Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras

Ming Chen
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China

Jiefeng Liu
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China

Yao Ma
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China

[0]Keywords: $n$-Lie algebra, Lie $n$-algebra, relative Rota-Baxter operator, cohomology, deformation.
[0]MSC: 17A42, 17A60,17B38, 18G60.

Based on the differential graded Lie algebra controlling deformations of an $n$-Lie algebra with a representation (called an $n$-pair), we construct a Lie $n$-algebra, whose Maurer-Cartan elements  characterize relative Rota-Baxter operators on $n$-pairs.  The notion of an $n$-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator.  We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations  and extensions of order $m$ deformations to order $m+1$ deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on $n$-pairs and those on $(n+1)$-pairs by certain linear functions.

§ INTRODUCTION

In this paper, we use Maurer-Cartan elements of Lie $n$-algebras to characterize the relative Rota-Baxter operators on $n$-pairs and study the cohomology and deformations of  relative Rota-Baxter operators on $n$-pairs.

§.§ $n$-Lie algebras,Lie $n$-algebras and relative Rota-Baxter operators on $n$-pairs

The notion of an $n$-Lie algebra, or a Filippov algebra was introduced in [20]. It is the algebraic structure corresponding to Nambu mechanics ([42, 52]). $n$-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 ([2, 3, 9, 11, 15, 16, 18, 19, 31, 45]).  For example, the structure of $n$-Lie algebras are applied to the study of the 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 Pl${\rm\ddot{u}}$cker relation in the physics literature. See the review article [13] for more details.

The notion of a Lie $n$-algebra was introduced by Hanlon and Wachs in [29]. Lie $n$-algebra is a special $L_\infty$-algebra, in which only the $n$-ary bracket is nonzereo. See [14, 47, 49] for more details on Lie $n$-algebras and  $L_\infty$-algebras. One useful method for constructing $L_\infty$-algebras is given by Vornov's higher derived bracket ([54]) and another one is by twisting with the Maurer-Cartan elements of a given $L_\infty$-algebra ([27]). In this paper, we will use these methods to construct Lie $n$-algebras and $L_\infty$-algebras that characterize relative Rota-Baxter operators on $n$-Lie algebras as Maurer-Cartan elements and deformations of relative Rota-Baxter operators.

The classical Yang-Baxter equation plays a significant role in many fields in mathematics and mathematical physics such as quantum groups ([10, 17]) and integrable systems ([48]). In order to gain better understanding of the relationship between the classical Yang-Baxter equation and the related integrable systems, the more general notion of a relative Rota-Baxter operator (also called $\huaO$-operator) on a pair was introduced by Kupershmidt ([35]). To study solutions of $3$-Lie classical Yang-Baxter equation, the notion of a relative Rota-Baxter operator on a $3$-pair was introduced in [4]. A relative Rota-Baxter operator on a $3$-pair $(\g;\ad)$, where $\ad$ is the adjoint representation of the $3$-Lie algebra $\g$, is exactly the Rota-Baxter operator on the $3$-Lie algebra $\g$ introduced in  [5]. See the book [28] for more details and applications about Rota-Baxter operators.

In [50], the authors showed that a relative Rota-Baxter operator on a Lie algebra is a Maurer-Cartan element of a graded Lie algebra. Recently, it was shown in [51]  that a relative Rota-Baxter operator on a $3$-Lie algebra is a Maurer-Cartan element of a Lie $3$-algebra. The first purpose in this paper is to realize the relative Rota-Baxter operators on $n$-Lie algebras as Maurer-Cartan elements of Lie $n$-algebras.

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 [8] and the references therein for more details. The beauty 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 ([26, 35]) and thus the pre-Lie algebra can be seen as the underlying algebraic structure of a relative Rota-Baxter operator. In this paper, we introduce the notion of an $n$-pre-Lie algebra, which gives an $n$-Lie algebra naturally and its left multiplication operator gives rise to a representation of this $n$-Lie algebra. An $n$-pre-Lie algebra can also be obtained through the action of a relative Rota-Baxter operator on an $n$-Lie algebra.

§.§ Deformations and cohomology theories

The theory of deformation plays a prominent role in mathematics and physics. In physics, the ideal of deformation appears in the perturbative quantum field theory and quantizing classical mechanics. The idea of treating deformation as a tool to study the algebraic structures was introduced by Gerstenhaber in his work of associative algebras ([24, 25]) and then was extended to Lie algebras by Nijenhuis and Richardson ([43, 44]).  Deformations of $3$-Lie algebras and $n$-Lie algebras are studied in [1, 19, 38, 53]. See the review paper [13, 39] for more details. Recently,  people pay more attention on the deformations of morphisms ([1, 21, 22, 23, 40, 55]), relative Rota-Baxter operators ([12, 50, 51]) and diagram of algebras ([7, 26, 41]).

Usually the cohomology theory is an important tool in the study of deformations of a mathematical structure.  Typically, infinitesimal deformations are classified by a suitable second cohomology group and the obstruction of an order $n$ deformation extending to an order $n+1$ deformation can be controlled by the third cohomology group. Cohomology and deformations of relative Rota-Baxter operators ($\huaO$-operators) on associative algebras, Lie algebras and $3$-Lie algebras were studied in [12, 50, 51].

In the present paper, we study the cohomology theory of relative Rota-Baxter operators on $n$-pairs. By using the underlying $n$-Lie algebra of a relative Rota-Baxter operator, we introduce the cohomology of a relative Rota-Baxter operator on an $n$-pair. Then we study infinitesimal deformations and order $n$ deformations of a relative Rota-Baxter operator on an $n$-pair. We show that infinitesimal deformations of relative Rota-Baxter operators are classified by the first cohomology group and that a higher order deformation of a relative Rota-Baxter operator is extendable if and only if its obstruction class in the second cohomology group of the relative Rota-Baxter operator is trivial.

§.§ Outline of the paper

In Section <ref>, we first recall representations and cohomologies
of $n$-Lie algebras and then construct a graded Lie algebra whose Maurer-Cartan elements are precisely $n$-pairs.  In Section <ref>, we use Voronov's higher derived brackets to construct a Lie $n$-algebra from an $n$-pair and show that relative Rota-Baxter operators on $n$-pairs can be characterized by Maurer-Cartan elements of the constructed Lie $n$-algebra. We give the notion of an $n$-pre-Lie algebra and show that a relative Rota-Baxter operator on an $n$-pair can give rise to an $n$-pre-Lie algebra naturally. In Section <ref>, we define the cohomology of
a relative Rota-Baxter operator on an $n$-pair using a certain $n$-pair constructed by the relative Rota-Baxter operator. In Section <ref>, we use the cohomology theory of relative Rota-Baxter operators to study deformations of relative Rota-Baxter operators. In Section <ref>, we study the relation between the cohomology groups of relative Rota-Baxter operators on $n$-pairs and those on $(n+1)$-pairs by certain linear functions.

In this paper, all the vector spaces are over an algebraically closed field $\mathbb K$ of characteristic $0$, and finite dimensional.

§ MAURER-CARTAN CHARACTERIZATIONS OF $N$-PAIRS

§.§ Representations and cohomology of $n$-Lie algebras

An $n$-Lie algebra is a vector space $\g$ together with a skew-symmetric linear map $[-,\cdots,-]_{\frak g}:\wedge^n{\frak g}\rightarrow {\frak g}$ such that for $x_i, y_j\in {\frak g}, 1\leq i\leq n-1,1\leq j\leq n$, the following identity holds:

\begin{equation}\label{1}
[x_1,\cdots,x_{n-1},[y_1,\cdots,y_n]_{\frak g}]_{\frak g}=\sum\limits_{i=1}^n[y_1,\cdots,y_{i-1},[x_1,\cdots,x_{n-1},y_i]_{\frak g},y_{i+1},\cdots,y_n]_{\frak g}.
\end{equation}

For $x_1,x_2,\cdots,x_{n-1}\in \frak g$, define $\ad_{x_1,x_2,\cdots,x_{n-1}}\in \frak \gl(\frak g)$ by $\ad_{x_1,x_2,\cdots,x_{n-1}}y:=[x_1,x_2,\cdots,x_{n-1},y]_\g,  \forall  y \in \frak g.$
Then $\ad_{x_1,x_2,\cdots,x_{n-1}}$ is a derivation, i.e.
$$\ad_{x_1,x_2,\cdots,x_{n-1}}[y_1,\cdots,y_n]_{\frak g}=\sum\limits_{i=1}^n[y_1,\cdots,y_{i-1},\ad_{x_1,x_2,\cdots,x_{n-1}}y_i,y_{i+1},\cdots,y_n]_{\frak g}.$$

A representation of an $n$-Lie algebra $(\frak g,[-,\cdots,-]_{\frak g})$ on a vector space $V$ is a linear map: $\rho:\wedge^{n-1}{\frak g}\rightarrow {\frak gl(V)},$ such that for all $x_1, x_2,\cdots,x_{n-1},y_1,\cdots,y_n\in \frak g,$ the following equalities hold:

\begin{eqnarray}
\label{eq:rep1}[\rho(\mathfrak{X}),\rho(\frkY)]&=&\rho(\mathfrak{X}\circ \frkY),\\
\label{eq:rep2}\rho(x_1,\cdots,x_{n-2},[y_1,\cdots,y_n])&=&\sum\limits_{i=1}^n(-1)^{n-i}\rho(y_1,\cdots,\widehat{y_{i}},\cdots,y_n)
\rho(x_1,\cdots,x_{n-2},y_i),
\end{eqnarray}

where $\mathfrak{X}=x_1\wedge\cdots\wedge x_{n-1},
\frkY=y_1\wedge\cdots\wedge y_{n-1}$ and  $\mathfrak{X}\circ \frkY=\sum\limits_{i=1}^{n-1}(y_1,\cdots,y_{i-1},[x_1,\cdots,x_{n-1},y_i]_\g,y_{i+1},\cdots,y_{n-1})$.

Let $(V;\rho)$ be a representation of an $n$-Lie algebra $(\frak g,[-,\cdots,-]_{\frak g}).$  Denote by
$$C^{m}_{\nl}(\g;V)=\Hom(\otimes^{m-1}(\wedge^{n-1}\g)\wedge\frak{g},V),\quad (m\geq 1),$$

which is the space of $n$-cochains. Define $\partial_\rho:C^{m}_{\nl}(\g;V)\rightarrow C^{m+1}_{\nl}(\g;V)$ by

\begin{eqnarray*}
\nonumber(\partial_\rho f)(\mathfrak{X}_1,\cdots,\mathfrak{X}_m,x_{m+1})&=&\sum\limits_{1\leq j< k\leq m}(-1)^jf(\mathfrak{X}_1,\cdots,\widehat{\mathfrak{X}_j},\cdots,\mathfrak{X}_{k-1},\mathfrak{X}_j\circ \mathfrak{X}_k,\mathfrak{X}_{k+1},\cdots,\mathfrak{X}_{m},x_{m+1})\\
[\mathfrak X_{j},x_{m+1}]_\g)\\
&&+\sum\limits_{j=1}^m(-1)^{j+1}\rho(\mathfrak X_{j})f(\mathfrak{X}_1,\cdots,\widehat{\mathfrak{X}_j},\cdots,\mathfrak{X}_{m},
\end{eqnarray*}

for any $\mathfrak{X}_i=x_i^1\wedge\cdots\wedge x_i^{n-1}\in \wedge^{n-1}\frak \g,i=1,2,\cdots,m,x_{m+1}\in \g.$

It was proved in [9] that $\partial_\rho\circ \partial_\rho=0.$ Thus, $(\oplus^{+\infty}_{m=1}C^{m}_{\nl}(\g;V),\partial_\rho)$ is a cochain complex.

The cohomology of the $n$-Lie algebra $\mathfrak{g}$ with coefficients in $V$ is the cohomology of the cochain complex  $(\oplus^{+\infty}_{m=1}C^{m}_{\nl}(\g;V),\partial_\rho)$.
The corresponding $n$-th cohomology group is denoted by $H^m_{\nl}(\g;V)$.

§.§ Bidegrees and graded Lie algebras on $C^{*}_{\nl}(\frak g;\frak g)$

Let $\g_1$ and $\g_2$ be two vector spaces. Denote by $\g^{l,k}$ the subspace of
$\otimes^{m}(\wedge^{n-1}(\g_1\oplus \g_2))\wedge(\g_1\oplus \g_2)$
which contains the number of $\g_1$ (resp. $\g_2$) is $l$ (resp. $k$).

