The Weil algebra of a double Lie algebroid
aa r X i v : . [ m a t h . DG ] D ec THE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID
ECKHARD MEINRENKEN AND JEFFREY PIKE
Abstract.
Given a double vector bundle D → M , we define a bigraded bundle of algebras W ( D ) → M called the ‘Weil algebra bundle’. The space W ( D ) of sections of this algebrabundle ‘realizes’ the algebra of functions on the supermanifold D [1 , D and those of the double vector bundles D ′ , D ′′ obtained from D by duality operations. We show that VB -algebroid structures on D are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gersten-haber bracket on the third. Furthermore, Mackenzie’s definition of a double Lie algebroid isequivalent to compatibilities between two such structures on any one of the three Weil alge-bras. In particular, we obtain a ‘classical’ version of Voronov’s result characterizing doubleLie algebroid structures. In the case that D = T A is the tangent prolongation of a Lie alge-broid, we find that W ( D ) is the Weil algebra of the Lie algebroid, as defined by Mehta andAbad-Crainic. We show that the deformation complex of Lie algebroids, the theory of IMforms and IM multivector fields, and 2-term representations up to homotopy, all have naturalinterpretations in terms of our Weil algebras. Contents
1. Introduction 12. Double vector bundles 53. Double-linear functions 104. The Weil algebra W ( D ) 165. Linear and core sections of ∧ A D Introduction
A well-known result of Vaintrob [62] states that the Lie algebroid structures on a given vectorbundle A → M are equivalent to homological vector fields Q on the graded supermanifold A [1]. In classical language, the smooth functions on A [1] are the sections of the exterior bundle ∧ A ∗ , and Vaintrob’s observation says that Lie algebroids A are completely determined by theassociated Chevalley-Eilenberg complex (Γ( ∧ A ∗ ) , d CE ) . If A is obtained by applying the Lie functor to a Lie groupoid G ⇒ M , then the Chevalley-Eilenberg complex is the infinitesimal counterpart to the complex of groupoid cochains on G ;the van Est map of Weinstein-Xu [66] gives a cochain map from the groupoid complex to theLie algebroid complex.We are interested in generalizations of this theory to double Lie algebroids , as introduced byMackenzie [43, 44, 49]. A double Lie algebroid is a double vector bundle D / / (cid:15) (cid:15) B (cid:15) (cid:15) A / / M for which all the sides are equipped with Lie algebroid structures, and with a certain com-patibility condition between the horizontal and vertical Lie algebroid structures. As a firstexample, the tangent bundle of any Lie algebroid is a double Lie algebroid [44, Example 4.6].Double Lie algebroids are of importance in second order differential geometry [45, 47]. Theyarise as the infinitesimal counterparts of double Lie groupoids, and as such appear in Poissongeometry and related areas of mathematics. Voronov [64] proved that a double Lie algebroidstructure on D is equivalent to two commuting homological vector fields Q h , Q v , of bidegrees(1 ,
0) and (0 , D [1 ,
1] obtained from D by a parity shift inboth vector bundle directions. Put differently, the algebra of functions on D [1 ,
1] is a doublecomplex, generalizing the Chevalley-Eilenberg complex of a Lie algebroid. One of the aims ofthis paper to give a coordinate-free, ‘classical’ description of this double complex, avoiding theuse of super-geometry. If the double Lie algebroid is obtained by applying the Lie functor toa double Lie groupoid or LA -groupoid, then this double complex will serve as the codomainof the corresponding van Est map. This application to van Est theory will be developed in aforthcoming paper; for the tangent prolongation of a Lie groupoid, one recovers the van Estmap of Abad-Crainic [1].To explain our construction, let D be any double vector bundle with side bundles A, B . Byresults of Grabowski-Rotkiewicz [21], the compatibility of the two vector bundle structuresreduces to the condition that the horizontal and vertical scalar multiplications commute. Thesubmanifold on which the two scalar multiplications coincide is itself a vector bundle over M ,called the core of D . We denote by E = core( D ) ∗ its dual bundle. There is a vector bundle b E → M whose space of sections consists of thesmooth functions on D that are double-linear , i.e., linear both horizontally and vertically. Itfits into an exact sequence 0 → A ∗ ⊗ B ∗ i b E −→ b E → E → , where the map b E → E is given by the restriction of double-linear functions to the core, whilethe map i b E is given by the multiplication of linear functions on A, B . Our definition of theWeil algebra bundle is as follows:
Definition.
The
Weil algebra bundle of the double vector bundle D is the bundle (over M )of bigraded super-commutative algebras W ( D ) = ( ∧ A ∗ ⊗ ∧ B ∗ ⊗ ∨ b E ) / ∼ , HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 3 taking the quotient by the (fiberwise) ideal generated by elements of the form αβ − i b E ( α ⊗ β )for ( α, β ) ∈ A ∗ × M B ∗ . Here, generators α ∈ A ∗ have bidegree (1 , β ∈ B ∗ havebidegree (0 , b e ∈ b E have bidegree (1 , W ( D ) = Γ( W ( D )) will be called the Weil algebra of D .Double vector bundles come in triples, with cyclic permutation of the roles of the vectorbundles A, B, E over M : D / / (cid:15) (cid:15) B (cid:15) (cid:15) A / / M D ′ / / (cid:15) (cid:15) E (cid:15) (cid:15) B / / M D ′′ / / (cid:15) (cid:15) A (cid:15) (cid:15) E / / M The double vector bundle D ′ is (essentially) obtained by taking the dual of D as a vector bundleover B and interchanging the roles of horizontal and vertical structures; similarly D ′′ ∼ = ( D ′ ) ′ .(One has ( D ′′ ) ′ = D .) See Section 2 for a more precise description. Accordingly, we have threeWeil algebras W ( D ) , W ( D ′ ) , W ( D ′′ ) . A linear Lie algebroid structure of D as a vector bundle over A , also known as a VB -algebroid structure of D over A , is equivalent to a double-linear Poisson structure on D ′′ , and also toa VB -algebroid structure of D ′ over E . (See [45].) Section 6 explains in detail how thesestructures are expressed in terms of the Weil algebras. In particular, one finds: Theorem I.
Let D be a double vector bundle. Then the following are equivalent:(a) a VB -algebroid structure of D over A ,(b) a vertical differential d v on W ( D ),(c) a horizontal differential d ′ h on W ( D ′ ),(d) a Gerstenhaber bracket (of bidegree ( − , − W ( D ′′ ).There is a wealth of identities relating the differentials and brackets on the Weil algebras,through duality pairings and contraction operators. Aspects of this ‘Cartan calculus’ are ex-plored in Section 7.Using cyclic permutations of D, D ′ , D ′′ , one has similar results when starting out with a VB -algebroid structure of D over B or with a double-linear Poisson structure on D . Section 8deals with the situation that D has any two of these structures; in particular we prove: Theorem II.
Let D be a double vector bundle, with VB -algebroid structures over B as wellas over A . Then the following are equivalent:(a) D is a double Lie algebroid,(b) the horizontal and vertical differentials d h , d v on W ( D ) commute,(c) the horizontal differential d ′ h on W ( D ′ ) is a derivation of the Gerstenhaber bracket,(d) the vertical differential d ′′ v on W ( D ′′ ) is a derivation of the Gerstenhaber bracket. ECKHARD MEINRENKEN AND JEFFREY PIKE
If one uses the identification of W ( D ) with functions on the supermanifold D [1 , ⇔ (b) translates into Voronov’s result [64] mentioned above; however, we willgive a direct proof of this result, not using any super-geometry. More precisely, given verticaland horizontal VB -algebroid structures, the proof will give an explicit relationship betweentheir compatibility (or lack thereof) and the super-commutator of the two differentials.If D = T g is the tangent bundle of a Lie algebra, viewed as a double Lie algebroid with A = g , B = 0 , E = g ∗ , then the three Weil algebras are W ( T g ) = ∧ g ∗ ⊗ ∨ g ∗ , W (( T g ) ′ ) = ∧ g ⊗ ∨ g , W (( T g ) ′′ ) = ∧ g ∗ ⊗ ∧ g . Here W ( T g ) is the standard Weil algebra [65], with d h the Chevalley-Eilenberg differentialfor the g -module ∨ g ∗ and d v the Koszul differential. The differential d ′ h on W (( T g ) ′ ) is theKoszul differential, and the Gerstenhaber bracket is a natural extension of the Lie bracket ofthe semi-direct product g ⋉ g for the adjoint action. The differential d ′′ v on W (( T g ) ′′ ) is theChevalley-Eilenberg differential for the g -module ∧ g , and the Gerstenhaber bracket extendsthe pairing between g ∗ and g (this is a special case of Kosmann-Schwarzbach’s big bracket [32, 34]). More generally, if D = T A is the tangent bundle of a Lie algebroid A , the doublecomplex W ( T A ) coincides with the
Weil algebra of the Lie algebroid A , as defined by Mehta[55] using super-geometry, and by Abad-Crainic [1] in a classical framework.Returning to a general double vector bundle, let ∧ A D and ∧ B D be the exterior bundles of D viewed as vector bundles over A and B , respectively. By considering the homogeneity ofsections in the A -direction, one can define distinguished subspaces of of linear sections of thesebundles. In Section 5, we show that these have descriptions in terms of the Weil algebras:Γ lin ( ∧ • A D, A ) = W • , ( D ′′ ) , Γ lin ( ∧ • B D, B ) = W , • ( D ′ ) . As a consequence of Theorem I above, a double-linear Poisson structure on D determines adegree 1 differential on these spaces, while a VB -algebroid structure on D over A (respectivelyover B ) determines a Gerstenhaber bracket. For the cotangent and tangent bundles of a vectorbundle V → M (with the DVB structures as in Section 2.2), some of these spaces have well-known interpretations: W , • ( T ∗ V ) = X • lin ( V ) , W , • ( T V ) = Ω • lin ( V ) . Here X lin ( V ) are linear multi-vector fields with the Schouten bracket, while Ω lin ( V ) are lineardifferential forms with the de Rham differential. If V is a Lie algebroid over M , one also hashorizontal differentials on W ( T ∗ V ) and on W ( T V ), coming from the VB -algebroid structuresof T ∗ V over V ∗ and T V over
T M , respectively. In Section 9, we will see that W , • ( T ∗ V ) ∩ ker(d h ) = X • IM ( V ) , W , • ( T V ) ∩ ker(d h ) = Ω • IM ( V ) , the space of infinitesimally multiplicative multi-vector fields [29] and infinitesimally multiplica-tive differential forms [5], respectively. On the other hand, the Lie algebroid structure of V over M also induces a VB -algebroid structure on T ∗ V ∗ over V ∗ , and the corresponding dif-ferential d v on W , • ( T ∗ V ∗ ) = X lin ( V ∗ ) is the Poisson differential for the resulting Poissonstructure on V ∗ , identifying this space with the deformation complex of Crainic-Moerdijk [15].(This may also be seen as a consequence of a result of Cabrera-Drummond [9] for the VB -algebroid T ∗ V ∗ .) Further applications relate the Weil algebra to the Fr¨olicher-Nijenhuis [19] HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 5 and Nijenhuis-Richardson [59] brackets, to matched pairs of Lie algebroids [57], and to thenotion of representations up to homotopy [2, 23, 25].Let us finally remark that there is no principal difficulty in generalizing the constructions ofthis paper to n -fold vector bundles, however, the theory requires careful bookkeeping due tothe presence of multiple cores and a more complicated structure group (see e.g. [24]).After completion of this work, Madeleine Jotz Lean informed us that she had independentlyobtained a geometric construction of the Weil algebra for ‘split’ double Lie algebroids, usingthe ideas from [23, 38]. Acknowledgments.
It is a pleasure to thank Francis Bischoff, Henrique Bursztyn, MadeleineJotz Lean, Yvette Kosmann-Schwarzbach, Kirill Mackenzie, and Luca Vitagliano for fruitfuldiscussions and comments related to this paper. We also thank the referees for numeroushelpful suggestions. 2.
Double vector bundles
Definitions.
The concept of a double vector bundle (
DVB ) was introduced by Pradines[60, 61] in terms of local charts, and later reformulated as manifolds with ‘commuting’ vectorbundle structures [45]. We shall work with an elegant approach due to Grabowski-Rotkiewicz[21], who observed that vector bundle structures on manifolds V are completely determinedby their scalar multiplications κ t : V → V, t ∈ R , and vector bundle morphisms V → V ′ areexactly the smooth maps intertwining scalar multiplications. Similarly, consider a manifold D with two vector bundle structures, referred to as vertical and horizontal, respectively. Thecorresponding scalar multiplications are denoted κ vt , κ hs . Then D is called a double vectorbundle if(1) κ vt κ hs = κ hs κ vt for all t, s ∈ R . A morphism of double vector bundles ( DVB morphism) ϕ : D → D ′ is a smoothmap intertwining both the horizontal and the vertical scalar multiplications. The condition (1)implies a list of compatibilities with the horizontal and vertical addition operations, such asthe interchange property [45], however, we will not need these in this paper.Let A = κ v ( D ) and B = κ h ( D ) be the base submanifolds for the two vector bundle structures;each of these is a vector bundle over the submanifold M = κ h κ v ( D ). The situation is depictedby a diagram(2) D / / (cid:15) (cid:15) B (cid:15) (cid:15) A / / M One calls
A, B the side bundles , and M the base of the double vector bundle. The projectionsto the side bundles combine into a DVB morphism(3) ϕ : D → A × M B, where A × M B has the double vector structure given by the two obvious scalar multiplications.(See also Section 2.2 below.) As stated in [21, Section 4], this map is a surjective submersion.(Proof: At points m ∈ M ⊆ D , the tangent map T m ϕ : T m D → T m ( A × M B ) = T m A ⊕ T m B ECKHARD MEINRENKEN AND JEFFREY PIKE is surjective, with right inverse given by the tangent maps to the inclusions
A ֒ → D, B ֒ → D .Hence ϕ is a submersion on a neighborhood of M in D , and by equivariance with respect to κ ht κ vs it is a submersion everywhere.) It follows that the preimage of the base under this mapis a submanifold core( D ) = ϕ − ( M ) ⊆ D called the core of the double vector bundle. It admits an alternative characterization as thesubmanifold on which the two scalar multiplications coincide [21]:(4) core( D ) = { d ∈ D | ∀ t ∈ R : κ ht ( d ) = κ vt ( d ) } . The restriction of κ ht (or, equivalently, of κ vt ) is the scalar multiplication for a vector bundlestructure on core( D ) → M . The core may be regarded as a subbundle of D , for each of thetwo vector bundle structures. From now on, we will reserve the notation E = core( D ) ∗ for the dual bundle of the core. Remark . We would prefer the letter C , since we will make extensive use of a cyclic symmetryinterchanging the bundles A, B , and core( D ) ∗ ; see Section 2.5 below. However, since C iscommonly used to denote the core itself, this might cause confusion with the existing literature.2.2. Examples.
Here are some examples of double vector bundles:(a) If
A, B, E are vector bundles over M , then A × M B × M E ∗ is a double vector bundle A × M B × M E ∗ / / (cid:15) (cid:15) B (cid:15) (cid:15) A / / M with core given by E ∗ . The horizontal and vertical scalar multiplications are given by κ ht ( a, b, ε ) = ( ta, b, tε ) and κ vt ( a, b, ε ) = ( a, tb, tε ), respectively. In particular, any vectorbundle V → M can be regarded as a double vector bundle in three ways, by playingthe role of A, B or E ∗ .(b) If V → M is any vector bundle, then its tangent bundle and cotangent bundle aredouble vector bundles T V / / (cid:15) (cid:15) T M (cid:15) (cid:15) V / / M T ∗ V / / (cid:15) (cid:15) V ∗ (cid:15) (cid:15) V / / M with core( T V ) = V (thought of as the vertical bundle of T V | M ) and core( T ∗ V ) = T ∗ M .The DVB structure on
T V appeared in [61]; the
DVB structure on T ∗ V was firstdiscussed in [51].(c) Suppose V → M is a subbundle of a vector bundle W → Q . Then the normal andconormal bundle of V in W are double vector bundles ν ( W, V ) / / (cid:15) (cid:15) ν ( Q, M ) (cid:15) (cid:15) V / / M ν ∗ ( W, V ) / / (cid:15) (cid:15) ( W | M /V ) ∗ (cid:15) (cid:15) V / / M HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 7 with core( ν ( W, V )) = W | M /V and core( ν ∗ ( W, V )) = ν ∗ ( Q, M ).(d) Let M , M be submanifolds of a manifold Q , with clean intersection [27]. Then thereis a double normal bundle with base M = M ∩ M , ν ( Q, M , M ) / / (cid:15) (cid:15) ν ( M , M ) (cid:15) (cid:15) ν ( M , M ) / / M with core T Q | M / ( T M | M + T M | M ). (Note that the core is trivial if and only if theintersection is transverse.) The double normal bundle can be defined as an iteratednormal bundle ν (cid:0) ν ( Q, M ) , ν ( M , M ) (cid:1) , or more symmetrically, as follows: Give C ∞ ( Q )the bi-filtration by the order of vanishing on the two submanifolds, and let S be theassociated bi-graded algebra. Then the double normal bundle is recovered as the char-acter spectrum ν ( Q, M , M ) = Hom alg ( S , R ). In other words, S is the algebra S ( D ) ofdouble-polynomial functions on D = ν ( Q, M , M ). More details on this example willbe given in a forthcoming work.Given a double vector bundle D as in (2), one can switch the roles of the horizontal and verticalscalar multiplications. This defines the flip (5) flip( D ) / / (cid:15) (cid:15) A (cid:15) (cid:15) B / / M A more interesting way of obtaining new double vector bundles is by taking dual bundles,horizontally or vertically. We will denote these by D h and D v , respectively: D h / / (cid:15) (cid:15) B (cid:15) (cid:15) E / / M D v / / (cid:15) (cid:15) E (cid:15) (cid:15) A / / M Also common are the notations D B B, D B A (e.g., [48]) or D ∗ B , D ∗ A (e.g., [25]). The horizontaland vertical scalar multiplication on D h are characterized by the property h κ ht ( φ ) , d i = t h φ, d i = h κ vt ( φ ) , κ vt ( d ) i , for φ ∈ D h , d ∈ D in the same fiber over B . Similarly, for ψ ∈ D v , d ∈ D in the same fiberover A , h κ ht ( ψ ) , κ ht ( d ) i = t h ψ, d i = h κ vt ( ψ ) , d i . One finds core( D h ) = A ∗ , core( D v ) = B ∗ . Clearly, flip( D v ) = flip( D ) h . In Example 2.2(b), T ∗ V is the vertical dual of T V , while in Example 2.2(c), ν ∗ ( W, V ) is the vertical dual of ν ( W, V ). ECKHARD MEINRENKEN AND JEFFREY PIKE
Splittings.
Let D be a double vector bundle over M , with side bundles A, B and withcore( D ) = E ∗ . A splitting (or decomposition ) of D is a DVB isomorphism D → A × M B × M E ∗ , inducing the identity on A, B, E ∗ . Here the DVB structure on the right hand side is as inExample 2.2(a).
Example . Let V → M be a vector bundle, and T V its tangent bundle regarded as a
DVB .A splitting of
T V is equivalent to a linear connection ∇ on V . (Cf. [25, Example 2.12].) Theorem 2.3.
Every double vector bundle admits a splitting.
This result was stated in [25] with a reference to [21]; a detailed proof was given in the Ph.D.thesis of del Carpio-Marek [16]. (The recent paper [26] by Heuer and Jotz Lean generalizesthis result to n -fold vector bundles.) In the appendix, we will present a somewhat shorterargument.Combining the existence of splittings with local trivializations of A, B, E ∗ , we see in partic-ular that every double vector bundle D is a fiber bundle over its base manifold M , with bundleprojection q = κ h κ v : D → M . Its fibers D m = q − ( m ) are double vector spaces (i.e., doublevector bundles over a point).2.4. The associated principal bundle.
