VB-Courant algebroids, E-Courant algebroids and generalized geometry

TL;DR

This paper explores the relationships between VB-Courant and E-Courant algebroids, introduces generalized complex structures on E-Courant algebroids, and connects these structures to complex Lie algebras, unifying various geometric frameworks.

Contribution

It establishes the connection between VB-Courant and E-Courant algebroids, introduces generalized complex structures on E-Courant algebroids, and relates these to complex Lie algebra structures.

Findings

Constructed examples of E-Courant algebroids.

Unified generalized complex structures on different manifolds.

Showed correspondence between structures on omni-Lie algebras and complex Lie algebras.

Abstract

In this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids and construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\gl(V)\oplus V$ correspond to complex Lie algebra structures on V.

Equations

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathfrak{J}E\cong \left\{\nu \in \mathrm{Hom}(\mathfrak{D}E,E)|\nu (\Phi )=\Phi \circ \nu ({\mathsf{id}}_E),\forall \Phi \in \mathrm{gl}(E)\right\}.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\text{mathbbm}du(\mathfrak{d}):=\mathfrak{d}(u),\forall u\in \Gamma (E),\mathfrak{d}\in \Gamma (\mathfrak{D}E).$$

$$\text{mathbbm}d(fu)=\mathrm{d}f\otimes u+f\text{mathbbm}du,\forall u\in \Gamma (E),f\in C^{\infty }(M).$$

$${⟨\mu ,\mathfrak{d}⟩}_E\triangleq \mathfrak{d}(u),\forall \mu \in (\mathfrak{J}E{)}_m,\mathfrak{d}\in (\mathfrak{D}E{)}_m,$$

$${⟨\mu ,\Phi ⟩}_E$$

$${⟨\mathfrak{y},\mathfrak{d}⟩}_E$$

$${\rho }^{\star }:\mathrm{Hom}(Q,E)\to \mathrm{Hom}(P,E),{\rho }^{\star }(\nu )(k)=\nu (\rho (k)),\forall k\in P,\nu \in \mathrm{Hom}(Q,E).$$

$$[X,{\rho }^{\star }\text{mathbbm}du{]}_{\mathcal{K}}=2{\rho }^{\star }\text{mathbbm}d{\left(X,{\rho }^{\star }\text{mathbbm}du\right)}_E,[{\rho }^{\star }\text{mathbbm}du,X{]}_{\mathcal{K}}=0,\forall X\in \Gamma (\mathcal{K}),u\in \Gamma (E).$$

$$\left(\mathfrak{d}+\mu,\mathfrak{t}+\nu\right)_{E}=\frac{1}{2}\big{(}\left\langle\mu,\mathfrak{t}\right\rangle_{E}+\left\langle\nu,\mathfrak{d}\right\rangle_{E}\big{)},\quad\forall~{}\mathfrak{d}+\mu,~{}\mathfrak{t}+\nu\in\mathfrak{D}E\oplus\mathfrak{J}E.$$

$${⟨{\mathfrak{L}}_{\mathfrak{d}}\mu ,{\mathfrak{d}}^{\prime }⟩}_E$$

$$\displaystyle\left\llbracket\mathfrak{d}+\mu,\mathfrak{r}+\nu\right\rrbracket$$

$$\mathbb{T}M:=TM\oplus T^{\ast }M.$$

$$\left\llbracket X+\xi,Y+\eta\right\rrbracket=[X,Y]+\mathcal{L}_{X}\eta-i_{Y}\mathrm{d}\xi,\quad\forall~{}X+\xi,~{}Y+\eta\in\Gamma(\mathbb{T}M).$$

$$\left(X+\xi,Y+\eta\right)_{+}=\frac{1}{2}\big{(}\eta(X)+\xi(Y)\big{)}.$$

$${\mathcal{J}}^2=-\mathsf{id}\text{mbox}and{\left(\mathcal{J}(u),\mathcal{J}(v)\right)}_{+}={\left(u,v\right)}_{+}$$

$$\left\llbracket\mathcal{J}(u),\mathcal{J}(v)\right\rrbracket-\left\llbracket u,v\right\rrbracket-\mathcal{J}\big{(}\left\llbracket\mathcal{J}(u),v\right\rrbracket+\left\llbracket u,\mathcal{J}(v)\right\rrbracket\big{)}=0,\quad\forall~{}u,~{}v\in\Gamma(\mathbb{T}M).$$

$$(\mathcal{I}\alpha ,\beta {)}_L+(\alpha ,\mathcal{I}\beta {)}_L=0,\forall \alpha ,\beta \in \Gamma (\mathfrak{D}L\oplus \mathfrak{J}L),$$

$$[[\mathcal{I}\alpha ,\mathcal{I}\beta ]]-[[\alpha ,\beta ]]-\mathcal{I}[[\mathcal{I}\alpha ,\beta ]]-\mathcal{I}[[\alpha ,\mathcal{I}\beta ]]=0,\forall \alpha ,\beta \in \Gamma (\mathfrak{D}L\oplus \mathfrak{J}L).$$

$${\mathcal{I}}=\left(\begin{array}[]{cc}\phi&J^{\sharp}\\ \omega_{\flat}&-\phi^{\dagger}\end{array}\right),$$

$$D\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$c^{†}(b_m)={\tilde{0}}_{b_m}{+}_A\overline{c(m)},\forall m\in M,b_m\in B_m,$$

$$\tilde{\psi }(b_m)={\tilde{0}}_{b_m}{+}_A\overline{\psi (b_m)}.$$

$$0\to B^{\ast }\otimes C⟶\hat{A}⟶A\to 0.$$

$$\mathbb{E}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Theta :(\mathbb{E};A,B;M)⟶(TB;TM,B;M)$$

$$\left\llbracket\Gamma_{B}^{l}(\mathbb{E}),\Gamma_{B}^{l}(\mathbb{E})\right\rrbracket\subseteq\Gamma_{B}^{l}(\mathbb{E}),\ \ \ \left\llbracket\Gamma_{B}^{l}(\mathbb{E}),\Gamma_{B}^{c}(\mathbb{E})\right\rrbracket\subseteq\Gamma_{B}^{c}(\mathbb{E}),\quad\left\llbracket\Gamma_{B}^{c}(\mathbb{E}),\Gamma_{B}^{c}(\mathbb{E})\right\rrbracket=0.$$

$$T{E}^{\ast }\oplus T^{\ast }{E}^{\ast }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\to \mathrm{gl}(E)\oplus T^{\ast }M\otimes E⟶\hat{A}⟶TM\oplus E\to 0.$$

$$T\mathcal{C}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1 Introduction

The theory of Courant algebroids was first introduced by Liu, Weinstein and Xu [17] providing an extension of Drinfeld’s double for Lie bialgebroids. Precisely, the double of a Lie bialgebroid is a special Courant algebroid [17, 20]. Jacobi algebroids are natural extensions of Lie algebroids and Courant-Jacobi algebroids. They were considered by Grabowski and Marmo [7] and they can be viewed as generalizations of Courant algebroids. Both Courant algebroids and Courant-Jacobi algebroids have been extensively studied in the last decade since these are crucial geometric tools in Poisson geometry and mathematical physics. It is known that they both belong to a more general framework, namely that of $\mathsf{E}$-Courant algebroids. Indeed, $\mathsf{E}$-Courant algebroids were introduced by Chen, Liu and the second author in [5] as a differential geometric object encompassing Courant algebroids [17], Courant-Jacobi algebroids [7], omni-Lie algebroids [4], conformal Courant algebroids [2] and $AV$-Courant algebroids [14]. It turns out that $\mathsf{E}$-Courant algebroids are related to more geometric structures such as $\mathsf{VB}$-Courant algebroids [15].