The vector space
$\otimes^m(\wedge^{n-1}(\g_1\oplus \g_2))\wedge(\g_1\oplus \g_2)$ can be extended into the direct sum of $\g^{l,k},l+k=m(n-1)+1$. Furthermore, we have the following isomorphism

\begin{equation}\label{1234qwer}
C^m(\g_1\oplus \g_2,\g_1\oplus \g_2)\cong\sum\limits_{l+k=m(n-1)+1}\Hom(\g^{l,k},\g_1)\oplus\sum\limits_{l+k=m(n-1)+1}\Hom(\g^{l,k},\g_2).
\end{equation}

An element $f\in \Hom(\g^{l,k},\g_1)$(resp. $f\in \Hom(\g^{l,k},\g_2)$) naturally gives an element $\hat{f}\in C^m(\g_1\oplus \g_2,\g_1\oplus \g_2)$, which is called its lift. For example, the lifts of linear maps $\mu: \wedge^n\g_1\rightarrow\g_1, \rho: \wedge^{n-1}\g_1\otimes\g_2\rightarrow\g_2$ are defined by

\begin{eqnarray}
\hat{\mu}((x_1,u_1),\cdots,(x_n,u_n))&=& (\mu(x_1,\cdots, x_n),0),\\
\hat{\rho}((x_1,u_1),\cdots,(x_n,u_n))&=&(0,\sum_{i=1}^n(-1)^{n-i}\rho(x_1,\cdots,\hat{x_i},\cdots,x_n)(u_i)),
\end{eqnarray}

respectively. Let $H:\g_2\rightarrow \g_1$ be a linear map. Its lift is given by $\hat{H}(x,u)=(H(u),0)$.

A linear map $f\in \Hom(\otimes^m(\wedge^{n-1}(\g_1\oplus \g_2))\wedge(\g_1\oplus \g_2),\g_1\oplus \g_2)$ has a bidegree $k|l$, which
is denoted by $\|f\|=k|l$, if f satisfies the following four conditions:

* $k+l=m(n-1)$

* If $X$ is an element in $\g^{k+1,l}$, then $f(X)\in \g_1$

* If $X$ is an element in $\g^{k,l+1}$, then $f(X)\in \g_2$

* All the other case, $f(X)=0$.

We denote the set of linear maps of bidegree $k|l$ by $C^{k\mid l}(\g_1\oplus \g_2,\g_1\oplus \g_2)$. A linear map $f$ is said to be homogeneous if $f$ has a bidegree.

Let $\frak g$ be a vector space. We consider the graded vector space

$$C^{*}_{\nl}(\frak g;\frak g)=\oplus_{m\geq 0}C^{m}_{\nl}(\frak g;\frak g)=\oplus_{m\geq 0} \Hom(\otimes^m(\wedge^{n-1}\g)\wedge\frak{g},\g).$$

Then the graded vector space $C^{*}_{\nl}(\frak g;\frak g)$ equipped with the graded commutator bracket

\begin{equation}\label{14}
[P,Q]_{\nl}=P\circ Q-(-1)^{pq}Q\circ P, \qquad\forall ~P\in C^p(\frak g,\frak g),Q\in C^q(\g,\g)
\end{equation}

is a graded Lie algebra ([46]), where $P\circ Q\in C^{p+q}(\frak g,\frak g)$ is defined by

\begin{eqnarray*}
&&(P\circ Q)(\mathfrak{X}_1,\cdots,\mathfrak{X}_{p+q},x)\\
&=&\sum\limits^{p}_{k=1}(-1)^{(k-1)q}\sum\limits_{\sigma\in S(k-1,q)}(-1)^{\sigma}\sum\limits^{n-1}_{i=1}P(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x^1_{k+q}\wedge\cdots\wedge x^{i-1}_{k+q}\wedge\\
&&Q(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},x^i_{k+q})\wedge x^{i+1}_{k+q}\wedge\cdots\wedge x^{n-1}_{k+q},\mathfrak{X}_{k+q+1},\cdots,
\mathfrak{X}_{p+q},x)\\
&& +\sum\limits_{\sigma\in S(p,q)}(-1)^{pq}(-1)^{\sigma}P(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(p)},Q(\mathfrak{X}_{\sigma(p+1)},\cdots,\mathfrak{X}_{\sigma(p+q-1)},
\mathfrak{X}_{\sigma(p+q)},x)),
\end{eqnarray*}

for all $\mathfrak{X}_i=x^1_i\wedge\cdots\wedge x^{n-1}_i\in\wedge^{n-1}\frak g, i=1,2,\cdots,p+q$ and $x\in \g$.  In particular, $\mu:\wedge^n\g\rightarrow \g$ defines an $n$-Lie structure on $\g$ if and only if $[\mu,\mu]_{\nl}=0$, i.e. $\mu$ is a Maurer-Cartan element of
the graded Lie algebra $(C^{*}_{\nl}(\frak g;\frak g),[-,-]_{\nl})$. Moreover, the coboundary operator $\partial$ of the $n$-Lie algebra with the coefficients in the adjoint representation can be given by

$$\partial_{\ad}  f=(-1)^{p}[\mu ,f]_{\nl}, \quad \forall f \in C^p(\g,\g).$$

The graded vector space $C^{*}_{\nl}(\frak g;\frak g)$ equipped with the graded commutator bracket

\begin{equation}\label{14}
[P,Q]_{\nl}=P\circ Q-(-1)^{pq}Q\circ P, \qquad\forall ~P\in C^p(\frak g,\frak g),Q\in C^q(\g,\g)
\end{equation}

is a graded Lie algebra, where $P\circ Q\in C^{p+q}(\frak g,\frak g)$ is defined by

\begin{eqnarray*}
&&(P\circ Q)(\mathfrak{X}_1,\cdots,\mathfrak{X}_{p+q},x)\\
&=&\sum\limits^{p}_{k=1}(-1)^{(k-1)q}\sum\limits_{\sigma\in S(k-1,q)}(-1)^{\sigma}\sum\limits^{n-1}_{i=1}P(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x^1_{k+q}\wedge\cdots\wedge x^{i-1}_{k+q}\wedge\\
&&Q(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},x^i_{k+q})\wedge x^{i+1}_{k+q}\wedge\cdots\wedge x^{n-1}_{k+q},\mathfrak{X}_{k+q+1},\cdots,
\mathfrak{X}_{p+q},x)\\
&& +\sum\limits_{\sigma\in S(p,q)}(-1)^{pq}(-1)^{\sigma}P(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(p)},Q(\mathfrak{X}_{\sigma(p+1)},\cdots,\mathfrak{X}_{\sigma(p+q-1)},
\mathfrak{X}_{\sigma(p+q)},x)),
\end{eqnarray*}

for all $\mathfrak{X}_i=x^1_i\wedge\cdots\wedge x^{n-1}_i\in\wedge^{n-1}\frak g, i=1,2,\cdots,p+q$ and $x\in \g$.
Then $\mu:\wedge^n\g\rightarrow \g$ defines an $n$-Lie structure on $\g$ if and only if $[\mu,\mu]_{\nl}=0$, i.e. $\mu$ is a Maurer-Cartan element of
the graded Lie algebra $(C^{*}_{\nl}(\frak g;\frak g),[-,-]_{\nl})$. Moreover, the coboundary map $\partial$ of the $n$-Lie algebra with the coefficients in the adjoint representation can be given by

$$\partial  f=(-1)^{p-1}[\partial ,f]_{\rm nLie}, \quad \forall f \in C^p(\g,\g).$$

The following lemma shows that the graded Lie algebra structure on $C^{*}_{\nl}(\frak g;\frak g)$ is compatible with the bigrading.

Let $f\in C^p(\g_1\oplus \g_2,\g_1\oplus \g_2)$ and $g\in C^q(\g_1\oplus \g_2,\g_1\oplus \g_2)$ be the homogeneous linear maps with bidegrees $k_f|l_f$ and $k_g|l_g$ respectively. Then the graded Lie bracket $[f,g]_{\nl}\in C^{p+q}(\g_1\oplus \g_2,\g_1\oplus \g_2)$ is a homogeneous linear map of bidegree $(k_f+k_g)|(l_f+l_g)$.

It follows by a direct calculation.

§.§ Maurer-Cartan characterizations of $n$-pairs

An $n$-pair is a pair of an $n$-Lie algebra $(\g,[-,\cdots,-]_\g)$ together with a representation $\rho$ on $V$. We denote an $n$-pair by $(\g,[-,\cdots,-]_{\g};\rho)$, or simply $(\g;\rho)$.

For $2$-pair, we usually call it  pair for short.

Let $(\frak g,[-,\cdots,-]_{\frak g};\rho)$ be an $n$-LieRep pair. Usually we will also use $\mu$ to indicate the $n$-Lie bracket $[-,\cdots,-]_{\frak g}$. Then $\mu+\rho$ corresponds to the semidirect product $n$-Lie algebra structure on $\g\oplus V$ given by

\begin{equation}\label{15}
[x_1+u_1,\cdots,x_n+u_n]_{\rho}=[x_1,\cdots,x_n]_{\frak g}+\sum\limits^{n}_{i=1}(-1)^{n-i}\rho(x_1,\cdots,\widehat{x_i},\cdots,x_n)u_i.
\end{equation}

We denote the semidirect product $n$-Lie algebra  by $\g\ltimes_{\rho} V$. Since $\mu\in \Hom(\wedge^n \g,\g)$ and $\rho\in \Hom(\wedge^{n-1} \g\otimes V,V)$, we have $\|\mu\|=n-1|0$ and $\|\rho\|=n-1|0$. Thus $\mu+\rho\in C^{n-1\mid 0}_{\nl}(\g\oplus V,\g\oplus V)$.

Let $\g$ and $V$ be two vector spaces. Then $(\oplus_{k=0}^{+\infty}C^{k(n-1)\mid 0}(\g\oplus V,\g\oplus V),[-,-]_{\nl})$ is a graded Lie algebra. Its Maurer-Cartan element are precisely $n$-pairs.

For $f\in C^{k(n-1)\mid 0}(\g_1\oplus \g_2,\g_1\oplus \g_2)$ and $g\in C^{l(n-1)\mid 0}(\g_1\oplus \g_2,\g_1\oplus \g_2)$, by Lemma <ref>, we have $||[f,\g]_{\nl}||=(k+l)(n-1)| 0$. Thus $(\oplus_{k=0}^{+\infty}C^{k(n-1)\mid 0}(\g\oplus V,\g\oplus V),[-,-]_{\nl})$ is a graded Lie subalgebra of $(C^*(\g\oplus V,\g\oplus V),[-,-]_{\nl})$.

By a direct calculation, we have

\begin{eqnarray*}
\rho(x_1,\cdots,x_{n-1})(v_k)-\sum\limits_{k=1}^n(-1)^{n-k}\rho(x_1,\cdots,x_{n-1})\rho(y_1,\cdots,\hat{y_k},\cdots,y_n)(v_k)\\
&&+\sum\limits_{i=1}^n\sum\limits_{k\neq i,k=1}^n(-1)^{n-k}\rho(y_1,\cdots,y_{i-1},[x_1,\cdots,x_{n-1},y_i]_\g,\cdots,\hat{y_k},\cdots,y_n)(v_k)\\
\end{eqnarray*}

Thus $[\mu+\rho,\mu+\rho]=0$ if and only if $\mu$ defines an $n$-Lie algebra structure and $\rho$ is a representation of $(\g,\mu)$ on $V$.

Let $(\g,\mu;\rho)$ be an $n$-pair. Note that $\pi=\mu+\rho$ is the Maurer-Cartan element of the graded Lie algebra $(\oplus_{k=0}^{+\infty}C^{k(n-1)\mid 0}(\g\oplus V,\g\oplus V),[-,-]_{\nl})$, by the graded Jacobi identity,  $\dM_\pi:=[\pi,-]_{\nl}$ is a graded derivation of the graded Lie algebra $(\oplus_{k=0}^{+\infty}C^{k(n-1)\mid 0}(\g\oplus V,\g\oplus V),[-,-]_{\nl})$ and satisfies $\dM_\pi^2=0$. Thus we have

Let $(\g,\mu;\rho)$ be an $n$-pair. Then $(\oplus_{k=0}^{+\infty}C^{k(n-1)\mid 0}(\g\oplus V,\g\oplus V),[-,-]_{\nl},\dM_\pi)$ is a differential graded Lie algebra. Furthermore, $(\g,\mu+\mu';\rho+\rho')$ is also an $n$-pair for $\mu'\in \Hom(\wedge^n \g,\g)$ and $\rho'\in \Hom(\wedge^{n-1} \g\otimes V,V)$ if and only if $\mu'+\rho'$ is a Maurer-Cartan element of the differential graded Lie algebra $(\oplus_{k=0}^{+\infty}C^{k(n-1)\mid 0}(\g\oplus V,\g\oplus V),[-,-]_{\nl},\dM_\pi)$.

It follows by a direct calculation.

§ MAURER-CARTAN CHARACTERIZATION OF RELATIVE ROTA-BAXTER OPERATORS ON $N$-PAIRS

In this section, we apply higher derived brackets introduced by Voronov in [54] to construct the Lie $n$-algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements.

