Representations and cohomolgies of Rota-Baxter 3-Lie algebras

Qinxiu Sun

Department of Mathematics, Zhejiang University of Science and Technology,
Hangzhou, China, 310023

E-mail<EMAIL_ADDRESS>
Shan Chen

Department of Mathematics, Zhejiang University of Science and Technology,
Hangzhou, China, 310023

E-mail<EMAIL_ADDRESS>
###### Abstract

The goal of the present paper is to investigate representations and
cohomologies of Rota-Baxter 3-Lie algebras with any weight. We introduce
representations, matched pairs and Manin triples of Rota-Baxter 3-Lie
algebras. Furthermore, we discuss cohomology theory of Rota-Baxter 3-Lie
algebras. The deformations and central extensions of Rota-Baxter 3-Lie
algebras are also studied.

MR Subject Classification 2010: 17B38,17B62, 16T10, 81R60.

††Keywords: 3-Lie algebra, Rota-Baxter operator, representation, cohomology,
matched pair, Manin triple,$\mathcal{O}$-operator

##  1 Introduction

Rota-Baxter operators are an algebraic abstraction of the integral operator,
which originated from Baxter’s study of the fluctuation theory in probability
[5]. Since then, they have been related to many topics in mathematics and
mathematical physics. It has close connection to combinatorics [21, 23],
renormalization in quantum field theory [9], multiple zeta values in number
theory [25], Yang Baxter equations [1, 3], algebraic operad [6] and so on.

Cohomologies and deformations are vital roles both in mathematics and physics.
Algebraic deformation theory arose initially in Gerstenhaber’s works for rings
and algebras in [19, 20]. Furthermore, Nijenhuis and Richardson considered the
deformation theory of Lie algebras [29, 30]. And followed by these extension
to n-Lie algebras[2, 10, 18, 31, 34]. More generally, Balavoine [4]
investigated deformation theory of quadratic operads.

Due to the importance of Rota-Baxter algebras, it is inevitable to investigate
their cohomologies with any weight. Recently, a great deal of works have been
devoted to this subject. The deformation and cohomology theories of
$\mathcal{O}$-operators (also named relative Rota-Baxter operators) on Lie
algebras were investigated in [35]. The authors explored deformation and
cohomology theories of relative Rota-Baxter Lie algebras of weight zero and
found applications to triangular Lie bialgebras in [28]. In the same
direction, Das gave cohomology and deformation theories of associative Rota-
Baxter operators of weight 0 in [11, 16] and relative Rota-Baxter operators
with any weight [14]. The cohomology and deformation of twisted Rota-Baxter
operators were considered in [12, 13]. Another breakthrough gained by Wang and
Zhou in [37] is cohomology and homotopy theories of Rota-Baxter algebras with
any weight. Das constructed cohomologies of Rota-Baxter operators of any
weight on Lie algebras [15]. Guo, Lang and Sheng in [22] introduced the
notions of Rota-Baxter Lie groups and Rota-Baxter Lie bialgebras [27], and
cohomologies of Rota-Baxter operators of weight 1 on Lie groups was
investigated in [26]. There are some other works on differential algebras with
any weight [24, 32, 33].

Inspired by the previous works, we would like to study the representation and
cohomological theory of Rota-Baxter 3-Lie algebras.

The paper is organized as follows. In Section 2, we consider representations,
matched pairs and Manin triples of Rota-Baxter 3-Lie algebras. The relation
between matched pairs and Manin triples are also characterized. In Section 3,
we investigate the cohomologies of a Rota-Baxter 3-Lie algebra with
coefficients in a representation. In Section 4, we study the deformations of
Rota-Baxter 3-Lie algebras. We also introduce the notions of a Nijenhuis
operator and $\mathcal{O}$-operator on Rota-Baxter 3-Lie algebras. Finally, we
consider the the central extensions of Rota-Baxter 3-Lie algebras.

##  2 Representations and matched pairs of Rota-Baxter 3-Lie algebras

A 3-Lie algebra [17] is a vector space $A$ together with a skew-symmetric
linear map $[\cdot,\cdot,\cdot]:\wedge^{3}A\longrightarrow A$ satisfying

$[x_{1},x_{2},[y_{1},y_{2},y_{3}]]=[[x_{1},x_{2},y_{1}],y_{2},y_{3}]+[y_{1},[x_{1},x_{2},y_{2}],y_{3}]+[y_{1},y_{2},[x_{1},x_{2},y_{3}]]$

for all $x_{i},y_{i}\in A$.

Let $A$ be a 3-Lie algebra over field $k$ and $\lambda\in k$. A linear
operator $R:A\longrightarrow A$ is said to be a Rota-Baxter operator of weight
$\lambda$ if it satisfies

$\displaystyle[R(x),R(y),R(z)]$ $\displaystyle=$ $\displaystyle
R([R(x),R(y),z]+[R(x),y,R(z)]+[x,R(y),R(z)]$
$\displaystyle+\lambda([R(x),y,z]+[x,R(y),z]+[x,y,R(z)])+\lambda^{2}[x,y,z]),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.1)$

for any $x,y,z\in A$. We call $(A,R)$ a Rota-Baxter 3-Lie algebra of weight
$\lambda$.

In view of [8], we know that if $(A,R)$ is a Rota-Baxter 3-Lie algebra of
weight $\lambda$, then $(A,[\ ,\ ,\ ]_{R},R)$ is also a Rota-Baxter 3-Lie
algebra of weight $\lambda$, where $[\ ,\ ,\ ]_{R}$ is given by

$\displaystyle[x,y,z]_{R}$ $\displaystyle=$
$\displaystyle[R(x),R(y),z]+[R(x),y,R(z)]+[x,R(y),R(z)]$
$\displaystyle+\lambda([R(x),y,z]+[x,R(y),z]+[x,y,R(z)])+\lambda^{2}[x,y,z],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.2)$

for any $x,y,z\in A$. Denote it by $(A_{R},R)$.

Example 2.1. Let $({\mathfrak{g}},[\ ,\ ,\ ])$ be 3-dimensional 3-Lie algebra
with basis ${\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$ and the nonzero
multiplication is given by

$[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}]=\varepsilon_{1}.$

Define linear map $R:{\mathfrak{g}}\longrightarrow{\mathfrak{g}}$ by the
matrix $\begin{bmatrix}r_{11}&r_{12}&r_{13}\\\ r_{21}&r_{22}&r_{23}\\\
r_{31}&r_{32}&r_{33}\end{bmatrix}$ with respect to the basis
${\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$. Then the linear map
$R:A\longrightarrow A$ given by the matrix $\begin{bmatrix}1&a&b\\\ 0&1&c\\\
0&d&1\end{bmatrix}$ with respect to the basis
${\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$ is a Rota-Baxter operator
of weight $-1$ and the linear map $R:A\longrightarrow A$ given by the matrix
$\begin{bmatrix}4&a&b\\\ 0&1&c\\\ 0&d&3\end{bmatrix}$ with respect to the
basis ${\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}$ is a Rota-Baxter
operator of weight $-2$.

A representation of a 3-Lie algebra $A$ consisting of a vector space $V$
together with a skew-symmetric bilinear map $\rho:A\wedge
A\longrightarrow\mathfrak{gl}(V)$ satisfying

$\rho(x_{1},x_{2})\rho(x_{3},x_{4})-\rho(x_{3},x_{4})\rho(x_{1},x_{2})=\rho([x_{1},x_{2},x_{3}],x_{4})-\rho([x_{1},x_{2},x_{4}],x_{3}),$
$\rho([x_{1},x_{2},x_{3}],x_{4})=\rho(x_{1},x_{2})\rho(x_{3},x_{4})+\rho(x_{2},x_{3})\rho(x_{1},x_{4})+\rho(x_{3},x_{1})\rho(x_{2},x_{4}),$

for all $x_{i}\in A,~{}1\leq i\leq 4.$ We also denote $\rho(x_{1},x_{2})$ by
$\rho(x_{1}\wedge x_{2})$ in the following.

One can recall the knowledge on representation of a 3-Lie algebra in [7].

Definition 2.2. Let $(A,R)$ be a Rota-Baxter 3-Lie algebra of weight
$\lambda$. A representation on Rota-Baxter 3-Lie algebra $(A,R)$ is a triple
$(V,R_{V},\rho)$, where $(V,\rho)$ is a representation of 3-Lie algebra $A$
and satisfies

$\displaystyle\rho(R(x_{1}),R(x_{2}))R_{V}$ $\displaystyle=$ $\displaystyle
R_{V}\rho(R(x_{1}),R(x_{2}))+R_{V}(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2}))R_{V}$
$\displaystyle+\lambda
R_{V}(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2})),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.3)$

where $x_{1},x_{2}\in A$.

Let $(V,\rho)$ be a representation of 3-Lie algebra $A$. Define
$\rho^{*}:\wedge^{2}A\longrightarrow\mathfrak{gl}(V)$ by

$\langle\rho^{*}(x,y)v^{*},w\rangle=-\langle v^{*},\rho(x_{1},x_{2})w\rangle$

for any $v^{*}\in V^{*},w\in V,x,y\in A.$ Then $(V^{*},\rho^{*})$ is again a
representation of $A$. For the case of Rota-Baxter 3-Lie algebras, we have the
following result.

