# Dirac structures and Nijenhuis operators

Henrique Bursztyn  IMPA, Estrada Dona Castorina 110, Rio de Janeiro,
22460-320, Brazil<EMAIL_ADDRESS>,  Thiago Drummond  Departamento de
Matemática, Instituto de Matemática, Universidade Federal do Rio de Janeiro,
Caixa Postal 68530, Rio de Janeiro, RJ, 21941-909, Brasil.
<EMAIL_ADDRESS>and  Clarice Netto  IMPA, Estrada Dona Castorina 110, Rio
de Janeiro, 22460-320, Brazil<EMAIL_ADDRESS>
###### Abstract.

We introduce a notion of compatibility between (almost) Dirac structures and
$(1,1)$-tensor fields extending that of Poisson-Nijenhuis structures. We study
several properties of the “Dirac-Nijenhuis” structures thus obtained,
including their connection with holomorphic Dirac structures, the geometry of
their leaves and quotients, as well as the presence of hierarchies. We also
consider their integration to Lie groupoids, which includes the integration of
holomorphic Dirac structures as a special case.

##  1\. Introduction

There are several situations of interest in geometry that involve the
compatibility of a given geometric structure with a suitable $(1,1)$-tensor
field. An illustrative example is the study of holomorphic objects on a
complex manifold. In this paper we introduce a notion of compatibility between
(almost) Dirac structures and general $(1,1)$-tensor fields that extends, and
is inspired by, the theory of Poisson-Nijenhuis structures [31, 37]. In this
context the compatibility of a Poisson structure with a Nijenhuis operator has
remarkable consequences, such as the presence of bi-hamiltonian structures
[24, 35], Poisson hierarchies and integrals of motion that underpin many
integrable systems, see e.g. [3, 19, 20, 36, 37]. In Poisson-Lie theory,
Poisson-Nijenhuis structures (see [29]) play a central role in certain
quantization schemes [1, 2]. Poisson-Nijenhuis structures also emerge
naturally in the characterization of holomorphic Poisson structures in terms
of their “real parts” [32, $\S$ 2.3], and this viewpoint is key to establish
the holomorphic version of the integration of Poisson structures by symplectic
groupoids [33, 41] (see also [17]). The notion of Dirac-Nijenhuis structure
introduced in this paper provides a similar viewpoint to holomorphic Dirac
structures and leads to more general integration results.

Dirac structures [16] are common generalizations of Poisson structures and
presymplectic forms, originally motivated by constrained mechanics (as the
intrinsic geometry of submanifolds of Poisson manifolds), with a wide array of
recent applications, see e.g. [4, 38] and references therein. A Dirac
structure on a manifold $M$ is a subbundle $L\subset\mathbb{T}M:=TM\oplus
T^{*}M$ that is lagrangian (for the canonical symmetric pairing on
$\mathbb{T}M$) and involutive with respect to the Courant-Dorfman bracket; any
Poisson structure $\pi$ on $M$ may be regarded as a Dirac structure $L_{\pi}$
given by the graph of $\pi^{\sharp}:T^{*}M\to TM$, $\alpha\mapsto
i_{\alpha}\pi$. Dirac structures have also been considered in the holomorphic
category, e.g. in connection with generalized Kähler geometry [25] and
Poisson-Lie theory [9].

To better explain our results, recall that, in Poisson-Nijenhuis theory, one
finds a notion of compatibility between a bivector field
$\pi\in\mathfrak{X}^{2}(M)$ and a $(1,1)$-tensor field $r:TM\to TM$ that is
expressed by two conditions:

  * (1)

$\pi^{\sharp}\circ r^{*}=r\circ\pi^{\sharp}$,

  * (2)

For $X\in\mathfrak{X}(M)$ and $\alpha\in\Omega^{1}(M)$,

$R_{\pi}^{r}(X,\alpha):=\pi^{\sharp}(\mathcal{L}_{X}r^{*}(\alpha)-\mathcal{L}_{r(X)}\alpha)-(\mathcal{L}_{\pi^{\sharp}(\alpha)}r)(X)=0.$

The expression in (2) is known as the concomitant of $\pi$ and $r$. The
starting point of this paper is a new interpretation of these conditions that
clarifies how to extend them to Dirac structures.

By considering the map $(r,r^{*}):\mathbb{T}M\to\mathbb{T}M$, the purely
algebraic condition (1) can be rephrased in terms of $L_{\pi}$ as
$(r,r^{*})(L_{\pi})\subset L_{\pi}$. Finding a parallel reformulation of the
vanishing of the concomitant is less evident. For that, a central role is
played by two “connection-like” operators,
$D^{r}:\mathfrak{X}(M)\to\Omega^{1}(M,TM)$ and
$D^{r,*}:\Omega^{1}(M)\to\Omega^{1}(M,T^{*}M)$, given by

$\displaystyle D^{r}_{X}(Y)=i_{X}(D^{r}(Y)):=(\mathcal{L}_{Y}r)(X),$
$\displaystyle
D^{r,*}_{X}(\alpha)=i_{X}(D^{r,*}(\alpha)):=\mathcal{L}_{X}r^{*}(\alpha)-\mathcal{L}_{r(X)}\alpha$

for $X,Y\in\mathfrak{X}(M)$, $\alpha\in\Omega^{1}(M)$. Setting
$\mathbb{D}^{r}:=(D^{r},D^{r,*})$, condition (2) is equivalent to
$\mathbb{D}^{r}_{X}(\Gamma(L_{\pi}))\subseteq\Gamma(L_{\pi})$ for all
$X\in\mathfrak{X}(M)$ [21, Sec. 2]. This leads to the following definition: a
lagrangian subbundle $L\subset\mathbb{T}M$ is said to be compatible with a
$(1,1)$-tensor field $r$ if

(1.1) $(r,r^{*})(L)\subset
L,\qquad\mathbb{D}^{r}_{X}(\Gamma(L))\subseteq\Gamma(L),$

for all $X\in\mathfrak{X}(M)$. When $L$ is a Dirac structure and $r$ is a
Nijenhuis operator, the pair $(L,r)$ is called a Dirac-Nijenhuis structure.
Besides recovering Poisson-Nijenhuis structures when $L$ is the graph of a
Poisson structure, Dirac-Nijenhuis structures agree with the so-called $\Omega
N$-structures111In this paper, we will refer to $\Omega N$-structures as
presymplectic-Nijenhuis structures. of [37] when $L$ is the graph of a closed
2-form (see $\S$ 3). We stress that the notion of compatibility in (1.1) does
not rely on the Courant-integrability of $L$ (or even the fact that it is
lagrangian) nor the Nijenhuis-integrability of $r$. This degree of generality
makes it possible to accommodate, with no extra effort, weaker forms of
integrability of $L$ and $r$, such as the “quasi-Nijenhuis” structures of [41]
(see [22] for recent applications), see Remark 3.14. Other approaches to
Dirac-Nijenhuis structures can be found in the literature, see [12, 14, 26,
27]; we discuss how they compare with ours in Appendix B. As we will discuss
in [8], our Dirac-Nijenhuis structures fit into a more general theory of
“Courant-Nijenhuis algebroids”.

We pass to the description of the main results and structure of the paper. We
start by explaining in $\S$ 2 the general context in which the operators
$D^{r}$ and $D^{r,*}$ naturally arise, namely that of generalized derivations
of degree 1 [21, Sec. 2], or 1-derivations for simplicity. Just as usual
derivations of a vector bundle are in one-to-one correspondence with its
linear vector fields (see e.g. [34, Thm. 3.4.5]), 1-derivations on a vector
bundle $E\to M$ are derivation-like objects that correspond to linear
$(1,1)$-tensor fields on the total space $E$, see Theorem 2.1. For a
$(1,1)$-tensor field $r$ on $M$, the triples $(D^{r},r,r)$ and
$(D^{r,*},r^{*},r)$ are examples of (dual) 1-derivations on $TM$ and $T^{*}M$,
respectively; their corresponding linear $(1,1)$-tensor fields are the so-
called tangent and cotangent lifts [43] of $r$, denoted by

$r^{tg}:T(TM)\to T(TM),\qquad r^{cotg}:T(T^{*}M)\to T(T^{*}M).$

Similarly, $(\mathbb{D}^{r},(r,r^{*}),r)$ is a 1-derivation on $\mathbb{T}M$
associated with $(r^{tg},r^{cotg}):T(\mathbb{T}M)\to T(\mathbb{T}M)$. With
this correspondence at hand, the two conditions (1.1) in the definition of a
Dirac-Nijenhuis structure $(L,r)$ are seen to be equivalent to the single
condition

$(r^{tg},r^{cotg})(TL)\subset TL.$

In the particular case of Poisson-Nijenhuis structures $(\pi,r)$, this last
condition becomes

$T\pi^{\sharp}\circ r^{cotg}=r^{tg}\circ T\pi^{\sharp},$

which provides a re-formulation of the compatibility conditions (1) and (2) as
one natural equation (see Proposition 3.3). It is also recalled in Example 2.3
that holomorphic structures on (real) vector bundles (regarded as partial flat
connections as in [39]) can be conveniently seen as particular 1-derivations.
When $r$ is a complex structure on $M$, the $1$-derivations determined by
$D^{r}$ and $D^{r,*}$ codify the holomorphic structures on $TM$ and $T^{*}M$
arising from their usual identifications with $T^{1,0}M$ and $(T^{1,0}M)^{*}$,
respectively (see Example 2.6). By means of these identifications, we prove in
Proposition 3.16 that Dirac-Nijenhuis structures $(L,r)$, with
$L\subset\mathbb{T}M$, are equivalent to holomorphic Dirac structures
$\mathcal{L}\subset\mathbb{T}^{1,0}M=T^{1,0}M\oplus(T^{1,0}M)^{*}$.

After the definition and basic examples of Dirac-Nijenhuis structures are
presented in §3, in §4 we discuss some of their fundamental properties, such
as their behavior with respect to backward and forward Dirac maps
(Propositions 4.1 and 4.4), the geometry of their presymplectic leaves and
Poisson quotients, as well as the existence of admissible functions in
involution ($\S$ 4.3) and hierarchies of Dirac structures ($\S$ 4.4).

The last two sections of the paper concern integration to Lie groupoids. A
central result in Poisson geometry is the infinitesimal-global correspondence
relating Poisson manifolds to objects known as symplectic groupoids, see e.g.
[13, 15, 18] and references therein; the extension of this fact to Dirac
geometry is a correspondence between Dirac manifolds and presymplectic
groupoids [6]. In this paper we enhance this picture by considering additional
compatible $(1,1)$-tensor fields at infinitesimal and global levels.

As observed in $\S$ 5, the notion of compatibility of a 2-form with a given
$(1,1)$-tensor field $r$ naturally extends to differential forms of arbitrary
degrees (one can in fact consider the compatibility of $r$ with arbitrary
tensor fields through suitable extensions of the operators $D^{r}$ and
$D^{r,*}$, though this is not discussed here), see Definition 5.1. We consider
Lie groupoids $\mathcal{G}$ equipped with a compatible pair $(\omega,K)$,
where $\omega$ is a multiplicative differential form and $K$ is a
multiplicative $(1,1)$-tensor field, and provide their description at Lie-
algebroid level in Theorem 5.3. As a consequence, when $K$ is a complex
structure making $\mathcal{G}$ into a holomorphic Lie groupoid, we obtain in
Theorem 5.5 a holomorphic version of [5, Thm. 4.6] describing the
infinitesimal counterparts of multiplicative differential forms. By means of
these general results, we derive in $\S$ 6 the infinitesimal-global
correspondence between Dirac-Nijenhuis structures and presymplectic-Nijenhuis
groupoids in Theorem 6.3 (see also Remark 6.4), as well as the special case of
this correspondence relating holomorphic Dirac structures and holomorphic
presymplectic groupoids, see Theorems 6.8 and 6.9. (Concrete constructions of
holomorphic presymplectic groupoids integrating holomorphic Dirac structures
arising in Poisson-Lie theory can be found in [9].) Our differentiation-
integration results are schematically illustrated in the next diagram.

(1,1)-tensor fieldCompatibilitywith Dirac structuresIntegrationDirac-
NijenhuisStructuresPresymplectic-NijenhuisgroupoidsIM forms compatiblewith
1-derivationsMultiplicative formscompatible with multiplicative
(1,1)-tensorsDifferentiationCompatibilitywith differential formsHolomorphic IM
formsHolomorphic Dirac
structuresHolomorphicpresymplecticgroupoidsHolomorphicmultiplicative forms
Figure 1. Compatibility vs. integration

Acknowledgments. We thank Pedro Frejlich, Jiang-Hua Lu, Juan Carlos Marrero
and Igor Mencattini for stimulating discussions. We are grateful to CNPq,
Capes and Faperj for financial support.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries: linear (1,1)-tensors and 1-derivations
  3. 3 Compatibility of Dirac structures and $(1,1)$-tensor fields
  4. 4 Properties of Dirac-Nijenhuis structures
  5. 5 Compatibility of multiplicative forms and $(1,1)$-tensor fields
  6. 6 Integration of Dirac-Nijenhuis structures
  7. A Proof of Theorem 2.1
  8. B Other notions of Dirac-Nijenhuis structures

##  2\. Preliminaries: linear (1,1)-tensors and 1-derivations

In this section we explain the correspondence between linear $(1,1)$-tensor
fields on vector bundles and objects that we call $1$-derivations, a higher-
degree analogue of the equivalence between linear vector fields and
derivations on vector bundles. We then explain how the operators $D^{r}$ and
$D^{r,*}$ naturally arise in this context.

###  2.1. The correspondence

Let $q:E\to M$ be a (real) vector bundle, and consider its tangent bundle
$TE\to E$. Recall that $TE$ carries a second vector-bundle structure $Tq:TE\to
TM$ (its “tangent prolongation”) making it a double vector bundle [34]. We
denote by $V_{E}=\mathrm{Ker}(Tq)\subset TE$ the vertical bundle of $E$.

A $(1,1)$-tensor field $K\in\Gamma(T^{*}E\otimes TE)=\Omega^{1}(E,TE)$ on the
total space of $E$ is called linear if, regarded as a map $K:TE\to TE$, it is
a vector-bundle morphism

$\textstyle{TE\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K}$$\textstyle{TE\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TM\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{TM,}$

for some $r\in\Omega^{1}(M,TM)$. Note that linear $(1,1)$-tensors are higher-
degree analogs of linear vector fields on $E$, which are elements in
$\Omega^{0}(E,TE)=\Gamma(TE)$ that, viewed as maps $E\to TE$, are vector-
bundle morphisms from $E\to M$ to $TE\to TM$. (Equivalently, linear vector
fields are characterized as those with flows given by vector-bundle
automorphisms.)

Linear vector fields on $E$ are known to be in one-to-one correspondence with
derivations of $E$, i.e., pairs $(\Delta,X)$, where
$\Delta:\Gamma(E)\to\Gamma(E)$ is $\mathbb{R}$-linear and
$X\in\mathfrak{X}(M)$ is a vector field, satisfying the Leibniz rule

$\Delta(fu)=f\Delta(u)+(\mathcal{L}_{X}f)u,$

for $u\in\Gamma(E)$ and $f\in C^{\infty}(M)$. One direct way to define the
derivation $(\Delta,X)$ corresponding to a linear vector field $Z$ is via the
relations

(2.1) $\Delta(u)^{\uparrow}=[u^{\uparrow},Z],\qquad
q^{*}\langle\alpha,X\rangle=\langle q^{*}\alpha,Z\rangle,$

for $u\in\Gamma(E),\,\alpha\in\Omega^{1}(M)$; here
$u^{\uparrow}\in\mathfrak{X}(E)$ is the vertical lift  of $u\in\Gamma(E)$,

$u^{\uparrow}(e)=\frac{d}{d\tau}\Big{|}_{\tau=0}(e+\tau u(x)),\qquad\mbox{ for
}e\in E|_{x}.$

As proven in [7, Sec. 4.4], this correspondence between linear vector fields
and derivations is part of a general characterization of linear tensor fields
on vector bundles, which establishes, in particular, a bijection between
linear tensors of type $(1,p)$ (i.e., vector-valued forms) on $E$ and objects
called generalized derivations, see [21, Sec. 2]. More specifically, linear
$(1,1)$-tensor fields on $E$ bijectively correspond to generalized derivations
of degree 1 (or 1-derivations for simplicity), which are triples $(D,l,r)$,
where $D:\Gamma(E)\to\Omega^{1}(M,E)$ is $\mathbb{R}$-linear,
$l\in\mathrm{End}(E)$, $r\in\Omega^{1}(M,TM)$, satisfying the Leibniz-type
rule

(2.2) $D(fu)=fD(u)+df\wedge l(u)-r^{*}(df)\wedge u,$

for $u\in\Gamma(E)$ and $f\in C^{\infty}(M)$. In particular, each
$X\in\mathfrak{X}(M)$ gives rise to an operator $D_{X}:\Gamma(E)\to\Gamma(E)$,
$D_{X}(u):=i_{X}(D(u))$, satisfying

(2.3) $D_{X}(fu)=fD_{X}(u)+(\mathcal{L}_{X}f)l(u)-(\mathcal{L}_{r(X)}f)u.$

We collect the main facts that we will need regarding linear $(1,1)$-tensors
in the next result. Consider the map
$\mathcal{V}:\Omega^{1}(M,E)\to\Omega^{1}(E,TE)$, $\alpha\otimes u\mapsto
q^{*}\alpha\otimes u^{\uparrow}$, for $\alpha\in\Omega^{1}(M)$ and
$u\in\Gamma(E)$.

######  Theorem 2.1.

  * (i)

Any linear $(1,1)$-tensor $K$ on $E\to M$ defines a 1-derivation $(D,l,r)$ via

(2.4)
$\mathcal{L}_{u^{\uparrow}}K=\mathcal{V}(D(u)),\;\;\;K(u^{\uparrow})=l(u)^{\uparrow},\;\;\;\langle
K,q^{*}\beta\rangle=q^{*}\langle\beta,r\rangle$

for $u\in\Gamma(E)$ and $\beta\in\Omega^{1}(M)$, and this correspondence
establishes a bijection between linear $(1,1)$-tensor fields and
1-derivations.

  * (ii)

For $i=1,2$, let $E_{i}\to M$ be a vector bundle equipped with a linear
$(1,1)$-tensor field $K_{i}$, with corresponding 1-derivation
$(D^{i},l_{i},r_{i})$. Let $\Phi:E_{1}\to E_{2}$ be a vector-bundle map over
the identity (and we keep the same notation for the induced map between spaces
of sections). Then $T\Phi\circ K_{1}=K_{2}\circ T\Phi$ if and only if

(2.5) $\Phi\circ D^{1}_{X}=D^{2}_{X}\circ\Phi,\;\;\;\Phi\circ
l_{1}=l_{2}\circ\Phi,\;\;\;r_{1}=r_{2}.$

for all $X\in\mathfrak{X}(M)$.

Part (i) follows from the general results in[7]. We present a proof of part
(ii) in Appendix A. A consequence of the theorem (keeping the notation of part
(i)) is that a subbundle $(F\to M)\subseteq(E\to M)$ satisfies $K(TF)\subseteq
TF$ if and only if $l(F)\subseteq F$ and $D(\Gamma(F))\subseteq\Gamma(F)$.

######  Example 2.2 (Connections).

The one-to-one correspondence between linear $(1,1)$-tensors and
$1$-derivations on $E\to M$ in Theorem 2.1 (i) restricts to a bijection
between linear $(1,1)$-tensors $\Theta:TE\to TE$ satisfying
$\Theta^{2}=\Theta$ and $\mathrm{Im}(\Theta)=V_{E}$ and 1-derivations
$(D,l,r)$ such that $l=\mathrm{id}_{E}$ and $r=0$. Note that any such $\Theta$
is completely determined by the choice of a linear subbundle (with respect to
both linear structures on $E$) $H=\mathrm{Ker}(\Theta)\subset TE$ such that
$TE=V_{E}\oplus H$, whereas $D:\Gamma(E)\to\Omega^{1}(M,E)$ defines a
1-derivation of the type $(D,\mathrm{id_{E}},0)$ if and only if it is a
connection operator on $E$. In other words, we recover the equivalence between
two viewpoints to linear connections on $E$: via horizontal bundles or
covariant derivatives, see e.g. [28, $\S$11.11]. (The operators $\Theta$ above
are closely related to the so-called “connectors” of linear connections [28,
$\S$11.10].)

