Omni $n$-Lie algebras and linearization of higher analogues of Courant algebroids

TL;DR

This paper introduces omni n-Lie algebras as linearizations of higher Courant algebroids and explores their nonabelian variants related to Nambu-Poisson manifolds, advancing the understanding of higher geometric structures.

Contribution

It defines omni n-Lie algebras and nonabelian omni n-Lie algebras, linking them to higher Courant algebroids and Nambu-Poisson structures, respectively.

Findings

Omni n-Lie algebras serve as linearizations of higher Courant algebroids.

Nonabelian omni n-Lie algebras relate to higher Courant algebroids on Nambu-Poisson manifolds.

The paper establishes new connections between algebraic and geometric higher structures.

Abstract

In this paper, we introduce the notion of an omni $n$-Lie algebra and show that they are linearization of higher analogues of standard Courant algebroids. We also introduce the notion of a nonabelian omni $n$-Lie algebra and show that they are linearization of higher analogues of Courant algebroids associated to Nambu-Poisson manifolds.

Equations

$$(X+\alpha ,Y+\beta {)}_{+}=i_X\beta +i_Y\alpha ,\forall X+\alpha ,Y+\beta \in \mathfrak{X}(M)\oplus {\Omega }^{n-1}(M),$$

$$\left\llbracket X+\alpha,Y+\beta\right\rrbracket=[X,Y]+L_{X}\beta-i_{Y}d\alpha.$$

$$(A+u,B+v{)}_{+}=Av+Bu,\forall A+u,B+v\in \mathfrak{gl}(V)\oplus V,$$

$$\{A+u,B+v\}=[A,B]+Av.$$

$$[u_1,u_2,\cdots ,u_{n-1},[v_1,v_2,\cdots ,v_n{]}_{\mathfrak{g}}{]}_{\mathfrak{g}}=\sum _{i=1}^n[v_1,v_2,\cdots ,[u_1,u_2,\cdots ,u_{n-1},v_i{]}_{\mathfrak{g}},\cdots ,v_n{]}_{\mathfrak{g}}.$$

$${\mathrm{ad}}_{u_1,u_2,\cdots ,u_{n-1}}v=[u_1,u_2,\cdots ,u_{n-1},v{]}_{\mathfrak{g}},\forall v\in \mathfrak{g}.$$

$${\mathrm{ad}}_{\mathfrak{u}}[v_1,v_2,\cdots ,v_n{]}_{\mathfrak{g}}=\sum _{i=1}^n[v_1,v_2,\cdots ,{\mathrm{ad}}_{\mathfrak{u}}{v}_i,\cdots ,v_n{]}_{\mathfrak{g}},\forall \mathfrak{u}=u_1\wedge u_2\wedge \cdots \wedge u_{n-1}\in {\wedge }^{n-1}\mathfrak{g}.$$

$$\mathfrak{u}\circ \mathfrak{v}=\sum _{i=1}^{n-1}{v}_1\wedge \cdots \wedge v_{i-1}\wedge \mathfrak{u}\circ v_i\wedge v_{i+1}\wedge \cdots \wedge v_{n-1},$$

$$\mathfrak{u}\circ (\mathfrak{v}\circ w)-\mathfrak{v}\circ (\mathfrak{u}\circ w)=(\mathfrak{u}\circ \mathfrak{v})\circ w,\forall \mathfrak{u},\mathfrak{v}\in {\wedge }^{n-1}\mathfrak{g},w\in \mathfrak{g}.$$

$$\{f_1,\cdots ,f_n\}=(-1{)}^{|\sigma |}\{f_{\sigma (1)},\cdots ,f_{\sigma (n)}\};$$

$$\{f_{1}g,\cdots ,f_n\}=f_1\{g,\cdots ,f_n\}+g\{f_1,\cdots ,f_n\};$$

$$\{f_1,\cdots ,f_{n-1},\{g_1,\cdots ,g_n\}\}=\sum _{i=1}^n\{g_1,\cdots ,\{f_1,\cdots ,f_{n-1},g_i\},\cdots ,g_n\}.$$

$$\{f_1,\cdots ,f_n\}=\pi (d{f}_1,\cdots ,d{f}_n).$$

$$L_{{\pi }^{♯}(d{f}_1\wedge \cdots \wedge d{f}_{n-1})}\pi =0,$$

$$⟨{\pi }^{♯}({\xi }_1\wedge \cdots \wedge {\xi }_{n-1}),{\xi }_n⟩=\pi ({\xi }_1\wedge \cdots \wedge {\xi }_{n-1}\wedge {\xi }_n),\forall {\xi }_1,\cdots ,{\xi }_n\in {\Omega }^1(M).$$

$$[\alpha ,\beta {]}_{\pi }=L_{{\pi }^{♯}(\alpha )}\beta -L_{{\pi }^{♯}(\beta )}\alpha +d{i}_{{\pi }^{♯}(\beta )}\alpha ,\forall \alpha ,\beta \in {\Omega }^{n-1}(M)$$

