\mu \rangle \xi _i=\langle \xi ^i,\partial _Az_\mu \rangle \xi _i=\partial _A(z_\mu )$ because of the choices of the dual basis $\{\xi _i\}$ and $\{\xi ^i\}$, and $\{\eta _\alpha \}$ and $\{\eta ^\alpha \}$. Hence, Equation (V 50) now reads $\rho _A(\partial _A(z_\mu ))(x_a)=\rho _B(\partial _B(z_\mu ))(x_a)$ for all $\mu$, a. Since $x_a$ are coordinates on the base manifold and $\{z_\mu \}$ is a basis for C, we get $\rho _A\circ \partial _A=\rho _B\circ \partial _B$.

Equation (V 51): $0=-Q^i_\mu Q^\lambda _{\nu i}+Q_\mu ^\alpha Q_{\nu \alpha }^\lambda -Q_\nu ^i Q_{\mu i}^\lambda +Q_\nu ^\alpha Q_{\mu \alpha }^\lambda$ for all $\mu ,\nu ,\lambda$ is

$$\begin{aligned} 0=&\langle \partial _A^*\xi ^i,z_\mu \rangle \langle z^\lambda ,\nabla _{\xi _i}z_\nu \rangle -\langle \partial _B^*\eta ^\alpha ,z_\mu \rangle \langle z^\lambda ,\nabla _{\eta _\alpha }z_\nu \rangle \\&+\langle \partial _A^*\xi ^i,z_\nu \rangle \langle z^\lambda ,\nabla _{\xi _i}z_\mu \rangle -\langle \partial _B^*\eta ^\alpha ,z_\nu \rangle \langle z^\lambda ,\nabla _{\eta _\alpha }z_\mu \rangle \end{aligned}$$

for all $\mu ,\nu ,\lambda$. As before, this is

$$\begin{aligned} 0&=\langle z^\lambda ,\nabla _{\partial _Az_\mu }z_\nu \rangle -\langle z^\lambda ,\nabla _{\partial _Bz_\mu }z_\nu \rangle +\langle z^\lambda ,\nabla _{\partial _Az_\nu }z_\mu \rangle -\langle z^\lambda ,\nabla _{\partial _Bz_\nu }z_\mu \rangle \\&=\langle z^\lambda ,\nabla _{\partial _Az_\mu }z_\nu -\nabla _{\partial _Bz_\mu }z_\nu +\nabla _{\partial _Az_\nu }z_\mu -\nabla _{\partial _Bz_\nu }z_\mu \rangle \end{aligned}$$

for all $\mu ,\nu ,\lambda$. This implies $\nabla _{\partial _Az_\mu }z_\nu -\nabla _{\partial _Bz_\nu }z_\mu =-(\nabla _{\partial _Az_\nu }z_\mu -\nabla _{\partial _Bz_\mu }z_\nu )$ for all $\mu ,\nu$, 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 52): $Q_{\alpha j}^{\beta }\,Q_{\beta }^a+Q_j^b\,\partial _bQ_{\alpha }^a-Q_{j\alpha }^i\,Q_i^a = Q_{\alpha }^b\,\partial _bQ_j^a$ is our (M0).

Equation (V 53): $Q_j^a\partial _a\,Q_{\mu }^{i}+Q_{\mu }^k\,Q_{jk}^i- Q_{\mu }^\alpha \,Q_{j\alpha }^i = -Q_{\mu j}^{\lambda }\,Q_{\lambda }^i$ is (M2).

Equation (V 54):

$$\begin{aligned}&Q_{\mu }^i\,Q_{\beta ji}^{{\lambda }}-Q_{\beta j}^{\alpha }\,Q_{\mu \alpha }^{{\lambda }} +Q_{\nu j}^{{\lambda }}\,Q_{\mu \beta }^{\nu }-Q_j^a\,\partial _aQ_{\mu \beta }^{{\lambda }} +Q_{\mu i}^{{\lambda }}\,Q_{j\beta }^i-Q_{\mu }^{\alpha }\,Q_{j\beta \alpha }^{{\lambda }}\\&\quad = -Q_{\beta }^a\,\partial _aQ_{\mu j}^{{\lambda }}+Q_{\mu j}^{\nu }\,Q_{\nu \beta }^{{\lambda }} \end{aligned}$$

is (M4).

Equation (V 55): $-Q_{\mu }^i\,Q_{\alpha i}^{\gamma } +Q_{{\lambda }}^{\gamma }\,Q_{\mu \alpha }^{{\lambda }} -Q_{\mu }^{\beta }\,Q_{\beta \alpha }^{\gamma }= -Q_{\alpha }^a\,\partial _aQ_{\mu }^{\gamma }$ is (M3).

Equation (V 56):

$$\begin{aligned}&Q_{\alpha j}^{\beta }\,Q_{i\beta }^k +Q_{j l}^k\,Q_{i\alpha }^l +Q_{j}^a\,\partial _aQ_{i\alpha }^k -Q_{\alpha i}^{\beta }\,Q_{j\beta }^k -Q_{i l}^k\,Q_{j\alpha }^l -Q_{i}^a\,\partial _aQ_{j\alpha }^k\\&\quad = Q_{\alpha }^a\,\partial _aQ_{ij}^k -Q_{ij}^l\,Q_{l\alpha }^k -Q_{\alpha ij}^{\mu }\,Q_{\mu }^k \end{aligned}$$

is (M5), providing the last term in (V 56) is changed to a positive sign.

Equation (V 57):

$$\begin{aligned}&Q_{\alpha ik}^{\lambda }Q_{j\beta }^k - Q_{\beta ik}^{\lambda }Q_{j\alpha }^k - Q_{\alpha i}^{\gamma }Q_{j\beta \gamma }^{\lambda } + Q_{\beta i}^{\gamma }Q_{j\alpha \gamma }^{\lambda } + Q_{\mu i}^{\lambda } Q_{j \beta \alpha }^{\mu } - Q_i^a\partial _aQ_{j\beta \alpha }^{\lambda }\\&\qquad - Q_{\alpha jk}^{\lambda }Q_{i\beta }^k + Q_{\beta jk}^{\lambda }Q_{i\alpha }^k + Q_{\alpha j}^{\gamma }Q_{i\beta \gamma }^{\lambda } - Q_{\beta j}^{\gamma }Q_{i\alpha \gamma }^{\lambda } - Q_{\mu j}^{\lambda } Q_{i \beta \alpha }^{\mu } + Q_j^a\partial _aQ_{i\beta \alpha }^{\lambda }\\&\quad = - Q_{ij}^kQ_{k\beta \alpha }^{\lambda } - Q_{\mu \alpha }^{\lambda }Q_{\beta ij}^{\mu } + Q_{\mu \beta }^{\lambda }Q_{\alpha ij}^{\mu } + Q_{\alpha }^a\partial _aQ_{\beta ij}^{\lambda } - Q_{\beta }^a\partial _aQ_{\alpha ij}^{\lambda } + Q_{\gamma ij}^{\lambda }Q_{\beta \alpha }^{\gamma } \end{aligned}$$

is (M7).

