On the other hand, the definition (4.2) of the curvature 3-form gives:

$G|_{U_{i}}=\operatorname{d}F_{i}.$ (4.5)

Equations (4.3), (4.4) and (4.5) show that $[G]\in H^{3}(N,\mathbb{R})$
corresponds to the class $[\operatorname{d}f_{ijk}]\in
H^{2}(N,\underline{\Omega^{1}_{\textrm{cl}}})$, so using that
$c_{ijk}=\exp(f_{ijk})$ the result follows. ∎

##  5\. IM Ehresmann connections

###  5.1. IM connection for a surjective algebroid morphism

We now turn to the infinitesimal version of multiplicative Ehresmann
connections. If one starts with a surjective Lie algebroid morphism $\phi:A\to
B$ covering the identity, its differential is a morphism of VB algebroids:

---
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{d}\phi}$$\textstyle{TB\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Id}}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Id}}$$\textstyle{M}$

We recall that double arrows $\implies$ represent Lie algebroids. The kernel
of $\operatorname{d}\phi$ is a VB subalgebroid of $TA$:

$\textstyle{\ker(\operatorname{d}\phi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

This leads naturally to the infinitesimal version of a multiplicative
Ehresmann connection:

######  Definition 5.1.

An IM Ehresmann connection for $\phi:A\to B$ is a VB subalgebroid
$E\Rightarrow TM$ of $TA\Rightarrow TM$ which is “horizontal”:

$TA=\ker(\operatorname{d}\phi)\oplus E.$

###  5.2. Infinitesimal partially split bundles of ideals

A surjective Lie algebroid morphism $\phi:A\to B$ covering the identity is
completely determined by its kernel:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.2778pt\raise 0.0pt\hbox{\hbox{\kern
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3.0pt\raise
0.0pt\hbox{$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
140.3646pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
140.3646pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

So the notion of IM Ehresmann connection for $\phi$ can be rephrased in terms
of the bundle of ideals $\mathfrak{k}$, without any mentioning of $\phi$ and
$B$. To do this we need to express $\ker(\operatorname{d}\phi)$ in terms of
$\mathfrak{k}$, which can be done as follows.

Let $A\Rightarrow M$ be a Lie algebroid and $\mathfrak{k}\subset A$ a bundle
of ideals, i.e., $\mathfrak{k}\to M$ is a vector subbundle included in
$\ker\rho_{A}$ and satisfying

$\alpha\in\Gamma(A),\
\gamma\in\Gamma(\mathfrak{k})\quad\Longrightarrow\quad[\alpha,\gamma]_{A}\in\Gamma(\mathfrak{k}).$

Then we have the canonical representation of $A$ on $\mathfrak{k}$:

$\nabla^{\mathfrak{k}}_{\alpha}\gamma:=[\alpha,\gamma]_{A},\quad\alpha\in\Gamma(A),\gamma\in\Gamma(\mathfrak{k}).$
(5.1)

This is the infinitesimal version of the groupoid representation (2.1) and it
gives rise to the infinitesimal version of the groupoid
$\mathcal{G}\times_{M}\mathfrak{k}\rightrightarrows M$. Namely, we have the
semi-direct product Lie algebroid $A\times_{M}\mathfrak{k}\Rightarrow M$, with
Lie bracket

$[(\alpha_{1},\gamma_{1}),(\alpha_{2},\gamma_{2})]:=([\alpha_{1},\alpha_{2}]_{A},\nabla^{\mathfrak{k}}_{\alpha_{1}}\gamma_{2}-\nabla^{\mathfrak{k}}_{\alpha_{2}}\gamma_{1}),$

and anchor $\rho_{A}\circ\operatorname{pr}_{A}$. This is a VB algebroid:

$\textstyle{A\times_{M}\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

and, in fact, we have:

######  Lemma 5.2.

Given a bundle of ideals $\mathfrak{k}\subset A$ the image of the inclusion

$A\times_{M}\mathfrak{k}\hookrightarrow
TA,\quad(a,\gamma)\mapsto\frac{\operatorname{d}}{\operatorname{d}t}\Big{|}_{t=0}(a+t\gamma),$
(5.2)

coincides with $\ker(\operatorname{d}\phi)$, where $\phi:A\to
B:=A/\mathfrak{k}$ is the quotient map.

###### Proof.

Since $\phi:A\to B$ is surjective and covers the identity,
$\operatorname{d}_{a}\phi:T_{a}A\to T_{\phi(a)}B$ is surjective, for any $a\in
A$. Also, if $(a,\gamma)\in A\times_{M}\mathfrak{k}$ then
$\phi(a+t\gamma)=\phi(a)$ so under the natural inclusion (5.2)
$A\times_{M}\mathfrak{k}$ is mapped to the kernel of $\operatorname{d}\phi$.
Since

$\operatorname{rank}_{A}TA-\operatorname{rank}_{B}TB=\operatorname{rank}_{M}\mathfrak{k}=\operatorname{rank}_{A}(A\times_{M}\mathfrak{k}),$

the result follows. ∎

Hence, we can rephrase Definition 5.1 in terms of bundle of ideals as follows:

######  Definition 5.3.

A bundle of ideals $\mathfrak{k}\subset A$ is called partially split if there
is a VB subalgebroid $E\Rightarrow TM$ of $TA\Rightarrow TM$ such that:

$TA=(A\times_{M}\mathfrak{k})\oplus E.$

We emphasize that, unlike the groupoid case, IM Ehresmann connections for a
surjective Lie algebroid morphism and partially split bundle of ideals are
equivalent notions. From now on we will mostly assume the perspective of
bundle of ideals.

###  5.3. The infinitesimal partially split condition

We now look for alternative characterizations of (infinitesimal) partially
split ideals.

If we represent the inclusion (5.2) by the diagram of VB algebroids:

---
$\textstyle{A\times_{M}\mathfrak{k}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{M}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

then the VB dual to this inclusion is the projection:

---
$\textstyle{A\ltimes\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

Here $A\ltimes\mathfrak{k}^{*}\Rightarrow\mathfrak{k}^{*}$ is the action
algebroid associated with the representation $\nabla^{\mathfrak{k}^{*}}$ of
$A$ on $\mathfrak{k}^{*}$ dual to the representation (5.1), which is defined
by:

$\mathscr{L}_{\rho_{A}(\alpha)}(\langle\beta,X\rangle)=\langle\nabla^{\mathfrak{k}}_{\alpha}\beta,X\rangle+\langle\beta,\nabla^{\mathfrak{k}^{*}}_{\alpha}X\rangle.$

We have the following infinitesimal analogue of Proposition 2.8 giving
alternative characterizations of partial splittings (for the terminology, see
the appendix):

######  Proposition 5.4.

Given a bundle of ideals $\mathfrak{k}\subset A$, the following structures are
in 1-to-1 correspondence:

  1. (i)

VB subalgebroids $E\subset TA$ that are complementary to
$A\times_{M}\mathfrak{k}$:

$TA=(A\times_{M}\mathfrak{k})\oplus E;$

  2. (ii)

VB algebroid morphisms $\theta:A\ltimes\mathfrak{k}^{*}\to T^{*}A$ that are
splittings of the natural projection $p:T^{*}A\to A\ltimes\mathfrak{k}^{*}$:

$p\circ\,\theta=\mathrm{Id}_{T^{*}A};$

  3. (iii)

$\mathfrak{k}$-valued, IM 1-forms
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$ that restrict to the
identity on $\mathfrak{k}$:

$l|_{\mathfrak{k}}=\mathrm{Id}_{\mathfrak{k}};$

  4. (iv)

linear, closed, IM 2-forms
$\mu\in\Omega^{2}_{\mathrm{IM}}(A\ltimes\mathfrak{k}^{*})$ that along
$\mathfrak{k}\subset(A\ltimes\mathfrak{k}^{*})|_{M}$ satisfy:

$\big{(}\operatorname{pr}_{\mathfrak{k}}\circ\mu|_{M}\big{)}|_{\mathfrak{k}}=\mathrm{Id}_{\mathfrak{k}}.$

###### Proof.

The equivalence between (i) and (ii) follows by passing from the VB algebroid
$TA$ to its dual VB algebroid $T^{*}A$, as explained above.

A vector bundle map

---
$\textstyle{A\ltimes\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{T^{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Id}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

is a VB algebroid morphism covering the identity if and only if its dual is a
VB algebroid morphism:

---
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\vee}}$$\textstyle{A\oplus\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Id}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

But a Lie algebroid morphism $\theta^{\vee}:TA\to A\oplus\mathfrak{k}$ is the
same thing as an $\mathfrak{k}$-valued, IM 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$ (see Example A.5). This
establishes the equivalence between (ii) and (iii), since the extra conditions
along $M$ correspond to each other.

Next, a closed IM 2-form $\mu:A\ltimes\mathfrak{k}^{*}\to
T^{*}\mathfrak{k}^{*}$ can be viewed, alternatively, as a VB algebroid
morphism:

$\textstyle{T(A\ltimes\mathfrak{k}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\omega_{\mu})^{\flat}}$$\textstyle{T^{*}(A\ltimes\mathfrak{k}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(A\ltimes\mathfrak{k}^{*})^{*}}$

where $\omega_{\mu}\in\Omega^{2}(A\ltimes\mathfrak{k}^{*})$ is the fiberwise
linear 2-form:

$\omega_{\mu}=\mu^{*}\omega_{\mathrm{can}}.$

The algebroid morphism $(\omega_{\mu})^{\flat}$ restricts to a VB algebroid
morphism:

$\theta^{\vee}:TA\to A\oplus\mathfrak{k}$

covering the identity $\mathrm{Id}:A\to A$, such that the following diagram
commutes:

---
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\vee}}$$\textstyle{A\oplus\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(A\ltimes\mathfrak{k}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\omega_{\mu})^{\flat}}$$\textstyle{T^{*}(A\ltimes\mathfrak{k}^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\mathfrak{k}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(A\ltimes\mathfrak{k}^{*})^{*}}$

Moreover, $\theta^{\vee}$ (or equivalently $\theta$) completely determines
$\mu$: using the fact that $\mu$ is a linear IM form, one finds that it
coincides with the pullback under the algebroid morphism
$\theta:A\ltimes\mathfrak{k}^{*}\to T^{*}A$ of the canonical IM 2-form
$\mu_{\mathrm{can}}$:

$\mu=\theta^{*}\mu_{\mathrm{can}}.$

This proves the equivalence between (ii)-(iv) since the additional conditions
along $M$ correspond to each other. ∎

######  Definition 5.5.

Given a partially split bundle of ideals $\mathfrak{k}$ with an IM Ehresmann
connection $E\subset TA$, the corresponding $\mathfrak{k}$-valued IM 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$, given by Proposition 5.4
(iii), is called the IM connection 1-form.

We can now make precise the statement that IM Ehresmann connections are the
infinitesimal counterpart of multiplicative Ehresmann connections:

######  Theorem 5.6.

Let $\mathcal{G}\rightrightarrows M$ be a target 1-connected Lie groupoid with
Lie algebroid $A\Rightarrow M$. Given a bundle of ideals $\mathfrak{k}\subset
A$ there is a 1:1 correspondence:

$\left\\{\hbox{\hbox{\kern
0.0pt\raise-12.48547pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
multiplicative Ehresmann}\\\\\hbox{{\ignorespaces connections $E\subset
T\mathcal{G}$\, }\\\\\hbox{{\ignorespaces for
$\mathcal{G}\times_{M}\mathfrak{k}$}\crcr}}}}\right\\}\tilde{\longleftrightarrow}\left\\{\hbox{\hbox{\kern
0.0pt\raise-11.96916pt\vbox{\halign{\relax\hfil\txtline@@{#}\hfil\cr\hbox{{\ignorespaces\leavevmode
IM Ehresmann}\\\\\hbox{{\ignorespaces connections $E_{*}\subset TA$\,
}\\\\\hbox{{\ignorespaces for $A\times_{M}\mathfrak{k}$
}\crcr}}}}\right\\}}}}}}}$

###### Proof.

The proof follows from the description of multiplicative and IM Ehresmann
connections in terms of their connections 1-forms given by Propositions 2.8
and 5.4, and by observing that for a target 1-connected Lie groupoid one has a
canonical isomorphism [14]

$\Omega^{1}_{\mathrm{M}}(\mathcal{G};\mathfrak{k})\simeq\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k}),$

under which the extra conditions on the 1-forms correspond to each other. ∎

If $\mathcal{G}$ is not target 1-connected the differentiation of
multiplicative forms still exists, and one can still associate to a
multiplicative Ehresmann connection a IM Ehresmann connection, but not
conversely.

Note that for a IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$, the symbol
$l:A\to\mathfrak{k}$ satisfies $l|_{\mathfrak{k}}=\mathrm{Id}$ and hence gives
a splitting of the short exact sequence:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
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0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.2778pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.2778pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
140.3646pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
140.3646pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

In general, the induced splitting $B\to A$ is not a Lie algebroid morphism.

The partially split condition implies certain properties of the bundle of
ideals. A first consequence is the following infinitesimal versions of
Corollaries 2.12 and 2.13:

######  Corollary 5.7.

Let $\mathfrak{k}\subset A$ be a partially split bundle of ideals with IM
connection 1-form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$, then

$\nabla^{L}_{X}\xi:=i_{X}L(\xi),$ (5.3)

defines a connection on $\mathfrak{k}$ which preserves the Lie bracket:

$\nabla^{L}_{X}[\xi,\eta]_{\mathfrak{k}}=[\nabla^{L}_{X}\xi,\eta]_{\mathfrak{k}}+[\xi,\nabla^{L}_{X}\eta]_{\mathfrak{k}},\quad
X\in\mathfrak{X}(S),\xi,\eta\in\Gamma(\mathfrak{k}).$

In particular, $\mathfrak{k}$ is a locally trivial bundle of Lie algebras.

###### Proof.

Since $l|_{\mathfrak{k}}=\mathrm{Id}$, the symbol equation (A.7) implies that
(5.3) defines a linear connection $\nabla^{L}$ on $\mathfrak{k}$. On the other
hand, the third equation in the IM condition (A.2) shows that one has

$\mathscr{L}_{\xi}\eta=[\xi,\eta],\quad\xi,\eta\in\Gamma(\mathfrak{k}).$

Then the second equation in (A.2) shows that $\nabla^{L}$ preserves the Lie
bracket. ∎

######  Corollary 5.8.

Let $\mathfrak{k}\subset A$ be a partially split bundle of ideals and fix
$x\in M$. The symbol of any IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$ gives a decomposition of
the isotropy Lie algebra $\mathfrak{g}_{x}:=\ker\rho_{A}|_{x}$ into a direct
sum of ideals:

$\mathfrak{g}_{x}\simeq(\mathfrak{g}_{x}\cap\ker l)\oplus\mathfrak{k}_{x}.$

###### Proof.

If $\alpha\in\Gamma(A)$ and $\beta\in\Gamma(\ker l)$ are such that
$\alpha_{x},\beta_{x}\in\mathfrak{g}_{x}$, then the third equation in the IM
condition (A.2) shows that

$[\alpha_{x},\beta_{x}]=[\alpha,\beta]|_{x}\in\ker l.$

This proves that $\mathfrak{g}_{x}\cap\ker l$ is an ideal in
$\mathfrak{g}_{x}$, so the corollary follows. ∎

An interesting characterization of the partial split condition can be obtained
as follows. Given a (usual) connection $\nabla$ on a Lie algebroid
$A\Rightarrow M$ one has the following associated $A$-connections on $A$ and
$TM$:

$\displaystyle\overline{\nabla}_{\alpha}\beta$
$\displaystyle:=\nabla_{\rho(\beta)}\alpha+[\alpha,\beta],$
$\displaystyle\overline{\nabla}_{\alpha}X$
$\displaystyle:=\rho(\nabla_{X}\alpha)+[\rho(\alpha),X]$

They satisfy

$\rho(\overline{\nabla}_{\alpha}\beta)=\overline{\nabla}_{\alpha}\rho(\beta).$

One defines the basic curvature of $\nabla$ to be the tensor

$R^{\mathrm{bas}}_{\nabla}(\alpha,\beta)(X):=\nabla_{X}([\alpha,\beta])-[\nabla_{X}\alpha,\beta]-[\alpha,\nabla_{X}\beta]-\nabla_{\overline{\nabla}_{\beta}X}\alpha+\nabla_{\overline{\nabla}_{\alpha}X}\beta,$
(5.4)

where $X\in\mathfrak{X}(M)$ and $\alpha,\beta\in\Gamma(A)$. When
$R^{\mathrm{bas}}_{\nabla}\equiv 0$ one calls $\nabla$ a Cartan connection and
this is precisely the infinitesimal version of a Cartan connection on a Lie
groupoid (see, e.g., [1, 3, 4]).

