# On $n$-pre-Lie algebras and dendrification of $n$-Lie algebras

T. Chtioui, A. Hajjaji, S. Mabrouk, A. Makhlouf Taoufik Chtioui1 111 E-mail:
<EMAIL_ADDRESS>, Atef Hajjaji1 222E-mail<EMAIL_ADDRESS>,
Sami Mabrouk2 333 E-mail<EMAIL_ADDRESS>and Abdenacer Makhlouf3 444
Corresponding author, E-mail<EMAIL_ADDRESS>
1\. University of Sfax, Faculty of Sciences Sfax, BP 1171, 3038 Sfax, Tunisia
2\. University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, Tunisia
3\. IRIMAS - Département de Mathématiques, 6, rue des frères Lumière, F-68093
Mulhouse, France

###### Abstract

The main purpose of this paper is to introduce the notion of $n$-L-dendriform
algebra which can be seen as a dendrification of $n$-pre-Lie algebras by means
of $\mathcal{O}$-operators. We investigate the representation theory of
$n$-pre-Lie algebras and provide some related constructions. Furthermore, we
introduce the notion of phase space of a $n$-Lie algebra and show that a
$n$-Lie algebra has a phase space if and only if it is sub-adjacent to a
$n$-pre-Lie algebra. Moreover, we present a procedure to construct
$(n+1)$-pre-Lie algebras from $n$-pre-Lie algebras equipped with a generalized
trace function.

Key words: $n$-pre-Lie algebra, $n$-L-dendriform, representation, symplectic
structure.

Mathematics Subject Classification: 17A40,17A42,17B15.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and Basics
  3. 3 Representations of $n$-pre-Lie algebras
  4. 4 Constructions on $n$-pre-Lie algebras
    1. 4.1 Symplectic structures and phase spaces of $n$-Lie algebras
    2. 4.2 $(n+1)$-pre-Lie algebras induced by $n$-pre-Lie algebras
  5. 5 $n$-L-dendriform algebras

##  1 Introduction

The notion of an $n$-Lie algebra, or a Filippov algebra was introduced in
[24]. It is the algebraic structure corresponding to Nambu mechanics [36, 40].
Notice that $n$-Lie algebras, or more generally, $n$-Leibniz algebras, have
attracted attention from several fields of mathematics and physics due to
their close connection with dynamics, geometries as well as string theory [13,
14], see also [15, 16, 25, 29, 38]. For example, the structure of $n$-Lie
algebra is applied to study supersymmetry and gauge symmetry transformations
of the word-volume theory of multiple $M2$-branes and the generalized identity
for $n$-Lie algebras can be regarded as a generalized Plucker relation in the
physics literature. See the review article [22] for more details.

A dendriform algebra is a module equipped with two binary products whose sum
is associative. This concept was introduced by Loday in the late 1990s in the
study of periodicity in algebraic K-theory [32]. Several years later, Loday
and Ronco introduced the concept of tridendriform algebra from their study of
algebraic topology [33]. It is a module with three binary operations whose sum
is associative. Afterward, quite few similar algebraic structures were
introduced, such as the quadri-algebra and ennea algebra [1]. The notion of
splitting of associativity was introduced by Loday to describe this phenomena
in general, for the associative operation (see [23, 28, 34], for more
details).

Pre-Lie algebras are a class of nonassociative algebras coming from the study
of convex homogeneous cones, affine manifolds and affine structures on Lie
groups, and the cohomologies of associative algebras. See the survey [10] and
the references therein for more details. The main properties of a pre-Lie
algebra is that the commutator gives rise to a Lie algebra and the left
multiplication gives rise to a representation of the commutator Lie algebra.
Conversely, a relative Rota-Baxter operator action on a Lie algebra gives rise
to a pre-Lie algebra [27, 31] and thus the pre-Lie algebra can be seen as the
underlying algebraic structure of a relative Rota-Baxter operator. In [18],
the authors introduce the notion of $n$-pre-Lie algebra, which gives a $n$-Lie
algebra naturally and its left multiplication operator gives rise to a
representation of this $n$-Lie algebra. A $n$-pre-Lie algebra can also be
obtained through the action of a relative Rota-Baxter operator on a $n$-Lie
algebra. For ($n=3$), see [7] for more details.

Note that the notion of a symplectic structure on a $n$-Lie algebra was
introduced in [18] and it is shown that the underlying structure of a
symplectic $n$-Lie algebra is a quadratic $n$-pre-Lie algebra. In this paper,
we introduce the notion of a phase space of a $n$-Lie algebra $A$, which is a
symplectic $n$-Lie algebra $A\oplus A^{*}$ satisfying some conditions, and
show that a $n$-Lie algebra has a phase space if and only if it is sub-
adjacent to a $n$-pre-Lie algebra. We also introduce the notion of Manin
triple of $n$-pre-Lie algebras and show that there is a one-to-one
correspondence between Manin triples of $n$-pre-Lie algebras and phase spaces
of $n$-Lie algebras.

In [5], the authors introduce a realization of the quantum Nambu bracket in
terms of matrices (using the commutator and the trace of matrices). This
construction was generalized in [2] to the case of any Lie algebra where the
commutator is replaced by the Lie bracket, and the matrix trace is replaced by
linear forms having similar properties, that led to a so called $3$-Lie
algebras induced by Lie algebras. This construction is generalized to the case
of $n$-Lie algebras in [3], see also [4].

The following is an outline of the paper. In Section 2, we summarize some
definitions and known results about $n$-Lie algebras and $n$-pre-Lie algebras
which will be useful in the sequel. Section 3 is dedicated to study
representations of $n$-pre-Lie algebras. In Section 4, we introduce the notion
of phase space of a $n$-Lie algebra and show that a $n$-Lie algebra has a
phase space if and only if it is sub-adjacent to a $n$-pre-Lie algebra. We
also introduce the notion of Manin triple of $n$-pre-Lie algebras and study
its relationship with phase spaces of $n$-Lie algebras. Next, we provide a
construction procedure of $(n+1)$-pre-Lie algebras starting from an $n$-pre-
Lie algebra and a trace map. In Section 5, we introduce the notion of
$n$-L-dendriform algebra. We establish that it has two associated $n$-pre-Lie
algebras (horizontal and vertical). They have the same sub-adjacent $n$-Lie
algebra. In addition, the left multiplication operator of the first operation
(called north-west: $\nwarrow$) and the right multiplication operator of the
second operation (called north-east:$\nearrow$ ) lead to a bimodule of the
associated horizontal $n$-pre-Lie algebra.

Throughout this paper $\mathbb{K}$ is a field of characteristic $0$ and all
vector spaces are over $\mathbb{K}$ and finite dimensional.

##  2 Preliminaries and Basics

In this section, we give some preliminaries and basic results on $n$-Lie
algebras and $n$-pre-Lie algebras.

######  Definition 2.1.

[24] A $n$-Lie algebra consists of a vector space $A$ equipped with a skew-
symmetric $n$-linear map $[\cdot,\cdots,\cdot]:A^{n}\rightarrow A$ such that
the following Filippov-Jocobi identity holds (for $x_{i},y_{i}\in A,1\leq
i\leq{n}$)

$\displaystyle[x_{1},\cdots,x_{n-1},[y_{1},\cdots,y_{n}]]=\sum_{i=1}^{n}[y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}],y_{i+1},\cdots,y_{n}].$
(2.1)

Recall that the $n$-linear bracket is said to be skew-symmetric if for any
permutation $\sigma$ in the permutation group $S_{n}$ and
$x_{1},\cdots,x_{n}\in A$, we have

$[x_{\sigma(1)},\cdots,x_{\sigma(n)}]=Sgn(\sigma)[x_{1},\cdots,x_{n}],$

where $Sgn(\sigma)$ is the signature of $\sigma$.

The Filippov-Jacobi identity can expressed as follows. For
$X=(x_{1},\cdots,x_{n-1})\in A^{n-1}$, the operator

$\displaystyle ad(X):A\to A,\quad
ad(X)(y):=[x_{1},\cdots,x_{n-1},y],\quad\forall y\in A,$ (2.2)

is a derivation in the sense that

$ad(X)([y_{1},\cdots,y_{n}])=\sum_{i=1}^{n}[y_{1},\cdots,y_{i-1},ad(X)(y_{i}),y_{i+1},\cdots,y_{n}].$
(2.3)

A morphism between $n$-Lie algebras is a linear map that preserves the $n$-Lie
brackets.

######  Example 2.1.

[24] Consider $(n+1)$-dimensional $n$-Lie algebra $A$ generated by
$(e_{1},\cdots,e_{n+1})$ with the following multiplication

$[e_{1},\cdots,\widehat{e}_{i},\cdots,e_{n}]=e_{i},$

where $\widehat{e}_{i}$ means that the element $e_{i}$ is omitted.

The notion of a representation of an $n$-Lie algebra was introduced in [30],
see also [21].

######  Definition 2.2.

A representation of a $n$-Lie algebra $(A,[\cdot,\cdots,\cdot])$ on a vector
space $V$ is a skew-symmetric linear map $\rho:\wedge^{n-1}A\rightarrow gl(V)$
satisfying

$\displaystyle\rho([x_{1},\cdots,x_{n}],y_{1},\cdots,y_{n-2})=\sum_{i=1}^{n}(-1)^{n-i}\rho(x_{1},\cdots,\widehat{x}_{i},\cdots,x_{n})\rho(x_{i},y_{1},\cdots,y_{n-2}),$
(2.4)
$\displaystyle[\rho(x_{1},\cdots,x_{n-1}),\rho(y_{1},\cdots,y_{n-1})]=\sum_{i=1}^{n-1}\rho(y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}],y_{i+1},\cdots,y_{n-1}),$
(2.5)

for $x_{i},y_{i}\in A,1\leq i\leq n$.
We refer to the representation by the pair $(V,\rho)$. We say also that $V$ is
an $A$-module. Let $\rho(x_{1},\cdots,x_{n-1})=ad(x_{1},\cdots,x_{n-1})$ for
$x_{1},\cdots,x_{n-1}\in A$. Then $(A,ad)$ is an $A$-module and is called the
adjoint module of $A$.

######  Lemma 2.2.

$(V,\rho)$ is a representation of a $n$-Lie algebra $(A,[\cdot,\cdots,\cdot])$
if and only if there is a $n$-Lie algebra structure on the direct sum $A\oplus
V$ of the underlying vector spaces $A$ and $V$ given by

$[x_{1}+u_{1},\cdots,x_{n}+u_{n}]_{A\oplus
V}=[x_{1},\cdots,x_{n}]+\sum_{i=1}^{n}(-1)^{n-i}\rho(x_{1},\cdots,\widehat{x}_{i},\cdots,x_{n})(u_{i}),$
(2.6)

for $x_{i}\in A,u_{i}\in V,1\leq i\leq n$, and $\widehat{x}_{i}$ means that
$x_{i}$ is omitted. We denote it by $A\ltimes_{\rho}V.$

Let $(V,\rho)$ be a representation of a $n$-Lie algebra $A$ and $V^{*}$ be the
dual space of $V$. Define $\rho^{*}:\wedge^{n-1}A\rightarrow gl(V^{*})$ by

$\langle\rho^{*}(x_{1},\cdots,x_{n-1})\xi,v\rangle=-\langle\xi,\rho(x_{1},\cdots,x_{n-1})v\rangle,$
(2.7)

$\forall\xi\in V^{*},\;x_{i}\in A,1\leq i\leq{n-1},\;v\in V.$

######  Lemma 2.3.

Let $(V,\rho)$ be a representation of a $n$-Lie algebra
$(A,[\cdot,\cdots,\cdot])$. Then $(V^{*},\rho^{*})$ is a representation of
$(A,[\cdot,\cdots,\cdot])$, which is called the dual representation.

######  Example 2.4.

Let $A$ be a $n$-Lie algebra. Let $(A,ad)$ be the adjoint representation of
$A$ given by Eq. (2.2). The dual representation of the adjoint representation
$(A,ad)$ is denoted by $(A^{*},ad^{*})$, where $A^{*}$ is the dual space of
$A$, and $ad^{*}:\wedge^{n-1}A\rightarrow End(A^{*})$, for all
$x_{1},\cdots,x_{n-1}\in A$, $x\in A$ and $\xi\in A^{*}$,

$\langle
ad^{*}(x_{1},\cdots,x_{n-1})\xi,x\rangle=-\langle\xi,ad(x_{1},\cdots,x_{n-1})x\rangle,$

and is called the coadjoint representation. For any $n$-Lie algebra $A$, we
have the semi-direct product $n$-Lie algebra $(A\oplus
A^{*},[\cdot,\cdots,\cdot]^{*})$, where $[\cdot,\cdots,\cdot]^{*}:(A\oplus
A^{*})^{n}\rightarrow A\oplus A^{*}$, for all $x_{i}\in A$, $\xi_{i}\in
A^{*}$, $1\leq i\leq n$,

$[x_{1}+\xi_{1},\cdots,x_{n}+\xi_{n}]^{*}=[x_{1},\cdots,x_{n}]+\sum_{i=1}^{n}(-1)^{n-i}ad^{*}(x_{1},\cdots,\widehat{x}_{i},\cdots,x_{n})(\xi_{i}),$

where $\widehat{x}_{i}$ means that $x_{i}$ is omitted.

Now we recall the definition of $n$-pre-Lie algebra and exhibit construction
results in terms of $\mathcal{O}$-operators on $n$-Lie algebras (see [18] for
more details)

######  Definition 2.3.

[35] Let $(A,[\cdot,\cdots,\cdot])$ be a $n$-Lie algebra and $(V,\rho)$ a
representation. A linear operator $T:V\rightarrow A$ is called an
$\mathcal{O}$-operator associated to $(V,\rho)$ if $T$ satisfies

$[Tu_{1},\cdots,Tu_{n}]=T\Big{(}\sum_{i=1}^{n}(-1)^{n-i}\rho(Tu_{1},\cdots,\widehat{Tu_{i}},\cdots,Tu_{n})u_{i}\Big{)},\quad\forall
u_{i}\in V,1\leq i\leq n.$ (2.8)

An $\mathcal{O}$-operator associated to the adjoint representation $(A,ad)$ is
called a Rota-Baxter operator of weight $\lambda=0$.