Equation (V 58):

$$\begin{aligned}&Q_{\beta k}^{\gamma }\,Q_{j\alpha }^k -Q_{\alpha k}^{\gamma }\,Q_{j\beta }^k +Q_{j\beta \alpha }^{{\lambda }}\,Q_{{\lambda }}^{\gamma } -Q_{\alpha j}^{\varepsilon }\,Q_{\beta \varepsilon }^{\gamma } +Q_{\beta j}^{\varepsilon }\,Q_{\alpha \varepsilon }^{\gamma } -Q_j^a\,\partial _aQ_{\beta \alpha }^{\gamma }\nonumber \\&\quad = -Q_{\beta \alpha }^{\varepsilon }\,Q_{\varepsilon j}^{\gamma } -Q_{\alpha }^a\,\partial _aQ_{\beta j}^{\gamma }+Q_{\beta }^a\,\partial _aQ_{\alpha j}^{\gamma } \end{aligned}$$

(15)

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.

3.3 The tangent double vector bundle of a Lie algebroid

Let $A\rightarrow M$ be a Lie algebroid with anchor $\rho$. We have seen in Sect. 2.3.2 that

is endowed with two LA–vector bundle structures. The vertical structure $(TA\rightarrow A,TM\rightarrow M)$ is the standard tangent bundle Lie algebroid (Example 2.5) and $(TA\rightarrow TM, A\rightarrow M)$ is the tangent prolongation of $A\rightarrow M$ as in Sect. 2.3.2. We refer to TA loosely as the tangent double Lie algebroid.

Recall from Sect. 2.1.2 that a linear connection $\nabla :\mathfrak {X}(M)\times \Gamma (A)\rightarrow \Gamma (A)$ defines a decomposition $\Sigma ^\nabla :A\times _M TM\rightarrow TA$. This decomposition induces the two following 2-representations:

• (i) The LA–vector bundle $(TA\rightarrow A,TM\rightarrow M)$ is described by the 2-representation of TM on ${{\mathrm{Id}}}_A:A\rightarrow A$ via $(\nabla ,\nabla ,R_\nabla )$ (Example 2.11). The anchor of TM is ${{\mathrm{Id}}}_{TM}$ and the bracket is the Lie bracket of vector fields.

• (ii) The LA–vector bundle $(TA\rightarrow TM, A\rightarrow TM)$ is described by the 2-representation of A on $\rho :A\rightarrow TM$ via $(\nabla ^\mathrm{bas},\nabla ^\mathrm{bas},R_\nabla ^\mathrm{bas})$ as in Sect. 2.3.2.

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]. Conditions (M1) and (M2) in Definition 3.1 are just two times the definition of $\nabla ^\mathrm{bas}:\Gamma (A)\times \Gamma (A)\rightarrow \Gamma (A)$ and (M3) is the definition of $\nabla ^\mathrm{bas}:\Gamma (A)\times \mathfrak {X}(M)\rightarrow \mathfrak {X}(M)$. Condition (M5) is the definition of $R^\mathrm{bas}_\nabla$. Hence, we only need to check (M4), (M6) and (M7).

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

$$\begin{aligned}&\nabla _{X}\nabla ^\mathrm{bas}_{a_1}a_2-\nabla ^\mathrm{bas}_{a_1}\nabla _Xa_2 -\nabla ^\mathrm{bas}_{\nabla _Xa_1}a_2+\nabla _{\nabla ^\mathrm{bas}_{a_1}X}a_2\\&\quad \quad = \nabla _{X}[a_1,a_2]+\nabla _X\nabla _{\rho (a_2)}a_1 -[a_1,\nabla _Xa_2]-\nabla _{\rho (\nabla _Xa_2)}a_1\\&\qquad \quad -[\nabla _Xa_1,a_2]-\nabla _{\rho (a_2)}\nabla _Xa_1 +\nabla _{[\rho (a_1),X]}a_2+\nabla _{\rho (\nabla _Xa_1)}a_2. \end{aligned}$$

The second and sixth terms add up to $R(X,\rho (a_2))a_1+\nabla _{[X,\rho (a_2)]}a_1$ and the first, third, and fifth terms to $-R_\nabla ^\mathrm{bas}(a_1,a_2)X+\nabla _{\nabla ^\mathrm{bas}_{a_2}X}a_1-\nabla _{\nabla ^\mathrm{bas}_{a_1}X}a_2$. The definition of $\nabla ^\mathrm{bas}:\Gamma (A)\times \mathfrak {X}(M)\rightarrow \mathfrak {X}(M)$ yields then immediately the right hand side of (M4), namely

$$\begin{aligned} R(X,\rho (a_2))a_1-R_\nabla ^\mathrm{bas}(a_1,a_2)X. \end{aligned}$$

(M6) This equation is easily verified:

$$\begin{aligned}&-\nabla ^\mathrm{bas}_a[X_1,X_2] + [\nabla ^\mathrm{bas}_aX_1,X_2] +[X_1,\nabla ^\mathrm{bas}_aX_2] +\nabla ^\mathrm{bas}_{\nabla _{X_2}a}X_1-\nabla ^\mathrm{bas}_{\nabla _{X_1}a}X_2\\&\quad =-[\rho (a),[X_1,X_2]]-\rho (\nabla _{[X_1,X_2]}a) +[[\rho (a),X_1] +\rho (\nabla _{X_1}a),X_2]\\&\qquad +[X_1,[\rho (a),X_2]+\rho (\nabla _{X_2}a)] +[\rho (\nabla _{X_2}a),X_1]\\&\qquad +\rho (\nabla _{X_1}\nabla _{X_2}a) -[\rho (\nabla _{X_1}a),X_2]-\rho (\nabla _{X_2}\nabla _{X_1}a)\\&\quad =\rho (R_\nabla (X_1,X_2)a) \end{aligned}$$