§.§ $L_{\infty}$-algebras, Lie $n$-algebras and higher derived brackets

A permutation $\sigma\in S_n$ is called an $(i,n-i)$-shuffle if $\sigma(1)<\cdots<\sigma(i)$ and $\sigma(i+1)<\cdots<\sigma(n)$. If $i=0$ or $n$ we assume $\sigma=id$. The set of all $(i,n-i)$-shuffles will be denoted by $S(i,n-i)$.
An $L_{\infty}$-algebra is a $\Integ$-graded vector space $\frak g=\oplus_{k\in Z}\frak g^k$ equipped with a collection $(k\geq1)$ of linear maps $l_k:\otimes^k\frak g\rightarrow\frak g$ of degree 1 with the property that, for any  homogeneous elements $x_1,\cdots,x_n\in \frak g$, we have

(i) for every $\sigma\in S_n$,

(ii)for all $n\geq 1$,

\begin{equation}\label{eq:general-JI}
\sum\limits_{i=1}^n\sum\limits_{\sigma\in S(i,n-i)}\varepsilon(\sigma)l_{n-i+1}(l_i(x_{\sigma(1)},\cdots,x_{\sigma(i)}),x_{\sigma(i+1)},\cdots,x_{\sigma(n)})=0.
\end{equation}

The notion of a Lie $n$-algebra was introduced in [30]. A Lie $n$-algebra is a special $L_{\infty}$-algebra, in which only $n$-ary bracket is nonzero.

A Lie $n$-algebra is a $\Integ$-graded vector space $\frak g=\oplus_{k\in \Integ}\frak g^k$ equipped with an $n$-multilinear bracket $\{-,\cdots,-\}$ of degree $1$ satisfying,

(i)for all homogeneous elements $x_1,\cdots,x_n\in \frak g,$

\begin{equation}\label{6}
{\{x_1,x_2,\cdots,x_n\}_{\frak g}}=\varepsilon(\sigma)\{x_{\sigma(1)},\cdots,x_{\sigma(n-1)},x_{\sigma(n)}\}_{\frak g},
\end{equation}

(ii) for all homogeneous elements $x_i\in\frak g, 1\leq i\leq 2n-1$

\begin{equation}\label{7}\sum\limits_{\sigma\in S(n,n-1)}\varepsilon(\sigma)\{\{x_{\sigma(1)},\cdots,x_{\sigma(n)}\}_{\frak g},x_{\sigma(n+1)},\cdots,x_{\sigma(2n-1)}\}_{\frak g}=0.
\end{equation}

Recall that the desuspension operator $s^{-1}$ is defined by mapping a graded vector space to a copy of itself shifted down  by $1$, i.e. $(s^{-1}\g)^i:=\g^{i+1}$.

Let $(\g,[-,-])$ be a graded Lie algebra. Then $(s^{-1}\g,\{-,-\})$ is a Lie $2$-algebra, where $\{s^{-1}x,s^{-1}y\}=(-1)^{|x|}s^{-1}[x,y]$ for homogenous elements $x,y\in\g$.

* A Maurer-Cartan element of an $L_{\infty}$-algebra $(\frak g,\{l_i\}^{+\infty}_{i=1})$ is an element $\alpha\in \frak g^{0}$ satisfying the Maurer-Cartan equation
\begin{equation}\label{9}\sum\limits^{+\infty}_{n=1}\frac{1}{n!}l_n(\alpha,\cdots,\alpha)=0.\end{equation}

* A Maurer-Cartan element of an Lie $n$-algebra $(\frak g,\{-,\cdots,-\})$ is an element $\alpha\in \frak g^{0}$ satisfying the Maurer-Cartan equation
\begin{equation}\label{10}
\frac{1}{n!}\{\alpha,\cdots,\alpha\}_\g=0.
\end{equation}

Let $\alpha$ be a Maurer-Cartan element of a Lie $n$-algebra $(\frak g,\{-,\cdots,-\})$. For all $k\geq1$ and $x_1,\cdots,x_n\in\frak g$. Define $l^{\alpha}_k:\otimes^k\frak g\rightarrow \frak g$ by

\begin{eqnarray}
l^{\alpha}_k(x_1,\cdots,x_k) &=&  \frac{1}{(n-k)!}\{\underbrace{\alpha,\cdots,\alpha}_{n-k},x_1,\cdots,x_k\}_{\frak g},  \quad \forall~k\leq n,  \\
l_k &=& 0, \quad \forall~k\geq n+1.
\end{eqnarray}

With the above notation, $(\frak g,l^{\alpha}_1,\cdots,l^{\alpha}_n)$ is an $L_{\infty}$-algebra, obtained from the Lie $n$-algebra $(\frak g,\{-,\cdots,-\})$ by twisting with the Maurer-Cartan element $\alpha$. Moreover, $\alpha+\alpha'$ is a Maurer-Cartan element of $(\frak g,\{-,\cdots,-\})$ if and only if $\alpha'$ is a Maurer-Cartan element of the twisted $L_{\infty}$-algebra $(\frak g,l^{\alpha}_1,\cdots,l^{\alpha}_n)$.

A $V$-data is a quadruple $((L,[-,-]),\mathfrak h,{\rm P},\Delta)$, where $(L,[-,-])$ is a graded Lie algebra, $\mathfrak h$ is an abelian graded Lie subalgebra of $(L,[-,-])$, ${\rm P}: L\rightarrow L$ is a projection whose image is $\mathfrak h$ and kernel is a graded Lie subalgebra of $(L,[-,-])$ and $\Delta$ is an element in $\Ker~({\rm P})^1$ satisfying  $[\Delta,\Delta]=0$.

Let $((L,[-,-]),\mathfrak h,{\rm P},\Delta)$ be a V-data. Then $(\mathfrak h,\{l_k\}^{+\infty}_{k=1})$ is an $L_{\infty}$-algebra where

\begin{equation}\label{12}
l_k(a_1,\cdots,a_k)={\rm P}[\cdots[[\Delta,a_1],a_2],\cdots,a_k]\end{equation}

for homogeneous $a_1,\cdots,a_k\in \mathfrak h$.

We call $\{l_k\}^{+\infty}_{k=1}$ the higher derived brackets of V-data $(L,\mathfrak h,{\rm P},\Delta)$.

§.§ Maurer-Cartan characterization of relative Rota-Baxter operators on $n$-pairs

The notion of relative Rota-Baxter operators on $n$-pairs are generalization of relative Rota-Baxter operators on both Lie algebras introduced in [35] and $3$-Lie algebras introduced in [4].

Let $(\frak g,[-,\cdots,-]_{\frak g};\rho)$ be an $n$-pair.
A linear operator $T: V \rightarrow \frak g$ is called a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$ if T satisfies

\begin{equation}\label{13}
[Tv_1,\cdots,Tv_n]_{\frak g}=\sum\limits_{i=1}^n(-1)^{n-i}T(\rho(Tv_1,\cdots,\widehat{Tv}_i,\cdots,Tv_n)(v_i)),
\end{equation}

where $v_1,v_2,\cdots,v_n\in V$.

A Rota-Baxter operator $T:\g\rightarrow\g$ on an $n$-Lie algebra $\g$ is a relative Rota-Baxter operator on an $n$-pair $(\g;\ad)$. Furthermore, if  the $n$-Lie algebra reduces to a Lie algebra $(\g,[-,-])$, then the resulting linear operator $T:\g\rightarrow \g$ is a Rota-Baxter operator on the Lie algebra $\g$ introduced by the physicists C.-N. Yang and R. Baxter and if  the $n$-Lie algebra reduces to a $3$-Lie algebra $(\g,[-,-,-])$, then the resulting linear operator $T:\g\rightarrow \g$ is a Rota-Baxter operator on the $3$-Lie algebra $\g$ given in [5].

Consider the graded vector space

$$C^*(V,\frak g)=\oplus_{m\geq0}C^{m}(V,\frak g)=\Hom(\otimes^m(\wedge^{n-1}V)\wedge V,\frak g).$$

Define an $n$-linear operator $\{-,\cdots,-\}:C^{m_1}(V,\frak g)\times C^{m_2}(V,\frak g)\times\cdots \times C^{m_n}(V,\frak g) \rightarrow C^{m_1+\cdots+m_n+1}(V,\frak g)$ by

\begin{equation}\label{eq16}
\{P_1,P_2,\cdots,P_n\}=[[[\mu+\rho,P_1]_{\nl},P_2]_{\nl},\cdots,P_n]_{\nl}.
\end{equation}

For the formula $f(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x^1_{k+q}\wedge\cdots\wedge x^{i-1}_{k+q}\wedge
\g(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},x^i_{k+q})$
$\wedge x^{i+1}_{k+q}\wedge\cdots\wedge x^{n-1}_{k+q},\mathfrak{X}_{k+q+1},\cdots,
\mathfrak{X}_{p+q},x)$, we let $X=(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)}), Y=x^1_{k+q}\wedge\cdots\wedge x^{i-1}_{k+q}, Z=(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},x^i_{k+q}), W= x^{i+1}_{k+q}\wedge\cdots\wedge x^{n-1}_{k+q},T=(\mathfrak{X}_{k+q+1},\cdots,
\mathfrak{X}_{p+q},x)$. The condition (i) holds, because $l_f+k_f=p(n-1),l_\g+k_\g=q(n-1), (l_f+l_\g)+(k_f+k_\g)=(p+q)(n-1)$.

Take an element $X\otimes Y\otimes Z\otimes W\otimes  T\in \g^{l_f+l_\g+1,k_f+k_\g}$,

\begin{equation}\label{eq:fsz19}
f(X,Y\wedge \g(Z)\wedge W,T)
\end{equation}

If Eq. ($\ref{eq:fsz19}$) is zero, then it is in $\g_1$. Next we assume Eq. ($\ref{eq:fsz19}) \neq 0$. We consider the case of $\g(Z)\in \g_1$. In this case, $Z$ is in $\g^{l_\g+1,k_\g}$ and
$X\otimes Y\otimes W\otimes T$ is in $\g^{l_f,k_f}$. Thus $X\otimes Y\wedge \g(Z)\wedge W\otimes T$ is an element in $\g^{l_f+1,k_f}$ which implies
$f(X\otimes Y\wedge \g(Z)\wedge W\otimes T) \in \g_1$.

When the case of  $\g(Z)\in \g_2$, $Z$ is in $\g^{l_\g,k_\g+1}$ and $X\otimes Y\otimes W\otimes T$ is in $\g^{l_f+1,k_f-1}$. Thus $X\otimes Y\wedge \g(Z)\wedge W\otimes T$ is an element in $\g^{l_f+1,k_f}$ which implies
$f(X\otimes Y\wedge \g(Z)\wedge W\otimes T) \in \g_1$.

Similarly, take an element $X\otimes Y\otimes Z\otimes W\otimes  T\in \g^{l_f+l_\g,k_f+k_\g+1}$,
We consider the case of $g(Z)\in \g_2$. In this case, $Z$ is in $g^{l_\g,k_\g+1}$ and
$X\otimes Y\otimes W\otimes T$ is in $\g^{l_f,k_f}$. Thus $X\otimes Y\wedge \g(Z)\wedge W\otimes T$ is an element in $\g^{l_f,k_f+1}$ which implies
$f(X\otimes Y\wedge \g(Z)\wedge W\otimes T) \in \g_2$.

When the case of  $\g(Z)\in \g_1$, $Z$ is in $\g^{l_f+1,k_\g}$ and $X\otimes Y\otimes W\otimes T$ is in $\g^{l_f-1,k_f+1}$. Thus $X\otimes Y\wedge \g(Z)\wedge W\otimes T$ is an element in $\g^{l_f,k_f+1}$, which implies
$f(X\otimes Y\wedge \g(Z)\wedge W\otimes T) \in \g_2$.

Finally, we show the condition $(4)$. If $X\otimes Y\otimes Z\otimes W\otimes  T$ is an element in
$\g^{l_f+l_\g+i+1,k_f+k_\g-i}$, where $i\neq -1,0$ and $\g(Z)\neq 0$, then $Z \in \g^{l_\g+1,k_\g}$ or $Z \in \g^{l_\g,k_\g+1}$.
Thus we have $X\otimes Y\otimes W\otimes T\in \g^{l_f+i,k_f-i}$ or $\in \g^{l_f+i+1,k_f-i-1}$, which implies
$X\otimes Y\wedge \g(Z)\wedge W\otimes T\in \g^{l_f+i+1,k_f-i}$, where $i\neq -1,0$, thus $f(X\otimes Y\wedge \g(Z)\wedge W\otimes T)=0$.

With the above notations, $(C^*(V,\frak g),\{-,\cdots,-\})$ is a Lie $n$-algebra. Moreover, its Maurer-Cartan elements are precisely relative Rota-Baxter operators on the $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$.