######  Proposition 5.9.

A bundle of ideals $\mathfrak{k}\subset A$ is partially split if and only if
there is a vector bundle splitting $l:A\to\mathfrak{k}$ of the associated
short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and a (usual) connection $\nabla$ on $A$ such that:

$\overline{\nabla}l=0,\qquad l(R^{\mathrm{bas}}_{\nabla})=0.$ (5.5)

In this case, the pair $(l,\nabla)$ determines a IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$ by setting:

$L:\Gamma(A)\to\Omega^{1}(M;\mathfrak{k}),\quad
i_{X}L(\alpha):=l(\nabla_{X}\alpha).$

Moreover, every IM connection 1-form $(L,l)$ takes this form for some pair
$(l,\nabla)$ satisfying (5.5).

###### Proof.

Let $l:A\to\mathfrak{k}$ be any splitting of the short exact sequence. Given a
map $L:\Gamma(A)\to\Omega^{1}(M;\mathfrak{k})$, we claim that the pair $(L,l)$
satisfies the IM-condition (A.7):

$L(f\alpha)=fL(\alpha)+\operatorname{d}f\wedge l(\alpha)$

if and only if

$i_{X}L(\alpha)=l(\nabla_{X}\alpha)$

for some connection $\nabla$ in $A$. Clearly, if $L$ takes this form then
(A.7) holds. Conversely, given $(L,l)$ satisfying (A.7) choose any connection
$\nabla^{\prime}$ on $A$ and consider the difference:

$D(\alpha,X):=i_{X}L(\alpha)-l(\nabla^{\prime}_{X}\alpha).$

This is $C^{\infty}$-linear in both entries and takes values in
$\mathfrak{k}\subset A$. Therefore, we can correct the connection
$\nabla^{\prime}$ by setting:

$\nabla_{X}\alpha:=\nabla^{\prime}_{X}\alpha+D(\alpha,X),$

so that $(L,l)$ takes the desired form.

It remains to check that a pair $(L,l)$ defined by $(l,\nabla)$ satisfies the
IM-conditions (A.2) if and only if (5.5) hold. Applying the definition of
$\overline{\nabla}$, we obtain:

$\displaystyle(\overline{\nabla}_{\alpha}l)(\beta)$
$\displaystyle=\overline{\nabla}_{\alpha}(l(\beta))-l(\overline{\nabla}_{\alpha}\beta)$
$\displaystyle=[\alpha,l(\beta)]-l(\nabla_{\rho(\beta)}\alpha+[\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\alpha}l(\beta)-i_{\rho(\beta)}L(\alpha)-l([\alpha,\beta]),$

for any $\alpha,\beta\in\Gamma(A)$. This shows that the condition
$\overline{\nabla}l=0$ amounts to the last IM condition in (A.2). It also
shows that this condition can be written as:

$l([\alpha,\beta])=[\alpha,l(\beta)]+l(\nabla_{\rho(\beta)}\alpha),$

for any $\alpha,\beta\in\Gamma(A)$. Using this last identity, the definition
of $L$ and the expression for the basic curvature, we now find:

$\displaystyle i_{X}L([\alpha,$
$\displaystyle\beta])-l(R^{\mathrm{bas}}_{\nabla}(\alpha,\beta)(X))=$
$\displaystyle=l(\nabla_{X}[\alpha,\beta])-l(R^{\mathrm{bas}}_{\nabla}(\alpha,\beta)(X))$
$\displaystyle=l([\nabla_{X}\alpha,\beta])+l([\alpha,\nabla_{X}\beta])+l(\nabla_{\overline{\nabla}_{\beta}X}\alpha)-l(\nabla_{\overline{\nabla}_{\alpha}X}\beta)$
$\displaystyle=l([\nabla_{X}\alpha,\beta])+l([\alpha,\nabla_{X}\beta])+l(\nabla_{\rho(\nabla_{X}\beta)+[\rho(\beta),X]}\alpha)-l(\nabla_{\rho(\nabla_{X}\alpha)+[\rho(\alpha),X]}\beta)$
$\displaystyle=[\alpha,l(\nabla_{X}\beta)]-l(\nabla_{[\rho(\alpha),X]}\beta)-[\beta,l(\nabla_{X}\alpha)]+l(\nabla_{[\rho(\beta),X]}\alpha)$
$\displaystyle=i_{X}\big{(}\mathscr{L}_{\alpha}L(\beta)-\mathscr{L}_{\beta}L(\alpha)\big{)}.$

So the second IM condition in (A.2) holds if and only if
$l(R^{\mathrm{bas}}_{\nabla})=0$. ∎

######  Remark 5.10.

The connection $\nabla$ in the previous proposition is not uniquely determined
by the partial IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$.

###  5.4. Couplings

In this section we give a coupling description of algebroids with a partially
split bundle of ideals. This is inspired by (and in fact generalizes) the
classical coupling description of symplectic fibrations [28] and horizontally
non-degenerate Poisson structures [33, 34].

To state the main result in this section we consider a partially split bundle
of ideals $\mathfrak{k}\subset A$ with a fixed choice of IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$. The base map gives a
vector bundle splitting of the associated short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\scriptstyle{\phi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

which we use to identify $A\simeq B\oplus\mathfrak{k}$. Then we associate the
following data:

  1. (i)

A linear connection $\nabla^{L}$ on the vector bundle $\mathfrak{k}\to M$, by
setting:

$\nabla^{L}_{X}\xi:=i_{X}L(\xi).$

  2. (ii)

A tensor $U\in\Gamma(B^{*}\otimes T^{*}M\otimes\mathfrak{k})$ given by:

$U(\alpha,X):=-i_{X}L(\alpha).$

That this is a connection and a tensor follows from the symbol equation (A.7)
and the fact that $l|_{\mathfrak{k}}=\mathrm{Id}$. Notice that the connection
$\nabla^{L}$ has already appeared in Corollary 5.7.

######  Proposition 5.11.

The data $(\nabla^{L},U)$ associated with a IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$ satisfies the structure
equations:

  1. (S1)

the connection $\nabla^{L}$ preserves the Lie bracket
$[\cdot,\cdot]_{\mathfrak{k}}$, i.e.,

$\nabla^{L}_{X}[\xi,\eta]_{\mathfrak{k}}=[\nabla^{L}_{X}\xi,\eta]_{\mathfrak{k}}+[\xi,\nabla^{L}_{X}\eta]_{\mathfrak{k}};$

  2. (S2)

the curvature of $\nabla^{L}$ is related to $\mathrm{ad}_{U}$ as follows:

$\nabla^{L}_{\rho_{B}(\alpha)}\nabla^{L}_{X}-\nabla^{L}_{X}\nabla^{L}_{\rho_{B}(\alpha)}-\nabla^{L}_{[\rho_{B}(\alpha),X]}=[U(\alpha,X),\cdot]_{\mathfrak{k}};$

  3. (S3)

$U$ satisfies the “mixed” cocyle-type equation:

$\displaystyle\nabla^{L}_{\rho_{B}(\alpha)}U(\beta,X)$
$\displaystyle-\nabla^{L}_{\rho_{B}(\beta)}U(\alpha,X)+\nabla^{L}_{X}U(\alpha,\rho_{B}(\beta))$
$\displaystyle+U(\alpha,[\rho_{B}(\beta),X])-U(\beta,[\rho_{B}(\alpha),X])=U([\alpha,\beta]_{B},X),$

for all $X\in\mathfrak{X}^{1}(M)$, $\alpha,\beta\in\Gamma(B)$,
$\xi,\eta\in\Gamma(\mathfrak{k})$.

The proof of this proposition follows immediately from the IM conditions (A.2)
and is left to the reader.

######  Definition 5.12.

Let $B\Rightarrow M$ be a Lie algebroid and
$(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})$ a Lie algebra bundle over $M$. A
coupling data is a pair $(\nabla^{L},U)$, where $\nabla^{L}$ is a connection
on $\mathfrak{k}$ and $U\in\Gamma(B^{*}\otimes T^{*}M\otimes\mathfrak{k})$ is
a tensor field satisfying the structure equations (S1), (S2) and (S3).

The following result shows that the coupling data gives a way of codifying
algebroids with a partially split bundle of ideals.

######  Proposition 5.13.

Let $B\Rightarrow M$ be a Lie algebroid,
$(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})$ a Lie algebra bundle over $M$
and $(U,\nabla^{L})$ coupling data. Then $A:=B\oplus\mathfrak{k}$ is a Lie
algebroid, with anchor $\rho_{A}(\alpha,\xi):=\rho_{B}(\alpha)$ and Lie
bracket:

$[(\alpha,\xi),(\beta,\eta)]_{A}:=([\alpha,\beta]_{B},U(\alpha,\rho_{B}(\beta))+\nabla^{L}_{\rho_{B}(\alpha)}\eta-\nabla^{L}_{\rho_{B}(\beta)}\xi+[\xi,\eta]_{\mathfrak{k}}).$
(5.6)

Then $\mathfrak{k}\subset A$ is partially split, with IM connection 1-form
$(L,l)$ given by:

$l=\mathrm{pr}_{\mathfrak{k}},\quad
i_{X}L(\alpha,\xi)=\nabla^{L}_{X}\xi-U(\alpha,X).$ (5.7)

Moreover, any Lie algebroid $A\Rightarrow M$ together with a partially split
bundle of ideals $\mathfrak{k}\subset A$ is isomorphic to one of this type.

###### Proof.

Assume we are given coupling data $(U,\nabla^{L})$. The skew-symmetrization
with respect to $\alpha$ and $\beta$ of (S3) gives:

$\nabla^{L}_{X}\big{(}U(\alpha,\rho_{B}(\beta))+U(\beta,\rho_{B}(\alpha))\big{)}=0,$

which yields:

$U(\alpha,\rho_{B}(\beta))=-U(\beta,\rho_{B}(\alpha)).$ (5.8)

Hence (5.6) defines a skew-symmetric bracket. It is immediate to check that it
satisfies the Leibniz identity and that its Jacobiator is given by:

$\displaystyle J((0,\xi),(0,\eta),(0,\nu))$
$\displaystyle=(0,J_{\mathfrak{k}}(\alpha,\eta,\nu))$ $\displaystyle
J((\alpha,0),(0,\xi),(0,\eta))$ $\displaystyle=(0,C_{1}(\alpha,\xi,\eta))$
$\displaystyle J((\alpha,0),(\beta,0),(0,\xi))$
$\displaystyle=(0,C_{2}(\alpha,\beta,\xi)$ $\displaystyle
J((\alpha,0),(\beta,0),(\gamma,0))$
$\displaystyle=(J_{B}(\alpha,\beta,\gamma),C_{3}(\alpha,\beta,\gamma))$

where $J_{\mathfrak{k}}$ and $J_{B}$ are the Jacobiators of the brackets
$[\cdot,\cdot]_{\mathfrak{k}}$ and $[\cdot,\cdot]_{B}$ and

$\displaystyle C_{1}(\alpha,\xi,\eta)$
$\displaystyle:=\nabla^{L}_{\rho_{B}(\alpha)}[\xi,\eta]_{\mathfrak{k}}-[\nabla^{L}_{\rho_{B}(\alpha)}\xi,\eta]_{\mathfrak{k}}-[\xi,\nabla^{L}_{\rho_{B}(\alpha)}\eta]_{\mathfrak{k}}$
$\displaystyle C_{2}(\alpha,\beta,\xi)$
$\displaystyle:=\nabla^{L}_{\rho_{B}(\alpha)}\nabla^{L}_{\rho_{B}(\beta)}\xi-\nabla^{L}_{\rho_{B}(\beta)}\nabla^{L}_{\rho_{B}(\alpha)}\xi-\nabla^{L}_{[\rho_{B}(\alpha),\rho_{B}(\beta)]}\xi-[U(\alpha,\rho_{B}(\beta)),\xi]_{\mathfrak{k}}$
$\displaystyle C_{3}(\alpha,\beta,\gamma)$
$\displaystyle:=\nabla^{L}_{\rho_{B}(\alpha)}U(\beta,\rho_{B}(\gamma))+U(\alpha,[\rho_{B}(\beta),\rho_{B}(\gamma)])-\nabla^{L}_{\rho_{B}(\beta)}U(\alpha,\rho_{B}(\gamma))$
$\displaystyle\qquad\quad-U(\beta,[\rho_{B}(\alpha),\rho_{B}(\gamma)])+\nabla^{L}_{\rho_{B}(\gamma)}U(\alpha,\rho_{B}(\beta))-U([\alpha,\beta]_{B},\rho_{B}(\gamma))$

If in the structure equations (S1), (S2) and (S3) we replace $X$ by
$\rho_{B}(\alpha)$, $\rho_{B}(\beta)$ and $\rho_{B}(\gamma)$, respectively, we
obtain that these 3 expressions vanish. Hence, the bracket (5.7) satisfies the
Jacobi identity, so we obtain a Lie algebroid. If we now define
$l:A\to\mathfrak{k}$ and $L:\Gamma(A)\to\Omega^{1}(M,\mathfrak{k})$ by (5.7),
one checks that the symbol equation (A.7) holds and that the IM conditions
(A.2) are equivalent to (S1)-(S3).

Finally, if we start with a partially split bundle of ideals
$\mathfrak{k}\subset A$, as we saw above, a choice of IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$ determines coupling data
satisfying the structure equations and a vector bundle isomorphism $A\simeq
B\oplus\mathfrak{k}$ where $B\simeq\ker l$. The anchor of $A$ is then given by
$\rho_{A}(\alpha,\xi):=\rho_{B}(\alpha)$ and all we have to check is that,
under this isomorphism, the Lie bracket becomes (5.7). For this we use the
third equation in the IM condition (A.2) for $(L,l)$ and the definitions of
$\nabla^{L}$ and $U$ to conclude that:

  * $\bullet$

If $\alpha\in\Gamma(B)\simeq\Gamma(\ker l)$ and $\xi\in\Gamma(\mathfrak{k})$
then:

$[\alpha,\xi]_{A}=\nabla^{L}_{\rho_{B}(\alpha)}\xi;$

  * $\bullet$

If $\alpha,\beta\in\Gamma(B)\simeq\Gamma(\ker l)$ then:

$[\alpha,\beta]_{A}=[\alpha,\beta]_{B}+U(\alpha,\rho_{B}(\beta)).$

Formula (5.7) for the bracket follows from these since $\mathfrak{k}$ is a
bundle of ideals. ∎

###  5.5. Flat IM Ehresmann connections

Let $\mathfrak{k}\subset A$ be a bundle of ideals and let $B=A/\mathfrak{k}$.
As we saw in Section 5.4, a choice of IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$ gives rise to the coupling
data $(\nabla^{L},U)$, where $U\in\Gamma(B^{*}\otimes
T^{*}M\otimes\mathfrak{k})$ is a tensor field and $\nabla^{L}$ is a usual
connection on $\mathfrak{k}$. Using Proposition 5.13, it is not hard to check
that we have a $\mathfrak{k}$-valued IM 2-form
$(\tilde{L},\tilde{l})\in\Omega^{2}_{\mathrm{IM}}(A,\mathfrak{k})$, where:

$\tilde{L}(\alpha,\xi)=R^{\nabla^{L}}\cdot\xi-\operatorname{d}^{\nabla^{L}}U(\alpha),\qquad\tilde{l}(\alpha,\xi)=-U(\alpha),$
(5.9)

for $\alpha\in\Gamma(B)$ and $\xi\in\Gamma(\mathfrak{k})$. This is the
infinitesimal version of the curvature of the multiplicative Ehresmann
connection from Proposition 2.22. Namely, in the case where $\phi:A\to B$ is
induced by a groupoid morphism $\Phi:\mathcal{G}\to\mathcal{H}$ and the IM
Ehresmann connection is induced by a multiplicative Ehresmann $E$ connection,
(5.9) is the IM form corresponding to curvature 2-form $\Omega$ of $E$.

######  Definition 5.14.

An IM Ehresmann connection $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$
with associated coupling data $(\nabla^{L},U)$ is called:

  1. (i)

totally flat if $R^{\nabla^{L}}=0$ and $U=0$;

  2. (ii)

leafwise flat if the splitting $B\to A$, $\alpha\mapsto(\alpha,0)$, is a Lie
algebroid map, i.e., $U(\alpha,\rho_{B}(\beta))=0$;

  3. (iii)

kernel flat if the associated linear connection $\nabla^{L}$ is flat:
$R^{\nabla^{L}}=0$.

