Lie theory of multiplicative tensors

TL;DR

This paper characterizes multiplicative tensors on Lie groupoids using infinitesimal data, unifying various structures like forms, multivectors, and holomorphic structures, and provides a comprehensive treatment of vector-valued forms and Nijenhuis operators.

Contribution

It offers a complete infinitesimal description of multiplicative tensors on Lie groupoids, unifying multiple geometric structures under a common framework.

Findings

Complete infinitesimal description of multiplicative tensors

Unified treatment of forms, multivectors, and holomorphic structures

Analysis of multiplicative vector-valued forms and Nijenhuis operators

Abstract

We study tensors on Lie groupoids suitably compatible with the groupoid structure, called {\em multiplicative}. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the infinitesimal counterparts of multiplicative forms, multivector fields and holomorphic structures, obtained through a unifying and conceptual method. We also give a full treatment of multiplicative vector-valued forms, particularly Nijenhuis operators and related structures.

Equations

$$\mathbb{G}:=({\oplus }^{p}T\mathcal{G})\oplus ({\oplus }^q{T}^{\ast }\mathcal{G}),$$

$$c_{\tau }(U_1,\dots ,U_p,{\xi }_1,\dots ,{\xi }_q)=\tau (U_1,\dots ,U_p,{\xi }_1,\dots ,{\xi }_q),$$

$$a\cdot (\beta \otimes \mathfrak{X})={\mathcal{L}}_{\rho (a)}\beta \otimes \mathfrak{X}+\beta \otimes [a,\mathfrak{X}],$$

$$D(fa)=fD(a)+df\wedge l(a)-a\wedge r(df),f\in C^{\infty }(M),a\in \Gamma (A),$$

$$D([a,b])$$

$$l([a,b])$$

$$r({\mathcal{L}}_{\rho (a)}\alpha )$$

$$i_{\rho (a)}l(b)$$

$$i_{{\rho }^{\ast }\alpha }r(\beta )$$

$$i_{\rho (a)}r(\alpha )$$

$$\mathcal{T}:\Gamma ({\wedge }^{\bullet }{T}^{\ast }M\otimes {\wedge }^{\bullet }A)\to \Gamma ({\wedge }^{\bullet }{T}^{\ast }\mathcal{G}\otimes {\wedge }^{\bullet }T\mathcal{G}),$$

$$i_{\overrightarrow{a}}\tau =\mathcal{T}(l(a)),i_{{\mathsf{t}}^{\ast }\alpha }\tau =\mathcal{T}(r(\alpha )),{\mathcal{L}}_{\overrightarrow{a}}\tau =\mathcal{T}(D(a)).$$

$$\mathcal{B}a(X)=T1(X)+a(x),$$

$$0⟶A\hookrightarrow 1^{\ast }T\mathcal{G}\stackrel{T\mathsf{s}}{⟶}TM⟶0.$$

$$⟨\tilde{\mathsf{s}}({\xi }_g),a⟩=⟨{\xi }_g,\overleftarrow{a}(g)⟩\text{ and }⟨\tilde{\mathsf{t}}({\xi }_g),b⟩=⟨{\xi }_g,\overrightarrow{b}(g)⟩,$$

$$⟨{\tilde{1}}_{\phi },T1(X)+a⟩=⟨\phi ,a⟩,$$

$$⟨{\xi }_1\bullet {\xi }_2,U_1\bullet U_2⟩={\xi }_1(U_1)+{\xi }_2(U_2),$$

$$0⟶T^{\ast }M\stackrel{(T\mathsf{t}{)}^{\ast }}{\hookrightarrow }{1}^{\ast }{T}^{\ast }\mathcal{G}\stackrel{\tilde{\mathsf{s}}}{⟶}{A}^{\ast }⟶0.$$

$$⟨\mathcal{B}\alpha (\phi ),T1(X)+a⟩=⟨\alpha (x),\rho (a)+X⟩+⟨\phi ,a⟩,$$

$${\mathbb{G}}^{(p,q)}=({\oplus }^{p}T\mathcal{G})\oplus ({\oplus }^q{T}^{\ast }\mathcal{G})\text{ and }{\mathbb{M}}^{(p,q)}=({\oplus }^{p}TM)\oplus ({\oplus }^q{A}^{\ast }),$$

$${\ell }_{\mu }(e):=⟨\mu (y),e⟩,\forall e\in E_y,$$

$$c_{\tau }={\ell }_{{\mu }_1}\circ {\mathrm{pr}}^1\dots {\ell }_{{\mu }_p}\circ {\mathrm{pr}}^p,$$

$$c_{f\tau }=(f\circ \pi )c_{\tau },\forall f\in C^{\infty }(N),$$

$$c_{\tau }=\sum _{\sigma \in S(p)}sgn(\sigma )({\ell }_{{\mu }_{\sigma (1)}}\circ {\mathrm{pr}}^1)\cdots ({\ell }_{{\mu }_{\sigma (p)}}.\circ {\mathrm{pr}}^p).$$

$$F\mapsto \frac{1}{p!}\sum _{\sigma \in S(p)}sgn(\sigma )F\circ \sigma ,$$

$$f(h_1{h}_2)=f(h_1)+f(h_2),\forall (h_1,h_2)\in {\mathcal{H}}_{\mathsf{s}}{\times }_{\mathsf{t}}\mathcal{H}.$$

$$⟨Af,\chi (y)⟩=\left({\mathcal{L}}_{\overrightarrow{\chi }}f\right)(y),$$

$$d_A(Af)=0,$$

$$⟨Af,e⟩=df\left({\frac{d}{dϵ}|}_{ϵ=0}ϵe\right)={\frac{d}{dϵ}|}_{ϵ=0}⟨\mu ,ϵe⟩=⟨\mu ,e⟩,\forall e\in E.$$

$${\mathcal{L}}_{\overrightarrow{\chi }}f={\mathsf{t}}^{\ast }⟨Af,\chi ⟩,\forall \chi \in \Gamma (A\mathcal{H}).$$

$$df(U\bullet V)=df(U)+df(V),\forall (U,V)\in T{\mathcal{H}}_{T\mathsf{s}}{\times }_{T\mathsf{t}}T\mathcal{H}.$$

$$({\mathcal{L}}_{\overrightarrow{\chi }}f)(h)=df(\overrightarrow{\chi }(h))=df(\chi (\mathsf{t}(h))\bullet 0_h)=df(\chi (\mathsf{t}(h)))=⟨Af,\chi ⟩(\mathsf{t}(h)),$$

$${\mathcal{L}}_{\overrightarrow{\chi }}f={\mathsf{t}}^{\ast }⟨\mu ,\chi ⟩,\forall \chi \in \Gamma (A\mathcal{H}).$$

Full text

1. Introduction

Lie groupoids permeate several areas of mathematics, including foliations, group actions and Poisson geometry. In these various contexts, Lie groupoids often come equipped with additional geometric structures, suitably compatible with the groupoid multiplication; such structures, referred to as multiplicative, are the main object of interest in this paper.

The study of multiplicative geometric structures on Lie groupoids has by now a long and rich history (see [38] for a recent survey). A basic example of multiplicative structure arises in the definition of complex Lie groups, regarded as real Lie groups endowed with a “compatible” complex structure; here, compatibility means that the group multiplication map is holomorphic. Another class of multiplicative structures on Lie groups arose in the early 80s with the emergence of Poisson-Lie groups, introduced by Drinfel’d [19] (later extended to Poisson groupoids by Weinstein [60]). Around the same time, the first examples of multiplicative differential forms on Lie groupoids came to notice with the advent of symplectic groupoids [32, 59] (and in their connections with the theory of hamiltonian actions [53] and equivariant cohomology; see [6, 7, 62]). Multiplicative structures now abound in the literature, where one finds multi-vector fields [28, 46, 49, 50], differential forms [1, 3, 5, 14], contact and Jacobi structures [16, 29, 33], holomorphic structures [42, 43], as well as distributions, foliations [14, 21, 27, 30], among others (e.g. [23, 44, 55]).

Any Lie groupoid has an underlying Lie algebroid, which linearizes it at the units. As in classical Lie theory, a central issue when considering multiplicative geometric structures is identifying their infinitesimal versions, i.e., finding their description solely in terms of Lie-algebroid data. This problem has been studied in numerous settings, through different approaches, leading to various “infinitesimal-global” correspondence results. Examples include the correspondences between symplectic groupoids and Poisson structures [9, 13, 51], Poisson groupoids and Lie bialgebroids [46, 49, 51], contact groupoids and Jacobi structures [15, 16], presymplectic groupoids and Dirac structures [7], complex Lie groupoids and holomorphic Lie algebroids [43], to mention a few (see also [1, 10, 28, 57]). All these results rely on defining a “Lie functor”, taking global to infinitesimal objects, and on an “integration” step, which reconstructs multiplicative structures from infinitesimal data.

In spite of these various results in specific settings, the theory of multiplicative geometric structures still lacks a complete treatment. The very notion of “multiplicativity” seems to be adapted to each case at hand, and different techniques have been employed to handle seemingly analogous results. In this paper, we introduce the concept of multiplicative tensor on Lie groupoids, which agrees with the existing notions of multiplicativity in known situations, and devise a general method to obtain their complete infinitesimal description. As a consequence, all aforementioned “infinitesimal-global” correspondences can be naturally derived from our main result (Theorem 3.19), with a unified proof and conceptual approach, and new applications are obtained. Although we focus on ordinary tensors, our method adapts to more general contexts (such as the study of 1-cocycles on VB-groupoids), including tensors with values in representations (up to homotopy), see [20, 22].

Main results. Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with source and target maps $\mathsf{s}$ and $\mathsf{t}$. We denote its Lie algebroid by $A$, equipped with anchor map $\rho$ and bracket $[\cdot,\cdot]$. Consider a $(q,p)$-tensor field111The assumption of skew symmetry lead to some simplifications of the results, but it is by no means essential. $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$. Let

[TABLE]

which carries a natural groupoid structure over $\mathbb{M}:=(\oplus^{p}TM)\oplus(\oplus^{q}A^{*})$ induced from the tangent and cotangent groupoids; see Section 2. We say that the tensor $\tau$ is multiplicative if the map $c_{\tau}:\mathbb{G}\to\mathbb{R}$,

[TABLE]

is a groupoid morphism, where $\mathbb{R}$ is viewed as an abelian group; in other words, $c_{\tau}$ is a differentiable 1-cocycle on $\mathbb{G}$.

To state our main result, consider the action of $\Gamma(A)$ on $\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}A)$ by

[TABLE]

where $[\cdot,\cdot]$ is the Schouten bracket on $\Gamma(\wedge^{\bullet}A)$. The following theorem gives a full description of the infinitesimal counterparts of multiplicative tensors:

Theorem 1.1**.**

If $\mathcal{G}\rightrightarrows M$ is source 1-connected, then there is a natural one-to-one correspondence between multiplicative $(q,p)$-tensors $\tau$ on $\mathcal{G}$ and triples $(D,l,r)$, where $l:A\rightarrow\wedge^{p-1}T^{*}M\otimes\wedge^{q}A$ and $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}A$ are vector bundle maps covering the identity, $D:\Gamma(A)\rightarrow\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}A)$ is an $\mathbb{R}$-linear map satisfying the Leibniz-type condition

[TABLE]

and the following equations hold: for $a,b\in\Gamma(A)$ and $\alpha,\beta\in\Omega^{1}(M)$,

[TABLE]

We refer to the equations (IM1)–(IM6) as cocycle equations, or IM equations (where IM stands for “infinitesimally multiplicative”). A more detailed formulation of the previous result can be found in Theorem 3.19 below.

The definition of the “Lie functor”, taking multiplicative tensors $\tau$ to triples $(D,l,r)$, relies on a useful characterization of multiplicativity proven in Theorem 3.11; it says, in particular, that for any $\alpha\in\Omega^{1}(M)$, $a\in\Gamma(A)$ and $\overrightarrow{a}$ the corresponding right-invariant vector field on $\mathcal{G}$, the tensors $i_{\overrightarrow{a}}\tau$, $i_{\mathsf{t}^{*}\alpha}\,\tau$, and $\mathcal{L}_{\overrightarrow{a}}\tau$ lie in the image of the map

[TABLE]

defined on homogeneous elements by ${\mathcal{T}}(\alpha\otimes\mathfrak{X})=\mathsf{t}^{*}\alpha\otimes\overrightarrow{\mathfrak{X}}$. The maps $D$, $l$, and $r$ arise from the equations

[TABLE]

The IM-equations, combined with the 1-connectedness of the source fibers of $\mathcal{G}$, permit the reconstruction of $\tau$ out of $(D,l,r)$. Theorem 1.1, when restricted to tensors of types $(0,p)$ or $(q,0)$, directly recovers the infinitesimal descriptions of multiplicative differential forms and multivector fields proven in [1, 6, 28], but using other methods. For $(1,1)$-tensors, it encompasses the correspondence of complex Lie groupoids and holomorphic Lie algebroids of [43]. In these special cases, the operator $D$ takes different guises, codifying $k$-differentials [28], IM-forms [6] (or the “Spencer operators” of [14]), or flat partial connections defining holomorphic structures.

The content of Theorem 1.1 is discussed in Section 3, and its proof is presented in Section 4, heavily based on our viewpoint to multiplicative tensors $\tau$ on $\mathcal{G}$ as multiplicative functions $c_{\tau}$ on the “big” Lie groupoid $\mathbb{G}\rightrightarrows\mathbb{M}$. Multiplicative functions are simple to describe infinitesimally: they correspond to Lie-algebroid 1-cocycles, i.e., sections of the dual of the Lie algebroid which are closed under the Lie-algebroid differential. So the proof follows from a detailed analysis of 1-cocycles of the Lie algebroid $\mathbb{A}$ of $\mathbb{G}$. The key fact that $\mathbb{A}\to\mathbb{M}$ has a natural VB-algebroid structure over $A\to M$ allows us to identify a special set of generators of the $C^{\infty}(\mathbb{M})$-module $\Gamma(\mathbb{A})$, parametrized by $\Gamma(A)$ and $\Omega^{1}(M)$. We use these generators to describe 1-cocycles of $\mathbb{A}$ by means of triples $(D,l,r)$ as in Theorem 1.1, and we resort to classical lifting operations to realize the cocycle condition as the equations (IM1)–(IM6).

In Section 5, we specialize our main result to the case of $(1,p)$-tensors, i.e., multiplicative vector-valued forms. As observed in [8], the usual Frölicher-Nijenhuis bracket preserves the multiplicativity condition, so it makes the space of multiplicative vector-valued forms into a graded Lie algebra. One of our key results is the identification of the corresponding graded Lie algebra at the infinitesimal level, in Prop. 5.4. In Section 6, we focus on multiplicative vector-valued 1-forms, i.e., (1,1)-tensors. In this case, we obtain an explicit infinitesimal description of their Nijenhuis torsions (Cor. 6.3), which provides a broader viewpoint to the results in [43, 57] concerning multiplicative (almost) complex structures and Poisson (quasi-) Nijenhuis structures. We also treat multiplicative projections and product structures, interpreting their infinitesimal versions in terms of matched pairs.

Acknowledgments: We thank CNPq, Capes and Faperj for financial support. We are grateful to several institutions for hosting us during various stages of this project, including IST (Lisbon), Utrecht University, ESI (Vienna) and the Fields Institute. Special thanks go to Y. Kosmann-Schwarzbach for her comments and interest. We also thank A. Cabrera, M. Crainic, M. del Hoyo, L. Egea, C. Ortiz, L. Vitagliano and M. Zambon for helpful advice; we are especially indebted to Noah Kieserman for his collaboration in the early stages of this project.

Contents

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Tangent Lie groupoids

4. 2.2 Cotangent Lie groupoids

5. 2.3 Whitney sums

6. 2.4 Functions on the Whitney sum of vector bundles

7. 3 Multiplicative tensors

8. 3.1 Multiplicative functions

9. 3.2 Definition and examples

10. 3.3 The main results: statements and first examples

11. 4 Proof of the Theorems

12. 4.1 Lifting operations

13. 4.2 The Lie algebroid of $\mathbb{G}\rightrightarrows\mathbb{M}$

14. 4.3 Proof of Theorem 3.11

15. 4.4 Linear tensor fields

16. 4.5 Proof of Theorem 3.19

17. 5 Multiplicative vector-valued forms

18. 5.1 The graded Lie algebra of vector-valued forms

19. 5.2 Infinitesimal description

20. 6 Multiplicative $(1,1)$-tensor fields

21. 6.1 Infinitesimal components

22. 6.2 Nijenhuis torsion

23. 6.3 Poisson quasi-Nijenhuis structures

24. 6.4 Multiplicative (almost) complex structures

25. 6.5 Multiplicative projections

26. A Lie theory of componentwise linear functions

27. B Cotangent Lie algebroid.

2. Preliminaries

This section reviews some preliminary material, including tangent and cotangent Lie groupoids; see e.g. [47, 49]. We also discuss a convenient viewpoint to classical tensor fields, regarded as real-valued functions on Whitney sums of vector bundles.

2.1. Tangent Lie groupoids

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. We use the following notation: $\mathsf{s},\mathsf{t}:\mathcal{G}\rightarrow M$ are the source and target maps, $\mathsf{1}:M\rightarrow\mathcal{G}$ is the unit map, $\mathsf{i}:\mathcal{G}\rightarrow\mathcal{G}$ is the inversion map, and $\mathsf{m}:\mathcal{G}\,_{\mathsf{s}}\!\times_{\mathsf{t}}\mathcal{G}\rightarrow\mathcal{G}$ is the multiplication map. We will often identify $M$ with its image under the unit map.

The tangent groupoid of $\mathcal{G}$ is the Lie groupoid $T\mathcal{G}\rightrightarrows TM$ whose structural maps are all obtained by taking the derivatives of the structural maps of $\mathcal{G}$; e.g., its multiplication map is $T\mathsf{m}:T\mathcal{G}\,_{T\mathsf{s}}\!\times_{T\mathsf{t}}T\mathcal{G}\rightarrow T\mathcal{G}$, where we have identified $T(\mathcal{G}\,_{\mathsf{s}}\!\times_{\mathsf{t}}\mathcal{G})\cong T\mathcal{G}\,_{T\mathsf{s}}\!\times_{T\mathsf{t}}T\mathcal{G}$. We shall denote the multiplication on $T\mathcal{G}$ by $\bullet$.

We denote the Lie algebroid of $\mathcal{G}$ by $A\rightarrow M$, or $A\mathcal{G}$ if there is any risk of confusion. We identify $A$ with $\ker(T\mathsf{s})|_{M}$, so the Lie bracket on $\Gamma(A)$ is induced by right-invariant vector fields on $\mathcal{G}$, and the anchor $\rho:A\to TM$ is given by $T\mathsf{t}|_{A}$. For $a\in\Gamma(A)$, we denote by $\overrightarrow{a}\in\mathfrak{X}(\mathcal{G})$ the corresponding right-invariant vector field, and by $\overleftarrow{a}$ the left-invariant vector field induced by $a-\rho(a)\in\Gamma(\ker(T\mathsf{t})|_{M})$.

Note that each section $a\in\Gamma(A)$ defines a bisection $\mathcal{B}a:TM\rightarrow T\mathcal{G}$ of $T\mathcal{G}\rightrightarrows TM$,

[TABLE]

for $X\in T_{x}M$, covering the map $TM\to TM$, $X\mapsto X+\rho(a)(x)$. This bisection splits the exact sequence

[TABLE]

We refer to $\mathcal{B}a$ as the translation bisection associated with $a$.

2.2. Cotangent Lie groupoids

The cotangent bundle of a Lie groupoid $\mathcal{G}\rightrightarrows M$ also carries a natural Lie groupoid structure, $T^{*}\mathcal{G}\rightrightarrows A^{*}$, where $A^{*}$ is the dual vector bundle to $A$. The source and target maps $\widetilde{\mathsf{s}},\widetilde{\mathsf{t}}:T^{*}\mathcal{G}\rightarrow A^{*}$ are defined by the restriction of covectors to the subspaces tangent to the $\mathsf{s}$- and $\mathsf{t}$-fibers, respectively:

[TABLE]

for $\xi_{g}\in T^{*}_{g}\mathcal{G},\,a\in A_{\mathsf{s}(g)},b\in A_{\mathsf{t}(g)}$. The unit map $\widetilde{\mathsf{\mathsf{1}}}:A^{*}\rightarrow T^{*}\mathcal{G}$ is the vector bundle morphism covering $\mathsf{1}:M\rightarrow\mathcal{G}$, determined by

[TABLE]

for $(X,\varphi,a)\in TM\times_{M}A^{*}\times_{M}A$.

The multiplication on $T^{*}\mathcal{G}$ is defined as follows: for $\xi_{1}\in T_{g_{1}}^{*}\mathcal{G}$ and $\xi_{2}\in T_{g_{2}}^{*}\mathcal{G}$ such that $\widetilde{\mathsf{s}}(\xi_{1})=\widetilde{\mathsf{t}}(\xi_{2})$, their product $\xi_{1}\bullet\xi_{2}\in T_{g_{1}g_{2}}^{*}\mathcal{G}$ is determined by

[TABLE]

for composable $U_{1}\in T_{g_{1}}\mathcal{G},U_{2}\in T_{g_{2}}\mathcal{G}$.