Let $(V;\rho)$ be a representation of $n$-Lie algebra $(\frak g,\mu)$. Then the following quadruple gives a $V$-data:

* the graded Lie algebra $(L,[-,-])$ is given by $(C^*_{\nl}(\frak g\oplus V,\frak g\oplus V),[-,-]_{\nl})$;

* the abelian graded Lie subalgebra $\mathfrak h$ is given by $\mathfrak h=C^*(V,\frak g)=\oplus_{m\geq0}\Hom(\otimes^m(\wedge^{n-1}V)\wedge V,\g)$;

* ${\rm P}:L\rightarrow L$ is the projection onto the subspace $\mathfrak h$;

* $\Delta=\mu+\rho\in \Ker~({\rm P})^1,~[\Delta,\Delta]_{\nl}=0$.

By Theorem <ref>, $(\mathfrak h,\{l_k\}^{+\infty}_{k=1})$ is an $L_{\infty}$-algebra, where $l_k$ is given by Eq. (<ref>).
Note that $\|\mu\|=n-1|0$ and $\|\rho\|=n-1|0$ for $\mu\in \Hom(\wedge^n \g,\g)$ and $\rho\in \Hom(\wedge^{n-1} \g\otimes V,V)$.
For any $P_i\in \Hom(\otimes^{m_i}(\wedge^{n-1} V)\wedge V,\g)$, we have $\|P_i\|=-1|(n-1)m_i+1, 1\leq i\leq n$.
By Theorem <ref>, we have

\begin{gather*}
\|[\mu+\rho,P_1]_{\nl}\|=n-2|(n-1)m_1+1,  \\
\|[[\mu+\rho,P_1]_{\nl},P_2]_{\nl}\|=n-3|(n-1)(m_1+m_2)+2,\\
\vdots\\
\|[[[\mu+\rho,P_1]_{\nl},P_2]_{\nl},\cdots,P_{n-1}]_{\nl}\|=0|(n-1)(m_1+\cdots+m_{n-1}+1),\\
\|[[[\mu+\rho,P_1]_{\nl},P_2]_{\nl},\cdots,P_n]_{\nl}\|=-1|(n-1)(m_1+\cdots+m_n+1)+1,
\end{gather*}

which imply that
\begin{gather*}
[\mu+\rho,P_1]_{\nl}\in \Ker~({\rm P}),\\
[[\mu+\rho,P_1]_{\nl},P_2]_{\nl}\in \Ker~({\rm P}), \\
\vdots\\
[[[\mu+\rho,P_1]_{\nl},P_2]_{\nl},\cdots,P_{n-1}]_{\nl}\in\mathfrak \Ker~({\rm P}),\\
[[[\mu+\rho,P_1]_{\nl},P_2]_{\nl},\cdots,P_n]_{\nl}\in\mathfrak h.
\end{gather*}

Thus, we deduce $l_k=0$ for all $k\geq 1, k\neq n$. Therefore $(C^*(V,\g),\{-,\cdots,-\}=l_n)$ is a Lie $n$-algebra.

By a direct calculation, we have

\begin{eqnarray*}
&=&(n-1)!\sum\limits_{1\leq i_1<\cdots<i_{n-1}\leq n}[\mu+\rho,T]_{\nl}(\cdots,Tv_{i_1},\cdots,Tv_{i_{n-1}},\cdots)\\
&&-(n-1)!T\sum\limits_{1\leq i_1<\cdots<i_{n-2}\leq n}[\mu+\rho,T]_{\nl}(\cdots,Tv_{i_1},\cdots\,Tv_{i_{n-2}},\cdots)\\
&=&n![Tv_1,\cdots,Tv_n]_{\frkg}-[(n-1)!+(n-1)!C^{n-2}_{n-1}]T\sum\limits_{1\leq i_1<\cdots<i_{n-1}\leq n}(\mu+\rho)(\cdots,Tv_{i_1},\cdots,Tv_{i_{n-1}},\cdots)\\
&=&n![Tv_1,\cdots,Tv_n]_{\frkg}-n!T\sum\limits_{1\leq i_1<\cdots<i_{n-1}\leq n}(\mu+\rho)(\cdots,Tv_{i_1},\cdots,Tv_{i_{n-1}},\cdots)\\
\end{eqnarray*}

which implies that the linear map $T\in \Hom(V,\g)$ is a Maurer-Cartan element of the Lie $n$-algebra $(C^*(V,\g),\{-,\cdots,-\})$ if and only if $T$ is a relative Rota-Baxter operator on the $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$.

Consider the graded vector space $C^*(V,\frak g)=\Hom((\otimes^m V)\wedge V,\frak g).$ By Theorem <ref>, $(C^*(V,\frak g),\{-,-\})$ is a Lie $2$-algebra and its Maurer-Cartan elements are precisely relative Rota-Baxter operators on the pair $(\frak g,[-,-]_{\frak g};\rho)$. In [50], the authors showed that the graded vector $\tilde{C}^*(V,\frak g)=\Hom(\wedge^{m+1} V,\frak g)$  equipped with the bracket $\{-,-\}$ given by

$$\{P,Q\}=[[\mu+\rho,P]_{\rm NR},Q]_{\rm NR},\quad \forall~P\in \Hom(\wedge^{m_1+1},\g),Q\in\Hom(\wedge^{m_2+1},\g)$$

is also a  Lie $2$-algebra and its Maurer-Cartan elements are also precisely relative Rota-Baxter operators on the pair $(\frak g,[-,-]_{\frak g};\rho)$, where the bracket $[-,-]_{\rm NR}$ is the Nijenhuis-Richardson bracket on $\tilde{C}^*(V,\frak g)$. Therefore, we give a new Lie $2$-algebra whose Maurer-Cartan elements are precisely relative Rota-Baxter operators on pairs.

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. Since $T$ is a Maurer-Cartan element of the Lie $n$-algebra $(C^*(V,\frak g),\{-,\cdots,-\})$ given by Theorem <ref>, we have the twisted $L_{\infty}$-algebra structure on $C^*(V,\frak g)$ as follows:

\begin{eqnarray}\label{eq:l^T}
l^{T}_k(P_1,\cdots,P_k) &=&  \frac{1}{(n-k)!}\{\underbrace{T,\cdots,T}_{n-k},P_1,\cdots,P_k\},  \quad \forall~k\leq n,  \\
l^T_k &=& 0, \quad \forall~k\geq n+1.
\end{eqnarray}

Let $T:V\rightarrow \g$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. Then for a linear map $T':V\rightarrow \g$, $T+T'$ is a relative Rota-Baxter operator if and only if $T'$ is a Maurer-Cartan element of the twisted $L_{\infty}$-algebra $(C^*(V,\frak g),l^{T}_1,\cdots,l^{T}_n)$, that is, $T'$ satisfies the Maurer-Cartan equation:

By Theorem <ref>, $T+T'$ is a relative Rota-Baxter operator if and only if

Applying $\underbrace{\{T,\cdots,T\}}_{n}=0$, the above condition is equivalent to

\begin{equation*}
\frac{1}{n!}(C_n^1\{\underbrace{T,\cdots,T}_{n-1},T'\}+\cdots
\end{equation*}

That is, $l^{T}_1(T')+\frac{1}{2}l^{T}_2(T',T')+\frac{1}{3!}l^{T}_3(T',T',T')+\cdots+\frac{1}{n!}l^{T}_n(\underbrace{T',\cdots,T'}_{n})=0$, which implies that $T'$ is a Maurer-Cartan element of the twisted $L_{\infty}$-algebra $(C^*(V,\frak g),l^{T}_1,\cdots,l^{T}_n)$.

§.§ $n$-pre-Lie algebras and relative Rota-Baxter operators on $n$-pairs

Now we give the notion of an $n$-pre-Lie algebra, which is a generalization of $3$-pre-Lie algebra introduced in [4].

Let $\g$ be a vector space with a multilinear map $\{-,\cdots,-\}:\wedge^{n-1}\g\otimes \g\rightarrow\g$. The pair $(\g,\{-,\cdots,-\})$ is called an $n$-pre-Lie algebra if for $x_1,\cdots,x_n,y_1,\cdots,y_n\in\g$, the following identities hold:

\begin{eqnarray}\label{eq:n-pre1}
\{x_1,\cdots,x_{n-1},\{y_1,\cdots,y_{n-1},y_n\}\}&=&\sum\limits_{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\}\\
\nonumber&&+\{y_1,\cdots,y_{n-1},\{x_1,\cdots,x_{n-1},y_n\}\},\\
\label{eq:n-pre2}\{[y_1,\cdots,y_n]_C,x_1,\cdots,x_{n-2},x_{n-1}\}&=&\sum\limits_{i=1}^{n}(-1)^{n-i}\{y_1,\cdots,\hat{y_i},\cdots,y_n,\{y_i,x_1,\cdots,x_{n-2},x_{n-1}\}\},
\end{eqnarray}

\begin{equation}\label{eq:npreC}
\end{equation}

Recall that a pre-Lie algebra is a pair $(\g,\star)$, where $\g$ is a vector space and  $\star:\g\otimes \g\longrightarrow \g$ is a bilinear multiplication

\begin{equation}\label{eq:pre-Lie algebra}
(x\star y)\star z-x\star(y\star z)=(y\star x)\star
z-y\star(x\star z),\quad \forall~x,y,z\in \g.
\end{equation}

For  a $2$-pre-Lie algebra $(\g,\{-,-\})$, we set $x\star y=\{x,y\}$ for $x,y\in\g$. It is obvious that  Eq. $\eqref{eq:n-pre1}$ is  equivalent to

$$x_1\star(y_1\star y_2)=(x_1\star y_1)\star y_2-(y_1\star x)\star y_2+y_1\star(x_1\star y_2)$$

and Eq. $\eqref{eq:n-pre2}$ is equivalent to

$$(y_1\star y_2)\star x_1-(y_2\star y_1)\star x_1=-y_2\star(y_1\star x_1)+y_1\star(y_2\star x_1).$$

Then we have
$$\mbox{Eq. } \eqref{eq:n-pre1}\Leftrightarrow \mbox{Eq. }\eqref{eq:n-pre2} \Leftrightarrow \mbox{Eq. }\eqref{eq:pre-Lie algebra}.$$

Thus $2$-pre-Lie algebra is a pre-Lie algebra. See the survey
[8] and the references therein for
more details on pre-Lie algebras.

Let $(\g,\{-,\cdots,-\})$ be an $n$-pre-Lie algebra. Then the induced $n$-bracket $[-,\cdots,-]_C$ given by Eq. $(\ref{eq:npreC})$ defines an $n$-Lie algebra.

By the skew-symmetry of the first $n-1$ variables, the induced $n$-bracket $[-,\cdots,-]_C$ given by Eq. $(\ref{eq:npreC})$ is skew-symmetric.
For $x_1,\cdots,x_{n-1},y_1,\cdots,y_n\in \g$, by Eq. (<ref>) and Eq. (<ref>), we have

\begin{eqnarray*}
\cdots,x_{n-1},[y_1,\cdots,y_n]_C,x_i\}\\
\sum\limits_{j=1}^{n-1}(-1)^{n-j}\{x_1,\cdots,\hat{x_j},\cdots,x_{n-1},y_i,x_j\}+\{x_1,\cdots,x_{n-1},y_i\}\}\\
&&-\sum\limits_{i=1}^n\sum\limits_{k=1,k\neq i}^{n}(-1)^{n-k}\{y_1,\cdots,\hat{y_k},\cdots,y_{i-1},
&&-\sum\limits_{i=1}^{n}\sum\limits_{k=1,k\neq i}^{n}(-1)^{n-i}\{y_1,\cdots,\hat{y_i},\cdots,y_{k-1},[x_1,\cdots,x_{n-1},y_k]_C,y_{k+1},\cdots,y_n,y_i\}\\
\end{eqnarray*}

Thus $(\g,[-,\cdots,-]_C)$ is an $n$-Lie algebra.

Let $(\g,\{-,\cdots,-\})$ be an $n$-pre-Lie algebra. The $n$-Lie algebra $(\g,[-,\cdots,-]_C)$ is called the sub-adjacent $n$-Lie algebra of  $(\g,\{-,\cdots,-\})$ and $(\g,\{-,\cdots,-\})$ is called a compatible $n$-pre-Lie algebra of the $n$-Lie algebra $(\g,[-,\cdots,-]_C)$ .

Let $(\g,\{-,\cdots,-\})$ be an $n$-pre-Lie algebra. Define a skew-symmetric multi-linear map $L:\wedge^{n-1}\g\rightarrow \gl(\g)$

\begin{equation}\label{40}
L(x_1,\cdots,x_{n-1})(x_n)=\{x_1,\cdots,x_{n-1},x_n\}, \qquad\forall~ x_1,\cdots,x_n\in \g.
\end{equation}

With the above notations, $(\g;L)$ is a representation of the $n$-Lie algebra $(\g,[-,\cdots,-]_C)$.

By (<ref>), we have

\begin{eqnarray*}
\end{eqnarray*}

which implies that $[L(\frkX),L(\frkY)](y_n)=L(\frkX\circ\frkY)(y_n)$ holds.