$${\mathcal{L}}_A(v_1\otimes \cdots \otimes v_{n-1})=\sum _{i=1}^{n-1}{v}_1\otimes \cdots \otimes A{v}_i\otimes \cdots \otimes v_{n-1}.$$

$$\{A+\mathfrak{u},B+\mathfrak{v}\}=[A,B]+{\mathcal{L}}_A\mathfrak{v},$$

$$(A+\mathfrak{u},B+\mathfrak{v})_{+}=\sum^{n-1}_{i=1}(-1)^{i+1}\big{(}Av_{i}\otimes v_{1}\wedge\cdots\wedge\hat{v}_{i}\wedge\cdots\wedge v_{n-1}+Bu_{i}\otimes u_{1}\wedge\cdots\wedge\hat{u}_{i}\wedge\cdots\wedge u_{n-1}\big{)},$$

$$(\{e_1,e_2\},e_3{)}_{+}+(e_2,\{e_1,e_3\}{)}_{+}={\rho }_V(e_1)(e_2,e_3{)}_{+},$$

$${\rho }_V(A+\mathfrak{u})(\mathfrak{w})={\mathcal{L}}_A\mathfrak{w},\forall \mathfrak{w}\in V\otimes {\wedge }^{n-2}V.$$

$$(\{A+\mathfrak{u},B+\mathfrak{v}\},C+\mathfrak{w}{)}_{+}+(B+\mathfrak{v},\{A+\mathfrak{u},C+\mathfrak{w}\}{)}_{+}$$

$${\rho }_V(A+u)(B+v,C+w{)}_{+}$$

$$F^{♯}(\mathfrak{u})(u)=F(\mathfrak{u},u),\forall \mathfrak{u}\in {\wedge }^{n-1}V,u\in V.$$

$$F^{♯}({\mathcal{L}}_{F^{♯}(\mathfrak{u})}\mathfrak{v})=[F^{♯}(\mathfrak{u}),F^{♯}(\mathfrak{v})].$$

$$F^{♯}(\mathfrak{u}\circ \mathfrak{v})=[F^{♯}(\mathfrak{u}),F^{♯}(\mathfrak{v})],$$

$$\Phi (A+\mathfrak{u})=\hat{A}+\hat{\mathfrak{u}}.$$

$$\overline{v\otimes \mathfrak{u}}=l_v\hat{\mathfrak{u}}.$$

$$(\hat{A},\hat{\mathfrak{u}}{)}_{+}$$

$$d(\hat{A},\hat{\mathfrak{u}}{)}_{+}$$

$$L_{\hat{A}}\hat{\mathfrak{u}}$$

Full text

1 Introduction

Courant algebroids were introduced in [20] (see also [22]), and have many applications. See [17] and references therein for more details. On $\mathcal{T}^{n-1}M\triangleq TM\oplus\wedge^{n-1}T^{*}M$, define a symmetric nondegenerate $\wedge^{n-2}T^{*}M$-valued pairing $(\cdot,\cdot)_{+}:\mathcal{T}^{n-1}M\times\mathcal{T}^{n-1}M\longrightarrow\wedge^{n-2}T^{*}M$ by

[TABLE]

and define a bracket operation \left\llbracket\cdot,\cdot\right\rrbracket:\Gamma(\mathcal{T}^{n-1}M)\times\Gamma(\mathcal{T}^{n-1}M)\longrightarrow\Gamma(\mathcal{T}^{n-1}M) by

[TABLE]

The quadruple $(TM\oplus\wedge^{n-1}T^{*}M,(\cdot,\cdot)_{+},\{\cdot,\cdot\},\mathrm{pr}_{TM})$ is called the higher analogue of the standard Courant algebroid. In particular, if $n=2$, we obtain the standard Courant algebroid. Recently, due to applications in multisymplectic geometry, Nambu-Poisson geometry, $L_{\infty}$-algebra theory and topological field theory, higher analogues of Courant algebroids are widely studied. See [2, 3, 9, 10, 11, 12, 14, 27] for more details.

The notion of an omni-Lie algebra was introduced by Weinstein in [26] to study the linearization of the standard Courant algebroid. Then it was studied from several aspects [15, 23, 25]. An omni-Lie algebra associated to a vector space $V$ is a triple $(\mathfrak{gl}(V)\oplus V,(\cdot,\cdot)_{+},\{\cdot,\cdot\})$, where $(\cdot,\cdot)_{+}$ is the $V$-valued pairing given by

[TABLE]

and $\{\cdot,\cdot\}$ is the bilinear bracket operation given by

[TABLE]

Even though $(\mathfrak{gl}(V)\oplus V,\{\cdot,\cdot\})$ is not a Lie algebra, its Dirac structures characterize all Lie algebra structures on $V$. We can construct a Lie 2-algebra from an omni-Lie algebra. See [23] for more details.