The first condition is equivalent to involutivity of the distribution
$E\subset TA$ corresponding to the IM Ehresmann connection. This condition is
very restrictive, as the following infinitesimal analogue of Proposition 2.42
shows:

######  Proposition 5.15.

If a bundle of ideals $\mathfrak{k}\subset A$ admits a totally flat IM
Ehresmann connection, then there is a covering space of the base
$p:\widetilde{M}\to M$ such that the pullback of $A$ to $\widetilde{M}$ is
isomorphic to a product $p^{*}A\simeq p^{*}B\times\mathfrak{g}$, where
$p^{*}\mathfrak{k}\simeq\widetilde{M}\times\mathfrak{g}$.

###### Proof.

Let $(\nabla^{L},U=0)$ be a totally flat coupling data. Let
$p:\widetilde{M}\to M$ denote the holonomy cover of $\nabla^{L}$. Parallel
transport along $\nabla^{L}$ induces a trivialization of the bundle of Lie
algebras $p^{*}\mathfrak{k}\simeq\widetilde{M}\times\mathfrak{g}$, where
$\mathfrak{g}=\mathfrak{k}_{x}$, for some $x\in M$. Since $\nabla^{L}$ becomes
the trivial connection on $\widetilde{M}\times\mathfrak{g}$, formulas (5.6)
show that the algebroid $p^{*}A$ is a product (see also the example from
Subsection 6.2). ∎

The second flatness condition can be understood as a leafwise version of the
first:

######  Proposition 5.16.

Given a bundle of ideals $\mathfrak{k}\subset A$, an IM Ehresmann connection
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$ is leafwise flat if and
only if, for any leaf $\mathcal{O}\subset M$ of the Lie algebroid $A$, the
pullback IM Ehresmann connection
$(L,l)|_{\mathcal{O}}\in\Omega^{1}_{\mathrm{IM}}(A|_{\mathcal{O}};\mathfrak{k}|_{\mathcal{O}})$
is totally flat.

###### Proof.

Given a leaf $\mathcal{O}$ of $A$, the coupling data associated to the
pullback of $(L,l)$ to $A|_{\mathcal{O}}$ is
$(\nabla^{\mathcal{O}},U^{\mathcal{O}})$, where $\nabla^{\mathcal{O}}$ is the
pullback of the connection $\nabla^{L}$ to $\mathfrak{k}|_{\mathcal{O}}$ and
$U^{\mathcal{O}}$ is the restriction of $U$ to $B|_{\mathcal{O}}\otimes
T\mathcal{O}$. Now, $(L,l)$ being leafwise flat, i.e.,

$U(\alpha,\rho_{B}(\beta))=0,\quad\textrm{for all}\ \alpha,\beta\in B,$

is clearly equivalent to $U^{\mathcal{O}}=0$ for all leaves
$\mathcal{O}\subset M$. On the other hand, using that $A|_{\mathcal{O}}$ is
transitive and (S2), we see that $U^{\mathcal{O}}=0$ implies that
$\nabla^{\mathcal{O}}$ is flat. ∎

Note that a leafwise flat IM Ehresmann connection turns $A$ into a semi-direct
product, in the sense of the following.

Let $B\Rightarrow M$ be a Lie algebroid which acts on a bundle of Lie algebras
$(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})\to M$ by infinitesimal Lie
algebra automorphisms. The semi-direct product is the Lie algebroid
$A:=B\times_{M}\mathfrak{k}\Rightarrow M$ with Lie bracket:

$[(\alpha,\xi),(\beta,\eta)]=([\alpha,\beta],\nabla^{\mathfrak{k}}_{\alpha}\eta-\nabla^{\mathfrak{k}}_{\beta}\xi+[\xi,\eta]_{\mathfrak{k}})$

and anchor $\rho_{B}\circ\operatorname{pr}_{B}$. Note that the short exact
sequence

${\mathfrak{k}}$${B\times_{M}\mathfrak{k}}$${B}$$\scriptstyle{\operatorname{pr}_{1}}$

is canonically split by a Lie algebroid morphism:

$\sigma:B\longrightarrow
B\times_{M}\mathfrak{k},\quad\alpha\mapsto(\alpha,0).$

Conversely, given a surjective Lie algebroid map $\phi:A\to B$ covering the
identity, which admits a Lie algebroid splitting:

${\mathfrak{k}}$${A}$${B}$$\scriptstyle{\phi}$$\scriptstyle{\sigma}$

we obtain an action of $B\Rightarrow M$ on the bundle of Lie algebras
$(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})\to M$:

$\nabla_{\alpha}^{\mathfrak{k}}\xi:=[\sigma(\alpha),\xi]_{A},$

preserving the Lie algebra structure, and a Lie algebroid isomorphism:

$B\times_{M}\mathfrak{k}\simeq A,\quad(\alpha,\xi)\mapsto\sigma(\alpha)+\xi.$

Notice that the Lie algebroid splitting $\sigma:B\to A$ allows to pull back
$\mathfrak{k}$-valued IM forms on $A$ to IM forms on $B$:

$\sigma^{*}:\Omega_{\mathrm{IM}}^{\bullet}(A;\mathfrak{k})\longrightarrow\Omega_{\mathrm{IM}}^{\bullet}(B;\mathfrak{k}).$

Therefore we conclude the following:

######  Corollary 5.17.

Given a bundle of ideals $\mathfrak{k}\subset A$, with an IM Ehresmann
connection $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$ that is leafwise
flat, we obtain a IM 1-form
$(U,0)\in\Omega_{\mathrm{IM}}^{1}(B;\mathfrak{k})$.

The following infinitesimal version of Proposition 2.39 gives yet another
notion of flatness, weaker than totally flat and stronger than leafwise flat:

######  Proposition 5.18.

Let $\mathfrak{k}\subset A$ be a bundle of ideals with quotient
$B:=A/\mathfrak{k}$. The following are equivalent:

  1. (i)

There is an IM Ehresmann connection $(L,l)\in\Omega^{1}(A;\mathfrak{k})$ on
$A$ and $\operatorname{pr}:A\to B$ whose associated splitting $\sigma:B\to A$
is a horizontal algebroid splitting: i.e., $\sigma^{*}(L,l)=0$;

  2. (ii)

There exists an IM Ehresmann connection
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;\mathfrak{k})$ with coupling data
$(\nabla^{L},U=0)$;

  3. (iii)

$A$ is isomorphic to a semi-direct product $A\simeq B\times_{M}\mathfrak{k}$
and $\mathfrak{k}$ admits an $B$-invariant connection $\nabla^{L}$ preserving
the bracket $[\cdot,\cdot]_{\mathfrak{k}}$ (see Definition A.9).

###### Proof.

Note that $\sigma^{*}(L,l)=(U,0)$, so (i) and (ii) are equivalent.

If $U=0$, then the formula for the Lie bracket on $A$ (5.6) together with
(S1), (S2) are equivalent to $\nabla^{L}$ being a $B$ invariant connection on
$\mathfrak{k}$ which preserves $[\cdot,\cdot]_{\mathfrak{k}}$.

Conversely, if $A\simeq B\times_{M}\mathfrak{k}$, and the representation of
$B$ on $\mathfrak{k}$ admits a $B$-invariant connection $\nabla^{L}$ which
preserves $[\cdot,\cdot]_{\mathfrak{k}}$, then $(\nabla^{L},U=0)$ is a
coupling data for an IM Ehresmann connection. ∎

Finally, we look at the kernel flat IM Ehresmann connections. For this,
observe that the representation $\nabla^{\mathfrak{k}}$ of the Lie algebroid
$A$ restricts to an $A$-representation on the center $z(\mathfrak{k})$. While
the representation of $A$ on the whole $\mathfrak{k}$, in general, does not
factor to a representation of $B$, its restriction to the center
$z(\mathfrak{k})$ does factor. We still denote this representation by
$\nabla^{\mathfrak{k}}$. In terms of the coupling the representation of $B$ on
the center $z(\mathfrak{k})$ is given by:

$\nabla^{\mathfrak{k}}_{\alpha}\xi=\nabla^{L}_{\rho_{B}(\alpha)}\xi,$

for all $\alpha\in\Gamma(B)$, $\xi\in\Gamma(z(\mathfrak{k}))$. This follows
from the expression of the bracket given in Proposition 5.13.

In the kernel flat case, the structure equations (S1)-(S3) of Proposition 5.11
simplify to:

######  Proposition 5.19.

Let $\mathfrak{k}\subset A$ be a partially split bundle of ideals with
coupling data $(\nabla^{L},U)$. If $\nabla^{L}$ is flat, then:

  * (S1’)

The connection $\nabla^{L}$ preserves the bracket
$[\cdot,\cdot]_{\mathfrak{k}}$;

  * (S2’)

The tensor $U$ takes values in the center of $\mathfrak{k}$;

  * (S3’)

The pair $(\operatorname{d}^{\nabla^{L}}U,U)$ is a $z(\mathfrak{k})$-valued IM
2-form on $B$.

Conversely, given a Lie algebroid $B\Rightarrow M$, a Lie algebra bundle
$\mathfrak{k}\to M$ with a flat connection $\nabla^{L}$, and a tensor field
$U\in\Gamma(B^{*}\otimes T^{*}M\otimes\mathfrak{k})$ satisfying (S1’)-(S3’),
then $\mathfrak{k}$ is a partially split bundle of ideals in Lie algebroid
$A=B\oplus\mathfrak{k}$ with kernel flat coupling data $(\nabla^{L},U)$.

###### Proof.

If $\nabla^{L}$ is flat, it is obvious that (S1)-(S2) are equivalent to
(S1’)-(S2’).

Now consider the pair $(\operatorname{d}^{\nabla^{L}}U,U)$. The properties of
$\operatorname{d}^{\nabla^{L}}$ give immediately that:

$\operatorname{d}^{\nabla^{L}}U(f\alpha)=f\operatorname{d}^{\nabla^{L}}U(\alpha)+\operatorname{d}f\wedge
U(\alpha),$

so $U:B\to T^{*}M\otimes\mathfrak{k}$ is the symbol of
$\operatorname{d}^{\nabla^{L}}U:\Gamma(B)\to\Omega^{2}(M,\mathfrak{k})$, i.e.,
(A.7) holds. The skew-symmetry of $U$, given by (5.8), can be written as:

$i_{\rho_{B}(\beta)}U(\alpha)=-i_{\rho_{B}(\alpha)}U(\beta),$

which is the first equation in the IM conditions (A.9). Then (S3) can be
rewritten using the connection $\nabla^{L}$ as:

$U([\alpha,\beta]_{B})=\mathscr{L}^{\nabla^{L}}_{\rho_{B}(\alpha)}U(\beta)-i_{\rho_{B}(\beta)}\operatorname{d}^{\nabla^{L}}U(\alpha),$

and since $\mathscr{L}_{\alpha}=\mathscr{L}^{\nabla^{L}}_{\rho_{B}(\alpha)}$
this gives the third equation in the IM conditions (A.9).

Finally, applying $\operatorname{d}^{\nabla}$ to this last equation, using
that its square is zero and “Cartan’s magic formula” holds for
$\mathscr{L}^{\nabla^{L}}$, we obtain:

$\displaystyle\operatorname{d}^{\nabla^{L}}U([\alpha,\beta]_{B})$
$\displaystyle=\operatorname{d}^{\nabla^{L}}\mathscr{L}_{\alpha}U(\beta)-\operatorname{d}^{\nabla^{L}}i_{\rho_{B}(\beta)}\operatorname{d}^{\nabla^{L}}U(\alpha)$
$\displaystyle=\mathscr{L}_{\alpha}\operatorname{d}^{\nabla^{L}}U(\beta)-\mathscr{L}^{\nabla^{L}}_{\rho_{B}(\beta)}\operatorname{d}^{\nabla^{L}}U(\alpha)$
$\displaystyle=\mathscr{L}_{\alpha}\operatorname{d}^{\nabla^{L}}U(\beta)-\mathscr{L}_{\beta}\operatorname{d}^{\nabla^{L}}U(\alpha),$

which is the second equation in the IM conditions (A.9). Hence,
$(\operatorname{d}^{\nabla^{L}}U,U)$ is a $z(\mathfrak{k})$-valued IM 2-form
on $B$.

It should clear from this that the converse also holds: a pair
$(\nabla^{L},U)$, with $\nabla^{L}$ flat and satisfying (S1’)-(S3’), also
satisfies (S1)-(S3). Hence, the last part follows from Proposition 5.13. ∎

######  Remark 5.20.

For a kernel flat partial splitting of $\mathfrak{k}\subset A$, the
proposition yields the IM valued 2-form:

$(\operatorname{d}^{\nabla^{L}}U,U)\in\Omega^{2}_{\mathrm{IM}}(B;z(\mathfrak{k})),$
(5.10)

which is the infinitesimal analogue of the class from Proposition 2.29.
Namely, the IM form corresponding the multiplicative form $\underline{\Omega}$
is $(-\operatorname{d}^{\nabla^{L}}U,-U)$ – see (5.9).

Since $\nabla^{L}$ is flat, by Proposition A.10 we have an induced
differential:

$\operatorname{d}^{\nabla^{L}}_{\mathrm{IM}}:\Omega^{\bullet}_{\mathrm{IM}}(B;z(\mathfrak{k}))\longrightarrow\Omega^{\bullet+1}_{\mathrm{IM}}(B;z(\mathfrak{k})),$

for which the element (5.10) is clearly closed. Applying the chain map from
Lemma A.11 we obtain a $B$-cocycle with values in $z(\mathfrak{k})$:

$\lambda\in\Omega^{2}(B;z(\mathfrak{k})),\quad\lambda(\alpha,\beta):=U(\alpha,\rho_{B}(\beta)).$

That $\lambda$ is a cocycle can be seen directly also from (S3). Note that the
classes

$[(\operatorname{d}^{\nabla^{L}}U,U)]\in
H^{2}_{\mathrm{IM}}(B;z(\mathfrak{k}))\quad\textrm{and}\quad[\lambda]\in
H^{2}(B;z(\mathfrak{k}))$

depend on the choice of the IM connection.

The remark can be developed into a criterion for the existence of kernel flat
IM connection for abelian bundle of ideals $\mathfrak{k}\subset A$. In this
case, the canonical action of $B:=A/\mathfrak{k}$ is on
$\mathfrak{k}=z(\mathfrak{k})$ and, after choosing a splitting $A\simeq
B\oplus\mathfrak{k}$, the Lie bracket takes the form:

$[(\alpha,\xi),(\beta,\eta)]_{A}:=([\alpha,\beta]_{B},\lambda(\alpha,\beta)+\nabla^{\mathfrak{k}}_{\alpha}\eta-\nabla^{\mathfrak{k}}_{\beta}\xi).$

Then $\lambda\in\Omega^{2}(B;\mathfrak{k})$ is a 2-cocycle, whose cohomology
class:

$c_{2}(A):=[\lambda]\in H^{2}(B;\mathfrak{k})$

is independent of the splitting, and it determines the extension
$\mathfrak{k}\to A\to B$ up to isomorphism. We have the following:

######  Proposition 5.21.

An abelian bundle of ideals $\mathfrak{k}\subset A$, with $B:=A/\mathfrak{k}$,
admits a kernel flat partial splitting if and only if $\mathfrak{k}$ admits a
flat connection $\nabla$ inducing $\nabla^{\mathfrak{k}}$ and the class
$c_{2}(A)$ is in the image of the map (from Lemma A.11):

$H_{\mathrm{IM}}^{2}(B;\mathfrak{k})\to H^{2}(B;\mathfrak{k}).$

###### Proof.

Given a flat coupling data $(\nabla^{L},U)$, the connection $\nabla^{L}$
induces $\nabla^{\mathfrak{k}}$, and, as remarked, $c_{2}(A)$ is the image of
the class $[(\operatorname{d}^{\nabla^{L}}U,U)]\in
H_{\mathrm{IM}}^{2}(B;\mathfrak{k})$.