By (<ref>), we have

\begin{equation*}
\{[y_1,\cdots,y_n]_C,x_1,\cdots,x_{n-2},x_{n-1}\}=\sum\limits_{i=1}^{n}(-1)^{n-i}\{y_1,\cdots,\hat{y_i},\cdots,y_n,\{y_i,x_1,\cdots,x_{n-2},x_{n-1}\}\},
\end{equation*}

which implies

Thus $(\g;L)$ is a representation of the $n$-Lie algebra $(\g,[-,\cdots,-]_C)$.

By Proposition <ref> and Proposition <ref>, we have

Let $\g$ be a vector space with a linear map $\{-,\cdots,-\}:\wedge^{n-1}\g\otimes \g\rightarrow \g$. Then $(\g,\{-,\cdots,-\})$ is an $n$-pre-Lie algebra if and only if the bracket $[-,\cdots,-]_C$ defined by Eq. (<ref>) is an $n$-Lie algebra structure on $\g$ and the left multiplication operation $L$ defined by Eq. (<ref>) gives a representation of this $n$-Lie algebra.

Let $T:V\rightarrow \g$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. Then there exists an $n$-pre-Lie algebra structure on $V$ given by

\begin{equation}
\{u_1,\cdots,u_n\}_T=\rho(Tu_1,\cdots,Tu_{n-1})(u_n), \quad\forall~ u_1,\cdots,u_n\in V.
\end{equation}

Furthermore, $(V,[-,\cdots,-]_T)$ is the sub-adjacent $n$-Lie algebra of the $n$-pre-Lie algebra $(\g,\{-,\cdots,-\}_T)$, where the bracket $[-,\cdots,-]_T:\wedge^n\g\rightarrow \g$ is given by

\begin{equation}\label{eqkj}
\end{equation}
and $T$ is an $n$-Lie algebra morphism from $(V,[-,\cdots,-]_T)$ to $(\g,[-,\cdots,-]_\g)$.

It is obvious that  $\{-,\cdots,-\}_T\in\Hom(\wedge^{n-1}V\otimes V, V)$. Since $T:V\rightarrow \g$ is a relative Rota-Baxter operator and $(V;\rho)$ is a representation, by Eq. (<ref>) in Definition <ref>,  we have

\begin{eqnarray*}
\end{eqnarray*}

Similarly, by Eq. (<ref>) in Definition <ref>, we have

\begin{eqnarray*}
\sum\limits_{i=1}^{n}(-1)^{n-i}\rho(Tv_1,\cdots,\widehat{Tv_i},\cdots,Tv_n)\rho(Tv_i,Tu_1,\cdots,Tu_{n-2})(u_{n-1})\\
\end{eqnarray*}

The rest follows immediately.

Let  $(\g,[-,\cdots,-]_\g)$ be an $n$-Lie algebra. Then there is a compatible $n$-pre-Lie algebra if and only if there exists an invertible relative Rota-Baxter operator $T:V\rightarrow\g$ on an $n$-pair $(\g;\rho)$. Moreover, the compatible $n$-pre-Lie structure on $\g$ is given by

\begin{equation}
\{x_1,\cdots,x_n\}_\g=T\rho(x_1,\cdots,x_{n-1})(T^{-1}x_n),\quad\forall~x_1,\cdots,x_n\in\g.
\end{equation}

Let $T$ be an invertible relative Rota-Baxter operator on an $n$-pair $(\g;\rho)$. Then there exists an $n$-pre-Lie algebra structure on $V$ defined by
$$\{u_1,\cdots,u_n\}_T=\rho(Tu_1,\cdots,Tu_{n-1})(u_n), ~\forall~ u_1,\cdots,u_n\in V.$$

Moreover, there is an induced $n$-pre-Lie algebra structure $\{-,\cdots,-\}_\g$ on $\g=T(V)$ given by

for all $x_1,\cdots,x_n\in \g$. Since $T$ is a relative Rota-Baxter operator, we have
\sum\limits_{i=1}^{n}(-1)^{n-i}\{x_1,\cdots,\hat{x_i},\cdots,x_n,x_i\}_\g.$$

Therefore $(\g,\{-,\cdots,-\}_\g)$ is a compatible $n$-pre-Lie algebra of $n$-Lie algebra $(\g,[-,\cdots,-]_\g)$.

Conversely, the identity map $\id:\g\rightarrow\g$ is an invertible relative Rota-Baxter operator on the $n$-pair $(\g;L)$.

A symplectic structure  on  an $n$-Lie algebra $(\g,[-,\cdots,-]_\g)$ is a nondegenerate skew-symmetric bilinear form $\omega\in\wedge^2\g^*$ satisfies the following condition:

\begin{equation}\label{eq:relationB}
\omega([x_1,\cdots,x_n]_\g,y)=-\sum\limits_{i=1}^{n}(-1)^{n-i}\omega(x_i,[x_1,\cdots,\hat{x_i},\cdots,x_n,y]_\g),\quad\forall~x_1,\cdots,x_n,y\in A.
\end{equation}

Let $(\g,[-,\cdots,-]_\g,\omega)$ be a symplectic $n$-Lie algebra. Then there exists a compatible $n$-pre-Lie algebra structure $\{-,\cdots,-\}_\g$ on $\g$ defined by
\begin{equation}
\omega(\{x_1,\cdots,x_n\}_\g,y)=-\omega(x_n,[x_1,\cdots,x_{n-1},y]_\g),\quad\forall~x_1,\cdots,x_n,y\in A.
\end{equation}

Define a linear map $T: \g^*\rightarrow A$ by $\langle T^{-1}x,y\rangle=\omega(x,y)$ for all $x,y\in \g$. It is straightforward to check  that $\omega$ is a symplectic structure on the $n$-Lie algebra $\g$ if and only if $T$ is an invertible relative Rota-Baxter operator on the $n$-pair $(\g;\ad^*)$. Then by Proposition <ref>, there exists a compatible $n$-pre-Lie algebra on $\g$ given by $\{x_1,\cdots,x_n\}_\g=T(\ad_{x_1,\cdots,x_{n-1}}^*T^{-1}x_n)$ for $x_1,\cdots,x_n\in \g$.
Thus we have

\begin{eqnarray*}
&=&\langle \ad_{x_1,\cdots,x_{n-1}}^*T^{-1}x_n,y\rangle=-\langle T^{-1}x_n,\ad_{x_1,\cdots,x_{n-1}}y\rangle=-\omega(x_n,[x_1,\cdots,x_{n-1},y]_\g).
\end{eqnarray*}

This completes the proof.

§ COHOMOLOGY OF RELATIVE ROTA-BAXTER OPERATORS ON $N$-PAIRS

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. By Proposition <ref>, $(V,[-,\cdots,-]_T)$ is an $n$-Lie algebra, where the bracket $[-,\cdots,-]_T$ is given by (<ref>). Furthermore, we have

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$.
Define $\rho_T: \wedge^{n-1}V \rightarrow \gl(\g)$ by

\begin{equation}\label{eqlem4.2}
\rho_T(u_1,\cdots,u_{n-1})x=[Tu_1,\cdots,Tu_{n-1},x]_\g-\sum\limits^{n-1}_{i=1}(-1)^{n-i}T\rho(Tu_1,\cdots,\widehat{Tu_i},\cdots,Tu_{n-1},x)(u_i),
\end{equation}

where $u_1,\dots,u_{n-1}\in V$ and $x\in\g$. Then $(\g;\rho_{T})$ is a representation of the $n$-Lie algebra $(V,[-,\cdots,-]_{T})$.

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$.
Define $\bar{T}: \g\oplus V\rightarrow \g\oplus V$ by

$$\bar{T}(x+v)=Tv, \qquad \forall x\in \g, v\in V.$$

It is obvious that $\bar{T}\circ\bar{T}=0$. Furthermore, $\bar{T}$ is a Nijenhuis operator on the semidirect product $n$-Lie algebra $\g\ltimes_{\rho}V$ ([38]).

\begin{eqnarray*}
[Tv_1,\cdots,Tv_n]_{\g} &=& [\bar{T}(x_1+v_1),\cdots,\bar{T}(x_n+v_n)]_{\ltimes_{\rho}}\\
\end{eqnarray*}

where $j\in n/ \{i_1,i_2,\cdots,i_{n-1}\}$, which is satisfied with Definition <ref>.
Thus $(\g\oplus V,[-,\cdots,-]_{\bar{T}})$ is an $n$-Lie algebra, where the bracket $[-,\cdots,-]_{\bar{T}})$ is given
\sum\limits_{i_1<\cdots<i_{n-2}}[\cdots,\bar{T}x_{i_1},\cdots,\bar{T}x_{i_{n-2}},\cdots]_{\ltimes_{\rho}}.$$

It was shown in [38] that if $N$ is a Nijenhuis operator on an $n$-Lie algebra $(\g,[-,\cdots,-]_{\frkg})$, $(\g,[-,\cdots,-]^{n-1}_{N})$ is an $n$-Lie algebra, where

\begin{eqnarray*}
\label{25}\{-,\cdots,-\}^{j}_{N}&=&\sum\limits_{i_1<\cdots<i_{j}}[\cdots,Nx_{i_1},\cdots,Nx_{i_{j}},\cdots]_\g-N[x_1,\cdots,x_n]^{j-1}_{N},\\
\label{26}\{x_1,\cdots,x_n\}^1_{N}&=&\sum\limits^{n}_{i=1}[x_1,\cdots,Nx_i,\cdots,x_n]_\g-N[x_1,\cdots,x_n]_\g,\quad 2\leq j\leq n-1.
\end{eqnarray*}

By a direct calculation, we have

\begin{eqnarray*}
&&[x_1+u_1,\cdots,x_n+u_n]_{\bar{T}} \\
&=& \sum\limits_{i_1<\cdots<i_{n-1}}[\cdots,\bar{T}(x_{i_1}+u_{i_1}),\cdots,\bar{T}(x_{i_{n-1}}+u_{i_{n-1}}),\cdots]_{\rho}\\
\sum\limits_{i_1<\cdots<i_{n-2}}[\cdots,\bar{T}(x_{i_1}+u_{i_1}),\cdots,\bar{T}(x_{i_{n-2}}+u_{i_{n-2}}),\cdots]_{\rho}\\
&=& \sum\limits^{n}_{i=1}[Tu_1,\cdots,x_i+u_i,\cdots,Tu_n]_{\rho}-\sum\limits^{n}_{i=1}\sum\limits_{j> i}(-1)^{n-j}T\rho(Tu_1,\cdots,x_i,\cdots,
\widehat{Tu_j},\cdots,Tu_n)(u_j)\\
\cdots,Tu_n)(u_j)\\
\widehat{Tu_j},\cdots,Tu_n,x_i)(u_j)\\
\widehat{Tu_i},\cdots,\widehat{Tu_j},\cdots,Tu_n,x_i)(u_j)\\
\end{eqnarray*}

which implies that $(\g;\rho_{T})$ is a representation of the $n$-Lie algebra $(V,[-,\cdots,-]_{T})$.

Let $\partial_T:C^{m}_{\nl}(V;\g)\rightarrow C^{m+1}_{\nl}(V;\g)~(m\geq1)$ be the corresponding coboundary operator of the $n$-Lie algebra $(V,[-,\cdots,-]_{T})$ with coefficients in the representation $(\g;\rho_{T})$. More precisely, $\partial_T:C^{m}_{\nl}(V;\g)\rightarrow C^{m+1}_{\nl}(V;\g) ~(m\geq 1)$ is given by

\begin{eqnarray*}
\nonumber&&(\partial_T f)(\mathfrak{U}_1,\cdots,\mathfrak{U}_m,u_{m+1})\\
&=&\sum\limits_{1\leq j< k\leq m}(-1)^jf(\mathfrak{U}_1,\cdots,\widehat{\mathfrak{U}_j},\cdots,\mathfrak{U}_{k-1},\mathfrak{U}_j\circ \mathfrak{U}_k,\mathfrak{U}_{k+1},\cdots,\mathfrak{U}_{m},u_{m+1})\\
[\mathfrak U_{j},u_{m+1}]_{T})\\
&&+\sum\limits_{j=1}^m(-1)^{j+1}\rho_T(\mathfrak U_{j})f(\mathfrak{U}_1,\cdots,\widehat{\mathfrak{U}_j},\cdots,\mathfrak{U}_{m},
\end{eqnarray*}

for any $\mathfrak{U}_i=u_i^1\wedge\cdots\wedge u_i^{n-1}\in \wedge^{n-1}V, ~i=1,2,\cdots,m, u_{m+1}\in V.$

For any $\frkX\in \wedge^{n-1}\g$, we define $\delta_T(\frkX):V\rightarrow \g$ by

\begin{equation}\label{eq:0-cocycle}
\delta_T(\frkX)v=T\rho(\frkX)v-[\frkX,Tv]_{\g}, ~v\in V.
\end{equation}

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. Then $\delta_T(\frkX)$ is a $1$-cocycle on the $n$-Lie algebra $(V,[-,\cdots,-]_T)$ with coefficients in $(\g;\rho_{T})$.