The source map $\widetilde{\mathsf{s}}$ fits into the following exact sequence of vector bundles over $M$:

[TABLE]

Given a differential 1-form $\alpha\in\Omega^{1}(M)$ on $M$, the map $\mathcal{B}\alpha:A^{*}\rightarrow\mathsf{1}^{*}T^{*}\mathcal{G}$ given by

[TABLE]

for $\varphi\in A_{x}^{*},\,X\in T_{x}M,\,a\in A_{x}$, provides a splitting of the sequence (2.5). It is a bisection of $T^{*}\mathcal{G}\rightrightarrows A^{*}$ covering $\varphi\mapsto\varphi+\rho^{*}(\alpha(x)),\,\,\varphi\in A_{x}^{*}.$ We call $\mathcal{B}\alpha$ the translation bisection corresponding to $\alpha$.

2.3. Whitney sums

Tangent and cotangent groupoids satisfy the property that their Whitney sums as vector bundles again carry natural Lie groupoid structures, defined componentwise222This property holds, more generally, for $\mathcal{VB}$-groupoids, of which tangent and cotangent groupoids are special cases (see e.g. [4]). In this paper, we will be interested in Lie groupoids of the form $\mathbb{G}^{(p,q)}\rightrightarrows\mathbb{M}^{(p,q)}$, where

[TABLE]

for positive integers $p$ and $q$. When there is no risk of confusion, we omit the indices $(q,p)$ and write just $\mathbb{G}\rightrightarrows\mathbb{M}$. We denote the source and target maps by $\mathbbm{s},\mathbbm{t}:\mathbb{G}\rightarrow\mathbb{M}$ and the unit map by $\mathbbm{1}:\mathbb{M}\rightarrow\mathbb{G}$. We keep the notation $\bullet$ for the multiplication.

We shall denote by $(\underline{X\vphantom{\xi}},\underline{\varphi})$ and $(\underline{U\vphantom{\xi}},\underline{\xi})$ the elements $(X_{1},\dots,X_{p},\varphi_{1},\dots,\varphi_{q})\in\mathbb{M}$ and $(U_{1},\dots,U_{p},\xi_{1},\dots,\xi_{q})\in\mathbb{G}$, respectively.

Remark 2.1**.**

An important observation is that each source-fiber of $\mathbb{G}$ is an affine bundle over a source-fiber of $\mathcal{G}$, see e.g. [4, Rem. 3.1.1(a)]; it follows that $\mathbb{G}$ is source connected, or source 1-connected, if and only if so is $\mathcal{G}$.

2.4. Functions on the Whitney sum of vector bundles

A key viewpoint pursued in this work is that tensor fields should be regarded as functions on the Whitney sum of the tangent and cotangent bundles. We now explain this point of view from a general perspective. Let $E_{1},\dots,E_{p}$ be vector bundles over $N$ and consider their Whitney sum $\pi:E_{1}\oplus\dots\oplus E_{p}\to N$.

Definition 2.2**.**

A function $F:E_{1}\oplus\dots\oplus E_{p}\rightarrow\mathbb{R}$ is said to be componentwise linear if $F_{y}:(E_{1})_{y}\times\dots\times(E_{p})_{y}\rightarrow\mathbb{R}$ is multi-linear, for each $y\in N$.

Example 2.3**.**

For $p=1$, componentwise linear functions on a vector bundle $E\rightarrow N$ are fiberwise linear functions, i.e., those of the form $\ell_{\mu}$, where

[TABLE]

for $\mu\in\Gamma(E^{*})$.

Every tensor field $\tau\in\Gamma(E_{1}^{*}\otimes\dots\otimes E_{p}^{*})$ defines a componentwise linear function $c_{\tau}:E_{1}\oplus\dots\oplus E_{p}\rightarrow\mathbb{R}$ by pulling back the linear function $\ell_{\tau}:E_{1}\otimes\dots\otimes E_{p}\rightarrow\mathbb{R}$ by the natural map $E_{1}\oplus\dots\oplus E_{p}\rightarrow E_{1}\otimes\dots\otimes E_{p}$. The letter “$c$” in our notation stands for “componentwise”, and it is used to distinguish $c_{\tau}$ from the linear functions on $E_{1}\oplus\dots\oplus E_{p}$ defined by sections of its dual. In case $\tau=\mu_{1}\otimes\dots\otimes\mu_{p}$, for $\mu_{i}\in\Gamma(E_{i}^{*})$, $i=1,\dots,p$,

[TABLE]

where ${\rm pr}^{j}:E_{1}\oplus\dots\oplus E_{p}\to E_{j}$ is the projection on the $j$-th component.

The next result is a direct consequence of the properties of tensor products.

Lemma 2.4**.**

The map $\tau\mapsto c_{\tau}$ defines a bijection between $\Gamma(E_{1}^{*}\otimes\dots\otimes E_{p}^{*})$ and the space of componentwise linear functions on $E_{1}\oplus\dots\oplus E_{p}$ satisfying

[TABLE]

where $\pi:E_{1}\oplus\dots\oplus E_{p}\to N$ is the natural projection.

When $E_{1}=\dots=E_{p}=E$, we say that a function $F\in C^{\infty}(\oplus_{i=1}^{p}E)$ is skew-symmetric if $F(e_{\sigma(1)},\dots,e_{\sigma(p)})=sgn(\sigma)F(e_{1},\dots,e_{p}),$ for every permutation $\sigma\in S(p)$. Under the correspondence of Lemma 2.4, the componentwise linear functions on $\oplus^{p}E$ which are skew-symmetric correspond to $\Gamma(\wedge^{p}E^{*})$. For $\tau=\mu_{1}\wedge\dots\wedge\mu_{p}$, with $\mu_{i}\in\Gamma(E^{*})$,

[TABLE]

There is a natural projection $C^{\infty}(\oplus^{p}E)\rightarrow C^{\infty}(\oplus^{p}E)$ on the space of skew-symmetric functions, defined by

[TABLE]

where $\sigma:\oplus^{p}E\rightarrow\oplus^{p}E$ is given by $\sigma(e_{1},\dots,e_{p})=(e_{\sigma(1)},\dots,e_{\sigma(p)})$.

Remark 2.5**.**

Functions which are skew-symmetric only on some components can be defined similarly by considering $E_{1}=\dots=E_{p^{\prime}}=E$, $p^{\prime}<p$. Extending the previous observations to this case is straightforward.

3. Multiplicative tensors

In this section, we introduce our main object of study, multiplicative tensors, and state our main theorem, which gives their full infinitesimal description. As we follow the idea of regarding tensors as functions on Whitney sums of the tangent and cotangent bundles, we start by discussing multiplicative functions in general.

3.1. Multiplicative functions

Let $\mathcal{H}\rightrightarrows N$ be a Lie groupoid with Lie algebroid $A\mathcal{H}$.

Definition 3.1**.**

A smooth function $f\in C^{\infty}(\mathcal{H})$ is said to be multiplicative if

[TABLE]

In other words, a multiplicative function $f:\mathcal{H}\rightarrow\mathbb{R}$ is a groupoid morphism from $\mathcal{H}\rightrightarrows N$ to the abelian Lie group $\mathbb{R}$. As such, it defines a Lie-algebroid morphism $Af:A\mathcal{H}\rightarrow\mathbb{R}$ given by $\langle Af,\,\chi\rangle=df(\chi),$ for $\chi\in A_{y}\mathcal{H}$ and $y\in N$; equivalently,

[TABLE]

for $\chi\in\Gamma(A\mathcal{H})$. When we view $Af$ as a section of the dual bundle $A^{*}\mathcal{H}$, the condition for $Af$ to be a Lie-algebroid morphism is expressed by the cocycle equation

[TABLE]

where $d_{A}:\Gamma(\wedge^{\bullet}A^{*}\mathcal{H})\to\Gamma(\wedge^{\bullet+1}A^{*}\mathcal{H})$ is the Lie-algebroid differential.

Example 3.2**.**

The function $f=\mathsf{t}^{*}\psi-\mathsf{s}^{*}\psi$ is multiplicative, for any $\psi\in C^{\infty}(N)$. The associated cocycle $Af$ is $d_{A}\psi\in\Gamma(A^{*}\mathcal{H})$, which is exact, i.e., a coboundary.

Example 3.3**.**

Let $\pi:E\rightarrow N$ be a vector bundle, viewed as a Lie groupoid. A function $f\in C^{\infty}(E)$ is multiplicative if and only if it is fiberwise linear. Let $\mu\in\Gamma(E^{*})$ be such that $f=\ell_{\mu}$. The cocycle $Af\in\Gamma(A^{*}E)=\Gamma(E^{*})$ is given by

[TABLE]

So $Af=\mu$ (which agrees with $f$ itself, if seen as a function $E\rightarrow\mathbb{R}$). The equation $d_{A}\mu=0$ is trivially satisfied since $d_{A}=0$ in this case.

The next result gives a useful formula relating the cocycle $Af$ and the function $f$.

Lemma 3.4**.**

For a multiplicative function $f:\mathcal{H}\rightarrow\mathbb{R}$, its corresponding cocycle $Af\in\Gamma(A^{*}\mathcal{H})$ satisfies

[TABLE]

Proof.

First, by differentiating equation (3.1), one sees that

[TABLE]

Also, $\overrightarrow{\chi}(h)=\chi(\mathsf{t}(h))\bullet 0_{h}$ (where $0_{h}$ is the zero vector field on $\mathcal{H}$ at $h$). Hence

[TABLE]

as we wanted. ∎

It turns out that when $\mathcal{H}$ is source connected, equation (3.3) essentially characterizes multiplicative functions:

Proposition 3.5**.**

Let $\mathcal{H}\rightrightarrows N$ be a source-connected Lie groupoid. A function $f\in C^{\infty}(\mathcal{H})$ is multiplicative if and only if $f|_{N}=0$ and there exists $\mu\in\Gamma(A^{*}\mathcal{H})$ such that

[TABLE]

In this case, $Af=\mu$.

Proof.

If $f$ is multiplicative, then $f|_{N}=0$ as a consequence of (3.1). Also, setting $\mu=Af$, (3.4) follows from Lemma 3.3.

In the other direction, fix $\chi_{1},\chi_{2}\in\Gamma(A\mathcal{H})$. Then

[TABLE]

which implies that

[TABLE]

So $\mu$ is a Lie-algebroid cocycle. If $\widetilde{\mathcal{H}}$ is a source 1-connected integration of $A\mathcal{H}$, then there exists a multiplicative function $f_{\mu}\in C^{\infty}(\widetilde{\mathcal{H}})$ such that $Af_{\mu}=\mu$. The Lie groupoids $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ correspond to the same Lie algebroid, so they are related by a groupoid map $\sigma:\widetilde{\mathcal{H}}\to\mathcal{H}$ which is a local diffeomorphism (and restricts to the identity map on $N$). To check that $f$ is multiplicative, it is enough to check that $\sigma^{*}f$ is multiplicative. We will verify that $\sigma^{*}f=f_{\mu}$.

By (3.3), since $\sigma$ is a groupoid morphism (hence commutes with structure maps and preserves invariant vector fields), one has that

[TABLE]

where we have kept the same notation for the structure maps on $\widetilde{\mathcal{H}}$ and $\mathcal{H}$. Since $\widetilde{\mathcal{H}}$ has source-connected fibers, it follows that $\sigma^{*}f-f_{\mu}$ is constant along the $\mathsf{s}$-fibers. Finally, since $(\sigma^{*}f-f_{\mu})|_{N}=0$, one has that $\sigma^{*}f-f_{\mu}=0$ everywhere. ∎

3.2. Definition and examples

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid and consider a $(q,p)$-tensor field $\tau\in\Gamma(\wedge^{p}\,T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$. Let $c_{\tau}:\mathbb{G}^{(p,q)}\rightarrow\mathbb{R}$ be the corresponding componentwise linear function,

[TABLE]

as in Definition 2.2, where $\mathbb{G}^{(p,q)}\rightrightarrows\mathbb{M}^{p,q)}$ is the Lie groupoid (2.7). In the following, we shall omit the $(p,q)$-indices.

Definition 3.6**.**

A $(q,p)$-tensor field $\tau\in\Gamma({\wedge^{p}\,T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G}})$ on $\mathcal{G}\rightrightarrows M$ is multiplicative if the function $c_{\tau}:\mathbb{G}\rightarrow\mathbb{R}$ is multiplicative.

Note that the very same definition of multiplicativity makes sense for elements of $\Gamma((\otimes^{p}T^{*}\mathcal{G})\otimes(\otimes^{q}T\mathcal{G}))$ (or for their symmetric versions). Along the paper, we will make some comments on how to adapt our results to this more general case.

The next examples relate our definition with known notions of multiplicativity for special types of tensors.

Example 3.7**.**

A differential form $\omega\in\Omega^{p}(\mathcal{G})$ is multiplicative if it satisfies

[TABLE]

where ${\rm pr}_{i}:\mathcal{G}\,_{\mathsf{s}}\!\times_{\mathsf{t}}\mathcal{G}\rightarrow\mathcal{G}\rightarrow\mathcal{G}$, $i=1,2$, are the natural projections. One can directly check (see e.g. [3]) that this is equivalent to $c_{\omega}$ being multiplicative.

Example 3.8**.**

A multivector field $\Pi\in\mathfrak{X}^{q}(\mathcal{G})$ is said to be multiplicative [28] if the graph of the multiplication map is coisotropic with respect to $\Pi\oplus\Pi\oplus(-1)^{q+1}\Pi$: for $\xi_{i}\in\mathrm{Ann}({\mathrm{graph}}(\mathsf{m}))\subset T^{*}(\mathcal{G}\times\mathcal{G}\times\mathcal{G})$, $i=1,\ldots,q$, we have

[TABLE]

It is shown in [28] that this condition is equivalent to $c_{\Pi}$ being multiplicative.

The next result shows that our definition of multiplicativity for $(1,p)$-tensor fields $K\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes T\mathcal{G})$ agrees with the one given in [43]. Note that $K$ can be seen as a map $\overline{K}:\oplus^{p}T\mathcal{G}\to T\mathcal{G}$.

Proposition 3.9**.**

A $(1,p)$-tensor field $K\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes T\mathcal{G})$ is multiplicative if and only if there is a vector-bundle map $\overline{r}:\oplus^{p}TM\to TM$ covering the identity such that

[TABLE]

is a groupoid morphism.

Proof.

It is straightforward to check that if (3.5) is a groupoid morphism, then $K$ is multiplicative. Conversely, let us assume that $K$ is a multiplicative $(1,p)$-tensor field. From Proposition 3.5, one knows that $c_{K}|_{\mathbb{M}}=0$. This implies that, for $(X_{1},\dots,X_{p})\in TM\oplus\dots\oplus TM$, $\overline{K}(X_{1},\dots,X_{p})\in T\mathcal{G}|_{M}$ has zero component on $A$ under the decomposition $T\mathcal{G}|_{M}=TM\oplus A$. So, define $\overline{r}=\overline{K}|_{\oplus^{p}TM}.$

It is straightforward to see that once

[TABLE]

$\overline{K}$ will preserve the multiplication as a direct consequence of the multiplicativity of $K$. So, let $\alpha\in T_{\mathsf{s}(g)}^{*}M$.

[TABLE]

Here, we have used the equality $(d\mathsf{s})_{g}^{*}\,\alpha=0_{g}\bullet(d\mathsf{s})_{\mathsf{s}(g)}^{*}\,\alpha,$ which follows from (2.4). The other equality follows similarly. ∎

We now consider a special class of multiplicative tensor fields on $\mathcal{G}$, analogous to the multiplicative functions in Example 3.2. Let $\mathcal{S},\,\mathcal{T}:\Gamma(\wedge^{p}\,T^{*}M\otimes\wedge^{q}A)\rightarrow\Gamma(\wedge^{p}\,T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$ be the $\mathbb{R}$-linear maps defined on homogeneous elements as

[TABLE]

Proposition 3.10**.**

The maps $\mathcal{S}$ and ${\mathcal{T}}$ satisfy the following properties:

1. (1)

${\mathcal{T}}(f\Phi)=(\mathsf{t}^{*}f)\,{\mathcal{T}}(\Phi)$,   $\mathcal{S}(f\Phi)=(\mathsf{s}^{*}f)\,\mathcal{S}(\Phi)$, 2. (2)

$\mathbbm{t}^{*}c_{\Phi}=c_{\mathcal{T}(\Phi)}$, 3. (3)

$\mathbbm{s}^{*}c_{\Phi}=c_{\mathcal{S}(\Phi)}$,

for $\Phi\in\Gamma(\wedge^{p}\,T^{*}M\otimes\wedge^{q}A)$ and $f\in C^{\infty}(M)$. In particular,

[TABLE]

is a multiplicative $(q,p)$-tensor field on $\mathcal{G}$.

Proof.

The formulas in (1) can be verified directly.

For $\Phi$ of the form $\alpha\otimes\mathfrak{X}$ and $\xi_{1},\dots,\xi_{q}\in T_{g}^{*}\mathcal{G}$, formulas (2) and (3) boil down to

[TABLE]

respectively, and these identities are direct consequences of the definitions of the source and target maps of the cotangent groupoid (see (2.3)). The case of arbitrary $\Phi$ follows by linearity.

The last assertion follows from the fact that $\mathbbm{t}^{*}c_{\Phi}-\mathbbm{s}^{*}c_{\Phi}$ is always a multiplicative function (see Example 3.2). ∎

3.3. The main results: statements and first examples

We now present a complete infinitesimal characterization of multiplicative tensor fields. Our first main theorem is an analog of Proposition 3.5 for general tensor fields.

Theorem 3.11**.**

Let $\mathcal{G}\rightrightarrows M$ be a source-connected Lie groupoid. A $(q,p)$-tensor field $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$ is multiplicative if and only if

[TABLE]

and there exist vector-bundle maps $l:A\rightarrow\wedge^{p-1}T^{*}M\otimes\wedge^{q}A$ and $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}A$, covering the identity map on $M$, and an $\mathbb{R}$-linear map $D:\Gamma(A)\rightarrow\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}A)$, such that

[TABLE]

for $a\in\Gamma(A)$, $\alpha\in\Gamma(T^{*}M)$ and $f\in C^{\infty}(M)$.

We refer to the triple $(D,l,r)$ as the infinitesimal components of the multiplicative tensor $\tau$ and to equation (3.9) as the Leibniz condition for $D$.

Remark 3.12**.**

Regarding the notation in Theorem 3.11 above, in (3.9) we view $\Gamma(\wedge^{\bullet}T^{*}M\otimes\wedge^{\bullet}A)$ as a left module for the exterior algebras $\Gamma(\wedge^{\bullet}A)$ and $\Gamma(\wedge^{\bullet}T^{*}M)$, and both actions are denoted by $\wedge$: for $\mathfrak{Y}\in\Gamma(\wedge^{\bullet}A)$, $\eta\in\Gamma(\wedge^{\bullet}T^{*}M)$, and $\omega\otimes\mathfrak{X}\in\Gamma(\wedge^{\bullet}T^{*}M\otimes\wedge^{\bullet}A)$,

[TABLE]

In (3.10), the contraction operators are defined as follows: for $U\in\Gamma(T\mathcal{G})$, $\xi\in\Gamma(T^{*}\mathcal{G})$ and $\tau=\lambda\otimes W$,

[TABLE]

We mention some particular cases of interest.

Corollary 3.13**.**

On a source-connected Lie groupoid $\mathcal{G}\rightrightarrows M$, a $p$-form $\omega\in\Omega^{p}(\mathcal{G})$ is multiplicative if and only if there is an $\mathbb{R}$-linear map $D:\Gamma(A)\rightarrow\Omega^{p}(M)$ and a vector-bundle morphism $l:A\rightarrow\wedge^{p-1}T^{*}M$ such that the following holds:

[TABLE]

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

The next result recovers [28, Thm. 2.19] (showing that some conditions there are redundant, cf. [11, Lem. 2.3]).

Corollary 3.14**.**

Let $\mathcal{G}\rightrightarrows M$ be a source connected Lie groupoid. A $q$-vector field $\Pi\in\mathfrak{X}^{q}(\mathcal{G})$ is multiplicative if and only there is an $\mathbb{R}$-linear map $D:\Gamma(A)\rightarrow\Gamma(\wedge^{q}A)$ and a vector bundle morphism $r:T^{*}M\rightarrow\wedge^{q-1}A$ such that the following holds:

[TABLE]

for all $(\varphi_{1},\dots,\varphi_{q})\in A^{*}\times_{M}\dots\times_{M}A^{*}$, $a\in\Gamma(A)$ and $\alpha\in\Omega^{1}(M)$. (In the terminology of [28, 51], the first condition above says that $M$ is coisotropic with respect to $\Pi$, while the other conditions say that $\Pi$ is an affine multivector field (see also [18, 35]).)

Following the previous corollary, Theorem 3.11 indicates a notion of affine tensors on Lie groupoids, in which we keep all properties of multiplicative tensors described in the theorem except for (3.8) (cf. [61, Thm. 4.5]).

In the next example, we identify the infinitesimal components of a multiplicative tensor field of type (3.7). To do so, let us consider, for a Lie algebroid $(A,\rho,[\cdot,\cdot])$, the action of the Lie algebra $\Gamma(A)$ on $\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}A)$ given by

[TABLE]

where $[\cdot,\cdot]$ is the Schouten bracket on $\Gamma(\wedge^{\bullet}A)$.