######  Definition 2.4.

Let $A$ be a vector space with a multilinear map
$\\{\cdot,\cdots,\cdot\\}:(\wedge^{n-1}A)\otimes A\rightarrow A$. The pair
$(A,\\{\cdot,\cdots,\cdot\\})$ is called a $n$-pre-Lie algebra if for all
$x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\in A$, the following identities hold:

$\displaystyle\\{x_{1},\cdots,x_{n-1},\\{y_{1},\cdots,y_{n}\\}\\}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\\{y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}]^{C},y_{i+1},\cdots,y_{n}\\}$
(2.9)
$\displaystyle+\\{y_{1},\cdots,y_{n-1},\\{x_{1},\cdots,x_{n-1},y_{n}\\}\\},$
$\displaystyle\\{[x_{1},\cdots,x_{n}]^{C},y_{1},\cdots,y_{n-1}\\}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{n-i}\\{x_{1},\cdots,\widehat{x}_{i},\cdots,x_{n},\\{x_{i},y_{1},\cdots,y_{n-1}\\}\\},$
(2.10)

where $[\cdot,\cdots,\cdot]^{C}$ is defined by

$[x_{1},\cdots,x_{n}]^{C}=\sum_{i=1}^{n}(-1)^{n-i}\\{x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i}\\},\quad\forall
x_{i}\in A,1\leq i\leq n.$ (2.11)

######  Proposition 2.5.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra. Then the induced
$n$-commutator given by Eq. (2.11) defines a $n$-Lie algebra.

######  Definition 2.5.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra. The $n$-Lie
algebra $(A,[\cdot,\cdots,\cdot]^{C})$ is called the sub-adjacent $n$-Lie
algebra of $(A,\\{\cdot,\cdots,\cdot\\})$, and denoted by $A^{c}$.
$(A,\\{\cdot,\cdots,\cdot\\})$ is called a compatible $n$-pre-Lie algebra of
the $n$-Lie algebra $A^{c}$.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra. Define the left
multiplication $L:\wedge^{n-1}A\rightarrow gl(A)$ by

$L(x_{1},\cdots,x_{n-1})x_{n}=\\{x_{1},\cdots,x_{n-1},x_{n}\\},\quad\forall
x_{i}\in A,1\leq i\leq{n}.$ (2.12)

Moreover, we define the right multiplication $R:\otimes^{n-1}A\rightarrow
gl(A)$ by

$R(x_{1},\cdots,x_{n-1})x_{n}=\\{x_{n},x_{1},\cdots,x_{n-1}\\},\quad\forall
x_{i}\in A,1\leq i\leq{n}.$ (2.13)

If there is a $n$-pre-Lie algebra structure on its dual space $A^{*}$, we
denote the left multiplication and right multiplication by $\mathcal{L}$ and
$\mathcal{R}$ respectively.

By the definitions of a $n$-pre-Lie algebra and a representation of a $n$-Lie
algebra, we immediately obtain :

######  Proposition 2.6.

With the above notations, $(A,L)$ is a representation of the $n$-Lie algebra
$(A,[\cdot,\cdots,\cdot]^{C})$. On the other hand, let $A$ be a vector space
with a $n$-linear map $\\{\cdot,\cdots,\cdot\\}:(\wedge^{n-1}A)\otimes
A\rightarrow A$ . Then $(A,\\{\cdot,\cdots,\cdot\\})$ is a $n$-pre-Lie algebra
if $[\cdot,\cdots,\cdot]^{C}$ defined by Eq. (2.11) is a $n$-Lie algebra and
the left multiplication $L$ defined by Eq. (2.12) gives a representation of
this $n$-Lie algebra.

######  Proposition 2.7.

Let $(A,[\cdot,\cdots,\cdot])$ be a $n$-Lie algebra and $(V,\rho)$ be a
representation. Suppose that the linear map $T:V\rightarrow A$ is an
$\mathcal{O}$-operator associated to $(V,\rho)$. Then there exists a $n$-pre-
Lie algebra structure on $V$ given by

$\\{u_{1},\cdots,u_{n}\\}=\rho(Tu_{1},\cdots,Tu_{n-1})u_{n},\quad\forall~{}u_{i}\in
V,1\leq i\leq n.$ (2.14)

In particular; If $V=A$, let $P:A\rightarrow A$ be a Rota-Baxter operator of
weight zero associated to $(A,ad)$. Then the compatible $n$-pre-Lie algebra on
$A$ is given by

$\\{x_{1},\cdots,x_{n}\\}=[P(x_{1}),\cdots,P(x_{n-1}),x_{n}],$ (2.15)

for any $x_{1},\cdots,x_{n}\in A$.

######  Corollary 2.8.

With the above conditions, $(V,[\cdot,\cdots,\cdot]^{C})$ is a $n$-Lie algebra
as the sub-adjacent $n$-Lie algebra of the $n$-pre-Lie algebra given in
Proposition 2.7, and $T$ is a $n$-Lie algebra morphism from
$(V,[\cdot,\cdots,\cdot]^{C})$ to $(A,[\cdot,\cdots,\cdot])$. Furthermore,
$T(V)=\\{Tv\;|\;v\in V\\}\subset A$ is a $n$-Lie subalgebra of $A$ and there
is an induced $n$-pre-Lie algebra structure $\\{\cdot,\cdots,\cdot\\}_{T(V)}$
on $T(V)$ given by

$\\{Tu_{1},\cdots,Tu_{n}\\}_{T(V)}:=T\\{u_{1},\cdots,u_{n}\\},\quad\;\forall
u_{i}\in V,1\leq i\leq n.$ (2.16)

######  Proposition 2.9.

Let $(A,[\cdot,\cdots,\cdot])$ be a $n$-Lie algebra. Then there exists a
compatible $n$-pre-Lie algebra if and only if there exists an invertible
$\mathcal{O}$-operator $T:V\rightarrow A$ associated to a representation
$(V,\rho)$. Furthermore, the compatible $n$-pre-Lie structure on $A$ is given
by

$\\{x_{1},\cdots,x_{n}\\}_{A}=T\rho(x_{1},\cdots,x_{n-1})T^{-1}(x_{n}),\;\forall
x_{i}\in A,1\leq i\leq n.$ (2.17)

##  3 Representations of $n$-pre-Lie algebras

In this section, we introduce the notion of a representation of a $n$-pre-Lie
algebra, construct the corresponding semi-direct product $n$-pre-Lie algebra
and give the dual representation.

######  Definition 3.1.

A representation of a $n$-pre-Lie algebra $(A,\\{\cdot,\cdots,\cdot\\})$ on a
vector space $V$ consists of a pair $(l,r)$, where $l:\wedge^{n-1}A\rightarrow
gl(V)$ is a representation of the $n$-Lie algebra $A^{c}$ on $V$ and
$r:(\wedge^{n-2}A)\otimes A\rightarrow gl(V)$ is a linear map such that for
all $x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\in A$, the following identities
holds:

$\displaystyle\bullet
l(x_{1},\cdots,x_{n-1})r(y_{1},\cdots,y_{n-1})=r(y_{1},\cdots,y_{n-1})\mu(x_{1},\cdots,x_{n-1})$
$\displaystyle+\sum_{i=1}^{n-2}r(y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}]^{C},y_{i+1},\cdots,y_{n-1})+r(y_{1},\cdots,y_{n-2},\\{x_{1},\cdots,x_{n-1},y_{n-1}\\}),$
(3.1) $\displaystyle\bullet
r([x_{1},\cdots,x_{n}]^{C},y_{1},\cdots,y_{n-2})=\sum_{i=1}^{n}(-1)^{n-i}l(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})r(x_{i},y_{1},\cdots,y_{n-2}),$
(3.2) $\displaystyle\bullet
r(x_{1},\cdots,x_{n-2},\\{y_{1},\cdots,y_{n}\\})=l(y_{1},\cdots,y_{n-1})r(x_{1},\cdots,x_{n-2},y_{n})$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i+1}r(y_{1},\cdots,\widehat{y_{i}},\cdots,y_{n})\mu(x_{1},\cdots,x_{n-2},y_{i}),$
(3.3) $\displaystyle\bullet
r(y_{1},\cdots,y_{n-1})\mu(x_{1},\cdots,x_{n-1})=l(x_{1},\cdots,x_{n-1})r(y_{1},\cdots,y_{n-1})$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i}r(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},\\{x_{i},y_{1},\cdots,y_{n-1}\\}),$
(3.4)

where
$\quad\mu(x_{1},\cdots,x_{n-1})=l(x_{1},\cdots,x_{n-1})+\sum_{i=1}^{n-1}(-1)^{i}r(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i})$.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra and $l$ a
representation of the sub-adjacent $n$-Lie algebra $A^{c}$ on $V$ . Then
$(l,0)$ is a representation of the $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\})$ on the vector space $V$. It is obvious that
$(A,L,R)$ is a representation of a $n$-pre-Lie algebra on itself, which is
called the adjoint representation.

By straightforward computations, we have

######  Proposition 3.1.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra, $V$ a vector
space and $l,r:\otimes^{n-1}A\rightarrow gl(V)$ two linear maps. Then
$(V,l,r)$ is a representation of $A$ if and only if there is a $n$-pre-Lie
algebra structure $($called semi-direct product$)$ on the direct sum $A\oplus
V$ of vector spaces, defined by

$\displaystyle\\{x_{1}+u_{1},\cdots,x_{n}+u_{n}\\}_{A\oplus V}=$
$\displaystyle\\{x_{1},\cdots,x_{n}\\}+l(x_{1},\cdots,x_{n-1})(u_{n})$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i+1}r(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})(u_{i}),$
(3.5)

for $x_{i}\in A,u_{i}\in V,1\leq i\leq n$. We denote this semi-direct product
$n$-pre-Lie algebra by $A\ltimes_{l,r}V.$

Let $V$ be a vector space and $(V,l,r)$ be a representation of the $n$-pre-Lie
algebra $(A,\\{\cdot,\cdots,\cdot\\})$ on $V$. Define
$\widetilde{\rho}:\wedge^{n-1}A\rightarrow gl(V)$ by

$\widetilde{\rho}(x_{1},\cdots,x_{n-1})=l(x_{1},\cdots,x_{n-1})+\sum_{i=1}^{n-1}(-1)^{i}r(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i}),$
(3.6)

for all $x_{1},\cdots,x_{n-1}\in A.$

######  Proposition 3.2.

With the above notation, $(V,\widetilde{\rho})$ is a representation of the
sub-adjacent $n$-Lie algebra $(A^{c},[\cdot,\cdots,\cdot]^{C})$ on the vector
space $V$.

###### Proof.

By Proposition 3.1, we have the semi-direct product $n$-pre-Lie algebra
$A\ltimes_{l,r}V.$ Consider its sub-adjacent $n$-Lie algebra structure
$[\cdot,\cdots,\cdot]^{C}$, we have for any $x_{i}\in A,\;u_{i}\in V$

$\displaystyle\quad\;[x_{1}+u_{1},\cdots,x_{n}+u_{n}]_{A\oplus
V}^{C}=\sum_{i=1}^{n}(-1)^{n-i}\\{x_{1}+u_{1},\cdots,\widehat{x_{i}+u_{i}},\cdots,x_{n}+u_{n},x_{i}+u_{i}\\}_{A\oplus
V}$
$\displaystyle=\sum_{i=1}^{n}(-1)^{n-i}\\{x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i}\\}+\sum_{i=1}^{n}(-1)^{n-i}l(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})(u_{i})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{n-i}\Big{(}\sum_{1\leq i<j\leq
n}(-1)^{j}r(x_{1},\cdots,\widehat{x_{i}},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{i})(u_{j})$
$\displaystyle+\sum_{1\leq j<i\leq
n}(-1)^{j+1}r(x_{1},\cdots,\widehat{x_{j}},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i})(u_{j})\Big{)}$
$\displaystyle=[x_{1},\cdots,x_{n}]^{C}+\sum_{i=1}^{n}(-1)^{n-i}\Big{(}l(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})(u_{i})+\sum_{1\leq
i<j\leq
n}(-1)^{j}r(x_{1},\cdots,\widehat{x_{i}},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{i})(u_{j})$
$\displaystyle+\sum_{1\leq j<i\leq
n}(-1)^{j+1}r(x_{1},\cdots,\widehat{x_{j}},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i})(u_{j})$
$\displaystyle=[x_{1},\cdots,x_{n}]^{C}+\sum_{k=1}^{n}(-1)^{n-k}\;\widetilde{\rho}(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n})(u_{k}).$
(3.7)

By Lemma 2.2, $(V,\widetilde{\rho})$ is a representation of the sub-adjacent
$n$-Lie algebra $(A^{c},[\cdot,\cdots,\cdot]^{C})$ on the vector space $V$.
The proof is finished. ∎

If $(l,r)=(L,R)$ is a representation of a $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\})$, then $\widetilde{\rho}=ad$ is the adjoint
representation of the sub-adjacent $n$-Lie algebra
$(A^{c},[\cdot,\cdots,\cdot]^{C})$ on itself.

######  Corollary 3.3.

Let $(V,l,r)$ be a representation of a $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\})$ on $V$. Then the semi-product $n$-pre-Lie
algebra $A\ltimes_{l,r}V$ and $A\ltimes_{\widetilde{\rho}}V$ given by the
representations $(V,l,r)$ and $(V,\widetilde{\rho},0)$ respectively have the
same sub-adjacent $n$-Lie algebra $A^{c}\ltimes_{\widetilde{\rho}}V$ given by
(3.7).

######  Proposition 3.4.

Let $(V,l,r)$ be a representation of a $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\})$ on $V$. Then
$(V^{*},\widetilde{\rho}^{*},-r^{*})$ is a representation of the $n$-pre-Lie
algebra $(A,\\{\cdot,\cdots,\cdot\\})$ on $V^{*}$, which is called the dual
representation of the representation $(V,l,r)$.

###### Proof.

By Proposition 3.2, $(V,\widetilde{\rho})$ is a representation of the sub-
adjacent $n$-Lie algebra $(A^{c},[\cdot,\cdots,\cdot]^{C})$ on $V$. By Lemma
2.3, $(V^{*},\widetilde{\rho}^{*})$ is a representation of the sub-adjacent
$n$-Lie algebra $(A^{c},[\cdot,\cdots,\cdot]^{C})$ on the dual vector space
$V^{*}$. It is straightforward to deduce that other conditions in Definition
3.1 also holds. We leave details to readers. ∎

######  Corollary 3.5.

Let $(V,l,r)$ be a representation of $(A,\\{\cdot,\cdots,\cdot\\})$ on $V$.
Then the semi-product $n$-pre-Lie algebra $A\ltimes_{l^{*},0}V^{*}$ and
$A\ltimes_{\widetilde{\rho}^{*},-r^{*}}V^{*}$ given by the representations
$(V^{*},l^{*},0)$ and $(V^{*},\widetilde{\rho}^{*},-r^{*})$ respectively have
the same sub-adjacent $n$-Lie algebra $A^{c}\ltimes_{l^{*}}V^{*}$, which is
the representation of the $n$-Lie algebra $(A^{c},[\cdot,\cdots,\cdot]^{C})$
and its representation $(V^{*},l^{*})$.

######  Definition 3.2.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra and $(V,l,r)$ be a
representation. A linear operator $T:V\rightarrow A$ is called an
$\mathcal{O}$-operator associated to $(V,l,r)$ if $T$ satisfies

$\\{Tu_{1},\cdots,Tu_{n}\\}=T\Big{(}l(Tu_{1},\cdots,Tu_{n-1})(u_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}r(Tu_{1},\cdots,\widehat{Tu_{i}},\cdots,Tu_{n})(u_{i})\Big{)},$
(3.8)

$\forall u_{i}\in V,\;1\leq i\leq n$. If $(V,l,r)=(A,L,R)$, then $T$ is called
a Rota-Baxter operator on $A$ of weight zero denoted by $P$.