The aim of this paper is two-fold. Firstly, we illuminate the relationship between $\mathsf{VB}$-Courant algebroids and $\mathsf{E}$-Courant algebroids. Secondly, we study generalized complex structures on $\mathsf{E}$-Courant algebroids. Recall that a generalized almost complex structure on a manifold $M$ is an endomorphism ${\mathcal{J}}$ of the generalized tangent bundle $\mathbb{T}M:=TM\oplus T^{\ast}M$ which preserves the natural pairing on $\mathbb{T}M$ and such that ${\mathcal{J}}^{2}=-\operatorname{id}$. If, additionally, the $\sqrt{-1}$-eigenbundle of ${\mathcal{J}}$ in the complexification $\mathbb{T}M\otimes\mathbb{C}$ is involutive relative to the Dorfman (equivalently, the Courant) bracket, then ${\mathcal{J}}$ is said to be integrable, and $(M,\mathcal{J})$ is called a generalized complex manifold. See [3, 6, 8, 9, 23] for more details.

Given a vector bundle $E\stackrel{{\scriptstyle q}}{{\,\rightarrow\,}}M$, we consider its gauge Lie algebroid $\mathfrak{D}E$, i.e. the gauge Lie algebroid of the frame bundle $\mathcal{F}(E)$. It is known that $\mathfrak{D}{E}$ is a transitive Lie algebroid over $M$ and the first jet bundle $\mathfrak{J}E$ is its $E$-dual bundle. In fact, $\mathfrak{o}\mathfrak{l}(E)=\mathfrak{D}E\oplus\mathfrak{J}E$ is called an omni-Lie algebroid [4], which is a generalization of Weinstein’s concept of an omni-Lie algebra [26]. In particular, the line bundle case where $E$ comes from a contact distribution brings us to the concept of a generalized contact bundle. To have a better grasp of the concept of a generalized contact bundle, we briefly review the line bundle approach to contact geometry. By definition, a contact structure on an odd-dimensional manifold $M$ is a maximal non-integrable hyperplane distribution $H\subset TM$. In a dual way, any hyperplane distribution $H$ on $M$ can be regarded as a nowhere vanishing $1$-form $\theta:TM\to L$ (its structure form) with values in the line bundle $L=TM/H$, such that $H=\ker\theta$. Replacing the tangent algebroid with the Atiyah algebroid of a line bundle in the definition of a generalized complex manifold, we obtain the notion of a generalized contact bundle. In this paper, we extend the concept of a generalized contact bundle to the context of $\mathsf{E}$-Courant algebroids.

The paper is organized as follows. Section 2 contains basic definitions used in the sequel. Section 3 highlights the importance and naturality of the notion of $\mathsf{E}$-Courant algebroids. Explicitly, *the fat Courant algebroid associated to a $\mathsf{VB}$-Courant algebroid * (see the definition of a $\mathsf{VB}$-Courant algebroid below) is an $\mathsf{E}$-Courant algebroid. We observe the following facts:

• $\bullet$

Given a crossed module of Lie algebras $(\mathfrak{m},\mathfrak{g})$, we get an $\mathfrak{m}$-Courant algebroid $\mathrm{Hom}(\mathfrak{g},\mathfrak{m})\oplus\mathfrak{g}$, which was given in [13] as a generalization of an omni-Lie algebra.

• $\bullet$

The omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)=\mathfrak{D}E\oplus\mathfrak{J}E$ is the linearization of the $\mathsf{VB}$-Courant algebroid $TE^{*}\oplus T^{*}E^{*}$. This generalizes the fact that an omni-Lie algebra is the linearization of the standard Courant algebroid.

• $\bullet$

For a Courant algebroid ${\mathcal{C}}$, $T{\mathcal{C}}$ is a $\mathsf{VB}$-Courant algebroid. The associated fat Courant algebroid $\mathfrak{J}{\mathcal{C}}$ is a $T^{*}M$-Courant algebroid. The fact that $\mathfrak{J}{\mathcal{C}}$ is a $T^{*}M$-Courant algebroid was first obtained in [5, Theorem 2.13].

In Section 4, we introduce generalized complex structures on $\mathsf{E}$-Courant algebroids and provide examples. In Sections 5, we describe generalized complex structures on omni-Lie algebroids. In Section 6, we show that generalized complex structures on the omni-Lie algebra $\mathfrak{o}\mathfrak{l}(V)$ are in one-to-one correspondence with complex Lie algebra structures on $V$.

2 Preliminaries

Throughout the paper, $M$ is a smooth manifold, $\mathrm{d}$ is the usual differential operator on forms, and $E\longrightarrow M$ is a vector bundle. In this section, we recall the notions of $\mathsf{E}$-Courant algebroids introduced in [5], omni-Lie algebroids introduced in [4], generalized complex structures introduced in [8, 9] and generalized contact structures introduced in [24].

2.1 $\mathsf{E}$-Courant algebroids and omni-Lie algebroids

For a vector bundle $E{\,\rightarrow\,}M$, its gauge Lie algebroid $\mathfrak{D}E$ with the commutator bracket $[\cdot,\cdot]_{\mathfrak{D}}$ is just the gauge Lie algebroid of the frame bundle $\mathcal{F}(E)$, which is also called the covariant differential operator bundle of $E$ (see [18, Example 3.3.4]). The corresponding Atiyah sequence is as follows:

[TABLE]

In [4], the authors proved that the jet bundle $\mathfrak{J}E$ may be considered as an $E$-dual bundle of $\mathfrak{D}E$:

[TABLE]

Associated to the jet bundle $\mathfrak{J}E$, there is a jet sequence given by:

[TABLE]

Define the operator $\mathbbm{d}:\Gamma(E)\rightarrow\Gamma(\mathfrak{J}E)$ by:

[TABLE]

An important formula which will be often used is

[TABLE]

In fact, there is an $E$-valued pairing between $\mathfrak{J}{E}$ and $\mathfrak{D}{E}$ by setting:

[TABLE]

where $u\in\Gamma(E)$ satisfies $\mu=[u]_{m}$. In particular, one has

[TABLE]

For vector bundles $P,Q$ over $M$ and a bundle map $\rho:P\longrightarrow Q$, we denote the induced $E$-dual bundle map by $\rho^{\star}$, i.e.