In [19], the authors introduced the notion of a nonabelian omni-Lie algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\mathfrak{g},(\cdot,\cdot)_{+},\{\cdot,\cdot\}_{\mathfrak{g}})$ associated to a Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$, which originally comes from the study of homotopy Poisson manifolds [18]. In particular, they showed that it is the linearization of the Courant algebroid $T\mathfrak{g}^{*}\oplus T^{*}_{\pi_{\mathfrak{g}}}\mathfrak{g}^{*}$ associated to the linear Poisson manifold $(\mathfrak{g}^{*},\pi_{\mathfrak{g}})$, where $\pi_{\mathfrak{g}}$ is the Lie-Poisson structure on $\mathfrak{g}^{*}$.

The purpose of this paper is to extend the above results to the $n$-ary case. First we introduce the notion of an omni $n$-Lie algebra, which is a triple $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,(\cdot,\cdot)_{+},\{\cdot,\cdot\})$ including a bracket operation $\{\cdot,\cdot\}$ and a $(V\otimes\wedge^{n-2}V)$-valued pairing $(\cdot,\cdot)_{+}$. Similar to the classical case, $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,\{\cdot,\cdot\})$ is a Leibniz algebra. We show that a linear map $F:\wedge^{n}V\longrightarrow V$ defines an $n$-Lie algebra structure on $V$ if and only if the graph of $F^{\sharp}:\wedge^{n-1}V\longrightarrow\mathfrak{gl}(V)$ is a sub-Leibniz algebra of $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,\{\cdot,\cdot\})$. Note that this result is not totally parallel the classical case. Namely the condition that $F$ being skew-symmetric can not be simply described by being isotropic with respect to the $(V\otimes\wedge^{n-2}V)$-valued pairing $(\cdot,\cdot)_{+}$. We further show that an omni $n$-Lie algebra $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,(\cdot,\cdot)_{+},\{\cdot,\cdot\})$ can be viewed as the linearization of the higher analogue of the standard Courant algebroid (TM\oplus\wedge^{n-1}T^{*}M,(\cdot,\cdot)_{+},\left\llbracket\cdot,\cdot\right\rrbracket,\mathrm{pr}_{TM}) via letting $M=V^{*}$. Then we introduce the notion of a nonabelian omni $n$-Lie algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},(\cdot,\cdot)_{+},\{\cdot,\cdot\}_{\mathfrak{g}})$ associated to an $n$-Lie algebra $\mathfrak{g}$ and study its algebraic properties. Finally, we give the notion of higher analogues of Courant algebroids associated to Nambu-Poisson manifolds and study their properties. Furthermore, we show that nonabelian omni $n$-Lie algebras are linearization of higher analogues of Courant algebroids associated to Nambu-Poisson manifolds.

The paper is organized as follows. In Section 2, we recall $n$-Lie algebras and Nambu-Poisson manifolds. In Section 3, we introduce the notion of an omni $n$-Lie algebra associated to a vector space $V$ and characterize $n$-Lie algebra structures on $V$ via sub-Leibniz algebra structures of the omni $n$-Lie algebra. In Section 4, we show that an omni $n$-Lie algebra is the linearization of the higher analogue of the standard Courant algebroid. In Section 5, we introduce the notion of a nonabelian omni $n$-Lie algebra and study its algebraic properties. In Section 6, we introduce the notion of higher analogues of Courant algebroids associated to Nambu-Poisson manifolds and show that nonabelian omni $n$-Lie algebras are their linearization.

2 Preliminaries

In this section, we briefly recall the notions of $n$-Lie algebras and Nambu-Poisson manifolds. The notion of an $n$-Lie algebra, or a Filippov algebra, was introduced in [8] and have many applications in mathematical physics. See the review article [6] for more details. Nambu-Poisson structures were introduced in [24] by Takhtajan in order to find an axiomatic formalism for Nambu-mechanics which is a generalization of Hamiltonian mechanics

Definition 2.1**.**

An $n$-Lie algebra is a vector space $\mathfrak{g}$ together with an $n$-multilinear skew-symmetric bracket $[\cdot,\cdots,\cdot]_{\mathfrak{g}}:\wedge^{n}\mathfrak{g}\longrightarrow\mathfrak{g}$ such that for all $u_{i},v_{i}\in\mathfrak{g}$, the following Fundamental Identity is satisfied:

[TABLE]

For $u_{1},u_{2},\cdots,u_{n-1}\in\mathfrak{g}$, define $\mathrm{ad}:\wedge^{n-1}\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$ by

[TABLE]

Then Eq. (5) is equivalent to that $\mathrm{ad}_{u_{1},u_{2},\cdots,u_{n-1}}$ is a derivation, i.e.

[TABLE]

Elements in $\wedge^{n-1}\mathfrak{g}$ are called fundamental objects of the $n$-Lie algebra $(\mathfrak{g},[\cdot,\cdots,\cdot]_{\mathfrak{g}})$. In the sequel, we will denote $\mathrm{ad}_{\mathfrak{u}}v$ simply by $\mathfrak{u}\circ v$.