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

$$\begin{aligned} (\mathbf {d}_{\nabla }R_\nabla ^\mathrm{bas})(X_1,X_2)(a_1,a_2)&= -R_\nabla ^\mathrm{bas}(a_1,a_2)[X_1,X_2]\\&\quad +\nabla _{X_1}(R_\nabla ^\mathrm{bas}(a_1,a_2)X_2)-\nabla _{X_2}(R_\nabla ^\mathrm{bas}(a_1,a_2)X_1)\\&\quad -R_\nabla ^\mathrm{bas}(\nabla _{X_1}a_1,a_2)X_2 -R_\nabla ^\mathrm{bas}(a_1,\nabla _{X_1}a_2)X_2\\&\quad +R_\nabla ^\mathrm{bas}(\nabla _{X_2}a_1,a_2)X_1 +R_\nabla ^\mathrm{bas}(a_1,\nabla _{X_2}a_2)X_1. \end{aligned}$$

This expands out to

Twelve terms of this equation cancel pairwise as shown, and a reordering of the remaining terms yields

$$\begin{aligned}&-R_\nabla (X_1,X_2)[a_1,a_2]+[R_\nabla (X_1,X_2)a_1,a_2]+[a_1,R_\nabla (X_1,X_2)a_2]\\&\quad +R_\nabla (X_2,\nabla ^\mathrm{bas}_{a_1}{X_1})a_2+\nabla _{[X_2,\nabla ^\mathrm{bas}_{a_1}X_1]}a_2 -R_\nabla (X_2,\nabla ^\mathrm{bas}_{a_2}{X_1})a_1-\nabla _{[X_2,\nabla ^\mathrm{bas}_{a_2}X_1]}a_1\\&\quad -R_\nabla (X_1,\nabla ^\mathrm{bas}_{a_1}{X_2})a_2-\nabla _{[X_1,\nabla ^\mathrm{bas}_{a_1}X_2]}a_2 +R_\nabla (X_1,\nabla ^\mathrm{bas}_{a_2}{X_2})a_1+\nabla _{[X_1,\nabla ^\mathrm{bas}_{a_2}X_2]}a_1\\&\quad +\nabla _{\nabla ^\mathrm{bas}_{a_1}{[X_1,X_2]}}a_2-\nabla _{\nabla ^\mathrm{bas}_{a_2}{[X_1,X_2]}}a_1-\nabla _{\nabla ^\mathrm{bas}_{\nabla _{X_2}a_1}{X_1}}a_2\\&\quad +\nabla _{\nabla ^\mathrm{bas}_{\nabla _{X_2}a_2}{X_1}}a_1 +\nabla _{\nabla ^\mathrm{bas}_{\nabla _{X_1}a_1}{X_2}}a_2-\nabla _{\nabla ^\mathrm{bas}_{\nabla _{X_1}a_2}{X_2}}a_1. \end{aligned}$$

By (M6), this equals

$$\begin{aligned}&-R_\nabla (X_1,X_2)[a_1,a_2]+[R_\nabla (X_1,X_2)a_1,a_2]+[a_1,R_\nabla (X_1,X_2)a_2]\\&\quad +R_\nabla (X_2,\nabla ^\mathrm{bas}_{a_1}{X_1})a_2-R_\nabla (X_2,\nabla ^\mathrm{bas}_{a_2}{X_1})a_1 -R_\nabla (X_1,\nabla ^\mathrm{bas}_{a_1}{X_2})a_2\\&\quad +R_\nabla (X_1,\nabla ^\mathrm{bas}_{a_2}{X_2})a_1 +\nabla _{\rho (R_\nabla (X_1,X_2)a_2)}a_1-\nabla _{\rho (R_\nabla (X_1,X_2)a_1)}a_2, \end{aligned}$$

which is

$$\begin{aligned}&-R_\nabla (X_1,X_2)[a_1,a_2]-\nabla ^\mathrm{bas}_{a_2}R_\nabla (X_1,X_2)a_1+\nabla ^\mathrm{bas}_{a_1}R_\nabla (X_1,X_2)a_2\\&\quad -R_\nabla (\nabla ^\mathrm{bas}_{a_1}{X_1},X_2)a_2+R_\nabla (\nabla ^\mathrm{bas}_{a_2}{X_1},X_2)a_1 -R_\nabla (X_1,\nabla ^\mathrm{bas}_{a_1}{X_2})a_2\\&\quad +R_\nabla (X_1,\nabla ^\mathrm{bas}_{a_2}{X_2})a_1 =(\mathbf {d}_{\nabla ^\mathrm{bas}}R_\nabla )(a_1,a_2)(X_1,X_2). \end{aligned}$$

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 (TA; A, TM; M) of the form $(F;A,F_M;M)$, such that $(F\rightarrow A, F_M\rightarrow M)$ and $(F\rightarrow F_M, A\rightarrow M)$ are sub-LA–vector bundles of $(TA\rightarrow A, TM\rightarrow M)$ and $(TA\rightarrow TM, A\rightarrow M)$, respectively.

More explicitly, suppose given a decomposition of TA that is adapted to the double subbundle F in the sense that the LA–vector bundle $(F\rightarrow A, F_M\rightarrow M)$ is described by the 2-representation $({{\mathrm{Id}}}_A:A\rightarrow A, \bar{\nabla },\bar{\nabla },R_{\bar{\nabla }})$, where $\bar{\nabla }:\Gamma (F_M)\times \Gamma (A)\rightarrow \Gamma (A)$ is the restriction to $F_M$ of the connection $\nabla$ corresponding to the decomposition [2]. Then the LA–vector bundle $(F\rightarrow F_M, A\rightarrow TM)$ is described by the 2-representation of A on $\rho :J\rightarrow F_M$ via $(\overline{\nabla ^\mathrm{bas}},\overline{\nabla ^\mathrm{bas}},\overline{R_\nabla ^\mathrm{bas}})$, where the $\overline{\nabla ^\mathrm{bas}}$ are the (well-defined!) restrictions of the basic connections to $\Gamma (A)\times \Gamma (J)\rightarrow \Gamma (J)$ and $\Gamma (A)\times \Gamma (F_M)\rightarrow \Gamma (F_M)$ [2].

It is easy to see that (M1)–(M7) for the tangent double (TA; A, TM; M) restrict to (M1)–(M7) for this pair of 2-representations. This refines the result in [2] to a bijective correspondence between infinitesimal ideal systems and sub-double Lie algebroids of (TA; A, TM; M).

3.4 The cotangent double of a Lie bialgebroid

Let $(A,\rho , [\cdot \,,\cdot ])$ be a Lie algebroid over a smooth manifold M, and assume that $A^*$ also has a Lie algebroid structure, with anchor denoted by $\rho _\star$ and with bracket $[\cdot \,,\cdot ]_\star$.

