Modules and representations up to homotopy of Lie n-algebroids

This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}$$\end{document}. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles.


Introduction
Lie n-algebroids, for n ∈ N, are graded geometric structures which generalise the notion of Lie algebroids. They have become a field of much interest in mathematical physics, since they form a nice framework for higher analogues of Poisson and symplectic structures.
Courant algebroids [29] give an important example of such higher structures. The work of Courant and Weinstein [12] and of Hitchin and Gualtieri [19,20,22] shows that Courant algebroids serve as a convenient framework for Hamiltonian systems with constraints, as well as for generalised geometry. A significant result from Roytenberg [40] and Ševera [42] shows that Courant algebroids are in one-to-one correspondence with Lie 2-algebroids equipped with a compatible symplectic structure.
The standard super-geometric description of a Lie n-algebroid generalises the differential algebraic way of defining a usual Lie algebroid as a vector bundle A over a smooth manifold M together with a degree 1 differential operator on the space • (A) := (∧ • A * ). In the language of graded geometry, this is equivalent to a graded manifold of degree 1 equipped with a homological vector field [45], i.e. a degree 1 derivation on its sheaf of functions which squares to zero and satisfies the graded Leibniz rule. A Lie n-algebroid is then defined as a graded manifold M of degree n, whose sheaf of functions C ∞ (M) is equipped with a homological vector field Q. In more "classical" geometric terms, a (split) Lie n-algebroid can also be defined as a graded vector bundle A = n i=1 A i [i] over a smooth manifold M together with some multi-brackets on its space of sections (A) which satisfy some higher Leibniz and Jacobi identities [43]. A Lie n-algebroid (M, Q) is called Poisson if its underlying graded manifold carries a degree −n Poisson structure {· , ·} on its sheaf of functions C ∞ (M), such that the homological vector field is a derivation of the Poisson bracket.
A well-behaved representation theory of Lie n-algebroids for n ≥ 2 has not been developed yet. In the case n = 1, i.e. in the case of usual Lie algebroids, Gracia-Saz and Mehta [18], and independently Abad and Crainic [2], showed that the notion of representation up to homotopy is a good notion of representation, which includes the adjoint representation. Roughly, the idea is to let the Lie algebroid act via a differential on Lie algebroid forms which take values on a cochain complex of vector bundles instead of just a single vector bundle. This notion is essentially a Z-graded analogue of Quillen's super-representations [39]. After their discovery, representations up to homotopy have been extensively studied in other works, see e.g. [3-5, 8, 10, 15, 17, 25, 35, 36, 44, 48]. In particular, the adjoint representation up to homotopy of a Lie algebroid is proving to be as fundamental in the study of Lie algebroids as the adjoint representation of a Lie algebra is in the study of Lie algebras. As is well known, the adjoint representation controls deformations and symmetries of Lie algebras (see e.g. [13] and references therein), and it is a key to the classification and the algebraic integration of Lie algebras [46,47]. Similarly, the deformations of a Lie algebroid are controlled by the cohomology with values in its adjoint representation up to homotopy [35,45], and an ideal in a Lie algebroid is a subrepresentation of the adjoint representation up to homotopy [15]. While a Lie bialgebra is a matched pair of the adjoint and coadjoint representations, a Lie bialgebroid is a matched pair of the adjoint and coadjoint representations up to homotopy [17]. From another point of view, 2-term representations up to homotopy, which are equivalent to decompositions of VB-algebroids [18], have proved to be a powerful tool in the study of multiplicative structures on Lie groupoids (se e.g. [1,9,15,27]), which, at the infinitesimal level, can be described as linear structures on algebroids.
One of the authors proved in [35] that representations up to homotopy of Lie algebroids are equivalent, up to isomorphism, to Lie algebroid modules in the sense of [45]. This paper extends this notion of modules, and consequently of representations up to homotopy, to the context of higher Lie algebroids. The definition is the natural generalisation of the case of usual Lie algebroids explained above, i.e. differential graded modules over the space of smooth functions of the underlying graded manifold. The obtained notion is analysed in detail, including the two most important examples of representations, namely, the adjoint and the coadjoint representations (up to homotopy). An equivalent geometric point of view of a special class of representations is given by split VB-Lie n-algebroids, i.e. double vector bundles with a graded side and a linear split Lie n-algebroid structure over a split Lie n-algebroid.
In addition to the impact of representations up to homotopy in the study of Lie algebroids in the last ten years, our general motivation for studying representations up to homotopy of higher Lie n-algebroids comes from the case n = 2, and in particular from Courant algebroids. In light of AKSZ theory, it seems reasonable to expect that the category of representations (up to homotopy) of Courant algebroids might have interesting connections to 3-dimensional topology. The results in this paper should be useful in the study of such representations. The first step is the search for a good notion not only of the adjoint representation of a Courant algebroid, but also of its ideals, similar to the work done in [27]. Since Courant algebroids are equivalent to Lie 2-algebroids with a compatible symplectic structure [40,42], the following question arises naturally: Question Is a compatible Poisson or symplectic structure on a Lie n-algebroid encoded in its adjoint representation?
The answer to this question is positive, since it turns out that a Poisson bracket on a Lie n-algebroid gives rise to a natural map from the coadjoint to the adjoint representation which is a morphism of right representations (see Theorem 4.13, Corollary 4.14 and Sect. 7.2), i.e. it anti-commutes with the differentials of their structure as left representations and commutes with the differentials of their structure as right representations. Further, the Poisson structure is symplectic if and only if this map is in fact a right isomorphism. This result is already known in some special cases, including Poisson Lie 0-algebroids, i.e. ordinary Poisson manifolds (M, {· , ·}), and Courant algebroids over a point, i.e. quadratic Lie algebras (g, [· , ·], · , · ). In the former case the map reduces to the natural map : T * M → T M obtained from the Poisson bracket on M, and in the latter case it is the inverse of the map defined by the nondegenerate pairing g → g * , x → x, · .
Let us conclude by explaining why the study of representations up to homotopy of split Lie n-algebroids is prominent in this paper. Our approach in this paper emphasises the similarity of the formulas in the split case with the usual formulas for the now well-known representations up to homotopy of Lie algebroids [2,18]. More precisely, we construct objects evidently generalising this "classical" theory, and we employ techniques and constructions that are similar to those that are well-known. The correspondence between decomposed split VB-Lie n-algebroids and (n + 1)representations of Lie n-algebroids is an example of this, since it is a generalisation of the correspondence of decomposed VB-algebroids with 2-representations of Lie 1-algebroids [18].
In addition, some examples naturally have the split form and are easier to work with in this setting. For instance, the symplectic Lie 2-algebroids corresponding to Courant algebroids [40,42] are often given as split Lie 2-algebroids, after a choice of metric connection on the Courant algebroid.

