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\title{Cohomology and deformations of modified Rota--Baxter Lie--Yamaguti algebras}
\alttitle{Cohomologie et dÃ©formations des algÃ¨bres de Lie--Yamaguti de Rota--Baxter modifiÃ©es}

\author{\firstname{Khaled} \lastname{Basdouri}}
\address{University of Sfax, Faculty of Sciences of Sfax, BP 1171, 3038 Sfax, Tunisia}
\email{khaled.basdouri2016@gmail.com}

\author{\firstname{Sami} \lastname{Benabdelhafidh}}
\address[1]{University of Sfax, Faculty of Sciences of Sfax, BP 1171, 3038 Sfax, Tunisia}
\email{abdelhafidhsami41@gmail.com}

\begin{abstract}
In this paper, first we introduce a notion of modified Rota--Baxter Lie--Yamaguti algebra of any weight. Also, we introduce the concept of a representation modified Rota--Baxter Lie--Yamaguti algebra of any weight. Then, we define a cohomology theory for modified Rota--Baxter Lie--Yamaguti algebras of any weight. As applications of the cohomology, we study formal deformations of modified Rota--Baxter Lie--Yamaguti algebras of arbitrary weights.
\end{abstract}

\begin{altabstract}
Dans cet article, nous introduisons d'abord la notion d'algÃ¨bre de Rota--Baxter Lie--Yamaguti modifiÃ©e de poids quelconque. Nous introduisons Ã©galement le concept de reprÃ©sentation d'algÃ¨bre de Rota--Baxter Lie--Yamaguti modifiÃ©e de poids quelconque. Nous dÃ©finissons ensuite une thÃ©orie de cohomologie pour les algÃ¨bres de Rota--Baxter Lie--Yamaguti modifiÃ©es de poids quelconque. Comme applications de la cohomologie, nous Ã©tudions les dÃ©formations formelles d'algÃ¨bres de Rota--Baxter Lie--Yamaguti modifiÃ©es de poids quelconque.
\end{altabstract}

\keywords{\kwd{Lie--Yamaguti algebra} \kwd{modified Rota--Baxter operator} \kwd{representation} \kwd{cohomology} \kwd{deformation}}
\altkeywords{\kwd{AlgÃ¨bre de Lie--Yamaguti} \kwd{opÃ©rateur de Rota--Baxter modifiÃ©} \kwd{reprÃ©sentation} \kwd{cohomologie} \kwd{dÃ©formation}}

\begin{document}

\maketitle

\section{Introduction}\label{sec:intr}

\subsection{Lie--Yamaguti algebras}

The concept of Lie--Yamaguti algebra was introduced after to Nomizu's work on the affine invariant connections on homogeneous spaces in 1950's~\cite{KN1}. Later in 1960's, Yamaguti considered an algebraic structure and called it a general Lie triple system~\cite{KY1,KY2,KY3}, since it can be showed as a generalization of a Lie algebra and a Lie triple system. Kinyon and Weinstein first called this object a Lie--Yamaguti algebra when studying courant algebroids in the earlier 21st century~\cite{MK1}. Since a Lie--Yamaguti algebra originates from differential geometry completely and it is a key higher structure in mathematical physics, recently, it has attracted much attention and is widely developed. For instance, Benito and his helpers deeply searched irreducible Lie--Yamaguti algebras and their relations with orthogonal Lie algebras~\cite{PB1,PB3,PB4}. Deformations and Abelian extensions of Lie--Yamaguti algebras were studied in~\cite{JL1,YM1,TZ1,TZ2}. Takahashi considered modules over quandles using representations of Lie--Yamaguti algebras in~\cite{NT1}. Sheng and Zhou analyzed product structures and complex structures on Lie--Yamaguti algebras by means of Nijenhuis operators in~\cite{YS1}. Many important results about Lie algebra have been developed to Lie--Yamaguti algebra.

\subsection{Modified Rota--Baxter operators}

In~\cite{MA1} Semenov-Tian-Shansky showed that under certain conditions, a Rota--Baxter operator of weight 0 on a Lie algebra is exactly the operator form of the classical Yang--Baxter equation. In the same paper, the Rota--Baxter operator also introduced a closely connected notion of modified classical Yang--Baxter equation whose solutions are said to be modified $r$-matrices~\cite{JI1}. The modified classical Yang--Baxter equation play an important role in studying solutions of Lax equations, affine geometry on Lie groups, factorization problems in Lie algebras and compatible Poisson structures~\cite{CB1,LC1,AG1,MA1,MS1,NY1}.

Rota--Baxter operator first introduced in fluctuation theory in probability~\cite{GB1}. It was further extended by Rota~\cite{GC1} and Cartier~\cite{PC1}. The author has many applications in mathematical physics. Das, Hazra, and~Mishra considered Rota--Baxter Lie algebras in~\cite{AD4}. The Rota--Baxter operator of arbitrary weights on Lie algebra studied in~\cite{AD2,AD3,SG1,KW1}. Rota--Baxter Lie triple systems of any weight established in~\cite{SC1}.

The notion of a modified Rota--Baxter algebra obtained from the combination of those of a Rota--Baxter operator and a modified Yang--Baxter equation. A Rota--Baxter operator gives a modified Rota--Baxter operator by a linear transformation~\cite{KE1}, a close relationship between the above two operators is showed in~\cite{XZ1,XZ2}, here, X.~Zhang, X.~Gao and L.~Guo studied the Rota--Baxter type algebra and constructed the free object uniformly in that category.

Recently, the modified Rota--Baxter operator on an associative algebra studied in~\cite{AD1}, the cohomology, deformations theory and Abelian extensions of modified Rota--Baxter Leibniz algebras are given in~\cite{YL1,BM1}. Cohomology and deformations of modified $\lambda$-differential Jacobi--Jordan algebras established in~\cite{IB}. The method of this paper is to follow the recent work.

\subsection{Motivation}

Our main objective is to consider modified Rota--Baxter Lie--Yamaguti algebras. More precisely, we introduce the concept of a modified Rota--Baxter Lie--Yamaguti algebra, which includes a Lie--Yamaguti algebra and a modified Rota--Baxter operator. We define a cochain map $\psi$, and then give the cohomology of modified Rota--Baxter Lie--Yamaguti algebras with coefficients in a representation by using a Yamaguti coboundary operator and a cochain map $\psi$. Finally we study formal one-parameter deformations of a modified Rota--Baxter Lie--Yamaguti algebra and we prove that if the second cohomology group $\operH^2_{\mRBLYl}(A)=0$, then the modified Rota--Baxter Lie--Yamaguti algebra is rigid.

\subsection{Outline of the paper}

In Section~\ref{sec:Pre}, we recall some preliminary results about Lie--Yamaguti algebra and their cohomology theory. In Section~\ref{sec:Induced}, we consider modified Rota--Baxter Lie--Yamaguti algebras of arbitrary weights and introduce their representations. We also provide a new construction. In Section~\ref{sec:Cohom}, we define a cohomology theory for modified Rota--Baxter Lie--Yamaguti algebras. In Section~\ref{sec:Defor}, we justify this cohomology theory by interpreting lower degree cohomology groups as formal deformations.

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

\section{Preliminaries}\label{sec:Pre}

In this section, we start with the background of Lie--Yamaguti algebras and their cohomology that we refer the reader to~\cite{MK1,KY2,KY3} for more details.

\begin{defi}[\cite{MK1}]\label{def1}
A Lie--Yamaguti algebra is a vector space $A$ together with a bilinear bracket $[\,\cdot\,,\cdot\,] \colon \wedge^{2} A \to A$ and a trilinear bracket $\{\,\cdot\,,\cdot\,,\cdot\,\} \colon \wedge^{2} A \otimes A \to A$, if the following conditions hold: for all $a,b,c,u,v \in A$,
\begin{align}
	\label{eq1}	&amp; \bracks[\big]{[a,b],c}+\bracks[\big]{[b,c],a}+\bracks[\big]{[c,a],b}+\{ a,b,c\}+\{ b,c,a\} +\{c,a,b\} =0,			
\\  \label{eq2}	&amp; \braces[\big]{[a,b],c,u}+\braces[\big]{[b,c],a,u}+\braces[\big]{[c,a],b,u}=0, 								
\\	\label{eq3}	&amp; \braces[\big]{a,b,[c,u]}=\bracks[\big]{\{a,b,c\},u}+\bracks[\big]{c,\{a,b,u\}}, 									
\\ 	\label{eq4}	&amp; \braces[\big]{ a,b,\{c,u,v\}}=\braces[\big]{\{a,b,c\},u,v}+\braces[\big]{c,\{a,b,u\},v}+\braces[\big]{c,u,\{a,b,v\}}.
\end{align}
\end{defi}

\begin{exem}\label{ex1}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]}$ be a Lie algebra. Define a trilinear bracket
\[
\{\,\cdot\,,\cdot\,,\cdot\,\} \colon \wedge^{2} A \otimes A \to A,
\]
by
\[
\{ a,b,c\}=\bracks[\big]{[a,b],c}, \quad \forall a,b,c \in A.
\]
Then, by a direct computation, we know that $\parens[\big]{A,[\,\cdot\,,\cdot\,],\{\,\cdot\,,\cdot\,,\cdot\,\}}$ is a Lie--Yamaguti algebra.
\end{exem}

\begin{exem}\label{ex2}
Let $F$ be a closed manifold with an affine connection, and denote by $\mathfrak{X}(F)$ the set of vector fields on $F$. For all $a,b,c \in \mathfrak{X}(F)$, set
\begin{align*}
[a,b]		&amp; = -K_1(a,b),	\\
\{a,b,c\}	&amp; = -K_2(a,b)c,
\end{align*}
where $K_1$ and $K_2$ are torsion tensor and curvature tensor respectively. It turns out that the triple $\parens[\big]{\mathfrak{X}(F),[\,\cdot\,,\cdot\,],\{\,\cdot\,,\cdot\,,\cdot\,\}}$ is a Lie--Yamaguti algebra. See~\cite{KN1} for more details.
\end{exem}

\begin{defi}[\cite{KY2}]\label{def2}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,],\{\,\cdot\,,\cdot\,,\cdot\,\}}$ be a Lie--Yamaguti algebra. A representation of $A$ is a vector space $V$ together with a linear map $\rho \colon A \to \opergl(V)$ and a bilinear map\/ $\theta \colon \otimes^2A \to \opergl(V)$, if the following equalities hold: for all $a,b,c,u \in A$,
\begin{align}
	\label{eq5}	&amp; \theta\parens[\big]{[a,b],c}-\theta(a,c)\rho(b)+\theta(b,c)\rho(a)=0,														
\\	\label{eq6}	&amp; \theta\parens[\big]{a,[b,c]}-\rho(b)\theta(a,c)+\rho(c)\theta(a,b)=0, 														
\\	\label{eq7}	&amp; \rho\parens[\big]{\{a,b,c\}}=\bracks[\big]{\mathfrak{D}_{\rho,\theta}(a,b),\rho(c)}, 														
\\	\label{eq8}	&amp; \theta(c,u)\theta(a,b)-\theta(b,u)\theta(a,c)-\theta\parens[\big]{a,\{b,c,u\}}+\mathfrak{D}_{\rho,\theta}(b,c)\theta(a,u)=0, 	
\\	\label{eq9}	&amp; \theta\parens[\big]{\{a,b,c\},u}+\theta\parens[\big]{c,\{a,b,u\}}=\bracks[\big]{\mathfrak{D}_{\rho,\theta}(a,b),\theta(c,u)},							
\end{align}
where the bilinear map $\mathfrak{D}_{\rho,\theta} \colon \otimes^2A\to \opergl(A)$ is defined by
\begin{equation}
\mathfrak{D}_{\rho,\theta}(a,b)= \theta(b,a)-\theta(a,b)+\bracks[\big]{\rho(a),\rho(b)}-\rho\parens[\big]{[a,b]}, \quad \forall a,b \in A. \label{eq10}
\end{equation}
It is obvious that $\mathfrak{D}_{\rho,\theta}$ is skew-symmetric, and we write $\mathfrak{D}$ in the sequel without ambiguities. We denote a representation of $A$ by $(V;\rho,\theta)$.
\end{defi}

\begin{exem}\label{ex3}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,],\{\,\cdot\,,\cdot\,,\cdot\,\}}$ be a Lie--Yamaguti algebra. We define two linear maps by
\begin{align*}
	&amp; \ad \colon A\to \opergl(A) 			
\\	&amp; \ad_a c = [a,c], \quad \forall c \in A,
\end{align*}
and
\begin{align*}
	&amp; K \colon \otimes^2A \to \opergl(A) 			
\\	&amp; K_{a,b}c=\{c,a,b\}, \quad \forall c \in A.
\end{align*}
Then, $(\ad_a,K)$ is a representation of $A$ on itself, where $\mathcal{D}=\mathfrak{D}_{\ad,K}$ is given by
\[
\mathcal{D}_{a,b}=K_{b,a}-K_{a,b}+[\ad_a,\ad_b]-\ad_{[a,b]}, \quad \forall a,b \in A.
\]
By~\eqref{eq1}, we have
\[
\mathcal{D}_{a,b}=\{a,b,c\}, \quad \forall c \in A.
\]
Thus, $(A,\ad,K)$ is said to be the adjoint representation of $A$.
\end{exem}