Given non-negative integers n , n , n ∈ Z ≥ , put A = R n , B = R n , E = R n , and let D be the double vector space(6) D = A × B × E ∗ with κ ht ( a, b, ε ) = ( a, tb, tε ) and κ vt ( a, b, ε ) = ( ta, b, tε ). (Cf. Example 2.2(a).) For any vectorspace V , we denote by GL( V ) its general linear group; it comes with standard representationson V and on the dual space V ∗ . Thus, GL( A ) × GL( B ) × GL( E ) has a standard action onthe double vector space (6). Lemma 2.4. [24]
The group of
DVB automorphisms of D = A × B × E ∗ is a semi-directproduct (7) Aut( D ) = (cid:0) GL( A ) × GL( B ) × GL( E ) (cid:1) ⋉ ( A ∗ ⊗ B ∗ ⊗ E ∗ ) . with the standard action of GL( A ) × GL( B ) × GL( E ) , and with ω ∈ A ∗ ⊗ B ∗ ⊗ E ∗ acting as (8) ( a, b, ε ) ( a, b, ε + ω ( a, b )) . Given a double vector bundle D , take n , n , n to be the ranks of the bundles A, B, E . Anisomorphism of double vector spaces D m → D will be called a frame of D at m ∈ M . Clearly,any two frames are related by the action of Aut( D ). We define the frame bundle of D to bethe principal Aut( D )-bundle P → M whose fibers P m are the set of frames at m . Remark . In [36], the semi-direct product (7) is regarded as a double Lie group , and a generaltheory of double principal bundles for double Lie groups is developed.Many constructions with double vector bundles may be expressed in terms of bundles asso-ciated to P . In particular, D is itself an associated bundle for the action (8),(9) D = ( P × D ) / Aut( D ) . HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 9 (Pradines’ original definition [60] of double vector bundles was in terms of local trivializationswith Aut( D )-valued transition functions.) A splitting of D amounts to a reduction of thestructure group to GL( A ) × GL( B ) × GL( E ) ⊆ Aut( D ); the fibers Q m of the reduction Q ⊆ P are all those DVB isomorphisms D m → D that also preserve the splittings.Let D − be equal to D as a double vector space, but with the new action of Aut( D ), where ω ∈ A ∗ ⊗ B ∗ ⊗ E ∗ acts by ( a, b, ε ) ( a, b, ε − ω ( a, b )), while GL( A ) × GL( B ) × GL( E ) actsin the standard way. The resulting double vector bundle D − = ( P × D − ) / Aut( D ) . will feature in some of the constructions below. Lemma 2.6.
There is a canonical
DVB isomorphism D − → D that is the identity on the sidebundles but minus the identity on the core.Proof. The isomorphism is induced by the Aut( D )-equivariant isomorphism of double vectorspaces D − → D , ( a, b, ε ) ( a, b, − ε ). (cid:3) Triality of double vector bundles.
By cyclic permutation of the roles of A , B , E , theaction (8) of Aut( D ) on D = A × B × E ∗ gives rise to similar actions on D ′ = B × E × A ∗ and D ′′ = E × A × B ∗ . The bilinear pairings(10) D × B D ′ → R , (cid:0) ( a, b, ε ) , ( b, e, α ) (cid:1) α ( a ) − ε ( e ) , and similar maps given by cyclic permutation, are Aut( D )-equivariant. Taking associatedbundles, we obtain three double vector bundles D / / (cid:15) (cid:15) B (cid:15) (cid:15) A / / M D ′ / / (cid:15) (cid:15) E (cid:15) (cid:15) B / / M D ′′ / / (cid:15) (cid:15) A (cid:15) (cid:15) E / / M with bilinear pairings(11) D × B D ′ → R , D ′ × E D ′′ → R , D ′′ × A D → R The bundles D ′ , D ′′ are closely related to the horizontal and vertical duals: Proposition 2.7.
There are canonical
DVB isomorphisms D h ∼ = flip( D ′ ) − , D v ∼ = flip( D ′′ ) − that are the identity on the side bundles and on the core.Proof. We give the proof for D h (the argument for D v is similar). It suffices to consider thedouble vector space D . Write D as an associated bundle (9). Then D h = ( P × D h ) / Aut( D )where the Aut( D )-action on D h = E × B × A ∗ is given by the standard action of GL( A ) × GL( B ) × GL( E ), while ω ∈ A ∗ ⊗ B ∗ ⊗ E ∗ acts as( e, b, α ) ( e, b, α − ω ( b, e )) . This action is dictated by invariance of the duality pairing D × B D h → R , (cid:0) ( a, b, ε ) , ( e, b, α ) (cid:1) α ( a ) + ε ( e ) . The
DVB -isomorphism D h → flip( D ′ ) , ( e, b, α ) ( e, b, − α ) is Aut( D )-equivariant, andinduces a DVB -isomorphism D h → flip( D ′ ) that is the identity on the sides but minus theidentity on the core. Now use Lemma 2.6. (cid:3) Remark . Using the isomorphisms from Proposition 2.7, the second pairing in (11) translatesinto Mackenzie’s pairing D v × E D h → R [46, Theorem 3.1]. We also recover the result ofMackenzie [46] and Konieczna and Urba´nski [31], giving a canonical DVB isomorphism(( D h ) v ) h ∼ = (( D v ) h ) v that is the identity on the side bundles and on the core; indeed, by iteration of Proposition 2.7we see that both are identified with D − . As a special case, if D = T ∗ V we have that D v = T V ,( D v ) h = T V ∗ , (( D v ) h ) v = T ∗ V ∗ , and we recover the canonical DVB isomorphism(12) T ∗ ( V ∗ ) ∼ = flip( T ∗ V ) − of Mackenzie-Xu [51]. Example . Consider D = T V as a double vector bundle with sides A = V and B = T M .The natural pairing between tangent and cotangent vectors identifies D v = T ∗ V , while thetangent prolongation of the pairing V × M V ∗ → R identifies D h = T V ∗ . The three doublevector bundles D, D ′ , D ′′ are therefore, T V / / (cid:15) (cid:15) T M (cid:15) (cid:15) V / / M flip( T ( V ∗ )) − / / (cid:15) (cid:15) V ∗ (cid:15) (cid:15) T M / / M flip( T ∗ V ) − / / (cid:15) (cid:15) V (cid:15) (cid:15) V ∗ / / M Put differently, there is a canonical
DVB isomorphism (
T V ) ′ → flip( T ( V ∗ )) that is the identityon the side bundles and minus identity on the core V ∗ , and a canonical DVB isomorphism(
T V ) ′′ → flip( T ∗ V ) that is the identity on the side bundles and minus the identity on the core T ∗ M . 3. Double-linear functions
For any vector bundle V → M , the fiberwise linear functions on V are identified with thesections of the dual bundle V ∗ . We will similarly associate to any double vector bundle D a vector bundle whose space of sections are the functions on D that are double-linear , i.e.,linear for both scalar multiplications. Throughout this discussion, we will find it convenient topresent D as an associated bundle D = ( P × D ) / Aut( D ) with D = A × M B × M E ∗ .3.1. Double-linear functions.
Consider the Aut( D )-action on b E = ( A ∗ ⊗ B ∗ ) ⊕ E , where GL( A ) × GL( B ) × GL( E ) acts in the standard way, while elements ω ∈ A ∗ ⊗ B ∗ ⊗ E ∗ ∼ =Hom( E , A ∗ ⊗ B ∗ ) act as ( ν, e ) ( ν − ω ( e ) , e ) . The projection b E → E is Aut( D )-equivariant, with kernel A ∗ ⊗ B ∗ . Taking associatedbundles, we obtain a vector bundle b E = ( P × b E ) / Aut( D ) HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 11 with an exact sequence of vector bundles over M ,(13) 0 −→ A ∗ ⊗ B ∗ i b E −→ b E −→ E −→ . Using cyclic permutation of A , B , E , we obtain three such bundles b A, b B, b E , with inclusionmaps(14) i b E : A ∗ ⊗ B ∗ → b E, i b A : B ∗ ⊗ E ∗ → b A, i b B : E ∗ ⊗ A ∗ → b B, and exact sequences similar to (13). In Section 3.3 below we will identify the bundles b A, b B withthose introduced by Gracia-Saz and Mehta [25]; the corresponding exact sequences appear asEquation (26) in that reference. Proposition 3.1.
The space of sections of b E is canonically isomorphic to the space of double-linear functions on D . Under this identification, the quotient map to E is given by restriction ofdouble-linear functions to core( D ) = E ∗ , and the inclusion map i b E is given by the multiplicationof pull-backs of linear functions on A and on B .Proof. It suffices to prove these claims for the double vector space D . Using a Taylor expansion,we see that the double-linear functions on D = A × B × E ∗ are b E = ( A ∗ ⊗ B ∗ ) ⊕ E , where E is interpreted as linear functions on E ∗ and A ∗ ⊗ B ∗ as linear combinations of products oflinear functions on A , B . (cid:3) In terms of this interpretation through double-linear functions, the exact sequence (13) wasdiscussed by Chen-Liu-Sheng [12] as the dual of the
DVB sequence . A central result of theirpaper is that the double vector bundle may be recovered from this sequence:
Proposition 3.2. [12]
The double vector bundle D is the sub-double vector bundle of b D = A × M B × M b E ∗ consisting of all ( a, b, b ε ) ∈ A × M B × M b E ∗ such that i ∗ b E ( b ε ) = a ⊗ b . A splitting of D is equivalentto a splitting of the exact sequence (13) . An alternative way of proving this result is to verify the analogous statement for the doublevector space D = A × B × E ∗ . Remark . A direct consequence is that every double vector bundle D comes with a map(15) D → b E ∗ given by the inclusion D ֒ → ˆ D followed by projection to b E ∗ . This map is a DVB -morphism ifthe vector bundle b E ∗ is regarded as a double vector bundle (with zero sides). In terms of theassociated bundle construction, this is induced by the map D = A × B × E ∗ → b E ∗ = ( A ⊗ B ) ⊕ E ∗ , ( a, b, ε ) a ⊗ b + ε. Remark . The inclusion
D ֒ → b D dualizes to a surjective DVB morphism b D ′ = B × M b E × M A ∗ → D ′ . Replacing D ′ with D , this shows that every double vector bundle also arises as a quotient of asplit double vector bundle. Remark . Since a splitting of D is equivalent to a splitting of D ′ , D ′′ , we see that a splittingof D is equivalent to a splitting of any one of the three vector bundle maps b A → A, b B → B or b E → E .3.2. The three pairings.
In what follows, we will denote elements of the bundles b A, b B, b E by b a, b b, b e , and their images in A, B, E by a, b, e . Proposition 3.6.
There are canonical bilinear pairings h· , ·i E ∗ : b B × M b A → E ∗ , h· , ·i A ∗ : b E × M b B → A ∗ , (16) h· , ·i B ∗ : b A × M b E → B ∗ , with the properties (cid:10)b b, i b A ( µ ) (cid:11) E ∗ = µ ( b ) , µ ∈ B ∗ ⊗ E ∗ , b b ∈ b B, (17) (cid:10) i b B ( ν ) , b a (cid:11) E ∗ = − ν ( a ) , ν ∈ E ∗ ⊗ A ∗ , b a ∈ b A, (18) and similar properties obtained by cyclic permutations of A, B, E . The pairings are related bythe identity (19) (cid:10)b b, b a (cid:11) E ∗ ( e ) + (cid:10)b e, b b (cid:11) A ∗ ( a ) + (cid:10)b a, b e (cid:11) B ∗ ( b ) = 0 . Proof.
Using the associated bundle construction, it suffices to define the corresponding pairingsfor the double vector space D , and check that they are Aut( D )-equivariant. We have b A = ( B ∗ ⊗ E ∗ ) ⊕ A , b B = ( E ∗ ⊗ A ∗ ) ⊕ B , b E = ( A ∗ ⊗ B ∗ ) ⊕ E . Put(20) h· , ·i E ∗ : b B × b A → E ∗ , h ( ν, b ) , ( µ, a ) i E ∗ = µ ( b ) − ν ( a );This is clearly equivariant for the actions of GL( A ) × GL( B ) × GL( E ). For the action of ω ∈ A ∗ ⊗ B ∗ ⊗ E ∗ , observe that in the pairing between ω. ( µ, a ) = ( µ − ω ( a ) , a ) , ω. ( ν, b ) = ( ν − ω ( b ) , b ) , the terms involving ω cancel. The properties (17) and (18) hold by definition. Furthermore,given b a = ( µ, a ) ∈ b A , b b = ( ν, b ) ∈ b B , b e = ( ρ, e ) ∈ b E the three terms in (19) are µ ( e, a ) − ν ( b, e ), ρ ( a, b ) − µ ( e, a ) and ν ( b, e ) − ρ ( a, b ), hence their sum is zero. (cid:3) As we shall explain in Remark 3.9 below, the pairing h· , ·i E ∗ is equivalent to Mackenzie’snotion of ‘warp’ [50].Replacing D with flip( D ) reverses the role of A and B . Hence, the three inclusion maps(14) are unchanged, but the three pairings (16) all change sign. On the other hand, for D − wehave: Proposition 3.7.
The bundles b A, b B, b C for D are canonically isomorphic to those for D − .Under this identification, replacing D with D − changes the signs of the inclusion maps (14) and also of the pairings (16) . HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 13
Proof.
Let b A − , b B − , b E − be the corresponding bundles for D − . The Aut( D )-equivariant iso-morphism b E − → b E , ( ν, c ) ( − ν, c )gives the desired isomorphism b E − → b E , and similar for b A − , b B − . One readily checks that theseisomorphisms give sign changes for the inclusions and pairings. (cid:3) Geometric interpretations.
The bundles b A, b B, b E and the pairings between them havevarious geometric interpretations, in terms of functions and vector fields on D .We begin by recalling analogous interpretations for vector bundles V → M . The space X ( V ) [ r ] of vector fields on V that are homogeneous of degree r for the scalar multiplication(i.e., κ ∗ t X = t r X for t = 0) is trivial if r < −
1, while the core and linear vector fields(21) X ( V ) [ − := X core ( V ) , X ( V ) [0] =: X lin ( V )are identified with sections of V (via the vertical lift , taking a section σ ∈ Γ( V ) to the corre-sponding fiberwise constant vector field σ ♯ ), and infinitesimal automorphisms of V , respectively.On the other hand, C ∞ ( V ) [0] = C ∞ ( M ) and C ∞ ( V ) [1] = Γ( V ∗ ).The pairing V × M V ∗ → R is realized as the map X ( V ) [ − ⊗ C ∞ ( V ) [1] → C ∞ ( V ) [0] given byLie derivative, X ⊗ f L X f . We can also take a dual viewpoint (‘Fourier transform’), usingthe identifications C ∞ ( V ∗ ) [0] = C ∞ ( M ) , C ∞ ( V ∗ ) [1] = Γ( V ) , X ( V ∗ ) [ − = Γ( V ∗ ). Here, werealize the pairing V × M V ∗ → R as minus the Lie derivative, h ⊗ Z
7→ − L Z h . (Working withmulti-vector fields, it is convenient to think of these pairings as Schouten brackets between1-vector fields and 0-vector fields.)For a double vector bundle, let X ( D ) [ k,l ] be the space of vector fields on D that are ho-mogeneous of degree k horizontally and of degree l vertically. Similar notation will be usedfor smooth functions, differential forms, and so on. Let Γ( D, A ) be the sections of D as avector bundle over A , and Γ lin ( D, A ) the subspace of sections that are homogeneous of degree0 horizontally, i.e., such that the corresponding map A → D is κ ht -equivariant. Linear sectionsof D over B are defined similarly. We have(22) Γ lin ( D, A ) ∼ = X ( D ) [0 , − , Γ lin ( D, B ) ∼ = X ( D ) [ − , , Γ( E ∗ ) ∼ = X ( D ) [ − , − . To verify (22), note that vector fields X ∈ X ( D ) [ k,l ] with k = − l = − C ∞ ( D ) [0 , = C ∞ ( M ), and are thus vertical for the bundle projection D → M . Hence, itsuffices to check for the double vector space D = A × B × E ∗ . But X ( D ) [0 , − ∼ = B ⊕ ( A ∗ ⊗ E ∗ ) , X ( D ) [ − , = A ⊕ ( B ∗ ⊗ E ∗ ) , X ( D ) [ − , − ∼ = E ∗ , where elements of a vector space are seen as constant vector fields on the vector space, andelements of the dual space as linear functions. Indeed, with this interpretation the elements of Recall that the Schouten bracket on multi-vector fields on a manifold Q makes X • ( Q )[1] into a gradedsuper-Lie algebra, in such a way that the bracket extends the usual Lie bracket of vector fields and satisfies[[ X, f ]] = L X ( f ) for vector fields X and functions f . In particular we have[[ X, Y ]] = − ( − ( k − l − [[ Y, X ]] , [[ X, [[ Y, Z ]]]] = [[[[
X, Y ]] , Z ]] + ( − ( k − l − [[ Y, [[ X, Z ]]]]for X ∈ X k ( D ) , Y ∈ X l ( D ) , Z ∈ X m ( D ). The map X [[ X, · ]] is by graded derivation of the wedge product:[[ X, Y ∧ Z ]] = [[ X, Y ]] ∧ Z + ( − ( k − l Y ∧ [[ X, Z ]] . A , B , E ∗ have homogeneity bidegrees ( − , , (0 , − , ( − , −
1) respectively, while elements of A ∗ , B ∗ , E have homogeneity bidegrees (1 , , (0 , , (1 , Proposition 3.8.
The space of sections of b E is canonically isomorphic to (a) the space C ∞ ( D ) [1 , of double-linear functions on D , (b) the space Γ lin ( D ′ , B ) of linear sections of D ′ over B , (c) the space Γ lin ( D ′′ , A ) of linear sections of D ′′ over A , (d) the space X ( D ′ ) [0 , − of vector fields on D ′ of homogeneity (0 , − , (e) the space X ( D ′′ ) [ − , of vector fields on D ′′ of homogeneity ( − , .Similar descriptions hold for sections of b A, b B .Proof. It suffices to prove these descriptions for the double vector spaces D = A × B × E ∗ .We have already remarked that b E is the space of double-linear functions on D . For (b), notethat sections of D ′ = B × E × A ∗ over B are smooth functions B → E × A ∗ ; such a functiondefines a linear section if and only if its first component is a constant map B → E , while itssecond component is a linear map B → A ∗ . Hence, we obtain Γ lin ( D ′ , B ) = ( A ∗ ⊗ B ∗ ) ⊕ E = b E . The proof of (c) is similar, and by the counterparts of (22) for D ′ , D ′′ the properties (b),(c)are equivalent to (d),(e). (cid:3) Remark . In the work of Gracia-Saz and Mehta [25, Section 2.4], the isomorphisms Γ( b A ) ∼ =Γ lin ( D, B ) and Γ( b B ) ∼ = Γ lin ( D, A ) are used as the definition of b A, b B . With these identifications,the pairing h· , ·i E ∗ : Γ lin ( D, A ) × Γ lin ( D, B ) → Γ( E ∗ ) becomes Mackenzie’s warp of two linearsections [50].Using these geometric interpretations, the three inclusion maps (14) are realized as thebilinear maps i b E : C ∞ ( D ) [1 , × C ∞ ( D ) [0 , → C ∞ ( D ) [1 , , ( f, g ) f gi b A : C ∞ ( D ) [0 , × X ( D ) [ − , − → X ( D ) [ − , , ( g, Z ) gZ (23) i b B : X ( D ) [ − , − × C ∞ ( D ) [1 , → X ( D ) [0 , − , ( Z, f ) f Z while the three pairings (16) are h· , ·i E ∗ X ( D ) [0 , − × X ( D ) [ − , → X ( D ) [ − , − , ( X, Y ) [[ X, Y ]] h· , ·i A ∗ C ∞ ( D ) [1 , × X ( D ) [0 , − → C ∞ ( D ) [1 , , ( h, X )
7→ − L X h (24) h· , ·i B ∗ X ( D ) [ − , × C ∞ ( D ) [1 , → C ∞ ( D ) [0 , , ( Y, h ) L Y h. The identity (19) just amounts to L [ X,Y ] = [ L X , L Y ]. To verify (23) and (24), it is enough toconsider the double vector space D , but there it follows by a routine check from the definitions.3.4. Applications to vector bundles I.