Define a bilinear operation on the set of fundamental objects $\circ:(\wedge^{n-1}\mathfrak{g})\otimes(\wedge^{n-1}\mathfrak{g})\longrightarrow\wedge^{n-1}\mathfrak{g}$ by

[TABLE]

for all $\mathfrak{u}=u_{1}\wedge u_{2}\wedge\cdots\wedge u_{n-1}$ and $\mathfrak{v}=v_{1}\wedge v_{2}\wedge\cdots\wedge v_{n-1}.$ In [5], the authors proved that $(\wedge^{n-1}\mathfrak{g},\circ)$ is a Leibniz algebra. See [21] for details about Leibniz algebras, which are also called Loday algebras. Moreover, the Fundamental Identity (5) is equivalent to

[TABLE]

Definition 2.2**.**

[24]* A Nambu-Poisson structure of order $n-1$ on $M$ is an $n$-linear map $\{\cdot,\cdots,\cdot\}:C^{\infty}(M)\times\cdots\times C^{\infty}(M)\longrightarrow C^{\infty}(M)$ satisfying the following properties:*

• (1)

skewsymmetry, i.e. for all $f_{1},\cdots,f_{n}\in C^{\infty}(M)$ and $\sigma\in\mathrm{Sym}(n),$

[TABLE]

• (2)

the Leibniz rule, i.e. for all $g\in C^{\infty}(M)$, we have

[TABLE]

• (3)

integrability condition, i.e. for all $f_{1},\cdots,f_{n-1},g_{1},\cdots,g_{n}\in C^{\infty}(M)$,

[TABLE]

In particular, a Nambu-Poisson structure of order $1$ is exactly a usual Poisson structure. Similar to the fact that a Poisson structure is determined by a bivector field, a Nambu-Poisson structure of order $n-1$ is determined by an $n$-vector field $\pi\in\mathfrak{X}^{n}(M)$ such that

[TABLE]

An $n$-vector field $\pi\in\mathfrak{X}^{n}(M)$ is a Nambu-Poisson structure if and only if for all $f_{1},\cdots,f_{n-1}\in C^{\infty}(M)$, we have

[TABLE]

where $\pi^{\sharp}:\wedge^{n-1}T^{*}M\longrightarrow TM$ is defined by

[TABLE]

Let $\pi$ be a Nambu-Poisson structure on a manifold $M$. Then there is a natural operation $[\cdot,\cdot]_{\pi}$ on $\Omega^{n-1}(M)$ given by

[TABLE]

such that $(\wedge^{n-1}T^{*}M,[\cdot,\cdot]_{\pi},\pi^{\sharp})$ is a Leibniz algebroid. See [2, 13] for more details.

3 Omni $n$-Lie algebras

Let $V$ be a finite dimensional vector space. For all $A\in\mathfrak{gl}(V)$, define $\mathcal{L}_{A}:\otimes^{n-1}V\longrightarrow\otimes^{n-1}V$ by

[TABLE]

Definition 3.1**.**

An omni $n$-Lie algebra associated to a vector space $V$ is a triple $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,(\cdot,\cdot)_{+},\{\cdot,\cdot\})$, where $\{\cdot,\cdot\}$ is the bilinear bracket operation given by

[TABLE]

and $(\cdot,\cdot)_{+}$ is the $(V\otimes\wedge^{n-2}V)$-valued pairing given by

[TABLE]

where $\mathfrak{u}=u_{1}\wedge u_{2}\wedge\cdots\wedge u_{n-1}$ and $\mathfrak{v}=v_{1}\wedge v_{2}\wedge\cdots\wedge v_{n-1}$.

Remark 3.2**.**

When $n=2$, we recover Weinstein’s omni-Lie algebras [26].

Proposition 3.3**.**

With the above notations, $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,\{\cdot,\cdot\})$ is a Leibniz algebra. Furthermore, the pairing $(\cdot,\cdot)_{+}$ and the bracket $\{\cdot,\cdot\}$ are compatible in the sense that

[TABLE]

where $e_{i}\in\mathfrak{gl}(V)\oplus\wedge^{n-1}V,~{}i=1,2,3$, and $\rho_{V}:\mathfrak{gl}(V)\oplus\wedge^{n-1}V\longrightarrow\mathfrak{gl}(V\otimes\wedge^{n-2}V)$ is given by

[TABLE]

Proof. Since $\mathcal{L}_{[A,B]}=\mathcal{L}_{A}\mathcal{L}_{B}-\mathcal{L}_{B}\mathcal{L}_{A}$, we can deduce that $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,\{\cdot,\cdot\})$ is a Leibniz algebra directly.

For all $A+\mathfrak{u},B+\mathfrak{v},C+\mathfrak{w}\in\mathfrak{gl}(V)\oplus\wedge^{n-1}V$, on one hand, we have

[TABLE]

On the other hand, we have

[TABLE]

which implies that (12) holds.

Let $F:\wedge^{n}V\longrightarrow V$ be a linear map. Then $F$ induces a linear map $F^{\sharp}:\wedge^{n-1}V\longrightarrow\mathfrak{gl}(V)$ by

[TABLE]

Denote by $G_{F}\subset\mathfrak{gl}(V)\oplus\wedge^{n-1}V$ the graph of $F^{\sharp}$.

