# Maurer-Cartan characterization, $L_{\infty}$-algebras, and cohomology of
relative Rota-Baxter operators on Lie-Yamaguti algebras

Jia Zhao  Jia Zhao, School of Sciences, Nantong University, Nantong 226019,
Jiangsu, China<EMAIL_ADDRESS>and  Yu Qiao  Yu Qiao (corresponding
author), School of Mathematics and Statistics, Shaanxi Normal University,
Xi’an 710119, Shaanxi, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first construct a differential graded Lie algebra that
controls deformations of a Lie-Yamaguti algebra. Furthermore, a relative Rota-
Baxter operator on a Lie-Yamaguti algebra is characterized as a Maurer-Cartan
element in an appropriate $L_{\infty}$-algebra that we build through the
graded Lie bracket of Lie-Yamaguti algebra’s controlling algebra, and gives
rise to a twisted $L_{\infty}$-algebra that controls its deformation. Next we
establish the cohomology theory of relative Rota-Baxter operators on Lie-
Yamaguti algebras via the Yamaguti cohomology. Then we clarify the
relationship between the twisted $L_{\infty}$-algebra and the cohomology
theory. Finally as byproducts, we classify certain deformations on Lie-
Yamaguti algebras using the cohomology theory.

###### Key words and phrases:

Lie-Yamaguti algebra, Maurer-Cartan element, relative Rota-Baxter operator,
$L_{\infty}$-algebra, cohomology, deformation

Mathematics Subject Classification (2020): 17A30,17B37,17B56

###### Contents

  1. 1 Introduction
    1. 1.1 Maurer-Cartan characterizations and deformations
    2. 1.2 Rota-Baxter operators
    3. 1.3 Lie-Yamaguti algebras
    4. 1.4 Outline of the paper
  2. 2 Preliminaries
  3. 3 Maurer-Cartan characterizations for Lie-Yamaguti algebras
  4. 4 Maurer-Cartan characterizations and Cohomology of relative Rota-Baxter operators
    1. 4.1 Maurer-Cartan characterizations for relative Rota-Baxter operators
    2. 4.2 Cohomology of relative Rota-Baxter operators
  5. 5 Deformations of relative Rota-Baxter operators
    1. 5.1 Linear deformations of relative Rota-Baxter operators
    2. 5.2 Higher order deformations of relative Rota-Baxter operators

##  1\. Introduction

In this paper, we determine a differential graded Lie algebra as controlling
algebra for Lie-Yamaguti algebras and an $L_{\infty}$-algebra for their
relative Rota-Baxter operators. Then we establish the cohomology theory to
classify deformations of relative Rota-Baxter operators.

###  1.1. Maurer-Cartan characterizations and deformations

In mathematics, informally speaking, a deformation of an object is another
object that shares the same structure of the original object after a
perturbation. Motivated by the foundational work of Kodaira and Spencer [20]
for complex analytic structures, the generalization in algebraic geometry of
deformation theory was founded [18]. Recently, a number of works referring to
deformation quantization turn up in the context of mathematical physics [21,
22]. As an application in algebra, Gerstenhaber first studied the deformation
theory on associative algebras [10, 11, 12, 13, 14]. Then Nijenhuis and
Richardson extended this idea and established the similar results on Lie
algebras [29, 30]. Deformations of other algebraic structures such as pre-Lie
algebras have also been developed [8]. In general, deformation theory was set
up for binary quadratic operads by Balavoine [2].

As we all know, for an algebraic object, deformation theory and cohomology
theory are connected tightly. More precisely, a suitable deformation of an
object should obey the following rule: on one hand, there exists a
differential graded Lie algebra (or generally an $L_{\infty}$-algebra), to
whose Maurer-Cartan elements the algebraic structures correspond; on the other
hand, there should be a suitable cohomology theory that characterizes its
deformations. Deligne, Drinfeld, and Kontsevich proposed a celebrated slogan
that every reasonable deformation is controlled by a differential graded Lie
algebra, up to quasi-isomorphisms. More precisely, this slogan means that for
an algebraic structure, Maurer-Cartan elements in a suitable graded Lie
algebra are used to characterize realizations of its algebraic structure on a
vector space. And moreover, for a given realization of the algebraic
structure, a Maurer-Cartan element induces a differential making the graded
Lie algebra into a differential graded Lie algebra (this graded Lie algebra is
also called the controlling algebra). See [16, 39, 40, 41] for Maurer-Cartan
characterizations of some algebraic structures and their relative Rota-Baxter
operators.

###  1.2. Rota-Baxter operators

Baxter introduced the notion of Rota-Baxter operators on associative algebras
when studying fluctuation theory [3]. Then Kupershmidt introduced the notion
of ${\mathcal{O}}$-operators (called relative Rota-Baxter operators in the
present paper) on Lie algebras when he found that a relative Rota-Baxter
operator is a solution to the classical Yang-Baxter equation in [23], whereas
the classical Yang-Baxter equation plays an important role in many fields such
as in integration systems [9]. For more details about the classical Yang-
Baxter equation and Rota-Baxter operators, one can see [34]. Recently, people
found that Rota-Baxter operators have many applications. For eaxmple, the
first work about Rota-Baxter algebras by Guo had been published [17]. From
then on, Rota-Baxter algebra open up a new branch in Lie theory and in
mathematical physics. Bai and his collaborators examined Rota-Baxter operators
in the context of binary operads in [1, 32]. Moreover, Rota-Baxter operators
are closely connected with Hopf algebras.

###  1.3. Lie-Yamaguti algebras

A Lie-Yamaguti algebra is a generalization of a Lie algebra and a Lie triple
system, which can be traced back to Nomizu’s work on the invariant affine
connections on homogeneous spaces in 1950’s [31]. Inspired by thoughts of
Nomizu, in 1960’s Yamaguti introduced an algebraic object called a general Lie
triple system, defined its representation and established its cohomology
theory in [43, 44]. Kinyon and Weinstein first called this object a Lie-
Yamaguti algebra when studying Courant algebroids in the earlier 21st century
[19]. Since then, this system is called a Lie-Yamaguti algebra, which has
attracted much attention and is widely investigated recently. For instance,
Benito, Draper, and Elduque investigated Lie-Yamaguti algebras related to
simple Lie algebras of type $G_{2}$ [5]. Afterwards, Benito, Elduque, and
Mart$\acute{i}$n-Herce explored irreducible Lie-Yamaguti algebras in [6, 7].
More recently, Benito, Bremmer, and Madariaga examined orthogonal Lie-Yamaguti
algebras in [4]. Deformations and extensions of Lie-Yamaguti algebras were
explored in [26, 45]. Takahashi studied modules over quandles using
representations of Lie-Yamaguti algebras in [38].

Since a Lie-Yamaguti algebra owns two algebraic structures, by the virtue of
this object, the first author has been working on the deformation theory of
Lie-Yamaguti algebras these years. For example, Sheng, the first author, and
Zhou analyzed product structures and complex structures on Lie-Yamaguti
algebras by means of Nijenhuis operators in [35]. Afterwards, Sheng and the
first author introduced the notion of relative Rota-Baxter operators on Lie-
Yamaguti algebras and revealed the fact that a pre-Lie-Yamaguti algebra is the
underlying algebraic structure of relative Rota-Baxter operators [36]. Since
then, thanks to importance of deformations and relative Rota-Bxter operators
on Lie-Yamaguti algebras, we established Lie-Yamaguti bialgebra theory and
clarified the relationship between the solution to the classical Lie-Yamaguti
Yang-Baxter equation and relative Rota-Baxter operators in [46], and studied
cohomology and deformations of $\mathsf{LieYRep}$ pairs and then explored
several properties of relative Rota-Baxter-Nijenhuis structures in [47].

Motivation. Since Sheng and the first author defined relative Rota-Baxter
operators on Lie-Yamaguti algebras [36], according to Deligne, Drinfeld, and
Kontsevich’s slogan, it is natural to consider Maurer-Cartan characterizations
for relative Rota-Baxter operators on Lie-Yamaguti algebras and its
relationship between cohomology and the Maurer-Cartan elements. However, the
first and foremost, we have to consider Maurer-Cartan characterization for
Lie-Yamaguti algebras because on the one hand, unlike associative algebras,
Lie algebras, Leibniz algebras, or even $3$-Lie algebras, no works of
controlling algebras of Lie-Yamaguti algebras appeared before; on the other
hand, it is inevitable, for we construct the controlling algebra for relative
Rota-Baxter operators. Thus, this step of constructing controlling algebra for
Lie-Yamaguti algebras also promotes development of deformation theory for Lie-
Yamaguti algebras. Anyway, our project can be transferred to the following
questions:

  * •

Does there exist a suitable graded Lie algebra whose Maurer-Cartan elements
correspond to the Lie-Yamaguti algebra structures on a vector space?

  * •

Does there exist a suitable algebra whose Maurer-Catan elemnts are precisely
the relative Rota-Baxter operators on Lie-Yamaguti algebras?

  * •

Does there exist an appropriate cohomology theory of relative Rota-Baxter
operators on Lie-Yamaguti algebras, which can be used to classify certain
types of deformations?

After solving these problems, we are able to consider deformations
spontaneously, perfecting researches about cohomology and deformation theory
on Lie-Yamaguti algebras and on their relative Rota-Baxter operators.

###  1.4. Outline of the paper

We tackle these problems as follows. After some preliminaries introduced in
Section 2, Section 3 answers the first question. As was stated before, no
works of controlling algebras of Lie-Yamaguti algebras appeared, thus
constructing a graded Lie algebra whose Maurer-Cartan elements correspond to
the Lie-Yamaguti algebra structures is a difficulty to overcome. Once the
graded Lie algebra is constructed (see Theorem 3.1), consequently a Maurer-
Cartan element induces a differential making the graded Lie algebra into a
differential graded Lie algebra as the controlling algebra for Lie-Yamaguti
algebras (see Theorem 3.3). Moreover, we establish the relation between the
controlling algebra and the coboundary operator associated to the adjoint
representation (see Theorem 3.5).

Section 4 is the solution to the last two questions, which is the core of the
present paper. Immediately after the constructing controlling algebra for Lie-
Yamaguti algebras, we make use of the graded Lie brackets by the method of
derived brackets to construct an $L_{\infty}$-algebra whose Maurer-Cartan
elements are precisely relative Rota-Baxter operators (see Theorem 4.6).
Moreover, from a relative Rota-Baxter operator (as a Maurer-Cartan element),
we obtain a twisted $L_{\infty}$-algebra that controls deformations of
relative Rota-Baxter operators (see Theorem 4.8). As an application, we endow
a continuous function space and a series ring with Lie-Yamaguti algebra
structure respectively and give two integral operators which are treated as
relative Rota-Baxter operators on them. And then we build cohomology of
relative Rota-Baxter operators as follows. Given a relative Rota-Baxter
operator $T:V\longrightarrow\mathfrak{g}$ with respect to a representation
$(V;\rho,\mu)$, we mentioned in [36] that there is a Lie-Yamaguti algebra
structure
$([\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$ on
$V$, in which case the Lie-Yamaguti algebra
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$
is called the sub-adjacent Lie-Yamaguti algebra. In order to establish the
cohomology theory, we first should build the cohomology of this sub-adjacent
Lie-Yamaguti algebra using Yamaguti cohomology. Thus, we have to construct a
representation of the sub-adjacent Lie-Yamaguti algebra (see Theorem 4.13).
However, note that the cochain complex of Yamaguti cohomology starts only from
$1$-cochians, not from $0$-cochains. We have to point out that $0$-cochains
play an import role in our study of deformations. Thus the next step is to
choose $0$-cochains appropriately and build a proper coboundary map from the
set of $0$-cochains to that of $1$-cochains, which is another difficulty (see
Proposition 4.14). In this way, we obtain a cochain complex (associated to
$V$) starting from $0$-cochains, which gives rise to the cohomology of the
relative Rota-Baxter operator $T$ on Lie-Yamaguti algebras
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
(see Definition 4.15). We also establish the relation between the differential
of the twisted $L_{\infty}$-algebra and the coboundary operator of cohomology
of relative Rota-Baxter operators parallel to the case of Lie-Yamaguti
algebras (see Theorem 4.16).

Finally in Section 5, as an application, we make use of the cohomology theory
to investigate deformations of relative Rota-Baxter operators on Lie-Yamaguti
algebras. We intend to consider two kinds of deformations: linear and higher
order deformations. It turns out that our cohomology theory satisfies the rule
that is mentioned above and works well (see Theorem 5.6 and Theorem 5.14).

A Lie-Yamaguti algebra owns two algebraic operations, which makes the
operation of its controlling algebra and cochain complex much more complicated
than other algebras, such as Lie algebras, pre-Lie algebras, Leibniz algebras
or even $3$-Lie algebras, one of which owns only one operation. As a result,
the computation is technically difficult in constructing the controlling
algebras and defining the cohomology of relative Rota-Baxter operators. Note
that a Lie triple system is a spacial case of a Lie-Yamaguti algebra [28], so
the conclusions in the present paper can also be adapted to the Lie triple
system context.

In this paper, all the vector spaces are over $\mathbb{K}$, a field of
characteristic $0$.

##  2\. Preliminaries

In this section, we recall some basic notions such as Lie-Yamaguti algebras,
representations and their cohomology theory. The notion of Lie-Yamaguti
algebras was first defined by Yamaguti in [43, 44].

######  Definition 2.1.

[19] A Lie-Yamaguti algebra is a vector space $\mathfrak{g}$, together with a
bilinear bracket $[\cdot,\cdot]:\wedge^{2}\mathfrak{g}\to\mathfrak{g}$ and a
trilinear bracket
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket:\wedge^{2}\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$
such that the following equations are satisfied for all
$x,y,z,w,t\in\mathfrak{g}$,

(1) $\displaystyle[[x,y],z]+[[y,z],x]+[[z,x],y]+\left\llbracket
x,y,z\right\rrbracket+\left\llbracket y,z,x\right\rrbracket+\left\llbracket
z,x,y\right\rrbracket=0,$ (2)
$\displaystyle\left\llbracket[x,y],z,w\right\rrbracket+\left\llbracket[y,z],x,w\right\rrbracket+\left\llbracket[z,x],y,w\right\rrbracket=0,$
(3) $\displaystyle\left\llbracket x,y,[z,w]\right\rrbracket=[\left\llbracket
x,y,z\right\rrbracket,w]+[z,\left\llbracket x,y,w\right\rrbracket],$ (4)
$\displaystyle\left\llbracket x,y,\left\llbracket
z,w,t\right\rrbracket\right\rrbracket=\left\llbracket\left\llbracket
x,y,z\right\rrbracket,w,t\right\rrbracket+\left\llbracket z,\left\llbracket
x,y,w\right\rrbracket,t\right\rrbracket+\left\llbracket z,w,\left\llbracket
x,y,t\right\rrbracket\right\rrbracket.$

######  Example 2.2.

[31] Let $M$ be a closed manifold with an affine connection, and denote by
$\mathfrak{X}(M)$ the set of vector fields on $M$. For all
$x,y,z\in\mathfrak{X}(M)$, set

$\displaystyle[x,y]=-T(x,y),\quad\left\llbracket
x,y,z\right\rrbracket=-R(x,y)z,$

where $T$ and $R$ are torsion tensor and curvature tensor respectively. It
turns out that the triple
$(\mathfrak{X}(M),[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
forms an (infinite-dimensional) Lie-Yamaguti algebra.

######  Definition 2.3.

[44] Let
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
be a Lie-Yamaguti algebra and $V$ a vector space. A representation of
$\mathfrak{g}$ on $V$ consists of a linear map
$\rho:\mathfrak{g}\to\mathfrak{gl}(V)$ and a bilinear map
$\mu:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(V)$ such that for all
$x,y,z,w\in\mathfrak{g}$,

(5) $\displaystyle\mu([x,y],z)-\mu(x,z)\rho(y)+\mu(y,z)\rho(x)=0,$ (6)
$\displaystyle\mu(x,[y,z])-\rho(y)\mu(x,z)+\rho(z)\mu(x,y)=0,$ (7)
$\displaystyle\rho(\left\llbracket
x,y,z\right\rrbracket)=[D_{\rho,\mu}(x,y),\rho(z)],$ (8)
$\displaystyle\mu(z,w)\mu(x,y)-\mu(y,w)\mu(x,z)-\mu(x,\left\llbracket
y,z,w\right\rrbracket)+D_{\rho,\mu}(y,z)\mu(x,w)=0,$ (9)
$\displaystyle\mu(\left\llbracket
x,y,z\right\rrbracket,w)+\mu(z,\left\llbracket
x,y,w\right\rrbracket)=[D_{\rho,\mu}(x,y),\mu(z,w)],$

where the bilinear map
$D_{\rho,\mu}:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(V)$ is given by

$\displaystyle
D_{\rho,\mu}(x,y):=\mu(y,x)-\mu(x,y)+[\rho(x),\rho(y)]-\rho([x,y]),\quad\forall
x,y\in\mathfrak{g}.$

It is obvious that $D_{\rho,\mu}$ is skew-symmetric. We write $D$ in the
sequel without ambiguities and we denote a representation of $\mathfrak{g}$ on
$V$ by $(V;\rho,\mu)$.

