# Weighted relative Rota-Baxter operators on Leibniz algebras and Post-Leibniz
algebra structures

and  Apurba Das  Department of Mathematics, Indian Institute of Technology,
Kharagpur 721302, West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

Leibniz algebras are non-skewsymmetric analogue of Lie algebras. In this
paper, we consider weighted relative Rota-Baxter operators on Leibniz
algebras. We define cohomology of such operators and as an application, we
study their deformations. Finally, we introduce and study post-Leibniz
algebras as the structure behind weighted relative Rota-Baxter operators.

###### Key words and phrases:

Leibniz algebras, Rota-Baxter operators with weight, Cohomology, Deformations,
Post-Leibniz
algebras.

###### 2010 Mathematics Subject Classification:

17A32, 17A36, 17B56, 16S80.

###### Contents

  1. 1 Introduction
  2. 2 Background on Leibniz algebras
  3. 3 Weighted relative Rota-Baxter operators
  4. 4 Maurer-Cartan characterization and cohomology of weighted relative Rota-Baxter operators
  5. 5 Deformations of weighted relative Rota-Baxter operators
  6. 6 Post-Leibniz algebras

##  1\. Introduction

Rota-Baxter operators on algebras first appeared in the study of fluctuation
theory in probability [11]. Later, they were found important applications in
splitting of algebras [1], renormalizations in quantum field theory [6],
combinatorial study of rooted trees [4], noncommutative analogue of Poisson
geometry [15], also in many areas of mathematics and mathematical physics.
Relative Rota-Baxter operators are generalization of Rota-Baxter operators in
the presence of a representation of the underlying algebra. Recenely,
cohomology and deformation theory of relative Rota-Baxter operators are
studied in [13, 7]. Relative Rota-Baxter operators on Leibniz algebras, their
deformation theory, and relation with Leibniz Yang-Baxter equation, Leibniz
bialgebras are extensively considered in [12, 14]. Such operators induce
Leibniz-dendriform algebras (also called pre-Leibniz algebras) which are split
object for Leibniz algebras.

Our aim in this paper is to consider weighted relative Rota-Baxter operators
on Leibniz algebras. Let $\mathfrak{g},\mathfrak{h}$ be two Leibniz algebras
and let $\mathfrak{h}$ be a Leibniz $\mathfrak{g}$-representation. A linear
map $T:\mathfrak{h}\rightarrow\mathfrak{g}$ is said to be a $\lambda$-weighted
relative Rota-Baxter operator if $T$ satisfies some identity that involves
linear and quadratic expressions of $T$ (see Definition 3.3). Here
$\lambda\in{\bf k}$ is a fixed scalar. We construct a differential graded Lie
algebra whose Maurer-Cartan elements are precisely $\lambda$-weighted relative
Rota-Baxter operators. We observe that a $\lambda$-weighted relative Rota-
Baxter operator $T$ induces a new Leibniz algebra structure on $\mathfrak{h}$,
denoted by $\mathfrak{h}_{T}$. There is a representation of the Leibniz
algebra $\mathfrak{h}_{T}$ on the vector space $\mathfrak{g}$. The
corresponding Leibniz cohomology groups are called the cohomology of $T$. As
applications of the cohomology, we study formal and finite order deformations
of $T$. We observe that the linear term (coefficients of the formal parameter
$t$) in a formal deformation is a $1$-cocycle in the cohomology complex of
$T$, whose cohomology class depends only on the equivalence class of the
deformation. Given a fnite order deformation of $T$, we associate a second
cohomology class which is the obstruction for the extensibility of the
deformation.

In the last part, we introduce a notion of post-Leibniz algebras. They are
non-skewsymmetric analogue of post-Lie algebras introduced by Vallette [16].
Further, any pre-Leibniz algebra considered in [12] is a particular type of
post-Leibniz algebras. Any $\lambda$-weighted relative Rota-Baxter operator
$T:\mathfrak{h}\rightarrow\mathfrak{g}$ induces a post-Leibniz algebra
structure on $\mathfrak{h}$. We further show that a post-Leibniz algebra gives
rise to a Leibniz algebra structure on the underlying vector space, called the
total Leibniz algebra. We also find a necessaray and sufficient for a Leibniz
algebra to have a compatible post-Leibniz algebra structure.

