Vector bundles over Lie groupoids and algebroids
aa r X i v : . [ m a t h . DG ] O c t VECTOR BUNDLES OVER LIE GROUPOIDS AND ALGEBROIDS
HENRIQUE BURSZTYN, ALEJANDRO CABRERA, AND MATIAS DEL HOYO
Abstract.
We study VB-groupoids and VB-algebroids, which are vector bundlesin the realm of Lie groupoids and Lie algebroids. Through a suitable reformulationof their definitions, we elucidate the Lie theory relating these objects, i.e., theirrelation via differentiation and integration. We also show how to extend our tech-niques to describe the more general Lie theory underlying double Lie algebroidsand LA-groupoids.
Contents
1. Introduction 12. Preliminaries 32.1. A characterization of vector bundles 32.2. Double vector bundles 52.3. Linear Poisson structures 83. VB-groupoids and VB-algebroids as regular actions 93.1. VB-groupoids 93.2. Characterization as multiplicative actions 113.3. VB-algebroids 133.4. Characterization as IM actions 144. Lie theory for vector bundles 164.1. Differentiation of VB-groupoids 174.2. The vertical lift for multiplicative and IM actions 194.3. Integration 205. Applications to double Lie algebroids 235.1. Interlude: Lie bialgebroids and Poisson groupoids 235.2. LA-groupoids and double Lie algebroids: the dual viewpoint 265.3. Lie theory 28Appendix A. Fibred products of Lie groupoids and Lie algebroids 30A.1. The groupoid case 31A.2. The algebroid case 33A.3. Fibred products and the Lie functor 35References 361.
Introduction
Lie groupoids arise in various areas of geometry and topology, such as group ac-tions, foliations and Poisson geometry [7, 25, 29], often serving as models for singularspaces (see e.g. [10] and references therein). They provide a unifying viewpoint toseemingly unrelated questions that has led to important extensions of classical geo-metrical results. Lie algebroids are their infinitesimal counterparts, and both arerelated by a rich Lie theory [9], with many applications beyond the classical theoryof Lie algebras and Lie groups. The main objects of study in this paper are the so-called
VB-groupoids and
VB-algebroids [13, 20, 21, 32], which can be thought of as (categorified) vector bundlesin the realm of Lie groupoids and Lie algebroids. Paradigmatic examples include thetangent and cotangent bundles of Lie groupoids and Lie algebroids. These objectshave been the subject of extensive study in the last years [20, 23, 26], partly motivatedby their deep ties with Poisson geometry [22, 27, 28].From another perspective, VB-groupoids and VB-algebroids are intimately relatedto the study of representations. Indeed, representations of Lie groupoids on vectorbundles provide a wealth of examples of VB-groupoids through the construction ofsemi-direct products, and analogously for Lie algebroids. More generally, it is shownin [13, 14] that VB-groupoids and VB-algebroids provide an intrinsic approach torepresentations up to homotopy [1, 2], a “higher” notion of representation needed tomake sense e.g. of the adjoint representation of a Lie groupoid or algebroid.Our main goal in this paper is to describe the Lie theory relating VB-algebroidsand VB-groupoids, i.e., to elucidate how they are related via differentiation andintegration. A key step in our work relies on finding simpler formulations of VB-groupoids and VB-algebroids, explained in Theorems 3.2.3 and 3.4.3. We show thatVB-groupoids (resp. VB-algebroids) can be described as Lie groupoids (resp. Liealgebroids) equipped with an additional action of the monoid ( R , · ), with naturalcompatibility conditions, in the spirit of the characterization of vector bundles in[11]. From this viewpoint, we can study the Lie functor relating VB-groupoidsand VB-algebroids by differentiating and integrating these actions. We prove inTheorem 4.3.4 that, if the total algebroid of a VB-algebroid is integrable, then itsvector-bundle structure can be lifted to the source-simply-connected Lie groupoidintegrating it, which then becomes a VB-groupoid. This result finds applications e.g.in the study of Dirac structures on Lie groupoids [31] and also provides informationabout integration of representations up to homotopy [4] (see also [5]).Our techniques to handle VB-algebroids and VB-groupoids allow us to go furtherand explain the Lie theory relating more general objects, known as double Lie alge-broids and LA-groupoids [20, 23, 24, 26]. One can think of them as Lie algebroidsdefined over Lie algebroids and Lie groupoids, or as generalizations of VB-algebroidsand VB-groupoids in which the vector-bundle structures are enhanced to be Lie al-gebroids. In this more general context, we prove in Theorem 5.3.5 that if the top Liealgebroid in a double Lie algebroid is integrable, then its source-simply-connectedintegration naturally becomes an LA-groupoid; this provides the reverse procedureto the differentiation in [23]. Our approach to establish this result heavily relies onthe well-known duality between Lie algebroids and linear Poisson structures. Ratherthan treating double Lie algebroids and LA-groupoids directly, we focus on theirdual objects. Building on our previous results for VB-groupoids and VB-algebroids,we describe these duals as Lie bialgebroids and Poisson groupoids carrying an ex-tra compatible ( R , · )-action. The result then follows from the differentiation andintegration properties of these actions, along with a natural integration result formorphisms of Lie bialgebroids (see Prop. 5.1.3).Throughout the paper, several arguments rely on the construction of fibred prod-ucts in the categories of Lie algebroids and Lie groupoids; we collect the necessaryresults in the appendix, organizing and extending previous discussions about fibredproducts in the literature. Organization.
After preliminaries in Section 2, which include the description of(double) vector bundles and linear Poisson structures in terms of ( R , · )-actions, wepresent new characterizations of VB-groupoids and VB-algebroids in Section 3. TheirLie theory is explained in Section 4. In Section 5, we consider double Lie algebroidsand LA-groupoids, describe their dual objects, and extend the Lie theory of Section4 to this context. Acknowledgments.
We are grateful to several institutes for hosting us duringdifferent stages of this project, especially IST (Lisbon) and U. Utrecht. We thankseveral people for useful discussions and helpful advice, including M. Crainic, T.Drummond, R. Fernandes, M. Jotz, D. Li-Bland, K. Mackenzie, R. Mehta, E. Mein-reken, and C. Ortiz. We thank CNPq and FAPERJ for financial support.
Notation and conventions.
A Lie groupoid with arrows manifold G and units M will be denoted by G ⇒ M, or simply by G . We write s G and t G for the source andtarget maps. The set G (2) = { ( g, h ) : s G ( g ) = t G ( h ) } ⊂ G × G is the domain of themultiplication m G : G (2) → G , m ( g, h ) = gh . The unit map u G : M → G is oftenused to identify M with its image in G , and we write inv G : G → G , inv G ( g ) = g − ,for the inversion. We often suppress the subscript G and simply write s , t , m, u, inv.We denote Lie-groupoid maps by (Φ , φ ) : ( G ⇒ M ) → ( G ⇒ M ), or simply byΦ : G → G , having in mind that φ = Φ | M .Given a Lie groupoid G ⇒ M , its Lie algebroid Lie( G ) = A G has underlyingvector bundle A G = ker(d s ) | M → M , with anchor map d t | A G : A G → M and Liebracket on sections of A G induced by right-invariant vector fields. A general Liealgebroid with underlying vector bundle A → M will be usually denoted by A ⇒ M. We use the notation ρ A for the anchor map and [ · , · ] A for the bracket, or simply ρ and [ · , · ], if there is no risk of confusion.2. Preliminaries
We start by discussing a characterization of vector bundles via actions of themultiplicative monoid ( R , · ) as in [11], where details can be found. We will alsoconsider double vector bundles and linear Poisson structures from this viewpoint.2.1. A characterization of vector bundles.
Let D be a smooth manifold, anddenote by ( R , · ) the multiplicative monoid of real numbers. An action h : ( R , · ) y D of ( R , · ) on D is a smooth map h : R × D → D, h ( λ, x ) = h λ ( x ) , satisfying the usual action axioms: h = id D and h λ h λ ′ = h λλ ′ for all λ, λ ′ ∈ R .Assume that D is connected. Since the map h is a projection, i.e. h ◦ h = h , itfollows that h ( D ) ⊂ D is an embedded submanifold with T h ( x ) h ( D ) = d x h ( T x D )for all x ∈ D , see e.g. [17, Thm 1.13]. Using that h is an action one may checkthat the rank of the map h : D → h ( D ) is constant, and hence it is a surjectivesubmersion. When D is not connected, the rank of h is only locally constant, i.e.,it is constant on each connected component, but may vary from one component toanother. When considering ( R , · )-actions on disconnected manifolds, we will alwaysassume that h has constant rank . This guarantees that h ( D ) is an embeddedsubmanifold of D . The key example of an action of ( R , · ) is the fibrewise scalar multiplication (homo-theties) on a vector bundle E → M , in which case h ( E ) = M . This action satisfiesan additional property: if x ∈ E is a non-zero vector then the curve λ h λ ( x ) hasnon-zero velocity at the origin. This motivates the following definition. Definition 2.1.1.
We call an action h : ( R , · ) y D regular if the following equationholds at all points in x ∈ D :(2.1) ddλ (cid:12)(cid:12)(cid:12) λ =0 h λ ( x ) = 0 ⇒ x = h ( x ) . It turns out that an action is regular if and only if it can be realized as thehomotheties of a vector bundle. Let us recall a construction from [11] that explainsthis fact and plays a key role in this paper.Given an action h : ( R , · ) y D , there is always a vector bundle over h ( D )canonically associated with it, the so-called vertical bundle , defined by(2.2) V h D = ker(d h ) | h ( D ) . Note that its underlying ( R , · )-action is the restriction of the homotheties on T D → D . This passage from ( R , · )-actions to vector bundles is functorial, i.e., an ( R , · )-equivariant map D → D yields a canonical vector bundle map V h D → V h D , andthis assignment respects identities and compositions.The vertical lift V h : D → V h D is the smooth map that associates to each point x ∈ D the velocity at time 0 of the curve λ h λ ( x ):(2.3) V h ( x ) = ddλ (cid:12)(cid:12)(cid:12) λ =0 h λ ( x ) , so the action is regular if and only if the zeroes of V h are exactly its fixed points. Onemay readily verify (through the chain rule) that the vertical lift is ( R , · )-equivariant.When h is defined by homotheties of a vector bundle, the vertical lift is the standardidentification with its associated vertical bundle.The regularity of an action is clearly a necessary condition for the vertical lift tobe a diffeomorphism. The less evident fact is that it is also sufficient: Theorem 2.1.2 ([11]) . An action h : ( R , · ) y D is regular if and only if the verticallift V h is a diffeomorphism onto V h D . In this case D → h ( D ) inherits a naturalvector-bundle structure for which V h is a vector-bundle isomorphism. This theorem sets an equivalence between regular ( R , · )-actions and vector bun-dles, and allows the theory of vector bundles to be rephrased in terms of ( R , · )-actions.For instance, as immediate consequences of Theorem 2.1.2, we see that a vector sub-bundle is the same as an invariant submanifold, and that a vector-bundle map is thesame as a smooth equivariant map. (Note that, by continuity, it is enough to checkequivariance for λ = 0.) For more details, see [11]. Remark 2.1.3.
Denote by VB the category of vector bundles and by ACT that of( R , · )-actions. By considering the vertical bundles associated with actions and theactions by homotheties underlying vector bundles, we obtain a pair of functorsACT V / / VB U o o . The vertical lift V : id Act → U ◦ V is a natural transformation that is invertible overthe image of U . It easily follows that V is a left adjoint for U , that U is fully faithful,and hence VB is a co-reflective subcategory of ACT. From this perspective, V is aprojection that associates to any action a regular one, so we may think of it as a“regularization functor”. Remark 2.1.4. If D is a manifold equipped with an action h : ( R , · ) y D , regular ornot, the vertical lift map V h : D → T D can be expressed as the following composition:(2.4) D V h / / l $ $ ❏❏❏❏❏❏❏❏❏❏ T DT D × T R d h ssssssssss where l ( x ) = (( x, , (0 , ∂ λ )) is the map whose first component is the zero section of T D and whose second component is a constant map. The factorization V h = d h ◦ l will be useful in subsequent sections.2.2. Double vector bundles. A double vector bundle ( D, E, A, M ) is a commuta-tive diagram(2.5) D → E ↓ ↓ A → M in which every arrow is a vector bundle and so that the two vector-bundle structureson D are compatible , in the sense that the structural maps of one (projection, zerosection, fibrewise addition and multiplication by scalars) are vector-bundle mapswith respect to the other (see [13, Prop. 2.1]). Whenever we need to specify thestructure maps involved in a double vector bundle, we use the notation q DE for thebundle projection D → E , 0 DE for the corresponding zero section, and similarly forthe structure maps of the other vector bundles.For double vector bundles ( D, E, A, M ) and ( ˜ D, ˜ E, ˜ A, ˜ M ), a map Ψ : D → ˜ D is a morphism if it gives rise to vector-bundle maps (Ψ , ψ A ) : ( D → A ) → ( ˜ D → ˜ A ) and(Ψ , ψ E ) : ( D → E ) → ( ˜ D → ˜ E ). It follows that ψ A : A → ˜ A and ψ E : E → ˜ E arealso vector-bundle maps, covering the same map ψ M : M → ˜ M . Identifying M, A, E with submanifolds of D via the corresponding zero sections, the maps ψ E , ψ A , ψ M are just the restrictions Ψ | E , Ψ | A , Ψ | M .For a double vector bundle as in (2.5) the bundles E → M and A → M are calledthe side bundles . The intersection of the kernels of the projections q DA : ( D → E ) → ( A → M ) and q DE : ( D → A ) → ( E → M ) defines another vector bundle C → M ,known as the core bundle of D . The core and side bundles are central ingredients inthe structure of D : there always exists a (non-canonical) splitting D ∼ −→ A ⊕ C ⊕ E, i.e., an isomorphism inducing the identity on the sides and core, where the triplesum is regarded as a double vector bundle in the obvious way (see e.g. [13]). Example 2.2.1.