By a direct computation, we have

######  Proposition 2.4.

If $(V;\rho,\mu)$ is a representation of a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$,
then we have the following equalities:

(11) $\displaystyle D([x,y],z)+D([y,z],x)+D([z,x],y)=0;$ $\displaystyle
D(\left\llbracket x,y,z\right\rrbracket,w)+D(z,\left\llbracket
x,y,w\right\rrbracket)=[D(x,y),D(z,w)];$ (12)
$\displaystyle\mu(\left\llbracket
x,y,z\right\rrbracket,w)=\mu(x,w)\mu(z,y)-\mu(y,w)\mu(z,x)-\mu(z,w)D(x,y).$

######  Example 2.5.

Let
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
be a Lie-Yamaguti algebra. We define linear maps
$\mathrm{ad}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$ and
$\mathfrak{R}:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$ to be
$x\mapsto\mathrm{ad}_{x}$ and $(x,y)\mapsto\mathfrak{R}_{x,y}$, where
$\mathrm{ad}_{x}z=[x,z]$ and $\mathfrak{R}_{x,y}z=\left\llbracket
z,x,y\right\rrbracket$ for all $z\in\mathfrak{g}$ respectively. Then
$(\mathfrak{g};\mathrm{ad},\mathfrak{R})$ forms a representation of
$\mathfrak{g}$ on itself, called the adjoint representation. In this case,
$\mathfrak{L}\triangleq D_{\mathrm{ad},\mathfrak{R}}$ is given by for all
$x,y\in\mathfrak{g}$,

$\displaystyle\mathfrak{L}_{x,y}=\mathfrak{R}_{y,x}-\mathfrak{R}_{x,y}+[\mathrm{ad}_{x},\mathrm{ad}_{y}]-\mathrm{ad}_{[x,y]}.$

By (1), we have

$\displaystyle\mathfrak{L}_{x,y}z=\left\llbracket
x,y,z\right\rrbracket,\quad\forall z\in\mathfrak{g}.$

Representations of a Lie-Yamaguti algebra can be characterized by the
semidirect product Lie-Yamaguti algebras.

######  Proposition 2.6.

[45] Let
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
be a Lie-Yamaguti algebra and $V$ a vector space. Let
$\rho:\mathfrak{g}\to\mathfrak{gl}(V)$ and
$\mu:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(V)$ be linear maps. Then
$(V;\rho,\mu)$ is a representation of
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
if and only if there is a Lie-Yamaguti algebra structure
$([\cdot,\cdot]_{\rho,\mu},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\rho,\mu})$
on the direct sum $\mathfrak{g}\oplus V$ which is defined by for all
$x,y,z\in\mathfrak{g},~{}u,v,w\in V$,

$\displaystyle[x+u,y+v]_{\rho,\mu}$ $\displaystyle=$
$\displaystyle[x,y]+\rho(x)v-\rho(y)u,$ $\displaystyle~{}\left\llbracket
x+u,y+v,z+w\right\rrbracket_{\rho,\mu}$ $\displaystyle=$
$\displaystyle\left\llbracket
x,y,z\right\rrbracket+D(x,y)w+\mu(y,z)u-\mu(x,z)v,$

This Lie-Yamaguti algebra $(\mathfrak{g}\oplus
V,[\cdot,\cdot]_{\rho,\mu},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{\rho,\mu})$
is called the semidirect product Lie-Yamaguti algebra, and denoted by
$\mathfrak{g}\ltimes_{\rho,\mu}V$.

The cohomology theory of Lie-Yamaguti algebras was established in [44]. Let
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
be a Lie-Yamaguti algebra and $(V;\rho,\mu)$ a representation. We denote by
$C^{p}_{\rm LieY}(\mathfrak{g},V)~{}(p\geqslant 1)$ the set of $p$-cochains,
where

$\displaystyle C^{n+1}_{\rm
LieY}(\mathfrak{g},V)\triangleq\begin{cases}\mathrm{Hom}(\underbrace{\wedge^{2}\mathfrak{g}\otimes\cdots\otimes\wedge^{2}\mathfrak{g}}_{n},V)\times\mathrm{Hom}(\underbrace{\wedge^{2}\mathfrak{g}\otimes\cdots\otimes\wedge^{2}\mathfrak{g}}_{n}\otimes\mathfrak{g},V),&n\geqslant
1,\\\ \mathrm{Hom}(\mathfrak{g},V),&n=0.\end{cases}$

For $p\geqslant 1$, the coboundary operator $\delta:C^{p}_{\rm
LieY}(\mathfrak{g},V)\to C^{p+1}_{\rm LieY}(\mathfrak{g},V)$ is defined as
follows:

  * •

If $n\geqslant 1$, for any $F=(f,g)\in C^{n+1}_{\rm LieY}(\mathfrak{g},V)$,
the coboundary map

$\delta=(\delta_{\rm I},\delta_{\rm II}):C^{n+1}_{\rm LieY}(\mathfrak{g},V)\to
C^{n+2}_{\rm LieY}(\mathfrak{g},V),$ $\qquad\qquad\qquad\qquad\qquad\quad
F\mapsto(\delta_{\rm I}(F),\delta_{\rm II}(F)),$

is given by:

$\displaystyle\Big{(}\delta_{\rm
I}(F)\Big{)}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1})$ (13)
$\displaystyle=$
$\displaystyle(-1)^{n}\Big{(}\rho(x_{n+1})g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},y_{n+1})-\rho(y_{n+1})g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},x_{n+1})$
$\displaystyle-g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},[x_{n+1},y_{n+1}])\Big{)}$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k+1}D(\mathfrak{X}_{k})f(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{n+1})$
$\displaystyle+\sum_{1\leqslant k<l\leqslant
n+1}(-1)^{k}f(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{k}\circ\mathfrak{X}_{l},\cdots,\mathfrak{X}_{n+1}),$

and

$\displaystyle\Big{(}\delta_{\rm
II}(F)\Big{)}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1},z)$ (14)
$\displaystyle=$
$\displaystyle(-1)^{n}\Big{(}\mu(y_{n+1},z)g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},x_{n+1})-\mu(x_{n+1},z)g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},y_{n+1})\Big{)}$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k+1}D(\mathfrak{X}_{k})g(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{n+1},z)$
$\displaystyle+\sum_{1\leqslant k<l\leqslant
n+1}(-1)^{k}g(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{k}\circ\mathfrak{X}_{l},\cdots,\mathfrak{X}_{n+1},z)$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k}g(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{n+1},\left\llbracket
x_{k},y_{k},z\right\rrbracket),$

where $\mathfrak{X}_{i}=x_{i}\wedge
y_{i}\in\wedge^{2}\mathfrak{g},~{}(i=1,\cdots,n+1),~{}z\in\mathfrak{g}$ and
the operation $\circ$ means that

$\mathfrak{X}_{k}\circ\mathfrak{X}_{l}\triangleq\left\llbracket
x_{k},y_{k},x_{l}\right\rrbracket\wedge y_{l}+x_{l}\wedge\left\llbracket
x_{k},y_{k},y_{l}\right\rrbracket.$

  * •

For the case that $n=0$, any element $f\in C^{1}_{\rm LieY}(\mathfrak{g},V)$
given, the coboundary map

$\delta:C^{1}_{\rm LieY}(\mathfrak{g},V)\to C^{2}_{\rm LieY}(\mathfrak{g},V),$
$\qquad\qquad\qquad f\mapsto(\delta_{\rm I}(f),\delta_{\rm II}(f)),$

is given by

(15) $\displaystyle(\delta_{\rm I}(f))(x,y)$ $\displaystyle=$
$\displaystyle\rho(x)f(y)-\rho(y)f(x)-f([x,y]),$ (16)
$\displaystyle~{}(\delta_{\rm II}(f))(x,y,z)$ $\displaystyle=$ $\displaystyle
D(x,y)f(z)+\mu(y,z)f(x)-\mu(x,z)f(y)-f(\left\llbracket
x,y,z\right\rrbracket),\quad\forall x,y,z\in\mathfrak{g}.$

Yamaguti showed in [44] that $\delta$ is a differential, i.e.,
$\delta\circ\delta=0$. More precisely, for any $f\in C^{1}_{\rm
LieY}(\mathfrak{g},V)$, we have

$\delta_{\rm I}\Big{(}\delta_{\rm I}(f),\delta_{\rm
II}(f)\Big{)}=0\quad\text{and}\quad\delta_{\rm II}\Big{(}\delta_{\rm
I}(f),\delta_{\rm II}(f)\Big{)}=0.$

Moreover, for all $F\in C^{p}_{\rm LieY}(\mathfrak{g},V)~{}(p\geqslant 2)$, we
have

$\delta_{\rm I}\Big{(}\delta_{\rm I}(F),\delta_{\rm
II}(F)\Big{)}=0\quad\text{and}\quad\delta_{\rm II}\Big{(}\delta_{\rm
I}(F),\delta_{\rm II}(F)\Big{)}=0.$

Thus the cochain complex $(C^{\bullet}_{\rm
LieY}(\mathfrak{g},V)=\bigoplus_{p=1}^{\infty}C^{p}_{\rm
LieY}(\mathfrak{g},V),\delta)$ is well defined, whose cohomology is called the
Yamaguti cohomology in this paper.

######  Definition 2.7.

An $p$-cocycle $(p\geqslant 1)$ is an element $F=(f,g)$ in $C^{p}_{\rm
LieY}(\mathfrak{g},V)~{}(p\geqslant 2)$ (resp. $f\in C^{1}_{\rm
LieY}(\mathfrak{g},V)$) such that $\delta(F)=0$ (resp. $\delta(f)=0$). The set
of $p$-cocycles is denoted by $Z^{p}_{\rm LieY}(\mathfrak{g},V)$; for an
element $F$ in $C^{p}_{\rm LieY}(\mathfrak{g},V)~{}(p\geqslant 2)$, if there
exists $G=(h,s)\in C^{p-1}_{\rm LieY}(\mathfrak{g},V)$ (resp. $t\in
C^{1}(\mathfrak{g},V)$, if $p=2$) such that $F=\delta(G)$ (resp.
$F=\delta(t)$), then $F$ is called an $p$-coboundary. The set of
$p$-coboundaries is denoted by $B^{p}_{\rm LieY}(\mathfrak{g},V)$. The
resulting $p$-cohomology group is defined by the factor space

$H^{p}_{\rm LieY}(\mathfrak{g},V)=Z^{p}_{\rm LieY}(\mathfrak{g},V)/B^{p}_{\rm
LieY}(\mathfrak{g},V)~{}\quad(p\geqslant 2).$

##  3\. Maurer-Cartan characterizations for Lie-Yamaguti algebras

In this section, for a preparation of constructing controlling algebra for
relative Rota-Baxter operators, we have to construct the controlling algebra
for Lie-Yamaguti algebras first. Let us recall some notions and basic facts in
[27].

A degree $1$ element $x\in\mathfrak{g}_{1}$ is called a Maurer-Cartan element
of a differential graded Lie algebra
$(\mathfrak{g}=\oplus_{k\in\mathbb{Z}}\mathfrak{g}_{k},[\cdot,\cdot],d)$ if it
satisfies the Maurer-Cartan equation: $dx+\frac{1}{2}[x,x]=0.$ Note that a
graded Lie algebra is a special differential graded Lie algebra with $d=0$.
Correspondingly, for a graded Lie algebra
$(\mathfrak{g}=\oplus_{k\in\mathbb{Z}}\mathfrak{g}_{k},[\cdot,\cdot])$, an
element $x\in\mathfrak{g}_{1}$ satisfying $[x,x]=0$ is a Maurer-Cartan element
of $\mathfrak{g}$.

A permutation $\sigma\in S_{n}$ is called an $(i,n-i)$-shuffle if
$\sigma(1)<\cdots<\sigma(i)$ and $\sigma(i+1)<\cdots<\sigma(n)$. If $i=0$ or
$i=n$, we assume $\sigma={\rm{Id}}$. The set of $(i,n-i)$-shuffles is denoted
by $\mathbb{S}_{(i,n-i)}$.

Let $\mathfrak{g}$ be a vector space. Denote by

$\displaystyle\mathfrak{C}^{p}(\mathfrak{g},\mathfrak{g}):=\begin{cases}\mathrm{Hom}(\underbrace{(\wedge^{2}\mathfrak{g})\otimes\cdots(\wedge^{2}\mathfrak{g})}_{p},\mathfrak{g})\oplus\mathrm{Hom}(\underbrace{(\wedge^{2}\mathfrak{g})\otimes\cdots(\wedge^{2}\mathfrak{g})}_{p}\otimes\mathfrak{g},\mathfrak{g}),&p\geqslant
1,\\\ \mathrm{Hom}(\mathfrak{g},\mathfrak{g}),&p=0.\end{cases}$

Then $\mathfrak{C}^{\bullet}(\mathfrak{g},\mathfrak{g})=\oplus_{p\geqslant
0}\mathfrak{C}^{p}(\mathfrak{g},\mathfrak{g})$ is a graded vector space, in
which the degree of elements in $\mathfrak{C}^{p}(\mathfrak{g},\mathfrak{g})$
is assumed to be $p$. For $P=(P_{\rm I},P_{\rm
II})\in\mathfrak{C}^{p}(\mathfrak{g},\mathfrak{g}),~{}Q=(Q_{\rm I},Q_{\rm
II})\in\mathfrak{C}^{q}(\mathfrak{g},\mathfrak{g})~{}(p,q\geqslant 1)$, we
define $P\circ Q=\Big{(}(P\circ Q)_{\rm I},(P\circ Q)_{\rm
II}\Big{)}\in\mathfrak{C}^{p+q}(\mathfrak{g},\mathfrak{g})$ to be

$\displaystyle\Big{(}P\circ Q\Big{)}_{\rm
I}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p+q})$ (17) $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(p,q)}\atop\sigma(p+q)=p+q}(-1)^{pq}sign(\sigma)P_{\rm
II}(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(p)},Q_{\rm
I}(\mathfrak{X}_{\sigma(p+1)},\cdots,\mathfrak{X}_{\sigma(p+q)}))$
$\displaystyle+\sum_{k=1}^{p}(-1)^{(k-1)q}\sum_{\sigma\in\mathbb{S}_{(k-1,q)}}sign(\sigma)P_{\rm
I}(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x_{q+k}\wedge
Q_{\rm
II}(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},y_{k+q}),\mathfrak{X}_{k+q+1},\cdots,\mathfrak{X}_{p+q})$
$\displaystyle+\sum_{k=1}^{p}(-1)^{(k-1)q}\sum_{\sigma\in\mathbb{S}_{(k-1,q)}}sign(\sigma)P_{\rm
I}(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},Q_{\rm
II}(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},x_{k+q})\wedge
y_{k+q},\mathfrak{X}_{k+q+1},\cdots,\mathfrak{X}_{p+q}),$

and

$\displaystyle\Big{(}P\circ Q\Big{)}_{\rm
II}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p+q},x)$ (18) $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(p,q)}}(-1)^{pq}sign(\sigma)P_{\rm
II}(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(p)},Q_{\rm
II}(\mathfrak{X}_{\sigma(p+1)},\cdots,\mathfrak{X}_{\sigma(p+q)},x))$
$\displaystyle+\sum_{k=1}^{p}(-1)^{(k-1)q}\sum_{\sigma\in\mathbb{S}_{(k-1,q)}}sign(\sigma)P_{\rm
II}(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x_{q+k}\wedge
Q_{\rm
II}(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},y_{k+q}),\mathfrak{X}_{k+q+1},\cdots,\mathfrak{X}_{p+q},x)$
$\displaystyle+\sum_{k=1}^{p}(-1)^{(k-1)q}\sum_{\sigma\in\mathbb{S}_{(k-1,q)}}sign(\sigma)P_{\rm
II}(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},Q_{\rm
II}(\mathfrak{X}_{\sigma(k)},\cdots,\mathfrak{X}_{\sigma(k+q-1)},x_{k+q})\wedge
y_{k+q},\mathfrak{X}_{k+q+1},\cdots,\mathfrak{X}_{p+q},x),$

In particular, for
$f,g\in\mathfrak{C}^{0}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\mathfrak{g},\mathfrak{g})$
and $P=(P_{\rm I},P_{\rm
II})\in\mathfrak{C}^{p}(\mathfrak{g},\mathfrak{g})~{}(p\geqslant 1)$, we
define

(19) $\displaystyle\Big{(}P\circ f\Big{)}_{\rm
I}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{p}P_{\rm
I}\Big{(}\mathfrak{X}_{1},\cdots,\mathfrak{X}_{k-1},x_{k}\wedge
f(y_{k}),\mathfrak{X}_{k+1},\cdots,\mathfrak{X}_{p}\Big{)}$
$\displaystyle+\sum_{k=1}^{p}P_{\rm
I}\Big{(}\mathfrak{X}_{1},\cdots,\mathfrak{X}_{k-1},f(x_{k})\wedge
y_{k},\mathfrak{X}_{k+1},\cdots,\mathfrak{X}_{p}\Big{)},$ (20)
$\displaystyle\Big{(}f\circ P\Big{)}_{\rm
I}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p})$ $\displaystyle=$ $\displaystyle
f\Big{(}P_{\rm I}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p})\Big{)},$ (21)
$\displaystyle\Big{(}P\circ f\Big{)}_{\rm
II}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p},x)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{p}P_{\rm
II}\Big{(}\mathfrak{X}_{1},\cdots,\mathfrak{X}_{k-1},x_{k}\wedge
f(y_{k}),\mathfrak{X}_{k+1},\cdots,\mathfrak{X}_{p},x\Big{)}$
$\displaystyle+\sum_{k=1}^{p}P_{\rm
II}\Big{(}\mathfrak{X}_{1},\cdots,\mathfrak{X}_{k-1},f(x_{k})\wedge
y_{k},\mathfrak{X}_{k+1},\cdots,\mathfrak{X}_{p},x\Big{)}$
$\displaystyle+P_{\rm
II}\Big{(}\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p},f(x)\Big{)},$ (22)
$\displaystyle\Big{(}f\circ P\Big{)}_{\rm
II}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p},x)$ $\displaystyle=$
$\displaystyle f\Big{(}P_{\rm
II}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p},x)\Big{)}.$

Moreover, $f\circ g$ means the composition of $f$ and $g$.