We illustrate the concepts above with various doublevector bundles associated to a vector bundle V → M . In this context, we will encounter the jet bundle J ( V ) and the Atiyah algebroid
At( V ) (often denoted by D ( V ), or similar). For σ ∈ Γ( V ), we denote by j ( σ ) ∈ Γ( J ( V )) its jet prolongation. The jet bundle comes with aquotient map J ( V ) → V taking sections of the form f j ( σ ) to f σ ; this defines a short exactsequence(25) 0 → T ∗ M ⊗ V i J V ) −→ J ( V ) → V → HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 15 with i J ( V ) (d f ⊗ σ ) = j ( f σ ) − f j ( σ ). On the other hand, the Atiyah algebroid comes with ashort exact sequence(26) 0 → V ⊗ V ∗ i At( V ) −→ At( V ) a −→ T M → a is the anchor. We shall find it convenient to use the identification Γ(At( V )) ∼ = X lin ( V )(cf. (21)) to interpret sections δ of the Atiyah algebroid in terms of the corresponding lin-ear vector field e a ( δ ) on V ; its restriction to the zero section is a ( δ ). From this perspective, e a (cid:0) i At( V ) ( σ ⊗ τ ) (cid:1) = φ τ σ ♯ , where φ τ ∈ C ∞ ( V ) is the linear function defined by τ ∈ Γ( V ∗ ), and σ ♯ ∈ X ( V ) [ − denotes the vertical lift of σ ∈ Γ( V ). The representation of At( V ) on V is givenby the Lie bracket, ( ∇ δ σ ) ♯ = [[ e a ( δ ) , σ ♯ ]], and the dual representation ∇ ∗ on V ∗ , defined as(27) L a ( δ ) h τ, σ i = h τ, ∇ δ σ i + h∇ ∗ δ τ, σ i , is realized by the Lie derivative of e a ( δ ) on linear functions, φ τ L e a ( δ ) φ τ .3.4.1. Tangent bundle of V . For D = T V we have A = V, B = T M, E = V ∗ . One finds that(28) b A = J ( V ) , b B = At( V ) , b E = J ( V ∗ ) . In terms of Γ( b A ) ∼ = X ( D ) [ − , , Γ( b B ) ∼ = X ( D ) [0 , − , Γ( b E ) ∼ = C ∞ ( D ) [1 , , these identificationsare given by j ( σ ) ( σ ♯ ) T , δ e a ( δ ) ♯ , j ( τ ) ( φ τ ) T , where X X T is the tangent lift of a (multi-)vector field. Using (23) and (24), we obtain thethree inclusions i b E ( τ ⊗ d f ) = i J ( V ∗ ) (d f ⊗ τ ) , i b A (d f ⊗ σ ) = i J ( V ) (d f ⊗ σ ) , i b B ( σ ⊗ τ ) = i At( V ) ( σ ⊗ τ )and the three pairings h δ, j ( σ ) i V = ∇ δ σ, h j ( τ ) , δ i V ∗ = −∇ ∗ δ τ, h j ( σ ) , j ( τ ) i T ∗ M = d h τ, σ i for σ ∈ Γ( V ) , τ ∈ Γ( V ∗ ) , δ ∈ Γ(At( V )). As a sample computation, note that (23) gives i b E ( τ ⊗ d f ) = ( φ τ ) ♯ f T = ( f φ τ ) T − f ♯ ( φ τ ) T ∈ C ∞ ( T V ) [1 , (here the vertical lift f ♯ of a functionis simply the pullback, while the vertical lift f T is the exterior differential, regarded as a functionon the tangent bundle). This coincides with the image of i J ( V ∗ ) (d f ⊗ τ ) = j ( f τ ) − f j ( τ ).The exact sequences (13) are just the standard exact sequences for the jet bundles and theAtiyah algebroid. Remark . The E ∗ = V -valued pairing between b A = J ( V ) and b B = At( V ) was observedby Chen-Liu in [11, Section 2].Let us also note that by Remark 3.5, a splitting of D = T V is equivalent to a splitting ofany one of the exact sequences for J ( V ∗ ) , J ( V ) or At( V ); in turn, these are equivalent to alinear connection on the vector bundle V . The vertical lift of vector fields extends to an algebra morphism on multi-vector fields. On the other hand,the usual tangent lift X X T of vector fields extends uniquely to a linear map on multivector fields, in such away that ( X ∧ Y ) T = X T ∧ Y ♯ + X ♯ ∧ Y T . One has the Schouten bracket relations [[ X T , Y T ]] = [[ X, Y ]] T , [[ X T , Y ♯ ]] =[[ X, Y ]] ♯ , [[ X ♯ , Y ♯ ]] = 0. See e.g. [13, 22]. Cotangent bundle of V . We will use the following notations for cotangent bundles T ∗ Q .Given X ∈ X ( Q ), let φ X ∈ C ∞ ( T ∗ Q ) be the corresponding linear function, defined by thepairing with covectors. The standard Poisson structure on T ∗ Q is described by the condition { φ X , φ Y } = φ [ X,Y ] for any two such vector fields. For any H ∈ C ∞ ( T ∗ Q ), the derivation { H, ·} is its Hamiltonian vector field ; in particular, X T ∗ = { φ X , ·} is the cotangent lift of the vectorfield X . One has the identity ( f X ) T ∗ = f X T ∗ − φ X (d f ) ♯ for f ∈ C ∞ ( Q ).For D = T ∗ V , we have that A = V, B = V ∗ , E = T M with b A = J ( V ) , b B = J ( V ∗ ) , b E = At( V ) . In terms of the identifications of their spaces of sections with X ( D ) [ − , , X ( D ) [0 , − , C ∞ ( D ) [1 , ,these isomorphisms are given by j ( σ )
7→ { φ σ ♯ , ·} , j ( τ )
7→ { ( φ τ ) ♯ , ·} , δ φ δ . Using (23) and (24), we find that the three pairings are the same as for
T V (with the orderof the two entries interchanged), while each of the three inclusion maps changes sign. This isconsistent with T ∗ V = flip( T V ) − , see Proposition 3.7. The three exact sequences (13) are thestandard exact sequences for the jet bundles and the Atiyah algebroid, up to a sign change ofthe three inclusion maps. 4. The Weil algebra W ( D )Throughout this section, we consider a fixed double vector bundle D over M , with sidebundles A, B and with E = core( D ) ∗ . It will be convenient to regard D as an associatedbundle ( P × D ) / Aut( D ) with D = A × B × E ∗ .4.1. Overview.
Recall (e.g., [10]) that a graded supermanifold M is given by a base manifold M together with a structure sheaf of algebras, admitting local trivializations C ∞ ( U ) × ∨ E for U ⊆ M . Here E is a Z -graded vector space, and the symmetric algebra is defined usingthe super-sign conventions. One formally thinks of global sections of the structure sheaf asthe algebra of functions C ∞ ( M ) on the ‘space’ M , even though the latter is not an actualtopological space. If V → M is a vector bundle, the algebra Γ( ∧ V ∗ ) is regarded as the algebraof functions on the graded manifold V [1], where the [1] signifies a degree shift: Here E = ( R k [1]) ∗ with k = rank( V ).The definition of graded supermanifold has a straightforward generalization to bigraded su-permanifold. Given a double vector bundle D , there is a bigraded supermanifold D [1 ,
1] (witha parity shift for both vector bundle directions), defined in terms of a structure sheaf. InVoronov’s work [64], the structure sheaf was obtained from local coordinates on D , compatiblewith a given splitting, by declaring coordinates for the two side bundle directions to be odd.The corresponding bigraded vector space is the direct sum E = A ∗ ⊕ B ∗ ⊕ E , where the threesummands reside in bidegrees (1 , , (0 , , (1 ,
1) respectively. In particular, one obtains analgebra of functions C ∞ ( D [1 , Weil algebra since it generalizes the Weil algebra [65] from equivariantcohomology.
HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 17
As pointed out to us by a referee, the work of Grabowski-J´o´zwikowski-Rotkievicz [20] sug-gests analogous constructions in more general contexts. The algebra of double-polynomial functions.
A smooth function on a double vectorbundle D will be called a (homogeneous) double-polynomial of bidegree ( r, s ) if it is homogeneousof degree r for the horizontal scalar multiplication, and of degree s for the vertical scalarmultiplication. The space of such functions is denoted S r,s ( D ) = C ∞ ( D ) [ r,s ] ; their direct sumover all r, s ≥ S ( D ). Lemma 4.1.
The space S r,s ( D ) of double-polynomial functions on D of bidegree ( r, s ) is thespace of sections of a vector bundle S r,s ( D ) → M. Proof.
The space S r,s ( D ) of double-polynomial functions of bidegree ( r, s ) on the double vectorspace D is a subspace of the space of ordinary polynomials of degree at most r + s , and inparticular is finite-dimensional. Clearly, this space is Aut( D )-invariant, and the sections ofthe vector bundle S r,s ( D ) = ( P × S r,s ( D )) / Aut( D ) are the double-polynomial functions ofbidegree r, s . (cid:3) We hence obtain a bigraded algebra bundle S ( D ) = L r,s S r,s ( D ). By applying a similarconstruction to the double vector bundles D ′ and D ′′ , we also have the bigraded algebra bundles S ( D ′ ) → M and S ( D ′′ ) → M . Proposition 4.2.
The algebra bundle S ( D ) is the bundle of bigraded commutative algebras S ( D ) = ( ∨ A ∗ ⊗ ∨ B ∗ ⊗ ∨ b E ) / ∼ where the generators α ∈ A ∗ , β ∈ B ∗ , b e ∈ b E have bidegrees (1 , , (0 , , (1 , , respectively.Here the kernel of the quotient map is the ideal generated by elements of the form αβ − i b E ( α ⊗ β ) for ( α, β ) ∈ A ∗ × M B ∗ .Proof. The claim is immediate for the double vector space D = A × B × E ∗ ; the generalcase follows by the associated bundle construction. (cid:3) The construction of S ( D ) is functorial: a DVB morphism D → D , with base map F : M → M , defines algebra morphisms S ( D ) F ( m ) → S ( D ) m , hence a comorphism of bigraded algebrabundles S ( D ) S ( D ) . The induced pullback on sections is simply the pullback of double-polynomial functions. If M = M = M , with base map the identity, then the comorphism of algebra bundles maybe seen as an ordinary morphism of algebra bundles S ( D ) → S ( D ). For example, thepresentation of S ( D ) as a quotient of S ( b D ) = ∨ A ∗ ⊗ ∨ B ∗ ⊗ ∨ b E is functorially induced by theinclusion D ֒ → b D . Remark . Just as vector bundles may be recovered from their algebra of polynomial functionsas a character spectrum V ∼ = Hom alg ( S ( V ) , R ), double vector bundles are recovered as D ∼ =Hom alg ( S ( D ) , R ). Note however that the ‘Weil algebra bundles’ in [20] involve a different notion of Weil algebra, as in [30,Chapter 8].
Definition and basic properties of W ( D ) . The Weil algebra bundle is obtained fromthe description of S ( D ), given in Proposition 4.2, by replacing commutativity with super-commutativity: Definition 4.4.
The
Weil algebra bundle W ( D ) is the bundle of bigraded super-commutativealgebras given as W ( D ) = ( ∧ A ∗ ⊗ ∧ B ∗ ⊗ ∨ b E ) / ∼ , where the generators α ∈ A ∗ , β ∈ B ∗ , b e ∈ b E have bidegrees (1 , , (0 , , (1 ,
1) respectively.Here the kernel of the quotient map is the ideal generated by elements of the form αβ − i b E ( α ⊗ β )with ( α, β ) ∈ A ∗ × M B ∗ .In the definition above, ⊗ denotes the usual tensor product of superalgebras. For x ∈ W p,q ( D ) we write | x | = p + q for the total degree; thus super-commutativity means x x = ( − | x || x | x x . The bigradedalgebra of sections W ( D ) = Γ( W ( D )) is called the Weil algebra of D . In super-geometricterms, it is the algebra of smooth functions on the supermanifold D [1 , S ( D ), a DVB morphism D → D with base map F : M → M inducesa comorphism of bigraded superalgebra bundles W ( D ) W ( D ), hence a morphism ofbigraded superalgebras W ( D ) → W ( D ).The definition gives a number of straightforward properties of W ( D ):(a) In degree p ≤ , q ≤ W p,q ( D ) coincides with S p,q ( D ): W , , ( D ) = M, W , ( D ) = A ∗ , W , ( D ) = B ∗ , W , ( D ) = b E. (b) A choice of splitting D ∼ = A × M B × M E ∗ gives an algebra bundle isomorphism W ( D ) ∼ = ∧ A ∗ ⊗ ∧ B ∗ ⊗ ∨ E. (c) W ( D ) = ( P × W ( D )) / Aut( D ). Since W ( D ) = ∧ A ∗ ⊗ ∧ B ∗ ⊗ ∨ E , this may be usedas an alternative definition of W ( D ).Replacing D with D ′ and D ′′ , we have three bigraded algebra bundles W ( D ) , W ( D ′ ) , W ( D ′′ )over M , where the roles of A, B , and E are cyclically permuted. In particular, W , ( D ) = b E, W , ( D ′ ) = b A, W , ( D ′′ ) = b B. The pairings (16) between these bundles extend to h· , ·i E ∗ : W p, ( D ′′ ) × M W ,q ( D ′ ) → ∧ p + q − E ∗ , h· , ·i A ∗ : W p, ( D ) × M W ,q ( D ′′ ) → ∧ p + q − A ∗ , (29) h· , ·i B ∗ : W p, ( D ′ ) × M W ,q ( D ) → ∧ p + q − B ∗ . Here h· , ·i E ∗ is the unique extension of the given pairing such that h α, b a i E ∗ = − α ( a ) , h b b, β i E ∗ = β ( b ) HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 19 for the cases p = 0 , q = 1 and p = 1 , q = 0, and such that the following bilinearity propertyholds: h λx, y i E ∗ = λ h x, y i E ∗ , h x, yλ i E ∗ = h x, y i E ∗ λ for λ ∈ ∧ E ∗ , x ∈ W • , ( D ′′ ) , y ∈ W , • ( D ′ ) (with the same base points). The discussion for thepairings h· , ·i A ∗ , h· , ·i B ∗ is similar. In Section 5, we will give geometric interpretations of thesepairings. Remark . The description of the Weil algebra bundle for D − is obtained from that for D byreplacing the sign of the inclusion map i b E . That is, W ( D − ) has the same generators, but thedefining relation becomes αβ = − i b E ( α ⊗ β ). The map on generators α α, β β, b e
7→ − b e extends to an isomorphism of algebra bundles W ( D − ) → W ( D ).4.4. Derivations.
For a vector bundle V → M , the graded bundle Der( ∧ V ∗ ) of fiberwise superderivations of ∧ V ∗ is the free ∧ V ∗ -module generated by contractions. ThusDer( ∧ V ∗ ) = ∧ V ∗ ⊗ V as a bundle of graded super-Lie algebras, where the elements 1 ⊗ v have degree − D , we are interested in the structure of the bigraded bundleDer( W ( D )) = M r,s Der r,s ( W ( D )) . Here Der r,s ( W ( D )) → M is the bundle of fiberwise superderivations of bidegree ( r, s ) of thealgebra bundle W ( D ) → M : its space of sections consists of bundle maps δ : W ( D ) → W ( D )of bidegree ( r, s ) with the superderivation property δ ( xy ) = δ ( x ) y + ( − | δ || x | xδ ( y )for homogeneous elements x, y , where | δ | = r + s and | x | are the total degrees of δ and x . Thefollowing result describes the structure of Der( W ( D )) as a W ( D )-module and as a bundle ofgraded Lie algebras. Theorem 4.6.
Let m ∈ M and b a ∈ b A m , b b ∈ b B m , and ε ∈ E ∗ m . There are unique contractionoperators ι h ( b a ) ∈ Der − , ( W ( D )) m , ι v ( b b ) ∈ Der , − ( W ( D )) m , ι ( ε ) ∈ Der − , − ( W ( D )) m such that (30) ι h ( b a ) v = h b a, v i B ∗ , ι v ( b b ) u = ( − | u | h u, b b i A ∗ , ι ( ε ) e = − ε ( e ) for all u ∈ W , • ( D ) m , v ∈ W • , ( D ) m , e ∈ W , ( D ) m = b E m . The contraction operatorssatisfy the commutation relations (31) [ ι v ( b b ) , ι h ( b a )] = − ι (cid:0) h b b, b a i E ∗ (cid:1) , b a ∈ b A, b b ∈ b B, while all other commutations of contractions are zero. The W ( D ) m -module Der( W ( D )) m isgenerated by the three types of contraction operators, subject to the relations (32) ι h (cid:0) i b A ( β ⊗ ε ) (cid:1) = βι ( ε ) , ι v (cid:0) i b B ( ε ⊗ α ) (cid:1) = − αι ( ε ) , α ∈ A ∗ m , β ∈ B ∗ m , ε ∈ E ∗ m . Proof.
For degree reasons, the proposed expressions for the contractions determine the formulason generators of W ( D ) m . Specifically, ι h ( b a ) is given on generators α ∈ A ∗ m , β ∈ B ∗ m , b e ∈ b E m by α α ( a ) , β , b e
7→ h b a, b e i B ∗ , while ι v ( b b ) is given by α , β β ( b ) , b e
7→ h b e, b b i A ∗ ,and ι h ( ε ) is given by α , β , b e
7→ − ε ( e ). One readily checks that these formulas arecompatible with the defining relation of the Weil algebra, and hence extend to a derivationon all of W ( D ) m . Furthermore, the super-commutation relation between these contractionsoperators are verified by evaluating on generators.The three types of contraction operators define a W ( D ) m -module morphism(33) W ( D ) m ⊗ ( b A m ⊕ b B m ⊕ E ∗ m ) → Der( W ( D )) m whose kernel contains elements of the form(34) 1 ⊗ i b A ( β ⊗ ε ) − β ⊗ ε, ⊗ i b B ( ε ⊗ α ) + α ⊗ ε with α ∈ A ∗ m , β ∈ B ∗ m , ε ∈ E ∗ m . We have to show that (33) is surjective, with kernel thesubmodule generated by elements of the form (34).It suffices to prove this for the double vector space D = A × B × E ∗ . Here W ( D ) issimply a tensor product ∧ A ∗ ⊗ ∧ B ∗ ⊗ ∨ E , and hence Der( W ( D )) = W ( D ) ⊗ ( A ⊕ B ⊕ E ∗ ).Since b A = A ⊕ ( B ∗ ⊗ E ∗ ) and b B = B ⊕ ( E ∗ ⊗ A ∗ ), it is immediate that the module map(33) (with D replaced by D ) is surjective. Its kernel contains elements of the form (34); henceit also contains the W ( D )-submodule generated by elements of this form. But this submoduleis a complement to the submodule W ( D ) ⊗ ( A ⊕ B ⊕ E ∗ ), and is therefore the entire kernelof (33). (cid:3) In particular, we see that the bundle Der r,s ( W ( D )) is zero if r < − s < −
1, while(35) Der − , ( W ( D )) = b A, Der , − ( W ( D )) = b B, Der − , − ( W ( D )) = E ∗ . Proposition 4.7.
The horizontal contractions extend to an isomorphism of left ∧ B ∗ -modules (36) ι h : W • , ( D ′ ) → Der − , − • ( W ( D )) , x ι h ( x ) such that (37) ι h ( x ) z = h x, z i B ∗ , x ∈ W • , ( D ′ ) , z ∈ W , • ( D ) . The vertical contractions extend to an isomorphism of left ∧ A ∗ -modules (38) ι v : W , • ( D ′′ ) → Der − • , − ( W ( D )) , y ι v ( y ) given by (39) ι v ( y ) z = − ( − ( | y | +1)( | z | +1) h z, y i A ∗ , y ∈ W , • ( D ′′ ) , z ∈ W • , ( D )The sign in (38) comes from the fact that we are using the left ∧ A ∗ -module structures,whereas the pairing is bilinear for the right ∧ A ∗ -module structure in the second argument.Note that ι h ( ε ) = ι v ( ε ) = ι ( ε ) for ε ∈ E ∗ . Proof.