[TABLE]

Definition 2.1**.**

([5])* An * $\mathsf{E}$-Courant algebroid* is a quadruple $(\mathcal{K},[\cdot,\cdot]_{\mathcal{K}},\left(\cdot,\cdot\right)_{E},\rho)$, where $\mathcal{K}$ is a vector bundle over $M$ such that $(\Gamma(\mathcal{K}),[\cdot,\cdot]_{\mathcal{K}})$ is a Leibniz algebra, $\left(\cdot,\cdot\right)_{E}:~{}\mathcal{K}\otimes\mathcal{K}\,\rightarrow\,E$ is a nondegenerate symmetric $E$-valued pairing, which induces an embedding: $\mathcal{K}\hookrightarrow\mathrm{Hom}(\mathcal{K},E)$ via $Y(X)=2\left(X,Y\right)_{E}$, and $\rho:~{}\mathcal{K}\,\rightarrow\,\mathfrak{D}E$ is a bundle map called the anchor, such that for all $~{}X,Y,Z\in\Gamma(\mathcal{K})$, the following properties hold:*

• (EC-1)

$\rho[X,Y]_{\mathcal{K}}=[\rho(X),\rho(Y)]_{\mathfrak{D}}$;

• (EC-2)

$[X,X]_{\mathcal{K}}=\rho^{\star}\mathbbm{d}\left(X,X\right)_{E}$;

• (EC-3)

$\rho(X)\left(Y,Z\right)_{E}=\left([X,Y]_{\mathcal{K}},Z\right)_{E}+\left(Y,[X,Z]_{\mathcal{K}}\right)_{E}$;

• (EC-4)

$\rho^{\star}(\mathfrak{J}E)\subset\mathcal{K}$, i.e. $\left(\rho^{\star}(\mu),X\right)_{E}=\frac{1}{2}\mu(\rho(X)),~{}~{}~{}\forall~{}\mu\in\mathfrak{J}E;$

• (EC-5)

$\rho\circ\rho^{\star}=0$.

Obviously, a Courant algebroid is an $\mathsf{E}$-Courant algebroid, where $E=M\times\mathbb{R}$, the trivial line bundle. Similar to the proof for Courant algebroids ([20, Lemma 2.6.2]), we have

Lemma 2.2**.**

For an $\mathsf{E}$-Courant algebroid $\mathcal{K}$, one has

[TABLE]

An omni-Lie algebroid, which was introduced in [4], is a very interesting example of $\mathsf{E}$-Courant algebroids. Let us recall it briefly. There is an $E$-valued pairing $\left(\cdot,\cdot\right)_{E}$ on $\mathfrak{D}E\oplus\mathfrak{J}E$ defined by

[TABLE]

Furthermore, $\Gamma(\mathfrak{J}E)$ is invariant under the Lie derivative $\mathfrak{L}_{\mathfrak{d}}$ for any $\mathfrak{d}\in\Gamma(\mathfrak{D}E)$ which is defined by the Leibniz rule:

[TABLE]

On the section space $\Gamma(\mathfrak{D}E\oplus\mathfrak{J}E)$, we can define a bracket as follows

[TABLE]

Definition 2.3**.**

([4])* The quadruple (\mathfrak{D}E\oplus\mathfrak{J}E,\left\llbracket\cdot,\cdot\right\rrbracket,\left(\cdot,\cdot\right)_{E},\rho) is called an * omni-Lie algebroid*, where $\rho$ is the projection from $\mathfrak{D}E\oplus\mathfrak{J}E$ to $\mathfrak{D}{E}$, $\left(\cdot,\cdot\right)_{E}$ and \left\llbracket\cdot,\cdot\right\rrbracket are given by $(\ref{eqn:pair omni})$ and $(\ref{eqn:backet omni})$ respectively.*

We will denote an omni-Lie algebroid by $\mathfrak{o}\mathfrak{l}(E)$.

2.2 Generalized complex structures and generalized contact

structures

The notion of a Courant algebroid was introduced in [17]. A Courant algebroid is a quadruple ({\mathcal{C}},\left\llbracket\cdot,\cdot\right\rrbracket,\left(\cdot,\cdot\right)_{+},\rho), where ${\mathcal{C}}$ is a vector bundle over $M$, \left\llbracket\cdot,\cdot\right\rrbracket is a bracket operation on $\Gamma({\mathcal{C}})$, $\left(\cdot,\cdot\right)_{+}$ is a nondegenerate symmetric bilinear form on ${\mathcal{C}}$, $\rho:{\mathcal{C}}\longrightarrow TM$ is a bundle map called the anchor, such that some compatibility conditions are satisfied. See [20] for more details. Consider the generalized tangent bundle

[TABLE]

On its section space $\Gamma(\mathbb{T}M)$, there is a Dorfman bracket:

[TABLE]

Furthermore, there is a canonical nondegenerate symmetric bilinear form on $\mathbb{T}M$:

[TABLE]

We call (\mathbb{T}M,\left\llbracket\cdot,\cdot\right\rrbracket,\left(\cdot,\cdot\right)_{+},\mathsf{pr}_{TM}) the standard Courant algebroid.

Definition 2.4**.**

A generalized complex structure on a manifold $M$ is a bundle map $\mathcal{J}:\mathbb{T}M\longrightarrow\mathbb{T}M$ satisfying the algebraic properties:

[TABLE]

and the integrability condition:

[TABLE]

Here, $\left(\cdot,\cdot\right)_{+}$ and \left\llbracket\cdot,\cdot\right\rrbracket are given by $(\ref{eqn:pair})$ and $(\ref{eqn:bracket})$ respectively.

See [8, 9] for more details. Note that only even-dimensional manifolds can have generalized complex structures. In [24], the authors give the odd-dimensional analogue of the concept of a generalized complex structures extending the definition given in [10]. We now recall the definition of a generalized contact bundle from [24]. A generalized contact bundle is a line bundle $L\to M$ equipped with a generalized contact structure, i.e. a vector bundle endomorphism ${\mathcal{I}}:\mathfrak{D}L\oplus\mathfrak{J}L\to\mathfrak{D}L\oplus\mathfrak{J}L$ such that

• •

${\mathcal{I}}$ is almost complex, i.e. ${\mathcal{I}}^{2}=-\mathsf{id}$,

• •

${\mathcal{I}}$ is skew-symmetric, i.e.

[TABLE]

• •

${\mathcal{I}}$ is integrable, i.e.

[TABLE]

Let $(L\to M,{\mathcal{I}})$ be a generalized contact bundle. Using the direct sum $\mathfrak{o}\mathfrak{l}(L)=\mathfrak{D}L\oplus\mathfrak{J}L$ and the definition, one can see that

[TABLE]

where $J$ is a Jacobi bi-derivation, $\phi$ is an endomorphism of $\mathfrak{D}L$ compatible with $J$, and the $2$-form $\omega:\wedge^{2}\mathfrak{D}L\to L$ and its associated vector bundle morphism $\omega_{\flat}:\mathfrak{D}L\to\mathfrak{J}L$ satisfy additional compatibility conditions [24].

3 $\mathsf{VB}$-Courant algebroids and $\mathsf{E}$-Courant algebroids

In this section, we highlight the relation between $\mathsf{VB}$-Courant algebroids and $\mathsf{E}$-Courant algebroids and give more examples of $\mathsf{E}$-Courant algebroids.

Denote a double vector bundle

[TABLE]

with core $C$ by $(D;A,B;M)$. The space of sections $\Gamma_{B}(D)$ is generated as a $C^{\infty}(B)$-module by core sections $\Gamma_{B}^{c}(D)$ and linear sections $\Gamma_{B}^{l}(D)$. See [19] for more details. For a section $c:M\rightarrow C$, the corresponding core section $c^{\dagger}:B\rightarrow D$ is defined as

[TABLE]

where $\bar{\cdot}$ means the inclusion $C\hookrightarrow D$. A section $\xi:B\rightarrow D$ is called linear if it is a bundle morphism from $B\rightarrow M$ to $D\rightarrow A$ over a section $a\in\Gamma(A)$. Given $\psi\in\Gamma(B^{*}\otimes C)$, there is a linear section $\tilde{\psi}:B\rightarrow D$ over the zero section $0^{A}:M\rightarrow A$ given by

[TABLE]

Note that $\Gamma_{B}^{l}(D)$ is locally free as a $C^{\infty}(M)$-module. Therefore, $\Gamma_{B}^{l}(D)$ is equal to $\Gamma(\hat{A})$ for some vector bundle $\hat{A}\rightarrow M$. Moreover, we have the following short exact sequence of vector bundles over $M$

[TABLE]

Example 3.1**.**

Let $E$ be a vector bundle over $M$.