Let us introduce some notations. Let $\mathfrak{g}$ and $V$ be vector spaces.
Denote a degree $1$ element
$(\pi,\omega)\in\mathfrak{C}^{1}(\mathfrak{g},\mathfrak{g})$ in
$\mathfrak{C}^{\bullet}(\mathfrak{g},\mathfrak{g})$ by $\Pi$. Correspondingly,
for another element
$\Pi^{\prime}=(\pi^{\prime},\omega^{\prime})\in\mathfrak{C}^{1}(\mathfrak{g},\mathfrak{g})$,
the element $(\pi+\pi^{\prime},\omega+\omega^{\prime})$ is written as
$\Pi+\Pi^{\prime}$. Moreover, recall that

$\mathfrak{C}^{1}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus
V)=\mathrm{Hom}(\wedge^{2}(\mathfrak{g}\oplus V),\mathfrak{g}\oplus
V)\oplus\mathrm{Hom}(\wedge^{2}(\mathfrak{g}\oplus
V)\otimes(\mathfrak{g}\oplus V),\mathfrak{g}\oplus V).$

For linear maps $\rho:\mathfrak{g}\to\mathfrak{gl}(V)$ and
$\mu:\otimes^{2}\mathfrak{g}\to\mathfrak{gl}(V)$, define the degree $1$
elements $\overline{\Pi}=(\bar{\pi},\bar{\omega})$ and
$\overline{\Theta}=(\bar{\rho},\bar{\mu})\in\mathfrak{C}^{1}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)$ to be for all $x,y,z\in\mathfrak{g}$ and $u,v,w\in
V$,

$\displaystyle\begin{cases}\bar{\pi}(x+u,y+v)=\pi(x,y),\\\
\bar{\omega}(x+u,y+v,z+w)=\omega(x,y,z),\end{cases}$

and

$\displaystyle\begin{cases}\bar{\rho}(x+u,y+v)=\rho(x)v-\rho(y)u,\\\
\bar{\mu}(x+u,y+v,z+w)=D(x,y)w+\mu(y,z)u-\mu(x,z)v,\end{cases}$

respectively. In the following, we denote $\bar{\rho}$ and $\bar{\mu}$ by
$\rho$ and $\mu$ respectively, then the degree $1$ element
$\overline{\Pi}+\overline{\Theta}=(\bar{\pi}+\bar{\rho},\bar{\omega}+\bar{\mu})\in\mathfrak{C}^{1}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)$ is written as $\Pi+\Theta=(\pi+\rho,\omega+\mu)$
correspondingly. In the sequel, we also denote a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot\right\rrbracket)$ by
$(\mathfrak{g},\pi,\omega)$, where $\pi=[\cdot,\cdot]$ and
$\omega=\left\llbracket\cdot,\cdot,\cdot\right\rrbracket$.

######  Theorem 3.1.

With the above notations, the graded vector space
$\mathfrak{C}^{\bullet}(\mathfrak{g},\mathfrak{g})$ equipped with the graded
commutator

$[P,Q]_{\mathsf{LieY}}=P\circ Q-(-1)^{pq}Q\circ P$

is a graded Lie algebra, where the operation $\circ$ is defined as (17)-(22).
Moreover, if $\Pi\in\mathfrak{C}^{1}(\mathfrak{g},\mathfrak{g})$ defines a
Lie-Yamaguti algebra structure on $\mathfrak{g}$, then $\Pi$ is a Maurer-
Cartan element of the graded Lie algebra
$(C^{\bullet}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]_{\mathsf{LieY}})$.

###### Proof.

We first prove the graded bracket is skew-symmetric. Indeed, for all
$P\in\mathfrak{C}^{p}(\mathfrak{g},\mathfrak{g}),~{}Q\in\mathfrak{C}^{q}(\mathfrak{g},\mathfrak{g})~{}(p,q\geqslant
0)$, we write

$[P,Q]_{\mathsf{LieY}}=([P,Q]_{\rm I},[P,Q]_{\rm II})$

where

$\displaystyle~{}[P,Q]_{\rm I}$ $\displaystyle=$ $\displaystyle\Big{(}P\circ
Q\Big{)}_{\rm I}-(-1)^{pq}\Big{(}Q\circ P\Big{)}_{\rm I}$
$\displaystyle~{}[P,Q]_{\rm II}$ $\displaystyle=$ $\displaystyle\Big{(}P\circ
Q\Big{)}_{\rm II}-(-1)^{pq}\Big{(}Q\circ P\Big{)}_{\rm II}.$

We have

$\displaystyle~{}[P,Q]_{\mathsf{LieY}}$ $\displaystyle=$
$\displaystyle([P,Q]_{\rm I},[P,Q]_{\rm II})$ $\displaystyle=$
$\displaystyle(-(-1)^{pq}[Q,P]_{\rm I},-(-1)^{pq}[Q,P]_{\rm II}$
$\displaystyle=$ $\displaystyle-(-1)^{pq}\Big{(}[Q,P]_{\rm I},[Q,P]_{\rm
II}\Big{)}$ $\displaystyle=$ $\displaystyle-(-1)^{pq}[Q,P]_{\mathsf{LieY}}.$

Next we prove that the bracket $[\cdot,\cdot]_{\mathsf{LieY}}$ satisfies the
graded Jacobi Identity, which means that both $[\cdot,\cdot]_{\rm I}$ and
$[\cdot,\cdot]_{\rm{II}}$ satisfy the graded Jacobi Identity. In fact, that
the bracket $[\cdot,\cdot]_{\rm{II}}$ satisfies the graded Jacobi Identity has
been proved in [33] using the weight rule. For any element
$P\in\mathfrak{C}^{p}(\mathfrak{g},\mathfrak{g})~{}(p\geqslant 1)$, fix a
nonzero element $x\in\mathfrak{g}$, and define a map

$\Phi:\mathrm{Hom}(\wedge^{2}\mathfrak{g}^{\otimes
p}\otimes\mathfrak{g},\mathfrak{g})\to\mathrm{Hom}(\wedge^{2}\mathfrak{g}^{\otimes
p},\mathfrak{g})$

to be

$\Phi(P)(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p})=P(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{p},x),\quad\forall
P\in\mathrm{Hom}(\wedge^{2}\mathfrak{g}^{\otimes
p}\otimes\mathfrak{g},\mathfrak{g}),$

where $\mathfrak{X}_{k}\in\wedge^{2}\mathfrak{g}~{}(k=1,2,\cdots,p)$. Then
since the graded vector space $\oplus_{p\geqslant
0}\mathrm{Hom}(\wedge^{2}\mathfrak{g}^{\otimes
p}\otimes\mathfrak{g},\mathfrak{g})$ endowed with $[\cdot,\cdot]_{\rm II}$ is
a graded Lie algebra, the graded vector space $\oplus_{p\geqslant
0}\mathrm{Hom}(\wedge^{2}\mathfrak{g}^{\otimes p},\mathfrak{g})$ is also a
graded Lie algebra via the map $\Phi$, whose graded Lie bracket is exactly the
bracket $[\cdot,\cdot]_{\rm I}$ for two variables whose degrees $\geqslant 1$.
Furthermore, by a direct computation, if either two variables is of degree
$0$, then the bracket $[\cdot,\dot{]}_{\rm I}$ is also a graded Lie bracket.
Thus the bracket $[\cdot,\cdot]_{\mathsf{LieY}}$ satisfies the graded Jacobi
Identity.

Finally, for all $x,y,z,w,t\in\mathfrak{g}$, we have

$\displaystyle[\Pi,\Pi]_{\rm I}(x,y,z,w)$ $\displaystyle=$ $\displaystyle
2\Big{(}\big{(}\Pi\circ\Pi\big{)}_{\rm I}(x,y,z,w)\Big{)}$ $\displaystyle=$
$\displaystyle
2\Big{(}\omega(x,y,\pi(z,w))-\pi(\omega(x,y,z),w)-\pi(z,\omega(x,y,w))\Big{)},$

and

$\displaystyle[\Pi,\Pi]_{\rm II}(x,y,z,w,t)$ $\displaystyle=$ $\displaystyle
2\Big{(}\big{(}\Pi\circ\Pi\big{)}_{\rm II}(x,y,z,w)\Big{)}$ $\displaystyle=$
$\displaystyle
2\Big{(}\omega(x,y,\omega(z,w,t))-\omega(z,w,\omega(x,y,t))-\omega(\omega(x,y,z),w,t)-\omega(z,\omega(x,y,w),t)\Big{)},$

which proves the last statement. This completes the proof. ∎

######  Remark 3.2.

We have to point out that any Maurer-Cartan element
$\Pi=(\pi,\omega)\in\mathfrak{C}^{1}(\mathfrak{g},\mathfrak{g})$ in the graded
Lie algebra $\mathfrak{C}^{\bullet}(\mathfrak{g},\mathfrak{g})$ does not
precisely correspond to a Lie-Yamaguti algebra structure on $\mathfrak{g}$. By
the proof of Theorem 3.1, we know that $\Pi$ is a Maurer-Cartan element if and
only if $\Pi$ satisfies Eqs. (3) and (4), i.e., for all
$x,y,z,w,t\in\mathfrak{g}$, $\Pi$ satisfies

$\displaystyle\omega(x,y,\pi(z,w))$ $\displaystyle=$
$\displaystyle\pi(\omega(x,y,z),w)+\pi(z,\omega(x,y,w)),$
$\displaystyle~{}\omega(x,y,\omega(z,w,t))$ $\displaystyle=$
$\displaystyle\omega(\omega(x,y,z),w,t)+\omega(z,\omega(x,y,w),t)+\omega(z,w,\omega(x,y,t)).$

This reveals to us that case of Lie-Yamaguti algebras is more complicated than
that of Lie algebras, Leibniz algebras or even $3$-Lie algebras. See [16] for
more details about Maurer-Cartan characterizations of other algebraic
structures.

Let $\Pi$ define a Lie-Yamaguti algebra structure on the vector space
$\mathfrak{g}$. It follows from the graded Jacobi identity that
$d_{\Pi}:=[\Pi,\cdot]_{\rm LieY}$ is a differential on
$(\mathfrak{C}^{\bullet}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]_{\rm
LieY})$. More precisely, we have

######  Theorem 3.3.

Let $(\mathfrak{g},\pi,\omega)$ be a Lie-Yamaguti algebra. Then with the above
notations, the triple
$(\mathfrak{C}^{\bullet}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]_{\rm
LieY},d_{\Pi})$ is a differential graded Lie algebra, where $d_{\Pi}$ is
defined to be

$d_{\Pi}:=[\Pi,\cdot]_{\rm LieY}.$

If moreover, for an element
$\Pi^{\prime}\in\mathfrak{C}^{1}(\mathfrak{g},\mathfrak{g})$,
$\Pi+\Pi^{\prime}$ defines a Lie-Yamaguti algebra structure on $\mathfrak{g}$,
then $\Pi^{\prime}$ is a Maurer-Cartan element of the differential graded Lie
algebra $(\mathfrak{C}^{\bullet}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]_{\rm
LieY},d_{\Pi})$.

From Proposition 2.6 and Theorem 3.1, representations of Lie-Yamaguti algebras
can be characterized as Maurer-Cartan elements in a graded Lie algebra. More
precisely, we have

######  Proposition 3.4.

With the above notations, if $(V;\rho,\mu)$ is a representation of Lie-
Yamaguti algebra $(\mathfrak{g},\pi,\omega)$, then
$\overline{\Theta}\in\mathfrak{C}^{1}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus
V)$ is a Maurer-Cartan element of the differential graded Lie algebra
$(\mathfrak{C}^{\bullet}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus
V),[\cdot,\cdot]_{\rm LieY},d_{\overline{\Pi}})$.

At the end of this section, we give the relationship between the coboundary
operator of a Lie-Yamaguti algebra associated to the adjoint representation
and the differential $d_{\Pi}$ induced by the Maurer-Cartan element.

######  Theorem 3.5.

Let $(\mathfrak{g},\pi,\omega)$ be a Lie-Yamaguti algebra and
$\delta:\mathfrak{C}^{n}_{\mathsf{LieY}}(\mathfrak{g},\mathfrak{g})\to\mathfrak{C}^{n+1}_{\mathsf{LieY}}(\mathfrak{g},\mathfrak{g})$
be the coboundary map associated to the adjoint representation. Then we have

(23) $\displaystyle\delta(F)$ $\displaystyle=$
$\displaystyle(-1)^{n}d_{\Pi}(F)=(-1)^{n}[\Pi,F]_{\rm LieY},\quad\forall
F\in\mathfrak{C}^{n}(\mathfrak{g},\mathfrak{g})~{}(n\geqslant 1),$ (24)
$\displaystyle\delta(f)$ $\displaystyle=$ $\displaystyle
d_{\Pi}(f)=[\Pi,f]_{\rm LieY},\quad\forall
f\in\mathfrak{C}^{0}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\mathfrak{g},\mathfrak{g}).$

###### Proof.

For all $F=(f,g)\in\mathfrak{C}^{n}(\mathfrak{g},\mathfrak{g})~{}(n\geqslant
1)$ and for all
$\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1}\in\wedge^{2}\mathfrak{g},~{}x\in\mathfrak{g}$,
we compute that

$\displaystyle[\Pi,F]_{\rm I}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1})$
$\displaystyle=$ $\displaystyle\Big{(}\big{(}\Pi\circ F\big{)}_{\rm
I}-(-1)^{n}\big{(}F\circ\Pi\big{)}_{\rm
I}\Big{)}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1})$
$\displaystyle\stackrel{{\scriptstyle\eqref{gradedbra1}}}{{=}}$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(1,n)}\atop\sigma(n+1)=n+1}(-1)^{n}sign(\sigma)\left\llbracket\mathfrak{X}_{\sigma(1)},f(\mathfrak{X}_{\sigma(2)},\cdots,\mathfrak{X}_{\sigma(n+1)})\right\rrbracket$
$\displaystyle+[x_{n+1},g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},y_{n+1})]-[y_{n+1},g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},x_{n+1})]$
$\displaystyle-(-1)^{n}\Big{(}(-1)^{n}g(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(n)},[x_{n+1},y_{n+1}])$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)f(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x_{k+1}\wedge\left\llbracket\mathfrak{X}_{\sigma(k)},y_{k+1}\right\rrbracket,\mathfrak{X}_{k+2},\cdots,\mathfrak{X}_{n+1})$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)f(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},\left\llbracket\mathfrak{X}_{\sigma(k)},x_{k+1}\right\rrbracket\wedge
y_{k+1},\mathfrak{X}_{k+2},\cdots,\mathfrak{X}_{n+1})\Big{)}$ $\displaystyle=$
$\displaystyle(-1)^{n}\Big{(}(-1)^{n}\big{(}[x_{n+1},g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},y_{n+1})]-[y_{n+1},g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},x_{n+1})]$
$\displaystyle-g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},[x_{n+1},y_{n+1}])\big{)}$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k+1}\left\llbracket\mathfrak{X}_{k},f(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{n+1})\right\rrbracket$
$\displaystyle+\sum_{1\leqslant k<l\leqslant
n+1}(-1)^{k}f(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{k}\circ\mathfrak{X}_{l},\cdots,\mathfrak{X}_{n+1})\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{cohomolo1}}}{{=}}$
$\displaystyle(-1)^{n}\delta_{\rm
I}(F)(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1}),$

and

$\displaystyle[\Pi,F]_{\rm II}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1},x)$
$\displaystyle=$ $\displaystyle\Big{(}\big{(}\Pi\circ F\big{)}_{\rm
II}-(-1)^{n}\big{(}F\circ\Pi\big{)}_{\rm
II}\Big{)}(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1},x)$
$\displaystyle\stackrel{{\scriptstyle\eqref{gradedbra3}}}{{=}}$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(1,n)}}(-1)^{n}sign(\sigma)\left\llbracket\mathfrak{X}_{\sigma(1)},g(\mathfrak{X}_{\sigma(2)},\cdots,\mathfrak{X}_{\sigma(n+1)},x)\right\rrbracket$
$\displaystyle+\left\llbracket
g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},x_{n+1}),y_{n+1},x\right\rrbracket+\left\llbracket
x_{n+1},g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},y_{n+1}),x\right\rrbracket$
$\displaystyle-(-1)^{n}\Big{(}\sum_{\sigma\in\mathbb{S}_{(n,1)}}(-1)^{n}sign(\sigma)g(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(n)},\left\llbracket\mathfrak{X}_{\sigma(n+1)},x\right\rrbracket)$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)g(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},\left\llbracket\mathfrak{X}_{\sigma(k)},x_{k+1}\right\rrbracket\wedge
y_{k+1},\mathfrak{X}_{k+2},\cdots,\mathfrak{X}_{n+1},x)$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)g(\mathfrak{X}_{\sigma(1)},\cdots,\mathfrak{X}_{\sigma(k-1)},x_{k+1}\wedge\left\llbracket\mathfrak{X}_{\sigma{(k)}},y_{k+1}\right\rrbracket,\mathfrak{X}_{k+2},\cdots,\mathfrak{X}_{n+1},x)\Big{)}$
$\displaystyle=$ $\displaystyle(-1)^{n}\Big{(}(-1)^{n}\big{(}\left\llbracket
g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},x_{n+1}),y_{n+1},x\right\rrbracket-\left\llbracket
g(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n},y_{n+1}),x_{n+1},x\right\rrbracket\big{)}$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k+1}\left\llbracket
x_{k},y_{k},g(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{n+1},x)\right\rrbracket$
$\displaystyle+\sum_{1\leqslant k<l\leqslant
n+1}(-1)^{k}g(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{k}\circ\mathfrak{X}_{l},\cdots,\mathfrak{X}_{n+1},x)$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k}g(\mathfrak{X}_{1},\cdots,\hat{\mathfrak{X}_{k}},\cdots,\mathfrak{X}_{n+1},\left\llbracket
x_{k},y_{k},x\right\rrbracket)\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{cohomolo2}}}{{=}}$
$\displaystyle(-1)^{n}\delta_{\rm
II}(F)(\mathfrak{X}_{1},\cdots,\mathfrak{X}_{n+1},x).$

Thus, we have

$[\Pi,F]_{\rm I}=(-1)^{n}\delta_{\rm I}(F)\quad\text{and}\quad[\Pi,F]_{\rm
II}=(-1)^{n}\delta_{\rm II}(F),\quad\forall
F\in\mathfrak{C}^{n}(\mathfrak{g},\mathfrak{g})~{}(n\geqslant 1),$

which proves (23). Similarly, for any
$f\in\mathrm{Hom}(\mathfrak{g},\mathfrak{g})$ and for all
$x,y,z\in\mathfrak{g}$, we have

$\displaystyle[\Pi,f]_{\rm I}(x,y)$ $\displaystyle=$
$\displaystyle\Big{(}\Pi\circ f\Big{)}_{\rm
I}(x,y)-\Big{(}f\circ\Pi\Big{)}_{\rm I}(x,y)$
$\displaystyle\stackrel{{\scriptstyle\eqref{gradedbra4},\eqref{gradedbra5}}}{{=}}$
$\displaystyle[x,f(y)]+[f(x),y]-f([x,y])$
$\displaystyle\stackrel{{\scriptstyle\eqref{1cochain}}}{{=}}$
$\displaystyle\delta_{\rm I}(f)(x,y),$

and

$\displaystyle[\Pi,f]_{\rm II}(x,y,z)$
$\displaystyle\stackrel{{\scriptstyle\eqref{gradedbra6},\eqref{gradedbra2}}}{{=}}$
$\displaystyle\left\llbracket f(x),y,z\right\rrbracket+\left\llbracket
x,f(y),z\right\rrbracket+\left\llbracket
x,y,f(z)\right\rrbracket-f(\left\llbracket x,y,z\right\rrbracket)$
$\displaystyle\stackrel{{\scriptstyle\eqref{2cochain}}}{{=}}$
$\displaystyle\delta_{\rm II}(f)(x,y,z).$

Thus we have

$[\Pi,f]_{\rm I}=\delta_{\rm I}(f)\quad\text{and}\quad[\Pi,f]_{\rm
II}=\delta_{\rm II}(f),\quad\forall
f\in\mathfrak{C}^{0}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\mathfrak{g},\mathfrak{g}),$

which proves (24). The conclusion thus follows. ∎

##  4\. Maurer-Cartan characterizations and Cohomology of relative Rota-Baxter
operators

In this section, we determine an $L_{\infty}$-algebra as controlling algebra
for relative Rota-Baxter operators and then establish their cohomology theory.
Finally, we reveal the relation between cohomology and differential in the
controlling algebra.