######  Example 2.3 (Holomorphic structures).

A holomorphic vector bundle can be defined as a complex vector bundle over a
complex manifold $M$ equipped with a flat $T^{0,1}M$-connection [39].
Equivalently, one may regard a holomorphic vector bundle as a real vector
bundle $E\to M$ together with a holomorphic structure consisting of (1) a
complex structure $r:TM\to TM$ on $M$, (2) a fiberwise complex structure on
$E$, i.e., an endomorphism $l:E\to E$ with $l^{2}=-\mathrm{id}_{E}$, and (3) a
flat $T^{0,1}M$-connection $\nabla$ on the complex vector bundle $(E,l)$. The
partial connection $\nabla:\Gamma(T^{0,1}M)\times\Gamma(E)\to\Gamma(E)$ is the
same as a Dolbeault operator
$\overline{\partial}:\Gamma(E)\to\Gamma((T^{0,1}M)^{*}\otimes E)$, so we refer
to it as a Dolbeault connection. A (local) section of $E$ is holomorphic if
and only if it is flat. The holomorphic structure on $E$ may be alternatively
represented by the triple $(D^{\mathrm{hol}},l,r)$, where
$D^{\mathrm{hol}}:\Gamma(E)\to\Omega^{1}(M,E)$ is given by

(2.6) $D^{\mathrm{hol}}_{X}(u):=l(\nabla_{X+\mathbf{i}\,r(X)}u).$

One may verify that $(D^{\mathrm{hol}},l,r)$ is in fact a 1-derivation on $E$,
that we will call the Dolbeault 1-derivation associated with a holomorphic
vector bundle. Such 1-derivations admit an intrinsic characterization as the
1-derivations $(D,l,r)$ on a vector bundle $E\to M$ satisfying the following
set of equations (see [7]):

(2.7)
$r^{2}=-\mathrm{id}_{TM},\;\;\;l^{2}=-\mathrm{id}_{E},\;\;\;l(D_{X}(u))+D_{r(X)}(u)=0,$

as well as

(2.8)
$\mathcal{N}_{r}(X,Y)=0,\;\;\,l(D_{X}(u)=D_{X}(l(u)),\;\;\,l(D_{[X,Y]}(u))=[D_{X},D_{Y}](u)+D_{[X,Y]_{r}}(u),$

for all $X,Y\in\mathfrak{X}(M)$ and $u\in\Gamma(E)$, where $\mathcal{N}_{r}$
is the Nijenhuis torsion of $r$ (see (2.16) below) and $[\cdot,\cdot]_{r}$ is
given by $[X,Y]_{r}=[r(X),Y]+[X,r(Y)]-r([X,Y]$. Note that these equations just
spell out conditions (1), (2) and (3) above, for $\nabla$ defined by
$\nabla_{X+\mathbf{i}\,r(X)}u=-l(D_{X}(u))$. The linear (1,1)-tensor on $E$
associated to a Dolbeault 1-derivation $(D^{\mathrm{hol}},l,r)$ via Theorem
2.1 is the complex structure $J$ on the total space of $E\to M$, see [7, §6]
and [21, §5] (here (2.7) codifies the fact that $J^{2}=-\mathrm{id}_{TE}$
while conditions (2.8) give the vanishing of its Nijenhuis torsion).

Analogously to ordinary derivations, any 1-derivation $(D,l,r)$ on $E$ gives
rise to a dual 1-derivation on $E^{*}$ defined by the triple
$(D^{*},l^{*},r)$, where $l^{*}:E^{*}\to E^{*}$ is dual to $l$, and $D^{*}$ is
defined by the equation

(2.9) $\langle
D^{*}_{X}(\xi),u\rangle=\mathcal{L}_{X}\langle\xi,l(u)\rangle-\mathcal{L}_{r(X)}\langle\xi,u\rangle-\langle\xi,D_{X}(u)\rangle,$

for $u\in\Gamma(E)$, $\xi\in\Gamma(E^{*})$, and $X\in\mathfrak{X}(M)$.

One can also describe this duality in terms of the corresponding linear
$(1,1)$-tensor fields. For such $K:TE\to TE$ with corresponding 1-derivation
$(D,l,r)$, we denote by $K^{\top}:T(E^{*})\to T(E^{*})$ the linear
$(1,1)$-tensor corresponding to $(D^{*},l^{*},r)$. Consider the non-degenerate
pairing
$\langle\\!\langle\cdot,\,\cdot\rangle\\!\rangle:TE\times_{TM}T(E^{*})\to
TM\times\mathbb{R}$ obtained by differentiation of the natural pairing between
$E$ and $E^{*}$. One can verify that, for every $(U,V)\in
TE\times_{TM}T(E^{*})$,

(2.10) $\langle\\!\langle K(U),K^{\top}(V)\rangle\\!\rangle=\langle\\!\langle
Tl(U),V\rangle\\!\rangle.$

###  2.2. The operators $D^{r}$ and $D^{r,*}$

For a given $(1,1)$-tensor field $r$ on $M$, consider the following nonlinear
operators: $D^{r}:\mathfrak{X}(M)\to\Omega^{1}(M,TM)$,

(2.11) $\displaystyle D^{r}_{X}(Y)=i_{X}(D^{r}(Y))$
$\displaystyle:=(\mathcal{L}_{Y}r)(X)$ $\displaystyle=[Y,r(X)]-r([Y,X]),$

for $X,Y\in\mathfrak{X}(M)$, and
$D^{r,*}:\Omega^{1}(M)\to\Omega^{1}(M,T^{*}M)$,

(2.12) $\displaystyle D^{r,*}_{X}(\alpha)=i_{X}(D^{r,*}(\alpha))$
$\displaystyle:=\mathcal{L}_{X}r^{*}(\alpha)-\mathcal{L}_{r(X)}\alpha$
$\displaystyle=i_{X}d(r^{*}\alpha)-i_{r(X)}d\alpha,$

for $X\in\mathfrak{X}(M)$ and $\alpha\in\Omega^{1}(M)$. (The last equality in
(2.12) follows from Cartan’s formula.) We now recall from [21, §3] how these
operators naturally arise as 1-derivations.

Any $(1,1)$-tensor field $r$ on $M$ gives rise to two $(1,1)$-tensor fields,
one on $TM$ and another on $T^{*}M$,

$r^{\scriptscriptstyle{tg}}\in\Omega^{1}(TM,T(TM)),\qquad
r^{\scriptscriptstyle{cotg}}\in\Omega^{1}(T^{*}M,T(T^{*}M)),$

known as the tangent and cotangent lifts of $r$, see e.g. [43]. The tangent
lift $r^{\scriptscriptstyle{tg}}:T(TM)\to T(TM)$ is obtained from the tangent
map $Tr$ via $J_{M}\circ Tr\circ J_{M}$, where $J_{M}:T(TM)\to T(TM)$ is the
canonical involution of the double tangent bundle [34, §9.6]. For the
definition of the cotangent lift $r^{\scriptscriptstyle{cotg}}$, consider the
canonical symplectic structure $\omega_{\mathrm{can}}$ on the cotangent bundle
$T^{*}M$, and let $\phi_{r}:=r^{*}:T^{*}M\to T^{*}M$. Then $r^{cotg}$ is
determined by

$i_{r^{cotg}(U)}\omega_{\mathrm{can}}=i_{U}(\phi_{r}^{*}\,\omega_{\mathrm{can}}),$

see also [11].

The next result is proven in [21, §3].

######  Proposition 2.4.

For $r\in\Omega^{1}(M,TM)$, the $(1,1)$-tensor fields
$r^{\scriptscriptstyle{tg}}$ and $r^{\scriptscriptstyle{cotg}}$ are linear and
satisfy $r^{cotg}=(r^{tg})^{\top}$. Their corresponding 1-derivations are
$(D^{r},r,r)$ and $(D^{r,*},r^{*},r)$.

In particular, the operators $D^{r}$ and $D^{r,*}$ in (2.11) and (2.12)
satisfy the Leibniz-type identity in (2.2) and are dual to one another as
1-derivations:

(2.13) $\langle
D^{r,*}_{X}(\alpha),Y\rangle=\mathcal{L}_{X}\langle\alpha,r(Y)\rangle-\mathcal{L}_{r(X)}\langle\alpha,Y\rangle-\langle\alpha,D^{r}_{X}(Y)\rangle.$

######  Remark 2.5 (Naturality properties).

Consider $r_{i}\in\Omega(M_{i},TM_{i})$, $i=1,2$, and a smooth map
$\psi:M_{1}\to M_{2}$. Suppose that $r_{1}$ and $r_{2}$ are $\psi$-related,
i.e., $r_{2}\circ d\psi=d\psi\circ r_{1}$. Let $X_{i}\in\mathfrak{X}(M_{i})$,
$i=1,2$, be vector fields which are also $\psi$-related; we use the notation
$X_{1}\sim_{\psi}X_{2}$. The following can be directly verified:

(2.14) $\displaystyle
Y_{1}\sim_{\psi}Y_{2}\;\;\Rightarrow\;\;D^{r_{1}}_{X_{1}}(Y_{1})\sim_{\psi}D^{r_{2}}_{X_{2}}(Y_{2}),$
(2.15) $\displaystyle
D^{r_{1},*}_{X_{1}}(\psi^{*}\beta)=\psi^{*}D^{r_{2},*}_{X_{2}}(\beta),\quad\forall\,\beta\in\Omega^{1}(M_{2}).$

$\diamond$

We will be particularly interested in Nijenhuis operators
$r\in\Omega^{1}(M,TM)$, i.e., endomorphisms $r:TM\to TM$ with vanishing
Nijenhuis torsion $\mathcal{N}_{r}\in\Omega^{2}(M,TM)$:

(2.16) $\mathcal{N}_{r}(X,Y):=[r(X),r(Y)]-r([r(X),Y]+[X,r(Y)]-r[X,Y])=0.$

We note that the Nijenhuis torsion of $r\in\Omega^{1}(M,TM)$ admits the
following expressions in terms of $D^{r}$ and $D^{r,*}$: for
$X,Y\in\mathfrak{X}(M)$ and $\alpha\in\Omega^{1}(M)$,

(2.17) $\displaystyle\mathcal{N}_{r}(X,Y)$
$\displaystyle=r(D_{X}^{r}(Y))-D^{r}_{X}(r(Y))$ (2.18)
$\displaystyle\langle\alpha,\mathcal{N}_{r}(X,Y)\rangle$
$\displaystyle=\langle
r^{*}(D_{X}^{r,*}(\alpha))-D_{X}^{r,*}(r^{*}(\alpha)),Y\rangle.$

The next example explains the meaning of the operators $D^{r}$ and $D^{r,*}$
when $r\in\Omega^{1}(M,TM)$ is a complex structure, i.e., $r$ is a Nijenhuis
operator satisfying $r^{2}=-\mathrm{id}$.

######  Example 2.6 (Tangent and cotangent holomorphic structures).

On a complex manifold $(M,r)$, $T^{1,0}M$ and $(T^{1,0}M)^{*}$ are holomorphic
vector bundles. Explicitly, the Dolbeault connection on $T^{1,0}M$ is given by

$\nabla_{X^{\scriptscriptstyle{0,1}}}(Y^{\scriptscriptstyle{1,0}})=\mathrm{pr}_{10}([X^{\scriptscriptstyle{0,1}},Y^{\scriptscriptstyle{1,0}}]),$

where $\mathrm{pr}_{10}$ is the projection of
$TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$ onto $T^{1,0}M$, while
$(T^{1,0}M)^{*}$ carries the dual Dolbeault connection, defined by the
condition

(2.19)
$\langle\nabla^{*}_{X^{\scriptscriptstyle{0,1}}}(\alpha^{\scriptscriptstyle{1,0}}),Y^{\scriptscriptstyle{1,0}}\rangle=\mathcal{L}_{X^{\scriptscriptstyle{0,1}}}\langle\alpha^{\scriptscriptstyle{1,0}},Y^{\scriptscriptstyle{1,0}}\rangle-\langle\alpha^{\scriptscriptstyle{1,0}},\nabla_{X^{\scriptscriptstyle{0,1}}}(Y^{\scriptscriptstyle{1,0}})\rangle,$

where $X^{\scriptscriptstyle{0,1}}\in\mathfrak{X}^{0,1}(M)$,
$Y^{\scriptscriptstyle{1,0}}\in\mathfrak{X}^{1,0}(M)$, and
$\alpha^{\scriptscriptstyle{1,0}}\in\Omega^{1,0}(M)$. Consider the usual
identifications (of real vector bundles)

(2.20) $\Phi_{T}:TM\to
T^{1,0}M,\;X\mapsto\frac{1}{2}(X-\mathbf{i}\,r(X)),\qquad\Phi_{T^{*}}:T^{*}M\to(T^{1,0}M)^{*},\;\alpha\mapsto\alpha-\mathbf{i}\,r^{*}(\alpha).$

One can then verify (see [21, Prop. 5.3]) that the Dolbeault 1-derivations
codifiying the holomorphic structures on $TM$ and $T^{*}M$ resulting from
these identifications are $(D^{r},r,r)$ and $(D^{r,*},r^{*},r)$; in
particular, by Proposition 2.4, the complex structures on $TM$ and $T^{*}M$
are $r^{\scriptscriptstyle{tg}}$ and $r^{\scriptscriptstyle{cotg}}$,
respectively.

##  3\. Compatibility of Dirac structures and $(1,1)$-tensor fields

We introduce in this section a notion of compatibility for (almost) Dirac
structures and $(1,1)$-tensor fields, extending the well-known case of
Poisson-Nijenhuis manifolds that we recall first.

###  3.1. The Poisson-Nijenhuis compatibility revisited

On a manifold $M$, let $\pi\in\mathfrak{X}^{2}(M)$ be a bivector field and
$r\in\Omega^{1}(M,TM)$. Consider the bundle map $\pi^{\sharp}:T^{*}M\to TM$,
$\alpha\mapsto i_{\alpha}\pi$.

######  Definition 3.1.

We say that the pair $(\pi,r)$ is compatible if the following conditions hold:

  * (1)

$\pi^{\sharp}\circ r^{*}=r\circ\pi^{\sharp}$;

  * (2)

For $X\in\mathfrak{X}(M)$, $\alpha\in\Omega^{1}(M)$, we have

(3.1)
$R_{\pi}^{r}(X,\alpha):=\pi^{\sharp}(\mathcal{L}_{X}r^{*}(\alpha)-\mathcal{L}_{r(X)}\alpha)-(\mathcal{L}_{\pi^{\sharp}(\alpha)}r)(X)=0.$

Note that by (1) we have another bivector field
$\pi_{r}\in\mathfrak{X}^{2}(M)$ defined by the condition

$\pi_{r}^{\sharp}=r\circ\pi^{\sharp}.$

We also notice that
$R_{\pi}^{r}:\mathfrak{X}(M)\times\Omega^{1}(M)\to\mathfrak{X}(M)$ is
$C^{\infty}(M)$-bilinear if and only if condition (1) holds; in this case
$R_{\pi}^{r}$ defines a tensor field called the Magri-Morosi concomitant [37].

######  Remark 3.2.

In the literature, condition (2) above is often expressed through the
vanishing of an alternative concomitant,
$C^{r}_{\pi}:\Omega^{1}(M)\times\Omega^{1}(M)\to\Omega^{1}(M)$, given by

(3.2)
$C_{\pi}^{r}(\alpha,\beta):=[\alpha,\beta]_{\pi_{r}}-([r^{*}(\alpha),\beta]_{\pi}+[\alpha,r^{*}(\beta)]_{\pi}-r^{*}([\alpha,\beta]_{\pi})),$

where $[\cdot,\cdot]_{\Lambda}$ is the bracket on $\Omega^{1}(M)$ defined by a
bivector field $\Lambda$ via
$[\alpha,\beta]_{\Lambda}:=\mathcal{L}_{\Lambda^{\sharp}(\alpha)}\beta-
i_{\Lambda^{\sharp}(\beta)}d\alpha$. The concomitants $R^{r}_{\pi}$ and
$C^{r}_{\pi}$ are related by

$\langle\beta,R_{\pi}^{r}(X,\alpha)\rangle=\langle
C_{\pi}^{r}(\alpha,\beta),X\rangle,$

so $R_{\pi}^{r}=0$ if and only if $C_{\pi}^{r}=0$. $\diamond$

A Poisson-Nijenhuis structure (or simply a PN structure) on a manifold $M$, as
introduced in [37] (see also [31]), is a compatible pair $(\pi,r)$, where
$\pi$ is a Poisson bivector field and $r$ is a Nijenhuis operator. In this
setting, a geometric interpretation of the compatibility conditions in terms
of Lie bialgebroids can be found in [30].

Our extension of the compatibility condition of PN structures to Dirac
structures relies on a suitable reformulation of condition (2) above, see [21,
§2.3]. Let $\pi$ be a bivector field on $M$ and $r\in\Omega^{1}(M,TM)$.
Consider the operators $D^{r}$ and $D^{r,*}$ defined in (2.11) and (2.12).
Then the Magri-Morosi concomitant can be written as

(3.3)
$R_{\pi}^{r}(X,\alpha)=\pi^{\sharp}(D^{r,*}_{X}(\alpha))-D^{r}_{X}(\pi^{\sharp}(\alpha)).$

As a consequence of Prop. 2.4 and Theorem 2.1, part (ii), we have the
following alternative description of the compatibility of the pair $(\pi,r)$
in terms of tangent and cotangent lifts:

######  Proposition 3.3.

The pair $(\pi,r)$ is compatible if and only if $T\pi^{\sharp}\circ
r^{\scriptscriptstyle{cotg}}=r^{\scriptscriptstyle{tg}}\circ T\pi^{\sharp}$.

###  3.2. The presymplectic-Nijenhuis compatibility revisited

Given $\omega\in\Omega^{2}(M)$ and $r\in\Omega^{1}(M,TM)$, we now introduce a
notion of compatibility that is parallel to the one in Section 3.1.

Consider $\omega^{\flat}:TM\to T^{*}M$, $X\mapsto i_{X}\omega$, and define (in
analogy with (3.3))

(3.4) $\displaystyle\widetilde{S}^{r}_{\omega}(X,Y)$
$\displaystyle:=D^{r,*}_{X}(\omega^{\flat}(Y))-\omega^{\flat}(D^{r}_{X}(Y))$
$\displaystyle=i_{X}d(r^{*}\circ\omega^{\flat}(Y))-i_{r(X)}d\omega^{\flat}(Y)-\omega^{\flat}([Y,r(X)]-r([Y,X])),$

for $X$, $Y\in\mathfrak{X}(M)$.

Suppose that $\omega^{\flat}\circ r=r^{*}\circ\omega^{\flat}$, so that we have
$\omega_{r}\in\Omega^{2}(M)$ given by

$i_{X}\omega_{r}:=i_{r(X)}\omega.$

Let $(d\omega)_{r}\in\Gamma(T^{*}M\otimes\wedge^{2}T^{*}M)$ be given by
$i_{X}(d\omega)_{r}=i_{r(X)}d\omega$.

######  Lemma 3.4.

If $\omega^{\flat}\circ r=r^{*}\circ\omega^{\flat}$, then
$\widetilde{S}^{r}_{\omega}=d(\omega_{r})-(d\omega)_{r}$.

###### Proof.

The expression for $\widetilde{S}^{r}_{\omega}$ in (3.4) can be re-written as
follows:

$\widetilde{S}^{r}_{\omega}(X,Y)=i_{X}di_{Y}\omega_{r}-i_{r(X)}di_{Y}\omega-
i_{[Y,r(X)]}\omega+i_{[Y,X]}\omega_{r}.$

From the identities

$\displaystyle
i_{X}di_{Y}\omega_{r}=i_{X}\mathcal{L}_{Y}\omega_{r}-i_{X}i_{Y}d(\omega_{r}),$
$\displaystyle i_{r(X)}di_{Y}\omega=i_{r(X)}\mathcal{L}_{Y}\omega-
i_{r(X)}i_{Y}d\omega$ $\displaystyle
i_{[Y,r(X)]}\omega=\mathcal{L}_{Y}i_{r(X)}\omega-
i_{r(X)}\mathcal{L}_{Y}\omega,$ $\displaystyle
i_{[Y,X]}\omega_{r}=\mathcal{L}_{Y}i_{X}\omega_{r}-i_{X}\mathcal{L}_{Y}\omega_{r},$

we see that

$\widetilde{S}^{r}_{\omega}(X,Y)=-i_{X}i_{Y}d(\omega_{r})+i_{r(X)}i_{Y}d\omega=i_{Y}i_{X}d(\omega_{r})-i_{Y}i_{X}(d\omega)_{r}.$

∎

######  Definition 3.5.

For $\omega\in\Omega^{2}(M)$ and $r\in\Omega^{1}(M,TM)$, we say that the pair
$(\omega,r)$ is compatible if

  * (1)

$\omega^{\flat}\circ r=r^{*}\circ\omega^{\flat}$,

  * (2)

$\widetilde{S}^{r}_{\omega}=0$ (equivalently, $d(\omega_{r})=(d\omega)_{r}$).

In analogy with Proposition 3.3, this notion of compatibility has the
following interpretation:

######  Proposition 3.6.

The pair $(\omega,r)$ is compatible if and only if
$r^{\scriptscriptstyle{cotg}}\circ T\omega^{\flat}=T\omega^{\flat}\circ
r^{\scriptscriptstyle{tg}}$.

Alongside Poisson-Nijenhuis structures, Magri and Morosi considered in [37]
presymplectic-Nijenhuis structures (called $\Omega N$-structures therein),
which were defined as compatible pairs $(\omega,r)$, where
$\omega\in\Omega^{2}(M)$ is closed and $r\in\Omega^{1}(M,TM)$ is Nijenhuis. In
this case, condition (2) above reduces to $d(\omega_{r})=0$.

######  Remark 3.7.

In the original formulation of [37], condition $(2)$ above was written as the
vanishing of a concomitant $S^{r}_{\omega}\in\Omega^{2}(M,T^{*}M)$ given by

$S_{\omega}^{r}(X,Y):=i_{X}\mathcal{L}_{r(Y)}\omega-
i_{Y}\mathcal{L}_{r(X)}\omega-i_{r([X,Y])}\omega+d(\omega(r(Y),X)).$

Using condition (1), a direct application of Cartan calculus shows that

$S^{r}_{\omega}(X,Y)=i_{Y}i_{X}d(\omega_{r})-i_{Y}i_{r(X)}d\omega-
i_{r(Y)}i_{X}d\omega.$

When $\omega$ is closed, we see that
$S^{r}_{\omega}=d(\omega_{r})=\widetilde{S}^{r}_{\omega}$. In general, the
concomitant $S^{r}_{\omega}$ does not agree with our
$\widetilde{S}^{r}_{\omega}$ in (3.7); they are related by
$\widetilde{S}^{r}_{\omega}(X,Y)=S^{r}_{\omega}(X,Y)+d\omega(X,r(Y))$.
$\diamond$

###  3.3. Dirac-Nijenhuis structures

Let $M$ be a smooth manifold, and consider the direct sum
$\mathbb{T}M=TM\oplus T^{*}M$ with natural projections
$\mathrm{pr}_{T}:\mathbb{T}M\to TM$ and $\mathrm{pr}_{T^{*}}:\mathbb{T}M\to
T^{*}M$. We equip $\mathbb{T}M$ with the nondegenerate fiberwise symmetric
pairing

$\langle(X,\alpha),(Y,\beta)\rangle=\beta(X)+\alpha(Y),$

and the Courant-Dorfman bracket $\left[\\!\left[\cdot,\cdot\right]\\!\right]$
on $\Gamma(\mathbb{T}M)=\mathfrak{X}(M)\oplus\Omega^{1}(M)$,

$\left[\\!\left[(X,\alpha),(Y,\beta)\right]\\!\right]=([X,Y],\mathcal{L}_{X}\beta-
i_{Y}d\alpha).$

A subbundle $L\subset\mathbb{T}M$ is called lagrangian if $L=L^{\perp}$ with
respect to $\langle\cdot,\cdot\rangle$. A Dirac structure [16] on $M$ is a
lagrangian subbundle $L\subset\mathbb{T}M$ which is involutive with respect to
$\left[\\!\left[\cdot,\cdot\right]\\!\right]$. Important examples of
lagrangian subbundles include bivector fields $\pi$ and 2-forms $\omega$,
viewed as subbundles of $\mathbb{T}M$ by means of the graphs of the
corresponding maps $\pi^{\sharp}:T^{*}M\to TM$ and $\omega^{\flat}:TM\to
T^{*}M$; the additional involutivity condition to make these subbundles into
Dirac structures amount to $\pi$ being Poisson and $\omega$ being closed.