Conversely, assume one is given a flat connection $\nabla^{L}$ inducing
$\nabla^{\mathfrak{k}}$ and that $c_{2}(A)$ is the image of $c\in
H^{2}_{\mathrm{IM}}(B;\mathfrak{k})$. Note that any representative of $c$ if
of the form $(\operatorname{d}^{\nabla^{L}}U,U)$. Choose a splitting inducing

$\lambda(\alpha,\beta)=U(\alpha,\rho_{B}(\beta)).$

Since the pair $(\nabla^{L},U)$ satisfies (S1’)-(S3’), it follows from
Proposition 5.13 that it is the coupling data of a kernel flat partial
splitting for $A$. ∎

We also have the infinitesimal version of Corollary 2.41:

######  Corollary 5.22.

A semi-direct product $A=B\times_{M}\mathfrak{k}$, with $\mathfrak{k}$ abelian
is partially split if and only if the representation of $B$ on $\mathfrak{k}$
admits a $B$-invariant connection (in the sense of Definition A.9).

###### Proof.

If $\mathfrak{k}$ admits a $B$-invariant connection $\nabla^{L}$, then
$(\nabla^{L},U=0)$ is the coupling data for an IM connection on
$B\times_{M}\mathfrak{k}$.

Conversely, let $(L,l)\in\Omega^{1}(B\times_{M}\mathfrak{k};\mathfrak{k})$ be
an IM Ehresmann connection. Both the projection
$\phi:B\times_{M}\mathfrak{k}\to B$ and the inclusion $\sigma:B\to
B\times_{M}\mathfrak{k}$ are Lie algebroid morphisms which intertwine the
actions on $\mathfrak{k}$. Therefore, $\phi^{*}\circ\sigma^{*}(L,l)$ is an IM
1-form. By subtracting it from $(L,l)$, we may assume that
$\sigma^{*}(L,l)=0$. By Proposition 5.18, this is equivalent to $(L,l)$ having
coupling data $(\nabla^{L},U=0)$, with $\nabla^{L}$ a $B$-invariant connection
on $\mathfrak{k}$. ∎

##  6\. Examples and applications (algebroids)

Using the correspondence from Theorem 5.6, many of the examples that follow
can be seen as infinitesimal counterparts of the examples discussed in Section
3. Note, however, that we will make no integrability assumption, so often we
obtain more general classes of examples.

###  6.1. Lie algebra bundles

Let $A\to M$ be a Lie algebra bundle. A bundle of ideals $\mathfrak{k}\subset
A$ is not necessarily partially split. The obstructions are precisely those
found in Corollaries 5.7 and 5.8:

######  Proposition 6.1.

A bundle of ideals $\mathfrak{k}\subset A$ in a Lie algebra bundle is
partially split if and only if the following two conditions hold:

  1. (i)

$\mathfrak{k}$ is a Lie algebra bundle;

  2. (ii)

there exists a splitting $A\simeq B\oplus\mathfrak{k}$, for which the
fiberwise Lie bracket is a direct product.

###### Proof.

By Corollaries 5.7 and 5.8, a partial splitting for $\mathfrak{k}\subset A$
yields a (usual) connection $\nabla^{L}$ on $\mathfrak{k}$ which preserves the
Lie bracket and a splitting $A\simeq B\oplus\mathfrak{k}$, for which the
fiberwise Lie bracket is a direct product.

Conversely, fix a connection $\nabla^{L}$ on $\mathfrak{k}$ which preserves
the Lie bracket and a splitting $A\simeq B\oplus\mathfrak{k}$ for which the
Lie bracket is a direct product. Then we can obtain a partial splitting from
Proposition 5.11 by setting $U:=0$. ∎

######  Remark 6.2.

If $M$ is connected, then for a bundle of Lie algebras
$(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})\to M$ to be a Lie algebra bundle,
it suffices that:

$H^{2}(\mathfrak{k}_{x},\mathfrak{k}_{x})=0,\quad\forall\,x\in M.$

Indeed, this condition implies that each $\mathfrak{k}_{x}$ is a rigid Lie
algebra: any small deformation of the Lie bracket on $\mathfrak{k}_{x}$ is
isomorphic to $\mathfrak{k}_{x}$ (see, e.g., [13]). Therefore, all Lie
algebras in $\mathfrak{k}$ are isomorphic to each other and it follows that
$\mathfrak{k}$ is a bundle of Lie algebras.

###  6.2. Products

Consider the Lie algebroid $A\Rightarrow M$ obtained as the product of a Lie
algebroid $B\Rightarrow M$ and a Lie algebra $\mathfrak{g}$:

$A=B\times\mathfrak{g}.$

Then $\mathfrak{k}:=M\times\mathfrak{g}\subset A$ is a bundle of Lie algebras
which is always partially split with a canonical, totally flat partial
splitting. This follows by observing that we have canonical isomorphisms:

$TA=TB\times(\mathfrak{g}\ltimes\mathfrak{g}),\quad
A\times_{M}\mathfrak{k}=B\times(\mathfrak{g}\ltimes\mathfrak{g}),$

in terms of which the inclusion (5.2) is

$B\times(\mathfrak{g}\ltimes\mathfrak{g})\to
TB\times(\mathfrak{g}\ltimes\mathfrak{g}),\quad(b,v,w)\mapsto(0_{b},v,w).$

Hence, we have the canonical IM Ehresmann connection $E\subset TA$ given by:

$E_{(b,v)}:=\\{(X,v,0):X\in T_{b}(B_{x}),v\in\mathfrak{g}\\}.$

It is also easy to determine the alternative data of specifying this
connection given by Proposition 5.4, namely:

  * $\bullet$

The canonical splitting $\theta:A\ltimes\mathfrak{k}^{*}\to T^{*}A$:

$\theta:B\times(\mathfrak{g}\ltimes\mathfrak{g}^{*})\to
T^{*}(B)\times(\mathfrak{g}\ltimes\mathfrak{g}^{*}),\quad(b,v,\xi)\mapsto(0_{b},v,\xi).$

  * $\bullet$

The IM connection 1-form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$:

$\displaystyle l:B\times\mathfrak{g}\to M\times\mathfrak{g},\quad
L(b_{x},v)=(x,v),$ $\displaystyle L:\Gamma(B)\times
C^{\infty}(M;\mathfrak{g})\to\Omega^{1}(M,\mathfrak{g}),\quad
L(\alpha,f)=\operatorname{d}f;$

  * $\bullet$

The linear, closed, IM 2-form
$\mu\in\Omega^{2}_{\mathrm{IM}}(A\ltimes\mathfrak{k}^{*})$:

$\mu:B\times(\mathfrak{g}\ltimes\mathfrak{g}^{*})\to
T^{*}(M\times\mathfrak{g}^{*})=T^{*}M\times(\mathfrak{g}\ltimes\mathfrak{g}^{*}),\quad(\alpha,v,\xi)\mapsto(0,v,\xi).$

###  6.3. Transitive algebroids

Let $A\Rightarrow M$ be a transitive Lie algebroid. Its isotropy
$\mathfrak{k}=\ker\rho$ is a bundle of ideals. We claim that this is always
partially split, and there is a 1-to-1 correspondence between splittings of
the anchor:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{l}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{0}$

and IM connection 1-forms $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$.

Indeed, given a splitting $l$, we can define a linear operator
$L:\Gamma(A)\to\Omega^{1}(M,\mathfrak{k})$ by setting:

$i_{X}L(\alpha):=l([\tau(X),\alpha]).$ (6.1)

One checks easily that the pair $(L,l)$ satisfies (A.2), so it is a
$\mathfrak{k}$-valued, IM 1-form, with $l|_{\mathfrak{k}}=\mathrm{Id}$.

Conversely, given any IM connection 1-form
$(L^{\prime},l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$, the bundle map
$l:A\to\mathfrak{k}$ determines a splitting of the anchor. We claim that
$L^{\prime}=L$, where $L$ is given by (6.1). Note that the difference
$L-L^{\prime}$ is a $\mathfrak{k}$-valued, IM 1-form whose symbol vanishes. It
follows from the last equation in (A.2) that:

$i_{\rho(\beta)}(L(\alpha)-L^{\prime}(\alpha))=0,\quad\forall\beta\in A.$

Since $\rho$ is surjective, this means that $L=L^{\prime}$.

###  6.4. Cartan connections

Recall that a _Cartan connection_ is a connection $\nabla$ on a Lie algebroid
$A$ whose basic curvature vanishes identically:

$R^{\mathrm{bas}}_{\nabla}\equiv 0.$

Hence, as an immediate consequence of Proposition 5.9, we obtain:

######  Corollary 6.3.

Let $\mathfrak{k}\subset A$ be a bundle of ideals. If $A$ admits a Cartan
connection $\nabla$ and a splitting $l:A\to\mathfrak{k}$ which is
$\overline{\nabla}$-invariant, then $\mathfrak{k}$ is partially split. In
particular, this holds when $A$ admits a fiberwise metric that is
$\overline{\nabla}$-invariant.

###### Proof.

The second part of the statement follows by choosing the splitting
$l:A\to\mathfrak{k}$ to be the orthogonal projection relative to the invariant
metric. ∎

In Subsection 3.8 we saw that, for a Lie groupoid admitting a bi-invariant
metric, every bundle of ideals is partially split. At the Lie algebroid level,
the corresponding notion of a bi-invariant metric (see [24]) is given by a
Cartan connection and a pair of metrics $(\eta^{A},\eta^{M})$ on $A$ and $M$
satisfying:

$\overline{\nabla}\eta^{A}=0,\quad\overline{\nabla}\eta^{M}=0.$

The previous corollary shows that for a Lie algebroid $A$ admitting a bi-
invariant metric any bundle of ideals $\mathfrak{k}\subset A$ is partially
split. Notice that the metric $\eta^{M}$ plays here no role.

###  6.5. Action Lie algebroids

Let $A=\mathfrak{g}\ltimes M\Rightarrow M$ be the action Lie algebroid
associated with a Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}(M)$.
The canonical flat connection $\nabla$ on $A$ has vanishing basic curvature,
hence it is a Cartan connection. Given a bundle of ideals $\mathfrak{k}\subset
A$ a splitting $l:A\to\mathfrak{k}$ is $\overline{\nabla}$-invariant if and
only if it is $\mathfrak{g}$-equivariant:

$l([v,w]_{\mathfrak{g}})=[v,l(w)]_{\mathfrak{g}\ltimes M},$

for all $v,w\in\mathfrak{g}$ (here we identify elements of $\mathfrak{g}$ with
constant sections of $A$). Hence, we deduce from Corollary 6.3 the
infinitesimal version of Proposition 3.1:

######  Corollary 6.4.

A bundle of ideals on an action algebroid $\mathfrak{g}\ltimes M\Rightarrow M$
admitting a $\mathfrak{g}$-equivariant splitting is partially split.

For example, if $\mathfrak{g}$ is a Lie algebra of compact type then it admits
an $\operatorname{ad}$-invariant scalar product $\langle\cdot,\cdot\rangle$.
Such an inner product defines a fiberwise metric $\eta$ on $A$ which is
$\overline{\nabla}$-invariant. Hence, in this case, any bundle of ideals on
$A=\mathfrak{g}\ltimes M$ admits a $\mathfrak{g}$-equivariant splitting, so it
is partially split.

###  6.6. Infinitesimal principal type

Let $A^{\prime}\Rightarrow M$ be a transitive Lie algebroid and $B\Rightarrow
M$ any Lie algebroid. Consider the Lie algebroid fibre product:

$A:=A^{\prime}\times_{TM}B:=\\{(\alpha,\beta)\,:\,\rho_{A^{\prime}}(\alpha)=\rho_{B}(\beta)\\},$

where the structure is such that the inclusion in the product
$A\hookrightarrow A^{\prime}\times B$ is a Lie algebroid morphism. The
projection $\phi:=\operatorname{pr}_{B}:A\to B$ is a Lie algebroid morphism,
which is surjective because we assume $B$ to be transitive. The kernel
$\mathfrak{k}:=\ker\phi$ is then a bundle of ideals which can be identified
with $\ker\rho_{A^{\prime}}$ via $\operatorname{pr}_{A^{\prime}}$. A bundle of
ideals $\mathfrak{k}\subset A$ obtained via this construction will be called
of principal type. This is of course the infinitesimal counterpart of
groupoids with bundle of ideals of principal type, discussed in Subsection
3.5.

Since $A^{\prime}\Rightarrow M$ is transitive, as discussed in Subsection 6.3,
a splitting $l_{A^{\prime}}:A^{\prime}\to\mathfrak{k}$ of $\rho_{A^{\prime}}$
determines an IM connection 1-form
$(L_{A^{\prime}},l_{A^{\prime}})\in\Omega^{1}_{\mathrm{IM}}({A^{\prime}},\mathfrak{k})$.
Pulling back this connection 1-form to $A$, we obtain a multiplicative
$\mathfrak{k}$-valued 1-form

$(L:=L_{A^{\prime}}\circ\operatorname{pr}_{A^{\prime}},l:=l_{A^{\prime}}\circ\mathrm{pr}_{A^{\prime}})\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k}),$

which satisfies $l|_{\mathfrak{k}}=\mathrm{Id}_{\mathfrak{k}}$. So bundles of
ideals of principal type are partially split.

The IM connection 1-form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,\mathfrak{k})$
can also be described using the fact that the splitting $l_{A^{\prime}}$ gives
an identification ${A^{\prime}}\simeq TM\oplus\mathfrak{k}$, where the anchor
becomes $\operatorname{pr}_{TM}$ and the bracket is given by:

$[(X,\xi),(Y,\eta)]_{{A^{\prime}}}=([X,Y],\Omega(X,Y)+\nabla_{X}\eta-\nabla_{Y}\xi+[\xi,\eta]_{\mathfrak{k}}),$
(6.2)

for all $X,Y\in\mathfrak{X}^{1}(M)$, $\xi,\eta\in\Gamma(\mathfrak{k})$. Here:

  1. -

$\Omega$ is $C^{\infty}(M)$-bilinear, so that
$\Omega\in\Omega^{2}(M;\mathfrak{k})$;

  2. -

$\nabla$ is a connection on $\mathfrak{k}$ preserving
$[\cdot,\cdot]_{\mathfrak{k}}$ with curvature
$R^{\nabla}=\operatorname{ad}(\Omega)$.

Then, one finds that, for $\alpha\in\Gamma(B)$ and
$\xi\in\Gamma(\mathfrak{k})$:

$i_{X}L(\alpha,\xi)=\nabla_{X}(\xi)+\Omega(X,\rho_{B}(\alpha)).$

In the notation of Proposition 5.13, we have that

$U(\alpha,X)=\Omega(\rho_{B}(\alpha),X),\qquad\nabla^{L}=\nabla.$

Here is a class of bundle of ideals which are of principal type:

######  Proposition 6.5.

Let $\mathfrak{k}\subset A$ be a bundle of ideals such that:

  1. (i)

$\mathfrak{k}$ is a locally trivial Lie algebra bundle;

  2. (ii)

the typical fiber $(\mathfrak{g},[\cdot,\cdot])$ of $\mathfrak{k}$ satisfies:

$H^{0}(\mathfrak{g},\mathfrak{g})=H^{1}(\mathfrak{g},\mathfrak{g})=0.$

Then $\mathfrak{k}\subset A$ is of principal type, and in particular it is
partially split.

######  Corollary 6.6.

If $\mathfrak{k}\subset A$ is a bundle of semi-simple Lie algebras, then it is
of principal type.

###### Proof.

For a semi-simple Lie algebra $\mathfrak{g}$, we have that
$H^{i}(\mathfrak{g},\mathfrak{g})=0$, for all $i$. Also, $\mathfrak{k}$ is
automatically a locally trivial Lie algebra (see Remark 6.2), hence
Proposition 6.5 applies. ∎

###### Proof of Proposition 6.5.

Consider the Lie algebroid $\operatorname{\mathfrak{gl}}(\mathfrak{k})$ whose
sections are the derivations of the vector bundle $\mathfrak{k}$, i.e., the
bundle maps $D_{X}:\Gamma(\mathfrak{k})\to\Gamma(\mathfrak{k})$ satisfying:

$D_{X}(f\xi)=fD_{X}(\xi)+X(f)\xi,$

for all $f\in C^{\infty}(M)$. The anchor of
$\operatorname{\mathfrak{gl}}(\mathfrak{k})$ assigns to the section $D_{X}$
its symbol $X\in\mathfrak{X}(M)$, and the Lie bracket is the commutator of
derivations. We let
$A^{\prime}\subset\operatorname{\mathfrak{gl}}(\mathfrak{k})$ be the Lie
subalgebroid whose sections are the derivations of the bracket
$[\cdot,\cdot]_{\mathfrak{k}}$

$\Gamma(A^{\prime})=\mathrm{Der}(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}}).$

The fact that $A^{\prime}$ is a Lie subalgebroid and transitive, follows from
(i).