• (a)

The tangent bundle $(TE;TM,E;M)$ is a double vector bundle with core $E$. Then $\hat{A}$ is the gauge bundle $\mathfrak{D}E$ and the exact sequence (9) is exactly the Atiyah sequence (1).

• (b)

The cotangent bundle $(T^{*}E;E^{*},E;M)$ is a double vector bundle with core $T^{*}M$. In this case, $\hat{A}$ is exactly the jet bundle $\mathfrak{J}E^{*}$ and the exact sequence (9) is indeed the jet sequence (3).

Definition 3.2**.**

([15])* A $\mathsf{VB}$-Courant algebroid is a metric double vector bundle*

[TABLE]

with core $C$ such that $\mathbb{E}\rightarrow B$ is a Courant algebroid and the following conditions are satisfied:

• (i)

The anchor map $\Theta:\mathbb{E}\rightarrow TB$ is linear. That is

[TABLE]

is a morphism of double vector bundles.

• (ii)

The Courant bracket is linear. That is

[TABLE]

For a $\mathsf{VB}$-Courant algebroid $\mathbb{E}$, we have the exact sequence $(\ref{fat})$. Note that the restriction of the pairing on $\mathbb{E}$ to linear sections of $\mathbb{E}$ defines a nondegenerate pairing on $\hat{A}$ with values in $B^{*}$, which is guaranteed by the metric double vector bundle structure; see [11]. Coupled with the fact that the Courant bracket is closed on linear sections, one gets the following result.

Proposition 3.3**.**

([11])* The vector bundle $\hat{A}$ inherits a Courant algebroid structure with the pairing taking values in $B^{*}$, which is called the fat Courant algebroid of this $\mathsf{VB}$-Courant algebroid.*

Alternatively, we have

Proposition 3.4**.**

For a $\mathsf{VB}$-Courant algebroid $(\mathbb{E};A,B;M)$, its associated fat Courant algebroid is a $B^{*}$-Courant algebroid.

Example 3.5**.**

(Standard $\mathsf{VB}$-Courant algebroid over a vector bundle)  For a vector bundle $E$, there is a standard $\mathsf{VB}$-Courant algebroid

[TABLE]

with base $E^{*}$ and core $E^{*}\oplus T^{*}M\rightarrow M$. The corresponding exact sequence is given by

[TABLE]

Actually, by Example 3.1, the corresponding fat Courant algebroid $\hat{A}$ here is exactly the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)=\mathfrak{D}E\oplus\mathfrak{J}E$. So the omni-Lie algebroid is the linearization of the standard $\mathsf{VB}$-Courant algebroid. **

Example 3.6**.**

(Tangent $\mathsf{VB}$-Courant algebroid)  The tangent bundle $T{\mathcal{C}}$ of a Courant algebroid ${\mathcal{C}}\rightarrow M$

[TABLE]

carries a $\mathsf{VB}$-Courant algebroid structure with base $TM$ and core ${\mathcal{C}}\rightarrow M$. The associated exact sequence is

[TABLE]

Actually the fat Courant algebroid $\hat{{\mathcal{C}}}$ is $\mathfrak{J}{\mathcal{C}}$, which is a $T^{*}M$-Courant algebroid by Proposition 3.4. So we get that on the jet bundle of a Courant algebroid, there is a $T^{*}M$-Courant algebroid structure. This result was first given in [5]. **

A crossed module of Lie algebras consists of a pair of Lie algebras $(\mathfrak{m},\mathfrak{g})$, an action $\triangleright$ of $\mathfrak{g}$ on $\mathfrak{m}$ and a Lie algebra morphism $\phi:\mathfrak{m}\rightarrow\mathfrak{g}$ such that

[TABLE]

Given a crossed module, there is an action $\rho:\mathfrak{g}\ltimes\mathfrak{g}^{*}\longrightarrow\mathfrak{X}(\mathfrak{m}^{*})$ of the natural quadratic Lie algebra $\mathfrak{g}\ltimes\mathfrak{g}^{*}$ on $\mathfrak{m}^{*}$ given by

[TABLE]

where $u\triangleright\cdot\in\mathrm{gl}(\mathfrak{m})$ is viewed as a linear vector field on $\mathfrak{m}^{*}$ and $\phi^{*}\alpha\in\mathfrak{m}^{*}$ is viewed as a constant vector field on $\mathfrak{m}^{*}$. Note that this action is coisotropic. We get the action Courant algebroid [16] $(\mathfrak{g}\ltimes\mathfrak{g}^{*})\times\mathfrak{m}^{*}$ over $\mathfrak{m}^{*}$ with the anchor given by $\rho$ and the Dorfman bracket given by

[TABLE]

for any $e_{1},e_{2}\in\Gamma((\mathfrak{g}\ltimes\mathfrak{g}^{*})\times\mathfrak{m}^{*})$. Here $\mathrm{d}e_{1}\in\Omega^{1}(\mathfrak{m}^{*},\mathfrak{g}\ltimes\mathfrak{g}^{*})$ is given by Lie derivatives: $(\mathrm{d}e_{1})(X)=\mathcal{L}_{X}e_{1}$ for $X\in\mathfrak{X}(\mathfrak{m}^{*})$. Moreover, it is a $\mathsf{VB}$-Courant algebroid

[TABLE]

with base $\mathfrak{m}^{*}$ and core $\mathfrak{g}^{*}$. See [15] for details. The associated exact sequence is

[TABLE]

Since the double vector bundle is trivial, we have $\hat{A}=\mathrm{Hom}(\mathfrak{g},\mathfrak{m})\oplus\mathfrak{g}$.

Moreover, applying (10), we get the Dorfman bracket on $\mathrm{Hom}(\mathfrak{g},\mathfrak{m})\oplus\mathfrak{g}$.

Proposition 3.7**.**

With the above notations, $(\mathrm{Hom}(\mathfrak{g},\mathfrak{m})\oplus\mathfrak{g},[\cdot,\cdot],(\cdot,\cdot)_{\mathfrak{m}},\rho=0)$ is an $\mathfrak{m}$-Courant algebroid, where the pairing $(\cdot,\cdot)_{\mathfrak{m}}$ is given by

[TABLE]

and the Dorfman bracket is given by

[TABLE]

for all $A,B\in\mathrm{Hom}(\mathfrak{g},\mathfrak{m}),u,v\in\mathfrak{g}.$ Here $\mathrm{ad}_{v}^{0}\in\mathrm{gl}(\mathfrak{g})$ and $\mathrm{ad}_{v}^{1}\in\mathrm{gl}(\mathfrak{m})$ are given by $\mathrm{ad}_{v}^{0}(u)=[v,u]_{\mathfrak{g}}$ and $\mathrm{ad}_{v}^{1}(a)=v\triangleright a$ respectively and $\cdot\triangleright Av\in\mathrm{Hom}(\mathfrak{g},\mathfrak{m})$ is defined by $(\cdot\triangleright Av)(u)=u\triangleright Av.$

Proof. By (10), it is obvious that

[TABLE]

For $A,B\in\mathrm{Hom}(\mathfrak{g},\mathfrak{m}),v\in\mathfrak{g}$, applying (10), we find

[TABLE]

Observe that $\mathcal{L}_{\rho(v)}A=\mathrm{ad}_{v}^{1}(A)=\mathrm{ad}_{v}^{1}\circ A$ and $[A,v]_{\mathfrak{g}\ltimes\mathfrak{g}^{*}}=-(\mathrm{ad}^{0})_{v}^{*}A=A\circ\mathrm{ad}_{v}^{0}$. We have

[TABLE]

where we have used

[TABLE]

Finally, we have

[TABLE]

This completes the proof.