The proposed formulas determine ι h ( x ) , ι v ( y ) on generators. To show that the formula(39) for ι v ( y ) gives a well-defined ∧ A ∗ -module homomorphism, it suffices to check that theright hand side is linear in the argument y for the left ∧ A ∗ -module structure and linear in the HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 21 argument z for the right ∧ A ∗ -module structure. Indeed, replacing y with αy changes the righthand side to( − | y | ( | z | +1) h z, αy i A ∗ = ( − | y | | z | h z, y i A ∗ α = ( − ( | y | +1)( | z | +1) α h z, y i A ∗ . Similarly, replacing z with zα for α ∈ A ∗ changes the right hand side to( − ( | y | +1) | z | h zα, y i A ∗ = ( − | y | | z | α h z, y i A ∗ = ( − ( | y | +1)( | z | +1) h z, y i A ∗ α as required. The argument for ι h ( x ) is similar. (cid:3) Applications to vector bundles II.
Continuing the discussion from Section 3.4, wehave the following description of the Weil algebras and contraction operators for the tangentbundles and cotangent bundles of vector bundles V → M .4.5.1. Tangent bundle of V . Consider D = T V , so that A = V, B = T M, C = V ∗ . The Weilalgebra W ( T V ) is generated by functions f ∈ C ∞ ( M ) (bidegree (0 , f (bidegree (0 , τ ∈ Γ( V ∗ ) (bidegree (1 , j ( τ ) ∈ Γ( J ( V ∗ )) (bidegree (1 , C ∞ ( M )-linearity and therelation that τ d f = j ( f τ ) − f j ( τ ) . Here we used that i b E ( τ ⊗ d f ) = i J ( V ∗ ) ( τ ⊗ d f ). The contraction operators are computed fromthe pairings, for example: ι v ( δ ) j ( τ ) = h j ( τ ) , δ i V ∗ = −∇ ∗ δ τ. Cotangent bundle of V . Consider D = T ∗ V , so that A = V, B = V ∗ , C = T M . TheWeil algebra W ( T ∗ V ) is generated by functions f ∈ C ∞ ( M ) (bidegree (0 , σ ∈ Γ( V ) (bidegree (0 , τ ∈ Γ( V ∗ ) (bidegree (1 , δ ∈ Γ(At( V )) (bidegree (1 , C ∞ ( M )-linearity and the relation thatthe product τ σ in the Weil algebra, for τ ∈ Γ( V ∗ ) and σ ∈ Γ( V ), equals the section i b E ( τ ⊗ σ ) = − i At( V ) ( σ ⊗ τ ) = − φ τ σ ♯ ∈ Γ(At( V )). Again, the various contraction operators may becomputed from the pairings.5. Linear and core sections of ∧ A D We have already encountered the linear and core sections of a double vector bundle D overits side bundles. We shall now consider the generalization to the exterior algebra bundles, andrelate it to the Weil algebra bundles. Throughout, D will denote a double vector bundle withsides A, B and with core( D ) = E ∗ .5.1. Linear and core sections of ∧ • A D → A . Given a double vector bundle D , we denoteby ∧ nA D → A its exterior powers as a vector bundle over A . The horizontal scalar multiplications κ ht : D → D are vector bundle endomorphisms of D → A , hence they extend to algebra bundle endomor-phisms ∧ • κ ht of ∧ • A D → A . A section σ : A → ∧ nA D is homogeneous of degree k if it satisfies σ ( κ ht ( a )) = t k ( ∧ n κ ht )( σ ( a ))for all t ∈ R ; the space of such sections is denoted Γ( ∧ nA D, A ) [ k ] . Definition 5.1.
The spaces of core sections and linear sections of ∧ nA D over A are defined asfollows: Γ core ( ∧ nA D, A ) = Γ( ∧ nA D, A ) [ − n ] , Γ lin ( ∧ nA D, A ) = Γ( ∧ nA D, A ) [ − n +1] . The spaces Γ core ( ∧ nB D, B ) and Γ lin ( ∧ nB D, B ) are defined similarly.The core sections Γ core ( ∧ • A D, A ) are a super-commutative graded algebra under the wedgeproduct, and Γ lin ( ∧ • A D, A ) is a graded module over this algebra. The significance of thesespaces is clarified by the following result.
Proposition 5.2.
The space Γ( ∧ nA D, A ) [ k ] is zero if k < − n , and for k = − n is given by (40) Γ core ( ∧ nA D, A ) ∼ = Γ( ∧ n E ∗ ) . The space of linear sections fits into a short exact sequence (41) 0 → Γ( ∧ n E ∗ ⊗ A ∗ ) → Γ lin ( ∧ nA D, A ) → Γ( ∧ n − E ∗ ⊗ B ) → . Proof.
Recall that when D is viewed as a vector bundle over A , its restriction to the submanifold M is the direct sum D = E ∗ ⊕ B . Hence, the restriction of sections to M ⊆ A gives a mapΓ( ∧ A D, A ) → Γ( ∧ ( E ∗ ⊕ B )) = Γ( ∧ E ∗ ⊗ ∧ B ) . We claim that the restriction of core sections gives the isomorphism (40), while restriction oflinear sections gives a map from Γ lin ( ∧ nA D, A ) onto Γ( ∧ n − E ∗ ⊗ B ), with kernel Γ( ∧ n E ∗ ⊗ A ∗ )spanned by products of core sections with linear functions on the base A . Using the associatedbundle construction, it suffices to prove these claims for the double vector space D = A × B × E ∗ . We have Γ( ∧ nA D , A ) = C ∞ ( A , M i + j = n ∧ j E ∗ ⊗ ∧ i B ) . The elements of ∧ j E ∗ ⊗ ∧ n − j B → M (regarded as constant sections of ∧ nA D ) are homoge-neous of degree − j . To obtain a section that is homogeneous of degree k , we must multiply bya polynomial on A of degree k + j . Thus,Γ( ∧ nA D , A ) [ k ] = M j ∧ j E ∗ ⊗ ∧ n − j B ⊗ ∨ k + j A ∗ where the sum is over all j with 0 ≤ j ≤ n and k + j ≥
0. In particular, this space is zero if k < − n , and is equal to ∧ n E ∗ for k = − n . Specializing to k = − n + 1 this shows(42) Γ lin ( ∧ nA D , A ) = ( ∧ n E ∗ ⊗ A ∗ ) ⊕ ( ∧ n − E ∗ ⊗ B ) . Hence, the map Γ lin ( ∧ nA D , A ) → ∧ n − E ∗ ⊗ B is surjective, with kernel ∧ n E ∗ ⊗ A ∗ . (cid:3) HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 23
Interpretation in terms of Weil algebras.
The linear and core sections of ∧ A D → A are graded subspaces of Weil algebras, as follows. Proposition 5.3.
There is a canonical isomorphism Γ core ( ∧ • A D, A ) ∼ = W • , ( D ′′ ) = Γ( ∧ • E ∗ ) as graded algebras, and an isomorphism of left modules over this algebra, Γ lin ( ∧ • A D, A ) ∼ = W • , ( D ′′ ) . Similarly, there is a canonical isomorphism of graded algebras, Γ core ( ∧ • B D, B ) ∼ = W , • ( D ′ ) =Γ( ∧ • E ∗ ) and an isomorphism of right modules over this algebra, Γ lin ( ∧ • B D, B ) ∼ = W , • ( D ′ ) . Proof.
It suffices to prove the claim for the double vector space D = A × B × E ∗ . But W • , ( D ′′ ) = ∧ • E ∗ , W , • ( D ′ ) = ∧ • E ∗ as graded algebras. Furthermore, the isomorphism ofgraded left ∧ E ∗ -modules W • , ( D ′′ ) = ( ∧ • E ∗ ⊗ A ∗ ) ⊕ ( ∧ •− E ∗ ⊗ B )is exactly the description of linear sections of ∧ nA D , see (42). Similarly for W , • ( D ′ ) = ( B ∗ ⊗ ∧ • E ∗ ) ⊕ ( A ⊗ ∧ •− E ∗ ) . as graded right ∧ E ∗ -modules. (cid:3) With these identifications, the pairing h· , ·i E ∗ : W p, ( D ′′ ) × M W ,q ( D ′ ) → ∧ p + q − E ∗ (cf. (29))translates into a Γ( ∧ E ∗ )-bilinear pairing(43) h· , ·i E ∗ : Γ lin ( ∧ pA D, A ) × Γ lin ( ∧ qB D, B ) → Γ( ∧ p + q − E ∗ ) . Application to vector bundles III.
Let V → M be a vector bundle.5.3.1. Linear Multi-vector fields on vector bundles.
Consider
T V as a double vector bundle asin Example 2.2b. The core n -vector fields X n core ( V ) ≡ Γ core ( ∧ nV T V, V ) are the sections of ∧ n V ,regarded as vertical ‘fiberwise constant’ multi-vector fields on V : X n core ( V ) = Γ( ∧ n V ) . The linear n -vector fields X n lin ( V ) = Γ lin ( ∧ nV T V, V ) may be defined by their property that theevaluation on linear 1-forms on V is a linear function on V [22]. The short exact sequence (41)specializes to(44) 0 → Γ( ∧ n V ⊗ V ∗ ) → X n lin ( V ) → Γ( ∧ n − V ⊗ T M ) → ∧ n V ⊗ V ∗ ) is as the subspace of linear n -vector fields on V that aretangent to the fibers of V → M . In local vector bundle coordinates, with x i the coordinateson the base and y j the coordinates on the fiber, the linear n -vector fields on V are of the form X a jj ··· j n ( x ) y j ∂∂y j ∧ · · · ∧ ∂∂y j n + X b i j ··· j n − ( x ) ∂∂y j ∧ · · · ∧ ∂∂y j n − ∧ ∂∂x i . The Schouten bracket of multi-vector fields defines a graded Lie algebra structure on X • lin ( V )[1],with a representation on X • core ( V ) = Γ( ∧ • V ). These are the multi-differentials in the work ofIglesias-Ponte, Laurent-Gengoux, and Xu [29]. Linear differential forms on vector bundles.
Consider next the double vector bundle T ∗ V from Example 2.2b. The space Ω n core ( V ) = Γ core ( ∧ nV T ∗ V, V ) is just Ω n ( M ), viewed as the spaceof basic n -forms on V via pullback. The spaceΓ lin ( ∧ nV T ∗ V, V ) = Ω n lin ( V )of linear n -forms on V consists of n-forms α with κ ∗ t α = tα where κ t is scalar multiplicationby t on V . (Note that the homogeneity of n -forms on V relative to pullback κ ∗ t is not the sameas homogeneity as sections of ∧ nV T ∗ V over V .) In local bundle bundle coordinates, with x i the coordinates on the base and y j the coordinates on the fiber, the 1-forms d x i , seen as localsections of ∧ • V T ∗ V , have homogeneity degree − y j have homogeneity 0. A generallinear n -form is locally given by an expression X a j i ··· i n − ( x ) d x i ∧ · · · ∧ d x i n − ∧ d y j + X b j i ··· i n ( x ) y j d x i ∧ · · · ∧ d x i n . The short exact sequence (41) becomes(45) 0 → Γ( ∧ n T ∗ M ⊗ V ∗ ) → Ω n lin ( V ) → Γ( ∧ n − T ∗ M ⊗ V ∗ ) → ∧ n T ∗ M ⊗ V ∗ ) is as the space of linear n -forms on V that are horizontalfor the projection to M , while the projection to Γ( ∧ n − T ∗ M ⊗ V ∗ ) is given by contraction withsections of V (regarded as the space X core ( V ) of fiberwise constant vector fields on V ). Theexact sequence (45) has a canonical splitting [5]: every element of Ω n lin ( V ) decomposes uniquelyas ν + d µ where ν ∈ Γ( ∧ n T ∗ M ⊗ V ∗ ) and µ ∈ Γ( ∧ n − T ∗ M ⊗ V ∗ ). Using the Mackenzie-Xuisomorphism (12) we obtain a similar interpretationΓ lin ( ∧ qV ∗ T ∗ V, V ∗ ) = Ω q lin ( V ∗ ) . Equation (43) defines an Ω( M )-bilinear pairing(46) h· , ·i T ∗ M : Ω p lin ( V ) × Ω q lin ( V ∗ ) → Ω p + q − ( M ) , (using the right Ω( M )-module structure in the second argument), given in low degrees by h φ τ , φ σ i T ∗ M = 0 , h φ τ , d φ σ i T ∗ M = −h d φ τ , φ σ i T ∗ M = h τ, σ i , h d φ τ , d φ σ i T ∗ M = d h τ, σ i for τ ∈ Γ( V ∗ ) , σ ∈ Γ( V ) (with φ τ , φ σ the corresponding linear functions).5.4. Multi-vector fields on D . The linear and core sections of ∧ A D → A and ∧ B D → B ,and their pairings, have a simple interpretation in terms of the space X • ( D ) of multi-vectorfields on D . Using the discussion from Example 5.3.1, the sections of ∧ nA D are identified with n -vector fields on D that are homogeneous of degree − n with respect to the vertical vectorbundle structure. A similar description applies to sections of ∧ nB D . As in Section 3.3, we let X n ( D ) [ k,l ] denote the space of n -vector fields that are homogeneous of degree k horizontally andof degree l vertically. This space is trivial if k < − n or l < − n , while X n ( D ) [ − n, − n ] ∼ = Γ( ∧ n E ∗ )is identified with Γ core ( ∧ nA D, A ) and also with Γ core ( ∧ nB D, B ). Furthermore, we have canonicalisomorphisms Γ lin ( ∧ pA D, A ) ∼ = X p ( D ) [1 − p, − p ] , Γ lin ( ∧ qB D, B ) ∼ = X q ( D ) [ − q, − q ] . The first isomorphism is compatible with the left module structure over Γ( ∧ E ∗ ), the secondisomorphism with the right module structure, realized as wedge product of the correspondingmultivector fields from the left or right, respectively. HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 25
Proposition 5.4.
With the above identifications, the pairing (43) is given by the Schoutenbracket h x, y i E ∗ = [[ x, y ]] for all x ∈ X p ( D ) [1 − p, − p ] and y ∈ X q ( D ) [ − q, − q ] .Proof. The Schouten bracket of elements λ ∈ X n ( D ) [ − n, − n ] with x ∈ X p ( D ) [1 − p, − p ] or with y ∈ X q ( D ) [ − q, − q ] is zero, for degree reasons. Hence, the derivation property of the Schoutenbracket shows that [[ x, y ]] is Γ( ∧ E ∗ )-bilinear, for the left module structure on X p ( D ) [1 − p, − p ] and the right module structure on X q ( D ) [ − q, − q ] . The pairing h x, y i E ∗ has the same bilinearityproperty. It therefore suffices to prove the formula for p, q ≤
1. If p = q = 1, we are dealing withthe pairing of vector fields X ∈ X ( D ) [0 , − = Γ( b B ) and Y ∈ X ( D ) [ − , = Γ( b A ), and the claimwas already noted in Section 3.3. If p = 0 , q = 1 we have x = α ∈ Γ( A ∗ ) , y = b a ∈ Γ( b A ) withthe pairing h α, b a i E ∗ = − α ( a ). After identification of b a with a vector field Y ∈ X ( D ) [ − , and α with a function f ∈ C ∞ ( D ) [1 , , this coincides with L Y f = − [[ f, Y ]], as required. Similarly,for p = 1 , q = 0 we have x = b b ∈ Γ( b B ) and y = β ∈ Γ( B ∗ ), with pairing h b b, β i A ∗ = β ( b ),which, after identification of b b with a vector field X ∈ X ( D ) [0 , − and β with a function g ∈ C ∞ ( D ) [0 , , coincides with − L X g = − [[ X, g ]]. (cid:3) Poisson double vector bundles
Reminder on Poisson vector bundles.
Given a vector bundle p : V → M , one knowsthat the following structures are equivalent:(i) a linear Poisson structure π on V → M ,(ii) a degree − {· , ·} on the algebra of polynomial functions on V ,(iii) a Lie algebroid structure on the dual bundle, V ∗ ⇒ M ,(iv) a degree − Schouten bracket ) on Γ( ∧ V ∗ ),(v) a degree 1 differential d CE on Γ( ∧ V ).Here (and from now on) we write A ⇒ M to indicate a Lie algebroid over M ; the notation(which we learned from [6]) suggests the differentiation of a Lie groupoid G ⇒ M when sourceand target become ‘infinitesimally close’. Let us briefly recall how these equivalences comeabout. Given a linear Poisson tensor π on V , the corresponding Poisson bracket {· , ·} on C ∞ ( V ) restricts to a bracket on the space of polynomial functions on V , and is uniquelydetermined by this restriction. The Poisson bivector π being linear is equivalent to the bracketof linear functions being again linear, thus to {· , ·} having degree −
1. Hence, (i) ⇔ (ii). ThePoisson bracket is in fact already determined by its restriction to linear and basic functionson V . Using the identification of linear functions with sections σ ∈ Γ( V ∗ ), this gives theequivalence (ii) ⇔ (iii), where the Lie bracket and anchor are expressed by(47) [[ σ , σ ]] = { σ , σ } , p ∗ ( L a ( σ ) f ) = { σ, p ∗ ( f ) } . This Lie algebroid bracket extends to a Schouten bracket on the algebra Γ( ∧ V ∗ ), with[[ σ, p ∗ f ]] = p ∗ ( L a ( σ ) f ) as the bracket between generators of degrees 1 and 0, hence (iii) ⇔ (iv).The Chevalley-Eilenberg differential d CE on Γ( ∧ V ) is the unique degree 1 derivation such that(48) ι ( σ )d CE f = L a ( σ ) ( f ) We shall directly regard σ as a linear function, rather than using our earlier notation φ σ . for f ∈ C ∞ ( M ) and σ ∈ Γ( V ∗ ), and such that(49) [ ι ( σ ) , [ ι ( σ ) , d CE ]] = ι ([[ σ , σ ]])for all σ , σ ∈ Γ( V ∗ ); one can recover the bracket and anchor from these identities, giving theequivalence (iii) ⇔ (v).6.2. Results for Poisson double vector bundles.
We are interested in the counterpartsof these correspondences for double vector bundles. A Poisson bivector field π ∈ X ( D ) on adouble vector bundle is called double-linear if it is linear for both vector bundle structures, i.e.,homogeneous of bidegree ( − , − π is called a Poisson double vector bundle . Theorem 6.1.