Let us recall the cohomology theory on Lie--Yamaguti algebras given in~\cite{KY2}. Let $\parens[\big]{A,[\,\cdot\,,\cdot\,],\{\,\cdot\,,\cdot\,,\cdot\,\}}$ be a Lie--Yamaguti algebra and $(V;\rho,\theta)$ a representation of $A$. We denote the set of $n$-cochains by $C^n_{\LY}(A,V)$ for ($n &gt; 1$), where
\[
C^{n+1}_{\LY}(A,V)=
\begin{cases}
	\Hom(\underbrace{\wedge^2A \otimes \dotsb \otimes \wedge^2A}_n,V) \times \Hom(\underbrace{\wedge^2A \otimes \dotsb \otimes \wedge^2A}_n \otimes A,V),	&amp; \forall n\geq 1
\\	\Hom(A,V),	&amp; \text{for $n=0$}.
\end{cases}
\]

If $n &gt; 1$ and for any $(f,g) \in C^{n+1}_{\LY}(A,V)$, the coboundary map is defined by
\[
\begin{aligned}
	\delta^{n+1}=(\delta^{n+1}_\coboundI,\delta^{n+1}_\coboundII) \colon C_{\LY}^{n+1}(A,V)	&amp; \to C_{\LY}^{n+2}(A,V) 						
\\	(f,g)	&amp; \mapsto \parens[\big]{\delta^{n+1}_\coboundI(f,g),\delta^{n+1}_\coboundII(f,g)}
\end{aligned}
\]
where $\delta^{n+1}_\coboundI(f,g)$ and $\delta^{n+1}_\coboundII(f,g)$ are given to be
\begin{equation}\label{eq11}
\begin{split}
	&amp; \delta^{n+1}_\coboundI(f,g)(\mathfrak{\mathfrak{A}}_1,\dotsc,\mathfrak{A}_{n+1}) 
\\	&amp; \hspace{1.5cm} = (-1)^n\parens[\big]{\rho(\mathfrak{a}_{n+1})g(\mathfrak{A}_1,\dotsc,\mathfrak{A}_n,\mathfrak{b}_{n+1}) -\rho(\mathfrak{b}_{n+1})g(\mathfrak{A}_1,\dotsc,\mathfrak{A}_n,\mathfrak{a}_{n+1}) 
\\	&amp; \hspace{1.5cm} \quad - g(\mathfrak{A}_1,\dotsc,\mathfrak{A}_n,[\mathfrak{a}_{n+1},\mathfrak{b}_{n+1}])} + \sum_{i=1}^n(-1)^{i+1}\mathfrak{D}(\mathfrak{A}_i)f(\mathfrak{A}_1,\dotsc,
\hat{\mathfrak{A}}_i,\dotsc,\mathfrak{A}_{n+1})
\\	&amp; \hspace{1.5cm} \quad + \sum_{1 \leq i &lt; j \leq n+1} (-1)^i f \parens[\big]{\mathfrak{A}_1,\dotsc,\hat{\mathfrak{A}}_i,\dotsc, \{\mathfrak{a}_i,\mathfrak{b}_i,\mathfrak{a}_j \} \wedge \mathfrak{b}_j + \mathfrak{a}_j \wedge \{\mathfrak{a}_i,\mathfrak{b}_i,\mathfrak{b}_j \},\dotsc,\mathfrak{A}_{n+1}},
\end{split}
\end{equation}
and
\begin{equation}\label{eq12}
\begin{split}
	&amp; \delta^{n+1}_\coboundII(f,g)(\mathfrak{\mathfrak{A}}_1,\dotsc,\mathfrak{A}_{n+1},c)
\\	&amp; \hspace{1.5cm} = (-1)^n\parens[\big]{\theta(\mathfrak{b}_{n+1},c)g(\mathfrak{A}_1,\dotsc,\mathfrak{A}_n,\mathfrak{a}_{n+1}) -\theta(\mathfrak{a}_{n+1},c)g(\mathfrak{A}_1,\dotsc,\mathfrak{A}_n,\mathfrak{b}_{n+1})}
\\	&amp; \hspace{1.5cm} \quad + \sum_{i=1}^{n+1}(-1)^{i+1}\mathfrak{D}(\mathfrak{A}_i)g(\mathfrak{A}_1,\dotsc,
\hat{\mathfrak{A}}_i,\dotsc,\mathfrak{A}_{n+1},c)  
\\	&amp; \hspace{1.5cm} \quad +  \sum_{1 \leq i &lt; j \leq n+1} (-1)^i g \parens[\big]{\mathfrak{A}_1,\dotsc,\hat{\mathfrak{A}}_i,\dotsc, \{\mathfrak{a}_i,\mathfrak{b}_i,\mathfrak{a}_j \} \wedge \mathfrak{b}_j + \mathfrak{a}_j \wedge \{\mathfrak{a}_i,\mathfrak{b}_i,\mathfrak{b}_j \},\dotsc,\mathfrak{A}_{n+1},c} 
\\	&amp; \hspace{1.5cm} \quad + \sum_{i=1}^{n+1} (-1)^i g \parens[\big]{\mathfrak{A}_1,\dotsc,\hat{\mathfrak{A}}_i,\dotsc,\mathfrak{A}_{n+1},\{\mathfrak{a}_i,\mathfrak{b}_i,c\}}, 
\end{split}
\end{equation}
where $\mathfrak{A}_i=\mathfrak{a}_i \wedge \mathfrak{b}_i \in \wedge^2A$ and $c \in A$ for $i=1,2,\dotsc, n$.

If $n=0$ and for any $f \in C^1_{\LY}(A,V)$, the coboundary map is defined by
\[
\begin{aligned}
	\delta^{1}=(\delta^{1}_\coboundI,\delta^{1}_\coboundII) \colon C_{\LY}^{1}(A,V) &amp; \to C_{\LY}^{2}(A,V) 
\\ 	f 	&amp; \mapsto \parens[\big]{\delta^{1}_\coboundI(f),\delta^{1}_\coboundII(f)}
\end{aligned}
\]
where $\delta^{1}_\coboundI(f,g)$ and $\delta^{1}_\coboundII(f,g)$ are given to be
\begin{align}
	\label{eq13}	\parens[\big]{\delta^{1}_\coboundI(f)}(a,b)		&amp; = \rho(a)f(b)-\rho(b)f(a)-f\parens[\big]{[a,b]}, 
\\	\label{eq14}	\parens[\big]{\delta^{1}_\coboundII(f)}(a,b,c)	&amp; = \mathfrak{D}_{\rho,\theta}(a,b)f(c)+\theta(b,c)f(a)-\theta(a,c)f(b)-f\parens[\big]{\{a,b,c\}}, \quad \forall a,b,c \in A. 
\end{align}

Yamaguti proved the following basic fact.

\begin{prop}[\cite{MK1}]\label{pro1}
With the notations above, for any $f \in C^1_{\LY}(A,V)$, we have
\[
\delta^{2}_\coboundI\parens[\big]{\delta^{1}_\coboundI(f),\delta^{1}_\coboundII(f)}=0 \qquad \text{and} \qquad \delta^{2}_\coboundII\parens[\big]{\delta^{1}_\coboundI(f),\delta^{1}_\coboundII(f)}=0
\]
Moreover, for all $(f,g) \in C^n_{\LY}(A,V)$ for ($n \geq 2$), we have
\[
\delta^{n+2}_\coboundI\parens[\big]{\delta^{n+1}_\coboundI(f,g),\delta^{n+1}_\coboundII(f,g)}=0 \qquad \text{and} \qquad \delta^{n+2}_\coboundII\parens[\big]{\delta^{n+1}_\coboundI(f,g),\delta^{n+1}_\coboundII(f,g)}=0.
\]
Thus, $\parens[\big]{C_{\LY}^{\bullet}(A,V)\coloneqq \bigoplus_{n=1}^{+\infty}C_{\LY}^{n}(A,V),\delta^{n+1}}$ is well-defined. We call this cohomology the Yamaguti cohomology and denoted by\/ $\operH^\bullet_{\LY}$.
\end{prop}

\section{Induced modified Rota--Baxter Lie--Yamaguti algebras}\label{sec:Induced}

In this section, we consider modified Rota--Baxter Lie--Yamaguti algebras and introduce their representations. We also give various examples and new constructions.

\begin{defi}\label{def4}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ be a Lie--Yamaguti algebra. A linear map $K \colon A \to A$ is called a modified Rota--Baxter of weight $\lambda$ on $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ if $K$ satisfies
\begin{equation}\label{eq15}
[Ka,Kb]_A = K\parens[\big]{[Ka,b]_A+[a,Kb]_A}+\lambda[a,b]_A 
\end{equation}
and
\begin{multline}\label{eq16}
	\{Ka,Kb,Kc\}_A = K\parens[\big]{\{Ka,Kb,c\}_A + \{Ka,b,Kc\}_A + \{a,Kb,Kc\}_A-\lambda\{a,b,c\}_A}
\\	+ \lambda\{Ka,b,c\}_A + \lambda\{a,Kb,c\}_A + \lambda\{a,b,Kc\}_A
\end{multline}
for all $a,b,c \in A$ and $\lambda \in \mathbb{K}$. A modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$ is a quadruple $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ consisting of a Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ equipped with a modified Rota--Baxter operator $K$ of weight $\lambda$ on it.
\end{defi}

\begin{exem}\label{ex4}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ be a $2$-dimensional Lie--Yamaguti algebra, whose non-zero bracket is defined by, with respect to a basis $\{e_1,e_2 \}$,
\[
[e_1,e_2]_A \ = \ e_1, \qquad \{e_1,e_2,e_2\}_A \ = \ e_1.
\]
Moreover,
\[
K=
\begin{pmatrix}
1 \;&amp;\; 0 \\
0 \;&amp;\; 0
\end{pmatrix}
\]
is a modified Rota--Baxter operator of weight $-1$ on $A$.
\end{exem}

\begin{exem}\label{ex5}
Let $A$ be a $3$-dimensional Lie--Yamaguti algebra with a basis $\{e_1,e_2,e_3\}$ defined by
\[
[e_1,e_2]_A=-[e_2,e_1]_A=e_3, \qquad \{e_1,e_2,e_1\}_A=-\{e_2,e_1,e_1\}_A=e_3.
\]
Then
\[
K=
\begin{pmatrix}
0 \;&amp;\; 1 \;&amp;\; 0 \\
1 - \lambda \;&amp;\; 1 \;&amp;\; 0 \\
1 + \lambda \;&amp;\; 0 \;&amp;\; 1
\end{pmatrix}
\]
is a modified Rota--Baxter operator of weight $\lambda$ on $A$.
\end{exem}

\begin{defi}\label{def5}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ and $\parens[\big]{A^\prime,[\,\cdot\,,\cdot\,]_{A^\prime},\{\,\cdot\,,\cdot\,,\cdot\,\}_{A^\prime},K^\prime}$ be two modified Rota--Baxter Lie--Yamaguti algebras. A homomorphism from $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ to $\parens[\big]{A^\prime,[\,\cdot\,,\cdot\,]_{A^\prime},\{\,\cdot\,,\cdot\,,\cdot\,\}_{A^\prime},K^\prime}$ is a Lie--Yamaguti algebra homomorphism $\phi \colon A \to A^\prime$ satisfying $K \circ \phi= \phi \circ K^\prime$. In particular, if $\phi$ is invertible, then a homomorphism $\phi$ becomes an isomorphism from $K$ to $K^\prime$.
\end{defi}

Denote by $\mRBLYl$ the category of modified Rota--Baxter Lie--Yamaguti algebras of weight $\lambda$ with obvious morphisms.