Proposition 2.3. Let $(V,\rho,R_{V})$ be a representation of Rota-Baxter 3-Lie
algebra $(A,R)$. Then $(V^{*},\rho^{*},-\lambda I-R_{V}^{*})$ is also a
representation of $(A,R)$, where $I$ is the identity map on $V^{*}$. We call
it the dual representation.

###### Proof.

We only need to check that $(\rho^{*},-\lambda I-R_{V}^{*})$ holds for (2.3).
In fact, for any $u^{*}\in V^{*},v\in V$, using (2.3), we get

$\displaystyle\langle(\lambda
I+R_{V}^{*})\rho^{*}(R(x_{1}),R(x_{2}))u^{*}-(\lambda
I+R_{V}^{*})(\rho^{*}(R(x_{1}),x_{2})+\rho^{*}(x_{1},R(x_{2}))$
$\displaystyle+\lambda\rho^{*}(x_{1},x_{2}))(\lambda
I+R_{V}^{*})u^{*}+\lambda(\lambda
I+R_{V}^{*})(\rho^{*}(R(x_{1}),x_{2})+\rho^{*}(x_{1},R(x_{2}))+\lambda\rho^{*}(x_{1},x_{2}))u^{*}$
$\displaystyle-\rho^{*}(R(x_{1}),R(x_{2}))(\lambda I+R_{V}^{*})u^{*},v\rangle$
$\displaystyle=$ $\displaystyle\langle u^{*},\rho(R(x_{1}),R(x_{2}))(\lambda
I+R_{V})v-(\lambda
I+R_{V})(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2}))$
$\displaystyle\times(\lambda
I+R_{V})v+\lambda(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2}))(\lambda
I+R_{V})v-(\lambda I+R_{V})\rho(R(X))v\rangle$ $\displaystyle=$ $\displaystyle
0,$

which yields that

$\displaystyle(\lambda I+R_{V}^{*})\rho^{*}(R(x_{1}),R(x_{2}))-(\lambda
I+R_{V}^{*})(\rho^{*}(R(x_{1}),x_{2})+\rho^{*}(x_{1},R(x_{2}))+\lambda\rho^{*}(x_{1},x_{2}))(\lambda
I+R_{V}^{*})$ $\displaystyle+\lambda(\lambda
I+R_{V}^{*})(\rho^{*}(R(x_{1}),x_{2})+\rho^{*}(x_{1},R(x_{2}))+\lambda\rho^{*}(x_{1},x_{2}))-\rho^{*}(R(x_{1}),R(x_{2}))(\lambda
I+R_{V}^{*})=0,$

that is, $(\rho^{*},-\lambda I-R_{V}^{*})$ holds for (2.3). ∎

Example 2.4. Let $(A,R)$ be a Rota-Baxter 3-Lie algebra of weight $\lambda$
and define ${ad}:A\wedge A\longrightarrow A$ by
${ad}(x_{1},x_{2})(x_{3})=[x_{1},x_{2},x_{3}]$. Then $(A,{ad},R)$ is a
representation of Rota-Baxter 3-Lie algebra $(A,R)$. We call it the adjoint
representation. Moreover, $(A^{*},{ad}^{*},-\lambda I-R^{*})$ is the dual
representation, where $I$ is the identity map on $A^{*}$..

Proposition 2.5. Let $(A,R)$ be a Rota-Baxter 3-Lie algebra of weight
$\lambda$ and $(V,R_{V},\rho)$ be representation of it. Define
$\tilde{\rho}:A\wedge A\longrightarrow\mathfrak{gl}(V)$ by

$\tilde{\rho}(x_{1},x_{2})=\rho(R(x_{1}),R(x_{2}))-R_{V}(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2})).$
$None$

Then $(V,R_{V},\tilde{\rho})$ is a representation of $(A_{R},R)$. We denote it
by $\tilde{V}$.

###### Proof.

By routine calculation, $(V,\tilde{\rho})$ is a representation of $A_{R}$. We
only need to check that (2.3) holds for $\tilde{\rho}$. In fact, due to (2.3)
and (2.4),

$\displaystyle\tilde{\rho}(R(x_{1}),R(x_{2}))R_{V}$ $\displaystyle=$
$\displaystyle\rho(R^{2}(x_{1}),R^{2}(x_{2}))R_{V}-R_{V}(\rho(R^{2}(x_{1}),R(x_{2}))+\rho(R(x_{1}),R^{2}(x_{2}))+\lambda\rho(R(x_{1}),R(x_{2})))R_{V}$
$\displaystyle=$ $\displaystyle
R_{V}\rho(R^{2}(x_{1}),R^{2}(x_{2}))+R_{V}(\rho(R^{2}(x_{1}),R(x_{2}))+\rho(R(x_{1}),R^{2}(x_{2}))+\lambda\rho(R(x_{1}),R(x_{2})))R_{V}$
$\displaystyle+\lambda
R_{V}(\rho(R^{2}(x_{1}),R(x_{2}))+\rho(R(x_{1}),R^{2}(x_{2}))+\lambda\rho(R(x_{1}),R(x_{2})))$
$\displaystyle-
R_{V}^{2}\rho(R^{2}(x_{1}),R(x_{2}))-R_{V}^{2}(\rho(R^{2}(x_{1}),x_{2})+\rho(R(x_{1}),R(x_{2}))+\lambda\rho(R(x_{1}),x_{2}))R_{V}$
$\displaystyle-\lambda
R_{V}^{2}(\rho(R^{2}(x_{1}),x_{2})+\rho(R(x_{1}),R(x_{2}))+\lambda\rho(R(x_{1}),x_{2}))-R_{V}^{2}\rho(R(x_{1}),R^{2}(x_{2}))$
$\displaystyle-
R_{V}^{2}(\rho(R(x_{1}),R(x_{2})+\rho(x_{1},R^{2}(x_{2}))+\lambda\rho(x_{1},R(x_{2})))R_{V}-\lambda
R_{V}^{2}(\rho(R(x_{1}),R(x_{2}))$
$\displaystyle+\rho(x_{1},R^{2}(x_{2}))+\lambda\rho(x_{1},R(x_{2})))-\lambda
R_{V}^{2}\rho(R(x_{1}),R(x_{2}))-\lambda R_{V}^{2}(\rho(R(x_{1}),x_{2})$
$\displaystyle+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2}))R_{V}-\lambda^{2}R_{V}^{2}(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2}))$
$\displaystyle=$ $\displaystyle
R_{V}\rho(R^{2}(x_{1}),R^{2}(x_{2}))-R_{V}^{2}(\rho(R^{2}(x_{1}),R(x_{2}))+\rho(R(x_{1}),R^{2}(x_{2}))+\lambda\rho(R(x_{1}),R(x_{2})))$
$\displaystyle+R_{V}\rho(R^{2}(x_{1}),R(x_{2}))-R_{V}^{2}(\rho(R^{2}(x_{1}),x_{2})+\rho(R(x_{1}),R(x_{2}))+\lambda\rho(R(x_{1}),x_{2}))R_{V}$
$\displaystyle+R_{V}\rho(R(x_{1}),R^{2}(x_{2}))-R_{V}^{2}(\rho(x_{1},R^{2}(x_{2}))+\rho(R(x_{1}),R(x_{2}))+\lambda\rho(x_{1},R(x_{2})))R_{V}$
$\displaystyle+\lambda
R_{V}\rho(R(x_{1}),R(x_{2}))-\lambda^{2}R_{V}^{2}(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},R(x_{2})))R_{V}$
$\displaystyle=$ $\displaystyle
R_{V}\tilde{\rho}(R(x_{1}),R(x_{2}))+R_{V}(\tilde{\rho}(R(x_{1}),x_{2})+\tilde{\rho}(x_{1},R(x_{2}))+\lambda\tilde{\rho}(x_{1},x_{2}))R_{V}$
$\displaystyle+\lambda
R_{V}(\tilde{\rho}(R(x_{1}),x_{2})+\tilde{\rho}(x_{1},R(x_{2}))+\lambda\tilde{\rho}(x_{1},x_{2})).$

∎

The above representation $\tilde{V}$ can be applied to cohomologies of Rota-
Baxter operators and Rota-Baxter 3-Lie algebras in the next Section. W e can
also construct another representation of $(A_{R},R)$ from the given
representation $(V,\rho)$ of Rota-Baxter 3-Lie algebra $(A,R)$.

Proposition 2.6. Let $(A,R)$ be a Rota-Baxter 3-Lie algebra of weight
$\lambda$ and $(V,R_{V},\rho)$ be representation of it. Define bilinear map
$\bar{\rho}:A\wedge A\longrightarrow\mathfrak{gl}(V)$ by

$\displaystyle\bar{\rho}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\rho(R(x_{1}),R(x_{2}))+(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2}))R_{V}$
$\displaystyle+\lambda(\rho(R(x_{1}),x_{2})+\rho(x_{1},R(x_{2}))+\lambda\rho(x_{1},x_{2})).$

Then $(V,R_{V},\bar{\rho})$ is a representation of $(A_{R},R)$. We denote it
by $\bar{V}$.