Let D be a double vector bundle with sides A, B and with core( D ) = E ∗ . Thefollowing are equivalent. (i) a double-linear Poisson structure π on D → M , (ii) a bidegree ( − , − Poisson bracket {· , ·} on the algebra S ( D ) of double polynomials, (iii) a VB -algebroid structure on D ′ over B , (iv) a VB -algebroid structure on D ′′ over A , (v) a Lie algebroid structure on b E , together with representations on A ∗ and B ∗ , and aninvariant bilinear pairing A ∗ × M B ∗ → R , with certain compatibility conditions (cf. The-orem 6.8 below), (vi) a bidegree ( − , − Gerstenhaber bracket on the Weil algebra W ( D ) , (vii) a bidegree (0 , differential d ′ v on the Weil algebra W ( D ′ ) , (viii) a bidegree (1 , differential d ′′ h on the Weil algebra W ( D ′′ ) . Some of these equivalences are already known: Given π , the corresponding Poisson bracket {· , ·} on C ∞ ( D ) restricts to the subalgebra S ( D ) of double-polynomial functions, and isuniquely determined by this restriction (since the differentials of functions in this subalgebraspan the cotangent bundle everywhere). The bivector π being double-linear means preciselythat this Poisson bracket has bidegree ( − , − ⇔ (ii). The equivalence with (iii),(iv) is due to Mackenzie [49] (see also [6]): Regarding D as a vector bundle over B , and us-ing the nondegenerate pairing D × B D ′ → R from (11), the Poisson structure π determinesa Lie algebroid structure D ′ ⇒ B . The bivector field π being linear in the vertical direction D → A implies that the horizontal scalar multiplication on D ′ is by Lie algebroid morphisms,which shows that D ′ is a VB -algebroid. Similarly, from the pairing D ′′ × A D → R we obtain a VB -algebroid structure on D ′′ ⇒ A . We depict these VB -algebroid structures on D ′ , D ′′ by(50) D ′ / / (cid:11) (cid:19) E (cid:11) (cid:19) B / / M D ′′ + (cid:15) (cid:15) A (cid:15) (cid:15) E + M where the double arrow indicates Lie algebroid directions. In particular, we see that the bundle E becomes a Lie algebroid E ⇒ M . (The Lie algebroid structures on E coming from the VB -algebroid structures on D ′ , D ′′ coincide; indeed, we will see below that both are induced froma VB -algebroid structure b E → M .) The characterizations (v), (vi), (vii) will be consequencesof Theorems 6.8, 6.11, and 7.2 below, while (viii) is obtained by applying (vii) to the flip of D . HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 27
Examples of Poisson double vector bundles.
As a preparation for the general situ-ation, let us consider some special cases.
Example . Any Poisson vector bundle V → M can be seen as a Poisson double vector bundlewith zero side bundles, thus A = B = M, E = V ∗ . In this case, D = V / / (cid:15) (cid:15) M (cid:15) (cid:15) M / / M D ′ = V ∗ / / (cid:11) (cid:19) V ∗ (cid:11) (cid:19) M / / M D ′′ = V ∗ + (cid:15) (cid:15) M (cid:15) (cid:15) V ∗ + M Example . Suppose D is a Poisson double vector bundle for which the side bundle A is zero.Then D = B × M E ∗ / / (cid:15) (cid:15) B (cid:15) (cid:15) M / / M D ′ = B × M E / / (cid:11) (cid:19) E (cid:11) (cid:19) B / / M D ′′ = E × M B ∗ + (cid:15) (cid:15) M (cid:15) (cid:15) E + M Here D ′ ⇒ B is the action Lie algebroid for a representation of E ⇒ M on B , and thesecond diagram describes D ′′ ⇒ M as the semi-direct product Lie algebroid for the dual E -representation on B ∗ . Example . Suppose D is a vacant Poisson double vector bundle, that is, with a zero core: D = A × M B / / (cid:15) (cid:15) B (cid:15) (cid:15) A / / M D ′ = A ∗ × M B / / (cid:11) (cid:19) M (cid:11) (cid:19) B / / M D ′′ = A × M B ∗ + (cid:15) (cid:15) A (cid:15) (cid:15) M + M We claim that double-linear Poisson structures π on D = A × M B are equivalent to bilinearpairings ( · , · ) : A ∗ × M B ∗ → R . To see this, note that the bigraded algebra S ( D ) is generatedby S , ( D ) = C ∞ ( M ), S , ( D ) = Γ( A ∗ ), and S , ( D ) = Γ( B ∗ ). Given π , it follows for degreereasons that the only non-trivial Poisson bracket of generators are between α ∈ Γ( A ∗ ) and β ∈ Γ( B ∗ ); the resulting pairing ( α, β ) = { α, β } is C ∞ ( M )-linear by the derivation property.Conversely, given the pairing, we define a bi-derivation by letting { α, β } = ( α, β ), and settingall other brackets between generators equal to zero. This bi-derivation satisfies the Jacobiidentity, since triple brackets between generators are always zero. This proves the claim. TheLie algebroid structure on D ′ is that of an action Lie algebroid for the translation action of A ∗ on B , given by the map Γ( A ∗ ) → Γ( B ) = X ( B ) [ − , α ( α, · ), and similarly for D ′′ . Since E = 0, we have b E = A ∗ ⊗ B ∗ . The sections of this bundle have a Lie bracket, coming from itsidentification with double-linear functions on D :[ α β , α β ] ≡ { α β , α β } = ( α , β ) α β − ( α , β ) α β ;thus b E becomes a Lie algebroid with zero anchor. Likewise, the Poisson bracket of suchfunctions with α ∈ Γ( A ∗ ) or β ∈ Γ( B ∗ ) defines representations of this Lie algebroid on A ∗ , B ∗ ,respectively. Example . Suppose that E ⇒ M is a Lie algebroid, together with representations ∇ A ∗ , ∇ B ∗ on A ∗ , B ∗ , and that ( · , · ) : A ∗ × M B ∗ → R is a bilinear pairing that is E -invariant in the sensethat ( ∇ A ∗ e α, β ) + ( α, ∇ B ∗ e β ) = L a ( e ) ( α, β )for α ∈ Γ( A ∗ ) , β ∈ Γ( B ∗ ) , e ∈ Γ( E ). Then D = A × M B × M E ∗ becomes a Poisson doublevector bundle, with the non-zero brackets on generators given as { α, β } = ( α, β ) , { e, α } = ∇ A ∗ e α, { e, β } = ∇ B ∗ e β, { e , e } = [[ e , e ]] , { e, f } = L a ( e ) f, for f ∈ C ∞ ( M ) = S , ( D ), α ∈ Γ( A ∗ ) = S , ( D ), β ∈ Γ( B ∗ ) = S , ( D ), and e, e , e ∈ Γ( E ) ⊆ S , ( D ). The Jacobi identity and the biderivation property of {· , ·} follow from the definitionof Lie algebroids and their representations, together with the invariance of the pairing ( · , · ).6.4. The Lie algebroid structure on b E . In this section, we will concentrate on the char-acterization (v) of Poisson double vector bundles from Theorem 6.1. Suppose D is a Pois-son double vector bundle, with corresponding Poisson bracket {· , ·} . Recall that the algebra S ( D ) = L S r,s ( D ) is generated by S , ( D ) = Γ( b E ) , S , ( D ) = Γ( A ∗ ) , S , ( D ) = Γ( B ∗ ) , S , ( D ) = C ∞ ( M );we will use these identifications without further comment, and for example think of α ∈ Γ( A ∗ ) as a function on D . The Poisson bracket gives bilinear maps S r,s ( D ) × S r ′ ,s ′ ( D ) → S r + r ′ − ,s + s ′ − ( D ), and is uniquely determined by the resulting maps on generators,[[ · , · ]] : S , ( D ) × S , ( D ) → S , ( D ) , [[ b e , b e ]] = { b e , b e } , (51) a : S , ( D ) × S , ( D ) → S , ( D ) , a ( b e, f ) = { b e, f } , (52) ∇ A ∗ : S , ( D ) × S , ( D ) → S , ( D ) , ∇ A ∗ b e α = { b e, α } , (53) ∇ B ∗ : S , ( D ) × S , ( D ) → S , ( D ) , ∇ B ∗ b e β = { b e, β } , (54) ( · , · ) : S , ( D ) × S , ( D ) → S , ( D ) , ( α, β ) = { α, β } (55)(the other brackets between generators are zero, for degree reasons). Lemma 6.6.
The formulas (51) – (55) define a Lie algebroid structure on b E , with representa-tions on A ∗ , B ∗ , and with an invariant bilinear pairing ( · , · ) : A ∗ × M B ∗ → R .Proof. The derivation property of the Poisson bracket shows that (52) is C ∞ ( M )-linear in thefirst argument and satisfies a Leibniz rule in the second argument; hence that a ( b e ) = a ( b e, · )comes from a bundle map a : b E → T M . The Jacobi identity for {· , ·} implies that (51) is aLie bracket on Γ( b E ), and the derivation property for {· , ·} shows that [[ · , · ]] satisfies the Leibnizrule for the anchor map a , hence that b E is a Lie algebroid. Further applications of the Jacobiidentity and derivation property of {· , ·} show that ∇ A ∗ , ∇ B ∗ are representations of the Liealgebroid b E on A ∗ , B ∗ , and that the pairing ( · , · ) is b E -invariant. (cid:3) Remark . The Lie algebroid structure b E ⇒ M , and its action on A ∗ , B ∗ were first observedby Gracia-Saz and Mehta, [25, Section 4.3] in terms of the VB -algebroid D ′ ⇒ B and theidentification Γ lin ( D ′ , B ) = Γ( b E ). In this approach, the Lie algebroid bracket of b E is therestriction of the bracket [[ · , · ]] D ′ to linear sections, the representation on A ∗ is the bracket of HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 29 linear sections with Γ core ( D ′ , B ) = Γ( A ∗ ). The representation of b E on B ∗ (respetively, thepairing ( · , · ) between A ∗ and B ∗ ) are given by the anchor a D ′ on linear (respectively, core)sections, applied to Γ( B ∗ ) viewed as linear functions on B .Similarly, we can describe the data (51)–(55) in terms of the VB -algebroid structure D ′′ ⇒ A .The Lie algebroid representations of b E on A ∗ , B ∗ and the bilinear form satisfy certain com-patibility conditions. Recall from Example 6.4 that the pairing ( · , · ) : A ∗ × M B ∗ → R defines aLie algebroid structure on A ∗ ⊗ B ∗ , with zero anchor, and that this Lie algebroid comes withnatural representations on A ∗ , B ∗ . The data for b E must ‘extend’ these data for its subbundle i b E ( A ∗ ⊗ B ∗ ): Theorem 6.8.
Let D be a double vector bundle. A Lie algebroid structure on the bundle b E → M , together with Lie algebroid representations on A ∗ and B ∗ and an invariant bilinearpairing ( · , · ) : A ∗ × M B ∗ → R , defines a double-linear Poisson structure on D if and only if thefollowing compatibility conditions are satisfied: (i) The image of i b E : Γ( A ∗ ⊗ B ∗ ) ֒ → Γ( b E ) is a Lie algebra ideal (in particular, i b E ( A ∗ ⊗ B ∗ ) is a Lie subalgebroid of b E ), (ii) the b E -representations on A ∗ , B ∗ extend those of its Lie subalgebroid i b E ( A ∗ ⊗ B ∗ ) , (iii) the b E -representation on i b E ( A ∗ ⊗ B ∗ ) is the tensor product of those on A ∗ , B ∗ . Condition (i) determines a Lie algebroid structure on E , in such a way that0 → A ∗ ⊗ B ∗ i b E −→ b E → E → Proof.
Throughout, we denote by α, α sections of A ∗ , by β, β sections of B ∗ , and by b e, b e sections of b E . Suppose first that a double-linear Poisson structure on D is given, determiningthe Lie algebroid structure on b E , representations on A ∗ , B ∗ , and a pairing ( · , · ). On the levelof sections, the inclusion i b E is just the multiplication map α ⊗ β αβ . Thus (i), (iii) amountto the derivation property { b e, αβ } = { b e, α } β + α { b e, β } , while (ii) corresponds to { αβ, α } = − ( α , β ) α and { αβ, β } = ( α, β ) β .Conversely, suppose (i),(ii),(iii) are satisfied. Recall from Proposition 3.2 that D is a sub-double vector bundle of b D = A × M B × M b E ∗ . The formulas of Example 6.5 (with E replaced by b E ) define a double-linear Poisson structure on b D . We will show that D is a Poisson submanifoldof b D , and hence is a Poisson double vector bundle. The ideal I ⊆ S ( b D ) of double-polynomialfunctions vanishing on D is generated by functions of the form(56) αβ − i b E ( α ⊗ β )with α ∈ Γ( A ∗ ) , β ∈ Γ( B ∗ ). To show that I is an ideal for the Poisson bracket, it suffices toshow that the Poisson bracket of functions (56) with any of the generators lies in the ideal. For b e ∈ Γ( b E ) we have(57) { b e, αβ − i b E ( α ⊗ β ) } = ( ∇ A ∗ b e α ) β − i b E ( ∇ A ∗ b e α ⊗ β ) + α ∇ B ∗ b e β − i b E ( α ⊗ ∇ B ∗ b e β ) ∈ I , where we used (i) and (iii). For α ∈ Γ( A ∗ ) we compute(58) { α , αβ − i b E ( α ⊗ β ) } = α ( α , β ) + ∇ A ∗ i b E ( α ⊗ β ) α = 0 , where we used (ii). A similar argument applies to generators β ∈ Γ( B ∗ ). Finally, for f ∈ C ∞ ( M )(59) { f, αβ − i b E ( α ⊗ β ) } = L a ( i b E ( α ⊗ β )) ( f ) = 0since a ◦ i b E = 0 by (i). (cid:3) Remark . As pointed out by one of the referees, the necessity of the conditions (i),(ii),(iii)may also be found in Section 4 of the paper [25]. The sufficiency is not noted there, but canbe proved using similar techniques. Luca Vitagliano has remarked that Theorem 6.8 can alsobe obtained as a consequence of [17, Theorem 2.33].Let us note the following consequences of the discussion above:
Proposition 6.10.
Let D be a Poisson double vector bundle, with side bundles A, B and with E = core( D ) ∗ . Then (a) A × M B inherits a double-linear Poisson structure, with ϕ : D → A × M B a Poissonmap. (b) The subbundle core( D ) is a Poisson-Dirac submanifold of D : every smooth function onthe core extends to a smooth function on D with Hamiltonian vector field tangent to thecore. (c) b D = A × M B × M b E ∗ acquires a double-linear Poisson structure, such that D ⊆ b D is aPoisson submanifold.Proof. For (a), observe that the image of the pullback map ϕ ∗ : S ( A × M B ) → S ( D ) is the sub-algebra generated by Γ( A ∗ ) , Γ( B ∗ ); by the bracket relations (51)–(55) it is a Poisson subalgebra.Part (c) is contained in the proof of Theorem 6.8. For (b), note that it is enough to prove theanalogous statement for b D , since D is a Poisson submanifold and core( D ) = D ∩ core( b D ). Butfunctions on b E ∗ extend canonically to functions on b D , by taking the pullback under b D → b E ∗ .The vanishing ideal of b E ∗ is generated by Γ( A ∗ ) , Γ( B ∗ ) ⊆ C ∞ ( b D ), and is preserved underPoisson brackets with pullbacks of functions on b E ∗ . This means that the Hamiltonian vectorfields of the latter are tangent to b E ∗ . (cid:3) Gerstenhaber brackets.
Our next aim is to interpret double-linear Poisson structureson D in terms of a ‘Gerstenhaber’ bracket on the Weil algebra W ( D ), as in item (vi) of Theorem6.1. We make the following definitions. Let A be a bigraded commutative superalgebra. A bidegree ( − , − Gerstenhaber bracket on A is a bilinear map [[ · , · ]] : A × A → A of bidegree( − , − A [1 ,
1] (i.e., the space A with bidegrees shifted down by (1 , x ∈ A p,q the map [[ x, · ]] is a superderivation of bidegree( p − , q −
1) of the algebra structure on A : In particular[[ x, y ]] = − ( − | x || y | [[ y, x ]] , [[ x, yz ]] = [[ x, y ]] z + ( − | x || y | y [[ x, z ]] . From now on, we we will omit the explicit mention of the bidegree ( − , − , · , · ]] for Gerstenhaber HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 31 brackets on bigraded superalgebras, to avoid confusion with various other Lie brackets andcommutators.
Theorem 6.11.
A double-linear Poisson structure π on a double vector bundle D is equivalentto a Gerstenhaber bracket [[ · , · ]] on the Weil algebra W ( D ) .Proof. The argument is similar to that for Theorem 6.8, hence we will be brief. Note that for r, s ≤
1, the spaces W r,s ( D ) coincide with S r,s ( D ): W , ( D ) = Γ( b E ) , W , ( D ) = Γ( A ∗ ) , W , ( D ) = Γ( B ∗ ) , W , ( D ) = C ∞ ( M ) . A Gerstenhaber bracket on W ( D ) gives bilinear maps W r,s ( D ) × W r ′ ,s ′ ( D ) → W r + r ′ − ,s + s ′ − ( D ). The following formulas define a Lie algebroid structure on b E , togetherwith representations of this Lie algebroid on A ∗ , B ∗ , and a bilinear pairing ( · , · ) between A ∗ and B ∗ :(60) [[ b e , b e ]] = [[ b e , b e ]] , [[ b e, f ]] = L a ( b e ) ( f ) , (61) [[ b e, α ]] = ∇ A ∗ b e α, [[ b e, β ]] = ∇ B ∗ b e β, [[ α, β ]] = − ( α, β ) . Note that these formulas are obtained from (51)–(55) by replacing Poisson brackets with Ger-stenhaber brackets, except for an extra minus sign in the last formula. The various compati-bilities in Theorem 6.8 follow from the Jacobi identity and derivation property of [[ · , · ]].Conversely, given a double-linear Poisson structure π and the associated data from Theorem6.8, we obtain a Gerstenhaber bracket as follows. Consider the Poisson double vector bundle b D = A × M B × M b E ∗ (cf. Example 6.5). On the super-commutative algebra W ( b D ), we definea super-biderivation [[ · , · ]] of bidegree ( − , − S ( D ), this Gerstenhaber bracket descends to W ( D ) = W ( b D ) / ∼ . In repeating the calculations(57) – (59), the second equation encounters a minus sign since [[ α , αβ ]] = − ( α , β ) α in contrastto { α , αβ } = ( α , β ) α ; this is compensated by the extra sign in the last equation of (61). (cid:3) Differentials.
Suppose D is a Poisson double vector bundle. The corresponding VB -algebroid structure D ′ ⇒ B (dual to the Poisson vector bundle D → B ) gives a Chevalley-Eilenberg differential d D ′ on Γ( ∧ • B D, B ). Since d D ′ commutes with the scalar multiplication κ vt on D and on B , it restricts to a differential on core and linear sections of ∧ B D over B .Since d D ′ is a derivation with respect to the wedge product, we see that the core sectionsbecome a differential graded algebra, and the linear sections a differential graded module overthis differential graded algebra.On the other hand, recall from Proposition 5.3 that the linear and core sections of ∧ B D over B are identified with W , • ( D ′ ) and W , • ( D ′ ), respectively. Hence, a bidegree (0 ,
1) differentialon W ( D ′ ) again restricts to differentials on the core and linear sections, making them intodifferential graded algebras and differential graded modules, respectively. Hence, to prove thecharacterization of double-linear Poisson structures on D in terms of differentials on W ( D ′ ), itsuffices to show that we can reverse these constructions. Proposition 6.12.