######  Proposition 3.6.

Let $(P_{1},P_{2})$ be a pair of commuting Rota-Baxter operators (of weight
zero) on a $n$-Lie algebra $(A,[\cdot,\cdots,\cdot])$. Then $P_{2}$ is a Rota-
Baxter operator (of weight zero) on the associated $n$-pre-Lie algebra defined
by

$\\{x_{1},\cdots,x_{n}\\}=[P_{1}(x_{1}),\cdots,P_{1}(x_{n-1}),x_{n}],\quad\forall
x_{i}\in A,\;1\leq i\leq n.$

###### Proof.

For any $x_{i}\in A,\;1\leq i\leq n$, we have

$\displaystyle\quad\;\\{P_{2}(x_{1}),\cdots,P_{2}(x_{n})\\}=[P_{1}(P_{2}(x_{1})),\cdots,P_{1}(P_{2}(x_{n-1})),P_{2}(x_{n})]$
$\displaystyle=[P_{2}(P_{1}(x_{1})),\cdots,P_{2}(P_{1}(x_{n-1})),P_{2}(x_{n})]$
$\displaystyle=P_{2}\Big{(}[P_{2}(P_{1}(x_{1})),\cdots,P_{2}(P_{1}(x_{n-1})),x_{n}]$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{n-i}[P_{2}(P_{1}(x_{1})),\cdots,\widehat{P_{2}(P_{1}(x_{i}))},\cdots,P_{2}(P_{1}(x_{n-1})),P_{1}(x_{i})]\Big{)}$
$\displaystyle=P_{2}\Big{(}\\{P_{2}(x_{1}),\cdots,P_{2}(x_{n-1}),x_{n}\\}$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i+1}\\{x_{i},P_{2}(x_{1}),\cdots,\widehat{P_{2}(x_{i})},\cdots,P_{2}(x_{n-1}),P_{2}(x_{n})\\}\Big{)}.$

Hence $P_{2}$ is a Rota-Baxter operator (of weight zero) on the $n$-pre-Lie
algebra $(A,\\{\cdot,\cdots,\cdot\\})$. ∎

If $(l,r)=(L,R)$ is the representation of a $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\})$, then
$(\widetilde{\rho}^{*},-r^{*})=(ad^{*},-R^{*})$ and the corresponding semi-
direct product $n$-Lie algebra is $A^{c}\ltimes_{L^{*}}A^{*}$, which is the
key object when we construct phase space of $n$-Lie algebras in the next
section.

##  4 Constructions on $n$-pre-Lie algebras

In this section, we introduce the notion of phase space of a $n$-Lie algebra
and show that a $n$-Lie algebra has a phase space if and only if it is sub-
adjacent to a $n$-pre-Lie algebra. Moreover, we introduce the notion of Manin
triple of $n$-pre-Lie algebras and show that there is a one-to-one
correspondence between Manin triples of $n$-pre-Lie algebras and perfect phase
spaces of $n$-Lie algebras. Finally, we give a procedure to construct a
$(n+1)$-pre-Lie algebra induced by a $n$-pre-Lie algebra using a trace map.

###  4.1 Symplectic structures and phase spaces of $n$-Lie algebras

######  Definition 4.1.

[18] An element $B\in\wedge^{2}A^{*}$ is a symplectic structure on a $n$-Lie
algebra $(A,[\cdot,\cdots,\cdot])$ if $B$ is a nondegenerate skew-symmetric
bilinear form satisfying the following equality

$B([x_{1},\cdots,x_{n}],y)=-\sum_{i=1}^{n}(-1)^{n-i}B(x_{i},[x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},y]),$
(4.1)

for any $x_{i},y\in A$ and $1\leq i\leq{n}$.

######  Definition 4.2.

[8] Let $(A,[\cdot,\cdots,\cdots])$ be a $n$-Lie algebra and let $B:A\times
A\rightarrow\mathbb{K}$ be a non-degenerate symmetric bilinear form on $A$. If
$B$ satisfies

$B([x_{1},\cdots,x_{n-1},x_{n}],x_{n+1})=-B([x_{1},\cdots,x_{n-1},x_{n+1}],x_{n}),\;\forall
x_{i}\in A,\;1\leq i\leq{n+1}.$ (4.2)

Then $B$ is called an invariant scalar product on $A$ or a metric on $A$, and
$(A,[\cdot,\cdots,\cdot],B)$ is a metric $n$-Lie algebra.

If there exists a metric $B$ and a symplectic structure $\omega$ on a $n$-Lie
algebra $(A,[\cdot,\cdots,\cdot])$, then
$(A,[\cdot,\cdots,\cdot],B,\omega)$ is called a metric symplectic $n$-Lie
algebra.

Let $(A,[\cdot,\cdots,\cdot],B)$ be a metric $n$-Lie algebra, we set

$Der_{B}(A)=\\{D\in Der(A)\;|\;B(Dx,y)+B(x,Dy)=0,\;\forall x,y\in A\\}.$

######  Theorem 4.1.

Let $(A,B)$ be a metric $n$-Lie algebra. Then there exists a symplectic
structure on $A$ if and only if there exists a skew-symmetric invertible
derivation $D\in Der_{B}(A)$.

###### Proof.

Let $(A,B,\omega)$ be a metric symplectic $n$-Lie algebra. Defines
$D:A\rightarrow A$ by

$B(Dx,y)=\omega(x,y),\;\forall x,y\in A.$ (4.3)

It is clear that $D$ is invertible. Next we will check that $D$ is a skew-
symmetric invertible derivation of $(A,[\cdot,\cdots,\cdot],B)$. From
Eq.(4.1), for $x_{i},y\in A,\;1\leq i\leq{n}$

$\displaystyle\quad\;\sum_{i=1}^{n}B([x_{1},\cdots,x_{i-1},D(x_{i}),x_{i+1}\cdots,x_{n}],y)-B(D([x_{1},\cdots,x_{n}]),y)$
$\displaystyle=-\sum_{i=1}^{n}(-1)^{n-i}B([x_{1},\cdots,\widehat{x}_{i},\cdots,x_{n},y],Dx_{i})-B(D([x_{1},\cdots,x_{n}]),y)$
$\displaystyle=-\sum_{i=1}^{n}(-1)^{n-i}\omega(x_{i},[x_{1},\cdots,\widehat{x}_{i},\cdots,x_{n},y])-\omega([x_{1},\cdots,x_{n}],y)=0,$

and by the definition of $D$, we have

$B(Dx,y)+B(x,Dy)=\omega(x,y)+\omega(y,x)=0,$

that is, $D\in Der_{B}(A)$. Conversely, if $D\in Der_{B}(A)$ is invertible.
Defines $\omega:A\times A\rightarrow\mathbb{K}$ by Eq.(4.3). Then by the above
discussion, $\omega$ is non-degenerate, and satisfies Eq.(4.1). The result
follows. ∎

######  Example 4.2.

Let $(A,[\cdot,\cdots,\cdot])$ be a $n$-Lie algebra and $(m>2)$ a positive
integer, we can construct a metric symplectic $n$-Lie algebra. Let $N$ be the
set of all non-negative integers

$\mathbb{K}[t]=\\{f(t)=\sum_{i=0}^{j}a_{i}t^{i}\;|\;a_{i}\in\mathbb{K},\;j\in
N\\},$

be the algebra of polynomials over $\mathbb{K}$. We consider the tensor
product of vector spaces

$A_{m}=A\otimes(t\mathbb{K}[t]\diagup t^{m}\mathbb{K}[t]),$ (4.4)

where $t\mathbb{K}[t]\diagup t^{m}\mathbb{K}[t]$ is the quotient space of
$t\mathbb{K}[t]$ module $t^{m}\mathbb{K}[t]$. Then $A_{m}$ is a nilpotent
$n$-Lie algebra with the following bracket

$[x_{1}\otimes t^{\overline{p_{1}}},\cdots,x_{n}\otimes
t^{\overline{p_{n}}}]=[x_{1},\cdots,x_{n}]_{A}\otimes
t^{\overline{p_{1}+\cdots+p_{n}}},\;x_{1},\cdots,x_{n}\in
A,\;p_{1},\cdots,p_{n}\in N\setminus\\{0\\}.$ (4.5)

Defines an endomorphism $D$ of $A_{m}$ by

$D(x\otimes t^{\overline{p}})=p(x\otimes t^{\overline{p}}),\;\forall x\in
A,\;p=1,\cdots,m-1.$

Then $D$ is an invertible derivation of the $n$-Lie algebra $A_{m}$. Let
$\widetilde{A}_{m}=A_{m}\oplus A^{*}_{m}$, where $A^{*}_{m}$ is the dual space
of $A_{m}$. Defines the bracket

$[x_{1}+\xi_{1},\cdots,x_{n}+\xi_{n}]_{A_{m}\oplus
A^{*}_{m}}=[x_{1},\cdots,x_{n}]_{A_{m}}+\sum_{k=1}^{n}(-1)^{n-k}ad^{*}(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n})(\xi_{k}),$
(4.6)

and the bilinear form

$B(x_{1}+\xi_{1},x_{2}+\xi_{2})=\xi_{1}(x_{2})+\xi_{2}(x_{1}),$ (4.7)

for $x_{1},\cdots,x_{n}\in A_{m},\;\xi_{1},\cdots,\xi_{n}\in A^{*}_{m}$. Then
$(\widetilde{A}_{m},B)$ is a metric $n$-Lie algebra. Define a linear map
$\widetilde{D}:\widetilde{A}_{m}\rightarrow\widetilde{A}_{m}$ by

$\widetilde{D}(x+\xi)=D(x)+D^{*}(\xi),\;\forall x\in A_{m},\;\xi\in
A^{*}_{m},$ (4.8)

where $D^{*}(\xi)=-\xi D$. Then $\widetilde{D}$ is an invertible derivation,
and by a direct computation, we have $\widetilde{D}\in Der_{B}(A^{*}_{m})$.
Hence the metric $n$-Lie algebra $(A^{*}_{m},B)$ admits a symplectic structure
$\omega$ as follows

$\omega(x_{1}+\xi_{1},x_{2}+\xi_{2})=B\Big{(}\widetilde{D}(x_{1}+\xi_{1}),x_{2}+\xi_{2}\Big{)}=-\xi_{1}(D(x_{2}))+\xi_{2}(D(x_{1})).$
(4.9)

######  Proposition 4.3.

[18] Let $(A,[\cdot,\cdots,\cdot],B)$ be a symplectic $n$-Lie algebra. Then
there exists a compatible $n$-pre-Lie algebra structure on $A$ given by

$B(\\{x_{1},\cdots,x_{n}\\},y)=-B(x_{n},[x_{1},\cdots,x_{n-1},y]),$ (4.10)

for any $x_{1},\cdots,x_{n+1}\in A$.

A quadratic $n$-pre-Lie algebra is a $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\})$ equipped with a nondegenerate skew-symmetric
bilinear form $B\in\wedge^{2}A^{*}$ such that the following invariant
condition holds:

$B(\\{x_{1},\cdots,x_{n}\\},x_{n+1})=-B(x_{n},[x_{1},\cdots,x_{n-1},x_{n+1}]^{C}),\;\forall
x_{1},\cdots,x_{n+1}\in A.$ (4.11)

Proposition 4.3 tells us that quadratic $n$-pre-Lie algebras are the
underlying structures of symplectic $n$-Lie algebras.

Let $V$ be a vector space and $V^{*}$ its dual space. Then there exists a
natural non-degenerate skew-symmetric bilinear form $B$ on $T^{*}V=V\oplus
V^{*}$ given by:

$B(x+f,y+g)=\langle f,y\rangle-\langle g,x\rangle,\;\forall x,y\in V,\;f,g\in
V^{*}.$ (4.12)

######  Definition 4.3.

Let $(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$ be a $n$-Lie algebra
and $\mathfrak{h}^{*}$ its dual space.

$\bullet$ If there is a $n$-Lie algebra structure $[\cdot,\cdots,\cdot]$ on
the direct sum vector space
$T^{*}\mathfrak{h}=\mathfrak{h}\oplus\mathfrak{h}^{*}$ such that
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$ is a symplectic
$n$-Lie algebra, where $B$ is given by (4.12), and
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$ and
$(\mathfrak{h}^{*},[\cdot,\cdots,\cdot]_{\mathfrak{h}^{*}})$ are $n$-Lie
subalgebras of $(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot])$,
then the symplectic $n$-Lie algebra
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$ is called a
phase space of the $n$-Lie algebra
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$.

$\bullet$ A phase space
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$ is called
perfect if the following conditions are satisfied:

$[x_{1},\cdots,x_{n-1},\alpha]\in\mathfrak{h}^{*},\quad[\alpha_{1},\cdots,\alpha_{n-1},x]\in\mathfrak{h},\;\forall
x,x_{1},\cdots,x_{n-1}\in\mathfrak{h},\;\alpha,\alpha_{1},\cdots,\alpha_{n-1}\in\mathfrak{h}^{*}.$
(4.13)

Next, we show the important role of $n$-pre-Lie algebras in the study of phase
spaces of $n$-Lie algebras.

######  Theorem 4.4.

A $n$-Lie algebra has a phase space if and only if it is sub-adjacent to a
$n$-pre-Lie algebra.