Outline of the paper
This paper consists of seven sections and is organised as follows. Section 2 sets the notation and conventions, and recalls the definitions and constructions of graded vector bundles and Lie algebroids. Section 3 offers a quick introduction to graded manifolds, (split) Lie n-algebroids, and Poisson and symplectic structures on Lie n-algebroids. In particular, it discusses the space of generalised functions of a Lie n-algebroid, gives the geometric description of a split Lie 2-algebroid [25] which is used in the rest of the paper, and defines the Weil algebra of a Lie n-algebroid-as it is done in [33] in the case n = 1. Sections 4 and 5 generalise the notions of Lie algebroid modules and representations up to homotopy to the setting of Lie n-algebroids. They offer a detailed explanation of the theory and give some useful examples, including the classes of the adjoint and coadjoint modules, whose properties are discussed thoroughly, especially in the case of Lie 2-algebroids. Section 4 provides the answer to the question expressed above about the connection between higher Poisson or symplectic structures and the adjoint and coadjoint modules. Section 6 recalls some basic definitions and examples from the theory of double vector bundles and defines VB-Lie n-algebroids together with the prototype example of the tangent prolongation of a Lie n-algebroid. It also shows that there is a 1-1 correspondence between split VB-Lie n-algebroids and representations up to homotopy of degree n + 1, which relates again the adjoint representation of a Lie algebroid with its tangent prolongation.
Finally, Sect. 7 discusses in the split case the results of this paper. It analyses the Weil algebra of a split Lie n-algebroid using vector bundles and connections, and it gives more details about the map between the coadjoint and adjoint representations for split Poisson Lie algebroids of degree n ≤ 2.

Relation to other work
During the preparation of this work, the authors learnt that Caseiro and Laurent-Gengoux also consider representations up to homotopy of Lie n-algebroids, in particular the adjoint representation, in their article [11], which was then also in preparation.
In [48], Vitagliano considers representations of strongly homotopy Lie-Rinehart algebras. Strongly homotopy Lie Rinehart algebras are the purely algebraic versions of graded vector bundles, over graded manifolds, equipped with a homological vector field that is tangent to the zero section. If the base manifold has grading concentrated in degree 0 and the vector bundle is negatively graded, the notion recovers the one of split Lie n-algebroids. In that case, Vitagliano's representations correspond to the representations up to homotopy considered in this paper.
In addition, since the DG M-modules considered in this paper are the sheaves of sections of Q-vector bundles, they are themselves also special cases of Vitagliano's strongly homotopy Lie-Rinehart algebras.

Preliminaries
This section recalls basic definitions and conventions that are used later on. In what follows, M is a smooth manifold and all the considered objects are supposed to be smooth even if not explicitly mentioned. Moreover, all (graded) vector bundles are assumed to have finite rank.

(Graded) vector bundles and complexes
Given two ordinary vector bundles E → M and F → N , there is a bijection between vector bundle morphisms φ : E → F covering φ 0 : M → N and morphisms of modules φ : Throughout the paper, underlined symbols denote graded objects. For instance, a graded vector bundle is a vector bundle q : E → M together with a direct sum decomposition of vector bundles E n over M. The finiteness assumption for the rank of E implies that E is both upper and lower bounded, i.e. there exists a n 0 ∈ Z such that E n = 0 for all |n| > n 0 . Here, an element e ∈ E n is (degree-)homogeneous of degree |e| = −n.
All the usual algebraic constructions from the theory of ordinary vector bundles extend to the graded setting. More precisely, for graded vector bundles E, F, the dual the direct sum E ⊕ F, the space of graded homomorphisms Hom(E, F), the tensor product E ⊗ F, and the symmetric and antisymmetric powers S(E) and A(E) are defined. A (cochain) complex of vector bundles is a graded vector bundle E over M equipped with a degree one 1 endomorphism over the identity on M which squares to zero; ∂ 2 = 0, and is called the differential.
Given two complexes (E, ∂) and (F, ∂), one may construct new complexes by considering all the constructions that were discussed before. Namely, the bundles S(E), A(E), E * , Hom(E, F) and E ⊗ F inherit a degree one operator that squares to 0. The basic principle for all the constructions is the graded derivation rule. For example, for φ ∈ Hom(E, F) and e ∈ E: This can also be expressed using the language of (graded) commutators as

Dull algebroids vs Lie algebroids
A dull algebroid [23] is a vector bundle Q → M endowed with an anchor ρ Q : Q → T M and a bracket (i.e. an R-bilinear map) [· , ·] : and the Leibniz identity is satisfied in both entries: A dull algebroid is a Lie algebroid if its bracket is also skew-symmetric and satisfies the Jacobi identity Given a skew-symmetric dull algebroid Q, there is an associated operator d Q of degree 1 on the space of Q-forms • (Q) = (∧ • Q * ), defined by the formula for τ ∈ k (Q) and q 1 , . . . , q k+1 ∈ (Q); the notationq means that q is omitted. The operator d Q satisfies as usual for τ 1 , τ 2 ∈ • (Q). In general, the operator d Q squares to zero only on 0-forms is equivalent to the compatibility of the anchor with the bracket (1). The identity d 2 Q = 0 on all forms is equivalent to (Q, ρ Q , [· , ·]) being a Lie algebroid.