The main examples of double vector bundles are the tangent andcotangent bundles of a vector bundle A → M (see e.g. [25, § T A → T M ↓ ↓ A → M, T ∗ A → A ∗ ↓ ↓ A → M. If h denotes the ( R , · )-action on A by homotheties, the action corresponding to thebundle structure T A → T M is λ d h λ . The action corresponding to T ∗ A → A ∗ ,sometimes referred to as the phase lift , will be described in terms of h after Prop. 2.2.5below. Note that the core of T A is the vertical bundle
V A → M , which is isomorphicto A → M . The core of T ∗ A can be identified with T ∗ M → M . Remark 2.2.2 (The reversal isomorphism) . For a vector bundle A → M , there isa canonical isomorphism of double vector bundles,(2.7) R A : T ∗ A → T ∗ A ∗ , known as the reversal isomorphism , preserving side bundles and restricting to − idon the cores. In local coordinates we have splittings T ∗ A ∼ = A ⊕ T ∗ M ⊕ A ∗ and T ∗ A ∗ ∼ = A ∗ ⊕ T ∗ M ⊕ A , with respect to which R A ( φ, ω, v ) = ( v, − ω, φ ), and thisturns out to be well defined globally. For a detailed discussion, see e.g. [25, § § D, E, A, M ) we have a horizontal dual and a vertical dual , D ∗ E → E ↓ ↓ C ∗ → M, D ∗ A → C ∗ ↓ ↓ A → M, which are double vector bundles containing the dual of the core bundle of D as sidebundles, and whose cores are A ∗ → M and E ∗ → M , respectively. For example,given a vector bundle A → M , the vertical dual of its tangent bundle is its cotangentbundle, as depicted in (2.6), while the horizontal dual is a new double vector bundle(( T A ) ∗ T M , T M, A ∗ , M ).It is often convenient to think of a double vector bundle (2.5) and its two dualsas parts of a larger object, the so-called cotangent cube (2.8) T ∗ D / / (cid:15) (cid:15) ❋❋❋❋ D ∗ E (cid:15) (cid:15) " " ❋❋❋ D (cid:15) (cid:15) / / E (cid:15) (cid:15) D ∗ A / / ❋❋❋ C ∗ " " ❋❋❋ A / / M. The horizontal and vertical duals of a double vector bundle are related by a naturalpairing D ∗ A × C ∗ D ∗ E → R , which induces an isomorphism of double vector bundles(2.9) Z D : D ∗ A → ( D ∗ E ) ∗ C ∗ , interchanging the side bundles and inducing − id on the cores (cf. [25, Thms. 9.2.2& 9.2.4]). This shows that, when considering the double vector bundles D , D ∗ A and D ∗ E , taking further (horizontal or vertical) duals basically interchanges them. Remark 2.2.3.
For later use, we recall the following compatibility between the iso-morphisms (2.7) and (2.9). Given D as in (2.5), consider the induced vector-bundles T ∗ D → D ∗ A and T ∗ D ∗ E → ( D ∗ E ) ∗ C ∗ . Then the reversal isomorphism associated with D → E preserves these bundle structures and covers (2.9); i.e., the following squarecommutes (see [24, Thm. 6.1]): T ∗ D R / / (cid:15) (cid:15) T ∗ D ∗ E (cid:15) (cid:15) D ∗ A Z / / ( D ∗ E ) ∗ C ∗ , The pair (
R, Z ) actually defines an isomorphism between the cotangent cubes of D and D ∗ E , that we may see as a higher analogue of the reversal isomorphism (2.7).Double vector bundles admit a simple characterization in terms of regular actions.In a double vector bundle ( D, E, A, M ), the actions h, k : ( R , · ) y D correspondingto D → A and D → E commute, i.e., h λ k µ = k µ h λ for all λ, µ ∈ R . Conversely, if amanifold D is endowed with two commuting regular actions h, k : ( R , · ) y D , thenin light of Theorem 2.1.2 we get a commutative diagram of vector bundles D → k ( D ) ↓ ↓ h ( D ) → h k ( D ) . To see that this is a double vector bundle, note that, since h and k commute, thevertical lift V h : D → T D intertwines k and d k , so we can embed the previous squareinto the double vector bundle ( T D, d k ( T D ) , D, k ( D )), from where it inherits therequired compatibility condition. Hence we conclude (see [11, Thm 3.1]): Proposition 2.2.4.
There is a one-to-one correspondence between double vectorbundle structures on D and pairs of commuting regular actions ( R , · ) y D . For double vector bundles D and ˜ D , defined by ( R , · )-actions h, k and ˜ h, ˜ k respec-tively, a map D → ˜ D is a morphism if and only if it is equivariant for both actions,i.e., it intertwines h and ˜ h as well as k and ˜ k .Let us now discuss the behavior of the ( R , · )-actions under duality of double vectorbundles. Given D as in (2.5) and its vertical dual, let h , k , ¯ h and ¯ k denote the actionscorresponding to D → A , D → E , D ∗ A → A and D ∗ A → C ∗ , respectively. Then therestrictions k | A and ¯ k | A agree, and k and ¯ k are related by the following equation(see [25, pp. 349]):(2.10) h ¯ k λ ( ξ ) , k λ ( v ) i = λ h ξ, v i , a ∈ A, ξ ∈ ( D ∗ A ) a , v ∈ D a . We will relate the homotheties of the dual, ¯ k λ , with the dual relation of thehomotheties k λ − . Recall that if φ : ( E → M ) → ( ˜ E → ˜ M ) is a map of vectorbundles, then its dual relation is the relation defined by(2.11) φ ∗ := { (cid:0) φ ∗ ( ξ ) , ξ (cid:1) , x ∈ M, ξ ∈ ˜ E ∗ φ ( x ) } ⊂ E ∗ × ˜ E ∗ , and if the map φ is invertible, then (2.11) is the graph of an actual vector bundlemap ˜ E ∗ → E ∗ , still denoted by φ ∗ , that agrees with φ − on the base. Proposition 2.2.5.
For λ = 0 , the following is an identity of vector-bundle maps: ¯ k λ = ( k λ − ) ∗ ¯ h λ = ( k − λ ) ∗ ¯ h λ : ( D ∗ A → A ) → ( D ∗ A → A ) . Proof.
Let us show that the maps agree over the base and on each fiber. If a ∈ A ,since k | A = ¯ k | A and ¯ h | A is trivial, we have ¯ k λ ( a ) = k λ ( a ) = ( k − λ ) ∗ ( a ) = ( k − λ ) ∗ ¯ h λ ( a ).Now let ξ ∈ ( D ∗ A ) a , so ξ : D a → R is a linear map. Since the linear structure on D a is given by h λ we have h ¯ h λ ( ξ ) , v i = λ h ξ, v i , for v ∈ D a . Therefore h ( k − λ ) ∗ ¯ h λ ( ξ ) , k λ ( v ) i = h ¯ h λ ( ξ ) , v i = λ h ξ, v i , which shows that ( k − λ ) ∗ ¯ h λ ( ξ ) must be ¯ k λ ( ξ ), by (2.10). (cid:3) As a corollary we obtain an explicit description of the phase lift (see Exam-ple 2.2.1): if h : ( R , · ) y E is a regular action, then the action ( R , · ) y T ∗ E corresponding to T ∗ E → E ∗ is given, for each λ = 0, by(2.12) λ · (d h λ − ) ∗ : T ∗ E → T ∗ E, where λ · ( − ) stands for the multiplication by λ in the canonical structure T ∗ E → E .2.3. Linear Poisson structures.
Given a vector bundle q : E → M , a function f ∈ C ∞ ( E ) is said to be linear (resp. basic ) if it is linear (resp. constant) whenrestricted to each fiber. Linear functions are in one-to-one correspondence withsections of the dual bundle,(2.13) Γ( E ∗ ) ∋ ξ ℓ ξ ∈ C ∞ ( E ) , ℓ ξ ( v ) = h ξ, v i , while basic functions correspond to pullbacks of functions defined over the base, C ∞ ( M ) ∋ f q ∗ f ∈ C ∞ ( E ) . A linear Poisson structure on E → M is a Poisson structure {· , ·} on the totalspace E of the vector bundle E → M which satisfies the following:( i ) f, g linear ⇒ { f, g } linear,( ii ) f linear, g basic ⇒ { f, g } basic,( iii ) f, g basic ⇒ { f, g } = 0.Linear Poisson structures can be also described by means of Poisson bivector fields π ∈ Γ( ∧ T E ): a Poisson structure π on E is linear if and only if π : T ∗ E → T E yields a map of double vector bundles (see e.g. [18, Sec. 7.2]):(2.14) T ∗ E → E ∗ ↓ ↓ E → M π −−→ T E → T M ↓ ↓ E → M. An example is given by the canonical Poisson structure on E = T ∗ M .The following is an alternative characterization of linear Poisson structures via( R , · )-actions: Proposition 2.3.1.
Let E → M be a vector bundle, with regular action h : ( R , · ) y E , and let π be a Poisson bivector field on E . Then the Poisson structure is linearif and only if h λ : ( E, π ) → ( E, λπ ) is a Poisson map for all λ = 0 . Proof.
The linearity of the Poisson structure is equivalent to the condition thatthe map π : T ∗ E → T E intertwines the actions corresponding to the bundles T ∗ E → E ∗ and T E → T M . This means that, for each λ = 0, π ( λ (d h λ − ) ∗ ) = (d h λ ) π , (see Example 2.2.1 and (2.12)), which can be re-written as λπ = (d h λ ) π (d h λ ) ∗ ,exactly the condition for h λ : ( E, π ) → ( E, λπ ) being a Poisson map. (cid:3)
One can immediately apply the previous proposition to characterize double linearPoisson structures , i.e., Poisson structures on double vector bundles which are linearwith respect to both vector-bundle structures (see [13, Sec. 3.4]).A key fact, to be used recurrently throughout the paper, is the duality betweenlinear Poisson structures and Lie algebroids. For a vector bundle A → M , there isa one-to-one correspondence between Lie algebroid structures A ⇒ M and linearPoisson structures on the dual A ∗ → M (see e.g. [25, Chp. 10]), via the relations { ℓ X , ℓ Y } = ℓ [ X,Y ] , { ℓ X , q ∗ f } = q ∗ ( ρ ( X ) f ) , for X, Y ∈ Γ( A ), f ∈ C ∞ ( M ). As for the behavior of maps under this correspon-dence, if φ : ( A → M ) → ( A → M ) is a map between the underlying vectorbundles of given Lie algebroids, then it is a Lie algebroid map if and only if its dualrelation φ ∗ (see (2.11)) is coisotropic in A ∗ × A ∗ (here − indicates that A ∗ is equippedwith the opposite Poisson structure). We will discuss more general instances of thisduality in Section 5. Remark 2.3.2.
Recall that a Poisson manifold has an induced Lie-algebroid struc-ture on its cotangent bundle, and hence a linear Poisson structure on its tangentbundle, known as the tangent lift . For a vector bundle E equipped with a linearPoisson structure, there are induced Lie-algebroid structures on E ∗ and T ∗ E , andthese are compatible in the sense that the natural projection T ∗ E → E ∗ is a Lie-algebroid map (cf. [25, Prop. 10.3.6]). Moreover, the tangent-lift Poisson structureon T E is double linear (cf. [25, Thm. 10.3.14]).3.
VB-groupoids and VB-algebroids as regular actions
Just as Lie groupoids generalize both smooth manifolds and Lie groups, VB-groupoids simultaneously encompass vector bundles and representations of Lie groups,see e.g. [25, § R , · )-actions. In spite of the clear analogy betweenthe results, we point out that the arguments justifying them will often differ in spirit,reflecting the structural differences in how Lie algebroids and groupoids are defined.3.1. VB-groupoids. A VB-groupoid (cf. [14, Def. 3.3]) consists of Lie groupoidsΓ ⇒ E , G ⇒ M , and vector bundles Γ → G , E → M , forming a diagram(3.1) Γ ⇒ E ↓ ↓ G ⇒ M that is compatible in the following sense: the groupoid structural maps of Γ (source,target, multiplication, unit, inverse) cover the corresponding ones of G and arevector-bundle maps. (To consider the compatibility for the multiplication we view Γ × E Γ → G × M G as a vector subbundle of the product.) We will refer to Γ ⇒ E as the total groupoid and to G ⇒ M as the base groupoid .Given a VB-groupoid as above, its (right) core C → M is defined as the kernel ofthe vector-bundle map ( u ∗ G Γ → M ) s Γ → ( E → M ). The core plays a key role in thestructure of a VB-groupoid: there is a short exact sequence of bundles over G ,0 → t ∗ G C → Γ s Γ −→ s ∗ G E → , called the (right) core exact sequence , and any splitting of this sequence induces adecomposition of the VB-groupoid into the base groupoid, the vector bundles C and E , and some extra algebraic data [14] (see Remark 3.1.5).A VB-groupoid map (or morphism )Γ ⇒ E ↓ ↓ G ⇒ M → ˜Γ ⇒ ˜ E ↓ ↓ ˜ G ⇒ ˜ M is defined by a Lie groupoid map Φ between the total Lie groupoids which is linear.It follows that the restriction Φ | E : E → ˜ E is also linear, Φ | G : G → ˜ G is a Liegroupoid map, and core bundles are preserved.We denote by VB( G ⇒ M ) the category of VB-groupoids over G ⇒ M , withmorphisms being VB-groupoid maps restricting to the identity on G ⇒ M . Remark 3.1.1. (a) The core exact sequence implies that each source-fiber of Γ is an affine bundleover a source-fiber of G , and hence Γ is source-connected, or source-simply-connected, if and only if so is G .(b) The core exact sequence is natural with respect to maps, from where one verifiesthat a VB-groupoid map Φ as above is fiberwise injective (resp. surjective) ifand only if it is so when restricted to the side bundle E and the core C . Example 3.1.2 (Tangent groupoid) . Given a Lie groupoid G ⇒ M , its tangentgroupoid T G ⇒ T M is defined by differentiating the structural maps of G ⇒ M . Itis naturally a VB-groupoid over G ⇒ M , whose core A G → M is the vector bundleof the Lie algebroid of G . One can readily check that the passage from Lie groupoidsto their tangent groupoids is functorial. Example 3.1.3 (Cotangent groupoid) . Given a Lie groupoid G ⇒ M , it is alsopossible to define its cotangent groupoid . Its space of arrows consists of the cotangentbundle T ∗ G , and its objects are given by the dual of the core, A ∗ G . The structuralmaps of T ∗ G ⇒ A ∗ G are described, e.g., in [25, § T ∗ G ⇒ A ∗ G into a VB-groupoid over G ⇒ M , with core T ∗ M → M . Example 3.1.4 (Representations) . Given a representation of a Lie groupoid G ⇒ M on a vector bundle E → M , the corresponding action groupoid G × M E ⇒ E isnaturally a VB-groupoid over G ⇒ M , whose core is trivial. One may directly verifythat every VB-groupoid with trivial core arises in this way.There is a duality construction for VB-groupoids, for which the cotangent groupoid T ∗ G ⇒ A ∗ G is the dual of T G ⇒ T M . Given a VB-groupoid Γ as in (3.1), its dualVB-groupoid has the same base as (3.1) and total groupoid with arrows Γ ∗ and objects C ∗ , the total space of the dual of the core bundle C → M ,(3.2) Γ ∗ ⇒ C ∗ ↓ ↓ G ⇒ M. For the definition of the groupoid structural maps on Γ ∗ ⇒ C ∗ , we refer to [25, § → ˜Γ covering the identity G → G , thedual map Φ ∗ : ˜Γ ∗ → Γ is a VB-groupoid map, so taking duals defines an involutivefunctor on VB( G ⇒ M ). Remark 3.1.5.