Example 3.15**.**

For $\Phi\in\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}A)$, consider the multiplicative tensor given by $\tau=(\mathcal{T}-\mathcal{S})(\Phi)$ (see Prop. 3.10). As particular cases, for $q=0$, $\tau=\mathsf{t}^{*}\Phi-\mathsf{s}^{*}\Phi$, while for $p=0$, $\tau=\overrightarrow{\Phi}-\overleftarrow{\Phi}$. The infinitesimal components $(D,l,r)$ corresponding to $\tau$ are

[TABLE]

Remark 3.16**.**

For $\tau\in\Gamma((\otimes^{p}T^{*}\mathcal{G})\otimes(\otimes^{q}T\mathcal{G}))$, there is a result similar to Theorem 3.11 characterizing multiplicativity. In this case we have infinitesimal components $(D,l_{1},\dots,l_{p},r_{1},\dots,r_{q})$, where $D:\Gamma(A)\to\Gamma((\otimes^{p}T^{*}M)\otimes(\otimes^{q}A))$, $l_{i}:A\to(\otimes^{p-1}T^{*}M)\otimes(\otimes^{q}A)$, and $r_{j}:T^{*}M\to(\otimes^{p}T^{*}M)\otimes(\otimes^{q-1}A)$, defined by

[TABLE]

The Leibniz equation for $D$ will change accordingly. For instance, for a multiplicative $\tau\in\Gamma(T\mathcal{G}\otimes T\mathcal{G})$, the infinitesimal components $(D,r_{1},r_{2})$ satisfy

[TABLE]

We now formulate our main result, which concerns the correspondence between multiplicative tensors $\tau$ and their infinitesimal components $(D,l,r)$. The multiplicativity of $\tau$ is expressed by a set of equations satisfied by $(D,l,r)$, described in the next definition.

Definition 3.17**.**

Let $(A,[\cdot,\cdot],\rho)$ be a Lie algebroid. An IM $(q,p)$-tensor on $A$ is a triple $(D,l,r)$, where $l:A\rightarrow\wedge^{p-1}T^{*}M\otimes\wedge^{q}A$ and $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}A$ are vector-bundle maps covering the identity map on $M$, $D:\Gamma(A)\rightarrow\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}A)$ is $\mathbb{R}$-linear and satisfies the Leibniz rule

[TABLE]

such that the following equations hold:

[TABLE]

for $a,b\in\Gamma(A)$ and $\alpha,\beta\in\Omega^{1}(M)$.

We refer to equations (IM1) – (IM6) as the IM-equations of an IM $(q,p)$-tensor.

Remark 3.18** (Redundancies).**

We observe that, in many cases, there are some redundancies in the IM-equations. Note first that, if $p>\dim(M)+1$ or $q>\mathrm{rank}(A)+1$, then any IM $(p,q)$-tensor is trivial. On the other hand, when $p<\dim(M)+1$ and $q<\mathrm{rank}(A)+1$, we claim that $\{\eqref{IM1},\eqref{IM2},\eqref{IM6}\}$ and $\{\eqref{IM1},\eqref{IM3},\eqref{IM6}\}$ are minimal sets of independent IM-equations. Indeed, in this case, (IM1) implies that

[TABLE]

for all $f\in C^{\infty}(M),a,b\in\Gamma(A)$. This follows from the Leibniz rule for the Lie bracket $[\cdot,\cdot]$, the Leibniz formula for $D$, and Proposition 4.3 below. Therefore,

[TABLE]

Similarly, one can prove that (IM2) (resp. (IM3)) implies that

[TABLE]

As a result, $\eqref{IM2}+\eqref{IM6}\Rightarrow\eqref{IM4}\text{ and }\eqref{IM3}+\eqref{IM6}\Rightarrow\eqref{IM5}.$

Our main theorem can now be stated as follows:

Theorem 3.19**.**

Let $\mathcal{G}\rightrightarrows M$ be a source 1-connected Lie groupoid, and let $A$ be its Lie algebroid. There is a one-to-one correspondence between multiplicative $(q,p)$-tensor fields $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$ and IM $(q,p)$-tensors $(D,l,r)$ on $A$ satisfying

[TABLE]

where ${\mathcal{T}}$ is the map given by (3.6).

Remark 3.20**.**

The correspondence in Theorem 3.19 can be naturally extended to multiplicative tensors $\tau\in\Gamma((\otimes^{p}T^{*}\mathcal{G})\otimes(\otimes^{q}T\mathcal{G}))$; in this case, following Rem. 3.16, we have more infinitesimal components $(D,l_{1},\dots,l_{p},$ $r_{1},\dots,r_{q})$, obtained as in (3.12), satisfying analogous IM-equations.

We now show how Theorem 3.19, when restricted to $(0,p)$ and $(q,0)$ tensors, directly recovers the infinitesimal descriptions of multiplicative differential forms and multivector fields, proven in [1, 6, 28]. It also recovers the correspondences in [43, 57], but we will leave this discussion to Section 6, where we present a more general treatment of multiplicative $(1,1)$-tensor fields.

Multiplicative differential forms

According to Theorem 3.19, multiplicative $p$-forms correspond, infinitesimally, to IM $(0,p)$-tensors. Explicitly, on a given Lie algebroid $A\to M$, IM $(0,p)$-tensors are pairs $(D,l)$, where $D:\Gamma(A)\rightarrow\Gamma(\wedge^{p}T^{*}M)$, $l:A\rightarrow\wedge^{p-1}T^{*}M$, and these maps satisfy

[TABLE]

for $a\in\Gamma(A)$ and $f\in C^{\infty}(M)$, and

[TABLE]

Note that IM $(0,p)$-tensors agree with Spencer operators with values in the trivial representation, as considered in [14].

Example 3.21**.**

On a Poisson manifold $(M,\Pi)$, the cotangent bundle $T^{*}M$ has a Lie algebroid structure whose anchor is given by the contraction of covectors with $\Pi$, $\Pi^{\sharp}:T^{*}M\to TM$, and the Lie bracket is given by

[TABLE]

There exists a canonical IM $(0,2)$-tensor on $T^{*}M$ given by $D=d$, the de Rham differential, and $l=\mathrm{id}_{T^{*}M}$.

We have the following alternative way to express IM $(0,p)$-tensors, see [1, 3, 5]. We consider pairs $(\mu,\nu)$ with $\mu:A\rightarrow\wedge^{p-1}T^{*}M$, $\nu:A\rightarrow\wedge^{p}T^{*}M$ bundle maps (covering the identity) satisfying

[TABLE]

for all $a,b\in\Gamma(A)$. Such a pair $(\mu,\nu)$ is called an IM $p$-form in [3]. The equivalence between IM $(0,p)$-tensors $(D,l)$ and IM $p$-forms $(\mu,\nu)$ is given by the following explicit relations:

[TABLE]

When $(D,l)$ are the infinitesimal components of a multiplicative $(0,p)$ tensor field (i.e. a differential $p$-form) $\omega\in\Omega^{p}(\mathcal{G})$, we have that

[TABLE]

for all $a\in\Gamma(A)$. It follows that

[TABLE]

In this way, we can see that Theorem 3.19 immediately recovers the main result of [3], relating multiplicative and IM differential forms:

Corollary 3.22**.**

For a source 1-connected groupoid $\mathcal{G}\rightrightarrows M$, there is a one-to-one correspondence between multiplicative $p$-forms $\omega\in\Omega^{p}(\mathcal{G})$ and IM $p$-forms $(\mu,\nu)$, given by

[TABLE]

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

For a Poisson manifold $(M,\Pi)$, if $(T^{*}M,[\cdot,\cdot]_{\Pi})$ is the Lie algebroid of a source 1-connected groupoid $\mathcal{G}\rightrightarrows M$, then $\mathcal{G}$ has a canonical multiplicative 2-form integrating the canonical IM $(0,2)$-tensor of Example 3.21. In this case, note that $\nu=0$, and this implies that $d\omega=0$. One can also verify that $l=\mu$ being an isomorphism implies that $\omega$ is non-degenerate. So $(\mathcal{G},\omega)$ is the symplectic groupoid integrating $T^{*}M$.

Remark 3.23**.**

Multiplicative differential forms on a Lie groupoid $\mathcal{G}$ form a subcomplex of the de Rham complex. On the other hand, if $(\mu,\nu)$ is an IM $p$-form, one can directly verify that $(\nu,0)$ is an IM $(p+1)$-form. So $(\mu,\nu)\mapsto(\nu,0)$ defines a differential on the space of all IM-forms, in such a way that the correspondence in Cor. 3.22 is an isomorphism of complexes.

Multiplicative multivector fields

Let us now consider the case of multiplicative $(q,0)$-tensor fields, i.e., multiplicative $q$-vector fields. Following Theorem 3.19, their infinitesimal counterparts are IM $(q,0)$-tensors: on a given Lie algebroid $A\to M$, these are pairs $(D,r)$, where $r:T^{*}M\to\wedge^{q-1}A$ is a vector-bundle map (covering the identity), $D:\Gamma(A)\to\Gamma(\wedge^{q}A)$ is $\mathbb{R}$-linear and satisfies

[TABLE]

for $a\in\Gamma(A)$ and $f\in C^{\infty}(M)$, and the following compatibility conditions hold:

[TABLE]

for $a,b\in\Gamma(A)$, $\alpha,\beta\in\Omega^{1}(M)$.

In order to make the connection with the work in [28], recall that a $q$-differential on a Lie algebroid $A$ is an $\mathbb{R}$-linear map $\delta:\Gamma(\wedge^{\bullet}A)\rightarrow\Gamma(\wedge^{\bullet+q-1}A)$ satisfying

[TABLE]

where $\mathfrak{X}_{i}\in\Gamma(\wedge^{k_{i}}A)$, $i=1,2$, and $[\cdot,\cdot]$ is the Schouten bracket on $\Gamma(\wedge^{\bullet}A)$ (which makes it into a Gerstenhaber algebra). In other words, $\delta$ is a derivation of degree $(q-1)$ of $\Gamma(\wedge^{\bullet}A)$ which is also a derivation of the Schouten bracket. We denote the space of $q$-differentials by $\mathcal{A}_{q}$. The space $\mathcal{A}=\oplus_{q\geq 0}\mathcal{A}_{q}$ is naturally a Gerstenhaber algebra with respect to the bracket given by the commutator

[TABLE]

where $\delta\in\mathcal{A}_{q}$ and $\widetilde{\delta}\in\mathcal{A}_{\widetilde{q}}$.

Note that a $q$-differential $\delta$ is determined by its restrictions

[TABLE]

For this reason, we may denote a $q$-differential by the pair $(\delta_{0},\delta_{1})$.

Before studying the relationship between $q$-differentials and IM $(q,0)$-tensors, we list here some properties of the Schouten bracket that we need (see e.g. [40, 41]):

1. (1)

For $f\in C^{\infty}(N)$, $\mathfrak{X}\in\Gamma(\wedge^{q}A)$,

[TABLE] 2. (2)

For $\mathfrak{X}_{i}\in\Gamma(\wedge^{q_{i}}A)$, $i=1,2$,

[TABLE] 3. (3)

For $\mathfrak{X}_{i}\in\Gamma(\wedge^{q_{i}}A)$, $i=1,2,3$,

[TABLE]

Lemma 3.24**.**

There is a one-to-one correspondence between $q$-differentials $(\delta_{0},\delta_{1})$ and IM $(q,0)$-tensors $(D,r)$ via

[TABLE]

where $d$ is the de Rham differential.

Proof.

Note that (3.15), (3.16) give rise to five equations involving $\delta_{0}$ and $\delta_{1}$, to be compared with the four equations characterizing IM $(q,0)$-tensors.

For $k_{1}=k_{2}=0$, (3.15) is equivalent to the existence of a vector-bundle map $r_{0}:T^{*}M\rightarrow\wedge^{q-1}A$ such that $r_{0}(df)=\delta_{0}(f)$. This guarantees that we can always assume that $\delta_{0}$ is of the form described in (3.17), i.e., we set $r=(-1)^{q}r_{0}$ and $D=\delta_{1}$.

For $k_{1}=1$ and $k_{2}=0$, (3.15) becomes

[TABLE]

which is just the Leibniz rule for $(D,r)$.

Next, when $k_{1}=k_{2}=0$ (3.16) reads

[TABLE]

Using that the Schouten bracket satisfies $[\mathfrak{X},f]=(-1)^{k-1}i_{\rho^{*}df}\mathfrak{X}$, for $f\in C^{\infty}(M)$ and $\mathfrak{X}\in\Gamma(\wedge^{k}A)$, we see that this last equation boils down to $i_{\rho^{*}dg}r(df)=-i_{\rho^{*}df}r(dg)$, which is equivalent to the condition

[TABLE]

for $\alpha,\beta\in\Omega^{1}(M)$.

It is immediate that, for $k_{1}=k_{2}=1$, (3.16) becomes

[TABLE]

Finally, when $k_{1}=1$, $k_{2}=0$, (3.16) amounts to

[TABLE]

which is equivalent to the condition

[TABLE]

∎

For a multiplicative $q$-vector $\Pi$ on a Lie groupoid $\mathcal{G}\rightrightarrows M$, its infinitesimal components are written in terms of the corresponding $q$-differential as follows:

[TABLE]

Since $[\overrightarrow{a},\Pi]=\mathcal{L}_{\overrightarrow{a}}\Pi$ and $[\Pi,\mathsf{t}^{*}f]=(-1)^{q-1}i_{\mathsf{t}^{*}df}\Pi$, and using the fact that $\delta$ is a $q$-differential if and only if so is $-\delta$, we see that Theorem 3.19 recovers the following correspondence (which is the central result in [28]; see also [6]):

Corollary 3.25**.**

For a source 1-connected Lie groupoid $\mathcal{G}\rightrightarrows M$, there is a one-to-one correspondence between multiplicative $q$-vector fields $\Pi\in\mathfrak{X}^{q}(\mathcal{G})$ and $q$-differentials $(\delta_{0},\delta_{1})$, given by

[TABLE]

for $f\in C^{\infty}(M)$ and $a\in\Gamma(A)$.

Remark 3.26**.**

*In [28], it is verified that the space of multiplicative multivector fields on a Lie groupoid $\mathcal{G}$ is closed under the Schouten bracket, so it is a Gerstenhaber subalgebra of the space of all multivector fields on $\mathcal{G}$. The correspondence in the previous corollary is proven to give rise to an isomorphism of Gerstenhaber algebras. *

The next example shows how the correspondence between Poisson groupoids and Lie bialgebroids [51] fits into the framework of IM $(2,0)$-tensors.

Example 3.27**.**

For the case $q=2$, there is a one-to-one correspondence between pairs $(D,r)$ satisfying (3.14) and pre-Lie algebroid 333A pre-Lie algebroid structure on a vector bundle $E\to M$ consist of an anchor map $\rho_{E}:E\to TM$ together with a skew-symmetric bilinear bracket $[\cdot,\cdot]$ on $\Gamma(E)$ such that the Leibniz equation $[u,fv]=f[u,v]+(\mathcal{L}_{\rho_{E}(u)}f)\,v$ holds (see [26]). structures on $A^{*}$ given as follows: $\rho_{*}:A^{*}\to TM$ is the dual map to $r:A\to T^{*}M$, and the bracket $[\cdot,\cdot]_{*}$ is determined by the Koszul formula:

[TABLE]

for $a\in\Gamma(A)$ and $\mu_{1},\mu_{2}\in\Gamma(A^{*})$. This is just another incarnation of the known correspondence between pre-Lie algebroids structures on $A^{*}$ and linear bivector fields on $A$ [26] (see also Corollary 4.11).

Recall that a Poisson groupoid is a Lie groupoid $\mathcal{G}\rightrightarrows M$ endowed with a multiplicative 2-vector field $\Pi\in\mathfrak{X}^{2}(\mathcal{G})$ such that $[\Pi,\Pi]=0$. The IM $(2,0)$-tensor $(D,r)$ on its Lie algebroid $A$ corresponding to $\Pi$ defines a pre-Lie algebroid structure on $A^{*}$. It has an associated operator $\delta:\Gamma(\wedge^{\bullet}A)\to\Gamma(\wedge^{\bullet+1}A)$ which is the 2-differential defined by (3.17). Now, the fact that $[\Pi,\Pi]=0$ implies that $\delta^{2}=0$, which says that the Jacobi identity holds for the pre-Lie bracket $[\cdot,\cdot]_{*}$; so $A^{*}$ is a Lie algebroid. Finally, the IM-equation

[TABLE]

gives the compatibility condition for $(A,A^{*})$ to be a Lie bialgebroid. This is the only relevant IM-equation because of the redundancies explained in Rem. 3.18 (see [36]).

4. Proof of the Theorems

Before delving into the proofs of Theorems 3.11 and 3.19, let us briefly sketch the general strategy to obtain the infinitesimal description of multiplicative tensors.

Given a multiplicative $(q,p)$-tensor field $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$, our main object of analysis is formula (3.3) applied to $c_{\tau}$, the corresponding multiplicative function on the groupoid $\mathbb{G}\rightrightarrows\mathbb{M}$ in (2.7):

[TABLE]

Our goal is to have a concrete description of the infinitesimal cocycle $Ac_{\tau}\in\Gamma(\mathbb{A}^{*})$, which codifies the infinitesimal information of $\tau$.

The first key observation is that it is enough to check the identity (4.1) when $\chi$ varies within a special set of generators for the $C^{\infty}(\mathbb{M})$-module of sections of $\mathbb{A}$. These generators will be parametrized by $\Gamma(A)$ and $\Omega^{1}(M)$, and their pairing with $Ac_{\tau}$ will give rise to maps from the space of parameters into $C^{\infty}(\mathbb{M})$; more precisely, we will obtain three maps, $D$, $l$, and $r$, taking values in the subspace $\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}A)\subseteq C^{\infty}(\mathbb{M})$ of componentwise linear functions. These maps, which completely determine $Ac_{\tau}$, will agree with the infinitesimal components of $\tau$.

Considering the left-hand side of (4.1), we will see that the Lie derivatives of $c_{\tau}$ can be expressed in terms of contraction and Lie-derivative operations on the tensor field $\tau$ itself. In this way, the equality (4.1) is re-written as the relations involving $\tau$ and $(D,l.r)$ in Theorem 3.19. The last step is expressing the cocycle condition $d_{\mathbb{A}}(Ac_{\tau})=0$, where $d_{\mathbb{A}}$ is the Lie algebroid differential on $\Gamma(\mathbb{A}^{*})$, in terms of $(D,l,r)$. This will lead to the IM-equations.

We will need a few technical tools to carry out this strategy, including the study of lifting operations (Section 4.1) and an analysis of linear tensor fields (Section 4.4).

4.1. Lifting operations

As we now see, classical lifting operations (see e.g. [63]) of vector fields are essential ingredients in relating Lie derivatives of tensor fields $\tau$ with Lie derivatives of the corresponding componentwise linear functions $c_{\tau}$.

Let $\pi_{E}:E\to N$ be a vector bundle over a smooth manifold $N$. Given a section $u\in\Gamma(E)$, its vertical lift is the vector field $u^{{\mathrm{v}}}:E\rightarrow TE$ on $E$ defined by

[TABLE]

For a section $\psi\in\Gamma(E^{*})$, we recall that the Lie derivative of its corresponding linear function $\ell_{\psi}\in C^{\infty}(E)$ along $u^{{\mathrm{v}}}$ is given by

[TABLE]

In this paper, we are mostly interested in the cases where $E=TN$ or $E=A^{*}$, the dual of a Lie algebroid $(A,[\cdot,\cdot],\rho)$ over $N$. We denote the bundle projections by $\pi:TN\rightarrow N$ and $\pi_{*}:A^{*}\rightarrow N$. In these cases, besides the vertical lifting, there are two other important lifting constructions that we need to recall. For a vector field $X\in\mathfrak{X}(N)$, consider its (local) flow $\phi_{\epsilon}:N\rightarrow N$. The tangent lift of $X$ is the vector field $X^{T}$ on $TN$ with flow given by $\epsilon\mapsto T\phi_{\epsilon}$. For $\alpha\in\Omega^{1}(N)$ and $f\in C^{\infty}(N)$, the Lie derivatives of the functions $\ell_{\alpha}$ and $f\circ\pi$ in $C^{\infty}(TN)$ along $X^{T}$ are given by

[TABLE]

The Hamiltonian lift of a section $a\in\Gamma(A)$ of a Lie algebroid is the vector field $H_{a}$ on $A^{*}$ defined by

[TABLE]

where $\Pi_{lin}\in\Gamma(\wedge^{2}TA^{*})$ is the linear Poisson structure on $A^{*}$ (dual to the Lie algebroid structure on $A$) and $\ell_{a}\in C^{\infty}(A^{*})$ is the linear function corresponding to $a$. For $b\in\Gamma(A)$ and $f\in C^{\infty}(N)$,

[TABLE]

Note that (4.4) and (4.6) completely characterize $X^{T}$ and $H_{a}$, respectively. When $A=TN$, the linear Poisson structure on $T^{*}N$ comes from the canonical symplectic form, and the Hamiltonian lift $H_{X}$ of a vector field $X\in\mathfrak{X}(N)$ coincides with the cotangent lift $X^{T^{*}}\in\mathfrak{X}(T^{*}N)$, which is the vector field with flow $\epsilon\mapsto(T\phi_{-\epsilon})^{*}$.