Given $r\in\Omega^{1}(M,TM)$, consider the operator

(3.5)
$\mathbb{D}^{r}:\Gamma(\mathbb{T}M)\to\Omega^{1}(M,\mathbb{T}M),\qquad\mathbb{D}^{r}=(D^{r},D^{r,*}).$

For each vector field $X\in\mathfrak{X}(M)$ we then have an
$\mathbb{R}$-linear operator
$\mathbb{D}^{r}_{X}:\Gamma(\mathbb{T}M)\rightarrow\Gamma(\mathbb{T}M)$,

(3.6) $\mathbb{D}^{r}_{X}(Y,\alpha)=(D^{r}_{X}(Y),D^{r,*}_{X}(\alpha)),$

for $(Y,\alpha)\in\Gamma(\mathbb{T}M)=\mathfrak{X}(M)\oplus\Omega^{1}(M)$.
From the Leibniz-type equations for $D^{r}$ and $D^{r,*}$, we see that
$\mathbb{D}^{r}$ satisfies

(3.7)
$\mathbb{D}^{r}_{X}(f\,\sigma)=f\,\mathbb{D}(\sigma)+(\mathcal{L}_{X}f)(r,r^{*})(\sigma)-(\mathcal{L}_{r(X)}f)\,\sigma,$

for $\sigma\in\Gamma(\mathbb{T}M)$, and $f\in C^{\infty}(M)$, so the triple
$(\mathbb{D}^{r},(r,r^{*}),r)$ is a 1-derivation on $\mathbb{T}M$.

######  Definition 3.8.

Let $L\subset\mathbb{T}M$ be a lagrangian subbundle and
$r\in\Omega^{1}(M,TM)$. We say that the pair $(L,r)$ is compatible if

  * $(1)$

$(r,r^{*})(L)\subseteq L$

  * $(2)$

$\mathbb{D}^{r}_{X}(\Gamma(L))\subseteq\Gamma(L),\>\forall
X\in\mathfrak{X}(M).$

(Note that this notion of compatibility makes sense for any subbundle of
$\mathbb{T}M$, not necessarily lagrangian.)

The previous definition has an interpretation that naturally extends
Propositions 3.3 and 3.6: a pair $(L,r)$ is compatible if and only if

$TL\subset T(TM\oplus T^{*}M)=T(TM)\times_{TM}T(T^{*}M)$

satisfies
$(r^{\scriptscriptstyle{tg}},r^{\scriptscriptstyle{cotg}})(TL)\subset TL$.

######  Remark 3.9.

Condition (2) in Definition 3.8 can be also written in terms of the vanishing
of a concomitant: for a lagrangian subbundle $L\subset\mathbb{T}M$ satisfying
$(r,r^{*})(L)\subseteq L$, the expression

(3.8)
$C^{r}_{L}(\sigma_{1},\sigma_{2}):=\langle\mathbb{D}^{r}_{(\cdot)}(\sigma_{1}),\sigma_{2}\rangle\in\Omega^{1}(M),\qquad\sigma_{1},\,\sigma_{2}\in\Gamma(L),$

defines an element in $\Gamma(\wedge^{2}L^{*}\otimes T^{*}M)$. (The skew-
symmetry of $C^{r}_{L}$ is a consequence of the duality (2.13), and the
$C^{\infty}(M)$-linearity follows from (3.7).) The fact that $L$ is lagrangian
implies that (2) holds if and only if $C^{r}_{L}=0$. $\diamond$

######  Definition 3.10.

A Dirac-Nijenhuis structure on $M$ is a compatible pair $(L,r)$ where $L$ is a
Dirac structure and $r$ is a Nijenhuis operator.

More general Dirac-Nijenhuis structures in abstract “Courant-Nijenhuis
algebroids” will be the subject of [8].

######  Example 3.11.

Let $\pi\in\mathfrak{X}^{2}(M)$ and consider the lagrangian subbundle
$L_{\pi}:=\mathrm{graph}(\pi^{\sharp})\subset\mathbb{T}M$. Given
$r\in\Omega^{1}(M,TM)$, then $(r,r^{*})(L_{\pi})\subseteq L_{\pi}$ if and only
if $\pi^{\sharp}\circ r^{*}=r\circ\pi^{\sharp}$. It follows from (3.3) that
$\mathbb{D}^{r}_{X}(\Gamma(L_{\pi}))\subset\Gamma(L_{\pi})$ if and only if
$R_{\pi}^{r}=0$. Hence the pair $(L_{\pi},r)$ is compatible if and only if so
is the pair $(\pi,r)$. (Under the natural identification $L_{\pi}\cong
T^{*}M$, the concomitants $C_{L_{\pi}}^{r}$ and $C_{\pi}^{r}$, see Remarks 3.2
and 3.9, are in fact identified.) In particular, $(\pi,r)$ is a Poisson-
Nijenhuis structure if and only if $(L_{\pi},r)$ is Dirac-Nijenhuis.

Other examples of Dirac-Nijenhuis structures can be found on submanifolds of
Poisson-Nijenhuis structures, see Corollary 4.2.

######  Example 3.12.

For $\omega\in\Omega^{2}(M)$, let
$L_{\omega}=\mathrm{graph}(\omega^{\flat})\subset\mathbb{T}M$ be the
corresponding lagrangian subbundle. Let $r\in\Omega^{1}(M,TM)$. Then condition
$(r,r^{*})(L_{\omega})\subseteq L_{\omega}$ is equivalent to
$\omega^{\flat}\circ r=r^{*}\circ\omega^{\flat}$, while condition
$\mathbb{D}^{r}_{X}(\Gamma(L))\subseteq\Gamma(L)$ is equivalent to
$\widetilde{S}^{r}_{\omega}=0$, see (3.4). So the pair $(\omega,r)$ is
compatible if and only if so is the pair $(L_{\omega},\omega)$. When $\omega$
is closed, $\widetilde{S}^{r}_{\omega}=d(\omega_{r})$ by Lemma 3.4, so
$\mathbb{D}^{r}_{X}(\Gamma(L))\subseteq\Gamma(L)$ if and only if
$d(\omega_{r})=0$. In particular, $(\omega,r)$ is a presymplectic-Nijenhuis
structure if and only if $(L_{\omega},r)$ is a Dirac-Nijenhuis structure.

Recall that any involutive distribution $F\subseteq TM$ defines a Dirac
structure $F\oplus\mathrm{Ann}(F)\subseteq\mathbb{T}M$. The next example
illustrates Dirac-Nijenhuis structures of this type (cf. [27, Sec. 6]).

######  Example 3.13.

Let $r$ be a Njenhuis operator of constant rank, and consider the distribution
$F=\mathrm{Im}(r)\subseteq TM$. The Nijenhuis property implies that $F$ is
involutive, so $L_{r}:=F\oplus\mathrm{Ann}(F)$ is a Dirac structure. Since
$\mathrm{Ann}(\mathrm{Im}(r))=\ker(r^{*})$, it is simple to check that
$(r,r^{*})(L_{r})\subseteq L_{r}$. Condition (2) in Definition 3.8 follows
from the identities 2.17 and (2.18), so $(L_{r},r)$ is Dirac-Nijenhuis.

######  Remark 3.14 (Quasi-Nijenhuis).

There are variants of the definition of Dirac-Nijenhuis structure obtained by
weakening the integrability conditions on $L$ or $r$. For example, a Dirac
quasi-Nijenhuis structure is a triple $(L,r,\phi)$, where $L$ is a Dirac
structure compatible with $r\in\Omega^{1}(M,TM)$, $\phi\in\Omega^{3}(M)$ is a
closed 3-form, and

(3.9)
$\langle\alpha,\mathcal{N}_{r}(Y,Z)\rangle=-\phi(X,Y,Z),\qquad\forall\,(X,\alpha)\in
L.$

Notice that even when $\phi=0$ this condition is weaker than the vanishing of
the Nijenhuis torsion $\mathcal{N}_{r}$: it just says that the image of
$\mathcal{N}_{r}$ lies in $L\cap TM=\mathrm{Ann}(\mathrm{pr}_{T^{*}}(L))$.
When $L$ is the graph of a Poisson structure $\pi$, then (3.9) becomes

$\mathcal{N}_{r}(X,Y)=\pi^{\sharp}(\phi(X,Y,\cdot)),$

hence recovering the Poisson quasi-Nijenhuis structures of [41, Def. 3.3]
$\diamond$

###  3.4. Holomorphic Dirac structures

Suppose that $r\in\Omega^{1}(M,TM)$ is a complex structure on $M$. It is known
(see e.g. [32, Sec. 3]) that holomorphic Poisson structures on the complex
manifold $(M,r)$ are characterized by the fact that their real parts are
Poisson-Nijenhuis structures with respect to $r$. We will now extend this
result to Dirac structures.

On a complex manifold $(M,r)$, we consider the holomorphic vector bundle
$\mathbb{T}^{1,0}M=T^{1,0}M\oplus(T^{1,0}M)^{*}$ equipped with its natural
holomorphic Courant algebroid structure, consisting of the non-degenerate
symmetric bilinear pairing $\langle\cdot,\cdot\rangle$ on $\mathbb{T}^{1,0}M$
(obtained from the duality of $T^{1,0}M$ and $(T^{1,0}M)^{*}$), the projection
$\mathbb{T}^{1,0}M\to T^{1,0}M$ and the $\mathbb{C}$-bilinear bracket on the
sheaf of holomorphic sections $\Gamma_{\mathrm{hol}}(\cdot,\mathbb{T}^{1,0}M)$
given by

$\left[\\!\left[(\mathsf{X},\xi),(\mathsf{Y},\zeta)\right]\\!\right]=([\mathsf{X},\mathsf{Y}],\mathcal{L}_{\mathsf{X}}\zeta-\textit{i}_{\mathsf{Y}}\partial\xi),$

where $(\mathsf{X},\xi),(\mathsf{Y},\zeta)$ are local holomorphic sections of
$\mathbb{T}^{1,0}M$. A holomorphic Dirac structure on $M$ (see e.g. [25]) is a
holomorphic subbundle $\mathcal{L}\subset\mathbb{T}^{1,0}M$ which is
lagrangian with respect to $\langle\cdot,\cdot\rangle$ and whose sheaf of
holomorphic sections is involutive with respect to
$\left[\\!\left[\cdot,\cdot\right]\\!\right]$.

To explain the sense in which holomorphic Dirac structures are equivalent to
Dirac-Nijenhuis structures, note that, from Example 2.6, the holomorphic
structure on $\mathbb{T}M$ coming from its identification with
$\mathbb{T}^{1,0}M$ via

(3.10)
$\Phi:\mathbb{T}M\to\mathbb{T}^{1,0}M,\qquad\Phi(X,\alpha)=\Big{(}\frac{1}{2}(X-\mathbf{i}\,r(X)),\alpha-\mathbf{i}\,r^{*}(\alpha)\Big{)}$

is given by the Dolbeault 1-derivation $(\mathbb{D}^{r},(r,r^{*}),r)$. Another
key property of the map $\Phi$ is that it preserves Courant brackets of
holomorphic sections.

######  Lemma 3.15.

For (local) holomorphic sections $(X,\alpha),\,(Y,\beta)$ of $\mathbb{T}M$, we
have

$\Phi(\left[\\!\left[(X,\alpha),(Y,\beta)\right]\\!\right])=\left[\\!\left[\Phi(X,\alpha),\Phi(Y,\beta)\right]\\!\right].$

###### Proof.

Recall that $\left[\\!\left[\Phi(X,\alpha),\Phi(Y,\beta)\right]\\!\right]$ has
$T^{1,0}M$-component given by

(3.11) $\frac{1}{4}[X-\mathbf{i}\,r(X),Y-\mathbf{i}\,r(Y)]$

and $(T^{1,0}M)^{*}$-component given by

(3.12)
$\frac{1}{2}(\mathcal{L}_{X-\mathbf{i}\,r(X)}(\beta-\mathbf{i}\,r^{*}\beta)-i_{Y-\mathbf{i}\,r(Y)}\,\partial(\alpha-\mathbf{i}\,r^{*}\alpha)).$

On the other hand,

$\Phi(\left[\\!\left[(X,\alpha),(Y,\beta)\right]\\!\right])=\left(\frac{1}{2}([X,Y]-\mathbf{i}\,r[X,Y]),\mathcal{L}_{X}\beta-
i_{Y}d\alpha-\mathbf{i}\,r^{*}(\mathcal{L}_{X}\beta-i_{Y}d\alpha)\right)$

Recall (see (2.6)) that $(X,\alpha)$ is holomorphic if and only if
$\mathbb{D}^{r}_{Z}(X,\alpha)=0$, for all $Z\in\mathfrak{X}(M)$, and this
condition implies the following identities for $(X,\alpha)$ (similarly for
$(Y,\beta)$):

$\displaystyle[X,r(Z)]=r([X,Z]),\qquad\mathcal{L}_{X}r^{*}(\gamma)=r^{*}(\mathcal{L}_{X}\gamma),$
$\displaystyle\mathcal{L}_{r(Z)}\alpha=\mathcal{L}_{Z}r^{*}(\alpha),\qquad\;\;\;\;i_{r(Z)}d\alpha=i_{Z}d(r^{*}(\alpha)),$

for all $Z\in\mathfrak{X}(M)$, and $\gamma\in\Omega^{1}(M)$. It follows that
(3.11) equals

$\frac{1}{4}([X,Y]-[r(X),r(Y)]-\textbf{i}([r(X),Y]+[X,r(Y)]))\\\
=\frac{1}{2}([X,Y]-\textbf{i}r([X,Y])).$

Similarly, (3.12) equals (recall that $d=\partial$ on holomorphic forms)

$\displaystyle\frac{1}{2}\left(\mathcal{L}_{X}\beta-\mathcal{L}_{r(X)}r^{*}(\beta)-{\mathbf{i}\,}(\mathcal{L}_{r(X)}\beta+\mathcal{L}_{X}r^{*}(\beta))-2(i_{Y}d\alpha-\mathbf{i}\,i_{r(Y)}d\alpha)\right)$
$\displaystyle=\mathcal{L}_{X}\beta-
i_{Y}d\alpha-\mathbf{i}\,r^{*}(\mathcal{L}_{X}\beta-i_{Y}d\alpha).$

∎

We now verify that holomorphic Dirac structures correspond to Dirac-Nijenhuis
structures under the isomorphism (3.10).

######  Proposition 3.16.

Let $r\in\Omega^{1}(M,TM)$ be a complex structure on $M$ and
$L\subset\mathbb{T}M$ be a lagrangian subbundle. The pair $(L,r)$ is
compatible if and only if $\Phi(L)$ is a holomorphic lagrangian subbundle of
$\mathbb{T}^{1,0}M$, and $(L,r)$ is a Dirac-Nijenhuis structure if and only if
$\Phi(L)$ is a holomorphic Dirac structure.

###### Proof.

A real subbundle $L\subset\mathbb{T}M$ is a complex subbundle if and only if
$(r,r^{*})(L)\subseteq L$, and such a subbundle is holomorphic if and only if
$\mathbb{D}^{r}_{X}(\Gamma(L))\subseteq\Gamma(L)$ for all
$X\in\mathfrak{X}(M)$. Also, for a complex subbundle $L\subseteq\mathbb{T}M$
one can directly check that $\Phi(L)^{\perp}=\Phi(L^{\perp})$, so $L$ is
lagrangian in $\mathbb{T}M$ if and only if $\Phi(L)$ is lagrangian in
$\mathbb{T}^{1,0}M$. This proves the first assertion of the proposition.

For the second assertion, we note that the involutivity of $L$ is equivalent
to the vanishing of $T\in\Gamma(\wedge^{3}L^{*})$ given by

$T(\sigma_{1},\sigma_{2},\sigma_{3}):=\langle\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right],\sigma_{3}\rangle.$

For holomorphic sections $\sigma_{1},\sigma_{2},\sigma_{3}\in\Gamma(L)$, we
have

$\displaystyle\langle\left[\\!\left[\Phi(\sigma_{1}),\Phi(\sigma_{2})\right]\\!\right],\Phi(\sigma_{3})\rangle$
$\displaystyle=\langle\Phi(\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right]),\Phi(\sigma_{3})\rangle=T(\sigma_{1},\sigma_{2},\sigma_{3})-\mathbf{i}\,T(\sigma_{1},\sigma_{2},(r,r^{*})(\sigma_{3})).$

The condition $\Phi(L)=\Phi(L)^{\perp}$ implies that the sheaf of holomorphic
sections of $\Phi(L)$ is involutive if and only if $T$ vanishes on holomorphic
sections of $L$, which is in turn equivalent to the vanishing of $T$ (since
any smooth section $\sigma$ of $L$ can be written locally as
$\sigma=\sum_{i}f_{i}\sigma_{i}$, where $f_{i}\in{C}^{\infty}(M)$ and
$\sigma_{i}$ is holomorphic). ∎

For a complex lagrangian subbundle $\mathcal{L}\subset\mathbb{T}^{1,0}M$, we
define its real part as the real lagrangian subbundle $L\subset\mathbb{T}M$
such that $\Phi(L)=\mathcal{L}$.

######  Example 3.17.

Let $\Pi=\pi+\mathbf{i}\,\pi_{1}\in\Gamma(\wedge^{2}TM\otimes\mathbb{C})$ be a
complex bivector field. Then $\Pi\in\Gamma(\wedge^{2}T^{1,0}M)$ if and only if
$\pi_{1}^{\sharp}=-\pi^{\sharp}\circ r^{*}$, i.e., $\pi_{1}=-\pi_{r}$. In this
case, the lagrangian subbundle $\mathcal{L}$ of $\mathbb{T}^{1,0}M$ defined as
the graph of $\Pi^{\sharp}:(T^{1,0}M)^{*}\to T^{1,0}M$ has real part given by
$L=\mathrm{graph}(4\pi^{\sharp})$. By Proposition 3.16 , we have a bijective
correspondence between Poisson-Nijenhuis structures $(\pi,r)$ and holomorphic
Poisson structure $\Pi=\pi-\mathbf{i}\,\pi_{r}$, see [32, Prop. 2.6].

######  Example 3.18.

Let
$\Omega=\omega+\mathbf{i}\,\omega_{1}\in\Gamma(\wedge(TM\otimes\mathbb{C})$ be
a complex 2-form. Then $\Omega\in\Gamma(\wedge^{2}(T^{1,0}M)^{*})$ if and only
if $\omega_{1}=-\omega_{r}$, and the real part of the lagrangian subbundle
$\mathcal{L}:=\mathrm{graph}(\Omega^{\flat})\subset\mathbb{T}^{1,0}M$, where
$\Omega^{\flat}:T^{1,0}M\to(T^{1,0}M)^{*}$, is given by
$L=\mathrm{graph}(\omega^{\flat})$. Proposition 3.16 thens gives a bijective
correspondence between presymplectic-Nijenhuis structures $(\omega,r)$ and
holomorphic closed 2-forms $\Omega=\omega-\mathbf{i}\,\omega_{r}$.

We will generalize Example 3.18 to higher-degree forms in Proposition 5.4
below.

##  4\. Properties of Dirac-Nijenhuis structures

In this section we discuss presymplectic foliations and Poisson quoients of
Dirac-Nijenhuis structures, as well as hierarchies that extend those of
Poisson-Nijenhuis structures.

###  4.1. Backward maps and presymplectic leaves

Let $N$ be a smooth manifold, and let $\psi:N\to M$ be a smooth map. If
$L_{N}$, $L_{M}$ are Dirac structures on $N$ and $M$, respectively, recall
that $\psi$ is a backward Dirac map (see e.g. [4]) if

$(L_{N})_{x}=\\{(X,(T_{x}\psi)^{*}(\beta))\in\mathbb{T}_{x}N\,\,|\,\,(T_{x}\psi(X),\beta)\in(L_{M})_{\psi(x)}\\}.$

Recall that $r_{N}\in\Omega^{1}(N,TN)$ and $r_{M}\in\Omega^{1}(M,TM)$ are
$\psi$-related if $r_{M}\circ d\psi=d\psi\circ r_{N}$.

######  Proposition 4.1.

Suppose that $\psi:(N,L_{N})\to(M,L_{M})$ is a backward Dirac map and that
$r_{N}$ and $r_{M}$ are $\psi$-related. If $L_{M}$ is compatible with $r_{M}$,
then $L_{N}$ is compatible with $r_{N}$.