###  4.1. Maurer-Cartan characterizations for relative Rota-Baxter operators

In this subsection, we use the Voronov’s derived bracket to construct an
$L_{\infty}$-algebra, which characterizes the relative Rota-Baxter operators
on a Lie-Yamaguti algebra as Maurer-Cartan elements. At the beginning, we
recall some notions and conclusions. The notion of an $L_{\infty}$-algebra was
introduced by Stasheff in [37]. See also [24, 25] for more details.

######  Definition 4.1.

An $L_{\infty}$-algebra is a $\mathbb{Z}$-graded vector space
$\mathfrak{g}=\oplus_{k\in\mathbb{Z}}\mathfrak{g}_{k}$ equipped with a
collection of multilinear maps
$l_{k}:\otimes^{k}\mathfrak{g}\to\mathfrak{g}~{}(k\geqslant 1)$ of degree $1$
with the property that, for all homogeneous elements
$x_{1},\cdots,x_{n}\in\mathfrak{g}$, we have

  * (i)

for all $\sigma\in S_{n}$,

$l_{n}(x_{\sigma(1)},\cdots,x_{\sigma(n)})=\varepsilon(\sigma)l_{n}(x_{1},\cdots,x_{n}).$

  * (ii)

for all $n\geqslant 1$,

$\sum_{i=1}^{n}\sum_{\sigma\in\mathbb{S}_{(i,n-1)}}\varepsilon(\sigma)l_{n-i+1}(l_{i}(x_{\sigma(1)},\cdots,x_{\sigma(i)}),x_{\sigma(i+1)},\cdots,x_{\sigma(n)})=0,$

where $\varepsilon(\sigma)$ means the Koszul sign:
$l_{n}(x_{1},\cdots,x_{i},x_{i+1},\cdots,x_{n})=(-1)^{|x_{i}||x_{i+1}|}l_{n}(x_{1},\cdots,x_{i+1},x_{i},\cdots,x_{n})$.
In the sequel, we denote an $L_{\infty}$-algebra by
$(\mathfrak{g},\\{l_{k}\\}_{k=1}^{\infty})$.

######  Definition 4.2.

Let $(\mathfrak{g},\\{l_{k}\\}_{k=1}^{\infty})$ be an $L_{\infty}$-algebra.

  * (i)

A degree $0$ element $\alpha\in\mathfrak{g}_{0}$ is called a Maurer-Cartan
element of the $L_{\infty}$-algebra
$(\mathfrak{g},\\{l_{k}\\}_{k=1}^{\infty})$ if $\alpha$ satisfies the
following Maurer-Cartan equation:

$\displaystyle\sum_{k=1}^{\infty}\frac{1}{k!}l_{k}(\underbrace{\alpha,\cdots,\alpha}_{k})=0.$

  * (ii)

A Maurer-Cartan element $\alpha\in\mathfrak{g}_{0}$ of
$(\mathfrak{g},\\{l_{k}\\}_{k=1}^{\infty})$ is called strict if

$l_{k}(\alpha,\cdots,\alpha)=0,\quad k\geqslant 1.$

Let us recall the method by which we use the Voronov’s derived brackets to
construct an $L_{\infty}$-algebra.

######  Definition 4.3.

([42]) A $V$-data is a quadruple $(L,\mathfrak{h},P,\Delta)$ consisting of the
following items:

  * •

$(L,[\cdot,\cdot])$ is a graded Lie algebra;

  * •

$\mathfrak{h}$ is an abelian graded Lie subalgebra of $(L,[\cdot,\cdot])$;

  * •

$P:L\to L$ is a projection, i.e. $P\circ P=P$, whose image is $\mathfrak{h}$
and its kernel is a graded Lie subalgebra of $(L,[\cdot,\cdot])$;

  * •

$\Delta$ is an element in $\ker(P)^{1}$ such that $[\Delta,\Delta]=0$.

######  Theorem 4.4.

([42]) Let $(L,\mathfrak{h},P,\Delta)$ be a $V$-data. Define multilinear maps
$l_{k}:\otimes^{k}\mathfrak{h}\to\mathfrak{h}$ to be

(25) $\displaystyle
l_{k}(x_{1},\cdots,x_{k})=P[\cdots[[\Delta,x_{1}],x_{2}],\cdots,x_{k}],$

for all homogeneous elements $x_{1},\cdots,x_{k}\in\mathfrak{h}$. Then
$(\mathfrak{h},\\{l_{k}\\}_{k=1}^{\infty})$ is an $L_{\infty}$-algebra, where
$\\{l_{k}\\}_{k=1}^{\infty}$ is called the higher derived brackets of the
$V$-data $(L,\mathfrak{h},P,\Delta)$.

######  Definition 4.5.

[36] Let
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
be a Lie-Yamaguti algebra and $(V;\rho,\mu)$ a representation of
$\mathfrak{g}$. A linear map $T:V\to\mathfrak{g}$ is called a relative Rota-
Baxter operator on $\mathfrak{g}$ with respect to $(V;\rho,\mu)$ if $T$
satisfies

(26) $\displaystyle~{}[Tu,Tv]$ $\displaystyle=$ $\displaystyle
T\Big{(}\rho(Tu)v-\rho(Tv)u\Big{)},$ (27) $\displaystyle~{}\left\llbracket
Tu,Tv,Tw\right\rrbracket$ $\displaystyle=$ $\displaystyle
T\Big{(}D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big{)},\quad\forall
u,v,w\in V.$

In the sequel, let $\mathfrak{g}$ and $V$ be vector spaces, and we denote the
elements in $\mathfrak{g}$ by $x,y,z$, and we denote the elements in $V$ by
$u,v,w$. Consider the graded vector subspace
$C^{*}(V,\mathfrak{g})=\oplus_{p\geqslant 0}C^{p}(V,\mathfrak{g})$ of
$\mathfrak{C}^{\bullet}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus V)$, where

$\displaystyle
C^{p}(V,\mathfrak{g})=\begin{cases}\mathrm{Hom}(\underbrace{\wedge^{2}V\otimes\cdots\otimes\wedge^{2}V}_{p},\mathfrak{g})\oplus\mathrm{Hom}(\underbrace{\wedge^{2}V\otimes\cdots\otimes\wedge^{2}V}_{p}\otimes
V,\mathfrak{g}),&p\geqslant 1,\\\
\mathrm{Hom}(V,\mathfrak{g}),&p=0.\end{cases}$

Recall that
$\Pi+\Theta=(\pi+\rho,\omega+\mu)\in\mathfrak{C}^{1}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)$. For all $P\in C^{p}(V,\mathfrak{g}),~{}Q\in
C^{q}(V,\mathfrak{g})$, and $R\in C^{r}(V,\mathfrak{g})~{}(p,q,r\geqslant 0)$,
we define a bilinear operation

$l_{2}:C^{p}(V,\mathfrak{g})\times C^{q}(V,\mathfrak{g})\to
C^{p+q+1}(V,\mathfrak{g})$

and a trilinear operation

$l_{3}:C^{p}(V,\mathfrak{g})\times C^{q}(V,\mathfrak{g})\times
C^{r}(V,\mathfrak{g})\to C^{p+q+r+1}(V,\mathfrak{g})$

to be

(28) $\displaystyle l_{2}(P,Q)$ $\displaystyle=$
$\displaystyle[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}},$

and

(29) $\displaystyle l_{3}(P,Q,R)$ $\displaystyle=$
$\displaystyle[[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}},R]_{\mathsf{LieY}},$

respectively. Note that the standard symbols for $l_{2}$ and $l_{3}$ are
$\Big{(}(l_{2})_{\rm I},(l_{2})_{\rm II}\Big{)}$ and $\Big{(}(l_{3})_{\rm
I},(l_{3})_{\rm II}\Big{)}$ respectively, if we emphasize their components.

Now we give the main theorem in this section which gives the Maurer-Cartan
characterization for relative Rota-Baxter operators on Lie-Yamaguti algebras.

######  Theorem 4.6.

With the above notations, $(C^{*}(V,\mathfrak{g}),l_{2},l_{3})$ is an
$L_{\infty}$-algebra. Moreover, its strict Maurer-Cartan elements are
precisely the relative Rota-Baxter operators on Lie-Yamaguti algebra
$(\mathfrak{g},\pi,\omega)$ with respect to the representation $(V;\rho,\mu)$.

###### Proof.

Let $(V;\rho,\mu)$ be a representation of the Lie-Yamaguti algebra
$(\mathfrak{g},\pi,\omega)$. Then we have

  * •

a graded Lie algebra $(\mathfrak{C}^{\bullet}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V),[\cdot,\cdot]_{\mathsf{LieY}})$;

  * •

an abelian subalgebra $C^{*}(V,\mathfrak{g})$;

  * •

$\mathrm{pr}_{\mathfrak{g}}$ is a projection onto $C^{*}(V,\mathfrak{g})$;

  * •

$\Delta=\Pi+\Theta$.

These items listed above forms a $V$-data. By Theorem 4.4,
$(C^{*}(V,\mathfrak{g}),\\{l_{k}\\}_{k=1}^{\infty})$ is an
$L_{\infty}$-algebra, where multilinear maps ${l_{k}}^{\prime}s$ are given by
(25). But we note that

$\displaystyle~{}\mathrm{pr}_{\mathfrak{g}}[\Pi+\Theta,P]_{\mathsf{LieY}}=0,$
$\displaystyle~{}\mathrm{pr}_{\mathfrak{g}}[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}}\in
C^{*}(V,\mathfrak{g}),$
$\displaystyle~{}\mathrm{pr}_{\mathfrak{g}}[[[\Pi+\Theta,P]_{\mathsf{LieY}},Q]_{\mathsf{LieY}},R]_{\mathsf{LieY}}\in
C^{*}(V,\mathfrak{g}),$

for all $P\in C^{p}(V,\mathfrak{g}),~{}Q\in C^{q}(V,\mathfrak{g})$ and $R\in
C^{r}(V,\mathfrak{g})$. Thus, the nonzero multilinear maps of
$L_{\infty}$-algebra $(C^{*}(V,\mathfrak{g}),\\{l_{k}\\}_{k=1}^{\infty})$ are
only $l_{2}$ and $l_{3}$.

For a degree $0$ element $T\in\mathrm{Hom}(V,\mathfrak{g})$, by (28) and (29),
we have for all $u,v,w\in V$,

$\displaystyle(l_{2})_{\rm I}(T,T)(u,v)$ $\displaystyle=$
$\displaystyle[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm I}(u,v)$
$\displaystyle=$ $\displaystyle\Big{(}[\Pi+\Theta,T]_{\mathsf{LieY}}\circ
T\Big{)}_{\rm I}(u,v)-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
I}(u,v)$ $\displaystyle=$ $\displaystyle[\Pi+\Theta,T]_{\rm
I}(Tu,v)+[\Pi+\Theta,T]_{\rm I}(u,Tv)$ $\displaystyle=$
$\displaystyle(\Pi+\Theta)(Tu,Tv)-\Big{(}T\circ(\Pi+\Theta)\Big{)}_{\rm
I}(Tu,v)$
$\displaystyle+(\Pi+\Theta)(Tu,Tv)-\Big{(}T\circ(\Pi+\Theta)\Big{)}_{\rm
I}(u,Tv)$ $\displaystyle=$ $\displaystyle
2\Big{(}[Tu,Tv]-T\big{(}\rho(Tu)v-\rho(Tv)u\big{)}\Big{)},$

and

$\displaystyle(l_{3})_{\rm II}(T,T,T)(u,v,w)$ $\displaystyle=$
$\displaystyle[[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}},T]_{\rm
II}(u,v,w)$ $\displaystyle=$
$\displaystyle[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm
II}(Tu,v,w)+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(u,Tv,w)$
$\displaystyle+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm
II}(u,v,Tw)-\Big{(}T\circ[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}}\Big{)}_{\rm
II}(u,v,w)$ $\displaystyle=$ $\displaystyle[\Pi+\Theta,T]_{\rm
II}(Tu,Tv,w)+[\Pi+\Theta,T]_{\rm II}(Tu,v,Tw)$
$\displaystyle-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
II}(Tu,v,w)+[\Pi+\Theta,T]_{\rm II}(Tu,Tv,w)$
$\displaystyle+[\Pi+\Theta,T]_{\rm
II}(u,Tv,Tw)-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
II}(u,Tv,w)$ $\displaystyle+[\Pi+\Theta,T]_{\rm
II}(Tu,v,Tw)+[\Pi+\Theta,T]_{\rm II}(u,Tv,Tw)$
$\displaystyle-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
II}(u,v,Tw)-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
II}(Tu,v,w)$
$\displaystyle-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
II}(u,Tv,w)-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
II}(u,v,Tw)$ $\displaystyle=$ $\displaystyle 6\Big{(}\left\llbracket
Tu,Tv,Tw\right\rrbracket-T\big{(}D(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\big{)}\Big{)}.$

To sum up, we compute the nonzero multilinear maps $l_{2}$ and $l_{3}$ as
follows:

$\displaystyle\begin{cases}(l_{2})_{\rm
I}(T,T)(u,v)=2\Big{(}[Tu,Tv]-T\big{(}\rho(Tu)v-\rho(Tv)u\big{)}\Big{)},\\\
(l_{2})_{\rm II}(T,T)(u,v,w)=0,\\\ (l_{3})_{\rm I}(T,T,T)(u,v)=0,\\\
(l_{3})_{\rm II}(T,T,T)(u,v,w)=6\Big{(}\left\llbracket
Tu,Tv,Tw\right\rrbracket-T\big{(}D(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\big{)}\Big{)}.\end{cases}$

Thus we have that a linear map $T:V\to\mathfrak{g}$ is a relative Rota-Bxter
operator on the Lie-Yamaguti algebra $(\mathfrak{g},\pi,\omega)$ with respect
to the representation $(V;\rho,\mu)$ if and only if $T$ is a strict Maurer-
Cartan element of the $L_{\infty}$-algebra
$(C^{*}(V,\mathfrak{g}),l_{2},l_{3})$. This completes the proof. ∎

######  Remark 4.7.

From Theorem 4.6, we notice that for a relative Rota-Baxter operator on a Lie-
Yamaguti algebra, on the one hand, its realization of Maurer-Cartan
characterization is an $L_{\infty}$-algebra owning two nonzero multilinear
maps: $l_{2}$ and $l_{3}$. In [39, 40], authors showed the realizations of
Maurer-Cartan characterization for relative Rota-Baxter operators on Lie
algebras and $3$-Lie algebras are graded Lie algebras and Lie $3$-algebras
respectively, both of which can be seen as a special $L_{\infty}$-algebra
(whose nonzero multilinear maps are $l_{2}$ and $l_{3}$ respectively).
Whichever is a graded Lie algebra or a Lie $3$-algebra, it (treated as
$L_{\infty}$-algebra) owns one nonzero bracket. On the other hand, a relative
Rota-Baxter operator on a Lie-Yamaguti algebra corresponds to a strict Maurer-
Cartan element in its $L_{\infty}$-algebra, not a Maurer-Cartan element. The
above two points distinguish from the case of relative Rota-Baxter operators
on Lie algebras and on $3$-Lie algebras [39, 40, 41], and from proof of
Theorem 4.6, it reveals the fact that the case of Lie-Yamaguti algebras is
more complicated than other algebras.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},\pi,\omega)$ with respect to a representation $(V;\rho,\mu)$.
Since $T\in C^{0}(V,\mathfrak{g})=\mathrm{Hom}(V,\mathfrak{g})$ is a (strict)
Maurer-Cartan element of the $L_{\infty}$-algebra
$(C^{*}(V,\mathfrak{g}),l_{2},l_{3})$, by [15], we have a twisted
$L_{\infty}$-algebra $(C^{*}(V,\mathfrak{g}),l_{1}^{T},l_{2}^{T},l_{3}^{T})$
as follows:

(30) $\displaystyle~{}l_{1}^{T}(P)$ $\displaystyle=$ $\displaystyle
l_{2}(T,P)+\frac{1}{2}l_{3}(T,T,P),$ (31) $\displaystyle~{}l_{2}^{T}(P,Q)$
$\displaystyle=$ $\displaystyle l_{2}(P,Q)+l_{3}(T,P,Q),$ (32)
$\displaystyle~{}l_{3}^{T}(P,Q,R)$ $\displaystyle=$ $\displaystyle
l_{3}(P,Q,R),$ (33) $\displaystyle~{}l_{k}^{T}$ $\displaystyle=$
$\displaystyle 0,\quad k\geqslant 4,$

for all $P\in C^{p}(V,\mathfrak{g}),~{}Q\in C^{q}(V,\mathfrak{g})$ and $R\in
C^{r}(V,\mathfrak{g})$.