Basic connections and basic curvature
Let Q → M be a skew-symmetric dull algebroid and E → M another vector bundle. A Q-connection on E is defined similarly as usual, as a map ∇ : for all q ∈ (Q), e ∈ (E) and f ∈ C ∞ (M). The dual connection ∇ * is the Q-connection on E * defined by the formula for all ε ∈ (E * ), e ∈ (E) and q ∈ (Q), where · , · is the natural pairing between E and its dual E * . A Q-connection on a graded vector bundle (E = n∈Z E n [n], ∂) is a family of Qconnections ∇ n , n ∈ N, on each of the bundles E n . If E is a complex with differential ∂, then the Q-connection is a connection on the complex (E, ∂) if it commutes with ∂, i.e. ∂(∇ n q e) = ∇ n−1 q (∂e) for q ∈ (Q) and e ∈ (E n ). The curvature R ∇ of a Q-connection on a vector bundle E is defined by for all q 1 , q 2 ∈ (Q) and e ∈ (E), and generally, it is an element of (Q * ⊗ Q * ⊗ E * ⊗ E). In this situation (where we are assuming Q is skew-symmetric), the curvature is a 2-form with values in the endomorphism bundle End(E) = E * ⊗ E, i.e. R ∇ ∈ 2 (Q, End(E)). A connection is called as usual flat if its curvature R ∇ vanishes identically.
A Q-connection ∇ on E induces an operator d ∇ on the space of E-valued Q-forms for all τ ∈ k (Q, E) and q 1 , . . . , q k+1 ∈ (Q). It satisfies for all τ 1 ∈ k (Q) and τ 2 ∈ • (Q, E), and squares to zero if and only if Q is a Lie algebroid and ∇ is flat.
Suppose that ∇ : X(M) × (Q) → (Q) is a T M-connection on the vector bundle Q. The induced basic connections on Q and T M are defined similarly as the ones associated to Lie algebroids [2,18]: The basic curvature is the form R bas ∇ ∈ 2 (Q, Hom(T M, Q)) defined by Simple computations show that the basic connections and the basic curvature satisfy

(Split) Lie n-algebroids and NQ-manifolds
This section recalls basic results about N-manifolds and Lie n-algebroids (based on [24]), and describes the Weil algebra of a Lie n-algebroid for general n (see [34] for n = 1). It focuses on the category of split N-manifolds, which is equivalent to the category of N-manifolds ( [7,40]).

(Split) N-manifolds and homological vector fields
Graded manifolds of degree n ∈ N are defined as follows, in terms of sheaves over ordinary smooth manifolds. For short, "[n]-manifold" means "N-manifold of degree n". The degree of a (degree-) homogeneous element ξ ∈ C ∞ (M) is written |ξ |. Note that the degree 0 elements of C ∞ (M) are just the smooth functions of the manifold M. By definition, a [1]-manifold M is a locally free and finitely generated sheaf C ∞ (M) of C ∞ (M)modules. That is, C ∞ (M) = (∧E * ) for a vector bundle E → M. In that case, M =: E [1]. Recall that this means that the elements of E have degree −1, and so the sections of E * have degree 1.
Consider now a (non-graded) vector bundle E of rank r over the smooth manifold M of dimension m. Similarly as before, assigning the degree n to the fibre coordinates of E defines an [n]-manifold of dimension (m; r 1 = 0, . . . , r n−1 = 0, r n = r ) denoted by E[n], with C ∞ (E[n]) n = (E * ). More generally, let E 1 , . . . , E n be vector bundles of ranks r 1 , . . . , r n , respectively, and assign the degree i to the fibre coordinates of E i , for each i = 1, . . . , n. The direct sum E = E 1 [1] ⊕ · · · ⊕ E n [n] is a graded vector bundle with grading concentrated in degrees −1, . . . , −n. When seen as an [n]-manifold, E 1 [1] ⊕ · · · ⊕ E n [n] has the local basis of sections of E * i as local generators of degree i and thus its dimension is (m; r 1 , . . . , r n ).

Definition 3.2 An
The relation between [n]-manifolds and split [n]-manifolds is explained by the following theorem, which is implicit in [40] and explicitly proved in [7].

Theorem 3.3 Any [n]-manifold is non-canonically diffeomorphic to a split [n]manifold.
Note that under the above correspondence, the structure sheaf of an [n]-manifold and a different choice of splitting leaves the bundles unchanged, up to isomorphism. In particular, for the case of a split [2] where the grading is defined such that Using the language of graded derivations, the usual notion of vector field can be generalized to a notion of vector field on an [n]-manifold M. Definition 3.4 A vector field of degree j on M is a degree j (graded) derivation of C ∞ (M), i.e. a map X : C ∞ (M) → C ∞ (M) such that |X(ξ )| = j + |ξ | and X(ξ ζ ) = X(ξ )ζ + (−1) j|ξ | ξ X(ζ ), for homogeneous elements ξ, ζ ∈ C ∞ (M).
As usual, |X| denotes the degree of a homogeneous vector field X. The Lie bracket of two vector fields X, Y on M is the graded commutator The following relations hold: Note that in the case of a split [n]-manifold E 1 [1] ⊕ · · · ⊕ E n [n], each section e ∈ (E j ) defines a derivationê of degree − j on M by the relations: The vector fields of the form ∇ 1 X ⊕· · ·⊕∇ n X are of degree 0 and are understood to send The negative degree vector fields are generated by those of the formê.