###### Proof.

The fact that $r_{N}$ and $r_{M}$ are $\psi$-related directly implies that
$(r_{N},r_{N}^{*})(L_{N})\subset L_{N}$. To show that $L_{N}$ verifies
condition (2) in Definition 3.8, we can equivalently verify the vanishing of
the concomitant $C^{r_{N}}_{L_{N}}\in\Gamma(\wedge^{2}L_{N}^{*}\otimes
T^{*}N)$ of Remark 3.9.

We say that (local) sections $a=(X,\alpha)$ of $\mathbb{T}N$ and $b=(Y,\beta)$
of $\mathbb{T}M$ are $\psi$-related if $X$ and $Y$ are $\psi$-related, and
$\alpha=\psi^{*}\beta$. Note that if $a\in\Gamma(\mathbb{T}N)$ is
$\psi$-related to $b\in\Gamma(L_{M})$, then it follows from the definition of
backward Dirac map that $a\in\Gamma(L_{N})$.

Suppose that $a\in\Gamma(L_{N})$ is $\psi$-related to $b\in\Gamma(L_{M})$, and
let $Z_{1}\in\mathfrak{X}(N)$ and $Z_{2}\in\mathfrak{X}(M)$ be $\psi$-related.
It directly follows from the naturality properties (2.14) and (2.15) that
$\mathbb{D}_{Z_{1}}^{r_{N}}(a)$ is $\psi$-related to
$\mathbb{D}_{Z_{2}}^{r_{M}}(b)\in\Gamma(L_{M})$ (since we are assuming that
$L_{M}$ is $r_{M}$-compatible), and hence
$\mathbb{D}_{Z_{1}}^{r_{N}}(a)\in\Gamma(L_{N})$. Now suppose that $x\in N$
satisfies the following property: any $a_{x}\in L_{N}|_{x}$ and
$(Z_{1})_{x}\in TN|_{x}$ can be extended to local sections $a$ of $L_{N}$ and
$Z_{1}$ of $TN$ which are $\psi$-related to some local sections $b$ of $L_{M}$
and $Z_{2}$ on $TM$, respectively; in this case, $C^{r_{N}}_{L_{N}}|_{x}=0$.
We finally observe that the points $x\in N$ with this property form an open
dense subset of $N$ (see the proof of Prop. 5.6 in [4]), hence
$C^{r_{N}}_{L_{N}}=0$. ∎

On a Dirac manifold $(M,L)$, let $\psi:C\hookrightarrow M$ be a submanifold.
It is known (see e.g. [4, Sec. 5]) that if the lagrangian distribution

(4.1)
$\mathcal{B}_{\psi}(L)=\\{(X,\psi^{*}\alpha)\in\mathbb{T}C\>|\>T\psi(X),\alpha)\in
L\\}\subset TC\oplus T^{*}C$

defined by pointwise pullback of $L$ to $C$ fits into a smooth vector bundle
(this is ensured by suitable clean-intersection conditions [16], see also
[4]), then it defines a Dirac structure on $C$ for which $\psi$ is a backward
Dirac map.

For $r\in\Omega^{1}(M,TM)$, we call a submanifold $C\hookrightarrow M$
$r$-invariant if $r(TC)\subseteq TC$; we define $r_{C}\in\Omega^{1}(C,TC)$ by
$r|_{TC}$.

######  Corollary 4.2.

Let $L$ be a Dirac structure on $M$ compatible with $r\in\Omega^{1}(M,TM)$.
Let $\psi:C\hookrightarrow M$ be an $r$-invariant submanifold for which
$\mathcal{B}_{\psi}(L)$ in (4.1) is smooth, hence a Dirac structure on $C$.
Then $\mathcal{B}_{\psi}(L)$ is compatible with $r_{C}$.

Note that $r_{C}$ is Nijenhuis if $r$ is Nijenhuis. So the previous corollary
illustrates how Dirac-Nijenhuis structures naturally arise on submanifolds of
Dirac-Nijenhuis manifolds.

A special case is that of presymplectic leaves of a Dirac manifold $(M,L)$.
Recall that $L$ defines a (singular) foliation on $M$ whose leaves are tangent
to the distribution given by the projection of $L$ on $TM$; moreover, each
leaf $\mathcal{O}$ carries a closed $2$-form
$\omega_{\mathcal{O}}\in\Omega^{2}(\mathcal{O})$, which can be seen as the
pullback of $L$ to $\mathcal{O}$. Note also that the condition
$(r,r^{*})(L)\subset L$ implies that $T\mathcal{O}=\mathrm{pr}_{T}(L)$ is
preserved by $r$.

######  Corollary 4.3.

Let $L$ be a Dirac structure on $M$ compatible with $r\in\Omega^{1}(M,TM)$.
Then each leaf $\mathcal{O}$ is $r$-invariant, and its presymplectic form
$\omega_{\mathcal{O}}$ is compatible with the restriction $r_{\mathcal{O}}$.
In particular, if $(M,L,r)$ is a Dirac-Nijenhuis manifold, then its leaves are
presymplectic-Nijenhuis manifolds.

###  4.2. Forward maps and Poisson quotients

A map $\psi:(M,L_{M})\to(N,L_{N})$ between Dirac manifolds is a forward Dirac
map if

$(L_{N})_{\psi(x)}=\\{(T_{x}\psi(X),\beta)\in\mathbb{T}_{\psi(x)}N\,\,|\,\,(X,(T_{x}\psi)^{*}\beta)\in(L_{M})_{x}\\}.$

As in the proof of Proposition 4.1, one can use the naturality properties of
Remark 2.5 to show the following result:

######  Proposition 4.4.

Let $\psi:(M,L_{M})\to(N,L_{N})$ be a surjective submersion that is a forward
Dirac map. Suppose that $r_{M}\in\Omega^{1}(M,TM)$ is $\psi$-related to
$r_{N}\in\Omega^{1}(N,TN)$. Then $L_{N}$ is compatible with $r_{N}$ provided
$L_{M}$ is compatible with $r_{M}$.

Note that if $r_{M}$ is Nijenhuis, then so is $r_{N}$, hence the previous
result illustrates how Dirac-Nijenhuis structures may arise on quotients. The
following is a special case.

Recall [16] (see also [4]) that the null distribution of a Dirac structure $L$
on $M$ is

$K:=L\cap TM\subseteq TM.$

At each point $x\in M$, $K$ agrees with the kernel of the presymplectic form
on the leaf through $x$. Whenever $K$ has constant rank, it is integrable and
defines the so-called null foliation $\mathcal{K}$. If the null foliation is
simple, then $Q:=M/\mathcal{K}$ inherits a Poisson structure $\pi_{Q}$
uniquely determined by the fact that the quotient map $M\to Q$ is a forward
Dirac map.

Given $r\in\Omega^{1}(M,TM)$ and a surjective submersion $\psi:M\to N$, we say
that $r$ descends to $N$ if there exists $r_{N}\in\Omega^{1}(Q,TN)$ that is
$\psi$-related to $r$, in which case $r_{N}$ is uniquely determined by this
property.

######  Corollary 4.5.

Let $L$ be a Dirac structure on $M$ whose null foliation $\mathcal{K}$ is
simple, let $Q$ be the leaf space with induced Poisson structure $\pi_{Q}$,
and suppose that $L$ is compatible with $r\in\Omega^{1}(M,TM)$. Then $r$
descends to $r_{Q}\in\Omega^{1}(Q,TQ)$ which is compatible with $\pi_{Q}$. In
particular, if $(M,L,r)$ is a Dirac-Nijenhuis manifold, its quotient
$Q=M/\mathcal{K}$ is a Poisson-Nijenhuis manifold.

###### Proof.

Once we prove that $r$ descends to $Q$, the result follows directly from
Proposition 4.4.

The condition $(r,r^{*})(L)\subset L$ implies that $r(K)\subseteq K$ and
$r^{*}(\mathrm{Ann}(K))\subseteq\mathrm{Ann}(K)$ (since
$\mathrm{Ann}(K)=\mathrm{pr}_{T^{*}}(L)$). To verify that $r$ descends to $Q$,
we must check that $r$ preserves projectable vector fields, i.e.,
$r(\mathfrak{X}(M)_{pr})\subseteq\mathfrak{X}(M)_{pr}$, where

$\mathfrak{X}(M)_{pr}=\\{Y\in\mathfrak{X}(M)\,\,|\,\,[X,Y]\in\Gamma(K),\,\,\forall
X\in\Gamma(K)\\}.$

So let $\alpha\in\Gamma(\mathrm{Ann}(K))$, $Y\in\mathfrak{X}(M)_{pr}$ and
$X\in\Gamma(K)$, and note that

$\displaystyle\langle\alpha,[X,r(Y)]\rangle$
$\displaystyle=i_{X}\mathcal{L}_{r(Y)}\alpha-\cancel{\mathcal{L}_{r(Y)}i_{X}\alpha}^{=0}=i_{X}(-D^{r,*}_{Y}(\alpha)+\mathcal{L}_{Y}(r^{*}\alpha))$
$\displaystyle=-\cancel{\langle
D^{r,*}_{Y}(\alpha),X\rangle}^{=0}+\cancel{i_{[X,Y]}r^{*}(\alpha)}^{=0}+\cancel{\mathcal{L}_{Y}i_{X}r^{*}(\alpha)}^{=0}=0,$

where for the last equality, besides the fact that $r^{*}$ preserves
$\mathrm{Ann}(K)$, we used that $D^{r,*}_{Y}$ preserves
$\Gamma(\mathrm{Ann}(K))$ (which follows from
$\mathbb{D}^{r}_{Y}(\Gamma(L))\subseteq\Gamma(L)$ and
$\mathrm{Ann}(K)=\mathrm{pr}_{T^{*}}(L)$). ∎

###  4.3. Traces of powers of $r$.

An important property of a PN manifold $(\pi,r)$ is that

(4.2) $\phi_{j}:=\frac{1}{j}\mathrm{trace}(r^{j}),\,\,\,j\geq 1,$

are functions in involution: $\\{\phi_{i},\phi_{j}\\}=0$, $\forall\,i,\,j\geq
1$, see [37]. We now present a generalization of this result for Dirac-
Nijenhuis manifolds.

Recall that, for a Dirac structure $L$ on $M$, the space of admissible
functions on $M$, denoted by $C^{\infty}_{adm}(M)$, is the space of functions
$f\in C^{\infty}(M)$ for which there exists $X_{f}\in\mathfrak{X}(M)$
satisfying $(X_{f},df)\in\Gamma(L)$. Although such $X_{f}$ may not be uniquely
defined, there is a Poisson bracket on $C^{\infty}_{adm}(M)$ defined by the
usual formula

$\\{f,g\\}=dg(X_{f}),$

see [16]. In case the the null distribution is regular, a function is
admissible if and only if it is constant along the leaves of the null
foliation $\mathcal{K}$. If $\mathcal{K}$ is simple, the natural
identification $C^{\infty}(M/\mathcal{K})\cong C^{\infty}_{adm}(M)$ is an
isomorphism of Poisson algebras.

######  Proposition 4.6.

Let $(L,r)$ be a Dirac-Nijenhuis manifold. If $\phi_{1}=\mathrm{trace}(r)$ is
admissible, then $\phi_{i}$ is admissible for every $i\geq 2$ and

$\\{\phi_{i},\phi_{j}\\}=0,\,\,\forall\,i,j\geq 1.$

###### Proof.

Recall the useful formula $r^{*}(d\phi_{i})=d\phi_{i+1}$ (see [37, Eq. 4.2]).
So, if $\phi_{1}$ is admissible, there exists $X\in\mathfrak{X}(M)$ such that
$(X,d\phi_{1})\in\Gamma(L)$. In this case, since $L$ and $r$ are compatible,

$\Gamma(L)\ni(r,r^{*})^{i-1}(X,\phi_{1})=(r^{i-1}(X),d\phi_{i})\Longrightarrow\text{
$\phi_{i}$ is admissible. }$

Also, a simple calculation in local coordinates shows that $\langle
d\,\mathrm{trace}(r),X\rangle=\mathrm{trace}(D^{r}_{(\cdot)}(X)).$ Now, using
that $\mathrm{trace}(AB)=\mathrm{trace}(BA)$ and

$D^{r^{j}}_{X}(Y)=\sum_{k=0}^{j}r^{j-k}(D^{r}_{r^{k}(X)}(Y)),$

one shows that

$\displaystyle\\{\phi_{i},\phi_{j}\\}$
$\displaystyle=d\phi_{j}(r^{i-1}(X))=\mathrm{trace}(r^{j}(D_{\cdot}^{r}(r^{i-1}(X)))),$
$\displaystyle\\{\phi_{j},\phi_{i}\\}$
$\displaystyle=d\phi_{i}(r^{j-1}(X))=\mathrm{trace}(r^{i}(D_{\cdot}^{r}(r^{j-1}(X)))).$

As $D^{r}(r(X))=r(D^{r}(X))$ (see (2.17)), one obtains that
$\\{\phi_{i},\phi_{j}\\}=\\{\phi_{j},\phi_{i}\\}$, which can only happen if
$\\{\phi_{i},\phi_{j}\\}=0$. ∎

The following simple example shows that the admissibility condition on
$\mathrm{trace}(r)$ is not automatic. Moreover, even when it holds and the
Poisson-Nijenhuis quotient $Q=M/\mathcal{K}$ is well defined (see Cor. 4.5),
we will see that the relationship between $\mathrm{trace}(r)$ and the pull-
back of $\mathrm{trace}(r_{Q})$ is not straightforward.

######  Example 4.7.

In $\mathbb{R}^{2}$ with coordinates $(x_{1},x_{2})$, consider the
distribution $F$ generated by $\frac{\partial}{\partial x_{2}}$. Consider
$L=F\oplus\mathrm{Ann}(F)$ and

$\displaystyle r$ $\displaystyle=a(x_{1})\frac{\partial}{\partial
x_{1}}\otimes dx_{1}+b(x_{2})\frac{\partial}{\partial x_{2}}\otimes dx_{2}$
$\displaystyle\widetilde{r}$ $\displaystyle=a(x_{1})\frac{\partial}{\partial
x_{1}}\otimes dx_{1}+c(x_{1})\frac{\partial}{\partial x_{2}}\otimes dx_{2}$

for $a,b,c:\mathbb{R}\to\mathbb{R}$ smooth functions. One can check that
$(L,r)$ is Dirac-Nijenhuis, but $\mathrm{trace}(r)$ is admissible if and only
if $b$ is a constant function.

On the other, the pair $(L,\widetilde{r})$ is compatible,
$\mathrm{trace}(\widetilde{r})$ is admissible, but $\widetilde{r}$ is
Nijenhuis if and only if

(4.3) $(a-c)\frac{dc}{dx_{1}}=0.$

Then the relationship between $\mathrm{trace}(\widetilde{r})=a+c$ and
$\mathrm{trace}(\widetilde{r}_{Q})=a$ is that

$\mathrm{trace}(\widetilde{r})=2\mathrm{trace}(\widetilde{r}_{Q})\;\;\mbox{ or
}\;\;\mathrm{trace}(\widetilde{r})=\mathrm{trace}(\widetilde{r}_{Q})+\lambda,$

for $\lambda\in\mathbb{R}$, depending on whether $a=c$ or $dc/dx_{1}=0$,
respectively. But it is possible to have a mixture of these situations if
there exist points $\bar{x}_{1}$ where $d^{n}a/dx_{1}^{n}=0$ for all $n\geq
1$. In this case, $c$ defined as $a$ on $(-\infty,\bar{x}_{1})$ and as
$\lambda=a(\bar{x}_{1})$ on $[\bar{x}_{1},+\infty)$ is smooth and satisfies
(4.3).

###  4.4. Hierarchies of Dirac-Nijenhuis structures

A fundamental aspect in the theory of Poisson-Nijenhuis structures $(\pi,r)$
is the existence of a hierarchy of Poisson-Nijenhuis structures $(\pi_{n},r)$,
$n=1,2,\ldots$, recursively defined by

(4.4) $\displaystyle\pi_{0}:=\pi,\qquad$
$\displaystyle\pi_{n}^{\sharp}=r\circ\pi_{n-1}^{\sharp}=r^{n}\circ\pi^{\sharp},$

see e.g. [37, 31]. This hierarchy is a key ingredient in the construction of
integrals for bihamiltonian systems admitting a Poisson-Nijenhuis description.
Similarly, a presymplectic-Nijenhuis structure $(\omega,r)$ gives rise to a
hierarchy of presymplectic-Nijenhuis structures $(\omega_{n},r)$,
$n=1,2,\ldots$, where

(4.5) $\omega_{n}^{\flat}=\omega^{\flat}\circ r^{n}.$

In particular, symplectic-Nijenhuis structures give rise to two hierarchies,
one of Poisson structures, and the other of presymplectic forms. We now see
how to extend these hierarchies to Dirac-Nijenhuis structures.

Let $(L,r)$ be a Dirac-Nijenhuis structure on $M$. For each $n\in\mathbb{N}$,
let

(4.6) $L_{(n,0)}:=(r^{n},\mathrm{id}_{T^{*}M})(L),\qquad
L_{(0,n)}:=(\mathrm{id}_{TM},(r^{*})^{n})(L).$

######  Proposition 4.8.

(1) If $\ker(r,\mathrm{id}_{T^{*}M})|_{L}=0$, then $(L_{(n,0)},r)$ is a Dirac-
Nijenhuis structure for all $n\in\mathbb{N}$; (2) if
$\ker(\mathrm{id}_{TM},r^{*})|_{L}=0$, then $(L_{(0,n)},r)$ is a Dirac-
Nijenhuis structure for all $n\in\mathbb{N}$.

###### Proof.

We will prove (1); the proof of (2) is similar.

Suppose that $\ker(r,\mathrm{id}_{T^{*}M})|_{L}=0$. In this case, $L_{(1,0)}$
is a smooth subbundle of $\mathbb{T}M$ with the same rank as $L$. We will
prove that $L_{(1,0)}$ is isotropic (hence lagrangian) and involutive with
respect the Courant bracket. The result for a general $L_{(n,0)}$ will follow
by induction.

To prove that $L_{(1,0)}$ is isotropic, note that, for
$(X,\alpha),(Y,\beta)\in\Gamma(L)$,

$\langle(r(X),\alpha),(r(Y),\beta)\rangle=\langle(X,\alpha),(r,r^{*})(Y,\beta)\rangle=0$

since $(r,r^{*})(L)\subset L$. To prove that involutivity of $L_{(1,0)}$, we
will show that

$\langle\left[\\!\left[(r(X),\alpha),(r(Y),\beta)\right]\\!\right],(r(Z),\gamma)\rangle=0$

for any $(X,\alpha),(Y,\beta),(Z,\gamma)\in\Gamma(L)$. From the definitions of
$D^{r}$ and $D^{r,*}$, see (2.11) and (2.12), one can check that

(4.7)
$\displaystyle\langle\left[\\!\left[(r(X),\alpha),(r(Y),\beta)\right]\\!\right],(r(Z),\gamma)\rangle$
$\displaystyle=\langle\,\left[\\!\left[(r(X),r^{*}(\alpha)),(Y,\beta)\right]\\!\right],(r(Z),r^{*}(\gamma))\,\rangle$
$\displaystyle+\langle\mathbb{D}^{r}_{Y}(X,\alpha),(r(Z),r^{*}(\gamma))\rangle+\langle\gamma,\mathcal{N}_{r}(X,Y)\rangle.$

Since $(r,r^{*})(L)\subset L$, $\mathbb{D}^{r}_{Y}(\Gamma(L))\subset\Gamma(L)$
and $\mathcal{N}_{r}=0$, we conclude that $L_{(1,0)}$ is a Dirac structure on
$M$. The fact that $L_{(1,0)}$ and $r$ are compatible is a direct consequence
of (2.17). For the inductive step, notice that, if $L_{(n-1,0)}$ is a Dirac
structure, then

$\ker(r^{n},\mathrm{id}_{T^{*}M})|_{L}=\ker(r,\mathrm{id}_{T^{*}M})|_{L_{(n-1,0)}}\,\,\,\text{
and }L_{(n,0)}=(r,\mathrm{id}_{T^{*}M})(L_{(n-1,0)}).$

∎

######  Remark 4.9.

For a Dirac structure $L$ compatible with a (1,1)-tensor field $r$, it follows
from (4.7) that a weaker condition on $\mathcal{N}_{r}$ is sufficient to
ensure the involutivity of $L_{(1,0)}$. Indeed, one only needs that

$\mathcal{N}_{r}(X,Y)\in L\cap TM,\;\;\;\forall\,X,Y\in\mathrm{pr}_{T}(L).$

(In other words., $(L,r,0)$ is Dirac-quasi Nijenhuis, see Remark 3.14). On the
other hand, $L_{(0,1)}$ is always involutive as a consequence of the identity

$\langle\left[\\!\left[(X,r^{*}(\alpha)),(Y,r^{*}(\beta))\right]\\!\right],(Z,r^{*}(\gamma))\rangle=\langle\left[\\!\left[(X,\alpha),(Y,\beta)\right]\\!\right],(r(Z),r^{*}(\gamma))\rangle+\langle\mathbb{D}^{r}_{Z}(X,\alpha),(Y,\beta)\rangle.$

$\diamond$

######  Remark 4.10.

The conditions appearing in Proposition 4.8 admit the following natural
interpretations. For a Dirac-Nijenhuis structure $(L,r)$, recall that $r$
preserves $K=L\cap TM$ and $r^{*}$ preserves $L\cap T^{*}M$. One can check
that

  * (a)

$\ker(r,\mathrm{id}_{T^{*}M})|_{L}=0\,\Leftrightarrow\,r|_{K}:K\to K\text{ is
invertible}$,

  * (b)

$\ker(\mathrm{id}_{TM},r^{*})|_{L}=0\,\Leftrightarrow\,r^{*}|_{L\cap
T^{*}M}:L\cap T^{*}M\to L\cap T^{*}M\text{ is invertible}.$

Since the leaves $\mathcal{O}$ of $L$ are $r$-invariant, we have an induced
map $\bar{r}:TM/T\mathcal{O}\to TM/T\mathcal{O}$. Hence (b) above is also
equivalent to the invertibility of $\bar{r}$ (since $r^{*}|_{L\cap
T^{*}M}=\bar{r}^{*}$ with the identification $L\cap
T^{*}M=\mathrm{Ann}(T\mathcal{O})=(TM/T\mathcal{O})^{*}$). $\diamond$

In the context of Proposition 4.8, the following properties of $L_{(n,0)}$ and
$L_{(0,n)}$ are simple to check.