Remark 3.8**.**

This bracket can be viewed as a generalization of an omni-Lie algebra. See [13, Example 5.2] for more details.

More generally, since the category of Lie 2-algebroids and the category of $\mathsf{VB}$-Courant algebroids are equivalent (see [15]), we get an $\mathsf{E}$-Courant algebroid from a Lie 2-algebroid. This construction first appeared in [11, Corollary 6.9]. Explicitly, let $(A_{0}\oplus A_{-1},\rho_{A_{0}},l_{1},l_{2}=l_{2}^{0}+l_{2}^{1},l_{3})$ be a Lie 2-algebroid. Then we have an $A_{-1}$-Courant algebroid structure on

[TABLE]

where the paring is given by

[TABLE]

the anchor is

[TABLE]

and the Dorfman bracket is given by

[TABLE]

4 Generalized complex structures on $\mathsf{E}$-Courant algebroids

In this section, we introduce the notion of a generalized complex structure on an $\mathsf{E}$-Courant algebroid. We will see that it unifies the usual generalized complex structure on an even-dimensional manifold and the generalized contact structure on an odd-dimensional manifold.

Definition 4.1**.**

A bundle map $\mathcal{J}:\mathcal{K}\longrightarrow\mathcal{K}$ is called a generalized almost complex structure on an $\mathsf{E}$-Courant algebroid $(\mathcal{K},[\cdot,\cdot]_{\mathcal{K}},\left(\cdot,\cdot\right)_{E},\rho)$ if it satisfies the algebraic properties

[TABLE]

Furthermore, $\mathcal{J}$ is called a generalized complex structure if the following integrability condition is satisfied:

[TABLE]

Proposition 4.2**.**

Let $\mathcal{J}:\mathcal{K}\longrightarrow\mathcal{K}$ be a generalized almost complex structure on an $\mathsf{E}$-Courant algebroid $(\mathcal{K},[\cdot,\cdot]_{\mathcal{K}},\left(\cdot,\cdot\right)_{E},\rho)$. Then we have

[TABLE]

Proof. By (11), for all $X,Y\in\Gamma(\mathcal{K})$, we have

[TABLE]

Since $X\in\Gamma(\mathcal{K})$ is arbitrary, we have

[TABLE]

For any $Z\in\Gamma(\mathcal{K})$, let $Y=-\mathcal{J}(Z)$. By (11), we have $Z=\mathcal{J}(Y)$. Then we have

[TABLE]

which implies that $\mathcal{J}^{\star}|_{\mathcal{K}}=-\mathcal{J}.$

Remark 4.3**.**

Generalized complex structures on an $\mathsf{E}$-Courant algebroid $(\mathcal{K},[\cdot,\cdot]_{\mathcal{K}},\left(\cdot,\cdot\right)_{E},\rho)$ are in one-to-one correspondence with Dirac sub-bundles $S\subset\mathcal{K}\otimes\mathbb{C}$ such that $\mathcal{K}\otimes\mathbb{C}=S\oplus\overline{S}$. By a Dirac sub-bundle of $\mathcal{K}$, we mean a sub-bundle $S\subset\mathcal{K}$ which is closed under the bracket $[\cdot,\cdot]_{\mathcal{K}}$ and satisfies $S=S^{\bot}$. The pair $(S,\overline{S})$ is an $\mathsf{E}$-Lie bialgebroid in the sense of [5].

Remark 4.4**.**

Obviously, the notion of a generalized contact bundle associated to $L$, which was introduced in [24], is a special case of Definition 4.1 where $E$ is the line bundle $L$. In particular, if $E$ is the trivial line bundle $L^{\circ}=M\times\mathbb{R}$, we have

[TABLE]

Therefore, $\mathcal{E}^{1}(M)=\mathfrak{D}L^{\circ}\oplus\mathfrak{J}L^{\circ}$. Thus, a generalized complex structure on an $\mathsf{E}$-Courant algebroid unifies generalized complex structures on even-dimensional manifolds and generalized contact bundles on odd-dimensional manifolds

Example 4.5**.**

Consider the $\mathsf{E}$-Courant algebroid $A^{*}\otimes E\oplus A$ given in [5, Example 2.9] for any Lie algebroid $(A,[\cdot,\cdot]_{A},a)$ and an $A$-module $E$. Twisted by a 3-cocycle $\Theta\in\Gamma(\wedge^{3}A^{*},E)$, one obtains the AV-Courant algebroid introduced in [14] by Li-Bland. Consider $\mathcal{J}$ of the form \mathcal{J}_{D}=\Big{(}\begin{array}[]{cc}-R_{D}&0\\ 0&D\end{array}\Big{)}, where $D\in\mathrm{gl}(A)$ and $R_{D}:A^{*}\otimes E\longrightarrow A^{*}\otimes E$ is given by $R_{D}(\phi)=\phi\circ D.$ We get that $\mathcal{J}$ is a generalized complex structure on the $\mathsf{E}$-Courant algebroid $A^{*}\otimes E\oplus A$ if and only if $D$ is a Nijenhuis operator on the Lie algebroid $A$ and $D^{2}=-1$.

Actually, $D^{2}=-1$ ensures that condition (11) holds. The Dorfman bracket on $\mathcal{K}=A^{*}\otimes E\oplus A$ is given by

[TABLE]

where $\rho^{\star}:\mathfrak{J}E\to A^{*}\otimes E$ is the dual of the $A$-action $\rho:A\to\mathfrak{D}E$ on $E$. Then it is straightforward to see that the integrability condition (12) holds if and only if $D$ is a Nijenhuis operator on $A$.

Any generalized complex structure on a Courant algebroid induces a Poisson structure on the base manifold (see e.g. [1]). Similarly, any generalized complex structure on an $\mathsf{E}$-Courant algebroid induces a Lie algebroid or a local Lie algebra structure ([12]) on $E$ .

Theorem 4.6**.**

Let $\mathcal{J}:\mathcal{K}\longrightarrow\mathcal{K}$ be a generalized complex structure on an $\mathsf{E}$-Courant algebroid $(\mathcal{K},[\cdot,\cdot]_{\mathcal{K}},\left(\cdot,\cdot\right)_{E},\rho)$. Define a bracket operation $[\cdot,\cdot]_{E}:\Gamma(E)\wedge\Gamma(E)\longrightarrow\Gamma(E)$ by

[TABLE]

Then $(E,[\cdot,\cdot]_{E},\mathbbm{j}\circ\rho\circ\mathcal{J}\circ\rho^{\star}\circ\mathbbm{d})$ is a Lie algebroid when ${\rm rank}(E)\geq 2$ and $(E,[\cdot,\cdot]_{E})$ is a local Lie algebra when ${\rm rank}(E)=1$ .