###### Proof.

($\Leftarrow$) Assume $(A,\\{\cdot,\cdots,\cdot\\})$ is a $n$-pre-Lie algebra.
By Lemma 2.3 and Proposition 2.5, the left multiplication $L$ is a
representation of the sub-adjacent $n$-Lie algebra $A^{c}$ on $A$, $L^{*}$ is
a representation of the sub-adjacent $n$-Lie algebra $A^{c}$ on $A^{*}$, then
we have a $n$-Lie algebra structure $A^{c}\ltimes_{L^{*}}A^{*}$=$(A\oplus
A^{*},[\cdot,\cdots,\cdot]_{L^{*}})$. For all $x_{1},\cdots,x_{n+1}\in A$ and
$\alpha_{1},\cdots,\alpha_{n+1}\in A^{*}$, by Eq.(4.12) we have

$\displaystyle
B\Big{(}[x_{1}+\alpha_{1},\cdots,x_{n}+\alpha_{n}]_{L^{*}},x_{n+1}+\alpha_{n+1}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n}(-1)^{n-i}B\Big{(}x_{i}+\alpha_{i},[x_{1}+\alpha_{1},\cdots,\widehat{x_{i}+\alpha_{i}},\cdots,,x_{n}+\alpha_{n},x_{n+1}+\alpha_{n+1}]_{L^{*}}\Big{)}$
$\displaystyle=B\Big{(}[x_{1},\cdots,x_{n}]^{C}+\sum_{j=1}^{n}(-1)^{n-j}L^{*}(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n})\alpha_{j},x_{n+1}+\alpha_{n+1}\Big{)}$
$\displaystyle+\sum_{i=1}^{n}(-1)^{n-i}B\Big{(}x_{i}+\alpha_{i},[x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1}]^{C}+\sum_{1\leq
j<i\leq{n}}(-1)^{n-j}L^{*}(x_{1},\cdots,\widehat{x_{j}},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1})\alpha_{j}$
$\displaystyle+\sum_{1\leq
i<j\leq{n}}(-1)^{n-j+1}L^{*}(x_{1},\cdots,\widehat{x_{i}},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{n+1})\alpha_{j}+L^{*}(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})\alpha_{n+1}\Big{)}$
$\displaystyle=\langle\sum_{j=1}^{n}(-1)^{n-j}L^{*}(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n})\alpha_{j},x_{n+1}\rangle-\langle\alpha_{n+1},[x_{1},\cdots,x_{n}]^{C}\rangle$
$\displaystyle+\sum_{i=1}^{n}(-1)^{n-i}\Big{(}\langle\alpha_{i},[x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1}]^{C}\rangle-\langle\sum_{1\leq
j<i\leq{n}}(-1)^{n-j}L^{*}(x_{1},\cdots,\widehat{x_{j}},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1})\alpha_{j}$
$\displaystyle+\sum_{1\leq
i<j\leq{n}}(-1)^{n-j+1}L^{*}(x_{1},\cdots,\widehat{x_{i}},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{n+1})\alpha_{j}+L^{*}(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})\alpha_{n+1},x_{i}\rangle\Big{)}.$

And by Eq. (2.7), we have

$\displaystyle
B\Big{(}[x_{1}+\alpha_{1},\cdots,x_{n}+\alpha_{n}]_{L^{*}},x_{n+1}+\alpha_{n+1}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n}(-1)^{n-i}B\Big{(}x_{i}+\alpha_{i},[x_{1}+\alpha_{1},\cdots,\widehat{x_{i}+\alpha_{i}},\cdots,,x_{n}+\alpha_{n},x_{n+1}+\alpha_{n+1}]_{L^{*}}\Big{)}$
$\displaystyle=-\langle\alpha_{j},\sum_{j=1}^{n}(-1)^{n-j}L(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n})x_{n+1}\rangle-\langle\alpha_{n+1},[x_{1},\cdots,x_{n}]^{C}\rangle$
$\displaystyle+\sum_{i=1}^{n}(-1)^{n-i}\Big{(}\langle\alpha_{i},[x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1}]^{C}\rangle+\langle\alpha_{j},\sum_{1\leq
j<i\leq{n}}(-1)^{n-j}L(x_{1},\cdots,\widehat{x_{j}},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1})x_{i}$
$\displaystyle+\sum_{1\leq
i<j\leq{n}}(-1)^{n-j+1}L(x_{1},\cdots,\widehat{x_{i}},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{n+1})x_{i}\rangle+\langle\alpha_{n+1},L(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})x_{i}\rangle\Big{)}$
$\displaystyle=-\langle\alpha_{j},\sum_{j=1}^{n}(-1)^{n-j}\\{x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{n+1}\\}\rangle-\langle\alpha_{n+1},[x_{1},\cdots,x_{n}]^{C}\rangle$
$\displaystyle+\sum_{i=1}^{n}(-1)^{n-i}\Big{(}\langle\alpha_{i},[x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1}]^{C}\rangle+\langle\alpha_{j},\sum_{1\leq
j<i\leq{n}}(-1)^{n-j}\\{x_{1},\cdots,\widehat{x_{j}},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1},x_{i}\\}$
$\displaystyle+\sum_{1\leq
i<j\leq{n}}(-1)^{n-j+1}\\{x_{1},\cdots,\widehat{x_{i}},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{n+1},x_{i}\\}\rangle+\langle\alpha_{n+1},\\{x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i}\\}\rangle\Big{)}=0.$

Since
$\\{x_{\sigma(1)},\cdots,x_{\sigma(n-1)},x_{n}\\}=Sgn(\sigma)\\{x_{1},\cdots,x_{n-1},x_{n}\\}$
and by Eq.(2.11), we deduce that $B$ is a symplectic structure on the semi-
direct product $n$-Lie algebra $A^{c}\ltimes_{L^{*}}A^{*}$. Moreover,
$(A^{c},[\cdot,\cdots,\cdot]^{C})$ is a subalgebra of
$A^{c}\ltimes_{L^{*}}A^{*}$ and $A^{*}$ is an abelian subalgebra of
$A^{c}\ltimes_{L^{*}}A^{*}$. Thus, the symplectic $n$-Lie algebra
$(A^{c}\ltimes_{L^{*}}A^{*},B)$ is a phase space of the sub-adjacent $n$-Lie
algebra $(A^{c},[\cdot,\cdots,\cdot]^{C})$.
($\Rightarrow$) Let
$(T^{*}\mathfrak{h}=\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$
be a phase space of a $n$-Lie algebra
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$. By Proposition 4.3,
there exists a compatible $n$-pre-Lie algebra structure
$\\{\cdot,\cdots,\cdot\\}$ on $T^{*}\mathfrak{h}$ given by Eq. (4.10). Since
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$ is a subalgebra of
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot])$, we have

$\displaystyle
B(\\{x_{1},\cdots,x_{n}\\},x_{n+1})=-B(x_{n},[x_{1},\cdots,x_{n-1},x_{n+1}])$
$\displaystyle-B(x_{n},[x_{1},\cdots,x_{n-1},x_{n+1}]_{\mathfrak{h}})=0,\;\forall
x_{1},\cdots,x_{n+1}\in\mathfrak{h}.$

Thus, $\\{x_{1},\cdots,x_{n}\\}\in\mathfrak{h}$, which implies that
$(\mathfrak{h},\\{\cdot,\cdots,\cdot\\}_{\mathfrak{h}})$ is a subalgebra of
the $n$-pre-Lie algebra
$(\mathfrak{h}\oplus\mathfrak{h}^{*},\\{\cdot,\cdots,\cdot\\})$. Its sub-
adjacent $n$-Lie algebra $(\mathfrak{h}^{c},[\cdot,\cdots,\cdot]^{C})$ is
exactly the original $n$-Lie algebra
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$. ∎

######  Corollary 4.5.

If $(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$ is a phase
space of a $n$-Lie algebra
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$ and
$(\mathfrak{h}\oplus\mathfrak{h}^{*},\\{\cdot,\cdots,\cdot\\})$ is the
associated $n$-pre-Lie algebra. Then both
$(\mathfrak{h},\\{\cdot,\cdots,\cdot\\}_{\mathfrak{h}})$ and
$(\mathfrak{h}^{*},\\{\cdot,\cdots,\cdot\\}_{\mathfrak{h}^{*}})$ are
subalgebras of the $n$-pre-Lie algebra
$(\mathfrak{h}\oplus\mathfrak{h}^{*},\\{\cdot,\cdots,\cdot\\})$.

######  Proposition 4.6.

Let $\rho$ be a representation of
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$ on $\mathfrak{h}$ and
$\rho^{*}$ is its dual representation. If
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$ is a phase space
of a $n$-Lie algebra $(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$ such
that the $n$-Lie algebra
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot])$ is a semi-direct
product $\mathfrak{h}\ltimes_{\rho^{*}}\mathfrak{h}^{*}$, then

$\\{x_{1},\cdots,x_{n}\\}=\rho(x_{1},\cdots,x_{n-1})x_{n},\;\forall
x_{1},\cdots,x_{n}\in\mathfrak{h}.$ (4.14)

defines a $n$-pre-Lie algebra structure on $\mathfrak{h}$.

###### Proof.

For all $x_{1},\cdots,x_{n}\in\mathfrak{h}$, $\alpha\in\mathfrak{h}^{*}$, we
have

$\displaystyle\langle\alpha,\\{x_{1},\cdots,x_{n}\\}\rangle=-B(\\{x_{1},\cdots,x_{n}\\},\alpha)=B(x_{n},[x_{1},\cdots,x_{n-1},\alpha]_{\mathfrak{h}\oplus\mathfrak{h}^{*}})$
$\displaystyle
B(x_{n},\rho^{*}(x_{1},\cdots,x_{n-1})\alpha)=-\langle\rho^{*}(x_{1},\cdots,x_{n-1})\alpha,x_{n}\rangle=\langle\alpha,\rho(x_{1},\cdots,x_{n-1})x_{n}\rangle.$

Therefore, $\\{x_{1},\cdots,x_{n}\\}=\rho(x_{1},\cdots,x_{n-1})x_{n}$. ∎

######  Example 4.7.

Let $(A,\\{\cdot,\cdots,\cdot\\}_{A}$ be a $n$-pre-Lie algebra. Since there is
a semi-direct product $n$-pre-Lie algebra
$(A\ltimes_{L^{*},0}A^{*},\\{\cdot,\cdots,\cdot\\}_{L^{*},0})$ on the phase
space $T^{*}A^{c}=A^{c}\ltimes_{L^{*}}A^{*}$, one can construct a new phase
space $T^{*}A^{c}\ltimes_{L^{*}}(T^{*}A^{c})^{*}$. This process can be
continued indefinitely. Hence, there exist a series of phase spaces
$\\{A_{(m)}\\}_{m\geq 2}:$

$A_{(1)}=A^{c},\;\;\;A_{(2)}=T^{*}A_{(1)}=A^{c}\ltimes_{L^{*}}A^{*},\cdots,\;\;\;A_{(m)}=T^{*}A_{(m-1)},\cdots.$

$A_{(m)}\;(m\geq 2)$ is called the symplectic double of $A_{(m-1)}$.

At the end of this section, we introduce the notion of Manin triple of
$n$-pre-Lie algebras.

######  Definition 4.4.

A Manin triple of $n$-pre-Lie algebras is a triple
$(\mathcal{A};A,A^{\prime})$, where
$\bullet$ $(\mathcal{A},\\{\cdot,\cdots,\cdot\\},B)$ is a quadratic $n$-pre-
Lie algebra,
$\bullet$ both $A$ and $A^{\prime}$ are isotropic subalgebras of
$(\mathcal{A},\\{\cdot,\cdots,\cdot\\})$,
$\bullet$ $\mathcal{A}=A\oplus A^{\prime}$ as vector spaces,
$\bullet$ for all $x,x_{1},\cdots,x_{n-1}\in A$ and
$\alpha,\alpha_{1},\cdots,\alpha_{n-1}\in A^{\prime}$, holds:

$\\{x_{1},\cdots,x_{n-1},\alpha\\}\in
A^{\prime},\;\\{\alpha,x_{1},\cdots,x_{n-1}\\}\in
A^{\prime},\;\\{\alpha_{1},\cdots,\alpha_{n-1},x\\}\in
A,\;\\{x,\alpha_{1},\cdots,\alpha_{n-1}\\}\in A.$ (4.15)

In a Manin triple of $n$-pre-Lie algebras, since the skew-symmetric bilinear
form $B$ is nondegenerate, $A^{\prime}$ can be identified with $A^{*}$ via

$\langle\alpha,x\rangle\triangleq B(\alpha,x),\;\forall x\in A,\;\alpha\in
A^{\prime}.$

Thus, $\mathcal{A}$ is isomorphic to $A\oplus A^{*}$ naturally and the
bilinear form $B$ is exactly given by (4.12). By the invariant condition
(4.11), we can obtain the precise form of the $n$-pre-Lie structure
$\\{\cdot,\cdots,\cdot\\}$ on $A\oplus A^{*}$.

######  Proposition 4.8.

Let $(A\oplus A^{*};A,A^{*})$ be a Manin triple of $n$-pre-Lie algebras, where
the nondegenerate skew-symmetric bilinear form $B$ on the $n$-pre-Lie algebra
is given by (4.12). Then we have

$\displaystyle\\{x_{1},\cdots,x_{n-1},\alpha\\}=\Big{(}L^{*}(x_{1},\cdots,x_{n-1})+\sum_{i=1}^{n-1}(-1)^{i}R^{*}(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i})\Big{)}\alpha,$
(4.16)
$\displaystyle\\{\alpha,x_{1},\cdots,x_{n-1}\\}=-R^{*}(x_{1},\cdots,x_{n-1})\alpha,$
(4.17)
$\displaystyle\\{\alpha_{1},\cdots,\alpha_{n-1},x\\}=\Big{(}\mathcal{L}^{*}(\alpha_{1},\cdots,\alpha_{n-1})+\sum_{i=1}^{n-1}(-1)^{i}\mathcal{R}^{*}(\alpha_{1},\cdots,\widehat{\alpha_{i}},\cdots,\alpha_{n-1},\alpha_{i})\Big{)}x,$
(4.18)
$\displaystyle\\{x,\alpha_{1},\cdots,\alpha_{n-1}\\}=-\mathcal{R}^{*}(\alpha_{1},\cdots,\alpha_{n-1})x.$
(4.19)

###### Proof.

$\forall x_{1},\cdots,x_{n}\in A,\;\alpha\in A^{*}$, we have

$\displaystyle\quad\;\langle\\{x_{1},\cdots,x_{n-1},\alpha\\},x_{n}\rangle=B(\\{x_{1},\cdots,x_{n-1},\alpha\\},x_{n})$
$\displaystyle=-B(\alpha,[x_{1},\cdots,x_{n}]^{C})=-B(\alpha,\sum_{i=1}^{n}(-1)^{n-i}\\{x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i}\\})$
$\displaystyle=-B(\alpha,\Big{(}L(x_{1},\cdots,x_{n-1})+\sum_{i=1}^{n-1}(-1)^{i}R(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i})\Big{)}x_{n})$
$\displaystyle=-\langle\alpha,\Big{(}L(x_{1},\cdots,x_{n-1})+\sum_{i=1}^{n-1}(-1)^{i}R(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i})\Big{)}x_{n}\rangle$
$\displaystyle=\langle\Big{(}L^{*}(x_{1},\cdots,x_{n-1})+\sum_{i=1}^{n-1}(-1)^{i}R^{*}(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i})\Big{)}\alpha,x_{n}\rangle,$

which implies that (4.16) holds. We have

$\displaystyle\quad\;\langle\\{\alpha,x_{1},\cdots,x_{n-1}\\},x_{n}\rangle=B(\\{\alpha,x_{1},\cdots,x_{n-1}\\},x_{n})=-B(x_{n-1},[\alpha,x_{1},\cdots,x_{n-2},x_{n}]^{C})$
$\displaystyle=B(x_{n-1},[x_{n},x_{1},\cdots,x_{n-2},\alpha]^{C})=-B(\\{x_{n},x_{1},\cdots,x_{n-1}\\},\alpha)=B(\alpha,\\{x_{n},x_{1},\cdots,x_{n-1}\\})$
$\displaystyle=\langle\alpha,R(x_{1},\cdots,x_{n-1})x_{n}\rangle=-\langle
R^{*}(x_{1},\cdots,x_{n-1})\alpha,x_{n}\rangle,$

which implies that (4.17) holds. Similarly, we can deduce that (4.18) and
(4.19) hold. ∎

######  Theorem 4.9.

There is a one-to-one correspondence between Manin triples of $n$-pre-Lie
algebras and perfect phase spaces of $n$-Lie algebras. More precisely, if
$(A\oplus A^{*};A,A^{*})$ is a Manin triple of $n$-pre-Lie algebras, then
$(A\oplus A^{*},[\cdot,\cdots,\cdot]^{C},B)$ is a symplectic $n$-Lie algebra,
where $B$ is given by (4.12). Conversely, if
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$ is a perfect
phase space of a $n$-Lie algebra
$(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$, then
$(\mathfrak{h}\oplus\mathfrak{h}^{*};\mathfrak{h},\mathfrak{h}^{*})$ is a
Manin triple of $n$-pre-Lie algebras, where the $n$-pre-Lie algebra structure
on $\mathfrak{h}\oplus\mathfrak{h}^{*}$ is given by (4.10).