###### Proof.

The proof is similar to Proposition 2.5. ∎

Definition 2.7. Let $(A,R_{A})$ and $(B,R_{B})$ be two Rota-Baxter $3$-Lie
algebras of weight $\lambda$ such that $(B,\rho,R_{B})$ is a representation of
$(A,R_{A})$ and $(A,\varrho,R_{A})$ is a representation of $(B,R_{B})$ with
bilinear maps $\rho:\wedge^{2}A\longrightarrow\mathfrak{gl}(B)$ and
$\varrho:\wedge^{2}B\longrightarrow\mathfrak{gl}(A)$. Moreover,
$(A,B,\rho,\varrho)$ is a matched pair of 3-Lie algebras with the 3-Lie
algebra structure on $A\bowtie B$ given by

$\displaystyle[x+a,y+b,z+c]$ $\displaystyle=$
$\displaystyle[x,y,z]+\rho(a,b)z+\rho(b,c)x+\rho(c,a)y+[a,b,c]$
$\displaystyle+\varrho(x,y)c+\varrho(y,z)a+\varrho(z,x)b.$

Then $(A\bowtie B,R_{A}+R_{B})$ is a Rota-Baxter $3$-Lie algebra with

$(R_{A}+R_{B})(x+a)=R_{A}(x)+R_{B}(a)$

for any $x,y\in A,a,b\in B$. We call $(A,B,R_{A},R_{B},\rho,\varrho)$ the
matched pair of Rota-Baxter $3$-Lie algebras.

Proposition 2.8. Let $(A,R_{A})$ and $(B,R_{B})$ be two Rota-Baxter $3$-Lie
algebras of weight $\lambda$, $(B,\rho)$ is a representation of $A$ and
$(A,\varrho)$ is a representation of $B$ with bilinear maps
$\rho:\wedge^{2}A\longrightarrow{gl}(B)$ and
$\varrho:\wedge^{2}B\longrightarrow{gl}(A)$. Then the following conditions are
equivalent

(i) $(A,B,R_{A},R_{B},\rho,\varrho)$ is a matched pair of Rota-Baxter $3$-Lie
algebras.

(ii) $(A,B,\rho,\varrho)$ is a matched pair of 3-Lie algebras and satisfying

$\displaystyle\rho(R_{A}(x_{1}),R_{A}(x_{2}))R_{B}$ $\displaystyle=$
$\displaystyle
R_{B}(R_{A}(x_{1}),R_{A}(x_{2}))+R_{B}\rho(R_{A}(x_{1}),x_{2})+\rho(x_{1},R_{A}(x_{2}))+\lambda\rho(x_{1},x_{2}))R_{B}$
$\displaystyle+\lambda
R_{B}(\rho(R_{A}(x_{1}),x_{2})+\rho(x_{1},R_{A}(x_{2}))+\lambda\rho(x_{1},x_{2})),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.5)$
$\displaystyle\varrho(R_{B}(a_{1}),R_{B}(a_{2}))R_{A}$ $\displaystyle=$
$\displaystyle
R_{A}(\varrho(R_{B}(a_{1}),R_{B}(a_{2}))+R_{A}\varrho(R_{B}(a_{1}),a_{2})+\varrho(a_{1},R_{B}(a_{2}))+\lambda\varrho(a_{1},a_{2}))R_{A}$
$\displaystyle+\lambda
R_{A}(\varrho(R_{B}(a_{1}),a_{2})+\varrho(a_{1},R_{B}(a_{2}))+\lambda\varrho(a_{1},a_{2})),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.6)$

for any $x_{1},x_{2}\in A,a_{1},a_{2}\in B$.

(iii) $(A,B,\rho,\varrho)$ is a matched pair of 3-Lie algebras and

$R_{A}+R_{B}:A\bowtie B\longrightarrow A\bowtie B$

is a Rota-Baxter operator of weight $\lambda$ on the $3$-Lie algebras
$A\bowtie B$ with

$(R_{A}+R_{B})(x+a)=R_{A}(x)+R_{B}(a).$

###### Proof.

In view of the definitions of representations and matched pairs of Rota-Baxter
$3$-Lie algebras, we get that (i) is equivalent to (iii). Subsequently, we
will prove that $(iii)$ is equivalent to (ii).

If $(A,B,\rho,\varrho)$ is a matched pair of 3-Lie algebras, then
$R_{A}+R_{B}$ is a Rota-Baxter operator of weight $\lambda$ on the $3$-Lie
algebra $A\bowtie B$ if and only if

$\displaystyle[(R_{A}+R_{B})(x+a),(R_{A}+R_{B})(y+b),(R_{A}+R_{B})(z+c)]$
$\displaystyle=$
$\displaystyle(R_{A}+R_{B})([(R_{A}+R_{B})(x+a),(R_{A}+R_{B})(y+b),z+c]$
$\displaystyle+[(R_{A}+R_{B})(x+a),y+b,(R_{A}+R_{B})(z+c)]$
$\displaystyle+[x+a,(R_{A}+R_{B})(y+b),(R_{A}+R_{B})(z+c)]+\lambda[(R_{A}+R_{B})(x+a),y+b,z+c]$
$\displaystyle+\lambda[x+a,(R_{A}+R_{B})(y+b),z+c]+\lambda[x+a,y+b,(R_{A}+R_{B})(z+c)]$
$\displaystyle+\lambda^{2}[x+a,y+b,z+c]).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2.7)$

By direct calculation, (2.7) holds if and only if (2.5), (2.6) hold. It
follows that $(iii)$ is equivalent to (ii). ∎

Proposition 2.9. Let $(A,B,R_{A},R_{B},\rho,\varrho)$ be a matched pair of
Rota-Baxter $3$-Lie algebras. Then $(A_{R},B_{R},\bar{\rho},\bar{\varrho})$ is
a matched pair of $3$-Lie algebras. Furthermore, $A_{R}\bowtie
B_{R}\simeq(A\bowtie B)_{R}$ as $3$-Lie algebras.

###### Proof.

The $3$-Lie algebra structure on $(A\bowtie B)_{R}$ is given by

$\displaystyle[x+a,y+b,z+c]_{R}$ $\displaystyle=$
$\displaystyle[R_{A}x+R_{B}a,R_{A}y+R_{B}b,z+c]+[R_{A}x+R_{B}a,y+b,R_{A}z+R_{B}c]$
$\displaystyle+[x+a,R_{A}y+R_{B}b,,R_{A}z+R_{B}c]+\lambda([R_{A}x+R_{B}a,y+b,z+c]+[x+a,R_{A}y+R_{B}b,z+c]$
$\displaystyle+[x+a,y+b,R_{A}z+R_{B}c]+\lambda[x+a,y+b,z+c])$ $\displaystyle=$
$\displaystyle[R_{A}x,R_{A}y,z]+\rho(R_{B}a,R_{B}b)z+\rho(R_{B}b,c)R_{A}x+\rho(c,R_{B}a)R_{A}y+[R_{B}a,R_{B}b,c]$
$\displaystyle+\varrho(R_{A}x,R_{A}y)c+\varrho(R_{A}y,z)R_{B}a+\varrho(z,R_{A}x)R_{B}b+[R_{A}x,y,R_{A}z]+\rho(R_{B}a,b)R_{A}z$
$\displaystyle+\rho(b,R_{B}c)R_{A}x+\rho(R_{B}c,R_{B}a)y+[R_{B}a,b,R_{B}c]+\varrho(R_{A}x,y)R_{B}c+\varrho(y,R_{A}z)R_{B}a$
$\displaystyle+\varrho(R_{A}z,R_{A}x)b+[x,R_{A}y,R_{A}z]+\rho(a,R_{B}b)R_{A}z+\rho(R_{B}b,R_{B}c)x$
$\displaystyle+\rho(R_{B}c,a)R_{A}y+[a,R_{B}b,R_{B}c]+\varrho(x,R_{A}y)R_{B}c+\varrho(R_{A}y,R_{A}z)a+\varrho(R_{A}z,x)R_{B}b$
$\displaystyle+\lambda([R_{A}x,y,z]+\rho(R_{B}a,b)z+\rho(b,c)R_{A}x+\rho(c,R_{B}a)y+[R_{B}a,b,c]$
$\displaystyle+\varrho(R_{A}x,y)c+\varrho(y,z)R_{B}a+\varrho(z,R_{A}x)b+[x,R_{A}y,z]+\rho(a,R_{B}b)z+\rho(R_{B}b,c)x$
$\displaystyle+\rho(c,a)R_{A}y+[a,R_{B}b,c]+\varrho(x,R_{A}y)c+\varrho(R_{A}y,z)a+\varrho(z,x)R_{B}b+[x,y,R_{A}z]$
$\displaystyle+\rho(a,b)R_{A}z+\rho(b,R_{B}c)x+\rho(R_{B}c,a)y+[a,b,R_{B}c]+\varrho(x,y)R_{B}c+\varrho(y,R_{A}z)a$
$\displaystyle+\varrho(R_{A}z,x)b+\lambda[x,y,z]+\lambda\rho(a,b)z+\lambda\rho(b,c)x+\lambda\rho(c,a)y+\lambda[a,b,c]$
$\displaystyle+\lambda\varrho(x,y)c+\lambda\varrho(y,z)a+\lambda\varrho(z,x)b)$
$\displaystyle=$
$\displaystyle[x,y,z]_{R}+\bar{\rho}(a,b)z+\bar{\rho}(b,c)x+\bar{\rho}(c,a)y+[a,b,c]_{R}+\bar{\varrho}(x,y)c+\bar{\varrho}(y,z)a+\bar{\varrho}(z,x)b,$

which implies that $(A_{R},B_{R},\bar{\rho},\bar{\varrho})$ is a matched pair
of $3$-Lie algebras. Furthermore, $A_{R}\bowtie B_{R}\simeq(A\bowtie B)_{R}$
as $3$-Lie algebras. ∎

Proposition 2.10. Let $(A,B,R_{A},R_{B},\rho,\varrho)$ be a matched pair of
Rota-Baxter $3$-Lie algebras of weight $\lambda$. Then
$(A_{R},B_{R},R_{A},R_{B},\bar{\rho},\bar{\varrho})$ is a matched pair of
Rota-Baxter $3$-Lie algebras of weight $\lambda$.