Now, to prove the proposition, let $B=A/\mathfrak{k}$ and consider the map

$\psi:A\to A^{\prime}\times
B,\quad\alpha\mapsto(\nabla^{\mathfrak{k}}_{\alpha},[\alpha]).$

We claim that this is an injective Lie algebroid morphism, with image the
fiber product $A^{\prime}\times_{TM}B$. That $\psi$ is a Lie algebroid
morphism follows the fact that both components $\nabla^{\mathfrak{k}}:A\to
A^{\prime}$ and $\operatorname{pr}:A\to B$ are Lie algebroid maps. On the
other hand, if $\alpha\in A$ is such that $\psi(\alpha)=0$, then
$\alpha\in\mathfrak{k}_{x}$ satisfies:

$\nabla^{\mathfrak{k}}_{\alpha}=[\alpha,-]_{\mathfrak{k}_{x}}=0.$

But (ii) is equivalent to
$\operatorname{ad}:\mathfrak{k}_{x}\to\operatorname{Der}(\mathfrak{k}_{x})$
being an isomorphism of Lie algebras. Injectivity of this maps shows that
$\alpha=0$ and so $\psi$ is injective. Finally, surjectivity of this map
implies that the map $\psi$ is onto $A^{\prime}\times_{TM}B$. ∎

######  Remark 6.7.

As observed in the proof, condition (ii) of Proposition 6.5 is equivalent to
$\operatorname{ad}:\mathfrak{g}\to\operatorname{Der}(\mathfrak{g})$ being a
Lie algebra isomorphism. Therefore, $G:=\operatorname{Aut}(\mathfrak{g})$ is a
Lie group integrating $\mathfrak{g}$ and we have the principal $G$-bundle of
$\mathfrak{g}$-frames:

$P:=\\{\varphi:\mathfrak{g}\to\mathfrak{k}_{x}\,:\,x\in
M,\,\varphi\,\textrm{is a Lie algebra isomorphism}\\}.$

The transitive Lie algebroid $A^{\prime}\to M$ in the proof is isomorphic to
the Atiyah Lie algebroid of $P$ and hence is integrable.

We also have a global version of Proposition 6.5.

######  Proposition 6.8.

Let $\Phi:\mathcal{G}\to\mathcal{H}$ be a groupoid map covering the identity
which is a surjective submersion, and let $\phi:A\to B$ be the induced Lie
algebroid map. If $\mathfrak{k}:=\ker\phi$ satisfies the conditions of
Proposition 6.5 then $\Phi$ is partially split.

###### Proof.

The assumption implies that

$\mathcal{G}^{\prime}:=\big{\\{}\textrm{Lie algebra isomorphisms}\ \
\mathfrak{k}_{x}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}_{y},\
x,y\in M\big{\\}}\rightrightarrows M$

is a transitive Lie groupoid. The map (2.1)
$\operatorname{Ad}:\mathcal{G}\to\mathcal{G}^{\prime}$ together with the map
$\Phi$ induce a groupoid map into the fiber product:

$\Psi:=(\operatorname{Ad},\Phi):\mathcal{G}\to\mathcal{G}^{\prime}\times_{M\times
M}\mathcal{H}.$

Then $\Psi$ integrates the Lie algebroid map $\psi$ from the proof of
Proposition 6.5. Since $\psi$ is an isomorphism, it follows that $\Psi$ is a
local diffeomorphism. On the other hand, since $\mathcal{G}^{\prime}$ is
transitive, it follows that
$\operatorname{pr}_{2}:\mathcal{G}^{\prime}\times_{M\times
M}\mathcal{H}\to\mathcal{H}$ is of principal type, so by the example of
Subsection 3.5, $\operatorname{pr}_{2}$ admits a multiplicative Ehresmann
connection $E$. The preimage of $E$ under $\Psi$ is a multiplicative Ehresmann
connection for $\Phi$. ∎

###  6.7. Bundles of ideals of principal type with kernel flat IM connections

Let $\mathfrak{k}\subset A$ be an ideal of principal type, for which the
2-form (6.2) is center-valued:

$\Omega\in\Omega^{2}(M,z(\mathfrak{k})).$

Then the corresponding connection $\nabla$ is flat, so we are in the case of
an IM Ehresmann connection which is kernel flat. This kind of partially split
ideals can be explicitly described as follows.

Let $B\Rightarrow M$ be a Lie algebroid and let $\mathfrak{k}\to M$ be a Lie
algebra bundle with a flat connection $\nabla^{L}$ preserving the bracket
$[\cdot,\cdot]_{\mathfrak{k}}$. Given any center-valued,
$\operatorname{d}^{\nabla^{L}}$-closed, 2-form
$\Omega\in\Omega^{2}(M,z(\mathfrak{k}))$ one can define a center-valued tensor
$U$ by:

$U(\alpha,X):=\Omega(\rho_{B}(\alpha),X).$

One checks that $U$ satisfies (S3), so that (S1’)-(S3’) hold. Therefore, we
obtain a partially split bundle of ideals $\mathfrak{k}\subset A$ with flat
coupling data. Moreover, it is of principal type: we have $A\simeq
A^{\prime}\times_{TM}B$, where $A^{\prime}=TM\oplus\mathfrak{k}$ is the
transitive Lie algebroid with Lie bracket:

$[(X,\xi),(Y,\eta)]_{B}=([X,Y],\Omega(X,Y)+\nabla^{L}_{X}\eta-\nabla^{L}_{Y}\xi+[\xi,\eta]_{\mathfrak{k}}).$

Note that in this case the $B$-cocycle $\lambda$ in Remark 5.20 is given by
$(\rho_{B})^{*}(\Omega)$, so its class is the image of the map:

$(\rho_{B})^{*}:H^{2}(M,z(\mathfrak{k}))\to H^{2}(B,z(\mathfrak{k})).$

When $\Omega=\operatorname{d}^{\nabla_{L}}\theta$, with
$\theta\in\Omega^{1}(M,z(\mathfrak{k}))$ we have $[\lambda]=0$ and the
resulting bundle of ideals $\mathfrak{k}\subset A$ is isomorphic to a
“trivial” coupling data $(\nabla^{L},0)$, via the map:

$A\to
B\oplus\mathfrak{k},\quad(\alpha,\xi)\mapsto(\alpha,\theta(\alpha)+\xi).$

On the other hand, given any partially split bundle of ideals
$\mathfrak{k}\subset A$ with flat coupling data $(\nabla^{L},U)$ and a center-
valued, $\operatorname{d}^{\nabla^{L}}$-closed, 2-form
$\Omega\in\Omega^{2}(M,z(\mathfrak{k}))$, we obtain a “gauge transformed”
bundle of ideals with coupling data $(\nabla^{L},e^{\Omega}U)$ where:

$e^{\Omega}U(\alpha,X):=U(\alpha,X)+\Omega(\rho_{B}(\alpha),X).$

The bracket of the underlying algebroid $B$ changes by replacing the cocycle
$\lambda$ with the cocycle $\lambda+(\rho_{B})^{*}(\Omega)$.

###  6.8. Bundles of ideals of rank one

We treat in detail the case of bundle of ideals of rank one. In this setting
the existence of specific types of IM connections have cohomological
interpretations.

Let $\mathfrak{k}\subset A$ be a bundle of ideals of rank one, which, for
simplicity, we assume to be orientable. As usual, we set $B=A/\mathfrak{k}$,
and after choosing a splitting of the short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

we obtain an identification

$A\simeq B\oplus\mathfrak{k},$

where the anchor and the Lie bracket on $A$ become:

$\displaystyle\rho_{A}(\alpha,\xi)=\rho_{B}(\alpha),$
$\displaystyle[(\alpha,\xi),(\beta,\eta)]_{A}=([\alpha,\beta]_{B},\lambda(\alpha,\beta)+\nabla^{\mathfrak{k}}_{\alpha}\eta-\nabla^{\mathfrak{k}}_{\beta}\xi).$

Notice that:

  * $\bullet$

The flat $B$-connection $\nabla^{\mathfrak{k}}$ is independent of the choice
of splitting of $A$;

  * $\bullet$

Up to isomorphism, the representation of $B$ on $\mathfrak{k}$ is determined
by its characteristic class (see, e.g., Section 11.1 in [11]):

$c_{1}(\mathfrak{k})\in H^{1}(B).$

Explicitly, if we choose a trivialization $\mathfrak{k}\simeq
M\times\mathbb{R}$, then

$\nabla^{\mathfrak{k}}_{\alpha}f=\mathscr{L}_{\rho_{B}(\alpha)}f+i_{V}(\alpha)f,\quad(f\in
C^{\infty}(M))$

for a cocycle $V\in\Omega^{1}(B)$, which is a representative of
$c_{1}(\mathfrak{k})$. If we change the trivialization $\mathfrak{k}\simeq
M\times\mathbb{R}$ by multiplying with a non-zero function $h$, then $V$
changes by
$\rho_{B}^{*}(\operatorname{d}\log|h|)=\operatorname{d}_{B}\log|h|$.

  * $\bullet$

The $\mathfrak{k}$-valued 2-form $\lambda\in\Omega^{2}(B;\mathfrak{k})$ is
$\operatorname{d}_{B}$-closed, so it determines a class:

$c_{2}(A):=[\lambda]\in H^{2}(B;\mathfrak{k}).$

This class is well-defined: a different splitting of $A$ is determined by an
element $Z\in\Omega^{1}(B;\mathfrak{k})$, and this has the effect of changing
$\lambda$ by $\operatorname{d}_{B}Z$.

The anchor $\rho_{B}:B\to TM$, being a Lie algebroid morphism, defines a map
in cohomology

$\rho_{B}^{*}:H^{1}(M)\longrightarrow H^{1}(B).$

The class $c_{1}(\mathfrak{k})$ belongs to the image of this map if and only
if there exists a flat connection $\nabla$ on $\mathfrak{k}$ such that, for
all $\alpha\in\Gamma(B)$ and $\xi\in\Gamma(\mathfrak{k})$:

$\nabla^{\mathfrak{k}}_{\alpha}\xi=\nabla_{\rho_{B}(\alpha)}\xi.$ (6.3)

To see this, choose a trivialization $\mathfrak{k}\simeq M\times\mathbb{R}$,
giving rise to the representative $V$ of $c_{1}(\mathfrak{k})$. We have a one-
to-one correspondence between flat connections $\nabla$ on $M\times\mathbb{R}$
and closed forms $\theta\in\Omega^{1}(M)$, given by:

$\nabla_{X}f=\mathscr{L}_{X}f+\theta(X)f,\quad(f\in C^{\infty}(M))$

Under this correspondence, we have that (6.3) is equivalent to
$V=\rho_{B}^{*}\theta$. The class

$c_{1}(\nabla):=[\theta]\in H^{1}(M).$

is just the characteristic class of the representation of $TM$ on
$\mathfrak{k}$ given by $\nabla$, and we have by definition:

$\rho_{B}^{*}(c_{1}(\nabla))=c_{1}(\mathfrak{k}).$

Below, whenever $c_{1}(\mathfrak{k})$ belongs to the image of $\rho^{*}_{B}$,
and we choose a flat connection satisfying 6.3, we will say that “$\nabla$ is
a flat connection with $\rho_{B}^{*}\nabla=\nabla^{\mathfrak{k}}$”. In
particular, when this happens, $\nabla$ is a $B$-invariant connection (see
Definition A.9). Also, we have a map in cohomology with coefficients in
$\mathfrak{k}$, for the representations defined by $\nabla$ and
$\nabla^{\mathfrak{k}}$:

$\rho_{B}^{*}:H^{\bullet}(M;\mathfrak{k})\longrightarrow
H^{\bullet}(B;\mathfrak{k}),$

and another map in IM cohomology, for the differential
$\operatorname{d}^{\nabla}_{\mathrm{IM}}$ (see Lemma A.11):

$H^{\bullet}_{\mathrm{IM}}(B;\mathfrak{k})\longrightarrow
H^{\bullet}(B;\mathfrak{k}).$

After these preliminaries, we can now state a result about the various special
IM Ehresmann connections one can have for a bundle of ideals of rank one:

######  Proposition 6.9.

Let $\mathfrak{k}\subset A$ be a bundle of ideals of rank one, which is
orientable.

  1. (i)

The Lie algebroid $A$ is isomorphic to a product, $A\simeq B\times\mathbb{R}$,
with $\mathfrak{k}\simeq 0_{M}\times\mathbb{R}$ (see Subsection 6.2), if and
only if

$c_{1}(\mathfrak{k})=0\quad\textrm{and}\quad c_{2}(A)=0.$

  2. (ii)

There exists a totally flat IM Ehresmann connection for $\mathfrak{k}\subset
A$ if and only if

$c_{1}(\mathfrak{k})\in\operatorname{Im}(H^{1}(M)\to
H^{1}(B))\quad\textrm{and}\quad c_{2}(A)=0.$

  3. (iii)

There exists a leafwise flat IM Ehresmann connection for $\mathfrak{k}\subset
A$ if and only if $\mathfrak{k}$ admits a $B$-invariant connection $\nabla$
and $c_{2}(A)=0$.

  4. (iv)

There exists a kernel flat IM Ehresmann connection for $\mathfrak{k}\subset A$
if and only if

$c_{1}(\mathfrak{k})\in\operatorname{Im}(H^{1}(M)\to
H^{1}(B))\quad\textrm{and}\quad
c_{2}(A)\in\operatorname{Im}(H^{2}_{\mathrm{IM}}(B;\mathfrak{k})\to
H^{2}(B;\mathfrak{k})),$

where the last map is for a flat connection $\nabla$ with
$\rho_{B}^{*}\nabla=\nabla^{\mathfrak{k}}$.

  5. (v)

The Lie bundle of ideals $\mathfrak{k}\subset A$ is of principal type if and
only if

$c_{1}(\mathfrak{k})\in\operatorname{Im}(H^{1}(M)\to
H^{1}(B))\quad\textrm{and}\quad
c_{2}(A)\in\operatorname{Im}(H^{2}(M;\mathfrak{k})\to H^{2}(B;\mathfrak{k})),$

where the last map is for a flat connection $\nabla$ with
$\rho_{B}^{*}\nabla=\nabla^{\mathfrak{k}}$.

Before we look at the proof, let us discuss the general question of existence
of partial splittings for ideals of rank one:

######  Proposition 6.10.

A rank one bundle of ideals $\mathfrak{k}\subset A$ admits an IM Ehresmann
connection if and only if:

  1. (i)

$\mathfrak{k}$ admits a $B$-invariant connection $\nabla^{L}$, i.e., for all
$\alpha\in B$:

$\nabla^{\mathfrak{k}}_{\alpha}=\nabla^{L}_{\rho_{B}(\alpha)}\quad\textrm{and}\quad
i_{\rho_{B}(\alpha)}R^{\nabla^{L}}=0,$

  2. (ii)

and $c_{2}(A)$ has a representative of the form

$\lambda(\alpha,\beta)=i_{\rho_{B}(\beta)}U(\alpha),$

where $U:B\to T^{*}M\otimes\mathfrak{k}$ is a linear map satisfying the
structure equation:

$U([\alpha,\beta]_{B})=\mathscr{L}^{\nabla^{L}}_{\rho_{B}(\alpha)}U(\beta)-i_{\rho_{B}(\beta)}\operatorname{d}^{\nabla^{L}}U(\alpha),$

for all $\alpha,\beta\in\Gamma(B)$.