Since $A^*$ is a Lie algebroid, A has a linear Poisson structure, and so $(T^*A\rightarrow A,A^*\rightarrow M)$ has a LA–vector bundle structure. The Lie algebroid bracket is given by

$$\begin{aligned}{}[\mathbf {d}\ell _{\alpha _1}, \mathbf {d}\ell _{\alpha _2}]=\mathbf {d}\ell _{[\alpha _1,\alpha _2]}, \quad [\mathbf {d}\ell _\alpha , q_A^*\theta ]=q_A^*({{\pounds }}_{\rho _\star (\alpha )}\theta ), \quad [q_A^*\theta _1, q_A^*\theta _2]=0 \end{aligned}$$

(16)

for all $\alpha ,\alpha _1,\alpha _2\in \Gamma (A^*)$ and $\theta ,\theta _1,\theta _2\in \Omega ^1(M)$. The anchor is given by

$$\begin{aligned} \Theta _{A}(\mathbf {d}\ell _\alpha )=\widehat{{{\pounds }}_{\rho _\star (\alpha )}}\in \mathfrak {X}(A), \quad \Theta _{A}(q_A^*\theta )=(\rho _\star ^*\theta )^\uparrow \in \mathfrak {X}(A) \end{aligned}$$

(17)

(see Sect. 2.1.2 for the notation). Likewise the Lie algebroid structure on A induces a LA–vector bundle structure on $(T^*(A^*)\rightarrow A^*,A\rightarrow M)$, satisfying corresponding equations. Using the natural diffeomorphism $T^*(A^*)\rightarrow T^*A$ of [17], this LA–vector bundle structure may be transferred to $(T^*A\rightarrow A^*,A\rightarrow M)$ and equips the double vector bundle

(18)

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 $\nabla :\mathfrak {X}(M)\times \Gamma (A)\rightarrow \Gamma (A)$, is equivalent to a decomposition $\Sigma _\nabla$ of TA and so to a decomposition $\Sigma ^\star _\nabla$ of (18). Using (8), one can check that $\Sigma _\nabla ^\star :A\times A^*\rightarrow T^*A$ sends $(a_m,\alpha _m)$ to $\mathbf {d}_{a_m}\ell _\alpha -(T_{a_m}q)^*\langle \nabla _\cdot \alpha ,a_m\rangle$ for any $\alpha \in \Gamma (A^*)$ such that $\alpha (m)=\alpha _m$ (see also [6]).

A computation shows that the representation up to homotopy defined by $(T^*A\rightarrow A,A^*\rightarrow M)$ and the decomposition $\Sigma ^\star$ is given by the morphism $\rho _\star ^*:T^*M\rightarrow A$, by the connections

$$\begin{aligned} {(\nabla ^*)^\mathrm{bas}}^*:\Gamma (A^*)\times \Gamma (A)\rightarrow \Gamma (A),\quad {(\nabla ^*)^\mathrm{bas}}^*:\Gamma (A^*)\times \Omega ^1(M)\rightarrow \Omega ^1(M), \end{aligned}$$

(19)

and by

$$\begin{aligned} {R_{\nabla ^*}^\mathrm{bas}}^*\in \Omega ^2(A^*, {{\mathrm{Hom}}}(A,T^*M)). \end{aligned}$$

(20)

(Recall from the previous example that since we have an ordinary connection $\nabla ^*$—the dual connection to $\nabla$—on $A^*$, we can define the basic $A^*$-connections on $A^*$ and TM. The two connections above are their duals.)

Next the LA–vector bundle $(TA\rightarrow TM, A\rightarrow M)$ dualises over A to , which is $(T^*A\rightarrow A^*, A\rightarrow M)$. Recall that the decomposition $\Sigma ^\nabla$ of TA and this LA–vector bundle structure define the 2-representation $(\rho :A\rightarrow TM, \nabla ^\mathrm{bas}, \nabla ^\mathrm{bas}, R_\nabla ^\mathrm{bas})$. The discussion in Sect. 2.3.1 yields then that the 2-representation of A describing $(T^*A\rightarrow A^*, A\rightarrow M)$ in terms of the decomposition $\Sigma _\nabla ^\star$ is on the complex $\rho ^*:T^*M\rightarrow A^*$ and is defined by the connections

$$\begin{aligned} {\nabla ^\mathrm{bas}}^*:\Gamma (A)\times \Gamma (A^*)\rightarrow \Gamma (A^*), \quad {\nabla ^\mathrm{bas}}^*:\Gamma (A)\times \Omega ^1(M)\rightarrow \Omega ^1(M), \end{aligned}$$

(21)

and the curvature term

$$\begin{aligned} -{R_{\nabla }^\mathrm{bas}}^*\in \Omega ^2(A, {{\mathrm{Hom}}}(A^*,T^*M)). \end{aligned}$$

(22)

A long, but straightforward computation shows that (M1)–(M7) for the two 2-representations in (19)–(20) and in (21)–(22) are equivalent to (B1)–(B3) below, and hence to $(A,A^*)$ being a Lie bialgebroid. Hence we recover the fact that $(T^*A;A^*,A;M)$ is a double Lie algebroid if and only if $(A,A^*)$ is a Lie bialgebroid. Conversely, (M1)–(M7) provide interesting identities relating the basic curvatures and connections defined by two dual connections on a Lie bialgebroid. As a summary, we have the following corollary of our main theorem.

Theorem 3.11

Let $(A,\rho ,[\cdot \,\cdot ])$ and $(A,\rho _\star ,[\cdot \,\cdot ]_\star )$ be two Lie algebroids in duality. Then $(A,A^*)$ is a Lie bialgebroid if and only if for any connection $\nabla :\mathfrak {X}(M)\times \Gamma (A)\rightarrow \Gamma (A)$, the 2-representation $(\rho _\star ^*:T^*M\rightarrow A,{(\nabla ^*)^\mathrm{bas}}^*,{(\nabla ^*)^\mathrm{bas}}^*, {R_{\nabla ^*}^\mathrm{bas}}^*)$ of $A^*$ and the 2-representation $(\rho ^*:T^*M\rightarrow A, {\nabla ^\mathrm{bas}}^*, {\nabla ^\mathrm{bas}}^*, -{R_\nabla ^\mathrm{bas}}^*)$ of A form a matched pair.

4 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.

4.1 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 $A^*$.

Definition 4.1

Let $q_A:A\rightarrow M$ and $q_{A^*}:A^*\rightarrow M$ be a pair of dual vector bundles, and suppose each has a Lie algebroid structure, with anchors $\rho :A\rightarrow TM$ and $\rho _*:A^*\rightarrow TM$ respectively, and brackets $[\cdot \,,\cdot ]$ and $[\cdot \,,\cdot ]_*$.