The paper is organized as follows. In Section 2, we recall some basic
definitions and notations regarding Leibniz algebras. We define
$\lambda$-weighted relative Rota-Baxter operators and study some basic
properties in Section 3. Cohomology and deformations of $\lambda$-weighted
relative Rota-Baxter operators are respectively considered in Section 4 and 5.
Finally, post-Leibniz algebras are defined and studied in Section 6.

All vector spaces, linear and multilinear maps, tensor products are over a
field k of characteristic $0$. Throughout the paper, $\lambda\in{\bf k}$ is a
fixed scalar.

##  2\. Background on Leibniz algebras

In this section, we recall Leibniz algebras and their cohomology with
coefficients in a representation. Our main references are [2, 10].

######  2.1 Definition.

A Leibniz algebra is a pair $(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$ consists
of a vector space $\mathfrak{g}$ together with a bilinear bracket
$[~{},~{}]_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$
satisfying the following Leibniz identity

(1)
$\displaystyle[x,[y,z]_{\mathfrak{g}}]_{\mathfrak{g}}=[[x,y]_{\mathfrak{g}},z]_{\mathfrak{g}}+[y,[x,z]_{\mathfrak{g}}]_{\mathfrak{g}},~{}\text{
for }x,y,z\in\mathfrak{g}.$

Sometimes we denote a Leibniz algebra as above simply by $\mathfrak{g}$ when
the bracket is understood.

######  2.2 Remark.

(i) The identity (1) is equivalent to the fact that the left multiplication
map $[x,-]_{\mathfrak{g}}$ is a derivation for the bracket
$[~{},~{}]_{\mathfrak{g}}$ on $\mathfrak{g}$. Thus, the Leibniz algebras
defined in Definition 2.1 are also called left Leibniz algebras. The
definition of right Leibniz algebras can be similarly defined. Throughout this
paper, we shall always work on left Leibniz algebras.

(ii) Any Lie algebra is always a Leibniz algebra. In the presence of skew-
symmetry, the Jacobi identity is equivalent to the identity (1). Thus, Leibniz
algebras can be thought of as a non-skewsymmetric analogue of Lie algebras.

######  2.3 Definition.

Let $(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$ be a Leibniz algebra. A
representation of it consists of a triple $(V,\rho^{L},\rho^{R})$ of a vector
space $V$ together with two bilinear maps $\rho^{L}:\mathfrak{g}\otimes
V\rightarrow V$ and $\rho^{R}:V\otimes\mathfrak{g}\rightarrow V$ (called the
left and right $\mathfrak{g}$-actions)satisfying for $x,y\in\mathfrak{g}$,
$v\in V$,

(2) $\displaystyle\rho^{L}(x,\rho^{L}(y,v))=~{}$
$\displaystyle\rho^{L}([x,y]_{\mathfrak{g}},v)+\rho^{L}(y,\rho^{L}(x,v)),$ (3)
$\displaystyle\rho^{L}(x,\rho^{R}(v,y))=~{}$
$\displaystyle\rho^{R}(\rho^{L}(x,v),y)+\rho^{R}(v,[x,y]_{\mathfrak{g}}),$ (4)
$\displaystyle\rho^{R}(v,[x,y]_{\mathfrak{g}})=~{}$
$\displaystyle\rho^{R}(\rho^{R}(v,x),y)+\rho^{L}(x,\rho^{R}(v,y)).$

We often denote a representation simply by $V$ when the bilinear maps
$\rho^{L}$ and $\rho^{R}$ are clear from the context. It follows from the
above definition that any Leibniz algebra
$(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$ can be considered as a
representation of itself with left and right $\mathfrak{g}$-actions are given
by $\rho^{L}(x,y)=\rho^{R}(x,y)=[x,y]_{\mathfrak{g}}$, for
$x,y\in\mathfrak{g}$. This is called the adjoint representation.