Theorem 3.4**.**

Let $F:\wedge^{n}V\longrightarrow V$ be a linear map. Then $(V,F)$ is an $n$-Lie algebra if and only if $G_{F}$ is a Leibniz subalgebra of the Leibniz algebra $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,\{\cdot,\cdot\})$.

Proof. $G_{F}$ is a Leibniz subalgebra of the Leibniz algebra $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,\{\cdot,\cdot\})$ if and only if for all $\mathfrak{u},\mathfrak{v}\in\wedge^{n-1}V$, $\{F^{\sharp}(\mathfrak{u})+\mathfrak{u},F^{\sharp}(\mathfrak{v})+\mathfrak{v}\}\in G_{F}$, which is equivalent to

[TABLE]

Since $\mathcal{L}_{F^{\sharp}(\mathfrak{u})}\mathfrak{v}=\sum_{i=1}^{n-1}v_{1}\wedge\cdots\wedge F(\mathfrak{u},v_{i})\wedge\cdots\wedge v_{n-1}$, thus the above equality can be written as

[TABLE]

which is equivalent to that $(V,F)$ is an $n$-Lie algebra.

4 Linearization of the higher analogue of the standard Courant algebroid

Let $V$ be an $m$-dimensional vector space and $V^{*}$ its dual space. We consider the direct sum bundle $\mathcal{T}^{n-1}V^{*}=TV^{*}\oplus\wedge^{n-1}T^{*}V^{*}$. Denote the vector spaces of linear vector fields and constant $(n-1)$-forms on $V^{*}$ by $\mathfrak{X}_{\mathsf{lin}}(V^{*})$ and $\Omega^{n-1}_{\mathsf{con}}(V^{*})$ respectively. It is obvious that $\mathfrak{X}_{\mathsf{lin}}(V^{*})\oplus\Omega^{n-1}_{\mathsf{con}}(V^{*})\cong\mathfrak{gl}(V)\oplus\wedge^{n-1}V$. To make this explicit, for any $x\in V$, denote by $l_{x}$ the corresponding linear function on $V^{*}$. Let $\{x^{i}\}$ be a basis of the vector space $V$. Then $\{l_{x^{i}}\}$ forms a coordinate chart for $V^{*}$. So $\{\frac{\partial}{\partial l_{x^{i}}}\}$ constitutes a basis of vector fields on $V^{*}$ and $\{dl_{x^{i}}\}$ constitutes a basis of $1$-forms on $V^{*}$. For $A=(a_{ij})\in\mathfrak{gl}(V)$, we get a linear vector field $\widehat{A}=\sum_{i}l_{A(x^{i})}\frac{\partial}{\partial l_{x^{i}}}$ on $V^{*}$. Also $\mathfrak{u}=\sum_{1\leq i_{1}<\cdots<i_{n-1}\leq m}\xi_{i_{1}\cdots i_{n-1}}x^{i_{1}}\wedge\cdots\wedge x^{i_{n-1}}$ defines a constant ${(n-1)}$-form $\widehat{\mathfrak{u}}=\sum_{1\leq i_{1}<\cdots<i_{n-1}\leq m}\xi_{i_{1}\cdots i_{n-1}}dl_{x^{i_{1}}}\wedge\cdots\wedge dl_{x^{i_{n-1}}}$.

Define $\Phi:\mathfrak{gl}(V)\oplus\wedge^{n-1}V\longrightarrow\mathfrak{X}_{\mathsf{lin}}(V^{*})\oplus\Omega^{n-1}_{\mathsf{con}}(V^{*})$ by

[TABLE]

Obviously, $\Phi$ is an isomorphism between vector spaces.

Any element $v\otimes\mathfrak{u}\in V\otimes\wedge^{n-2}V$ will give rise to a linear $(n-2)$-form $\overline{v\otimes\mathfrak{u}}$ defined by

[TABLE]

We give some useful formulas first.

Lemma 4.1**.**

With the above notations, for all $A\in\mathfrak{gl}(V)$ and $\mathfrak{u}\in\wedge^{n-1}V$, we have

[TABLE]

Proof. On one hand, for $\mathfrak{u}=\sum_{1\leq i_{1}<\cdots<i_{n-1}\leq m}\xi_{i_{1}\cdots i_{n}}x^{i_{1}}\wedge\cdots\wedge x^{i_{n-1}}\in\wedge^{n-1}V$, we have

[TABLE]

On the other hand, we have

[TABLE]

which implies that (14) holds.

By direct calculation, we have

[TABLE]

On the other hand, we have

[TABLE]

Thus (15) follows immediately.

(16) follows from

[TABLE]

(17) is direct. We omit the details.

Now we are ready to show that an omni $n$-Lie algebra can be viewed as linearization of the higher analogue of the standard Courant algebroid.