Recall~\cite{YS2} that a linear map $T \colon A\to A$ is called a Rota--Baxter operator of weight $\lambda$ on the Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ if $T$ satisfies
\[
[Ta,Tb]_A = T\parens[\big]{[Ta,b]_A + [a,Tb]_A + \lambda[a,b]_A}
\]
and
\begin{multline*}
	\{Ta,Tb,Tc\}_A = T\parens[\big]{\{Ta,Tb,c\}_A + \{Ta,b,Tc\}_A + \{a,Tb,Tc\}_A
\\	+ \lambda\{Ta,b,c\}_A+ \lambda\{a,Tb,c\}_A+ \lambda\{a,b,Tc\}_A + \lambda^2\{a,b,c\}_A}
\end{multline*}
for all $a,b,c \in A$. A Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$ is a quadruple $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,T}$ consisting of a Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ together with a Rota--Baxter operator $T$ of weight $\lambda$ on it. Furthermore, we have

\begin{lemm}\label{lem1}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ be a Lie--Yamaguti algebra. Then $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A, T}$ is a Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$ if and only if $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,2T+\lambda \Id_A}$ is a modified Rota--Baxter Lie--Yamaguti algebra of weight $-\lambda^2$.
\end{lemm}

\begin{defi}\label{def6}
A representation of the modified Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ is a quadruple $(V; \rho, \theta, K_V)$ such that the following conditions are satisfied:
\begin{enumerate}\romanenumi
\item $(V; \rho, \theta)$ is a representation of the Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$;
\item $K_V \colon V\to V$ is a linear map satisfying the following equations
\begin{equation}\label{eq17}
\rho(Ka)(K_Vu) = K_V\parens[\big]{\rho(Ka)u+\rho(a)K_Vu}+\lambda\rho(a)u, 
\end{equation}
and
\begin{multline}\label{eq19}
	\theta(Ka,Kb)K_Vu = K_V\parens[\big]{\theta(Ka,Kb)u+\theta(Ka,b)K_Vu+\theta(a,Kb)K_Vu-\lambda\theta(a,b)u)} 
\\	+ \lambda\theta(Ka,b)u+\lambda\theta(a,Kb)u+\lambda\theta(a,b)K_Vu,
\end{multline}
for all $a,b \in A$ and $u \in V$.
\end{enumerate}
\end{defi}

It can be concluded from~\eqref{eq19} that
\begin{multline}\label{eq18}
	\mathfrak{D}(Ka,Kb)K_Vu = K_V\parens[\big]{\mathfrak{D}(Ka,Kb)u+\mathfrak{D}(Ka,b)K_Vu+\mathfrak{D}(a,Kb)K_Vu-\lambda\mathfrak{D}(a,b)u}
\\	+ \lambda\mathfrak{D}(Ka,b)u+\lambda\mathfrak{D}(a,Kb)u+\lambda\mathfrak{D}(a,b)K_Vu.
\end{multline}

\begin{exem}\label{ex6}
It is ready to observe that $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ itself is a representation over the modified Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$ of weight $\lambda$.
\end{exem}

The representation of modified Rota--Baxter Lie--Yamaguti algebras is closely related to the representation of Rota--Baxter Lie--Yamaguti algebras of weight $\lambda$.

\begin{defi}
A representation of the Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,T}$ of weight $\lambda$ is a quadruple $(V; \rho, \theta, T_V)$ such that the following conditions are satisfied:
\begin{enumerate}\romanenumi
\item $(V; \rho, \theta)$ is a representation of the Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A}$;
\item $T_V \colon V\to V$ is a linear map satisfying the following equations
\begin{equation}\label{4.4}
\rho(Ta)T_V(u) = T_V\parens[\big]{\rho(Ta)u+\rho(a)T_V(u)+\lambda\rho(a)u}
\end{equation}
and
\begin{multline}\label{4.5}
	\theta(Ta, Tb)T_Vu = T_V\parens[\big]{\theta(Ta, Tb) u +\theta(Ta, b)T_Vu+\theta(a, Tb)T_Vu 
\\	+ \lambda\theta(Ta, b) u +\lambda\theta(a, Tb)u+\lambda\theta(a, b)T_Vu-\lambda\theta(a, b)u}
\end{multline}
for any $a,b\in A$ and $ u\in V$.
\end{enumerate}
\end{defi}

\begin{prop}
If $(V; \rho, \theta, T_V)$ is a representation of the Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A, [\,\cdot\,, \cdot\,]_A, \{\,\cdot\,, \cdot\,, \cdot\,\}_A, T}$ of weight $\lambda$, then $(V; \rho,\theta, 2T_V+\lambda\Id_V)$ is a representation of the modified Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A, [\,\cdot\,, \cdot\,]_A, \{\,\cdot\,, \cdot\,, \cdot\,\}_A, 2T+\lambda\Id_A}$ of weight $-\lambda^2$.
\end{prop}

In the following, we construct the semidirect product in the context of modified Rota--Baxter Lie--Yamaguti algebras of arbitrary weights.

\begin{prop}\label{pro2}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ be a modified Rota--Baxter Lie--Yamaguti algebra and $(V; \rho, \theta, K_V)$ be a representation of it. Then $(A \oplus V,K \oplus K_V)$ is a modified Rota--Baxter Lie--Yamaguti algebra, where the Lie--Yamaguti bracket on $A \oplus V$ is given by the semidirect product
\begin{align}
	\label{eq20}	[a+u,b+v]_{\ltimes}			&amp; =[a,b]_A+\rho(a)v-\rho(b)u, 
\\	\label{eq21}	\{a+u,b+v,c+w\}_{\ltimes}	&amp; =\{a,b,c\}_A+\mathfrak{D}(a,b)w+\theta(b,c)u-\theta(a,c)v, 
\\	\label{eq22}	K\oplus K_V(a+u) 			&amp; = Ka + K_Vu, 
\end{align}
for all $a,b,c \in A$ and $u,v\in V$.
\end{prop}

\begin{prop}\label{pro3}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ be a modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$. Define a new binary and a new ternary operation as
\begin{align}
	\label{eq23}	[a,b]_K		&amp; = [Ka,b]_A+[a,Kb]_A, 
\\	\label{eq24}	\{a,b,c\}_K	&amp; = \{Ka,Kb,c\}_A + \{Ka,b,Kc\}_A + \{a,Kb,Kc\}_A, 
\end{align}
for any $a,b,c \in A$. Then:
\begin{enumerate}\romanenumi
\item $\parens[\big]{A,[\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K}$ is a new Lie--Yamaguti algebra, we denote this Lie--Yamaguti algebra by~$A_K$;
\item the quadruple $\parens[\big]{A_K, [\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K,K}$ is a modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$.
\end{enumerate}
\end{prop}

\begin{proof}
For all $a,b,c \in A$, we have
\begin{enumerate}\romanenumi
\item \eqref{eq1}--\eqref{eq4} show that $\parens[\big]{A, [\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K}$ is a Lie--Yamaguti algebra.
\item For any $a,b \in A$, we have
\[
\begin{split}
[Ka,Kb]_K	
	&amp; = [K^2a,Kb]_A + [Ka,K^2b]_A
\\	&amp; = K\parens[\big]{[K^2a,b]_A + [Ka,Kb]_A} + \lambda[Ka,b]_A + K\parens[\big]{[Ka,Kb]_A + [a,K^2b]_A} + \lambda[a,Kb]_A
\\	&amp; = K\parens[\big]{[Ka,b]_K + [a,Kb]_K} + \lambda[a,b]_K,
\end{split}
\]
and
\[
\begin{split}
\{Ka,Kb,Kc\}_K	
	&amp; = \{K^2a,K^2b,Kc\}_A + \{K^2a,Kb,K^2c\}_A + \{Ka,K^2b,K^2c\}_A,
\\	&amp; = K\parens[\big]{\{K^2a,K^2b,c\}_A + \{K^2a,Kb,Kc\}_A + \{Ka,K^2b,Kc\}_A- \lambda\{Ka,Kb,c\}_K}
\\	&amp; \quad +\lambda\{K^2a,Kb,c\}_A + \lambda\{Ka,K^2b,c\}_A +\lambda\{Ka,Kb,Kc\}_A
\\	&amp; \quad +K\parens[\big]{\{K^2a,Kb,Kc\}_A + \{K^2a,b,K^2c\}_A + \{Ka,Kb,K^2c\}_A- \lambda\{Ka,b,Kc\}_A}
\\	&amp; \quad +\lambda\{K^2a,b,Kc\}_A + \lambda\{Ka,Kb,Kc\}_A +\lambda\{Ka,b,K^2c\}_A
\\	&amp; \quad +K\parens[\big]{\{Ka,K^2b,Kc\}_A + \{Ka,Kb,K^2c\}_A + \{a,K^2b,K^2c\}_A- \lambda\{a,Kb,Kc\}_A}
\\	&amp; \quad +\lambda\{Ka,Kb,Kc\}_A + \lambda\{a,K^2b,Kc\}_A +\lambda\{a,Kb,K^2c\}_A
\\	&amp; = K\parens[\big]{\{Ka,Kb,c\}_K + \{Ka,b,Kc\}_K + \{a,Kb,Kc\}_K - \lambda\{a,b,c\}_K}
\\	&amp; \quad +\lambda\{Ka,b,c\}_K + \lambda\{a,Kb,c\}_K +\lambda\{a,b,Kc\}_K.
\end{split}
\]
\end{enumerate}
This proves that the quadruple $\parens[\big]{A_K, [\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K, K}$ is a modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$.
\end{proof}

\begin{theo}\label{th1}
Let $(V; \rho, \theta, K_V)$ be a representation of the modified Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ of weight $\lambda$. Define linear maps $\rho_K \colon A \to \opergl(V)$, $\theta_K \colon A \times A \to \opergl(V) $ by
\begin{align}
	\label{eq25}	\rho_K(a)u		&amp; = \rho(Ka)u+\rho(a)(K_Vu), 
\\	\label{eq27}	\theta_K(a,b)u	&amp; = \theta(Ka,Kb)u-K_V\parens[\big]{\theta(Ka,b)u+\theta(a,Kb)u}, 
\end{align}
for any $a,b \in A$ and $u \in V$. It can be concluded from~\eqref{eq27} that
\begin{equation}\label{eq26}
\mathfrak{D}_K(a,b)u = \mathfrak{D}(Ka,Kb)u - K_V\parens[\big]{\mathfrak{D}(Ka,b)u+\mathfrak{D}(a,Kb)u}.
\end{equation}
Then, $(V;\rho_K,\theta_K)$ is a representation of the Lie--Yamaguti algebra $\parens[\big]{A,[\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K}$. Moreover, $(V;\rho_K,\theta_K,K_V)$ is a representation of the modified Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A_K,[\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K,K}$ of weight $\lambda$.
\end{theo}

\begin{proof}
For all $a,b \in A$, $u \in V$, we have
\[
\begin{split}
\rho_K(Ka)(K_Vu)
	&amp; = \rho(K^2a)(K_Vu)+\rho(Ka)\parens[\big]{K^2_V(u)}
\\	&amp; = K_V\parens[\big]{\rho(K^2a)u+\rho(Ka)(K_Vu)}+\lambda\rho(Ka)u 
\\	&amp; \quad + K_V\parens[\big]{\rho(Ka)(K_Vu)+\rho(a)\parens[\big]{K^2_V(u)}}+\lambda\rho(a)(K_Vu)
\\	&amp; = K_V\parens[\big]{\rho_K(Ka)u+\rho_K(a)(K_Vu)}+\lambda\rho_K(a)u,
\end{split}
\]
and
\[
\begin{split}
\theta_K(Ka,Kb)(K_Vu)
	&amp; = \theta(K^2a,K^2b)(K_Vu)-K_V\parens[\big]{\theta(K^2a,Kb)K_Vu+\theta(Ka,K^2b)K_Vu}
\\	&amp; = K_V\parens[\big]{\theta(K^2a,K^2b)u+\theta(K^2a,Kb)K_Vu+\theta(Ka,K^2b)(K_Vu)-\lambda\theta(Ka,Kb)u}
\\	&amp; \quad +\lambda\theta(K^2a,Kb)u+\lambda\theta(Ka,K^2b)u+\lambda\theta(Ka,Kb)K_Vu
\\	&amp; \quad -K_V\parens[\Big]{K_V\parens[\big]{\theta(K^2a,Kb)u+\theta(K^2a,b)(K_Vu)+\theta(Ka,Kb)K_Vu-\lambda\theta(Ka,b)u}
\\	&amp; \quad +\lambda\theta(K^2a,b)u+\lambda\theta(Ka,Kb)u+\lambda\theta(Ka,b)K_Vu}
\\	&amp; \quad -K_V\parens[\Big]{K_V\parens[\big]{\theta(Ka,K^2b)u+\theta(Ka,Kb)K_Vu+\theta(a,K^2b)K_Vu-\lambda\theta(a,Kb)u}
\\	&amp; \quad +\lambda\theta(Ka,Kb)u+\lambda\theta(a,K^2b)u+\lambda\theta(a,Kb)K_Vu}
\\	&amp; = K_V\parens[\Big]{\theta_K(Ka,Kb)u+\theta_K(Ka,b)K_Vu+\theta_K(a,Kb)K_Vu-\lambda\theta_K(a,b)u}
\\	&amp; \quad + \lambda\theta_K(Ka,b)u+\lambda\theta_K(a,Kb)u+\lambda\theta_K(a,b)K_Vu.
\end{split}
\]
Hence, $(V;\rho_K,\theta_K,K_V)$ is a representation of the modified Rota--Baxter Lie--Yamaguti algebra $\parens[\big]{A_K,[\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K,K}$ of weight $\lambda$.
\end{proof}

\section{Cohomology of modified Rota--Baxter Lie--Yamaguti algebras }\label{sec:Cohom}

In this section, we will define a cohomology theory for modified Rota--Baxter Lie--Yamaguti algebras. This cohomology will be used in Section~\ref{sec:Defor} to study their deformations.