Our aim is to extend formulas (4.3), (4.4) and (4.6) to elements of the space $\Gamma(\wedge^{p}T^{*}N\otimes\wedge^{q}A)$. Let us first introduce some notation. For $Y\in\mathfrak{X}(M)$, we define vector fields on $\oplus^{p}TM$ as follows:

[TABLE]

Similarly, for $a\in\Gamma(A)$ and $\mu\in\Gamma(A^{*})$, we define vector fields on $\oplus^{q}A^{*}$ by

[TABLE]

Define

[TABLE]

to be the projections

[TABLE]

for $1\leq i\leq p$, $1\leq j\leq q$. When there is no risk of confusion, we simplify the notation by omitting the superscripts $(q,p)$ on the projections.

Proposition 4.1**.**

Let $\tau\in\Gamma(\wedge^{p}T^{*}N\otimes\wedge^{q}A)$, and consider the corresponding componentwise linear function $c_{\tau}:(\oplus^{p}TN)\oplus(\oplus^{q}A^{*})\rightarrow\mathbb{R}$. For $a\in\mathfrak{X}(A)$, $\mu\in\Gamma(A^{*})$ and $Y\in\Gamma(TN)$, one has that

[TABLE]

where $\cdot$ is the action (3.11).

Proof.

Let us consider the case $\tau=\omega\otimes\mathfrak{X}$, for $\omega\in\Omega^{p}(N)$ and $\mathfrak{X}\in\mathfrak{X}^{q}(A)$. First note that $c_{\tau}=(c_{\omega}\circ{\rm pr}_{TN})(c_{\mathfrak{X}}\circ{\rm pr}_{A^{*}}),$ where ${\rm pr}_{TN}$ and ${\rm pr}_{A^{*}}$ are the projections of $(\oplus^{p}TN)\oplus(\oplus^{q}A^{*})$ onto $\oplus^{p}TN$ and $\oplus^{q}A^{*}$, respectively. Using the Leibniz rule, it suffices to prove that

[TABLE]

Let us simplify matters once more by assuming that $\omega=\alpha_{1}\wedge\cdots\wedge\alpha_{p}$, for $\alpha_{1},\dots,\alpha_{p}\in\Gamma(T^{*}N)$. On the one hand, we have that

[TABLE]

On the other hand, by (2.8) and (4.3),

[TABLE]

where ${\rm pr}^{j}_{TN}:\oplus^{p}TN\rightarrow TN$ is the projection on the $j$-component, for $1\leq j\leq p$. Hence,

[TABLE]

The other equations follow similarly using (4.4) and (4.6). The case where $\tau$ is arbitrary follows from linearity of the Lie derivative. ∎

Remark 4.2**.**

When $A=TN$, the action $\cdot$ of $X\in\mathfrak{X}(N)$ on $\tau\in\Gamma(\wedge^{p}T^{*}N\otimes\wedge^{q}TN)$ is the Lie derivative of $\tau$ along $X$, $X\cdot\tau=\mathcal{L}_{X}\tau.$ In this case, Proposition 4.1 says that

[TABLE]

where $X^{T}$ and $X^{T^{*}}$ are the tangent and cotangent lifts of $X$.

For our next result, we keep the notation as in Remark 3.12.

Proposition 4.3**.**

Let $\tau\in\Gamma(\wedge^{p}T^{*}N\otimes\wedge^{q}A)$ and $a\in\Gamma(A)$. For $f\in C^{\infty}(N)$,

[TABLE]

Proof.

By linearity, it suffices to prove the result for $\tau=\omega\otimes\mathfrak{X}$, where $\omega\in\Gamma(\wedge^{p}T^{*}N)$ and $\mathfrak{X}\in\Gamma(\wedge^{q}A)$. First,

[TABLE]

and, by the properties of the Schouten bracket,

[TABLE]

Hence, $(fa)\cdot(\omega\otimes\mathfrak{X})=f\,(a\cdot\tau)+df\wedge\left((i_{\rho(a)}\omega)\otimes\mathfrak{X}\right)-a\wedge\left(\omega\otimes(i_{df}\mathfrak{X})\right)$, as we wanted. ∎

4.2. The Lie algebroid of $\mathbb{G}\rightrightarrows\mathbb{M}$

In this subsection, we discuss the Lie algebroid $\mathbb{A}\to\mathbb{M}$ of the Lie groupoid $\mathbb{G}\rightrightarrows\mathbb{M}$ introduced in (2.7) and describe a special set of generators for the $C^{\infty}(\mathbb{M})$-module $\Gamma(\mathbb{A})$.

Prolongations of vector bundles

Given a vector bundle $\pi_{E}:E\rightarrow M$, we may view it as a Lie groupoid whose source and target maps are equal to $\pi_{E}$, the unit map is the zero section $0:M\rightarrow E$, and the multiplication is fiberwise addition. In this case, the tangent Lie groupoid is the vector bundle $T\pi_{E}:TE\rightarrow TM$, called the tangent prolongation of $E$. Similarly, the cotangent Lie groupoid is the vector bundle $\widetilde{\pi}_{E}:T^{*}E\rightarrow E^{*}$, called the cotangent prolongation. The projection $\widetilde{\pi}_{E}$ has the following description: for $\xi\in T^{*}_{e}E$, $\widetilde{\pi}_{E}(\xi)\in E^{*}_{\pi_{E}(e)}$ is the element defined by

[TABLE]

The Lie algebroid $AE$ is identified with $E\rightarrow M$ itself, with the zero anchor and zero bracket; the right-invariant vector field corresponding to $u\in\Gamma(E)$ is the vertical lift $u^{{\mathrm{v}}}\in\mathfrak{X}(E)$, see (4.2). The exact sequence (2.2) becomes

[TABLE]

For $e\in E_{x}$,

[TABLE]

defines the inclusion $E\hookrightarrow 0^{*}TE$ in the exact sequence above. The translation bisection associated to $u\in\Gamma(E)$ is given by

[TABLE]

In the cotangent prolongation, the exact sequence (2.5) becomes the dual of (4.13),

[TABLE]

For $\beta\in T_{x}^{*}M$, its image under the first map of the exact sequence is

[TABLE]

For a 1-form $\alpha\in\Omega^{1}(M)$, its translation bisection $\mathcal{B}\alpha:E^{*}\rightarrow T^{*}E$ is given by

[TABLE]

Note that since $\mathsf{t}=\mathsf{s}=\pi_{E}$, $\mathcal{B}u$ and $\mathcal{B}\alpha$ are sections of $T\pi_{E}:TE\rightarrow TM$ and $\widetilde{\pi}_{E}:T^{*}E\rightarrow E^{*}$, respectively. They are known as core sections in the general theory of double vector bundles (see e.g. [47]).

Sections of prolongations

For a section $u:M\rightarrow E$, its derivative $Tu:TM\rightarrow TE$ defines a section of the tangent prolongation. There is also an induced section $\mathcal{R}_{u}:E^{*}\to T^{*}E$ of the cotangent prolongation, as we now explain.

Recall the reversal isomorphism $\mathcal{R}:T^{*}(E^{*})\rightarrow T^{*}E$ (see e.g. [47] for details): in local coordinates,

[TABLE]

where we are locally writing $T^{*}(E^{*})\cong E^{*}\oplus T^{*}M\oplus E$ and $T^{*}E\cong E\oplus T^{*}M\oplus E^{*}$. Globally, $\mathcal{R}$ is both a vector bundle morphism from the cotangent bundle of $E^{*}$, $T^{*}(E^{*})\rightarrow E^{*}$, to the cotangent prolongation of $E$, $T^{*}E\rightarrow E^{*}$, and from the cotangent prolongation of $E^{*}$, $T^{*}(E^{*})\rightarrow E$, to the cotangent bundle of $E$, $T^{*}E\rightarrow E$. It fits into the following commutative diagram of vector bundle morphisms:

[TABLE]

For $u\in\Gamma(E)$ and $\mu\in\Gamma(E^{*})$, let $\ell_{u}\in C^{\infty}(E^{*})$ and $\ell_{\mu}\in C^{\infty}(E)$ be the corresponding fiberwise linear functions. Then the composition

[TABLE]

defines a section of the cotangent prolongation. Note that the identities

[TABLE]

and

[TABLE]

completely determine $\mathcal{R}$, where $\pi_{E^{*}}:E^{*}\to M$ is the projection of the dual bundle.

Let us fix positive integers $p,q$ and consider the Lie groupoid (2.7), with $\mathcal{G}=E$. This is actually a vector bundle

[TABLE]

As before, we will simplify the notation by dropping the superindices.

For a section $u\in\Gamma(E)$, we denote by $(T^{p}u,\mathcal{R}_{u}^{\,q}):\mathbb{M}\rightarrow\mathbb{E}$ the section given by

[TABLE]

and by $\mathcal{B}u_{(i)}:\mathbb{M}\rightarrow\mathbb{E}$ the section given by

[TABLE]

for $i=1,\dots,p$. Similarly, for $\alpha\in\Omega^{1}(M)$, denote by $\mathcal{B}\alpha_{(j)}:\mathbb{M}\rightarrow\mathbb{E}$ the section defined by

[TABLE]

for $j=1,\dots,q$. The following result is proven in [48].

Proposition 4.4**.**

The $C^{\infty}(\mathbb{M})$-module of section $\Gamma(\mathbb{E})$ is generated by $(T^{p}u,\mathcal{R}_{u}^{q})$, $\mathcal{B}v_{(i)}$ and $\mathcal{B}\alpha_{(j)}$, for $u,v\in\Gamma(E)$ and $\alpha\in\Omega^{1}(M)$, $i=1,\dots,p$, $j=1,\dots,q$.

Remark 4.5**.**

[$C^{\infty}(M)$-linearity] One can check that

[TABLE]

where we used the notation $+_{\scriptscriptstyle{\mathfrak{p}}}$ and $\,\cdot\,$ for the sum and scalar multiplication on the fibers of the tangent prolongation $TE\rightarrow TM$, respectively, and $\ell_{df}\in C^{\infty}(TM)$ is the linear function corresponding to $df\in\Omega^{1}(M)$. Similarly,

[TABLE]

where $+_{\scriptscriptstyle{\mathfrak{p}}}$ and $\,\cdot\,$ denote the sum and scalar multiplication on the fibers of the cotangent prolongation $T^{*}E\rightarrow E^{*}$, respectively.

Brackets and anchors.

We now recall the main features of the tangent and cotangent Lie algebroids, see e.g. [49] for details.

For a Lie algebroid $(A,[\cdot,\cdot],\rho)$, consider its tangent prolongation $TA\to TM$. It has the structure of a Lie algebroid, where the Lie bracket $[\cdot,\cdot]$ on the space of sections of $TA\to TM$ is determined by the conditions

[TABLE]

for $a,\,b\in\Gamma(A)$, while the anchor $\rho_{T}:TA\rightarrow T(TM)$ is determined by

[TABLE]

where $(\cdot)^{T}$ and $(\cdot)^{{\mathrm{v}}}$ are the tangent and vertical lifts, respectively.

The cotangent Lie algebroid is the Lie algebroid structure on $T^{*}A\to A^{*}$ defined as follows: the anchor $\rho_{T^{*}}:T^{*}(A)\rightarrow T(A^{*})$ is determined by

[TABLE]

where $H_{a}\in\mathfrak{X}(A^{*})$ is the Hamiltonian lift of $a\in\Gamma(A)$, see (4.5). The Lie bracket on the space of sections of $T^{*}A\to A^{*}$ is determined by

[TABLE]

where $a,b\in\Gamma(A)$ and $\alpha,\beta\in\Omega^{1}(M)$.

The Whitney sum

[TABLE]

inherits a Lie algebroid structure which is determined componentwise by the tangent and the cotangent Lie algebroids. We give a detailed description here for convenience. The anchor map is determined by

[TABLE]

for $\alpha\in\Omega^{1}(M)$. The Lie bracket on the space of sections of $\mathbb{A}\to M$ is determined by what it does on generators according to the following formulas:

[TABLE]

For a Lie groupoid $\mathcal{G}\rightrightarrows M$, let $A\to M$ be its Lie algebroid. There are natural Lie-algebroid identifications $A(T\mathcal{G})\cong TA$ as well as $A(T^{*}\mathcal{G})\cong T^{*}A$ (see [49]). Hence, the Lie algebroid $A(\mathbb{G}^{(p,q)})\rightarrow\mathbb{M}^{(p,q)}$ of the Lie groupoid $\mathbb{G}^{(p,q)}\rightrightarrows\mathbb{M}^{(p,q)}$ is naturally isomorphic to

[TABLE]

The right-invariant vector fields on $T\mathcal{G}$ corresponding to the sections of type $Ta$, $\mathcal{B}a$ are given by

[TABLE]

see e.g. [49] for a proof. Similarly, the right-invariant vector fields on $T^{*}\mathcal{G}$ corresponding to the sections of type $\mathcal{R}_{a}$, $\mathcal{B}\alpha$ are

[TABLE]

The proof of these last formulas can be found in Appendix B.

It is now a straightforward consequence of (4.27) and (4.28) that the right-invariant vector fields $\overrightarrow{(T^{p}a,\mathcal{R}_{a}^{q})}$, $\overrightarrow{\mathcal{B}a_{(i)}}$ and $\overrightarrow{\mathcal{B}\alpha_{(j)}}\in\mathfrak{X}(\mathbb{G}^{p,q})$ are given by

[TABLE]

for $i=1,\dots,p$, $j=1,\dots,q$.

4.3. Proof of Theorem 3.11

Let us begin with two important lemmas. For the first one, we need to introduce some notation. Define $\pi^{(p,q)}_{(i,0)}:\mathbb{G}^{(p,q)}\rightarrow\mathbb{G}^{(p-1,q)}$, $\pi^{(p,q)}_{(0,j)}:\mathbb{G}^{(p,q)}\rightarrow\mathbb{G}^{(p,q-1)}$, as the natural projections

[TABLE]

for $1\leq i\leq p$, $1\leq j\leq q$. When there is no risk of confusion, we will omit the superscripts $(q,p)$ on the projections. Observe that $\pi_{(i,0)}$ and $\pi_{(0,j)}$ are groupoid morphisms (covering (4.11) and (4.12), respectively).

Lemma 4.6**.**

For any multiplicative tensor field $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$, there exist vector bundle maps $l:A\rightarrow\wedge^{p-1}T^{*}M\otimes\wedge^{q}A$ and $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}A$ covering the identity map such that

[TABLE]

Proof.

By formulas (3.3) and (4.30), we have that

[TABLE]

for $a\in\Gamma(A)$, where the last equality is a consequence of Prop. 4.1. Similarly, for $\alpha\in\Omega^{1}(M)$, we check that

[TABLE]

Note that, for $(\underline{X\vphantom{\xi}},\underline{\varphi})\in\mathbb{M}$,

[TABLE]

which shows that $\langle Ac_{\tau},\mathcal{B}a_{(i)}\rangle$ is a componentwise linear function of $\gamma_{(i,0)}(\underline{X\vphantom{\xi}},\underline{\varphi})$. Hence (see Lemma 2.4) there exists $l(a)\in\Gamma(\wedge^{p-1}T^{*}M\otimes\wedge^{q}A)$ such that

[TABLE]

Now, note that

[TABLE]

Formula (4.34) follows from the injectivity of the correspondence between tensors and componentwise linear functions (Lemma 2.4).

To prove that $l$ is $C^{\infty}(M)$-linear, we use Proposition 3.10 to see that

[TABLE]

for $f\in C^{\infty}(M)$, so that $C^{\infty}(M)$-linearity follows from the injectivity of ${\mathcal{T}}$.

Similarly, we can prove the existence of $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}A$ by checking that $\langle Ac_{\tau},\,\mathcal{B}\alpha_{(j)}\rangle$ is componentwise linear, so that (4.35) follows from

[TABLE]

∎

Lemma 4.7**.**

For any multiplicative tensor $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$, there exists an $\mathbb{R}$-linear map $D:\Gamma(A)\rightarrow\Gamma({\wedge^{p}\,T^{*}M\otimes\wedge^{q}A})$ satisfying the Leibniz condition (3.9) and such that

[TABLE]

Proof.

From (3.3) and (4.29), we see that

[TABLE]

where the last equality relies on Proposition 4.1. In particular, for $(\underline{X\vphantom{\xi}},\underline{\varphi})\in\mathbb{M}$,

[TABLE]

which proves that $\langle Ac_{\tau},\,(T^{p}a,\mathcal{R}_{a}^{q})\rangle\in C^{\infty}(\mathbb{M})$ is a componentwise linear function. By Lemma 2.4, there exists $D(a)\in\Gamma({\wedge^{p}\,T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G}})$ such that

[TABLE]

Now (4.38) follows from Proposition 3.10 (and the injectivity part of Lemma 2.4):

[TABLE]

To prove the Leibniz condition (3.9), we use Proposition 4.3 to see that

[TABLE]

The conclusion follows from the injectivity of ${\mathcal{T}}$. ∎

We are now in position to present the proof of Theorem 3.11.

Proof of Theorem 3.11.

If $\tau$ is multiplicative, then the existence of the triple $(D,l,r)$ is guaranteed by Lemmas 4.6 and 4.7; condition (3.8) follows from the fact that, since $c_{\tau}$ is a multiplicative function, it must vanish along groupoid units (see Prop. 3.5).

Conversely, assume the existence of $(D,l,r)$. We claim that there is a unique $\mu\in\Gamma(A^{*}\mathbb{G})$ defined by the following conditions:

[TABLE]

for $a\in\Gamma(A)$ and $\alpha\in\Omega^{1}(M)$. Uniqueness follows from Prop. 4.4, so it remains to verify that $\mu$ is indeed well-defined. For local frames $(a^{k})_{k=1,\ldots,\mathrm{rank}(A)}$ of $A$ and $(\alpha^{s})_{s=1,\ldots,\dim(M)}$ of $T^{*}M$, we observe that the collection of local sections $(T^{p}a^{k},\mathcal{R}^{q}_{a^{k}})$, $\mathcal{B}a^{k}_{(i)}$, and $\mathcal{B}\alpha^{s}_{(j)}$ form a local frame for $\mathbb{A}$. We first define $\mu$ on this local frame using the formulas above (and extend it by linearity). In the following, we use Einstein notation. To show that $\mu$ is globally well defined, it suffices to verify that (4.40), (4.41), and (4.42) hold for $a=f_{k}a^{k}$ and $\alpha=g_{s}\alpha^{s}$, where $f_{k}$, $g_{s}\in C^{\infty}(M)$. One can check that

[TABLE]

where ${\rm pr}:\mathbb{M}\to M$ is the bundle projection. Using (4.22) and (4.23), one also verifies that (see Remark 4.5 for notation)

[TABLE]

where $\beta^{k}=df_{k}$, and ${\rm pr}_{TM}^{i}:\mathbb{M}\to TM$, ${\rm pr}_{A^{*}}^{j}:\mathbb{M}\to A^{*}$ are given by ${\rm pr}_{TM}^{i}(\underline{X\vphantom{\xi}},\underline{\varphi})=X_{i}$, ${\rm pr}_{A^{*}}^{j}(\underline{X\vphantom{\xi}},\underline{\varphi})=\varphi_{j}$. Using Lemma 2.4, we see that

[TABLE]

which is (4.40). A similar argument verifies (4.41). For (4.42), note that

[TABLE]

By (2.8), it follows that the right-hand side above agrees with

[TABLE]

where the Leibniz condition for $D$ is used in the last equality. Hence (4.42) holds and $\mu$ is well defined.

Now consider the componentwise linear function $c_{\tau}\in C^{\infty}(\mathbb{G})$. It follows from the third equality in (3.10), Lemma 3.10 and Theorem 4.1 that

[TABLE]

Similarly, we see that

[TABLE]

By linearity, one has that $\mathcal{L}_{\overrightarrow{\chi}}c_{\tau}=\mathbbm{t}^{*}\langle\mu,\chi\rangle$, for every $\chi\in\Gamma(A\mathbb{G})$. As $\mathbb{G}$ is source connected (because $\mathcal{G}$ is, see Remark 2.1), the result follows from Proposition 3.5. ∎

4.4. Linear tensor fields

For the proof of Theorem 3.19, we will need to specialize our study of multiplicative tensor fields to Lie groupoids given by vector bundles $\pi_{E}:E\rightarrow M$. As we already saw, in this case the groupoid $\mathbb{E}=(\oplus^{p}TE)\oplus(\oplus^{q}T^{*}E)$ in (2.7) is a vector bundle over $\mathbb{M}=(\oplus^{p}TM)\oplus(\oplus^{q}E^{*})$, with Lie algebroid $A\mathbb{E}$ given by $\mathbb{E}\rightarrow\mathbb{M}$ itself with zero anchor and zero bracket. A tensor field $\tau\in\Gamma(\wedge^{p}T^{*}E\otimes\wedge^{q}TE)$ is multiplicative if and only if the associated function $c_{\tau}:\mathbb{E}\to\mathbb{R}$ is fiberwise linear on $\mathbb{E}\rightarrow\mathbb{M}$. For this reason, we refer to multiplicative tensor fields on $E\to M$ also as linear. Our goal here is to show how one can reconstruct linear tensor fields on $E\to M$ explicitly from their infinitesimal components.