######  Theorem 4.8.

Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. Then for a linear map
$T^{\prime}:V\to\mathfrak{g}$, $T+T^{\prime}$ is still a relative Rota-Baxter
operator on $\mathfrak{g}$ if and only if $T^{\prime}$ is a Maurer-Cartan
element of the twisted $L_{\infty}$-algebra
$(C^{*}(V,\mathfrak{g}),l_{1}^{T},l_{2}^{T},l_{3}^{T})$, i.e., $T^{\prime}$
satisfies the following Maurer-Cartan equation:

$l_{1}^{T}(T^{\prime})+\frac{1}{2}l_{2}^{T}(T^{\prime},T^{\prime})+\frac{1}{3!}l_{3}^{T}(T^{\prime},T^{\prime},T^{\prime})=0.$

###### Proof.

By Theorem 4.6, $T+T^{\prime}$ is a relative Rota-Baxter operator if and only
if it is a strict Maurer-Cartan element, i.e.,

(34) $\displaystyle\begin{cases}l_{2}(T+T^{\prime},T+T^{\prime})=0,\\\
l_{3}(T+T^{\prime},T+T^{\prime},T+T^{\prime})=0.\end{cases}$

Since $T$ is a relative Rota-Baxter operator on
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$,
then we have

$\displaystyle\begin{cases}l_{2}(T,T)=0,\\\ l_{3}(T,T,T)=0.\end{cases}$

Since the degrees of both $T$ and $T^{\prime}$ are $0$, expanding Eqs.(34)
yields that

(35)
$\displaystyle\begin{cases}2l_{2}(T,T^{\prime})+l_{2}(T^{\prime},T^{\prime})=0,\\\
3l_{3}(T,T,T^{\prime})+3l_{3}(T,T^{\prime},T^{\prime})+l_{3}(T^{\prime},T^{\prime},T^{\prime})=0.\end{cases}$

Then we obtain that

$\displaystyle
l_{1}^{T}(T^{\prime})+\frac{1}{2}l_{2}^{T}(T^{\prime},T^{\prime})+\frac{1}{3!}l_{3}^{T}(T^{\prime},T^{\prime},T^{\prime})$
$\displaystyle\stackrel{{\scriptstyle\eqref{diffs},\eqref{bracket},\eqref{homotopy}}}{{=}}$
$\displaystyle
l_{2}(T,T^{\prime})+\frac{1}{2}l_{3}(T,T,T^{\prime})+\frac{1}{2}\Big{(}l_{2}(T^{\prime},T^{\prime})+l_{3}(T,T^{\prime},T^{\prime})\Big{)}+\frac{1}{3!}l_{3}(T^{\prime},T^{\prime},T^{\prime})$
$\displaystyle=$
$\displaystyle\Big{(}l_{2}(T,T^{\prime})+\frac{1}{2}l_{2}(T^{\prime},T^{\prime})\Big{)}+\frac{1}{2}\Big{(}l_{3}(T,T,T^{\prime})+l_{3}(T,T^{\prime},T^{\prime})+\frac{1}{3}l_{3}(T^{\prime},T^{\prime},T^{\prime})\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{MC}}}{{=}}$ $\displaystyle 0,$

which implies that $T^{\prime}$ is a Maurer-Cartan element of the twisted
$L_{\infty}$-algebra $(C^{*}(V,\mathfrak{g}),l_{1}^{T},l_{2}^{T},l_{3}^{T})$.
This completes the proof. ∎

At the end of this subsection, we give two examples of Rota-Baxter operators
on Lie-Yamaguti algebras.

######  Example 4.9.

Let $\mathfrak{g}=C[0,1]$ endowed with the following operations

$\displaystyle[f,g](x)$ $\displaystyle=$ $\displaystyle f(x)g(x)-g(x)f(x),$
$\displaystyle\left\llbracket f,g,h\right\rrbracket(x)$ $\displaystyle=$
$\displaystyle
f(x)g(x)h(x)-g(x)f(x)h(x)-h(x)f(x)g(x)+h(x)g(x)f(x),\quad\forall x\in[0,1],$

for all $f,g,h\in\mathfrak{g}$. Then
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
forms a Lie-Yamaguti algebra. For all $f\in C[0,1]$, the integral operator
$R:\mathfrak{g}\longrightarrow\mathfrak{g}$ defined to be

$R(f)(x):=\int_{0}^{x}f(t){\mathrm{d}}t,\quad\forall x\in[0,1]$

is a Rota-Baxter operator on $\mathfrak{g}$. For the integral operator $R$,
define $l_{2}$ and $l_{3}$ to be

$\begin{cases}(l_{2})_{\rm
I}(R,R)(f,g)=2\Big{(}[R(f),R(g)]-R([R(f),g]-[f,R(g)])\Big{)},\\\ (l_{2})_{\rm
II}(R,R)(f,g,h)=0,\\\ (l_{3})_{\rm I}(R,R,R)(f,g)=0,\\\ (l_{3})_{\rm
II}(R,R,R)(f,g,h)=6\Big{(}\left\llbracket
R(f),R(g),R(h)\right\rrbracket-R\big{(}\left\llbracket
R(f),R(g),h\right\rrbracket-\left\llbracket
f,R(g),R(h)\right\rrbracket-\left\llbracket
R(f),g,R(h)\right\rrbracket\big{)}\Big{)},\end{cases}$

for all $f,g,h\in\mathfrak{g}$. Obviously, $R$ is a strict Maurer-Cartan
element of the $L_{\infty}$-algebra
$(C^{*}(\mathfrak{g},\mathfrak{g}),l_{2},l_{3})$ by Theorem 4.6, where $l_{2}$
and $l_{3}$ are defined by (28) and (29). Moreover, for another linear
operator $R^{\prime}:\mathfrak{g}\longrightarrow\mathfrak{g}$, $R+R^{\prime}$
is still a Rota-Baxter on $\mathfrak{g}$ if and only if $R^{\prime}$ satisfies
the following Maurer-Cartan equation:

$l_{1}^{R}(R^{\prime})+\frac{1}{2}l_{2}^{R}(R^{\prime},R^{\prime})+\frac{1}{3!}l_{3}^{R}(R^{\prime},R^{\prime},R^{\prime})=0,$

where $l_{1}^{R},~{}l_{2}^{R}$ and $l_{3}^{R}$ are defined by Eqs. (30)-(32)
respectively.

######  Example 4.10.

Let $(A,\cdot)$ be an associative algebra and $A[[\nu]]$ an algebra of formal
series with coefficients in $A$. Define operations $[\cdot,\cdot]$ and
$\left\llbracket\cdot,\cdot,\cdot\right\rrbracket$ to be

$\displaystyle[a_{i}\nu^{i},a_{j}\nu^{j}]$ $\displaystyle:=$
$\displaystyle[a_{i},a_{j}]\nu^{i+j},$ $\displaystyle~{}\left\llbracket
a_{i}\nu^{i},a_{j}\nu^{j},a_{k}\nu^{k}\right\rrbracket$ $\displaystyle:=$
$\displaystyle[[a_{i},a_{j}],a_{k}]\nu^{i+j+k},$

where $a_{i},a_{j},a_{k}\in A$ and $[\cdot,\cdot]$ on the right hand side
means the commutator: $[x,y]=x\cdot y-y\cdot x$ for all $x,y\in A$, then
$(A[[\nu]],[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
forms a Lie-Yamaguti algebra. Define a formal integral operator
$\Omega:A[[\nu]]\longrightarrow A[[\nu]]$ to be

$\Omega(a_{i}\nu^{i}):=\int
a_{i}\nu^{i}d\nu=\frac{1}{i+1}a_{i}\nu^{i+1},\quad\forall a_{i}\in A,$

then $\Omega$ is a Rota-Baxter operator on $A[[\nu]]$. Then we define $l_{2}$
and $l_{3}$ by Eqs. (28) and (29) respectively to achieve an
$L_{\infty}$-algebra $(C^{*}(A[[\nu]],A[[\nu]]),l_{2},l_{3})$, and
consequently we obtain that $\Omega$ is a strict Maurer-Cartan element, i.e.,

$l_{2}(\Omega,\Omega)=0,\quad\text{and}\quad l_{3}(\Omega,\Omega,\Omega)=0.$

###  4.2. Cohomology of relative Rota-Baxter operators

In this subsection, we establish the cohomology of relative Rota-Baxter
operators on Lie-Yamaguti algebras, and then we show the relation between the
differential of the twisting $L_{\infty}$-algebra and the coboundary operator.
We write $D_{\rho,\mu}$ to emphasize that it is relative with $\rho$ and $\mu$
in this subsection.

######  Proposition 4.11.

[36] Let $T:V\longrightarrow\mathfrak{g}$ be a relative Rota-Baxter operator
on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to $(V;\rho,\mu)$. Define

$\displaystyle[u,v]_{T}$ $\displaystyle=$ $\displaystyle\rho(Tu)v-\rho(Tv)u,$
$\displaystyle\left\llbracket u,v,w\right\rrbracket_{T}$ $\displaystyle=$
$\displaystyle D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v,\quad\forall
u,v,w\in V.$

Then
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$
is a Lie-Yamaguti algebra, which is called the sub-adjacent Lie-Yamaguti
algebra. Thus $T$ is a Lie-Yamaguti algebra homomorphism111A Lie-Yamaguti
homomorphism is a linear map between two Lie-Yamaguti algebras that preserves
the Lie-Yamaguti algebra structure. from
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$
to
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$.

Next, we give a representation of the sub-adjacent Lie-Yamaguti algebra
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$
on the vector space $\mathfrak{g}$. Define two linear maps
$\varrho:V\to\mathfrak{gl}(\mathfrak{g})$ and
$\varpi:\otimes^{2}V\to\mathfrak{gl}(\mathfrak{g})$ to be

(36) $\displaystyle\varrho(u)x$ $\displaystyle:=$
$\displaystyle[Tu,x]+T\big{(}\rho(x)u\big{)},$ (37)
$\displaystyle\varpi(u,v)x$ $\displaystyle:=$ $\displaystyle\left\llbracket
x,Tu,Tv\right\rrbracket-T\big{(}D_{\rho,\mu}(x,Tu)v-\mu(x,Tv)u\big{)},\quad\forall
x\in\mathfrak{g},~{}u,v\in V.$

Consequently, we give the precise formula of $D_{\varrho,\varpi}$ first.

######  Proposition 4.12.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to $(V;\rho,\mu)$. Then with the notations above, we have

(38) $\displaystyle D_{\varrho,\varpi}(u,v)x=\left\llbracket
Tu,Tv,x\right\rrbracket-T\Big{(}\mu(Tv,x)u-\mu(Tu,x)v\Big{)},\quad\forall
u,v\in V,~{}x\in\mathfrak{g}.$

###### Proof.

Since $T$ is a relative Rota-Baxter operator, for all $u,v\in V$ and
$x\in\mathfrak{g}$, by a direct computation, we have

$\displaystyle D_{\varrho,\varpi}(u,v)x$
$\displaystyle\stackrel{{\scriptstyle\eqref{rep}}}{{=}}$
$\displaystyle\varpi(v,u)x-\varpi(u,v)x+[\varrho(u),\varrho(v)]x-\varrho([u,v]_{T})x$
$\displaystyle\stackrel{{\scriptstyle\eqref{repre1},\eqref{repre2}}}{{=}}$
$\displaystyle\left\llbracket
x,Tv,Tu\right\rrbracket-T\Big{(}D_{\rho,\mu}(x,Tv)u-\mu(x,Tu)v\Big{)}-\left\llbracket
x,Tu,Tv\right\rrbracket+T\Big{(}D_{\rho,\mu}(x,Tu)v-\mu(x,Tv)u\Big{)}$
$\displaystyle+[Tu,[Tv,x]]+[Tu,T(\rho(x)v)]+T(\rho([Tv,x])u)+T(\rho(T(\rho(x)v)u))$
$\displaystyle-[Tv,[Tu,x]]-[Tv,T(\rho(x)u)]-T(\rho([Tu,x])v)-T(\rho(T(\rho(x)u)v))$
$\displaystyle-[T(\rho(Tu)v-\rho(Tv)u),x]-T(\rho(x)\rho(Tu)v)+T(\rho(x)\rho(Tv)u)$
$\displaystyle\stackrel{{\scriptstyle\eqref{Ooperator1}}}{{=}}$
$\displaystyle\left\llbracket x,Tv,Tu\right\rrbracket-\left\llbracket
x,Tu,Tv\right\rrbracket+[Tu,[Tv,x]]-[Tv,[Tu,x]]-[[Tu,Tv],x]$
$\displaystyle-T\Big{(}D_{\rho,\mu}(x,Tv)u-\mu(x,Tu)v\Big{)}+T\Big{(}D_{\rho,\mu}(x,Tu)v-\mu(x,Tv)u\Big{)}$
$\displaystyle+T(\rho(Tu)\rho(x)v-\rho(T(\rho(x)v)u)-T(\rho(Tv)\rho(x)u-\rho(T(\rho(x)u)v)$
$\displaystyle+T(\rho([Tv,x])u)+T(\rho(T(\rho(x)v)u))-T(\rho([Tu,x])v)-T(\rho(T(\rho(x)u)v))$
$\displaystyle-T(\rho(x)\rho(Tu)v)+T(\rho(x)\rho(Tv)u)$
$\displaystyle\stackrel{{\scriptstyle\eqref{LY1},\eqref{rep}}}{{=}}$
$\displaystyle\left\llbracket
Tu,Tv,x\right\rrbracket-T\Big{(}\mu(Tv,x)u-\mu(Tu,x)v\Big{)}.$

The conclusion thus follows. ∎

######  Theorem 4.13.

With the above notations, then $(\mathfrak{g};\varrho,\varpi)$ is a
representation of the sub-adjacent Lie-Yamaguti algebra
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$,
where linear maps $\varrho,~{}\varpi$, and $D_{\varrho,\varpi}$ are given by
Eqs. (36)-(38) respectively.

We may prove Theorem 4.13 by checking that linear maps $\varrho$, $\varpi$,
and $D_{\varrho,\varpi}$ satisfy conditions in Definition 2.3, but here we
prove this proposition by another way. In order to do this, we should go back
some notions in [35]. Recall that a Nijenhuis operator on a Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
is a linear map $N:\mathfrak{g}\to\mathfrak{g}$ satisfying

$\displaystyle[Nx,Ny]$ $\displaystyle=$ $\displaystyle
N\big{(}[Nx,y]+[x,Ny]-N[x,y]\big{)},$ $\displaystyle~{}\left\llbracket
Nx,Ny,Nz\right\rrbracket$ $\displaystyle=$ $\displaystyle
N\Big{(}\left\llbracket Nx,Ny,z\right\rrbracket+\left\llbracket
Nx,y,Nz\right\rrbracket+\left\llbracket x,Ny,Nz\right\rrbracket$
$\displaystyle-N\left\llbracket Nx,y,z\right\rrbracket-N\left\llbracket
x,Ny,z\right\rrbracket-N\left\llbracket
x,y,Nz\right\rrbracket+N^{2}\left\llbracket
x,y,z\right\rrbracket\Big{)},\quad\forall x,y,z\in\mathfrak{g}.$

Then we get a pair of deformed brackets
$([\cdot,\cdot]_{N},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{N})$:

(39) $\displaystyle[x,y]_{N}$ $\displaystyle=$
$\displaystyle[Nx,y]+[x,Ny]-N[x,y],$ (40) $\displaystyle~{}\left\llbracket
x,y,z\right\rrbracket_{N}$ $\displaystyle=$ $\displaystyle\left\llbracket
Nx,Ny,z\right\rrbracket+\left\llbracket
Nx,y,Nz\right\rrbracket+\left\llbracket x,Ny,Nz\right\rrbracket$
$\displaystyle-N\left\llbracket Nx,y,z\right\rrbracket-N\left\llbracket
x,Ny,z\right\rrbracket-N\left\llbracket
x,y,Nz\right\rrbracket+N^{2}\left\llbracket x,y,z\right\rrbracket,\quad\forall
x,y,z\in\mathfrak{g}.$

In [35], authors showed that
$(\mathfrak{g},[\cdot,\cdot]_{N},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{N})$
is a Lie-Yamaguti algebra and thus $N$ is a Lie-Yamaguti homomorphism from
$(\mathfrak{g},[\cdot,\cdot]_{N},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{N})$
to
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$.