As suggested by Example 3.1.4, one can think of VB-groupoids asgeneralized representations [14]. By choosing a splitting of the core exact sequenceof (3.1), one can associate to every arrow in G ⇒ M a linear map between the fibersof the complex C t Γ −→ E , and this yields a representation up to homotopy [2]. It isproven in [14] that there is a one-to-one correspondence between isomorphism classesof VB-groupoids and of 2-term representations up to homotopy.3.2. Characterization as multiplicative actions.
We will now use Thm. 2.1.2 toestablish a characterization of VB-groupoids by means of ( R , · )-actions, simplifyingtheir original definition. Definition 3.2.1.
An ( R , · )-action h on the space of arrows of a Lie groupoid Γ ⇒ E is called multiplicative if h λ : Γ → Γ is a groupoid map for each λ ∈ R .Note that the restriction h | E is an ( R , · )-action on E , which is regular if h is. Itis often convenient to think of h as a pair of actions on Γ and E , in which case weuse the notation ( h Γ , h E ).Recall that the fixed points of an ( R , · )-action h , i.e., the image of h , define anembedded submanifold. For a multiplicative action, we have the following: Lemma 3.2.2.
Let h = ( h Γ , h E ) be a multiplicative ( R , · ) -action on Γ ⇒ E . Thenits fixed points define an embedded Lie subgroupoid h Γ0 (Γ) ⇒ h E ( E ) of Γ ⇒ E .Moreover, this Lie subgroupoid is source-simply-connected whenever Γ is.Proof. Write G = h Γ0 (Γ) and M = h E ( E ). Since h Γ and h E are ( R , · )-actions, wehave that G ⊂ Γ and M ⊂ E are embedded submanifolds. Using that ( h Γ0 , h E ) is agroupoid map, one may verify that G and M define a set-theoretic subgroupoid ofΓ ⇒ E . It remains to check that the source map of G ⇒ M is a submersion (cf.Section A.1), which follows from the identity s G ◦ h Γ0 = h E ◦ s Γ . As for the secondassertion in the lemma, for each x ∈ M the map h Γ0 defines a retraction s − ( x ) → s − G ( x ), so it induces a surjective map at the level of fundamental groups. (cid:3) The next theorem characterizes VB-groupoids by means of regular actions. Wewill provide a direct proof of this result now, and discuss it from a broader viewpointin the next section.
Theorem 3.2.3.
There is a one-to-one correspondence between VB-groupoids withtotal groupoid Γ ⇒ E and regular multiplicative actions ( R , · ) y (Γ ⇒ E ) .Proof. Clearly every VB-groupoid has an underlying multiplicative ( R , · )-action thatis regular. Conversely, if we start with a regular multiplicative action h : ( R , · ) y (Γ ⇒ E ), and we write G = h Γ0 (Γ) and M = h E ( E ), then by Thm. 2.1.2 and Lemma 3.2.2 we obtain a diagram of Lie groupoids and vector bundles as in (3.1).It only remains to check the compatibility between these structures. The fact that h λ is a groupoid map for every λ implies that the structural maps of the groupoidstructure are equivariant, hence maps of vector bundles, and the result follows. (cid:3) Using this characterization we see that a Lie subgroupoid of a VB-groupoid that isinvariant for the ( R , · )-action is also a VB-groupoid, and that a VB-groupoid map isthe same as a groupoid map between the total Lie groupoids that is ( R , · )-equivariant.It is also clear that the direct product of VB-groupoids is a VB-groupoid: for VB-groupoids Γ i ⇒ E i over G i ⇒ M i , i = 1 ,
2, the direct product of the ( R , · )-actions onthe groupoid Γ × Γ ⇒ E × E makes it into a VB-groupoid over G × G ⇒ M × M .Next we list a few other useful consequences of Thm. 3.2.3. Corollary 3.2.4. (a) Let Γ , Γ and Γ be VB-groupoids, and let Φ i : Γ i → Γ , i = 1 , , be VB-groupoidmaps forming a good pair (cf. Lemma A.1.3). Then their fibred product is aVB-groupoid.(b) Given a VB-groupoid Γ ⇒ E over G ⇒ M and a Lie-groupoid map (Φ , φ ) :( ˜ G ⇒ ˜ M ) → ( G ⇒ M ) , the fibred product (Φ ∗ Γ ⇒ φ ∗ E ) (cid:15) (cid:15) / / (Γ ⇒ E ) (cid:15) (cid:15) ( ˜ G ⇒ ˜ M ) (Φ ,φ ) / / ( G ⇒ M ) endows the pullback vector bundles Φ ∗ Γ and φ ∗ E with a VB-groupoid structureover ˜ G ⇒ ˜ M .Proof. We know by Prop. A.1.4 that the fibred product of Φ i : Γ i → Γ, i = 1 ,
2, is aLie subgroupoid Γ × Γ Γ ⇒ E × E E of Γ × Γ ⇒ E × E . The fact that the mapsΦ i are vector-bundle morphisms implies that this Lie subgroupoid is ( R , · )-invariantwith respect to the direct-product action, so it inherits a VB-groupoid structure. Itis also evident that the projections Γ × Γ Γ → Γ i , i = 1 ,
2, are VB-groupoid maps.Finally, ( b ) follows from ( a ), since both ˜ G Φ −→ G and the projection Γ → G areVB-groupoid maps with respect to the VB-groupoids over G and ˜ G with zero fibers,and the projection is a submersion, hence transverse to any map, thus ensuring thegood pair property. (cid:3) Remark 3.2.5.
For a VB-groupoid map (Φ , φ ) : (Γ ⇒ E ) → (˜Γ ⇒ ˜ E ), assume thatthe underlying vector-bundle map Φ : (Γ → G ) → (˜Γ → ˜ G ) has constant rank. Then φ : ( E → M ) → ( ˜ E → ˜ M ) also has constant rank, and one can directly check thatker(Φ) and ker( φ ) define a VB-groupoid ker(Φ) ⇒ ker( φ ) over G ⇒ M , cf. [19, App.C]. (Note that this VB-groupoid may be seen, following Cor. 3.2.4(a), as the fibredproduct of the good pair given by Φ and the zero section ˜ G → ˜Γ.) Remark 3.2.6.
Just as the usual category of vector bundles over a manifold, thecategory VB( G ⇒ M ) of VB-groupoids over G ⇒ M is an additive category (it isequipped with linear structures on hom-sets, zero objects and direct sums). More-over, we can take kernels and cokernels of VB-groupoid maps that have constant rank and consider short exact sequences of VB-groupoids. (The construction of ker-nels is discussed in the previous remark, and for cokernels one may compute thekernels of the dual maps, and dualize again.) Remark 3.2.7.
Given a Lie-groupoid map Φ : ( ˜ G ⇒ ˜ M ) → ( G ⇒ M ), the pullbackconstruction from Cor. 3.2.4(b) defines a base-change functor(3.3) VB( G ⇒ M ) Φ ∗ −−→ VB( ˜ G ⇒ ˜ M ) , which preserves short exact sequences and duals (i.e., the natural vector-bundleidentifications (Φ ∗ Γ) ∗ ∼ = Φ ∗ Γ ∗ are isomorphisms of VB-groupoids). Moreover, theproperty that any vector-bundle map Ψ : (Γ → G ) → (˜Γ → ˜ G ) induces a vector-bundle map Γ → (Ψ | G ) ∗ ˜Γ over the identity G → G also extends: when Γ and ˜Γ areVB-groupoids and Ψ is a VB-groupoid map, so is this induced map.3.3. VB-algebroids. A VB-algebroid [23, 13] consists of a double vector bundle(Ω , E, A, M ) equipped with a Lie-algebroid structure Ω ⇒ E that is compatiblewith the second vector-bundle structure on Ω in the following sense: we require thelinear Poisson structure on Ω ∗ E , corresponding to Ω ⇒ E , to be also linear withrespect to the fibration Ω ∗ E → C ∗ (see [13, Sec. 3.4]); in other words, it is a doublelinear Poisson structure on the horizontal dual,(3.4) Ω ∗ E → E ↓ ↓ C ∗ → M. One can check that a double linear Poisson structure on Ω ∗ E induces a unique Lie-algebroid structure on A → M so that the projection q Ω A (and also the zero section0 Ω A ) is a Lie-algebroid map, see [13, Sec. 3] (cf. Lemma 3.4.2 and Thm. 3.4.3 below).For this reason, we depict a VB-algebroid as follows:(3.5) Ω ⇒ E ↓ ↓ A ⇒ M. We refer to Ω ⇒ E as the total algebroid , while A ⇒ M is the base algebroid , andwe say that Ω ⇒ E is a VB-algebroid over A ⇒ M . The core of a VB-algebroid isthat of the underlying double vector bundle.Other formulations of VB-algebroids and their equivalences can be found in [13].A VB-algebroid map is a Lie-algebroid morphism Φ : (Ω ⇒ E ) → ( ˜Ω ⇒ ˜ E ) thatis also linear, i.e., it defines a vector-bundle map (Ω → A ) → ( ˜Ω → ˜ A ). As aconsequence, one can check that Φ restricts to a Lie-algebroid map ( A ⇒ M ) → ( ˜ A ⇒ ˜ M ). As in the case of VB-groupoids, a VB-algebroid map determines foursmooth maps relating each of the involved manifolds and preserving all structures.We denote by VB( A ⇒ M ) the category of VB-algebroids over A ⇒ M , with VB-groupoid maps covering the identity as morphisms. Example 3.3.1 (Tangent algebroid) . If A ⇒ M is a Lie algebroid, then it induces aLie-algebroid structure on T A → T M , often referred to as the tangent prolongation of A ; in this example, the anchor and the bracket between tangent sections areobtained by differentiating the corresponding structures in A , see [25, § T A ⇒ T M is a VB-algebroid over A ⇒ M . Example 3.3.2 (Cotangent algebroid) . For a Lie algebroid A ⇒ M , the correspond-ing linear Poisson structure on A ∗ induces a Lie-algebroid structure T ∗ A ∗ ⇒ A ∗ ,which is a VB-algebroid over A ⇒ M (c.f. Remark 2.3.2). Using the reversal iso-morphism (2.7), we then obtain a VB-algebroid T ∗ A ⇒ A ∗ over A ⇒ M . Example 3.3.3 (Representations) . Any representation of a Lie algebroid A ⇒ M gives rise to a VB-algebroid over A , for which the ranks of Ω → A and E → M are equal; in fact there is a one-to-one correspondence between representationsof A and VB-algebroids with this additional property. Just as VB-groupoids (c.f.Remark 3.1.5), VB-algebroids are thought of as generalized representations, sincethey are similarly related to 2-term representations up to homotopy, see [13].Given a VB-algebroid (3.5), while its horizontal dual (3.4) carries a double linearPoisson structure, its vertical dual is again a VB-algebroid [13, Sec. 3.4]:(3.6) Ω ∗ A ⇒ C ∗ ↓ ↓ A ⇒ M. Indeed, using the linearity of the Poisson structure relative to Ω ∗ E → C ∗ , we see that(Ω ∗ E ) ∗ C ∗ ⇒ C ∗ is a VB-algebroid over A ⇒ M . The VB-algebroid (3.6) is definedby means of the isomorphism Z in (2.9), and we refer to it as the dual VB-algebroid of Ω ⇒ E . As in the case of VB-groupoids, duality defines an involution in thecategory VB( A ⇒ M ). Example 3.3.4.
The tangent and cotangent Lie algebroids, see Examples 3.3.1and 3.3.2, are dual VB-algebroids in the sense just described. In other words, Z :( T ∗ A ⇒ A ∗ ) → (( T A ) ∗ T M ) ∗ A ∗ ⇒ A ∗ ) is a Lie-algebroid map. This can be verifiedby noticing that the composition R ◦ Z − , where R is the reversal isomorphism (see[25, Prop. 9.3.2 & § R isa Lie-algebroid map by definition, so is Z .3.4. Characterization as IM actions.
We now discuss the Lie-algebroid version ofTheorem 3.2.3. We start with the infinitesimal counterpart of multiplicative actions:
Definition 3.4.1.