###### Proof.

By Proposition 2.9, $A_{R}\bowtie B_{R}\simeq(A\bowtie B)_{R}$. Thus,
$R_{A}+R_{B}$ is a Rota-Baxter operator of $A_{R}\bowtie B_{R}$. By
Proposition 2.8, $(A_{R},B_{R},R_{A},R_{B},\bar{\rho},\bar{\varrho})$ is a
matched pair of Rota-Baxter $3$-Lie algebras of weight $\lambda$. ∎

Definition 2.11. Let $(A,R)$ be a Rota-Baxter $3$-Lie algebra of weight
$\lambda$. A bilinear form $B$ on $(A,R)$ is called invariant if it satisfies

$B([x_{1},x_{2},x_{3}],x_{4})+B([x_{1},x_{2},x_{4}],x_{3})$

and

$B(R(x),y)+B(x,R(y)+\lambda B(x,y)=0.$

A Rota-Baxter $3$-Lie algebra is called pseudo-metric if there is a
nondegenerate symmetric invariant bilinear form on $(A,R)$.

Definition 2.12. A Manin triple of Rota-Baxter 3-Lie algebras of weight
$\lambda$ consists of a pseudo-metric Rota-Baxter $3$-Lie algebra $(A,R)$ and
Rota-Baxter 3-Lie algebras $(A_{1},R_{1})$, $(A_{2},R_{2})$ such that

(i) $(A_{1},R_{1})$ and $(A_{2},R_{2})$ are Rota-Baxter Lie subalgebras of
$(A,R)$.

(ii) $A=A_{1}\oplus A_{2}$ as vector spaces;

(iii) For all $x_{1},x_{2}\in A_{1},~{}a_{1},a_{2}\in A_{2}$, we have
$P_{1}[x_{1},x_{2},a_{1}]=0$ and $P_{2}[a_{1},a_{2},x_{1}]=0$ with
$P_{1},P_{2}$ are projections from $A_{1}\oplus A_{2}$ to $A_{1}$ and $A_{2}$
respectively.

More details on matched pairs and Manin triples of 3-Lie algebras can be found
in [7].

Proposition 2.13. Let $(A,R)$ and $(A^{*},-\lambda I-R^{*})$ be Rota-Baxter
3-Lie algebras of weight $\lambda$. Then $(A\oplus A^{*},B,A,A^{*})$ is a
Manin triple of Rota-Baxter 3-Lie algebras if and only if
$(A,A^{*},ad^{*},\mathfrak{ad}^{*},R,-\lambda I-R^{*})$ is a matched pair of
Rota-Baxter 3-Lie algebras, where

$B(x+\xi,y+\eta)=\langle x,\eta\rangle+\langle\xi,y\rangle.$

###### Proof.

According to Proposition 4.7 [7], we easily get the result. ∎

##  3 Cohomology of Rota-Baxter 3-Lie algebras

In this section, we first recall the cohomogies of 3-Lie algebras [2, 31].
Then, we combine the cohomologies of 3-Lie algebras $A$ and $A_{R}$ to
construct cohomologies of Rota-Baxter 3-Lie algebras.

Let $A$ be a $3$-Lie algebra and $L=\wedge^{2}A$ be the associated Leibniz
algebra. Suppose that $(\rho,V)$ is a representation of $A$, the space
$C_{3-{\hbox{Lie}}}^{p}(A,V)$ of $p$-cochains ($p\geq 1$) is the set of
multilinear maps of the form

$f:\wedge^{p-1}L\wedge A\longrightarrow V,$

and the coboundary operator
$\partial:C_{3-{\hbox{Lie}}}^{p}(A,V)\longrightarrow
C_{3-{\hbox{Lie}}}^{p+1}(A,V)$ is as follows:

$\displaystyle(\partial f)(X_{1},\cdot\cdot\cdot,X_{p},z)$ $\displaystyle=$
$\displaystyle\sum_{1\leq
i<k\leq{p}}(-1)^{i}f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{k-1},[X_{i},X_{k}]_{F},X_{k+1},\cdot\cdot\cdot,X_{p},z)$
$\displaystyle+\sum_{i=1}^{p}(-1)^{i}f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{p},[X_{i},z])+\sum_{i=1}^{p}(-1)^{i+1}\rho(X_{i})f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{p},z)$
$\displaystyle+(-1)^{p+1}(\rho(y_{p},z)f(X_{1},\cdot\cdot\cdot,X_{p-1},x_{p})+\rho(z,x_{p})f(X_{1},\cdot\cdot\cdot,X_{p-1},y_{p}))$

for all $X_{i}=x_{i}\wedge y_{i}\in L$ and $z\in A$.

In the following, we are ready to define cohomology of Rota-Baxter $3$-Lie
algebras.

Let $(A,R)$ be a Rota-Baxter $3$-Lie algebra of any weight $\lambda$ and
$(V,\rho,R_{V})$ be its representation. It is known that $(A_{R},[\ ,\ ,\
]_{R},R)$ is also a $\lambda$-weighted Rota-Baxter $3$-Lie algebra and
$(V,\tilde{\rho},R_{V})$ is a representation of $(A_{R},R)$ by Proposition
2.5.

It is natural to study the cohomologies of $3$-Lie algebra $(A_{R},[\ ,\ ,\
]_{R})$ with coefficients in $(\tilde{V},\tilde{\rho})$.

The coboundary operator
$\partial_{R}:C_{3-{\hbox{Lie}}}^{p}(A_{R},\tilde{V})\longrightarrow
C_{3-{\hbox{Lie}}}^{p+1}(A_{R},\tilde{V})$ is given by:

$\displaystyle(\partial_{R}f)(X_{1},\cdot\cdot\cdot,X_{p},z)$ $\displaystyle=$
$\displaystyle\sum_{1\leq
i<k\leq{p}}(-1)^{i}f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{k-1},[X_{i},X_{k}]_{F},X_{k+1},\cdot\cdot\cdot,X_{p},z)$
$\displaystyle+\sum_{i=1}^{p}(-1)^{i}f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{p},[X_{i},z]_{R})+\sum_{i=1}^{p}(-1)^{i+1}\tilde{\rho}(X_{i})f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{p},z)$
$\displaystyle+(-1)^{p+1}(\tilde{\rho}(y_{p},z)f(X_{1},\cdot\cdot\cdot,X_{p-1},x_{p})+\tilde{\rho}(z,x_{p})f(X_{1},\cdot\cdot\cdot,X_{p-1},y_{p}))$

for all $X_{i}=x_{i}\wedge y_{i}\in L$ and $z\in A$.

We obtain a complex $(C_{\hbox{3-Lie}}^{*}(A_{R},\tilde{V}),\partial_{R})$.
Denote the set of closed $n$-cochains by
$\mathcal{Z}^{n}_{\hbox{3-Lie}}(A_{R},\tilde{V})$ and the set of exact
$n$-cochains by $\mathcal{B}^{n}_{\hbox{3-Lie}}(A_{R},\tilde{V})$. We define
the corresponding cohomology group by

$\mathcal{H}^{n}_{\hbox{3-Lie}}(A_{R},\tilde{V})=\mathcal{Z}^{n}_{\hbox{3-Lie}}(A_{R},\tilde{V})/\mathcal{B}^{n}_{\hbox{3-Lie}}(A_{R},\tilde{V}).$

Definition 3.1. Let $(A,R)$ be a Rota-Baxter $3$-Lie algebra of weight
$\lambda$ and $(V,\rho,R_{V})$ be its representation. Then the cochain complex
$(C_{\hbox{3-Lie}}^{*}(A_{R},\tilde{V}),\partial_{R})$ is called the cochain
complex of Rota-Baxter operator $R$ with coefficients in $V$, the
corresponding cohomology group
$\mathcal{H}^{n}_{\hbox{3-Lie}}(A_{R},\tilde{V})$ is called the $n$\-
cohomology of Rota-Baxter operator $R$ with coefficients in $V$.