Theorem 4.2**.**

The omni $n$-Lie algebra $(\mathfrak{gl}(V)\oplus\wedge^{n-1}V,(\cdot,\cdot)_{+},\{\cdot,\cdot\})$ is induced from the higher analogue of the standard Courant algebroid (\mathcal{T}^{n-1}(V^{*}),\left(\cdot,\cdot\right)_{+},\left\llbracket\cdot,\cdot\right\rrbracket,\mathrm{pr}_{TV^{*}}) via restriction to $\mathfrak{X}_{\mathsf{lin}}(V^{*})\oplus\Omega^{n-1}_{\mathsf{con}}(V^{*})$. More precisely, we have

[TABLE]

Proof. By (14), we have

[TABLE]

By (16) and (17), we have

[TABLE]

Finally, for all $\mathfrak{w}=w_{1}\otimes w_{2}\wedge\cdots\wedge w_{n-1}\in V\otimes\wedge^{n-2}V$, we have

[TABLE]

The proof is finished.

5 Nonabelian omni $n$-Lie algebras

Definition 5.1**.**

A nonabelian omni $n$-Lie algebra associated to an $n$-Lie algebra $(\mathfrak{g},[\cdot,\cdots,\cdot]_{\mathfrak{g}})$ is a triple $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},(\cdot,\cdot)_{+},\{\cdot,\cdot\}_{\mathfrak{g}})$, where $(\cdot,\cdot)_{+}$ is the $(\mathfrak{g}\otimes\wedge^{n-1}\mathfrak{g})$-valued pairing given by (11) and $\{\cdot,\cdot\}_{\mathfrak{g}}$ is the bilinear bracket operation given by

[TABLE]

Theorem 5.2**.**

Let $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},(\cdot,\cdot)_{+},\{\cdot,\cdot\}_{\mathfrak{g}})$ be a nonabelian omni $n$-Lie algebra. Then we have

• (i)

$(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\}_{\mathfrak{g}})$* is a Leibniz algebra;*

• (ii)

$\{A+\mathfrak{u},A+\mathfrak{u}\}_{\mathfrak{g}}=-\mathrm{ad}_{\mathcal{L}_{A}\mathfrak{u}}+\mathcal{L}_{A}\mathfrak{u}+\mathfrak{u}\circ\mathfrak{u}$;

• (iii)

the pairing $(\cdot,\cdot)_{+}$ and the bracket $\{\cdot,\cdot\}_{\mathfrak{g}}$ are compatible in the sense that

[TABLE]

where $e_{1}=A+\mathfrak{u},e_{2}=B+\mathfrak{v},e_{3}=C+\mathfrak{w}\in\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g}$ and $\rho_{\mathfrak{g}}:\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g}\otimes\wedge^{n-2}\mathfrak{g})$ is given by

[TABLE]

Proof. (i) We can prove that $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\}_{\mathfrak{g}})$ is a Leibniz algebra directly by a complicated computation. In the sequel, we will show that $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\}_{\mathfrak{g}})$ is a trivial deformation of the omni $n$-Lie algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\})$. Thus, we omit details here.

(ii) It follows from (21) directly.

(iii) By straightforward computation, we have

[TABLE]

On the other hand, we have

[TABLE]

Thus we have

[TABLE]

The proof is finished.

Obviously, for all $A\in\mathrm{Der}(\mathfrak{g})$, we have $[A,\mathrm{ad}_{\mathfrak{v}}]-\mathrm{ad}_{\mathcal{L}_{A}\mathfrak{v}}=0$. Thus, we have

Corollary 5.3**.**

For all $e_{1},e_{2},e_{3}\in\mathrm{Der}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g}$, we have

[TABLE]

In the sequel we show that a nonabelian omni $n$-Lie algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\}_{\mathfrak{g}})$ can be viewed as a trivial deformation of the omni $n$-Lie algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\})$. For details of deformations of Leibniz algebras, see [4, 16].

Let $(\mathfrak{L},[\cdot,\cdot]_{\mathfrak{L}})$ be a Leibniz algebra. For an endomorphism $N$ of $\mathfrak{L}$, define

[TABLE]

and set

[TABLE]

The endomorphism $N$ is called a Nijenhuis operator if $TN=0$.

A Nijenhuis operator gives a trivial deformation of the Leibniz algebra $(\mathfrak{L},[\cdot,\cdot]_{\mathfrak{L}})$.

Proposition 5.4**.**

[4]* Let $N$ be a Nijenhuis operator on the Leibniz algebra $(\mathfrak{L},[\cdot,\cdot]_{\mathfrak{L}})$. Then we have*

• (1)

$(\mathfrak{L},[\cdot,\cdot]_{N})$* is a Leibniz algebra;*

• (2)

$N$* is a morphism of Leibniz algebras from $(\mathfrak{L},[\cdot,\cdot]_{N})$ to $(\mathfrak{L},[\cdot,\cdot]_{\mathfrak{L}})$;*

• (3)

$(\mathfrak{L},[\cdot,\cdot]_{\mathfrak{L}}+[\cdot,\cdot]_{N})$* is a Leibniz algebra.*

Let $(\mathfrak{g},[\cdot,\cdots,\cdot]_{\mathfrak{g}})$ be an $n$-Lie algebra. Then we can define a linear map $N:\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g}$ by

[TABLE]

Lemma 5.5**.**

The linear map $N$ given by (24) is a Nijenhuis operator on the Leibniz algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\})$, where the Leibniz bracket $\{\cdot,\cdot\}$ is given by $(\ref{eq:omnin2})$.

