# From relative Rota-Baxter operators and relative averaging operators on Lie
algebras to relative Rota-Baxter operators on Leibniz algebras: a uniform
approach

Rong Tang  Department of Mathematics, Jilin University, Changchun 130012,
Jilin, China<EMAIL_ADDRESS>,  Yunhe Sheng  Department of Mathematics,
Jilin University, Changchun 130012, Jilin, China<EMAIL_ADDRESS>and
Friedrich Wagemann  Laboratoire de Mathematiques Jean Leray, University de
Nantes, France<EMAIL_ADDRESS>
###### Abstract.

In this paper, first we construct two subcategories (using symmetric
representations and antisymmetric representations) of the category of relative
Rota-Baxter operators on Leibniz algebras, and establish the relations with
the categories of relative Rota-Baxter operators and relative averaging
operators on Lie algebras. Then we show that there is a short exact sequence
describing the relation between the controlling algebra of relative Rota-
Baxter operators on a Leibniz algebra with respect to a symmetric (resp.
antisymmetric) representation and the controlling algebra of the induced
relative Rota-Baxter operators (resp. averaging operators) on the canonical
Lie algebra associated to the Leibniz algebra. Finally, we show that there is
a long exact sequence describing the relation between the cohomology groups of
a relative Rota-Baxter operator on a Leibniz algebra with respect to a
symmetric (resp. antisymmetric) representation and the cohomology groups of
the induced relative Rota-Baxter operator (resp. averaging operator) on the
canonical Lie algebra.

###### Key words and phrases:

Rota-Baxter operator, averaging operator, cohomology, Lie algebra, Leibniz
algebra

###### Contents

  1. 1 Introduction
    1. 1.1 Relative Rota-Baxter operators on Lie algebras
    2. 1.2 Averaging operators on Lie algebras
    3. 1.3 Relative Rota-Baxter operators on Leibniz algebras
    4. 1.4 Outline of the paper
  2. 2 Relations between the categories $\mathsf{AO}\mathsf{Lie}$, $\mathsf{RBO}\mathsf{Lie}$ and $\mathsf{RBO}\mathsf{Leibniz}$
    1. 2.1 Relative Rota-Baxter operators and relative averaging operators
    2. 2.2 Relation to crossed modules
    3. 2.3 Relations between the categories $\mathsf{RBO}\mathsf{Lie}$ and $\mathsf{RBO}\mathsf{LeibnizS}$
    4. 2.4 Relations between the categories $\mathsf{AO}\mathsf{Lie}$ and $\mathsf{RBO}\mathsf{LeibnizA}$
  3. 3 Relations between the controlling algebras
    1. 3.1 The controlling algebra of relative Rota-Baxter operators on a Leibniz algebra
    2. 3.2 The controlling algebra of relative Rota-Baxter operators on a Lie algebra
    3. 3.3 The controlling algebras of relative averaging operators on a Lie algebra
  4. 4 Relations between the cohomologies
    1. 4.1 Cohomology of relative Rota-Baxter operators on Leibniz algebras
    2. 4.2 The Loday-Pirashvili cohomology of relative Rota-Baxter operators on Lie algebras
    3. 4.3 Cohomology of relative averaging operators on Lie algebras
    4. 4.4 Some computations of cohomologies

##  1\. Introduction

The purpose of this paper is to provide a unified approach to study relative
Rota-Baxter operators and relative averaging operators on Lie algebras.

###  1.1. Relative Rota-Baxter operators on Lie algebras

G. Baxter introduced the concept of a Rota-Baxter commutative algebra [8] in
his study of fluctuation theory in probability. A Rota-Baxter operator of
weight zero on an associative algebra $(A,\cdot)$ is a linear map
$\mathcal{R}:A\,\rightarrow\,A$ such that

$(\mathcal{R}x)\cdot(\mathcal{R}y)=\mathcal{R}((\mathcal{R}x)\cdot
y+x\cdot(\mathcal{R}y)),\quad\forall x,y\in A.$

These operators with general weights have been found many applications in
recent years, including the algebraic approach of Connes-Kreimer [13] to
renormalization of perturbative quantum field theory, quantum analogue of
Poisson geometry [48], twisting on associative algebras [49], dendriform
algebras and associative Yang-Baxter equations [1]. The notion of Rota-Baxter
operators on Lie algebras was introduced independently in the 1980s as the
operator form of the classical Yang-Baxter equation. A linear operator
$\mathcal{R}:\mathfrak{g}\longrightarrow\mathfrak{g}$ on a Lie algebra
$\mathfrak{g}$ is called a Rota-Baxter operator of weight zero if the
following condition is satisfied:

$[\mathcal{R}(x),\mathcal{R}(y)]_{\mathfrak{g}}=\mathcal{R}([\mathcal{R}(x),y]_{\mathfrak{g}}+[x,\mathcal{R}(y)]_{\mathfrak{g}}),\quad\forall~{}x,y\in\mathfrak{g}.$

Moreover, Kupershmidt introduced the notion of a relative Rota-Baxter operator
(also called an ${\mathcal{O}}$-operator) on a Lie algebra $\mathfrak{g}$ with
respect to arbitrary representation in [29]. Note that a skew-symmetric
classical $r$-matrix is a relative Rota-Baxter operator on a Lie algebra with
respect to the coadjoint representation. Relative Rota-Baxter operators play
important roles in the study of integrable systems [6, 40], provide solutions
of the classical Yang-Baxter equation in the semidirect product Lie algebra
and give rise to pre-Lie algebras [5]. See [21] for more details. Recently,
the deformation and homotopy theory of relative Rota-Baxter Lie algebras and
relative Rota-Baxter associative algebras were established in [15, 30, 44,
45].

###  1.2. Averaging operators on Lie algebras

An averaging operator on an associative algebra $(A,\cdot)$ is a linear
operator $\mathcal{R}:A\,\rightarrow\,A$ such that

$(\mathcal{R}x)\cdot(\mathcal{R}y)=\mathcal{R}((\mathcal{R}x)\cdot
y)=\mathcal{R}(x\cdot(\mathcal{R}y)),\quad\forall x,y\in A.$

In the last century, many studies on averaging operators were done for various
special algebras, such as function spaces, Banach algebras, and the topics and
methods were largely analytic [7, 12, 24, 39]. Recently, it was found that
commutative double algebra structures [20] on a vector space $V$ can be
described by symmetric averaging operators on the associative algebra
$\mathrm{End}(V)$. Averaging operators can be defined on algebras over
arbitrary binary operad [1]. In particular, a linear operator
$\mathcal{R}:\mathfrak{g}\longrightarrow\mathfrak{g}$ on a Lie algebra
$\mathfrak{g}$ is called an averaging operator if the following condition is
satisfied:

$[\mathcal{R}(x),\mathcal{R}(y)]_{\mathfrak{g}}=\mathcal{R}([\mathcal{R}(x),y]_{\mathfrak{g}}),\quad\forall~{}x,y\in\mathfrak{g}.$

There is a close relation between the classical Yang-Baxter equation,
conformal algebras and averaging operators on Lie algebras [26].

Averaging operators on Lie algebras are also called embedding tensors in some
mathematical physics literatures. Embedding tensors and their associated
tensor hierarchies provide an algebraic and efficient way to construct
supergravity theories and further to construct higher gauge theories (see e.g.
[10, 23, 28, 42]). Recently the controlling algebra and the cohomology theory
of embedding tensors were established in [41] using the higher derived
bracket. Meanwhile the cohomology and homotopy theory of averaging associative
algebras were studied in [52].

The concept of Rota-Baxter operators and averaging operators were further
studied on the level of algebraic operads in [3, 35, 36, 37, 50, 51]. The
action of the Rota-Baxter operator on a binary quadratic operad splits the
operad, and the action of the averaging operator on a binary quadratic operad
duplicates the operad. Since the splitting and duplication processes are in
Koszul duality, one can regard the Rota-Baxter operator and averaging operator
to be Koszul dual to each other.

###  1.3. Relative Rota-Baxter operators on Leibniz algebras

Leibniz algebras were first discovered by Bloh who called them D-algebras [9].
Then Loday rediscovered this algebraic structure and called them Leibniz
algebras with the motivation in the study of the periodicity in algebraic
K-theory [31, 33, 34]. Averaging operators on Lie algebras give rise to
Leibniz algebras, which can be viewed as the duplication of Lie algebras. From
the viewpoint of the operad theory, leibniz algebras are defined as the
algebras over the $Leibniz$ operad which is the duplicator of the $Lie$ operad
[36]. More intrinsically, let $\mathcal{P}$ be a binary quadratic operad. The
algebra duplicates by an averaging operator on the $\mathcal{P}$-algebra is a
$\mathrm{Du}(\mathcal{P})$-algebra. On the other hand, the algebra splits by a
Rota-Baxter operator on the $\mathcal{P}$-algebra is a
$\mathrm{BSu}(\mathcal{P})$-algebra. Since $\mathrm{Du}(\mathcal{P})$ is the
Koszul dual of $\mathrm{BSu}(\mathcal{P}^{!})$ [36], the duplication
construction of an averaging operator is a kind of Koszul dual of the
splitting construction of a Rota-Baxter operator [3, 36]. The (co)homology and
homotopy theories of Leibniz algebras were established in [18, 33, 38, 2].
Recently Leibniz algebras were studied from different aspects due to
applications in both mathematics and physics. In particular, integration of
Leibniz algebras were studied in [11, 14] and deformation quantization of
Leibniz algebras was studied in [16]. As the underlying algebraic structures
of embedding tensors and Courant algebroids, Leibniz algebras also have
application in higher gauge theories [10, 28, 43] and homogeneous spaces [25].
The notion of relative Rota-Baxter operators on Leibniz algebras was
introduced in [46] to study Leibniz bialgebras. Moreover, the cohomology
theory of relative Rota-Baxter operators on Leibniz algebras was given in [47]
and used to classify linear deformations and formal deformations.

###  1.4. Outline of the paper

In this paper, we propose a unified approach to study relative Rota-Baxter
operators and relative averaging operators on Lie algebras using relative
Rota-Baxter operators on Leibniz algebras. First we observe that there are
forgetful functors from the categories of relative Rota-Baxter operators and
relative averaging operators on Lie algebras to the category of relative Rota-
Baxter operators on Leibniz algebras. Conversely, we construct two functors
from certain subcategories of the category of relative Rota-Baxter operators
on Leibniz algebras to the categories of relative Rota-Baxter operators and
relative averaging operators on Lie algebras, and show that they are left
adjoint for the aforementioned forgetful functors. Then we show that the
controlling algebras of relative Rota-Baxter operators and relative averaging
operators on Lie algebras can be derived from the controlling algebra of
relative Rota-Baxter operators on Leibniz algebras. Finally, we give a long
exact sequence to describe the relation between the cohomology groups of a
relative Rota-Baxter operator on a Leibniz algebra with respect to a symmetric
(resp. antisymmetric) representation and the cohomology groups of the induced
relative Rota-Baxter operator (resp. averaging operator) on the canonical Lie
algebra. The concrete relations can be summarized by the following diagrams:

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The paper is organized as follows. In Section 2, using symmetric
representations and antisymmetric representations, we construct two
subcategories of the category of relative Rota-Baxter operators on Leibniz
algebras, and establish the relations with the categories of relative Rota-
Baxter operators and relative averaging operators on Lie algebras. In Section
3, we show that there is a short exact sequence describing the relation
between the controlling algebra of relative Rota-Baxter operators on a Leibniz
algebra with respect to a symmetric representation and the controlling algebra
of the induced relative Rota-Baxter operators on the canonical Lie algebra.
For relative averaging operators, we establish a similar result. In Section 4,
first we introduce the Loday-Pirashvili cohomology of a relative Rota-Baxter
operator on a Lie algebra and then use a long exact sequence to describe the
relation between the cohomology groups of a relative Rota-Baxter operator on a
Leibniz algebra with respect to a symmetric representation and the Loday-
Pirashvili cohomology groups of the induced relative Rota-Baxter operator on
the canonical Lie algebra. Similarly, there is a long exact sequence
describing the relation between the cohomology groups of a relative Rota-
Baxter operator on a Leibniz algebra with respect to an antisymmetric
representation and the cohomology groups of the induced relative averaging
operator on the canonical Lie algebra.

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

##  2\. Relations between the categories $\mathsf{AO}\mathsf{Lie}$,
$\mathsf{RBO}\mathsf{Lie}$ and $\mathsf{RBO}\mathsf{Leibniz}$

In this section, we establish the relations between certain subcategories of
the category of relative Rota-Baxter operators on Leibniz algebras and the
category of relative Rota-Baxter operators on Lie algebras as well as the
category of relative averaging operators on Lie algebras.

###  2.1. Relative Rota-Baxter operators and relative averaging operators

######  Definition 2.1.

([29]) A relative Rota-Baxter operator on a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to a representation
$(V;\rho)$ is a linear map $T:V\longrightarrow\mathfrak{g}$ satisfying the
following quadratic constraint:

(1)
$[Tu,Tv]_{\mathfrak{g}}=T\big{(}\rho(Tu)(v)-\rho(Tv)(u)\big{)},\quad\forall
u,v\in V.$

Let $T:V\longrightarrow\mathfrak{g}$ (resp.
$T^{\prime}:V^{\prime}\longrightarrow\mathfrak{g}^{\prime}$) be a relative
Rota-Baxter operator on the Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$
(resp. $(\mathfrak{g}^{\prime},\\{-,-\\}_{\mathfrak{g}^{\prime}})$) with
respect to the representation $(V;\rho)$ (resp. $(V^{\prime},\rho^{\prime})$).
A homomorphism from $T$ to $T^{\prime}$ is a pair $(\phi,\varphi)$, where
$\phi:\mathfrak{g}\longrightarrow\mathfrak{g}^{\prime}$ is a Lie algebra
homomorphism, $\varphi:V\longrightarrow V^{\prime}$ is a linear map such that

(2) $\displaystyle T^{\prime}\circ\varphi$ $\displaystyle=$
$\displaystyle\phi\circ T,$ (3) $\displaystyle\varphi\rho(x)(u)$
$\displaystyle=$ $\displaystyle\rho^{\prime}(\phi(x))(\varphi(u)),\quad\forall
x\in\mathfrak{g},u\in V.$

In particular, if $\phi$ and $\varphi$ are invertible, then $(\phi,\varphi)$
is called an isomorphism.

We denote by $\mathsf{RBO}\mathsf{Lie}$ the category of relative Rota-Baxter
operators on Lie algebras.

######  Definition 2.2.

([1, 39]) A relative averaging operator on a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to a representation
$(V;\rho)$ is a linear map $T:V\longrightarrow\mathfrak{g}$ satisfying the
following quadratic constraint:

(4) $[Tu,Tv]_{\mathfrak{g}}=T\big{(}\rho(Tu)(v)\big{)},\quad\forall u,v\in V.$

Let $T:V\longrightarrow\mathfrak{g}$ (resp.
$T^{\prime}:V^{\prime}\longrightarrow\mathfrak{g}^{\prime}$) be a relative
averaging operator on the Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$
(resp. $(\mathfrak{g}^{\prime},\\{-,-\\}_{\mathfrak{g}^{\prime}})$) with
respect to the representation $(V;\rho)$ (resp. $(V^{\prime},\rho^{\prime})$).
A homomorphism from $T$ to $T^{\prime}$ is a pair $(\phi,\varphi)$, where
$\phi:\mathfrak{g}\longrightarrow\mathfrak{g}^{\prime}$ is a Lie algebra
homomorphism, $\varphi:V\longrightarrow V^{\prime}$ is a linear map such that

(5) $\displaystyle T^{\prime}\circ\varphi$ $\displaystyle=$
$\displaystyle\phi\circ T,$ (6) $\displaystyle\varphi\rho(x)(u)$
$\displaystyle=$ $\displaystyle\rho^{\prime}(\phi(x))(\varphi(u)),\quad\forall
x\in\mathfrak{g},u\in V.$

In particular, if $\phi$ and $\varphi$ are invertible, then $(\phi,\varphi)$
is called an isomorphism.

We denote by $\mathsf{AO}\mathsf{Lie}$ the category of relative averaging
operators on Lie algebras.