Proposition 3.2. Define linear map
$\delta:C_{\hbox{3-Lie}}^{n+1}(A,V)\longrightarrow
C_{\hbox{3-Lie}}^{n+1}(A_{R},\tilde{V})$ by

$\delta
f(x_{1},\cdot\cdot\cdot,x_{2n+1})=f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n+1}))-R_{V}\sum_{k=0}^{2n}\lambda^{2n-k}f^{k}(x_{1},\cdot\cdot\cdot,x_{2n+1})$

where

$f^{k}(x_{1},\cdot\cdot\cdot,x_{n})=f(\underbrace{I\otimes\cdot\cdot\cdot\otimes
R\otimes\cdot\cdot\cdot\otimes
I}_{R~{}~{}\hbox{appears}~{}~{}k~{}~{}\hbox{times}})(x_{1},\cdot\cdot\cdot,x_{n}).$

Then $\delta$ is a cochain map from the cochain complex
$C_{\hbox{3-Lie}}^{*}(A,V)$ to $C_{\hbox{3-Lie}}^{*}(A_{R},\tilde{V})$, that
is, the following diagram is commutative:

$\textstyle{C_{\hbox{3-Lie}}^{n}(A,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\delta}$$\textstyle{C_{\hbox{3-Lie}}^{n}(A_{R},\tilde{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{R}}$$\textstyle{C_{\hbox{3-Lie}}^{n+1}(A,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C_{\hbox{3-Lie}}^{n+1}(A_{R},\tilde{V}).}$

###### Proof.

See the Appendix. ∎

Define the set of $(n+1)$-cochains by

$C_{\hbox{RB}}^{n+1}(A,V)=C_{\hbox{3-Lie}}^{n+1}(A,V)\times
C_{\hbox{3-Lie}}^{n}(A_{R},\tilde{V}),~{}(n\geq 1)$

and

$C_{\hbox{RB}}^{1}(A,V)=C_{\hbox{3-Lie}}^{1}(A,V).$

Given the linear map $\partial_{RB}:C_{\hbox{RB}}^{n}(A,V)\longrightarrow
C_{\hbox{RB}}^{n+1}(A,V)$ by

$\partial_{RB}(f,g)=(\partial f,\partial_{R}g+(-1)^{n}\delta
f)~{}~{}\hbox{for~{}~{} any}~{}~{}(f,g)\in C_{\hbox{RB}}^{n}(A,V).$

In view of Proposition 3.2, we have

Theorem 3.3. The operator $\partial_{RB}$ is a coboundary operator, that is,
$\partial_{RB}\partial_{RB}=0$.

Associated to the representation $(V,R_{V},\rho)$, we obtain a cochain complex
$(C_{\hbox{RB}}^{*}(A,V),\partial_{RB})$. Denote the cohomology group of this
cochain complex by $H^{*}_{RB}(A,V)$, which is called the cohomology of the
Rota-Baxter 3-Lie algebra $(A,R)$ with coefficients in the representation
$(V,R_{V},\rho)$.

In view of the definitions of $C_{\hbox{3-Lie}}^{n}(A,V)$ and
$C_{\hbox{3-Lie}}^{n}(A_{R},\tilde{V})$, we easily get an exact sequence of
cochain complexes,

$0\longrightarrow C_{\hbox{3-Lie}}^{n}(A_{R},\tilde{V})\longrightarrow
C_{\hbox{RB}}^{n+1}(A,V)\longrightarrow
C_{\hbox{3-Lie}}^{n+1}(A,V)\longrightarrow 0.$

Remark 3.4. If $A$ is a 3-Lie algebra, then $(L(A)=A\wedge A,[\ ,\ ]_{F})$ is
the associated Leibniz algebra [31] with $[\ ,\ ]_{F}$ given by

$[X,Y]_{F}=[x_{1},x_{2},y_{1}]\wedge y_{2}+y_{1}\wedge[x_{1},x_{2},y_{2}]$

for any $X=x_{1}\wedge x_{2},Y=y_{1}\wedge y_{2}\in L(A)$. Moreover, the
cohomology complex of a 3-Lie algebra $A$ with coefficients in $V$ coincides
with the Loday-Pirashvili cohomology complex of Leibniz algebra $L(A)$ with
coefficients in $\hbox{Hom}(A,V)$ with the same coboundary operator [10]. But,
in the case of Rota-Baxter 3-Lie algebras, we can not get a similar result,
since it is difficult to gain a Rota-Baxter operator on $L(A)=A\wedge A$.

##  4 Deformation of Rota-Baxter 3-Lie algebras

Let $(A,\pi,R)$ be a Rota-Baxter 3-Lie algebra,
$\pi_{i}:\wedge^{3}A\longrightarrow A$ be trilinear map and
$R_{i}:A\longrightarrow A$ be linear map. Consider the space $A[[t]]$ of
formal power series in $t$ with coefficients in $A$ and a $t$-parametrized
family of trilinear operations

$\pi_{t}(x,y,z)=\sum_{i=0}^{n-1}t^{i}\pi_{i}(x,y,z),$ $None$

and linear operations

$R_{t}(x)=\sum_{i=0}^{n-1}t^{i}R_{i}(x),$ $None$

where $R_{0}=R$ and $\pi_{0}=\pi$.

If all $(A[[t]],\pi_{i},R_{i})$ are Rota-Baxter 3-Lie algebras, we say that
$(\pi_{i},R_{i})~{}(i=0,1,\cdot\cdot\cdot,)$ generate a deformation of the
Rota-Baxter 3-Lie algebra $(A,\pi,R)$.

If all $(A[[t]],\pi_{i},R_{i})$ are Rota-Baxter 3-Lie algebras, then we have

$\displaystyle\sum_{i+j=k,~{}i,~{}j\geq
0}(\pi_{i}(\pi_{j}(v,w,x),y,z)+\pi_{i}(x,\pi_{j}(v,w,y),z)+\pi_{i}(x,y,\pi_{j}(v,w,z))$
$\displaystyle-\pi_{i}(v,w,\pi_{j}(x,y,z)))=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(4.3)$

and

$\displaystyle\sum_{i+j+k+l=n,~{}i,~{}j,~{}k,~{}l\geq
0}\pi_{i}(R_{j}(x),R_{k}(y),R_{l}(z))$ $\displaystyle=$
$\displaystyle\sum_{i+j+k+l=n,~{}i,~{}j,~{}k,~{}l\geq
0}R_{i}(\pi_{j}(R_{k}(x),R_{l}(y),z)+\pi_{j}(x,R_{k}(y),R_{l}(z))+\pi_{j}(R_{k}(x),y,R_{l}(z)))$
$\displaystyle+\lambda\sum_{i+j+k=n,~{}i,~{}j,~{}k\geq
0}R_{i}(\pi_{j}(R_{k}(x),y,z)+\pi_{j}(x,R_{k}(y),z)+\pi_{j}(x,y,R_{k}(z)))$
$\displaystyle+\lambda^{2}\sum_{i+j=n,~{}i,~{}j\geq
0}R_{i}\pi_{j}(x,y,z).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(4.4)$

Proposition 4.1. Let $(A[[t]],\pi_{i},R_{i})$ be a one-parameter deformation
of the Rota-Baxter 3-Lie algebra $(A,\pi,R)$. Then $(\pi_{1},R_{1})$ is a
2-cocycle in the cohomology of the Rota-Baxter 3-Lie algebra $(A,\pi,R)$ with
coefficients in itself.

###### Proof.

If $(\pi_{i},R_{i})$ is a one-parameter deformation of the Rota-Baxter 3-Lie
algebra $(A,\pi,R)$, then $(4.3)$ and $(4.4)$ hold. On the other hand, if
$(\pi_{1},R_{1})$ is a 2-cocycle, then $\partial\pi_{1}=0$ and
$\partial_{R}R_{1}+\delta\pi_{1}=0$.

On the basis of deformation theory of 3-Lie algebras [39, 40],
$\partial\pi_{1}=0$ is equivalent to (4.3) when $k=1$.