###### Proof.

From the theory of couplings, discussed in Section 5.4, $\mathfrak{k}\subset
A$ is partially split if and only if there exists a pair $(\nabla^{L},U)$ and
a splitting $A\simeq B\oplus\mathfrak{k}$ such that
$\nabla^{\mathfrak{k}}_{\alpha}=\nabla^{L}_{\rho_{B}(\alpha)}$,
$\lambda(\alpha,\beta)=U(\alpha,\rho_{B}(\beta))$, and the structure equations
(S2) and (S3) hold. It follows easily that (S2) is equivalent to invariance of
the connection, and that (S3) is equivalent to the equation in (ii) – as was
mentioned already in the proof of Proposition 5.19. ∎

Assume then that we have an IM Ehresmann connection with coupling data
$(\nabla^{L},U)$, as in the previous proposition. After fixing a
trivialization $\mathfrak{k}\simeq M\times\mathbb{R}$, we can write the
connection as

$\nabla^{L}_{X}=\mathscr{L}_{X}+\theta(X),$

for a unique 1-form $\theta\in\Omega^{1}(M)$. Then $c_{1}(\mathfrak{k})=[V]$
where $V\in\Omega^{1}(B)$ satisfies:

$V(\alpha)=i_{\rho_{B}(\alpha)}\theta.$

Moreover, regarding $U$ as a map $U:B\to T^{*}M$, $c_{2}(A)=[\lambda]$ where
$\lambda\in\Omega^{2}(B)$ satisfies:

$\lambda(\alpha,\beta)=i_{\rho_{B}(\beta)}U(\alpha).$

The structure equations from (i) and (ii) become:

$\displaystyle i_{\rho_{B}(\alpha)}\operatorname{d}\theta=0,$ (S2”)
$\displaystyle
U([\alpha,\beta]_{B})=\mathscr{L}_{\rho_{B}(\alpha)}U(\beta)-i_{\rho_{B}(\beta)}\operatorname{d}U(\alpha)$
(S3”)
$\displaystyle\qquad\qquad\quad+\theta(\rho_{B}(\alpha))U(\beta)-i_{\rho_{B}(\beta)}\big{(}\theta\wedge
U(\alpha)\big{)},$

for all $\alpha,\beta\in\Gamma(B)$. This follows from the relations:

$R^{\nabla^{L}}(X,Y)=\operatorname{d}\theta(X,Y),\quad\operatorname{d}^{\nabla^{L}}\omega=\operatorname{d}\omega+\theta\wedge\omega,\quad\mathscr{L}^{\nabla^{L}}_{X}\omega=\mathscr{L}_{X}\omega+\theta(X)\omega.$

Recall that a class $c\in H^{k}(B;V)$ is said to be tangential if it has some
representative $P\in\Omega^{k}(B;V)$ satisfying

$\rho_{B}(\alpha)=0\quad\Longrightarrow\quad i_{\alpha}P=0.$

The previous discussion implies the following obstruction for existence of IM
Ehresmann connection:

######  Corollary 6.11.

If a rank one bundle of ideals $\mathfrak{k}\subset A$ admits an IM Ehresmann
connection, then the classes $c_{1}(\mathfrak{k})\in H^{1}(B)$ and
$c_{2}(A)\in H^{2}(B;\mathfrak{k})$ are tangential.

###### Proof of Proposition 6.9.

(i) If $A$ is isomorphic to a product, then clearly $c_{1}(\mathfrak{k})=0$
and $c_{2}(A)=0$. Conversely, $c_{2}(A)=0$ means that there exists a splitting
$A\simeq B\times\mathfrak{k}$ such that $\lambda=0$, and
$c_{1}(\mathfrak{k})=0$ means that there exists a trivialization
$\mathfrak{k}\simeq M\times\mathbb{R}$ such that $V=0$. Thus $A$ is a product.

(ii) Assume that $\mathfrak{k}\subset A$ admits a totally flat IM connection,
with coupling data $(\nabla^{L},U=0)$. Then, as remarked above
$c_{1}(\mathfrak{k})$ belongs to the image of $\rho_{B}^{*}$, and $\lambda=0$,
so clearly $c_{2}(A)=0$. Conversely, under the assumptions, there exists a
splitting $A\simeq B\oplus\mathfrak{k}$, under which $\lambda=0$ and there is
a flat connection $\nabla^{L}$ on $\mathfrak{k}$ inducing
$\nabla^{\mathfrak{k}}$. Then $(\nabla^{L},U)$ is the coupling data of a
totally flat IM connection for $\mathfrak{k}$.

(iii) Assume that $\mathfrak{k}\subset A$ admits a leafwise flat IM
connection, with coupling data $(\nabla^{L},U)$. This means that $\lambda=0$,
so clearly, $c_{2}(A)=0$, and as remarked before $\nabla^{L}$ is
$B$-invariant. Conversely, under the assumptions, there exists a splitting
$A\simeq B\times\mathbb{R}$, under which $\lambda=0$, and there is a
connection $\nabla^{L}$ which is $B$-invariant. Then the pair
$(\nabla^{L},U=0)$ is the coupling data of a leafwise flat IM connection for
$\mathfrak{k}$.

(iv) Assume that $\mathfrak{k}\subset A$ admits a kernel flat IM connection,
with coupling data $(\nabla^{L},U)$. This means that $\nabla^{L}$ is flat, so
$c_{1}(\mathfrak{k})$ is in the image of $\rho_{B}^{*}$. By Proposition 5.19,
the pair $(\operatorname{d}^{\nabla^{L}}U,U)$ is an IM 2-form on $B$ with
values in $\mathfrak{k}$, which is clearly closed for the differential
$\operatorname{d}_{\mathrm{IM}}^{\nabla^{L}}$ from Proposition A.10, and which
is sent under the map from Lemma A.11 to $\lambda$. So $c_{2}(A)$ it the image
of $[\gamma]$. Conversely, the assumptions imply the existence of a flat
connection $\nabla^{L}$ on $\mathfrak{k}$ which is $B$-invariant, and of a
splitting $A\simeq B\oplus\mathfrak{k}$, under which $\lambda$ is in the image
of $\Omega_{\mathrm{IM}}^{2}(B;\mathfrak{k})\to\Omega^{2}(B;\mathfrak{k})$
(see Lemma A.11) of an element $\gamma$ which is
$\operatorname{d}^{\nabla^{L}}_{\mathrm{IM}}$-closed. This implies that
$\gamma=(\operatorname{d}^{\nabla^{L}}_{\mathrm{IM}}U,U)$. The second part of
Proposition 5.19 implies that $(\nabla^{L},U)$ is the coupling data of a
kernel flat IM connection.

(v) Assume that $\mathfrak{k}\subset A$ is of principal type, i.e., there
exists a transitive Lie algebroid $\mathfrak{k}\hookrightarrow
A^{\prime}\twoheadrightarrow TM$ so that $A\simeq A^{\prime}\times_{TM}B$. As
discussed in Subsection 6.6, after choosing a splitting $A^{\prime}\simeq
TM\oplus\mathfrak{k}$, the Lie bracket on $A^{\prime}$ is given by

$[(X,\xi),(Y,\eta)]_{{A^{\prime}}}=([X,Y],\Omega(X,Y)+\nabla_{X}\eta-\nabla_{Y}\xi),$
(6.4)

where $\nabla$ is a flat connection on $\mathfrak{k}$ and
$\Omega\in\Omega^{2}(M;\mathfrak{k})$ is $\operatorname{d}^{\nabla}$-closed.
Using the induced splitting $A\simeq B\oplus\mathfrak{k}$, we obtain that
$\nabla^{\mathfrak{k}}_{\alpha}=\nabla_{\rho_{B}(\alpha)}$ and
$\lambda=\rho_{B}^{*}(\Omega)$; so $c_{1}(\mathfrak{k})$ and $c_{2}(A)$ are in
the image of $\rho^{*}_{B}$. Conversely, if this happens, we have a flat
connection $\nabla$ on $\mathfrak{k}$ such that
$\nabla^{\mathfrak{k}}_{\alpha}=\nabla_{\rho_{B}(\alpha)}$, and a
$\operatorname{d}^{\nabla}$-closed 2-form
$\Omega\in\Omega^{2}(M;\mathfrak{k})$ such that
$\lambda=\rho_{B}^{*}(\Omega)$, for some splitting. This means precisely that
$A\simeq A^{\prime}\times_{TM}B$, where $A^{\prime}=TM\oplus\mathfrak{k}$ is
the transitive Lie algebroid with bracket (6.4). ∎

##  Appendix A Multiplicative and infinitesimal multiplicative forms

We recall here a few results about multiplicative and infinitesimal
multiplicative (IM) forms that are required throughout the paper. For us, it
will be specially important forms with coefficients, since they appear as
“connection 1-forms” for Ehresmann connections. More details on the results
mentioned in this section can be found in [6, 7, 10, 14, 18] and we only
provide proofs of the results that cannot be found in those references.

###  A.1. Multiplicative forms and IM forms

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with source/target
$\mathbf{s},\mathbf{t}:\mathcal{G}\to M$ and multiplication
$m:\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{G}\to\mathcal{G}$.

######  Definition A.1.

A differential form $\omega\in\Omega^{k}(\mathcal{G})$ is called
multiplicative if it lies in the kernel of the simplicial differential:

$\delta\omega:=\operatorname{pr}_{1}^{*}\omega-m^{*}\omega+\operatorname{pr}_{2}^{*}\omega=0$

where
$\operatorname{pr}_{i}:\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{G}\to\mathcal{G}$
are the projections on the factors.

The differential of a multiplicative form is again a multiplicative form, so
we have a complex of multiplicative differential forms
$(\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G}),\operatorname{d})$. A basic fact
is that two multiplicative forms
$\omega,\omega^{\prime}\in\Omega^{k}_{\mathrm{M}}(\mathcal{G})$ such that
$\omega|_{M}=\omega^{\prime}|_{M}$ and
$(\operatorname{d}\omega)|_{M}=(\operatorname{d}\omega^{\prime})|_{M}$
actually coincide, assuming that $\mathcal{G}$ has connected target fibers.

Denote by $(A,[\cdot,\cdot],\rho)$ the Lie algebroid of
$\mathcal{G}\rightrightarrows M$. Given a multiplicative form
$\omega\in\Omega^{k}_{\mathrm{M}}(\mathcal{G})$ one defines two vector bundle
maps $\mu:A\to\wedge^{k-1}T^{*}M$ and $\zeta:A\to\wedge^{k}T^{*}M$ by setting:

$\mu(a)=i_{a}\omega|_{TM},\quad\zeta(a)=i_{a}\operatorname{d}\omega|_{TM}.$
(A.1)

The pair $(\mu,\zeta)$ satisfies the following set of equations for any
$\alpha,\beta\in\Gamma(A)$:

$\displaystyle i_{\rho(\beta)}\mu(\alpha)$
$\displaystyle=-i_{\rho(\alpha)}\mu(\beta),$
$\displaystyle\mu([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\rho(\alpha)}\mu(\beta)-i_{\rho(\beta)}\operatorname{d}\mu(\alpha)-i_{\rho(\beta)}\zeta(\alpha),$
(A.2) $\displaystyle\zeta([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\rho(\alpha)}\zeta(\beta)-i_{\rho(\beta)}\operatorname{d}\zeta(\alpha).$

This leads to the notion of infinitesimal multiplicative form on an arbitrary
Lie algebroid $A\Rightarrow M$, integrable or not:

######  Definition A.2.

An IM $k$-form on a Lie algebroid $(A,[\cdot,\cdot],\rho)$ is a pair
$(\mu,\zeta)$, where $\mu:A\to\wedge^{k-1}T^{*}M$ and
$\zeta:A\to\wedge^{k}T^{*}M$ are bundle maps satisfying (A.1).

The space of IM forms is denoted by $\Omega_{\mathrm{IM}}^{k}(A)$ and it
becomes a cochain complex with differential given by

$\operatorname{d}_{\mathrm{IM}}:\Omega_{\mathrm{IM}}^{k}(A)\to\Omega_{\mathrm{IM}}^{k+1}(A),\quad\operatorname{d}_{\mathrm{IM}}(\mu,\zeta):=(\zeta,0).$

For a _target 1-connected_ Lie groupoid $\mathcal{G}\rightrightarrows M$ with
Lie algebroid $A\Rightarrow M$ the assignment $\omega\mapsto(\mu,\zeta)$ given
by (A.1) is an isomorphism of complexes:

$(\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G}),\operatorname{d})\simeq(\Omega^{\bullet}_{\mathrm{IM}}(A),\operatorname{d}_{\mathrm{IM}}).$

###  A.2. Multiplicative forms and IM forms with coefficients

This subsection follows [14], where all claims are proven. Let $\mathcal{G}$
be a Lie groupoid and $V$ a $\mathcal{G}$-representation. We will work with
differential forms on $\mathcal{G}$ with coefficients in $\mathbf{s}^{*}V$
(instead of $\mathbf{t}^{*}V$, as in [14]), which we denote by

$\Omega^{\bullet}(\mathcal{G};V):=\Omega^{\bullet}(\mathcal{G};\mathbf{s}^{*}V).$

Similarly, denote by $\Omega^{\bullet}(\mathcal{G}^{(k)};V)$ the space of
differential forms on the manifold $\mathcal{G}^{(k)}$ of composable
$k$-strings of arrows, with values in the vector bundle
$(\mathbf{s}\circ\operatorname{pr}_{k})^{*}(V)\to\mathcal{G}^{(k)}$, where
$\operatorname{pr}_{i}:\mathcal{G}^{(k)}\to\mathcal{G}$ is the projection onto
the $i$-th component.

######  Definition A.3.

A form $\omega\in\Omega^{\bullet}(\mathcal{G};V)$ is called multiplicative if

$\delta\omega=0,$

where
$\delta:\Omega^{\bullet}(\mathcal{G};V)\to\Omega^{\bullet}(\mathcal{G}^{(2)};V)$
is the simplicial differential defined by:

$\delta\omega|_{(g_{1},g_{2})}=g_{2}^{-1}\cdot\operatorname{pr}_{1}^{*}\omega|_{g_{1}}-m^{*}\omega|_{g_{1}g_{2}}+\operatorname{pr}_{2}^{*}\omega|_{g_{2}}.$
(A.3)

We denote by $\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G};V)$ the space of
$V$-valued multiplicative $k$-forms.

A simple way to produce multiplicative forms is by using the simplicial
differential in degree zero
$\delta:\Omega^{\bullet}(M;V)\to\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G};V)$
given by:

$\delta\omega|_{g}:=g^{-1}\cdot\mathbf{t}^{*}\omega-\mathbf{s}^{*}\omega,\quad
g\in\mathcal{G}.$ (A.4)

We will see later other examples.

Let us pass to the infinitesimal level, so denote by $A\Rightarrow M$ the Lie
algebroid of $\mathcal{G}\rightrightarrows M$. Given a representation $p:V\to
M$ of $\mathcal{G}$, at the algebroid level we obtain a flat $A$-connection on
$V$:

$\nabla:\Gamma(A)\times\Gamma(V)\to\Gamma(V),\quad(\alpha,s)\mapsto\nabla_{\alpha}s.$

This means that for all $\alpha,\beta\in\Gamma(A)$, $s\in\Gamma(V)$, and $f\in
C^{\infty}(M)$, we have:

$\nabla_{f\alpha}s=f\nabla_{\alpha}s,\ \ \
\nabla_{\alpha}(fs)=f\nabla_{\alpha}s+(\mathscr{L}_{\rho(\alpha)}f)s,\ \ \
\nabla_{[\alpha,\beta]}=[\nabla_{\alpha},\nabla_{\beta}].$

We also call $(V,\nabla)$ a representation of the Lie algebroid $A$.

For a $V$-valued multiplicative $k$-form
$\omega\in\Omega^{k}_{\mathrm{M}}(\mathcal{G};V)$, define a vector bundle map
$l:A\to\wedge^{k-1}T^{*}M\otimes V$ by setting

$l(a):=(i_{a}\omega)|_{TM},$ (A.5)

and a linear operator $L:\Gamma(A)\to\Omega^{k}(M;V)$ by letting

$L(\alpha)_{x}(v_{1},\dots,v_{k}):=\frac{\operatorname{d}}{\operatorname{d}\epsilon}\Big{|}_{\epsilon=0}\phi_{\alpha^{L}}^{\epsilon}(x)\cdot\omega(\operatorname{d}_{x}\phi_{\alpha^{L}}^{\epsilon}(v_{1}),\dots,\operatorname{d}_{x}\phi_{\alpha^{L}}^{\epsilon}(v_{k}))$
(A.6)

where $\phi_{\alpha^{L}}^{\epsilon}$ denotes the flow of the left-invariant
vector field $\alpha^{L}$. This operator is a kind of differential operator
with symbol $l$, in the sense that for any $f\in C^{\infty}(M)$ and
$\alpha\in\Gamma(A)$ it satisfies:

$L(f\alpha)=fL(\alpha)+\operatorname{d}f\wedge l(\alpha).$ (A.7)

Furthermore, the pair $(L,l)$ satisfies the following set of equations:

$\displaystyle i_{\rho(\alpha)}l(\beta)$
$\displaystyle=-i_{\rho(\beta)}l(\alpha),$ $\displaystyle L([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\alpha}L(\beta)-\mathscr{L}_{\beta}L(\alpha),$
(A.8) $\displaystyle l([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\alpha}l(\beta)-i_{\rho(\beta)}L(\alpha),$

where, for $\alpha\in\Gamma(A)$ and $\gamma\in\Omega^{k}(M;V)$, we denoted:

$\mathscr{L}_{\alpha}\gamma(X_{1},\dots,X_{k}):=\nabla_{\alpha}(\gamma(X_{1},\dots,X_{k}))-\sum_{i=1}^{k}\gamma(X_{1},\dots,[\rho(\alpha),X_{i}],\dots,X_{k}).$
(A.9)

Now observe that all this makes sense for an arbitrary algebroid $A\Rightarrow
M$, integrable or not, provided one has a representation $(V,\nabla)$:

######  Definition A.4.

Let $(V,\nabla)$ be a representation of the Lie algebroid $A\Rightarrow M$. An
$V$-valued IM $k$-form is a pair $(L,l)$, where
$L:\Gamma(A)\to\Omega^{k}(M,V)$ is a linear map and
$l:A\to\wedge^{k-1}T^{*}M\otimes V$ is a vector bundle map satisfying (A.7)
and (A.2).