\vspace{-.4\baselineskip}

\subsection{Cohomology of modified Rota--Baxter operators} 

\vspace{-.1\baselineskip}

Firstly, let's study the cohomology of modified Rota--Baxter operators. Let $\parens[\big]{A, [\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ be a modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$ and $(V;\rho,\theta,K_V)$ be a representation over it. Recall that Proposition~\ref{pro3} and Theorem~\ref{th1} give a new Lie--Yamaguti algebra $A_K$ and a new representation $V$ over $A_K$. Consider the cochain complex of $A_K$ with coefficients in $V$:
\[
C^{n+1}_{\LY}(A_K,V)=
\begin{cases}
\Hom(\underbrace{\wedge^2A_K \otimes \dotsb \otimes \wedge^2A_K}_n,V) 
\times \Hom(\underbrace{\wedge^2A_K \otimes \dotsb \otimes \wedge^2A_K}_n \otimes A_K,V),	&amp; \forall n\geq 1, \\[-1.5ex]
\Hom(A_{K},V), 																			&amp; \text{for $n=0$},
\end{cases}
\]
and its differential is defined as follows.

If $n \geq 1$ and for any $(f,g) \in C^{n+1}_{\LY}(A_K,V)$, the coboundary map is given by
\[
\begin{aligned}
	\partial^{n+1}=(\partial_\coboundI^{n+1},\partial^{n+1}_\coboundII) \colon C_{\LY}^{n+1}(A_K,V) 	&amp; \to C_{\LY}^{n+2}(A_K,V)
\\	(f,g)	&amp; \mapsto \parens[\big]{\partial^{n+1}_\coboundI(f,g),\partial_\coboundII^{n+1}(f,g)},
\end{aligned}
\]
where $\partial^{n+1}_\coboundI(f,g)$ and $\partial^{n+1}_\coboundII(f,g)$ are defined to be
\[
\begin{split}
	&amp; \partial^{n+1}_\coboundI(f,g)(\mathfrak{\mathfrak{U}}_1,\dotsc,\mathfrak{U}_{n+1})
\\	&amp; \quad = (-1)^n\parens[\Big]{\rho\parens[\big]{K(\mathfrak{u}_{n+1})}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{v}_{n+1}) +\rho(\mathfrak{u}_{n+1})K_Vg(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{v}_{n+1})
\\	&amp; \quad\quad -\rho\parens[\big]{K(\mathfrak{v}_{n+1})}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{u}_{n+1}) -\rho(\mathfrak{v}_{n+1})K_Vg(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{u}_{n+1})
\\	&amp; \quad\quad -g\parens[\big]{\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,[\mathfrak{u}_{n+1},\mathfrak{v}_{n+1}]_K}} + \sum_{i=1}^n(-1)^{i+1}\mathfrak{D}\parens[\big]{K(\mathfrak{u}_i),K(\mathfrak{v}_i)}f(\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_i,\dotsc,\mathfrak{U}_{n+1})
\\	&amp; \quad\quad - \sum_{i=1}^n(-1)^{i+1}K_V\parens[\Big]{\mathfrak{D}\parens[\big]{K(\mathfrak{u}_i),\mathfrak{v}_i}f(\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_i,\dotsc,\mathfrak{U}_{n+1}) +\mathfrak{D}(\mathfrak{u}_i,K\mathfrak{v}_i)f(\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_i,\dotsc,\mathfrak{U}_{n+1})}
\\	&amp; \quad\quad + \sum_{1 \leq i &lt; j \leq n+1} (-1)^i f \parens[\big]{\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_i,\dotsc,\{\mathfrak{u}_i,\mathfrak{v}_i,\mathfrak{u}_j\}_K\wedge\mathfrak{v}_j+\mathfrak{u}_j \wedge \{\mathfrak{u}_i,\mathfrak{v}_i,\mathfrak{v}_j\}_K,\dotsc,\mathfrak{U}_{n+1}}
\end{split}
\]
and
\[
\begin{split}
	&amp; \partial^{n+1}_\coboundII(f,g)(\mathfrak{\mathfrak{U}}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad = (-1)^n\parens[\Bigg]{\theta\parens[\big]{K(\mathfrak{v}_{n+1}),K(w)}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{u}_{n+1}) - K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{v}_{n+1}),w}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{u}_{n+1})
\\	&amp; \quad\quad +\theta\parens[\big]{\mathfrak{v}_{n+1},K(w)}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{u}_{n+1})} -\theta\parens[\big]{K(\mathfrak{u}_{n+1}),K(w)}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{v}_{n+1})
\\	&amp; \quad\quad + K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{u}_{n+1}),w}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{v}_{n+1}) +\theta\parens[\big]{\mathfrak{u}_{n+1},K(w)}g(\mathfrak{U}_1,\dotsc,\mathfrak{U}_n,\mathfrak{v}_{n+1})}}
\\	&amp; \quad\quad + \sum_{i=1}^{n+1}(-1)^{i+1}\mathfrak{D}\parens[\big]{K(\mathfrak{u}_i),K(\mathfrak{v}_i)}g(\mathfrak{U}_1,\dotsc, \hat{\mathfrak{U}}_i,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad\quad - \sum_{i=1}^{n+1}(-1)^{i+1} \! K_V\parens[\Big]{\mathfrak{D}\parens[\big]{K(\mathfrak{u}_i),\mathfrak{v}_i}g(\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_i,\dotsc,\mathfrak{U}_{n+1},w) +\mathfrak{D}\parens[\big]{\mathfrak{u}_i,K(\mathfrak{v}_i)}g(\mathfrak{U}_1,\dotsc,
\hat{\mathfrak{U}}_i,\dotsc,\mathfrak{U}_{n+1},w)}
\\	&amp; \quad\quad + \sum_{1 \leq i &lt; j \leq n+1} (-1)^i g \parens[\big]{\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_i,\dotsc,\{\mathfrak{u}_i,\mathfrak{v}_i,\mathfrak{u}_j\}_K\wedge\mathfrak{v}_j+
 \mathfrak{u}_j \wedge \{\mathfrak{u}_i,\mathfrak{v}_i,\mathfrak{v}_j\}_K,\dotsc,\mathfrak{U}_{n+1},w}
\\	&amp; \quad\quad + \sum_{i=1}^{n+1} (-1)^i g \parens[\big]{\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_i,\dotsc,\mathfrak{U}_{n+1},\{\mathfrak{u}_i,\mathfrak{v}_i,w\}_K},
\end{split}
\]
where $\mathfrak{U}_i=\mathfrak{u}_i \wedge \mathfrak{v}_i \in \wedge^2A_K$ and $w \in A_K$ for $i=1,2,\dotsc, n$.

If $n=0$ and for any $f \in C^1_{\LY}(A_K,V)$, the coboundary map is given by
\[
\begin{aligned}
	\partial^1=(\partial_\coboundI^1,\partial^1_\coboundII) \colon C_{\LY}^{1}(A_K,V)	&amp; \to C_{\LY}^{2}(A_K,V)
\\	f &amp;  \mapsto \parens[\big]{\partial^1_\coboundI(f),\partial^1_\coboundII(f)}
\end{aligned}
\]
where
\begin{align*}
	\partial^1_\coboundI(f)(u,v) 		
	&amp; = \rho(Ku)f(v)+\rho(u)\parens[\big]{K_V\parens[\big]{f(v)}} -\rho(Kv)f(u)-\rho(v)\parens[\big]{K_V\parens[\big]{f(u)}}-f\parens[\big]{[u,v]_K}, 
\\	\partial^1_\coboundII(f)(u,v,w) 	
	&amp; = \mathfrak{D}(Ku,Kv)f(w) -K_V\parens[\big]{\mathfrak{D}(Ku,v)f(w)+\mathfrak{D}(u,Kv)f(w)} +\theta(Kv,Kw)f(u)
\\	&amp; \quad - K_V\parens[\big]{\theta(Ku,w)f(v)+\theta(v,Kw)f(u)} -\theta(Ku,Kw)(v) 
\\	&amp; \quad - K_V\parens[\big]{\theta(Ku,w)f(v) +\theta(u,Kw)f(v)}-f\parens[\big]{\{u,v,w\}_K}.
\end{align*}

\begin{defi}\label{def7}
Let $\parens[\big]{A,[\,\cdot\,,\cdot\,]_A,\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ be a modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$ and $(V;\rho,\theta,K_V)$ be a representation over it. Then the cochain complex $\parens[\big]{C^\bullet_{\LY} (A_K,V),\partial}$ is called the cochain complex of modified Rota--Baxter operator $K$ of weight $\lambda$ with coefficients in $(V;\rho_K,\theta_K, K_V)$, denoted by $C^\bullet_{\mRBOl}(A,V)$. The cohomology of $\parens[\big]{C^\bullet_{\mRBOl}(A,V),\partial}$, denoted by\/ $\operH^\bullet_{\mRBOl}(A,V)$, is called the cohomology of modified Rota--Baxter operator $K$ of weight $\lambda$ with coefficients in $(V;\rho_K,\theta_K,K_V)$. In particular, when $(V;\rho_K,\theta_K,K_V)$ is the regular representation $\parens[\big]{A_K,[\,\cdot\,,\cdot\,]_K,\{\,\cdot\,,\cdot\,,\cdot\,\}_K,K}$, we denote $\parens[\big]{C^\bullet_{\mRBOl}(A,A),\partial}$ by $\parens[\big]{C^\bullet_{\mRBOl}(A),\partial}$ and call it the cochain complex of modified Rota--Baxter operator $K$ of weight $\lambda$, and denote\/ $\operH^\bullet_{\mRBOl}(A,A)$ by\/ $\operH^\bullet_{\mRBOl}(A)$ and call it the cohomology of modified Rota--Baxter operator $K$ of weight $\lambda$.
\end{defi}

\subsection{Cohomology of modified Rota--Baxter Lie--Yamaguti algebras}

In this subsection, we will combine the cohomology of Lie--Yamaguti algebras and the cohomology of modified Rota--Baxter operators of any weight to define a cohomology theory for modified Rota--Baxter Lie--Yamaguti algebras of any weight. Let $V= (V;\rho,\theta,K_V)$ be a representation of a modified Rota--Baxter Lie--Yamaguti algebra $A = \parens[\big]{A, [\,\cdot\,,\cdot\,]_A, \{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$ of weight $\lambda$. Then, we construct a chain map
\[
\begin{aligned}
	\psi^{n+1}=(\psi_\coboundI^{n+1},\psi^{n+1}_\coboundII) \colon C^{n+1}_{\LY}(A,V)	&amp; \to C^{n+1}_{\mRBOl}(A,V)
\\	(f,g)	&amp; \mapsto\parens[\big]{\psi^{n+1}_\coboundI(f),\psi^{n+1}_\coboundII(g)}
\end{aligned}
\]
i.e., the following commutative diagram:
\[
\xymatrix{ C^1_{\LY}(A,V)\ar[r]^{\delta^1} \ar[d]^{\psi^{1}}&amp; C^2_{\LY}(A,V) \ar@{.}[r] \ar[d]^{\psi^2} &amp;C^{n+1}_{\LY}(A,V)\ar[r]^{\delta^{n+1}} \ar[d]^{\psi^{n+1}}&amp;C^{n+2}_{\LY}(A,V)\ar[d]^{\psi^{n+2}} \ar@{.}[r]&amp; \\
C^1_{\mRBOl}(A,V)\ar[r]^{\partial^1} &amp; C^2_{\mRBOl}(A,V) \ar@{.}[r]&amp;C^{n+1}_{\mRBOl} (A,V)\ar[r]^{\partial^{n+1}}&amp;C^{n+2}_{\mRBOl}(A,V) \ar@{.}[r]&amp; }
\]define $\psi^{n+1}$ as
\[
\begin{split}
	&amp; \psi^{n+1}_\coboundI(f)
\\	&amp; \quad = f \circ (K, K, \dots, K, K) \circ \parens[\big]{\Id \wedge \Id)^{\otimes n}}
\\	&amp; \quad\quad - \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_s\leqslant {2n} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}f\circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n-i_{s}})\circ \parens[\big]{\Id \wedge \Id)^{\otimes n}}
\\	&amp; \quad\quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1} K_V\circ f\circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n-i_{s}})\circ\parens[\big]{\Id \wedge \Id)^{\otimes n}}
\end{split}
\]
and
\[
\begin{split}
	&amp; \psi^{n+1}_\coboundII(g)
\\	&amp; \quad = g \circ (K, K, \dots, K, K,K) \circ \parens[\big]{(\Id \wedge \Id)^{\otimes n-1} \otimes \Id^{\wedge 3}}
\\	&amp; \quad\quad - \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}g\circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+1-i_{s}})\circ\parens[\big]{(\Id \wedge \Id)^{\otimes n-1} \otimes \Id^{\wedge 3}}
\\	&amp; \quad\quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1}
\\	&amp; \hspace{2cm} K_V\circ g\circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+1-i_{s}}) \circ\parens[\big]{(\Id \wedge \Id)^{\otimes n-1} \otimes \Id^{\wedge 3}},
\end{split}
\]
where the operators $K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+1-i_{s}}$ successively operate on the element $u_1,v_1,\dotsc,u_{n+1},v_{n+1},w$. In particular, when $n=0$, define $\psi^{1} \colon C^{1}_{\LY}(A,V)\to C^{1}_{\mRBOl}(A,V)$ by
\[
\psi^1(f)=f \circ K -K_V \circ f.
\]

\begin{prop}\label{pro4}
The map $\psi^{\bullet} \colon C^{\bullet}_{\LY}(A,V)\to C^{\bullet}_{\mRBOl}(A,V)$ is a chain map, i.e, $\psi^{n+2} \circ \delta^{n+1} = \partial^{n+1} \circ \psi^{n+1}$.
\end{prop}