For any $u_1,\cdots,u_n\in V$, by the fact that $T$ is a relative Rota-Baxter operator, we have

\begin{eqnarray*}
&&-T(\sum\limits_{i=1}^n\sum\limits_{j=1,j\neq i}^n(-1)^{n-j}\rho(Tu_1,\cdots,\widehat{Tu_j},\cdots,Tu_{i-1},T\rho(\frkX)(u_i)-[\frkX,Tu_i]_\g,Tu_{i+1},\cdots,Tu_n)(u_j))\\
&&-T(\sum\limits_{i=1}^n\sum\limits_{j=1,j\neq i}^n(-1)^{n-j}\rho(Tu_1,\cdots,\widehat{Tu_j},\cdots,Tu_{i-1},T\rho(\frkX)(u_i)-[\frkX,Tu_i]_\g,Tu_{i+1},\cdots,Tu_n)(u_j))\\
&&-T(\sum\limits_{i=1}^n\sum\limits_{j=1,j\neq i}^n(-1)^{n-j}\rho(Tu_1,\cdots,\widehat{Tu_j},\cdots,Tu_{i-1},T\rho(\frkX)(u_i)-[\frkX,Tu_i]_\g,Tu_{i+1},\cdots,Tu_n)(u_j))\\
&&+\sum\limits_{i=1}^n\sum\limits_{j=1,j\neq i}
&&-\sum\limits_{i=1}^n\sum\limits_{j=1,j\neq i}
&&+\sum\limits_{i=1}^n\sum\limits_{j=1,j\neq i}^n(-1)^{n-j}T\rho(Tu_1,\cdots,\widehat{Tu_j},\cdots,Tu_{i-1},[\frkX,Tu_i]_\g,Tu_{i+1},\cdots,Tu_n)(u_j)\\
&&+\sum\limits_{i=1}^n\sum\limits_{j=1,j\neq i}^n(-1)^{n-j}T\rho(Tu_1,\cdots,\widehat{Tu_j},\cdots,Tu_{i-1},[\frkX,Tu_i]_\g,Tu_{i+1},\cdots,Tu_n)(u_j)\\
\end{eqnarray*}

which implies that $\delta_T(\frkX)$ is a $1$-cocycle on the $n$-Lie algebra $(V,[-,\cdots,-]_T)$ with coefficients in $(\g;\rho_{T})$.

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$.
Define the set of $m$-cochains by

\begin{eqnarray}
\begin{cases}
C^m_{\nl}(V;\g) & m\geq1,\\
\wedge^{n-1}\g&m=0.
\end{cases}
\end{eqnarray}

Define $d:C^m_T(V;\g)\rightarrow C_T^{m+1}(V;\g)$ by

\begin{eqnarray}
\begin{cases}
\partial_T & m\geq1,\\
\delta _T& m=0,
\end{cases}
\end{eqnarray}

where $\delta_T$ is given by Eq. (<ref>).

Denote the set of $m$-cocycles by $\huaZ^m(V;\g)$ and the set of $m$-coboundaries by $\huaB^m(V;\g)$. The $m$-th cohomology group for the relative Rota-Baxter operator $T$ is denoted by
\begin{equation}\label{31}
\huaH^m(V;\g)=\huaZ^m(V;\g)/\huaB^m(V;\g),~m\geq 0.
\end{equation}

The relation between the coboundary operator $d$ and the differential $l^T_1$ defined by Eq. (<ref>) using the Maurer-Cartan element $T$ of the Lie $n$-algebra $(C^*(V,\g),\{-,\cdots,-\})$ is given by the following theorem.

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$.
Then we have $df=(-1)^{m-1}l^T_1f, ~\forall~ f \in C^m_T(V;\g)$.

For all $x_1,\cdots,x_n\in\g,u_1,\cdots,u_n\in V$, by a direct calculation, we have

\begin{eqnarray*}
&&-(n-1)!\sum\limits^n_{i\neq j}(-1)^{n-j}T\rho(Tu_1,\cdots,x_i,\cdots,\widehat{Tu_j},\cdots,Tu_n)(u_j).
\end{eqnarray*}

Thus, we have

\begin{eqnarray}
\label{eq:RB-coboundary1}&&{[\cdots,[[\mu+\rho,\underbrace{T]_{\nl},T]_{\nl},\cdots,T}_{n-1}]_{\nl}}(u_1,\cdots,x_i,\cdots,u_n)\\
\nonumber &=&(n-1)!(-1)^{n-i}\rho_T(u_1,\cdots,\widehat{u_i},\cdots,u_n)x_i,\quad  1\leq i\leq n;\\
\label{eq:RB-coboundary2} &&{[\cdots,[[\mu+\rho,\underbrace{T]_{\nl},T]_{\nl},\cdots,T}_{n-1}]_{\nl}}(u_1,\cdots,u_n)\\
\nonumber&=&(n-1)![u_1,\cdots,u_n]_T.
\end{eqnarray}

Moreover, for all $\frkU_i=u^1_i\wedge \cdots \wedge u_i^{n-1}\in \wedge^{n-1}V,~i=1,2\cdots,m$ and $u\in V$, we have

\begin{eqnarray*}
&=&\sum\limits_{i=1}^{n-1}[\cdots,[[\mu+\rho,\underbrace{T]_{\nl},T]_{\nl},\cdots,T}_{n-1}]_{\nl}(u_m^1\wedge\cdots\wedge u_m^{i-1}\wedge f(\frkU_1,\cdots,\frkU_{m-1},u_m^i)\wedge \cdots\wedge u_m^{n-1},u)\\
&&+\sum\limits_{\sigma\in S(1,m-1)}(-1)^{m-1}(-1)^{\sigma}[\cdots,[[\mu+\rho,\underbrace{T]_{\nl},T]_{\nl},\cdots,T}_{n-1}]_{\nl}(\frkU_{\sigma(1)},f(\frkU_{\sigma(2)},\cdots,\frkU_{\sigma(m)},u))\\
&&-\sum\limits_{j=1}^{n-1}\sum\limits_{k=1}^{m-1}(-1)^{m+k}\sum\limits_{\sigma\in S(k-1,1)}
(-1)^{\sigma}f(\frkU_{\sigma(1)},\cdots,\frkU_{\sigma(k-1)},u^1_{k+1}\wedge\cdots\wedge u_{k+1}^{j-1}\wedge\\
&&[\cdots,[[\mu+\rho,\underbrace{T]_{\nl},T]_{\nl},\cdots,T}_{n-1}]_{\nl}(\frkU_{\sigma(k)}, u_{k+1}^{j})\wedge u_{k+1}^{j+1}\wedge\cdots u_{k+1}^{n-1},\frkU_{k+2},\cdots,\frkU_m,u)\\
&&-\sum\limits_{\sigma\in S(m-1,1)}(-1)^{\sigma}f(\frkU_{\sigma(1)},\cdots,\frkU_{\sigma(m-1)},[\cdots,[[\mu+\rho,\underbrace{T]_{\nl},T]_{\nl},\cdots,T}_{n-1}]_{\nl}(\frkU_{\sigma(m)},u))\\
\rho_T(u_m^1,\cdots,u_m^{i-1},\hat{u_m^i},u_m^{i+1},\cdots,u_m^{n-1},u)f(\frkU_1,\cdots,\frkU_{m-1},u_m^i)\\
u_{k+1}^1\wedge\cdots\wedge u_{k+1}^{j-1}\\
&&\wedge[\cdots,[[\mu+\rho,\underbrace{T]_{\nl},T]_{\nl},\cdots,T}_{n-1}]_{\nl}(\frkU_{i},u_{k+1}^j)\wedge u_{k+1}^{j+1}\wedge\cdots\wedge u_{k+1}^{n-1},\frkU_{k+2},
\cdots,\frkU_{m},u)\\
&&+\sum\limits_{i=1}^m(-1)^{i+1}\rho_T(\mathfrak U_{i})f(\mathfrak{U}_1,\cdots,\hat{\mathfrak{U}_i},\cdots,\mathfrak{U}_{m},u)+
\sum\limits_{i=1}^{m}(-1)^if(\frkU_1,\cdots,\widehat{\frkU_i},\cdots,\frkU_{m},[\frkU_{i},u]_T))\\
&&+(n-1)!(-1)^{m-1}\sum\limits_{1\leq i\leq k\leq m-1}(-1)^if(\frkU_1,\cdots,\widehat{\frkU_i},\cdots,\frkU_{k},
\sum\limits_{j=1}^{n-1}u_{k+1}^1\wedge\cdots\wedge u_{k+1}^{j-1}\\
\wedge\cdots\wedge u_{k+1}^{n-1},\frkU_{k+2},
\cdots,\frkU_{m},u)\\
\end{eqnarray*}

Thus, we deduce that $df=(-1)^{m-1}l^T_1f$.

§ DEFORMATIONS OF RELATIVE ROTA-BAXTER OPERATORS ON $N$-PAIRS

Let $V[[t]]$ denote the vector space of formal power series in $t$ with coefficients in $V$. If in addition, $(\g,[-,\cdots,-]_{\g})$ is an $n$-Lie algebra over $\K$, then there is an $n$-Lie algebra structure over the ring $\K[[t]]$ on $\g[[t]]$ given by

\begin{eqnarray}\label{eq:33defor}
\end{eqnarray}

where $a_1^{j_1},\cdots,a_n^{j_n}\in \g$.

For any representation $(V;\rho)$ of $(\g,[-,\cdots,-]_{\g})$, there is a natural representation of the $n$-Lie algebra $\g[[t]]$ on the $\K[[t]]$-module $V[[t]]$ given by

\begin{eqnarray}\label{eq:34defor}
\rho(\sum\limits_{j_1=0}^{+\infty}a_1^{j_1}t^{j_1},\cdots,\sum\limits_{j_{n-1}=0}^{+\infty}a_{n-1}^{j_{n-1}}t^{j_{n-1}})(\sum\limits_{k=0}^{+\infty}v_kt^k)
\end{eqnarray}

where $a_1^{j_1},\cdots,a_{n-1}^{j_{n-1}}\in \g, v_k\in V$.
Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. Consider a power series

\begin{equation}\label{eq:35defor}
\end{equation}

that is, $T_t\in \Hom_{\K}(V,\g[[t]])$. Furthermore,  $T_t$ can be extended to be a $\K[[t]]$-module map from $V[[t]]$ to $\g[[t]]$.

If $T_t=\sum_{i=0}^{+\infty}\frkT_it^i$ with $\frkT_0=T$ satisfies

\begin{equation}\label{Eq:def5.136}
\end{equation}

we say that $T_t$ is a formal deformation of relative Rota-Baxter operator $T$.

If $T_t=\sum_{i=0}^{+\infty}\frkT_it^i$ is a formal deformation of a relative Rota-Baxter operator $T$ on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$, then $[-,\cdots,-]_{T_t}$ defined by

\begin{equation*}
(-1)^{n-i}\rho(\frkT_{j_1}u_1,\cdots,\hat{u_i},\frkT_{j_i}u_{i+1},\cdots,\frkT_{j_{n-1}}u_n)(u_i))t^k,~\forall~u_1,\cdots,u_n\in V
\end{equation*}

is a formal deformation of the  $n$-Lie algebra $(V,[-,\cdots,-]_{T})$.

Applying Eqs. (<ref>)-(<ref>) to expand Eq. (<ref>) and comparing coefficients of $t^s$,  Eq. (<ref>) is equivalent to the following equations

\begin{equation}\label{eqpro5.237}
\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_n=s\\
i_1,\cdots,i_n\geq 0\end{array}$}}[\frkT_{i_1}u_1,\cdots,\frkT_{i_n}u_n]_{\g}=\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_n=s\\
i_1,\cdots,i_n\geq 0\end{array}$}}\frkT_{i_1}(\sum\limits_{k=1}^n(-1)^{n-k}\rho(\frkT_{i_2}u_1,\cdots,\frkT_{i_k}u_{k-1},\hat{u_k},\frkT_{i_{k+1}}u_{k+1},\cdots,\frkT_{i_n}u_n)(u_k)),
\end{equation}

for all $s\geq 0$ and $u_1,\cdots,u_n\in V$.

Let $T$ be a relative Rota-Baxter operator on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. An order $m$ deformation of the relative Rota-Baxter operator $T$  is a sequence of linear maps $\frkT_i\in \Hom_{\K}(V,\g)$ for $0\leq i\leq m$ with $\frkT_0=T$ such that $\K[t]/(t^{m+1})$-module map $T_t=\sum_{i=0}^{m}\frkT_it^i$ from $V[t]/(t^{m+1})$ to the $n$-Lie algebra $\g[t]/(t^{m+1})$ satisfies

\begin{equation}\label{40}
\end{equation}

where $u_1,u_2,\cdots u_n\in V$.