###### Proof.

($\Rightarrow$) Let $(A\oplus A^{*};A,A^{*})$ be a Manin triple of $n$-pre-Lie
algebras. Denote by $\\{\cdot,\cdots,\cdot\\}_{A}$ and
$\\{\cdot,\cdots,\cdot\\}_{A^{*}}$ the $n$-pre-Lie algebra structure on $A$
and $A^{*}$ respectively, and denote by $[\cdot,\cdots,\cdot]_{A}$ and
$[\cdot,\cdots,\cdot]_{A^{*}}$ the corresponding sub-adjacent $n$-Lie algebra
structure on $A$ and $A^{*}$ respectively. By Proposition 4.8, it is
straightforward to deduce that the corresponding $n$-Lie algebra structure
$[\cdot,\cdots,\cdot]^{C}$ on $A\oplus A^{*}$ is given by

$\displaystyle[x_{1}+\alpha_{1},\cdots,x_{n}+\alpha_{n}]^{C}=[x_{1},\cdots,x_{n}]_{A}+\sum_{i=1}^{n}(-1)^{n-i}\mathcal{L}^{*}(\alpha_{1},\cdots,\widehat{\alpha_{i}},\cdots,\alpha_{n})(x_{i})$
$\displaystyle+[\alpha_{1},\cdots,\alpha_{n}]_{{A}^{*}}+\sum_{i=1}^{n}(-1)^{n-i}L^{*}(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})(\alpha_{i}).$

For all $x_{1},\cdots,x_{n+1}\in A$ and $\alpha_{1},\cdots,\alpha_{n+1}\in
A^{*}$, similarly to the Proof of Theorem 4.4, we have

$\displaystyle
B\Big{(}[x_{1}+\alpha_{1},\cdots,x_{n}+\alpha_{n}]^{C},x_{n+1}+\alpha_{n+1}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n}(-1)^{n-i}B\Big{(}x_{i}+\alpha_{i},[x_{1}+\alpha_{1},\cdots,\widehat{x_{i}+\alpha_{i}},\cdots,,x_{n}+\alpha_{n},x_{n+1}+\alpha_{n+1}]^{C}\Big{)}=0.$

Since

$\\{x_{1},\cdots,x_{n}\\}_{A}=Sgn(\sigma)\\{x_{\sigma(1)},\cdots,x_{\sigma(n-1)},x_{n}\\},\;\forall\sigma\in
S_{n},\;and\;\sigma(n)=n$

and

$\\{\alpha_{1},\cdots,\alpha_{n}\\}_{A^{*}}=Sgn(\sigma)\\{\alpha_{\sigma(1)},\cdots,\alpha_{\sigma(n-1)},\alpha_{n}\\},\;\forall\sigma\in
S_{n},\;and\;\sigma(n)=n,$

then we deduce that $B$ is a symplectic structure on the $n$-Lie algebra
$(A\oplus A^{*},[\cdot,\cdots,\cdot]^{C})$. Therefore, it is a phase space.
($\Leftarrow$) Conversely, let
$(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdots,\cdot],B)$ be a phase space
of the $n$-Lie algebra $(\mathfrak{h},[\cdot,\cdots,\cdot]_{\mathfrak{h}})$.
By Proposition 4.3, there exists a $n$-pre-Lie algebra structure
$\\{\cdot,\cdots,\cdot\\}$ on $\mathfrak{h}\oplus\mathfrak{h}^{*}$ given by
(4.10) such that
$(\mathfrak{h}\oplus\mathfrak{h}^{*},\\{\cdot,\cdots,\cdot\\},B)$ is a
quadratic $n$-pre-Lie algebra. By Corollary 4.5,
$(\mathfrak{h},\\{\cdot,\cdots,\cdot\\}_{\mathfrak{h}})$ and
$(\mathfrak{h}^{*},\\{\cdot,\cdots,\cdot\\}_{\mathfrak{h}^{*}})$ are $n$-pre-
Lie subalgebras of
$(\mathfrak{h}\oplus\mathfrak{h}^{*},\\{\cdot,\cdots,\cdot\\})$. It is obvious
that both $\mathfrak{h}$ and $\mathfrak{h}^{*}$ are isotropic. Thus, we only
need to show that (4.15) holds. By (4.13), for all
$x_{1},\cdots,x_{n-1}\in\mathfrak{h}$ and
$\alpha_{1},\cdots,\alpha_{n-1}\in\mathfrak{h}^{*}$, we have

$B(\\{x_{1},\cdots,x_{n-1},\alpha_{1}\\})=-B(\alpha_{1},[x_{1},\cdots,x_{n-1},\alpha_{2}]^{C})=0,$

which implies that $\\{x_{1},\cdots,x_{n-1},\alpha_{1}\\}\in\mathfrak{h}^{*}$.
Similarly, we can show that the other conditions in (4.15) also hold. The
proof is finished. ∎

###  4.2 $(n+1)$-pre-Lie algebras induced by $n$-pre-Lie algebras

In this subsection we show how to derive $(n+1)$-pre-Lie algebras from
$n$-pre-Lie algebras and a trace map, extending to pre-Lie structures the
construction on Lie structures provided in [3]. Let us start by defining a map
which is skew-symmetric on the first $n$ variables that will be used to induce
the higher order algebra.

######  Definition 4.5.

Let $\phi:\otimes^{n}A\rightarrow A$ be a $n$-linear map and let $\tau$ be a
map from $A$ to $\mathbb{K}$.
Define $\phi_{\tau}:\otimes^{n+1}A\rightarrow A$ by

$\phi_{\tau}(x_{1},\cdots,x_{n},x_{n+1})=\sum_{k=1}^{n}(-1)^{k+1}\tau(x_{k})\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},x_{n+1})$
(4.20)

where the hat over $\widehat{x_{k}}$ on the right hand side means that $x_{k}$
is excluded, that is $\phi$ is calculated on
$(x_{1},\cdots,x_{k-1},x_{k+1},\cdots,x_{n+1})$.

######  Definition 4.6.

Let $\phi:\otimes^{n}A\rightarrow A$ be a $n$-linear map. A linear map
$\tau:A\rightarrow\mathbb{K}$ is called a $\phi$-trace if it satisfies for all
$x_{1},\cdots,x_{n}\in A$

$\tau(\phi(x_{1},\cdots,x_{n}))=0,\forall x_{1},\cdots,x_{n}\in A.$

######  Lemma 4.10.

Let $\phi:(\wedge^{n-1}A)\otimes A\rightarrow A$ be a n-linear map and $\tau$
a linear map $A\rightarrow\mathbb{K}$. Then $\phi_{\tau}$ is a $(n+1)$-linear
map such that is skew-symmetric on the first $n$ variables. Furthermore, if
$\tau$ is a $\phi$-trace then $\tau$ is a $\phi_{\tau}$-trace.

###### Proof.

The $(n+1)$-linearity property of $\phi_{\tau}$ follows from $n$-linearity of
$\phi$ and linearity of $\tau$ as it is a linear combination of $(n+1)$-linear
maps. To prove the skew-symmetry on the first $n$ variables one simply notes
that

$\phi_{\tau}(x_{1},\cdots,x_{n},x_{n+1})=\frac{1}{(n-1)!}\sum_{\sigma\in
S_{n}}Sgn(\sigma)\tau(x_{\sigma(1)})\phi(x_{\sigma(2)},\cdots,x_{\sigma(n)},x_{n+1}),$

which is clearly skew-symmetric on the first $n$ variables. Since each term in
$\phi_{\tau}$ is proportional to $\phi$ and since $\tau$ is linear and a
$\phi$-trace, $\tau$ will also be a $\phi_{\tau}$-trace. ∎

######  Theorem 4.11.

Suppose that $(A,\phi)$ is a $n$-pre-Lie algebra and
$\tau:A\rightarrow\mathbb{K}$ is a $\phi$-trace, then $(A,\phi_{\tau})$ is a
$(n+1)$-pre-Lie algebra. We shall say that $(A,\phi_{\tau})$ is induced by
$(A,\phi)$.

###### Proof.

Since $\phi_{\tau}$ is skew-symmetric on the first $n$ variables and
multilinear by Lemma 4.10, one only has to check that the two identity of
$(n+1)$-pre-Lie algebra $(A,\phi_{\tau})$ are fulfilled. The first identity is
written as

$\displaystyle\phi_{\tau}\Big{(}x_{1},\cdots,x_{n},\phi_{\tau}(y_{1},\cdots,y_{n+1})\Big{)}-\phi_{\tau}\Big{(}y_{1},\cdots,y_{n},\phi_{\tau}(x_{1},\cdots,x_{n},y_{n+1})\Big{)}$
$\displaystyle-\sum_{i=1}^{n}\phi_{\tau}\Big{(}y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n},y_{i}]^{C},y_{i+1},\cdots,y_{n},y_{n+1}\Big{)},$

where
$[x_{1},\cdots,x_{n+1}]^{C}=\sum_{i=1}^{n+1}(-1)^{(n+1)-i}\phi_{\tau}(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n+1},x_{i})$.
Let us write the left-hand-side of this equation as $B-C-D$. Furthermore, we
expand $B$ into terms $B_{kl}$ and $C$ into terms $C_{lk}$ such that

$\displaystyle
B_{kl}=(-1)^{k+l}\tau(x_{k})\tau(y_{l})\phi\Big{(}x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},\phi(y_{1},\cdots,\widehat{y_{l}},\cdots,y_{n},y_{n+1})\Big{)},$
$\displaystyle B=\sum_{k=1}^{n}\sum_{l=1}^{n}B_{kl},$ $\displaystyle
C_{lk}=(-1)^{l+k}\tau(y_{l})\tau(x_{k})\phi\Big{(}y_{1},\cdots,\widehat{y_{l}},\cdots,y_{n},\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},y_{n+1})\Big{)},$
$\displaystyle C=\sum_{l=1}^{n}\sum_{k=1}^{n}C_{lk}.$

Taking into account that $\tau$ is a $\phi-$trace form, we expand $D$ as

$\displaystyle
D=\sum_{i=1}^{n}\phi_{\tau}\Big{(}y_{1},\cdots,[x_{1},\cdots,x_{n},y_{i}]^{C},\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle=\sum_{i=1}^{n}\phi_{\tau}\Big{(}y_{1},\cdots,\sum_{j=1}^{n}(-1)^{(n+1)-j}\phi_{\tau}(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n},y_{i},x_{j})+\phi_{\tau}(x_{1},\cdots,x_{n},y_{i}),\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle=\sum_{i=1}^{n}\phi_{\tau}\Big{(}y_{1},\cdots,\phi_{\tau}(x_{1},\cdots,x_{n},y_{i}),\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle+\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{(n+1)-j}\phi_{\tau}\Big{(}y_{1},\cdots,\phi_{\tau}(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n},y_{i},x_{j}),\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle=D^{(1)}+D^{(2)},$

where

$\displaystyle
D^{(1)}=\sum_{i=1}^{n}\sum_{k=1}^{n}(-1)^{k+1}\tau(x_{k})\phi_{\tau}\Big{(}y_{1},\cdots,\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},y_{i}),\cdots,y_{n},y_{n+1}\Big{)},$
$\displaystyle
D^{(2)}=\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{(n+1)-j}\sum_{k=1,k\neq
j}^{n}(-1)^{k+1}\tau(x_{k})\phi_{\tau}\Big{(}y_{1},\cdots,\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,\widehat{x_{j}},\cdots,x_{n},y_{i},x_{j}),\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle+(-1)^{n+1}\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{(n+1)-j}\tau(y_{i})\phi_{\tau}\Big{(}y_{1},\cdots,\phi(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{j}),\cdots,y_{n},y_{n+1}\Big{)}.$

Let us now consider the expansion of $D^{(1)}$ via $D^{(1)}_{kl}$

$\displaystyle
D^{(1)}_{kl}=\sum_{i=1}^{l-1}(-1)^{k+l}\tau(x_{k})\tau(y_{l})\phi\Big{(}y_{1},\cdots,\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},y_{i}),\cdots,\widehat{y_{l}},\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle+\sum_{i=l+1}^{n}(-1)^{k+l}\tau(x_{k})\tau(y_{l})\phi\Big{(}y_{1},\cdots,\widehat{y_{l}},\cdots,\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},y_{i}),\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle
D^{(1)}=\sum_{i=1}^{n}\sum_{k=1}^{n}\sum_{l=1}^{n}D^{(1)}_{kl},$

and we expand $D^{(2)}$ as

$\displaystyle
D^{(2)}=\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{(n+1)-j}\sum_{k=1,k\neq
j}^{n}\sum_{l=1}^{n}$
$\displaystyle\times\Big{(}\sum_{i=1}^{l-1}(-1)^{k+l}\tau(x_{k})\tau(y_{l})\phi\Big{(}y_{1},\cdots,\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,\widehat{x_{j}},\cdots,x_{n},y_{i},x_{j}),\cdots,\widehat{y_{l}},\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle+\sum_{i=l+1}^{n}(-1)^{k+l}\tau(x_{k})\tau(y_{l})\phi\Big{(}y_{1},\cdots,\widehat{y_{l}},\cdots,\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,\widehat{x_{j}},\cdots,x_{n},y_{i},x_{j}),\cdots,y_{n},y_{n+1}\Big{)}\Big{)}$
$\displaystyle+\sum_{i=1}^{n}\sum_{j=1}^{n}(-1)^{(n+1)-j}\sum_{l=1}^{n}\Big{(}\sum_{i=1}^{l-1}(-1)^{n+l}\tau(y_{i})\tau(y_{l})\phi\Big{(}y_{1},\cdots,\phi(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{j}),\cdots,\widehat{y_{l}},\cdots,y_{n},y_{n+1}\Big{)}$
$\displaystyle+\sum_{i=l+1}^{n}(-1)^{n+l}\tau(y_{i})\tau(y_{l})\phi\Big{(}y_{1},\cdots,\widehat{y_{l}},\cdots,\phi(x_{1},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{j}),\cdots,y_{n},y_{n+1}\Big{)}\Big{)}.$