Definition 3.5 A homological vector field Q on an
is a differential d E associated to a Lie algebroid structure on the vector bundle E over M [45]. The following definition generalizes this to arbitrary degrees. Definition 3.6 A Lie n-algebroid is an [n]-manifold M endowed with a homological vector field Q-the pair (M, Q) is also called NQ-manifold of degree n. A split Lie n-algebroid is a split [n]-manifold M endowed with a homological vector field Q. A morphism of (split) Lie n-algebroids is a morphism μ of the underlying [n]-manifolds such that μ commutes with the homological vector fields.
The homological vector field associated to a split Lie n-algebroid A = A 1 [1] ⊕ · · · ⊕ A n [n] → M can be equivalently described by a family of brackets which satisfy some Leibniz and higher Jacobi identities [43]. More precisely, a homological vector field on A is equivalent to an L ∞ -algebra structure 4 on (A) that is anchored by a vector bundle morphism ρ : A 1 → T M. Such a structure is given by multibrackets ] i is graded alternating: for a permutation σ ∈ S i and for all a 1 , . . . , a i ∈ (A) degree-homogeneous sections and (4) (strong homotopy Jacobi identity) for k ∈ N and a 1 , . . . , a k ∈ (A) sections of homogeneous degree: This gives the following alternative geometric description of a split Lie 2-algebroid , Q), see [25]. For consistency with the notation in [25], set A 1 := Q and A * 2 =: B.
is given by a pair of an anchored vector bundle (Q → M, ρ Q ) and a vector bundle B → M, together with a vector bundle map : To pass from the definition above to the homological vector field Q, set On the other hand we may obtain the data of Definition 3.7 from a given homological vector field Q as follows. Define the vector bundle map to be the 1-bracket and ρ to be the anchor. The 2-bracket induces the skew-symmetric dull bracket on Q and the Q-connection on B * via the formula Finally, the 3-bracket induces the 3-form ω via the formula

Example 3.8 (Lie 2-algebras)
If we consider a Lie 2-algebroid over a point, then we recover the notion of Lie 2-algebra [6]. Specifically, a Lie 2-algebroid over a point consists of a pair of vector spaces g 0 , g 1 , a linear map : g 0 → g 1 , a skew-symmetric bilinear bracket [· , ·] : g 1 × g 1 → g 1 , a bilinear action bracket [· , ·] : g 1 × g 0 → g 0 , and an alternating trilinear bracket [· , · , ·] : ] for x, y ∈ g 1 and z ∈ g 0 , (5) and the higher Jacobi identity holds for x, y, z, w ∈ g 1 . Example 3.9 (Derivation Lie 2-algebr(oid)) For any Lie algebra (g, [· , ·] g ), the derivation Lie 2-algebra is defined as the complex , and the basic curvature is given by where P : E → E * is the isomorphism defined by the pairing, for all e, e ∈ (E) and The kind of split Lie 2-algebroids that arise in this way are the split symplectic Lie 2-algebroids [40]. They are splittings of the symplectic Lie 2-algebroid which is equivalent to the tangent prolongation of E, which is an LA-Courant algebroid [25,26].

Generalized functions of a Lie n-algebroid
In the following, (M, Q) is a Lie n-algebroid with underlying manifold M. Consider That is, these tensor products in the rest of the paper are always of

natural grading given by
It is well-known (see [2]) that any degree preserving vector bundle map h : E ⊗ F → G induces a wedge product operation for all a 1 , . . . , a p+q ∈ (A).
In particular, the above rule reads and e and f are homogeneous sections of E and F of degree i and j, respectively. Some notable cases for special choices of the map h are given by the identity, the composition of endomorphisms, the evaluation and the 'twisted' evaluation maps, the graded commutator of endomorphisms and the natural pairing of a graded vector bundle with its dual. In particular, the evaluation ( , e) → (e) and the twisted evaluation In the general case of a Lie n-algebroid has a natural grading, where the homogeneous elements of degree p are given by Similarly as in the case of a Lie algebroid, given a degree preserving map one obtains the multiplication In particular, for elements of the form where on the right hand side the multiplication ξζ is the one in C ∞ (M). The special cases above are defined similarly for the n-algebroid case. Moreover, Finally, the following fact will be useful later as it is a generalisation of [2, Lemma A.1], and gives the connection between the space C ∞ (M) ⊗ (Hom(E, F)) and the homomorphisms from There is a 1-1 correspondence between the degree n elements of C ∞ (M) ⊗ (Hom(E, F)) and the operators : ) induces the operator given by left multiplication by : Conversely, an operator of degree n must send a section e ∈ (E k ) into the sum Thus, this yields the (finite) sum Clearly, = and = .
Schematically, for a Lie n-algebroid M, the above discussion gives the following diagram: In particular, if E = F, then

The Weil algebra associated to a Lie n-algebroid
Let M be an [n]-manifold over a smooth manifold M and ξ 1 1 , . . . , ξ r 1 1 , ξ 1 2 , . . . , ξ r 2 2 , . . . , ξ 1 n , . . . , ξ r n n be its local generators over some open U ⊂ M with degrees 1, 2, . . . , n, respectively. By definition, the tangent bundle T M of M is an [n]-manifold over T M [33,34], whose local generators over T U ⊂ T M are given by The shifted tangent prolongation 6 T [1]M is an [n + 1]-manifold over M, with local generators over U given by It carries a bigrading ( p, q), where p comes from the grading of M and q is the grading of "differential forms". In other words, the structure sheaf of T [1]M assigns to every coordinate domain (U , 1) . Suppose now that (M, Q) is a Lie n-algebroid over M. Then T [1]M is an [n + 1]-manifold, which inherits the two commuting differentials £ Q and d defined as follows: • the de Rham differential d : and is extended to the whole algebra as a derivation of bidegree (0, 1).
, and is extended to the whole algebra as a derivation of bidegree (1, −1). The differential £ Q can be seen as a By checking their values on local generators, it is easy to see that £ 2 together with £ Q and d forms a double complex.

Definition 3.11
The Weil algebra of a Lie n-algebroid (M, Q) is the differential graded algebra given by the total complex of W p,q (M): In the case of a Lie 1-algebroid A → M, this is the Weil algebra from [33,34]. For the 1-algebroid case, see also [2] for an approach without the language of supergeometry.