Suppose Γ core ( ∧ • B D, B ) and Γ lin ( ∧ • B D, B ) are equipped with differentials d , for which they are a differential graded algebra and differential graded module respectively.Then there are unique extensions of d to (a) a bidegree (0 , differential d ′ v on the algebra W ( D ′ ) , (b) a degree differential d D ′ on the algebra Γ( ∧ B D, B ) ,as superderivations for the algebra structures.Proof. (a) By definition, W ( D ) is generated by elements in bidegree ( i, j ) with 0 ≤ i, j ≤ ′ v x = d x whenever x is one of these generators. To show that this definition ex-tends as a superderivation , we have to verify that it is compatible with the relations be-tween generators. The defining relation of W ( D ′ ) (aside from super-commutativity and C ∞ ( M )-linearity) is that for β ∈ Γ( B ∗ ) = Γ lin ( ∧ B D, B ) and ε ∈ E ∗ = Γ core ( ∧ BD, B ),the linear section i b A ( β ⊗ ε ) of b A = Γ lin ( ∧ B D, B ) coincides with the product βε . Thus,we need that d( i b A ( β ⊗ ε )) = (d β ) ε − β (d ε ) . But the linear section i b A ( β ⊗ ε ) is simply the product of the linear function β withthe core section ε ; hence the desired identity follows from the assumption that linearsections are a differential graded module over the core sections.(b) The algebra of sections of ∧ B D over B has a subalgebraΓ pol ( ∧ B D, B ) = M m,n ≥ Γ( ∧ nB D, B ) [ − n + m ] of polynomial sections , in the notation of Section 5.1. It is a super-commutative bigradedalgebra, with bigrading given by m, n , and with the for the Z -grading given as the mod2 reduction of n . It is generated by its components in degree m, n ≤
1, which coincidewith those for W ( D ′ ), and the relations between these generators are the same as for W ( D ′ ), with the exception that the relation β β = − β β for β , β ∈ Γ( B ∗ ) getsreplaced with β β = β β . The same argument as for W ( D ′ ) shows that d extends toa superderivation d D ′ of Γ pol ( ∧ B D, B ). By [25, Theorem 3.15], the latter determinesa Lie algebroid structure on D ′ over B , which, in turn, extends the differential to allsections of ∧ B D over B . (cid:3) Remark . In [25], the double complex Γ pol ( ∧ B D, B ) is denoted Ω • , • ( D ′ ). As explainedthere, it may be regarded as the space of double-polynomial functions on the supermanifold D ′ [1 , pol ( ∧ • B D, B ) [ k −• ] is the space of k -homogeneous cochainsfor the VB -algebroid D ′ ⇒ B .7. Relationships between brackets, differentials, and pairings
Throughout, D will denote a Poisson double vector bundle, with side bundles A, B and withcore( D ) ∗ = E . Equivalently, the double vector bundles D ′ , D ′′ are VB -algebroids D ′ ⇒ B , D ′′ ⇒ A , respectively. In the last section, we saw how these structures are equivalent toa Gerstenhaber bracket [[ · , · ]] on W ( D ), a vertical differential d ′ v on W ( D ′ ), and a horizontaldifferential d ′′ h on W ( D ′′ ). These data interact in many ways, via the contraction operators andthe various pairings between the Weil algebras. HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 33
Differential and contractions on W ( D ′ ) . Thinking of D as the dual bundle of D ′ over B , using once again the duality pairing (10), we have the usual contraction operators ι D ′ ( σ )on Γ( ∧ B D, B ), for any σ ∈ Γ( D ′ , B ). On the other hand, sections b e ∈ Γ( b E ) ∼ = Γ lin ( D ′ , B )define contraction operators ι ′ v ( b e ) of bidegree (0 , −
1) of W ( D ′ ), while sections α ∈ Γ( A ∗ ) definecontraction operators ι ′ v ( α ) of bidegree ( − , −
1) on W ( D ′ ). Recall again the identifications W , • ( D ′ ) ∼ = Γ lin ( ∧ B D, B ) and W , • ( D ′ ) ∼ = Γ core ( ∧ B D, B ). Checking on generators, one verifies:
Lemma 7.1. (a)
The isomorphism Γ( b E ) ∼ = Γ lin ( D ′ , B ) identifies the contraction operators ι ′ v ( b e ) : W p, • ( D ′ ) → W p, • ( D ′ ) for p = 0 , with the operator ι D ′ ( b e ) on core sections andlinear sections of ∧ B D . (b) The isomorphism Γ( A ∗ ) ∼ = Γ core ( D ′ , B ) identifies ι ′ v ( α ) : W , • ( D ′ ) → W , • ( D ′ ) with theoperator ι D ′ ( α ) : Γ lin ( ∧ B D, B ) → Γ core ( ∧ B D, B ) . Using these facts, we obtain:
Proposition 7.2.
The derivation d ′ v satisfies [ ι ′ v ( b e ) , [ ι ′ v ( b e ) , d ′ v ]] = ι ′ v ([[ b e , b e ]]) , (62) [ ι ′ v ( b e ) , [ ι ′ v ( α ) , d ′ v ]] = − ι ′ v ( ∇ A ∗ b e α ) , (63) [ ι ′ v ( α ) , [ ι ′ v ( α ) , d ′ v ]] = 0 , (64) for b e, b e , b e ∈ Γ( b E ) , α, α , α ∈ Γ( A ∗ ) . Furthermore, (65) ι ′ v ( b e )d ′ v f = L a ( b e ) f, ι ′ v ( b e )d ′ v β = ∇ B ∗ b e β, (66) ι ′ v ( α )d ′ v f = 0 , ι ′ v ( α )d ′ v β = − ( α, β ) , for all all b e ∈ Γ( b E ) , α ∈ Γ( A ∗ ) , f ∈ C ∞ ( M ) , β ∈ Γ( B ∗ ) .Proof. Equations (62)–(64) are equalities of derivations; hence it suffices to check that bothsides agree on W , • ( D ′ ) , W , • ( D ′ ). Since the identifications of these spaces with core and linearsections of ∧ B D takes ι ′ v , d ′ v to the contractions and Lie algebroid differential of Γ( ∧ B D, B ), andsince the right hand sides can be expressed in terms of Lie algebroid brackets (see Remark 6.7),the three equalities (62)–(64) correspond to the formula (49) for the bracket of Lie algebroids.Similarly, (65), (66) correspond to the formula (48) for the anchor of a Lie algebroid. (cid:3)
Relationship between Gerstenhaber bracket and differential.
The Gerstenhaberbracket on W ( D ) restricts to a bracket on W , • ( D ), making the latter into a graded Liealgebra with a representation x [[ x, · ]] on W , • ( D ) = Γ( ∧ B ∗ ). Likewise, W • , ( D ) is agraded Lie algebra with a representation on W • , ( D ) = Γ( ∧ A ∗ ) by graded superderivations x [[ x, · ]]. Proposition 7.3.
For x ∈ W , • ( D ) and y ∈ W , • ( D ) = Γ( ∧ B ∗ ) = W • , ( D ′ ) , (67) ι ′ v ( x )d ′ v y = [[ x, y ]] . Similarly, for x ∈ W , • ( D ) and y ∈ W , • ( D ) = Γ( ∧ B ∗ ) = W • , ( D ′ ) , (68) ι ′′ h ( x )d ′′ h y = [[ x, y ]] . Proof.
For x ∈ W ,q ( D ), both [[ x, · ]] and [ ι ′ v ( x ) , d ′ v ] define superderivations of degree q − ∧ B ∗ ). Since ι ′ v ( x ) vanishes on W • , ( D ′ ), Equation (67) amounts to the equality of these twosuperderivations. It suffices to verify this on generators y of Γ( ∧ B ∗ ), given by f ∈ C ∞ ( M ) or β ∈ Γ( B ∗ ). Furthermore, since ι ′ v is a left Γ( ∧ B ∗ )-module morphism, we only need to considerthe cases that x is a generator W , • ( D ), given by α ∈ Γ( A ∗ ) or b e ∈ Γ( b E ). These verificationsare as follows, using (66):[[ α, f ]] = 0 = ι ′ v ( α )d ′ v f, [[ α, β ]] = − ( α, β ) = ι ′ v ( α )d ′ v β, [[ b e, f ]] = L a ( b e ) f = ι ′ v ( b e )d ′ v f, [[ b e, β ]] = ∇ B ∗ b e β = ι ′ v ( b e )d ′ v β. Equation (68) may be proved along similar lines, or obtained from (67) by using the flipoperation. (cid:3)
Note that (67) can be written in various other ways:(69) [[ x, y ]] = − ( − ( | x | +1) | y | h d ′ v y, x i B ∗ = − ( − ( | x | +1) | y | ι h (d ′ v y ) x. Proposition 7.4.
For x , x ∈ W , • ( D ) , (70) [ ι ′ v ( x ) , [ ι ′ v ( x ) , d ′ v ]] = ( − | x | ι ′ v ([[ x , x ]]) For x , x ∈ W • , ( D ) , (71) [ ι ′′ h ( x ) , [ ι ′′ h ( x ) , d ′′ h ]] = ( − | x | ι ′′ h ([[ x , x ]]) . Proof.
Equation (70) holds for x i = b e i ∈ Γ( b E ) by (62) and for x = b e ∈ Γ( b E ) , x = α ∈ Γ( A ∗ )by (63). Since both sides change sign by ( − ( | x | +1)( | x | +1) under interchange of x , x , thisestablishes the identity for generators. The general case follows by induction: the statementfor x , x implies that for x , βx with β ∈ Γ( B ∗ ), as follows:[ ι ′ v ( x ) , [ ι ′ v ( β x ) , d ′ v ]] = (cid:2) ι ′ v ( x ) , β [ ι ′ v ( x ) , d ′ v ] + ( − | x | +1 (d ′ v β ) ι ′ v ( x ) (cid:3) = ( − | x | +1 β [ ι ′ v ( x ) , [ ι ′ v ( x ) , d ′ v ]] + ( − | x | +1 ( ι ′ v ( x )d ′ v β ) ι ′ v ( x )= ( − | x | + | x | +1 ι ′ v ( β [[ x , x ]]) + ( − | x | +1 ι ′ v ([[ x , β ]] x )= ( − | x | +1 ι ′ v [[ x , βx ]] . The arguments for d ′′ h , ι ′′ h ( x ) are analogous; alternatively, one can use the flip operation. (cid:3) For later reference, observe the following consequence of Proposition 7.4.
Corollary 7.5.
For φ ∈ W , • ( D ′′ ) , and x i ∈ W • , ( D ) , ( − | x | ι ′′ h ( x ) ι ′′ h ( x )d ′′ h φ = ι ′′ h ([[ x , x ]]) φ − [[ x , ι ′′ h ( x ) φ ]] + ( − | x || x | [[ x , ι h ( x ) φ ]] . Proof.
The left hand side ( − | x | ι ′′ h ( x ) ι ′′ h ( x )d ′′ h φ may be written as a sum of three terms,( − | x | [ ι ′′ h ( x ) , [ ι ′′ h ( x ) , d ′′ h ]] φ − ι ′′ h ( x ) d ′′ h ι ′′ h ( x ) φ + ( − | x | | x | ι ′′ h ( x ) d ′′ h ι ′′ h ( x ) φ. By Proposition 7.4, the first term is ι ′′ h ([[ x , x ]]) φ . For the second term, note that ι ′′ h ( x ) φ ∈ W , • ( D ′′ ) = Γ( ∧ A ∗ ). But on sections of ∧ A ∗ , the composition ι ′′ h ( x ) ◦ d ′′ h acts as [[ x , · ]], againusing Proposition 7.4. Hence, the second term is − [[ x , ι ′′ h ( x ) φ ]]. The third term is dealt withsimilarly. (cid:3) HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 35
Relationships between the differentials.
The differential d ′ v on W ( D ′ ) restricts to adifferential on W , • ( D ′ ) = Γ lin ( ∧ • B D, B ) ≡ Γ lin ( ∧ • B ( D ′ ) v , B ) . On the other hand, we also have the horizontal differential on W • , ( D ′′ ) = Γ lin ( ∧ A D, A ) ≡ Γ lin ( ∧ • A ( D ′′ ) h , A )coming from the horizontal VB -algebroid structure D ′′ ⇒ A under the identification of D with the flip of the horizontal dual ( D ′′ ) h , it is again the restriction of the Chevalley-Eilenbergdifferential. In other words, it is the space CE • VB ( D ′′ ) in the notation of [9]. The Lie algebroiddifferential d on Γ( ∧ E ∗ ) may be interpreted as d ′ v on W , • ( D ′ ) or as d ′′ h on W • , ( D ′′ ), and both W , • ( D ′ ) and W • , ( D ′′ ) are differential graded modules over this algebra. Proposition 7.6.
The ∧ E ∗ -valued pairing h· , ·i E ∗ (cf. (29) ) satisfies d h x, y i E ∗ = h d ′′ h x, y i E ∗ + ( − | x | +1 h x, d ′ v y i E ∗ , for all x ∈ W p, ( D ′′ ) and y ∈ W ,q ( D ′ ) . Here | x | = p + 1 . Before proving the proposition, recall that for any Poisson manifold (
Q, π ), the Schoutenbracket defines a degree 1 differential on the graded algebra X • ( Q ) of multi-vector fields on Q :[[ π, · ]] : X p ( Q ) → X p +1 ( Q ) . If Q = V is a Poisson vector bundle, thus π is homogeneous of degree −
1, then this differentialpreserves the graded subalgebra X • core ( V ) of core multi-vector fields, as well as the module X • lin ( V ) of linear multi-vector fields. It is well-known that the identification X • core ( V ) = Γ( ∧ • V )intertwines the differential [[ π, · ]] with the Lie algebroid differential for V ∗ ⇒ M . Proof of Proposition 7.6.
As explained in Section 5.4, X q ( D ) [ − q, − q ] ∼ = W ,q ( D ′ ) , X p ( D ) [1 − p, − p ] ∼ = W p, ( D ′′ ) , X n ( D ) [ − n, − n ] ∼ = Γ( ∧ n E ∗ ) . By a check on generators, one finds that the differential d ′ v is realized as − [[ π, · ]], whiled ′′ h , d are realized as [[ π, · ]]. Furthermore, according to Proposition 5.4 the pairing between x ∈ X p ( D ) [1 − p, − p ] and y ∈ X q ( D ) [ − q, − q ] is expressed in terms of the Schouten bracket as h x, y i E ∗ = [[ x, y ]]. The proposition thus translates into the Jacobi identity[[ π, [[ x, y ]]]] = [[[[ π, x ]] , y ]] + ( − | x | [[ x, [[ π, y ]]]] . (cid:3) A consequence of Proposition 7.6 is the following result about contraction operators.
Proposition 7.7.
For x ∈ W p, ( D ′′ ) , we have the following equality of superderivations of W ( D ′ ) , (72) [d ′ v , ι ′ h ( x )] = ι ′ h (d ′′ h x ) . Similarly, for y ∈ W ,q ( D ′ ) we have the equality of superderivations of W ( D ′′ ) , (73) [d ′′ h , ι ′′ v ( y )] = ι ′′ v (d ′ v y ) . Proof.
Both sides of (72) are superderivations of bidegree ( − , p ). Hence, they both vanish on W , • ( D ′ ). On sections y ∈ W ,q ( D ′ ), the identity becomesd ′ v ι ′ h ( x ) y + ( − | x | ι ′ h ( x )d ′ v y = ι ′ h (d ′′ h x ) y. After expressing the horizontal contractions in terms of the pairing h· , ·i E ∗ , this identity readsas d h x, y i E ∗ + ( − | x | h x, d ′ v y i E ∗ = h d ′′ h x, y i E ∗ . which is just the statement of Proposition 7.6. Similarly, the two sides of (73) are superderiva-tions of bidegree ( q, − x ∈ W p, ( D ′′ ), the identity becomesd ′′ h ι ′′ v ( y ) x + ( − | y | ι ′′ v ( y )d ′′ h x = ι ′′ v (d ′ v y ) x, which may be written( − ( | x | +1)( | y | +1) d h x, y i E ∗ + ( − | y | ( − | x | ( | y | +1) h d ′′ h x, y i E ∗ = ( − ( | x | +1) | y | h x, d ′ v y i E ∗ . After multiplying by the sign ( − ( | x | +1)( | y | +1) , this becomesd h x, y i E ∗ − h d ′′ h x, y i E ∗ = ( − | x | +1 h x, d ′ v y i E ∗ which again is a reformulation of Proposition 7.6. (cid:3) Application to vector bundles IV.
We continue the discussion from Sections 3.4, 4.5and 5.3.7.4.1.
Tangent bundle
T V . Recall from 4.5 that W ( T V ) is C ∞ ( M )-linearly generated by dif-ferentials d f , together with sections τ ∈ Γ( V ∗ ) and their jets j ( τ ), subject to the relation j ( f τ ) − f j ( τ ) = τ d f . The VB -algebroid structure T V ⇒ V determines a vertical differen-tial d v on the Weil algebra. This differential is given on the generators byd v ( f ) = d f, d v (d f ) = 0 , d v ( τ ) = − j ( τ ) , d v ( j ( τ )) = 0 . On W , • ( T V ) = Γ( ∧ T ∗ M ), it agrees with the de Rham differential; on W , • ( T V ) =Γ lin ( ∧ V T ∗ V, V ) it is the restriction of the de Rham differential to linear forms. For exam-ple, we may verify that ι v ( δ )d v τ = −∇ ∗ δ τ = −h j ( τ ) , δ i V ∗ = − ι v ( δ ) j ( τ )as required; similarly ι v ( δ )d v f = L a ( δ ) f = −h d f, δ i V ∗ = ι v ( δ )d f .7.4.2. Cotangent bundle T ∗ V . The Weil algebra W ( T ∗ V ) is C ∞ ( M )-linearly generated by sec-tions τ ∈ Γ( V ∗ ) , σ ∈ Γ( V ) , δ ∈ Γ(At( V )), subject to the relation that the product τ σ inthe Weil algebra equals − i At( V ) ( τ ⊗ σ ). Using the Poisson brackets between these generators(viewed as functions τ, φ σ ♯ , φ e a ( δ ) on T ∗ V , as in 3.4), we obtain the formulas for the Gersten-haber bracket,[[ τ, σ ]] = h τ, σ i , [[ δ, σ ]] = ∇ δ σ, [[ δ, τ ]] = ∇ ∗ δ τ, [[ δ, f ]] = L a ( δ ) f, [[ δ , δ ]] = [[ δ , δ ]] . The first equality follows from { τ, φ σ ♯ } = − L σ ♯ τ = −h τ, σ i , the second from { φ e a ( δ ) , φ σ ♯ } = φ [ e a ( δ ) ,σ ♯ ] = φ ( ∇ δ σ ) ♯ , and so on. HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 37 Double Lie algebroids
Definition, basic properties.
The concept of a double Lie algebroid was introduced byMackenzie in [44, 45, 47], as the infinitesimal counterpart to double Lie groupoids. It is givenby a double vector bundle with compatible horizontal and vertical VB -algebroid structures(74) D + (cid:11) (cid:19) B (cid:11) (cid:19) A + M To formulate the compatibility condition, recall that a vertical VB -algebroid structure makes D ′′ into a Poisson double vector bundle; hence D ′ becomes a VB -algebroid horizontally, D ′ + (cid:15) (cid:15) E (cid:15) (cid:15) B + M Similarly, a horizontal VB -algebroid structure on D makes D ′ into a double Poisson vectorbundle, and hence D ′′ becomes a VB -algebroid vertically, D ′′ / / (cid:11) (cid:19) A (cid:11) (cid:19) E / / M The compatibility condition is that the two Lie algebroids D ′ ⇒ E and D ′′ ⇒ E , with theirnatural duality pairing, form a Lie bialgebroid , as defined by Mackenzie-Xu [51]. In the for-mulation of Kosmann-Schwarzbach [33], this means that the Chevalley-Eilenberg differentiald CE on Γ( ∧ E D ′ , E ), defined by the identification of D ′ → E with the dual of the Lie algebroid D ′′ ⇒ E , is a derivation of the Schouten bracket [[ · , · ]] for the Lie algebroid D ′ ⇒ E :(75) d CE [[ λ , λ ]] = [[d CE λ , λ ]] + ( − n − [[ λ , d CE λ ]] , for λ i ∈ Γ( ∧ n i E D ′ , E ) , i = 1 , Examples . (a) Mackenzie arrived at the definition of a double Lie algebroid by applyingthe Lie functor to an LA -groupoid. For instance, any Poisson Lie groupoid G ⇒ M [51], with Lie algebroid A ⇒ M , gives rise to a double Lie algebroid structure on T ∗ A ,by applying the Lie functor to its cotangent Lie algebroid T ∗ G ⇒ A ∗ . Similarly, for a double Lie groupoid [4, 45, 47], applying the Lie functor twice produces a double Liealgebroid.(b) The tangent bundle of a Lie algebroid V ⇒ M is a double Lie algebroid [44, Example4.6] T V + (cid:11) (cid:19) T M (cid:11) (cid:19) V + M One may regard
T V as being obtained by applying the Lie functor to the LA -groupoid V × V ⇒ V (the pair groupoid of V ). (c) Matched pairs of Lie algebroids , due to Lu [41] and Mokri [57], are a generalization ofa matched pair of Lie algebras [54] (also known as double Lie algebras [42] or twilledextensions [35]). Two Lie algebroids A ⇒ M, B ⇒ M , with actions of A on B and of B on A , are a matched pair if the brackets and actions define a Lie algebroid structure onthe direct sum A ⊕ B ⇒ M . Mackenzie [44] proved that matched pairs of Lie algebroidsare equivalent to vacant double Lie algebroids D = A × M B + (cid:11) (cid:19) B (cid:11) (cid:19) A + M i.e. such that core( D ) = 0.8.2. Weil algebra of a double Lie algebroid.