Then $(A,A^*)$ is a Lie bialgebroid if for all $a_1,a_2\in \Gamma (A)$:

$$\begin{aligned} \mathbf {d}_{A^*}[a_1,a_2]=[\mathbf {d}_{A^*}a_1,a_2]+[a_1,\mathbf {d}_{A^*}a_2]. \end{aligned}$$

(23)

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 ${\mathcal S}\subseteq \Gamma (A)$ which spans $\Gamma (A)$ as a $C^\infty (M)$-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 $a_1,a_2\in \Gamma (A)$, $\alpha _1,\alpha _2\in \Gamma (A^*)$ and $f\in C^\infty (M)$, we have

$$\begin{aligned}&(\mathbf {d}_{A^*}[a_1,fa_2]-[\mathbf {d}_{A^*}a_1,fa_2]-[a_1,\mathbf {d}_{A^*}(fa_2)])(\alpha _1,\alpha _2)\nonumber \\&\quad = f\cdot \left( \mathbf {d}_{A^*}[a_1,a_2]-[\mathbf {d}_{A^*}a_1,a_2]-[a_1,\mathbf {d}_{A^*}a_2] \right) )(\alpha _1,\alpha _2)\nonumber \\&\qquad -\langle a_2,\alpha _2\rangle \cdot \left( [\rho (a_1),\rho _*(\alpha _1)](f)-\rho _*({{\pounds }}_{a_1}\alpha _1)(f)+\rho ({{\pounds }}_{\alpha _1}a_1)(f)-\rho _*(\mathbf {d}_{A}f)\langle a_1,\alpha _1\rangle \right) \nonumber \\&\qquad +\langle a_2,\alpha _1\rangle \cdot \left( [\rho (a_1),\rho _*(\alpha _2)](f)-\rho _*({{\pounds }}_{a_1}\alpha _2)(f)+\rho ({{\pounds }}_{\alpha _2}a_1)(f)-\rho _*(\mathbf {d}_{A}f)\langle a_1,\alpha _2\rangle \right) . \end{aligned}$$

(24)

Now assume that $(A,A^*)$ is a Lie bialgebroid. Take any $a_1\in \Gamma A$ and any nonvanishing local section $\alpha _1\in \Gamma (A^*)$. Choose a (local) nonvanishing $a_2\in \Gamma (A)$ and an $\alpha _2\in \Gamma (A^*)$ such that $\langle a_2,\alpha _1\rangle =0$ and $\langle a_2,\alpha _2\rangle =1$. (If A has rank 1 then (25) below is vacuously true.) Equation (24) now reduces to

$$\begin{aligned}{}[\rho (a_1),\rho _*(\alpha _1)](f)-\rho _*({{\pounds }}_{a_1}\alpha _1)(f)+\rho ({{\pounds }}_{\alpha _1}a_1)(f)-\rho _*(\mathbf {d}_{A}f)\langle a_1,\alpha _1\rangle =0 \end{aligned}$$

(25)

for all $a_1\in \Gamma (A)$, $f\in C^\infty (M)$, and local nonvanishing $\alpha _1\in \Gamma (A^*)$. A straightworward computation shows that the left-hand side of (25) is tensorial in the term $\alpha _1$. Hence, (25) holds for all $\alpha _1\in \Gamma (A^*)$. (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 $g\in C^\infty (M)$ in this equation, expand out, and subtract

$$\begin{aligned} g\cdot \left( [\rho (a_1),\rho _*(\alpha _1)](f)-\rho _*({{\pounds }}_{a_1}\alpha _1)(f)+\rho ({{\pounds }}_{\alpha _1}a_1(f)-\rho _*(\mathbf {d}_{A}f)\langle a_1,\alpha _1\rangle \right) =0. \end{aligned}$$

We get that

$$\begin{aligned} \langle a_1, \alpha _1\rangle \cdot \left( -\rho _*(\mathbf {d}_Ag)(f)-\rho _*(\mathbf {d}_Af)(g) \right) =0. \end{aligned}$$

Again, since $a_1$ and $\alpha _1$ were arbitrary, we have found

$$\begin{aligned} -\rho _*(\mathbf {d}_Ag)(f)=\rho _*(\mathbf {d}_Af)(g)\quad \text {for all } \; f,g\in C^\infty (M), \end{aligned}$$

which is easily seen to be equivalent to

$$\begin{aligned} -\rho \circ \rho _*^*=\rho _*\circ \rho ^*, \end{aligned}$$

(26)

see also [17], [15, §12.1]. The map $\rho _*\circ \rho ^*:T^*M\rightarrow TM$ 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.

Proposition 4.3

Let $q_A:A\rightarrow M$ and $q_{A^*}:A^*\rightarrow M$ be a pair of dual vector bundles, and suppose each has a Lie algebroid structure, with anchors $\rho :A\rightarrow TM$ and $\rho _*:A^*\rightarrow TM$ respectively, and brackets $[\cdot \,,\cdot ]$ and $[\cdot \,,\cdot ]_*$. Let ${\mathcal S}$ be a subset of $\Gamma (A)$ which spans $\Gamma (A)$ as a $C^\infty (M)$-module.

Then $(A,A^*)$ is a Lie bialgebroid if and only if the following three conditions hold.

• (B1) $\mathbf {d}_{A^*}[a_1,a_2]=[\mathbf {d}_{A^*}a_1,a_2]+[a_1,\mathbf {d}_{A^*}a_2]$ for all $a_1,a_2\in {\mathcal S}$,

• (B2) $[\rho (a),\rho _*(\alpha )](f)-\rho _*({{\pounds }}_{a}\alpha )(f)+\rho ({{\pounds }}_{\alpha }a)(f)-\rho _*(\mathbf {d}_{A}f)\langle a,\alpha \rangle =0$ for all $a\in {\mathcal S}$, $\alpha \in \Gamma (A^*)$ and $f\in C^\infty (M)$, and

• (B3) $-\rho \circ \rho _*^*=\rho _*\circ \rho ^*$.