We leave the long proof of this result to Appendix~\ref{appendix_A}.

\begin{defi}\label{def8}
Let $V = (V;\rho,\theta,K_V)$ be a representation of a modified Rota--Baxter Lie--Yamaguti algebra $A=\parens[\big]{A,\mu=[\,\cdot\,,\cdot\,]_A,\nu=\{\,\cdot\,,\cdot\,,\cdot\,\}_A,K}$. We define the cochain complex $\parens[\big]{C^\bullet_{\mRBLYl}(A,V), \dd^\bullet}$ of modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ of weight $\lambda$ with coefficients in $V = (V;\rho,\theta,K_V)$ to the negative shift of the mapping cone of $\psi^{\bullet}$, that is, let
\[
C^{1}_{\mRBLYl}(A,V)=C^{1}_{\LY}(A,V) \qquad \text{and} \qquad C^{n+1}_{\mRBLYl}(A,V) = C^{n+1}_{\LY}(A,V) \oplus C^{n}_{\mRBOl}(A,V), \quad \forall n\geq 1,
\]
and the differential\/ $\dd^{n+1} \colon C_{\mRBLYl}^{n+1}(A,V)\to C_{\mRBLYl}^{n+2}(A,V)$ is given by
\begin{align*}
&amp;	\dd^1(f_1)=\parens[\big]{\delta^1(f_1),-\psi^1(f_1)}, 											\quad &amp;&amp; \text{if $n=0$},\\
&amp;	\dd^2\parens[\big]{(f_1,g_1),f_2}=\parens[\big]{\delta^2(f_1,g_1),-\partial^1(f_2)-\psi^2(f_1,g_1)}, 	\quad &amp;&amp; \text{if $n=1$},\\
&amp;	\dd^{n+1}\parens[\big]{(f_1,g_1),(f_2,g_2)}=\parens[\big]{\delta^{n+1}(f_1,g_1),-\partial^{n}(f_2,g_2) -\psi^{n+1}(f_1,g_1)}, \quad &amp;&amp; \text{if $n\geq2$}.
\end{align*}
\end{defi}

\begin{theo}\label{th2}
The map\/ $\dd^{n+1} \colon C^{n+1}_{\mRBLYl}(A,V)\to C^{n+2}_{\mRBLYl}(A,V)$ satisfies $\dd^{n+2} \circ \dd^{n+1} = 0$, hence, it is a coboundary map.
\end{theo}

\begin{proof}
Let $\parens[\big]{(f_1,g_1),(f_2,g_2)} \in C^{n+1}_{\mRBLYl}$, then, we have
\begin{align*}
	&amp; \dd^{n+2} \circ \dd^{n+1}\parens[\big]{(f_1,g_1),(f_2,g_2)}
\\	&amp; \hspace{2cm} = \dd^{n+2}\parens[\big]{\delta^{n+1}(f_1,g_1),-\partial^{n}(f_2,g_2)-\psi^{n+1}(f_1,g_1)}
\\	&amp; \hspace{2cm} = \parens[\Big]{\delta^{n+2}\parens[\big]{\delta^{n+1}(f_1,g_1)},-\partial^{n+1}\parens[\big]{-\partial^{n}(f_2,g_2)-\psi^{n+1}(f_1,g_1)}-\psi^{n+2}\parens[\big]{\delta^{n+1}(f_1,g_2)}} 
\\	&amp; \hspace{2cm} = \parens[\Big]{0,\partial^{n+1}\parens[\big]{\psi^{n+1}(f_1,g_1)}-\psi^{n+2}\parens[\big]{\delta^{n+1}(f_1,g_1)}} \quad \text{(Proposition~\ref{pro4})} 
\\	&amp; \hspace{2cm} = 0. \qedhere
\end{align*}
\let\qed\relax
\end{proof}

The cohomology of $\parens[\big]{C^\bullet_{\mRBLYl}(A,V),\dd^\bullet}$, denoted by $\operH^\bullet_{\mRBLYl}(A,V)$, is called the cohomology of the modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ of weight $\lambda$ with coefficients in $(V;\rho,\theta,K_V)$. When $(V;\rho,\theta,K_V)=(A,\mu,\nu,K)$, we just denote $C^\bullet_{\mRBLYl}(A,A)$ and $\operH^\bullet_{\mRBLYl}(A,A)$ by $C^\bullet_{\mRBLYl}(A)$ and $\operH^\bullet_{\mRBLYl}(A)$ respectively, and call them the cochain complex, the cohomology of modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ of weight $\lambda$ respectively.

There is an obvious short exact sequence of complexes:
\begin{equation}\label{eq28}
0 \to C^{\bullet-1}_{\mRBOl}(A,V)\to C^\bullet_{\mRBLYl}(A,V)
\to C^\bullet_{\LY}(A,V)\to 0 
\end{equation}
which induces a long exact sequence of cohomology groups
\[
\dotsb \to \operH^p_{\LY}(A,V) \to \operH^P_{\mRBOl}(A,V) \to \operH^{p+1}_{\mRBLYl}(A,V)
\to \operH^{p+1}_{\LY}(A,V) \to \dotsb
\]

\section{Deformations of modified Rota--Baxter Lie--Yamaguti algebras}\label{sec:Defor}

In this section, we will study formal one-parameter deformation of modified Rota--Baxter Lie--Yamaguti algebra of arbitrary weights and interpret them via lower degree cohomology groups of modified Rota--Baxter Lie--Yamaguti algebras defined in last section. Let $\mathbb{K}\bbracks{t}$ be a ring of power series of one variable $t$, and let $A\bbracks{t}$ be the set of formal power series over $A$. If $\parens[\big]{A, [\,\cdot\,, \cdot\,]_A, \{\,\cdot\,, \cdot\,, \cdot\,\}_A}$ is a Lie--Yamaguti algebra, then there is a Lie--Yamaguti algebra structure over the ring $\mathbb{K}\bbracks{t}$ on $A\bbracks{t}$ given by
\begin{align*}
	&amp; \bracks*{\sum_{i=0}^{\infty}a_it^i,\sum_{i=0}^{\infty}b_jt^j}_A=\sum_{s=0}^{\infty}\sum_{i+j=s}[a_i,b_j]_At^s,
\\	&amp; \braces*{\sum_{i=0}^{\infty}a_it^i,\sum_{i=0}^{\infty}b_jt^j,\sum_{k=0}^{\infty}c_kt^k}_A=\sum_{s=0}^{\infty}\sum_{i+j+k=s}\braces{a_i,b_j,c_k}_At^s, \quad \forall a_i,b_j,c_k \in A.
\end{align*}

In the sequel, let us denote the brackets $[\,\cdot\,,\cdot\,]_A$ and $\{\,\cdot\,,\cdot\,,\cdot\,\}_A$ by $\mu$ and $\nu$, respectively.

\begin{defi}\label{def9}
A formal one-parameter deformation of a modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ of weight $\lambda$ is a triple of power series $(\mu_t,\nu_t,K_t)$
\begin{gather*}
	\mu_t= \sum_{i=0}^\infty\mu_it^i, \quad \nu_t= \sum_{i=0}^\infty\nu_it^i, \quad (\mu_i,\nu_i)\in C^2_{\LY}(A), \quad (\mu_0,\nu_0)=(\mu,\nu),
\\	K_t= \sum_{i=0}^\infty K_it^i, \quad K_i \in C^1_{\mRBOl}(A), \quad K_0=K,
\end{gather*}
such that $(A\bbracks{t},\mu_t,\nu_t,K_t)$ is a modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$, over $A\bbracks{t}$.

The triple $(\mu_t,\nu_t,K_t)$ generates a formal one-parameter deformation of modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ of weight $\lambda$ if and only if for all $a,b,c,u,v,w \in A$, the following equations hold:
\begin{align*}
&amp; \mu_t(a,b)+\mu_t(b,a)=0, \\
&amp; \nu_t(a,b,c)+\nu_t(b,a,c)=0, \\
&amp; \mu_t\parens[\big]{\mu_t(a,b),c}+\mu_t\parens[\big]{\mu_t(b,c),a}+\mu_t\parens[\big]{\mu_t(c,a),b}+\nu_t(a,b,c)+\nu_t(b,c,a) +\nu_t(c,a,b)=0,\\
&amp; \nu_t\parens[\big]{\mu_t(a,b),c,u}+\nu_t\parens[\big]{\mu_t(b,c),a,u}+\nu_t\parens[\big]{\mu_t(c,a),b,u}=0,\\
&amp; \nu_t\parens[\big]{a,b,\mu_t(u,v)}=\mu_t\parens[\big]{\nu_t(a,b,u),v}+\mu_t\parens[\big]{u,\nu_t(a,b,v)},\\
&amp; \nu_t\parens[\big]{a,b,\nu_t(u,v,w)}=\nu_t\parens[\big]{\nu_t(a,b,u),v,w}+\nu_t\parens[\big]{u,\nu_t(a,b,v),w} +\nu_t\parens[\big]{u,v,\nu_t(a,b,w)},\\
&amp; \mu_t\parens{K_ta,K_tb}=K_t\parens[\big]{\mu_t(K_ta,b)+\mu_t(a,K_tb)}+\lambda \mu_t(a,b),\\
&amp; \nu_t(K_ta,K_tb,K_tc)=K_t\parens[\big]{\nu_t(K_ta,K_tb,c)+\nu(K_ta,b,K_tc)+\nu_t(a,K_tb,K_tc) -\lambda\nu_t(a,b,c)}\\
&amp; \hspace{2.75cm} + \lambda\nu_t(K_ta,b,c)+\lambda\nu_t(a,K_tb,c)+\lambda\nu_t(a,b,K_tc).
\end{align*}
Expanding these equations and collecting coefficients of $t^n$, we prove that $\{\mu_i \}_{i\geq 0}$, $\{\nu_i \}_{i\geq 0}$ and $\{K_i \}_{i\geq 0}$, and~\eqref{eq15}--\eqref{eq16} are equivalent to the system of equations:
\begingroup
\allowdisplaybreaks
\begin{align}
	\label{eq29}	&amp; \mu_n(a,b)+\mu_n(b,a)=0, 
\\	\label{eq30}	&amp;\nu_n(a,b,c)+\nu_n(b,a,c)=0, 
\\	\nonumber	&amp; \sum_{i+j=n}\mu_i\parens[\big]{\mu_j(a,b),c}+ \sum_{i+j=n}\mu_i\parens[\big]{\mu_j(b,c),a} + \sum_{i+j=n}\mu_i\parens[\big]{\mu_j(c,a),b}
\\*	\label{eq31}	&amp; \hspace{1cm} + \nu_n(a,b,c)+\nu_n(b,c,a)+\nu_n(c,a,b)=0, 
\\	\label{eq32}	&amp; \sum_{i+j=n}\nu_i\parens[\big]{\mu_j(a,b),c,u}+ \sum_{i+j=n}\nu_i\parens[\big]{\mu_j(b,c),a,u} + \sum_{i+j=n}\nu_i\parens[\big]{\mu_j(c,a),b,u}=0, 
\\	\label{eq33}	&amp; \sum_{i+j=n}\nu_i\parens[\big]{a,b,\mu_j(u,v)}= \sum_{i+j=n}\mu_i\parens[\big]{\nu_j(a,b,u),v} + \sum_{i+j=n}\mu_i\parens[\big]{u,\nu_j(a,b,v)}, 
\\	\nonumber	&amp; \sum_{i+j=n}\nu_i\parens[\big]{a,b,\nu_j(u,v,w)}= \sum_{i+j=n}\nu_i\parens[\big]{\nu_j(a,b,u),v,w} + \sum_{i+j=n}\nu_i\parens[\big]{u,\nu_j(a,b,v),w}
\\*	\label{eq34}	&amp; \hspace{1cm} + \sum_{i+j=n}\nu_i\parens[\big]{u,v,\nu_j(a,b,w)}, 
\\	\label{eq35}	&amp; \sum_{i+j+k=n}\mu_i(K_ja,K_kb)= \sum_{i+j+k=n}K_i\parens[\big]{\mu_j(K_ka,b) +\mu_j(a,K_kb)} +\lambda\mu_k(a,b), 
\\	\nonumber	&amp; \sum_{i+j+k+m=n}\nu_i(K_ja,K_kb,K_mc)= \sum_{i+j+k+m=n}K_i\parens[\big]{\nu_j(K_ka,K_mb,c) +\nu_j(K_ka,b,K_mc)
\\*	\label{eq36}	&amp; \hspace{1cm} + \nu_j(a,K_kb,K_mc) - \lambda \nu_m(a,b,c)} +\lambda\nu_k(K_ma,b,c)+\lambda\nu_k(a,K_mb,c)+\lambda\nu_k(a,b,K_mc).
\end{align}
\endgroup
\end{defi}