We call an order $1$ deformation of a relative Rota-Baxter operator $T$ on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$ an infinitesimal deformation and denote it by $(\frak g,\frkT_1)$.

By direct calculations, $(\g,\frkT_1)$ is an infinitesimal deformation of a relative Rota-Baxter operator $T$ on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$  if and only if for all $u_1,u_2,\cdots,u_n\in \g$,

\begin{eqnarray*}
\label{eq:1-cocycle}&&[\frkT_1u_1,Tu_2,\cdots,Tu_n]_\g+\cdots+[Tu_1,\cdots,\frkT_1u_n]_\g \\
\nonumber&=& T(\sum\limits_{i,j=1,i\neq j}^n(-1)^{n-j}\rho(Tu_1,\cdots,\widehat{Tu_j},\cdots,\frkT_1 u_i,Tu_{i+1},\cdots,Tu_n)(u_j))\\
\nonumber&&+\frkT_1(\sum\limits_{i=1}^n(-1)^{n-i}\rho(Tu_1,\cdots,\widehat{Tu_i},\cdots,Tu_n)(u_i)),
\end{eqnarray*}

which implies that $\frkT_1$ is a 1-cocycle for the relative Rota-Baxter operator $T$, i.e. $d\frkT_1=0$.

Two infinitesimal deformations  $(\g,\frkT_1)$ and $(\g,\frkT'_1)$  of a
relative Rota-Baxter operator $T$ are said to be equivalent if there exists $\frkX\in \wedge^{n-1}\g$, such that for $\phi_t=\Id_{\g}+t\ad_{\frkX}$ and $\varphi_t=\Id_{V}+t\rho(\frkX)$, the following conditions hold:

(i) $[\phi_t(x_1),\cdots,\phi_t(x_{n})]_{\g}=\phi_t[x_1,\cdots,x_n]_{\g}$ modulo $t^2$ for all $x_1,\cdots,x_n$;

(ii) $\varphi_t\rho(x_1,\cdots,x_{n-1})(u)=\rho(\phi_t(x_1),\cdots,\phi_t(x_{n-1}))(\varphi_t(u)))$ modulo $t^2$ for all $x_1,\cdots,x_{n-1}\in \g,u\in V$;

(iii) $T_t\circ \varphi_t=\phi_t\circ{T'_t}$  modulo $t^2$, where $T_t=T+t \frkT_1$ and $T'_t=T+t \frkT'_1$.

By Eq. (<ref>) in the definition of $n$-Lie algebra and Eq. (<ref>) in the definition of representation of an $n$-Lie algebra, the conditions (i) and (ii) follow.

By the relation $T_t\circ \varphi_t=\phi_t\circ{T'_t}$, we have

$${\frkT'_1}(v)=\frkT_1(v)+T\rho(\frkX)(v)-[\frkX,Tv]_\g=\frkT_1(v)+(d\frkX)(v), ~v\in V,$$

which implies that ${\frkT'_1}-\frkT_1=d\frkX$. Thus we have

There is a one-to-one correspondence between the space of equivalence classes of infinitesimal deformations of a relative Rota-Baxter operator $T$ and the first cohomology group $\huaH^1(V,\g)$.

Let $T_t=\sum_{i=0}^{m}\frkT_it^i$ be an order $m$ deformation of a relative Rota-Baxter operator $T$ on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. $T_t$ is said to be extendable if there exists a $1$-cochain $\frkT_{m+1}\in \Hom(V,\g)$ such that $\tilde{T_t}=T_t+\frkT_{m+1}t^{m+1}$ is an order $m+1$ deformation of the relative Rota-Baxter operator $T$.

Let $T_t=\sum_{i=0}^{m}\frkT_it^i$ be an order $m$ deformation of a relative Rota-Baxter operator $T$ on an  $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. Define $\Theta_m\in C_{T}^2(V;\g)$ by

\begin{eqnarray}
\label{eq37}\Theta_m(u_1,u_2,\cdots,u_n)&=&\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_n=m+1\\
0\leq i_1,\cdots,i_n\leq m\end{array}$}}\Big([\frkT_{i_1}u_1,\cdots,\frkT_{i_n}u_n]_{\g}\\
\nonumber&&-\frkT_{i_1}(\sum\limits_{k=1}^n(-1)^{n-k}\rho(\frkT_{i_2}u_1,
\cdots,\frkT_{i_k}u_{k-1},\frkT_{i_{k+1}}u_{k+1},\cdots,\frkT_{i_n}u_n)(u_k))\Big),
\end{eqnarray}

where $u_1,\cdots,u_n\in V$.

The $2$-cochain $\Theta_m$ is a $2$-cocycle, that is, $d\Theta_m=0$.

Using Eq. (<ref>), we have

\begin{eqnarray}\label{eq:2-cochain1}
&&\{\frkT_{i_1},\cdots,\frkT_{i_n}\}(u_1,\cdots,u_n)=\sum\limits_{\sigma\in S_n}[\frkT_{i_{\sigma(1)}}(u_1),\cdots,\frkT_{i_{\sigma(n)}}(u_n)]_\g\\
\nonumber&& -\sum\limits_{\sigma\in S_n}\frkT_{i_{\sigma(1)}}\sum\limits_{k=1}^n(-1)^{n-k}\rho\big(\frkT_{i_{\sigma(2)}}(u_1),\cdots,\frkT_{i_{\sigma(k)}}(u_{k-1}),\frkT_{i_{\sigma(k+1)}}(u_{k+1}),\cdots,\frkT_{i_{\sigma(n)}}(u_n)\big) (u_k).
\end{eqnarray}

By Eq. (<ref>) and Eq. (<ref>), we deduce that

\begin{equation}\label{eq:2-cochain2}
\Theta_m=1/n!\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_n=m+1\\
0\leq i_1,\cdots,i_n\leq m\end{array}$}}\{\frkT_{i_1},\cdots,\frkT_{i_n}\}=\sum\limits_{j=0}^{n-2}\frac{C_n^j}{n!}\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_{n-j}=m+1\\
1\leq i_1,\cdots,i_{n-j}\leq m\end{array}$}}\{\underbrace{T,\cdots,T}_{j},\frkT_{i_1},\cdots,\frkT_{i_{n-j}}\}.
\end{equation}

Since $T_t$ is an order $m$ deformation of the relative Rota-Baxter operator $T$, by (<ref>) and (<ref>), we have

\begin{eqnarray}
\label{eq:(42)s+1}&=&\sum\limits_{j=0}^{n-2}\frac{C_n^j}{n!}\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_{n-j}=s\\
1\leq i_1,\cdots,i_{n-j}\leq s-1\end{array}$}}\{\underbrace{T,\cdots,T}_{j},\frkT_{i_1},\cdots,\frkT_{i_{n-j}}\}.
\end{eqnarray}

By Theorem <ref> and Eq. (<ref>), we have

\begin{eqnarray*}
d \Theta_m&=&-\frac{1}{(n-1)!}\{\underbrace{T,\cdots,T}_{n-1},\Theta_m\}\\
i_1+\cdots i_{n-j}=m+1\\
1\leq i_1,\cdots,i_{n-j}\leq m\end{array}$}}\{\{\frkT_{i_1},\cdots,\frkT_{i_{n-j}},\underbrace{T,\cdots,T}_{j}\},T,\cdots,T\}\\
&\stackrel{\eqref{eq:general-JI}}=& \frac{1}{(n-1)!}\sum\limits_{j=0}^{n-2}\frac{C_n^j}{n!}\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_{n-j}=m+1\\
1\leq i_1,\cdots,i_{n-j}\leq m\end{array}$}}\sum\limits_{k=j+1}^{n-1}\frac{C_{n-1+j}^k\cdot C_{n-j}^{n-k}}{C_{n-1+j}^{j}}\{\{\frkT_{i_1},\cdots,\frkT_{i_{n-k}},\underbrace{T,\cdots,T}_{k}\},\frkT_{i_{n-k+1}},\cdots,\frkT_{i_{n-j}},T,\cdots,T\}\\
&=&\frac{1}{(n-1)!}\sum\limits_{0\leq p<k\leq n-2}^{n-2}\frac{C_n^{p}}{n!}\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_{n-p}=m+1\\
1\leq i_1,\cdots,i_{n-p}\leq m\end{array}$}}\frac{C_{n-1+p}^k\cdot C_{n-p}^{n-k}}{C_{n-1+p}^{p}}\{\{\frkT_{i_1},\cdots,\frkT_{i_{n-k}},\underbrace{T,\cdots,T}_{k}\},\frkT_{i_{n-k+1}},\cdots,\frkT_{i_{n-p}},T,\cdots,T\}\\
i_1+\cdots i_{n-j}=m+1\\
1\leq i_1,\cdots,i_{n-j}\leq m\end{array}$}}C_{n-j}^1\{\{\frkT_{i_1},T,\cdots,T\},\frkT_{i_2},\cdots,\frkT_{i_{n-j}},T,\cdots,T\}\\
&\stackrel{\eqref{eq:(42)s+1}}=&\frac{1}{(n-1)!}\sum\limits_{0\leq {p}<k\leq n-2}^{n-2}\frac{C_n^{p}}{n!}\sum\limits_{\mbox{\tiny$\begin{array}{c}
i_1+\cdots i_{n-p}=m+1\\
1\leq i_1,\cdots,i_{n-p}\leq m\end{array}$}}\frac{C_{n-1+p}^k\cdot C_{n-p}^{n-k}}{C_{n-1+p}^{p}}\{\{\frkT_{i_1},\cdots,\frkT_{i_{n-k}},\underbrace{T,\cdots,T}_{k}\},\frkT_{i_{n-k+1}},\cdots,\frkT_{i_{n-p}},T,\cdots,T\}\\
i^{'}_1+\cdots i^{'}_{n-k}+i_2+\cdots+i_{n-j}=m+1\\
1\leq i^{'}_1,\cdots,i^{'}_{n-k},i_2,\cdots,i_{n-j}\leq m\end{array}$}}\frac{C_n^{j}}{n!}C_{n-j}^1\{\{\frkT_{i_1^{'}},\cdots,\frkT_{i_{n-k}^{'}},\underbrace{T,\cdots,T}_{k}\},\frkT_{i_2},\cdots,
\frkT_{i_{n-j}},T,\cdots,T\}\\
0< k+j\leq n-2\\
0\leq k\leq n-2\end{array}$}}\frac{C_n^{j}}{n!}\frac{C_n^{k}}{n!}C_{n-j}^1\sum\limits_{\mbox{\tiny$\begin{array}{c}
i^{'}_1+\cdots i^{'}_{n-k}+i_2+\cdots+i_{n-j}=m+1\\
1\leq i^{'}_1,\cdots,i^{'}_{n-k},i_2,\cdots,i_{n-j}\leq m\end{array}$}}\{\{\frkT_{i_1^{'}},\cdots,\frkT_{i_{n-k}^{'}},\underbrace{T,\cdots,T}_{k}\},\frkT_{i_2},\cdots,\frkT_{i_{n-j}},T,\cdots,T\}\\
\end{eqnarray*}

which implies that the $2$-cochain $\Theta_m$ is a cocycle.

Moreover, we have

Let $T_t=\sum_{i=0}^m\frkT_it^i$ be an order $m$ deformation of a relative Rota-Baxter operator $T$ on an $n$-pair $(\frak g,[-,\cdots,-]_{\frak g};\rho)$. Then $T_t$ is extendable if and only if the cohomology group ${[\Theta_m]}$ in $\huaH^2(V,\g)$ is trivial.

Assume that an order $m$ deformation $T_t$ of the relative Rota-Baxter operator $T$ can be extended to an order $m+1$ deformation. By Eq. (<ref>) with $s=m+1$, Eq. (<ref>) holds. Thus, we have $\Theta_m=-d\frkT_{m+1}$, which implies that the cohomology group ${[\Theta_m]}$ is trivial.

Conversely, if the cohomology group  ${[\Theta_m]}$ is trivial, then there exists a $1$-cochain $\frkT_{m+1} \in \Hom_{\K}(V,\g)$ such that ${\Theta_m}=d\frkT_{m+1}$.
Set $\tilde{T_t}=T_t+\frkT_{m+1}t^{m+1}$. Then $\tilde{T_t}$ satisfies Eq. (<ref>) for $0\leq s\leq m+1$. Thus $\tilde{T_t}$ is an order $m+1$ deformation, which implies that $T_t$ is extendable.