######  Proposition 4.11.

  * (1)

The Dirac structures $L_{(n,0)}$ have the same null distribution as $L$. If
the null foliation is simple and $Q$ is the leaf space, then

(4.8) $(\pi_{Q})_{n}^{\sharp}=r_{Q}^{n}\circ\pi_{Q}^{\sharp},$

where $(\pi_{Q})_{n}$ is the Poisson structure on $Q$ corresponding to
$L_{(n,0)}$.

  * (2)

The Dirac structures $L_{(0,n)}$ have the same leaves as $L$, and we have the
relation

(4.9) $(\omega_{\mathcal{O}})_{n}^{\flat}=\omega_{\mathcal{O}}^{\flat}\circ
r_{\mathcal{O}}^{n},$

where $(\omega_{\mathcal{O}})_{n}$ is the presymplectic form defined by
$L_{(0,n)}$ on a leaf $\mathcal{O}$.

######  Example 4.12 (Poisson hierarchies).

Let $(M,\pi,r)$ be a Poisson-Nijenhuis manifold, and let
$L=\mathrm{graph}(\pi^{\sharp})$. Since $L\cap TM=0$, the hierarchy
$L_{(n,0)}$ is always well defined and is given by

$L_{(n,0)}=\mathrm{graph}(\pi_{n}^{\sharp}),$

where the Poisson structure $\pi_{n}$ is defined by (4.4). Regarding the other
hierarchy, the condition $\ker(\mathrm{id}_{TM},(r^{*})^{n})|_{L}=0$ is
equivalent to $\ker(\pi^{\sharp})\cap\ker((r^{*})^{n})=0$. If this holds, as
we saw in Prop. 4.11, part (2), $L_{(0,n)}$ is a Dirac structure whose leaves
$\mathcal{O}$ agree with the leaves of $\pi$, and the leafwise presymplectic
form $(\omega_{\mathcal{O}})_{n}$ is related to the symplectic form
$\omega_{\mathcal{O}}$ on $\mathcal{O}$ corresponding to $\pi$ by (4.9).
Notice that $L_{(0,n)}$ is the graph of a Poisson structure if and only if
each $r_{\mathcal{O}}$ is invertible, which is equivalent to $r$ being
invertible by Remark 4.10 (see also Remark 4.15).

The next example shows how gauge transformations of Poisson structures [40]
may be seen as elements of a hierarchy.

######  Example 4.13 (Gauge transformations).

Let $(M,\pi)$ be a Poisson manifold and $B\in\Omega^{2}(M)$. Define
$r:=\mathrm{id}+\pi^{\sharp}\circ B^{\flat}:TM\to TM$, noticing that
$\pi^{\sharp}\circ r^{*}=r\circ\pi^{\sharp}$. Also, as shown in [37] (see
equation B.3.9 therein),

$R_{\pi}^{r}(X,\alpha)=\pi^{\sharp}(i_{X}i_{\pi^{\sharp}(\alpha)}dB).$

By assuming that $B$ is closed, one then has that the pair $(\pi,r)$ is
compatible and (cf. Remark 4.9)

$L_{(0,1)}=\\{(\pi^{\sharp}(\alpha),\alpha+i_{\pi^{\sharp}(\alpha)}B),\,\;\,\alpha\in
T^{*}M\\}$

is a Dirac structure. The Dirac structure $L_{(0,1)}$ is known as the gauge
transformation of $\pi$ by $B$. Note that $L_{(0,1)}$ is again the graph of a
Poisson structure if and only if $r$ is invertible. The vanishing of the
Nijenhuis torsion of $r$, making the pair $(\pi,r)$ into a Poisson-Nijenhuis
structure, is equivalent to the 2-form defined by
$B^{\flat}\circ\pi^{\sharp}\circ B^{\flat}:TM\to T^{*}M$ being closed, which
is the same as saying that the pair $(B,r)$ is presymplectic-Nijenhuis [37].

######  Example 4.14 (Presymplectic hierarchies).

Let $(M,\omega,r)$ be a presymplectic-Nijenhuis manifold, and let
$L=\mathrm{graph}(\omega^{\flat})$. Since $L\cap T^{*}M=0$, the hierarchy
$L_{(0,n)}$ is always well defined and

$L_{(0,n)}=\mathrm{graph}(\omega^{\flat}\circ r^{n}).$

As for the other hierarchy, the condition
$\ker(r^{n},\mathrm{id}_{T^{*}M})|_{L}=0$ is equivalent to
$\ker(r^{n})\cap\ker(\omega^{\flat})=0$. Following Prop. 4.11 (1), in this
case $L_{(n,0)}$ is a Dirac structure whose null distribution coincides the
kernel $K$ of $\omega$. When $K$ is tangent to a simple foliation with leaf
space $Q$, the Poisson structure $(\pi_{Q})_{n}$ on $Q$ corresponding to
$L_{(n,0)}$ satisfies (4.8).

######  Remark 4.15.

For a Dirac-Nijenhuis structure $(L,r)$ on $M$ with invertible $r:TM\to TM$,
both hierarchies $L_{(n,0)}$ and $L_{(0,n)}$ are well defined. In this case
one can verify that $(L,r^{-1})$ is also a Dirac-Nijenhuis structure using the
relation

$D^{r^{-1}}_{X}(Y)=r^{-1}(D_{r^{-1}(X)}^{r}(Y)).$

Moreover, the “$(0,n)$” hierachy of $(L,r)$ coincides with the “$(n,0)$”
hierarchy of $(L,r^{-1})$.  $\diamond$

The next example describes the hierarchy associated to a holomorphic Dirac
structure. It will be convenient to define the operation on Dirac structures,
$L\mapsto L^{-}$, given by

$L^{-}=(-\mathrm{id}_{TM},\mathrm{id}_{T^{*}M})(L).$

######  Example 4.16 (Holomorphic hierarchy).

Let $(M,r)$ be a complex manifold. Let $\mathcal{L}\subset\mathbb{T}^{1,0}M$
be a holomorphic Dirac structure, with corresponding Dirac-Nijenhuis structure
$(L,r)$, see Prop. 3.16. Recall that $L$ is referred to as the real part of
$\mathcal{L}$. One can check that

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(r,\mathrm{id})}$$\textstyle{L_{(1,0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id},r^{*})}$$\scriptstyle{(r,\mathrm{id})}$$\textstyle{L^{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id},r^{*})}$$\scriptstyle{(r,\mathrm{id})}$$\textstyle{L_{(1,0)}^{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id},r^{*})}$$\scriptstyle{(r,\mathrm{id})}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id},r^{*})}$

We define the imaginary part of $\mathcal{L}$ as $L_{(0,1)}=L_{(1,0)}^{-}$.
When $\mathcal{L}=\mathrm{graph}(\Pi)$ (resp. $\mathrm{graph}(\Omega)$) for a
holomorphic Poisson structure $\Pi=\pi-\mathbf{i}\,\pi_{r}$ (resp. holomorphic
closed 2-form $\Omega=\omega-\mathbf{i}\,\omega_{r}$), the imaginary part of
$\mathcal{L}$ is $\mathrm{graph}(-4\pi_{r})$ (resp.
$\mathrm{graph}(-\omega_{r})$), see Examples 3.17 and 3.18.

A central fact concerning the Poisson structures $\pi_{n}$, $n=1,2,\ldots$, in
the hierarchy (4.4) of a Poisson-Nijenhuis structure $(\pi,r)$ is that, for
any $i$, $j$ , $\pi_{i}+\pi_{j}$ is a Poisson structure; equivalently,
$[\pi_{i},\pi_{j}]=0$, where $[\cdot,\cdot]$ is the Schouten bracket. We
briefly explain how this compatibility property can be generalized to Dirac-
Nijenhuis hierarchies.

Given Dirac structures $L_{1}$ and $L_{2}$ on a manifold $M$, following [23]
we define their “cotangential product” as

$L_{1}\circledast L_{2}=\\{(X_{1}+X_{2},\alpha)\,|\,(X_{1},\alpha)\in
L_{1},\,(X_{2},\alpha)\in L_{2}\\}.$

In general, this product results in a (possibly non-smooth) lagrangian
distribution in $\mathbb{T}M$; we say that the Dirac structures concur if
$L_{1}\circledast L_{2}$ is a smooth involutive subbundle, i.e., a Dirac
structure. For Poisson structures $\pi_{1}$ and $\pi_{2}$, it is a direct
verification that $L_{i}=\mathrm{graph}(\pi_{i})$, $i=1,2$, concur if and only
if $\pi_{1}+\pi_{2}$ is a Poisson structure.

For a Dirac-Nijenhuis structure $(L,r)$ on $M$, consider its hierarchy
$L_{(n,0)}$, $n=1,2,\ldots$, as in Prop. 4.8, part (1). The fact that all
these Dirac structures have the same null distribution $K$ (and hence the same
projections on $T^{*}M$, which are $\mathrm{Ann}(K)$) ensures that their
pairwise cotangential products are smooth lagrangian subbundles of
$\mathbb{T}M$. Moreover, using a local version of Prop. 4.11, part (1), around
points where $\mathrm{Ann}(K)$ has locally constant rank (which form an open
dense subset of $M$) and the fact that the quotient Poisson structures
commute, one can verify that any pair $L_{(n,0)}$ and $L_{(m,0)}$ concurs.

##  5\. Compatibility of multiplicative forms and $(1,1)$-tensor fields

In this section we extend the compatibility in Def. 3.5 to higher-degree
forms. We refine the infinitesimal description of multiplicative forms in [5]
to include their compatibility with $(1,1)$-tensor fields (Thm. 5.3), leading
to a holomorphic version of [5, Thm. 4.6] (Thm. 5.5).

###  5.1. Review of multiplicative structures

We begin with a brief review of multiplicative differential forms and
$(1,1)$-tensor fields on Lie groupoids and their infinitesimal descriptions,
following [5, 7].

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. We denote its source
and targent maps by $s,t:\mathcal{G}\to M$, and its multiplication map by
$m:\mathcal{G}^{(2)}\to\mathcal{G}$, where $\mathcal{G}^{(2)}$ is the fiber
product $\mathcal{G}_{\,s}\\!\\!\times_{t}\mathcal{G}$. We will often identify
$M$ with its image under the unit map. The Lie algebroid of $\mathcal{G}$ is
$A_{\mathcal{G}}:=\ker(Ts)|_{M}$, with anchor $Tt|_{A_{\mathcal{G}}}$ and
bracket defined by the identification of $\Gamma(A_{\mathcal{G}})$ with right-
invariant vector fields on $\mathcal{G}$.

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

$m^{*}\omega=\mathrm{pr}_{1}^{*}\omega+\mathrm{pr}_{2}^{*}\omega,$

where $\mathrm{pr}_{i}:\mathcal{G}^{(2)}\rightarrow\mathcal{G}$ is the natural
projection, for $i=1,2$. These objects have infinitesimal counterparts given
as follows. An IM p-form on a Lie algebroid $A\to M$ is defined by a pair
$(\mu,\nu)$, where

$\mu:A\rightarrow\wedge^{p-1}T^{*}M,\qquad\nu:A\rightarrow\wedge^{p}T^{*}M$

are vector-bundle maps (over the identity on $M$) satisfying the so-called IM
equations:

(5.1)
$\begin{cases}\mu([a,b])=\mathcal{L}_{\rho(a)}\mu(b)-i_{\rho(b)}(d\mu(a)+\nu(a))\\\
\nu([a,b])=\mathcal{L}_{\rho(a)}\nu(b)-i_{\rho(b)}d\nu(a)\\\
i_{\rho(a)}\mu(b)=-i_{\rho(b)}\mu(a)\end{cases}$

for all $a,b\in\Gamma(A)$. Any multiplicative $p$-form $\omega$ on a Lie
groupoid $\mathcal{G}$ gives rise to an IM $p$-form $(\mu,\nu)$ on the
corresponding Lie algebroid $A=A_{\mathcal{G}}$ via

(5.2)
$i_{\overrightarrow{a}}\,\omega=t^{*}\mu(a),\,\;\;\;\;\;i_{\overrightarrow{a}}\,d\omega=t^{*}\nu(a).$

Here $\overrightarrow{a}$ denotes the right-invariant vector field on
$\mathcal{G}$ corresponding to $a\in\Gamma(A)$. When $\mathcal{G}$ is source
1-connected, (5.2) defines a one-to-one correspondence between multiplicative
$p$-forms on $\mathcal{G}$ and IM $p$-forms on $A$, see [5]. Under this
bijection, closed multiplicative forms correspond to IM-forms with $\nu=0$,
that we will also refer to as closed.

A $(1,1)$-tensor field $K\in\Gamma(T^{*}\mathcal{G}\otimes T\mathcal{G})$ is
called multiplicative if there is a vector bundle map $r:TM\rightarrow TM$
such that

${T\mathcal{G}}$${T\mathcal{G}}$${TM}$${TM\par}$$\scriptstyle{K}$$\scriptstyle{r}$

is a groupoid morphism. Their infinitesimal counterparts are described as
follows [7, Sec. 6]. On a Lie algebroid $A$, an IM $(1,1)$-tensor is a triple
$(D,l,r)$, where $l:A\rightarrow A$ and $r:TM\to TM$ are vector-bundle maps
(over the identity on $M$) and $D:\Gamma(A)\rightarrow\Gamma(T^{*}M\otimes A)$
is a $\mathbb{R}$-linear map satisfying the Leibniz-type rule

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

and such that the following IM-equations hold:

(5.3) $\left\\{\begin{array}[]{rl}r\circ\rho\\!\\!&=\rho\circ l,\\\
\rho(D_{X}(a))\\!\\!&=D_{X}^{r}(\rho(a)),\vspace{2pt}\\\
l([a,b])\\!\\!&=[a,l(b)]-D_{\rho(b)}(a)\vspace{2pt}\\\
D_{X}([a,b])\\!\\!&=[a,D_{X}(b)]-[b,D_{X}(a)]+D_{[\rho(b),X]}(a)-D_{[\rho(a),X]}(b).\end{array}\right.$

In the language of $\S$ 2.1, $(D,l,r)$ is a 1-derivation on the vector bundle
$A$ satisfying the additional compatibilities (5.3) with respect to its Lie-
algebroid structure.

The IM $(1,1)$-tensor $(D,l,r)$ corresponding to a multiplicative
$(1,1)$-tensor field $K$ is defined by the conditions

(5.4) $K(\overrightarrow{a})=\overrightarrow{l(a)},\,\,\quad
Tt(K(U))=r(Tt(U)),\,\,\quad(\mathcal{L}_{\overrightarrow{a}}K)(U)=\overrightarrow{D_{Tt(U)}(a)},$

for $a\in\Gamma(A)$ and $t$-projectable $U\in\mathfrak{X}(\mathcal{G})$. With
the identification $T\mathcal{G}|_{M}=TM\oplus A$, we directly see that
$r=K|_{TM}$ and $l=K|_{A}.$

As before, when $\mathcal{G}$ is a source 1-connected Lie groupoid, we have a
bijective correspondence between multiplicative $(1,1)$-tensor fields and IM
$(1,1)$-tensors on $A$ via (5.4), see [7, Thm. 3.19]. By regarding vector
bundles as Lie groupoids (with respect to fiberwise addition), one can check
that multiplicative $(1,1)$-tensors agree with linear ones, and Theorem 2.1
follows as a particular case of this result.

For a multiplicative $(1,1)$-tensor field $K$ on a source-connected Lie
groupoid $\mathcal{G}\rightrightarrows M$, it is also shown in [7, $\S$ 6.2]
that $K$ is a Nijenhuis operator if and only if the following equations hold
for its corresponding IM $(1,1)$-tensor $(D,l,r)$ (cf. (2.8)):

(5.5) $\begin{cases}\mathcal{N}_{r}(X,Y)=0,\\\ l(D_{X}(a)-D_{X}(l(a))=0,\\\
D^{2}_{(X,Y)}(a):=l(D_{[X,Y]}(a))-[D_{X},D_{Y}](a)-D_{[X,Y]_{r}}(a)=0,\end{cases}$

for all $X,Y\in\mathfrak{X}(M)$ and $a\in\Gamma(A)$. Here $[D_{X},D_{Y}]$ is
the commutator of the operators on $\Gamma(A)$ and $[\cdot,\cdot]_{r}$ is the
deformed bracket

(5.6) $[X,Y]_{r}=[r(X),Y]+[X,r(Y)]-r([X,Y]).$

The notation $D^{2}$ is motivated by a natural extension of $D$ as an operator
on $\Gamma(\wedge^{\bullet}T^{*}M\otimes A)$ studied in [7].

###  5.2. Global and infinitesimal compatibility

We will deal here with a natural extension of the compatibility in Definition
3.5 to differential forms of arbitrary degrees. Given a differential form
$\omega\in\Omega^{p}(M)$ and $r\in\Omega^{1}(M,TM)$, we denote by
$\omega_{r}\in\Gamma(T^{*}M\otimes\wedge^{p-1}T^{*}M)$ the tensor field given
by

$\omega_{r}(X_{1};X_{2},\dots,X_{p})=\omega(r(X_{1}),X_{2},\dots,X_{p}).$

######  Definition 5.1.

We say that the pair $(\omega,r)$ is compatible if

  * $(a)$

$\omega_{r}$ is skew-symmetric, i.e. $\omega_{r}\in\Omega^{p}(M)$,

  * $(b)$

$d(\omega_{r})=(d\omega)_{r}$.

A motivating example for this notion of compatibility is given in Proposition
5.4 below.

Denote by $\Omega_{r}^{p}(M)$ the space of differential $p$-forms $\omega$
such that $\omega_{r}\in\Omega^{p}(M)$. Then the operator $D^{r,*}$ (2.12) can
be extended to
$D^{r,*}:\mathfrak{X}(M)\times\Omega_{r}^{p}(M)\to\Gamma(T^{*}M\otimes\wedge^{p-1}T^{*}M)$
via

(5.7)
$D^{r,*}_{Y}(\omega)=i_{Y}d(\omega_{r})-i_{r(Y)}d\omega,\qquad\forall\,Y\in\mathfrak{X}(M).$

So a pair $(\omega,r)$ is compatible if and only if
$\omega\in\Omega_{r}^{p}(M)$ and $D^{r,*}(\omega)=0$.

######  Definition 5.2.

We say that an IM 2-form $(\mu,\nu)$ and an IM $(1,1)$-tensor $(D,l,r)$ on a
Lie algebroid $A$ are compatible if

(5.8)
$\displaystyle\mu(l(a))=\mu(a)_{r},\,\,\qquad\quad\;\;\;\nu(l(a))=\nu(a)_{r}$
(5.9)
$\displaystyle\mu(D_{X}(a))=D_{X}^{r,*}(\mu(a)),\,\,\,\;\nu(D_{X}(a))=D_{X}^{r,*}(\nu(a)),$

for all $X\in\mathfrak{X}(M)$ and $a\in\Gamma(A)$.

######  Theorem 5.3.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid endowed with
multiplicative $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ and
$\omega\in\Omega^{p}(\mathcal{G})$, with corresponding infinitesimal data
$(D,l,r)$ and $(\mu,\nu)$. If the pair $(\omega,K)$ is compatible then
$(D,l,r)$ and $(\mu,\nu)$ are compatible, and the converse holds if
$\mathcal{G}$ is source-connected. Moreover, the IM $p$-form corresponding to
$\omega_{K}$ is $(\mu\circ l,\nu\circ l)$.

###### Proof.

Let $\tau_{1},\tau_{2}\in\Gamma(\otimes^{p}T^{*}\mathcal{G})$ be tensor fields
on $\mathcal{G}$ defined by

$\tau_{1}(U_{1},\dots,U_{p})=\omega(K(U_{1}),U_{2},\dots,U_{p}),\,\,\,\tau_{2}(U_{1},\dots,U_{p})=\omega(U_{1},K(U_{2}),U_{3},\dots,U_{p}),$

for $U_{1},\dots,U_{p}\in\mathfrak{X}(\mathcal{G})$. Note that
$\omega\in\Omega_{K}^{p}(\mathcal{G})$ if and only if $\tau_{1}=\tau_{2}$.
Since $\tau_{1},\tau_{2}$ are multiplicative tensor fields, assuming that
$\mathcal{G}$ is source connected they will be equal if and only if (see [7,
Rem. 3.16])

(A) $\displaystyle\tau_{1}(\cdot,\overbrace{\overrightarrow{a}}^{i\text{th-
entry}},\cdot)$
$\displaystyle=\tau_{2}(\cdot,\overbrace{\overrightarrow{a}}^{i\text{th-
entry}},\cdot),\,\,\quad\forall\,i=1,\dots,p,$ (B)
$\displaystyle\mathcal{L}_{\overrightarrow{a}}\tau_{1}$
$\displaystyle=\mathcal{L}_{\overrightarrow{a}}\tau_{2}.$

We first claim that (A) is equivalent to the first equation in (5.8). Indeed,
it follows from (5.2) and (5.4) that

$\displaystyle\tau_{1}(\overrightarrow{a},U_{1},...,U_{p-1})$
$\displaystyle=\omega(K(\overrightarrow{a}),U_{1},\dots,U_{p-1})=\omega(\overrightarrow{l(a)},U_{1},\dots,U_{p-1})=\mu(l(a))(X_{1},\dots,X_{p-1}),$

where $X_{j}=Tt(U_{j})$, $j=1,\dots,p$. Similarly,

$\displaystyle\tau_{2}(\overrightarrow{a},U_{1},...,U_{p-1})=\mu(a)(r(X_{1}),X_{2},\dots,X_{p-1}).$

Therefore $\mu(l(a))=\mu(a)_{r}$ is equivalent to (A) for $i=1$. Other values
of $i$ work similarly.