The following theorem gives equivalent for-mulations of Mackenzie’s definition of a double Lie algebroid in terms of the Weil algebras ofthe three double vector bundles
D, D ′ , D ′′ . Theorem 8.2.
Let D be a double vector bundle. The following are equivalent: (a) A double Lie algebroid structure on D ; (b) a Gerstenhaber bracket on the bigraded superalgebra W ( D ′ ) , together with a differential d ′ h of bidegree (1 , that is a derivation of the Gerstenhaber bracket; (c) a Gerstenhaber bracket on the bigraded superalgebra W ( D ′′ ) , together with a differential d ′′ v of bidegree (0 , that is a derivation of the Gerstenhaber bracket; (d) commuting differentials d h , d v on W ( D ) , of bidegrees (1 , and (0 , , respectively, We will break up the proof into several steps. Consider first equivalence (a) ⇔ (c). Lemma 8.3.
A double Lie algebroid structure on D is equivalent to a Gerstenhaber bracket on W ( D ′′ ) , together with a differential d ′′ v of bidegree (0 , that is a derivation of the Gerstenhaberbracket.Proof. As discussed in Section 5.2, there are canonical identifications W , • ( D ′′ ) ∼ = Γ core ( ∧ • E D ′ , E ) , W , • ( D ′′ ) ∼ = Γ lin ( ∧ • E D ′ , E ) . These spaces generate W ( D ′′ ) as an algebra, and also Γ( ∧ E D ′ , E ) as a module over C ∞ ( M ).Furthermore, by definition, the restriction of the Gerstenhaber bracket on W ( D ′′ ) to thesespaces agrees with the Lie algebroid bracket for D ′ ⇒ E , while the differential d ′′ v coincideswith the Chevalley-Eilenberg differential for D ′′ ⇒ E . Hence, d ′′ v being a derivation of theGerstenhaber bracket amounts to the defining compatibility condition of a double Lie algebroid. (cid:3) Theorem 6.1 shows that for a horizontal VB -algebroid structure D ⇒ B on a double vectorbundle D is equivalent to a vertical differential d v on W ( D ), and also to a horizontal differentiald ′ h on W ( D ′ ). By Proposition 7.7, these are related by(76) d v y = d ′ h y, [d v , ι h ( x )] = ι h (d ′ h x ) , for all y ∈ Γ( ∧ • B ∗ ) and all x ∈ W • , ( D ′ ). (The first identity uses that both W , • ( D ) and W • , ( D ′ ) are identified with sections of ∧ B ∗ .) On the other hand, using Theorem 6.1 again, avertical VB -algebroid structure D ⇒ A is equivalent to a horizontal differential d h on W ( D ), HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 39 and also to a Gerstenhaber bracket [[ · , · ]] on W ( D ′ ). According to Propositions 7.3 and 7.4,these are related by(77) ι h ( x ) d h y = [[ x, y ]] , [ ι h ( x ) , [ ι h ( x ) , d h ]] = ( − | x | ι h ([[ x , x ]]) , for all y ∈ Γ( ∧ • B ∗ ) and all x, x , x ∈ W • , ( D ′ ). Consider now the situation that both ahorizontal and a vertical VB -algebroid structure are given. Lemma 8.4.
Given a horizontal VB -algebroid structure D ⇒ B and a vertical VB -algebroidstructure D ⇒ A , the super-commutator [d h , d v ] satisfies (78) ι h ( x )[d h , d v ] y = ( − | x | +1 (cid:0) d ′ h [[ x, y ]] − [[d ′ h x, y ]] − ( − | x | [[ x, d ′ h y ]] (cid:1) for x ∈ W • , ( D ′ ) and y ∈ Γ( ∧ • B ∗ ) , as well as (79) [ ι h ( x ) , [ ι h ( x ) , [d h , d v ]]] = ( − | x | ι h (cid:0) d ′ h [[ x , x ]] − [[d ′ h x , x ]] − ( − | x | [[ x , d ′ h x ]] (cid:1) for x , x ∈ W • , ( D ′ ) .Proof. The two identities follow from the calculations, using (76) and (77), ι h ( x )[d h , d v ] y = ι h ( x )d h d v y + ι h ( x )d v d h y = [[ x, d v y ]] + ( − | x | [d v , ι h ( x )]d h y − ( − | x | d v ι h ( x )d h y = [[ x, d ′ h y ]] + ( − | x | ι h (d ′ h x )d h y − ( − | x | d ′ h [[ x, y ]]= ( − | x | +1 (cid:0) d ′ h [[ x, y ]] − [[d ′ h x, y ]] − ( − | x | [[ x, d ′ h y ]] (cid:1) and ι h (d ′ h [[ x , x ]]) = [d v , ι h ([[ x , x ]])]= ( − | x | [d v , [ ι h ( x ) , [ ι h ( x ) , d h ]]]= ( − | x | [[d v , ι h ( x )] , [ ι h ( x ) , d h ]] − ( − | x | + | x | [ ι h ( x ) , [[d v , ι h ( x )] , d h ]]+ ( − | x | [ ι h ( x ) , [ ι h ( x ) , [d v , d h ]]]= ( − | x | [ ι h (d ′ h x ) , [ ι h ( x ) , d h ]] − ( − | x | + | x | [ ι h ( x ) , [ ι h (d ′ h x ) , d h ]]+ ( − | x | [ ι h ( x ) , [ ι h ( x ) , [d v , d h ]]]= ι h ([[d ′ h x , x ]]) + ( − | x | ι h ([[ x , d ′ h x ]]) + ( − | x | [ ι h ( x ) , [ ι h ( x ) , [d v , d h ]]] . (cid:3) We now have all the tools we need to establish Theorem 8.2:
Proof of Theorem 8.2.
The equivalence (a) ⇔ (c) was already established in Lemma 8.4. Con-sider now the implication (b) ⇒ (d). Using (78) and (79), we see that if d ′ h is a derivation ofthe Gerstenhaber bracket, then ι h ( x )[d h , d v ] y = 0 , [ ι h ( x ) , [ ι h ( x ) , [d h , d v ]]] = 0for all x, x , x ∈ W • , ( D ′ ) and y ∈ Γ( ∧ • B ∗ ) = W • , ( D ′ ). The first equation shows that[d h , d v ] y = 0, so that [d h , d v ] vanishes on W • , ( D ′ ). Using the second equation, and inductionon q , it then follows that [d h , d v ] vanishes on all W • ,q ( D ′ ), hence that d h , d v super-commute. For the reverse implication (d) ⇒ (b), suppose [d h , d v ] = 0. Equations (78) and (79) tell usthatd ′ h [[ x, y ]] − [[d ′ h x, y ]] − ( − | x | [[ x, d ′ h y ]] = 0 , d ′ h [[ x , x ]] − [[d ′ h x , x ]] − ( − | x | [[ x , d ′ h x ]] = 0for all x, x , x ∈ W • , ( D ′ ) and y ∈ W • , ( D ′ ). This means that d ′ h acts as a derivation of theGerstenhaber bracket on generators, and hence in general. We have thus shown (b) ⇔ (d).The equivalence (c) ⇔ (d) follows by applying a flip, which interchanges the horizontal andvertical structures. (cid:3) The core of a double Lie algebroid.
It was pointed out in [44, Section 4] that for anydouble Lie algebroid D , the core E ∗ acquires the structure of a Lie algebroid. This fact maybe seen as a consequence of the fact that the base of any Lie bialgebroid has a natural Poissonstructure [51, Proposition 3.6]. It may also be obtained using the Weil algebras, as follows.Recall that W ( D ′ ) has a vertical differential and W ( D ′′ ) a horizontal differential, which arederivations of the Gerstenhaber brackets on these algebras. Proposition 8.5.
The core E ∗ of a double Lie algebroid D has a Lie algebroid structure, withbracket given in terms of the identification Γ( ∧ E ∗ ) = W • , ( D ′′ ) by [[ ε , ε ]] = [[ ε , d ′′ v ε ]] , L a ( ε ) ( f ) = [[ ε, d ′′ v f ]] , or in terms of the identification Γ( ∧ E ∗ ) = W , • ( D ′ ) by [[ ε , ε ]] = − [[ ε , d ′ h ε ]] , L a ( ε ) ( f ) = − [[ ε, d ′ h f ]] . Proof.
Note that [[ ε , d ′′ v ε ]] = [[d ′′ v ε , ε ]], which implies skew-symmetry of the bracket [[ · , · ]]. TheJacobi identity for [[ · , · ]] follows from that for the Gerstenhaber bracket:[[ ε , [[ ε , ε ]]]] = [[ ε , d ′′ v [[ ε , d ′′ v ε ]]]] = [[ ε , [[d ′′ v ε , d ′′ v ε ]]]]= [[[[ ε , d ′′ v ε ]] , d ′′ v ε ]] + [[d ′′ v ε , [[ ε , d ′′ v ε ]]]] = [[[[ ε , ε ]] , ε ]] + [[ ε , [[ ε , ε ]]]] . Furthermore, if f ∈ C ∞ ( M ),[[ ε , d ′′ v ( f ε )]] = f [[ ε , d ′′ v ε ]] + [[ ε , d ′′ v f ]] ε , so that L a ( ε ) ( f ) = [[ ε, d ′′ v f ]] defines an anchor map for this bracket. The expression of the Liealgebroid structure in terms of D ′ follows from[[d ′′ v ε , ε ]] = ι ′ h (d ′′ v ε )d ′ h ε = h d ′′ v ε , d ′ h ε i E ∗ = ι ′′ v (d ′ h ε )d ′′ v ε = [[d ′ h ε , ε ]] = − [[ ε , d ′ h ε ]] . (cid:3) The Lie algebroid bracket [[ ε , ε ]] = [[ ε , d ′′ v ε ]] on Γ( E ∗ ) extends to a Schouten bracket onΓ( ∧ E ∗ ), by [[ λ , λ ]] = [[ λ , d ′′ v λ ]] . Application to Lie algebroids.
Continuing the discussion from Sections 3.4, 4.5, 5.3,7.4, suppose that the vector bundle V → M carries the structure of a Lie algebroid, V ⇒ M .Then we obtain VB -algebroid structures T V ⇒ T M and T ∗ V ⇒ V ∗ , as well as a Poissonstructure on T V ∗ , to be described below. Let us discuss the resulting structures on the Weilalgebras. HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 41
Tangent bundle of V . The VB -algebroid T V ⇒ T M is the tangent prolongation of theLie algebroid V ⇒ M [13, 51]: its anchor is the tangent map to the anchor of V composedwith the canonical involution of T T M , and the Lie bracket bracket is such that the tangent liftΓ(
V, M ) → Γ( T V, T M ) , σ T σ is bracket preserving. The resulting Lie algebroid structureon b A = J ( V ) is the jet prolongation of the Lie algebroid; the bracket is uniquely characterizedby [[ j ( σ ) , j ( σ )]] = j ([[ σ , σ ]]), and its representations on B = T M and on E ∗ = V are givenby ∇ T Mj ( σ ) µ = L a ( σ ) µ, ∇ Vj ( σ ) σ = [[ σ , σ ]]for µ ∈ Ω ( M ) , σ, σ , σ ∈ Γ( V ). See [14] for a detailed discussion. The invariant bilinearpairing ( · , · ) : B ∗ × M E ∗ → R is given by the anchor a : V → T M . The resulting horizontaldifferential d h on the Weil algebra W ( T V ) is uniquely determined by its properties that d h agrees with the Lie algebroid differential d CE on W • , ( T V ) = Γ( ∧ V ∗ ) and satisfies [d h , d v ] = 0,where d v was described in 7.4.8.4.2. Tangent bundle of V ∗ . The Weil algebra of (
T V ) ′ = flip(( T V ∗ ) − ) is C ∞ ( M )-linearlygenerated by differentials d f ∈ Γ( T ∗ M ) (identified with the tangent lifts f T ), sections σ ∈ Γ( V )(identified with vertical lifts σ v ) and their jets j ( σ ) ∈ Γ( J ( V )) (identified with σ T ), subjectto the relation j ( f σ ) − f j ( σ ) = d f σ . It has a horizontal differential, characterized byd ′ h ( f ) = d f, d ′ h ( σ ) = j ( σ ), that is a derivation of the Gerstenhaber bracket. To describe thelatter, note that the Lie algebroid structure on V defines a linear Poisson structure (47) on V ∗ ;its tangent lift is a double-linear Poisson structure on T V ∗ . By definition of the tangent lift ofPoisson structures, { ( σ ) T , ( σ ) T } = [[ σ , σ ]] T , { ( σ ) T , ( σ ) v } = [[ σ , σ ]] v , { σ T , f v } = ( L a ( σ ) f ) v , { σ T , f T } = ( L a ( σ ) f ) T , { σ v , f T } = ( L a ( σ ) f ) v . We read off the Gerstenhaber brackets as[[ j ( σ ) , j ( σ )]] = j ([[ σ , σ ]]) , [[ j ( σ ) , σ ]] = [[ σ , σ ]] , [[ j ( σ ) , f ]] = L a ( σ ) f, [[ j ( σ ) , d f ]] = d L a ( σ ) f, [[ σ, d f ]] = − L a ( σ ) f. Note that d ′ h is a derivation of the Gerstenhaber bracket, as required.8.4.3. Cotangent bundle of V . The Weil algebra of (
T V ) ′′ = flip( T ∗ V ) − ∼ = T ∗ V ∗ (using theMackenzie-Xu isomorphism (12)) is C ∞ ( M )-linearly generated by sections of V, V ∗ , At( V ),subject to the relation that for σ ∈ Γ( V ), τ ∈ Γ( V ∗ ) the product στ equals i At( V ) ( σ ⊗ τ ).From the Poisson bracket relations between the corresponding functions φ σ ♯ , τ, φ δ on T ∗ V (see Section 3.4), we read off the Gerstenhaber brackets[[ δ , δ ]] = [[ δ , δ ]] , [[ δ, σ ]] = ∇ δ σ, [[ δ, τ ]] = ∇ ∗ δ τ, [[ δ, f ]] = L a ( δ f, [[ σ, τ ]] = −h τ, σ i . On the other hand, the Lie algebroid structure on V determines a VB -algebroid structure T ∗ V ∗ ⇒ V ∗ , and hence vertical differential d ′′ v . The latter agrees with the Chevalley-Eilenbergdifferential on W , • ( T ∗ V ∗ ) = Γ( ∧ • V ∗ ), whiled ′′ v σ = − δ ( σ ) ∈ Γ(At( V )) where δ ( σ ) ∈ Γ(At( V )) is the infinitesimal automorphism given in terms of its representationon V by ∇ δ ( σ ) ( σ ) = [[ σ , σ ]]. This follows from the formulas of Theorem 7.2: ι ′′ v ( j ( σ ))d ′′ v σ = ∇ Vj ( σ ) σ = [[ σ , σ ]] = −∇ δ ( σ ) σ = − ι ′′ v ( j ( σ )) δ ( σ ) . Finally, for δ ∈ Γ(At( V )) the differential d ′′ v δ ∈ W , (( T V ) ′′ ) is described by the formula ι ′′ v ( j ( σ )) ι ′′ v ( j ( σ ))d ′ v δ = ∇ δ [[ σ , σ ]] − [[ σ , ∇ δ σ ]] + [[ ∇ δ σ , σ ]] , which may be deduced from Corollary 7.5. One finds that the differential on W , • ( T ∗ V ∗ ) ∼ =Γ lin ( ∧ • T M ( T V ∗ ) , T M ) coincides with the restriction of the Chevalley-Eilenberg differential ofthe tangent prolongation T V ⇒ T M . (Recall that the dual of the tangent prolongation is thebundle
T V ∗ → T M .)9.
Applications, connections with other work
In this section, we indicate connections between the results and constructions presentedabove and various ideas appearing in the literature.9.1.
Matched pairs of Lie algebroids.
Consider a matched pair of Lie algebroids
A, B ,corresponding to a vacant double Lie algebroid D = A × M B , as in Example 8.1(c). Thus D = A × M B + (cid:11) (cid:19) B (cid:11) (cid:19) A + M D ′ = B × M A ∗ + (cid:15) (cid:15) M (cid:15) (cid:15) B + M D ′′ = A × M B ∗ / / (cid:11) (cid:19) A (cid:11) (cid:19) M / / M with corresponding Weil algebra bundles W ( D ) = ∧ A ∗ ⊗ ∧ B ∗ , W ( D ′ ) = ∧ B ∗ ⊗ ∨ A, W ( D ′′ ) = ∧ A ∗ ⊗ ∨ B. The double Lie algebroid structure defines commuting differentials d h , d v on W ( D ). Thisdouble complex was described in the work of Laurent-Gengoux, Stienon and Xu [37, Section4.2]. Identifying W ( D ) = ∧ ( A ⊕ B ) ∗ (with the total grading), the sum d h + d v is a degree 1differential, in such a way that the bundle maps to ∧ A ∗ , ∧ B ∗ give cochain maps on sections.We hence see that A ⊕ B becomes a Lie algebroid, with A, B as Lie subalgebroids. The Weilalgebra W ( D ′ ) has a Gerstenhaber bracket [[ · , · ]] and a compatible horizontal differential d ′ h .The restriction of the differential to W • , ( D ′ ) = Γ( ∧ • B ∗ ) gives the Lie algebroid structure on B , and the restriction to W • , ( D ′ ) = Γ( ∧ •− B ∗ ⊗ A ) gives the action of this Lie algebroid on A .On the other hand, the restriction of the Gerstenhaber bracket to W , ( D ′ ) = Γ( A ) recoversthe Lie algebroid bracket of A , and the bracket with elements of W , ( D ′ ) = Γ( B ∗ ) recoversthe A -action on B ∗ . The fact that the differential d ′ h on Γ( ∧ B ∗ ⊗ A ) is a derivation of the(Gerstenhaber) bracket on this space is thus an equivalent formulation of the compatibilitycondition. A similar discussion applies to D ′′ .9.2. Multi-derivations.