By direct calculation,

$\displaystyle\partial_{R}R_{1}(x,y,z)$ $\displaystyle=$ $\displaystyle-
R_{1}([x,y,z]_{R})+\tilde{\rho}(x,y)R_{1}(z)+\tilde{\rho}(y,z)R_{1}(x)+\tilde{\rho}(z,x)R_{1}(y)$
$\displaystyle=$ $\displaystyle-
R_{1}([x,y,z]_{R})+(\rho(R(x),R(y))-R(\rho(R(x),y)+\rho(x,R(y))+\lambda\rho(x,y)))R_{1}(z)$
$\displaystyle+(\rho(R(y),R(z))-R(\rho(R(y),z)+\rho(y,R(z))+\lambda\rho(y,z)))R_{1}(x)$
$\displaystyle+(\rho(R(z),R(x))-R(\rho(R(z),x)+\rho(z,R(x))+\lambda\rho(z,x)))R_{1}(y)$
$\displaystyle=$ $\displaystyle-
R_{1}([x,y,z]_{R})+[R(x),R(y),R_{1}(z)]-R([R(x),y,R_{1}(z)]+[x,R(y),R_{1}(z)]$
$\displaystyle+\lambda[x,y,R_{1}(z)])+[R(y),R(z),R_{1}(x)]-R([R(y),z,R_{1}(x)]+[y,R(z),R_{1}(x)]$
$\displaystyle+\lambda[y,z,R_{1}(x)])+[R(z),R(x),R_{1}(y)]-R([R(z),x,R_{1}(y)]+[z,R(x),R_{1}(y)]$
$\displaystyle+\lambda[z,x,R_{1}(y)]),$

and

$\displaystyle\delta\pi_{1}(x,y,z)$ $\displaystyle=$
$\displaystyle\pi_{1}(R(x),R(y),R(z))-\lambda^{2}R\pi_{1}(x,y,z)-\lambda
R(\pi_{1}(Rx,y,z)+\pi_{1}(x,Ry,z)$
$\displaystyle+\pi_{1}(x,y,Rz))-R(\pi_{1}(Rx,Ry,z)+\pi_{1}(x,Ry,Rz)+\pi_{1}(Rx,y,Rz)).$

Hence, $\partial_{R}R_{1}+\delta\pi_{1}=0$ is equivalent to (4.4) when $n=1$.
∎

Definition 4.2. A deformation of the Rota-Baxter 3-Lie algebra $(A,\pi,R)$ is
said to be trivial if there is a linear map $N:A\longrightarrow A$ such that
$K_{t}=I+tN~{}(\forall~{}t)$ satisfies

$K_{t}R=RK_{t}$

and

$K_{t}[x,y,z]_{t}=[K_{t}x,K_{t}y,K_{t}z].$

Based on the case of 3-Lie algebras, we give the definition of a Nijenhuis
operator of on Rota-Baxter 3-Lie algebras.

Definition 4.3. Let $(A,R)$ be a Rota-Baxter 3-Lie algebra. A linear map
$N:A\longrightarrow A$ is called a Nijenhuis operator if $NR=RN$ and $N$ is a
Nijenhuis operator of $A$, that is,

$\displaystyle[Nx,Ny,Nz]$ $\displaystyle=$ $\displaystyle
N([Nx,Ny,z]+[x,Ny,Nz]+[Nx,y,Nz])$
$\displaystyle-N^{2}([Nx,y,z]+[x,Ny,z]+[x,y,Nz])+N^{3}([x,y,z])$

for any $x,~{}y,~{}z\in A$.

Remark 4.4. $(A,[\ ,\ ,]_{N},R)$ is also a Rota-Baxter 3-Lie algebra, where

$\displaystyle[x,y,z]_{N}$ $\displaystyle=$
$\displaystyle[Nx,Ny,z]+[x,Ny,Nz]+[Nx,y,Nz]$
$\displaystyle-N([Nx,y,z]+[x,Ny,z]+[x,y,Nz])+N^{2}([x,y,z]).$

Definition 4.5. A linear map $K:V\longrightarrow A$ is called an
$\mathcal{O}$-operator on the Rota-Baxter 3-Lie algebra $A$ associated with
the representation $(V,\rho,R_{V})$ if $K:V\longrightarrow A$ is an
$\mathcal{O}$-operator on the 3-Lie algebra $A$ associated to the
representation $(V,\rho)$, that is, for all $u,v,w\in V$,

$[K(u),K(v),K(w)]=K(\rho(K(u),K(v))w+\rho(K(v),K(w))u+\rho(K(w),K(u))v).$

and $KR_{V}=RK$. Then $K$ is called an $\mathcal{O}$-operator on Rota-Baxter
3-Lie algebras.

Proposition 4.6. Let $K:V\longrightarrow A$ be an $\mathcal{O}$-operator on
the Rota-Baxter 3-Lie algebra $(A,R)$ associated to the representation
$(V,\rho,R_{V})$. Then $(V,[\ ,\ ,\ ]_{K},R_{V})$ is Rota-Baxter 3-Lie
algebra, where

$[u,v,w]_{K}=\rho(K(u),K(v))w+\rho(K(v),K(w))u+\rho(K(w),K(u))v$

for all $u,v,w\in V$.

###### Proof.

Due to Proposition 4.6 in [36], $(V,[\ ,\ ,\ ]_{K})$ is a 3-Lie algebra. In
view of (2.3), by direct computation, we can check that $R_{V}$ is a Rota-
Baxter operator of $(V,[\,\,\ ]_{K})$. ∎

Proposition 4.7. Let $K:V\longrightarrow A$ be an $\mathcal{O}$-operator on
Rota-Baxter 3-Lie algebra $(A,R)$ associated to the representation
$(V,\rho,R_{V})$. Define bilinear map $\varrho_{K}:V\wedge
V\longrightarrow\mathfrak{gl}(A)$ by

$\varrho_{K}(u,v)x=[K(u),K(v),x]-K(\rho(Kv,x)u+\rho(x,Ku)v).$

Then $(A,\varrho_{K},R)$ is a representation of $(V,[\,\,\ ]_{K},R_{V})$.

###### Proof.

In the light of Proposition 4.7 in [36], we know that $(A,\varrho_{K})$ is a
representation of $(V,[\,\,\ ]_{K})$. In the following, we verify that (2.3)
holds for $\varrho_{K}$. In fact,

$\displaystyle\varrho_{K}(R_{V}u\wedge R_{V}v)R(x)$ $\displaystyle=$
$\displaystyle[KR_{V}(u),KR_{V}(v),R(x)]-K\rho(KR_{V}v\wedge
Rx)R_{V}(u)-K\rho(Rx\wedge KR_{V}u)R_{V}v$ $\displaystyle=$ $\displaystyle
R([RK(u),KR_{V}(v),x]+[RK(u),K(v),R(x)]+[K(u),RK(v),R(x)])$
$\displaystyle+\lambda
R([RK(u),K(v),x]+[K(u),RK(v),x]+[K(u),K(v),R(x)]+\lambda+[K(u),K(v),x])$
$\displaystyle-KR_{V}(\rho(RK(v)\wedge
R(x))u+R_{V}(\rho(K(v),R(x))+\rho(RK(v),x)+\lambda\rho(Kv,x))R_{V}(u)$
$\displaystyle+\lambda
R_{V}(\rho(K(v),R(x))+\rho(RK(v),x)+\lambda\rho(Kv,x))u-KR_{V}(\rho(R(x),RK(u))v$
$\displaystyle-
R_{V}(\rho(R(x),K(u))+\rho(x,RK(u))+\lambda\rho(x,Ku))R_{V}(v)-\lambda
R_{V}(\rho(R(x),K(u))$ $\displaystyle+\rho(x,RK(u))+\lambda\rho(x,Ku))v$
$\displaystyle=$ $\displaystyle
R\varrho_{K}(R_{V}(u),R_{V}(v))x+R(\varrho_{K}(R_{V}(u),v)+\varrho_{K}(u,R_{V}(v))+\lambda\varrho_{K}(u,v))R(x)$
$\displaystyle+\lambda
R(\varrho_{K}(R_{V}(u),v)+\varrho_{K}(u,R_{V}(v))+\lambda\varrho_{K}(u,v))x,$

which follows that

$\displaystyle\varrho_{K}(R_{V}u,R_{V}v)R$ $\displaystyle=$ $\displaystyle
R\varrho_{K}(R_{V}(u),R_{V}(v))+R(\varrho_{K}(R_{V}(u),v)+\varrho_{K}(u,R_{V}(v))+\lambda\varrho_{K}(u,v))R$
$\displaystyle+\lambda
R(\varrho_{K}(R_{V}(u),v)+\varrho_{K}(u,R_{V}(v))+\lambda\varrho_{K}(u,v)).$

Hence, $(A,\varrho_{K},R)$ is a representation of $(V,[\ ,\ ,\ ]_{K},R_{V})$.
∎

It is natural to study the cohomology of the Rota-Baxter 3-Lie algebra $(V,[\
,\ ,\ ]_{K},R_{V})$ with coefficients in the representation
$(A,\varrho_{K},R)$. We do not intend to discuss it here and leave it to the
readers.

##  5 Extension of Rota-Baxter 3-Lie algebras

Let $(A,R)$ be a Rota-Baxter 3-Lie algebra and $(V,R_{V})$ an abelian Rota-
Baxter 3-Lie algebra with the trivial product.

Definition 5.1. A central extension of Rota-Baxter 3-Lie algebra $(A,R)$ by
the abelian Rota-Baxter 3-Lie algebra $(V,R_{V})$ is an exact sequence of
Rota-Baxter 3-Lie algebras

$0\longrightarrow(V,R_{V})\stackrel{{\scriptstyle
i}}{{\longrightarrow}}(\hat{A},R_{\hat{A}})\stackrel{{\scriptstyle
p}}{{\longrightarrow}}(A,R)\longrightarrow 0$

such that $[V,V,\hat{A}]=[V,\hat{A},V]=[\hat{A},V,V]=0$.