\begin{rema}\label{r1}
For $n = 0$, the conditions~\eqref{eq29}--\eqref{eq34} are precisely the Lie--Yamaguti-algebra $(\mu,\nu)=(\mu_0,\nu_0)$ and~\eqref{eq35}--\eqref{eq36} are equal to the fact that $K=K_0$ is a modified Rota--Baxter Lie--Yamaguti algebra of weight $\lambda$.
\end{rema}

\begin{prop}\label{pro5}
Let $(\mu_t,\nu_t,K_t)$ be a formal one-parameter deformation of modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ of weight $\lambda$. Then $(\mu_1,\nu_1,K_1)$ is a 2-cocycle in the cochain complex $\parens[\big]{C^\bullet_{\mRBLYl}(A),\dd^\bullet}$.
\end{prop}

\begin{proof}
For $n = 1$, \eqref{eq29}--\eqref{eq34} become
\begin{align*}
&amp; \mu_1(a,b)+\mu_1(b,a)=0, \\
&amp; \nu_1(a,b,c)+\nu_1(b,a,c)=0, \\
&amp; \bracks[\big]{\mu_1(a,b),c}_A+\mu_1\parens[\big]{[a,b]_A,c}+\bracks[\big]{\mu_1(b,c),a}_A+\mu_1\parens[\big]{[b,c]_A,a}+\bracks[\big]{\mu_1(c,a),b}_A + \mu_1\parens[\big]{[c,a]_A,b} \\
&amp; \hspace{1.5cm} + \{a,b,c\}_A+\{b,c,a\}_A+\{c,a,b\}_A=0,\\
&amp; \braces[\big]{\mu_1(a,b),c,u}_A + \nu_1\parens[\big]{[a,b]_A,c,u} + \braces[\big]{\mu_1(b,c),a,u}_A + \nu_1\parens[\big]{[b,c]_A,a,u}\\
&amp; \hspace{1.5cm} + \braces[\big]{\mu_1(c,a),b,u}_A + \nu_1\parens[\big]{[c,a]_A,b,u} = 0,\\
&amp; \braces[\big]{a,b,\mu_1(u,v)}_A + \nu_1\parens[\big]{a,b,[u,v]_A} = \bracks[\big]{\nu_1(a,b,u),v}_A + \mu_1\parens[\big]{\{a,b,u\}_A,v}\\
&amp; \hspace{1.5cm} +\bracks[\big]{u,\nu_1(a,b,v)}_A+\mu_1\parens[\big]{u,\{a,b,v\}_A}, \\
&amp; \braces[\big]{a,b,\nu_1(u,v,w)}_A+\nu_1\parens[\big]{a,b,\{u,v,w\}}_A= \braces[\big]{\nu_1(a,b,u),v,w)}_A + \nu_1\parens[\big]{\{a,b,u\}_A,v,w}\\
&amp; \hspace{1.5cm} + \braces[\big]{u,\nu_1(a,b,v),w}_A + \nu_1\parens[\big]{u,\{a,b,v\}_A,w} + \braces[\big]{u,v,\nu_1(a,b,w)}_A + \nu_1\parens[\big]{u,v,\{a,b,w\}_A}.
\end{align*}
Note that the above equations are exactly $\partial^1(\mu_1,\nu_1)=0$.

For $n = 1$, \eqref{eq35} and~\eqref{eq36} become
\begin{multline*}
[K_1a,Kb]_A+[Ka,K_1b]_A+\mu_1(Ka,Kb) = K_1\parens[\big]{[Ka,b]_A+[a,Kb]_A}+\lambda[a,b]_A\\
+ K\parens[\big]{\mu_1(Ka,b)+\mu_1(a,Kb)}+\lambda[a,b]_A + K\parens[\big]{[K_1a,b]_A+[a,K_1b]_A}+\lambda\mu_1(a,b),
\end{multline*}
and
\[
\begin{split}
&amp; \{K_1a,Kb,Kc\}_A+\{Ka,K_1b,Kc\}_A+\{Ka,Kb,K_1c\}_A +\nu_1(Ka,Kb,Kc)\\
&amp; \hspace{4cm} = K_1\parens[\big]{\{Ka,Kb,c\}_A + \{Ka,b,Kc\}_A + \{a,Kb,Kc\}_A-\lambda\{a,b,c\}_A}\\
&amp; \hspace{4cm} \quad + \lambda\{Ka,b,c\}_A + \lambda\{a,Kb,c\}_A + \lambda\{a,b,Kc\}_A\\
&amp; \hspace{4cm} \quad + K\parens[\big]{\nu_1(Ka,Kb,c)+ \nu_1(Ka,b,Kc) + \nu_1(a,Kb,Kc)-\lambda\{a,b,c\}_A}\\
&amp; \hspace{4cm} \quad + \lambda\{Ka,b,c\}_A + \lambda\{a,Kb,c\}_A + \lambda\{a,b,Kc\}_A \\
&amp; \hspace{4cm} \quad + K\parens[\big]{\{K_1a,Kb,c\}_A + \{K_1a,b,Kc\}_A + \{a,K_1b,Kc\}_A-\lambda\{a,b,c\}_A}\\
&amp; \hspace{4cm} \quad + \lambda\nu_1(Ka,b,c) + \lambda\nu_1(a,Kb,c) + \lambda\nu_1(a,b,Kc)\\
&amp; \hspace{4cm} \quad + K\parens[\big]{\{Ka,K_1b,c\}_A+ \{Ka,b,K_1c\}_A + \{a,Kb,K_1c\}_A-\lambda\nu_1(a,b,c)} \\
&amp; \hspace{4cm} \quad + \lambda\{K_1a,b,c\}_A + \lambda\{a,K_1b,c\}_A + \lambda\{a,b,K_1c\}_A.
\end{split}
\]
Note that the above equations are exactly to $-\partial^1(K_1)-\psi^2(\mu_1,\nu_1)=0$, so $\dd^2\parens[\big]{(\mu_1,\nu_1),K_1}=0$.
\end{proof}

\begin{defi}\label{def10}
Two formal one-parameter deformations $(A\bbracks{t},\mu_t,\nu_t, K_t)$ and $(A\bbracks{t},\bar{\mu}_t,\bar{\nu}_t, \bar{K}_t)$ of a modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ are called equivalent if there exists a formal isomorphism from $(A\bbracks{t},\bar{\mu}_t,\bar{\nu}_t, \bar{K}_t)$ to $(A\bbracks{t},\mu_t,\nu_t, K_t)$ is a power series $\Phi_t=\sum_{i\geq0}\phi_it^i \colon A\bbracks{t} \to A\bbracks{t}$, where $\phi_i \colon A\to A$ are a linear maps with $\phi_0 = \Id_A$, such that
\begin{align}
\phi_t \circ \mu_t &amp;=\bar{\mu}_t\circ(\phi_t \otimes \phi_t), \label{eq37}
\\
\phi_t \circ \nu_t &amp;=\bar{\nu}_t \circ (\phi_t \otimes \phi_t \otimes \phi_t), \label{eq38}
\\
\phi_t \circ K_t &amp;= \bar{K}_t \circ \phi_t. \label{eq39}
\end{align}
\end{defi}

Given a modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ of weight $\lambda$, then the power series $\mu_t$, $\nu_t$, $K_t$ with $\mu_i=\delta_{i,0}\mu$, $\nu_i=\delta_{i,0}\nu$ and $K_i=\delta_{i,0}K$ make $(A\bbracks{t},\mu_t,\nu_t,K_t)$ into a formal one-parameter deformation of $(A,\mu,\nu,K)$. A formal one-parameter deformation is called trivial if it is equivalent to $(A,\mu,\nu,K)$.

\begin{theo}\label{th3}
The infinitesimals of two equivalent formal one-parameter deformations of the modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ are in the same cohomology class in $\operH^2_{\mRBLYl}(A)$.
\end{theo}
\begin{proof}
Let $\phi_t \colon (A\bbracks{t},\bar{\mu}_t,\bar{\nu}_t,\bar{K}_t) \to (A\bbracks{t},\mu_t,\nu_t,K_t)$ be a formal isomorphism. Expanding these equalities and collecting coefficients of $t$, we have from~\eqref{eq37}--\eqref{eq39}:
\begin{align*}
\bar{\mu}_1 &amp;= \mu_1 -\phi_1 \circ \mu + \mu \circ (\phi_1 \otimes \Id_A)+ \mu \circ (\Id_A \otimes \phi_1),\\
\bar{\nu}_1 &amp;= \nu_1 - \phi_1 \circ \nu + \nu \circ (\phi_1 \otimes \Id_A \otimes \Id_A) +
\nu \circ (\Id_A \otimes \phi_1 \otimes \Id_A) + \nu \circ (\Id_A \otimes \Id_A \otimes \phi_1), \\
\bar{K}_1 &amp;= K_1 - \phi_1 \circ K + K \circ \phi_1,
\end{align*}
that is, we have
\[
(\bar{\mu}_1,\bar{\nu}_1,\bar{K}_1)-(\mu_1,\nu_1,K_1) = \parens[\big]{\delta^1(\phi_1),-\psi^1(\phi_1)} = \dd^1(\phi_1) \in B^2_{\mRBLYl}(A). \qedhere
\]
\let\qed\relax
\end{proof}

\begin{defi}\label{def11}
A modified Rota--Baxter Lie--Yamaguti algebra $(A,\mu,\nu,K)$ is called rigid if every formal one-parameter deformation is trivial.
\end{defi}

\begin{theo}\label{th4}
Let $(A,\mu,\nu,K)$ be a modified Rota--Baxter Lie--Yamaguti algebra. Then $(A,\mu,\nu,K)$ is rigid if\/ $\operH^2_{\mRBLYl}(A)=0$.
\end{theo}
\begin{proof}
Let $(A\bbracks{t},\mu_t,\nu_t,K_t)$ be a formal one-parameter deformation of $(A,\mu,\nu,K)$. By Proposition~\ref{pro5} $(\mu_1,\nu_1,K_1)$ is a 2-cocycle and as $\operH^2_{\mRBLYl}(A)=0$, there exists a 1-cochain $\phi_1 \in C^1_{\mRBLYl}(A)$ such that $(\mu_1,\nu_1,K_1)=\dd^1(\phi_1)$. Hence, $(\mu_1,\nu_1)=\delta^1(\phi_1)=\parens[\big]{\delta_\coboundI^1(\phi_1),\delta_\coboundII^1(\phi_1)}$, and $K_1=-\psi^1(\phi_1)$. Then setting $\phi_t = \Id_A - t\phi_1 \colon A\bbracks{t} \to A\bbracks{t} $, we have a deformation $(A\bbracks{t},\bar{\mu}_t,\bar{\nu}_t,\bar{K}_t)$, where
\begin{align*}
\bar{\mu}_t	&amp;=\phi^{-1}_t \circ \mu_t\circ(\phi_t \otimes \phi_t), \\
\bar{\nu}_t 	&amp;= \phi^{-1}_t \circ \nu_t \circ (\phi_t \otimes \phi_t \otimes \phi_t), \\
\bar{K}_t 	&amp;= \phi^{-1}_t \circ K_t \circ \phi_t.
\end{align*}
Thus, $(\bar{\mu}_t,\bar{\nu}_t,\bar{K}_t)$ is equivalent to $(\mu_t,\nu_t,K_t)$ and it is easy to verify that $\bar{\mu}_1=0, \bar{\nu}_1=0, \bar{K}_1=0$. Furthermore, we have
\begin{align*}
\bar{\mu}_t&amp;=\mu+t^2\bar{\mu}_2+ \dots, \\
\bar{\nu}_t&amp;=\nu+t^2\bar{\nu}_2+ \dots, \\
\bar{K}_t&amp;=K+t^2\bar{K}_2+ \dots.
\end{align*}
By~\eqref{eq29}--\eqref{eq36}, we show that $(\bar{\mu}_2,\bar{\nu}_2,\bar{K}_2)$ is still a 2-cocycle. Then by induction, we can see that $(\mu_t,\nu_t,K_t)$ is equivalent to the trivial deformation $(A\bbracks{t},\mu,\nu,K)$. Thus, $(A,\mu,\nu,K)$ is rigid.
\end{proof}

\appendix

\section{Proof of Proposition~\ref{pro4}}\label{appendix_A}

The purpose of this appendix is to prove Proposition~\ref{pro4}.