In [15], motivated by the study of deformations of Lie algebroids,Crainic and Moerdijk associate to any vector bundle V → M a graded vector space Der • ( V )of multi-derivations of V , equipped with a Gerstenhaber bracket. Its simplest description is interms of the isomorphism (see [15, Section 4.9])Der • ( V ) ∼ = X • lin ( V ∗ ) , HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 43 with bracket the usual Schouten bracket of multivector fields. A Lie algebroid structure on V defines a compatible degree 1 differential on this space, given by Schouten bracket [[ π, · ]]with the Poisson bivector field π ∈ X ( V ∗ ) dual to the Lie algebroid structure. As shownin [15], the Maurer-Cartan elements of this deformation complex describe the deformations ofthe Lie algebroid structure. See also the work of Esposito-Tortorella-Vitagliano [17], wherethe deformation complex is generalized further to the setting of VB -algebroids. For any vectorbundle V → M , the cotangent bundle T ∗ V is a Poisson double vector bundle, hence W ( T ∗ V )inherits a Gerstenhaber bracket. By Proposition 5.3 and Example 5.3.1, the isomorphism W • , ( T ∗ V ) ∼ = Γ lin ( ∧ • V ∗ T V ∗ , V ∗ ) ∼ = X • lin ( V ∗ )takes this to the Schouten bracket of multivector fields. Given a Lie algebroid structure on V ,the resulting horizontal differential on W • , ( T ∗ V ) is d h = [[ π, · ]], which is the differential on thedeformation complex. In conclusion, the deformation complex is identified with W • , ( T ∗ V ).Alternatively, this follows from the fact (Remark 6.13) that W • , ( T ∗ V ) for T ∗ V ⇒ V ∗ is thelinear Chevalley-Eilenberg complex CE • VB ( T ∗ V ) of Cabrera-Drummond [9]; the isomorphism ofthe latter with the deformation complex was observed in [9, page 312].9.3. Abad-Crainic’s Weil algebra of a vector bundle V . As already mentioned, the Weilalgebra for a vector bundle V → M was first discussed in the work of Mehta [55, Section4.2.4] in super-geometric terms. Our Weil algebra W ( T V ) is a classical description of thisalgebra. On the other hand, Abad and Crainic [1] defined a bigraded algebra W ( V ), alsocalled Weil algebra , as follows. An element of W p,q ( V ) is given by a sequence of R -multilinearskew-symmetric maps c i : Γ( V ) × · · · × Γ( V ) | {z } p − i times → Ω q − i ( M, ∨ i V ∗ ) . Here c is considered the ‘leading term’, c measures the failure of c to be multi-linear, c measures the failure of c to be multi-linear, and so on. (See [1] for details.) We claim thatevery w ∈ W p,q ( T V ) gives rise to such a sequence, thereby identifying W ( V ) ∼ = W ( T V ).To see this, note that for any double vector bundle D as in (2), there is a canonical surjectivemorphism of bigraded algebra bundles Π : W ( D ) → W ( B × M E ∗ ) = ∧ B ∗ ⊗ ∨ E , induced bythe DVB morphism B × M E ∗ ֒ → D . Explicitly, the maps Π : W p,q ( D ) → ∧ q − p B ∗ ⊗ ∨ p E aregiven by p -fold contractions with elements ε ∈ E ∗ . In the case of D = T V , with A = V, B = T M, E = V ∗ , we obtain projection mapsΠ : W p,q ( T V ) → ∧ q − p T ∗ M ⊗ ∨ p V. On the other hand, for σ ∈ Γ( V ), its jet prolongation j ( σ ) ∈ Γ( J ( V )) = Γ( b A ) defines acontraction operator ι ( j ( σ )) of bidegree ( − , c i corresponding to w ∈ W p,q ( T V )is given by c i ( σ i +1 , . . . , σ p ) = Π (cid:16) ι ( j ( σ p )) · · · ι ( j ( σ i )) w (cid:17) ∈ Ω q − i ( M, ∨ i V ∗ ) . The relation j ( f σ ) − f j ( σ ) = i J ( V ) (d f ⊗ σ ) = σ d f implies that ι (cid:0) j ( f σ ) (cid:1) − f ι (cid:0) j ( σ ) (cid:1) = − d f ι ( σ ), where ι ( σ ) is the contraction operator of bidegree ( − , −
1) defined by σ (regardedas a section of E ∗ ). Consequently, the failure of C ∞ ( M )-linearity of c i is expressed in terms of c i +1 , leading to the conditions in [1]. IM-forms and IM-multivector fields.
Let V → M be a vector bundle. In Section 5.3,we discussed the spaces X • lin ( V ) of linear multivector fields and Ω • lin ( V ) of linear differentialforms on V . The Schouten bracket of multivector fields and the de Rham differential of formsrestrict to these linear subspaces.Given a Lie algebroid structure V ⇒ M , there are notions of infinitesimally multiplicative (IM) multi-vector fields and differential forms, X • IM ( V ) ⊆ X • lin ( V ) , Ω • IM ( V ) ⊆ Ω • lin ( V ) . These are designed to be the infinitesimal versions of multiplicative multivector fields or formson Lie groupoids.IM-multivector fields were introduced by Iglesias-Ponte, Laurent-Gengoux and Xu [29] underthe name of k -differentials . To define them, note that for any vector bundle V , the graded Liealgebra X • ( V ) acts on Γ( ∧ V ) by derivations. Using the identification Γ( ∧ V ) ∼ = X • core ( V ),this action is just the Schouten bracket of multi-vector fields. In particular, for δ ∈ X k lin ( V )and σ ∈ Γ( ∧ l V ) we have that δ.σ ∈ Γ( ∧ k + l − V ). If V is a Lie algebroid, then the bracket [[ · , · ]]on Γ( V ) extends to the exterior algebra. The element δ is called an IM-multivector field if itis a derivation of this Lie bracket:(80) δ. [[ σ , σ ]] = [[ δ.σ , σ ]] + ( − | δ | ( | σ | +1) [[ σ , δ.σ ]]for all σ i ∈ Γ( ∧ l i V ), i = 1 ,
2. Here | δ | = k + 1. Using the derivation property with respectto wedge product, it is actually enough to have this condition for l = l = 1. The universallifting theorem [29], generalizing earlier results of Mackenzie-Xu [52, 53], integrates any such δ to a multiplicative vector field on the corresponding (local) Lie groupoid.To describe the IM condition for forms, recall (cf. Example 5.3.2) that any φ ∈ Ω k lin ( V ) canbe uniquely written as φ = ν + d Rh µ where ν ∈ Γ( V ∗ ⊗ ∧ k T ∗ M ) and µ ∈ Γ( V ∗ ⊗ ∧ k − T ∗ M )(viewed as linear differential forms), and d Rh denotes the de Rham differential on linear forms.If V is a Lie algebroid, then φ = ν + d Rh µ is called an IM form if the following three conditionsare satisfied ι a ( σ ) µ ( σ ) = − ι a ( σ ) µ ( σ ) , (81) µ ([[ σ , σ ]]) = L a ( σ ) µ ( σ ) − ι a ( σ ) d Rh µ ( σ ) − ι a ( σ ) ν ( σ ) , (82) ν ([[ σ , σ ]]) = L a ( σ ) ν ( σ ) − ι a ( σ ) d Rh ν ( σ ) , (83)for all σ , σ ∈ Γ( V ). These conditions are due to Bursztyn and Cabrera [5] (see [7, 8] for thecase of 2-forms); as shown in [5], these are exactly the conditions ensuring that φ integrates toa multiplicative form on the associated (local) Lie groupoid.We will now give interpretations of IM multivector fields and IM forms in terms of the Weilalgebras. Recall from 5.3 that for any vector bundle V → M , X • lin ( V ) = W , • ( T ∗ V ) , Ω • lin ( V ) = W , • ( T V ) . The first isomorphism is compatible with the Gerstenhaber bracket [[ · , · ]] on W ( T ∗ V ) definedby the Poisson structure on T ∗ V , the second isomorphism with the vertical differential d v on W ( T V ) defined by the VB -algebroid structure T V ⇒ V . A Lie algebroid structure V ⇒ M gives VB -algebroid structures T ∗ V ⇒ V ∗ and T V ⇒ T M , resulting in horizontal differentialsd h on both W ( T ∗ V ) and W ( T V ). The second part of the following result is due to Bursztynand Cabrera [5]; for a more restrictive notion of IM forms it was observed in [1, Section 6].
HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 45
Theorem 9.1.
For any Lie algebroid V ⇒ M , X • IM ( V ) = W , • ( T ∗ V ) ∩ ker(d h ) , Ω • IM ( V ) = W , • ( T V ) ∩ ker(d h ) . Proof.
Consider a VB -algebroid D ⇒ B over A ⇒ M , so that the Weil algebra W ( D ) carriesa horizontal differential d h . Then W ( D ′ ) has a Gerstenhaber bracket. By Corollary 7.5, anelement φ ∈ W , • ( D ) is d h -closed if and only if(84) ι h ([[ x , x ]]) φ = [[ x , ι h ( x ) φ ]] + ( − | x || x | +1 [[ x , ι h ( x ) φ ]]for all x i ∈ W p i , ( D ′ ). (It suffices to verify this on generators.)Suppose now that D also carries a double-linear Poisson structure; thus D ′′ is a double Liealgebroid. In particular B ∗ = core( D ′′ ) is a Lie algebroid, with the bracket[[ σ , σ ]] := [[ σ , d ′ v σ ]]for σ , σ ∈ Γ( ∧ B ∗ ). (We have in mind the case D = T ∗ V ; the Lie algebroid structureon B ∗ = V being the standard one.) The space W , • ( D )[1] is a graded Lie algebra for theGerstenhaber bracket, with a representation on Γ( ∧ B ∗ ) given by (cf. (69)) φ.σ := [[ φ, σ ]] = − ( − ( | φ | +1) | σ | ι h (d ′ v σ ) φ for σ ∈ Γ( ∧ B ∗ ). Let x i = d ′ v σ i with σ i ∈ Γ( ∧ B ∗ ). Then [[ x , x ]] = d ′ v [[ σ , σ ]], and therefore(85) φ. [[ σ , σ ]] = − ( − ( | φ | +1)( | σ | + | σ | +1) ι h ([[ x , x ]]) φ. On the other hand,( − | φ | ( | σ | +1) [[ σ , φ.σ ]] + [[ φ.σ , σ ]]= − ( − ( | φ | +1)( | σ | + | σ | +1) (cid:16) [[ x , ι h ( x ) φ ]] + ( − ( | φ | +1)( | σ | +1) [[ ι h ( x ) φ, x ]] (cid:17) = − ( − ( | φ | +1)( | σ | + | σ | +1) (cid:16) [[ x , ι h ( x ) φ ]] − ( − ( | σ | +1)( | σ | +1) [[ x , ι h ( x ) φ ]] (cid:17) . By (84), if φ is d h -closed then this expression coincides with (85), proving that φ.σ = [[ φ.σ , σ ]]+( − | φ | ( | σ | +1) [[ σ , φ.σ ]]. For D = T ∗ V , the converse is true, because in that case the space W • , ( D ′ ) is spanned, as a C ∞ ( M )-module, by d ′ v W • , ( D ′ ). The case of IM-differential formscan be discussed similarly; in terms of the Abad-Crainic approach to the Weil algebra W ( T V )this is done in [5]. (cid:3)
Fr¨olicher-Nijenhuis and Nijenhuis-Richardson brackets.
Suppose V → M is a Liealgebroid, so that V ∗ has a linear Poisson structure. The double-linear Poisson structure on T V ∗ defines a Gerstenhaber bracket on W ( T V ∗ ), compatible with the vertical differential d v .Hence,(86) W , • ( T V ∗ ) ∼ = Ω • +1lin ( V ∗ ) ∼ = Ω • +1 ( M, V ) ⊕ Ω • ( M, V )becomes a differential graded Lie algebra. It comes with a morphism of graded Lie algebras(also Ω( M )-module morphism)(87) Ω • +1 ( M, V ) ⊕ Ω • ( M, V ) → Der • (Ω( M ))given by Gerstenhaber bracket with elements of W , • ( T V ∗ ) ∼ = Ω( M ). One verifies that the firstsummand in (87) acts as contractions via the anchor V → T M , the second as Lie derivatives.
For the Lie algebroid V = T M , the map (87) is an isomorphism, hence we recover the bracketon Ω • +1 ( M, T M ) ⊕ Ω • ( M, T M ) given by the Fr¨olicher-Nijenhuis bracket on the first summand,the Nijenhuis-Richardson bracket on the second summand, and a cross term. See [30, ChapterII.8] for a detailed discussion; see also [22, 58] for related brackets and generalizations to Liealgebroids.9.6.
Representations up to homotopy.
Representations up to homotopy were introducedby Evens-Lu-Weinstein [18] and Abad and Crainic [2] as generalizations of the usual concept ofrepresentations of a Lie algebroid. Among other things, they give a notion of the adjoint actionof a Lie algebroid on itself, which is generally not possible using ordinary representations. Theessential idea is to represent Lie algebroids on complexes of vector bundles rather than justsingle vector bundles. We will adopt the definition from [2].
Definition 9.2.
Let A → M be a Lie algebroid. A representation up to homotopy of A is a Z -graded vector bundle U • over M along with a degree 1 differential δ on sections of ∧ A ∗ ⊗ U (using the graded tensor product) satisfying δ ( ωη ) = d A ( ω ) η + ( − k ωδ ( η )for all ω ∈ Γ( ∧ k A ∗ ), η ∈ Γ( ∧ A ∗ ⊗ U ).Given a Lie algebroid A → M , Gracia-Saz and Mehta [25] showed how to construct a 2-steprepresentation up to homotopy of A from a horizontal VB -algebroid D + (cid:15) (cid:15) B (cid:15) (cid:15) A + M having A as its horizontal side bundle. The construction depends on the choice of a splittingof D , and the resulting graded vector bundle is U = E ∗ [1] ⊕ B ;that is, U − = E ∗ and U = B . We briefly review their construction, making use of some ofour observations in Section 6. By Lemma 6.6, the double-linear Poisson structure on D ′ givesthe following data: • a Lie algebroid structure on b A , • b A -representations b ∇ B ∗ , b ∇ E ∗ on B ∗ and on E ∗ , • an invariant pairing ( · , · ) : B ∗ × E ∗ → R .A choice of splitting of the double vector bundle D is equivalent to a choice of splitting s : A → b A . In general, s need not preserve Lie brackets, and so we can consider its curvature tensorΩ ∈ Γ( ∧ A ∗ ⊗ ( B ∗ ⊗ E ∗ )) defined byΩ( a , a ) = s ([[ a , a ]]) − [ s ( a ) , s ( a )] , a , a ∈ Γ( A ) . The U – U − -component of δ is the linear map Γ( ∧ • A ∗ ⊗ B ) → Γ( ∧ • +2 A ∗ ⊗ E ∗ ) given by Ω(with the identification B ∗ ⊗ E ∗ ∼ = Hom( B, E ∗ )). The U − – U -component of δ is the linear mapΓ( ∧ • A ∗ ⊗ E ∗ ) → Γ( ∧ • A ∗ ⊗ B ) defined by the pairing ( · , · ) viewed as a bundle map E ∗ → B .The connection b ∇ E ∗ pulls back under s to a non-flat A -connection ∇ E ∗ on E ∗ ; its extensionto a map ∇ E ∗ : Γ( ∧ • A ∗ ⊗ E ∗ ) → Γ( ∧ • +1 A ∗ ⊗ E ∗ ) is the U − – U − -component of δ . Similarly, HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 47 the flat b A -connection on B pulls back to a non-flat A -connection, and the resulting map onsections gives the U – U -component. See [25, Theorem 4.10]. This establishes a one-to-onecorrespondence: Theorem 9.3. [25, Theorem 4.11 (2)]
Let D be a double vector bundle with side bundles A, B and core E ∗ , such that A ⇒ M is a Lie algebroid. After choosing a splitting s : A → ˆ A , thereis a one-to-one correspondence between horizontal VB -algebroid structures D ⇒ B extending A ⇒ M , and representations up to homotopy of A on U = E ∗ [1] ⊕ B . This correspondence has a simple interpretation in terms of the Weil algebras. Recall fromSection 6 that horizontal VB -algebroid structures D ⇒ B are in one-to-one correspondencewith vertical differentials d v on W ( D ′′ ). This restricts to a differential d v on W , • ( D ′′ ) ∼ =Γ lin ( ∧ • E D ′ , E ). Once we choose a splitting of D , or equivalently a vector bundle splitting s : A → b A , we obtain a decomposition (see Proposition 5.2) W , • ( D ′′ ) ∼ = Γ( ∧ • A ∗ ⊗ B ) ⊕ Γ( ∧ • +1 A ∗ ⊗ E ∗ ) . With U − = E ∗ and U = A ∗ , the differential d defines a representation up to homotopyof A on U . To see that this correspondence is bijective, we note that a vertical differentialon the bigraded algebra W ( D ′′ ) is uniquely determined by its restrictions to W , • ( D ′′ ) and W , • ( D ′′ ) = Γ( ∧ • A ∗ ) thanks to the Leibniz rule. Remark . Horizontal VB -algebroid structures on D are also equivalent to differentials d h of bidegree (1 ,
0) on W ( D ). After a choice of splitting, this induces a degree 1 operator on W • , ( D ) ∼ = Γ( ∧ • A ∗ ⊗ E ) ⊕ Γ( ∧ • +1 A ∗ ⊗ B ∗ ), giving a representation up to homotopy of A onthe bundle B ∗ [1] ⊕ E . This representation up to homotopy is dual to the one on E ∗ [1] ⊕ B , asdiscussed in [25, Section 4.5].Let us turn now to the case where D has both a horizontal and a vertical VB -algebroidstructure. Then both D ′ and D ′′ are Poisson double vector bundles, which is equivalent to thefollowing structures: • Lie algebroid structures on b A and b B , • b A -representations on B ∗ , E ∗ and b B -representations on A ∗ , E ∗ , • an b A -invariant pairing B ∗ × E ∗ → R and a b B -invariant pairing A ∗ × E ∗ → R subject to certain compatibility conditions (Theorem 6.8). It is natural to ask what additionalcompatibility conditions on these data ensure that D is a double Lie algebroid. This questionwas answered in the work of Gracia-Saz, Jotz Lean, Mackenzie, and Mehta [23, Theorem 3.4]using a splitting and a notion of matched pair for representations up to homotopy.9.7. Van Est maps.
Recall that the classical van Est map [63] is a morphism from the cochaincomplex of a Lie group G to the Chevalley-Eilenberg cochain complex of its Lie algebra g . Thismap was extended by Weinstein and Xu [66] to the case of Lie groupoids G and their Liealgebroids A , thus obtaining a morphism of cochain complexes(88) VE : e C ∞ ( B • G ) → Γ( ∧ • A ∗ ) . Here B p G is the space of p -arrows in G , and the tilde signifies the normalized complex for thesimplicial manifold B • G . In [55, Chapter 6], Mehta generalized (88) to a van Est map from‘Q-groupoids’ into the double complex of the corresponding ‘ Q -algebroid’; in particular this gives a version of (88) for differential forms. A construction in more classical terms was givenby Abad and Crainic [1], in terms of the Weil algebra W ( A ) = W ( T A ). The van Est map (88)described in [1] is a morphism of double complexes(89) VE : e Ω • ( B • G ) → W • , • ( T A ) . A geometric construction of this map was provided in [39]. In an upcoming work, we willgeneralize this construction to the setting of LA -groupoids, with Mehta’s double complex ofthe LA -groupoid [56] as the domain and the Weil algebra W ( D ) of the corresponding doubleLie algebroid D as the codomain. Here (89) corresponds to the case of the tangent groupoid T G ⇒ T M . This is related to work of Cabrera and Drummond in [9]; as mentioned earlier(Remark 6.13), if D is a (horizontal) VB -algebroid, then the codomain CE • VB ( D ) of their vanEst map for homogeneous cochains for VB -groupoids is our W , • ( D ). Going one step further,we will describe a van Est map for double Lie groupoids H , a morphism from a certain doublecomplex e C ∞ ( B • , • H ) to the double complex defined by the Weil algebra of its associated doubleLie algebroid. The recent thesis of Angulo [3] considers similar questions in the context of Lie2-algebras and Lie 2-groups. Appendix A. Splitting of double vector bundles
In this appendix, we give a quick proof of the existence of splittings of double vector bundles D (as in (2)). Note that if D is regarded as a vector bundle over A , then its restriction to M ⊆ A is the vector bundle ( κ v ) − ( M ) = B × M E ∗ ≡ B ⊕ E ∗ over M . Theorem A.1.
Every double vector bundle admits a splitting.Proof.
Regard D and A × M B as vector bundles over A ; their restrictions to the submanifold M ⊆ A are canonically B ⊕ E ∗ and B , respectively. The surjective submersion ϕ : D → A × M B from (3), regarded as a morphism of vector bundles over A , restricts along M to the obviousprojection B ⊕ E ∗ → B . This restriction has a canonical splitting B → B ⊕ E ∗ , b ( b, A , ψ : A × M B → D. Then ψ intertwines the vertical scalar multiplications κ vt , but not necessarily the horizontalscalar multiplications. Applying the normal bundle functor, we obtain a DVB morphism ν ( ψ ) : ν ( A × M B, B ) → ν ( D, B ) . But recall that for any vector bundle V → M , one has a canonical isomorphism T V | M = V ⊕ T M , giving rise to an isomorphism of vector bundles ν ( V, M ) ∼ = V . In a similar fashion,we have canonical DVB isomorphisms ν ( D, B ) ∼ = D and ν ( A × M B, B ) ∼ = A × M B . Underthese identifications, ν ( ψ ) =: ψ is the desired splitting A × M B → D . (cid:3) HE WEIL ALGEBRA OF A DOUBLE LIE ALGEBROID 49
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