By a direct simplification and using the two identity of $n$-pre-Lie algebras
$(A,\phi)$, we get that the first identity of $(n+1)$-pre-Lie algebras is
satisfied for $\phi_{\tau}$. Also, by the same way we can prove that
$\phi_{\tau}$ satisfies the second identity of $(n+1)$-pre-Lie algebras. ∎

In particular, one may derive from a pre-Lie algebra a 3-pre-Lie algebra using
a trace map. Recall that a pre-Lie algebra is a pair $(A,\circ)$, where $A$ is
a vector space and $\circ:A\times A\rightarrow A$ is a bilinear multiplication
satisfying, for all $x,y,z\in A$,

$(x\circ y)\circ z-x\circ(y\circ z)=(y\circ x)\circ z-y\circ(x\circ z).$
(4.21)

It is obvious that all associative algebras are pre-Lie algebras and for a
pre-Lie algebra $(A,\circ)$, the commutator $[x,y]^{C}=x\circ y-y\circ x$,
defines a Lie algebra. By Theorem 4.11, suppose that $(A,\circ)$ is a pre-Lie
algebra and $\tau:A\rightarrow\mathbb{K}$ is a linear map, such that $\tau$ is
a $\circ$-trace. Then $(A,\\{\cdot,\cdot,\cdot\\}_{\tau})$ is a $3$-pre-Lie
algebra, and we say that it is induced by $(A,\circ)$ where
$\\{\cdot,\cdot,\cdot\\}_{\tau}:(\wedge^{2}A)\otimes A\rightarrow A$ is
defined by

$\\{x,y,z\\}_{\tau}=\tau(x)y\circ z-\tau(y)x\circ z,\;\forall x,y,z\in A.$
(4.22)

######  Example 4.12.

Let $(A,\circ)$ be a three-dimensional pre-Lie algebra with a basis
$\\{e_{1},e_{2},e_{3}\\}$ whose nonzero brackets are given as follows:

$e_{3}\circ e_{2}=e_{2},\quad e_{3}\circ e_{3}=-e_{3}.$

We define a trace map as

$\tau(e_{1})=a,\;\tau(e_{2})=0,\;\tau(e_{3})=0,$

for any $a\in\mathbb{K}$. Then, according to Theorem 4.11, we obtain a three-
dimensional $3$-pre-Lie algebra with a basis $\\{e_{1},e_{2},e_{3}\\}$ whose
nonzero brackets are given as follows:

$\\{e_{1},e_{3},e_{2}\\}=ae_{2},\quad\\{e_{1},e_{3},e_{3}\\}=-ae_{3}.$

Since $\tau$ is also a $\phi_{\tau}$-trace, one can repeat the procedure in
Theorem 4.11 to induce a $(n+2)$-pre-Lie algebra from an $n$-pre-Lie algebra.
However, the result is an abelian algebra.

######  Proposition 4.13.

Let $\mathcal{A}=(A,\phi)$ be a $n$-pre-Lie algebra. Let
$\mathcal{A}^{{}^{\prime}}$ be any $(n+1)$-pre-Lie algebra induced by
$\mathcal{A}$ via the $\phi$-trace $\tau$. If
$\mathcal{A}^{{}^{\prime\prime}}$ is a $(n+2)$-pre-Lie algebra induced by
$\mathcal{A}^{{}^{\prime}}$ using the same $\tau$ again, then
$\mathcal{A}^{{}^{\prime\prime}}$ is abelian.

###### Proof.

By the definition of $\phi_{\tau}$ , the bracket on the algebra
$\mathcal{A}^{{}^{\prime\prime}}$ can be written as

$(\phi_{\tau})_{\tau}=\sum_{k=1}^{k+1}(-1)^{k+1}\tau(x_{k})\phi_{\tau}(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n+1},x_{n+2})$

Expanding the bracket $\phi_{\tau}$ , there will be, for every choice of
integers $k<l$, two terms which are proportional to $\tau(x_{k})\tau(x_{l})$.
For $l=1,\cdots,n+1$, their sum becomes

$\tau(x_{k})\tau(x_{l})\phi(x_{1},\cdots,\widehat{x_{k}},\cdots,\widehat{x_{l}},\cdots,x_{n+1},x_{n+2})\Big{(}(-1)^{k+l}+(-1)^{k+l-1}\Big{)}=0.$

Hence $(\phi_{\tau})_{\tau}(x_{1},\cdots,x_{n+2})=0$, for
$x_{1},\cdots,x_{n+2}\in A$. ∎

######  Remark 4.1.

Note that one might choose a different $\phi_{\tau}$-trace when repeating the
construction, and this can lead to non-abelian algebras.

######  Proposition 4.14.

Let $(V,l,r)$ be a representation of a $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\})$ and $\tau$ be a trace map of $A$. Then
$(V,l_{\tau},r_{\tau})$ is a representation of the $(n+1)$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\}_{\tau})$ where $l_{\tau}:\wedge^{n}A\rightarrow
gl(V)$, $r_{\tau}:\otimes^{n}A\rightarrow gl(V)$ are defined by

$\displaystyle
l_{\tau}(x_{1},\cdots,x_{n})=\sum_{k=1}^{n}(-1)^{k+1}\tau(x_{k})l(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n}),$
(4.23) $\displaystyle
r_{\tau}(x_{1},\cdots,x_{n})=\sum_{k=1}^{n-1}(-1)^{k}\tau(x_{k})r(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n-1},x_{n}),\quad\forall
x_{i}\in A.$ (4.24)

###### Proof.

Let $\tau$ be a trace map of $(A,\\{\cdot,\cdots,\cdot\\})$ and $(V,l,r)$ be a
representation of $A$. We define $\overline{\tau}:A\oplus
V\rightarrow\mathbb{K}$ by

$\overline{\tau}(x+u)=\tau(x),\;\forall x\in A,\;\forall u\in V.$

Then $\overline{\tau}$ is a trace map on the semidirect product $n$-pre-Lie
algebra $A\ltimes_{(l,r)}V$. Then by (4.20) , there is a $(n+1)$-pre-Lie
algebra structure $\\{\cdot,\cdots,\cdot\\}_{\overline{\tau}}$ on the vector
space $A\oplus V$ given by

$\displaystyle\\{x_{1}+u_{1},\cdots,x_{n+1}+u_{n+1}\\}_{\overline{\tau}}=\sum_{k=1}^{n}(-1)^{k+1}\overline{\tau}(x_{k}+u_{k})\\{x_{1}+u_{1},\cdots,\widehat{x_{k}+u_{k}},\cdots,x_{n}+u_{n},x_{n+1}+u_{n+1}\\}_{A\oplus
V}$
$\displaystyle=\sum_{k=1}^{n}(-1)^{k+1}\tau(x_{k})\Big{(}\\{x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},x_{n+1}\\}+l(x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n})u_{n+1}$
$\displaystyle+\sum_{1\leq k<j\leq
n}(-1)^{j}r(x_{1},\cdots,\widehat{x_{k}},\cdots,\widehat{x_{j}},\cdots,x_{n},x_{n+1})u_{j}+\sum_{1\leq
j<k\leq
n}(-1)^{j+1}r(x_{1},\cdots,\widehat{x_{j}},\cdots,\widehat{x_{k}},\cdots,x_{n},x_{n+1})u_{j}$
$\displaystyle=\\{x_{1},\cdots,x_{n+1}\\}_{\tau}+l_{\tau}(x_{1},\cdots,x_{n})u_{n+1}+\sum_{i=1}^{n}(-1)^{i+1}r_{\tau}(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{n+1})(u_{i}).$

Since a semidirect product of $(n+1)$-pre-Lie algebras is equivalent to
representations of a $(n+1)$-pre-Lie algebras. Thus, we deduce that
$(V,l_{\tau},r_{\tau})$ is a representation of the $(n+1)$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\}_{\tau})$. ∎

######  Lemma 4.15.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra and $\tau$ be a
trace map. Let $D:A\rightarrow A$ be a $n$-pre-Lie derivation, then $\tau\circ
D$ is a trace map.

###### Proof.

For any $x_{i}\in A$, we have

$\tau(D(\\{x_{1},\cdots,x_{n}\\}))=\tau\Big{(}\sum_{i=1}^{n}\\{x_{1},\cdots,D(x_{i}),\cdots,x_{n}\\}\Big{)}=\sum_{i=1}^{n}\tau(\\{x_{1},\cdots,D(x_{i}),\cdots,x_{n}\\})=0.$

∎

######  Theorem 4.16.

Let $D:A\rightarrow A$ be a derivation of a $n$-pre-Lie algebra $A$, then $D$
is a derivation of the induced $(n+1)$-pre-Lie algebra if and only if:

$\\{x_{1},\cdots,x_{n+1}\\}_{\tau\circ D}=0,\quad\forall x_{i}\in A.$

###### Proof.

Let $D$ be a derivation of $A$ and $x,y,z\in A$, we obtain

$\displaystyle
D(\\{x_{1},\cdots,x_{n+1}\\}_{\tau})=D(\sum_{k=1}^{n}(-1)^{k+1}\tau(x_{k})\\{x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},x_{n+1}\\})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\;\;=\sum_{k=1}^{n}(-1)^{k+1}\tau(x_{k})\Big{(}\sum_{i=1,i\neq
k}^{n+1}\\{x_{1},\cdots,\widehat{x_{k}},\cdots,Dx_{i},\cdots,x_{n},x_{n+1}\\}\Big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\;\;\quad+\sum_{k=1}^{n}(-1)^{k+1}(\tau\circ
D)(x_{k})\\{x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n},x_{n+1}\\}-\\{x_{1},\cdots,x_{n+1}\\}_{\tau\circ
D}$
$\displaystyle\quad\quad\quad\quad\quad\quad\;\;\quad=\sum_{k=1}^{n+1}\\{x_{1},\cdots,Dx_{k},\cdots,x_{n+1}\\}_{\tau}-\\{x_{1},\cdots,x_{n+1}\\}_{\tau\circ
D}.$

∎

######  Remark 4.2.

This construction fits also with sub-adjacent $n$-Lie algebras. The sub-
adjacent $(n+1)$-Lie algebra of the induced $(n+1)$-pre-Lie algebra is the
induced $(n+1)$-Lie algebra of the sub-adjacent $n$-Lie algebra of the
$n$-pre-Lie algebra with respect to the same trace map.

##  5 $n$-L-dendriform algebras

In this section, we introduce the notion of a $n$-L-dendriform algebra which
is exactly the $n$-ary version of a L-dendriform algebra. We provide some
construction results in terms of $\mathcal{O}$-operator and symplectic
structure.

######  Definition 5.1.

Let $A$ be a vector space with two $n$-linear maps
$\nwarrow:(\wedge^{n-1}A)\otimes A\rightarrow A$ and
$\nearrow:A\otimes(\wedge^{n-2}A)\otimes A\rightarrow A$. The tuple
$(A,\nwarrow,\nearrow)$ is called a $n$-L-dendriform algebra if the following
identities hold

$\displaystyle\nwarrow(x_{1},\cdots,x_{n-1},\nwarrow(y_{1},\cdots,y_{n}))-\nwarrow(y_{1},\cdots,y_{n-1},\nwarrow(x_{1},\cdots,x_{n-1},y_{n}))$
$\displaystyle\hskip
56.9055pt=\sum_{i=1}^{n-1}\nwarrow(y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}]^{C},y_{i+1},\cdots,y_{n}),$
(5.1)
$\displaystyle\nwarrow([x_{1},\cdots,x_{n}]^{C},y_{1},\cdots,y_{n-1})=\sum_{i=1}^{n}(-1)^{n-i}\nwarrow(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},\nwarrow(x_{i},y_{1},\cdots,y_{n-1})),$
(5.2)
$\displaystyle\nwarrow(x_{1},\cdots,x_{n-1},\nearrow(y_{n},y_{1},\cdots,y_{n-1}))-\nearrow(y_{n},y_{1},\cdots,y_{n-2},\\{x_{1},\cdots,x_{n-1},y_{n-1}\\}^{h})$
$\displaystyle\hskip
56.9055pt=\nearrow(\\{x_{1},\cdots,x_{n-1},y_{n}\\}^{v},y_{1},\cdots,y_{n-1})$
$\displaystyle\hskip
56.9055pt+\sum_{i=1}^{n-2}\nearrow(y_{n},y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}]^{C},y_{i+1},\cdots,y_{n-1}),$
(5.3)
$\displaystyle\nearrow(y_{n-1},[x_{1},\cdots,x_{n}]^{C},y_{1},\cdots,y_{n-2})$
$\displaystyle\hskip
56.9055pt=\sum_{i=1}^{n}(-1)^{n-i}\nwarrow(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},\nearrow(y_{n-1},x_{i},y_{1},\cdots,y_{n-2})),$
(5.4)
$\displaystyle\nearrow(x_{n-1},x_{1},\cdots,x_{n-2},\\{y_{1},\cdots,y_{n}\\}^{h})-\nwarrow(y_{1},\cdots,y_{n-1},\nearrow(x_{n-1},x_{1},\cdots,x_{n-2},y_{n}))$
$\displaystyle\hskip
56.9055pt=\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(\\{x_{1},\cdots,x_{n-2},y_{i},x_{n-1}\\}^{v},y_{1},\cdots,\widehat{y_{i}},\cdots,y_{n}),$
(5.5)
$\displaystyle\nearrow(\\{x_{1},\cdots,x_{n-1},y_{n}\\}^{v},y_{1},\cdots,y_{n-1})-\nwarrow(x_{1},\cdots,x_{n-1},\nearrow(y_{n},y_{1},\cdots,y_{n-1}))$
$\displaystyle\hskip
56.9055pt=\sum_{i=1}^{n-1}(-1)^{i}\nearrow(y_{n},x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},\\{x_{i},y_{1},\cdots,y_{n-1}\\}^{h}),$
(5.6)

for all $x_{i},y_{i}\in A$, $1\leq i\leq n$, where

$\displaystyle\\{x_{1},\cdots,x_{n}\\}^{h}=\nwarrow(x_{1},\cdots,x_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(x_{i},x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n}),$
(5.7)
$\displaystyle\\{x_{1},\cdots,x_{n}\\}^{v}=\nwarrow(x_{1},\cdots,x_{n})+\sum_{i=1}^{n-1}(-1)^{i}\nearrow(x_{n},x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i}),$
(5.8)
$\displaystyle[x_{1},\cdots,x_{n}]^{C}=\sum_{i=1}^{n}(-1)^{n-i}\\{x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i}\\}^{h}=\sum_{i=1}^{n}(-1)^{n-i}\\{x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n},x_{i}\\}^{v},$
(5.9)

for any $x_{i}\in A$, $1\leq i\leq n$.

######  Remark 5.1.

Let $(A,\nwarrow,\nearrow)$ be a $n$-L-dendriform algebra. If $\nearrow=0$,
then $(A,\nwarrow)$ is a $n$-pre-Lie algebra.