Next, using that
$\mathcal{L}_{\overrightarrow{a}}\omega=t^{*}(\nu(a)+d\mu(a))$ and (5.4), we
see that

$\displaystyle(\mathcal{L}_{\overrightarrow{a}}\tau_{1})(U_{1},U_{2},...,U_{p})=(\nu(a)+d\mu(a))_{r}(X_{1},\dots,X_{p})+\mu(D^{r}_{X_{1}}(a))|_{(X_{2},\dots,X_{p})}.$

Similarly,

$\displaystyle(\mathcal{L}_{\overrightarrow{a}}\tau_{2})(U_{1},U_{2},...U_{p})$
$\displaystyle=-(\nu(a)+d\mu(a))_{r}(X_{2},X_{1},X_{3},\dots,X_{p})-\mu(D^{r}_{X_{2}}(a))|_{(X_{1},X_{3},...,X_{p})}.$

So, if we define $\zeta:\Gamma(A)\to\Gamma(T^{*}M\otimes\wedge^{p-1}T^{*}M)$
as

$\zeta(a)(Y_{1};Y_{2},\dots,Y_{p})=(\nu(a)+d\mu(a))_{r}(Y_{1},\dots,Y_{p})+\mu(D^{r}_{Y_{1}}(a))|_{(Y_{2},\dots,Y_{p})},$

we see that (B) holds if and only if $\zeta$ takes values in $\Omega^{p}(M)$
(i.e. $\zeta(a)$ is skew-symmetric) for all $a\in\Gamma(A)$.

When (A) and (B) hold, we have that
$\tau_{1}=\tau_{2}=\omega_{K}\in\Omega^{p}(\mathcal{G})$ is a multiplicative
$p$-form, and the corresponding IM $p$-form $(\mu_{K},\nu_{K})$ on $A$ is
given by

$\mu_{K}(a)=\mu(l(a)),\qquad\nu_{K}(a)=\zeta(a)-d(\mu(l(a))),$

where for the second equality we used that
$i_{\overrightarrow{a}}d\omega_{K}=\mathcal{L}_{\overrightarrow{a}}\omega_{K}-di_{\overrightarrow{a}}\omega_{K}$.
Using that $\mu(l(a))=\mu(a)_{r}$ and (5.7), the second equality can be
rewritten as

$i_{X}\nu_{K}(a)=i_{X}\nu(a)_{r}+\mu(D^{r}_{X}(a))-D^{r,*}_{X}(\mu(a)).$

Arguing similarly with $d\omega$ in place of $\omega$, we conclude that
$d\omega\in\Omega_{K}^{p+1}(\mathcal{G})$ if and only if

$\nu(l(a))=\nu(a)_{r}$

and $\zeta^{d}:\Gamma(A)\to\Gamma(T^{*}M\otimes\wedge^{p}T^{*}M)$ defined by

$\zeta^{d}(a)(Y_{1};Y_{2},\dots,Y_{p+1})=(d\nu(a))_{r}(Y_{1},Y_{2},\dots,Y_{p+1})+\nu(D^{r}_{Y_{1}}(a))|_{(Y_{2},\dots,Y_{p+1})}$

is skew-symmetric. In this case, the IM $(p+1)$-form
$(\mu_{K}^{d},\nu_{K}^{d})$ corresponding to $(d\omega)_{K}$ satisfies

$\mu_{K}^{d}(a)=\nu(l(a)),\,\,\quad
i_{X}\nu_{K}^{d}(a)=\nu(D^{r}_{X}(a))-D^{r,*}_{X}(\nu(a)).$

When $\mathcal{G}$ is source-connected, we conclude that
$(d\omega)_{K}=d(\omega_{K})$ if and only if $\mu^{d}_{K}(a)=\nu_{K}(a)$ and
$\nu_{K}^{d}=0$, which in turn is equivalent to

$\mu(D^{r}(a))=D^{r,*}(\mu(a)),\qquad\nu(D^{r}(a))=D^{r,*}(\nu(a)).$

∎

###  5.3. Holomorphic multiplicative forms

In this section we specialize the results from the previous section to the
holomorphic setting.

We start with a general observation relating the compatibility in Definition
5.1 with holomorphic forms. Let $r:TM\to TM$ be a complex structure on $M$.

######  Proposition 5.4.

A complex form
$\Omega=\omega+\mathbf{i}\,\omega_{1}\in\Omega^{p}(M,\mathbb{C})$ is a
holomorphic $p$-form if and only if the pair $(\omega,r)$ is compatible and
$\omega_{1}=-\omega_{r}$.

###### Proof.

The condition $\Omega\in\Omega^{(p,0)}(M)$ is equivalent to
$i_{X+\mathbf{i}\,r(X)}\Omega=0$ for all $X\in\mathfrak{X}(M)$, which in turn
holds if and only if $\omega_{1}=-\omega_{r}$. We now verify that condition
$(b)$ in Definition 5.1 is equivalent to $\bar{\partial}\Omega=0$. For
$Y\in\mathfrak{X}(M)$, using that $d=\partial+\bar{\partial}$ and Cartan’s
formula, we see that

$i_{Y+\mathbf{i}\,r(Y)}\bar{\partial}\Omega=\mathcal{L}_{Y+\mathbf{i}\,r(Y)}\Omega=\mathcal{L}_{Y+\mathbf{i}\,r(Y)}(\omega-\mathbf{i}\,\omega_{r})=A+\mathbf{i}\,B,$

where $A=\mathcal{L}_{Y}\omega+\mathcal{L}_{r(Y)}(\omega_{r})$ and
$B=\mathcal{L}_{r(Y)}\omega-\mathcal{L}_{Y}(\omega_{r})$. Again by Cartan’s
formula, we have

$A=i_{r(Y)}(d(\omega_{r})-(d\omega)_{r}),\qquad
B=i_{Y}((d\omega)_{r}-d(\omega_{r}))$

So the result follows. ∎

####  5.3.1. Infinitesimal description of holomorphic multiplicative forms.

A holomorphic Lie groupoid is a Lie groupoid $\mathcal{G}\rightrightarrows M$,
where both $\mathcal{G}$ and $M$ are complex manifolds and all the structures
maps ${m},{\epsilon},{\textit{i}},{s}$ and ${t}$ are holomorphic. A
holomorphic Lie algebroid is a holomorphic vector bundle $\mathcal{A}\to M$
endowed with a holomorphic bundle map $\widehat{\rho}:\mathcal{A}\to T^{1,0}M$
and a complex Lie algebra structure on the sheaf of holomorphic sections
$\Gamma_{\mathrm{hol}}(\cdot,\mathcal{A})$ such that $\widehat{\rho}$ induces
a morphism of sheaves of complex Lie algebras from
$\Gamma_{\mathrm{hol}}(\cdot,\mathcal{A})$ to
$\Gamma_{\mathrm{hol}}(\cdot,T^{1,0}M)$ and

$[\sigma_{1},f\sigma_{2}]=f[\sigma_{1},\sigma_{2}]+(\mathcal{L}_{\widehat{\rho}(\sigma_{1})}f)\sigma_{2},$

for $\sigma_{1}$, $\sigma_{2}$ (local) holomorphic sections of $\mathcal{A}$
and $f$ a (local) holomorphic function.

A holomorphic Lie groupoid $\mathcal{G}\rightrightarrows M$ gives rise to a
holomorphic Lie algebroid just as in the smooth real case. Since $s$ is
holomorphic, $\mathcal{A}_{\mathcal{G}}:=\ker(Ts:T^{1,0}\mathcal{G}|_{M}\to
T^{1,0}M)$ is a holomorphic vector bundle over $M$. It becomes a holomorphic
Lie algebroid with Lie bracket $[\cdot,\,\cdot]$ induced by the one on (local)
right-invariant holomorphic vector fields and
$\widehat{\rho}=Tt|_{\mathcal{A}_{\mathcal{G}}}$. A holomorphic Lie algebroid
$\mathcal{A}\to M$ is said to be integrable if
$\mathcal{A}=\mathcal{A}_{\mathcal{G}}$ for a holomorphic Lie groupoid
$\mathcal{G}\rightrightarrows M$.

The theory of multiplicative forms and IM-forms can be naturally extended to
the holomorphic context. A holomorphic $p$-form $\Omega$ is multiplicative if

$m^{*}\Omega=\mathrm{pr}_{1}^{*}\Omega+\mathrm{pr}_{2}^{*}\Omega.$

Similarly, a holomorphic IM $p$-form on a holomorphic Lie algebroid
$\mathcal{A}$ is a pair $(\widehat{\mu},\widehat{\nu})$, where
$\widehat{\mu}:\mathcal{A}\rightarrow\wedge^{(p-1,0)}T^{*}M$ and
$\widehat{\nu}:\mathcal{A}\rightarrow\wedge^{(p,0)}T^{*}M$ are holomorphic
vector-bundle maps satisfying the IM equations

(5.10)
$\left\\{\begin{array}[]{rl}\widehat{\mu}([u,v])\\!\\!&=\mathcal{L}_{\widehat{\rho}(u)}\widehat{\mu}(v)-i_{\widehat{\rho}(v)}(\partial\widehat{\mu}(u)+\widehat{\nu}(u)),\\\
\widehat{\nu}([u,v])\\!\\!&=\mathcal{L}_{\widehat{\rho}(u)}\widehat{\nu}(v)-i_{\widehat{\rho}(v)}\partial\widehat{\nu}(u),\\\
i_{\widehat{\rho}(u)}\widehat{\mu}(v)\\!\\!&=-i_{\widehat{\rho}(v)}\widehat{\mu}(u),\end{array}\right.$

for all $u,v$ (local) holomorphic sections of $\mathcal{A}$.

The following result is a holomorphic version of [5, Thm. 4.6] describing the
infinitesimal counterparts of multiplicative forms.

######  Theorem 5.5.

Let $\mathcal{G}\rightrightarrows M$ be a holomorphic source 1-connected Lie
groupoid and $\mathcal{A}$ its holomorphic Lie algebroid. There is a one-to-
one correspondence between multiplicative holomorphic $p$-forms $\Omega$ on
$\mathcal{G}$ and holomorphic IM $p$-forms $(\widehat{\mu},\widehat{\nu})$ on
$\mathcal{A}$ given by

(5.11) $i_{\overrightarrow{u}}\Omega=t^{*}\widehat{\mu}(u),\,\,\qquad
i_{\overrightarrow{u}}\partial\Omega=t^{*}\widehat{\nu}(u),$

where $\overrightarrow{u}$ is the (local) right-invariant vector field on
$\mathcal{G}$ corresponding to a (local) holomorphic section $u$ of
$\mathcal{A}$.

As in the real smooth case, we say that a holomorphic IM $p$-form is closed
when $\widehat{\nu}$ is zero, since these are in correspondence with closed
multiplicative holomorphic forms in Theorem 5.5.

We will prove this theorem in $\S$ 5.3.3 below using Theorem 5.3 and the
characterization of holomorphic $p$-forms and IM $p$-forms in terms of their
real parts that we discuss next (Prop. 5.8).

####  5.3.2. The underlying real geometry

We now expressing various holomorphic objects in terms of their underlying
real structures.

As shown in [32, Prop. 3.3], a holomorphic Lie algebroid $\mathcal{A}\to M$
can always be viewed as a real Lie algebroid $A\to M$ together with a
holomorphic structure on its underlying vector bundle such that the Lie
bracket on $\Gamma(A)$ restricts to a $\mathbb{C}$-linear Lie bracket on the
sheaf of holomorphic sections. Equivalently, if one expresses the holomorphic
structure on $A$ as a Dolbeault 1-derivation $(D^{\mathrm{hol}},l,r)$ (see
Example 2.3), it is shown in [7, Sec. 6] that its compatibility with the Lie-
algebroid structure amounts to $(D^{\mathrm{hol}},l,r)$ being an IM
$(1,1)$-tensor on $A$ (i.e., the equations in (5.3) hold).

A holomorphic Lie groupoid, in turn, is equivalent to a Lie groupoid
$\mathcal{G}\rightrightarrows M$ equipped with a complex structure $J$ that is
multiplicative [33, Prop. 3.16]. For a holomorphic Lie groupoid
$(\mathcal{G},J)$, its real Lie algebroid $A_{\mathcal{G}}$ inherits an IM
$(1,1)$-tensor $(D,l,r)$ defined by $J$ as well as a holomorphic structure
$(D^{\mathrm{hol}},l,r)$ induced by the the isomorphism of real vector bundles

$\Psi:A_{\mathcal{G}}\to\mathcal{A}_{\mathcal{G}},\qquad\Psi(a)=\frac{1}{2}(a-\mathbf{i}\,l(a)),$

coming from $T\mathcal{G}\ni U\mapsto\frac{1}{2}(U-\mathbf{i}\,J(U))\in
T^{1,0}\mathcal{G}$. Notice that

(5.12) $\widehat{\rho}(\Psi(a))=\frac{1}{2}(\rho(a)-\mathbf{i}\,r(\rho(a))).$

######  Lemma 5.6.

We have that $D=D^{\mathrm{hol}}$, so $(D,l,r)$ coincides with the Dolbeault
1-derivation on $A_{\mathcal{G}}$ induced by $\Psi$.

###### Proof.

Note that $\Psi:A_{\mathcal{G}}\to\mathcal{A}_{\mathcal{G}}$ is the
restriction of the usual identification $T\mathcal{G}\to T^{1,0}\mathcal{G}$,
which induces the holomorphic structure on $T\mathcal{G}$ given by the
Dolbeault 1-derivation $(D^{J},J,J)$, see Example 2.6. Hence, for
$a\in\Gamma(A)$, we have that

$D_{X}^{\mathrm{hol}}(a)=D^{J}_{U}(\overrightarrow{a})|_{M}=(\mathcal{L}_{\overrightarrow{a}}J)(U)|_{M}=D_{X}(a),$

where $U\in\mathfrak{X}(\mathcal{G})$ is any vector field such that
$U|_{M}=X$, and we used (2.11) and (5.4) in the last two equalities. ∎

######  Remark 5.7 (Integration of holomorphic Lie algebroids).

It is shown in [32] that a holomorphic Lie algebroid is integrable if and only
if so is its underlying real Lie algebroid. This can be readily seen as
follows. Consider a holomorphic Lie algebroid $\mathcal{A}$, given by a real
Lie algebroid $A\to M$ with Dolbeault IM $(1,1)$-tensor
$(D^{\mathrm{hol}},l,r)$. If $\mathcal{G}$ is a source 1-connected Lie
groupoid integrating $A$, then $(D^{\mathrm{hol}},l,r)$ integrates to a
multiplicative $(1,1)$-tensor field $J$ on $\mathcal{G}$ that is a complex
structure, making $\mathcal{G}$ into a holomorphic Lie groupoid integrating
$\mathcal{A}$ (by Lemma 5.6). $\diamond$

Let $\mathcal{A}$ be a holomorphic Lie algebroid defined by a real Lie
algebroid $A\to M$ with holomorphic structure $(D^{\mathrm{hol}},l,r)$.

######  Proposition 5.8.

There is a one-to-one correspondence between IM $p$-forms $(\mu,\nu)$ on $A$
compatible with $(D^{\mathrm{hol}},l,r)$ (as in Def. 5.2) and holomorphic IM
$p$-forms $(\widehat{\mu},\widehat{\nu})$ on $\mathcal{A}$ given by

(5.13) $\widehat{\mu}=\mu-\mathbf{i}\,\mu\circ
l,\qquad\widehat{\nu}=\nu-\mathbf{i}\,\nu\circ l.$

###### Proof.

We start with some general observations. Let
$\widehat{\gamma}:\mathcal{A}\to\wedge^{(p,0)}T^{*}M$ be a $\mathbb{C}$-linear
vector-bundle map, and recall that the complex vector bundle underlying
$\mathcal{A}$ is $(A,l)$. Writing
$\widehat{\gamma}=\gamma+\mathbf{i}\,\gamma_{1}$, for
$\gamma,\,\gamma_{1}:A\to\wedge^{p}T^{*}M$, the $\mathbb{C}$-linearity of
$\widehat{\gamma}$ is equivalent to $\gamma_{1}=-\gamma\circ l$, while the
fact that $\widehat{\gamma}$ takes values in $\wedge^{(p,0)}T^{*}M$ is
equivalent to $\gamma(l(a))=(\gamma(a))_{r}$. We also note that
$\widehat{\gamma}$ is holomorphic if and only if

$\gamma(D^{\mathrm{hol}}_{X}(a))=D^{r,*}_{X}(\gamma(a)).$

Indeed, $\widehat{\gamma}$ is holomorphic if and only if it takes holomorphic
sections to holomorphic sections, which by Proposition 5.4 is equivalent to
$D^{r,*}_{X}(\gamma(a))=0$ for all $X\in\mathfrak{X}(M)$ and $a$ (local)
holomorphic section of $A$. Since a (local) section $a$ is holomorphic if and
only if $D^{\mathrm{hol}}(a)=0$, we conclude that $\gamma$ is holomorphic if
and only if

(5.14) $\gamma(D^{\mathrm{hol}}_{X}(a))-D^{r,*}_{X}(\gamma(a))=0$

for all $X\in\mathfrak{X}(M)$ and $a$ (local) holomorphic section. Now note
that the map $a\mapsto\gamma(D^{\mathrm{hol}}_{X}(a))-D^{r,*}_{X}(\gamma(a))$
is $C^{\infty}(M)$-linear, so (5.14) holds for all smooth sections $a$ since
$\Gamma(A)$ is locally generated by holomorphic sections as a
$C^{\infty}(M)$-module.

It immediately follows that any holomorphic IM $p$-form
$(\widehat{\mu},\widehat{\nu})$ is determined via (5.13) by a pair of vector
bundle maps $\mu:A\to\wedge^{p-1}T^{*}M$, $\nu:A\to\wedge^{p}T^{*}M$
satisfying (5.8) and (5.9). So it remains to prove that the pair
$(\widehat{\mu},\widehat{\nu})$ satisfies the holomorphic IM-equations (5.10)
if and only if $(\mu,\nu)$ satisfies the IM-equations (5.1). For $a$ and $b$
(local) holomorphic sections of $A$, define

$\Upsilon(a,b):=\widehat{\mu}([a,b])-\mathcal{L}_{\widehat{\rho}(a)}\widehat{\mu}(b)+i_{\widehat{\rho}(b)}\left(\partial\widehat{\mu}(a)+\widehat{\nu}(a)\right).$

From (5.12) and (5.13), it follows that
$\Upsilon(a,b)=\Upsilon_{0}(a,b)-\mathbf{i}\,\Upsilon_{1}(a,b)$, where

$\displaystyle\Upsilon_{0}(a,b)=$
$\displaystyle\mu([a,b])-\frac{1}{2}\left(\mathcal{L}_{\rho(a)}\mu(b)-i_{\rho(b)}(d\mu(a)+\nu(b))\,\right)$
$\displaystyle+\frac{1}{2}\left(\mathcal{L}_{r(\rho(a))}\mu(l(b))+i_{r(\rho(b))}(d\mu(l(a))+\nu(l(a)))\,\right),$
$\displaystyle\Upsilon_{1}(a,b)=$
$\displaystyle\mu(l([a,b])+\frac{1}{2}\left(\mathcal{L}_{r(\rho(a))}\mu(b)-i_{r(\rho(b))}(d\mu(a)+\nu(a))\,\right)$
$\displaystyle+\frac{1}{2}\left(\mathcal{L}_{\rho(a)}\mu(l(b))-i_{\rho(b)}(d\mu(l(a))+\nu(l(a)))\,\right).$

Since $D^{r,*}(\mu(a))=\mu(D^{\mathrm{hol}}(a))=0$ for $a$ holomorphic, we
have that $d(\mu(l(a)))=d(\mu(a)_{r})=(d\mu(a))_{r}$, which implies that

$\displaystyle i_{r(\rho(b))}d\mu(l(a))=-i_{\rho(b)}d\mu(a),$
$\displaystyle\mathcal{L}_{r(\rho(a))}\mu(l(b))=-\mathcal{L}_{\rho(a)}\mu(b).$

Hence, since $\nu(l(a))=\nu(a)_{r}$, we obtain that

$\Upsilon_{0}(a,b)=\mu([a,b])-\mathcal{L}_{\rho(a)}\mu(b)+i_{\rho(b)}(d\mu(a)+\nu(a)).$

By a similar argument, and using the fact that $l([a,b])=[a,l(b)]$ for $a$,
$b$ holomoprhic (as a consequence of (5.3)), one has that

$\Upsilon_{1}(a,b)=\mu([a,l(b)])-\mathcal{L}_{\rho(a)}\mu(l(b))+i_{\rho(l(b))}(d\mu(a)+\nu(a)).$

It follows that $\Upsilon(a,b)=0$ for all $a$, $b$ (local) holomorphic
sections if and only if

$\mu([a,b])=\mathcal{L}_{\rho(a)}\mu(b)-i_{\rho(b)}(d\mu(a)+\nu(a)),\,\,\;\;\forall\,a,b\in\Gamma(A),$

since $\Upsilon$ is $C^{\infty}(M)$-bilinear as a function of $a,\,b$ and
$\Gamma(A)$ is locally generated by holomorphic sections as a
$C^{\infty}(M)$-module. The analogous result for $\widehat{\nu}$ is proven
similarly. ∎

####  5.3.3. Proof of Theorem 5.5

Write a holomorphic $p$-form $\Omega$ on $(\mathcal{G},J)$ as
$\Omega=\omega-\mathbf{i}\,\omega_{J}$, with $(\omega,J)$ a compatible pair,
see Proposition 5.4. Notice that $\Omega$ is multiplicative if and only if
$\omega$ is multiplicative, and assume that this holds. Let $(\mu,\nu)$ be the
IM-form corresponding to $\omega$ via (5.2). For a local holomorphic section
$a$ of $A_{\mathcal{G}}$, we have

(5.15) $\displaystyle i_{\overrightarrow{\Psi(a)}}\Omega$
$\displaystyle=i_{\overrightarrow{a}}\omega-\mathbf{i}\,i_{\overrightarrow{l(a)}}\omega=t^{*}(\mu(a)-\mathbf{i}\,\mu(l(a))),$
(5.16) $\displaystyle i_{\overrightarrow{\Psi(a)}}\partial\Omega$
$\displaystyle=i_{\overrightarrow{a}}d\omega-\mathbf{i}\,i_{\overrightarrow{a}}d(\omega_{J})=t^{*}(\nu(a)-\mathbf{i}\,\nu(l(a))),$

where we have used that $J(\overrightarrow{a})=\overrightarrow{l(a)}$ and the
compatibility $d(\omega_{J})=(d\omega)_{J}$. Define
$(\widehat{\mu},\widehat{\nu})$ by (5.13). It is clear that (5.11) holds for
$(\widehat{\mu},\widehat{\nu})$ and it is a holomorphic IM $p$-form as
consequence of the compatibility of $(\omega,J)$ via Theorem 5.3 and
Proposition 5.8. Conversely, given a holomorphic IM $p$-form
$(\widehat{\mu},\widehat{\nu})$, write it as in (5.13) for a real IM $p$-form
$(\mu,\nu)$. When $\mathcal{G}$ is source 1-connected, there is a real
multiplicative $p$-form $\omega$ integrating $(\mu,\nu)$. From Theorem 5.3 and
Lemma 5.6, one has that $(\omega,J)$ is compatible, so
$\Omega=\omega-\mathbf{i}\,\omega_{J}$ is a holomorphic $p$-form by
Proposition 5.4. Moreover, it follows from (5.15) and (5.16) that (5.11)
holds.