We will now recall the cohomology of a Leibniz algebra with coefficients in a
representation. Let $(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$ be a Leibniz
algebra and $(V,\rho^{L},\rho^{R})$ be a representation of it. For each $n\geq
0$, we define the $n$-th cochain group as
$C^{n}_{\mathrm{Leib}}(\mathfrak{g},V):=\mathrm{Hom}(\mathfrak{g}^{\otimes
n},V)$ and the differential
$\delta_{\mathrm{Leib}}:C^{n}_{\mathrm{Leib}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{Leib}}(\mathfrak{g},V)$ by

(5) $\displaystyle(\delta_{\mathrm{Leib}}f)(x_{1},\ldots,x_{n+1})=~{}$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}~{}\rho^{L}\big{(}x_{i},f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})\big{)}+(-1)^{n+1}~{}\rho^{R}\big{(}f(x_{1},\ldots,x_{n}),x_{n+1}\big{)}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}~{}f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{g}},x_{j+1},\ldots,x_{n+1}),$

for $x_{1},\ldots,x_{n+1}\in\mathfrak{g}$. Then
$\\{C^{\ast}_{\mathrm{Leib}}(\mathfrak{g},V),\delta_{\mathrm{Leib}}\\}$ is a
cochain complex. The corresponding cohomology groups are called the cohomology
of the Leibniz algebra $\mathfrak{g}$ with coefficients in the representation
$V$.

In the following, we recall the Balavoine’s bracket that characterizes Leibniz
algebras as its Maurer-Cartan elements. We will use this bracket in Section 4
in the Maurer-Cartan characterization of weighted relative Rota-Baxter
operators.

Let $\mathfrak{g}$ be a vector space (not necessarily a Leibniz algebra). Let
$L=\oplus_{n\geq 0}\mathrm{Hom}(\mathfrak{g}^{\otimes n+1},\mathfrak{g})$ be
the graded vector space of all multilinear maps on $\mathfrak{g}$ with a
degree shift. It carries a graded Lie bracket (the Balavoine’s bracket) given
by

$\displaystyle\llbracket
f,g\rrbracket_{B}=\sum_{i=1}^{m+1}(-1)^{(i-1)n}f\circ_{i}g-(-1)^{mn}\sum_{i=1}^{n+1}(-1)^{(i-1)m}g\circ_{i}f,~{}\text{
where }$ $\displaystyle(f\circ_{i}g)(x_{1},\ldots,x_{m+n+1})=\sum_{\sigma\in
Sh(i-1,n)}(-1)^{\sigma}f\big{(}x_{\sigma(1)},\ldots,x_{\sigma(i-1)},g(x_{\sigma(i)},\ldots,x_{\sigma(i+n-1)},x_{i+n}),\ldots,x_{m+n+1}\big{)},$

for $f\in\mathrm{Hom}(\mathfrak{g}^{\otimes m+1},\mathfrak{g})$ and
$g\in\mathrm{Hom}(\mathfrak{g}^{\otimes n+1},\mathfrak{g})$. In other words,
$(L,\llbracket~{},~{}\rrbracket_{B})$ is a graded Lie algebra. A bilinear
bracket
$\mu_{\mathfrak{g}}=[~{},~{}]_{\mathfrak{g}}\in\mathrm{Hom}(\mathfrak{g}^{\otimes
2},\mathfrak{g})$ defines a Leibniz algebra structure on $\mathfrak{g}$ if and
only if $\mu_{\mathfrak{g}}$ satisfies
$\llbracket\mu_{\mathfrak{g}},\mu_{\mathfrak{g}}\rrbracket_{B}=0$, i.e.,
$\mu_{\mathfrak{g}}$ is a Maurer-Cartan element in the graded Lie algebra
$(L,\llbracket~{},~{}\rrbracket_{B})$.

##  3\. Weighted relative Rota-Baxter operators

In this section, we introduce $\lambda$-weighted (relative) Rota-Baxter
operators on Leibniz algebras and study some basic properties.