Proof. The bracket is obviously skew-symmetric. By the integrability of $\mathcal{J}$, we have

[TABLE]

Pairing with $\rho^{\star}\mathbbm{d}w$ for $w\in\Gamma(E)$, by (EC-3) in Definition 2.1 and the first equation in Lemma 2.2, we have

[TABLE]

By (EC-1) and (EC-5) in Definition 2.1, we have

[TABLE]

Finally, using Lemma 2.2, we have

[TABLE]

Thus we get the Jacobi identity for $[\cdot,\cdot]_{E}$. To see the Leibniz rule, by definition, we have

[TABLE]

So it is a Lie algebroid structure if and only if $\mathbbm{j}\circ\rho\circ\mathcal{J}\circ\rho^{\star}\circ\mathbbm{d}:E\to TM$ is a bundle map, which is always true when ${\rm rank}(E)\geq 2$ (see the proof of [4, Theorem 3.11]).

5 Generalized complex structures on omni-Lie algebroids

In this section, we study generalized complex structures on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$. We view $\mathfrak{o}\mathfrak{l}(E)$ as a sub-bundle of $\mathrm{Hom}(\mathfrak{o}\mathfrak{l}(E),E)$ by the nondegenerate $E$-valued pairing $\left(\cdot,\cdot\right)_{E}$, i.e.

[TABLE]

By Proposition 4.2, we have

Corollary 5.1**.**

A bundle map $\mathcal{J}:\mathfrak{o}\mathfrak{l}(E)\longrightarrow\mathfrak{o}\mathfrak{l}(E)$ is a generalized almost complex structure on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$ if and only if the following conditions are satisfied:

[TABLE]

Since $\mathfrak{o}\mathfrak{l}(E)$ is the direct sum of $\mathfrak{D}E$ and $\mathfrak{J}E$, we can write a generalized almost complex structure $\mathcal{J}$ in the form of a matrix. To do that, we need some preparations.

Vector bundles $\mathrm{Hom}(\wedge^{k}\mathfrak{D}E,E)_{\mathfrak{J}E}$ and $\mathrm{Hom}(\wedge^{k}\mathfrak{J}E,E)_{\mathfrak{D}E}$ are introduced in [5] and [21] to study deformations of omni-Lie algebroids and deformations of Lie algebroids respectively. More precisely, we have

[TABLE]

in which $\mu_{\natural}:\wedge^{k-1}\mathfrak{D}E\longrightarrow\mathrm{Hom}(\mathfrak{D}E,E)$ is given by

[TABLE]

and $\mathfrak{d}^{\sharp}$ is defined similarly. By (2), for any $\mu\in\mathrm{Hom}(\wedge^{k}\mathfrak{D}E,E)_{\mathfrak{J}E}$, we have

[TABLE]

Furthermore, $(\Gamma(\mathrm{Hom}(\wedge^{\bullet}\mathfrak{D}E,E)_{\mathfrak{J}E}),\mathbbm{d})$ is a subcomplex of $(\Gamma(\mathrm{Hom}(\wedge^{\bullet}\mathfrak{D}E,E),\mathbbm{d})$, where $\mathbbm{d}$ is the coboundary operator of the gauge Lie algebroid $\mathfrak{D}E$ with the obvious action on $E$.

Proposition 5.2**.**

Any generalized almost complex structure $\mathcal{J}$ on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$ must be of the form

[TABLE]

where $N:\mathfrak{D}E\longrightarrow\mathfrak{D}E$ is a bundle map satisfying $N^{\star}(\mathfrak{J}E)\subset\mathfrak{J}E$, $\pi\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{J}E,E)_{\mathfrak{D}E})$, $~{}\sigma\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{D}E,E)_{\mathfrak{J}E})$ such that the following conditions hold:

[TABLE]

Proof. By Corollary 5.1, for any generalized almost complex structure $\mathcal{J}$, we have $\mathcal{J}^{\star}|_{\mathfrak{o}\mathfrak{l}(E)}=-\mathcal{J}$. Thus $\mathcal{J}$ must be of the form

[TABLE]

where $N:\mathfrak{D}E\longrightarrow\mathfrak{D}E$ is a bundle map satisfying $N^{\star}(\mathfrak{J}E)\subset\mathfrak{J}E$, $\phi:\mathfrak{J}E\longrightarrow\mathfrak{D}E$ and $\psi:\mathfrak{D}E\longrightarrow\mathfrak{J}E$ are bundle maps satisfying

[TABLE]

Therefore, we have $\phi=\pi^{\sharp}$, for some $\pi\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{J}E,E)_{\mathfrak{D}E})$, and $\psi=\sigma_{\natural}$ for some $~{}\sigma\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{D}E,E)_{\mathfrak{J}E})$. This finishes the proof of the first part. As for the second part, it is straightforward to see that the conditions follow from the fact that $\mathcal{J}^{2}=-\mathsf{id}$.

Remark 5.3**.**

For a line bundle $L$, we have $\mathfrak{J}L=\mathrm{Hom}(\mathfrak{D}L,L)$ and $\mathfrak{D}L=\mathrm{Hom}(\mathfrak{J}L,L)$. Therefore, the condition $N^{\star}(\mathfrak{J}L)\subset\mathfrak{J}L$ always holds.

Theorem 5.4**.**

A generalized almost complex structure $\mathcal{J}$ given by $(\ref{eqn:type J})$ is a generalized complex structure on the omni-Lie algebroid ${\mathfrak{o}\mathfrak{l}(E)}$ if and only if

• (i)

$\pi$* satisfies the equation:*

[TABLE]

where the bracket $[\cdot,\cdot]_{\pi}$ on $\Gamma(\mathfrak{J}E)$ is defined by

[TABLE]

• (ii)

$\pi$* and $N$ are related by the following formula:*

[TABLE]

• (iii)

$N$* satisfies the condition:*

[TABLE]

where $T(N)$ is the Nijenhuis tensor of $N$ defined by

[TABLE]

• (iv)

$N$* and $\sigma$ are related by the following condition*

[TABLE]

where $\sigma_{N}\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{D}E,E)_{\mathfrak{J}E})$ is defined by

[TABLE]

Proof. Consider the integrability condition (12). In fact, there are two equations since $\Gamma(\mathfrak{o}\mathfrak{l}(E))$ has two components $\Gamma(\mathfrak{D}E)$ and $\Gamma(\mathfrak{J}E)$. First let $e_{1}=\mu,~{}e_{2}=\nu$ be elements in $\Gamma(\mathfrak{J}E)$, we have $\mathcal{J}(\mu)=\pi^{\sharp}(\mu)-N^{\star}(\mu)$, $\mathcal{J}(\nu)=\pi^{\sharp}(\nu)-N^{\star}(\nu)$ and \left\llbracket\mu,\nu\right\rrbracket=0. Therefore, we obtain

[TABLE]

Thus we get conditions (17) and (19).

Then let $e_{1}=\mathfrak{d}\in\Gamma(\mathfrak{D}E)$ and $e_{2}=\mu\in\Gamma(\mathfrak{J}E)$, we have $\mathcal{J}(e_{1})=N(\mathfrak{d})+\sigma_{\natural}(\mathfrak{d})$ and $\mathcal{J}(e_{2})=\pi^{\sharp}(\mu)-N^{\star}(\mu)$. Therefore, we obtain

[TABLE]

Thus we have

[TABLE]

We claim that (22) is equivalent to (19). In fact, applying (22) to $\nu\in\Gamma(\mathfrak{J}E)$ and (19) to $\mathfrak{d}\in\Gamma(\mathfrak{D}E)$, we get the same equality.