######  Remark 5.9.

Note that a central fact shown in the proof of Theorem 5.5 is that a
multiplicative holomorphic $p$-form $\Omega\in\Omega^{(p,0)}(\mathcal{G})$
integrates a holomorphic IM $p$-form $(\widehat{\mu},\widehat{\nu})$ if and
only if $\omega$, the real part of $\Omega$, integrates $(\mu,\nu)$, the real
part of $(\widehat{\mu},\widehat{\nu})$. $\diamond$

##  6\. Integration of Dirac-Nijenhuis structures

In this section we refine the inifinitesimal-global correspondence between
Dirac structures and presymplectic groupoids to include compatible with
$(1,1)$-tensor fields. In this way we obtain the global counterparts of Dirac-
Nijenhuis structures and, as a particular case, the integration of holomorphic
Dirac structures (see [33, 41] for the special cases of Poisson (quasi-)
Nijenhuis and holomorphic Poisson structures).

###  6.1. Dirac structures and presymplectic groupoids

We now briefly recall the integration of Dirac structures to presymplectic
groupoids [6], as an application of the correspondence between multiplicative
and IM-forms in [5].

The starting point is the observation that Dirac structures can be identified
with special types of IM 2-forms. Any Dirac structure $L\subset\mathbb{T}M$ is
naturally a Lie algebroid with respect to the restriction of the Courant
bracket $\left[\\!\left[\cdot,\cdot\right]\\!\right]$ to $\Gamma(L)$ and
anchor $\rho=\mathrm{pr}_{T}|_{L}:L\to TM$. One can then directly verify that
the projection $\mathrm{pr}_{T^{*}}|_{L}:L\to T^{*}M$ defines a closed IM
2-form on $L$. Conversely, the closed IM 2-forms that arise in this way have a
simple characterization. Let $\mu:A\to T^{*}M$ be a closed IM 2-form on a Lie
algebroid $A$ such that

(6.1) $\begin{cases}\mathrm{rank}(A)=\dim(M),\\\
\ker(\rho)\cap\ker(\mu)=\\{0\\}.\end{cases}$

A consequence of the IM-equations (5.1) (with $\nu=0$) is that the image of
the map $(\rho,\mu):A\to\mathbb{T}M$,

(6.2) $L:=\mathrm{Im}(\rho,\mu),$

is a Dirac structure. The Lie algebroid $A$ is identified with $L$ under this
map, in such a way that $\mu=\mathrm{pr}_{T^{*}}|_{L}$.

A presymplectic groupoid is a Lie groupoid $\mathcal{G}\rightrightarrows M$
equipped with a closed multiplicative 2-form $\omega$ such that

(6.3) $\begin{cases}\dim(\mathcal{G})=2\dim(M)\\\
\ker(Ts)_{x}\cap\ker(Tt)_{x}\cap\ker(\omega)_{x}=\\{0\\},\,\,\,\forall\,x\in
M.\end{cases}$

It is shown in [5, Sec. 4] that conditions (6.3) on $\mathcal{G}$ and $\omega$
are equivalent to conditions (6.1) for the corresponding Lie algebroid $A$ and
closed IM 2-form $\mu:A\to T^{*}M$. It follows that if $(\mathcal{G},\omega)$
is a presymplectic groupoid, then $M$ inherits a Dirac structure $L$, as in
(6.2), together with an induced isomorphism of Lie algebroids between $L$ and
$A$. Moreover, the first equation in (5.2) implies that $L$ is uniquely
determined by the fact that $t:\mathcal{G}\to M$ is a forward Dirac map. When
this happens, we say that the presymplectic groupoid $(\mathcal{G},\omega)$
integrates the Dirac structure $L$.

The opposite direction, going from Dirac structures to presymplectic
groupoids, follows from the integration of IM-forms (see $\S$ 5.1): given a
Dirac structure on $L$ that is integrable as a Lie algebroid and its source
1-connected integration $\mathcal{G}$, the integration of the IM 2-form
$\mu=\mathrm{pr}_{T^{*}}|_{L}$ defines a unique (up to isomorphism) closed
multiplicative 2-form $\omega$ such that $(\mathcal{G},\omega)$ is a
presymplectic groupoid integrating $L$.

###  6.2. Integration of Dirac structures with compatible $(1,1)$-tensor
fields

We will now relate compatible $(1,1)$-tensor fields on presymplectic groupoids
and on their associated Dirac manifolds.

######  Lemma 6.1.

Let $L$ be a Dirac structure on $M$, $r\in\Omega^{1}(M,TM)$, so that the pair
$(L,r)$ is compatible. Then the triple
$(\mathbb{D}^{r}|_{\Gamma(L)},(r,r^{*})|_{L},r)$ is an IM $(1,1)$-tensor on
the Lie algebroid $L$. Moreover, if $r$ is Nijenhuis, then this IM
$(1,1)$-tensor satisfies the infinitesimal Nijenhuis equations (5.5).

###### Proof.

The Leibnz-type identity for $(\mathbb{D}^{r}|_{\Gamma(L)},(r,r^{*})|_{L},r)$
follows from (3.7). The IM-equations will be a consequence of the following
general equations relating $(\mathbb{D}^{r},(r,r^{*}),r)$ with the Courant-
algebroid structure on $\mathbb{T}M$:

(6.4) $\displaystyle\mathrm{pr}_{T}\circ(r,r^{*})=$ $\displaystyle
r\circ\mathrm{pr}_{T}$ (6.5)
$\displaystyle\mathrm{pr}_{T}(\mathbb{D}^{r}_{X}(\sigma_{1}))=$ $\displaystyle
D^{r}_{X}(\mathrm{pr}_{T}(\sigma_{1}))$ (6.6)
$\displaystyle(r,r^{*})(\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right])=$
$\displaystyle\left[\\!\left[\sigma_{1},(r,r^{*})(\sigma_{2})\right]\\!\right]-\mathbb{D}^{r}_{\mathrm{pr}_{T}(\sigma_{2})}(\sigma_{1})-C^{r}(\sigma_{1},\sigma_{2})$
(6.7)
$\displaystyle\mathbb{D}^{r}_{Z}(\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right])=$
$\displaystyle\left[\\!\left[\sigma_{1},\mathbb{D}^{r}_{Z}(\sigma_{2})\right]\\!\right]-\left[\\!\left[\sigma_{2},\mathbb{D}^{r}_{Z}(\sigma_{1})\right]\\!\right]+\mathbb{D}^{r}_{[\mathrm{pr}_{T}(\sigma_{2}),Z]}(\sigma_{1})$
$\displaystyle-\mathbb{D}^{r}_{[\mathrm{pr}_{T}(\sigma_{1}),Z]}(\sigma_{2})-i_{Z}dC^{r}(\sigma_{1},\sigma_{2})$

where $Z\in\mathfrak{X}(M)$, $\sigma_{1},\,\sigma_{2}\in\Gamma(\mathbb{T}M)$,
and
$C^{r}(\sigma_{1},\sigma_{2})=\langle\mathbb{D}^{r}_{(\cdot)}(\sigma_{1}),\sigma_{2}\rangle\in\Omega^{1}(M)$.
To obtain the IM equations (5.3), just observe that when $(L,r)$ is a Dirac
structure, $C^{r}(\sigma_{1},\sigma_{2})=0$ for
$\sigma_{1},\sigma_{2}\in\Gamma(L)$.

The first two equations above are straightforward. The other two can be
verified as follows. Let $\sigma_{1}=(X,\alpha)$, $\sigma_{2}=(Y,\beta)$. Note
that

$\langle\beta,D_{(\cdot)}^{r}(X)\rangle=\mathcal{L}_{X}(r^{*}(\beta))-r^{*}(\mathcal{L}_{X}\beta),\qquad\langle
D^{r,*}_{(\cdot)}(\alpha),Y\rangle=r^{*}(i_{Y}d\alpha)-i_{Y}dr^{*}(\alpha).$

Then

$\displaystyle(r,r^{*})(\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right])-\left[\\!\left[\sigma_{1},(r,r^{*})(\sigma_{2})\right]\\!\right]$
$\displaystyle=-(D^{r}_{Y}(X),\,\langle\beta,D^{r}_{(\cdot)}(X)\rangle+r^{*}(i_{Y}d\alpha)-i_{r(Y)}d\alpha)$
$\displaystyle=-\mathbb{D}^{r}_{Y}(X,\alpha)-\langle\beta,D^{r}_{(\cdot)}(X)\rangle-\langle
D^{r,*}_{(\cdot)}(\alpha),Y\rangle,$

which proves (6.6). To verify (6.7), define

$\Sigma(Z,\sigma_{1},\sigma_{2})=\mathbb{D}^{r}_{Z}(\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right])-\left[\\!\left[\sigma_{1},\mathbb{D}^{r}_{Z}(\sigma_{2})\right]\\!\right]+\left[\\!\left[\sigma_{2},\mathbb{D}^{r}_{Z}(\sigma_{1})\right]\\!\right]-\mathbb{D}^{r}_{[\mathrm{pr}_{T}(\sigma_{2}),Z]}(\sigma_{1})+\mathbb{D}^{r}_{[\mathrm{pr}_{T}(\sigma_{1}),Z]}(\sigma_{2}).$

It follows from the Jacobi identity for the Lie bracket of vector fields that
the $TM$-component of $\Sigma$ vanishes. The component on $T^{*}M$ can written
as

$\mathrm{pr}_{T^{*}}(\Sigma(Z,\sigma_{1},\sigma_{2}))=\Upsilon(Z,X,\beta)+\Upsilon(Z,Y,\alpha),$

where

$\displaystyle\Upsilon(Z,X,\beta)=$
$\displaystyle\mathcal{L}_{Z}(r^{*}(\mathcal{L}_{X}\beta))-\mathcal{L}_{r(Z)}\mathcal{L}_{X}\beta-\mathcal{L}_{X}\mathcal{L}_{Z}(r^{*}(\beta))+\mathcal{L}_{X}\mathcal{L}_{r(Z)}\beta-
i_{[X,r(Z)]}d\beta$
$\displaystyle+i_{r([X,Z])}d\beta+\mathcal{L}_{[X,Z]}(r^{*}(\beta))-\mathcal{L}_{r([X,Z])}\beta,$
$\displaystyle\Upsilon(Z,Y,\alpha)=$
$\displaystyle-\mathcal{L}_{Z}(r^{*}(i_{Y}d\alpha))+\mathcal{L}_{r(Z)}i_{Y}d\alpha+i_{[Y,r(Z)]}d\alpha-
i_{r([Y,Z])}d\alpha+\mathcal{L}_{Y}\mathcal{L}_{Z}r^{*}(\alpha)$
$\displaystyle-\mathcal{L}_{Y}\mathcal{L}_{r(Z)}\alpha-\mathcal{L}_{[Y,Z]}r^{*}(\alpha)+\mathcal{L}_{r([Y,Z])}\alpha.$

Using Cartan calculus and (2.13), one verifies that

$\displaystyle\Upsilon(Z,X,\beta)$
$\displaystyle=d\langle\beta,D_{Z}^{r}(X)\rangle+\mathcal{L}_{Z}\underbrace{(r^{*}(\mathcal{L}_{X}\beta)-\mathcal{L}_{X}r^{*}\beta)}_{-\langle\beta,D_{(\cdot)}^{r}(X)\rangle}=-i_{Z}d\langle\beta,D_{(\cdot)}^{r}(X)\rangle$
$\displaystyle\Upsilon(Z,Y,\alpha)$
$\displaystyle=d(\underbrace{\mathcal{L}_{Z}\langle\alpha,r(Z)\rangle-\mathcal{L}_{r(Z)}\langle\alpha,Y\rangle-\langle\alpha,D^{r}_{Z}(Y)\rangle}_{\langle
D^{r,*}_{Z}(\alpha),Y\rangle})-\mathcal{L}_{Z}\underbrace{(r^{*}(i_{Y}d\alpha)-i_{Y}dr^{*}(\alpha))}_{\langle
D^{r,*}_{(\cdot)}(\alpha),Y\rangle}$ $\displaystyle=-i_{Z}d\langle
D_{(\cdot)}^{r,*}(\alpha),Y\rangle.$

This proves the first part of the lemma.

For the second assertion regarding the infinitesimal Nijenhuis equations (5.5)
for the IM $(1,1)$-tensor $(\mathbb{D}^{r}|_{\Gamma(L)},(r,r^{*})|_{L},r)$,
first note that

$\displaystyle(r,r^{*})(\mathbb{D}^{r}_{X}(Z,\gamma))-\mathbb{D}^{r}_{X}((r,r^{*})(Z,\gamma))=(\mathcal{N}_{r}(X,Z),\langle\gamma,\mathcal{N}_{r}(X,\cdot)\rangle)$

for all $X,Z\in\mathfrak{X}(M),\,\gamma\in\Omega^{1}(M)$, as a consequence of
(2.17) and (2.18). Similarly, one can check, using (2.17), (2.18) and the
Jacobi identity for the Lie bracket of vector fields, that

(6.8)
$(\mathbb{D}^{r})^{2}_{(X,Y)}(Z,\gamma)=((\mathcal{L}_{Z}\mathcal{N}_{r})(X,Y),i_{X}i_{Y}d\mathcal{N}_{r}^{*}\gamma-
i_{\mathcal{N}_{r}(X,Y)}d\gamma),$

where $\mathcal{N}_{r}^{*}:T^{*}M\to\wedge^{*}T^{*}M$ is the dual of the
Nijenhuis torsion and $(\mathbb{D}^{r})^{2}$ is defined in (5.5). In
particular, if $\mathcal{N}_{r}=0$, then equations (5.5) hold for
$(\mathbb{D}^{r}|_{\Gamma(L)},(r,r^{*})|_{L},r)$.

∎

######  Remark 6.2 (Courant-Nijenhuis algebroids).

Equations (6.4)–(6.7) make sense for arbitrary Courant algebroids endowed with
a self-dual 1-derivation (in place of $(\mathbb{D}^{r},(r,r^{*}),r)$) and lead
to the general notion of Courant-Nijenhuis algebroid, studied in [8] (which
arise, in particular, as “doubles” of the Lie-Nijenhuis bialgebroids in [21]).
$\diamond$

We are now ready to show the infinitesimal-global correspondence of
presymplectic-Nijenhuis groupoids and Dirac-Nijenhuis structures.

######  Theorem 6.3.

Let $(\mathcal{G},\omega)$ be a presymplectic groupoid integrating a Dirac
structure $L$ on $M$.

  * (a)

Let $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ be multiplicative, and let
$r=K|_{TM}:TM\to TM$. If the pair $(\omega,K)$ is compatible (resp.
presymplectic-Nijenhuis) then the pair $(L,r)$ is compatible (resp. Dirac-
Nijenhuis).

  * (b)

If $\mathcal{G}$ is source 1-connected, the correspondence $K\mapsto
r=K|_{TM}$ is a bijection between multiplicative
$K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$ compatible with $\omega$ and
$r\in\Omega^{1}(M,TM)$ compatible with $L$ (moreover, $K$ is Nijenhuis if and
only if so is $r$).

###### Proof.

Suppose that the pair $(\omega,K)$ is compatible. Let $(D,l,r)$ be the IM
$(1,1)$-tensor corresponding to $K$ and $\mu$ the closed IM 2-form of
$\omega$, so that $L=(\rho,\mu)(A)$. It follows from the first two equalities
in (5.3),

$r\circ\rho=\rho\circ l,\qquad\rho(D_{X}(a))=D_{X}^{r}(\rho(a)),$

and the compatibility conditions (5.8) and (5.9) (which hold by Thm. 5.3,
recalling that $\nu=0$),

$r^{*}\mu(a)=\mu(l(a)),\qquad\mu(D_{X}(a))=D_{X}^{r,*}(\mu(a)),$

that the pair $(L,r)$ is compatible. Moreover, if $K$ is Nijenhuis then so is
$r$ (see (5.5)). This proves part (a). Note also that, under the Lie-algebroid
isomorphism $(\rho,\mu):A\stackrel{{\scriptstyle\sim}}{{\to}}L$, the IM
$(1,1)$-tensor $(D,l,r)$ is identified with
$(\mathbb{D}^{r}|_{\Gamma(L)},(r,r^{*}),r)$. As a consequence, if
$\mathcal{G}$ is source connected, any $K^{\prime}$ that is multiplicative,
compatible with $\omega$, and satisfies $K^{\prime}|_{TM}=r$, must agree with
$K$.

Suppose now that $(\mathcal{G},\omega)$ is a source 1-connected integration of
$L$. Let $r\in\Omega^{1}(M,TM)$ be compatible with $L$. By Lemma 6.1,
$(\mathbb{D}^{r}|_{\Gamma(L)},(r,r^{*}),r)$ is an IM $(1,1)$-tensor on $L$,
which is easily seen to be compatible with the closed IM 2-form
$\mathrm{pr}_{T^{*}}|_{L}$. It follows that the IM $(1,1)$-tensor $(D,l,r)$
corresponding to $(\mathbb{D}^{r}|_{\Gamma(L)},(r,r^{*}),r)$ through the
identification $(\rho,\mu):A\stackrel{{\scriptstyle\sim}}{{\to}}L$ is
compatible with $\mu$. Hence $K\in\Omega^{1}(\mathcal{G},T\mathcal{G})$
obtained by integration of $(D,l,r)$ is compatible with $\omega$ by Thm. 5.3.
Such $K$ satisfies $K|_{TM}=r$ and, as argued above, any multiplicative
$(1,1)$-tensor field on $\mathcal{G}$ compatible with $\omega$ with this
property must coincide with $K$. The fact that $K$ is Nijenhuis if $r$ is
follows from the second assertion in Lemma 6.1.

∎

When (a) holds, we say that $(\mathcal{G},\omega,K)$ integrates $(M,L,r)$.

######  Remark 6.4 (The quasi-Nijenhuis version).

The correspondence in Theorem 6.3 has the following version for quasi-
Nijenhuis structures (see Remark 3.14).

$(L,r,\phi)\text{ is Dirac quasi-Nijenhuis
}\Longleftrightarrow\omega(\cdot,\mathcal{N}_{K}(\cdot,\cdot))=s^{*}\phi-t^{*}\phi.$

Indeed, by considering $\tau(U,V,W)=\omega(U,\mathcal{N}_{K}(V,W))$ and its
infinitesimal description, one can show that $\tau$ is skew-symmetric (hence a
3-form) with corresponding IM 3-form $\tilde{\mu}:A\to\wedge^{2}T^{*}M$,
$\tilde{\nu}:A\to\wedge^{3}T^{*}M$ on $A$ given by

$\displaystyle\tilde{\mu}(a)$ $\displaystyle=\mathcal{N}_{r}^{*}(\mu(a))$
$\displaystyle\tilde{\nu}(a)|_{(X,Y,Z)}$
$\displaystyle=d\mu(a)(X,\mathcal{N}_{r}(Y,Z))-\langle\mu(D^{2}_{(Y,Z)}(a)),X\rangle-d(\mathcal{N}_{r}^{*}\mu(a))(X,Y,Z).$

Since the pair $(\omega,K)$ is compatible, it follows from Theorem 5.3 and
(6.8) that

$\mu(D^{2}_{(Y,Z)}(a))=(D^{r,*})^{2}_{(Y,Z)}(\mu(a))=i_{Y}i_{Z}d(\mathcal{N}_{r}^{*}\mu(a))-i_{\mathcal{N}_{r}(Y,Z)}d\mu(a),$

which implies that $\tilde{\nu}=0$. Also, the quasi-Nijenshui condition (3.9)
is equivalent to $\tilde{\mu}(a)=-i_{\rho(a)}\phi,$ which is the closed IM
3-form corresponding to $s^{*}\phi-t^{*}\phi$.