Proof. First by definition, we have

[TABLE]

Hence it is clear that

[TABLE]

which says that $N$ is a Nijenhuis operator.

It is straightforward to see that

[TABLE]

Therefore, by Proposition 5.4 and Lemma 5.5, we have

Theorem 5.6**.**

Let $(\mathfrak{g},[\cdot,\cdots,\cdot]_{\mathfrak{g}})$ be an $n$-Lie algebra. Then the bracket $\{\cdot,\cdot\}_{\mathfrak{g}}$ is a trivial deformation of the Leibniz bracket $\{\cdot,\cdot\}$. In particular, $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\}_{\mathfrak{g}})$ is a Leibniz algebra.

Remark 5.7**.**

If we view $(\mathfrak{gl}(\mathfrak{g}),[\cdot,\cdot])$ as a Leibniz algebra, then $(\mathfrak{gl}(\mathfrak{g}),[\cdot,\cdot])$ and $(\wedge^{n-1}\mathfrak{g},\circ)$ form a matched pair of Leibniz algebras and the Leibniz algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},\{\cdot,\cdot\}_{\mathfrak{g}})$ is exactly their double. See [1] for more details about matched pairs of Leibniz algebras.

6 Linearization of higher analogues of Courant algebroids associated to Nambu-Poisson structures

Let $(M,\pi)$ be a Nambu-Poisson manifold. We introduce a bracket \left\llbracket\cdot,\cdot\right\rrbracket_{\pi}:\Gamma(\mathcal{T}^{n-1}M)\times\Gamma(\mathcal{T}^{n-1}M)\longrightarrow\Gamma(\mathcal{T}^{n-1}M) by

[TABLE]

where $X,Y\in\mathfrak{X}(M),\alpha,\beta\in\Omega^{n-1}(M)$ and $[\cdot,\cdot]_{\pi}$ is given by (9).

Let $\rho_{\pi}:\mathcal{T}^{n-1}M\rightarrow TM$ be the bundle map defined by

[TABLE]

We call the quadruple (\mathcal{T}^{n-1}M,(\cdot,\cdot)_{+},\left\llbracket\cdot,\cdot\right\rrbracket_{\pi},\rho_{\pi}) the higher analogue of the Courant algebroid associated to a Nambu-Poisson manifold and denote it by $\mathcal{T}^{n-1}_{\pi}M$. In the sequel, we will see that even though we call it the higher analogue of a Courant algebroid, some important properties for Courant algebroids do not hold anymore.

Theorem 6.1**.**

Let (\mathcal{T}^{n-1}M,(\cdot,\cdot)_{+},\left\llbracket\cdot,\cdot\right\rrbracket_{\pi},\rho_{\pi}) be the higher analogue of the Courant algebroid associated to a Nambu-Poisson manifold. Then we have

• (i)

(\mathcal{T}^{n-1}M,\left\llbracket\cdot,\cdot\right\rrbracket_{\pi},\rho_{\pi})* is a Leibniz algebroid.*

• (ii)

The bracket \left\llbracket\cdot,\cdot\right\rrbracket_{\pi} is not skew-symmetric. Instead, we have

[TABLE]

• (iii)

The pairing (1) and the bracket \left\llbracket\cdot,\cdot\right\rrbracket_{\pi} are compatible in the following sense:

[TABLE]

where $e_{1}=X+\alpha,~{}e_{2}=Y+\beta,~{}e_{3}=Z+\gamma$.

Proof. (i) Let $\Psi:\mathcal{T}^{n-1}M\longrightarrow\mathcal{T}^{n-1}M$ be the invertible bundle map defined by

[TABLE]

By direct calculation, we have

[TABLE]

where the bracket \left\llbracket\cdot,\cdot\right\rrbracket is given by (2). Since (\Gamma(\mathcal{T}^{n-1}M),\left\llbracket\cdot,\cdot\right\rrbracket) is a Leibniz algebra, we deduce that (\Gamma(\mathcal{T}^{n-1}M),\left\llbracket\cdot,\cdot\right\rrbracket_{\pi}) is also a Leibniz algebra.

For all $f\in C^{\infty}(M)$, $X,Y\in\mathfrak{X}(M)$ and $\alpha,\beta\in\Omega^{n-1}(M)$, we have

[TABLE]

Thus, (\mathcal{T}^{n-1}M),\left\llbracket\cdot,\cdot\right\rrbracket_{\pi},\rho_{\pi}) is a Leibniz algebroid.

(ii) It is straightforward to obtain (27) by (25).

(iii) The left hand side of the above equality is equal to

[TABLE]

The right hand side is equal to

[TABLE]

The conclusion follows from

[TABLE]

This finishes the proof.