Next let $e_{1}=\mathfrak{d}$ and $e_{2}=\mathfrak{t}$ be elements in $\Gamma(\mathfrak{D}E)$, we have $\mathcal{J}(e_{1})=N(\mathfrak{d})+\sigma_{\natural}(\mathfrak{d})$ and $\mathcal{J}(e_{2})=N(\mathfrak{t})+\sigma_{\natural}(\mathfrak{t})$. Therefore, we have

[TABLE]

Thus we have

[TABLE]

We claim that (23) and (24) are equivalent. In fact, applying (23) and (24) to $\mathfrak{t}\in\Gamma(\mathfrak{D}E)$ and $\mu\in\Gamma(\mathfrak{J}E)$ respectively, we get the same equality

[TABLE]

By the equality $\pi^{\sharp}\circ\sigma_{\natural}+N^{2}=-\mathsf{id}$ and (24), we have

[TABLE]

which implies that $T(N)(\mathfrak{d},\mathfrak{t})=\pi^{\sharp}(i_{\mathfrak{d}\wedge\mathfrak{t}}\mathbbm{d}\sigma)$. Thus (24) is equivalent to (20).

At last, we consider condition (25). Acting on an arbitrary $\mathfrak{k}\in\Gamma(\mathfrak{D}E)$, we have

[TABLE]

Note that the following equality holds:

[TABLE]

Therefore, we have

[TABLE]

which implies that (25) is equivalent to (21).

Remark 5.5**.**

Let \mathcal{J}=\Big{(}\begin{array}[]{cc}N&\pi^{\sharp}\\ \sigma_{\natural}&-N^{\star}\end{array}\Big{)} be a generalized complex structure on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$. Then $\pi$ satisfies (17). On one hand, in [4], the authors showed that such $\pi$ will give rise to a Lie bracket $[\cdot,\cdot]_{E}$ on $\Gamma(E)$ via

[TABLE]

On the other hand, by Theorem 4.6, the generalized complex structure $\mathcal{J}$ will also induce a Lie algebroid structure on $E$ by $\eqref{eq:LiebracketJ}$. By the equality

[TABLE]

these two Lie algebroid structures on $E$ are the same.

Remark 5.6**.**

Recall that any $b\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{D}E,E)_{\mathfrak{J}E})$ defines a transformation $e^{b}:\mathfrak{o}\mathfrak{l}(E)\,\rightarrow\,\mathfrak{o}\mathfrak{l}(E)$ defined by

[TABLE]

Thus, $e^{b}$ is an automorphism of the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$ if and only if $\mathbbm{d}b=0$. In this case, $e^{b}$ is called a $B$-field transformation. Genuinely, an automorphism of the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$ is just the composition of an automorphism of the vector bundle $E$ and a $B$-field transformation. In fact $B$-field transformations map generalized complex structures on $\mathfrak{o}\mathfrak{l}(E)$ into new generalized complex structures as follows:

[TABLE]

Example 5.7**.**

Let $D:E\to E$ be a bundle map satisfying $D^{2}=-\mathsf{id}$. Define $R_{D}:\mathfrak{D}E\to\mathfrak{D}E$ by $R_{D}(\mathfrak{d})=\mathfrak{d}\circ D$ and $\hat{D}:\mathfrak{J}E\to\mathfrak{J}E$ by $\hat{D}(\mathbbm{d}u)=\mathbbm{d}(Du)$ for $u\in\Gamma(E)$. Then

[TABLE]

is a generalized complex structure on $\mathfrak{o}\mathfrak{l}(E)$. In fact, since

[TABLE]

we have $R_{D}^{\star}=\hat{D}$. It is direct to check that the Nijenhuis tensor $T(R_{D})$ vanishes and the condition $D^{2}=-\mathsf{id}$ ensures that $R_{D}^{2}=-\mathsf{id}$. **

Let $\pi\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{J}E,E)_{\mathfrak{D}E})$ and suppose that the induced map $\pi^{\sharp}:\mathfrak{J}E\to\mathfrak{D}E$ is an isomorphism of vector bundles. Then the rank of $E$ is $1$ or equals to the dimension of $M$. We denote by $(\pi^{\sharp})^{-1}$ the inverse of $\pi^{\sharp}$ and by $\pi^{-1}$ the corresponding element in $\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{D}E,E)_{\mathfrak{J}E})$.

Lemma 5.8**.**

With the above notations, the following two statements are equivalent:

• (i)

$\pi\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{J}E,E)_{\mathfrak{D}E})$* satisfies (17);*

• (ii)

$\pi^{-1}$* is closed, i.e. $\mathbbm{d}\pi^{-1}=0$.*

Proof. The conclusion follows from the following equality:

[TABLE]

which can be obtained by straightforward computations.

Let $(E,[\cdot,\cdot]_{E},a)$ be a Lie algebroid. Define $\pi^{\sharp}:\mathfrak{J}E\to\mathfrak{D}E$ by

[TABLE]

Then $\pi^{\sharp}$ satisfies (17). Furthermore, $(\mathfrak{J}E,[\cdot,\cdot]_{\pi},\mathbbm{j}\circ\pi^{\sharp})$ is a Lie algebroid, where the bracket $[\cdot,\cdot]_{\pi}$ is given by (18). By Theorem 5.4 and Lemma 5.8, we have

Corollary 5.9**.**

Let $(E,[\cdot,\cdot]_{E},a)$ be a Lie algebroid such that the induced map $\pi^{\sharp}:\mathfrak{J}E\to\mathfrak{D}E$ is an isomorphism. Then

[TABLE]

is a generalized complex structure on $\mathfrak{o}\mathfrak{l}(E)$.

Example 5.10**.**

Let $(TM,[\cdot,\cdot]_{TM},\mathsf{id})$ be the tangent Lie algebroid. Define $\pi^{\sharp}:\mathfrak{J}(TM)\to\mathfrak{D}(TM)$ by

[TABLE]

Then $\pi^{\sharp}$ is an isomorphism. See [4, Corollary 3.9] for details. Then

[TABLE]

is a generalized complex structure on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(TM)$.

Example 5.11**.**

Let $(M,\omega)$ be a symplectic manifold and $(T^{*}M,[\cdot,\cdot]_{\omega^{-1}},(\omega^{\sharp})^{-1})$ the associated natural Lie algebroid. Define $\pi^{\sharp}:\mathfrak{J}(T^{*}M)\to\mathfrak{D}(T^{*}M)$ by

[TABLE]

which is an isomorphism (see [4, Corollary 3.10]). Then

[TABLE]

is a generalized complex structure on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(T^{*}M)$.

At the end of this section, we introduce the notion of an algebroid-Nijenhuis structure, which can give rise to generalized complex structures on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$.

Definition 5.12**.**

Let $(E,[\cdot,\cdot]_{E},a)$ be a Lie algebroid, $N:\mathfrak{D}E\longrightarrow\mathfrak{D}E$ a Nijenhuis operator on the Lie algebroid $(\mathfrak{D}E,[\cdot,\cdot]_{\mathfrak{D}},\mathbbm{j})$ satisfying $N^{\star}(\mathfrak{J}E)\subset\mathfrak{J}E$ and $\pi:\mathfrak{J}E\longrightarrow\mathfrak{D}E$ given by (28). $N$ and $\pi$ are said to be compatible if

[TABLE]

where

[TABLE]

Here $\pi_{N}\in\Gamma(\mathrm{Hom}(\wedge^{2}\mathfrak{J}E,E)_{\mathfrak{D}E})$ is given by

[TABLE]

If $N$ and $\pi$ are compatible, we call the pair $(\pi,N)$ an algebroid-Nijenhuis structure on the Lie algebroid $(E,[\cdot,\cdot]_{E},a)$.