Differential graded modules
This section defines the notion of a differential graded module over a Lie n-algebroid (M, Q) and gives the two fundamental examples of modules which come canonically with any Lie n-algebroid, namely the adjoint and the coadjoint modules. Note that the case of differential graded modules over a Lie 1-algebroid A → M is studied in detail in [35].

The category of differential graded modules
Let A → M be a Lie 1-algebroid. A Lie algebroid module [45] over A is defined as a sheaf E of locally freely and finitely generated graded (A)-modules over M together with a map D : E → E which squares to zero and satisfies the Leibniz rule for α ∈ (A) and η ∈ E . For a Lie n-algebroid (M, Q) over M, this is generalised to the following definitions.
Note that a right differential graded module of (M, Q) is a sheaf E of right graded modules as above together with a map D : E → E of degree 1, such that D 2 = 0 and for all ξ ∈ C ∞ (M) and η ∈ E (M). Any left-module can be made into a right-module (and vice versa) by setting η · ξ := (−1) |η|·|ξ | ξ · η for ξ ∈ C ∞ (M) and η ∈ E .
A differential graded bimodule of (M, Q) is then a sheaf E as above together with left and right differential graded module structures such that the gradings and the differentials coincide, and the two module structures commute: (ξ 1 η)ξ 2 = ξ 1 (ηξ 2 ) for all ξ 1 , ξ 2 ∈ C ∞ (M) and η ∈ E . Occasionally, a module structure naturally arises in a given direction and so, although left and right modules are essentially equivalent, considering them distinctly helps to minimize the signs in the formulas. If there is no danger of confusion, the prefixes "left" and "right", as well as the subscripts "L" and "R", will be omitted. D 1 ) and (E 2 , D 2 ) be two differential graded modules over the Lie n-algebroids (M, Q M ) and (N, Q N ), respectively, and let k ∈ Z. A degree 0 morphism, or simply a morphism, from E 1 to E 2 consists of a morphism of Lie nalgebroids φ : N → M and a degree preserving map μ : , and commutes with the differentials D 1 and D 2 .

Remark 4.3
The sheaves E 1 and E 2 in the definition above are equivalent to sheaves of linear functions on Q-vector bundles over M [33]. From this point of view, it is natural that the definition of a morphism of differential graded modules has a contravariant nature.
As in the case of Lie algebroids, new examples of DG M-modules of Lie nalgebroids are obtained by considering the usual algebraic constructions. In the following, we describe these constructions only for left DG modules but the case of right DG modules can be deduced from this.

Definition 4.4 (Dual module)
Given a DG M-module E with differential D E , one defines a right-DG M-module structure on the dual sheaf E * := Hom(E , C ∞ ) with differential D E * defined via the property for all ψ ∈ E * (M) and η ∈ E (M), where · , · is the pairing of E * and E [33].
for all η 1 , . . . , η k ∈ E (M). A similar formula gives also the characterisation for the operator D A(E ) of the antisymmetric powers A q (E ).  N, Q N ) be Lie n-algebroids, and suppose that E 1 and E 2 are DG-modules over M and N, respectively. A degree k-morphism, for k ∈ Z, from E 1 to E 2 is defined as a degree 0 morphism μ : E 1 → E 2 [k]; that is, a map sending elements of degree i in E 1 to elements of degree i + k in E 2 , such that it is linear over a Lie n-algebroid morphism φ : N → M and commutes with the differentials. A k-isomorphism is a k-morphism with an inverse.

Adjoint and coadjoint modules
Recall that every The corresponding right-DG module has the structure operator −£ Q .

Poisson Lie n-algebroids: coadjoint vs adjoint modules
This section shows that a compatible pair of a homological vector field and a Poisson bracket on an [n]-manifold gives rise to a degree −n map from the coadjoint to the adjoint module which is an morphism of right DG M-modules.
Let k ∈ Z.
for homogeneous elements ξ 1 , ξ 2 , ξ 3 ∈ C ∞ (M). We remark that the role of k in the above formulas can be explained by viewing the comma in the bracket as having degree k.
A morphism between two Poisson As is the case for ordinary Poisson manifolds, a degree k Poisson bracket on M induces a degree k map  A Poisson (symplectic) Lie 0-algebroid is a usual Poisson (symplectic) manifold M. A Poisson Lie 1-algebroid corresponds to a Lie bialgebroid (A, A * ) and a symplectic Lie 1-algebroid is again a usual Poisson manifold-Sect. 7 explains this in detail. A result due to Ševera [42] and Roytenberg [40] shows that symplectic Lie 2-algebroids are in one-to-one correspondence with Courant algebroids.
In [31], it was shown that a Lie algebroid A with a linear Poisson structure satisfies the Lie bialgebroid compatibility condition if and only if the map T * A → T A induced by the Poisson bivector is a Lie algebroid morphism from T * A = T * A * → A * to T A → T M. This is now generalized to give a characterisation of Poisson Lie n-algebroids.
for all ξ 1 , ξ 2 ∈ C ∞ (M), and extended odd linearly by the rules Proof From (5), In other words, the compatibility of Q with {· , ·} is equivalent to A detailed analysis of this map in the cases of Poisson Lie algebroids of degree n ≤ 2 is given in Section 7.2. The two following corollaries can be realised as obstructions for a Lie n-algebroid with a Poisson bracket to be symplectic. In particular, for n = 2 one obtains the corresponding results for Courant algebroids.

Representations up to homotopy
This section generalises the notion of representation up to homotopy of Lie algebroids from [2,18] to representations of higher Lie algebroids. Some basic examples are given, and 3-term representations of a split Lie 2-algebroid are described in detail. The adjoint and coadjoint representations of a split Lie 2-algebroid are special examples, which this section describes with explicit formulas for their structure objects and their coordinate transformation. Lastly, it shows how to define these two representations together with their objects for general Lie n-algebroids for all n.