Definition 5.2. Let $(\hat{A_{1}},R_{\hat{A_{1}}})$ and
$(\hat{A_{2}},R_{\hat{A_{2}}})$ be two central extensions of $(A,R)$ by
$(V,R_{V})$ . They are said to be equivalent if there is a homomorphism of
Rota-Baxter 3-Lie algebras
$\varphi:(\hat{A_{1}},R_{\hat{A_{1}}})\longrightarrow(\hat{A_{2}},R_{\hat{A_{2}}})$
such that the following commutative diagram holds:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V,R_{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{i}$$\textstyle{(\hat{A_{1}},R_{\hat{A_{1}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{p}$$\textstyle{(A,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V,R_{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{(\hat{A_{2}},R_{\hat{A_{2}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{(A,R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Similarly to the case of 3-Lie algebra [38, 39, 40], we easily get the
following result:

Lemma 5.3. With the above notions, $(V,\rho,R_{V})$ is a representation of
$(A,R)$ and is independent on the choice of the section $s$. Moreover,
equivalent abelian extensions give the same representation.

A section of a central extension $(\hat{A},R_{\hat{A}})$ of $(A,R)$ by
$(V,R_{V})$ is a linear map $s:A\longrightarrow\hat{A}$ such that $ps=I$.

Let $s:A\longrightarrow\hat{A}$ be any section of $p$. Define two multilinear
map $\psi:\wedge^{3}A\longrightarrow V$ and linear map $\chi:A\longrightarrow
V$ by

$\psi(x,y,z)=[s(x),s(y),s(z)]-s[x,y,z]$

and

$\chi(x)=R_{\hat{A}}(s(x))-s(R(x)).$

Proposition 5.4. The vector space $(A\oplus V,[\ ,\ ,\ ]_{\psi},R_{\chi})$
with the bracket

$[x+a,y+b,z+c]_{\psi}=[x,y,z]+\psi(x,y,z)$

and

$R_{\chi}(x+a)=R(x)+R_{V}(a)+\chi(x)$

is a Rota-Baxter 3-Lie algebra if and only if $(\psi,\chi)$ is a 2-cocycle of
$(A,R)$ with coefficients in the trivial representation $(V,R_{V})$.

###### Proof.

If $(\psi,\chi)\in C_{\hbox{RB}}^{2}(A,V)=C_{\hbox{3-Lie}}^{2}(A,V)\times
C_{\hbox{3-Lie}}^{1}(A_{R},\tilde{V})$ is a 2-cocycle, then

$\partial_{RB}(\psi,\chi)=(\partial\psi,\partial_{R}\chi+(-1)^{2}\delta\psi)=0.$

Based on deformation theory of 3-Lie algebras [38, 39, 40], we only need to
check that $R_{\chi}$ is a Rota-Baxter operator of the 3-Lie algebra $A\oplus
V$ if and only if $\partial_{R}\chi+\delta\psi=0$.

On the one hand, by computation,

$\displaystyle
R_{\chi}([R_{\chi}(x+a),R_{\chi}(y+b),z+c]_{\psi}+[R_{\chi}(x+a),y+b,R_{\chi}(z+c)]_{\psi}$
$\displaystyle+[x+a,R_{\chi}(y+b),R_{\chi}(z+c)]_{\psi}+\lambda([R_{\chi}(x+a),y+b,z+c]_{\psi}+[x+a,R_{\chi}(y+b),z+c]_{\psi}$
$\displaystyle+[x+a,y+b,R_{\chi}(z+c)]_{\psi}+\lambda[x+a,y+b,z+c]_{\psi}))-[R_{\chi}(x+a),R_{\chi}(y+b),R_{\chi}(z+c)]_{\psi}$
$\displaystyle=$ $\displaystyle
R([Rx,Ry,z]+[Rx,y,Rz]+[x,Ry,Rz]+\lambda([Rx,y,z]+[x,Ry,z]+[x,y,Rz])$
$\displaystyle+\lambda^{2}[x,y,z])+R_{V}(\psi(Rx,Ry,z)+\psi(Rx,y,Rz)+\psi(x,Ry,Rz)+\lambda(\psi(Rx,y,z)$
$\displaystyle+\psi(x,Ry,z)+\psi(x,y,Rz))+\lambda^{2}\psi(x,y,z))-[Rx,Ry,Rz]-\psi(Rx,Ry,Rz).~{}~{}~{}~{}~{}~{}(5.1)$

On the other hand, since $(V,R_{V})$ is a trivial representation,

$\displaystyle(\partial_{R}\chi+\delta\psi)(x,y,z)$ $\displaystyle=$
$\displaystyle-\chi([x,y,z]_{R})+\psi(Rx,Ry,Rz)-R_{V}\lambda^{2}\psi(x,y,z)-R_{V}\lambda(\psi(Rx,y,z)+\psi(x,Ry,z)$
$\displaystyle+\psi(x,y,Rz))-R_{V}(\psi(Rx,Ry,Rz)+\psi(Rx,Ry,z)+\psi(Rx,y,Rz)).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(5.2)$

In the light of (5.1) and (5.2), we get the conclusion. ∎

##  6 Appendix: Proof of Proposition 3.2

###### Proof.

Denote $f_{i}^{k}$ by

$f_{i}^{k}(x_{1},\cdot\cdot\cdot,x_{n})=f(\underbrace{I\otimes\cdot\cdot\cdot\otimes
R\otimes\cdot\cdot\cdot\otimes
I}_{R~{}~{}\hbox{appears}~{}~{}k~{}~{}\hbox{times but not in the ith
place}})(x_{1},\cdot\cdot\cdot,x_{n}),$

and $\tilde{R}=R\wedge R$.

For any $f\in C_{\hbox{3-Lie}}^{n}(A,V)$ and $X_{i}=x_{i}\wedge y_{i}\in L$
and $z\in A$. For convenient, we denote
$(X_{1},\cdot\cdot\cdot,X_{n},z)=(x_{1},x_{2},\cdot\cdot\cdot,x_{2n+1})\in
L^{\otimes n}\wedge A$.

On the one hand, by direct calculation,

$\displaystyle\delta\partial f(X_{1},\cdot\cdot\cdot,X_{n},z)$
$\displaystyle=$ $\displaystyle\partial
f(\tilde{R}(X_{1}),\cdot\cdot\cdot,\tilde{R}(X_{n}),R(z))-R_{V}\sum_{k=0}^{2n}\lambda^{2n-k}(\partial
f)^{k}(X_{1},\cdot\cdot\cdot,X_{n},z)$ $\displaystyle=$ $\displaystyle
B_{1}+B_{2}+B_{3}+B_{4}+B_{5}-C_{1}-C_{2}-C_{3}-C_{4}-C_{5},$

with

$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle\sum_{1\leq
i<k\leq{n}}(-1)^{i}f(R(x_{1}),\cdot\cdot\cdot,R(x_{2j-1}),[R(x_{2i-1}),R(x_{2i}),R(x_{2j})],R(x_{2j+1}),\cdot\cdot\cdot,R(x_{2n+1}))$
$\displaystyle+\sum_{1\leq
i<j\leq{n}}(-1)^{i}f(R(x_{1}),\cdot\cdot\cdot,R(x_{2j-2}),[R(x_{2i-1}),R(x_{2i}),R(x_{2j-1})],R(x_{2j}),\cdot\cdot\cdot,R(x_{2n+1})),$
$B_{2}=\sum_{i=1}^{n}(-1)^{i}f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n}),[R(x_{2i-1}),R(x_{2i}),R(x_{2n+1})]),$
$B_{3}=\sum_{i=1}^{n}(-1)^{i+1}\rho(R(x_{2i-1}),R(x_{2i}))f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n+1})),$
$\displaystyle B_{4}$ $\displaystyle=$
$\displaystyle(-1)^{n+1}(\rho(R(x_{2n}),R(x_{2n+1}))f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n-2}),R(x_{2n-1}))$
$\displaystyle+\rho(R(x_{2n+1}),R(x_{2n-1})f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n-2}),R(x_{2n}))),$
$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle R_{V}\sum_{1\leq
i<j\leq{n}}(-1)^{i}\sum_{k=0}^{2n-2}(\lambda^{2n-k}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[x_{2i-1},x_{2i},x_{2j}],\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-1}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[R(x_{2i-1}),x_{2i},x_{2j}],\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-1}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[x_{2i-1},R(x_{2i}),x_{2j}],\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-1}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[x_{2i-1},x_{2i},R(x_{2j})],\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-2}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[R(x_{2i-1}),R(x_{2i}),x_{2j}],\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-2}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[R(x_{2i-1}),x_{2i},R(x_{2j})],\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-2}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[x_{2i-1},R(x_{2i}),R(x_{2j})],\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-3}\lambda^{2n-k-3}f_{2j-2}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-1},[R(x_{2i-1}),R(x_{2i}),R(x_{2j})],\cdot\cdot\cdot,x_{2n+1})),$
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle R_{V}\sum_{1\leq
i<j\leq{n}}(-1)^{i}(\sum_{k=0}^{2n-2}\lambda^{2n-k}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[x_{2i-1},x_{2i},x_{2j-1}],x_{2j}\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-1}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[R(x_{2i-1}),x_{2i},x_{2j-1}],x_{2j}\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-1}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[x_{2i-1},R(x_{2i}),x_{2j-1}],x_{2j}\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-1}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[x_{2i-1},x_{2i},R(x_{2j-1})],x_{2j}\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-2}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[R(x_{2i-1}),R(x_{2i}),x_{2j-1}],x_{2j}\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-2}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[R(x_{2i-1}),x_{2i},R(x_{2j-1})],x_{2j}\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-k-2}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[x_{2i-1},R(x_{2i}),R(x_{2j-1})],x_{2j}\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-3}\lambda^{2n-k-3}f_{2j-3}^{k}(x_{1},\cdot\cdot\cdot,x_{2j-2},[R(x_{2i-1}),R(x_{2i}),R(x_{2j-1})],x_{2j}\cdot\cdot\cdot,x_{2n+1})),$
$\displaystyle C_{3}$ $\displaystyle=$ $\displaystyle
R_{V}\sum_{i=1}^{n}(-1)^{i}(\sum_{k=0}^{2n-2}\lambda^{2n-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2i-1},x_{2i},x_{2n+1}])$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-1-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[R(x_{2i-1}),x_{2i},z])$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-1-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2i-1},R(x_{2i}),z])$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-1-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2i-1},x_{2i},R(z)])$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[R(x_{2i-1}),R(x_{2i}),z])$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[R(x_{2i-1}),x_{2i},R(z)])$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2i-1},R(x_{2i}),R(z)])$
$\displaystyle+\sum_{k=0}^{2n-3}\lambda^{2n-3-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[R(x_{2i-1}),R(x_{2i}),R(z)]),$
$\displaystyle C_{4}$ $\displaystyle=$ $\displaystyle
R_{V}\sum_{i=1}^{n}(-1)^{i+1}(\sum_{k=0}^{2n-1}\lambda^{2n-k}\rho(x_{2i-1},x_{2i})f^{k}(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-1}\lambda^{2n-1-k}\rho(R(x_{2i-1}),x_{2i})f^{k}(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-1}\lambda^{2n-1-k}\rho(x_{2i-1},R(x_{2i}))f^{k}(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-2-k}\rho(R(x_{2i-1}),R(x_{2i}))f^{k}(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+1})),$