We denote by $\Omega^{\bullet}_{\mathrm{IM}}(A;V)$ the space of $V$-valued IM
$k$-forms. For historical reasons, these are also sometime called _Spencer
operators_. For a _target 1-connected_ Lie groupoid
$\mathcal{G}\rightrightarrows M$ with Lie algebroid $A\Rightarrow M$ the
assignment $\omega\mapsto(L,l)$ given by (A.5) and (A.6) gives an isomorphism:

$\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G};V)\simeq\Omega^{\bullet}_{\mathrm{IM}}(A;V).$

######  Example A.5 ($V$-valued forms of degree 1).

For us the $V$-valued multiplicative forms of degree 1 will be important. Such
a form $\omega\in\Omega^{1}_{\mathrm{M}}(\mathcal{G};V)$ is the same thing as
a morphism of VB groupoids
$\omega:T\mathcal{G}\to\mathbf{s}^{*}V=\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{p}}V$
covering the identity morphism $\mathrm{Id}:\mathcal{G}\to\mathcal{G}$. Let us
explain this in more detail.

First, the tangent bundle of a Lie groupoid $\mathcal{G}\rightrightarrows M$
is a VB-groupoid:

$\textstyle{T\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

Next, if $p:V\to M$ is a representation of $\mathcal{G}$, then we have the
groupoid extension
$\mathcal{G}\times_{M}V:=\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{p}}V\rightrightarrows
M$, where composition of arrows is given by:

$(g,v)\cdot(h,w)=(gh,h^{-1}\cdot v+w).$

It can also be viewed as a VB-groupoid:

$\textstyle{\mathcal{G}\times_{M}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

Now an $V$-valued multiplicative differential form
$\omega\in\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G};V)$ is just a map:

$\omega:T\mathcal{G}\to\mathbf{s}^{*}V=\mathcal{G}\times_{M}V,\quad
v_{g}\mapsto(g,\omega(v_{g})),$

which is a morphism of VB groupoids covering the identity. In other words, it
is a groupoid morphism, which is a vector bundle map making the diagram
commute:

---
$\textstyle{T\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\mathcal{G}\times_{M}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Id}}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

A similar discussion holds at the infinitesimal level for $V$-valued IM forms
of degree 1: a 1-form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;V)$ is the same
thing as a VB algebroid morphism $\theta^{\vee}:TA\to A\oplus V$ covering the
identity morphism $\mathrm{Id}:A\to A$, as we now explain.

First, the tangent bundle of a Lie algebroid $A\Rightarrow M$ is a VB-
algebroid:

$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

where the double arrow is used to distinguish the algebroid projections from
the algebroid morphisms. Next, given a representation $(V,\nabla)$ of $A$,
then we have the algebroid extension $A\oplus V$ with anchor
$\rho\circ\operatorname{pr}_{A}$ and bracket:

$[(\alpha,s),(\beta,t)]=([\alpha,\beta],\nabla_{\alpha}t-\nabla_{\beta}s).$

It can also be viewed as a VB-algebroid:

$\textstyle{A\oplus
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

Now, given a morphism of VB algebroids $\theta^{\vee}:TA\to A\oplus V$
covering the identity morphism $\mathrm{Id}:A\to A$:

---
$\textstyle{TA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\vee}}$$\textstyle{A\oplus
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Id}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

we can define an $V$-valued IM form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A,V)$ by
setting for any section $\alpha\in\Gamma(A)$ and $v\in TM$:

$\displaystyle L(\alpha)(v)$
$\displaystyle:=\operatorname{pr}_{V}(\theta^{\vee}(\operatorname{d}\alpha(v))),$
$\displaystyle l(\alpha)$
$\displaystyle:=\operatorname{pr}_{V}\theta^{\vee}(\hat{\alpha})|_{M},$

where $\hat{\alpha}\in\mathfrak{X}(A)$ is the vertical lift of the section
$\alpha$. Conversely, given an $V$-valued IM form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A;V)$ there exists a unique VB algebroid
morphism $\theta^{\vee}:TA\to A\oplus V$ which covers the identity and
corresponds to $(L,l)$ under the above assignment.

###  A.3. Differentiation of multiplicative forms with coefficients

In this section we discuss differentiation of multiplicative forms with
coefficients. The material here seems to be new, so we include detailed
proofs.

On a manifold $M$, in order to be able to differentiate forms with
coefficients in a vector bundle $p:V\to M$ one needs to choose a connection
$\nabla$. This gives a $\nabla$-de Rham operator
$\operatorname{d}^{\nabla}:\Omega^{k}(M;V)\to\Omega^{k+1}(M;V)$ defined by

$\displaystyle\operatorname{d}^{\nabla}\omega(X_{0},\dots,X_{k})$
$\displaystyle:=\sum_{i}(-1)^{i}\nabla_{X_{i}}\omega(x_{0},\dots,\widehat{X_{i}},\dots,X_{k})+$
$\displaystyle\qquad+\sum_{i<j}(-1)^{i+j}\omega([X_{i},X_{j}],X_{0},\dots,\widehat{X_{i}},\dots,\widehat{X_{j}},\dots,X_{k}).$

Note that $\operatorname{d}^{\nabla}$ is a differential if and only if
$\nabla$ is flat.

Now, consider a representation $p:V\to M$ of a Lie groupoid
$\mathcal{G}\rightrightarrows M$, with Lie algebroid $A\Rightarrow M$. If we
fix a connection $\nabla$ on $V$, we can take the pullback connection
$\nabla^{\mathbf{s}}$ on $\mathbf{s}^{*}V\to\mathcal{G}$ and use it to
differentiate $V$-valued forms. However, in general, given a multiplicative
form $\omega\in\Omega^{k}_{\mathrm{M}}(\mathcal{G};V)$, its differential
$\operatorname{d}^{\nabla^{\mathbf{s}}}\omega$ will not be a multiplicative
form. We need the connection $\nabla$ to be compatible with the
representation.

######  Definition A.6.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid and $p:V\to M$ a
$\mathcal{G}$-representation. A (usual) connection $\nabla$ on $V$ is called
$\mathcal{G}$-invariant if the isomorphism of vector bundles:

$\mathbf{s}^{*}V\to\mathbf{t}^{*}V,\quad(g,v)\mapsto(g,g\cdot v),$

preserves the connections $\nabla^{\mathbf{s}}:=\mathbf{s}^{*}\nabla$ and
$\nabla^{\mathbf{t}}:=\mathbf{t}^{*}\nabla$.

The $\mathcal{G}$-invariance of a connection is equivalent to the commutation
relation:

$\operatorname{d}^{\nabla^{\mathbf{s}}}\delta=\delta\,\operatorname{d}^{\nabla},$

where
$\delta:\Omega^{\bullet}(M;V)\to\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G};V)$
is the degree zero simplicial differential (A.4). It turns out that this is
equivalent to require this commutation relation to hold for any simplicial
degree. We only need the case of degree 1:

######  Proposition A.7.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid, let $p:V\to M$ be a
$\mathcal{G}$-representation, and let $\nabla$ be a connection on $V$. We
denote by $\nabla^{\mathbf{s}}$ both the pullback connections on
$\mathbf{s}^{*}V\to\mathcal{G}$ and on
$(\mathbf{s}\circ\operatorname{pr}_{2})^{*}V\to\mathcal{G}^{(2)}$. The
$\nabla^{\mathbf{s}}$-de Rham differential
$\operatorname{d}^{\nabla^{\mathbf{s}}}$ and the simplicial differential (A.3)
commute:

$\operatorname{d}^{\nabla^{\mathbf{s}}}\delta=\delta\,\operatorname{d}^{\nabla^{\mathbf{s}}},$

if and only if the connection $\nabla$ is $\mathcal{G}$-invariant. In
particular, if this is the case, then $\operatorname{d}^{\nabla^{\mathbf{s}}}$
maps multiplicative forms to multiplicative forms.

###### Proof.

Let $E\to M$ be a vector bundle equipped with a linear connection $\nabla$.
Given a map $\phi:N\to M$, denote by $\nabla^{\phi}$ the pullback connection
on $\phi^{*}V$. Then:

$\operatorname{d}^{\nabla^{\phi}}\phi^{*}=\phi^{*}\operatorname{d}^{\nabla}.$

Now, we can write the simplicial differential with coefficients as:

$\delta=\Phi^{*}\operatorname{pr}_{1}^{*}-m^{*}+\operatorname{pr}_{2}^{*},$

where $\Phi$ is the vector bundle isomorphism covering the identity:

$\Phi:(\mathbf{s}\circ\operatorname{pr}_{1})^{*}V=(\mathbf{t}\circ\operatorname{pr}_{2})^{*}V\to(\mathbf{s}\circ\operatorname{pr}_{2})^{*}V,\quad(g,h,v)\mapsto(g,h,h^{-1}v).$

The condition that the connection $\nabla$ is $\mathcal{G}$-invariant is
easily seen to be equivalent to this map preserving the pullback connections:

$(\mathbf{t}\circ\operatorname{pr}_{2})^{*}\nabla=\operatorname{pr}_{2}^{*}\nabla^{\mathbf{t}},\quad(\mathbf{s}\circ\operatorname{pr}_{2})^{*}\nabla=\operatorname{pr}_{2}^{*}\nabla^{\mathbf{s}}.$

This condition, in turn, is equivalent to:

$\Phi^{*}\operatorname{pr}_{1}^{*}\operatorname{d}^{\nabla}=\operatorname{d}^{\nabla^{\mathbf{s}}}\Phi^{*}\operatorname{pr}_{1}^{*}.$

Hence, we conclude that $\nabla$ is $\mathcal{G}$-invariant if and only if
$\operatorname{d}^{\nabla^{\mathbf{s}}}\delta=\delta\,\operatorname{d}^{\nabla^{\mathbf{s}}}$.
∎

We now pass to infinitesimal level. First we deduce the infinitesimal analogue
of the $\mathcal{G}$-invariance condition on a connection:

######  Proposition A.8.

Let $\mathcal{G}\rightrightarrows M$ be an $\mathbf{t}$-connected Lie groupoid
with Lie algebroid $A\Rightarrow M$ and let $p:V\to M$ be a
$\mathcal{G}$-representation. Denote by $\nabla^{A}$ the corresponding flat
$A$-connection on $V$. A connection $\nabla$ is $\mathcal{G}$-invariant if and
only if

$\nabla^{A}_{a}=\nabla_{\rho(a)}\quad\textrm{and}\quad
R^{\nabla}(\rho(a),v)=0,$ (A.10)

for all $a\in A_{x}$, $v\in T_{x}M$, and $x\in M$.

###### Proof.

We claim that $\mathcal{G}$-invariance of $\nabla$ is equivalent to the
statement that for any path $g(t)\in\mathcal{G}$, one has a commutative
diagram:

$\textstyle{V_{\mathbf{s}(g(0))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{\nabla}_{\mathbf{s}(g(t))}}$$\scriptstyle{g(0)}$$\textstyle{V_{\mathbf{s}(g(t))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g(t)}$$\textstyle{V_{\mathbf{t}(g(0))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{\nabla}_{\mathbf{t}(g(t))}}$$\textstyle{V_{\mathbf{t}(g(t))}}$
(A.11)

where $\tau^{\nabla}$ denotes parallel transport for the connection $\nabla$.
To see this note that the $\mathcal{G}$-invariance condition on $\nabla$ is
equivalent to require the map $\Phi:\mathbf{s}^{*}V\to\mathbf{t}^{*}V$,
$(g,v)\mapsto(g,g\cdot v)$, to preserve parallel transport along any curve
$g(t)$:

$\Phi\circ\tau^{\nabla^{\mathbf{s}}}_{g(t)}=\tau^{\nabla^{\mathbf{t}}}_{g(t)}.$

The claim now follows by observing that since the connections are obtained by
pulling back $\nabla$ along $\mathbf{s}$ and $\mathbf{t}$, we have:

$\tau^{\nabla^{\mathbf{s}}}_{g(t)}=(g(t),\tau^{\nabla}_{\mathbf{s}(g(t))}),\qquad\tau^{\nabla^{\mathbf{t}}}_{g(t)}=(g(t),\tau^{\nabla}_{\mathbf{t}(g(t))}).$

Fix $g\in\mathcal{G}$, and let $x:=\mathbf{t}(g)$. Since $\mathcal{G}$ is
$\mathbf{t}$-connected we can choose a path $g(\varepsilon)$ in the target
fiber $\mathbf{t}^{-1}(x)$ with $g(0)=1_{x}$ and $g(1)=g$. Then we have the
$A$-path $a:[0,1]\to T^{*}M$,
$a(\varepsilon):=g^{-1}(\varepsilon)g^{\prime}(\varepsilon)$, and parallel
transport along $a(\varepsilon)$ for the $A$-connection $\nabla^{A}$ amounts
to acting by the inverse of $g$:

$\tau_{a}^{\nabla^{A}}:V_{\mathbf{t}(g)}\to
V_{\mathbf{s}(g)},\quad\tau_{a}^{\nabla^{A}}(v)=g^{-1}v.$

We prove the proposition by showing the equivalence between
$\mathcal{G}$-invariance in the form of the diagram (A.11) and conditions
(A.10). If we assume $\mathcal{G}$-invariance, then given any $a\in A_{x}$ we
choose a path $g(\varepsilon)$ in the target fiber $\mathbf{t}^{-1}(x)$ such
that $g(0)=1_{x}$ and $g^{\prime}(0)=a$. Then the $A$-path
$a(\varepsilon):=g^{-1}(\varepsilon)g^{\prime}(\varepsilon)$ satisfies

$\tau^{\nabla^{A}}_{a}=\tau^{\nabla}_{\mathbf{s}\circ g}.$

Since $\mathbf{s}\circ g$ is the base path of $a$, by differentiating at
$t=0$, we obtain:

$\nabla^{A}_{a}=\nabla_{\rho(a)}.$

Next, given $a\in A_{x}$ and $v\in T_{x}M$ we choose a smooth family of paths
$g(t,\varepsilon)$ where for each fixed $t$ the path $\varepsilon\mapsto
g(t,\varepsilon)$ lies in the target fiber $\mathbf{t}^{-1}(x(t))$ and starts
at an identity, $g(t,0)=1_{x(t)}$, and moreover:

$\frac{\operatorname{d}g}{\operatorname{d}\varepsilon}(0,0)=a,\quad\frac{\operatorname{d}}{\operatorname{d}t}(\mathbf{s}\circ
g)(0,0)=v.$

It follows from diagram (A.11) that for the family
$\gamma(t,\varepsilon):=\mathbf{s}(g(t,\varepsilon))$ the $\nabla$-parallel
transports commute:

$\tau^{\nabla}_{\varepsilon\mapsto\gamma(t,\varepsilon)}\circ\tau^{\nabla}_{t\mapsto\gamma(t,0)}=\tau^{\nabla}_{t\mapsto\gamma(t,\varepsilon)}\circ\tau^{\nabla}_{\varepsilon\mapsto\gamma(0,\varepsilon)}.$

By differentiation, this gives:

$R^{\nabla}(\rho(a),v)=R^{\nabla}\Big{(}\frac{\operatorname{d}\gamma}{\operatorname{d}\varepsilon}(0,0),\frac{\operatorname{d}\gamma}{\operatorname{d}t}(0,0)\Big{)}=0.$

This argument can be reversed, to prove the converse. If the curvature
condition in (A.10) holds for all $x\in M$, $a\in A_{x}$ and $v\in T_{x}M$,
then one obtains that for any smooth family of paths $g(t,\varepsilon)$, where
for each fixed $t$ the path $\varepsilon\mapsto g(t,\varepsilon)$ lies in a
target fiber, the $\nabla$-parallel transports commute:

$\tau^{\nabla}_{\varepsilon\mapsto\gamma(t,\varepsilon)}\circ\tau^{\nabla}_{t\mapsto\gamma(t,0)}=\tau^{\nabla}_{t\mapsto\gamma(t,\varepsilon)}\circ\tau^{\nabla}_{\varepsilon\mapsto\gamma(0,\varepsilon)},$

where $\gamma(t,\varepsilon):=\mathbf{s}(g(t,\varepsilon))$. Then if the
relationship between connections in (A.10) holds, we conclude that (A.11) must
hold for the path $g(t):=g(t,1)$. Since one can obtain any path $g(t)$ in
$\mathcal{G}$ in this way, the result follows. ∎

This motivates the following:

######  Definition A.9.

Let $A\Rightarrow M$ be a Lie algebroid and let $(V,\nabla^{A})$ be an
$A$-representation. We call a (usual) connection $\nabla$ on $V$ $A$-invariant
if (A.10) holds.