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

$$\begin{aligned}&(\mathbf {d}_{A^*}[ga_1,fa_2]-[\mathbf {d}_{A^*}(ga_1),fa_2]-[ga_1,\mathbf {d}_{A^*}(fa_2)])(\alpha _1,\alpha _2)\\&\quad = fg\cdot \left( \mathbf {d}_{A^*}[a_1,a_2]-[\mathbf {d}_{A^*}a_1,a_2]-[a_1,\mathbf {d}_{A^*}a_2] \right) )(\alpha _1,\alpha _2)\\&\qquad \, -fg\langle a_2,\alpha _2\rangle \cdot \left( [\rho (a_1),\rho _*(\alpha _1)](f)-\rho _*({{\pounds }}_{a_1}\alpha _1)(f)+\rho ({{\pounds }}_{\alpha _1}a_1)(f)-\rho _*(\mathbf {d}_{A}f)\langle a_1,\alpha _1\rangle \right) \\&\qquad \, +fg\langle a_2,\alpha _1\rangle \cdot \left( [\rho (a_1),\rho _*(\alpha _2)](f)-\rho _*({{\pounds }}_{a_1}\alpha _2)(f)+\rho ({{\pounds }}_{\alpha _2}a_1)(f)-\rho _*(\mathbf {d}_{A}f)\langle a_1,\alpha _2\rangle \right) \\&\qquad \, +fg\langle a_1,\alpha _1\rangle \cdot \left( [\rho (a_2),\rho _*(\alpha _2)](f)-\rho _*({{\pounds }}_{a_2}\alpha _2)(f)+\rho ({{\pounds }}_{\alpha _2}a_2)(f)-\rho _*(\mathbf {d}_{A}f)\langle a_2,\alpha _2\rangle \right) \\&\qquad \, -fg\langle a_1,\alpha _2\rangle \cdot \left( [\rho (a_2),\rho _*(\alpha _1)](f)-\rho _*({{\pounds }}_{a_2}\alpha _1)(f)+\rho ({{\pounds }}_{\alpha _1}a_2)(f)-\rho _*(\mathbf {d}_{A}f)\langle a_2,\alpha _1\rangle \right) . \end{aligned}$$

for all $a_1,a_2\in {\mathcal S}$, $\alpha _1,\alpha _2\in \Gamma (A^*)$ and $f,g\in C^\infty (M)$. This vanishes by (B1) and (B2). Since the Lie bialgebroid condition is additive and $\Gamma (A)$ is spanned as a $C^\infty (M)$-module by ${\mathcal S}$, we are done. $\square$

Remark 4.4

In Proposition 4.3, the first two conditions are $C^\infty (M)$-linear in the $\Gamma (A^*)$-argument, so it is sufficient to check them on a subset ${{\mathcal R}}\subseteq \Gamma (A^*)$ that spans $\Gamma (A^*)$ as a $C^\infty (M)$-module.

4.2 The Lie bialgebroid conditions on lifts and on core sections

We write here for the anchor of and for the anchor of . We set

and

Proposition 4.5

Condition (B3) on ${\mathcal S}$ and ${\mathcal R}$ is equivalent to $\rho _A\circ \partial _A=\rho _B\circ \partial _B$ and (M1).

Proof

Since $\Theta _A\circ \Theta _B^*$ and $\Theta _B\circ \Theta _A^*$ are vector bundle maps $T^*C^*\rightarrow TC^*$, it is sufficient to check (B3) on $\mathbf {d}F$ for $F\in C^\infty (C^*)$. In fact, it is even sufficient to check (B3) on $\mathbf {d}(q_{C^*}^*f)$ for $f\in C^\infty (M)$ and $\mathbf {d}\ell _c$ for $c\in \Gamma (C)$.

Choose first $f\in C^\infty (M)$ and consider $q_{C^*}^*f\in C^\infty (C^*)$. We have for any section $b\in \Gamma (B)$:

and for any $\alpha \in \Gamma (A^*)$:

This shows that

(27)

We get consequently $\Theta _A\circ \Theta _B^*(\mathbf {d}q_{C^*}^* f)=(\partial _B^*\rho _B^*\mathbf {d}f)^\uparrow \in \mathfrak {X}(C^*)$. In the same manner, we find $\Theta _A\circ \Theta _B^*(\mathbf {d}q_{C^*}^* f)=(-\partial _A^*\rho _A^*\mathbf {d}f)^\uparrow \in \mathfrak {X}(C^*)$. The equality of $\Theta _A\circ \Theta _B^*$ and $-\Theta _B\circ \Theta _A^*$ on pullbacks is hence equivalent to $\rho _A\circ \partial _A=\rho _B\circ \partial _B$.

We continue with linear functions. Choose $c\in \Gamma (C)$. Then for any section $b\in \Gamma (B)$, we get

and for any $\alpha \in \Gamma (A^*)$:

This shows

(28)

where $\langle \nabla _\cdot c,\cdot \rangle$ is seen as an element of $\Gamma ({\text {Hom}}(C^*,B^*))$. This leads to

$$\begin{aligned} \Theta _A\circ \Theta _B^*(\mathbf {d}\ell _c)(\ell _{c'})=-\ell _{\nabla _{\partial _A(c)}c'}+\ell _{\nabla _{\partial _B(c')}c} \end{aligned}$$

for all $c'\in \Gamma (C)$ and

$$\begin{aligned} \Theta _A\circ \Theta _B^*(\mathbf {d}\ell _c)(q_{C^*}^*f)=-q_{C^*}^*(\rho _A\circ \partial _A(c)f) \end{aligned}$$

for $f\in C^\infty (M)$. We find similar equations for $\Theta _B\circ \Theta _A^*(\mathbf {d}\ell _c)(\ell _{c'})$ and $\Theta _B\circ \Theta _A^*(\mathbf {d}\ell _c)(q^*_{C^*}f)$, and can conclude that $\Theta _A\circ \Theta _B^*=-\Theta _B\circ \Theta _A^*$ holds if and only if $\rho _A\circ \partial _A=\rho _B\circ \partial _B$ and (M1) are satisfied. $\square$

As a corollary of this proof, we find the following result. Recall that the map $\Theta _B \circ \Theta ^*_A :T^*C^*\rightarrow TC$ 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

$$\begin{aligned} \{\ell _{c_1}, \ell _{c_2}\}&= (\Theta _B\circ \Theta ^*_A)(q^*_{C^*}\mathbf {d}\ell _{c_1})(\ell _{c_2}) = \ell _{\nabla _{\partial _A(c_1)}(c_2) - \nabla _{\partial _B(c_2)}(c_1)} = \ell _{[c_1,c_2]},\nonumber \\ \{\ell _{c_1}, q^*_{C^*}f\}&= (\Theta _B\circ \Theta ^*_A)(q^*_{C^*}\mathbf {d}\ell _{c_1})(q^*_{C^*}f) = q^*_{C^*}(\rho _A(\partial _A(c))(f))\nonumber \\ \{q^*_{C^*}f_1, q^*_{C^*}f_2\}&= (\Theta _B\circ \Theta ^*_A)(q^*_{C^*}\mathbf {d}f_1)(q^*_{C^*}f) = 0. \end{aligned}$$

(29)

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 $\partial _A$ and $\partial _B$ 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 ${{\pounds }}_{}:\Gamma (A)\times \Gamma (A^*)\rightarrow \Gamma (A^*)$ is defined by

$$\begin{aligned} \langle {{\pounds }}_{a}\alpha , a'\rangle =\rho _A(a)\langle \alpha , a'\rangle -\langle \alpha , [a,a']\rangle \end{aligned}$$

for all $a,a'\in \Gamma (A)$ and $\alpha \in \Gamma (A^*)$.