An ( R , · )-action h on the total space of a Lie algebroid Ω ⇒ E is called infinitesimally multiplicative (or simply IM ) if each h λ : Ω → Ω defines aLie-algebroid map.As in the case of VB-groupoids, the restriction of h to E is an ( R , · )-action, whichis regular if h is, and we can think of h as a pair of actions ( h Ω , h E ), on Ω and E .We now consider fixed points of IM-actions. The discussion parallels Lemma 3.2.2,though its direct proof cannot be adapted to Lie algebroids. So we provide analternative argument using fibred products, which works in both contexts. Lemma 3.4.2. If h = ( h Ω , h E ) is an IM-action on Ω ⇒ E , then its fixed pointsdefine an embedded Lie subalgebroid h Ω0 (Ω) ⇒ h E ( E ) of Ω ⇒ E .Proof. Write A = h Ω0 (Ω) and M = h E ( E ). Since h Ω and h E are ( R , · )-actions, A ⊂ Ωand M ⊂ E are embedded submanifolds. One then verifies that the following pair of Lie-algebroid maps is good in the sense of the Appendix (see Lemma A.2.3):(3.7) (Ω ⇒ E ) ( h Ω0 ,h E ) × (id Ω , id E ) (cid:15) (cid:15) (Ω ⇒ E ) ∆ / / (Ω ⇒ E ) × (Ω ⇒ E ) , where ∆ is the diagonal map. It follows from Prop. A.2.4 that their algebroid-theoretic fibred product is well-defined. The natural identification of this fibredproduct with A → M makes it into a Lie subalgebroid of Ω ⇒ E . (cid:3) The following result characterizes VB-algebroids in terms of IM-actions:
Theorem 3.4.3.
There is a one-to-one correspondence between VB-algebroids withtotal Lie algebroid Ω ⇒ E and regular IM actions ( R , · ) y (Ω ⇒ E ) .Proof. We know that a regular action h on Ω → E by vector-bundle maps is the sameas a double vector bundle structure (Ω , E, A, M ), where A = h Ω0 (Ω) and M = h E ( E )(cf. Prop. 2.2.4). So what we need to show is that h acts by Lie-algebroid maps ifand only if the compatibility condition defining VB-algebroids is fulfilled.Let k be the action corresponding to Ω → E , and let ¯ h, ¯ k be the ( R , · )-actionsassociated to Ω ∗ E → C ∗ and Ω ∗ E → E , respectively. For each λ = 0, by Prop. 2.2.5we know that ¯ h λ = ( h λ − ) ∗ ◦ ¯ k λ . Letting π denote the Poisson bivector field onΩ ∗ E corresponding to the Lie-algebroid structure Ω ⇒ E , we consider the followingdiagram: (Ω ∗ E , π ) ¯ h λ % % ▲▲▲▲▲▲▲▲▲▲ ¯ k λ / / (Ω ∗ E , λπ ) h ∗ λ − x x qqqqqqqqqq (Ω ∗ E , λπ ) . First, we notice that it is commutative as a diagram of smooth maps. In addition,the map ¯ k λ is Poisson as a result of the linearity of π over E (see Prop. 2.3.1). Itthen follows that the left arrow is a Poisson map if and only if so is the right arrow.Note that ¯ h λ : (Ω ∗ E , π ) → (Ω ∗ E , λπ ) is a Poisson map for every λ = 0 if and only ifthe Poisson structure π on Ω ∗ E is linear with respect to Ω ∗ E → C ∗ (cf. Prop. 2.3.1),and this is by definition the same as Ω ⇒ E being a VB-algebroid. On the otherhand, the map h ∗ λ − : (Ω ∗ E , λπ ) → (Ω ∗ E , λπ ) is Poisson if and only if h ∗ λ − is a Poissonautomorphism of (Ω ∗ E , π ), or equivalently, h λ − is an algebroid map. Note that ifthis holds for every λ = 0 then it is also true for λ = 0, by passing to the limit. (cid:3) As previously mentioned, on a VB-algebroid the base Lie algebroid is determinedby the total algebroid. Note that Lemma 3.4.2 explains this fact from the point ofview of ( R , · )-actions.Just as in the discussion for VB-groupoids, we conclude that a Lie subalgebroidof a VB-algebroid that is invariant under the ( R , · )-action is a VB-algebroid itself. Itis also clear that a VB-algebroid map is a Lie-algebroid map between the total Liealgebroids that is ( R , · )-equivariant, and that the direct product of VB-algebroids isa VB-algebroid. In the remainder of this section we collect other consequences ofThm. 3.4.3, which are parallel to those for groupoids presented after Thm. 3.2.3. Corollary 3.4.4. (a) Given VB-algebroids Ω , Ω and Ω and VB-algebroid maps Ω i → Ω , i = 1 , ,if the maps form a good pair (cf. Lemma A.2.3), then their fibred product is aVB-algebroid.(b) Given a VB-algebroid Ω ⇒ E over A ⇒ M and a Lie-algebroid map (Φ , φ ) :( ˜ A ⇒ ˜ M ) → ( A ⇒ M ) , the fibred product (Φ ∗ Ω ⇒ φ ∗ E ) (cid:15) (cid:15) / / (Ω ⇒ E ) (cid:15) (cid:15) ( ˜ A ⇒ ˜ M ) (Φ ,φ ) / / ( A ⇒ M ) endows the pullback vector bundles Φ ∗ Ω and φ ∗ E with a VB-algebroid structureover ˜ A ⇒ ˜ M .Proof. The proofs are completely analogous to those in Corollary 3.2.4, but nowmaking use of Prop. A.2.4 rather than Prop. A.1.4. (cid:3)
Remark 3.4.5.
The category VB( A ⇒ M ) is additive, and one can also considerkernels and co-kernels: given a VB-algebroid map (Φ , φ ) : (Ω ⇒ E ) → ( ˜Ω ⇒ ˜ E ),if it has constant rank as a map of vector bundles Φ : (Ω → A ) → ( ˜Ω → ˜ A ) (thesame automatically holds for φ : ( E → M ) → ( ˜ E → ˜ M )), then its kernel naturallyinherits the structure of a VB-algebroid ker(Φ) ⇒ ker( φ ) over A ⇒ M . This maynot be as direct to check as in the groupoid case, but one may use the same argumentsketched in the end of Remark 3.2.5: the map Φ and the zero section ˜ A → ˜Ω form agood pair of VB-algebroid maps (considering the zero vector bundle over ˜ A ), so byCor. 3.4.4(a) one can realize the kernel of Φ as the fibred-product VB-algebroid(3.8) (ker(Φ) ⇒ ker( φ )) / / (cid:15) (cid:15) (Ω ⇒ E ) (Φ ,φ ) (cid:15) (cid:15) ( ˜ A ⇒ ˜ M ) (0 ˜Ω , ˜ E ) / / ( ˜Ω ⇒ ˜ E ) . The construction of cokernels follows from duality.
Remark 3.4.6.
Through the pullback construction of Cor. 3.4.4(b), each Lie-algebroidmap Φ : ( ˜ A ⇒ ˜ M ) → ( A ⇒ M ) gives rise to a base-change functor(3.9) VB( A ⇒ M ) Φ ∗ −−→ VB( ˜ A ⇒ ˜ M )preserving short exact sequences and duals. If Φ : Ω → ˜Ω is a VB-algebroid map,then the induced vector-bundle map Ω → (Φ | A ) ∗ ˜Ω, covering the identity map on A → M , is a also VB-algebroid map.4. Lie theory for vector bundles
A differentiation procedure (see e.g. [25, 29]) gives rise to a functor(4.1) Lie Groupoids
Lie −−→
Lie Algebroids . It is well known that this Lie functor is not an equivalence of categories, so a perfecttranslation between the global and infinitesimal pictures is not always possible. For a Lie groupoid G we use the notation A G = Lie( G ) and say that G integrates A . It is a fact that not every Lie algebroid comes from a Lie groupoid, see [9] for adiscussion of the integrability problem.For a morphism Φ : G → G we often write Φ ′ = Lie(Φ) : A G → A G to simplifythe notation. Upon an additional topological assumption, namely if G is a source-simply connected, the Lie functor sets a bijection between groupoid maps G → G and algebroid maps A G → A G . This is the content of Lie’s second theorem (seee.g. [29, Sec. 6.3]), that will be used recurrently in this paper.
Remark 4.0.7. If G is just source-connected, we still have injectivity: if Φ , Ψ : G → G are groupoid maps such that Φ ′ = Ψ ′ , then necessarily Φ = Ψ.In this section we use Theorems 3.2.3 and 3.4.3 to explain how VB-groupoids andVB-algebroids are related by differentiation and integration.4.1. Differentiation of VB-groupoids.
In order to relate VB-groupoids and VB-algebroids by the Lie functor, it will be convenient to consider the following alterna-tive formulations of multiplicative and IM actions.Denoting by R ⇒ R the unit groupoid of the real line R , one can directly see thata multiplicative action h : ( R , · ) y (Γ ⇒ E ) is the same as a Lie groupoid map(4.2) h : (Γ ⇒ E ) × ( R ⇒ R ) → (Γ ⇒ E )satisfying h = id and h λ h λ ′ = h λλ ′ , for all λ, λ ′ ∈ R . Analogously, if 0 R ⇒ R isthe zero Lie algebroid over the real line, then an IM action h : ( R , · ) y (Ω ⇒ E ) isequivalent to a Lie-algebroid map(4.3) h : (Ω ⇒ E ) × (0 R ⇒ R ) → (Ω ⇒ E )satisfying h = id and h λ h λ ′ = h λλ ′ , for all λ, λ ′ ∈ R . Proposition 4.1.1.
Let h : ( R , · ) y (Γ ⇒ E ) be a multiplicative action. Then:(a) The map A Γ × R → A Γ given by ( a, λ ) ( h λ ) ′ ( a ) defines an IM-action h ′ on A Γ ⇒ E .(b) If G ⇒ M denotes the Lie subgroupoid of Γ ⇒ E given by the fixed points of h ,then the fixed points of h ′ are identified with A G ⇒ M .(c) If h is a regular action, then so is h ′ .Proof. The Lie functor (4.1) preserves products and maps R ⇒ R to 0 R ⇒ R .So viewing the action h as a groupoid map as in (4.2), it is immediate that bydifferentiation we obtain a Lie-algebroid map h ′ : A Γ × R → A Γ . One can alsodirectly check that h ′ satisfies ( h ′ ) λ = ( h λ ) ′ for all λ ∈ R , from where we see that h ′ = id, h ′ λ h ′ µ = h ′ λµ . So ( a ) follows.Regarding ( b ), we can express the fixed points of h , i.e., the image of h , asthe good fibred product between the map ( h , id Γ ) : Γ → Γ × Γ and the diagonal∆ Γ : Γ → Γ × Γ (c.f. (3.7)). The same holds for the fixed points of h ′ , now consideringthe maps ( h ′ , id A Γ ) : A Γ → A Γ × A Γ and the diagonal ∆ A Γ . Note that these mapson A Γ correspond to the maps previously defined on Γ by the Lie functor. Theconclusion follows from the fact thet the Lie functor preserves fibred products, asshown in Prop. A.3.1.Finally, ( c ) holds because h ′ is a restriction of the tangent lift action d h : ( R , · ) y T Γ, which is regular if h is. (cid:3) The previous proposition, together with our characterizations of VB-groupoidsand VB-algebroids in Theorems 3.2.3 and 3.4.3, lead to:
Corollary 4.1.2. If Γ ⇒ E is a VB-groupoid over G ⇒ M , then A Γ ⇒ E inheritsa VB-algebroid structure over A G ⇒ M . Remark 4.1.3.
By viewing vector bundles as particular cases of Lie groupoids andLie algebroids, one may view VB-groupoids (resp. VB-algebroids) as special typesof double Lie groupoids (resp. LA-groupoids); Corollary 4.1.2 then also follows fromthe fact that double Lie groupoids can be differentiated to LA-groupoids [20].Corollary 4.1.2 is part of a more general observation: since VB-groupoid and VB-algebroid maps are characterized by ( R , · )-equivariance, it is a direct verification thatthe Lie functor (4.1) restricts to a functor(4.4) VB( G ⇒ M ) Lie −−→
VB( A G ⇒ M ) . Remark 4.1.4.
The functor (4.4) satisfies the following natural properties, thatwe explicitly describe for later use:(a) It commutes with the pullback functors defined in (3.3) and (3.9): If Φ : G → G is a Lie-groupoid map and Γ is a VB-groupoid over G , then the identificationin Prop. A.3.1 yields an isomorphism of VB-algebroids over A G , r Γ : A Φ ∗ Γ → (Φ ′ ) ∗ A Γ . which is natural, namely r Γ ◦ (Φ ∗ (Ψ)) ′ = ((Φ ′ ) ∗ (Ψ ′ )) ◦ r Γ for any Ψ : Γ → Γ .(b) It preserves duals as described in (3.2) and (3.6): given a VB-groupoid Γ over G , by differentiating the canonical pairing Γ ∗ × G Γ → R we obtain a naturalisomorphism of VB-algebroids (c.f.[25, Thm 11.5.12]) i Γ : A Γ ∗ → ( A Γ ) ∗ A G . For a VB-groupoid map Ψ : Γ → Γ over G ⇒ M we have (Ψ ′ ) ∗ ◦ i Γ = i Γ ◦ (Ψ ∗ ) ′ .(c) It maps tangent VB-groupoids to tangent VB-algebroids upon the identificationgiven by restriction of the natural involution of the double tangent bundle (see[25, Thm. 9.7.5]), j G : T A G → A T G . This fact, combined with the previous item (b), shows that the Lie functor (4.4)also maps cotangent VB-groupoids to cotangent VB-algebroids, via θ G = j ∗ G i T G : A T ∗ G → T ∗ A G . Given a Lie groupoid map Φ : G → G , the naturality of these identificationsis expressed by the following equations:(4.5) (dΦ) ′ ◦ j G = j G ◦ d(Φ ′ ) , (d(Φ ′ )) ∗ ◦ θ G = θ G ◦ ((dΦ) ∗ ) ′ . (d) The Lie functor preserves short exact sequences, in particular kernels and coker-nels. One can check that it preserves kernels by expressing the kernel of a mapΦ : Γ → Γ as the good fibred product between the map itself and the zerosection of Γ , and using Prop. A.3.1. The fact that it preserves cokernels follows,for instance, from this property for kernels and the behavior under duality. The vertical lift for multiplicative and IM actions.