In conclusion, we obtain a natural bijective correspondence between source
1-connected presymplectic-Nijenhuis groupoids and Dirac-Nijenhuis manifolds
(which are integrable as Lie algebroids),

$(\mathcal{G},\omega,K)\;\;\leftrightharpoons\;\;(M,L,r),$

where $L$ is the forward image of $\omega$ by $t$ and $r$ is $t$-related with
$K$ (and the same holds for the quasi-Nijenhuis condition). This restricts to
a correspondence between symplectic (quasi-)Nijenhuis groupoids and Poisson
(quasi-)Nijenhuis structures, recovering [41, Thm. 5.2].

We have the following simple relation between integration of Dirac-Nijenhuis
structures and the “$(0,n)$” hierarchies described in $\S$ 4.4.

######  Corollary 6.5.

Suppose that $(\mathcal{G},\omega,K)$ is a presymplectic-Nijenhuis groupoid
integrating the Dirac-Nijenhuis structure $(L,r)$. If
$\ker(\mathrm{id}_{TM},r^{*})|_{L})=0$, then for each $n=1,2,\ldots$
$(\mathcal{G},\omega_{n})$ is a presymplectic groupoid integrating the Dirac
structure $L_{(0,n)}$.

###### Proof.

The proof follows from the fact that, if
$\ker(\mathrm{id}_{TM},r^{*})|_{L})=0$, then
$(\mathrm{id}_{TM},(r^{*})^{n}):L\to L_{(0,n)}$ is a Lie algebroid
isomorphism. Hence, as $L=(\rho,\mu)(A)$, it follows that
$L_{(0,n)}=(\rho,(r^{*})^{n}\circ\mu)(A)=(\rho,\mu\circ l^{n})(A)$, which is
the Dirac structure corresponding to the presymplectic groupoid
$(\mathcal{G},\omega_{n})$. ∎

######  Example 6.6.

Let $(\mathcal{G},\omega)$ be a symplectic groupoid integrating a Poisson
manifold $(M,\pi)$. Consider a closed 2-form $B\in\Omega^{2}(M)$ and
$r=\mathrm{id}+\pi^{\sharp}\circ B^{\flat}:TM\to TM$, as in Example 4.13. One
can check that the IM $(1,1)$-form $(D^{r,*},r^{*},r)$ on the cotangent Lie
algebroid $(T^{*}M,[\cdot,\cdot]_{\pi},\pi^{\sharp})$ integrates to
$K=\mathrm{id}_{T\mathcal{G}}+\Pi^{\sharp}\circ(t^{*}B-s^{*}B)^{\flat}$. As
observed in Example 4.13, the gauge transformation of $\pi$ by $B$ is
$L_{(0,1)}$, and hence by Corollary 6.5 it integrates to
$(\mathcal{G},\omega_{1}=\omega+t^{*}B-s^{*}B)$, thus recovering [10, Thm.
4.1]. When $r$ is invertible, $K$ is also invertible and $\omega_{1}$ is the
symplectic form integrating the Poisson structure resulting from the gauge
transformation of $\pi$ by $B$.

###  6.3. Integration of holomorphic Dirac structures

We now introduce the global counterparts of holomorphic Dirac structures.

######  Definition 6.7.

A holomorphic presymplectic groupoid is a holomorphic Lie groupoid
$\mathcal{G}\rightrightarrows M$ endowed with a multiplicative holomorphic
$2$-form $\Omega$ such that $\partial\Omega=0$, $\dim(\mathcal{G})=2\dim(M)$,
and

(6.9)
$\ker(\Omega)_{x}\cap\ker(Ts)_{x}\cap\ker(Tt)_{x}=\\{0\\},\qquad\forall\,x\in
M.$

The intersection condition (6.9) takes place in $T^{1,0}\mathcal{G}|_{M}$. We
denote by $\widehat{\mu}:\mathcal{A}\to(T^{1,0}M)^{*}$ the holomorphic IM
2-form corresponding to $\Omega$, $\widehat{\mu}(u)=(i_{u}\Omega)|_{T^{1,0}M}$
(note that $\widehat{\nu}=0$ since $\partial\Omega=0$).

We will prove the infinitesimal-global correspondence relating holomorphic
Dirac structures and holomorphic presymplectic groupoids as a consequence of
Theorem 5.5, though it can also be derived from Theorem 6.3 by working with
real parts (see comments after Thm. 6.9 below).

Just as in the real smooth case, holomorphic presymplectic groupoids give rise
to holomorphic Dirac structures on their spaces of units.

######  Theorem 6.8.

Let $(\mathcal{G},\Omega)\rightrightarrows M$ be a holomorphic presymplectic
groupoid. (1) There exists a unique holomorphic Dirac structure
$\mathcal{L}\subset\mathbb{T}^{1,0}M$ for which $t:\mathcal{G}\to M$ is a
forward Dirac map. (2) The map
$(\widehat{\rho},\widehat{\mu}):\mathcal{A}\to\mathbb{T}^{1,0}M$ induces an
isomorphism of Lie algebroids
$\mathcal{A}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{L}$.

###### Proof.

The proof is analogous to the real case. Holomorphic Dirac structures
$\mathcal{L}\subset\mathbb{T}^{1,0}M$ can be identified with closed
holomorphic IM 2-forms $\widehat{\mu}:\mathcal{A}\to(T^{1,0}M)^{*}$ satisfying

(6.10)
$\mathrm{rank}(\mathcal{A})=\dim_{\mathbb{C}}(M),\qquad\ker(\widehat{\rho})\cap\ker(\widehat{\mu})=\\{0\\},$

(cf. $\S$ 6.1) as the image of
$(\widehat{\rho},\widehat{\mu}):\mathcal{A}\to\mathbb{T}^{1,0}M$,

(6.11) $\mathcal{L}=\mathrm{Im}(\widehat{\rho},\widehat{\mu}).$

The multiplicativity of $\Omega$ implies that

(6.12) $i_{\overrightarrow{u}}\Omega=t^{*}\widehat{\mu}(u),$

for any section of $\Gamma(\mathcal{A})$. By using this last property, one can
check that (6.9) boils down to the intersection condition in (6.10), so one
obtains a holomorphic Dirac structure $\mathcal{L}$ as in (6.11). Moreover,
$(\widehat{\rho},\widehat{\mu}):\mathcal{A}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{L}$
is an isomorphism of Lie algebroids.

Using the identity

(6.13) $Tt(\overrightarrow{u})=\widehat{\rho}(u),$

it also follows from (6.12) that $t:(\mathcal{G},\Omega)\to(M,L)$ is a forward
Dirac map.

∎

We say that a holomorphic pre-symplectic groupoid $(\mathcal{G},\Omega)$
integrates a holomorphic Dirac structure $\mathcal{L}$ if
$t:(\mathcal{G},\Omega)\to(M,\mathcal{L})$ is a forward Dirac map. The next
result gives the integration of holomorphic Dirac structures.

######  Theorem 6.9.

Let $\mathcal{L}\subset\mathbb{T}^{1,0}M$ be a holomorphic Dirac structure
whose underlying holomorphic Lie algebroid is integrable, and let
$\mathcal{G}\rightrightarrows M$ be its source 1-connected holomorphic
integration. Then there exists a unique holomorphic multiplicative $2$-form
$\Omega$ such that $(\mathcal{G},\Omega)$ is the pre-symplectic groupoid
integrating $\mathcal{L}$.

###### Proof.

Set $\widehat{\mu}:=\mathrm{pr}_{T^{*}}:\mathcal{L}\to(T^{1,0}M)^{*}$, the
projection of $\mathcal{L}$ on the cotangent component. The fact that
$\mathcal{L}$ is a holomorphic Dirac structure implies that $\hat{\mu}$ is a
holomorphic IM $2$-form on $\mathcal{L}$. By Theorem 5.5, there exists a
unique multiplicative holomorphic $2$-form $\Omega$ on $\mathcal{G}$
integrating $(\widehat{\mu},\widehat{\nu}=0)$. It is clear that
$\partial\Omega=0$ and $\dim(\mathcal{G})=2\dim(M)$, and
$t:(\mathcal{G},\Omega)\to(M,\mathcal{L})$ is a forward Dirac map (as a
consequence of (6.12) and (6.13)). Since $\widehat{\rho}=\mathrm{pr}_{T}$, we
have that $\ker(\widehat{\rho})\cap\ker(\widehat{\mu})=\\{0\\}$, which is
equivalent to $\ker(\Omega)\cap\ker(Tt)\cap\ker(Ts)|_{M}=\\{0\\}$.

If $\Omega$ and $\Omega^{\prime}$ are two presymplectic integrations of
$\mathcal{L}$, then by Theorem 6.8, part (2), there is a (holomorphic) Lie-
algebroid automorphism $\phi$ such that
$\widehat{\mu}^{\prime}=\widehat{\mu}\circ\phi$. Since $\mathcal{G}$ is source
1-connected, there is an automorphism of $F:\mathcal{G}\to\mathcal{G}$ such
that $F^{*}\Omega=\Omega^{\prime}$.

∎

Theorems 6.8 and 6.9 extend the correspondence between (integrable)
holomorphic Poisson structures and source 1-connected holomorphic symplectic
groupoids shown in [32].

Another approach to the previous two theorems relies on the underlying real
structures. A holomorphic Dirac structure $\mathcal{L}$ on a complex manifold
$(M,r)$ is integrable as a holomorphic Lie algebroid if and only if its real
part $L$ is integrable as a real Lie algebroid (see Remark 5.7). From the
bijective correspondence between holomorphic Dirac structures $\mathcal{L}$
and Dirac-Nijenhuis structures $(L,r)$ in Proposition 3.16, one obtains an
equivalence between presymplectic-Nijenhuis structures and closed holomorphic
2-forms on $M$ via $\omega\mapsto\omega-\mathbf{i}\,\omega_{r}$, see Example
3.18. As in Remark 5.9, one can check that if $(\mathcal{G},\Omega)$ is a
holomorphic presymplectic groupoid integrating $\mathcal{L}$, then
$(\mathcal{G},\omega,J)$ is a presymplectic-Nijenhuis groupoid integrating
$(L,r)$, so passing to real parts is compatible with differentiation and
integration:

Presymplectic-Nijenhuisgroupois ($\displaystyle\mathcal{G},\ \omega,\
J)$Holomorphic presymplecticgroupoids ($\displaystyle\mathcal{G},\
\Omega=\omega-i\ \omega_{J})$$\displaystyle\Phi$Real
part$\displaystyle\Phi$Real partDirac-Nijenhuismanifolds
$\displaystyle(M,L,r)$Holomorphic Diracmanifolds $\displaystyle(M,\
\mathcal{L})\ $IntegrationDifferentiationIntegrationDifferentiation

So one can derive Theorems 6.8 and 6.9 from Theorem 6.3. This viewpoint
generalizes [32, Thm. 3.22] in the case of holomorphic Poisson structures and
holomorphic symplectic groupoids, through different methods.

An interesting class of holomorphic Dirac structures is given by the affine
Dirac structures on complex Poisson Lie groups studied in [9], where an
explicit description of their holomorphic presymplectic groupoids is presented
together with an application to the construction of holomorphic symplectic
groupoids integrating holomorphic Poisson homogeneous spaces.

##  Appendix A Proof of Theorem 2.1

We shall assume some familiarity with double vector bundles (as e.g. in [34]).

Consider the identification $TM\oplus E\to TE|_{TM}\subseteq TE$ as vector
bundles over $TM$. Any element $U$ of $TE$ can be written (not uniquely) as

$U=Tu(X)+_{\scriptscriptstyle{T}}h(X,e),$

for some $u\in\Gamma(E),\,e\in E$, where $+_{\scriptscriptstyle{T}}$ stands
for the addition on the fibers of $TE\to TM$. The following lemma gives an
explicit formula relating a linear $(1,1)$-tensor with its corresponding
1-derivation.

######  Lemma A.1.

If $K:TE\to TE$ is a linear $(1,1)$-tensor with associated 1-derivation
$(D,l,r)$, then

(A.1) $\displaystyle K(Tu(X))$
$\displaystyle=Tu(r(X))+_{\scriptscriptstyle{T}}h(r(X),D_{X}(u))$ (A.2)
$\displaystyle K(h(X,v))$ $\displaystyle=h(r(X),l(v)),$

where $u,v\in\Gamma(E)$, $X\in\mathfrak{X}(M)$.

###### Proof.

The proof of (A.2) follows directly from (2.4). The proof of (A.1) is more
involved and we will need to recall some facts from [7]. Consider the reversal
isomorphism $\mathcal{R}:T^{*}(E^{*})\to T^{*}E$, defined as

$\mathcal{R}(\varphi,\beta,e)=(e,-\beta,\varphi),$

under the decompositions $T^{*}(E^{*})\cong E^{*}\oplus T^{*}M\oplus E$ and
$T^{*}E\cong E\oplus T^{*}M\oplus E^{*}$ coming from local trivializations
(see [34, § 9.5] for further details). From [7, Lemma 4.9(a)], one has that

$\displaystyle\langle\xi,D_{X}(u)\rangle$ $\displaystyle=\langle
K(Tu(X)),\mathcal{R}(d\ell_{u}(\xi))\rangle_{E}$

where $u\in\Gamma(E)$, $\xi\in\Gamma(E^{*})$, $X\in\mathfrak{X}(M)$,
$\ell_{u}\in C^{\infty}(E^{*})$ is the fiberwise linear function associated
with $u$, and $\langle\cdot,\cdot\rangle_{E}$ is the natural pairing between
$TE$ and $T^{*}E$. By writing $K=Tu(r(X))+_{\scriptscriptstyle{T}}h(r(X),c)$,
for some $c\in\Gamma(E)$, and using basic properties of the cotangent
prolongation bundle $T^{*}E\to E^{*}$ (see [34, § 9.4], one has that

$\displaystyle\langle\xi,D_{X}(u)\rangle$ $\displaystyle=\langle
Tu(r(X))+_{\scriptscriptstyle{T}}h(r(X),c),\mathcal{R}(d\ell_{u}(\xi))+_{\scriptscriptstyle{E^{*}}}0_{\xi}\rangle_{E}$
$\displaystyle=\langle Tu(r(X)),\mathcal{R}(d\ell_{u}(\xi))\rangle_{E}+\langle
h(r(X),c),0_{\xi}\rangle_{E},$

where $+_{\scriptscriptstyle{E^{*}}}$ is the fiberwise sum and $0_{\xi}$ is
the zero of the cotangent prolongation $T^{*}E\to E^{*}$. As $\langle
h(r(X),c),0_{\xi}\rangle_{E}=\langle\xi,c\rangle$, it only remains to prove
that $\langle Tu(r(X)),\mathcal{R}(d\ell_{u}(\xi))\rangle_{E}=0$. This will
follow from (see [34, Prop. 9.5.3])

$\mathcal{R}(d\ell_{u}(\xi(x)))=d\ell_{\xi}(u(x))-q^{*}(d\langle\xi,u\rangle)(u(x)),\,\,\,x\in
M.$

Indeed, this identity implies that

$\displaystyle\langle Tu(r(X)),\mathcal{R}(d\ell_{u}(\xi))\rangle_{E}$
$\displaystyle=\mathcal{L}_{X}\langle\xi,u\rangle-\mathcal{L}_{X}\langle\xi,u\rangle=0.$

∎

To complete the proof of Theorem 2.1, notice that $\Phi$ being a vector bundle
map implies that

$T\Phi(h(X,e_{1}))=h(X,\Phi(e_{1})),$

for $e_{1}\in E_{1}$. It follows from (A.2) that

$T\Phi\circ K_{1}(h(X,e_{1}))=K_{2}(T\Phi(h(X,e_{1})))\Longleftrightarrow
r_{1}=r_{2}=r,\,\,\,\ \Phi\circ l_{1}=l_{2}\circ\Phi.$

Similarly, from (A.1),

$\displaystyle T\Phi(K_{1}(Tu(X)))$
$\displaystyle=T(\Phi(u))(r(X))+_{\scriptscriptstyle{T}}h(r(X),\Phi(D^{1}_{X}(u))),$
$\displaystyle K_{2}(T\Phi(Tu(X)))$
$\displaystyle=T(\Phi(u))(r(X))+_{\scriptscriptstyle{T}}h(r(X),D^{2}_{X}(\Phi(u))).$

Therefore

$T\Phi(K_{1}(Tu(X)))=K_{2}(T\Phi(Tu(X)))\Longleftrightarrow\Phi\circ
D^{1}_{X}=D^{2}_{X}\circ\Phi.$

##  Appendix B Other notions of Dirac-Nijenhuis structures

In this section, we relate our notion of Dirac-Nijenhuis structure with other
existing definitions in the literature [12, 14, 26]. Our main tool to compare
the different definitions is the bracket on $\mathbb{T}M$ defined by

(B.1)
$\displaystyle\left[\\!\left[(X,\alpha),(Y,\beta)\right]\\!\right]_{\mathbb{D}^{r}}$
$\displaystyle=\left[\\!\left[(r(X),r^{*}(\alpha)),(Y,\beta)\right]\\!\right]+\mathbb{D}^{r}_{Y}(X,\alpha)$
$\displaystyle=([X,Y]_{r},\mathcal{L}_{r(X)}\beta-i_{r(Y)}d\alpha),$

for $r\in\Omega^{1}(M,TM)$ (and $[\cdot,\cdot]_{r}$ defined in (5.6)). The
properties of
$(\mathbb{T}M,\left[\\!\left[\cdot,\cdot\right]\\!\right]_{\mathbb{D}^{r}})$
will be discussed in [8]. Here, we only need two facts about the bracket
(B.1): (1) if $L$ is a Dirac structure that is compatible with $r$, then $L$
is involutive with respect to
$\left[\\!\left[\cdot,\cdot\right]\\!\right]_{\mathbb{D}^{r}}$; 2) the
following identity holds:

(B.2)
$\left[\\!\left[(X,\alpha),(Y,\beta)\right]\\!\right]_{\mathbb{D}^{r}}+\left[\\!\left[(Y,\beta),(X,\alpha)\right]\\!\right]_{\mathbb{D}^{r}}=d\langle(r(X),r^{*}(\alpha)),(Y,\beta)\rangle.$

### Dirac-Nijenhuis structures of contraction type

A vector-bundle morphism $N:\mathbb{T}M\rightarrow\mathbb{T}M$ (over the
identity on $M$) induces a bracket on sections of $\mathbb{T}M$ called the
contracted bracket:

(B.3)
$\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right]_{N}:=\left[\\!\left[N\sigma_{1},\sigma_{2}\right]\\!\right]+\left[\\!\left[\sigma_{1},N\sigma_{2}\right]\\!\right]-N(\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right])$

for all $\sigma_{1},\sigma_{2}\in\Gamma(\mathbb{T}M)$. One can also define a
Nijenhuis-type torsion by

(B.4)
$T_{N}(\sigma_{1},\sigma_{2})=N(\left[\\!\left[\sigma_{1},\sigma_{2}\right]\\!\right]_{N})-\left[\\!\left[N\sigma_{1},N\sigma_{2}\right]\\!\right].$

To distinguish from our definition of Dirac-Nijenhuis structure, we shall say
that a Dirac structure $L\subset\mathbb{T}M$ and $N$ as above form a Dirac-
Nijenhuis structure of contraction type if
$\left[\\!\left[\cdot,\cdot\right]\\!\right]_{N}$ restricts to skew-symmetric
bracket on $\Gamma(L)$ and $T_{N}|_{\Gamma(L)}=0$. Dirac-Nijenhuis structures
of contraction type were defined and studied in [12, 14].

######  Proposition B.1.

Let $L$ be a Dirac structure on $M$, $r\in\Omega^{1}(M,TM)$, and consider
$N=(r,0)$. Then $(L,N)$ is a Dirac-Nijenhuis structure of contraction type if
and only if

  * (i)

$(r,r^{*})(L)\subseteq L$,

  * (ii)

$\mathbb{D}^{r}_{Y}(\Gamma(L))\subseteq\Gamma(L)$, for every
$Y\in\mathrm{pr}_{T}(L)$,

  * (iii)

$\mathcal{N}_{r}(X,Y)=0$, for every $X,Y\in\mathrm{pr}_{T}(L)$.

In particular, if $(L,r)$ is Dirac-Nijenhuis, then $(L,N)$ is Dirac-Nijenhuis
of contraction type.

###### Proof.

It is a direct verification that, for $N=(r,0)$,
$\left[\\!\left[\cdot,\cdot\right]\\!\right]_{N}=\left[\\!\left[\cdot,\cdot\right]\\!\right]_{\mathbb{D}^{r}}.$
Hence, it follows from (B.2) that
$\left[\\!\left[\cdot,\cdot\right]\\!\right]_{N}$ is skew-symmetric on $L$ if
and only if

$d\langle(r,r^{*})(\sigma_{1}),\sigma_{2}\rangle=0,\,\,\;\forall\,\sigma_{1},\sigma_{2}\in\Gamma(L).$

We claim that this is equivalent to $(r,r^{*})(L)\subset L$. Indeed, write
$\mathbb{T}M=L\oplus L^{\prime}$, where $L^{\prime}$ is a lagrangian
complement of $L$, and take a local basis
$\\{\sigma_{1},\dots,\sigma_{m},\sigma^{1},\dots,\sigma^{m}\\}$, where

$\sigma_{i}\in\Gamma(L),\,\;\sigma^{j}\in\Gamma(L^{\prime}),\;\langle\sigma_{i},\sigma^{j}\rangle=\delta_{i}^{j},\,\,\;\;\;i,j=1,\dots,m.$

By writing $(r,r^{*})(\sigma_{k})=a_{k}^{i}\sigma_{i}+b_{kj}\sigma^{j}$, one
sees that $b_{kj}$ must be zero since
$d\langle\sigma_{k},f\sigma_{j}\rangle=0,\,\,\forall\,f\in