Lemma 4.8

The Lie derivative is given by the following identities:

$$\begin{aligned} {{\pounds }}_{\beta ^\dagger }\alpha ^\dagger&=0,\quad {{\pounds }}_{\beta ^\dagger }\sigma _B^\star (b)=-\langle b, \nabla ^*_\cdot \beta \rangle ^\dagger ,\quad {{\pounds }}_{\sigma _A^\star (a)}\alpha ^\dagger ={{\pounds }}_{a}\alpha ^\dagger ,\\ {{\pounds }}_{\sigma _A^\star (a)}\sigma _B^\star (b)&=\sigma _B^\star (\nabla _ab)+\widetilde{R(a,\cdot )b} \end{aligned}$$

for all $a\in \Gamma (A)$, $b\in \Gamma (B)$, $\alpha \in \Gamma (A^*)$ and $\beta \in \Gamma (B^*)$. The Lie derivative is given by:

$$\begin{aligned} {{\pounds }}_{\alpha ^\dagger }\beta ^\dagger&=0,\quad {{\pounds }}_{\alpha ^\dagger }\sigma _A^\star (a)=-\langle a, \nabla ^*_\cdot \alpha \rangle ^\dagger ,\quad {{\pounds }}_{\sigma _A^\star (b)}\beta ^\dagger ={{\pounds }}_{b}\beta ^\dagger \\ {{\pounds }}_{\sigma _A^\star (b)}\sigma _A^\star (a)&=\sigma _B^\star (\nabla _ba)+\widetilde{R(b,\cdot )a}. \end{aligned}$$

Note that in these equations, $R(a,\cdot )b$ is seen as a section of ${\text {Hom}}(C^*,A^*)$ and $R(b,\cdot )a$ is seen as a section of ${\text {Hom}}(C^*,B^*)$.

Proof

We have

for arbitrary $\beta _1,\beta _2\in \Gamma (B^*)$, $\alpha \in \Gamma (A^*)$ and $a\in \Gamma (A)$. This proves ${{\pounds }}_{\beta _1^\dagger }\alpha ^\dagger =0$.

Then we compute

which shows that ${{\pounds }}_{\beta _1^\dagger }\sigma _B^\star (b)$ is a section with values in the core, and

This proves ${{\pounds }}_{\beta _1^\dagger }\sigma _B^\star (b)=-\langle b, \nabla _\cdot ^*\beta _1\rangle ^\dagger$, with $\langle b, \nabla _\cdot ^*\beta _1\rangle \in \Gamma (A^*)$. We also find

for arbitrary $a_1,a_2\in \Gamma (A)$ and $\alpha \in \Gamma (A^*)$. This proves the equality ${{\pounds }}_{\sigma _A^\star (a_1)}\alpha ^\dagger ={{\pounds }}_{a_1}\alpha ^\dagger$.

The identity

shows that ${{\pounds }}_{\sigma _A^\star (a_1)}\sigma _B^\star (b)$ is the sum of $\sigma _B^\star (\nabla _{a_1}b)$ with a section with values in the core. To find out this core term, we finally compute

This shows that ${{\pounds }}_{\sigma _A^\star (a_1)}\sigma _B^\star (b)=\sigma _B^\star (\nabla _{a_1}b)+\widetilde{R(a_1,\cdot )b}$.

The formulas describing the Lie derivative can be verified in the same manner. $\square$

Proposition 4.9

Condition (B2) on ${\mathcal S}$ and ${\mathcal R}$ is equivalent to (M2), (M3), (M4) and (M0).

Proof

The idea of this proof is to check (B2) on linear and core sections in ${\mathcal S}$ and ${\mathcal R}$, and on linear and $q_{C^*}$-pullback functions on $C^*$. We start with core sections. Choose $\alpha \in \Gamma (A^*)$ and $\beta \in \Gamma (B^*)$. We have $[\Theta _B(\alpha ^\dagger ), \Theta _A(\beta ^\dagger )]=[(\partial _A^*\alpha )^\uparrow ,(\partial _B^*\beta )^\uparrow ]=0$. By Lemma 4.8 and with , we find that (B2) is trivially satisfied on $\alpha ^\dagger , \beta ^\dagger$ and any element of $C^\infty (C^*)$.

Now choose $a\in \Gamma (A)$, $\alpha \in \Gamma (A^*)$. Using Lemma 4.8 we find for all $F\in C^\infty (C^*)$

In particular, for $F=q_{C^*}^*f$, $f\in C^\infty (M)$, this is $0+(\partial _B^*\rho _B^*\mathbf {d}f)^\uparrow q_{C^*}^*\langle \alpha , a\rangle =0$ by (27) and for $F=\ell _c$, $c\in \Gamma (C)$, this is

$$\begin{aligned} q_{C^*}^*(-\langle \nabla _a^*(\partial _A^*\alpha )),c\rangle +\langle a, \nabla _{\partial _Bc}^*\alpha \rangle +\langle \partial _A^*{{\pounds }}_{a}\alpha , c\rangle -(\rho _B\circ \partial _B(c))\langle \alpha ,a\rangle ) \end{aligned}$$

by (28). But this equals $q_{C^*}^*\left( \left\langle \alpha , \partial _A\left( \nabla _ac\right) -\nabla _{\partial _Bc} a - [a, \partial _Ac]\right\rangle \right)$. This shows that (B2) is in this case equivalent to (M2). In the same manner, (B2) on $\beta ^\dagger \in {\mathcal S}$ for $\beta \in \Gamma (B^*)$, $\sigma _B^\star (b)\in {\mathcal R}$ for $b\in \Gamma (B)$ and $F\in C^\infty (C^*)$ is equivalent to (M3).

Now choose $a\in \Gamma (A)$ and $b\in \Gamma (B)$. (B2) on $\sigma _A^\star (a)$, $\sigma _B^\star (b)$ and $q_{C^*}^*f$, $f\in C^\infty (M)$, is

$$\begin{aligned} q_{C^*}^*\left( \left( [\rho _B(b),\rho _A(a)]-\rho _A(\nabla _ba)+\rho _B(\nabla _ab)\right) (f) \right) =0 \end{aligned}$$

by Lemma 4.8. This is (M0). Finally we compute (B2) on $\sigma _A^\star (a)$, $\sigma _B^\star (b)$ and $\ell _c$, for $c\in \Gamma (C)$. This is

$$\begin{aligned} \ell _{\nabla _b\nabla _ac-\nabla _a\nabla _bc}-\ell _{\nabla _{\nabla _ba}c+R_{BA}(b,\partial _Bc)a} +\ell _{\nabla _{\nabla _ab}c+R_{AB}(a,\partial _Ac)b}=0 \end{aligned}$$

by Lemma 4.8. We find hence that (B2) on $\sigma _A^\star (a)$, $\sigma _B^\star (b)$ and $\ell _c$ is equivalent to (M4). $\square$

We conclude the proof of Theorem 3.6 with the study of (B1) on linear and core sections.