Firstly, let's introduce some notations we will use in the following proof. For an element
\[
(\mathfrak{U}_1, \dots, \mathfrak{U}_{n+1 }, w)\in\underbrace{\wedge^2A\otimes \dotsb \otimes \wedge^2A}_{n}\wedge A
\]
with $\mathfrak{U}_i=u_i\wedge v_i$, we write it as an ordered sequence $(w_1,w_2,\dots,w_{2n+2},w_{2n+3})$, where
\[
w_1=u_1,\ w_2=v_1,\dots, w_{2k-1}=u_{k},\ w_{2k}=v_k,\dots, w_{2n+1}=u_{n+1},\ w_{2n+2}=v_{n+1},\ w_{2n+3}=w.
\]
For arbitrary permutation $\tau \in S_{2n+3}$, its action on the element $(w_1,\dots,w_{2n+3})$ is given as
\[
\tau(w_{1},w_2,\dots,w_{2n+2},w_{2n+3})= (w_{\tau^{-1}(1)},w_{\tau^{-1}(2)},\dots,w_{\tau^{-1}(2n+2)},w_{\tau^{-1}(2n+3)}).
\]
Define an operation $[i-1,i,i+1]_A$ from $\underbrace{\wedge^2A\otimes \dotsb \otimes \wedge^2A}_{n}\wedge A$ to $\underbrace{\wedge^2A\otimes \dotsb \otimes \wedge^2A}_{n-1}\wedge A$ as
\[
[i-1,i,i+1]_A\parens{w_{1},w_2,\dots,w_{2n+2},w_{2n+3}}=\parens[\big]{w_{1},w_2,\dots,w_{i-2},[w_{i-1},w_{i },w_{i+1}]_A,w_{i+2},\dots,w_{2n+3}}.
\]