######  Proposition 5.1.

Let $(A,\nwarrow,\nearrow)$ be a $n$-L-dendriform algebra.

  1. 1.

The bracket given in (5.7) defines a $n$-pre-Lie algebra structure on $A$
which is called the associated horizontal $n$-pre-Lie algebra of
$(A,\nwarrow,\nearrow)$ and $(A,\nwarrow,\nearrow)$ is also called a
compatible $n$-L-dendriform algebra structure on the $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\}^{h})$.

  2. 2.

The bracket given in (5.8) defines a $n$-pre-Lie algebra structure on $A$
which is called the associated vertical $n$-pre-Lie algebra of
$(A,\nwarrow,\nearrow)$ and $(A,\nwarrow,\nearrow)$ is also called a
compatible $n$-L-dendriform algebra structure on the $n$-pre-Lie algebra
$(A,\\{\cdot,\cdots,\cdot\\}^{v})$.

###### Proof.

We will just prove item 1. Note, first that by the skew-symmetry of the first
$n-1$ variables of $\nwarrow$ and since
$\nearrow(x_{1},x_{\sigma(2)},\cdots,x_{\sigma(n-1)},x_{n})=Sgn(\sigma)\nearrow(x_{1},x_{2},\cdots,x_{n-1},x_{n}),\;\forall\sigma\in
S_{n}$, with $\sigma(1)=1$ and $\sigma(n)=n$, the induced $n$-bracket
$\\{\cdot,\cdots,\cdot\\}^{h}$ given by Eq. (5.7) is skew-symmetric on the
first $n-1$ variables.
Let $x_{i},y_{i}\in A,\ 1\leq i\leq n$. Then

$\displaystyle\\{x_{1},\cdots,x_{n-1},\\{y_{1},\cdots,y_{n}\\}^{h}\\}^{h}-\\{y_{1},\cdots,y_{n-1},\\{x_{1},\cdots,x_{n-1},y_{n}\\}^{h}\\}^{h}$
$\displaystyle-\sum_{i=1}^{n-1}\\{y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}]^{C},y_{i+1},\cdots,y_{n}\\}^{h}$
$\displaystyle=\nwarrow(x_{1},\cdots,x_{n-1},\nwarrow(y_{1},\cdots,y_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(y_{i},y_{1},\cdots,\widehat{y_{i}},\cdots,y_{n-1},y_{n}))$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(x_{i},x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},\\{y_{1},\cdots,y_{n}\\}^{h})$
$\displaystyle-\nwarrow(y_{1},\cdots,y_{n-1},\nwarrow(x_{1},\cdots,x_{n-1},y_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(x_{i},x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},y_{n}))$
$\displaystyle-\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(y_{i},y_{1},\cdots,\widehat{y_{i}},\cdots,y_{n-1},\\{x_{1},\cdots,x_{n-1},y_{n}\\}^{h})$
$\displaystyle-\sum_{i=1}^{n-1}\Big{(}\nwarrow(y_{1},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}]^{C},y_{i+1},\cdots,y_{n-1},y_{n})$
$\displaystyle+\sum_{j=1,j\neq
i}^{n-1}(-1)^{j+1}\nearrow(y_{j},y_{1},\cdots,\widehat{y_{j}},\cdots,y_{i-1},[x_{1},\cdots,x_{n-1},y_{i}]^{C},y_{i+1},\cdots,y_{n-1},y_{n})\Big{)}$
$\displaystyle-\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(\sum_{k=1}^{n-1}\\{x_{1},\cdots,\widehat{x_{k}},\cdots,x_{n-1},y_{i},x_{k}\\}^{v}+\\{x_{1},\cdots,x_{n-1},y_{i}\\}^{v},y_{1},\cdots,\widehat{y_{i}},\cdots,y_{n-1},y_{n}).$

From identities (5.1),(5.1) and (5.1), we obtain immediately that (2.9) hold.
On the other hand, by the same way we can prove the second identity (2.10) of
$n$-pre-Lie algebra structure. ∎

######  Corollary 5.2.

Let $(A,\nwarrow,\nearrow)$ be a $n$-L-dendriform algebra. Then the bracket
defined in (5.9) defines a $n$-Lie algebra structure on $A$ which is called
the associated $n$-Lie algebra of $(A,\nwarrow,\nearrow)$.

The following Proposition is obvious.

######  Proposition 5.3.

Let $(A,\nwarrow,\nearrow)$ be a $n$-L-dendriform algebra. Define
$L_{\nwarrow},R_{\nearrow}:\otimes^{n-1}A\to gl(A)$ and
$\rho:\wedge^{n-1}\rightarrow gl(A)$ by

$L_{\nwarrow}(x_{1},\cdots,x_{n-1})x_{n}=\nwarrow(x_{1},\cdots,x_{n-1},x_{n}),\quad
R_{\nearrow}(x_{1},\cdots,x_{n-1})x_{n}=\nearrow(x_{n},x_{1},\cdots,x_{n-1}),$
$\rho(x_{1},\cdots,x_{n-1})x_{n}=\nwarrow(x_{1},\cdots,x_{n})+\sum_{i=1}^{n-1}(-1)^{i}\nearrow(x_{n},x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i})$

for all $x_{i}\in A$, $1\leq i\leq n.$ Then

  1. (1)

$(A,L_{\nwarrow},R_{\nearrow})$ is a representation of its horizontal
associated $n$-pre-Lie algebra $(A,\\{\cdot,\cdots,\cdot\\}^{h})$ and
$(A,L_{\nwarrow},-L_{\nearrow})$ is a representation of its vertical
associated $n$-pre-Lie algebra $(A,\\{\cdot,\cdots,\cdot\\}^{v})$.

  2. (2)

$(A,L_{\nwarrow})$ is a representation of its associated $n$-Lie algebra
$(A^{c},[\cdot,\cdots,\cdot]^{C})$.

  3. (3)

$(A,\rho)$ is a representation of its associated $n$-Lie algebra
$(A^{c},[\cdot,\cdots,\cdot]^{C})$.

######  Remark 5.2.

In the sense of the above conclusion (1), a $n$-L-dendriform algebra is
understood as a $n$-ary algebra structure whose left and right multiplications
give a bimodule structure on the underlying vector space of the $n$-pre-Lie
algebra defined by certain commutators. It can be regarded as the rule of
introducing the notion of $n$-L-dendriform algebra, which more generally, is
the rule of introducing the notions of $n$-pre-Lie algebras, the Loday
algebras and their Lie, Jordan and alternative algebraic analogues.

######  Theorem 5.4.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra and $(V,l,r)$ be a
representation. Suppose that $T:V\to A$ is an $\mathcal{O}$-operator
associated to $(V,l,r)$. Then there exists a $n$-L-dendriform algebra
structure on $V$ given by

$\displaystyle\nwarrow(u_{1},\cdots,u_{n})=l(Tu_{1},\cdots,Tu_{n-1})u_{n},\quad\nearrow(u_{1},\cdots,u_{n})=r(Tu_{2},\cdots,Tu_{n})u_{1},$
(5.10)

$\forall u_{i}\in V,\;1\leq i\leq n.$ Therefore, there exists two associated
$n$-pre-Lie algebra structures on $V$ and $T$ is a homomorphism of $n$-pre-Lie
algebras. Moreover, $T(V)=\\{T(v)|v\in V\\}$ is a $n$-pre-Lie subalgebra of
$(A,\\{\cdot,\cdots,\cdot\\})$ and there is an induced $n$-L-dendriform
algebra structure on $T(V)$ given by

$\displaystyle\nwarrow(Tu_{1},\cdots,Tu_{n})=T(\nwarrow(u_{1},\cdots,u_{n})),\quad\nearrow(Tu_{1},\cdots,Tu_{n})=T(\nearrow(u_{1},\cdots,u_{n})),$
(5.11)

$\forall u_{i}\in V,\;1\leq i\leq n.$

###### Proof.

Let $u_{i}\in V$,$1\leq i\leq n$ . Define
$\\{\cdot,\cdots,\cdot\\}_{V}^{h},\\{\cdot,\cdots,\cdot\\}_{V}^{v},[\cdot,\cdots,\cdot]^{C}_{V}:\otimes^{n}V\to
V$ by

$\displaystyle\\{u_{1},\cdots,u_{n}\\}_{V}^{h}=\nwarrow(u_{1},\cdots,u_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(u_{i},u_{1},\cdots,\widehat{u_{i}},\cdots,u_{n}),$
$\displaystyle\\{u_{1},\cdots,u_{n}\\}_{V}^{v}=\nwarrow(u_{1},\cdots,u_{n})+\sum_{i=1}^{n-1}(-1)^{i}\nearrow(u_{n},u_{1},\cdots,\widehat{u_{i}},\cdots,u_{n-1},u_{i}),$
$\displaystyle[u_{1},\cdots,u_{n}]^{C}_{V}=\sum_{i=1}^{n}(-1)^{n-i}\\{u_{1},\cdots,\widehat{u_{i}},\cdots,u_{n},u_{i}\\}_{V}^{h}.$

Using identity (3.8), we have

$\displaystyle T\\{u_{1},\cdots,u_{n}\\}_{V}^{h}$
$\displaystyle=T(\nwarrow(u_{1},\cdots,u_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(u_{i},u_{1},\cdots,\widehat{u_{i}},\cdots,u_{n}))$
$\displaystyle=T(l(Tu_{1},\cdots,Tu_{n-1})(u_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}r(Tu_{1},\cdots,\widehat{Tu_{i}},\cdots,Tu_{n})(u_{i}))$
$\displaystyle=\\{Tu_{1},\cdots,Tu_{n}\\}^{h}$

and

$\displaystyle
T[u_{1},\cdots,u_{n}]^{C}_{V}=T(\sum_{i=1}^{n}(-1)^{n-i}\\{u_{1},\cdots,\widehat{u_{i}},\cdots,u_{n},u_{i}\\}_{V}^{h})$
$\displaystyle=\sum_{i=1}^{n}(-1)^{n-i}\\{Tu_{1},\cdots,\widehat{Tu_{i}},\cdots,Tu_{n},Tu_{i}\\}^{h}=[Tu_{1},\cdots,Tu_{n}]^{C}.$

It is straightforward that

$\nwarrow(u_{\sigma(1)},\cdots,u_{\sigma(n-1)},u_{n})=l(Tu_{\sigma(1)},\cdots,Tu_{\sigma(n-1)})u_{n}=Sgn(\sigma)\nwarrow(u_{1},\cdots,u_{n-1},u_{n}),$

$\forall\sigma\in S_{n-1}$. Furthermore, for any $u_{i},v_{i}\in V,\ 1\leq
i\leq n$, we have

$\displaystyle\nwarrow(u_{1},\cdots,u_{n-1},\nwarrow(v_{1},\cdots,v_{n}))-\nwarrow(v_{1},\cdots,v_{n-1},\nwarrow(u_{1},\cdots,u_{n-1},v_{n}))$
$\displaystyle-\sum_{i=1}^{n-1}\nwarrow(v_{1},\cdots,v_{i-1},[u_{1},\cdots,u_{n-1},v_{i}]_{V}^{C},v_{i+1},\cdots,v_{n}),$
$\displaystyle=l(Tu_{1},\cdots,Tu_{n-1})l(Tv_{1},\cdots,Tv_{n-1})uv{n}-l(Tv_{1},\cdots,Tv_{n-1})l(Tu_{1},\cdots,Tu_{n-1})v_{n}$
$\displaystyle-\sum_{i=1}^{n-1}l(Tv_{1},\cdots,Tv_{i-1},T[u_{1},\cdots,u_{n-1},v_{i}]_{V}^{C},Tv_{i+1},\cdots,Tv_{n-1})v_{n}$
$\displaystyle=\Big{(}[l(Tu_{1},\cdots,Tu_{n-1}),l(Tv_{1},\cdots,Tv_{n-1})]$
$\displaystyle-\sum_{i=1}^{n-1}l(Tv_{1},\cdots,Tv_{i-1},[Tu_{1},\cdots,Tu_{n-1},Tv_{i}]^{C},Tv_{i+1},\cdots,Tv_{n-1})\Big{)}v_{n}$
$\displaystyle=0.$

This implies that (5.1) holds. Moreover, (5.1) holds. Indeed,

$\displaystyle\nwarrow(u_{1},\cdots,u_{n-1},\nearrow(v_{n},v_{1},\cdots,v_{n-1}))-\nearrow(v_{n},v_{1},\cdots,v_{n-2},\\{u_{1},\cdots,u_{n-1},v_{n-1}\\}_{V}^{h})$
$\displaystyle-\nearrow(\\{u_{1},\cdots,u_{n-1},v_{n}\\}_{V}^{v},v_{1},\cdots,v_{n-1})$
$\displaystyle-\sum_{i=1}^{n-2}\nearrow(v_{n},v_{1},\cdots,v_{i-1},[u_{1},\cdots,u_{n-1},v_{i}]_{V}^{C},v_{i+1},\cdots,v_{n-1}),$
$\displaystyle=l(Tu_{1},\cdots,Tu_{n-1})r(Tv_{1},\cdots,Tv_{n-1})v_{n}-r(Tv_{1},\cdots,Tv_{n-2},T\\{u_{1},\cdots,u_{n-1},v_{n-1}\\}^{h}_{V})v_{n}$
$\displaystyle-r(Tv_{1},\cdots,Tv_{n-1})\\{u_{1},\cdots,u_{n-1},v_{n}\\}^{v}_{V}$
$\displaystyle-\sum_{i=1}^{n-2}r(Tv_{1},\cdots,Tv_{i-1},T[u_{1},\cdots,u_{n-1},v_{i}]_{V}^{C},Tv_{i+1},\cdots,Tv_{n-1})v_{n}$
$\displaystyle=l(Tu_{1},\cdots,Tu_{n-1})r(Tv_{1},\cdots,Tv_{n-1})v_{n}-r(Tv_{1},\cdots,Tv_{n-2},\\{Tu_{1},\cdots,Tu_{n-1},Tv_{n-1}\\}^{h})v_{n}$
$\displaystyle-r(Tv_{1},\cdots,Tv_{n-1})\mu(Tu_{1},\cdots,Tu_{n-1})v_{n}$
$\displaystyle-\sum_{i=1}^{n-2}r(Tv_{1},\cdots,Tv_{i-1},[Tu_{1},\cdots,Tu_{n-1},Tv_{i}]^{C},Tv_{i+1},\cdots,Tv_{n-1})v_{n}$
$\displaystyle=0.$

The other conclusions follow immediately by the same way. ∎

######  Corollary 5.5.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra and $P:A\to A$ is
a Rota-Baxter operator of weight $0$. Then there exists a $n$-L-dendriform
algebra structure on $A$ given by

$\displaystyle\nwarrow(x_{1},\cdots,x_{n})=\\{P(x_{1}),\cdots,P(x_{n-1}),x_{n}\\},\quad\nearrow(x_{1},\cdots,x_{n})=\\{x_{1},P(x_{2}),\cdots,P(x_{n})\\},$
(5.12)

for all $x_{i}\in A$,$\;1\leq i\leq n.$

######  Proposition 5.6.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra. Then there exists
a compatible $n$-L-dendriform algebra if and only if there exists an
invertible $\mathcal{O}$-operator on $A$.