In the presence of an invariant connection we have an operator on IM-forms,
which is the infinitesimal version of the operator on multiplicative forms
that we saw before:

######  Proposition A.10.

Let $A\Rightarrow M$ be a Lie algebroid and $(V,\nabla^{A})$ an
$A$-representation. An $A$-invariant connection $\nabla$ on $V$ induces
operator on IM-forms, as follows:

$\operatorname{d}_{\mathrm{IM}}^{\nabla}:\Omega^{k}_{\mathrm{IM}}(A,V)\to\Omega^{k+1}_{\mathrm{IM}}(A,V),\quad(L,l)\mapsto(\operatorname{d}^{\nabla}L,L-\operatorname{d}^{\nabla}l).$

If $\nabla$ is flat, this operator is a differential.

###### Proof.

If $\nabla^{A}_{a}=\nabla_{\rho(a)}$ then we obtain:

$\mathscr{L}_{\alpha}=\mathscr{L}^{\nabla}_{\rho(\alpha)}.$

On the other, this together with the condition $R^{\nabla}(\rho(a),v)=0$,
gives:

$\displaystyle\operatorname{d}^{\nabla}\mathscr{L}_{\alpha}-\mathscr{L}_{\alpha}\operatorname{d}^{\nabla}$
$\displaystyle=\operatorname{d}^{\nabla}\mathscr{L}^{\nabla}_{\rho(\alpha)}-\mathscr{L}^{\nabla}_{\rho(\alpha)}\operatorname{d}^{\nabla}$
$\displaystyle=(\operatorname{d}^{\nabla})^{2}i_{\rho(\alpha)}-i_{\rho(\alpha)}(\operatorname{d}^{\nabla})^{2}$
$\displaystyle=R^{\nabla}\wedge(i_{\rho(\alpha)}\,\cdot)-i_{\rho(\alpha)}(R^{\nabla}\wedge\cdot)=-(i_{\rho(\alpha)}R^{\nabla})\wedge\cdot=0.$

It follows that if $(L,l)$ satisfies the IM conditions (A.2), so does
$(\operatorname{d}^{\nabla}L,L-\operatorname{d}^{\nabla}l)$. ∎

There is also a natural map:

$\displaystyle\Omega_{\mathrm{IM}}^{k}(A,V)\to\,$
$\displaystyle\Omega^{k}(A,V),\quad(L,l)\mapsto\omega_{l}$ (A.12)
$\displaystyle\text{where}\quad\omega_{l}(\alpha_{1},\dots,\alpha_{k}):=l(\alpha_{1})(\rho(\alpha_{2}),\dots,\rho(\alpha_{k})).$

The corresponding operation on the Lie groupoid level is the restriction of
multiplicative forms to the target fibers. It is immediate to check:

######  Lemma A.11.

Let $A\Rightarrow M$ be a Lie algebroid and $(V,\nabla^{A})$ an
$A$-representation. If $\nabla$ is an $A$-invariant connection then (A.12)
intertwines the differentials:

$(\Omega_{\mathrm{IM}}^{\bullet}(A,V),\operatorname{d}^{\nabla}_{\mathrm{IM}})\to(\Omega^{\bullet}(A,V),\operatorname{d}^{\nabla^{A}}).$

Hence, when $\nabla$ is flat we obtain a map of complexes, and so a map in
cohomology:

$H^{\bullet}_{\mathrm{IM}}(A,V)\to H^{\bullet}(A,V).$ (A.13)

######  Remark A.12.

In the flat case we also obtain a alternative way of expressing IM-forms with
coefficients in $V$, which is similar to (in fact extends) the case of trivial
coefficients. Namely, an $A$-invariant connection $\nabla$ gives a 1:1
correspondence between maps $L:\Gamma(A)\to\Omega^{k}(M;V)$ with symbol
$l:A\to\wedge^{k-1}T^{*}M\otimes V$ and pairs of tensors $(\mu,\zeta)$, where
$\mu:A\to\wedge^{k-1}T^{*}M\otimes V$ and $\zeta:A\to\wedge^{k}T^{*}M\otimes
V$, by setting:

$\mu:=l,\quad\zeta:=L-\operatorname{d}^{\nabla}l.$

When $\nabla$ is flat, one checks that $(L,l)$ is an IM-form if and only if
the pair $(\mu,\zeta)$ satisfies the $\nabla$-analogue of equations (A.1),
namely:

$\displaystyle i_{\rho(\beta)}\mu(\alpha)$
$\displaystyle=-i_{\rho(\alpha)}\mu(\beta),$
$\displaystyle\mu([\alpha,\beta])$
$\displaystyle=\mathscr{L}^{\nabla}_{\rho(\alpha)}\mu(\beta)-i_{\rho(\beta)}\operatorname{d}^{\nabla}\mu(\alpha)-i_{\rho(\beta)}\zeta(\alpha),$
$\displaystyle\zeta([\alpha,\beta])$
$\displaystyle=\mathscr{L}^{\nabla}_{\rho(\alpha)}\zeta(\beta)-i_{\rho(\beta)}\operatorname{d}^{\nabla}\zeta(\alpha).$

###  A.4. Linear forms

A differential form $\omega\in\Omega^{k}(F)$ on the total space of a vector
bundle $F\to M$ is called linear if

$m_{t}^{*}\omega=t\,\omega,\quad\forall t>0.$

where $m_{t}:F\to F$ denotes fiberwise multiplication by $t$. Let
$\Omega_{\mathrm{lin}}^{k}(F)$ the space of linear $k$-forms on $F$.

To any vector bundle map $\Theta:F\to\wedge^{k}T^{*}M$ covering the identity,
one can associate a linear $k$-form:

$\Theta^{*}(\alpha^{k}_{\mathrm{can}})\in\Omega^{k}_{\mathrm{lin}}(F),$

where $\alpha^{k}_{\mathrm{can}}\in\Omega^{k}(\wedge^{k}T^{*}M)$ is the
tautological $k$-form:

$\alpha^{k}_{\mathrm{can}}|_{\xi}:=\operatorname{pr}_{\wedge^{k}T^{*}M}^{*}(\xi).$

We have the following decomposition of linear forms:

######  Lemma A.13.

Any linear $k$-form $\omega\in\Omega^{k}_{\mathrm{lin}}(F)$ can be decomposed
as:

$\omega=\Xi^{*}(\alpha^{k}_{\mathrm{can}})+\operatorname{d}\Theta^{*}(\alpha^{k-1}_{\mathrm{can}}),$

for unique linear maps

$\Xi:F\to\wedge^{k}T^{*}M,\quad\Theta:F\to\wedge^{k-1}T^{*}M.$

Moreover, $\omega$ is closed if and only if
$\omega=\operatorname{d}\Theta^{*}(\alpha^{k-1}_{\mathrm{can}})$, i.e., if and
only if $\Xi\equiv 0$.

We have the corresponding notion on a VB groupoid:

######  Definition A.14.

Given a VB groupoid:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

a form $\omega\in\Omega^{k}_{\mathrm{M}}(F)$ is called a linear multiplicative
form if it is multiplicative for the groupoid $F\rightrightarrows E$ and
linear for the vector bundle $F\to\mathcal{G}$.

One important example is the canonical symplectic form
$\omega_{\mathrm{can}}\in\Omega^{2}_{\mathrm{M}}(T^{*}\mathcal{G})$ on the
cotangent VB groupoid

$\textstyle{T^{*}\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

where $A$ denotes the Lie algebroid of $\mathcal{G}$. The canonical primitive
of $\omega_{\mathrm{can}}$, i.e., the Liouville 1-form
$\alpha_{\mathrm{can}}^{1}\in\Omega^{1}_{\mathrm{M}}(T^{*}\mathcal{G})$, is
also a linear multiplicative 1-form.

We now turn to the infinitesimal version of these results.

######  Definition A.15.

A linear IM form on a VB algebroid

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

is an IM form $(\mu,\zeta)\in\Omega^{k}_{\mathrm{IM}}(B)$ on the algebroid
$B\to E$ satisfying

$m_{t}^{*}\mu=t\,\mu,\quad m_{t}^{*}\zeta=t\,\zeta,\quad\forall t>0,$

where $m_{t}:B\to B$ is the scalar multiplication on the vector bundle $B\to
A$.

Under the correspondence between multiplicative forms and IM forms, linear
multiplicative forms on a VB groupoid correspond to linear IM forms on the
associated VB algebroid.

For any Lie algebroid $A\to M$, its cotangent algebroid is a VB algebroid:

$\textstyle{T^{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

and carries a canonical closed, linear, multiplicative 2-form
$\mu_{\mathrm{can}}\in\Omega^{2}_{\mathrm{IM}}(T^{*}A)$, namely the canonical
isomorphism (called the reverse isomorphism in [26]):

$\mu_{\mathrm{can}}:T^{*}A\to T^{*}A^{*}.$

This IM 2-form is exact with canonical primitive the linear, multiplicative
1-form
$(f_{\mathrm{can}},\mu_{\mathrm{can}})\in\Omega^{1}_{\mathrm{IM}}(T^{*}A)$,
where $f_{\mathrm{can}}:T^{*}A\to\mathbb{R}$ is the tautological map

$f_{\mathrm{can}}(\alpha)=\alpha\Big{(}\frac{\operatorname{d}}{\operatorname{d}t}\big{|}_{t=0}e^{t}u\Big{)},\quad\text{
for }\quad\alpha\in T^{*}_{u}A.$

## References

  * [1] C. Arias Abad and M. Crainic, Representations up to homotopy and Bott’s spectral sequence for Lie groupoids, _Adv. Math._ 248 (2013), 416–452.
  * [2] K. Behrend, On the de Rham cohomology of differential and algebraic stacks, _Adv. Math._ 198 (2005), no. 2, 583–622.
  * [3] A. D. Blaom, Geometric structures as deformed infinitesimal symmetries, _Trans. Amer. Math. Soc._ 358 (2006), no. 8, 3651–3671.
  * [4] A. D. Blaom, Lie algebroids and Cartan’s method of equivalence, _Trans. Amer. Math. Soc._ 364 (2012), no. 6, 3071–3135.
  * [5] A. D. Blaom, Cartan connections on Lie groupoids and their integrability, _SIGMA Symmetry Integrability Geom. Methods Appl._ 12 (2016), No. 114, 26 pp.
  * [6] H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level, _Math. Ann._ 353 (2012), no. 3, 663–705.
  * [7] H. Bursztyn, A. Cabrera and M. del Hoyo, Vector bundles over Lie groupoids and algebroids, _Adv. Math._ 290 (2016), 163–207.
  * [8] H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, _Duke Math. J._ 123 (2004), no. 3, 549–607.
  * [9] H. Bursztyn and T. Drummond, Lie theory of multiplicative tensors, _Math. Ann._ 375 (2019), no. 3–4, 1489–1554.
  * [10] A. Cabrera, T. Drummond, Van Est isomorphism for homogeneous cochains, _Pacific J. Math._ 287 (2017), no. 2, 297–336.
  * [11] M. Crainic, R. L. Fernandes and I. Mărcu  t
---
,
, _Lectures on Poisson geometry_ , Graduate Studies in Mathematics, 217.
American Mathematical Society, Providence, RI, 2021, 479 pp.

  * [12] M. Crainic and J. N. Mestre, Orbispaces as differentiable stratified spaces, _Lett. Math. Phys._ 108 (2018), no. 3, 805–859.
  * [13] M. Crainic, F. Schätz and I. Struchiner, A survey on stability and rigidity results for Lie algebras, _Indag. Math. (N.S.)_ 25 (2014), no. 5, 957–976.
  * [14] M. Crainic, M.A. Salazar and I. Struchiner, Multiplicative forms and Spencer operators, _Mathematische Zeitschrift_ 279 (2015), 939–979.
  * [15] M. Crainic and I. Struchiner, On the linearization theorem for proper Lie groupoids; _Ann. Scient. Éc. Norm. Sup._ $4^{e}$ série, 46 (2013), 723–746.
  * [16] M. del Hoyo, Lie Groupoids and their Orbispaces, _Portugaliae Mathematica_ 70 (2013), 161–209.
  * [17] M. del Hoyo, Complete connections on fiber bundles, _Indag. Math. (N.S.)_ 27 (2016), no. 4, 985–990.
  * [18] T. Drummond and L. Egea, Differential forms with values in VB-groupoids and its Morita invariance, _J. Geometry and Physics_ 135 (2019), 42–69.
  * [19] R. L. Fernandes and I. Mărcu  t
---
,
, Poisson geometry around Poisson submanifolds, 2022, preprint
arXiv:2205.11457.

  * [20] P. Frejlich, Submersions by Lie algebroids, _J. Geom. Phys._ 137 (2019), 237–246.
  * [21] A. Gracia-Saz and R.A. Mehta, VB-groupoids and representation theory of Lie groupoids, _J. Symplectic Geom._ 15 (2017), no. 3, 741–783.
  * [22] N. Hitchin, Lectures on special Lagrangian submanifolds, in _Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds_ , AMS/IP Stud. Adv. Math., vol. 23, AMS, Providence, RI, 2001, 151–182.
  * [23] M. Jotz Lean, C. Ortiz, Foliated groupoids and infinitesimal ideal systems, _Indag. Math. (N.S.)_ 25 (2014), no. 5, 1019–1053.
  * [24] A. Kotov, T. Ströbl, Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids, _Lett. Math. Phys._ 108 (2018), n. 3, 737–756.
  * [25] C. Laurent-Gengoux, M. Stiénon and P. Xu, Non-abelian differentiable gerbes, _Adv. Math._ 220 (2009), no. 5, 1357–1427.
  * [26] K.C.H. Mackenzie, _General theory of Lie groupoids and Lie algebroids_. London Mathematical Society Lecture Note Series, 213. Cambridge University Press, Cambridge, 2005.
  * [27] I. Mărcu  t
---
,
, Rigidity around Poisson submanifolds, _Acta Math._ 213 (2014), no. 1,
137–198.

  * [28] D. McDuff, D. Salamon, _Introduction to symplectic topology_ , Oxford Graduate Texts in Mathematics, 3rd Edition, Oxford University Press, Oxford, 2017.
  * [29] I. Moerdijk and J. Mrčun, _Introduction to foliations and Lie groupoids_. Cambridge Studies in Advanced Mathematics, 91. Cambridge University Press, Cambridge, 2003.
  * [30] K.M. Murray. Bundle gerbes, _J. London Math. Soc._ 54 (1996) 403– 416.
  * [31] M.A. Salazar, _Pfaffian groupoids_ , PhD Thesis, Utrecht University (2013) https://dspace.library.uu.nl/handle/1874/275521.
  * [32] X. Tang, Deformation quantization of pseudo-symplectic (Poisson) groupoids, _Geom. Funct. Anal._ 3 (2006), 731–766.
  * [33] Y. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, _Banach Center Publ._ 54 (2001), 249–274.
  * [34] Y. Vorobjev, Poisson equivalence over a symplectic leaf, Quantum algebras and Poisson geometry in mathematical physics, _Amer. Math. Soc. Transl._ Ser. 2, 216 (2005), 241–277.
  * [35] A. Weinstein, Linearization of regular proper groupoids, _J. Inst. Math. Jussieu._ 1 (2002), 493–511.
  * [36] N.T. Zung, Proper groupoids and momentum maps: linearization, affinity and convexity, _Ann. Scient. Éc. Norm. Sup._ $4^{e}$ série, 39 (2006), 841-869.