A (left) Leibniz algebra is a vector space $\mathfrak{G}$ together with a
bilinear operation
$[-,-]_{\mathfrak{G}}:\mathfrak{G}\otimes\mathfrak{G}\,\rightarrow\,\mathfrak{G}$
such that

$\displaystyle[x,[y,z]_{\mathfrak{G}}]_{\mathfrak{G}}=[[x,y]_{\mathfrak{G}},z]_{\mathfrak{G}}+[y,[x,z]_{\mathfrak{G}}]_{\mathfrak{G}},\quad\forall
x,y,z\in\mathfrak{G}.$

A representation of a Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ is
a triple $(V;\rho^{L},\rho^{R})$, where $V$ is a vector space,
$\rho^{L},\rho^{R}:\mathfrak{G}\,\rightarrow\,\mathfrak{gl}(V)$ are linear
maps such that for all $x,y\in\mathfrak{G}$,

$\displaystyle\rho^{L}([x,y]_{\mathfrak{G}})$ $\displaystyle=$
$\displaystyle[\rho^{L}(x),\rho^{L}(y)],$
$\displaystyle\rho^{R}([x,y]_{\mathfrak{G}})$ $\displaystyle=$
$\displaystyle[\rho^{L}(x),\rho^{R}(y)],$
$\displaystyle\rho^{R}(y)\circ\rho^{L}(x)$ $\displaystyle=$
$\displaystyle-\rho^{R}(y)\circ\rho^{R}(x).$

Here $[-,-]$ is the commutator Lie bracket on $\mathfrak{gl}(V)$. There are
two important kinds of representations of Leibniz algebras:

  * •

a representation $(V;\rho^{L},\rho^{R})$ is called symmetric if
$\rho^{R}=-\rho^{L}$;

  * •

a representation $(V;\rho^{L},\rho^{R})$ is called antisymmetric if
$\rho^{R}=0$.

For any representation $(V;\rho^{L},\rho^{R})$ of a Leibniz algebra
$\mathfrak{G}$, we define a vector subspace $V_{\rm anti}$ which is spanned by
all elements $\rho^{L}(x)v+\rho^{R}(x)v$ with $x\in\mathfrak{G}$ and $v\in V$.
Directly from $\rho^{R}(y)\circ\rho^{L}(x)=-\rho^{R}(y)\circ\rho^{R}(x)$ for
all $x,y\in\mathfrak{G}$, we deduce that $(V_{\rm anti};\rho^{L},\rho^{R})$ is
an antisymmetric representation. The quotient representation $V/V_{\rm anti}$
is then symmetric, denoted by $V_{\rm sym}$, and we have an exact sequence

(7) $0\to V_{\rm anti}\to V\to V_{\rm sym}\to 0.$

Now we recall the notion of a relative Rota-Baxter operator on a Leibniz
algebra.

######  Definition 2.3.

([46]) Let $(V;\rho^{L},\rho^{R})$ be a representation of a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. A linear operator
$T:V\,\rightarrow\,\mathfrak{G}$ is called a relative Rota-Baxter operator on
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to $(V;\rho^{L},\rho^{R})$
if $T$ satisfies:

(8)
$\displaystyle[Tu,Tv]_{\mathfrak{G}}=T(\rho^{L}(Tu)v+\rho^{R}(Tv)u),\,\,\,\,\forall
u,v\in V.$

Let $T:V\longrightarrow\mathfrak{G}$ (resp.
$T^{\prime}:V^{\prime}\longrightarrow\mathfrak{G}^{\prime}$) be a relative
Rota-Baxter operator on the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ (resp.
$(\mathfrak{G}^{\prime},\\{-,-\\}_{\mathfrak{G}^{\prime}})$) with respect to
the representation $(V;\rho^{L},\rho^{R})$ (resp.
$(V;{\rho^{L}}^{\prime},{\rho^{R}}^{\prime})$). A homomorphism from $T$ to
$T^{\prime}$ is a pair $(\phi,\varphi)$, where
$\phi:\mathfrak{G}\longrightarrow\mathfrak{G}^{\prime}$ is a Leibniz algebra
homomorphism, $\varphi:V\longrightarrow V^{\prime}$ is a linear map such that
for all $x\in\mathfrak{G},u\in V$,

(9) $\displaystyle T^{\prime}\circ\varphi$ $\displaystyle=$
$\displaystyle\phi\circ T,$ (10) $\displaystyle\varphi{\rho^{L}}(x)(u)$
$\displaystyle=$ $\displaystyle{\rho^{L}}^{\prime}(\phi(x))(\varphi(u)),$ (11)
$\displaystyle\varphi{\rho^{R}}(x)(u)$ $\displaystyle=$
$\displaystyle{\rho^{R}}^{\prime}(\phi(x))(\varphi(u)).$

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

We denote by $\mathsf{RBO}\mathsf{Leibniz}$ the category of relative Rota-
Baxter operators on Leibniz algebras. Moreover, we denote by
$\mathsf{RBO}\mathsf{LeibnizS}$ and $\mathsf{RBO}\mathsf{LeibnizA}$ the
categories of relative Rota-Baxter operators on Leibniz algebras with respect
to symmetric representations and antisymmetric representations respectively.

###  2.2. Relation to crossed modules

Note that relative Rota-Baxter operators on Lie algebras and Leibniz algebras
$T:V\to\mathfrak{g}$ resp. $T:V\to\mathfrak{G}$ as well as relative averaging
operators on Lie algebras $T:V\to\mathfrak{g}$ give rise to a bracket on $V$.

For a relative Rota-Baxter operator $T:V\to\mathfrak{g}$ on a Lie algebra
$\mathfrak{g}$, the bracket

$[v,w]_{T}:=\rho(Tv)w-\rho(Tw)v$

for all $v,w\in V$ renders $V$ a Lie algebra such that $T:V\to\mathfrak{g}$
becomes a morphism of Lie algebras.

In the same vein, given a relative Rota-Baxter operator $T:V\to\mathfrak{G}$
on a Leibniz algebra $\mathfrak{G}$, the bracket

$[v,w]_{T}:=\rho^{L}(Tv)w+\rho^{R}(Tw)v$

for all $v,w\in V$ renders $V$ a Leibniz algebra such that
$T:V\to\mathfrak{G}$ becomes a morphism of Leibniz algebras.

Still in the same vein, given a relative averaging operator
$T:V\to\mathfrak{g}$ on a Lie algebra $\mathfrak{g}$, the bracket

$[v,w]_{T}:=\rho(Tv)w$

for all $v,w\in V$ renders $V$ a Leibniz algebra such that
$T:V\to\mathfrak{g}$ is a morphism of Leibniz algebras.

The next proposition shows that with an equivariance condition, the action of
$\mathfrak{g}$ (resp. $\mathfrak{G}$) on V becomes an action by derivations
with respect to these brackets. This kind of equivariance condition is
present, for example, in Loday-Pirashvili’s tensor category of linear maps,
see [32].

######  Proposition 2.4.

  1. (1)

Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie algebra
and suppose that $T$ is equivariant in the sense that
$T(\rho(x)v)=[x,Tv]_{\mathfrak{g}}$ for all $x\in\mathfrak{g}$ and all $v\in
V$. We deduce that $\rho(x)$ for all $x\in\mathfrak{g}$ is a derivation of the
Lie algebra $(V,[-,-]_{T})$. Thus, we obtain that
$\rho:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(V)$ is homomorphism of Lie
algebras.

  2. (2)

Let $T:V\to\mathfrak{G}$ be a relative Rota-Baxter operator on a Leibniz
algebra and suppose that $T$ is equivariant in the sense that
$T(\rho^{L}(x)v)=[x,Tv]_{\mathfrak{G}}$ and
$T(\rho^{R}(x)v)=[Tv,x]_{\mathfrak{G}}$ for all $x\in\mathfrak{g}$ and all
$v\in V$. Then $\mathfrak{G}$ acts on $(V,[-,-]_{T})$ by derivations (in the
Leibniz sense), i.e. for all $x\in\mathfrak{G}$ and all $v,w\in V$

$\displaystyle\rho^{L}(x)[v,w]_{T}=[\rho^{L}(x)v,w]_{T}+[v,\rho^{L}(x)w]_{T},$
$\displaystyle\rho^{R}(x)[v,w]_{T}=[v,\rho^{R}(x)w]_{T}-[w,\rho^{R}(x)v]_{T},$
$\displaystyle[\rho^{L}(x)v,w]_{T}+[\rho^{R}(x)v,w]_{T}=0.$

We deduce that the pair $(\rho^{L}(x),\rho^{R}(x))$ for all $x\in\mathfrak{G}$
is a biderivation [31] of the Leibniz algebra $(V,[-,-]_{T})$. For relative
averaging operators, equivariance will imply that the left action is by
derivations, but defining the right operation to be the opposite of the left
action or zero will not render this an action by derivations in the Leibniz
sense in general.

Note that this proposition does not give rise to crossed modules, because the
requirements $\rho(Tv)w=[v,w]$ in the Lie case and
$\rho^{L}(Tv)w=[v,w]=\rho^{R}(w)v$ in the Leibniz case for $v,w\in V$ are not
fullfilled. In the case of relative averaging operators, this requirement is
fullfilled, but the action is not by derivations in general. On the other
hand, it is well known that crossed modules of Lie algebras provide examples
of relative averaging operators, see Example 2.14 in [41].

###  2.3. Relations between the categories $\mathsf{RBO}\mathsf{Lie}$ and
$\mathsf{RBO}\mathsf{LeibnizS}$

First we construct a functor $F$ from the category $\mathsf{RBO}\mathsf{Lie}$
of relative Rota-Baxter operators on Lie algebras to the category
$\mathsf{RBO}\mathsf{LeibnizS}$ of relative Rota-Baxter operators on Leibniz
algebras with respect to symmetric representations.

On objects, the functor $F$ is defined as follows. Let
$T:V\longrightarrow\mathfrak{g}$ be a relative Rota-Baxter operator on the Lie
algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the
representation $(V;\rho)$. We view the Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ as a Leibniz algebra. Moreover the Lie
algebra representation $\rho$ gives rise to a symmetric representation
$(V;\rho,-\rho)$ of the Leibniz algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$.
Then the equality (1) means that $T:V\longrightarrow\mathfrak{g}$ is a
relative Rota-Baxter operator on the Leibniz algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the symmetric
representation $(V;\rho,-\rho)$.

On morphisms, the functor $F$ is defined as follows. Let
$T:V\longrightarrow\mathfrak{g}$ (resp.
$T^{\prime}:V^{\prime}\longrightarrow\mathfrak{g}^{\prime}$) be a relative
Rota-Baxter operator on the Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$
(resp. $(\mathfrak{g}^{\prime},\\{-,-\\}_{\mathfrak{g}^{\prime}})$) with
respect to the representation $(V;\rho)$ (resp. $(V^{\prime},\rho^{\prime})$).
Let $(\phi,\varphi)$ be a homomorphism from the relative Rota-Baxter operator
$T$ to $T^{\prime}$. Then it is obvious that $(\phi,\varphi)$ is also a
homomorphism between the above induced relative Rota-Baxter operators on
Leibniz algebras.

It is straightforward to see that $F$ defined above is indeed a functor.

In the sequel, we construct a functor $G$ from the category
$\mathsf{RBO}\mathsf{LeibnizS}$ to the category $\mathsf{RBO}\mathsf{Lie}$.

Let $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ be a Leibniz algebra. Denote by
$\mathsf{Leib}(\mathfrak{G})$ the ideal of squares spanned by all elements
$[x,x]_{\mathfrak{G}}$ for all $x\in\mathfrak{G}$. We call
$\mathsf{Leib}(\mathfrak{G})$ the Leibniz kernel of
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. Observe that
$\mathsf{Leib}(\mathfrak{G})=V_{\rm anti}$ for the adjoint representation
$(V=\mathfrak{G},\rho^{L},\rho^{R})$ with $\rho^{L}(x)=[x,-]$ and
$\rho^{R}(x)=[-,x]$. The following result is obvious.

######  Lemma 2.5.

Let $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ be a Leibniz algebra. Then

$\mathfrak{G}_{\mathrm{Lie}}:=\mathfrak{G}/\mathsf{Leib}(\mathfrak{G})$

is naturally a Lie algebra in which we denote the Lie bracket by
$[-,-]_{\mathfrak{G}_{\mathrm{Lie}}}$.

We call $\mathfrak{G}_{\mathrm{Lie}}$ the canonical Lie algebra associated to
the Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$. Observe that
$\mathfrak{G}_{\mathrm{Lie}}=V_{\rm sym}$ for the adjoint representation
$(V=\mathfrak{G},\rho^{L},\rho^{R})$. The equivalence class of
$x\in\mathfrak{G}$ in $\mathfrak{G}_{\mathrm{Lie}}$ will be denoted by
$\bar{x}$. Moreover, $\mathsf{Leib}(\mathfrak{G})$ is contained in the left
center of the Leibniz algebra $\mathfrak{G}.$ We denote by

(12) $\mathrm{pr}:\mathfrak{G}\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$

the natural projection from the Leibniz algebra $\mathfrak{G}$ to the
canonical Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$. Obviously, the map
$\mathrm{pr}$ preserves the bracket operation, i.e. the following equality
holds:

(13)
$\mathrm{pr}[x,y]_{\mathfrak{G}}=[\mathrm{pr}(x),\mathrm{pr}(y)]_{\mathfrak{G}_{\mathrm{Lie}}},\quad\forall
x,y\in\mathfrak{G}.$

Let $(V;\rho^{L},\rho^{R})$ be a representation of a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. Then
$\rho^{L}:\mathfrak{G}\,\rightarrow\,\mathfrak{gl}(V)$ is a Leibniz algebra
homomorphism. We deduce that

(14) $\displaystyle\rho^{L}(\mathsf{Leib}(\mathfrak{G}))=0.$

Thus, there is exactly one Lie algebra homomorphism
$\theta:\mathfrak{G}_{\mathrm{Lie}}\longrightarrow\mathfrak{gl}(V)$ which is
defined by

(15) $\displaystyle\theta(\bar{x})=\rho^{L}(x),\quad\forall x\in\mathfrak{G},$

such that following diagram of Leibniz algebra homomorphisms commute:

$\textstyle{\mathfrak{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}}$$\scriptstyle{\rho^{L}}$$\textstyle{\mathfrak{G}_{\mathrm{Lie}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\mathfrak{gl}(V).}$

It implies that $(V;\theta)$ is a representation of the canonical Lie algebra
$\mathfrak{G}_{\mathrm{Lie}}.$

######  Proposition 2.6.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to a
symmetric representation $(V;\rho^{L},\rho^{R}=-\rho^{L})$. Then
$\bar{T}:=\mathrm{pr}\circ T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ is a
relative Rota-Baxter operator on the canonical Lie algebra
$\mathfrak{G}_{\mathrm{Lie}}$ with respect to the representation $(V;\theta)$.

###### Proof.

Applying the projection
$\mathrm{pr}:\mathfrak{G}\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ to the
both sides of (8), we obtain

$\displaystyle[\mathrm{pr}\circ T(u),\mathrm{pr}\circ
T(v)]_{\mathfrak{G}_{\mathrm{Lie}}}$ $\displaystyle=$
$\displaystyle\mathrm{pr}\circ T(\rho^{L}(Tu)v-\rho^{L}(Tv)u)$
$\displaystyle=$ $\displaystyle\mathrm{pr}\circ T(\theta(\mathrm{pr}\circ
T(u))v-\theta(\mathrm{pr}\circ T(v))u),$

which implies that $\mathrm{pr}\circ
T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ is a relative Rota-Baxter
operator on the canonical Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$ with
respect to the representation $(V;\theta)$. ∎

######  Proposition 2.7.

Let $T:V\longrightarrow\mathfrak{G}$ (resp.
$T^{\prime}:V^{\prime}\longrightarrow\mathfrak{G}^{\prime}$) be a relative
Rota-Baxter operator on the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ (resp.
$(\mathfrak{G}^{\prime},\\{-,-\\}_{\mathfrak{G}^{\prime}})$) with respect to
the symmetric representation $(V;\rho^{L},\rho^{R}=-\rho^{L})$ (resp.
$(V;{\rho^{L}}^{\prime},{\rho^{R}}^{\prime}=-{\rho^{L}}^{\prime})$). Let
$(\phi,\varphi)$ be a homomorphism of relative Rota-Baxter operators on
Leibniz algebras from $T$ to $T^{\prime}$. Then $(\bar{\phi},\varphi)$ is a
homomorphism of relative Rota-Baxter operators on Lie algebras from $\bar{T}$
to $\bar{T^{\prime}}$, where the Lie algebra homomorphism
$\bar{\phi}:\mathfrak{G}_{\mathrm{Lie}}\,\rightarrow\,\mathfrak{G}^{\prime}_{\mathrm{Lie}}$
is defined by

$\bar{\phi}(\bar{x}):=\overline{\phi(x)},\quad\forall x\in\mathfrak{G}.$

###### Proof.

By
$\phi(\mathsf{Leib}(\mathfrak{G}))\subset\mathsf{Leib}(\mathfrak{G}^{\prime})$,
we obtain that $\phi$ induces the Lie algebra homomorphism
$\bar{\phi}:\mathfrak{G}_{\mathrm{Lie}}\,\rightarrow\,\mathfrak{G}^{\prime}_{\mathrm{Lie}}.$
Then it is straightforward to deduce that $(\bar{\phi},\varphi)$ is a
homomorphism of relative averaging operators from $\bar{T}$ to
$\bar{T^{\prime}}$. ∎

Now we are ready to give the main result in this subsection.