For a linear tensor $\tau\in\Gamma(\wedge^{p}T^{*}E\otimes\wedge^{q}TE)$,

[TABLE]

noticing that since $c_{\tau}$ is fiberwise linear on $\mathbb{E}\to\mathbb{M}$, it can be seen as a section of $\mathbb{E}^{*}\to\mathbb{M}$, cf. Example 3.3. In particular, from (4.40), (4.41) and (4.42), we see that the infinitesimal components $(D,l,r)$ of a linear tensor $\tau$ satisfy

[TABLE]

[TABLE]

for $u\in\Gamma(E)$, $\alpha\in\Omega^{1}(M)$, and where $\gamma_{(i,0)}$, $\gamma_{(o,j)}$ are the projections in (4.11) and (4.12).

Before presenting the main result of this subsection, we need two lemmas. We begin with a useful property of multiplicative tensors on Lie groupoids with $\mathsf{s}=\mathsf{t}$, so in particular linear ones.

Lemma 4.8**.**

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid such that $\mathsf{s}=\mathsf{t}$. If $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$ is a multiplicative $(q,p)$-tensor field, then

[TABLE]

for $a,b\in\Gamma(A)$ and $\alpha,\beta\in\Gamma(T^{*}M)$.

Proof.

As $\mathsf{s}=\mathsf{t}$, it follows from (3.6) that $i_{\overrightarrow{a}}{\mathcal{T}}(\Phi)=0$ and $i_{\mathsf{t}^{*}\alpha}{\mathcal{T}}(\Phi)=0$, for any $\Phi\in\Gamma(\wedge^{\bullet}T^{*}M\otimes\wedge^{\bullet}A)$. The result now follows from Theorem 3.11. ∎

The following lemma shows how the infinitesimal components of linear tensors can be obtained by means of pointwise evaluation of the tensor on special vectors and covectors .

Lemma 4.9**.**

Let $\tau\in\Gamma(\wedge^{p}T^{*}E\otimes\wedge^{q}TE)$ be a linear $(q,p)$ tensor field. For $(\underline{X\vphantom{\xi}},\underline{\varphi})\in\mathbb{M}_{x}$, $u\in\Gamma(E)$ and $\alpha\in\Omega^{1}(M)$, define

[TABLE]

The infinitesimal components $l:E\rightarrow\wedge^{p-1}T^{*}M\otimes\wedge^{q}E$, $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}E$ and $D:\Gamma(E)\rightarrow\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}E)$ satisfy

• (a)

$D(u)(\underline{X\vphantom{\xi}},\underline{\varphi})=\tau((Tu^{p},\mathcal{R}^{q})(\underline{X\vphantom{\xi}},\underline{\varphi}))$

• (b)

$l(u)(\gamma_{(i,0)}(\underline{X\vphantom{\xi}},\underline{\varphi}))=(-1)^{i-1}\tau(\underline{U}_{(i)},\widetilde{0}_{\varphi_{1}},\dots,\widetilde{0}_{\varphi_{q}})$**

• (c)

$r(\alpha)(\gamma_{(0,j)}(\underline{X\vphantom{\xi}},\underline{\varphi}))=(-1)^{j-1}\tau(T0(X_{1}),\dots,T0(X_{p}),\underline{\xi}_{(j)}).$**

Proof.

From (4.44), it is clear that

[TABLE]

which proves (a). Similarly, from the first equation in (4.43),

[TABLE]

But since $\mathcal{B}u(X_{i})=T0(X_{i})+\overline{u(m)}$, we have that this last term equals

[TABLE]

To conclude that (b) holds, note that $\tau(T0(X_{1}),\dots,T0(X_{p}),\widetilde{0}_{\varphi_{1}},\dots,\widetilde{0}_{\varphi_{q}})=0$, as $\tau$ is linear on $\mathbb{E}\rightarrow\mathbb{M}$. The verification of (c) is similar. ∎

We can now present the main result regarding linear tensor fields.

Proposition 4.10**.**

A tensor $\tau\in\Gamma(\wedge^{p}T^{*}E\otimes\wedge^{q}TE)$ is linear if and only if there exist vector-bundle maps $l:E\rightarrow\wedge^{p-1}T^{*}M\otimes\wedge^{q}E$ and $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}E$ covering the identity, and $D:\Gamma(E)\rightarrow\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}E)$ satisfying the Leibniz condition (3.9), such that, for $U_{1},\dots,U_{p}\in T_{e}E$, $\xi_{1},\dots,\xi_{q}\in T_{e}^{*}E$, $e\in E_{x}$,

[TABLE]

where $u\in\Gamma(E)$ is any section such that $u(x)=e$, $X_{i}=T\pi_{E}(U_{i})$, $\varphi_{j}=\widetilde{\pi_{E}}(\xi_{j})$, and $e_{i}\in E_{x}$, $\beta_{j}\in T_{x}^{*}M$ are defined by

[TABLE]

for $\psi\in E^{*}_{x}$, $Y\in T_{x}M$, and $i=1,\dots,p$, $j=1,\dots,q$. In this case, $(D,l,r)$ are the infinitesimal components of $\tau$.

Proof.

First note that the right-hand side of formula (4.48) is well defined, in the sense that it does not depend on the extension $u$. Indeed, let $(u^{k})_{k=1,\dots,\mathrm{rank}(E)}$, $(\alpha^{s})_{s=1,\dots,\dim(M)}$ be local frames of $E$ and $T^{*}M$ respectively. One can write (see notation in Remark 4.5)

[TABLE]

for $t_{k},h_{ik},g_{js}\in\mathbb{R}$, where $e=t_{k}u^{k}(x)$. For any section $u=f_{k}u^{k}$, $f^{k}\in C^{\infty}(M)$, $u(x)=e$ if and only if $f_{k}(x)=t_{k}$. Also,

[TABLE]

So, by using the Leibniz condition (3.9), one can rewrite (4.48) as

[TABLE]

Let us assume that $\tau\in\Gamma(\wedge^{p}T^{*}E\otimes\wedge^{q}TE)$ is a linear tensor, and let $(D,l,r)$ be its infinitesimal components. One may directly check that

[TABLE]

Since $\tau$ is linear, we have

[TABLE]

Now, using the multilinearity of the tensor $\tau$, one can expand $\tau(\underline{V\vphantom{\zeta}},\underline{\zeta})$ as a sum in which every term is $\tau$ evaluated on a string involving $T0(X_{i}),\overline{e_{i}}$ separatedly on the $TE$ part and $\widetilde{0}_{\varphi_{j}},\overline{\beta_{j}}$ separatedly on the $T^{*}E$ part.

Claim:

The only non-zero terms on the expansion of $\tau(\underline{V\vphantom{\zeta}},\underline{\zeta})$ as a sum are the ones in which the $\overline{(\cdot)}$ terms appear exactly once (counting both the $TE$ and $T^{*}E$ parts). Indeed, if they do not appear at all, one has $\tau(T0(X_{1}),\dots,T0(X_{p}),\widetilde{0}_{\varphi_{1}},\dots,\widetilde{0}_{\varphi_{q}})=0,$ because $\tau$ is linear on the fibers of $\mathbb{E}\rightarrow\mathbb{M}$. If they appear twice or more, note that $\overline{e_{i}}=u_{i}^{{\mathrm{v}}}(0_{m})$ and $\overline{\beta_{j}}=(\pi_{E}^{*}\alpha_{j})(0_{m})$, where $u_{i}\in\Gamma(E)$ and $\alpha_{j}\in\Gamma(T^{*}M)$ satisfy $u_{i}(m)=e_{i}$ and $\alpha_{j}(m)=\beta_{j}$. So, the claim follows from Lemma 4.8. Therefore,

[TABLE]

Formula (4.48) now follows from Lemma 4.9.

Conversely, let us assume $\tau\in\Gamma(\wedge^{p}T^{*}E\otimes\wedge^{q}TE)$ is a $(q,p)$-tensor field for which (4.48) holds. It is straightforward to check that (4.48) is linear on the fibers of $\mathbb{E}\rightarrow\mathbb{M}$, so $\tau$ is linear. To prove that $(D,l,r)$ are exactly the infinitesimal components of $\tau$, one proceeds as follows: first substitute $(\underline{U\vphantom{\xi}},\underline{\xi})$ with $(T^{p}u(\underline{X\vphantom{\xi}}),\mathcal{R}_{u}(\underline{\varphi}))$. In this case, $e_{i}=0$, $\beta_{j}=0$ and formula (4.48) becomes $\tau(T^{p}u(\underline{X\vphantom{\xi}}),\mathcal{R}_{u}(\underline{\varphi}))=D(u)(\underline{X\vphantom{\xi}},\underline{\varphi}).$ By substituting $(\underline{U\vphantom{\xi}},\underline{\xi})$ with $(\overline{e_{1}},T0(X_{2}),\dots,T0(X_{p}),\widetilde{0}_{\varphi_{1}},\dots,\widetilde{0}_{\varphi_{q}})$, one has that $X_{1}=0$, $e_{2}=\dots=e_{p}=0$, $\beta_{j}=0$ and, therefore, formula (4.48) becomes

[TABLE]

Finally, by substituting $(\underline{U\vphantom{\xi}},\underline{\xi})$ with $(T0(X_{1}),\dots,T0(X_{p}),\overline{\beta_{1}},\widetilde{0}_{\varphi_{2}},\dots,\widetilde{0}_{\varphi_{q}})$, formula (4.48) becomes

[TABLE]

The result now follows from Lemma 4.9. ∎

As an immediate consequence, we have

Corollary 4.11**.**

There is a one-to-one correspondence defined by (4.48) between linear tensors $\tau\in\Gamma(\wedge^{p}T^{*}E\otimes\wedge^{q}TE)$ and triples $(D,l,r)$, where $l:E\rightarrow\wedge^{p-1}T^{*}M\otimes\wedge^{q}E$ and $r:T^{*}M\rightarrow\wedge^{p}T^{*}M\otimes\wedge^{q-1}E$ are vector bundle maps covering the identity and $D:\Gamma(E)\rightarrow\Gamma(\wedge^{p}T^{*}M\otimes\wedge^{q}E)$ satisfies the Leibniz condition (3.9).

4.5. Proof of Theorem 3.19

We just saw in Cor. 4.11 how linear tensors $\tau$ on a vector bundle are described in terms of triples $(D,l,r)$. Let $(A,[\cdot,\cdot],\rho)$ be a Lie algebroid, and consider linear tensors $\tau$ on $A$ for which the corresponding fiberwise linear functions $c_{\tau}:\mathbb{A}\rightarrow\mathbb{R}$ are Lie-algebroid cocycles. We now see how to express this additional cocycle property in terms of $(D,l,r)$.

Proposition 4.12**.**

There is a one-to-one correspondence defined by (4.48) between linear $(q,p)$-tensors $\tau\in\Gamma(\wedge^{p}T^{*}A\otimes\wedge^{q}TA)$ for which $c_{\tau}:\mathbb{A}\rightarrow\mathbb{R}$ is a Lie-algebroid cocycle and IM $(q,p)$-tensors $(D,l,r)$ on $A$, as in Definition 3.17.

Proof.

By definition, the cocycle condition $d_{\mathbb{A}}\,c_{\tau}=0$ is equivalent to the equation

[TABLE]

all $U,V\in\Gamma(\mathbb{A})$. In order to prove that this equality holds, it suffices to consider $U,V$ varying on the set of generators given by Proposition 4.4, and this will be shown to be equivalent to the set of IM-equations (IM1)–(IM6) in Definition 3.17. Let us first fix $U=(T^{p}a,\mathcal{R}_{a}^{\,q})$, for $a\in\Gamma(A)$. In the following, we shall use repeatedly the anchor and Lie bracket equations (4.24), (4.25) for the Lie algebroid $\mathbb{A}\to\mathbb{M}$.

Equation (IM1):

Take $V=(T^{p}b,\mathcal{R}_{b}^{\,q})$, for $b\in\Gamma(A)$. It follows from (4.44) that the cocycle equation (4.49) is equivalent to

[TABLE]

where the last equality follows from Proposition 4.1. So, in this case (4.49) is equivalent to (IM1).

Equations (IM2) and (IM3):

Take $V=\mathcal{B}b_{(i)}$, $1\leq i\leq p$. From (4.43), it follows that the cocycle equation (4.49) for this pair $U,V$ can be rewritten as

[TABLE]

where $\gamma_{(i,0)}$ is the projection (4.11) and the last equality follows from Proposition 4.1. So, for the given choices of $U$ and $V$, (4.49) is equivalent to (IM2).

When $V=\mathcal{B}\alpha_{(j)}$, $1\leq j\leq q$, for $\alpha\in\Omega^{1}(M)$, one can prove analogously that (4.49) and (IM3) are equivalent.

Equations (IM4), (IM5) and (IM6):

Let $U=\mathcal{B}a_{(i)}$ and $V=\mathcal{B}b_{(k)}$, for $1\leq i<k\leq p$. As $[\mathcal{B}a_{(i)},\mathcal{B}b_{(k)}]=0$, it follows from (4.43) that the cocycle equation (4.49) can be rewritten as

[TABLE]

where in the second equality we have used Proposition 4.1. One can now directly check that $\gamma^{(p-1,q)}_{(i,0)}\circ\gamma_{(k,0)}$ and $\gamma^{(p-1,q)}_{(k-1,0)}\circ\gamma_{(i,0)}$ are the same projection from $\mathbb{M}^{(p,q)}$ to $\mathbb{M}^{(p-2,q)}$, which forgets the $i$-th and the $k$-th components on $TM$. Hence, for these choices of $U$ and $V$, (4.49) is equivalent to (IM4).

In a similar way, one checks that, for $U=\mathcal{B}\alpha_{(j)}$, $V=\mathcal{B}\beta_{(k)}$, $1\leq j<k\leq q$, one obtains the equivalence of (4.49) with (IM5), and for $U=\mathcal{B}a_{(i)}$, $V=\mathcal{B}\alpha_{(j)}$, $1\leq i\leq p$, $1\leq j\leq q$, one has the equivalence of (4.49) with (IM6).

These 6 cases cover all possibilities of $U,V$ varying in the set of generators, so the result follows. ∎

Let now $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with Lie algebroid $A\to M$. For a multiplicative $(q,p)$-tensor field $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$, consider the corresponding multiplicative function $c_{\tau}\in C^{\infty}(\mathbb{G})$ on the groupoid (2.7). Let $Ac_{\tau}\in\Gamma(\mathbb{A}^{*})\subseteq C^{\infty}(\mathbb{A})$ be its associated infinitesimal cocycle.

Lemma 4.13**.**

The following holds:

• (a)

There is a linear tensor field $\tau_{A}\in\Gamma(\wedge^{p}T^{*}A\otimes\wedge^{q}TA)$ such $Ac_{\tau}=c_{\tau_{A}}$.

• (b)

The infinitesimal components $(D_{A},l_{A},r_{A})$ of $\tau_{A}$ satisfy

[TABLE]

where $(\underline{X\vphantom{\xi}},\underline{\varphi})\in\mathbb{M}$, $a\in\Gamma(A)$, $\alpha\in\Omega^{1}(M)$ and $\gamma_{(i,0)}$, $\gamma_{(0,j)}$, $\pi_{(i,0)}$ and $\pi_{(0,j)}$ are the forgetful projections (4.11), (4.12), (4.32) and (4.33), respectively.

• (c)

The infinitesimal components $(D,l,r)$ of $\tau$ coincide with those of $\tau_{A}$.

Proof.

It follows from Proposition A.3 that $Ac_{\tau}:\mathbb{A}\rightarrow\mathbb{R}$ is a componentwise linear function which is antisymmetric on the $TA$ components as well as on the $T^{*}A$ components. Hence, there exists $\tau_{A}\in\Gamma(\wedge^{p}T^{*}A\otimes\wedge^{q}TA)$ such that $Ac_{\tau}=c_{\tau_{A}}$. This proves (a).

By (3.2), (4.44) and Proposition 4.1,

[TABLE]

Similarly,

[TABLE]

The equation involving $r_{A}$ follows similarly, and we conclude that (b) holds.

If we now let $(D,l,r)$ be the infinitesimal components of $\tau$, the equalities $D=D_{A}$, $l=l_{A}$ and $r=r_{A}$ follow from Theorem 3.11. ∎

We can now finally proceed to the proof of our main result.

Proof of Theorem 3.19.

By Proposition 4.12, there exists a Lie algebroid $(q,p)$ tensor field $\tau_{A}$ having $(D,l,r)$ as its infinitesimal components. The fact that $c_{\tau_{A}}:A\mathbb{G}\rightarrow\mathbb{R}$ is a Lie algebroid cocycle and $\mathbb{G}$ is source 1-connected implies that there exists a unique multiplicative function $F:\mathbb{G}\rightarrow\mathbb{R}$ satisfying $AF=c_{\tau_{A}}$. By Proposition A.3, $F$ is componentwise linear and anti-symmetric on both the $T\mathcal{G}$ and $T^{*}\mathcal{G}$ components. Therefore, $F=c_{\tau}$ for a (unique) multiplicative tensor field $\tau\in\Gamma(\wedge^{p}T^{*}\mathcal{G}\otimes\wedge^{q}T\mathcal{G})$. The fact that the infinitesimal components of $\tau$ are $(D,l,r)$ follows from Lemma 4.13. ∎

5. Multiplicative vector-valued forms

Given a manifold $N$, by a vector-valued form on $N$ we mean an element of $\Omega^{\bullet}(N,TN)=\Gamma(\wedge^{\bullet}T^{*}N\otimes TN)$. The space of vector-valued forms is a graded Lie algebra with respect to the Frölicher-Nijenhuis bracket [24]. On a Lie groupoid, the space of multiplicative vector-valued forms is closed under the Frölicher-Nijenhuis bracket [8], so it is also a graded Lie algebra. We now identify its infinitesimal counterpart, in the spirit of Remarks 3.23 and 3.26. Before discussing Lie groupoids, we briefly recall vector-valued forms on manifolds.

5.1. The graded Lie algebra of vector-valued forms

Let $\Omega^{\bullet}(N)$ be the graded algebra of differential forms on $N$. A degree $k$ derivation of $\Omega^{\bullet}(N)$ is a linear map $\Delta:\Omega^{\bullet}(N)\to\Omega^{\bullet+k}(N)$ such that $\Delta(\alpha\wedge\beta)=\Delta(\alpha)\wedge\beta+(-1)^{kj}\alpha\wedge\Delta(\beta)$, for $\alpha\in\Omega^{j}(N)$. Any vector-valued form $K\in\Gamma(\wedge^{p}T^{*}N\otimes TN)$ gives rise to a degree $(p-1)$ derivation of $\Omega^{\bullet}(N)$ by

[TABLE]

for $\omega\in\Omega^{j}(N)$, $X_{1},\ldots,X_{p+j-1}\in TN$. It also gives rise to a degree $p$ derivation of $\Omega^{\bullet}(N)$ via

[TABLE]

where $d$ is the exterior differential on $N$.

We extend $i_{K}$ to a contraction operation $i_{K}:\Omega^{\bullet}(N,TN)\rightarrow\Omega^{\bullet+p-1}(N,TN)$ by

[TABLE]

Given $K\in\Omega^{p}(N,TN)$ and $L\in\Omega^{p^{\prime}}(N,TN)$, their Frölicher-Nijenhuis bracket [24] (see also [34, Ch. 2]) is the vector-valued form $[K,L]\in\Omega^{p+p^{\prime}}(N,TN)$ uniquely defined by the condition

[TABLE]

When $K$ and $L$ have degree zero (i.e., they are vector fields on $N$), (5.4) agrees with the usual Lie bracket of vector fields. More generally, for $X\in\mathfrak{X}(N)$,

[TABLE]

The Frölicher-Nijenhuis bracket makes $\Omega^{\bullet}(N,TN)$ into a graded Lie algebra. It is a natural bracket in the following sense: for a smooth map $F:N_{1}\to N_{2}$ , let $K_{i}\in\Omega^{p}(N_{i},TN_{i})$, $L_{i}\in\Omega^{p^{\prime}}(N_{i},TN_{i})$, $i=1,2$, be such that $K_{1}$ is $F$-related to $K_{2}$ and $L_{1}$ is $F$-related to $L_{2}$ 444$K_{1}\in\Omega^{p}(N_{1},TN_{1})$ is $F$-related to $K_{2}\in\Omega^{p}(N_{2},TN_{2})$ if

$K_{2}(TF(X_{1}),\ldots,TF(X_{k}))=TF(K_{1}(X_{1},\ldots,X_{k})),$

for all $X_{1},\ldots,X_{k}\in T_{x}N$, and $x\in N$.. Then $[K_{1},L_{1}]$ is $F$-related to $[K_{2},L_{2}]$. There are other important properties of the Frölicher-Nijenhuis bracket which will be recalled in subsequent sections.