_Proof of Theorem 4.13:_ It is direct to see that if $T:V\to\mathfrak{g}$ is a
relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$, then
$N_{T}=\begin{pmatrix}0&T\\\ 0&0\end{pmatrix}$ is a Nijenhuis operator on the
semidirect product Lie-Yamaguti algebra $\mathfrak{g}\ltimes_{\rho,\mu}V$.
Then by (39) and (40), we deduce that there is a Lie-Yamaguti algebra
structure on $V\oplus\mathfrak{g}\cong\mathfrak{g}\oplus V$ given by for all
$x,y,z\in\mathfrak{g},~{}u,v,w\in V,$

$\displaystyle[x+u,y+v]_{N_{T}}$ $\displaystyle=$
$\displaystyle[N_{T}(x+u),y+v]_{\rho,\mu}+[x+u,N_{T}(y+v)]_{\rho,\mu}-N_{T}[x+u,y+v]_{\rho,\mu}$
$\displaystyle=$
$\displaystyle[Tu,y+v]_{\rho,\mu}+[x+u,Tv]_{\rho,\mu}-N_{T}([x,y]+\rho(x)v-\rho(y)u)$
$\displaystyle=$
$\displaystyle[Tu,y]+\rho(Tu)v+[x,Tv]-\rho(Tv)u-T(\rho(x)v-\rho(y)u)$
$\displaystyle=$ $\displaystyle[u,v]_{T}+\varrho(u)y-\varrho(v)x,$
$\displaystyle\left\llbracket x+u,y+v,z+w\right\rrbracket_{N_{T}}$
$\displaystyle=$ $\displaystyle\left\llbracket
N_{T}(x+u),N_{T}(y+v),z+w\right\rrbracket_{\rho,\mu}+\left\llbracket
N_{T}(x+u),y+v,N_{T}(z+w)\right\rrbracket_{\rho,\mu}+\left\llbracket
x+u,N_{T}(y+v),N_{T}(z+w)\right\rrbracket_{\rho,\mu}$ $\displaystyle-
N_{T}(\left\llbracket
N_{T}(x+u),y+v,z+w\right\rrbracket_{\rho,\mu}+\left\llbracket
x+u,N_{T}(y+v),z+w\right\rrbracket_{\rho,\mu})+\left\llbracket
x+u,y+v,N_{T}(z+w)\right\rrbracket_{\rho,\mu})$ $\displaystyle=$
$\displaystyle\left\llbracket
Tu,Tv,z+w\right\rrbracket_{\rho,\mu}+\left\llbracket
Tu,y+v,Tw\right\rrbracket_{\rho,\mu}+\left\llbracket
x+u,Tv,Tw\right\rrbracket_{\rho,\mu}$ $\displaystyle-N_{T}(\left\llbracket
Tu,y+v,z+w\right\rrbracket_{\rho,\mu}+\left\llbracket
x+u,Tv,z+w\right\rrbracket_{\rho,\mu}+\left\llbracket
x+u,y+v,Tw\right\rrbracket_{\rho,\mu})$ $\displaystyle=$
$\displaystyle\left\llbracket
Tu,Tv,z\right\rrbracket+D_{\rho,\mu}(Tu,Tv)w+\left\llbracket
Tu,y,Tw\right\rrbracket-\mu(Tu,Tw)v+\left\llbracket
x,Tv,Tw\right\rrbracket+\mu(Tv,Tw)u$
$\displaystyle-T\Big{(}D_{\rho,\mu}(Tu,y)w-\mu(Tu,z)v+D_{\rho,\mu}(x,Tv)w+\mu(Tv,z)u+\mu(y,Tw)u-\mu(x,Tw)v\Big{)}$
$\displaystyle=$ $\displaystyle\left\llbracket
u,v,w\right\rrbracket_{T}+D_{\varrho,\varpi}(u,v)z+\varpi(v,w)x-\varpi(u,w)y,$

which implies that $(\mathfrak{g};\varrho,\varpi)$ is a representation of Lie-
Yamaguti algebra
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$.
This finishes the proof.∎

Next, we construct the $0$-cochians and the corresponding coboundary map. For
all $\mathfrak{X}\in\wedge^{2}\mathfrak{g}$, define
$\delta(\mathfrak{X}):V\to\mathfrak{g}$ to be

(41)
$\displaystyle\delta(\mathfrak{X})v:=TD(\mathfrak{X})v-\left\llbracket\mathfrak{X},Tv\right\rrbracket,\quad\forall
v\in V.$

######  Proposition 4.14.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot,],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. Then $\delta(\mathfrak{X})$
is a $1$-cocycle on the Lie-Yamaguti algebra
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$
with coefficients in the representation $(\mathfrak{g};\varrho,\varpi)$.

###### Proof.

It is sufficient to show that both $\delta_{\rm I}^{T}(\delta(\mathfrak{X}))$
and $\delta_{\rm II}^{T}(\delta(\mathfrak{X}))$ all vanish. Indeed, for any
$u,v,w\in V$, we have

$\displaystyle\delta_{\rm II}^{T}\Big{(}\delta(\mathfrak{X})\Big{)}(u,v,w)$
$\displaystyle\stackrel{{\scriptstyle\eqref{2cochain}}}{{=}}$
$\displaystyle-\delta(\mathfrak{X})(\left\llbracket
u,v,w\right\rrbracket_{T})+D_{\varrho,\varpi}(u,v)\Big{(}\delta(\mathfrak{X})w\Big{)}+\varpi(v,w)\Big{(}\delta(\mathfrak{X})u\Big{)}-\varpi(u,w)\Big{(}\delta(\mathfrak{X})v\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{repre1},\eqref{repre2}}}{{=}}$
$\displaystyle\left\llbracket
Tu,Tv,TD_{\rho,\mu}(\mathfrak{X})w-\left\llbracket\mathfrak{X},Tw\right\rrbracket\right\rrbracket+\left\llbracket
TD_{\rho,\mu}(\mathfrak{X})u-\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tv,Tw\right\rrbracket$
$\displaystyle+\left\llbracket
Tu,TD_{\rho,\mu}(\mathfrak{X})v-\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tw\right\rrbracket$
$\displaystyle-
TD(\mathfrak{X})\Big{(}D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big{)}$
$\displaystyle+\left\llbracket\mathfrak{X},T\big{(}D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\big{)}\right\rrbracket$
$\displaystyle-T\Big{(}D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})u-\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tv\big{)}w-D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})v-\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tu\big{)}w\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})v-\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tw\big{)}u-\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})u-\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tw\big{)}v\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}Tv,TD_{\rho,\mu}(\mathfrak{X})w-\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}u-\mu\big{(}Tu,TD_{\rho,\mu}(\mathfrak{X})w-\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}v\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Ooperator2}}}{{=}}$
$\displaystyle\left\llbracket
Tu,Tv,TD_{\rho,\mu}(\mathfrak{X})w\right\rrbracket-\left\llbracket
Tu,Tv,\left\llbracket\mathfrak{X},Tw\right\rrbracket\right\rrbracket+\left\llbracket
TD_{\rho,\mu}(\mathfrak{X})u,Tv,Tw\right\rrbracket$
$\displaystyle-\left\llbracket\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tv,Tw\right\rrbracket+\left\llbracket
Tu,TD_{\rho,\mu}(\mathfrak{X})v,Tw\right\rrbracket-\left\llbracket
Tu,\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tw\right\rrbracket$
$\displaystyle-
TD_{\rho,\mu}(\mathfrak{X})\Big{(}D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big{)}+\left\llbracket\mathfrak{X},\left\llbracket
Tu,Tv,Tw\right\rrbracket\right\rrbracket$
$\displaystyle-T\Big{(}D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})u-\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tv\big{)}w-D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})v-\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tu\big{)}w\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})v-\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tw\big{)}u-\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})u-\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tw\big{)}v\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}Tv,\big{(}TD_{\rho,\mu}(\mathfrak{X})w-\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}u-\mu\big{(}Tu,TD_{\rho,\mu}(\mathfrak{X})w-\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}v\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{fundamental}}}{{=}}$
$\displaystyle\left\llbracket
Tu,Tv,TD_{\rho,\mu}(\mathfrak{X})w\right\rrbracket+\left\llbracket
TD_{\rho,\mu}(\mathfrak{X})u,Tv,Tw\right\rrbracket+\left\llbracket
Tu,TD_{\rho,\mu}(\mathfrak{X})v,Tw\right\rrbracket$ $\displaystyle-
TD_{\rho,\mu}(\mathfrak{X})\Big{(}D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big{)}$
$\displaystyle-T\Big{(}D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})u,Tv\big{)}w-D_{\rho,\mu}\big{(}\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tv\big{)}w-D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})v,Tu\big{)}w\Big{)}+D_{\rho,\mu}\big{(}\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tu\big{)}w\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})v,Tw\big{)}u-\mu\big{(}\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tw\big{)}u-\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})u,Tw\big{)}v\Big{)}+\mu\big{(}\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tw\big{)}v\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}Tv,TD_{\rho,\mu}(\mathfrak{X})w\big{)}u-\mu\big{(}Tv,\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}u-\mu\big{(}Tu,TD_{\rho,\mu}(\mathfrak{X})w\big{)}v+\mu\big{(}Tu,\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}v\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Ooperator2}}}{{=}}$ $\displaystyle
T\Big{(}D_{\rho,\mu}(Tu,Tv)D_{\rho,\mu}(\mathfrak{X})w+\mu(Tv,TD_{\rho,\mu}(\mathfrak{X})w)u-\mu(Tu,TD_{\rho,\mu}(\mathfrak{X})w)v\Big{)}$
$\displaystyle+T\Big{(}D_{\rho,\mu}(TD_{\rho,\mu}(\mathfrak{X})u,Tv)w+\mu(Tv,Tw)D_{\rho,\mu}(\mathfrak{X})u-\mu(TD_{\rho,\mu}(\mathfrak{X})u,Tw)v\Big{)}$
$\displaystyle+T\Big{(}D_{\rho,\mu}(Tu,TD_{\rho,\mu}(\mathfrak{X})v)w+\mu(TD_{\rho,\mu}(\mathfrak{X})v,Tw)u-\mu(Tu,Tw)D_{\rho,\mu}(\mathfrak{X})v\Big{)}$
$\displaystyle-
TD_{\rho,\mu}(\mathfrak{X})\Big{(}D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big{)}$
$\displaystyle-T\Big{(}D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})u,Tv\big{)}w-D_{\rho,\mu}\big{(}\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tv\big{)}w-D_{\rho,\mu}\big{(}TD_{\rho,\mu}(\mathfrak{X})v,Tu\big{)}w+D_{\rho,\mu}\big{(}\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tu\big{)}w\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})v,Tw\big{)}u-\mu\big{(}\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tw\big{)}u-\mu\big{(}TD_{\rho,\mu}(\mathfrak{X})u,Tw\big{)}v)+\mu\big{(}\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tw\big{)}v\Big{)}$
$\displaystyle-T\Big{(}\mu\big{(}Tv,TD_{\rho,\mu}(\mathfrak{X})w\big{)}u-\mu\big{(}Tv,\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}u-\mu\big{(}Tu,TD_{\rho,\mu}(\mathfrak{X})w\big{)}v+\mu\big{(}Tu,\left\llbracket\mathfrak{X},Tw\right\rrbracket\big{)}v\Big{)}$
$\displaystyle=$ $\displaystyle
T\Big{(}D_{\rho,\mu}(Tu,Tv)D_{\rho,\mu}(\mathfrak{X})w+\mu(Tv,Tw)D_{\rho,\mu}(\mathfrak{X})u-\mu(Tu,Tw)D_{\rho,\mu}(\mathfrak{X})v\Big{)}$
$\displaystyle-
TD_{\rho,\mu}(\mathfrak{X})\Big{(}D_{\rho,\mu}(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big{)}$
$\displaystyle+T\Big{(}D_{\rho,\mu}(\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tv)w-D_{\rho,\mu}(\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tu)w+\mu(\left\llbracket\mathfrak{X},Tv\right\rrbracket,Tw)u$
$\displaystyle-\mu(\left\llbracket\mathfrak{X},Tu\right\rrbracket,Tw)v+\mu(Tv,\left\llbracket\mathfrak{X},Tw\right\rrbracket)u-\mu(Tu,\left\llbracket\mathfrak{X},Tw\right\rrbracket)v\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{RLY5},\eqref{RLY5a}}}{{=}}$
$\displaystyle 0.$

Similarly, we deduce that $\delta^{T}_{\rm
I}\Big{(}\delta(\mathfrak{X})\Big{)}(u,v)=0$ for all $u,v\in V$. This finishes
the proof. ∎

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. Combining the Yamaguti
cohomology with Proposition 4.14, we obtain a well-defined cochain complex
$({\mathcal{C}}_{T}^{\bullet}(V,\mathfrak{g})=\bigoplus_{n=0}^{\infty}{\mathcal{C}}_{T}^{n}(V,\mathfrak{g}),\mathrm{d})$,
where the $n$-cochians ${\mathcal{C}}^{n}_{T}(V,\mathfrak{g})$ and the
coboundary map
$\mathrm{d}:{\mathcal{C}}^{n}_{T}(V,\mathfrak{g})\to{\mathcal{C}}^{n+1}_{T}(V,\mathfrak{g})$
are defined to be

$\displaystyle{\mathcal{C}}^{n}_{T}(V,\mathfrak{g}):=\begin{cases}C^{n}_{\rm
LieY}(V,\mathfrak{g}),&n\geqslant 1,\\\
\wedge^{2}\mathfrak{g},&n=0,\end{cases}$

and

$\displaystyle\mathrm{d}:=\begin{cases}\delta^{T},&n\geqslant 1,\\\
\delta,&n=0,\end{cases}$

respectively. Here $\delta^{T}:C_{\rm LieY}^{n}(V,\mathfrak{g})\to C_{\rm
LieY}^{n+1}(V,\mathfrak{g})~{}(n\geqslant 1)$ is the corresponding Yamaguti
coboundary operator on the sub-adjacent Lie-Yamaguti algebra
$(V,[\cdot,\cdot]_{T},\left\llbracket\cdot,\cdot,\cdot\right\rrbracket_{T})$
with coefficients in the representation $(\mathfrak{g};\varrho,\varpi)$.

######  Definition 4.15.

The cohomology of cochian complex
$({\mathcal{C}}_{T}^{\bullet}(V,\mathfrak{g})=\bigoplus_{n=0}^{\infty}{\mathcal{C}}_{T}^{n}(V,\mathfrak{g}),\mathrm{d})$
is called the cohomology of relative Rota-Baxter operator $T$ on Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to the representation $(V;\rho,\mu)$. Denote the set of
$n$-cocycles and $n$-coboundaries by $\mathcal{Z}^{n}_{T}(V,\mathfrak{g})$ and
$\mathcal{B}^{n}_{T}(V,\mathfrak{g})$ respectively. The $n$-th cohomology
group of relative Rota-Baxter operator $T$ is taken to be

$\displaystyle\mathcal{H}^{n}_{T}(V,\mathfrak{g}):=\mathcal{Z}^{n}_{T}(V,\mathfrak{g})/\mathcal{B}^{n}_{T}(V,\mathfrak{g}),\quad
n\geqslant 1.$

Let us give the formula of $\delta^{T}$ explicitly:

  * •

if $n\geqslant 1$, $\delta^{T}:C_{\rm LieY}^{n+1}(V,\mathfrak{g})\to C_{\rm
LieY}^{n+2}(V,\mathfrak{g})$ is given by for any $F=(f,g)\in C_{\rm
LieY}^{n+1}(V,\mathfrak{g})$,

$\displaystyle\Big{(}\delta^{T}_{\rm
I}(F)\Big{)}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$ (42) $\displaystyle=$
$\displaystyle(-1)^{n}\Big{(}[Tu_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1})]-[Tv_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1})]$
$\displaystyle-g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},\rho(Tu_{n+1})v_{n+1}-\rho(Tv_{n+1})u_{n+1})+T\big{(}\rho(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}))u_{n+1}\big{)}$
$\displaystyle-T\big{(}\rho(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}))v_{n+1}\big{)}\Big{)}$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k+1}\Big{(}\left\llbracket
Tu_{k},Tv_{k},f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1})\right\rrbracket+T\big{(}\mu(Tv_{k},f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1}))u_{k}\big{)}$
$\displaystyle-T\big{(}\mu(Tu_{k},f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1}))v_{k}\big{)}\Big{)}$
$\displaystyle+\sum_{1\leqslant k<l\leqslant
n+1}(-1)^{k}f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{k}\circ\mathcal{V}_{l},\cdots,\mathcal{V}_{n+1}),$

and

$\displaystyle\Big{(}\delta^{T}_{\rm
II}(F)\Big{)}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1},w)$ (43)
$\displaystyle=$ $\displaystyle(-1)^{n}\Big{(}\left\llbracket
g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}),Tv_{n+1},Tw\right\rrbracket-\left\llbracket
g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}),Tu_{n+1},Tw\right\rrbracket$
$\displaystyle+T\big{(}D(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}),Tv_{n+1})w-\mu(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}),Tw)v_{n+1}$
$\displaystyle+D(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}),Tu_{n+1})w-\mu(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}),Tw)u_{n+1}\big{)}\Big{)}$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k}\Big{(}\left\llbracket
Tu_{k},Tv_{k},g(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1},w)\right\rrbracket+T\big{(}\mu(Tv_{k},g(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1},w))u_{k}$
$\displaystyle-\mu(Tu_{k},g(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1},w))v_{k}\big{)}\Big{)}$
$\displaystyle+\sum_{1\leqslant k<l\leqslant
n+1}(-1)^{k}g(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{k}\circ\mathcal{V}_{l},\cdots,\mathcal{V}_{n+1},w)$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k}g(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1},\left\llbracket
u_{k},v_{k},w\right\rrbracket_{T}),$

where $\mathcal{V}_{i}=u_{i}\wedge v_{i}\in\wedge^{2}V,~{}(1\leqslant
i\leqslant n+1),~{}w\in V$ and
$\mathcal{V}_{k}\circ\mathcal{V}_{l}=\left\llbracket
u_{k},v_{k},u_{l}\right\rrbracket_{T}\wedge v_{l}+u_{l}\wedge\left\llbracket
u_{k},v_{k},v_{l}\right\rrbracket_{T}$.

  * •

if $n=0$, for any $f\in C_{\rm
LieY}^{1}(V,\mathfrak{g})=\mathrm{Hom}(V,\mathfrak{g})$,

$\delta^{T}:C_{\rm LieY}^{1}(V,\mathfrak{g})\to C_{\rm
LieY}^{2}(V,\mathfrak{g}),\quad f\mapsto(\delta^{T}_{\rm I}(f),\delta^{T}_{\rm
II}(f))$

is given by

$\displaystyle(\delta^{T}_{\rm I}(f))(u,v)$ $\displaystyle=$
$\displaystyle[Tu,f(v)]-[Tv,f(u)]+T\Big{(}\rho(f(v)u)-\rho(f(u)v)\Big{)}-f([u,v]_{T}),$
$\displaystyle(\delta^{T}_{\rm II}(f))(u,v,w)$ $\displaystyle=$
$\displaystyle\left\llbracket Tu,Tv,f(w)\right\rrbracket+\left\llbracket
f(u),Tv,Tw\right\rrbracket-\left\llbracket
f(v),Tu,Tw\right\rrbracket-f(\left\llbracket u,v,w\right\rrbracket_{T})$
$\displaystyle-T\Big{(}D(f(u),Tv)w-D(f(v),Tu)w+\mu(Tv,f(w))u-\mu(Tu,f(w))v$
$\displaystyle-\mu(f(u),Tw)v+\mu(f(v),Tw)u\Big{)},\qquad\forall u,v,w\in V.$

At the end of this section, we show that the coboundary map $\mathrm{d}$
coincides (up to sign) with the differential $l_{1}^{T}$ defined by Equation
(30) which involves the Maurer-Cartan element $T$ of the twisted
$L_{\infty}$-algebra $(C^{*}(V,\mathfrak{g}),l_{1}^{T},l_{2}^{T},l_{3}^{T})$.