######  Theorem 2.8.

There is a functor $G$ from the category $\mathsf{RBO}\mathsf{LeibnizS}$ of
relative Rota-Baxter operators on Leibniz algebras with respect to symmetric
representations to the category $\mathsf{RBO}\mathsf{Lie}$ of relative Rota-
Baxter operators on Lie algebras, such that $G$ is a left adjoint for $F$.

###### Proof.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to a
symmetric representation $(V;\rho^{L},\rho^{R}=-\rho^{L})$. By Proposition
2.6, $\bar{T}:=\mathrm{pr}\circ T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$
is a relative Rota-Baxter operator on the canonical Lie algebra
$\mathfrak{G}_{\mathrm{Lie}}$ with respect to the representation $(V;\theta)$.
Thus, on objects, we define

$G(T):=\bar{T}.$

Let $(\phi,\varphi)$ be a homomorphism of relative Rota-Baxter operators on
Leibniz algebras from $T$ to $T^{\prime}$. By Proposition 2.7,
$(\bar{\phi},\varphi)$ is a homomorphism of relative Rota-Baxter operators
from $\bar{T}$ to $\bar{T^{\prime}}$. Thus, on morphisms, we define

$G(\phi,\varphi):=(\bar{\phi},\varphi).$

Then it is straightforward to see that $G$ is indeed a functor. Moreover, let
$\mathfrak{g}$ be a Lie algebra, we have $\mathsf{Leib}(\mathfrak{g})=0$.
Thus, we obtain that $G\circ F={\rm{Id}}_{\mathsf{RBO}\mathsf{Lie}}.$ Thus, we
have the identity natural transformation $\varepsilon$ (the counit of the
adjunction)

$\varepsilon={\rm{Id}}_{{\rm{Id}}_{\mathsf{RBO}\mathsf{Lie}}}:{\rm{Id}}_{\mathsf{RBO}\mathsf{Lie}}=G\circ
F\,\rightarrow\,{\rm{Id}}_{\mathsf{RBO}\mathsf{Lie}}.$

Moreover, for any relative Rota-Baxter operator
$K:W\longrightarrow\mathfrak{h}$ on a Lie algebra $\mathfrak{h}$ with respect
to a representation $(W;\rho)$, relative Rota-Baxter operator
$T:V\longrightarrow\mathfrak{G}$ on a Leibniz algebra $\mathfrak{G}$ with
respect to a symmetric representation $(V;\rho^{L},\rho^{R}=-\rho^{L})$ and
relative Rota-Baxter operator homomorphism $(\chi,\xi)$ from $G(T)$ to $K$, we
deduce that $\chi\circ\mathrm{pr}$ is a Leibniz algebra homomorphism from
$\mathfrak{G}$ to $\mathfrak{h}$ and

$\displaystyle K\circ\xi=(\chi\circ\mathrm{pr})\circ T$
$\displaystyle\xi(\rho^{L}(x)u)\stackrel{{\scriptstyle\eqref{induce-
rep}}}{{=}}\xi(\theta(\bar{x})u)=\rho(\chi(\bar{x}))\xi(u)=\rho\big{(}(\chi\circ\mathrm{pr})(x)\big{)}\xi(u),~{}\forall
x\in\mathfrak{G},u\in V.$

Thus, there is exactly one relative Rota-Baxter operator homomorphism
$(\chi\circ\mathrm{pr},\xi)$ from $T$ to $F(K)$ such that the following
diagram commutes:

$\textstyle{K=GF(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({{\rm{Id}}}_{\mathfrak{h}},{{\rm{Id}}}_{W})}$$\textstyle{K}$$\textstyle{G(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{G(\chi\circ\mathrm{pr},\xi)}$$\scriptstyle{(\chi,\xi)}$

which implies that $G$ is a left adjoint for $F$. The proof is finished. ∎

######  Remark 2.9.

The functors $G:\mathsf{RBO}\mathsf{LeibnizS}\to\mathsf{RBO}\mathsf{Lie}$ and
$F:\mathsf{RBO}\mathsf{Lie}\to\mathsf{RBO}\mathsf{LeibnizS}$ are induced by
those establishing the equivalence of categories between modules over a Lie
algebra $\mathfrak{g}$ and symmetric Leibniz representations over
$\mathfrak{g}$, viewed as a Leibniz algebra. From this, one may deduce that
the pair $(F,G)$ constitutes almost an adjoint equivalence. In fact, $F$ is
fully faithful, because the counit of the adjunction $\epsilon:G\circ
F\to{\rm{Id}}$ is a natural isomorphism for each object of
$\mathsf{RBO}\mathsf{Lie}$. On the other hand, the unit of the adjunction
$\eta:{\rm{Id}}\to F\circ G$ is not an isomorphism. For an object
$T:V\to\mathfrak{G}$ in $\mathsf{RBO}\mathsf{LeibnizS}$, $(F\circ G)(T)$
corresponds to viewing $\mathrm{pr}\circ T:V\to\mathfrak{G}_{\mathrm{Lie}}$
again as an object in $\mathsf{RBO}\mathsf{LeibnizS}$. We have a natural
relative Rota-Baxter operator homomorphism $(\mathrm{pr},{\rm{Id}}_{V})$ from
$T$ to $\mathrm{pr}\circ T$

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm{Id}}_{V}}$$\scriptstyle{T}$$\textstyle{\mathfrak{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}\circ
T}$$\textstyle{\mathfrak{G}_{\mathrm{Lie}},}$

but it is not an isomorphism in general.

Note that one may add the data which arises from the passage from
$\mathfrak{G}_{\mathrm{Lie}}$ to $\mathfrak{G}$ to the category
$\mathsf{RBO}\mathsf{Lie}$ in order to make it an equivalence of categories.
The data consists of the isomorphism class of an antisymmetric representation
of $\mathfrak{G}_{\mathrm{Lie}}$ (corresponding to
$\mathsf{Leib}(\mathfrak{G})$) and of the Loday-Pirashvili cohomology class of
a $2$-cocycle characterizing the abelian extension

(16) $\displaystyle
0\to\mathsf{Leib}(\mathfrak{G})\to\mathfrak{G}\to\mathfrak{G}_{\mathrm{Lie}}\to
0.$

Given a relative Rota-Baxter operator $T:V\to\mathfrak{g}$ together with a
$\mathfrak{g}$-module $A$ (viewed as an antisymmetric representation of the
Leibniz algebra $\mathfrak{g}$) and a $2$-cocycle $\alpha\in
Z^{2}(\mathfrak{g},A)$, we obtain a relative Rota-Baxter operator
$T:V\to\mathfrak{G}$ for the Leibniz algebra $\mathfrak{G}$ constructed from
$(\mathfrak{g},A,\alpha)$ with $A=\mathsf{Leib}(\mathfrak{G})$ and $V$ viewed
as a symmetric representation of $\mathfrak{G}$, because
$\mathsf{Leib}(\mathfrak{G})$ acts trivially from the left and thus also
trivially from the right as the representation $V$ is symmetric. This
construction depends on the choice of the cocycle and the module $A$, but
becomes natural when passing to cohomology and isomorphism classes.

###  2.4. Relations between the categories $\mathsf{AO}\mathsf{Lie}$ and
$\mathsf{RBO}\mathsf{LeibnizA}$

First we construct a functor $\mathcal{F}$ from the category
$\mathsf{AO}\mathsf{Lie}$ of relative averaging operators on Lie algebras to
the category $\mathsf{RBO}\mathsf{LeibnizA}$ of relative Rota-Baxter operators
on Leibniz algebras with respect to antisymmetric representations.

On objects, the functor $\mathcal{F}$ is defined as follows. Let
$T:V\longrightarrow\mathfrak{g}$ be a relative averaging operator on a Lie
algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to a representation
$(V;\rho)$. We view the Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ as a
Leibniz algebra. Moreover the Lie algebra representation $\rho$ gives rise to
an antisymmetric representation $(V;\rho,0)$ of the Leibniz algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$. Then the equality (4) means that
$T:V\longrightarrow\mathfrak{g}$ is a relative Rota-Baxter operator on the
Leibniz algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the
antisymmetric representation $(V;\rho,0)$.

On morphisms, the functor $\mathcal{F}$ is defined as follows. Let
$T:V\longrightarrow\mathfrak{g}$ (resp.
$T^{\prime}:V^{\prime}\longrightarrow\mathfrak{g}^{\prime}$) be a relative
averaging operator on a Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$
(resp. $(\mathfrak{g}^{\prime},\\{-,-\\}_{\mathfrak{g}^{\prime}})$) with
respect to a representation $(V;\rho)$ (resp. $(V^{\prime},\rho^{\prime})$).
Let $(\phi,\varphi)$ be a homomorphism from the relative averaging operator
$T$ to $T^{\prime}$. Then it is obvious that $(\phi,\varphi)$ is also a
homomorphism between the above induced relative Rota-Baxter operators.

It is straightforward to see that $\mathcal{F}$ defined above is indeed a
functor.

In the sequel, we construct a functor $\mathcal{G}$ from the category
$\mathsf{RBO}\mathsf{LeibnizA}$ to the category $\mathsf{AO}\mathsf{Lie}$.

######  Proposition 2.10.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to an
antisymmetric representation $(V;\rho^{L},\rho^{R}=0)$. Then
$\bar{T}:=\mathrm{pr}\circ T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ is a
relative averaging operator on the Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$
with respect to the representation $(V;\theta)$.

###### Proof.

Applying the projection
$\mathrm{pr}:\mathfrak{G}\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ to the
both sides of (8), we obtain

$\displaystyle[\mathrm{pr}\circ T(u),\mathrm{pr}\circ
T(v)]_{\mathfrak{G}_{\mathrm{Lie}}}=\mathrm{pr}\circ
T(\rho^{L}(Tu)v)\stackrel{{\scriptstyle\eqref{induce-
rep}}}{{=}}\mathrm{pr}\circ T(\theta(\mathrm{pr}\circ T(u))v),$

which implies that $\mathrm{pr}\circ
T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ is an averaging operator on the
Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$ with respect to the representation
$(V;\theta)$. ∎

Similar to Proposition 2.7, we have the following result.

######  Proposition 2.11.

Let $T:V\longrightarrow\mathfrak{G}$ (resp.
$T^{\prime}:V^{\prime}\longrightarrow\mathfrak{G}^{\prime}$) be a relative
Rota-Baxter operator on the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ (resp.
$(\mathfrak{G}^{\prime},\\{-,-\\}_{\mathfrak{G}^{\prime}})$) with respect to
the antisymmetric representation $(V;\rho^{L},\rho^{R}=0)$ (resp.
$(V;{\rho^{L}}^{\prime},{\rho^{R}}^{\prime}=0)$). Let $(\phi,\varphi)$ be a
homomorphism of relative Rota-Baxter operators on Leibniz algebras from $T$ to
$T^{\prime}$. Then $(\bar{\phi},\varphi)$ is a homomorphism of relative
averaging operators from $\bar{T}$ to $\bar{T^{\prime}}$, where the Lie
algebra homomorphism
$\bar{\phi}:\mathfrak{G}_{\mathrm{Lie}}\,\rightarrow\,\mathfrak{G}^{\prime}_{\mathrm{Lie}}$
is defined by

$\bar{\phi}(\bar{x}):=\overline{\phi(x)},\quad\forall x\in\mathfrak{G}.$

Now we are ready to give the main result in this subsection.

######  Theorem 2.12.

There is a functor $\mathcal{G}$ from the category
$\mathsf{RBO}\mathsf{LeibnizA}$ of relative Rota-Baxter operators on Leibniz
algebras with respect to antisymmetric representations to the category
$\mathsf{AO}\mathsf{Lie}$ of relative averaging operators on Lie algebras,
such that $\mathcal{G}$ is a left adjoint for $\mathcal{F}$.

###### Proof.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to an
antisymmetric representation $(V;\rho^{L},\rho^{R}=0)$. By Proposition 2.10,
$\bar{T}:=\mathrm{pr}\circ T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ is a
relative averaging operator on the Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$
with respect to the representation $(V;\theta)$. Thus, on objects, we define

$\mathcal{G}(T):=\bar{T}.$

Let $(\phi,\varphi)$ be a homomorphism of relative Rota-Baxter operators on
Leibniz algebras from $T$ to $T^{\prime}$. By Proposition 2.11,
$(\bar{\phi},\varphi)$ is a homomorphism of relative averaging operators from
$\bar{T}$ to $\bar{T^{\prime}}$. Thus, on morphisms, we define

$\mathcal{G}(\phi,\varphi):=(\bar{\phi},\varphi).$

Then it is straightforward to see that $\mathcal{G}$ is indeed a functor.

By the similar argument as Theorem 2.8, we can show that $\mathcal{G}$ is a
left adjoint for $\mathcal{F}$. We omit details. ∎

######  Remark 2.13.

Like in Remark 2.9, the pair of functors $\mathcal{F}$ and $\mathcal{G}$ form
almost an adjoint equivalence.

##  3\. Relations between the controlling algebras

In this section, we establish the relation between the controlling algebra of
relative Rota-Baxter operators on a Leibniz algebra with respect to a
symmetric (resp. antisymmetric) representation and the controlling algebra of
the relative Rota-Baxter (resp. averaging) operators on the canonical Lie
algebra.

A permutation $\sigma\in\mathbb{S}_{n}$ is called an $(i,n-i)$-shuffle if
$\sigma(1)<\cdots<\sigma(i)$ and $\sigma(i+1)<\cdots<\sigma(n)$. If $i=0$ or
$n$, we assume $\sigma={\rm{Id}}$. The set of all $(i,n-i)$-shuffles will be
denoted by $\mathbb{S}_{(i,n-i)}$. The notion of an
$(i_{1},\cdots,i_{k})$-shuffle and the set $\mathbb{S}_{(i_{1},\cdots,i_{k})}$
are defined analogously.

A degree $1$ element $\theta\in\mathfrak{g}^{1}$ is called a Maurer-Cartan
element of a differential graded Lie algebra
$(\oplus_{k\in\mathbb{Z}}\mathfrak{g}^{k},[\cdot,\cdot],d)$ if it satisfies
the Maurer-Cartan equation: $d\theta+\frac{1}{2}[\theta,\theta]=0$. The set of
Maurer-Cartan elements in a dgla $\mathfrak{g}$ will be denoted by
$\operatorname{MC}(\mathfrak{g})$.

Let $\mathfrak{g}$ be a vector space. We consider the graded vector space
$C^{*}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq
1}C^{n}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq
1}\mathrm{Hom}(\otimes^{n}\mathfrak{g},\mathfrak{g})$. An element $P\in
C^{p+1}(\mathfrak{g},\mathfrak{g})$ is defined to have degree $p$. The
Balavoine bracket on the graded vector space
$C^{*}(\mathfrak{g},\mathfrak{g})$ is given by:

(17)
$\displaystyle[P,Q]_{\mathsf{B}}=P\bar{\circ}Q-(-1)^{pq}Q\bar{\circ}P,\,\,\,\,\forall
P\in C^{p+1}(\mathfrak{g},\mathfrak{g}),Q\in
C^{q+1}(\mathfrak{g},\mathfrak{g}),$

where $P\bar{\circ}Q\in C^{p+q+1}(\mathfrak{g},\mathfrak{g})$ is defined by

(18) $\displaystyle P\bar{\circ}Q=\sum_{k=1}^{p+1}P\circ_{k}Q,$

and $\circ_{k}$ is defined by

$\displaystyle(P\circ_{k}Q)(x_{1},\cdots,x_{p+q+1})$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(k-1,q)}}(-1)^{(k-1)q}(-1)^{\sigma}P(x_{\sigma(1)},\cdots,x_{\sigma(k-1)},Q(x_{\sigma(k)},\cdots,x_{\sigma(k+q-1)},x_{k+q}),x_{k+q+1},\cdots,x_{p+q+1}).$

It is well known that

######  Theorem 3.1.

([4, 19]) With the above notations,
$(C^{*}(\mathfrak{g},\mathfrak{g}),[-,-]_{\mathsf{B}})$ is a graded Lie
algebra. Its Maurer-Cartan elements (as a differential graded Lie algebra with
zero differential) are precisely the Leibniz algebra structures on
$\mathfrak{g}$.