5.2. Infinitesimal description

Let $\mathcal{G}$ be a Lie groupoid. As seen in Proposition 3.9, a multiplicative vector-valued form $K\in\Omega^{p}(\mathcal{G},T\mathcal{G})$ as defined by [43] is exactly a multiplicative $(1,p)$-tensor field. From Theorem 3.19, one obtains a bijective correspondence between multiplicative vector-valued $p$-forms on a Lie groupoid $\mathcal{G}$ and IM $(p,1)$-tensors on its Lie algebroid $A$. For this reason, we will refer to IM $(p,1)$-tensors also as IM vector-valued $p$-forms.

We denote by $\Omega^{\bullet}_{\scriptscriptstyle{mult}}(\mathcal{G},T\mathcal{G})$ the space of multiplicative vector-valued forms on $\mathcal{G}$. In the following, we will also need the following result proven in [8, Thm. 4.3]:

Proposition 5.1**.**

On a Lie groupoid $\mathcal{G}$, $\Omega^{\bullet}_{\scriptscriptstyle{mult}}(\mathcal{G},T\mathcal{G})$ is closed under the Frölicher-Nijenhuis bracket.

Hence $\Omega^{\bullet}_{\scriptscriptstyle{mult}}(\mathcal{G},T\mathcal{G})\subseteq\Omega^{\bullet}(\mathcal{G},T\mathcal{G})$ is a graded Lie subalgebra. We now describe the graded Lie bracket on IM vector-valued forms corresponding to the Frölicher-Nijenhuis bracket on multiplicative vector-valued forms.

For a multiplicative vector-valued form $K\in\Omega^{p}(\mathcal{G},T\mathcal{G})$, consider its infinitesimal components $D:\Gamma(A)\rightarrow\Gamma(\wedge^{p}T^{*}M\otimes A)$, $l:A\rightarrow\wedge^{p-1}T^{*}M\otimes A$ and $r:T^{*}M\to\wedge^{p}T^{*}M$. Note that $r$ can be seen alternatively as an element $r\in\Omega^{p}(M,TM)$. As such, Proposition 3.9 shows that $K$ is $\mathsf{s},\mathsf{t}$-related to $r$.

Using the $\Omega^{\bullet}(M)$-module structure of $\Omega^{\bullet}(M,A)=\Gamma(\wedge^{\bullet}T^{*}M\otimes A)$, we extend $l$ to an operator $l:\wedge^{\bullet}T^{*}M\otimes A\rightarrow\wedge^{\bullet+p-1}T^{*}M\otimes A$ by

[TABLE]

and $D$ to an operator $D:\Omega^{j}(M,A)\rightarrow\Omega^{p+j}(M,A)$ by

[TABLE]

with $\mathcal{L}_{r}$ as in (5.2). This extension for $D$ is well-defined as a consequence of the Leibniz rule (3.9). Moreover,

[TABLE]

for $\alpha\in\Omega^{i}(M)$, and $\eta\in\Omega^{j}(M,A)$.

Lemma 5.2**.**

For $\eta\in\Omega^{\bullet}(M,A)$, we have $i_{{\mathcal{T}}(\eta)}K={\mathcal{T}}(l(\eta))$.

Proof.

For homogeneous $\eta=\alpha\otimes a$, one has that ${\mathcal{T}}(\eta)=\mathsf{t}^{*}\alpha\otimes\overrightarrow{a}$. So, by definition of the contraction (5.3),

[TABLE]

∎

Let us consider the operation $[\cdot,K]:\Omega^{\bullet}(\mathcal{G},T\mathcal{G})\rightarrow\Omega^{\bullet+k}(\mathcal{G},T\mathcal{G})$, where $[\cdot,\cdot]$ is the Frölicher-Nijenhuis bracket. The following result shows that $[\cdot,K]$ preserves the image of ${\mathcal{T}}$ inside $\Omega^{\bullet}(\mathcal{G},T\mathcal{G})$.

Lemma 5.3**.**

For $\eta\in\Omega^{\bullet}(M,A)$, we have $[{\mathcal{T}}(\eta),K]={\mathcal{T}}(D(\eta))$.

Proof.

If $\eta=\alpha\otimes a$, for $\alpha\in\Omega^{j}(M)$, then one has (see [34, Sec. 8.7]):

[TABLE]

In the second equality, we have used the fact that $K$ and $r$ are $\mathsf{t}$-related. ∎

Proposition 5.4**.**

Let $K_{1}\in\Omega^{p_{1}}(\mathcal{G},T\mathcal{G})$, $K_{2}\in\Omega^{p_{2}}(\mathcal{G},T\mathcal{G})$ be multiplicative vector-valued forms, with infinitesimal components $(D_{1},l_{1},r_{1})$ and $(D_{2},l_{2},r_{2})$, respectively. The infinitesimal components $(D,l,r)$ of their Frölicher-Nijenhuis bracket $[K_{1},K_{2}]$ are

[TABLE]

where the brackets on the right-hand side are the (graded) commutators of endomorphisms of $\Omega^{\bullet}(M,A)$, and

[TABLE]

where the last bracket is the Frölicher-Nijenhuis bracket of $r_{i}\in\Omega^{p_{i}}(M,TM)$, $i=1,2$.

Proof.

The equation (5.8) follows from the naturality of the Frölicher-Nijenhuis bracket. As $K_{1}$ is $\mathsf{t}$-related to $r_{1}$ and $K_{2}$ is $\mathsf{t}$-related to $r_{2}$, it follows that $[K_{1},K_{2}]$ must be $\mathsf{t}$-related to $[r_{1},r_{2}]$. The identity (5.6) for $D$ follows from the Jacobi equation for the Frölicher-Nijenhuis bracket. Indeed, using Lemma 5.3:

[TABLE]

Now, recall the following general property of the Frölicher-Nijenhuis bracket (see Theorem 8.11 in [34]): for $K_{i}\in\Omega^{p_{i}}(N,TN)$ and $L\in\Omega^{p^{\prime}+1}(N,TN)$, $i=1,2$, we have

[TABLE]

Using this identity and the previous lemmas, we obtain

[TABLE]

as we wanted to prove. ∎

Corollary 5.5**.**

The space of IM vector-valued forms is a graded Lie algebra with the bracket defined by (5.6),(5.7) and (5.8):

[TABLE]

The correspondence established by Theorem 3.19 between multiplicative vector-valued forms and IM vector-valued forms is a graded Lie algebra isomorphism.

This last result should be regarded as parallel to Remarks 3.23 and 3.26.

6. Multiplicative $(1,1)$-tensor fields

We will now focus on multiplicative vector-valued 1-forms, or $(1,1)$-tensor fields.

6.1. Infinitesimal components

Let $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ be a multiplicative $(1,1)$-tensor field, with infinitesimal components

[TABLE]

Note that we have dualized the $r$ component. For $X\in TM$, we will use the notation

[TABLE]

The $IM$-equations satisfied by the triple $(D,l,r)$ take the form

[TABLE]

Proposition 6.1**.**

Let $K:T\mathcal{G}\rightarrow T\mathcal{G}$ be a multiplicative $(1,1)$-tensor field on the Lie groupoid $\mathcal{G}$, with infinitesimal componentes $(D,l,r)$. Then $K^{n}$ is also multiplicative, and its infinitesimal components $(D^{\prime},l^{\prime},r^{\prime})$ satisfy

[TABLE]

Proof.

The equations for $l^{\prime}$ and $r^{\prime}$ are straightforward to check. As for the equation for $D^{\prime}$, the proof follows from an induction on $n$ using the recursion formula

[TABLE]

∎

As a result, we obtain infinitesimal descriptions of multiplicative projections and almost complex/product structures.

Corollary 6.2**.**

Let $K$ be a multiplicative $(1,1)$-tensor field on a source-connected Lie groupoid $\mathcal{G}\rightrightarrows M$ with infinitesimal componentes $(D,l,r)$. Then

• (a)

$K$* satisfies $K^{2}=K$ if and only if*

[TABLE]

• (b)

$K$* satisfies $K^{2}=\pm\,{\rm id}_{T\mathcal{G}}$ if and only if*

[TABLE]

Proof.

In (a), the equations for $(D,l,r)$ guarantee that the multiplicative $(1,1)$-tensors $K^{2}$ and $K$ have the same infinitesimal components. As $\mathcal{G}$ has connected $\mathsf{s}$-fibers, this implies that $K^{2}=K$. The same argument holds for almost product and almost complex structures. ∎

We will now consider an additional integrability condition in terms of the Nijenhuis torsion.

6.2. Nijenhuis torsion

Given a (1,1)-tensor field $K$ on a manifold $N$, its Nijenhuis torsion is the vector-valued 2-form $\mathcal{N}_{K}\in\Omega^{2}(N,TN)$ given by

[TABLE]

for $X,Y\in TN$. The Nijenhuis torsion has a well-known relation with the Frölicher-Nijenhuis bracket via

[TABLE]

For a multiplicative (1,1)-tensor field on a Lie groupoid $\mathcal{G}\rightrightarrows M$, the following description of the infinitesimal components of its Nijenhuis torsion is an immediate consequence of this last formula and Propositions  5.1 and  5.4:

Corollary 6.3**.**

Let $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ be multiplicative with infinitesimal components $(D,l,r)$. Then $\mathcal{N}_{K}\in\Omega^{2}(\mathcal{G},T\mathcal{G})$ is multiplicative and its infinitesimal components $(D^{\prime},l^{\prime},r^{\prime})$ are

[TABLE]

It will be useful to have a more concrete expression for

[TABLE]

Recall [34, Cor. 8.12] the following expression for the Frölicher-Nijenhuis bracket of $K_{1},K_{2}\in\Omega^{1}(\mathcal{G},T\mathcal{G})$:

[TABLE]

Since $[\overrightarrow{a},\frac{1}{2}[K,K]]=[[\overrightarrow{a},K],K],$ by taking $K_{1}=[\overrightarrow{a},K]$ and $K_{2}=K$, and letting $U=X$, $V=Y$ be in $TM$, one readily obtains that

[TABLE]

where both sides are seen as maps $\Gamma(A)\to\Gamma(A)$.

Corollary 6.3 gives a complete infinitesimal description of general multiplicative Nijenhuis operators on Lie groupoids. In the next subsections, we will illustrate how this general result can be specialized to various cases of interest.

6.3. Poisson quasi-Nijenhuis structures

Poisson-Nijenhuis structures [40, 52] play a central role in the theory of integrable systems. Their recent connections with Lie groupoids arose in quantization schemes for Poisson manifolds, see e.g. [2]. In this section we revisit the more general Poisson quasi-Nijenhuis structures [57]. We establish a link with the theory of IM $(1,1)$-tensors, which leads to an extension of the integration of Poisson quasi-Nijenhuis structures in [57, Thm. 6.2] (originally based on the theory of Lie bialgebroids [37]) as a consequence of Theorem 3.19.

Given a Poisson manifold $(M,\Pi)$, consider its cotangent bundle $T^{*}M$ with the Lie bracket $[\cdot,\cdot]_{\Pi}$ as in (3.13). We say that a (1,1) tensor $r:TM\to TM$ is compatible with $\Pi$ if

[TABLE]

(equivalently, $r\circ\Pi^{\sharp}:T^{*}M\to TM$ is skew-symmetric) and the following equation holds: for all $\alpha,\beta\in\Omega^{1}(M)$,

[TABLE]

where $[\cdot,\cdot]_{\Pi_{r}}$ is the bracket (3.13) for the bivector field $\Pi_{r}$ defined by $r\circ\Pi^{\sharp}$.

Remark 6.4**.**

*The condition $C^{r}_{\Pi}=0$ implies that $[\Pi,\Pi_{r}]=0$ (here $[\cdot,\cdot]$ is the Schouten bracket), but the converse does not hold in general; it does if $\Pi$ is symplectic, see e.g. [58]. *

The following definition extends [57, Def. 3.3]:

Definition 6.5**.**

A Poisson quasi-Nijenhuis structure is a pair $(\Pi,r)$, where $\Pi$ is a Poisson bivector field, $r:TM\to TM$ is a $(1,1)$ tensor, such that $\Pi$ and $r$ are compatible and the following condition holds:

[TABLE]

where $\mathcal{N}_{r}^{*}:T^{*}M\to\wedge^{2}T^{*}M$ is the adjoint of the Nijenhuis torsion of $r$, given by $\mathcal{N}^{*}_{r}(\alpha)(X,Y)=\langle\alpha,\mathcal{N}_{r}(X,Y)\rangle$. We refer to a symplectic quasi-Nijenhuis structure when $\Pi$ is nondegenerate, i.e., a symplectic structure.

Note that (6.5) and (6.6) say that $(\mathcal{N}_{r}^{*},0)$ defines an IM $3$-form on $T^{*}M$.

Remark 6.6**.**

The Poisson quasi-Nijenhuis structures considered in [57, Def. 3.3] are slightly more restricted than in our definition: they are required to satisfy the condition $\mathcal{N}_{r}^{*}(\alpha)=-i_{\Pi^{\sharp}(\alpha)}\phi,$ or, equivalently,

[TABLE]

*for a given closed 3-form $\phi\in\Omega^{3}(M)$. One may verify that this implies that $\mathcal{N}_{r}^{*}$ automatically satisfies (6.5) and (6.6). This difference in the definitions will become more transparent when we talk about integration, see Thm. 6.11 and Cor. 6.12 below. *

Following the previous remark, we shall refer to the structures satisfying (6.7) as Poisson quasi-Nijenhuis structures relative to $\phi$; we will specify them by triples $(\Pi,r,\phi)$ to make the dependence on the closed 3-form $\phi$ explicit.

The following proposition gives an alternative way to express the compatibility between a (1,1)-tensor $r$ and a Poisson structure $\Pi$ in terms of IM tensors. Let $D^{r}:\Gamma(T^{*}M)\to\Gamma(T^{*}M\otimes T^{*}M)$ be defined by

[TABLE]

where $\mathcal{L}_{r}$ is the operator (5.2).

Proposition 6.7**.**

The $(1,1)$-tensor $r$ and the Poisson tensor $\Pi$ are compatible if and only if the triple $(D^{r},r^{*},r)$ is an IM $(1,1)$-tensor on $T^{*}M$.

Proof.

The Leibniz equation for $D^{r}$ follows from the properties of $d$ and $\mathcal{L}_{r}$. So one only needs to check that the IM-equations for $(D,r,r^{*})$ are equivalent to $\Pi$ and $r$ being compatible. The IM equations in this case are:

[TABLE]

The compatibility equation (6.3) is exactly (6.12). Note that (6.10) is equivalent, for $df,dg\in\Omega^{1}(M)$, to

[TABLE]

Using that $[dg,df]_{\Pi_{r}}=-\mathcal{L}_{\Pi^{\sharp}(r^{*}(df))}dg$, it follows that (6.10) is equivalent to (6.4). So, $\Pi$ and $r$ are compatible if and only if $\eqref{IM2_2}$ and $\eqref{IM6_2}$ hold. The remaining equations follow from these two. Indeed, a long but straightforward computation shows that (6.9) is equivalent, for $\alpha=df$ and $\beta=dg$, to

[TABLE]

Finally, the redundancies among the IM equations (see Remark 3.18) guarantee that (6.11) holds. ∎

From now on, let us assume that the Lie algebroid $(T^{*}M,\Pi^{\sharp},[\cdot,\cdot]_{\Pi})$ integrates to a source 1-connected symplectic groupoid $(\mathcal{G},\omega)$ (see Section 3.3). Let $\Pi_{\omega}$ be the Poisson structure defined by $\omega$, so that

[TABLE]

is the inverse map to $T\mathcal{G}\to T^{*}\mathcal{G}$, $U\mapsto i_{U}\omega$. It follows from Theorem 3.19 and Proposition 6.7 that any $(1,1)$ tensor $r:TM\to TM$ compatible with $\Pi$ corresponds to a multiplicative $(1,1)$ tensor $K:T\mathcal{G}\to T\mathcal{G}$ on $\mathcal{G}$ integrating $(D^{r},r^{*},r)$.

Lemma 6.8**.**

Let $r:TM\to TM$ be a $(1,1)$ tensor compatible with $\Pi$. One has that $(\mathcal{N}_{r}^{*},0)$ is an IM 3-form if and only if there exists a closed 3-form $\lambda\in\Omega^{3}(\mathcal{G})$ such that

[TABLE]

where $K:T\mathcal{G}\to T\mathcal{G}$ is the multiplicative $(1,1)$ tensor integrating $(D^{r},r^{*},r)$. In this case, $\lambda$ is the multiplicative 3-form integrating $(-\mathcal{N}_{r}^{*},0)$.

Proof.

Following Cor. 6.3, the infinitesimal components $((D^{r})^{2},[D^{r},r^{*}],\mathcal{N}_{r})$ of the Nijenhuis torsion of $K$ can be explicitly calculated using (6.2) and (6.8). Indeed, using Cartan calculus and the Jacobi identity repeatedly, one may verify that, for $\alpha\in\Omega^{1}(M)$, $X,Y\in\mathfrak{X}(M)$,

[TABLE]

Similarly, for any multiplicative closed 3-form $\lambda\in\Omega^{3}(\mathcal{G})$, consider the multiplicative $(1,2)$ tensor field $\tau\in\Omega^{2}(\mathcal{G},T\mathcal{G})$ defined by $\tau(U,V)=\Pi_{\omega}^{\sharp}(\lambda(U,V,\cdot))$. Its infinitesimal components $\widetilde{D}:\Gamma(T^{*}M)\to\Omega^{2}(M,T^{*}M)$, $\widetilde{l}:T^{*}M\to T^{*}M\otimes T^{*}M$ and $\widetilde{r}:T^{*}M\to\wedge^{2}T^{*}M$ are:

[TABLE]

where $\mu:T^{*}M\to\wedge^{2}T^{*}M$ is such that $(\mu,0)$ is the IM 3-form corresponding to $\lambda$. The result now follows from the fact that (6.14) holds if and only if $\widetilde{r}=\mathcal{N}_{r}$, $\widetilde{l}=-\mathcal{N}_{r}$, $\widetilde{D}=(D^{r})^{2}$. ∎

Our aim now is to prove that $\Pi_{\omega}$ and $K$ define a symplectic quasi-Nijenhuis structure on $\mathcal{G}$; the previous lemma shows that this structure will be of the more restricted type of Remark 6.6. As $\Pi_{\omega}$ is symplectic, following Remark 6.4, it suffices to show that

[TABLE]

where $\Pi_{K}$ is the bivector field defined by $K\circ\Pi_{\omega}^{\sharp}$.

Integration of Poisson quasi-Nijenhuis structures

We start by analyzing conditions (6.15) and (6.16) in general. Our setting will be:

• •

a source 1-connected Poisson groupoid $(\mathcal{H},\widetilde{\Pi})\rightrightarrows N$;

• •

a multiplicative $(1,1)$-tensor $K:T\mathcal{G}\to T\mathcal{G}$ with $(D,l,r)$ as infinitesimal components.

Let $(A^{*}\mathcal{H},\rho_{*},[\cdot,\cdot]_{*})$ be the corresponding Lie algebroid structure on $A^{*}\mathcal{H}$ and denote by $\delta:\Gamma(\wedge^{\bullet}A\mathcal{H})\to\Gamma(\wedge^{\bullet+1}A\mathcal{H})$ the 2-differential associated with $\widetilde{\Pi}$. Define $\delta_{K}:\Gamma(A\mathcal{H})\to\Gamma(A^{*}\mathcal{H}\otimes A^{*}\mathcal{H})$ by the expression

[TABLE]

Lemma 6.9**.**

One has that

[TABLE]

In this case, the IM $(0,2)$-tensor on $A\mathcal{H}$ corresponding to the multiplicative bivector field $\widetilde{\Pi}_{K}$ is given by $l\circ(\rho_{*})^{*}:T^{*}N\to A\mathcal{H}$ and $\delta_{K}:\Gamma(A\mathcal{H})\to\Gamma(\wedge^{2}A\mathcal{H})$.

Proof.

Let $\tau_{L},\tau_{R}\in\Gamma(T\mathcal{H}\otimes T\mathcal{H})$ be (not necessarily skew-symmetric) tensor fields on $\mathcal{H}$ defined by

[TABLE]

for $\xi_{1}$, $\xi_{2}\in T^{*}\mathcal{H}$. Note that $K\circ\widetilde{\Pi}^{\sharp}=\widetilde{\Pi}^{\sharp}\circ K^{*}$ if and only if $\tau_{L}=\tau_{R}$. Now $\tau_{L},\tau_{R}$ are multiplicative and, therefore, $\tau_{L}=\tau_{R}$ if and only if

[TABLE]

using that $\mathcal{H}$ is source 1-connected (see Remarks 3.16 and 3.20). Since

[TABLE]

it follows that (ii) and (iii) hold if and only if $r\circ\rho_{*}=\rho_{*}\circ l^{*}$. Finally, from

[TABLE]

one can substitute $\xi_{i}=\mu_{i}\in A^{*}\mathcal{H}\subset T^{*}\mathcal{H}$ and use that $\Pi^{\sharp}|_{A^{*}\mathcal{H}}=-\rho_{*}$ to show that (i) holds if and only if $\delta_{K}$ is skew-symmetric. ∎

Let us assume that $K\circ\widetilde{\Pi}^{\sharp}=\widetilde{\Pi}^{\sharp}\circ K^{*}$. The IM $(0,2)$-tensor on $A\mathcal{H}$ associated to the multiplicative bivector field $\widetilde{\Pi}_{K}$ defines a pre-Lie algebroid structure on $A^{*}\mathcal{H}$. It follows from (3.18) and (6.17) that the pre-Lie bracket is given by

[TABLE]

where $\Theta\in\Gamma(\wedge^{2}A\mathcal{H}\otimes A^{*}\mathcal{H})$ is defined by

[TABLE]

For any element $\Omega\in\Gamma(\wedge^{k}A\mathcal{H}\otimes A^{*}\mathcal{H})$, one can define a contraction operator $i_{\Omega}:\Gamma(\wedge^{\bullet}A\mathcal{H})\to\Gamma(\wedge^{\bullet+k-1}A\mathcal{H})$ exactly as in (5.1). It is a graded derivation of degree $k-1$ of the graded algebra $\Gamma(\wedge^{\bullet}A\mathcal{H})$. Note that

[TABLE]

where $[\cdot,\cdot]$ is the commutator of derivations of $\Gamma(\wedge^{\bullet}A\mathcal{H})$.