###### Proof.

Let $T$ be an invertible $\mathcal{O}$-operator of $A$ associated to a
representation $(V,l,r)$. Then there exists a $n$-L-dendriform algebra
structure on $V$ defined by

$\displaystyle\nwarrow(u_{1},\cdots,u_{n})=l(Tu_{1},\cdots,Tu_{n-1})u_{n},\quad\nearrow(u_{1},\cdots,u_{n})=r(Tu_{2},\cdots,Tu_{n})u_{1},\forall
u_{i}\in V.$

In addition there exists a $n$-L-dendriform algebra structure on $T(V)=A$
given by

$\displaystyle\nwarrow(Tu_{1},\cdots,Tu_{n})=T(l(Tu_{1},\cdots,Tu_{n-1})u_{n}),\quad\nearrow(Tu_{1},\cdots,Tu_{n})=T(r(Tu_{2},\cdots,Tu_{n})u_{1}),\forall
u_{i}\in V.$

If we put $x_{1}=Tu_{1},\cdots,x_{n}=Tu_{n}$, we get

$\displaystyle\nwarrow(x_{1},\cdots,x_{n})=T(l(x_{1},\cdots,x_{n-1})T^{-1}(x_{n})),\quad\nearrow(x_{1},\cdots,x_{n})=T(r(x_{2},\cdots,x_{n})T^{-1}(x_{1})),\forall
x_{i}\in A.$

It is a compatible $n$-L-dendriform algebra structure on $A$. Indeed,

$\displaystyle\nwarrow(x_{1},\cdots,x_{n})+\sum_{i=1}^{n-1}(-1)^{i+1}\nearrow(x_{i},x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})$
$\displaystyle=T(l(x_{1},\cdots,x_{n-1})T^{-1}(x_{n}))+\sum_{i=1}^{n-1}(-1)^{i+1}T(r(x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n})T^{-1}(x_{i}))$
$\displaystyle=\\{TT^{-1}(x_{1}),\cdots,TT^{-1}(x_{n})\\}=\\{x_{1},\cdots,x_{n}\\}.$

Conversely, let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra and
$(A,\nwarrow,\nearrow)$ its compatible $n$-L-dendriform algebra. Then the
identity map $id:A\to A$ is an $\mathcal{O}$-operator of
$(A,\\{\cdot,\cdots,\cdot\\})$ associated to $(A,L,R)$. ∎

######  Definition 5.2.

Let $(A,\\{\cdot,\cdots,\cdot\\})$ be a $n$-pre-Lie algebra and $B$ be a
symmetric bilinear form on $A$. We say that $B$ is closed if it satisfies

$\displaystyle
B(\\{x_{1},\cdots,x_{n}\\},w)+B(x_{n},[x_{1},\cdots,x_{n-1},w]^{C})$
$\displaystyle-\sum_{i=1}^{n-1}(-1)^{i+1}B(x_{i},\\{w,x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{n}\\})=0,$
(5.13)

for any $x_{i},w\in A$, $\;1\leq i\leq n$. If in addition $B$ is non-
degenerate, then $B$ is called a pseudo-Hessian structure on $A$.

A $n$-pre-Lie algebra $(A,\\{\cdot,\cdots,\cdot\\})$ is called pseudo-Hessian
if it is equipped with a symmetric, closed and nondegenerate form $B$. It is
denoted by $(A,\\{\cdot,\cdots,\cdot\\},B)$.

######  Proposition 5.7.

Let $(A,\\{\cdot,\cdots,\cdot\\},B)$ be a pseudo-Hessian $n$-pre-Lie algebra.
Then there exists a compatible $n$-L-dendriform algebra structure on $A$ given
by

$\displaystyle
B(\nwarrow(x_{1},\cdots,x_{n}),w)=-B(x_{n},[x_{1},\cdots,x_{n-1},w]^{C}),$
$\displaystyle
B(\nearrow(x_{1},\cdots,x_{n}),w)=B(x_{1},\\{w,x_{2},\cdots,x_{n}\\}),$ (5.14)

$\forall x_{i},w\in A,\;1\leq i\leq n$.

###### Proof.

Define the linear map $T:A^{*}\to A$ by $\langle T^{-1}x,y\rangle=B(x,y)$.
Using Eq. (5.2), we obtain that $T$ is an invertible $\mathcal{O}$-operator on
$A$ associated to the dual representation $(A^{*},ad^{*},-R^{*})$. By
Proposition 5.6, there exists a compatible $n$-L-dendriform algebra structure
given by

$\displaystyle\nwarrow(x_{1},\cdots,x_{n})=T(ad^{*}(x_{1},\cdots,x_{n-1})T^{-1}(x_{n})),\quad\nearrow(x_{1},\cdots,x_{n})=-T(R^{*}(x_{2},\cdots,x_{n})T^{-1}(x_{1})).$

Then we have

$\displaystyle
B(\nwarrow(x_{1},\cdots,x_{n}),w)=B(T(ad^{*}(x_{1},\cdots,x_{n-1})T^{-1}(x_{n})),w)=\langle
ad^{*}(x_{1},\cdots,x_{n-1})T^{-1}(x_{n}),w\rangle$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\;\;=-\langle
T^{-1}(x_{n}),[x_{1},\cdots,x_{n-1},w]^{C}\rangle=-B(x_{n},[x_{1},\cdots,x_{n-1},w]^{C})$

and

$\displaystyle
B(\nearrow(x_{1},\cdots,x_{n}),w)=-B(T(R^{*}(x_{2},\cdots,x_{n})T^{-1}(x_{1}))),w)=-\langle
R^{*}(x_{2},\cdots,x_{n})T^{-1}(x_{1}),w\rangle$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\;=\langle
T^{-1}(x_{1}),\\{w,x_{2},\cdots,x_{n}\\}\rangle=B(x_{1},\\{w,x_{2},\cdots,x_{n}\\}).$

The proof is finished. ∎

######  Corollary 5.8.

Let $(A,\\{\cdot,\cdots,\cdot\\},B)$ be a pseudo-Hessian $n$-pre-Lie algebra
and let $(A^{c},[\cdot,\cdots,\cdot]^{C})$ be its associated $n$-Lie algebra.
Then there exists a $n$-pre-Lie algebraic structure
$(A,\\{\cdot,\cdots,\cdot\\}^{\prime})$ on $A$ given by

$\displaystyle
B(\\{x_{1},\cdots,x_{n}\\}^{\prime},w)=-B(x_{n},[x_{1},\cdots,x_{n-1},w]^{C})+\sum_{i=1}^{n-1}(-1)^{i}B(x_{n},\\{w,x_{1},\cdots,\widehat{x_{i}},\cdots,x_{n-1},x_{i}\\}).$
(5.15)

######  Proposition 5.9.

Let $\\{P_{1},P_{2}\\}$ be a pair of commuting Rota-Baxter operators (of
weight zero) on a $n$-Lie algebra $(A,[\cdot,\cdots,\cdot])$. Then there
exists a $n$-L-dendriform algebra structure on $A$ defined by

$\displaystyle\nwarrow(x_{1},\cdots,x_{n})=[P_{1}P_{2}(x_{1}),\cdots,P_{1}P_{2}(x_{n-1}),x_{n}],$
$\displaystyle\nearrow(x_{1},\cdots,x_{n})=[P_{1}(x),P_{1}P_{2}(x_{2}),\cdots,P_{1}P_{2}(x_{n-1}),P_{2}(x_{n})],\forall
x_{i}\in A.$

###### Proof.

It follows immediately from Proposition 3.6 and Corollary 5.5. ∎

## References

  * [1] M. Aguiar and J.-L. Loday, Quadri-algebras, _J. Pure Applied Algebra_ , 191, (2004), 205-221.
  * [2] J. Arnlind, A. Makhlouf , S. Silvestrov, Ternary Hom-Nambu-Lie algebras induced by Hom- Lie algebras,_J. Math. Phys._ 51, 043515, 11 pp. (2010).
  * [3] J. Arnlind, A. Makhlouf, S. Silvestrov., Construction of $n$-Lie algebras and $n$-ary Hom-Nambu- Lie algebras,_J. Math. Phys._ 52, 123502, 13 pp. (2011).
  * [4] J. Arnlind, A. Kitouni, A. Makhlouf, S. Silvestrov, Structure and cohomology of 3-Lie algebras induced by Lie algebras. _In Algebra, Geometry and Mathematical physics_ (pp. 123-144) (2014). Springer, Berlin, Heidelberg.
  * [5] H. Awata, M. Li, D. Minic, Yoneya T., On the quantization of Nambu brackets, _J. High Energy Phys._ 2, Paper 13, 17 pp. (2001).
  * [6] C. Bai, X. Ni, Pre-alternative algebras and pre-alternative bialgebras, _Pacific J. Math._ 248 (2010) 355–390.
  * [7] C. Bai, G. Li and Y. Sheng, Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras._Advances in Theoretical and Mathematical Physics_ 23.1 (2019): 27–74.
  * [8] R. P. Bai , W. Qing Wu, and Z. Heng Li. ”Some results on metric $n$-Lie algebras.” _Acta Mathematica Sinica_ , English Series 28.6 (2012): 1209-1220.
  * [9] R. Bai, S. Chen, R. Cheng, Symplectic structures of 3-Lie algebras._(Chinese) Acta Math. Sinica (Chin. Ser.)_ 59(5) (2016) 711–720.
  * [10] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics. _Cent. Eur. J. Math._ 4, 323–357 (2006).
  * [11] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity,_Pac. J. Math._ 10, 731–742 (1960).
  * [12] A. Ben Hassine, T. Chtioui, and S. Mabrouk, Nijenhuis Operators on 3-Hom-L-dendriform algebras. _arXiv preprint arXiv:2007.10267_ (2020).
  * [13] J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes gauge theories. _Phys. Rev._ D 77 (2008), 065008.
  * [14] J. Bagger and N. Lambert, Three-algebras and $N=6$ Chern-Simons gauge theories. _Phys. Rev._ D 79 (2009), 025002.
  * [15] J.M. Casas, J.L. Loday and T. Pirashvili, Leibniz $n$-algebras._Forum Math._ 214 (2002), 189–207.
  * [16] S. Cherkis and C. Samann, Multiple M2-branes and generalized 3-Lie algebras._Phys. Rev._ D 78 (2008), 066019.
  * [17] T. Chtioui, and S. Mabrouk. $3$-L-dendriform algebras and generalized derivations._Filomat_ Vol 35, No 6 (2021).
  * [18] C. Ming, J. Liu, and Y. Ma. ”Lie $n$-algebras and cohomologies of relative Rota-Baxter operators on $n$-Lie algebras._” arXiv preprint arXiv:2108.04076_ (2021).
  * [19] H. Dongping and C. Bai, J-dendriform algebras,_Frontiers of Mathematics in China_ 7.1 (2012): 29–49.
  * [20] H. Dongping, X. Ni and C. Bai, Pre-Jordan algebras,_Mathematica Scandinavica_ (2013) 19–48.
  * [21] A. S. Dzhumadil′daev, Representations of vector product $n$-Lie algebras, _Comm. Alg._ 32 (2004), 3315–3326.
  * [22] J. A. de Azcarraga and J. M. Izquierdo, $n$-ary algebras: a review with applications._J. Phys. A: Math. Theor._ 43 (2010), 293001\.
  * [23] K. Ebrahimi-Fard, L. Guo, Unit actions on operads and Hopf algebras. _Theory Appl. Categ._ 18, 348–371 (2007).
  * [24] V. T. Filippov, $n$-Lie algebras, _Sib. Mat. Zh._ 26 (1985), 126–140.
  * [25] J. Figueroa-O’Farrill, Lorentzian Lie $n$-algebras._J. Math. Phys._ 49 (2008), 113509.
  * [26] J. Figueroa-O’Farrill : Metric Lie $N$-algebra and double extensions. _ArXiv:0806.3534v1 [math.RT]_
  * [27] I.Z. Golubchik and V.V. Sokolov, Generalized operator Yang-Baxter equations, integrable ODEs and nonassociative algebras. _J. Nonlinear Math. Phys._ , 7 (2000), 184–197.
  * [28] R. Holtkamp, On Hopf algebra structures over operads._Adv. Math._ 207, 544–565 (2006).
  * [29] P. Ho, R. Hou and Y. Matsuo, Lie 3-algebra and multiple M2-branes. _J. High Energy Phys. 6_ (2008), 020.
  * [30] S. M. Kasymov, Theory of $n$-Lie algebras._Algebra i Logika_ 26 (1987), no. 3, 277–297. (English translation: _Algebra and Logic_ 26 (1988), 155-166).
  * [31] B. A. Kupershmidt, What a classical $r$-matrix really is, _J. Nonlinear Math. Phys._ 6, 448–488 (1999).
  * [32] J.-L. Loday, Dialgebras, in Dialgebras and related operads, Lecture Notes in Math., 1763, _Springer, Berlin,_ (2001) 7–66.
  * [33] J.L. Loday, M.Ronco, Trialgebras and families of polytopes, Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory. _Comtep. Math._ 346, 369–398 (2004)
  * [34] J.-L. Loday, Scindement d’associativité et algèbres de Hopf,_Proceeding of the Conference in honor of Jean Leray, Nantes_ (2002), _Séminaire et Congrès (SMF)_ , 9 (2004) 155–172.
  * [35] Liu, Jie-Feng, et al. ”Nijenhuis operators on n-Lie algebras._” Communications in Theoretical Physics_ 65.6 (2016): 659.
  * [36] Y. Nambu, Generalized Hamiltonian dynamics. _Phys. Rev._ D 7 (1973), 2405-2412.
  * [37] J. Pei, C. Bai and G. Li. Splitting of operads and Rota-Baxter operators on operads._Applied Categorical Structures_ 25.4 (2017): 505–538.
  * [38] G. Papadopoulos, M2-branes, 3-Lie algebras and Plucker relations._J. High Energy Phys._ 5 (2008), 054.
  * [39] Y. Sheng and T. Rong, Symplectic, product and complex structures on 3-Lie algebras. _Journal of Algebra_ 508 (2018): 256–300.
  * [40] L. Takhtajan, On foundation of the generalized Nambu mechanics._Comm. Math. Phys._ 160 (1994), 295–315.
  * [41] B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture. _J. Reine Angew. Math._ 620, 105-164 (2008).
  * [42] G.W. Zinbiel, Encyclopedia of Types of Algebras 2010._Nankai Series in Pure, Applied Mathematics and Theoretical Physics, Operads and Universal Algebra, World Scientific_ , Singapore pp. 217–97 (2012).