###  3.1. The controlling algebra of relative Rota-Baxter operators on a
Leibniz algebra

Let $(V;\rho_{V}^{L},\rho_{V}^{R})$ be a representation of a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. Then there is a Leibniz algebra
structure on $\mathfrak{G}\oplus V$ given by

(19)
$\displaystyle[x+u,y+v]_{\ltimes}=[x,y]_{\mathfrak{G}}+\rho_{V}^{L}(x)v+\rho_{V}^{R}(y)u,\quad\forall
x,y\in\mathfrak{g},~{}u,v\in V.$

This Leibniz algebra is called the semidirect product of $\mathfrak{G}$ and
$(V;\rho_{V}^{L},\rho_{V}^{R})$, and denoted by
$\mathfrak{G}\ltimes_{\rho_{V}^{L},\rho_{V}^{R}}V.$ We denote the above
semidirect product Leibniz multiplication by $\mu.$ Consider the graded vector
space

$C^{*}(V,\mathfrak{G}):=\oplus_{n\geq 1}C^{n}(V,\mathfrak{G})=\oplus_{n\geq
1}\mathrm{Hom}(\otimes^{n}V,\mathfrak{G}),$

where an element $g\in C^{n}(V,\mathfrak{G})$ is defined to be of degree $n$.

######  Theorem 3.2.

([46]) With the above notations, $(C^{*}(V,\mathfrak{G}),\\{-,-\\}_{V})$ is a
graded Lie algebra, where the graded Lie bracket
$\\{-,-\\}_{V}:C^{m}(V,\mathfrak{G})\times
C^{n}(V,\mathfrak{G})\,\rightarrow\,C^{m+n}(V,\mathfrak{G})$ is given by the
derived bracket as following:

$\displaystyle\\{g_{1},g_{2}\\}_{V}$ $\displaystyle=$
$\displaystyle(-1)^{m-1}[[\mu,g_{1}]_{\mathsf{B}},g_{2}]_{\mathsf{B}},\quad\forall
g_{1}\in C^{m}(V,\mathfrak{G}),~{}g_{2}\in C^{n}(V,\mathfrak{G}).$

More precisely, we have

$\displaystyle\\{g_{1},g_{2}\\}_{V}(v_{1},v_{2},\cdots,v_{m+n})$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{m}\sum_{\sigma\in\mathbb{S}_{(k-1,n)}}(-1)^{(k-1)n+1}(-1)^{\sigma}g_{1}(v_{\sigma(1)},\cdots,v_{\sigma(k-1)},\rho_{V}^{L}(g_{2}(v_{\sigma(k)},\cdots,v_{\sigma(k+n-1)}))v_{k+n},v_{k+n+1},\cdots,v_{m+n})$
$\displaystyle+\sum_{k=2}^{m+1}\sum_{\sigma\in\mathbb{S}_{(k-2,n,1)}\atop\sigma(k+n-2)=k+n-1}(-1)^{kn}(-1)^{\sigma}g_{1}(v_{\sigma(1)},\cdots,v_{\sigma(k-2)},\rho_{V}^{R}(g_{2}(v_{\sigma(k-1)},\cdots,v_{\sigma(k+n-2)}))v_{\sigma(k+n-1)},v_{k+n},\cdots,v_{m+n})$
$\displaystyle+\sum_{k=1}^{m}\sum_{\sigma\in\mathbb{S}_{(k-1,n-1)}}(-1)^{(k-1)n}(-1)^{\sigma}[g_{2}(v_{\sigma(k)},\cdots,v_{\sigma(k+n-2)},v_{k+n-1}),g_{1}(v_{\sigma(1)},\cdots,v_{\sigma(k-1)},v_{k+n},\cdots,v_{m+n})]_{\mathfrak{G}}$
$\displaystyle+\sum_{\sigma\in\mathbb{S}_{(m,n-1)}}(-1)^{mn+1}(-1)^{\sigma}[g_{1}(v_{\sigma(1)},\cdots,v_{\sigma(m)}),g_{2}(v_{\sigma(m+1)},\cdots,v_{\sigma(m+n-1)},v_{m+n})]_{\mathfrak{G}}$
$\displaystyle+\sum_{k=1}^{n}\sum_{\sigma\in\mathbb{S}_{(k-1,m)}}(-1)^{m(k+n-1)}(-1)^{\sigma}g_{2}(v_{\sigma(1)},\cdots,v_{\sigma(k-1)},\rho_{V}^{L}(g_{1}(v_{\sigma(k)},\cdots,v_{\sigma(k+m-1)}))v_{k+m},v_{k+m+1},\cdots,v_{m+n})$
$\displaystyle+\sum_{k=1}^{n}\sum_{\sigma\in\mathbb{S}_{(k-1,m,1)}\atop\sigma(k+m-1)=k+m}(-1)^{m(k+n-1)+1}(-1)^{\sigma}g_{2}(v_{\sigma(1)},\cdots,v_{\sigma(k-1)},\rho_{V}^{R}(g_{1}(v_{\sigma(k)},\cdots,v_{\sigma(k-1+m)}))v_{\sigma(k+m)},v_{k+m+1},\cdots,v_{m+n}).$

Moreover, its Maurer-Cartan elements (where the differential is taken to be
zero again) are relative Rota-Baxter operators on the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to the representation
$(V;\rho_{V}^{L},\rho_{V}^{R})$.

In fact, this graded Lie algebra construction is functorial. More precisely,
we consider the category $\mathfrak{G}$-$\mathsf{Rep}$ of representations of
the Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$. Then, we have a
contravariant functor $\mathsf{C}_{\mathfrak{G}}$ from the category
$\mathfrak{G}$-$\mathsf{Rep}$ to the category $\mathsf{Gla}$ of graded Lie
algebras.

Let $(W;\rho_{W}^{L},\rho_{W}^{R})$ and $(V;\rho_{V}^{L},\rho_{V}^{R})$ be two
representations of the Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$.
A homomorphism from $(W;\rho_{W}^{L},\rho_{W}^{R})$ to
$(V;\rho_{V}^{L},\rho_{V}^{R})$ is a linear map $\phi:W\,\rightarrow\,V$ such
that

(20) $\displaystyle\phi(\rho_{W}^{L}(x)w)$ $\displaystyle=$
$\displaystyle\rho_{V}^{L}(x)\phi(w),$ (21)
$\displaystyle\phi(\rho_{W}^{R}(x)w)$ $\displaystyle=$
$\displaystyle\rho_{V}^{R}(x)\phi(w),\quad\forall x\in\mathfrak{G},w\in W.$

Let $\phi:W\,\rightarrow\,V$ be a homomorphism from the representation
$(W;\rho_{W}^{L},\rho_{W}^{R})$ to $(V;\rho_{V}^{L},\rho_{V}^{R})$. Define a
linear map
$\Phi:\mathrm{Hom}(\otimes^{n}V,\mathfrak{G})\,\rightarrow\,\mathrm{Hom}(\otimes^{n}W,\mathfrak{G}),n\geq
1,$ by

(22) $\displaystyle\Phi(f):=f\circ\phi^{\otimes n},\,\,\,\,\forall
f\in\mathrm{Hom}(\otimes^{n}V,\mathfrak{G}).$

######  Proposition 3.3.

With the above notations, $\Phi$ is a homomorphism from the graded Lie algebra
$(C^{*}(V,\mathfrak{G}),\\{-,-\\}_{V})$ to
$(C^{*}(W,\mathfrak{G}),\\{-,-\\}_{W})$.

###### Proof.

It follows directly from (20) and (21). We omit the details. ∎

Moreover, we have the following theorem.

######  Theorem 3.4.

Theorem 3.2 and Proposition 3.3 give us a contravariant functor
$\mathsf{C}_{\mathfrak{G}}$ from the category $\mathfrak{G}$-$\mathsf{Rep}$ to
the category $\mathsf{Gla}$ of graded Lie algebras,

  * •

on objects, the functor $\mathsf{C}_{\mathfrak{G}}:$
$\mathfrak{G}$-$\mathsf{Rep}$ $\to$ $\mathsf{Gla}$ is defined by

(23)
$\displaystyle\mathsf{C}_{\mathfrak{G}}\Big{(}(V;\rho_{V}^{L},\rho_{V}^{R})\Big{)}$
$\displaystyle=$ $\displaystyle(C^{*}(V,\mathfrak{G}),\\{-,-\\}_{V}),$

  * •

on morphisms, the functor $\mathsf{C}_{\mathfrak{G}}:$
$\mathfrak{G}$-$\mathsf{Rep}$ $\to$ $\mathsf{Gla}$ is defined by

(24)
$\displaystyle\mathsf{C}_{\mathfrak{G}}(W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V)$
$\displaystyle=$
$\displaystyle(C^{*}(V,\mathfrak{G}),\\{-,-\\}_{V})\stackrel{{\scriptstyle\Phi}}{{\,\rightarrow\,}}(C^{*}(W,\mathfrak{G}),\\{-,-\\}_{W}),$

where $(W;\rho_{W}^{L},\rho_{W}^{R})$ and $(V;\rho_{V}^{L},\rho_{V}^{R})$ are
representations of the Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$
and $\phi\in\mathrm{Hom}_{\mathfrak{G}\mbox{-}\mathsf{Rep}}(W,V)$.

###### Proof.

Let $(V;\rho_{V}^{L},\rho_{V}^{R})$ be a representation of the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. For the identity homomorphism
${\rm{Id}}_{V}:V\,\rightarrow\,V$, we have
$\mathsf{C}_{\mathfrak{G}}({\rm{Id}}_{V})={\rm{Id}}_{C^{*}(V,\mathfrak{G})}$.
Moreover, let $\phi:W\,\rightarrow\,V$ and $\psi:V\,\rightarrow\,U$ be two
homomorphisms of representations of the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. For $n\geq 1$ and
$\theta\in\mathrm{Hom}(U^{\otimes n},\mathfrak{G})$, we have

$\displaystyle\mathsf{C}_{\mathfrak{G}}(\psi\circ\phi)\theta$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:defiphi}}}{{=}}$
$\displaystyle\theta\circ(\psi\circ\phi)^{\otimes n}$ $\displaystyle=$
$\displaystyle(\theta\circ\psi^{\otimes n})\circ\phi^{\otimes n}$
$\displaystyle=$
$\displaystyle(\mathsf{C}_{\mathfrak{G}}(\phi)\circ\mathsf{C}_{\mathfrak{G}}(\psi))\theta.$

Thus, we deduce that $\mathsf{C}_{\mathfrak{G}}$ is a contravariant functor. ∎

As a consequence, the short exact sequence (7)

$0\to V_{\rm
anti}\stackrel{{\scriptstyle\phi}}{{\to}}V\stackrel{{\scriptstyle\psi}}{{\to}}V_{\rm
sym}\to 0$

associated to the $\mathfrak{G}$-representation
$(V;\rho_{V}^{L},\rho_{V}^{R})$ induces homomorphisms of graded Lie algebras

$(C^{*}(V_{\rm sym},\mathfrak{G}),\\{-,-\\}_{V_{\rm
sym}})\stackrel{{\scriptstyle\Psi}}{{\to}}(C^{*}(V,\mathfrak{G}),\\{-,-\\}_{V})\stackrel{{\scriptstyle\Phi}}{{\to}}(C^{*}(V_{\rm
anti},\mathfrak{G}),\\{-,-\\}_{V_{\rm anti}}),$

which do not necessarily form an exact sequence of graded Lie algebras. By
passage to the corresponding Maurer-Cartan sets, we obtain a sequence of
induced maps

(25) $\operatorname{MC}\big{(}C^{*}(V_{\rm
sym},\mathfrak{G})\big{)}\stackrel{{\scriptstyle\Psi}}{{\to}}\operatorname{MC}\big{(}C^{*}(V,\mathfrak{G})\big{)}\stackrel{{\scriptstyle\Phi}}{{\to}}\operatorname{MC}\big{(}C^{*}(V_{\rm
anti},\mathfrak{G})\big{)},$

which describe (loosely) how a relative Rota-Baxter operator on the Leibniz
algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to
$(V;\rho_{V}^{L},\rho_{V}^{R})$ is assembled from a relative Rota-Baxter
operator on the Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$ and a relative
averaging operator on $\mathfrak{G}_{\mathrm{Lie}}$, according to Theorems 2.8
and 2.12, see also Corollaries 3.6 and 3.9.

The (not necessarily exact) sequence (25) is the effect in the module-variable
of the standard short exact sequence. In order to investigate also the effect
in the Leibniz-algebra-variable of the standard short exact sequence (26)
(which will be the subject of Theorem 3.5), we need some more preparation.

###  3.2. The controlling algebra of relative Rota-Baxter operators on a Lie
algebra

Associated to any Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$, we
have an exact sequence of Leibniz algebras

(26) $\displaystyle
0\longrightarrow\mathsf{Leib}(\mathfrak{G})\stackrel{{\scriptstyle\mathfrak{i}}}{{\longrightarrow}}\mathfrak{G}\stackrel{{\scriptstyle\mathrm{pr}}}{{\longrightarrow}}\mathfrak{G}_{\mathrm{Lie}}\longrightarrow
0.$

It is an abelian extension of $\mathfrak{G}_{\mathrm{Lie}}$ by
$\mathsf{Leib}(\mathfrak{G})$. Moreover the induced representation of
$\mathfrak{G}_{\mathrm{Lie}}$ on $\mathsf{Leib}(\mathfrak{G})$ is an
antisymmetric representation. Observe that the sequence (26) is the special
case of the sequence (7) for the adjoint representation of $\mathfrak{G}$.

Let $(V;\rho_{V}^{L},\rho_{V}^{R})$ be a representation of a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. Thus, there is exactly one Lie algebra
homomorphism
$\theta_{V}:\mathfrak{G}_{\mathrm{Lie}}\longrightarrow\mathfrak{gl}(V)$, such
that following diagram of Leibniz algebra homomorphisms commutes:

(31)

Consider the category $\mathfrak{G}$-$\mathsf{SRep}$ of symmetric
representations of the Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$,
which is a subcategory of $\mathfrak{G}$-$\mathsf{Rep}$. Then we have the
following functors

$\displaystyle\mathcal{I}_{s}:\mathfrak{G}\mbox{-}\mathsf{SRep}\,\rightarrow\,\mathsf{Leib}(\mathfrak{G})\mbox{-}\mathsf{SRep}$
$\displaystyle\mathcal{P}_{s}:\mathfrak{G}\mbox{-}\mathsf{SRep}\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}\mbox{-}\mathsf{SRep}$

which are given by

  * •

the functor $\mathcal{I}_{s}:$ $\mathfrak{G}\mbox{-}\mathsf{SRep}$ $\to$
$\mathsf{Leib}(\mathfrak{G})\mbox{-}\mathsf{SRep}$, which is defined on
objects and on morphisms respectively by

(32)
$\displaystyle\mathcal{I}_{s}\Big{(}(V;\rho_{V}^{L},-\rho_{V}^{L})\Big{)}$
$\displaystyle=$
$\displaystyle(V;\rho_{V}^{L}\circ\mathfrak{i},-\rho_{V}^{L}\circ\mathfrak{i}),$
(33)
$\displaystyle\mathcal{I}_{s}(W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V)$
$\displaystyle=$ $\displaystyle
W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V,$

  * •

the functor $\mathcal{P}_{s}:$ $\mathfrak{G}\mbox{-}\mathsf{SRep}$ $\to$
$\mathfrak{G}_{\mathrm{Lie}}\mbox{-}\mathsf{SRep}$, which is defined on
objects and on morphisms respectively by

(34)
$\displaystyle\mathcal{P}_{s}\Big{(}(V;\rho_{V}^{L},-\rho_{V}^{L})\Big{)}$
$\displaystyle=$ $\displaystyle(V;\theta_{V},-\theta_{V}),$ (35)
$\displaystyle\mathcal{P}_{s}(W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V)$
$\displaystyle=$ $\displaystyle
W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V,$

for symmetric representations $(W;\rho_{W}^{L},-\rho_{W}^{L})$ and
$(V;\rho_{V}^{L},-\rho_{V}^{L})$ of the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ and
$\phi\in\mathrm{Hom}_{\mathfrak{G}\mbox{-}\mathsf{SRep}}(W,V)$.