Proposition 4.10

Condition (B1) on elements of ${\mathcal S}$ and ${\mathcal R}$ is equivalent to (M5), (M6) and (M7).

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 $a\in \Gamma (A)$ and $\alpha _1,\alpha _2\in \Gamma (A^*)$:

$$\begin{aligned} (\mathbf {d}_{A^*}a)(\alpha _1,\alpha _2)=\rho _{A^*}(\alpha _1)\langle \alpha _2, a\rangle -\rho _{A^*}(\alpha _2)\langle \alpha _1, a\rangle -\langle [\alpha _1,\alpha _2]_{A^*}, a\rangle . \end{aligned}$$

For all $a_1,a_2\in \Gamma (A)$ and $\alpha _1,\alpha _2\in \Gamma (A^*)$, we have

$$\begin{aligned}{}[\mathbf {d}_{A^*}a_1,a_2]_A(\alpha _1,\alpha _2)&=-({{\pounds }}_{a_2}\mathbf {d}_{A^*}a_1)(\alpha _1,\alpha _2)\\&=-{{\pounds }}_{\rho _A(a_2)}(\mathbf {d}_{A^*}a_1(\alpha _1,\alpha _2))+\mathbf {d}_{A^*}a_1({{\pounds }}_{a_2}\alpha _1, \alpha _2)\\&\quad +\mathbf {d}_{A^*}a_1(\alpha _1, {{\pounds }}_{a_2}\alpha _2). \end{aligned}$$

Proof

First choose $\alpha _1,\alpha _2\in \Gamma (A^*)$. We have . For $\beta _1,\beta _2\in \Gamma (B^*)$ and $a_1,a_2\in \Gamma (A)$, we find using Lemma 4.8

and

Thus, we have . Choose now $\alpha \in \Gamma (A^*)$ and $b\in \Gamma (B)$. We have , and so in particular , and

On the other hand, we can check as above that ,

that and that

Now using (10) and (11) we finally get ,

and

We hence find that

if and only if

$$\begin{aligned}&\mathbf {d}_{A}(\nabla _b^*\alpha )(a_1,a_2) +\langle \partial _A^*\alpha , R(a_1,a_2)b\rangle +\langle a_1,{\nabla _{\nabla _{a_2}b}}^*\alpha \rangle -\langle a_2,\nabla _{\nabla _{a_1}b}^*\alpha \rangle \\&\quad -\rho _B(b)(\mathbf {d}_A\alpha (a_1,a_2)) +\rho _A(\nabla _ba_1)\langle \alpha , a_2\rangle -\rho _A(a_2)\langle \nabla _ba_1,\alpha \rangle -\langle \alpha , [\nabla _ba_1, a_2]\rangle \\&\quad +\rho _A(a_1)\langle \nabla _ba_2,\alpha \rangle -\rho _A(\nabla _ba_2)\langle a_1,\alpha \rangle -\langle \alpha , [a_1, \nabla _ba_2]\rangle =0 \end{aligned}$$

for all $a_1,a_2\in \Gamma (A^*)$. This is

$$\begin{aligned}&\left\langle \alpha , \partial _AR(a_1,a_2)b+\nabla _b[a_1,a_2]-[\nabla _ba_1,a_2]-[a_1,\nabla _ba_2]+\nabla _{\nabla _{a_1}b}a_2-\nabla _{\nabla _{a_2}b}a_1\right\rangle \\&\quad +\left( [\rho _A(a_1),\rho _B(b)]-\rho _B(\nabla _{a_1}b)+\rho _A(\nabla _ba_1)\right) \langle \alpha , a_2\rangle \\&\quad -\left( [\rho _A(a_2),\rho _B(b)]-\rho _B(\nabla _{a_2}b)+\rho _A(\nabla _ba_2)\right) \langle \alpha , a_1\rangle =0. \end{aligned}$$

By replacing in this equation $a_2$ by $fa_2$ with $f\in C^\infty (M)$, we find (M0) since $a_1,a_2,b$ and $\alpha$ were arbitrary. Then, using (M0) twice, we obtain (M5).

We conclude the proof of the theorem with the most technical formula. Choose $b_1,b_2\in \Gamma (B)$. We want to study the equation

(30)

We have and we find easily that both sides of (30) vanish on $\beta _1^\dagger ,\beta _2^\dagger$, for $\beta _1, \beta _2\in \Gamma (B^*)$. We have for $a\in \Gamma (A)$ and $\beta \in \Gamma (B^*)$:

and

Now, using (11) this becomes,

Thus, we find that the two sides of (30) are equal on $(\sigma _A^\star (a),\beta ^\dagger )$ if and only if (M6) is satisfied.

Finally we consider $a_1,a_2\in \Gamma (A)$. We have

where $c= -\nabla _{a_1}(R(b_1,b_2)a_2)+\nabla _{a_2}(R(b_1,b_2)a_1)+R(b_1,b_2)[a_1,a_2] -R(a_1,a_2)[b_1,b_2]\in \Gamma (C)$. On the other hand,

and by (10) this is

where

$$\begin{aligned} c_1&= {\nabla _{b_2}(R(a_1,a_2)b_1)-\nabla _{a_2}(R(b_2,b_1)a_1)-R(\nabla _{b_2}a_1, a_2)b_1+\langle b_1, {\nabla _{a_2}^\mathrm{Hom}}(R(b_2,\cdot )a_1)\rangle }\\&=\nabla _{b_2}(R(a_1,a_2)b_1)+R(\nabla _{a_2} b_1,b_2)a_1-R(\nabla _{b_2}a_1, a_2)b_1,\\ c_2&= {\nabla _{a_1}(R(b_2,b_1)a_2)+R(\nabla _{b_2}a_2, a_1)b_1-\left\langle b_1, {\nabla _{a_1}^\mathrm{Hom}}R(b_2,\cdot )a_2 \right\rangle }\\&=R(\nabla _{a_1} b_1, b_2)a_2-R(a_1,\nabla _{b_2}a_2)b_1. \end{aligned}$$

Hence, we find that the two sides of (30) coincide on $(\sigma _A^\star (a_1),\sigma _A^\star (a_2))$ if and only if (M7) is satisfied. $\square$

This completes the proof of Theorem 3.6.