######  3.1 Definition.

Let $\mathfrak{g}$ be a Leibniz algebra. A linear map
$T:\mathfrak{g}\rightarrow\mathfrak{g}$ is said to be a $\lambda$-weighted
Rota-Baxter operator on $\mathfrak{g}$ if the linear map $T$ satisfies

$\displaystyle[Tx,Ty]_{\mathfrak{g}}=T([Tx,y]_{\mathfrak{g}}+[x,Ty]_{\mathfrak{g}}+\lambda[x,y]_{\mathfrak{g}}),~{}\text{
for }x,y\in\mathfrak{g}.$

######  3.2 Example.

(i) A Rota-Baxter operator on $\mathfrak{g}$ is simply a $0$-weighted Rota-
Baxter operator.

(ii) The identity map $\mathrm{id}:\mathfrak{g}\rightarrow\mathfrak{g}$ is a
$(-1)$-weighted Rota-Baxter operator.

(iii) If $f:\mathfrak{g}\rightarrow\mathfrak{g}$ is a Leibniz algebra morphism
and $f^{2}=f$ (idempotent condition), then $f$ is a $(-1)$-weighted Rota-
Baxter operator.

(iv) If $T$ is a $\lambda$-weighted Rota-Baxter operator, then for any
$\nu\in{\bf k}$, the map $\nu T$ is a $(\nu\lambda)$-weighted Rota-Baxter
operator.

(v) If $T$ is a $\lambda$-weighted Rota-Baxter operator, then
$-\lambda\mathrm{id}-T$ is a $\lambda$-weighted Rota-Baxter operator.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Leibniz algebras. We say that
$\mathfrak{h}$ is a Leibniz $\mathfrak{g}$-representation if there are
bilinear maps
$\rho^{L}:\mathfrak{g}\otimes\mathfrak{h}\rightarrow\mathfrak{h}$ and
$\rho^{R}:\mathfrak{h}\otimes\mathfrak{g}\rightarrow\mathfrak{h}$ that make
$(\mathfrak{h},\rho^{L},\rho^{R})$ into a representation of the Leibniz
algebra $\mathfrak{g}$ satisfying additionally for $u,v\in\mathfrak{h}$,
$x\in\mathfrak{g}$,

(6) $\displaystyle[u,\rho^{R}(v,x)]_{\mathfrak{h}}=~{}$
$\displaystyle\rho^{R}([u,v]_{\mathfrak{h}},x)+[v,\rho^{R}(u,x)]_{\mathfrak{h}},$
(7) $\displaystyle[u,\rho^{L}(x,v)]_{\mathfrak{h}}=~{}$
$\displaystyle[\rho^{R}(u,x),v]_{\mathfrak{h}}+\rho^{L}(x,[u,v]_{\mathfrak{h}}),$
(8) $\displaystyle\rho^{L}(x,[u,v]_{\mathfrak{h}})=~{}$
$\displaystyle[\rho^{L}(x,u),v]_{\mathfrak{h}}+[u,\rho^{L}(x,v)]_{\mathfrak{h}}.$

Note that, for any Leibniz algebra $\mathfrak{g}$, the adjoint representation
is a Leibniz $\mathfrak{g}$-representation.

######  3.3 Definition.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Leibniz algebras and
$\mathfrak{h}$ be a Leibniz $\mathfrak{g}$-representation. A linear map
$T:\mathfrak{h}\rightarrow\mathfrak{g}$ is said to be a $\lambda$-weighted
relative Rota-Baxter operator (on $\mathfrak{h}$ over the Leibniz algebra
$\mathfrak{g}$) if

(9)
$\displaystyle[Tu,Tv]_{\mathfrak{g}}=T(\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}),~{}\text{
for }u,v\in\mathfrak{h}.$

It follows that a $\lambda$-weighted Rota-Baxter operator on a Leibniz algebra
$\mathfrak{g}$ is a $\lambda$-weighted relative Rota-Baxter operator on
$\mathfrak{g}$ over itself.