Thus, we have three functors
$\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{s},~{}\mathsf{C}_{\mathfrak{G}}$
and $\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{s}$ from
$\mathfrak{G}\mbox{-}\mathsf{SRep}$ to $\mathsf{Gla}$, where the functor
$\mathsf{C}_{\mathfrak{G}}$ is given in Theorem 3.4. Moreover, for any
symmetric representation $(V;\rho_{V}^{L},-\rho_{V}^{L})$ of the Leibniz
algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$, we define

$\alpha_{V}:(\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{s})(V;\rho_{V}^{L},-\rho_{V}^{L})\,\rightarrow\,\mathsf{C}_{\mathfrak{G}}(V;\rho_{V}^{L},-\rho_{V}^{L})$

and

$\beta_{V}:\mathsf{C}_{\mathfrak{G}}(V;\rho_{V}^{L},-\rho_{V}^{L})\,\rightarrow\,(\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{s})(V;\rho_{V}^{L},-\rho_{V}^{L})$

as following:

(36) $\displaystyle\alpha_{V}(g)$ $\displaystyle=$ $\displaystyle
g,\quad\quad\quad\forall
g\in\mathrm{Hom}(\otimes^{n}V,\mathsf{Leib}(\mathfrak{G})),$ (37)
$\displaystyle\beta_{V}(f)$ $\displaystyle=$ $\displaystyle\mathrm{pr}\circ
f,\quad\forall f\in\mathrm{Hom}(\otimes^{n}V,\mathfrak{G}).$

######  Theorem 3.5.

With the above notations, $\alpha$ is a natural transformation from the
functor $\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{s}$ to
$\mathsf{C}_{\mathfrak{G}}$, and $\beta$ is a natural transformation from the
functor $\mathsf{C}_{\mathfrak{G}}$ to
$\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{s}$. Moreover, for
any symmetric representation $(V;\rho_{V}^{L},-\rho_{V}^{L})$ of the Leibniz
algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$, we have the following short
exact sequence of graded Lie algebras:

(38) $\displaystyle\qquad
0\,\rightarrow\,(\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{s})(V;\rho_{V}^{L},-\rho_{V}^{L})\stackrel{{\scriptstyle\alpha_{V}}}{{\,\rightarrow\,}}\mathsf{C}_{\mathfrak{G}}(V;\rho_{V}^{L},-\rho_{V}^{L})\stackrel{{\scriptstyle\beta_{V}}}{{\,\rightarrow\,}}(\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{s})(V;\rho_{V}^{L},-\rho_{V}^{L})\,\rightarrow\,0.$

###### Proof.

Since $\mathsf{Leib}(\mathfrak{G})$ is an ideal of the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$, it follows that
$C^{*}(V,\mathsf{Leib}(\mathfrak{G}))$ is a subalgebra of
$\mathsf{C}_{\mathfrak{G}}(V;\rho_{V}^{L},-\rho_{V}^{L})$. Thus the linear
embedding map $\alpha_{V}$ is a graded Lie algebra homomorphism. Let
$\phi:W\,\rightarrow\,V$ be a homomorphism of symmetric representations of
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. It is straightforward to obtain the
following commutative diagram:

which implies that $\alpha$ is a natural transformation.

Since $\mathrm{pr}$ is a Leibniz algebra homomorphism and the definition (15)
of $\theta_{V}$, we deduce that $\beta_{V}$ is a homomorphism of graded Lie
algebras. Moreover, let $\phi:W\,\rightarrow\,V$ be a homomorphism of
symmetric representations of $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ and
$f\in\mathrm{Hom}(\otimes^{n}V,\mathfrak{G})$. Then we have

$(\mathrm{pr}\circ f)\circ\phi^{\otimes
n}=\mathrm{pr}\circ(f\circ\phi^{\otimes n}),$

which implies that we have the following commutative diagram:

Thus, $\beta$ is a natural transformation.

Since $V$ is vector space over the field $\mathbb{K}$, for any positive
integer $n$, the functor $\mathrm{Hom}(\otimes^{n}V,-)$ is an exact functor
from the category of vector spaces over $\mathbb{K}$ to itself. Moreover, we
have $\alpha_{V}=\mathfrak{i}_{*}$ and $\beta_{V}=\mathrm{pr}_{*}$. By the
exact sequence of Leibniz algebras (26), we obtain the short exact sequence of
graded Lie algebras (38). The proof is finished. ∎

######  Corollary 3.6.

Let $(V;\rho_{V}^{L},-\rho_{V}^{L})$ be a symmetric representation of a
Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$. The Maurer-Cartan
elements of the graded Lie algebra
$(\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{s})(V;\rho_{V}^{L},-\rho_{V}^{L})$
are exactly relative Rota-Baxter operators on the Lie algebra
$\mathfrak{G}_{\mathrm{Lie}}$ with respect to the representation
$(V,\theta_{V})$ given in (15).

Let $(V;\rho)$ be a representation of a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$. Then $(V;\rho,-\rho)$ is a symmetric
representation of the Leibniz algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$.
By $\mathsf{Leib}(\mathfrak{g})=0$, we obtain a graded Lie algebra
$\mathsf{C}_{\mathfrak{g}}(V;\rho,-\rho)$ whose Maurer-Cartan elements are the
relative Rota-Baxter operators on the Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the representation
$(V;\rho)$. On the other hand, in [44] the authors construct another graded
Lie algebra on the graded vector subspace
${\mathcal{C}}^{*}(V,\mathfrak{g}):=\oplus_{k\geq
1}\mathrm{Hom}(\wedge^{k}V,\mathfrak{g}),$ where the graded Lie bracket is
given by

$\displaystyle\left\llbracket P,Q\right\rrbracket(v_{1},v_{2},\cdots,v_{m+n})$
$\displaystyle=$
$\displaystyle-\sum_{\sigma\in\mathbb{S}_{(n,1,m-1)}}(-1)^{\sigma}P(\rho(Q(v_{\sigma(1)},\cdots,v_{\sigma(n)}))v_{\sigma(n+1)},v_{\sigma(n+2)},\cdots,v_{\sigma(m+n)})$
$\displaystyle+(-1)^{mn}\sum_{\sigma\in\mathbb{S}_{(m,1,n-1)}}(-1)^{\sigma}Q(\rho(P(v_{\sigma(1)},\cdots,v_{\sigma(m)}))v_{\sigma(m+1)},v_{\sigma(m+2)},\cdots,v_{\sigma(m+n)})$
$\displaystyle-(-1)^{mn}\sum_{\sigma\in\mathbb{S}_{(m,n)}}(-1)^{\sigma}[P(v_{\sigma(1)},\cdots,v_{\sigma(m)}),Q(v_{\sigma(m+1)},\cdots,v_{\sigma(m+n)})]_{\mathfrak{g}}$

for all $P\in\mathrm{Hom}(\wedge^{m}V,\mathfrak{g})$ and
$Q\in\mathrm{Hom}(\wedge^{n}V,\mathfrak{g})$. Its Maurer-Cartan elements are
also relative Rota-Baxter operators on $\mathfrak{g}$ with respect to the
representation $(V;\rho)$.

######  Proposition 3.7.

The graded Lie algebra
$({\mathcal{C}}^{*}(V,\mathfrak{g}),\left\llbracket\cdot,\cdot\right\rrbracket)$
is a subalgebra of $\mathsf{C}_{\mathfrak{g}}(V;\rho,-\rho)$.

###### Proof.

Let $(V;\rho)$ be a representation of a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$. The semidirect product Lie algebra
$\mathfrak{g}\ltimes_{\rho}V$ is the same as the semidirect product Leibniz
algebra $\mathfrak{g}\ltimes_{\rho,-\rho}V$. We denote this semidirect product
multiplication by $\mu.$ Recall that the Nijenhuis-Richardson bracket
$[\cdot,\cdot]_{\mathsf{NR}}$ associated to the direct sum vector space
$\mathfrak{g}\oplus V$ gives rise to a graded Lie algebra $(\oplus_{k\geq
1}\mathrm{Hom}(\wedge^{k}(\mathfrak{g}\oplus V),\mathfrak{g}\oplus
V),[\cdot,\cdot]_{\mathsf{NR}})$. The graded Lie algebra
$({\mathcal{C}}^{*}(V,\mathfrak{g}),\left\llbracket\cdot,\cdot\right\rrbracket)$
is also obtained via the derived bracket [27]:

$\left\llbracket
P,Q\right\rrbracket=(-1)^{m-1}[[\mu,P]_{\mathsf{NR}},Q]_{\mathsf{NR}},\quad\forall
P\in\mathrm{Hom}(\wedge^{m}V,\mathfrak{g}),Q\in\mathrm{Hom}(\wedge^{n}V,\mathfrak{g}).$

Since $(\oplus_{k\geq 1}\mathrm{Hom}(\wedge^{k}(\mathfrak{g}\oplus
V),\mathfrak{g}\oplus V),[\cdot,\cdot]_{\mathsf{NR}})$ is a graded Lie
subalgebra of $(\oplus_{n\geq 1}\mathrm{Hom}(\otimes^{n}(\mathfrak{g}\oplus
V),\mathfrak{g}\oplus V),[\cdot,\cdot]_{\mathsf{B}})$, we deduce that

$\displaystyle\left\llbracket P,Q\right\rrbracket$ $\displaystyle=$
$\displaystyle(-1)^{m-1}[[\mu,P]_{\mathsf{NR}},Q]_{\mathsf{NR}}=(-1)^{m-1}[[\mu,P]_{\mathsf{B}},Q]_{\mathsf{B}}=\\{P,Q\\}_{V},$

which implies that
$({\mathcal{C}}^{*}(V,\mathfrak{g}),\left\llbracket\cdot,\cdot\right\rrbracket)$
is a subalgebra of
$(C^{*}(V,\mathfrak{g}),\\{-,-\\}_{V})=\mathsf{C}_{\mathfrak{g}}(V;\rho,-\rho)$.
∎

###  3.3. The controlling algebras of relative averaging operators on a Lie
algebra

Denote by $\mathfrak{G}$-$\mathsf{ASRep}$ the category of antisymmetric
representations of the Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$,
which is a subcategory of $\mathfrak{G}$-$\mathsf{Rep}$. Then we have the
following functors

$\displaystyle\mathcal{I}_{a}:\mathfrak{G}\mbox{-}\mathsf{ASRep}\,\rightarrow\,\mathsf{Leib}(\mathfrak{G})\mbox{-}\mathsf{ASRep}$
$\displaystyle\mathcal{P}_{a}:\mathfrak{G}\mbox{-}\mathsf{ASRep}\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}\mbox{-}\mathsf{ASRep}$

which are given by

  * •

the functor $\mathcal{I}_{a}:$ $\mathfrak{G}\mbox{-}\mathsf{ASRep}$ $\to$
$\mathsf{Leib}(\mathfrak{G})\mbox{-}\mathsf{ASRep}$, which is defined on
objects and on morphisms respectively by

(42) $\displaystyle\mathcal{I}_{a}\Big{(}(V;\rho_{V}^{L},0)\Big{)}$
$\displaystyle=$ $\displaystyle(V;\rho_{V}^{L}\circ\mathfrak{i},0),$ (43)
$\displaystyle\mathcal{I}_{a}(W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V)$
$\displaystyle=$ $\displaystyle
W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V,$

  * •

the functor $\mathcal{P}_{a}:$ $\mathfrak{G}\mbox{-}\mathsf{ASRep}$ $\to$
$\mathfrak{G}_{\mathrm{Lie}}\mbox{-}\mathsf{ASRep}$, which is defined on
objects and on morphisms respectively by

(44) $\displaystyle\mathcal{P}_{a}\Big{(}(V;\rho_{V}^{L},0)\Big{)}$
$\displaystyle=$ $\displaystyle(V;\theta_{V},0),$ (45)
$\displaystyle\mathcal{P}_{a}(W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V)$
$\displaystyle=$ $\displaystyle
W\stackrel{{\scriptstyle\phi}}{{\,\rightarrow\,}}V,$

for antisymmetric representations $(W;\rho_{W}^{L},0)$ and
$(V;\rho_{V}^{L},0)$ of the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ and
$\phi\in\mathrm{Hom}_{\mathfrak{G}\mbox{-}\mathsf{ASRep}}(W,V)$.

Thus, we have three functors
$\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{a},~{}\mathsf{C}_{\mathfrak{G}}$
and $\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{a}$ from the
category $\mathfrak{G}\mbox{-}\mathsf{ASRep}$ to the category $\mathsf{Gla}$.
Moreover, for any antisymmetric representation $(V;\rho_{V}^{L},0)$ of the
Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$, we define

$\alpha_{V}:(\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{a})(V;\rho_{V}^{L},0)\,\rightarrow\,\mathsf{C}_{\mathfrak{G}}(V;\rho_{V}^{L},0)$

and

$\beta_{V}:\mathsf{C}_{\mathfrak{G}}(V;\rho_{V}^{L},0)\,\rightarrow\,(\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{a})(V;\rho_{V}^{L},0)$

as following:

$\displaystyle\alpha_{V}(g)$ $\displaystyle=$ $\displaystyle
g,\quad\quad\quad\forall
g\in\mathrm{Hom}(\otimes^{n}V,\mathsf{Leib}(\mathfrak{G})),$
$\displaystyle\beta_{V}(f)$ $\displaystyle=$ $\displaystyle\mathrm{pr}\circ
f,\quad\forall f\in\mathrm{Hom}(\otimes^{n}V,\mathfrak{G}).$

Similar to Theorem 3.5, we have the following result.

######  Theorem 3.8.

With above notations, $\alpha$ is a natural transformation from the functor
$\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{a}$ to
$\mathsf{C}_{\mathfrak{G}}$, and $\beta$ is a natural transformation from the
functor $\mathsf{C}_{\mathfrak{G}}$ to
$\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{a}$. Moreover, for
any antisymmetric representation $(V;\rho_{V}^{L},0)$ of the Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$, we have the following short exact
sequence of graded Lie algebras:

$0\,\rightarrow\,(\mathsf{C}_{\mathsf{Leib}(\mathfrak{G})}\circ\mathcal{I}_{a})(V;\rho_{V}^{L},0)\stackrel{{\scriptstyle\alpha_{V}}}{{\,\rightarrow\,}}\mathsf{C}_{\mathfrak{G}}(V;\rho_{V}^{L},0)\stackrel{{\scriptstyle\beta_{V}}}{{\,\rightarrow\,}}(\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{a})(V;\rho_{V}^{L},0)\,\rightarrow\,0.$

###### Proof.

The proof is parallel to that of Theorem 3.5, so we omit the details. ∎

######  Corollary 3.9.

Let $(V;\rho_{V}^{L},0)$ be an antisymmetric representation of a Leibniz
algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$. The Maurer-Cartan elements of
the graded Lie algebra
$(\mathsf{C}_{\mathfrak{G}_{\mathrm{Lie}}}\circ\mathcal{P}_{a})(V;\rho_{V}^{L},0)$
are exactly relative averaging operators on the Lie algebra
$\mathfrak{G}_{\mathrm{Lie}}$ with respect to the representation
$(V,\theta_{V})$ given in (15).

Let $(V;\rho)$ be a representation of a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$. Then $(V;\rho,0)$ is an antisymmetric
representation of the Leibniz algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$.
By $\mathsf{Leib}(\mathfrak{g})=0$, we obtain a graded Lie algebra
$\mathsf{C}_{\mathfrak{g}}(V;\rho,0)$ whose Maurer-Cartan elements are the
relative averaging operators on the Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the representation
$(V;\rho)$. This graded Lie algebra is exactly the same as the one given in
[41].

##  4\. Relations between the cohomologies

In this section, we establish the relations between the cohomology groups of a
relative Rota-Baxter operator on a Leibniz algebra with respect to a symmetric
(resp. antisymmetric) representation and the cohomology groups of the induced
relative Rota-Baxter (resp. averaging) operator on the canonical Lie algebra.

###  4.1. Cohomology of relative Rota-Baxter operators on Leibniz algebras

######  Definition 4.1.

([33]) Let $(V;\rho^{L},\rho^{R})$ be a representation of a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. The Loday-Pirashvili cohomology of
$\mathfrak{G}$ with coefficients in $V$ is the cohomology of the cochain
complex
$(C^{\bullet}(\mathfrak{G},V)=\oplus_{k=0}^{+\infty}C^{k}(\mathfrak{G},V),\partial)$,
where $C^{k}(\mathfrak{G},V)=\mathrm{Hom}(\otimes^{k}\mathfrak{G},V)$ and the
coboundary operator $\partial:C^{k}(\mathfrak{G},V)\longrightarrow
C^{k+1}(\mathfrak{G},V)$ is defined by

$\displaystyle(\partial f)(x_{1},\cdots,x_{k+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(-1)^{i+1}\rho^{L}(x_{i})f(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1})+(-1)^{k+1}\rho^{R}(x_{k+1})f(x_{1},\cdots,x_{k})$
$\displaystyle+\sum_{1\leq i<j\leq
k+1}(-1)^{i}f(x_{1},\cdots,\hat{x_{i}},\cdots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{G}},x_{j+1},\cdots,x_{k+1}),$

for all $x_{1},\cdots,x_{k+1}\in\mathfrak{G}$. The resulting cohomology is
denoted by $\mathcal{HL}^{*}(\mathfrak{G},V)$.