In order to study theintegration of VB-algebroids, i.e., the inverse procedure to Corollary 4.1.2, it willbe convenient to reformulate Theorems 3.2.3 and 3.4.3 as a more general functorialconstruction, building on Theorem 2.1.2.Given a multiplicative action h : ( R , · ) y (Γ ⇒ E ), not necessarily regular, weknow that its fixed points define a Lie subgroupoid G ⇒ M (by Lemma 3.2.2). Theaction has an associated vertical bundle V h Γ → G (see (2.2)), while the restrictionof h to E gives rise to the vertical bundle V h E → M . A key observation is that V h Γ is a Lie groupoid over V h E , and V h Γ ⇒ V h E is a VB-groupoid over G ⇒ M ;these facts follow from the constructions of kernels and pullbacks in Remark 3.2.5 andCor. 3.2.4(b), since V h Γ is obtained by restricting to G the kernel of the VB-groupoidmap T h : T Γ → T Γ. We refer to the VB-groupoid(4.6) V h Γ ⇒ V h E over the fixed points G ⇒ M as the vertical bundle of the multiplicative action h .Note that, by construction, V h Γ naturally sits in the tangent VB-groupoid T Γ ⇒ T E as a VB-subgroupoid.The following result offers a more general viewpoint to Theorem 3.2.3.
Proposition 4.2.1.
Let Γ ⇒ E be a Lie groupoid endowed with a multiplicativeaction h : ( R , · ) y (Γ ⇒ E ) . Then:(a) The vertical lift V h : (Γ ⇒ E ) → ( V h Γ ⇒ V h E ) is a Lie-groupoid morphismwhich is ( R , · ) -equivariant;(b) The action h is regular if and only if V h is an isomorphism onto the verticalbundle V h Γ , in which case Γ ⇒ E inherits the structure of a VB-groupoid.Proof. For ( a ), since V h Γ ⇒ V h E is a Lie subgroupoid of T Γ ⇒ T E , and in light ofRemark 2.1.4, we just need to show that the composition d h ◦ l is a Lie-groupoid map,where l : Γ → T Γ × T R is defined as in (2.4). The tangent construction is functorial,so d h is a Lie-groupoid map (since so is (4.2)). Moreover, each component of l is aLie-groupoid map, since the first one is the zero section of the tangent VB-groupoidof Γ ⇒ E , while the second is a constant map into the unit groupoid T R ⇒ T R .The assertion in ( b ) is an immediate consequence of Theorem 2.1.2. (cid:3) There is an analogous result for IM actions on Lie algebroids. Given an IM action h : ( R , · ) y (Ω ⇒ E ), not necessarily regular, denote by A ⇒ M the Lie algebroiddefined by the fixed points of h (c.f. Lemma 3.4.2). We have a natural VB-algebroid V h Ω ⇒ V h E over A ⇒ M . As in the case of groupoids, it is a VB-subalgebroid of T Ω ⇒ T E .These results are direct consequences of our observations on pullbacks and kernelsin Cor 3.4.4(b) and Remark 3.4.5. We refer to the VB-algebroid V h Ω ⇒ V h E as the vertical bundle of the IM action h . Reasoning as in Proposition 4.2.1, we obtain: Proposition 4.2.2.
Let Ω ⇒ E be a Lie algebroid and h : ( R , · ) y (Ω ⇒ E ) an IMaction. Then:(a) The vertical lift V h : (Ω ⇒ E ) → ( V h Ω ⇒ V h E ) is Lie-algebroid morphism whichis ( R , · ) -equivariant;(b) The action h is regular if and only if V h is an isomorphism onto the verticalbundle V h Ω . In this case Ω ⇒ E inherits the structure of a VB-algebroid. As in Remark 2.1.3, the last two propositions define regularization functors fromthe categories of multiplicative actions and IM actions to the categories of VB-groupoids and VB-algebroids, respectively. The vertical bundle is the regular objectassociated to an action in each case.The following result clarifies the relation between vertical lifts and the Lie functor.
Proposition 4.2.3.
Let h : ( R , · ) y (Γ ⇒ E ) be a multiplicative action, and let h ′ :( R , · ) y (Ω ⇒ E ) be the corresponding IM action. Then the canonical isomorphism j Γ : T A Γ → A T Γ restricts to an isomorphism V h ′ A Γ ∼ −→ A V h Γ so that j Γ ◦ V h ′ = ( V h ) ′ : A Γ V h ′ / / ( V h ) ′ ❋❋❋❋❋❋❋❋❋ V h ′ A Γ j Γ ≀ (cid:15) (cid:15) A V h Γ . Proof.
Let us view the action h as in (4.2) and consider the factorization of V h asin Remark 2.1.4; we write V h = d h ◦ l Γ , recalling that l Γ : Γ → T Γ × T R is a Lie-groupoid map. There is an analogous factorization associated with h ′ , that we writeas V h ′ = d( h ′ ) ◦ l A , and l A : A Γ → T A Γ × T R is a Lie-algebroid map. By consideringeach component of the map l Γ : Γ → T Γ × T R (and recalling that both tangent andLie functors respect direct products), one may directly check that l ′ Γ = j Γ × R ◦ l A : A Γ → A T Γ × T R . By using the naturality of j (cf. Remark 4.1.4,(c)), we conclude that( V h ) ′ = (d h ) ′ ◦ l ′ Γ = (d h ) ′ ◦ j Γ × R ◦ l A = j Γ ◦ d( h ′ ) ◦ l A = j Γ ◦ V h ′ . (cid:3) Integration.
Assuming that the total Lie algebroid of a VB-algebroid is in-tegrable, the issue discussed in this subsection is whether it is integrated by aVB-groupoid. Following Theorems 3.2.3 and 3.4.3, the problem of integrating VB-algebroids can be viewed in two steps: first integrating IM actions to multiplicativeactions, and then dealing with the additional regularity condition.The first step, integration of IM actions, is handled by Lie’s second theorem,recalled in the beginning of the section.
Lemma 4.3.1.
Let Γ ⇒ E be a source-simply-connected Lie groupoid with Lie alge-broid Ω ⇒ E . Then any IM action ˜ h : ( R , · ) y (Ω ⇒ E ) integrates to a multiplicativeaction h : ( R , · ) y (Γ ⇒ E ) , in a way such that ( h λ ) ′ = ˜ h λ , for all λ .Proof. Viewing the IM action as a Lie-algebroid map ˜ h : Ω × R → Ω (c.f. (4.3)),and since the source-fibers of Γ × R are diffeomorphic to those of Γ, we can integrate˜ h via Lie’s second theorem to obtain a Lie-groupoid map h : Γ × R → R (as in(4.2)) such that h ′ = ˜ h . The uniqueness of the integration of maps implies that( h ′ ) λ = ( h λ ) ′ = ˜ h λ , from where the action axioms for h directly follow. (cid:3) The next example illustrates the relevance of the source-simply-connectednesshypothesis in the previous lemma.
Example 4.3.2.
Let R ⇒ ∗ be the 1-dimensional Lie algebra, viewed as a VB-algebroid over the point ∗ ⇒ ∗ . If we take Γ = S as the Lie group integrating R , then it is not possible to integrate the action by homotheties ˜ h as in Lemma 4.3.1,since its only fixed point h ( S ) = ∗ (see Prop. 4.1.1(b)) would have to be a retractof S , which cannot happen.We now address the second step, that of regularity, by proving the converse toProp. 4.1.1(c). Proposition 4.3.3.
Let Γ ⇒ E be a Lie groupoid equipped with a multiplicativeaction h : ( R , · ) y (Γ ⇒ E ) , and let h ′ the corresponding IM action on A Γ . If h ′ isregular then so is h .Proof. In the commutative triangle of Proposition 4.2.3, we know that V h ′ is anisomorphism because h ′ is regular, so ( V h ) ′ : A Γ → A V h Γ is an isomorphism as well.If we assume that Γ ⇒ E is source-simply connected, then by Lemma 3.2.2 weknow that G = h (Γ) is source-simply-connected, and hence so is V h Γ (since it isa VB-groupoid over G , see Remark 3.1.1). It then follows that V h : Γ → V h Γmust be an isomorphism, showing that h is regular. Not assuming source-simply-connectedness of Γ, we need a more elaborate argument.The key observation is that a groupoid map Φ : Γ → Γ is ´etale, i.e., its differ-ential is invertible at all points, if and only if Φ ′ : A Γ → A Γ is an isomorphism.To see that, consider the induced VB-groupoid map dΦ : T Γ → T Γ and use Re-mark 3.1.1(b). In our case, we conclude that the groupoid map V h : Γ → V h Γ is´etale. The proof ends with the observation that this cannot happen for the verticallift corresponding to a non-regular action. Indeed, if h is not regular, then thereexists z ∈ Γ such that h ( z ) = z = h ( z ) and V h ( z ) = 0. In particular, the curve λ h λ ( z ) is not constant, so there is a point with non-zero velocity vector X . But V h ( h λ ( z )) = 0 for all λ and therefore the differential of V h vanishes on X . (cid:3) For a VB-algebroid Ω ⇒ E over A ⇒ M , since A sits in Ω as a Lie subalgebroid,the assumption that Ω ⇒ E is integrable implies that so is A ⇒ M , see e.g. [29,Prop. 6.7] (the integrability of A also follows from Lemma 4.3.1 and Prop. 4.1.1(b)).Combining our last two results we have the integration of VB-algebroids: Theorem 4.3.4.
Let Ω ⇒ E be a VB-algebroid over A ⇒ M , so that Ω ⇒ E is integrable. Then its source-simply-connected integration Γ ⇒ E carries a VB-groupoid structure over the source-simply-connected Lie groupoid G ⇒ M integrating A ⇒ M , (4.7) Γ ⇒ E ↓ ↓ G ⇒ M, uniquely determined by the property that its differentiation is the given VB-algebroid.Proof. By Theorem 3.4.3, the given VB-algebroid is described by an IM action onΩ ⇒ E , and since Γ ⇒ E is its source-simply-connected integration, it acquires amultiplicative ( R , · )-action h by Lemma 4.3.1. By Prop. 4.3.3, we know that h isregular, and Theorem 3.2.3 concludes the proof. (cid:3) Though the integrability of the total algebroid in a VB-algebroid implies that ofthe base algebroid, the converse is not true. We illustrate this fact with an example. Example 4.3.5.
Let M be a connected manifold, and let ω ∈ Ω ( M ) be a closed2-form such that its group of periods, (cid:26)Z γ ω (cid:12)(cid:12)(cid:12) γ ∈ π ( M ) (cid:27) ⊂ R , is not trivial. Let E and C denote the trivial line bundle q : R M = M × R → M ,and consider the vector bundle Ω = T M ⊕ E ⊕ C → E . For X ∈ X ( M ) and f : R M → R M satisfying q ◦ f = q , let σ X,f be the section of Ω → E given by σ X,f ( e ) = ( X ( q ( e )) , e, f ( e )). Then Ω carries a unique Lie-algebroid structure suchthat its anchor map and bracket satisfy ρ ( X, e, c ) = ( X, ∈ T E | e ∼ = T M | q ( e ) × R , and [ σ X ,c , σ X ,c ] = σ [ X ,X ] ,f , where c i ∈ R (viewed as constant maps R M → R M ), and f ( e ) = eω ( X , X ) | q ( e ) .The ( R , · )-action on Ω defined by h λ ( X, e, c ) = (
X, λe, λc ) defines a VB-algebroidstructure on Ω ⇒ E over T M ⇒ M . Although the base Lie algebroid is clearlyintegrable, Ω ⇒ E is not. This follows from the obstruction theory for integrabilityof [9]: one can check that the monodromy group of Ω at e ∈ E | x = R corresponds tothe group of periods of eω ∈ Ω ( M ), so any of its non-trivial elements accumulateat 0 as e goes to 0 (c.f. [9, Thm. 4.1]).This example is the starting point for the study of obstructions to integrability ofVB-algebroids, further developed in [5].As shown by the next proposition, the expected relations between VB-algebroidand VB-groupoid maps via integration hold: Proposition 4.3.6.
Let Γ and Γ be VB-groupoids.(a) If Γ is source-connected, then Φ : Γ → Γ is a VB-groupoid map if and only if Φ ′ is a VB-algebroid map.(b) If Γ is source-simply-connected, then there is a one-to-one correspondence be-tween VB-groupoid maps Γ → Γ and VB-algebroid maps A Γ → A Γ .Proof. Item ( a ) follows from the characterization of VB-algebroid and VB-groupoidmaps by ( R , · )-equivariance, together with the uniqueness of integration of Lie-algebroid maps when the domain is a source-connected Lie groupoid, see Remark 4.0.7.Part ( b ) is a direct consequence of Lie’s second theorem. (cid:3) In general, given a source-simply-connected Lie groupoid G , Lie subalgebroids of A G may not integrate to Lie subgroupoids of G , see [30]. We can obtain informationabout the integration of VB-subalgebroids from the previous proposition: Corollary 4.3.7.