\begin{prop*}[Proposition~\ref{pro4}] The map $\psi^{\bullet} \colon C^{\bullet}_{\LY}(A,V)\to C^{\bullet}_{\mRBOl}(A,V)$ is a chain map.
\end{prop*}
\begin{proof}
Let $n\geqslant 1$. For arbitrary $ f\in C^{n+1}_{\LY}(A,V)$ and $\mathfrak{U}_1\otimes \mathfrak{U}_2\otimes \dotsb \otimes \mathfrak{U}_{n+1}\wedge w\in (A^{\wedge 2})^{\otimes n+1}\otimes A^{\wedge 3}$ with $\mathfrak{U}_i=u_i\wedge v_i$, $\forall 1\leqslant i\leqslant n+1$,
\begin{small}
%\begingroup
\allowdisplaybreaks
\begin{align*}
	&amp; \parens[\big]{\psi^{n+2}_\coboundII\delta^{n+1}_\coboundII(f,g)}(\mathfrak{\mathfrak{U}}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \,\,\,		= \parens[\big]{\delta^{n+1}_\coboundII(f,g) \circ (K, K,\dotsc, K, K,K)} \parens[\big]{\mathfrak{\mathfrak{U}}_1,\dotsc,\mathfrak{U}_{n+1},w } 
\\	&amp; \quad 		- \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}\parens[\big]{\delta^{n+1}_\coboundII(f,g)\circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})}\parens[\big]{\mathfrak{\mathfrak{U}}_1,\dotsc,\mathfrak{U}_{n+1},w}
\\	&amp; \quad 		- \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1}
\\*	&amp; \hspace{2cm} \parens[\big]{K_V\circ \delta^{n+1}_\coboundII(f,g)\circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})}\parens[\big]{\mathfrak{\mathfrak{U}}_1,\dotsc,\mathfrak{U}_{n+1},w}
\\	&amp; \,\,\,		= (-1)^{n}\theta\parens[\big]{K(\mathfrak{v}_{n+1}),K(w)}g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc,K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n}),K(\mathfrak{u}_{n+1})}
\\	&amp; \quad		-(-1)^{n}\theta\parens[\big]{K(\mathfrak{u}_{n+1}),K(w)}g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n}),K(\mathfrak{v}_{n+1})}
\\	&amp; \quad		+ \sum_{j=1}^{n+1}(-1)^{j+1}\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j) \wedge K(\mathfrak{v}_j) }g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_{j},\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1 }), K(w)} 
\\	&amp; \quad		+ \sum_{1 \leq j &lt;s \leq n+1} (-1)^j g \parens[\Big]{K(\mathfrak{u}_1)\wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_j,\dotsc, K(\mathfrak{u}_{s-1}) \wedge K(\mathfrak{v}_{s-1}), \parens[\big]{\braces[\big]{K(\mathfrak{u}_j), K(\mathfrak{v}_j), K(\mathfrak{u}_s)}_A \wedge K(\mathfrak{v}_s) 
\\*	&amp; \hspace{2cm}		+ K(\mathfrak{u}_s) \wedge \braces[\big]{K(\mathfrak{u}_j), K(\mathfrak{v}_j), K(\mathfrak{v}_s)}}, K(\mathfrak{u}_{s+1}) \wedge K(\mathfrak{v}_{s+1}),\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}),K(w) }
\\	&amp; \quad		+ \sum_{j=1}^{n+1} (-1)^j g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc,\hat{\mathfrak{U}}_j,\dotsc,K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}),\braces[\big]{K(\mathfrak{u}_j),K(\mathfrak{v}_j),K(w)}_A}
\\	&amp; \quad 		- \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}}(-1)^{n}(-\lambda)^{\frac{s-1}{2}} (\theta \circ g) \circ \parens[\big]{(1,\dotsc, 2n+3) + (1,\dotsc, 2n+3) \circ (1,\dotsc, 2n+1)}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		-\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}} (-1)^{n} (-\lambda)^{\frac{s}{2}+1} \parens[\Big]{K_V\circ (\theta \circ g) \circ \parens[\big]{(1,\dotsc, 2n+3) + (1,\dotsc, 2n+3) \circ (1,\dotsc, 2n+1)}
\\*	&amp; \hspace{2cm} \circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})}(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		- \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} \sum_{j=1}^{n+1} (-1)^{j+1}(-\lambda)^{\frac{s-1}{2}} (\mathfrak{D} \cdot g) \circ (1,\dotsc, 2s)^2
\\*	&amp; \hspace{2cm} \circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1}, w)
%%%%% Fin p. 15.
\\	&amp; \quad		-\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}} \sum_{j=1}^{n+1} (-1)^{j +1}(-\lambda)^{\frac{s}{2}+1} K_V (\mathfrak{D} \cdot g) \circ (1,\dotsc, 2s)^2
\\*	&amp; \hspace{2cm} \circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		- \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s-1}{2}} g  [2s-3,2s-2,2s-1]_A
\\*	&amp; \hspace{2cm} \circ(2j-1,\dotsc, 2s-2)^{-2}\circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		-\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}}\sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2s-3,2s-2,2s-1]_A 
\\*	&amp; \hspace{2cm} \circ(2j-1,\dotsc, 2s-2)^{-2}
\circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		- \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s-1}{2}} g \circ [2s-2,2s-1,2s]_A \circ (2j-1,\dotsc, 2s-1)^{-2}
\\*	&amp; \hspace{2cm} \circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		-\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}}\sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2s-2,2s-1,2s]_A \circ (2j-1,\dotsc, 2s-1)^{-2}
\\*	&amp; \hspace{2cm} \circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		- \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} \sum_{j=1}^{n+1} (-1)^j(-\lambda)^{\frac{s-1}{2}} g \circ [2n+1,2n+2,2n+3]_A 
\\*	&amp; \hspace{2cm} \circ(2j-1,\dotsc, 2n+2)^{-2}
\circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad		-\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}} \sum_{j=1}^{n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2n+1,2n+2,2n+3]_A 
\\*	&amp; \hspace{2cm} \circ(2j-1,\dotsc, 2n+2)^{-2} \circ(K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1}, w).
\end{align*}
%\endgroup
\end{small}
On the other hand, we have
\begin{small}
%\begingroup
\allowdisplaybreaks
\begin{align*}
	&amp; \partial^{n+1}_\coboundII\parens[\big]{\psi^{n+1}_\coboundI(f),\psi^{n+1}_\coboundII(g)}(\mathfrak{\mathfrak{U}}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \,\,\, = (-1)^{n}\theta\parens[\big]{K(\mathfrak{v}_{n+1}),K(w)}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{u}_{n+1})
\\	&amp; \quad -(-1)^{n}K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{v}_{n+1}),w}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{u}_{n+1}) +\theta\parens[\big]{\mathfrak{v}_{n+1},K(w)}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{u}_{n+1})}
\\	&amp; \quad -(-1)^{n}\theta\parens[\big]{K(\mathfrak{u}_{n+1}),K(w)}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{v}_{n+1})
\\	&amp; \quad +(-1)^{n} K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{u}_{n+1}),w}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{v}_{n+1})+\theta\parens[\big]{\mathfrak{u}_{n+1},K(w)}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{v}_{n+1})}
\\	&amp; \quad + \sum_{j=1}^{n+1}(-1)^{j+1}\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),K(\mathfrak{v}_j)}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,
\hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad - \sum_{j=1}^{n+1}(-1)^{j+1}
K_V\parens[\Big]{\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),\mathfrak{v}_j}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc,
\hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)
\\*	&amp; \hspace{2cm} +\mathfrak{D}\parens[\big]{\mathfrak{u}_j,K(\mathfrak{v}_j)}\psi^{n+1}_\coboundII(g)(\mathfrak{U}_1,\dotsc, \hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)}
\\	&amp; \quad + \sum_{1 \leq j &lt; s \leq n+1} (-1)^j \psi^{n+1}_\coboundII(g) \parens[\big]{\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{s-1},\{\mathfrak{u}_j,\mathfrak{v}_j,\mathfrak{u}_s\}_K\wedge\mathfrak{v}_s+\mathfrak{u}_s\wedge \{\mathfrak{u}_j,\mathfrak{v}_j,\mathfrak{v}_s\}_K,
\\*	&amp; \hspace{2cm} \mathfrak{U}_{s+1},\dotsc,\mathfrak{U}_{n+1},w}
\\	&amp; \quad + \sum_{j=1}^{n+1} (-1)^j \psi^{n+1}_\coboundII(g)\parens[\big]{\mathfrak{U}_1,\dotsc,\hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},\{\mathfrak{u}_j,\mathfrak{v}_j,w\}_K}
%%%%% Fin p. 16.
\\	&amp; \,\,\, = (-1)^{n}\theta\parens[\big]{K(\mathfrak{v}_{n+1}),K(w)} \cdot \parens[\Big]{g \circ (K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n})\wedge K(\mathfrak{u}_{n+1})}
\\*	&amp; \quad -(-1)^{n} \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}
\\*	&amp; \hspace{2cm} \theta\parens[\big]{K(\mathfrak{v}_{n+1}),K(w)} \cdot g\circ(\Id^{i_1-1},K,\dotsc, \Id^{2n+1-i_{s-1}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n}, \mathfrak{u}_{n+1})
\\	&amp; \quad -(-1)^{n}\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1} 
\\*	&amp; \hspace{2cm} \theta\parens[\big]{K(\mathfrak{v}_{n+1}),K(w)} \cdot \parens[\big]{K_V\circ g\circ(\Id^{i_1-1},K,\dotsc, \Id^{2n+1-i_{s-1}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{u}_{n+1})}
\\	&amp; \quad -(-1)^{n}K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{v}_{n+1}),w} \cdot g \parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n})\wedge K(\mathfrak{u}_{n+1})}}
\\	&amp; \quad +(-1)^{n} \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}
\\*	&amp; \hspace{2cm} K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{v}_{n+1}),w} \cdot g\circ(K^{i_1-1},\Id,\dotsc, K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n}, \mathfrak{u}_{n+1}) }
\\	&amp; \quad + (-1)^{n} \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1}
\\*	&amp; \hspace{2cm} K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{v}_{n+1}),w} \cdot K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{u}_{n+1})}
\\	&amp; \quad -(-1)^nK_V\parens[\bigg]{\theta\parens[\big]{\mathfrak{v}_{n+1},K(w)}\parens[\Big]{g \circ \parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n})\wedge K(\mathfrak{u}_{n+1})}}} %%%%% ParenthÃ¨se ouvrante extra p. 17 l. 11
\\	&amp; \quad +(-1)^n \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}} 
\\*	&amp; \hspace{2cm} K_V\parens[\Big]{\theta\parens[\big]{\mathfrak{v}_{n+1},K(w)}  \cdot \parens[\big]{g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n}, \mathfrak{u}_{n+1})}}
\\	&amp; \quad +(-1)^n\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1} 
\\*	&amp; \hspace{2cm} K_V\parens[\Big]{\theta\parens[\big]{\mathfrak{v}_{n+1},K(w)} \cdot \parens[\big]{K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{u}_{n+1})}}
\\	&amp; \quad -(-1)^{n}\theta\parens[\big]{K(\mathfrak{u}_{n+1}),K(w)}\parens[\Big]{g \circ \parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n})\wedge K(\mathfrak{v}_{n+1})}} %%%%% ParenthÃ¨se ouvrante extra p. 17 l. 16
\\	&amp; \quad +(-1)^{n}	\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s-1}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}
\\*	&amp; \hspace{2cm} \theta\parens[\big]{K(\mathfrak{u}_{n+1}),K(w)} \cdot g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s-1}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n}, \mathfrak{v}_{n+1})
\\	&amp; \quad +(-1)^{n}\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s-1}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1}  
\\*	&amp; \hspace{2cm} K_V\parens[\Big]{\theta\parens[\big]{K(\mathfrak{u}_{n+1}),K(w)} \cdot \parens[\big]{K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s-1}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{v}_{n+1})}}
\\	&amp; \quad +(-1)^{n} K_V\parens[\Big]{\theta \parens[\big]{K(\mathfrak{u}_{n+1}),w}\cdot g \parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n})\wedge K(\mathfrak{v}_{n+1})}}
\\	&amp; \quad -(-1)^{n} \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}
\\*	&amp; \hspace{2cm} K_V\parens[\Big]{\theta \parens[\big]{K(\mathfrak{u}_{n+1}),w}\cdot g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n}, \mathfrak{v}_{n+1})}
\\	&amp; \quad -(-1)^{n}\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1}K_V\parens[\Big]{\theta \parens[\big]{K(\mathfrak{u}_{n+1}),w}\cdot K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})}
\\	&amp; \quad +(-1)^nK_V\parens[\Big]{\theta\parens[\big]{\mathfrak{u}_{n+1},K(w)}\parens[\Big]{g \circ (K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n}) \wedge K(\mathfrak{v}_{n+1})}}
\\	&amp; \quad -(-1)^n \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}K_V\parens[\Big]{\theta\parens[\big]{\mathfrak{u}_{n},K(w)} \cdot g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n}, \mathfrak{v}_{n+1})}
%%%%% Fin p. 17.
\\	&amp; \quad -(-1)^n\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1} 
\\*	&amp; \hspace{2cm} K_V\parens[\Big]{\theta\parens[\big]{\mathfrak{u}_{n},K(w)} \cdot \parens[\big]{K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n},\mathfrak{v}_{n+1})}}
\\	&amp; \quad + \sum_{j=1}^{n+1}(-1)^{j+1}\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),K(\mathfrak{v}_j)}g \parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_j,\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}), K(w)}
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}} \mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),K(\mathfrak{v}_j)} \cdot g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc,
\hat{\mathfrak{U}}_j,\dotsc, \mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1} 
\\*	&amp; \hspace{2cm} \mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),K(\mathfrak{v}_j)} \cdot K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}})(\mathfrak{U}_1,\dotsc, \hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad - \sum_{j=1}^{n+1}(-1)^{j+1}K_V\parens[\Big]{\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),\mathfrak{v}_j} \cdot g \parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_j,\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}), K(w)}}
\\	&amp; \quad + \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}
 \sum_{j=1}^{n+1}(-1)^{j+1}
\\*	&amp; \hspace{2cm} K_V \circ \parens[\Big]{\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),\mathfrak{v}_j} \cdot g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}}) (\mathfrak{U}_1,\dotsc,
\hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)}
\\	&amp; \quad +\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1} \sum_{j=1}^{n+1}(-1)^{j+1}
\\*	&amp; \hspace{2cm} K_V \circ \parens[\Big]{\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j),\mathfrak{v}_j} \cdot K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s}}) (\mathfrak{U}_1,\dotsc, \hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)}
\\	&amp; \quad - \sum_{j=1}^{n+1}(-1)^{j+1}K_V\parens[\Big]{\mathfrak{D}\parens[\big]{\mathfrak{u}_j,K(\mathfrak{v}_j)}g \parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_j,\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}), K(w)}} %%%%% ParenthÃ¨se ouvrante extra p. 18 l. 12
\\	&amp; \quad + \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s-1}\leqslant {2n+1} \\ s~\text{odd}}}(-\lambda)^{\frac{s-1}{2}}
 \sum_{j=1}^{n+1}(-1)^{j+1}
\\*	&amp; \hspace{2cm} K_V \circ \parens[\Big]{\mathfrak{D}\parens[\big]{\mathfrak{u}_j,K(\mathfrak{v}_j)} \cdot g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s-1}}) (\mathfrak{U}_1,\dotsc,
\hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)}
\\	&amp; \quad +\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s-1}\leqslant {2n+1} \\ s~\text{even}}} (-\lambda)^{\frac{s}{2}+1} \sum_{j=1}^{n+1}(-1)^{j+1}
\\*	&amp; \hspace{2cm} K_V \circ \parens[\Big]{\mathfrak{D}\parens[\big]{\mathfrak{u}_j,K(\mathfrak{v}_j)} \cdot K_V\circ g\circ(K^{i_1-1},\Id,\dotsc,K^{2n+1-i_{s-1}})(\mathfrak{U}_1,\dotsc, \hat{\mathfrak{U}}_j,\dotsc,\mathfrak{U}_{n+1},w)}
\\	&amp; \quad + \sum_{1 \leq j &lt; s \leq n+1} (-1)^j g \parens[\Big]{K(\mathfrak{u}_1)\wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_j,\dotsc, K(\mathfrak{u}_{s-1}) \wedge K(\mathfrak{v}_{s-1}), \braces[\big]{K(\mathfrak{u}_j), K(\mathfrak{v}_j), K(\mathfrak{u}_s)}_A \wedge K(\mathfrak{v}_s) 
\\*	&amp; \hspace{2cm} + K(\mathfrak{u}_s) \wedge \braces[\big]{K(\mathfrak{u}_j), K(\mathfrak{v}_j), K(\mathfrak{v}_s)}_A, K(\mathfrak{u}_{s+1}) \wedge K(\mathfrak{v}_{s+1}),\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}),K(w)}
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} 
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s-1}{2}} g \circ [2s-3,2s-2,2s-1]_A \circ (2j-1,\dotsc, 2s-2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}}) (\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}}
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2s-3,2s-2,2s-1]_A \circ (2j-1,\dotsc, 2s-2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad - \sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} 
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s-1}{2}} g \circ [2s-2,2s-1,2s]_A \circ (2j-1,\dotsc, 2s-1)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
%%%%% Fin p. 18.
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}}
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2s-2,2s-1,2s]_A \circ (2j-1,\dotsc, 2s-1)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad + \sum_{j=1}^{n+1} (-1)^j g\parens[\Big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc,\hat{\mathfrak{U}}_j,\dotsc,K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}),\braces[\big]{K(\mathfrak{u}_j),K(\mathfrak{v}_j),K(w)}}
\\	&amp; \quad -\sum_{\substack{1\leqslant j_1&lt;j_2&lt;\dotsb&lt; j_{s}\leqslant {2n+3} \\ s~\text{odd}}} \sum_{j=1}^{n+1} (-1)^j(-\lambda)^{\frac{s-1}{2}} g \circ [2n+1,2n+2,2n+3]_A \circ (2j-1,\dotsc, 2n+2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant j_1&lt;j_2&lt;\dotsb&lt; j_{s}\leqslant {2n+3} \\ s~\text{even}}} \sum_{j=1}^{n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2n+1,2n+2,2n+3]_A \circ (2j-1,\dotsc, 2n+2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \,\,\, = (-1)^{n}\theta\parens[\big]{K(\mathfrak{v}_{n+1}),K(w)}g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc,K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n}),K(\mathfrak{u}_{n+1})}
\\	&amp; \quad -(-1)^{n}\theta\parens[\big]{K(\mathfrak{u}_{n+1}),K(w)}g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, K(\mathfrak{u}_{n}) \wedge K(\mathfrak{v}_{n}),K(\mathfrak{v}_{n+1})}
\\	&amp; \quad + \sum_{j=1}^{n+1}(-1)^{j+1}\mathfrak{D}\parens[\big]{K(\mathfrak{u}_j) \wedge K(\mathfrak{v}_j) }g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_{j},\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1 }), K(w)} 
\\	&amp; \quad + \sum_{1 \leq j &lt;s \leq n+1} (-1)^j g \parens[\Big]{K(\mathfrak{u}_1)\wedge K(\mathfrak{v}_1),\dotsc, \hat{\mathfrak{U}}_j,\dotsc, K(\mathfrak{u}_{s-1}) \wedge K(\mathfrak{v}_{s-1}), \parens[\big]{\braces[\big]{K(\mathfrak{u}_j), K(\mathfrak{v}_j), K(\mathfrak{u}_s)}_A \wedge K(\mathfrak{v}_s) 
\\*	&amp; \hspace{2cm} + K(\mathfrak{u}_s) \wedge \braces[\big]{K(\mathfrak{u}_j), K(\mathfrak{v}_j), K(\mathfrak{v}_s)}}, K(\mathfrak{u}_{s+1}) \wedge K(\mathfrak{v}_{s+1}),\dotsc, K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}),K(w) }
\\	&amp; \quad + \sum_{j=1}^{n+1} (-1)^j g\parens[\big]{K(\mathfrak{u}_1) \wedge K(\mathfrak{v}_1),\dotsc,\hat{\mathfrak{U}}_j,\dotsc,K(\mathfrak{u}_{n+1}) \wedge K(\mathfrak{v}_{n+1}),\braces[\big]{K(\mathfrak{u}_j),K(\mathfrak{v}_j),K(w)}_A}
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}}(-1)^{n}(-\lambda)^{\frac{s-1}{2}} (\theta \circ g) \circ \parens[\big]{(1,\dotsc, 2n+3) + (1,\dotsc, 2n+3) \circ (1,\dotsc, 2n+1)}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}} (-1)^{n} (-\lambda)^{\frac{s}{2}+1} \parens[\Big]{K_V\circ (\theta \circ g) \circ \parens[\big]{(1,\dotsc, 2n+3) + (1,\dotsc, 2n+3) \circ (1,\dotsc, 2n+1)}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})}(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} 
\sum_{j=1}^{n+1} (-1)^{j+1}(-\lambda)^{\frac{s-1}{2}} (\mathfrak{D} \cdot g) \circ (1,\dotsc, 2s)^2
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1}, w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}} 
\sum_{j=1}^{n+1} (-1)^{j +1}(-\lambda)^{\frac{s}{2}+1} K_V (\mathfrak{D} \cdot g) \circ (1,\dotsc, 2s)^2
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} 
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s-1}{2}} g \circ [2s-3,2s-2,2s-1]_A \circ (2j-1,\dotsc, 2s-2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}}
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2s-3,2s-2,2s-1]_A \circ (2j-1,\dotsc, 2s-2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
%%%%% Fin p. 19.
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} 
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s-1}{2}} g \circ [2s-2,2s-1,2s]_A \circ (2j-1,\dotsc, 2s-1)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}}
\; \sum_{1 \leq j &lt; s \leq n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2s-2,2s-1,2s]_A \circ (2j-1,\dotsc, 2s-1)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{odd}}} \sum_{j=1}^{n+1} (-1)^j(-\lambda)^{\frac{s-1}{2}} g \circ [2n+1,2n+2,2n+3]_A \circ(2j-1,\dotsc, 2n+2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1},w)
\\	&amp; \quad -\sum_{\substack{1\leqslant i_1&lt;i_2&lt;\dotsb&lt; i_{s}\leqslant {2n+3} \\ s~\text{even}}} \sum_{j=1}^{n+1} (-1)^j (-\lambda)^{\frac{s}{2}+1} K_V\circ g \circ [2n+1,2n+2,2n+3]_A \circ (2j-1,\dotsc, 2n+2)^{-2}
\\*	&amp; \hspace{2cm} \circ (K^{i_1-1},\Id, K^{i_2-i_1-1}, \Id,\dotsc, \Id, K^{2n+3-i_{s}})(\mathfrak{U}_1,\dotsc,\mathfrak{U}_{n+1}, w).
\end{align*}
So we have $\psi^{n+2}_\coboundII\circ\delta^{n+1}_\coboundII=\partial^{n+1}_\coboundII\circ (\psi^{n+1}_\coboundI,\psi^{n+1}_\coboundII)$. Similarly, we obtain $\psi^{n+2}_\coboundI \circ \delta^{n+1}_\coboundI=\partial^{n+1}_\coboundI \circ(\psi^{n+1}_\coboundI,\psi^{n+1}_\coboundII)$.
%\endgroup
\end{small}
\end{proof}

\section*{Acknowledgments}

The authors would like to thank the referee for valuable comments and suggestions on this article.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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