The following lemma is straightforward, we omit the proof.

Lemma 5.13**.**

Let $(E,[\cdot,\cdot]_{E},a)$ be a Lie algebroid, $\pi$ given by (28) and $N:\mathfrak{D}E\longrightarrow\mathfrak{D}E$ a Nijenhuis structure. Then $(\pi,N)$ is an algebroid-Nijenhuis structure on the Lie algebroid $(E,[\cdot,\cdot]_{E},a)$ if and only if $N\circ\pi^{\sharp}=\pi^{\sharp}\circ N^{\star}$ and

[TABLE]

By Theorem 5.4 and Lemma 5.13, we have

Theorem 5.14**.**

Let $(E,[\cdot,\cdot]_{E},a)$ be a Lie algebroid, $\pi$ given by (28) and $N:\mathfrak{D}E\longrightarrow\mathfrak{D}E$ a Nijenhuis structure. Then the following statements are equivalent:

• (a)

$(\pi,N)$* is an algebroid-Nijenhuis structure and $N^{2}=-\mathsf{id};$*

• (b)

\mathcal{J}=\Big{(}\begin{array}[]{cc}N&\pi^{\sharp}\\ 0&-N^{\star}\end{array}\Big{)}* is a generalized complex structure on the omni-Lie algebroid $\mathfrak{o}\mathfrak{l}(E)$.*

Remark 5.15**.**

An interesting special case is that where $E=L$ is a line bundle. Then $(\pi,N)$ becomes a Jacobi-Nijenhuis structure on $M$. Jacobi-Nijenhuis structures are studied by Luca Vitagliano and the third author in [25]. In this case, $\pi$ defines a Jacobi bi-derivation $\{\cdot,\cdot\}$ of $L$ (i.e. a skew-symmetric bracket which is a first order differential operator, hence a derivation, in each argument). Moreover, this bi-derivation is compatible with $N$ in the sense that $\pi^{\sharp}\circ N^{\star}=N\circ\pi^{\sharp}$ and $C(\pi,N)=0$. It defines a new Jacobi bi-derivation $\{\cdot,\cdot\}_{N}$. Furthermore, $(\{\cdot,\cdot\},\{\cdot,\cdot\}_{N})$ is a Jacobi bi-Hamiltonian structure, i.e. $\{\cdot,\cdot\}$, $\{\cdot,\cdot\}_{N}$ and $\{\cdot,\cdot\}+\{\cdot,\cdot\}_{N}$ are all Jacobi brackets.

6 Generalized complex structures on omni-Lie algebras

In this section, we consider the case that $E$ reduces to a vector space $V$. Then we have

[TABLE]

Furthermore, the pairing (4) reduces to

[TABLE]

Any $u\in V$ is a linear map from $\mathrm{gl}(V)$ to $V$,

[TABLE]

Therefore, an omni-Lie algebroid reduces to an omni-Lie algebra, which is introduced by Weinstein in [26] to study the linearization of the standard Courant algebroid.

Definition 6.1**.**

An omni-Lie algebra associated to $V$ is a triple (\mathrm{gl}(V)\oplus V,\left\llbracket\cdot,\cdot\right\rrbracket,\left(\cdot,\cdot\right)_{V}), where $\left(\cdot,\cdot\right)_{V}$ is a nondegenerate symmetric pairing given by

[TABLE]

and \left\llbracket\cdot,\cdot\right\rrbracket is a bracket operation given by

[TABLE]

We will simply denote an omni-Lie algebra associated to a vector space $V$ by $\mathfrak{o}\mathfrak{l}(V)$.

Lemma 6.2**.**

For any vector space $V$, we have

[TABLE]

Proof. In fact, for any $\phi\in\mathrm{Hom}(\wedge^{2}\mathrm{gl}(V),V)_{V}$ and $A,B\in\mathrm{gl}(V)$, by (15), we have

[TABLE]

Therefore, $\phi=0$, which implies that $\mathrm{Hom}(\wedge^{2}\mathrm{gl}(V),V)_{V}=0$. The second equality is obvious.

Proposition 6.3**.**

Any generalized almost complex structure $\mathcal{J}:\mathrm{gl}(V)\oplus V\longrightarrow\mathrm{gl}(V)\oplus V$ on the omni-Lie algebra $\mathfrak{o}\mathfrak{l}(V)$ is of the following form

[TABLE]

where $\pi\in\mathrm{Hom}(\wedge^{2}V,V)$, $D\in\mathrm{gl}(V)$ satisfying $D^{2}=-\mathsf{id}_{V}$ and $\pi(Du,v)=\pi(u,Dv)$, and $R_{D}:\mathrm{gl}(V)\longrightarrow\mathrm{gl}(V)$ is the right multiplication, i.e. $R_{D}(A)=A\circ D$.

Proof. By Proposition 5.2 and Lemma 6.2, we can assume that a generalized almost complex structure is of the form \Big{(}\begin{array}[]{cc}N&\pi^{\sharp}\\ 0&-N^{\star}\end{array}\Big{)}, where $N:\mathrm{gl}(V)\longrightarrow\mathrm{gl}(V)$ satisfies $N^{\star}\in\mathrm{gl}(V)$ and $N^{2}=-\mathsf{id}_{\mathrm{gl}(V)}$, and $\pi\in\mathrm{Hom}(\wedge^{2}V,V)$. Let $D=-N^{\star}$, then we have

[TABLE]

On the other hand, we have $(Dv)(A)=ADv$, which implies that $N(A)=-R_{D}(A)$. It is obvious that $N^{2}=-\mathsf{id}_{\mathrm{gl}(V)}$ is equivalent to that $D^{2}=-\mathsf{id}_{V}$. The proof is finished.

Theorem 6.4**.**

A generalized almost complex structure $\mathcal{J}:\mathrm{gl}(V)\oplus V\longrightarrow\mathrm{gl}(V)\oplus V$ on the omni-Lie algebra $\mathfrak{o}\mathfrak{l}(V)$ given by (33) is a generalized complex structure if and only if

• (i)

$\pi$* defines a Lie algebra structure $[\cdot,\cdot]_{\pi}$ on $V$;*

• (ii)

$D^{2}=-\mathsf{id}_{V}$* and $D[u,v]_{\pi}=[u,Dv]_{\pi}$ for $u,v\in V$.*

Thus, a generalized complex structure on the omni-Lie algebra $\mathfrak{o}\mathfrak{l}(V)$ gives rise to a complex Lie algebra structure on $V$.

Proof. By Theorem 5.4, we have

[TABLE]

Condition (17) implies that $[\cdot,\cdot]_{\pi}$ gives a Lie algebra structure on $V$. Condition (19) implies that $D[u,v]_{\pi}=[u,Dv]_{\pi}$. The other conditions are valid.

The conditions $D^{2}=-\mathsf{id}_{V}$ and $D[u,v]_{\pi}=[u,Dv]_{\pi}$ say by definition that $D$ is a complex Lie algebra structure on $(V,[\cdot,\cdot]_{\pi}).$ This finishes the proof.