Let Ω be a VB-subalgebroid of Ω defining, at the level of basisalgebroids, a Lie subalgebroid ( A ⇒ M ) ֒ → ( A ⇒ M ) . For i = 1 , , let Γ i and G i be source-simply-connected integrations of Ω i and A i , respectively. Then Γ is aVB-subgroupoid of Γ provided G is a Lie subgroupoid of G .Proof. Let (Φ , φ ) : (Γ ⇒ E ) → (Γ ⇒ E ) be the groupoid map such that Φ ′ :(Ω ⇒ E ) → (Ω ⇒ E ) is the subalgebroid inclusion. Since Φ ′ and d φ are injectiveon fibers, we see (from Remark 3.1.1(b), applied to dΦ : T Γ → T Γ ) that Φ is animmersion. So it remains to check that it is injective. By the previous proposition,Φ defines a vector-bundle map (Γ → G ) → (Γ → G ), so it can be identified with the restriction of its differential to the vertical bundles. The immersion propertythen implies that Φ is fibrewise injective. By assumption, Φ restricts to an injectivemap G → G , hence the result. (cid:3) This last corollary is used in the study of distributions [16] and Dirac structureson Lie algebroids and groupoids [31].
Example 4.3.8.
Let A ⇒ M be a Lie algebroid, and consider the VB-algebroid T A ⊕ T ∗ A ⇒ T M ⊕ A ∗ over it. Let L A ֒ → T A ⊕ T ∗ A be a VB-subalgebroidover A ⇒ M . If G is the source-simply-connected integration of A , then the VB-groupoid T G ⊕ T ∗ G ⇒ T M ⊕ A ∗ is the source-simply-connected integration of T A ⊕ T ∗ A . By Corollary 4.3.7 the source-simply-connected integration of L A definesa VB-subgroupoid L G ֒ → T G ⊕ T ∗ G over G ⇒ M . A special class of such VB-subalgebroids L A is given by those which are, additionally, Dirac structures. In thiscase, the VB-subgroupoids L G just defined are proven in [31, Thm. 5.2] to be Diracstructures as well.We end this section with comments on the relation between integration of VB-algebroids and representations up to homotopy. Remark 4.3.9.
Representations up to homotopy of a Lie groupoid can be differen-tiated to representations up to homotopy of its Lie algebroid, see [3]. The converseintegration procedure is considered in [4], but in a formal sense: representationsup to homotopy of a Lie algebroid are integrated to those of its ∞ -groupoid – aglobal object associated to any Lie algebroid. For an integrable Lie algebroid A , anatural question is whether, or under which conditions, representations up to homo-topy integrate to those of its source-simply-connected Lie groupoid G . Our resulton integration of VB-algebroids provides information about this problem: A repre-sentation of A on C → E is integrable to one of G if and only if the Lie algebroidΩ = A ⊕ E ⊕ C ⇒ E is integrable, where Ω is the VB-algebroid correspondingto C → E (in the sense of the results in [13, 14] mentioned in Example 3.3.3 andRemark 3.1.5). For example, the adjoint representation of a Lie algebroid is alwaysintegrable, and the representation up to homotopy of T M underlying Example 4.3.5is not integrable (c.f. [4, Prop 5.4]). More on this topic can be found in [5].5.
Applications to double Lie algebroids
In this section, following our previous results on VB-algebroids and VB-groupoids,we discuss the Lie theory relating more general objects, known as double Lie alge-broids and
LA-groupoids [20, 23, 26]. Rather than treating these objects directly,our approach is to focus on the dual picture, in which we trade Lie algebroids forlinear Poisson structures. From this viewpoint, the objects to be considered are VB-algebroids and VB-groupoids endowed with a compatible Poisson structure, and ourmain goal is to extend our integration result in Thm. 4.3.4 to this setting.From an alternative perspective, following Theorems 3.2.3 and 3.4.3, we will beconsidering regular actions on objects known as
Poisson groupoids and
Lie bialge-broids [27, 35]. We start the section by briefly discussing them.5.1.
Interlude: Lie bialgebroids and Poisson groupoids. A Lie bialgebroid isa pair of Lie-algebroid structures A ⇒ M and A ∗ ⇒ M which are compatible in the sense that d ∗ [ X, Y ] = [ d ∗ X, Y ] + [
X, d ∗ Y ] ∀ X, Y ∈ Γ( ∧ • A ) , where d ∗ is the differential in Γ( ∧ • A ) induced by the bracket of A ∗ , and [ · , · ] denotesthe Schouten bracket induced by the bracket of A . One may verify that the notionof Lie bialgebroid is symmetric in A and A ∗ , see e.g. [25, Sec. 12.1] for details.By the duality between Lie-algebroid structures and linear Poisson structures (seeSection 2.3), a Lie bialgebroid is the same as a Lie algebroid A ⇒ M equipped witha linear Poisson structure π on A satisfying the following compatibility condition[27]: the associated bundle map π : ( T ∗ A → A ) → ( T A → A ) is a Lie algebroidmap with respect to the tangent and cotangent Lie algebroids, T A ⇒ T M and T ∗ A ⇒ A ∗ , see Example 2.2.1. In other words, π defines a VB-algebroid map(5.1) T ∗ A ⇒ A ∗ ↓ ↓ A ⇒ M π −−→ T A ⇒ T M ↓ ↓ A ⇒ M. We will denote Lie bialgebroids by pairs ( A ⇒ M, π ). A map of Lie bialgebroids isa map of Lie algebroids which is also a Poisson map.If A ⇒ M is a Lie algebroid, then it becomes a bialgebroid with the trivial Poissonstructure. The following are less trivial examples. Example 5.1.1.
Lie bialgebras are Lie bialgebroids over a point. Other examples areassociated with Poisson manifolds (
P, π ): the tangent-lift π T on T P , correspondingto the Lie algebroid structure on T ∗ P , makes ( T P ⇒ P, π T ) into a bialgebroid.The global counterparts of Lie bialgebroids are Poisson groupoids [28]. A Poissongroupoid [27, 35] is a Lie groupoid G ⇒ M equipped with a Poisson structure π whichis multiplicative , in the sense that the bundle map π : T ∗ G → T G is a Lie-groupoidmap. In other words, π gives rise to a VB-groupoid map(5.2) T ∗ G ⇒ A ∗ ↓ ↓ G ⇒ M π −−→ T G ⇒ T M ↓ ↓ G ⇒ M. A map of Poisson groupoids is a Lie-groupoid map that is also a Poisson map.
Example 5.1.2.
Every Poisson-Lie group is a Poisson groupoid with a single object,and every symplectic groupoid [8] is a Poisson groupoid with non-degenerate Poissonstructure. These are two fundamental families of examples.If ( G ⇒ M, π G ) is a Poisson groupoid, then its Lie algebroid A G ⇒ M inherits aPoisson structure π A making it into a Lie bialgebroid. One may obtain π A from π G as follows: by applying the Lie functor to π G and using the canonical isomorphisms j G : T A G → A T G and θ G : A T ∗ G → T ∗ A G (see Remark 4.1.4(c)), we define thePoisson bivector π A on A G by(5.3) ( π G ) ′ = j G ◦ π A ◦ θ G . There is also an integration procedure going from Lie bialgebroids to Poissongroupoids [28]: if ( A ⇒ M, π A ) is a Lie bialgebroid and G ⇒ M is a source-simplyconnected Lie groupoid integrating A ⇒ M , then there exists a unique Poisson struc-ture π G on G that makes it into a Poisson groupoid and induces π A via (5.3). This follows from Lie’s second theorem: π G is obtained by integrating the Lie algebroidmap A T ∗ G → A T G defined by the right-hand-side of (5.3).We conclude this discussion with a direct proof of the integration of Lie-bialgebroidmaps, which completes the partial result in [36, Thm. 5.5.]:
Proposition 5.1.3.
Let ( G i ⇒ M i , π G i ) be Poisson groupoids, with Lie bialgebroids ( A i ⇒ M i , π A i ) , i = 1 , .(a) Let Φ : G → G be a Lie groupoid map. If it is a map of Poisson groupoids, then Φ ′ is a map of Lie bialgebroids, and the converse holds if G is source-connected;(b) When G is source-simply-connected, any Lie-bialgebroid map A → A inte-grates to a unique map of Poisson groupoids G → G .Proof. Notice that for ( a ), it is enough to assume that G is source-connected andshow that Φ is a Poisson map with respect to π G and π G if and only if Φ ′ is aPoisson map relative to π A and π A .From (5.3), we know that(5.4) ( π G ) ′ = j G ◦ π A ◦ θ G . By functoriality of pullbacks (see Remarks 3.2.7 and 3.4.6), π G : T ∗ G → T G gives rise to a VB-groupoid map Φ ∗ T ∗ G → Φ ∗ T G , that we keep denoting by π G . With this simplified notation, we write ( π G ) ′ : (Φ ′ ) ∗ A T ∗ G → (Φ ′ ) ∗ A T G , seeRemark 4.1.4(a). Similarly, one can apply the pullback functor (Φ ′ ) ∗ to all mapson the right-hand side of (5.4), and view (5.4) as an equality of VB-algebroid maps(Φ ′ ) ∗ A T ∗ G → (Φ ′ ) ∗ A T G .Consider the tangent VB-groupoid map dΦ : T G → Φ ∗ T G , and its dual (dΦ) ∗ :Φ ∗ T ∗ G → T ∗ G . Differentiating the composition dΦ ◦ π G ◦ (dΦ) ∗ : Φ ∗ T ∗ G → Φ ∗ T G leads to a VB-algebroid map(dΦ ◦ π G ◦ (dΦ) ∗ ) ′ = (dΦ) ′ ◦ j G ◦ π A ◦ θ G ◦ ((dΦ) ∗ ) ′ (5.5) = j G ◦ d(Φ ′ ) ◦ π A ◦ (dΦ ′ ) ∗ ◦ θ G , where we have used (5.3) and Remark 4.1.4(c).Note that Φ is a Poisson map if and only if π G = dΦ ◦ π G ◦ (dΦ) ∗ , as an equality of maps Φ ∗ T ∗ G → Φ ∗ T G . Both maps are Lie-groupoid maps andΦ ∗ T ∗ G is source-connected (since it is a VB-groupoid over G , and G is assumed tobe source-connected, see Remark 3.1.1(a)). By the uniqueness result in Remark 4.0.7,the last equation holds if and only if( π G ) ′ = (dΦ ◦ π G ◦ (dΦ) ∗ ) ′ . Comparing with (5.4) and (5.5), we see that this is equivalent to π A = d(Φ ′ ) ◦ π A ◦ (dΦ ′ ) ∗ , as an equality of maps (Φ ′ ) ∗ T ∗ A → (Φ ′ ) ∗ T A , which is the condition for Φ ′ beinga Poisson map.Finally, part ( b ) is an immediate consequence of ( a ). (cid:3) An alternative approach to this last result can be found in [34, Sec. 1.5]. LA-groupoids and double Lie algebroids: the dual viewpoint.
We nowconsider certain generalizations of VB-algebroids and VB-groupoids, in which thevector-bundle structures are enhanced to be Lie algebroids.An
LA-groupoid [20] consists of a VB-groupoid Γ ⇒ E over G ⇒ M , along withLie algebroid structures Γ ⇒ G and E ⇒ M , satisfying compatibility conditionssaying that the groupoid structure maps are Lie-algebroid morphisms (an alternativedefinition will be given below). We depict an LA-groupoid as(5.6) Γ ⇒ E ⇓ ⇓ G ⇒ M. The duality between Lie algebroids and linear Poisson structures provides an al-ternative viewpoint to LA-groupoids in terms of their duals, that we now recall.A
PVB-groupoid [22] consists of a VB-groupoid Γ ⇒ E over G ⇒ M and a Poissonstructure π on Γ which is multiplicative (i.e., (Γ , π ) is a Poisson groupoid) and linearwith respect to Γ → G . PVB-groupoids will be written as Γ ⇒ E ↓ ↓ G ⇒ M , π As proven in [22, Thm. 3.14], the compatibility conditions relating the groupoidand algebroid structures on an LA-groupoid (5.6) are equivalent to saying that thedual VB-groupoid Γ ∗ ⇒ C ∗ over G ⇒ M is a PVB-groupoid with respect to thelinear Poisson structure dual to Γ ⇒ G . So, through VB-groupoid duality, oneobtains a one-to-one correspondence between LA-groupoids and PVB-groupoids. Example 5.2.1.
Examples of PVB-groupoids include, e.g., the cotangent groupoidof any Lie groupoid (equipped with the Poisson structure of its canonical symplecticform) as well as the tangent groupoids to Poisson groupoids, equipped with thetangent-lift Poisson structure (see Remark 2.3.2).One advantage of passing from LA-groupoids to PVB-groupoids is that the latteradmit a simple characterization in terms of ( R , · )-actions, resulting from Prop. 2.3.1and Theorem 3.2.3: Proposition 5.2.2.
A PVB-groupoid is the same as a Poisson groupoid (Γ ⇒ E, π ) equipped with a regular action h : ( R , · ) y (Γ ⇒ E ) such that h λ : (Γ ⇒ E, π ) → (Γ ⇒ E, λπ ) is a map of Poisson groupoids for all λ = 0 . We now consider the analogous infinitesimal objects. A double Lie algebroid [26]consists of a VB-algebroid Ω ⇒ E over A ⇒ M equipped with additional Lie alge-broid structures Ω ⇒ A and E ⇒ M , depicted(5.7) Ω ⇒ E ⇓ ⇓ A ⇒ M, satisfying the following conditions:(i) Ω ⇒ A is a VB-algebroid over E ⇒ M ,(ii) the Lie-algebroid structure on the vertical dual Ω ∗ A ⇒ C ∗ (see (3.6)), togetherwith the linear Poisson structure π A induced by Ω ⇒ A , define a Lie bialgebroid. Note that (ii) can be equivalently stated in terms of Ω ∗ E → C ∗ , interchanging theroles of vertical and horizontal VB-algebroids in (5.7).Once again, it will be profitable to make use of the duality between Lie algebroidsand linear Poisson structures and pass to the dual picture.A PVB-algebroid consists of a VB-algebroid and a Poisson structure π on the totalspace Ω which is linear with respect to Ω → A and such that (Ω ⇒ E, π ) is a Liebialgebroid. We will use the notation Ω ⇒ E ↓ ↓ A ⇒ M , π . One can directly check that vertical duality of VB-algebroids establishes a one-to-one correspondence between double Lie algebroids and PVB-algebroids,(5.8) Ω ⇒ E ⇓ ⇓ A ⇒ M ←→ Ω ∗ A ⇒ C ∗ ↓ ↓ A ⇒ M , π A , analogously to what happens for LA-groupoids and PVB-groupoids [22, Thm. 3.14]. Remark 5.2.3.