Recall that a representation up to homotopy of a Lie algebroid A is given by an Amodule of the form (A, E) = (A) ⊗ (E) for a graded vector bundle E over M. In the same manner, a (left) representation up to homotopy of a Lie n-algebroid
is defined by the formula 9 If there is an element ξ ∈ C ∞ (M) k which is Q-cohomologous to ξ , i.e. ξ −ξ = Q(ξ ) for some ξ ∈ C ∞ (M) k−1 , then the representations E ξ and E ξ are isomorphic via the isomorphism μ : E ξ → E ξ defined in components by given by the formula Hence, one obtains a well-defined map H • (M) → Rep ∞ (M). In particular, if M is a Lie algebroid, the above construction recovers Example 3.5 in [2].

The case of (split) Lie 2-algebroids
Fix now a split Lie 2-algebroid M, and recall that from the analysis of Sect. 3.1, M is given by the sum Q[1] ⊕ B * [2] which forms the complex Unravelling the data of the definition of representations up to homotopy for the special case where E is concentrated only in degree 0 yields the following characterisation.

Proposition 5.3 A representation of the Lie 2-algebroid Q[1] ⊕ B * [2] consists of a (non-graded) vector bundle E over M, together with a Q-connection
∇ on E such that 10 :  We will be particularly interested in the case of 3-term representations of (split) Lie 2-algebroids. As we will see later, such representations correspond to VB-Lie 2-algebroids. In particular, the adjoint and coadjoint representations are 3-term representations.

Example 5.4 (Trivial line bundle) The trivial line bundle R[0] over M with Qconnection defined by
The reader should note the similarity of the following proposition with the description of 2-term representations of Lie algebroids from [2]. [2] is equivalent to the following data:

Remark 5.8 (1) If both of the bundles E 1 and E 2 are zero, the equations agree with
those of a 1-term representation. (2) The equations in the statement can be summarised as follows: and for all i: (3) Of course, there are similar descriptions of higher term representations up to homotopy of general split Lie n-algebroids. The proof below can easily be adapted to higher degrees. Since only the 3-term representations of split Lie 2-algebroids are explicitly needed later on, only this setting is worked out in detail here.

Proof It is enough to check that D acts on (E). Since D is of degree 1, it maps each
Considering the components of D, this translates to the following three equations: for e ∈ (E 0 ), for e ∈ (E 1 ), and for e ∈ (E 2 ). Due to the correspondence in (5)  A straightforward computation and a degree count in the expansion of the equation D 2 = 0 shows that (E, ∂) is a complex, ∇ commutes with ∂, and the equations in the statement hold.

Adjoint representation of a Lie 2-algebroid
This section shows that any split Lie 2-algebroid Q [1] ⊕ B * [2] admits a 3-term representation up to homotopy which is called the adjoint representation. It is a generalisation of the adjoint representation of a (split) Lie 1-algebroid studied in [2].
for q 1 , q 2 , q 3 ∈ (Q) and X ∈ X(M), (iv) the element ω 3 ∈ 3 (Q, Hom(T M, B * )) defined by for q 1 , q 2 , q 3 ∈ (Q) and X ∈ X(M), The proof can be done in two ways. First, one could check explicitly that all the conditions of a 3-representation of Q[1] ⊕ B * [2] are satisfied. This is an easy but long computation and it can be found in [37]. Instead, the following section shows that given a splitting and T M-connections on the vector bundles Q and B * , there exists an isomorphism of sheaves of C ∞ (M)-modules between the adjoint module X(M) and [2]), such that the objects defined above correspond to the differential £ Q . Another advantage of this approach is that it gives a precise recipe for the definition and the explicit formulas for the components of the adjoint representation of a Lie n-algebroid for general n. This data can be compared with the components of the representation up to homotopy of a Lie algebroid A → M, after the choice of a T M-connection on A, see [18]. The remaining terms are given by (iv), (v) and (vi) in Proposition 5.9, and as they do not seem more instructive than the general form in the proposition, they are not computed in more detail here.

Adjoint module vs adjoint representation
Recall that for a split These choices give as follows the adjoint representation ad ∇ , whose complex is given by T
A straightforward computation shows that and therefore, the objects in the statement of Proposition 5.9 define the differential , called the adjoint representation and denoted as (ad ∇ , D ad ∇ ). The adjoint representation is hence, up to isomorphism, independent of the choice of splitting and connections (see the following section for the precise transformations), and so gives a well-defined class ad ∈ Rep ∞ (M).
Due to the result above, one can also define the coadjoint representation of a Lie 2-algebroid (M, Q) as the isomorphism class ad * ∈ Rep ∞ (M). To find an explicit representative of ad * , suppose that Q[1] ⊕ B * [2] is a splitting of M, and consider its adjoint representation ad ∇ as above for some choice of T M-connections ∇ on B * and Q. Recall that given a representation up to homotopy (E, D) of (M, Q), its dual E * becomes a representation up to homotopy with operator D * characterised by the formula Here, ∧ = ∧ · ,· , with · , · the pairing of E with E * . Unravelling the definition of the dual for the representation ad ∇ , one finds that the structure differential of ad * ) is given by the following objects: (1) the coadjoint complex T * M → Q * → B obtained by −ρ * Q and − * , (2) the Q-connections ∇ on B and ∇ bas, * on Q * and T * M, for all q, q 1 , q 2 , q 3 ∈ (Q), τ ∈ (Q * ), b ∈ (B) and β ∈ (B * ).

Coordinate transformation of the adjoint representation
The adjoint representation up to homotopy of a Lie 2-algebroid was constructed after a choice of splitting and T M-connections. This section explains how the adjoint representation transforms under different choices. First, a morphism of 3-representations of a split Lie 2-algebroid can be described as follows.

Proposition 5.12 Let (E, D E ) and (F, D F ) be 3-term representations up to homotopy of the split Lie 2-algebroid Q[1]
⊕ B * [2]. A morphism μ : E → F is equivalent to the following data: The above objects are subject to the relations Proof As before it suffices to check how μ acts on (E), by the same arguments. Then it must be of the type F)). It is easy to see that the three equations in the statement come from the expansion of μ • D E = D F • μ when μ is written in terms of the components defined before.
for X ∈ X(M), q ∈ (Q) and β ∈ (B * ). The equations in Proposition 5.12 are automatically satisfied since by construction This yields the following result.