######  Theorem 4.16.

Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. Then we have

(44) $\displaystyle\mathrm{d}(F)$ $\displaystyle=$
$\displaystyle(-1)^{n-1}l_{1}^{T}(F),\quad\forall F\in
C^{n}(V,\mathfrak{g}),~{}n=1,2,3,\cdots,$ (45) $\displaystyle\mathrm{d}(f)$
$\displaystyle=$ $\displaystyle l_{1}^{T}(f),\quad\forall f\in
C^{0}(V,\mathfrak{g})=\mathrm{Hom}(V,\mathfrak{g}).$

###### Proof.

First, for all $x,y,z\in\mathfrak{g}$, $u,v,w\in V$, we compute that

$\displaystyle[\Pi+\Theta,T]_{\rm I}(x+u,y+v)$ $\displaystyle=$
$\displaystyle\Big{(}(\Pi+\Theta)\circ T\Big{)}_{\rm
I}(x+u,y+v)-\Big{(}T\circ(\Pi+\Theta)\Big{)}_{\rm I}(x+u,y+v)$
$\displaystyle=$
$\displaystyle[Tu,y]+\rho(Tu)v+[x,Tv]-\rho(Tv)u-T\Big{(}\rho(x)v-\rho(y)u\Big{)},$
$\displaystyle[\Pi+\Theta,T]_{\rm II}(x+u,y+v,z+w)$ $\displaystyle=$
$\displaystyle\Big{(}(\Pi+\Theta)\circ T\Big{)}_{\rm
II}(x+u,y+v,z+w)-\Big{(}T\circ(\Pi+\Theta)\Big{)}_{\rm II}(x+u,y+v,z+w)$
$\displaystyle=$ $\displaystyle\left\llbracket
Tu,y,z\right\rrbracket+D(Tu,y)w-\mu(Tu,z)v+\left\llbracket
x,Tv,z\right\rrbracket+D(x,Tv)w+\mu(Tv,z)u$ $\displaystyle+\left\llbracket
x,y,Tw\right\rrbracket+\mu(y,Tw)u-\mu(x,Tw)v-T\Big{(}D(x,y)w+\mu(y,z)u-\mu(x,z)v\Big{)},$
$\displaystyle[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm I}(x+u,y+v)$
$\displaystyle=$ $\displaystyle\Big{(}[\Pi+\Theta,T]_{\mathsf{LieY}}\circ
T\Big{)}_{\rm
I}(x+u,y+v)-\Big{(}T\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
I}(x+u,y+v)$ $\displaystyle=$ $\displaystyle
2\Big{(}[Tu,Tv]-T\big{(}\rho(Tu)v-\rho(Tv)u\big{)}\Big{)},$
$\displaystyle[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm II}(x+u,y+v,z+w)$
$\displaystyle=$ $\displaystyle[\Pi+\Theta,T]_{\rm
II}(Tu,y+v,z+w)+[\Pi+\Theta,T]_{\rm II}(x+u,Tv,z+w)$
$\displaystyle+[\Pi+\Theta,T]_{\rm II}(x+u,y+v,Tw)-T\Big{(}[\Pi+\Theta,T]_{\rm
II}(x+u,y+v,z+w)\Big{)}$ $\displaystyle=$ $\displaystyle
2\Big{(}\left\llbracket Tu,Tv,z\right\rrbracket+\left\llbracket
x,Tv,Tw\right\rrbracket+\left\llbracket
Tu,y,Tw\right\rrbracket+D(Tu,Tv)w+\mu(Tv,Tw)u-\mu(Tu,Tw)v\Big{)}$
$\displaystyle-2T\Big{(}D(Tu,y)w+D(x,Tv)w-\mu(Tu,z)v-\mu(x,Tw)v+\mu(Tv,z)u+\mu(y,Tw)u\Big{)}.$

Then, by (28) and (29), for all $F=(f,g)\in
C^{n}(V,\mathfrak{g})~{}(n\geqslant 1)$, let us compute that

$\displaystyle(l_{2})_{\rm
I}\Big{(}T,F\Big{)}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$
$\displaystyle=$
$\displaystyle\mathrm{pr}_{\mathfrak{g}}[[\Pi+\Theta,T]_{\mathsf{LieY}},F]_{\rm
I}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$ $\displaystyle=$
$\displaystyle\Big{(}[\Pi+\Theta,T]_{\mathsf{LieY}}\circ F\Big{)}_{\rm
I}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})-(-1)^{n}\Big{(}F\circ[\Pi+\Theta,T]_{\mathsf{LieY}}\Big{)}_{\rm
I}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(1,n)}\atop\sigma(n+1)=n+1}(-1)^{n}sign(\sigma)[\Pi+\Theta,T]_{\rm
II}(\mathcal{V}_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))$
$\displaystyle+[\Pi+\Theta,T]_{\rm
I}(u_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}))+[\Pi+\Theta,T]_{\rm
I}(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}),v_{n+1})$
$\displaystyle-(-1)^{n}\Big{(}(-1)^{n}g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},[\Pi+\Theta,T]_{\rm
I}(\mathcal{V}_{n+1}))$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},u_{k+1}\wedge[\Pi+\Theta,T]_{\rm
II}(\mathcal{V}_{\sigma(k)},v_{k+1}),\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},[\Pi+\Theta,T]_{\rm
II}(\mathcal{V}_{\sigma(k)},u_{k+1})\wedge
v_{k+1},\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\Big{)}$ $\displaystyle=$
$\displaystyle[Tu_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1})]+T(\rho(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}))u_{n+1})+[g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}),Tv_{n+1}]$
$\displaystyle-T(\rho(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}))v_{n+1})-g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},\rho(Tu_{n+1})v_{n+1}-\rho(Tv_{n+1})u_{n+1}),$

and

$\displaystyle(l_{3})_{\rm
I}\Big{(}T,T,F\Big{)}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$
$\displaystyle=$
$\displaystyle\mathrm{pr}_{\mathfrak{g}}[[[\Pi+\Theta,T]_{\mathsf{LieY}}],T]_{\mathsf{LieY}},F]_{\rm
I}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$ $\displaystyle=$
$\displaystyle\Big{(}[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}}\circ
F\Big{)}_{\rm
I}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})-(-1)^{n}\Big{(}F\circ[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}}\Big{)}_{\rm
I}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(1,n)}\atop\sigma(n+1)=n+1}(-1)^{n}sign(\sigma)[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm
II}(\mathcal{V}_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))$
$\displaystyle+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm
I}(u_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}))+[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm
I}(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}),v_{n+1})$
$\displaystyle-(-1)^{n}\Big{(}\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},u_{k+1}\wedge[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm
II}(\mathcal{V}_{\sigma(k)},v_{k+1}),\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})$
$\displaystyle+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}sign(\sigma)f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\rm
II}(\mathcal{V}_{\sigma(k)},u_{k+1})\wedge
v_{k+1},\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\Big{)}$ $\displaystyle=$
$\displaystyle
2\Big{(}\sum_{\sigma\in\mathbb{S}_{(1,n)}\atop\sigma(n+1)=n+1}(-1)^{n}sign(\sigma)\big{(}\left\llbracket
Tu_{\sigma(1)},Tv_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)})\right\rrbracket+T(\mu(Tv_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))u_{\sigma(1)})$
$\displaystyle-T(\mu(Tu_{\sigma(1)},f(\mathcal{V}_{\sigma(2)},\cdots,\mathcal{V}_{\sigma(n+1)}))v_{\sigma(1)})\big{)}+\sum_{k=1}^{n}(-1)^{k-1}\sum_{\sigma\in\mathbb{S}_{(k-1,1)}}f(\mathcal{V}_{\sigma(1)},\cdots,\mathcal{V}_{\sigma(k-1)},\mathcal{V}_{\sigma(k)}\circ\mathcal{V}_{k+1},\mathcal{V}_{k+2},\cdots,\mathcal{V}_{n+1})\Big{)}.$

Thus, we have that

$\displaystyle(-1)^{n-1}\Big{(}l_{1}^{T}\Big{)}_{\rm
I}(F)(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$
$\displaystyle\stackrel{{\scriptstyle\eqref{diffs}}}{{=}}$
$\displaystyle(-1)^{n-1}\Big{(}(l_{2})_{\rm I}(T,F)+\frac{1}{2}(l_{3})_{\rm
I}(T,T,F)\Big{)}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1})$
$\displaystyle\stackrel{{\scriptstyle\eqref{multi1},\eqref{multi2}}}{{=}}$
$\displaystyle(-1)^{n}\Big{(}[Tu_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1})]-[Tv_{n+1},g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1})]$
$\displaystyle-g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},\rho(Tu_{n+1})v_{n+1}-\rho(Tv_{n+1})u_{n+1})+T\big{(}\rho(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},v_{n+1}))u_{n+1}\big{)}$
$\displaystyle-T\big{(}\rho(g(\mathcal{V}_{1},\cdots,\mathcal{V}_{n},u_{n+1}))v_{n+1}\big{)}\Big{)}$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{k+1}\Big{(}\left\llbracket
Tu_{k},Tv_{k},f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1})\right\rrbracket+T\big{(}\mu(Tv_{k},f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1}))u_{k}\big{)}$
$\displaystyle-T\big{(}\mu(Tu_{k},f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{n+1}))v_{k}\big{)}\Big{)}$
$\displaystyle+\sum_{1\leqslant k<l\leqslant
n+1}(-1)^{k}f(\mathcal{V}_{1},\cdots,\hat{\mathcal{V}_{k}},\cdots,\mathcal{V}_{k}\circ\mathcal{V}_{l},\cdots,\mathcal{V}_{n+1}),$
$\displaystyle\stackrel{{\scriptstyle\eqref{Ocohomo1}}}{{=}}$
$\displaystyle\delta_{\rm
I}^{T}(F)(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1}).$

Hence, we have

(46) $\displaystyle\delta_{\rm
I}^{T}(F)=(-1)^{n-1}\Big{(}l_{1}^{T}\Big{)}_{\rm I}(F),\quad\forall F\in
C^{n}(V,\mathfrak{g})~{}(n\geqslant 1).$

Similarly, by a direct computation, we have that for all $F\in
C^{n}(V,\mathfrak{g})~{}(n\geqslant 1)$ and for all
$\mathcal{V}_{i}\in\wedge^{2}V~{}(i=1,\cdots,n+1)$ and $u\in V$

$\displaystyle~{}(l_{2})_{\rm
II}(F)(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1},u)$ $\displaystyle=$
$\displaystyle\mathrm{pr}_{\mathfrak{g}}[[\Pi+\Theta,T]_{\mathsf{LieY}},F]_{\rm
II}(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1},u)=0,$ $\displaystyle(l_{3})_{\rm
II}(T,T,F)(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1},u)$ $\displaystyle=$
$\displaystyle(-1)^{n-1}2\delta_{\rm
II}^{T}(F)(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1},u).$

Thus, we have that

(47) $\displaystyle(-1)^{n-1}\Big{(}l_{1}^{T}\Big{)}_{\rm
II}(F)(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1},u)=\delta_{\rm
II}^{T}(F)(\mathcal{V}_{1},\cdots,\mathcal{V}_{n+1},u).$

Combining Eqs. (46) and (47) yields (44). Moreover, for any
$f\in\mathrm{Hom}(V,\mathfrak{g})$, by a direct computation, we have that for
all $u,v,w\in V$,

$\displaystyle\Big{(}l_{2}\Big{)}_{\rm I}(T,f)(u,v)$ $\displaystyle=$
$\displaystyle\mathrm{pr}_{\mathfrak{g}}[[\Pi+\Theta,T]_{\mathsf{LieY}},f]_{\rm
I}(u,v)$ $\displaystyle=$
$\displaystyle[f(u),Tv]+[Tu,f(v)]-T\Big{(}\rho(f(u))v-\rho(f(v))u\Big{)}-f([u,v]_{T}),$
$\displaystyle\Big{(}l_{3}\Big{)}_{\rm
I}(T,T,f)(u,v)=\mathrm{pr}_{\mathfrak{g}}[[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}},f]_{\rm
I}=0.$

Thus we get

(48) $\displaystyle\Big{(}l_{1}^{T}\Big{)}_{\rm
I}(f)(u,v)=\Big{(}l_{2}\Big{)}_{\rm I}(T,f)(u,v)=\delta_{\rm I}^{T}(f)(u,v).$

Similarly,

$\displaystyle\Big{(}l_{2}\Big{)}_{\rm
II}(T,f)(u,v,w)=\mathrm{pr}_{\mathfrak{g}}[[\Pi+\Theta,T]_{\mathsf{LieY}},f]_{\rm
II}(u,v,w)=0,$ $\displaystyle\Big{(}l_{3}\Big{)}_{\rm
II}(T,T,f)(u,v,w)=\mathrm{pr}_{\mathfrak{g}}[[[\Pi+\Theta,T]_{\mathsf{LieY}},T]_{\mathsf{LieY}},f]_{\rm
II}(u,v,w)$ $\displaystyle=$ $\displaystyle 2\Big{(}\left\llbracket
f(u),Tv,Tw\right\rrbracket-T\big{(}D(f(u),Tv)w-\mu(f(u),Tw)v+c.p.\big{)}-f(\left\llbracket
u,v,w\right\rrbracket_{T})\Big{)},$

where the last equation is equivalent to $\Big{(}l_{3}\Big{)}_{\rm
II}(T,T,f)(u,v,w)=2\delta_{\rm II}^{T}(f)(u,v,w)$. Thus, for all $u,v,w\in V$,
we have that

(49) $\displaystyle\Big{(}l_{1}^{T}\Big{)}_{\rm II}(f)(u,v,w)=\delta_{\rm
II}^{T}(f)(u,v,w).$

Hence, Eqs. (48) and (49) give (45). This completes the proof. ∎

##  5\. Deformations of relative Rota-Baxter operators

After establishing cohomology theory of relative Rota-Baxter operators, we
study deformations of relative Rota-Baxter operators, i.e., we use this
cohomology to characterize deformations of relative Rota-Baxter operators on
Lie-Yamaguti algebras.

###  5.1. Linear deformations of relative Rota-Baxter operators

In this subsection, we explore linear deformations of relative Rota-Baxter
operators on Lie-Yamaguti algebras, and we show that the infinitesimals of two
equivalent linear deformations of a relative Rota-Baxter operator on Lie-
Yamaguti algebra are in the same cohomology classes of the first cohomology
group.

######  Definition 5.1.

Let $T$ and $T^{\prime}$ be two relative Rota-Baxter operators on a Lie-
Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. A homomorphism from
$T^{\prime}$ to $T$ consists of a Lie-Yamaguti homomorphism
$\phi_{\mathfrak{g}}:\mathfrak{g}\to\mathfrak{g}$ and a linear map
$\phi_{V}:V\to V$ such that

(50) $\displaystyle T\circ\phi_{V}$ $\displaystyle=$
$\displaystyle\phi_{\mathfrak{g}}\circ T^{\prime}$ (51)
$\displaystyle~{}\phi_{V}(\rho(x)v)$ $\displaystyle=$
$\displaystyle\rho(\phi_{\mathfrak{g}}(x))\phi_{V}(v),$ (52)
$\displaystyle~{}\phi_{V}\mu(x,y)(v)$ $\displaystyle=$
$\displaystyle\mu(\phi_{\mathfrak{g}}(x),\phi_{\mathfrak{g}}(y))(\phi_{V}(v)),\quad\forall
x,y\in\mathfrak{g},~{}v\in V.$

In particular, if $\phi_{\mathfrak{g}}$ and $\phi_{V}$ are invertible, then
$(\phi_{\mathfrak{g}},\phi_{V})$ is called an isomorphism from $T^{\prime}$ to
$T$.

######  Remark 5.2.

If two relative Rota-Baxter operators $T$ and $T^{\prime}$ are homomorphic,
that is, there exists a pair $(\phi_{\mathfrak{g}},\phi_{V})$ such that
(50)-(52) hold, then by a direct computation we have that

$\displaystyle\phi_{V}D(x,y)(v)$ $\displaystyle=$ $\displaystyle
D(\phi_{\mathfrak{g}}(x),\phi_{\mathfrak{g}}(y))(\phi_{V}(v)),\quad\forall
x,y\in\mathfrak{g},~{}v\in V.$

######  Definition 5.3.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$ and
$\mathfrak{T}:V\to\mathfrak{g}$ a linear map. If $T_{t}=T+t\mathfrak{T}$ are
still relative Rota-Baxter operators on
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
for all $t$, we say that $\mathfrak{T}$ generates a linear deformation of the
relative Rota-Baxter operator $T$.

######  Remark 5.4.

If $\mathfrak{T}$ generates a linear deformation of a relative Rota-Baxter
operator $T$, then we have that

  * (i)

$\mathfrak{T}$ is a relative Rota-Baxter operator on the Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to the representation $(V;\rho,\mu)$;

  * (ii)

$\mathfrak{T}\in{\mathcal{C}}^{1}(V,\mathfrak{g})$ is a $1$-cocycle of
$\delta^{T}$.

######  Definition 5.5.

Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$.

  * (i)

Two linear deformations $T_{t}^{1}=T+t\mathfrak{T}_{1}$ and
$T_{t}^{2}=T+t\mathfrak{T}_{2}$ are said to be equivalent if there exists an
element $\mathfrak{X}\in\wedge^{2}\mathfrak{g}$ such that
$({{\rm{Id}}}_{\mathfrak{g}}+t\mathfrak{L}_{\mathfrak{X}},{{\rm{Id}}}_{V}+tD(\mathfrak{X}))$
is a homomorphism from $T_{t}^{2}$ to $T_{t}^{1}$.