From the duality properties of VB-algebroids and the fact that Liebialgebroids are self-dual, one sees that both the horizontal and vertical duals of adouble Lie algebroid are PVB-algebroids. Thus, while the vertical dual of a PVB-algebroid is a double Lie algebroid, its horizontal dual is again a PVB-algebroid.
Example 5.2.4.
Analogously to Example 5.2.1, one can see that the cotangent Liealgebroid of any Lie algebroid is naturally a PVB-algebroid (with respect to thecanonical symplectic structure). The tangent Lie algebroid to any Lie bialgebroid(
A, π ) is a PVB-algebroid, with Poisson structure given by the tangent lift of π .Just as PVB-groupoids, PVB-algebroids admit a simple description in terms ofregular actions, following Prop. 2.3.1 and Theorem 3.4.3: Proposition 5.2.5.
PVB-algebroids are equivalently described as Lie bialgebroids (Ω ⇒ E, π ) endowed with a regular IM-actions h : ( R , · ) y (Ω ⇒ E ) such that h λ : (Ω ⇒ E, π ) → (Ω ⇒ E, λπ ) is a Poisson map ∀ λ = 0 . The characterizations of PVB-algebroids and PVB-groupoids in Props. 5.2.2 and5.2.5 will be useful in describing their relation via differentiation and integration.
Remark 5.2.6.
PVB-groupoids and PVB-algebroids can be characterized by meansof their Poisson-anchor maps π . Indeed, tangent and cotangent bundles of VB-groupoids and VB-algebroids inherit triple structures , which can be encoded in cu-bical diagrams as the cotangent cube (2.8) (see [22, Fig. 5]). Combining (2.14),(5.1) and (5.2), one can see that a Poisson structure defines a PVB-groupoid or aPVB-algebroid if and only if the map π preserves the structure of the underly-ing cubes. This characterization for PVB-algebroids is essentially [26, Thm. 3.9],and as explained there, it implies that π automatically yields an algebroid map(Ω ∗ A ⇒ C ∗ ) → ( T A ⇒ T M ), simplifying some redundacy in the original definitionof double Lie algebroids, see e.g. [21, Sec. 2]. Lie theory.
We finally explain how double Lie algebroids are related to LA-groupoids under differentiation and integration. We will do so by first studying thedual picture, i.e., the Lie theory relating PVB-algebroids and PVB-groupoids.
Proposition 5.3.1.
Consider a VB-groupoid Γ ⇒ E over G ⇒ M , and let π bea multiplicative Poisson structure on Γ . Consider the corresponding VB-algebroid A Γ ⇒ E over A G ⇒ M and Lie bialgebroid ( A Γ , π A Γ ) . If π is linear with respect to Γ → G , then π A Γ is linear with respect to A Γ → A G , and the converse holds provided Γ ⇒ E is source-connected.Proof. Note that if ( A ⇒ M, π ) is a Lie bialgebroid, then so is ( A ⇒ M, λπ ) for any λ ∈ R , since λπ is a Poisson structure and we can write ( λπ ) as the composition T ∗ A ⇒ A ∗ ↓ ↓ A ⇒ M π −−→ T A ⇒ T M ↓ ↓ A ⇒ M k λ −→ T A ⇒ T M ↓ ↓ A ⇒ M, where k denotes the regular action associated with the vector bundle T A → A .The analogous result holds for Poisson groupoids. Moreover, one may directly checkthat if ( G ⇒ M, π G ) integrates the bialgebroid ( A ⇒ M, π A ), then ( G ⇒ M, λπ G )integrates ( A ⇒ M, λπ A ), for λ ∈ R .Let h be the regular multiplicative action on Γ ⇒ E defining its VB-groupoidstructure, so that h ′ defines the VB-algebroid structure on A Γ ⇒ E (see Cor. 4.1.2).The linearity of π with respect to Γ → G is equivalent to h λ : (Γ , π ) → (Γ , λπ ) beinga map of Poisson groupoids for all λ = 0 (see Prop. 5.2.2), while the linearity of π A Γ with respect to A Γ → A G is equivalent to h ′ λ : ( A Γ , π A Γ ) → ( A Γ , λπ A Γ ) being a mapof Lie bialgebroids (see Prop. 5.2.5). The result now follows from the integration ofbialgebroid maps in Prop. 5.1.3. (cid:3) The previous proposition immediately gives rise to a Lie functor from PVB-groupoids to PVB-algebroids and implies the following integration result:
Corollary 5.3.2.
If the total algebroid of a PVB-algebroid is integrable, then itssource-simply-connected integration inherits a unique PVB-groupoid structure whosedifferentiation is the given PVB-algebroid.
Let us now consider LA-groupoids and double Lie algebroids. By applying the Liefunctor to the horizontal groupoid structures of an LA-groupoid (5.6), one obtainsa VB-algebroid A Γ ⇒ E over A G ⇒ M . A key observation is that there is also anatural Lie-algebroid structure A Γ ⇒ A G , described in [23, Thm. 2.14], so one canconsider the diagram of Lie algebroids(5.9) A Γ ⇒ E ⇓ ⇓ A G ⇒ M. This can be shown to be a double Lie algebroid [23], yielding a Lie functor from LA-groupoids to double Lie algebroids; we will revisit this fact below and complementit with the corresponding integration result. Remark 5.3.3.
The Lie algebroid A Γ ⇒ A G referred to above can be also directlydescribed as the Lie-algebroid fibred product (see Prop. A.2.4),(5.10) ( A Γ ⇒ A G ) / / (cid:15) (cid:15) ( T Γ ⇒ T G ) ( T s Γ ,T s G ) × ( q Γ ,q G ) (cid:15) (cid:15) ( E ⇒ M ) (0 E , M ) × ( u Γ ,u G ) / / ( T E ⇒ T M ) × (Γ ⇒ G ) . Indeed, the Lie algebroid resulting from this fibred product is uniquely characterizedby the fact that it sits in T Γ ⇒ T G as a Lie subalgebroid; since the one in [23,Thm. 2.14] also satisfies this property [31, Prop. 5.5], they must coincide.The following diagram illustrates our strategy to describe the Lie theory relatingdouble Lie algebroids and LA-groupoids:LA-groupoids
Lie (cid:15) (cid:15) ✤✤✤ o o duality / / PVB-groupoids
Lie (cid:15) (cid:15)
Double Lie algebroids o o duality / / PVB-algebroidsIn order to follow the dotted arrow backwards, we will pass to the dual frameworkand use the integration result in Corollary 5.3.2. But we first need to verify that theLie functors on each side correspond to one another under duality.
Proposition 5.3.4.
The square above commutes up to a canonical natural isomor-phism.Proof.
Starting with an LA-groupoid (5.6), let us consider its dual PVB-groupoid Γ ∗ ⇒ C ∗ ↓ ↓ G ⇒ M , π . By Prop. 5.3.1, after applying the Lie functor we get a PVB-algebroid A Γ ∗ ⇒ C ∗ ↓ ↓ A G ⇒ M , π A Γ ∗ . The pairing Γ × G Γ ∗ → R leads to a pairing A Γ × A G A Γ ∗ → R (via the Lie functor,see Remark 4.1.4(b)), and hence an identification of VB-algebroids over A G ⇒ M ,(5.11) φ : A Γ ∗ → ( A Γ ) ∗ A G . This induces a PVB-algebroid structure on the VB-algebroid ( A Γ ) ∗ A G ⇒ C ∗ over A G ⇒ M , and hence, by duality (5.8), a double Lie algebroid A Γ ⇒ E ⇓ ⇓ A G ⇒ M. It remains to check that the Lie algebroid A Γ ⇒ A G agrees with the one defined by(5.10). Equivalently, we should verify that (5.11) is a Poisson isomorphism φ : ( A Γ ∗ , π A Γ ∗ ) → (( A Γ ) ∗ A G , ¯ π ) , where ¯ π denotes the Poisson structure dual to the Lie algebroid defined in (5.10).To show that, recall that there is also a pairing (defined by the tangent functor) T Γ × T G T Γ ∗ → R , which leads to an identification Φ : T Γ ∗ → ( T Γ) ∗ T G . Denotingby ι A Γ : A Γ → T Γ and ι A Γ ∗ : A Γ ∗ → T Γ ∗ the natural inclusions, one may directlyverify that the maps Φ and φ are related by φ = ( ι A Γ ) ∗ ◦ Φ ◦ ι A Γ ∗ , where we view ( ι A Γ ) ∗ as the dual relation to ι A Γ and consider the composition ofrelations on the right-hand side. Note that the relation ( ι A Γ ) ∗ is one-to-one, anddefined over the whole image of the map Φ ◦ ι A Γ ∗ , so their composition is a map.Endowing T Γ ∗ with the tangent lift of π (cf. Remark 2.3.2) and ( T Γ) ∗ T G withthe Poisson structure dual to the tangent Lie algebroid T Γ ⇒ T G , it follows from[25, Thm. 10.3.14] that Φ is a Poisson isomorphism, so its graph is coisotropic. Theinclusion ι A Γ ∗ is also a Poisson map with respect to π A Γ ∗ and the tangent lift of π , seee.g. [6, Sec. 6.3] (cf. [25, Prop. 10.3.12 & Thm. 12.3.8]). Since the dual relation ( ι A Γ ) ∗ is also coisotropic, see Remark 5.3.3, and the composition of coisotropic relations iscoisotropic [35], the graph of φ is coisotropic, so it is a Poisson map. (cid:3) We conclude with the integration result.
Theorem 5.3.5.
Consider a double Lie algebroid (5.2) for which the horizontal Liealgebroid Ω ⇒ E is integrable. Then its source-simply-connected integration Γ ⇒ E fits into an LA-groupoid (5.12) Γ ⇒ E ⇓ ⇓ G ⇒ M, where G ⇒ M is the source-simply-connected integration of A ⇒ M , uniquely deter-mined by the property that its differentiation is the given double Lie algebroid.Proof. The dual VB-algebroid Ω ∗ A ⇒ C ∗ is integrated by the dual VB-groupoidΓ ∗ ⇒ C ∗ (see Prop. 5.3.4), and it is source-simply-connected (see Remark 3.1.1(a)).This VB-groupoid inherits a PVB-groupoid structure by Prop. 5.3.1. By dualizingit, we obtain an LA-groupoid (5.6) corresponding to (5.12). (cid:3) It would be interesting to use this theorem to extend the discussion in Remark 3.1.5to the context of representations up to homotopy encoded by double Lie algebroids,as studied in [12].A natural further step is the integration of LA-groupoids to double Lie groupoids[20], as considered in [33]. We hope to address this issue in a separate work.
Appendix A. Fibred products of Lie groupoids and Lie algebroids
We present in this appendix a criterion for the existence of fibred products inthe categories of Lie algebroids and Lie groupoids, extending and organizing someprevious results in the literature. We also study the behavior of fibred productsunder the Lie functor.Our criterion is based on the following notion. Two smooth maps f i : M i → M , i = 1 ,
2, form a good pair if their set-theoretic fibred product M := M × M M ⊂ M × M is an embedded submanifold with the expected tangent space, i.e., for all( x , x ) ∈ M with x = f ( x ) = f ( x ) the following sequence is exact:(A.1) 0 −→ T ( x ,x ) M −→ T x M × T x M f − d f −→ T x M. We refer to the resulting manifold M as a good fibred product , for it satisfies theuniversal property and behaves well with respect to the topologies and the tangentspaces (see e.g. [10]). The paradigmatic example of a good pair is given by transversemaps. Another example is given by embedded submanifolds with clean intersection.In this appendix, by a submanifold we mean an injective immersion. Submani-fold are usually identified with a subset of a manifold M , equipped with a smoothstructure, for which the inclusion map is an injective immersion.A.1. The groupoid case. A Lie subgroupoid of G ⇒ M is a Lie groupoid ˜ G ⇒ ˜ M along with a Lie-groupoid map ( ˜ G ⇒ ˜ M ) → ( G ⇒ M ) which is an injectiveimmersion on objects and on arrows.When studying fibred products, it is convenient to have an alternative viewpoint.Given a Lie groupoid G ⇒ M , let ( ˜ G ⇒ ˜ M ) ⊆ ( G ⇒ M ) be a set-theoreticsubgroupoid, defined by restrictions of the structure maps of G . Assume that thefollowing conditions hold:(1) ˜ G ⊆ G and ˜ M ⊆ M are submanifolds;(2) The restriction of the source map to ˜ G , s : ˜ G → ˜ M , is a submersion;(3) The structure maps of ˜ G ⇒ ˜ M are smooth.It is not hard to see that Lie subgroupoids are equivalent to set-theoretic sub-groupoids satisfying these three conditions. We remark that, in many situations, (3)automatically follows from (1) and (2), e.g. when ˜ G and ˜ M are embedded. We alsoobserve that there are set-theoretic subgroupoids satisfying (1) and (3), but whichfail to be Lie subgroupoids by not satisfying (2) (though this cannot happen whenthe subgroupoid is source-connected). We will give an example below. Remark A.1.1.
In order to consider the smoothness of the multiplication map, itis implicitly required in (3) that ˜ G (2) sits in G (2) as a submanifold. Condition (2)guarantees this fact, but the next example shows that it is not necessary. Example A.1.2.