Besides Example 3.2, we have the following examples.

######  3.4 Example.

(i) Let $\mathfrak{g},\mathfrak{h}$ be two Leibniz algebras and $\mathfrak{h}$
be a Leibniz $\mathfrak{g}$-representation. A linear map
$D:\mathfrak{g}\rightarrow\mathfrak{h}$ is a said to be a $\lambda$-weighted
crossed homomorphism (on $\mathfrak{g}$ with values in $\mathfrak{h}$) if

$\displaystyle
D([x,y]_{\mathfrak{g}})=\rho^{L}(x,Dy)+\rho^{R}(Dx,y)+\lambda[Dx,Dy]_{\mathfrak{h}},~{}\text{
for }x,y\in\mathfrak{g}.$

If $D$ is an invertible $\lambda$-weighted crossed homomorphism then
$D^{-1}:\mathfrak{h}\rightarrow\mathfrak{g}$ is a $\lambda$-weighted relative
Rota-Baxter operator.

(ii) Let $\mathfrak{g}$ be a Leibniz algebra. Any ideal $\mathfrak{h}$ of the
Leibniz algebra $\mathfrak{g}$ can be regarded as a Leibniz
$\mathfrak{g}$-representation. Then the inclusion map
$i:\mathfrak{h}\rightarrow\mathfrak{g}$ is a $(-1)$-weighted relative Rota-
Baxter operator.

######  3.5 Proposition.

(Weighted semidirect product) Let $\mathfrak{g}$ and $\mathfrak{h}$ be two
Leibniz algebras and $\mathfrak{h}$ be a Leibniz
$\mathfrak{g}$-representation. Then for any $\lambda\in{\bf k}$, the direct
sum $\mathfrak{g}\oplus\mathfrak{h}$ carries a new Leibniz algebra structure
with bracket

$\displaystyle[(x,u),(y,v)]_{\ltimes_{\lambda}}=\big{(}[x,y]_{\mathfrak{g}},~{}\rho^{L}(x,v)+\rho^{R}(u,y)+\lambda[u,v]_{\mathfrak{h}}\big{)},$

for $(x,u),(y,v)\in\mathfrak{g}\oplus\mathfrak{h}$. This is called the
$\lambda$-weighted semidirect product, often denoted by
$\mathfrak{g}\ltimes_{\lambda}\mathfrak{h}$.

We have the following characterization of $\lambda$-weighted relative Rota-
Baxter operators.

######  3.6 Proposition.

A linear map $T:\mathfrak{h}\rightarrow\mathfrak{g}$ is a $\lambda$-weighted
relative Rota-Baxter operator if and only if the graph
$Gr(T)=\\{(Tu,u)|~{}u\in\mathfrak{h}\\}$ is a subalgebra of the
$\lambda$-weighted semidirect product
$\mathfrak{g}\ltimes_{\lambda}\mathfrak{h}$.

###### Proof.

For any $u,v\in\mathfrak{h}$, we have

$\displaystyle[(Tu,u),(Tv,v)]_{\ltimes_{\lambda}}=([Tu,Tv]_{\mathfrak{g}},~{}\rho^{L}(Tu,u)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}).$

This is in $Gr(T)$ if and only if (9) holds. Hence the result follows. ∎

######  3.7 Remark.

Let $T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a $\lambda$-weighted relative
Rota-Baxter operator. Since $Gr(T)$ is isomorphism to $\mathfrak{h}$ as a
vector space, we get that $\mathfrak{h}$ inherits a new Leibniz algebra
structure with the bracket

$\displaystyle[u,v]_{T}:=\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}},~{}\text{
for }u,v\in\mathfrak{h}.$

In other words, $(\mathfrak{h},[~{},~{}]_{T})$ is a Leibniz algebra, denoted
by $\mathfrak{h}_{T}$ (called the induced Leibniz algebra). Moreover,
$T:\mathfrak{h}_{T}\rightarrow\mathfrak{g}$ is a morphism of Leibniz algebras.