######  Theorem 4.2.

([47]) Let $T$ be a relative Rota-Baxter operator on a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to $(V;\rho^{L},\rho^{R})$.
Define $[-,-]_{T}:\otimes^{2}V\,\rightarrow\,V$ by

(46) $\displaystyle[u,v]_{T}=\rho^{L}(Tu)v+\rho^{R}(Tv)u,\quad u,v\in V.$

Then $(V,[-,-]_{T})$ is a Leibniz algebra. Moreover, define
$\varrho^{L},\varrho^{R}:V\,\rightarrow\,\mathfrak{gl}(\mathfrak{G})$ by

(47)
$\displaystyle\varrho^{L}(u)x=[Tu,x]_{\mathfrak{G}}-T\rho^{R}(x)u,\quad\varrho^{R}(u)x=[x,Tu]_{\mathfrak{G}}-T\rho^{L}(x)u,\quad\forall
u\in V,x\in\mathfrak{G}.$

Then $(\mathfrak{G};\varrho^{L},\varrho^{R})$ is a representation of the
Leibniz algebra $(V,[-,-]_{T})$.

Let $\partial_{T}:C^{n}(V,\mathfrak{G})\longrightarrow
C^{n+1}(V,\mathfrak{G})$ be the corresponding Loday-Pirashvili coboundary
operator of the Leibniz algebra $(V,[-,-]_{T})$ with coefficients in the
representation $(\mathfrak{G};\varrho^{L},\varrho^{R})$. More precisely,
$\partial_{T}:C^{n}(V,\mathfrak{G})\longrightarrow C^{n+1}(V,\mathfrak{G})$ is
given by

$\displaystyle(\partial_{T}f)(v_{1},\cdots,v_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}[Tv_{i},f(v_{1},\cdots,\hat{v}_{i},\cdots,v_{n+1})]_{\mathfrak{G}}-\sum_{i=1}^{n}(-1)^{i+1}T\rho^{R}(f(v_{1},\cdots,\hat{v}_{i},\cdots,v_{n+1}))v_{i}$
$\displaystyle+(-1)^{n+1}[f(v_{1},\cdots,v_{n}),Tv_{n+1}]_{\mathfrak{G}}+(-1)^{n}T\rho^{L}(f(v_{1},\cdots,v_{n}))v_{n+1}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(v_{1},\cdots,\hat{v}_{i},\cdots,v_{j-1},\rho^{L}(Tv_{i})v_{j}+\rho^{R}(Tv_{j})v_{i},v_{j+1},\cdots,v_{n+1}).$

######  Definition 4.3.

([47]) Let $T$ be a relative Rota-Baxter operator on a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to a representation
$(V;\rho^{L},\rho^{R})$. The cohomology of the cochain complex

$(C^{*}(V,\mathfrak{G})=\oplus_{k=0}^{+\infty}C^{k}(V,\mathfrak{G}),\partial_{T})$

is taken to be the cohomology for the relative Rota-Baxter operator $T$.

We denote the set of $k$-cocycles by $\mathcal{Z}_{RB}^{k}(T)$, the set of
$k$-coboundaries by $\mathcal{B}_{RB}^{k}(T)$ and the $k$-th cohomology group
by

(48)
$\displaystyle\mathcal{H}_{RB}^{k}(T)=\mathcal{Z}_{RB}^{k}(T)/\mathcal{B}_{RB}^{k}(T).$

Up to a sign, the coboundary operators $\partial_{T}$ coincides with the
differential operator $\\{T,-\\}_{V}$ defined by using the Maurer-Cartan
element $T$.

######  Theorem 4.4.

([47]) Let $T$ be a relative Rota-Baxter operator on the Leibniz algebra
$(\mathfrak{G},[\cdot,\cdot]_{\mathfrak{G}})$ with respect to the
representation $(V;\rho^{L},\rho^{R})$. Then we have

${\partial}_{T}f=(-1)^{n-1}\\{T,f\\}_{V},\quad\forall
f\in\mathrm{Hom}(\otimes^{n}V,\mathfrak{G}),~{}n=1,2,\cdots.$

###  4.2. The Loday-Pirashvili cohomology of relative Rota-Baxter operators on
Lie algebras

In [44] the authors established a cohomology theory of a relative Rota-Baxter
operator on a Lie algebra as the Chevalley-Eilenberg cohomology of a Lie
algebra. Thus we will refer to it as the Chevalley-Eilenberg cohomology of a
relative Rota-Baxter operator on a Lie algebra. More precisely, let $T$ be a
relative Rota-Baxter operator on a Lie algebra $\mathfrak{g}$ with respect to
a representation $(V;\rho)$. Define
$\varrho:V\longrightarrow\mathfrak{gl}(\mathfrak{g})$ by

$\varrho(u)(x):=[Tu,x]+T\rho(x)(u),\;\;\forall x\in\mathfrak{g},u\in V.$

Then $(\mathfrak{g};\varrho)$ is a representation of the Lie algebra
$(V,[-,-]_{T})$ on the vector space $\mathfrak{g}$. We denote by
$(\mathfrak{C}^{*}(V,\mathfrak{g}),d_{\mathsf{CE}})$ the Chevalley-Eilenberg
cochain complex of the Lie algebra $(V,[-,-]_{T})$ with coefficients in the
representation $(\mathfrak{g};\varrho)$, where
$\mathfrak{C}^{*}(V,\mathfrak{g})=\oplus_{k\geq
0}\mathfrak{C}^{k}(V,\mathfrak{g})$ and
$\mathfrak{C}^{k}(V,\mathfrak{g})=\mathrm{Hom}(\wedge^{k}V,\mathfrak{g})$. The
cohomology of the cochain complex
$(\mathfrak{C}^{*}(V,\mathfrak{g}),d_{\mathsf{CE}})$ is called the Chevalley-
Eilenberg cohomology of the relative Rota-Baxter operator  $T$. Denote the set
of $k$-cocycles by $\mathcal{Z}^{k}_{\mathsf{CE}}(T)$ and the set of
$k$-coboundaries by $\mathcal{B}^{k}_{\mathsf{CE}}(T)$. Denote by

(49)
$\mathcal{H}^{k}_{\mathsf{CE}}(T)=\mathcal{Z}^{k}_{\mathsf{CE}}(T)/\mathcal{B}^{k}_{\mathsf{CE}}(T),\quad
k\geq 0,$

the corresponding $k$-th cohomology group.

As explained in Section 2, a relative Rota-Baxter operator
$T:V\,\rightarrow\,\mathfrak{g}$ on a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to a representation
$(V;\rho)$ can be viewed as a special relative Rota-Baxter operator on the
Leibniz algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the
symmetric representation $(V;\rho,-\rho)$. Thus, the above approach to define
the cohomology of a relative Rota-Baxter operator on a Leibniz algebra can be
applied to define a cohomology of a relative Rota-Baxter operator on a Lie
algebra. More precisely, let $T:V\,\rightarrow\,\mathfrak{g}$ be a relative
Rota-Baxter operator on a Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$
with respect to a representation $(V;\rho)$. Then $(V,[-,-]_{T})$ is a Leibniz
algebra, where the Leibniz bracket $[-,-]_{T}$ is defined by

$[u,v]_{T}=\rho(Tu)v-\rho(Tv)u,\quad\forall u,v\in V.$

Moreover, $(\mathfrak{g};\varrho^{L},\varrho^{R})$ is a symmetric
representation of the Leibniz algebra $(V,[-,-]_{T})$, where
$\varrho^{L},\varrho^{R}:V\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ is
defined by

$\varrho^{L}(u)x=[Tu,x]_{\mathfrak{g}}+T\rho(x)u,\quad\varrho^{R}(u)x=[x,Tu]_{\mathfrak{g}}-T\rho(x)u.$

Let $\partial_{T}:C^{n}(V,\mathfrak{g})\longrightarrow
C^{n+1}(V,\mathfrak{g})$ be the corresponding Loday-Pirashvili coboundary
operator of the Leibniz algebra $(V,[-,-]_{T})$ with coefficients in the
symmetric representation $(\mathfrak{g};\varrho^{L},\varrho^{R})$.

######  Definition 4.5.

Let $T:V\,\rightarrow\,\mathfrak{g}$ be a relative Rota-Baxter operator on a
Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to a
representation $(V;\rho)$. The cohomology of the cochain complex
$(\oplus_{n=0}^{+\infty}C^{n}(V,\mathfrak{g}),\partial_{T})$ is called the
Loday-Pirashvili cohomology of the relative Rota-Baxter operator $T$.

We denote the set of $k$-cocycles by $\mathcal{Z}^{k}_{\mathsf{LP}}({T})$, the
set of $k$-coboundaries by $\mathcal{B}^{k}_{\mathsf{LP}}({T})$ and the $k$-th
cohomology group by

(50)
$\displaystyle\mathcal{H}^{k}_{\mathsf{LP}}({T})=\mathcal{Z}^{k}_{\mathsf{LP}}({T})/\mathcal{B}^{k}_{\mathsf{LP}}({T}).$

######  Remark 4.6.

Note that the monomorphism
$\mathfrak{C}^{k}(V,\mathfrak{g})=\mathrm{Hom}(\wedge^{k}V,\mathfrak{g})\to
C^{k}(V,\mathfrak{g})=\mathrm{Hom}(\otimes^{k}V,\mathfrak{g})$ induced from
the natural epimorphism $\otimes^{k}V\to\wedge^{k}V$ induces isomorphisms

$\mathcal{H}^{0}_{\mathsf{CE}}({T})\cong\mathcal{H}^{0}_{\mathsf{LP}}({T})\,\,\,{\rm
and}\,\,\,\mathcal{H}^{1}_{\mathsf{CE}}({T})\cong\mathcal{H}^{1}_{\mathsf{LP}}({T})$

and a long exact sequence

$\displaystyle 0$ $\displaystyle\to$
$\displaystyle\mathcal{H}^{2}_{\mathsf{CE}}({T})\to\mathcal{H}^{2}_{\mathsf{LP}}({T})\to\mathcal{H}_{\rm
rel}^{0}({T})$ $\displaystyle\to$
$\displaystyle\mathcal{H}^{3}_{\mathsf{CE}}({T})\to\mathcal{H}^{3}_{\mathsf{LP}}({T})\to\mathcal{H}_{\rm
rel}^{1}({T}\to\cdots$

where $\mathcal{H}_{\rm rel}^{\bullet}({T})$ is the cohomology of the relative
complex $C_{\rm rel}^{\bullet}(V,\mathfrak{g})$ which is by definition (up to
a degree shift) the cokernel of the monomorphism
$\mathrm{Hom}(\wedge^{k}V,\mathfrak{g})\to\mathrm{Hom}(\otimes^{k}V,\mathfrak{g})$.
These matters as well as a spectral sequence linking $\mathcal{H}_{\rm
rel}^{\bullet}({T})$ and $\mathcal{H}_{\mathsf{LP}}^{\bullet}({T})$ in still
another way can be found in [18]. In particular, Theorem 2.6 in [18] shows how
the vanishing of Chevalley-Eilenberg cohomology implies the vanishing of
Loday-Pirashvili cohomology.

######  Remark 4.7.

Note furthermore that the comparison between Chevalley-Eilenberg cohomologies
and Loday-Pirashvili cohomologies makes sense more generally for any relative
Rota-Baxter operator $T:V\to\mathfrak{G}$ over a Leibniz algebra
$\mathfrak{G}$ as soon as the representation of $\mathfrak{G}$ on $V$ is
symmetric, because then the bracket $[-,-]_{T}$ on $V$ is a Lie bracket.

In case the induced representation of $(V,[-,-]_{T})$ on $\mathfrak{G}$ is not
symmetric, one may use the short exact sequence

$0\to\mathfrak{G}_{\rm anti}\to\mathfrak{G}\to\mathfrak{G}_{\rm sym}\to 0$

and the relation of Loday-Pirashvili cohomology with values in an
antisymmetric representation to the cohomology with values in a symmetric
representation (see [18]) in order to reduce the computations to the case of
symmetric coefficients.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $\mathfrak{G}$ with respect to a symmetric representation
$(V;\rho^{L},\rho^{R}=-\rho^{L})$. By Proposition 2.6,
$\bar{T}:=\mathrm{pr}\circ T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ is a
relative Rota-Baxter operator on the Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$
with respect to the representation $(V;\theta)$ given by (15). Now we
establish the relationship between the cohomology groups of the relative Rota-
Baxter operator $T$ and the Loday-Pirashvili cohomology groups of the relative
Rota-Baxter operator $\bar{T}.$

######  Lemma 4.8.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $\mathfrak{G}$ with respect to a symmetric representation
$(V;\rho^{L},\rho^{R}=-\rho^{L})$. Then
$(\mathsf{Leib}(\mathfrak{G});\varrho^{L},\varrho^{R})$ is a subrepresentation
of $(\mathfrak{G};\varrho^{L},\varrho^{R})$.

###### Proof.

Since $\rho^{R}=-\rho^{L}$, by (14), we have

$\displaystyle\varrho^{L}(u)[y,y]_{\mathfrak{G}}=[Tu,[y,y]_{\mathfrak{G}}]_{\mathfrak{G}}+\rho^{L}([y,y]_{\mathfrak{G}})u=[[Tu,y]_{\mathfrak{G}},y]_{\mathfrak{G}}+[y,[Tu,y]_{\mathfrak{G}}]_{\mathfrak{G}},$

for all $y\in\mathfrak{G}$, which implies that
$\varrho^{L}(u)(\mathsf{Leib}(\mathfrak{G}))\subset\mathsf{Leib}(\mathfrak{G}).$

By the fact that $\mathsf{Leib}(\mathfrak{G})$ is contained in the left center
of the Leibniz algebra $\mathfrak{G}$, we have

$\displaystyle\varrho^{R}(u)[y,y]_{\mathfrak{G}}=[[y,y]_{\mathfrak{G}},Tu]_{\mathfrak{G}}-T\rho^{L}([y,y]_{\mathfrak{G}})u=0,$

which implies that
$\varrho^{R}(u)(\mathsf{Leib}(\mathfrak{G}))=0\subset\mathsf{Leib}(\mathfrak{G})$.
Therefore, $(\mathsf{Leib}(\mathfrak{G});\varrho^{L},\varrho^{R})$ is a
subrepresentation of $(\mathfrak{G};\varrho^{L},\varrho^{R})$. ∎

Note that even though the Leibniz algebra $(V,[\cdot,\cdot]_{T})$ becomes a
Lie algebra if $T$ is a relative Rota-Baxter operator on a Leibniz algebra
$\mathfrak{G}$ with respect to a symmetric representation, the representation
given in Theorem 4.2 is still a representation of $(V,[\cdot,\cdot]_{T})$ as a
Leibniz algebra, and $\varrho^{R}\neq-\varrho^{L}.$ But the quotient
representation
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$ is
symmetric, where
$\bar{\varrho}^{L}:V\,\rightarrow\,\mathfrak{gl}(\mathfrak{G}_{\mathrm{Lie}})$
and
$\bar{\varrho}^{R}:V\,\rightarrow\,\mathfrak{gl}(\mathfrak{G}_{\mathrm{Lie}})$
are given by

(51)
$\bar{\varrho}^{L}(u)\bar{x}=\overline{\varrho^{L}(u)x},\quad\bar{\varrho}^{R}(u)\bar{x}=\overline{\varrho^{R}(u)x},\quad\forall
u\in V,x\in\mathfrak{G}.$

######  Proposition 4.9.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $\mathfrak{G}$ with respect to a symmetric representation
$(V;\rho^{L},\rho^{R}=-\rho^{L})$. Then the quotient representation
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$ is a
symmetric representation of $(V,[\cdot,\cdot]_{T})$ as a Leibniz algebra, i.e.
$\bar{\varrho}^{R}=-\bar{\varrho}^{L}$.

###### Proof.

For all $u\in V,x\in\mathfrak{G}$, we have

$\displaystyle\varrho^{L}(u)x+\varrho^{R}(u)x$ $\displaystyle=$
$\displaystyle[Tu,x]_{\mathfrak{G}}-T\rho^{R}(x)u+[x,Tu]_{\mathfrak{G}}-T\rho^{L}(x)u$
$\displaystyle=$
$\displaystyle[Tu,x]_{\mathfrak{G}}+[x,Tu]_{\mathfrak{G}}\in\mathsf{Leib}(\mathfrak{G}).$

Thus, we deduce that $\bar{\varrho}^{R}=-\bar{\varrho}^{L}$. ∎

Let
$\overline{\partial}_{{T}}:C^{n}(V,\mathfrak{G}_{\mathrm{Lie}})\longrightarrow
C^{n+1}(V,\mathfrak{G}_{\mathrm{Lie}})$ be the corresponding Loday-Pirashvili
coboundary operator of the Leibniz algebra $(V,[\cdot,\cdot]_{T})$ with
coefficients in the quotient representation
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$.