Proposition 5.13
Given two pairs of T M-connections on the bundles B * and Q, the isomorphism μ : ad ∇ → ad ∇ between the corresponding adjoint representations is The next step is to show how the adjoint representation transforms after a change of splitting of the Lie 2-algebroid. Fix a Lie 2-algebroid (M, Q) over the smooth manifold M and choose a splitting Q[1] ⊕ B * [2], with structure objects ( , ρ, [· , ·] 1 , ∇ 1 , ω 1 ) as before. Recall that a change of splitting does not change the vector bundles B * and Q, and it is equivalent to a section σ ∈ 2 (Q, B * ). The induced isomorphism of [2]-manifolds over the identity on M is given by: is the structure objects of the second splitting, then the compatibility of σ with the homological vector fields reads the following: • The skew-symmetric dull brackets are related by [q 1 , 1 , q 2 )). • The connections are related by ∇ 2 ·), b , or equivalently on the dual by ∇ 2 * q β = ∇ 1 * q β − σ (q, (β)). • The curvature terms are related by ω 2 = ω 1 + d 2,∇ 1 σ , where the operator is defined by the usual Koszul formula using the dull bracket [· , ·] 2 and the connection ∇ 1 * .
The above equations give the following identities between the structure data for the adjoint representations 13 ad 1 ∇ and ad 2 ∇ .
Consider now two Lie 2-algebroids M 1 and M 2 over M, and an isomorphism given by the maps F Q : Recall that a 0-morphism between two representations up to homotopy (E 1 , D 1 ) and (E 2 , D 2 ) of M 1 and M 2 , respectively, is given by a degree 0 map which is C ∞ (M 2 )-linear: μ(ξ ⊗e) = F ξ ⊗μ(e) for all ξ ∈ C ∞ (M 2 ) and e ∈ (E 2 ), and makes the following diagram commute The usual analysis as before implies that μ must be given by a morphism of complexes μ 0 : (E 2 , ∂ 2 ) → (E 1 , ∂ 1 ) and elements which satisfy equations similar to the set of equations in Proposition 5.12.
A change of splitting of the Lie 2-algebroid transforms as follows the adjoint representation. Since changes of choices of connections are now fully understood, choose the same connection for both splittings Suppose that σ ∈ 2 (Q, B * ) is the change of splitting and denote by F σ the induced isomorphism of the split Lie 2-algebroids whose components are given by F σ,Q = id Q * , F σ,B = id B , F σ,0 = σ . The composition map μ σ : ad 1 ∇ → X(M) → ad 2 ∇ is given in components by A similar argument as before implies that μ σ is a morphism between the two adjoint representations and therefore the following result follows.

Proposition 5.15
Given two splittings of a Lie 2-algebroid with induced change of splitting σ ∈ 2 (Q, B * ) and a pair of T M-connections on the vector bundles B * and Q, the isomorphism between the corresponding adjoint representations is given by μ = id ⊕ σ ⊕ ∇ · σ .

Adjoint representation of a Lie n-algebroid
The construction of the adjoint representation up to homotopy of a Lie n-algebroid (M, Q) for general n is similar to the n = 2 case. Specifically, choose a splitting M which at the level of generators is given by Then μ is used to transfer £ Q from X(M) to obtain the differential D ad ∇ :=

Split VB-Lie n-algebroids
This section gives a picture of representations up to homotopy in more "classical" geometric terms. That is, in terms of linear Lie n-algebroid structures on double vector bundles. It introduces the notion of split VB-Lie n-algebroids and explains how they correspond to (n+1)-representations of Lie n-algebroids. In particular, the tangent of a Lie n-algebroid is a VB-Lie n-algebroid which is linked to the adjoint representation. The main result in this section is a generalisation of the correspondence between decomposed VB-algebroids and 2-representations in [18].

Double vector bundles
Recall that a double vector bundle (D, V , F, M) is a commutative diagram such that all the arrows are vector bundle projections and the structure maps of the bundle D → V are vector bundle morphisms over the corresponding structure maps of F → M (see [30]). This is equivalent to the same condition holding for the structure maps of D → F over V → M. The bundles V and F are called the side bundles of D. The intersection of the kernels C := π −1 V (0 V ) ∩ π −1 F (0 F ) is the core of D and is naturally a vector bundle over M, with projection denoted by q C : , respectively (see [30]). The core section c † ∈ c V (D) corresponding to c ∈ (C) is defined as Finally, a section ψ ∈ (V * ⊗ C) defines a linear section ψ ∧ : V → D over the zero section 0 F : M → F by for all m ∈ M and v m ∈ V m . This type of linear section is called a core-linear section.
In terms of the generators θ ⊗ c ∈ (V * ⊗ C), the correspondence above reads (θ ⊗ c) ∧ = θ · c † , where θ is the linear function on V associated to θ ∈ (V * ).  [30]. The linear vector field which corresponds to the derivation δ is written X δ .