Let G ⇒ M be the Lie groupoid induced by the submersion π : R → R (projection on the first factor): in this case M = R and an arrow in G consists of a pair of points in R on the same vertical line. Define ˜ M = C ∪ C ⊂ M as the union of the two curves C = { ( t , t ) : − < t < } and C = { ( t,
2) : − < t < } , and define ˜ G ⊂ G as the space of arrows whose source and target lie in ˜ M . Then˜ G ⇒ ˜ M is a set-theoretic subgroupoid, and ˜ M ⊂ M and ˜ G ⊂ G are embeddedsubmanifolds. One may also directly verify that ˜ G (2) ⊂ G (2) is an embedded sub-manifold, so (1) and (3) above are satisfied. However, the differential of s : ˜ G → ˜ M is not surjective at the point ((0 , , (0 , ∈ ˜ G , so condition (2) above does not hold.The next lemma uses the following fact: Given a connected manifold M and asmooth map f : M → M such that f = f , its image f ( M ) ⊂ M is an embeddedsubmanifold and T x f ( M ) = d x f ( T x M ) for all x ∈ M , see [17, Thm. 1.13] for details. Lemma A.1.3.
Let ( F i , f i ) : ( G i ⇒ M i ) → ( G ⇒ M ) , i = 1 , , be two Lie-groupoidmaps. If F , F is a good pair, then so is f , f .Proof. Let us denote the set-theoretic fibred-product of F , F (resp. f , f ) by G (resp. M ), and consider the maps s := ( s , s ) : G → M and u := ( u , u ) : M → G , where s i , u i are the source and unit maps of G i , i = 1 ,
2. Then u s : G → G satisfies ( u s ) = u s , hence its image u ( M ) is an embedded submanifoldof G , hence of G × G , and then of u ( M × M ).Regarding the condition on the tangent spaces, we have to show that the sequence0 → T ( x ,x ) M → T x M × T x M → T x M is exact (cf. (A.1)). But it follows from s u = id that this last sequence is a directsummand of 0 → T ( x ,x ) G → T x G × T x G → T x G, which is exact by hypothesis, and hence the result. (cid:3) We are now ready to consider fibred products of Lie groupoids.
Proposition A.1.4.
Given a good pair of Lie groupoid maps as in Lemma A.1.3,the fibred-product manifolds G and M define an embedded Lie subgroupoid of theproduct groupoid, ( G ⇒ M ) ⊂ ( G × G ⇒ M × M ) . Moreover, this Lie groupoid satisfies the universal property of the fibred product inthe category of Lie groupoids.Proof.
Since G ⊂ G × G and M ⊂ M × M are embedded submanifolds, itremains to show that source map s of G × G restricts to a submersion ˜ s : G → M .Given g = ( g , g ) ∈ G with source x = ( x , x ) ∈ M , denote by K g and K ′ g the kernels of the mapsd s : T g ( G × G ) → T x ( M × M ) and d˜ s : T g G → T x M , respectively. Note that K ′ g = K g ∩ T g G . Since we know that ˜ s : G → M isa submersion close to the units (as a consequence of s u = id), it is enough to showthat dim K ′ g ≤ dim K ′ u ( x ) . We will show that d( R g − )( K ′ g ) ⊂ K ′ u ( x ) (here R g denotesright-translation), and since d( R g − ) is injective, the result follows.We have K ′ g = K g ∩ T g G ⊂ T g ( G × G ) = T g G × T g G and T g G = T g G × T F g G T g G . Given v = ( v , v ) ∈ K g , and given h = ( h , h ) ∈ G composable with g , we haved( R h )( v ) = (d( R h ) v , d( R h ) v ). If v ∈ K ′ g , then d F ( v ) = d F ( v ), andd F (d( R h )( v )) = d( R F ( h ) (d F ( v )) = d( R F ( h ) (d F ( v )) = d F (d( R h )( v )) , from where d( R h )( v ) ∈ K ′ gh , and hence d( R h )( K ′ g ) ⊂ K ′ gh as desired. (cid:3) Remark A.1.5.
Special cases of the last proposition have appeared in the literature.For instance, in the case of transverse maps (a particular type of good pair), theexistence of fibred products of Lie groupoids is stated in [29, pp. 123]. Under evenstronger assumptions, such fibred products were proven to exist e.g. in [33, Prop. 2.1](under an additional “source transversality condition”) and [25, Prop. 2.4.14] (for groupoid maps which are “fibrations”). A number of good pairs which are nottransverse maps appear in this paper, see e.g. Remark 3.2.5.A.2. The algebroid case.
By a vector subbundle of a vector bundle A → M wemean a vector bundle ˜ A → ˜ M and injective immersions ˜ A ֒ → A , ˜ M ֒ → M defininga vector-bundle map. We say that ˜ A → ˜ M is embedded if ˜ M ⊂ M is so, andconsequently also ˜ A ⊂ A . A Lie subalgebroid of A ⇒ M is a vector subbundleequipped with a Lie-algebroid structure for which the inclusion is a Lie-algebroidmap (c.f. [15, Def. 1.2]).In order to deal with fibred products, it is convenient to have a criterion to iden-tify vector subbundles which inherit the structure of a Lie subalgebroid. Given anembedded vector subbundle ˜ A → ˜ M of a Lie algebroid A ⇒ M , let us consider thefollowing compatibility conditions with the anchor and bracket:( i ) ρ ( ˜ A ) ⊂ T ˜ M , and( ii ) if X, Y ∈ Γ( A ) are such that X | ˜ M , Y | ˜ M ∈ Γ( ˜ A ), then [ X, Y ] | ˜ M ∈ Γ( ˜ A ). Remark A.2.1.
These compatibility conditions also make sense for non-embeddedsubbundles: in this case, the conditions ρ ( ˜ A ) ⊂ T ˜ M and X | ˜ M ∈ Γ( ˜ A ) implicitlyrequire that the induced set-theoretic functions ˜ A → T ˜ M and ˜ M → ˜ A are smooth,which is automatic in the embedded case.These compatibility conditions imply the following additional property: Lemma A.2.2.
Given a Lie algebroid A ⇒ M and a subbundle ˜ A → ˜ M satisfyingproperty (i) above, the following holds:(iii) If X, Y ∈ Γ( A ) satisfy X | ˜ M = 0 and Y | ˜ M ∈ Γ( ˜ A ) , then [ X, Y ] | ˜ M = 0 .Proof. We can work locally and assume that A → M is trivial, with a basis ofsections { e , . . . , e r } . Let[ e i , e j ] = X k c ijk e k , c ijk ∈ C ∞ ( M ) . Let
X, Y ∈ Γ( A ) be such that X | ˜ M = 0 and Y | ˜ M ∈ Γ( ˜ A ). We have to show that[ X, Y ]( x ) = 0 for all x ∈ ˜ M . We can write X = P i a i e i and Y = P j b j e j , with a i , b j ∈ C ∞ ( M ). Then their bracket is[ X, Y ] = X k X i,j a i b j c i,jk + ρ ( X ) b k − ρ ( Y ) a k e k . Given x ∈ ˜ M , X ( x ) = 0 and, equivalently, a i ( x ) = 0 for all i . The result now followsfrom ρ ( Y ) being tangent to ˜ M . (cid:3) It directly follows that a subbundle ˜ A → ˜ M satisfying ( i ) and ( ii ) above naturallyinherits a Lie-algebroid structure from A ⇒ M , where the bracket is defined bylocally extending sections of ˜ A to sections of A , using the bracket on A , and thenrestricting to ˜ M ; this operation is well defined by Lemma A.2.2. The structure on˜ A ⇒ ˜ M clearly makes the inclusion into a Lie-algebroid map. Conversely, any Liesubalgebroid satisfies ( i ) and ( ii ), and its Lie-algebroid structure agrees with theone induced from these properties. This notion of Lie subalgebroid appears in [25,Def. 4.3.14] (where condition ( iii ) is required as an extra axiom). A simple, but relevant, property of Lie subalgebroids, used recurrently, is that aLie-algebroid map ( B ⇒ N ) → ( A ⇒ M ) whose image lies in a Lie subalgebroid˜ A ⇒ ˜ M gives rise to a Lie-algebroid map ( B ⇒ N ) → ( ˜ A ⇒ ˜ M ).In order to study fibred products of Lie algebroids, we first discuss vector bundles. Lemma A.2.3.
Let ( F i , f i ) : ( E i → M i ) → ( E → M ) , i = 1 , , be two vector-bundlemaps. The smooth maps F , F form a good pair if and only if f , f form a goodpair and the vector-bundle map (A.2) ( F ) π − ( F ) π : E × E | M → E has constant rank. (Here π i : E × E → E i is the projection, i = 1 , .)Proof. Assuming that ( F , F ) is good, the same arguments used in Lemma A.1.3show that ( f , f ) is also good, hence M ⊂ M is embedded with the expectedtangent space. Since the kernel of the map (A.2) is the manifold E = E × M E ,it must have constant rank.Conversely, if ( f , f ) is a good pair and the rank of the map (A.2) is constant,then M ⊂ M × M is embedded with the expect tangent space, E (which is thekernel of the map) is also an embedded submanifold, and a vector subbundle. Itonly remains to show that it has the expected tangent space.For any vector bundle q : E → M , we can identify a fiber E q ( v ) with ker(d q ) v ⊂ T v E , and in this way we obtain a short exact sequence of complexes,0 / / (cid:15) (cid:15) ( E ) ( x ,x ) / / (cid:15) (cid:15) ( E ) x × ( E ) x / / (cid:15) (cid:15) E x (cid:15) (cid:15) / / (cid:15) (cid:15) T ( v ,v ) E / / (cid:15) (cid:15) T v E × T v E / / (cid:15) (cid:15) T v E (cid:15) (cid:15) / / T ( x ,x ) M / / T x M × T x M / / T x M. The top sequence is exact by assumption, the bottom one is exact because f , f isa good pair, so the middle one is also exact, proving the result. (cid:3) When the conditions in the previous lemma hold, the vector bundle E → M satisfies the universal property of the fibred product in the category of vector bundles.We now move to Lie algebroids. Proposition A.2.4.
Let ( F i , f i ) : ( A i ⇒ M i ) → ( A ⇒ M ) , i = 1 , , be two Lie-algebroid maps so that the pair ( F , F ) is good. Then the vector-bundle fibred product ( A → M ) ⊂ ( A × A ⇒ M × M ) is an embedded Lie subalgebroid of the product Lie algebroid. Moreover, the Liealgebroid ( A ⇒ M ) satisfies the universal property of the fibred product in thecategory of Lie algebroids.Proof. We have to show that A → M satisfies conditions ( i ) and ( ii ).Regarding ( i ), since ( F , f ) and ( F , f ) are Lie-algebroid maps, we have(d f ) ρ = ρF and (d f ) ρ = ρF , which implies that ( ρ , ρ )( A × A A ) ⊂ T M × T M
T M = T M , the last equalityfollowing from f , f being a good pair. Regarding ( ii ), given X, Y ∈ Γ( A × A ) such that X | M , Y | M ∈ Γ( A ), wehave to show that the same holds for [ X, Y ]. Consider the product map(
F, f ) = ( F × F , f × f ) : ( A × A ⇒ M × M ) → ( A × A ⇒ M × M ) . We can always write
F X = P i a i ( X i f ) for some functions a i ∈ C ∞ ( M × M ) andsections X i ∈ Γ( A × A ), as an identity between sections of the pullback bundle.Consider the diagonal subbundle ∆ A → ∆ M of A × A → M × M . Since X | M ∈ Γ( A ), we see that F X ( M ) ⊂ ∆ A ; also, we can choose the sections X i such that X i (∆ M ) ⊂ ∆ A . We can proceed analogously for Y .Since F is an algebroid map, we have the equation F [ X, Y ] = X i,j a i b j ([ X i , Y j ] f ) + ( ρ ( X ) b j )( Y j f ) − ( ρ ( Y ) a i )( X i f ) , and by using that ∆ A → ∆ M is a Lie subalgebroid of A × A → M × M , we concludethat F [ X, Y ]( M ) ⊂ ∆ A , and hence the result. (cid:3) A brief discussion on fibred products of Lie algebroids can be found in [15, pp.207],and further details (in the case of transverse maps) in [33, Prop. 2.3].A.3.
Fibred products and the Lie functor.
When passing from Lie groupoids toLie algebroids via the Lie functor, recall the notations A G = Lie( G ) and F ′ = Lie( F )for maps. For a Lie groupoid G ⇒ M , there is a natural splitting(A.3) T G | u ( M ) = A G ⊕ T M, so that, for a groupoid map (
F, f ) : G → ˜ G , we obtain a decomposition(A.4) d F = ( F ′ , d f ) : ( A G ) x ⊕ T M x → ( A ˜ G ) f ( x ) ⊕ T f ( x ) ˜ M .
Proposition A.3.1.
Let ( F i , f i ) : ( G i ⇒ M i ) → ( G ⇒ M ) , i = 1 , , be Lie-groupoidmaps such that F and F form a good pair. Then the induced pair F ′ i : A G i → A G , i = 1 , , is also good, and the canonical map (arising from the universal propertyof fibred products) A G × G G → A G × A G A G , is an isomorphism, which, upon theobvious identification T ( G × G ) = T G × T G , is just the identity.Proof. Denote by G the fibred-product Lie groupoid, which exists by Prop. A.1.4,and let A = Lie( G ). According to Lemma A.2.3, to see that F ′ , F ′ is a goodpair we need to show that f , f form a good pair and that the vector-bundle map(A.5) F ′ π − F ′ π : A G × A G | M → A has constant rank. The first condition follows from F , F forming a good pair, seeLemma A.1.3. Given ( x , x ) ∈ M , since F , F form a good pair, and using thesplitting (A.3), we have the following exact sequence (cf. (A.1)):0 → ( A ⊕ T M ) ( x ,x ) → ( A G × A G ) ( x ,x ) ⊕ ( T M × T M ) ( x ,x ) → ( A G ⊕ T M ) x . Using (A.4), we see that the kernel of ( F ′ π − F ′ π ) ( x ,x ) is exactly ( A ) ( x ,x ) , sothe map (A.5) has constant rank.Knowing that ( F ′ , F ′ ) is a good pair, we conclude that the fibred-product Liealgebroid is well-defined (by Prop. A.2.4), and by construction it agrees with A . (cid:3) References [1] Arias-Abad, C., Crainic, M.: Representations up to homotopy of Lie algebroids.
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