######  Proposition 4.10.

Let $T$ be a relative Rota-Baxter operator on a Leibniz algebra
$(\mathfrak{G},[\cdot,\cdot]_{\mathfrak{G}})$ with respect to a symmetric
representation $(V;\rho^{L},\rho^{R}=-\rho^{L})$. Then

$\overline{\partial}_{{T}}=\partial_{\bar{T}}.$

That is, the Loday-Pirashvili cochain complex
$(\oplus_{n=0}^{+\infty}C^{n}(V,\mathfrak{G}_{\mathrm{Lie}}),\partial_{\bar{T}})$
associated to the relative Rota-Baxter operator $\bar{T}$ and the cochain
complex
$(\oplus_{n=0}^{+\infty}C^{n}(V,\mathfrak{G}_{\mathrm{Lie}}),\overline{\partial}_{T})$
obtained by using the quotient representation
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$ are the
same.

###### Proof.

Obviously, the spaces of $n$-cochains are the same. Thus we only need to show
that the induced Leibniz algebra structure $[-,-]_{\bar{T}}$ on $V$ by the
relative Rota-Baxter operator $\bar{T}$ is the same as $[-,-]_{{T}}$, and the
representations $(\mathfrak{G}_{\mathrm{Lie}};\varrho^{L},\varrho^{R})$ and
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$ are the
same.

For all $u,v\in V$, by (15), we have

$[u,v]_{\bar{T}}=\theta(\bar{T}u)v-\theta(\bar{T}v)u=\rho^{L}(Tu)v-\rho^{L}(Tv)u=[u,v]_{T}.$

For any $u\in V,x\in\mathfrak{G}$, we have

$\displaystyle\varrho^{L}(u)(\bar{x})=[\bar{T}u,\bar{x}]_{\mathfrak{G}_{\mathrm{Lie}}}+\bar{T}\theta(\bar{x})u=\overline{[Tu,x]_{\mathfrak{G}}}+\overline{T\rho^{L}(x)u}=\bar{\varrho}^{L}(u)\bar{x},$
$\displaystyle\varrho^{R}(u)(\bar{x})=[\bar{x},\bar{T}u]_{\mathfrak{G}_{\mathrm{Lie}}}-\bar{T}\theta(\bar{x})u=\overline{[x,Tu]_{\mathfrak{G}}}-\overline{T\rho^{L}(x)u}=\bar{\varrho}^{R}(u)\bar{x}.$

Thus $\overline{\partial}_{{T}}=\partial_{\bar{T}}.$ ∎

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $\mathfrak{G}$ with respect to a symmetric representation
$(V;\rho^{L},\rho^{R}=-\rho^{L})$. By Lemma 4.8, we have the following short
exact sequence of representations of the Leibniz algebra
$(V,[\cdot,\cdot]_{T})$:

(52) $\displaystyle
0\,\rightarrow\,(\mathsf{Leib}(\mathfrak{G});\varrho^{L},\varrho^{R})\stackrel{{\scriptstyle\mathfrak{i}}}{{\,\rightarrow\,}}(\mathfrak{G};\varrho^{L},\varrho^{R})\stackrel{{\scriptstyle\mathrm{pr}}}{{\,\rightarrow\,}}(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})\,\rightarrow\,0.$

Moreover, we have the following result:

######  Theorem 4.11.

Let $T$ be a relative Rota-Baxter operator on a Leibniz algebra
$(\mathfrak{G},[\cdot,\cdot]_{\mathfrak{G}})$ with respect to a symmetric
representation $(V;\rho^{L},\rho^{R}=-\rho^{L})$. Then there is a short exact
sequence of the cochain complexes:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\longrightarrow\mathrm{Hom}(\otimes^{n}V,\mathsf{Leib}(\mathfrak{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{V}}$$\scriptstyle{\partial}$$\textstyle{\mathrm{Hom}(\otimes^{n+1}V,\mathsf{Leib}(\mathfrak{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\longrightarrow\cdots}$$\scriptstyle{\alpha_{V}}$$\textstyle{\cdots\longrightarrow\mathrm{Hom}(\otimes^{n}V,\mathfrak{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{V}}$$\scriptstyle{\partial_{T}}$$\textstyle{\mathrm{Hom}(\otimes^{n+1}V,\mathfrak{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\longrightarrow\cdots}$$\scriptstyle{\beta_{V}}$$\textstyle{\cdots\longrightarrow\mathrm{Hom}(\otimes^{n}V,\mathfrak{G}_{\mathrm{Lie}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{\bar{T}}}$$\textstyle{\mathrm{Hom}(\otimes^{n+1}V,\mathfrak{G}_{\mathrm{Lie}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\longrightarrow\cdots,}$$\textstyle{0}$$\textstyle{0}$

where $\alpha_{V}$ and $\beta_{V}$ are given by (36) and (37) respectively.

Consequently, there is a long exact sequence of the cohomology groups:

(53)
$\cdots\longrightarrow\mathcal{HL}^{n}(V,\mathsf{Leib}(\mathfrak{G}))\stackrel{{\scriptstyle\mathcal{H}^{n}(\alpha_{V})}}{{\longrightarrow}}\mathcal{H}_{RB}^{n}(T)\stackrel{{\scriptstyle\mathcal{H}^{n}(\beta_{V})}}{{\longrightarrow}}\mathcal{H}_{\mathsf{LP}}^{n}(\bar{T})\stackrel{{\scriptstyle
c^{n}}}{{\longrightarrow}}\mathcal{HL}^{n+1}(V,\mathsf{Leib}(\mathfrak{G}))\longrightarrow\cdots,$

where the connecting map $c^{n}$ is defined by
$c^{n}([h])=[\alpha_{V}^{-1}(\partial_{T}(\beta_{V}^{-1}(h)))],$ for all
$[h]\in\mathcal{H}_{\mathsf{LP}}^{n}(\bar{T}).$

###### Proof.

For any $g\in\mathrm{Hom}(\otimes^{n}V,\mathsf{Leib}(\mathfrak{G}))$ and
$v_{1},\cdots,v_{n+1}\in V$, we have

$\displaystyle\big{(}\alpha_{V}(\partial g)\big{)}(v_{1},\cdots,v_{n+1})$
$\displaystyle\stackrel{{\scriptstyle\eqref{natural-transformation-1}}}{{=}}$
$\displaystyle(\partial g)(v_{1},\cdots,v_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}[Tv_{i},g(v_{1},\cdots,\hat{v}_{i},\cdots,v_{n+1})]_{\mathfrak{G}}+(-1)^{n}T\rho^{L}(g(v_{1},\cdots,v_{n}))v_{n+1}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}g(v_{1},\cdots,\hat{v}_{i},\cdots,v_{j-1},\rho^{L}(Tv_{i})v_{j}+\rho^{R}(Tv_{j})v_{i},v_{j+1},\cdots,v_{n+1})$
$\displaystyle=$
$\displaystyle\big{(}\partial_{T}(\alpha_{V}(g))\big{)}(v_{1},\cdots,v_{n+1}),$

which implies that $\alpha_{V}$ is a homomorphism of cochain complexes. For
$n=0$ and $x\in\mathfrak{G},v\in V$, we have

$\displaystyle\big{(}\beta_{V}(\partial_{T}x)\big{)}(v)=\beta_{V}(-[x,Tv]_{\mathfrak{G}}+T\rho^{L}(x)v)\stackrel{{\scriptstyle\eqref{natural-
transformation-2}}}{{=}}-\bar{\varrho}^{R}(v)\bar{x}=\big{(}\partial_{\bar{T}}\beta_{V}(x)\big{)}(v).$

For $n\geq 1$, by Theorem 4.4 and Theorem 3.5, for all
$f\in\mathrm{Hom}(\otimes^{n}V,\mathfrak{G})$, we have

$\displaystyle\beta_{V}(\partial_{T}f)=(-1)^{n-1}\beta_{V}\\{T,f\\}_{V}=(-1)^{n-1}\\{\beta_{V}(T),\beta_{V}(f)\\}_{V}=\partial_{\bar{T}}\beta_{V}(f).$

Thus, we deduce that $\beta_{V}$ is a homomorphism of cochain complexes. Since
$V$ is vector space over the field $\mathbb{K}$, for any positive integer $n$,
the functor $\mathrm{Hom}(\otimes^{n}V,-)$ is an exact functor from the
category of vector spaces over $\mathbb{K}$ to itself. Moreover, we have
$\alpha_{V}=\mathfrak{i}_{*}$ and $\beta_{V}=\mathrm{pr}_{*}$. By the short
exact sequence (52) of representations of the Leibniz algebra $(V,[-,-]_{T})$,
we obtain the short exact sequence of the above cochain complexes. The proof
is finished. ∎

######  Remark 4.12.

In the situation of Theorem 4.11, the short exact sequence of cochain
complexes can be completed into a diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{C}^{\bullet}(V,\mathsf{Leib}(\mathfrak{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\bullet}(V,\mathsf{Leib}(\mathfrak{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{\rm
rel}^{\bullet}(V,\mathsf{Leib}(\mathfrak{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{C}^{\bullet}(V,\mathfrak{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\bullet}(V,\mathfrak{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{\rm
rel}^{\bullet}(V,\mathfrak{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{C}^{\bullet}(V,\mathfrak{G}_{\mathrm{Lie}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\bullet}(V,\mathfrak{G}_{\mathrm{Lie}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{\rm
rel}^{\bullet}(V,\mathfrak{G}_{\mathrm{Lie}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

which leads then to a corresponding exact diagram in cohomology (starting from
degree $2$, while there are isomorphisms in degree $0$ and $1$ as before).

###  4.3. Cohomology of relative averaging operators on Lie algebras

As explained in Section 2, a relative averaging operator
$T:V\,\rightarrow\,\mathfrak{g}$ on a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to a representation
$(V;\rho)$ can be viewed as a special relative Rota-Baxter operator on the
Leibniz algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the
antisymmetric representation $(V;\rho,0)$. Thus, the above approach to define
the cohomology of relative Rota-Baxter operators on Leibniz algebras can be
applied to define the cohomology of relative averaging operators. More
precisely, let $T:V\,\rightarrow\,\mathfrak{g}$ be a relative averaging
operator on a Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect
to a representation $(V;\rho)$. Then $(V,[-,-]_{T})$ is a Leibniz algebra,
where the Leibniz bracket $[-,-]_{T}$ is defined by

$[u,v]_{T}=\rho(Tu)v,\quad\forall u,v\in V.$

Moreover, $(\mathfrak{g};\varrho^{L},\varrho^{R})$ is a representation of the
Leibniz algebra $(V,[-,-]_{T})$, where
$\varrho^{L},\varrho^{R}:V\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ is
defined by

$\varrho^{L}(u)x=[Tu,x]_{\mathfrak{g}},\quad\varrho^{R}(u)x=[x,Tu]_{\mathfrak{g}}-T\rho(x)u.$

Let $\partial_{T}:C^{n}(V,\mathfrak{g})\longrightarrow
C^{n+1}(V,\mathfrak{g})$ be the corresponding Loday-Pirashvili coboundary
operator of the Leibniz algebra $(V,[-,-]_{T})$ with coefficients in the
representation $(\mathfrak{g};\varrho^{L},\varrho^{R})$. More precisely,
$\partial_{T}:C^{n}(V,\mathfrak{g})\longrightarrow C^{n+1}(V,\mathfrak{g})$ is
given by

$\displaystyle(\partial_{T}f)(v_{1},\cdots,v_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}[Tv_{i},f(v_{1},\cdots,\hat{v}_{i},\cdots,v_{n+1})]_{\mathfrak{g}}$
$\displaystyle+(-1)^{n+1}[f(v_{1},\cdots,v_{n}),Tv_{n+1}]_{\mathfrak{g}}+(-1)^{n}T\rho(f(v_{1},\cdots,v_{n}))v_{n+1}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(v_{1},\cdots,\hat{v}_{i},\cdots,v_{j-1},\rho(Tv_{i})v_{j},v_{j+1},\cdots,v_{n+1}).$

######  Definition 4.13.

Let $T:V\,\rightarrow\,\mathfrak{g}$ be a relative averaging operator on a Lie
algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to a representation
$(V;\rho)$. The cohomology of the cochain complex
$(\oplus_{n=0}^{+\infty}C^{n}(V,\mathfrak{g}),\partial_{T})$ is defined to be
the cohomology of the relative averaging operator $T$.

We denote the set of $k$-cocycles by $\mathcal{Z}^{k}_{AO}({T})$, the set of
$k$-coboundaries by $\mathcal{B}^{k}_{AO}({T})$ and the $k$-th cohomology
group by

(54)
$\displaystyle\mathcal{H}^{k}_{AO}({T})=\mathcal{Z}^{k}_{AO}({T})/\mathcal{B}^{k}_{AO}({T}).$

See [41] for more details about the cohomology theory of a relative averaging
operator on a Lie algebra.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on the
Leibniz algebra $\mathfrak{G}$ with respect to an antisymmetric representation
$(V;\rho^{L},\rho^{R}=0)$. By Proposition 2.10, $\bar{T}:=\mathrm{pr}\circ
T:V\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$ is a relative averaging
operator on the Lie algebra $\mathfrak{G}_{\mathrm{Lie}}$ with respect to the
representation $(V;\theta)$ given by (15). Now we establish the relationship
between the cohomology groups of the relative Rota-Baxter operator $T$ and the
cohomology groups of the relative averaging operator $\bar{T}.$

Similarly as Lemma 4.8, we have the following result.

######  Lemma 4.14.

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $\mathfrak{G}$ with respect to an antisymmetric representation
$(V;\rho^{L},\rho^{R}=0)$. Then
$(\mathsf{Leib}(\mathfrak{G});\varrho^{L},\varrho^{R})$ is a subrepresentation
of $(\mathfrak{G};\varrho^{L},\varrho^{R})$.

By Lemma 4.14, we have the quotient representation
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$, where
$\bar{\varrho}^{L}:V\,\rightarrow\,\mathfrak{gl}(\mathfrak{G}_{\mathrm{Lie}})$
and
$\bar{\varrho}^{R}:V\,\rightarrow\,\mathfrak{gl}(\mathfrak{G}_{\mathrm{Lie}})$
are given by (51). Let
$\overline{\partial}_{{T}}:C^{n}(V,\mathfrak{G}_{\mathrm{Lie}})\longrightarrow
C^{n+1}(V,\mathfrak{G}_{\mathrm{Lie}})$ be the corresponding Loday-Pirashvili
coboundary operator of the Leibniz algebra $(V,[-,-]_{T})$ with coefficients
in the quotient representation
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$. Similar
to Proposition 4.10, we have the following result.

######  Proposition 4.15.

Let $T$ be a relative Rota-Baxter operator on a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ with respect to an antisymmetric
representation $(V;\rho^{L},\rho^{R}=0)$. Then

$\overline{\partial}_{{T}}=\partial_{\bar{T}}.$

That is, the cochain complex
$(\oplus_{n=0}^{+\infty}C^{n}(V,\mathfrak{G}_{\mathrm{Lie}}),\partial_{\bar{T}})$
associated to the relative averaging operator $\bar{T}$ and the cochain
complex
$(\oplus_{n=0}^{+\infty}C^{n}(V,\mathfrak{G}_{\mathrm{Lie}}),\overline{\partial}_{T})$
obtained by using the quotient representation
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$ are the
same.

###### Proof.

Obviously, the spaces of $n$-cochains are the same. Thus we only need to show
that the induced Leibniz algebra structure $[-,-]_{\bar{T}}$ on $V$ by the
relative averaging operator $\bar{T}$ is the same as $[-,-]_{{T}}$, and the
representations $(\mathfrak{G}_{\mathrm{Lie}};\varrho^{L},\varrho^{R})$ and
$(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})$ are the
same.