Linear splittings, horizontal lifts and duals
A linear splitting of a double vector bundle (D, V , F, M) with core C is a double vector bundle embedding of the decomposed double vector bundle V × M F into D over the identities on V and F. It is well-known that every double vector bundle admits a linear splitting, see [14,16,38] or [21] for the general case. Moreover  Double vector bundles can be dualized in two ways, namely, as the dual of D either over V or over F [30]. Precisely, from a double vector bundle (D, V , F, M) with core C, one obtains the double vector bundles with cores F * and V * , respectively.
Given a linear splitting : V × M F → D, the dual splitting * :
As it is shown in [18], an interesting fact about the tangent prolongation of a Lie algebroid is that it encodes its adjoint representation. The same holds for a split Lie n-algebroid A 1 [1] ⊕ · · · ⊕ A n [n], since by definition the adjoint module is exactly the space of sections of the Q-vector bundle T ( Suppose that (D, V , A, M) is a VB-Lie n-algebroid with homological vector fields Q D and Q A , and choose a decomposition for each double vector bundle 14 , and consequently for (D, V , A, M). Consider the dual D * V and recall that the spaces V (D * i ) are generated as C ∞ (V )-modules by core and linear sections. For the latter, use the identification l . For all ψ ∈ (V * ), the 1-form d ψ is a linear section of T * V → V over ψ and the anchor ρ D 1 : D 1 → T V is a morphism of double vector bundles. This implies that the degree 1 function Q D ( ψ ) = ρ * D 1 d ψ is a linear section of V (D * ) and thus Moreover, due to the decomposition, A direct computation shows that the components of Q D (γ ) which lie in spaces with two or more sections of the form (q ! V C * i ) and (q ! V C * j ) vanish due to the bracket conditions of a VB-Lie n-algebroid. Therefore, define the representation D * of A on the dual complex E * by the equations for all ψ ∈ (V * ) and all γ ∈ (C * i ). Conversely, given a representation D * of A on E * , the above equations together with for all f ∈ C ∞ (M) and α ∈ (A * ), define a VB-Lie n-algebroid structure on the double vector bundle (D, V , A, M). As discussed in Remark 6.5, this yields the following theorem.

Constructions in terms of splittings
This section presents in terms of splittings two of the applications of the adjoint and coadjoint representations that were defined before. First, there is an explicit description of the Weil algebra of a split Lie n-algebroid together with its structure differentials, in terms of vector bundles and connections, similarly to [2]. Second, the map between the coadjoint and the adjoint representations in the case of a Poisson Lie n-algebroid for degrees n ≤ 2 is examined in detail.

The Weil algebra of a split Lie n-algebroid
is a split Lie 2-algebroid and consider two T M-connections on the vector bundles Q and B * , both denoted by ∇. Recall from Sect. 5.4 the (non-canonical) isomorphism of DG M-modules This implies that and thus the generators of the Weil algebra can be identified with , (S υ Q * ) (υ,υ) , (∧ w B) (2w,w) .
Using also that C ∞ (M) t = t=r +2s (∧ r Q * ) ⊗ (S s B), the space of ( p, q)-forms is decomposed as Therefore, after a choice of splitting and T M-connections ∇ on Q and B * , the total space of the Weil algebra of M can be written as The next step is to express the differentials £ Q and d on W (M, ∇) in terms of the two T M-connections ∇. For the horizontal differential, recall that by definition the q-th row of the double complex W (M, ∇) equals the space of q-forms q (M) on  . This then yields W p,q (M, ∇) = p=t+v 1 +2v 2 +... q=u+v 1 +v 2 +...
Similar considerations as before imply that the q-th row of W (M, ∇) is given by S q (ad * ∇ ) with £ Q = D S q (ad * ∇ ) , and that d is built again by the dualization of the 2representations of T M on the graded vector bundles E A i = A i [0] ⊕ A i [−1], for i = 1, . . . , n, whose differentials are given by (id A i , ∇, R ∇ ).
From this, it follows that 0 (β) = ρ (β). Using now this, the right-hand-side gives  [32]. Then the fact that : ad * ∇ → ad ∇ is an anti-morphism of 2-representations follows immediately [15], since ad * ∇ and ad ∇ are equivalent to decompositions of the VB-algebroids (T * A → A * , A → M) and (T A → T M, A → M), respectively. Now consider the case of 2-algebroids. First recall that a symplectic Lie 2-algebroid over a point, that is, a Courant algebroid over a point, is a usual Lie algebra (g, [· , ·]) together with a non-degenerate pairing · , · : g × g → g, such that [x, y], z + y, [x, z] = 0 for all x, y, z ∈ g.
Using the adjoint and coadjoint representations ad : g → End(g), x → [x, ·], and ad * : g → End(g * ), x → −ad(x) * , and denoting the canonical linear isomorphism induced by the pairing by P : g → g * , the equation above reads P(ad(x)y) = ad * (x)P(y) for all x, y ∈ g.
In other words, this condition is precisely what is needed to turn the vector space isomorphism P into an isomorphism of Lie algebra representations between ad and ad * . In fact, the map : ad * → ad for Poisson Lie 2-algebroids is a direct generalisation of this construction.
Let B → M be a usual Lie algebroid with a 2-term representation (∇ Q , ∇ Q * , R) on a complex ∂ Q : Q * → Q. The representation is called self dual [24] if it equals its dual, i.e. ∂ Q = ∂ * Q , the connections ∇ Q and ∇ Q * are dual to each other, and R * = −R ∈ 2 (B, Hom(Q, Q * )), i.e. R ∈ 2 (B, ∧ 2 Q * ). Ref. [24] Fix now a Poisson Lie 2-algebroid (M, Q, {· , ·}) together with a choice of a splitting Q [1] ⊕ B * [2] for M, a pair of T M-connections on B * and Q, and consider the representations ad ∇ and ad * ∇ . Similarly as before, we have that for f ∈ C ∞ (M), τ ∈ (Q * ), b ∈ (B), where we identify τ with dτ − d ∇ * τ and b with db − d ∇ * b. Then the map : ad * ∇ → ad ∇ consists of the (−2)-chain map given by the anti-commutative diagram and the elements 1 (q)τ = τ, ∇ Q · q − ∇ ρ B (·) q ∈ (B * ) for q ∈ (Q), τ ∈ (Q * ), b ∈ (B), for q 1 , q 1 ∈ (Q), b ∈ (B), where R is the component that comes from the self-dual 2-representation of B from the Poisson structure, for β ∈ (B * ), b ∈ (B). Suppose now that the split Lie 2-algebroid is symplectic, i.e. that it is of the form E [1] ⊕ T * M [2] for a Courant algebroid E → M. The only thing that is left from the construction in the Example 3.10 is a choice of a T M-connection on T M, and hence on the dual T * M. The (anti-)isomorphism : ad * ∇ → ad ∇ consists of the (-2)-chain map of the anti-commutative diagram