  * (ii)

A linear deformation $T_{t}=T+t\mathfrak{T}$ of a relative Rota-Baxter
operator $T$ is said to be trivial if there exists an element
$\mathfrak{X}\in\wedge^{2}\mathfrak{g}$ such that
$({{\rm{Id}}}_{\mathfrak{g}}+t\mathfrak{L}_{\mathfrak{X}},{{\rm{Id}}}_{V}+tD(\mathfrak{X}))$
is a homomorphism from $T_{t}$ to $T$.

Let
$({{\rm{Id}}}_{\mathfrak{g}}+t\mathfrak{L}_{\mathfrak{X}},{{\rm{Id}}}_{V}+tD(\mathfrak{X}))$
be a homomorphism from $T_{t}^{2}$ to $T_{t}^{1}$. Then
${{\rm{Id}}}_{\mathfrak{g}}+t\mathfrak{L}_{\mathfrak{X}}$ is a Lie-Yamaguti
algebra homomorphism of $\mathfrak{g}$, which is equivalent to the following
conditions:

(53)
$\displaystyle~{}[\left\llbracket\mathfrak{X},x\right\rrbracket,\left\llbracket\mathfrak{X},y\right\rrbracket]$
$\displaystyle=$ $\displaystyle 0,$ (54)
$\displaystyle~{}\left\llbracket\left\llbracket\mathfrak{X},x\right\rrbracket,\left\llbracket\mathfrak{X},y\right\rrbracket,z\right\rrbracket+\left\llbracket\left\llbracket\mathfrak{X},x\right\rrbracket,y,\left\llbracket\mathfrak{X},z\right\rrbracket\right\rrbracket+\left\llbracket
x,\left\llbracket\mathfrak{X},y\right\rrbracket,\left\llbracket\mathfrak{X},z\right\rrbracket\right\rrbracket$
$\displaystyle=$ $\displaystyle 0,$ (55)
$\displaystyle~{}\left\llbracket\left\llbracket\mathfrak{X},x\right\rrbracket,\left\llbracket\mathfrak{X},y\right\rrbracket,\left\llbracket\mathfrak{X},z\right\rrbracket\right\rrbracket$
$\displaystyle=$ $\displaystyle 0.$

By
$T_{t}^{1}\big{(}({{\rm{Id}}_{V}}+tD(\mathfrak{X}))v\big{)}=\big{(}{{\rm{Id}}}_{\mathfrak{g}}+t\mathfrak{L}_{\mathfrak{X}}\big{)}T_{t}^{2}(v)$,
we have

(56) $\displaystyle(\mathfrak{T}_{2}-\mathfrak{T}_{1})(v)$ $\displaystyle=$
$\displaystyle
T\Big{(}D(\mathfrak{X})v\Big{)}-\left\llbracket\mathfrak{X},Tv\right\rrbracket,$
(57) $\displaystyle\mathfrak{T}_{1}\Big{(}D(\mathfrak{X})v\Big{)}$
$\displaystyle=$
$\displaystyle\left\llbracket\mathfrak{X},\mathfrak{T}_{2}(v)\right\rrbracket$

Finally, by
$\Big{(}{{\rm{Id}}}_{V}+tD(\mathfrak{X})\Big{)}\mu(z,w)v=\mu\Big{(}({{\rm{Id}}}_{\mathfrak{g}}+t\mathfrak{L}_{\mathfrak{X}})z,({{\rm{Id}}}_{\mathfrak{g}}+t\mathfrak{L}_{\mathfrak{X}})w\Big{)}({{\rm{Id}}}_{V}+tD(\mathfrak{X}))v$,
we have

(58)
$\displaystyle\mu(z,\left\llbracket\mathfrak{X},w\right\rrbracket)D(\mathfrak{X})+\mu(\left\llbracket\mathfrak{X},z\right\rrbracket,w)D(\mathfrak{X})+\mu(\left\llbracket\mathfrak{X},z\right\rrbracket,\left\llbracket\mathfrak{X},w\right\rrbracket)$
$\displaystyle=$ $\displaystyle 0,$ (59)
$\displaystyle~{}\mu(\left\llbracket\mathfrak{X},z\right\rrbracket,\left\llbracket\mathfrak{X},w\right\rrbracket)D(\mathfrak{X})$
$\displaystyle=$ $\displaystyle 0.$

Note that (56) means that there exists
$\mathfrak{X}\in\wedge^{2}\mathfrak{g}$, such that
$\mathfrak{T}_{2}-\mathfrak{T}_{1}=\delta(\mathfrak{X})$. Thus we have the
following

######  Theorem 5.6.

Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. If two linear deformations
$T_{t}^{1}=T+t\mathfrak{T}_{1}$ and $T_{t}^{2}=T+t\mathfrak{T}_{2}$ of $T$ are
equivalent, then $\mathfrak{T}_{1}$ and $\mathfrak{T}_{2}$ are in the same
class of the cohomology group $\mathcal{H}^{1}_{T}(V,\mathfrak{g})$.

######  Definition 5.7.

Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. An element
$\mathfrak{X}\in\wedge^{2}\mathfrak{g}$ is called a Nijenhuis element with
respect to $T$ if $\mathfrak{X}$ satisfies (53)-(55), (58), (59) and the
following equation

$\displaystyle\left\llbracket\mathfrak{X},T(D(\mathfrak{X})v)-\left\llbracket\mathfrak{X},Tv\right\rrbracket\right\rrbracket=0,\quad\forall
v\in V.$

It is obvious that a trivial deformation of a relative Rota-Baxter operator on
a Lie-Yamaguti algebra gives rise to a Nijenhuis element. Indeed, the converse
is also true.

######  Proposition 5.8.

Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie-Yamaguti
algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. Then for any Nijenhuis
element $\mathfrak{X}\in\wedge^{2}\mathfrak{g}$, $T_{t}:=T+t\mathfrak{T}$ with
$\mathfrak{T}:=\delta(\mathfrak{X})$ is a trivial linear deformation of the
relative Rota-Baxter operator $T$.

###### Proof.

The proof is similar to the case of Lie algebras or Leibniz algebras etc. Thus
we omit the details. ∎

######  Example 5.9.

Let
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
be a 2-dimensional Lie-Yamaguti algebra, whose nontrivial brackets are given
by

$[e_{1},e_{2}]=e_{1},~{}\quad\left\llbracket
e_{1},e_{2},e_{2}\right\rrbracket=e_{1},$

where $\\{e_{1},e_{2}\\}$ is a basis for $\mathfrak{g}$. Moreover,

$R=\begin{pmatrix}0&a\\\ 0&b\end{pmatrix}$

is a Rota-Baxter operator on $\mathfrak{g}$. Then by a direct computation, any
element in $\wedge^{2}\mathfrak{g}$ is a Nijenhuis element of $R$.

######  Example 5.10.

Let $\mathfrak{g}$ be a 4-dimensional Lie-Yamaguti algebra with a basis
$\\{e_{1},e_{2},e_{3},e_{4}\\}$ defined by

$[e_{1},e_{2}]=2e_{4},~{}\quad\left\llbracket
e_{1},e_{2},e_{1}\right\rrbracket=e_{4}.$

And

$R=\begin{pmatrix}0&a_{12}&0&0\\\ 0&0&0&0\\\ a_{31}&a_{32}&a_{33}&a_{34}\\\
a_{41}&a_{42}&a_{43}&a_{44}\end{pmatrix}$

is a Rota-Baxter operator on $\mathfrak{g}$. Then any element in
$\wedge^{2}\mathfrak{g}$ is a Nijenhuis element of $R$. In particular,

$\displaystyle\mathfrak{X}_{1}=e_{1}\wedge
e_{2},\quad\mathfrak{X}_{2}=e_{1}\wedge
e_{3},\quad\mathfrak{X}_{3}=e_{1}\wedge e_{4},$
$\displaystyle\mathfrak{X}_{4}=e_{2}\wedge
e_{3},\quad\mathfrak{X}_{5}=e_{2}\wedge
e_{4},\quad\mathfrak{X}_{6}=e_{3}\wedge e_{4},$

are all Nijenhuis elements of $R$.

###  5.2. Higher order deformations of relative Rota-Baxter operators

In this subsection, we introduce a cohomology class associated to an order $n$
deformation of a relative Rota-Baxter operator, and show that an order $n$
deformation is extendable if and only if this cohomology class is trivial.
Thus we call this cohomology class the obstruction class of an order $n$
deformation being extendable.

######  Definition 5.11.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. If
$T_{t}=\sum_{i=0}^{n}\mathfrak{T}_{i}t^{i}$ with $\mathfrak{T}_{0}=T$,
$\mathfrak{T}_{i}\in\mathrm{Hom}_{\mathbb{K}}(V,\mathfrak{g})$,
$i=1,2,\cdots,n$, defines a $\mathbb{K}[t]/(t^{n+1})$-module from
$V[t]/(t^{n+1})$ to the Lie-Yamaguti algebra $\mathfrak{g}[t]/(t^{n+1})$
satisfying

$\displaystyle[T_{t}u,T_{t}v]$ $\displaystyle=$ $\displaystyle
T_{t}(\rho(T_{t})u-\rho(T_{t}v)u),$ $\displaystyle\left\llbracket
T_{t},T_{t}v,T_{t}w\right\rrbracket$ $\displaystyle=$ $\displaystyle
T_{t}(D_{\rho,\mu}(T_{t}u,T_{t}v)w+\mu(T_{t}v,T_{t}w)u-\mu(T_{t}u,T_{t}w)v),\quad\forall
u,v,w\in V,$

we say that $T_{t}$ is an order $n$ deformation of the relative Rota-Baxter
operator $T$.

######  Remark 5.12.

The left hand sides of the equations hold in the Lie-Yamaguti algebra
$\mathfrak{g}[t]/(t^{n+1})$ and the right hand sides of the equations above
make sense since $T_{t}$ is a $\mathbb{K}[t]/(t^{n+1})$-module map.

######  Definition 5.13.

Let $T_{t}=\sum_{i=0}^{n}\mathfrak{T}_{i}t^{i}$ be an order $n$ deformation of
a relative Rota-Baxter operator $T$ on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. If there exists a $1$-cochain
$\mathfrak{T}_{n+1}\in\mathrm{Hom}_{\mathbb{K}}(V,\mathfrak{g})$ such that
$\widetilde{T_{t}}=T_{t}+\mathfrak{T}_{n+1}t^{n+1}$ is an order $n+1$
deformation of $T$, then we say that $T_{t}$ is extensible.

######  Theorem 5.14.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$, and
$T_{t}=\sum_{i=0}^{n}\mathfrak{T}_{i}t^{i}$ be an order $n$ deformation of
$T$. Then $T_{t}$ is extensible if and only if the cohomology class
$[\mathsf{Ob}^{T}]\in\mathcal{H}_{T}^{2}(V,\mathfrak{g})$ is trivial, where
$\mathsf{Ob}^{T}=(\mathsf{Ob}_{I}^{T},\mathsf{Ob}_{II}^{T})\in{\mathcal{C}}_{T}^{2}(V,\mathfrak{g})$
is defined to be

$\displaystyle\mathsf{Ob}_{I}^{T}(v_{1},v_{2})$ $\displaystyle=$
$\displaystyle\sum_{i+j=n+1,\atop i,j\geqslant
1}\Big{(}[\mathfrak{T}_{i}v_{1},\mathfrak{T}_{j}v_{2}]-\mathfrak{T}_{i}(\rho(\mathfrak{T}_{j}v_{1})v_{2}-\rho(\mathfrak{T}_{j}v_{2})v_{1})\Big{)},$
$\displaystyle\mathsf{Ob}_{II}^{T}(v_{1},v_{2},v_{3})$ $\displaystyle=$
$\displaystyle\sum_{i+j+k=n+1,\atop i,j,k\geqslant
1}\Big{(}\left\llbracket\mathfrak{T}_{i}v_{1},\mathfrak{T}_{j}v_{2},\mathfrak{T}_{k}v_{3}\right\rrbracket-\mathfrak{T}_{i}(D(\mathfrak{T}_{j}v_{1},\mathfrak{T}_{k}v_{2})v_{3}$
$\displaystyle+\mu(\mathfrak{T}_{j}v_{2},\mathfrak{T}_{k}v_{3})v_{1}-\mu(\mathfrak{T}_{j}v_{1},\mathfrak{T}_{k}v_{3})v_{2}\Big{)},\quad\forall
v_{1},v_{2},v_{3}\in V.$

###### Proof.

Let $\widetilde{T_{t}}=\sum_{i=0}^{n+1}\mathfrak{T}_{i}t^{i}$ be the extension
of $T_{t}$, then for all $u,v,w\in V$,

(60) $\displaystyle[\widetilde{T_{t}}u,\widetilde{T_{t}}v]$ $\displaystyle=$
$\displaystyle\widetilde{T_{t}}\Big{(}\rho(\widetilde{T_{t}}u)v-\rho(\widetilde{T_{t}}v)u\Big{)},$
(61)
$\displaystyle\left\llbracket\widetilde{T_{t}}u,\widetilde{T_{t}}v,\widetilde{T_{t}}w\right\rrbracket$
$\displaystyle=$
$\displaystyle\widetilde{T_{t}}\Big{(}D(\widetilde{T_{t}}u,\widetilde{T_{t}}v)w+\mu(\widetilde{T_{t}}v,\widetilde{T_{t}}w)u-\mu(\widetilde{T_{t}}u,\widetilde{T_{t}}w)v\Big{)}.$

Expanding the Eq. (60) and comparing the coefficients of $t^{n}$ yields that

$\displaystyle\sum_{i+j=n+1,\atop i,j\geqslant
0}\Big{(}[\mathfrak{T}_{i}u,\mathfrak{T}_{j}v]-\mathfrak{T}_{i}\big{(}\rho(\mathfrak{T}_{j}u)v-\rho(\mathfrak{T}_{j}v)u\big{)}\Big{)}=0,$

which is equivalent to

$\displaystyle\sum_{i+j=n+1,\atop i,j\geqslant
1}\Big{(}[\mathfrak{T}_{i}u,\mathfrak{T}_{j}v]-\mathfrak{T}_{i}\big{(}\rho(\mathfrak{T}_{j}u)v-\rho(\mathfrak{T}_{j}v)u\big{)}\Big{)}+[\mathfrak{T}_{n+1}u,Tv]+[Tu,\mathfrak{T}_{n+1}v]$
$\displaystyle\quad-\big{(}T(\rho(\mathfrak{T}_{n+1}u)v-\rho(\mathfrak{T}_{n+1}v)u)+\mathfrak{T}_{n+1}(\rho(Tu)v-\rho(Tv)u)\big{)}=0,$

which is also equivalent to

(62) $\displaystyle\mathsf{Ob}_{I}^{T}+\delta_{I}^{T}(\mathfrak{T}_{n+1})=0.$

Similarly, expanding the Eq. (61) and comparing the coefficients of $t^{n}$
yields that

(63)
$\displaystyle\mathsf{Ob}_{II}^{T}+\delta_{II}^{T}(\mathfrak{T}_{n+1})=0.$

From (62) and (63), we get

$\mathsf{Ob}_{T}=-\delta^{T}(\mathfrak{T}_{n+1}).$

Thus, the cohomology class $[\mathsf{Ob}_{T}]$ is trivial.

Conversely, suppose that the cohomology class $[\mathsf{Ob}_{T}]$ is trivial,
then there exists
$\mathfrak{T}_{n+1}\in{\mathcal{C}}_{T}^{1}(V,\mathfrak{g})$, such that
$\mathsf{Ob}_{T}=-\delta^{T}(\mathfrak{T}_{n+1}).$ Set
$\widetilde{T_{t}}=T_{t}+\mathfrak{T}_{n+1}t^{n+1}$. Then for all $0\leqslant
s\leqslant n+1$, $\widetilde{T_{t}}$ satisfies

$\displaystyle\sum_{i+j=s}\Big{(}[\mathfrak{T}_{i}u,\mathfrak{T}_{j}v]-\mathfrak{T}_{i}\big{(}\rho(\mathfrak{T}_{j}u)v-\rho(\mathfrak{T}_{j}v)u\big{)}\Big{)}=0,$
$\displaystyle\sum_{i+j+k=s}\Big{(}\left\llbracket\mathfrak{T}_{i}u,\mathfrak{T}_{j}v,\mathfrak{T}_{k}w\right\rrbracket-\mathfrak{T}_{i}\big{(}D(\mathfrak{T}_{j}u,\mathfrak{T}_{k}v)w+\mu(\mathfrak{T}_{j}v,\mathfrak{T}_{k}w)u-\mu(\mathfrak{T}_{j}u,\mathfrak{T}_{k}w)v\big{)}\Big{)}=0,$

which implies that $\widetilde{T_{t}}$ is an order $n+1$ deformation of $T$.
Hence it is a extension of $T_{t}$. ∎

######  Definition 5.15.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$, and
$T_{t}=\sum_{i=0}^{n}\mathfrak{T}_{i}t^{i}$ be an order $n$ deformation of
$T$. Then the cohomology class
$[\mathsf{Ob}^{T}]\in\mathcal{H}_{T}^{2}(V,\mathfrak{g})$ defined in
Definition 5.14 is called the obstruction class  of $T_{t}$ being extensible.

######  Corollary 5.16.

Let $T$ be a relative Rota-Baxter operator on a Lie-Yamaguti algebra
$(\mathfrak{g},[\cdot,\cdot],\left\llbracket\cdot,\cdot,\cdot\right\rrbracket)$
with respect to a representation $(V;\rho,\mu)$. If
$\mathcal{H}_{T}^{2}(V,\mathfrak{g})=0$, then every $1$-cocycle in
$\mathcal{Z}_{T}^{1}(V,\mathfrak{g})$ is the infinitesimal of some formal
deformation of the relative Rota-Baxter operator $T$.

Acknowledgements: Qiao was partially supported by NSFC grant 11971282.

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