For all $u,v\in V$, by (15), we have

$[u,v]_{\bar{T}}=\theta(\bar{T}u)v=\rho^{L}(Tu)v=[u,v]_{T}.$

For any $u\in V,x\in\mathfrak{G}$, we have

$\displaystyle\varrho^{L}(u)\bar{x}=[\bar{T}u,\bar{x}]_{\mathfrak{G}_{\mathrm{Lie}}}=\overline{[Tu,x]_{\mathfrak{G}}}=\bar{\varrho}^{L}(u)\bar{x},$
$\displaystyle\varrho^{R}(u)\bar{x}=[\bar{x},\bar{T}u]_{\mathfrak{G}_{\mathrm{Lie}}}-\bar{T}\theta(\bar{x})u=\overline{[x,Tu]_{\mathfrak{G}}}-\overline{T\rho^{L}(x)u}=\bar{\varrho}^{R}(u)\bar{x}.$

Thus, $\overline{\partial}_{{T}}=\partial_{\bar{T}}.$ ∎

Let $T:V\,\rightarrow\,\mathfrak{G}$ be a relative Rota-Baxter operator on a
Leibniz algebra $\mathfrak{G}$ with respect to an antisymmetric representation
$(V;\rho^{L},\rho^{R}=0)$. By Lemma 4.14, we have the following short exact
sequence of representations of the Leibniz algebra $(V,[-,-]_{T})$:

(55) $\displaystyle
0\,\rightarrow\,(\mathsf{Leib}(\mathfrak{G});\varrho^{L},\varrho^{R})\stackrel{{\scriptstyle\mathfrak{i}}}{{\,\rightarrow\,}}(\mathfrak{G};\varrho^{L},\varrho^{R})\stackrel{{\scriptstyle\mathrm{pr}}}{{\,\rightarrow\,}}(\mathfrak{G}_{\mathrm{Lie}};\bar{\varrho}^{L},\bar{\varrho}^{R})\,\rightarrow\,0.$

Similar to Theorem 4.11, we have the following result describing the relation
among various cohomology groups.

######  Theorem 4.16.

Let $T$ be a relative Rota-Baxter operator on a Leibniz algebra
$(\mathfrak{G},[\cdot,\cdot]_{\mathfrak{G}})$ with respect to an antisymmetric
representation $(V;\rho^{L},\rho^{R}=0)$. Then we have a long exact sequence
of cohomology groups:

(56)
$\cdots\longrightarrow\mathcal{HL}^{n}(V,\mathsf{Leib}(\mathfrak{G}))\stackrel{{\scriptstyle\mathcal{H}^{n}(\alpha_{V})}}{{\longrightarrow}}\mathcal{H}_{RB}^{n}(T)\stackrel{{\scriptstyle\mathcal{H}^{n}(\beta_{V})}}{{\longrightarrow}}\mathcal{H}_{AO}^{n}(\bar{T})\stackrel{{\scriptstyle
c^{n}}}{{\longrightarrow}}\mathcal{HL}^{n+1}(V,\mathsf{Leib}(\mathfrak{G}))\longrightarrow\cdots.$

###  4.4. Some computations of cohomologies

######  Example 4.17.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the Lie algebra of all $n\times n$
matrices over $\mathbb{C}$. Then
$T={\rm{Id}}:\mathfrak{gl}(n,\mathbb{C})\,\rightarrow\,\mathfrak{gl}(n,\mathbb{C})$
is a relative Rota-Baxter operator on the Lie algebra
$\mathfrak{gl}(n,\mathbb{C})$ with respect to the representation
$(\mathfrak{gl}(n,\mathbb{C});\rho)$, where
$\rho:\mathfrak{gl}(n,\mathbb{C})\,\rightarrow\,\mathfrak{gl}(\mathfrak{gl}(n,\mathbb{C}))$
is defined by

$\rho(A)B=AB,\quad\forall A,B\in\mathfrak{gl}(n,\mathbb{C}).$

Then $(\mathfrak{gl}(n,\mathbb{C}),[-,-]_{T})$ is the original Lie algebra
$\mathfrak{gl}(n,\mathbb{C})$. Moreover,
$(\mathfrak{gl}(n,\mathbb{C});\varrho)$ is a representation of the Lie algebra
$\mathfrak{gl}(n,\mathbb{C})$, where
$\varrho:\mathfrak{gl}(n,\mathbb{C})\,\rightarrow\,\mathfrak{gl}(\mathfrak{gl}(n,\mathbb{C}))$
is defined by

(57) $\varrho(A)B:=[TA,B]+T\rho(B)(A)=AB=\rho(A)B,\;\;\forall
A,B\in\mathfrak{gl}(n,\mathbb{C}).$

Thus, the cohomology of the relative Rota-Baxter operator $T$ is the
Chevalley-Eilenberg cohomology of $\mathfrak{gl}(n,\mathbb{C})$ with
coefficients in the representation $(\mathfrak{gl}(n,\mathbb{C});\rho)$. In
fact, the Lie algebra $\mathfrak{gl}(n,\mathbb{C})$ is the direct sum
$\mathfrak{gl}(n,\mathbb{C})=\mathfrak{sl}(n,\mathbb{C})\oplus{\mathbb{C}}$,
where the factor ${\mathbb{C}}$ is the center and consists of diagonal
matrices. The Hochschild-Serre formula [22] reads then

$\displaystyle\mathcal{H}_{\mathsf{CE}}^{n}(\mathfrak{gl}(n,\mathbb{C}),\mathfrak{gl}(n,\mathbb{C}))=\bigoplus_{p+q=n}\mathcal{H}_{\mathsf{CE}}^{p}(\mathfrak{sl}(n,\mathbb{C}))\otimes\mathcal{H}_{\mathsf{CE}}^{q}({\mathbb{C}},\mathfrak{gl}(n,\mathbb{C}))^{\mathfrak{sl}(n,\mathbb{C})}$
$\displaystyle=\mathcal{H}_{\mathsf{CE}}^{n}(\mathfrak{sl}(n,\mathbb{C}))\otimes\mathcal{H}_{\mathsf{CE}}^{0}({\mathbb{C}},\mathfrak{gl}(n,\mathbb{C}))^{\mathfrak{sl}(n,\mathbb{C})}\oplus\mathcal{H}_{\mathsf{CE}}^{n-1}(\mathfrak{sl}(n,\mathbb{C}))\otimes\mathcal{H}_{\mathsf{CE}}^{1}({\mathbb{C}},\mathfrak{gl}(n,\mathbb{C}))^{\mathfrak{sl}(n,\mathbb{C})}.$

The invariant space
$\mathcal{H}_{\mathsf{CE}}^{0}({\mathbb{C}},\mathfrak{gl}(n,\mathbb{C}))^{\mathfrak{sl}(n,\mathbb{C})}$
is zero, because
$\mathcal{H}_{\mathsf{CE}}^{0}({\mathbb{C}},\mathfrak{gl}(n,\mathbb{C}))=0$.
Furthermore, we have
$\mathcal{H}_{\mathsf{CE}}^{1}({\mathbb{C}},\mathfrak{gl}(n,\mathbb{C}))=0$,
because all elements are coboundaries. Therefore all the cohomology groups are
zero.

######  Example 4.18.

Let $\mathfrak{t}(n,\mathbb{C})$ be the Lie algebras of all $n\times n$ upper
triangular matrices over $\mathbb{C}$. Then
$T={\rm{Id}}:\mathfrak{t}(n,\mathbb{C})\,\rightarrow\,\mathfrak{t}(n,\mathbb{C})$
is a relative Rota-Baxter operator on the Lie algebra
$\mathfrak{t}(n,\mathbb{C})$ with respect to the representation
$(\mathfrak{t}(n,\mathbb{C});\rho)$, where
$\rho:\mathfrak{t}(n,\mathbb{C})\,\rightarrow\,\mathfrak{gl}(\mathfrak{t}(n,\mathbb{C}))$
is defined by

$\rho(A)B=AB,\quad\forall A,B\in\mathfrak{t}(n,\mathbb{C}).$

Thus, the cohomology of the relative Rota-Baxter operator $T$ is the
Chevalley-Eilenberg cohomology of $\mathfrak{t}(n,\mathbb{C})$ with
coefficients in the representation $(\mathfrak{t}(n,\mathbb{C});\rho)$. Recall
that the Lie algebra $\mathfrak{t}(n,\mathbb{C})$ is an extension

$0\to\mathfrak{n}(n,\mathbb{C})\to\mathfrak{t}(n,\mathbb{C})\to\mathfrak{d}(n,\mathbb{C})\to
0,$

where $\mathfrak{n}(n,\mathbb{C})$ is the Lie algebras of all $n\times n$
upper triangular nilpotent matrices and $\mathfrak{d}(n,\mathbb{C})$ is the
Lie algebras of all $n\times n$ diagonal matrices.

The Hochschild-Serre spectral sequence [22] associated to this extension
converges towards
$\mathcal{H}_{\mathsf{CE}}^{\bullet}(\mathfrak{t}(n,\mathbb{C}),\mathfrak{t}(n,\mathbb{C}))$
and has as its second page

$E_{2}^{p,q}=\mathcal{H}_{\mathsf{CE}}^{p}(\mathfrak{d}(n,\mathbb{C}),\mathcal{H}_{\mathsf{CE}}^{q}(\mathfrak{n}(n,\mathbb{C}),\mathfrak{t}(n,\mathbb{C}))).$

Let us compute this $E_{2}$-term for $n=2$. The Lie algebra
$\mathfrak{n}(2,\mathbb{C})$ is $1$-dimensional and we have only non zero
contributions from $q=0,1$. We have
$\mathcal{H}_{\mathsf{CE}}^{0}(\mathfrak{n}(2,\mathbb{C}),\mathfrak{t}(2,\mathbb{C}))=\mathfrak{t}(2,\mathbb{C})^{\mathfrak{n}(2,\mathbb{C})}$
of dimension $2$ generated by

$\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right)\,\,\,\,{\rm
and}\,\,\,\,\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right).$

From there, we obtain using Dixmier’s Theorem 1 of [17] for the nilpotent Lie
algebra $\mathfrak{d}(2,\mathbb{C})$ with values in a module which does not
contains the trivial module

$\mathcal{H}_{\mathsf{CE}}^{p}(\mathfrak{d}(2,\mathbb{C}),\mathcal{H}_{\mathsf{CE}}^{0}(\mathfrak{n}(2,\mathbb{C}),\mathfrak{t}(2,\mathbb{C})))=0,$

for all $p$. On the other hand,
$\mathcal{H}_{\mathsf{CE}}^{1}(\mathfrak{n}(2,\mathbb{C}),\mathfrak{t}(2,\mathbb{C}))\cong\mathfrak{d}(2,\mathbb{C})$
as all maps in
$\mathrm{Hom}(\mathfrak{n}(2,\mathbb{C}),\mathfrak{t}(2,\mathbb{C}))\cong\mathfrak{t}(2,\mathbb{C})$
are cocycles and the coboundaries correspond to $\mathfrak{n}(2,\mathbb{C})$.
From there, we obtain again with Dixmier’s Theorem 1

$\mathcal{H}_{\mathsf{CE}}^{p}(\mathfrak{d}(2,\mathbb{C}),\mathcal{H}_{\mathsf{CE}}^{1}(\mathfrak{n}(2,\mathbb{C}),\mathfrak{t}(2,\mathbb{C})))=0,$

for all $p$. Thus again all the cohomology groups are zero. For general $n$,
the whole cohomology will still vanish in case the cohomology spaces
$\mathcal{H}_{\mathsf{CE}}^{q}(\mathfrak{n}(n,\mathbb{C}),\mathfrak{t}(n,\mathbb{C}))$
do not contain the trivial $\mathfrak{d}(n,\mathbb{C})$-module.

######  Example 4.19.

Let $\mathfrak{n}(n,\mathbb{C})$ be the Lie algebras of all $n\times n$ upper
triangular nilpotent matrices over $\mathbb{C}$. Then
$T={\rm{Id}}:\mathfrak{n}(n,\mathbb{C})\,\rightarrow\,\mathfrak{n}(n,\mathbb{C})$
is a relative Rota-Baxter operator on the Lie algebra
$\mathfrak{n}(n,\mathbb{C})$ with respect to the representation
$(\mathfrak{n}(n,\mathbb{C});\rho)$, where
$\rho:\mathfrak{n}(n,\mathbb{C})\,\rightarrow\,\mathfrak{gl}(\mathfrak{n}(n,\mathbb{C}))$
is defined by

$\rho(A)B=AB,\quad\forall A,B\in\mathfrak{n}(n,\mathbb{C}).$

Thus, the cohomology of the relative Rota-Baxter operator $T$ is the
Chevalley-Eilenberg cohomology of $\mathfrak{n}(n,\mathbb{C})$ with
coefficients in the representation $(\mathfrak{n}(n,\mathbb{C});\rho)$. Since
$\mathfrak{n}(n,\mathbb{C})$ is a nilpotent Lie algebra and the
$\mathfrak{n}(n,\mathbb{C})$-module $\mathfrak{n}(n,\mathbb{C})$ does contain
the trivial $\mathfrak{n}(n,\mathbb{C})$-module (namely the submodule
generated by the matrix with non-zero entry in the upper right corner), we
have by Dixmier’s Theorem 2 of [17] that

$\dim\mathcal{H}_{\mathsf{CE}}^{0}(\mathfrak{n}(n,\mathbb{C}),\mathfrak{n}(n,\mathbb{C}))\geq
1,\,\,\,\dim\mathcal{H}_{\mathsf{CE}}^{n(n-1)/2}(\mathfrak{n}(n,\mathbb{C}),\mathfrak{n}(n,\mathbb{C}))\geq
1,$

and

$\dim\mathcal{H}_{\mathsf{CE}}^{i}(\mathfrak{n}(n,\mathbb{C}),\mathfrak{n}(n,\mathbb{C}))\geq
2,\quad\,\,\forall 0<i<\frac{n(n-1)}{2}.$

######  Example 4.20.

([41]) Let $(\mathfrak{G},[\cdot,\cdot]_{\mathfrak{G}})$ be a Leibniz algebra.
Then the natural projection

$T=\mathrm{pr}:\mathfrak{G}\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$

gives a relative averaging operator on the Lie algebra
$\mathfrak{G}_{\mathrm{Lie}}$ with respect to the representation
$(\mathfrak{G};\rho)$, where
$\rho:\mathfrak{G}_{\mathrm{Lie}}\,\rightarrow\,\mathfrak{gl}(\mathfrak{G})$
is defined by

$\rho(\bar{x})y=[x,y]_{\mathfrak{G}},\quad\forall x,y\in\mathfrak{G}.$

Then $(\mathfrak{G},[-,-]_{T})$ is the original Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. Moreover,
$(\mathfrak{G}_{\mathrm{Lie}};\varrho^{L},\varrho^{R})$ is a representation of
the Leibniz algebra $(\mathfrak{G},[-,-]_{T})$, where
$\varrho^{L},\varrho^{R}:\mathfrak{G}\,\rightarrow\,\mathfrak{gl}(\mathfrak{G}_{\mathrm{Lie}})$
is defined by

$\varrho^{L}(x)\bar{y}=[\bar{x},\bar{y}]_{\mathfrak{G}_{\mathrm{Lie}}},\quad\varrho^{R}(x)\bar{y}=[\bar{y},\bar{x}]_{\mathfrak{G}_{\mathrm{Lie}}}-\overline{[y,x]_{\mathfrak{G}}}=0.$

Thus, the cohomology of the relative averaging operator $T$ is the Loday-
Pirashvili cohomology of $(\mathfrak{G},[\cdot,\cdot]_{\mathfrak{G}})$ with
coefficients in the antisymmetric representation
$(\mathfrak{G}_{\mathrm{Lie}};\varrho^{L},\varrho^{R})$.

Let now $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ be a finite-dimensional
semisimple Leibniz algebra over a field of characteristic zero. By Theorem 4.3
of [18], we obtain that
$\mathcal{HL}^{p}(\mathfrak{G},\mathfrak{G}_{\mathrm{Lie}})=\\{0\\}$ for
$p\geq 2$ and an exact sequence

$0\to(\mathfrak{G}_{\mathrm{Lie}})_{\mathrm{anti}}\to\mathcal{HL}^{0}(\mathfrak{G},\mathfrak{G}_{\mathrm{Lie}})\to\mathfrak{G}_{\mathrm{Lie}}^{\mathfrak{G}_{\mathrm{Lie}}}\to\mathrm{Hom}_{\mathfrak{G}}(\mathfrak{G},\mathfrak{G}_{\mathrm{Lie}})\to\mathcal{HL}^{1}(\mathfrak{G},\mathfrak{G}_{\mathrm{Lie}})\to
0.$

As $\mathfrak{G}_{\mathrm{Lie}}$ is already antisymmetric and
$\mathcal{HL}^{0}(\mathfrak{G},\mathfrak{G}_{\mathrm{Lie}})=\mathfrak{G}_{\mathrm{Lie}}$,
the first (non-trivial) map in this sequence is an isomorphism. Furthermore
$\mathfrak{G}_{\mathrm{Lie}}^{\mathfrak{G}_{\mathrm{Lie}}}=\\{0\\}$, because
$\mathfrak{G}_{\mathrm{Lie}}$ is semisimple. Therefore the last (non-trivial)
map in this sequence is also an isomorphism (as in Lemma 1.4 of [18]).

Acknowledgements.  This research is supported by NSFC (11922110,12001228).

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