Let $\mathfrak{X}_{H}(M)$ and $\Omega_{\mathsf{cl}}^{n-1}(M)$ denote the set of the Hamiltonian vector fields and closed $(n-1)$-forms respectively.

Corollary 6.2**.**

For all $e_{1},~{}e_{2},~{}e_{3}\in\mathfrak{X}_{H}(M)\oplus\Omega_{\mathsf{cl}}^{n-1}(M)$, we have

[TABLE]

Proof. For all $e_{1}=X+\alpha,~{}e_{2}=Y+\beta,~{}e_{3}=Z+\gamma\in\mathfrak{X}_{H}(M)\oplus\Omega_{\mathsf{cl}}^{n-1}(M)$, since $\alpha$ is closed, we have

[TABLE]

For all $\xi,\eta\in\Omega^{n-1}(M)$, we have the following formula

[TABLE]

Without loss of generality, let $X=\pi^{\sharp}(df_{1}\wedge\cdots\wedge df_{n-1})$, then we have

[TABLE]

We finishes the proof.

In the following, we show that the nonabelian omni $n$-Lie algebra is a linearization of the higher analogue of the Courant algebroid (\mathcal{T}^{n-1}M,(\cdot,\cdot)_{+},\left\llbracket\cdot,\cdot\right\rrbracket_{\pi},\rho_{\pi}) associated to a Nambu-Poisson manifold $(M,\pi)$.

Let $(\mathfrak{g},[\cdot,\cdots,\cdot]_{\mathfrak{g}})$ be an $m$-dimensional $n$-Lie algebra such that it induces a linear Nambu-Poisson structure111Not all $n$-Lie algebras can give linear Nambu-Poisson structures on dual spaces, see [7] for details. $\pi_{\mathfrak{g}}$ on $\mathfrak{g}^{*}$. Then we obtain the higher analogue of the Courant algebroid $\mathcal{T}^{n-1}_{\pi_{\mathfrak{g}}}\mathfrak{g}^{*}$. Let $\{x^{i}\}$ be a basis of the vector space $\mathfrak{g}$. Using the same notations as in Section 4, we have

[TABLE]

Lemma 6.3**.**

For all $A\in\mathfrak{gl}(\mathfrak{g})$ and $\mathfrak{u},\mathfrak{v}\in\wedge^{n-1}\mathfrak{g}$, we have

[TABLE]

Proof. For $\mathfrak{u}=\sum_{1\leq j_{1}<\cdots<j_{n-1}\leq m}\xi_{j_{1}\cdots j_{n-1}}x^{j_{1}}\wedge\cdots\wedge x^{j_{n-1}}\in\wedge^{n-1}\mathfrak{g}$ with the corresponding constant ${(n-1)}$-form $\widehat{\mathfrak{u}}=\sum_{1\leq j_{1}<\cdots<j_{n-1}\leq m}\xi_{j_{1}\cdots j_{n-1}}dl_{x^{j_{1}}}\wedge\cdots\wedge dl_{x^{j_{n-1}}}$, we have

[TABLE]

which implies that (31) holds.

Since $\widehat{\mathfrak{u}}$ is a constant $(n-1)$-form, by (16) and (31), we have

[TABLE]

which implies that (32) holds.

By (16) and (31), we have

[TABLE]

This ends the proof.

Theorem 6.4**.**

Let $(\mathfrak{g},[\cdot,\cdots,\cdot]_{\mathfrak{g}})$ be an $m$-dimensional $n$-Lie algebra such that it induces a linear Nambu-Poisson structure $\pi_{\mathfrak{g}}$ on $\mathfrak{g}^{*}$. Then the nonabelian omni $n$-Lie algebra $(\mathfrak{gl}(\mathfrak{g})\oplus\wedge^{n-1}\mathfrak{g},(\cdot,\cdot)_{+},\{\cdot,\cdot\}_{\mathfrak{g}})$ is induced from the higher analogue of the Courant algebroid (\mathcal{T}^{n-1}\mathfrak{g}^{*},(\cdot,\cdot)_{+},\left\llbracket\cdot,\cdot\right\rrbracket_{\pi_{\mathfrak{g}}},\rho_{\pi_{\mathfrak{g}}}) associated to the Nambu-Poisson manifold $(\mathfrak{g}^{*},\pi_{\mathfrak{g}})$ via restriction to $\mathfrak{X}_{\mathsf{lin}}(\mathfrak{g}^{*})\oplus\Omega^{n-1}_{\mathsf{con}}(\mathfrak{g}^{*})$. More precisely, we have

[TABLE]

Proof. (34) has been proved in Theorem 4.2. By (15)-(17) and (31)-(33), we deduce that

[TABLE]

which implies that (35) holds. By (17), we have $L_{\widehat{A}}\overline{\mathfrak{w}}=\overline{\rho_{\mathfrak{g}}(A)(\mathfrak{w})}$. Thus

[TABLE]

which says that (36) holds. This ends the proof.