Lemma 6.10**.**

The bivector fields $\widetilde{\Pi}$, $\widetilde{\Pi}_{K}$ satisfy $[\widetilde{\Pi},\widetilde{\Pi}_{K}]=0$ if and only $[\delta,i_{\Theta}]=0$.

Proof.

It is shown in [57] that $[\widetilde{\Pi},\widetilde{\Pi}_{K}]=0\Leftrightarrow[\delta,\delta_{K}]=0$. The result then follows from $[\delta,[i_{l^{*}},\delta]]=0$. ∎

We can now conclude the description of the integration of Poisson quasi-Nijenhuis structures using Thm. 3.19:

Theorem 6.11**.**

Let $(\mathcal{G},\omega)$ be the source 1-connected symplectic groupoid integrating a Poisson manifold $(M,\Pi)$. There is a one-to-one correspondencee between Poisson quasi-Nijenhuis structures $(\Pi,r)$ on $M$ and symplectic quasi-Nijenhuis structures $(\omega,K,\lambda)$ relative to $\lambda$, where $K:T\mathcal{G}\to T\mathcal{G}$ is the multiplicative $(1,1)$ tensor integrating $(D^{r},r^{*},r)$ and $\lambda\in\Omega^{3}(\mathcal{G})$ is the multiplicative closed 3-form integrating $(-\mathcal{N}_{r}^{*},0)$.

Proof.

We know that ${\Pi}_{\omega}$, the Poisson structure defined by $\omega$, makes $\mathcal{G}$ into a Poisson groupoid. The dual Lie algebroid in this case is $A^{*}=TM$, with anchor $\rho_{*}=\mathrm{id_{TM}}$ and bracket $[\cdot,\cdot]_{*}$ given by the Lie bracket of vector fields. Note that $\delta=d$ is the de Rham differential.

Let $K:T\mathcal{G}\to T\mathcal{G}$ be a multiplicative $(1,1)$ tensor on $\mathcal{G}$ and $\lambda$ a multiplicative closed 3-form. Consider the IM (1,1)-tensor $(D,l,r)$ and the IM 3-form $(\mu,0)$ associated to $K$ and $\lambda$, respectively. From Lemmas 6.9 and 6.10, one knows that $K$ and $\Pi_{\omega}$ are compatible if and only if $l=r^{*}$, $\delta_{K}$ in (6.17) is skew-symmetric and, by writing $\delta_{K}=[i_{r},d]+i_{\Theta}$, the condition $[d,i_{\Theta}]=0$ holds (notice that this last condition says that $\Theta=0$). Hence, using that $[d,i_{r}]=\mathcal{L}_{r}$,

[TABLE]

Therefore $K$ and $\Pi_{\omega}$ are compatible if and only if $\Pi$ and $r$ are compatible and, moreover, $K$ is the (1,1) tensor integrating $(D^{r},r^{*},r)$. Finally, from Lemma 6.8, the equation $\mathcal{N}_{K}(U,V)=\Pi^{\sharp}_{\omega}(\lambda(U,V,\cdot))$ holds if and if $(\mu,0)=(-\mathcal{N}_{r}^{*},0)$ is the IM 3-form associated to $\lambda$. ∎

By restricting the previous correspondence to Poisson quasi-Nijenhuis structures on $M$ relative to closed 3-forms, we recover [57, Thm. 6.2] with a different viewpoint:

Corollary 6.12**.**

Let $(\mathcal{G},\omega)$ be a source 1-connected symplectic groupoid integrating the Poisson manifold $(M,\Pi)$. There is one-to-one correspondence between Poisson quasi-Nijenhuis structures $(\Pi,r,\phi)$ on $M$ relative to a a closed 3-form $\phi$ and symplectic quasi-Nijenhuis structures $(\omega,K,\lambda)$ on $\mathcal{G}$ relative to $\lambda=\mathsf{t}^{*}\phi-\mathsf{s}^{*}\phi$ such that $K$ is multiplicative.

Proof.

It follows from the fact that $\mathsf{t}^{*}\phi-\mathsf{s}^{*}\phi$ is the multiplicative 3-form integrating the IM 3-form $(\alpha\mapsto i_{\Pi^{\sharp}(\alpha)}\phi,0)$ (see Example 3.15). ∎

Our methods also work, more generally, to describe the infinitesimal counterparts of multiplicative Poisson-Nijenhuis structures on Lie groupoids, not necessarily symplectic; here one obtains compatibilities between the IM (1,1)-tensor corresponding to Nijenhuis structure and the Lie bialgebroid associated with the Poisson groupoid. We will discuss this case elsewhere.

6.4. Multiplicative (almost) complex structures

Our general results on multiplicative (1,1) tensors in Sections 6.1 and 6.2 can be readily applied to the study of complex structures on Lie groupoids, giving another viewpoint to results in [43] concerning their infinitesimal versions.

Recall that, for a (real) vector bundle $E\to M$, a holomorphic structure is specified by a triple $(J_{E},J_{M},\nabla)$, where $J_{E}:E\to E$ is an endomorphism satisfying $J_{E}^{2}=-{\rm id}$ (which makes $E$ into a complex vector bundle), $J_{M}$ is a complex structure on $M$, and $\nabla$ is a flat $T^{(0,1)}$-connection on $E$, in such a way that holomorphic sections $u:M\rightarrow E$ are characterized by $\nabla u=0$; see [56]. We shall call $\nabla$ the Dolbeault connection on $E$. More generally, we will be interested in holomorphic structures on Lie groupoids and Lie algebroids.

A holomorphic structure on a Lie groupoid $\mathcal{G}\rightrightarrows M$ is a multiplicative complex structure; i.e., a multiplicative $J\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ such that $J^{2}=-{\rm id}$ and $\mathcal{N}_{J}=0$. One refers to $(\mathcal{G},J)$ as a complex Lie groupoid. A holomorphic structure on a Lie algebroid $A\rightarrow M$ is a holomorphic structure $(J_{A},J_{M},\nabla)$ on its underlying vector bundle such that the following compatibility conditions are satisfied:

1. (H1)

$[\cdot,\cdot]$ restricts to a Lie bracket $[\cdot,\cdot]_{\mathrm{hol}}$ on the holomorphic sections; 2. (H2)

$[\cdot,\cdot]_{\mathrm{hol}}$ is $\mathbb{C}$-linear.

As we will now see, the following correspondence, proven in [43, Thm. 3.17], is a consequence of Theorem 3.19, along with Corollaries 6.2 and 6.3.

Corollary 6.13**.**

Let $\mathcal{G}\rightrightarrows M$ be source 1-connected. Then holomorphic structures on $\mathcal{G}$ are in natural one-to-one correspondence with holomorphic structures on its Lie algebroid $A\to M$.

The remainder of this section proves this result. By Corollaries 6.2 and 6.3, we immediately see that the correspondence in Theorem 3.19 restricts to a bijective correspondence between holomorphic structures $J$ on $\mathcal{G}\rightrightarrows M$ and IM $(1,1)$-tensors $(D,l,r)$ on $A$ satisfying

[TABLE]

and

[TABLE]

So we must verify that an IM $(1,1)$-tensor $(D,l,r)$, for which (6.18) and (6.19) hold, is equivalent to a holomorphic structure on the Lie algebroid $A$. We start by checking that (6.18) and (6.19) exactly say that the triple $(D,l,r)$ defines a holomorphic structure on the vector bundle underlying $A$.

From (6.18), it is clear that $r$ is an almost complex structure on $M$ and $l$ is a complex structure on the fibres of $A$ (so we regard $A$ as a complex vector bundle). Defining $\nabla:\Gamma(T^{01})\times\Gamma(A)\to\Gamma(A)$ by

[TABLE]

we also see that the first equation in (6.18) says that

[TABLE]

Moreover, this last property along with the Leibniz rule for $D$ imply that

[TABLE]

for real functions $f\in C^{\infty}(M)$.

Let us now consider (6.19). The third condition says that $r$ is a complex structure on $M$, while the second says that $\nabla$ is complex linear in $A$. It follows that $T^{01}$ is a (complex) Lie algebroid and $\nabla$ is a (complex) $T^{01}$-connection. Finally, using (6.2), one can also check that the first equation in (6.19) amounts to $\nabla$ being flat. In conclusion, conditions (6.18) and (6.19) say that $(D,l,r)$ endow $A$ with the structure of a holomorphic vector bundle: $J_{A}=l$, $J_{M}=r$ and $\nabla$ given by (6.20). The result in Corollary 6.13 now follows from

Lemma 6.14**.**

The IM-equations for $(D,l,r)$ are equivalent to conditions (H1) and (H2) above.

Proof.

We saw that equations (6.18) and (6.19) say that $(D,l,r)$ defines a holomorphic structure on the vector bundle $A$; moreover, a section $a\in\Gamma(A)$ is holomorphic if and only if $Da=0$.

($\Rightarrow$): Note that (IM1*) implies that $[\cdot,\cdot]$ restricts to a Lie bracket $[\cdot,\cdot]_{\mathrm{h}ol}$ on the holomorphic sections of $A$, and (IM2*) implies that $[\cdot,\cdot]_{\mathrm{h}ol}$ is $\mathbb{C}$-linear.

($\Leftarrow$:) Assume that $(A,[\cdot,\cdot],\rho)$ is a holomorphic Lie algebroid, and consider

[TABLE]

One can check that $E_{l}$ and $E_{r}$ are $C^{\infty}(M)$-linear on both components and $E_{D}$ is anti-symmetric on the $\Gamma(A)$-components. Moreover, for $f\in C^{\infty}(M)$,

[TABLE]

Note that $[\cdot,\cdot]_{\mathrm{h}ol}$ being $\mathbb{C}$-linear implies that $E_{l}(a,b)=0$ for $a,b\in\Gamma(A)$ holomorphic. As $\Gamma(A)$ is generated as a $C^{\infty}(M)$-module by the holomorphic sections, it follows that $E_{l}\equiv 0$. The redundancies discussed in Remark 3.18 imply that $\rho\circ l=r\circ\rho$, so $\rho$ is a complex vector-bundle morphism. Furthermore, note that $\rho:A\rightarrow TM$ sends holomorphic sections to holomorphic sections. Indeed, if $h\in C^{\infty}(M,\mathbb{C})$ is a holomorphic function and $u_{1},u_{2}\in\Gamma(A)$ are arbitrary holomorphic sections,

[TABLE]

is a holomorphic section, which implies that $\rho(u_{2})$ is a holomorphic section of $TM$. Using the $C^{\infty}(M)$-linearity of $E_{r}$, one can argue as above to prove that $E_{r}(a,X)=0$ for all $a\in\Gamma(A)$ and $X\in\Gamma(TM)$ (use that $r=J_{M}$, the almost complex structure of $M$, and that $TM$ with the Lie bracket of vector fields and the identity as anchor is a holomorphic Lie algebroid). Finally, from (6.21) it follows that $E_{D}$ is tensorial, and the fact that $[\cdot,\cdot]$ restricts to $[\cdot,\cdot]_{\mathrm{h}ol}$ on holomorphic sections implies, as before, that $E_{D}\equiv 0$. ∎

As this section and Section 6.3 illustrate, Theorem 3.19 provides tools that can be directly applied to treat multiplicative geometric structures on holomorphic Lie groupoids, including holomorphic symplectic groupoids [42] or more general holomorphic Poisson groupoids, as well as multiplicative generalized complex structures [31], offering complementary information about the latter in terms of infinitesimal components. We will further discuss these cases in a separate work.

6.5. Multiplicative projections

For a Lie groupoid $\mathcal{G}\rightrightarrows M$ with Lie algebroid $A$, we consider a multiplicative $(1,1)$ tensor field $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ satisfying $K^{2}=K$, referred to as a multiplicative projection. In this section we apply our previous results to describe multiplicative projections infinitesimally, making connections with the theory of matched pairs [39, 45, 54].

We start by observing that projections can be used to treat other types of multiplicative $(1,1)$ tensors.

Example 6.15**.**

Suppose that $Q:T\mathcal{G}\to T\mathcal{G}$ satisfies $Q^{2}={\rm id}$ and is multiplicative; i.e., $Q$ is a multiplicative (almost) product structure on $\mathcal{G}$. Then $K:=(Q+{\rm id})/2$ is a multiplicative projection555By working with complexifications, one can also cast (almost) complex structures as projections..

We know that a multiplicative projection $K$ admits an infinitesimal description by its infinitesimal components $(D,l,r)$. We start by discussing alternative ways to express the operator $D$, that will be convenient when we consider the Nijenhuis torsion of $K$.

Since $r^{2}=r:TM\to TM$ and $l^{2}=l:A\to A$ (by Cor. 6.2), the bundles $TM$ and $A$ decompose as

[TABLE]

where $A^{0},T^{0}$ are the kernels of the maps $l,r$, and $A^{1},T^{1}$ are their images, respectively.

Lemma 6.16**.**

One has that

[TABLE]

Also, for $a\in\Gamma(A)$,

[TABLE]

Proof.

The conditions in (6.22) follow from (IM6*) (Section 6.1), whereas the statements about $D$ follow from Cor. 6.2, part (a). ∎

So, upon restriction, $D$ gives rise to two operators:

[TABLE]

Let us consider the operators

[TABLE]

Proposition 6.17**.**

$\Lambda^{+},\Lambda^{-}$* are tensorial, whereas $\nabla^{+},\nabla^{-}$ satisfy*

[TABLE]

for $f\in C^{\infty}(M)$, $a\in\Gamma(A^{0}),\,b\in\Gamma(A^{1})$.

Proof.

This follows immediately from the Leibniz rule for $D$ (3.9). ∎

Vanishing of the Nijenhuis torsion.

Let $\mathcal{N}_{K}\in\Omega^{2}(\mathcal{G},T\mathcal{G})$ be the Nijenhuis torsion of $K$. We say that $K$ is a flat projection if $\mathcal{N}_{K}=0$. The next result gives an equivalent description of the Nijenhuis vanishing condition.

Proposition 6.18**.**

Let $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ be a multiplicative projection on a source-connected Lie groupoid $\mathcal{G}\rightrightarrows M$. Then $\mathcal{N}_{K}=0$ if and only if

• •

$\Lambda^{+}=0$, $\Lambda^{-}=0$;

• •

$T^{0}$* and $T^{1}$ are involutive distributions;*

• •

$\nabla^{+}$* is a flat $T^{0}$-connection, and $\nabla^{-}$ is a flat $T^{1}$-connection.*

Proof.

As $\mathcal{G}$ is source connected, the Nijenhuis torsion $\mathcal{N}_{K}$ vanishes if and only if its infinitesimal components $(D^{2},[D,l],N_{r})$ are zero (see Corollary 6.3). A direct verification shows that

[TABLE]

Hence, $[D,l]=0$ if and only if both $\Lambda^{+}$ and $\Lambda^{-}$ vanish. Similarly, $N_{r}=0$ is equivalent to $T^{0}$ and $T^{1}$ being involutive666In general, for a projection $K\in\Omega^{1}(M,TM)$ a projection, the Nijenhuis torsion $\mathcal{N}_{K}$ can be written as $\mathcal{N}_{K}=R_{K}+\overline{R}_{K},$ where $R_{K},\overline{R}_{K}\in\Omega^{2}(M,TM)$ are the curvature and co-curvature of $K$ given, respectively, by

$\displaystyle R_{K}(X,Y)$ $\displaystyle=K([({\rm id}-K)(X),({\rm id}-K)(Y)])$

$\displaystyle\overline{R}_{K}(X,Y)$ $\displaystyle=({\rm id}-K)([K(X),K(Y)]).$

In particular, $\mathcal{N}_{K}=0$ if and only if both $\ker(K)$ and $\mathrm{im}(K)$ are involutive distributions.. By Proposition 6.17, $\nabla^{+}$ is a $T^{0}$-connection, and $\nabla^{-}$ is a $T^{1}$-connection. Using (6.2), we see that

[TABLE]

where $\mathrm{Curv}^{+}$ (resp. $\mathrm{Curv}^{-}$) is the curvature of $\nabla^{+}$ (resp. $\nabla^{-}$). Hence, $D^{2}=0$ if and only if both $\nabla^{+},\nabla^{-}$ are flat. This concludes the proof. ∎

Remark 6.19**.**

A distribution $\Delta\subset T\mathcal{G}$ is said to be multiplicative if $\Delta$ is a Lie subgroupoid. As observed in [8], a multiplicative projection is equivalent to a pair of multiplicative distributions $\Delta_{1}$, $\Delta_{2}$ such that $\Delta_{1}\oplus\Delta_{2}=T\mathcal{G}$. Also, $\mathcal{N}_{K}=0$ is equivalent to both distributions being involutive. In this context, Proposition 6.18 agrees with the integrability criteria for multiplicative distributions given in [14, 30].

Example 6.20**.**

Following Example 6.15, consider a multiplicative $Q:T\mathcal{G}\to T\mathcal{G}$ satisfying $Q^{2}={\rm id}$, and let $K=(Q+{\rm id})/2$ be the corresponding multiplicative projection. Let $(D,l,r)$ and $(D^{\prime},l^{\prime},r^{\prime})$ be the infinitesimal components of $K$ and $Q$, respectively. Then

[TABLE]

The bundles $T^{0}$, $A^{0}$ (resp. $T^{1}$, $A^{1}$) are now the $-1$ (resp. $+1$) eigenbundles of $r^{\prime}$ and $l^{\prime}$. Also, $D^{\prime}$ decomposes into tensors $\Lambda^{+}\in\Gamma(T^{0\,*}\otimes A^{0\,*}\otimes A^{1})$, $\Lambda^{-}\in\Gamma(T^{1\,*}\otimes A^{1\,*}\otimes A^{0})$ and connections $\nabla^{+}:\Gamma(A^{1})\to\Gamma(T^{0\,*}\otimes A^{1})$, $\nabla^{-}:\Gamma(A^{0})\to\Gamma(T^{1\,*}\otimes A^{0})$. Noticing that $\mathcal{N}_{K}=0\Leftrightarrow\mathcal{N}_{Q}=0$, we see that Prop. 6.18 directly applies to $Q$ instead of $K$.

We now illustrate the infinitesimal components of a multiplicative projection in the classical example of a projection defined by a connection on a principal bundle.

Example 6.21**.**

Let $P\to M$ be a principal bundle for a Lie group $G$, and consider its gauge groupoid $\mathcal{G}(P):=(P\times P)/G\rightrightarrows M$ (see e.g. [47, Sec, 1.1]). In [8], it is shown that there is a one-to-one correspondence between principal connections $\theta\in\Omega^{1}(P,\mathfrak{g})$ and multiplicative projections $K:T\mathcal{G}(P)\to T\mathcal{G}(P)$ such that $\mathrm{im}(K)=\ker(T\mathsf{s})\cap\ker(T\mathsf{t})$. To explicitly describe this projection it is useful to identify the tangent groupoid $T\mathcal{G}(P)$ with the gauge groupoid $\mathcal{G}(TP)=(TP\times TP)/TG\to TM$ of the principal $TG$-bundle $TP\to TM$. The quotient map $TP\times TP\to\mathcal{G}(TP)$ is denoted by

[TABLE]

The projection $K$ is now defined as

[TABLE]

where $u_{P}$ denotes the infinitesimal generator on $P$ corresponding to $u\in\mathfrak{g}$. One can check (see [8]) that the Nijenhuis torsion of $K$ is given by

[TABLE]

where $F_{\theta}\in\Omega^{2}(M,P\times_{G}\mathfrak{g})$ is the curvature of $\theta$, and the maps $\mathcal{T}$ and $\mathcal{S}$ are defined in (3.6); here $P\times_{G}\mathfrak{g}$ is the associated bundle with respect to the adjoint representation, and we are using the Atiyah sequence

[TABLE]

to view $P\times_{G}\mathfrak{g}$ as a subbundle of $A(\mathcal{G}(P))=TP/G$.