and

$\displaystyle C_{5}$ $\displaystyle=$ $\displaystyle
R_{V}(-1)^{n+1}(\sum_{k=0}^{2n-1}\lambda^{2n-k}\rho(x_{2n},x_{2n+1})f^{k}(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n-1})$
$\displaystyle+\sum_{k=0}^{2n-1}\lambda^{2n-1-k}\rho(R(x_{2n}),x_{2n+1})f^{k}(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n-1})$
$\displaystyle+\sum_{k=0}^{2n-1}\lambda^{2n-1-k}\rho(x_{2n},R(x_{2n+1}))f^{k}(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n-1})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-2-k}\rho(R(x_{2n}),R(x_{2n+1}))f^{k}(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n-1})$
$\displaystyle+\sum_{k=0}^{2n-1}\lambda^{2n-k}\rho(x_{2n+1},x_{2n-1})f(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n})$
$\displaystyle+\sum_{k=0}^{2n-1}\lambda^{2n-1-k}\rho(R(x_{2n+1}),x_{2n-1})f(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n})$
$\displaystyle+\sum_{k=0}^{2n-1}\lambda^{2n-1-k}\rho(x_{2n+1},R(x_{2n-1}))f(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n})$
$\displaystyle+\sum_{k=0}^{2n-2}\lambda^{2n-2-k}\rho(R(x_{2n+1}),R(x_{2n-1}))f(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n})).$

On the other hand, denote

$\partial_{R}\delta f(X_{1},\cdot\cdot\cdot,X_{n},z)=D_{1}-D_{2}+E+F+G,$

where

$\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle\sum_{1\leq
i<j\leq{n}}(-1)^{i}f(\tilde{R}(X_{1}),\cdot\cdot\cdot,\tilde{R}(X_{j-1}),\tilde{R}([X_{i},X_{j}]_{F}),\tilde{R}(X_{j+1}),\cdot\cdot\cdot,\tilde{R}(X_{n}),R(z))$
$\displaystyle=$ $\displaystyle\sum_{1\leq
i<j\leq{n}}(-1)^{i}f(R(x_{1}),\cdot\cdot\cdot,R(x_{2j-2}),R(x_{2j-1}),R[x_{2i-1},x_{2i},x_{2j}]_{R},R(x_{2j+1}),\cdot\cdot\cdot,R(x_{2n+1}))$
$\displaystyle+\sum_{1\leq
i<j\leq{n}}(-1)^{i}f(R(x_{1}),\cdot\cdot\cdot,R(x_{2j-2}),R[x_{2i-1},x_{2i},x_{2j-1}]_{R},R(x_{2j}),\cdot\cdot\cdot,R(x_{2n+1})),$
$\displaystyle D_{2}$ $\displaystyle=$ $\displaystyle
R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}(X_{1},\cdot\cdot\cdot,\hat{X_{i}},X_{j-1},[X_{i},X_{j}]_{F},X_{j+1},\cdot\cdot\cdot,X_{n},z)$
$\displaystyle=$ $\displaystyle
R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}(x_{1},\cdot\cdot\cdot,x_{2j-3},x_{2j-2},x_{2j-1},[x_{2i-1},x_{2i},x_{2j}]_{R},x_{2j+1},\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}(x_{1},\cdot\cdot\cdot,x_{2j-3},x_{2j-2},[x_{2i-1},x_{2i},x_{2j-1}]_{R},x_{2j},x_{2j+1},\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle=$ $\displaystyle
R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}_{2j-2}(x_{1},\cdot\cdot\cdot,x_{2j-3},x_{2j-2},x_{2j-1},[x_{2i-1},x_{2i},x_{2j}]_{R},x_{2j+1},\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+R_{V}\sum_{k=0}^{2n-3}\lambda^{2n-3-k}f^{k}_{2j-2}(x_{1},\cdot\cdot\cdot,x_{2j-3},x_{2j-2},x_{2j-1},R[x_{2i-1},x_{2i},x_{2j}]_{R},x_{2j+1},\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}_{2j-3}(x_{1},\cdot\cdot\cdot,x_{2j-3},x_{2j-2},[x_{2i-1},x_{2i},x_{2j-1}]_{R},x_{2j},x_{2j+1},\cdot\cdot\cdot,x_{2n+1})$
$\displaystyle+R_{V}\sum_{k=0}^{2n-3}\lambda^{2n-3-k}f^{k}_{2j-3}(x_{1},\cdot\cdot\cdot,x_{2j-3},x_{2j-2},R[x_{2i-1},x_{2i},x_{2j-1}]_{R},x_{2j},x_{2j+1},\cdot\cdot\cdot,x_{2n+1}),$
$\displaystyle E$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}(-1)^{i}\delta
f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{n},[X_{i},z]_{R})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i}f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n}),R[x_{2i-1},x_{2i},x_{2n+1}]_{R})$
$\displaystyle-
R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2i-1},x_{2i},x_{2n+1}]_{R})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i}f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n}),R[x_{2i-1},x_{2i},x_{2n+1}]_{R})$
$\displaystyle-
R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2i-1},x_{2i},x_{2n+1}]_{R})$
$\displaystyle-
R_{V}\sum_{k=0}^{2n-3}\lambda^{2n-3-k}f_{2n-1}^{k}(x_{1},\cdot\cdot\cdot,x_{2n},R[x_{2i-1},x_{2i},x_{2n+1}]_{R})$
$\displaystyle F$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}\tilde{\rho}(X_{i})\delta
f(X_{1},\cdot\cdot\cdot,\hat{X_{i}},\cdot\cdot\cdot,X_{n},z)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}\tilde{\rho}(X_{i})(f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n}),R(x_{2n+1}))-R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}(x_{1},\cdot\cdot\cdot,x_{2n+1})),$

and

$\displaystyle G$ $\displaystyle=$
$\displaystyle(-1)^{n+1}(\tilde{\rho}(x_{2n},x_{2n+1})\delta
f(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n-1})+\tilde{\rho}(x_{2n+1},x_{2n-1})\delta
f(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n}))$ $\displaystyle=$
$\displaystyle(-1)^{n+1}\tilde{\rho}(x_{2n},x_{2n+1})(f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n-2}),R(x_{2n-1})-R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}\times$
$\displaystyle
f^{k}(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n-1})+(-1)^{n+1}\tilde{\rho}(x_{2n+1},x_{2n-1})(f(R(x_{1}),\cdot\cdot\cdot,R(x_{2n-2}),R(x_{2n})$
$\displaystyle-
R_{V}\sum_{k=0}^{2n-2}\lambda^{2n-2-k}f^{k}(x_{1},\cdot\cdot\cdot,x_{2n-2},x_{2n}).$

Using 2.1 and 2.2, obviously, $B_{1}-C_{1}-C_{2}=D_{1}-D_{2}$. Similarly,
$B_{2}-C_{3}=E$, $B_{3}-C_{4}=F$ and $B_{4}-C_{5}=G$. These yield that
$\partial_{R}\delta=\delta\partial$. ∎

Acknowledgments

Project supported by the National Natural Science Foundation of China ( No.
11871421, No. 11401530 ) and the Natural Science Foundation of Zhejiang
Province of China ( No. LY19A010001 ).

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