§ FROM COHOMOLOGY GROUPS OF RELATIVE ROTA-BAXTER OPERATORS ON $N$-LIE ALGEBRAS TO THOSE ON $(N+1)$-LIE ALGEBRAS

Motivated by the construction of $3$-Lie algebras from the general linear Lie algebras with trace
forms in [32], the authors in [6]  provide a construction of $(n+1)$-Lie algebras from $n$-Lie algebras and certain linear functions.
Let $(\g,[-,\cdots,-]_\g)$ be an $n$-Lie algebra and $\g^*$ the dual space of $\g$. Suppose $f\in \g^*$ satisfies $f([x_1,\cdots,x_n]_\g)=0$ for all $x_i\in \g$. Then $(\g,\{-,\cdots,-\})$ is an $(n+1)$-Lie algebra, where the bracket is given by

\begin{equation}
\{x_1,\cdots,x_{n+1}\}=\sum\limits_{i=1}^{n+1}(-1)^{i-1}f(x_i)[x_1,\cdots,\hat{x_i},\cdots,x_{n+1}]_\g,\quad\forall~x_i\in\g.
\end{equation}

The $(n+1)$-Lie algebra constructed as above is denoted by $\g_f$.

Let $(\g,[-,\cdots,-]_{\g};\rho)$ be an $n$-pair. Define $\varrho:\wedge^{n}\g\rightarrow\gl(V)$ by

\begin{equation}
\varrho(x_1,\cdots,x_n)=\sum\limits_{i=1}^{n}(-1)^{i-1}f(x_i)\rho(x_1,\cdots,\hat{x_i},\cdots,x_n).
\end{equation}

Then $(\g_f;\varrho)$ is an $(n+1)$-pair.

Since $(\g,[-,\cdots,-]_{\g};\rho)$ is an $n$-pair and $f([x_1,\cdots,x_n])=0$ for all $x_i\in \g$, by a direct calculation, we have

\begin{eqnarray*}
\sum\limits_{j=1}^{n}(-1)^{j-1}f(y_j)\rho(y_1,\cdots,\hat{y_j},\cdots,y_n)]\\
\rho(y_1,\cdots,\hat{y_j},\cdots,y_n)]\\
\end{eqnarray*}

which implies that (<ref>) holds.

Similarly , we have

\begin{eqnarray*}
,\cdots,y_{n+1})\rho(x_1,\cdots,\hat{x_k},\cdots,x_{n-1},y_i) \\
,\cdots,y_{n+1})\rho(x_1,\cdots,\hat{x_k},\cdots,x_{n-1},y_i) \big)=0,
\end{eqnarray*}

which implies that (<ref>) holds. Thus $(\g_f;\varrho)$ is an $(n+1)$-pair.

Denote the cochain complexes of $n$-Lie algebra $\g$ associated to the representation $\rho$ and $(n+1)$-Lie algebra $\g_f$ associated to the representation $\varrho$ by $(\oplus^{+\infty}_{m=1}C^{m}_{\nl}(\g;V),\partial_\rho)$ and $(\oplus^{+\infty}_{m=1}C^{m}_{ {(n+1)-\rm{Lie}}}(\g_f;V),\tilde{\partial}_\varrho)$, respectively.

Let $(\frak g,[-,\cdots,-];\rho)$ be an $n$-pair and $(\g_f;\varrho)$ be the corresponding $(n+1)$-pair. For $P\in C^{m+1}(\g;V)~(m\geq 1)$, define $\widetilde{P}(\frkX_1,\cdots,\frkX_m,x)$ by

\begin{eqnarray}\label{eq:lift-cocycle}
&&\widetilde{P}(\frkX_1,\cdots,\frkX_m,x):=\sum\limits_{i_1,\cdots,i_m=1}^{n}(-1)^{i_1+\cdots+i_m-m}f(x_1^{i_1})\cdots f(x_m^{i_m})
\nonumber&&+\sum\limits_{i_1,\cdots,i_{m-1}=1}^{n}(-1)^{i_1+\cdots+i_{m-1}+n+1-m}f(x_1^{i_1})\cdots f(x_{m-1}^{i_{m-1}})f(x)
\end{eqnarray}

where $\frkX_j:=x_j^1\wedge\cdots \wedge x_j^n$ and $\frkX_j^{\widehat{{i_j}}}:=x_j^1\wedge \cdots \wedge\widehat{x_j^{i_j}}\cdots\wedge x_j^n$. Then $\tilde{P}\in {C^{m+1}}(\g_f;V)$, i.e., $\tilde{P}$ is an $(m+1)$-cochain of the $n$-Lie algebra $\g_f$. Thus we obtain a well-defined linear map

$$\Phi:\oplus^{+\infty}_{m=1}C^{m}_{\nl}(\g;V)\longrightarrow \oplus^{+\infty}_{m=1}C^{m}_{ {(n+1)-\rm{Lie}}}(\g_f;V)$$

defined by

\begin{eqnarray}
\Phi(P)=
\begin{cases}
\widetilde{P}, & \forall~P\in C^{m}(\g,V)~(m\geq2),\\
P,& \forall~P\in C^{1}(\g,V).
\end{cases}
\end{eqnarray}

Furthermore, we also have $\tilde{\partial}_\varrho \circ \Phi=\Phi\circ\partial_\rho $, i.e., $\Phi$ is a chain map between $(\oplus^{+\infty}_{m=1}C^{m}_{\nl}(\g;V),\partial_\rho)$ and $(\oplus^{+\infty}_{m=1}C^{m}_{ {(n+1)-\rm{Lie}}}(\g_f;V),\tilde{\partial}_\varrho)$. Thus $\Phi$ induces a map
$$\Phi_\ast:\oplus^{+\infty}_{m=1}H^m_{\nl}(\g;V)\longrightarrow\oplus^{+\infty}_{m=1}H^m_{ {(n+1)-\rm{Lie}}}(\g_f;V) $$

given by

$$\Phi_\ast([P])=[\Phi(P)],\quad \forall~\in [P]\in H^m_{\nl}(\g;V).$$

It follows by a long but straightforward computation.

Let $T:V\rightarrow \g$ be a relative Rota-Baxter operator on an $n$-pair $(\g,[-,\cdots,-]_{\g};\rho)$. Then $T$ is also a relative Rota-Baxter operator on an $(n+1)$-pair $(\g_f;\varrho)$.

By that the fact that $T:V\rightarrow \g$ is a relative Rota-Baxter operator on the $n$-pair $(\g;\rho)$, we have

\begin{eqnarray*}
\end{eqnarray*}

This implies that $T$ is a relative Rota-Baxter operator on an $(n+1)$-pair $(\g_f;\varrho)$.

If $T:V\rightarrow \g$ is a relative Rota-Baxter operator on an $n$-pair $(\g,[-,\cdots,-]_{\g};\rho)$, by Proposition <ref>, $T$ is a relative Rota-Baxter operator on an $(n+1)$-pair $(\g_f;\varrho)$. Thus by Proposition <ref>,  $(V,\Courant{-,\cdots,-}_T)$ is an $(n+1)$-Lie algebra, where the bracket $\Courant{-,\cdots,-}_T$ is given by

\begin{eqnarray}
\Courant{u_1,\cdots,u_n,u_{n+1}}_{T}&=&\sum\limits^{n+1}_{i=1}(-1)^{n+1-i}\varrho(Tu_1,\cdots,\widehat{Tu_i},\cdots,Tu_n,Tu_{n+1})(u_i),\quad \forall~u_1,\cdots,u_{n+1}\in V.
\end{eqnarray}

Furthermore, by a direct calculation, we have

\begin{equation}
\Courant{u_1,\cdots,u_n,u_{n+1}}_{T}=\sum\limits_{i=1}^{n+1}(-1)^{i-1}(f\circ T)(u_i)[u_1,\cdots,\hat{u_i},\cdots,u_n,u_{n+1}]_T,\quad \forall~u_1,\cdots,u_n\in V.
\end{equation}

By the fact that $T$ is a relative Rota-Baxter operator on an $n$-pair $(\g,[-,\cdots,-]_{\g};\rho)$ and $f([x_1,\cdots,x_n]_\g)=0$ for $x_1,\cdots,x_n\in\g$, we have

\begin{eqnarray}
(f\circ T)[u_1,\cdots,u_{n}]_T=f([T u_1,\cdots,T u_n]_\g)=0.
\end{eqnarray}

Thus we have

The $(n+1)$-Lie algebra $(V,\Courant{-,\cdots,-}_T)$ is just the $(n+1)$-Lie algebra constructed by the $n$-Lie algebra $(V,[-,\cdots,-]_T)$ with $ f\circ T\in V^*$ satisfying $(f\circ T)[u_1,\cdots,u_{n}]_T=0$ through the way given in Lemma <ref>.

Denote the cochain complexes of the relative Rota-Baxter operator $T$ on both  the $n$-pair $(\frak g;\rho)$  and the $(n+1)$-pair $(\frak g_f;\varrho)$ by $(\oplus^{+\infty}_{m=0}C^{m}_T(V;\g),d)$ and $(\oplus^{+\infty}_{m=0}C^{m}_{T}(V;\g_f),\tilde{d})$, respectively.

With the above notations, there is  a chain map

$$\Phi:(\oplus^{+\infty}_{m=0}C^{m}_T(V;\g),d)\longrightarrow (\oplus^{+\infty}_{m=0}C^{m}_{T}(V;\g_f),\tilde{d})$$

defined by

\begin{eqnarray}
\Phi(P)=
\begin{cases}
\widetilde{P}, & \forall~P\in C^{m}(V,\g)~(m\geq2),\\
P,& \forall~P\in C^{1}(V,\g),\\
P\wedge x_0, & \forall~P\in \wedge^{n-1} \g,
\end{cases}
\end{eqnarray}

where $x_0$ is an element in the center of the semi-direct product $n$-Lie algebra $\g\ltimes_{\rho} V$ and $\widetilde{P}$ is given by

\begin{eqnarray}\label{eq:RB-lift}
&&\widetilde{P}(\frkU_1,\cdots,\frkU_{m-1},v):=\sum\limits_{i_1,\cdots,i_{m-1}=1}^{n}(-1)^{i_1+\cdots+i_{m-1}-m+1}(f\circ T)(u_1^{i_1})\cdots (f\circ T)(u_{m-1}^{i_{m-1}})
\nonumber&&+\sum\limits_{i_1,\cdots,i_{m-2}=1}^{n}(-1)^{i_1+\cdots+i_{m-1}+n-m}(f\circ T)(u_1^{i_1})\cdots (f\circ T)(u_{m-2}^{i_{m-2}})(f\circ T)(v)
\end{eqnarray}

where $\frkU_j:=u_j^1\wedge\cdots \wedge u_j^n$ and $\frkU_j^{\widehat{{i_j}}}:=u_j^1\wedge \cdots \wedge\widehat{u_j^{i_j}}\cdots\wedge u_j^n$.

Thus $\Phi$ induces a map
$$\Phi_\ast:\oplus^{+\infty}_{m=0}\huaH^m_{\nl}(V;\g)\longrightarrow\oplus^{+\infty}_{m=0}\huaH^m_{ {(n+1)-\rm{Lie}}}(V;\g_f) $$

given by

$$\Phi_\ast([P])=[\Phi(P)],\quad \forall~\in [P]\in \huaH^m_{\nl}(V;\g).$$

Note that the cochain complex $(\oplus^{+\infty}_{m=1}C^{m}_T(V;\g),d)$ of the relative Rota-Baxter operator $T$ on  the $n$-pair $(\frak g;\rho)$  is just the cochain complexes of the $n$-Lie algebra $(V,[-,\cdots,-]_T)$ associated to the representation $(\g;\rho_T)$ and the cochain complexes $(\oplus^{+\infty}_{m=1}C^{m}_T(V;\g_f),\tilde{d})$ of the relative Rota-Baxter operator $T$ on  the $(n+1)$-pair $(\frak g_f;\varrho)$  is just the cochain complexes of the $(n+1)$-Lie algebra $(V,[-,\cdots,-]_T)$ associated to the representation $(\g;\varrho_T)$.  By Proposition <ref>,  Eq. (<ref>) is just Eq. (<ref>) by replacing  the $n$-Lie algebra $(\g,[-,\cdots,-]_\g)$ and $f\in \g^*$ with the  $n$-Lie algebra $(V,[-,\cdots,-]_T)$ and  $ f\circ T\in V^*$. Thus by Theorem <ref>,
$$\Phi:(\oplus^{+\infty}_{m=1}C^{m}_T(V;\g),d)\longrightarrow (\oplus^{+\infty}_{m=1}C^{m}_{T}(V;\g_f),\tilde{d})$$

given by (<ref>) is a chain map.

By the fact that $x_0$ is an element in the center of the semi-direct product $n$-Lie algebra $\g\ltimes_{\rho} V$, it is straightforward to check that

$$\Phi (d \frkX)=\tilde{d}\Phi (\frkX),\quad \forall~\frkX\in \wedge^{n-1}\g.$$

Thus $\Phi$ is a chain map. The rest is direct.

Acknowledgements.  This research is supported by NSFC (11771410,11801066,11901501). We give our warmest thanks to Yunhe Sheng and Rong Tang for very useful comments and discussions.

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