The infinitesimal components $(D,l,r)$ of the multiplicative projection $K$ are given as follows: $r=0$, while $l:TP/G\to P\times_{G}\mathfrak{g}\subset TP/G$ is the map induced by $\theta:TP\to\mathfrak{g}$, and $D:\Gamma(TP/G)\to\Gamma(T^{*}M\otimes TP/G)$ is given by

[TABLE]

where $X_{H}\in\mathfrak{X}(P)$ is the horizontal lift of $X$. Note that $T^{0}=TM$, $T^{1}=0$, $A^{0}=H/G\cong TM$, $A^{1}=P\times_{G}\mathfrak{g}$, where $H=\ker(\theta)\subset TP$ is the horizontal distribution. Under the splitting $D=D^{+}+D^{-}$, one may directly check that $D^{-}=0$ and $\nabla^{+}:\Gamma(P\times_{G}\mathfrak{g})\to\Gamma(T^{*}M\otimes P\times_{G}\mathfrak{g})$ is the natural connection on the associated bundle $P\times_{G}\mathfrak{g}$, whereas $\Lambda^{+}:\Gamma(TM)\to\Gamma(T^{*}M\otimes P\times_{G}\mathfrak{g})$ is

[TABLE]

Prop. 6.18 admits yet another geometric interpretation, that we discuss next.

Characterization via matched pairs

In the remainder of this section we provide a characterization of multiplicative flat projections using the theory of matched pairs of Lie algebroids [54].

Definition 6.22**.**

Let $A,B\rightarrow M$ be Lie algebroids. We say that $(A,B)$ is a matched pair if $A$ has a representation on $B$, and $B$ has a representation on $A$ such that

[TABLE]

Here we denote both representations by $\nabla$.

Definition 6.23**.**

A morphism of matched pairs from $(A_{1},B_{1})$ to $(A_{2},B_{2})$ is a pair of Lie algebroid morphisms $F_{A}:A_{1}\rightarrow A_{2}$, $F_{B}:B_{1}\rightarrow B_{2}$ such that

[TABLE]

A matched pair $(A,B)$ is equivalent to a Lie algebroid structure on the Whitney sum $A\oplus B$ such that $A$ and $B$ are Lie subalgebroids, see [48, 54]. From this viewpoint, the representations are determined by the Lie bracket:

[TABLE]

where ${\rm pr}_{A}:A\oplus B\rightarrow A$ and ${\rm pr}_{B}:A\oplus B\rightarrow B$ are the projections. In this context, a morphism of matched pairs from $(A_{1},A_{2})$ to $(B_{1},B_{2})$ is equivalent to a Lie-algebroid morphism $F:A_{1}\oplus B_{1}\rightarrow A_{2}\oplus B_{2}$ which restricts to Lie-algebroid morphisms from $A_{1}$ to $A_{2}$, and from $B_{1}$ to $B_{2}$.

For a flat multiplicative projection $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$, Proposition 6.18 implies that the decomposition $TM=T^{0}\oplus T^{1}$ defines a matched pair $(T^{0},T^{1})$. Also, for $a,b\in\Gamma(A^{0})$, the condition

[TABLE]

implies that $A^{0}\subset A$ is a subalgebroid. Similarly, one can check that $A^{1}\subset A$ is a subalgebroid. Thus, the decomposition $A=A^{0}\oplus A^{1}$ defines a matched pair $(A^{0},A^{1})$. For $a\in\Gamma(A^{0}),b\in\Gamma(A^{1})$, the representations are defined by

[TABLE]

We obtain the following infinitesimal characterization of flat multiplicative projections:

Theorem 6.24**.**

Let $\mathcal{G}$ be a source 1-connected groupoid. There is a one-to-one correspondence between flat multiplicative projections on $\mathcal{G}$ and decompositions $A=A^{0}\oplus A^{1}$ and $TM=T^{0}\oplus T^{1}$, where

• (i)

$A^{0},A^{1}\subset A$* and $T^{0},T^{1}\subset TM$ are Lie subalgebroids;*

• (ii)

$(A^{0},T^{1})$* and $(T^{0},A^{1})$ are matched pairs;*

• (iii)

The sides of the commutative square

[TABLE]

are morphisms of matched pairs.

Proof.

Consider decompositions $A=A^{0}\oplus A^{1}$ and $TM=T^{0}\oplus T^{1}$ for which (i), (ii) and (iii) hold. We will prove that they define the infinitesimal components of a flat multiplicative projection.

Define $l:A\rightarrow A$ (resp. $r:TM\rightarrow TM$) to be the projection on $A^{1}$ (resp. $T^{1}$) along $A^{0}$ (resp. $T^{0}$). Using that $\mathrm{Ann}(T^{1})\cong T^{0\,*}$ and $\mathrm{Ann}(T^{0})\cong T^{1\,*}$, define $D:\Gamma(A)\rightarrow\Gamma(T^{*}M\otimes A)$ to be the map

[TABLE]

where $\nabla^{+}:\Gamma(A^{1})\rightarrow\Gamma(T^{0\,*}\otimes A^{1})$, $\nabla^{-}:\Gamma(A^{0})\rightarrow\Gamma(T^{1\,*}\otimes A^{0})$ are the flat connections (i.e., representations) corresponding to the matched pairs $(A^{0},T^{1})$, $(T^{0},A^{1})$, respectively. One may directly verify that $D$ satisfies the Leibniz rule (3.9). We now prove that the triple $(D,l,r)$ satisfies the IM-equations (Section 6.1).

Equation (IM6*):

From (iii), one has that $\rho(A^{0})\subset T^{0}$, $\rho(A^{1})\subset T^{1}$, which is equivalent to $\rho\circ l=r\circ\rho$.

Equation (IM2*):

The subbundle $A^{0}\subset A$ is a Lie subalgebroid if and only if $l([a,b])=0,$ for $a,b\in\Gamma(A^{0})$. As $l(b)=0=D_{\rho(b)}(a)$, this implies that

[TABLE]

Similarly, one can check that (IM2*) holds for $a,b\in\Gamma(A^{1})$ using that $A^{1}\subset A$ is a Lie subalgebroid. It remains to verify that (IM2*) holds for crossed terms. Using (ii), let $\nabla^{+,\,bas}:\Gamma(A^{1})\times\Gamma(T^{0})\rightarrow\Gamma(T^{0})$, $\nabla^{-,\,bas}:\Gamma(A^{0})\times\Gamma(T^{1})\rightarrow\Gamma(T^{1})$ be the representations of $A^{1}$, $A^{0}$ on $T^{0}$, $T^{1}$, respectively. For equation (6.25) to hold for the matched pairs $(A^{0},T^{1})$, $(T^{0},A^{1})$, one must have that

[TABLE]

Now, $({\rm id}_{A^{0}},\,\rho):(A^{0},A^{1})\rightarrow(A^{0},T^{1})$ and $({\rm id}_{A^{1}},\,\rho):(A^{0},A^{1})\rightarrow(T^{0},A^{1})$ are morphisms of matched pairs if and only if, for all $a\in\Gamma(A^{0})$, $b\in\Gamma(A^{1})$,

[TABLE]

respectively. Altogether, this proves that (IM2*) holds.

Equation (IM3*):

Similarly to the previous case, one can check that (IM3*) follows from $T^{0}$, $T^{1}$ being involutive and $(\rho,\,{\rm id}_{T^{1}}):(A^{0},T^{1})\rightarrow(T^{0},T^{1})$ and $({\rm id}_{T^{0}},\rho):(T^{0},A^{1})\rightarrow(T^{0},T^{1})$ being morphisms of matched pairs.

Equation (IM1*):

We recall (IM1*) for convenience:

[TABLE]

There are 6 cases to check by taking $X$ in $T^{0}$ or $T^{1}$, and $a,b$ in $A^{0}$ or $A^{1}$.

1. (1)

$X\in\Gamma(T^{0})$ and $a,b\in\Gamma(A^{0})$, 2. (2)

$X\in\Gamma(T^{0})$ and $a,b\in\Gamma(A^{1})$, 3. (3)

$X\in\Gamma(T^{0})$ and $a\in\Gamma(A^{0}),\,b\in\Gamma(A^{1})$.

The other 3 cases where $X\in\Gamma(T^{1})$ work analogously. For (1), both sides of (IM1*) are trivially zero. For (2), using that $D|_{T^{1}}(a)=D|_{T^{1}}(b)=0$, one can show that (IM1*) is equivalent to (6.26) for the matched pair $(T^{0},A^{1})$. Finally, for (3), one has that

[TABLE]

On the other hand, using that $D|_{T^{0}}(a)=0$, the RHS of (IM1*) simplifies to

[TABLE]

Using (IM2*) once again, one can prove that

[TABLE]

so the case (3) follows from the flatness of $\nabla^{+}$. Putting everything together, we have proved that $(D,l,r)$ defines an IM $(1,1)$-tensor, so it integrates to a multiplicative $(1,1)$ tensor $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ by Theorem 3.19. It then follows from Corollary 6.2 and Proposition 6.18 that $K$ is a flat projection.

The converse, i.e., that the infinitesimal components of a multiplicative flat projection give rise to matched pairs as in the statement of the theorem, is proven by similar arguments and is left to the reader.

∎

Following Example 6.20, an analogous result holds for product structures; see [42, Thm. 4.8] for a parallel result in the context of holomorphic structures.

Appendix A Lie theory of componentwise linear functions

In this appendix, we study componentwise linear functions on Whitney sums of VB-groupoids. We start by presenting a useful characterization of these functions.

Let $E_{i}\to M$ be vector bundles, $i=1,\ldots,p$, and let $E=E_{1}\oplus\dots\oplus E_{p}$. Consider the multiplication by non-negative scalars $h:\mathbb{R}_{\geq 0}\times E\rightarrow E$, $h_{\lambda}(e)=\lambda e$. We shall refer to $h$ as the homogeneous structure on $E$. The next result gives a characterization of componentwise linear functions on $E$ in terms of its homogeneous structure. This is an extension of the characterization of vector bundle maps in [25], for $p=1$.

Proposition A.1**.**

A smooth function $F:E\rightarrow\mathbb{R}$ is componentwise linear if and only if

• (1)

$F\circ h_{\lambda}=\lambda^{p}F$, for all $\lambda\geq 0$,

• (2)

$F\circ 0_{i}=0$,

where $0_{i}:\oplus_{1\leq j\neq i\leq p}E_{j}\to\oplus_{1\leq j\leq p}E_{j}$ is the map

[TABLE]

for $i=1,\dots,p$.

Proof.

We will consider the case $p=2$, the general case being a direct generalization. By restricting $F$ to the fibers of $E_{1}$ and $E_{2}$, we can assume that $F:\mathbb{R}^{m}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ satisfies $F(\lambda x,\lambda y)=\lambda^{2}F(x,y)$, for all $\lambda\geq 0$, and $F(x,0)=F(0,y)=0$. Note that this implies that $F(0,0)=0$ and $DF(0,0)=0$. Now, one can use Taylor’s Theorem and the homogeneity of $F$ to prove that $F(z)=\sum_{1\leq i,j\leq m+n}\frac{\partial^{2}F}{\partial z_{i}\partial z_{j}}(0,0)z_{i}z_{j}$, for $z=(x,y)$ satisfying $|z|=1$. Using the homogeneity once more, it is possible to extend the equality to arbitrary $z$. The condition that $F(x,0)=F(0,y)=0$ implies that the terms $x_{i}x_{j}$ and $y_{i}y_{j}$ do not appear in the sum. Hence $F:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}$ is bilinear. This completes the proof. ∎

We now consider $\mathcal{VB}$-groupoids (see e.g. [47, Sec. 11.2] for details and original references), following the viewpoint of [4].

Definition A.2**.**

A $\mathcal{VB}$-groupoid is a square

[TABLE]

whose horizontal sides are vector bundles, the vertical sides are Lie groupoids, satisfying the following compatibility condition: denoting by $h$ and $h^{E}$ the homogeneous structures on $\mathcal{V}$ and $E$, then, for each $\lambda\geq 0$, $h_{\lambda}:\mathcal{V}\rightarrow\mathcal{V}$ is a groupoid morphism over $h_{\lambda}^{E}:E\to E$.

The Lie algebroid $A\mathcal{V}\rightarrow E$ of a $\mathcal{VB}$-groupoid inherits a vector bundle structure over $A$ by differentiation of $h_{\lambda}$. If $h^{A}$ is the corresponding homogeneous structure, then $h_{\lambda}^{A}:A\mathcal{V}\rightarrow A\mathcal{V}$ is a Lie algebroid morphism over $h_{\lambda}$ for each $\lambda$.

In the following, we consider VB-groupoids $\mathcal{V}_{1}\rightrightarrows E_{1}$, $\dots$, $\mathcal{V}_{p}\rightrightarrows E_{p}$ over $\mathcal{G}\rightrightarrows M$ and their Whitney sum $\mathcal{V}=\mathcal{V}_{1}\oplus\dots\oplus\mathcal{V}_{p}$ (as vector bundles over $\mathcal{G}$). This defines a $\mathcal{VB}$-groupoid $\mathcal{V}\rightrightarrows E$ over $\mathcal{G}\rightrightarrows M$, whose Lie algebroid $A\mathcal{V}\to E$ splits naturally as a Whitney sum $A\mathcal{V}_{1}\oplus\dots\oplus A\mathcal{V}_{p}$ over $A\mathcal{G}$.

Proposition A.3**.**

*For a source 1-connected Lie groupoid $\mathcal{G}\rightrightarrows M$, a multiplicative function $F:\mathcal{V}\rightarrow\mathbb{R}$ is componentwise linear if and only if so is $AF:A\mathcal{V}\to\mathbb{R}$. Moreover, in the case $\mathcal{V}_{1}=\dots=\mathcal{V}_{p}$, $F$ is skew-symmetric if and only if so is $AF$. *

Proof.

We will treat the case $p=2$, the general case follows similarly. Consider

[TABLE]

These are multiplicative functions and their associated infinitesimal cocycles are: $AF\circ h_{\lambda}^{A},\,\lambda^{2}AF:A\mathcal{V}\to\mathbb{R}$ and $AF\circ 0_{A\mathcal{V}_{i}}:A\mathcal{V}_{i}\to\mathbb{R}$, where $0_{A\mathcal{V}_{i}}:A\mathcal{V}_{i}\rightarrow A\mathcal{V}$ is the zero map (A.1) for the $A\mathcal{V}=A\mathcal{V}_{1}\oplus A\mathcal{V}_{2}$, $i=1,2$. Note that $0_{\mathcal{V}_{i}}$ is a Lie groupoid morphism and $A(0_{\mathcal{V}_{i}})=0_{A\mathcal{V}_{i}}$. The fact that $\mathcal{G}$ is source 1-connected implies that $\mathcal{V}$ is source 1-connected (see Remark 2.1) and, therefore, by uniqueness of integration, one has that

[TABLE]

So, it follows from Proposition A.1 that $F$ is componentwise linear if and only if so is $AF$.

When $\mathcal{V}_{1}=\dots=\mathcal{V}_{p}$, each permutation $\sigma\in S(p)$ acts by groupoid morphism on $\mathcal{V}$ via $\sigma(v_{1},\dots,v_{p})=(v_{\sigma(1)},\dots,v_{\sigma(p)})$. Applying the Lie functor, $A\sigma$ acts on $A\mathcal{V}=A\mathcal{V}_{1}\oplus\dots\oplus A\mathcal{V}_{1}$ permuting the elements of $A\mathcal{V}_{1}$ by $\sigma$ itself. Hence, for a function $F\in C^{\infty}(\mathcal{V})$,

[TABLE]

where $\mathfrak{p}_{\mathcal{V}}$, $\mathfrak{p}_{A\mathcal{V}}$ are the projections (2.9) for $\mathcal{V}$ and $A\mathcal{V}$, respectively. The result now follows exactly as before using the uniqueness of integration. ∎

Appendix B Cotangent Lie algebroid.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid, with Lie algebroid $\pi_{A}:A\to M$. Consider the cotangent Lie groupoid $T^{*}\mathcal{G}\rightrightarrows A^{*}$. It is well-known that the canonical symplectic form $\omega_{can}$ on $T^{*}\mathcal{G}$ is multiplicative and its infinitesimal component $l_{can}:A(T^{*}\mathcal{G})\rightarrow T^{*}(A^{*})$, which satisfies (see (4.34))

[TABLE]

is a Lie algebroid isomorphism, where the Lie algebroid structure on $T^{*}(A^{*})$ comes from the linear Poisson structure $\Pi_{lin}$ on $A^{*}$. In fact, $(T^{*}\mathcal{G},\omega_{can})$ is the symplectic groupoid integrating the linear Poisson structure of $A^{*}$, see [12].

The composition of $l_{can}$ with the reversal isomorphism $\mathcal{R}:T^{*}(A^{*})\rightarrow T^{*}A$ (see (4.15)) defines the isomorphism of Lie algebroids between $A(T^{*}G)$ and the cotangent Lie algebroid $T^{*}A$ mentioned in Section 4.2,

[TABLE]

(see [49, Theorem 5.3]). So, by $\overrightarrow{\mathcal{R}_{a}}$ and $\overrightarrow{\mathcal{B}\alpha}$ we mean the right-invariant vector fields of $T^{*}\mathcal{G}\rightrightarrows A^{*}$ corresponding to $\vartheta^{-1}(\mathcal{R}_{a})$ and $\vartheta^{-1}(\mathcal{B}\alpha)$, for $a\in\Gamma(A)$ and $\alpha\in\Omega^{1}(M)$, respectively. Note that, from (4.16) and (4.18), one has that

[TABLE]

where $\pi_{A^{*}}:A^{*}\to M$ is the projection of the dual bundle.

Lemma B.1**.**

For $a\in\Gamma(A)$, consider $\ell_{\overrightarrow{a}}\in C^{\infty}(T^{*}\mathcal{G})$. One has

[TABLE]

where $\overrightarrow{a}^{T^{*}}$ is the cotangent lift of $\overrightarrow{a}$ (4.6). In particular, $\overrightarrow{\mathcal{R}_{a}}=\overrightarrow{a}^{T^{*}}$.

Proof.

It is simple to check that the pull-back of the 1-form $d\ell_{a}\in\Omega^{1}(A^{*})$ by the target map $\widetilde{\mathsf{t}}:T^{*}\mathcal{G}\to A^{*}$ is exactly $d\ell_{\overrightarrow{a}}$, i.e., $d\ell_{\overrightarrow{a}}=\widetilde{\mathsf{t}}^{*}d\ell_{a}.$ Then, by (B.1),

[TABLE]

The second identity in (B.2) follows from the fact that the cotangent lift of a vector field is exactly its Hamiltonian lift with respect to $\omega_{can}$. The last statement follows from the non-degeneracy of $\omega_{can}$ ∎

In the following, we shall need the useful relationship between the sum and multiplication on the cotangent bundle known as interchange law777Interchange laws hold more generally for VB-groupoids, where they express the compatibility between the vector bundle and groupoid structures. See [47]:

[TABLE]

for $\xi_{1},\eta_{1}\in T_{g_{1}}^{*}\mathcal{G}$, $\xi_{2},\eta_{2}\in T_{g_{2}}^{*}\mathcal{G}$ such that $(\xi_{1},\xi_{2})$, $(\eta_{1},\eta_{2})$ are composable pairs.

Lemma B.2**.**

Given a 1-form $\alpha\in\Omega^{1}(M)$,

[TABLE]

the vertical lift of $\mathsf{t}^{*}\alpha\in\Omega^{1}(\mathcal{G})$.

Proof.

To simplify notation, denote $\mathsf{t}^{*}\alpha$ by $\eta$. First note that $\widetilde{\mathsf{s}}(\eta(g))=0$, $\forall\,g\in\mathcal{G}$. Indeed, for any $a\in A_{\mathsf{s}(g)}$, it follows from the definition of $\widetilde{\mathsf{s}}$ that

[TABLE]

Hence

[TABLE]

Also, one has that

[TABLE]

Indeed, for any $U\in T_{g}\mathcal{G}$, using (2.4), one obtains

[TABLE]

Let us now prove that $\eta^{{\mathrm{v}}}$ is a right-invariant vector field. For $\xi\in T_{g}^{*}\mathcal{G}$,

[TABLE]

This proves that $\eta^{{\mathrm{v}}}$ is right-invariant.

To conclude the proof, we just need to prove that $\omega_{can}(\eta^{\mathrm{v}}(\varphi),\Upsilon)=\omega_{can}(\overrightarrow{\mathcal{B}\alpha}(\varphi),\Upsilon)$ for any $\Upsilon\in T_{\varphi}A^{*}\subset T(T^{*}\mathcal{G})$, $\varphi\in A_{x}^{*}$, $x\in M$. Choose any projectable vector field $\widetilde{\Upsilon}\in\mathfrak{X}(T^{*}\mathcal{G})$ extending $\Upsilon$, with respect to the cotangent bundle projection ${\rm pr}:T^{*}\mathcal{G}\to\mathcal{G}$. Recall that ${\rm pr}$ is a groupoid morphism covering $\pi_{A^{*}}:A^{*}\to M$. As $\omega_{can}=-d\theta_{can}$ for the tautological 1-form $\theta_{can}\in\Omega^{1}(T^{*}\mathcal{G})$, one has that

[TABLE]

where we have used that $\theta_{can}(\eta^{{\mathrm{v}}})=\theta_{can}([\eta^{{\mathrm{v}}},\widetilde{\Upsilon}])=0$ and $T{\rm pr}(\widetilde{\Upsilon}(\varphi+\epsilon\eta(x)))=T{\rm pr}(\widetilde{\Upsilon}(\varphi)).$ The proof now follows from (B.1)

[TABLE]

∎