Let $T,T^{\prime}:\mathfrak{h}\rightarrow\mathfrak{g}$ be two
$\lambda$-weighted relative Rota-Baxter operators. A morphism from $T$ to
$T^{\prime}$ consists of a pair $(\phi,\psi)$ of Leibniz algebra morphisms
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}$ and
$\psi:\mathfrak{h}\rightarrow\mathfrak{h}$ satisfying

$\displaystyle\phi\circ
T=T^{\prime}\circ\psi,\quad\psi(\rho^{L}(x,u))=\rho^{L}(\phi(x),\psi(u))~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}\psi(\rho^{R}(u,x))=\rho^{R}(\psi(u),\phi(x)),~{}\text{ for
}x\in\mathfrak{g},u\in\mathfrak{h}.$

######  3.8 Proposition.

Let $(\phi,\psi)$ be a morphism of $\lambda$-weighted relative Rota-Baxter
operators from $T$ to $T^{\prime}$. Then
$\psi:\mathfrak{h}\rightarrow\mathfrak{h}$ is a morphism of induced Leibniz
algebras from $(\mathfrak{h},[~{},~{}]_{T})$ to
$(\mathfrak{h},[~{},~{}]_{T^{\prime}})$.

###### Proof.

For any $u,v\in\mathfrak{h}$, we have

$\displaystyle\psi([u,v]_{T})=~{}$
$\displaystyle\psi\big{(}\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}\big{)}$
$\displaystyle=~{}$ $\displaystyle\rho^{L}(\phi\circ
T(u),\psi(v))+\rho^{R}(\psi(u),\phi\circ
T(v))+\lambda[\psi(u),\psi(v)]_{\mathfrak{h}}$ $\displaystyle=~{}$
$\displaystyle\rho^{L}(T^{\prime}(\psi(u)),\psi(v))+\rho^{R}(\psi(u),T^{\prime}(\psi(v)))+\lambda[\psi(u),\psi(v)]_{\mathfrak{h}}=[\psi(u),\psi(v)]_{T^{\prime}}.$

This shows that
$\psi:(\mathfrak{h},[~{},~{}]_{T})\rightarrow(\mathfrak{h},[~{},~{}]_{T^{\prime}})$
is a morphism of Leibniz algebras. ∎

##  4\. Maurer-Cartan characterization and cohomology of weighted relative
Rota-Baxter operators

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Leibniz algebras and
$\mathfrak{h}$ be a Leibniz $\mathfrak{g}$-representation. With this data, we
first construct a differential graded Lie algebra whose Maurer-Cartan elements
are precisely $\lambda$-weighted relative Rota-Baxter operators. This
characterization allows us to define a cohomology associated to a
$\lambda$-weighted relative Rota-Baxter operator. Later, we interpret this as
the cohomology of the corresponding induced Leibniz algebra with coefficients
in a suitable representation.

Let $\mu_{\mathfrak{g}}\in\mathrm{Hom}(\mathfrak{g}^{\otimes 2},\mathfrak{g})$
and $\mu_{\mathfrak{h}}\in\mathrm{Hom}(\mathfrak{h}^{\otimes 2},\mathfrak{h})$
denote the Leibniz operations on $\mathfrak{g}$ and $\mathfrak{h}$,
respectively. Moreover, let
$\rho^{L}:\mathfrak{g}\otimes\mathfrak{h}\rightarrow\mathfrak{h}$ and
$\rho^{R}:\mathfrak{h}\otimes\mathfrak{g}\rightarrow\mathfrak{h}$ denote the
left and right $\mathfrak{g}$-actions on $\mathfrak{h}$, respectively. Take
$V=\mathfrak{g}\oplus\mathfrak{h}$ and consider the graded Lie algebra
$(L=\oplus_{n\geq 0}\mathrm{Hom}(V^{\otimes n+1},V),[~{},~{}]_{B})$ on the
shifted space of multilinear maps on $V$ with the Balavoine’s bracket. Then it
can be checked that the element
$\theta:=\mu_{\mathfrak{g}}+\rho^{L}+\rho^{R}\in\mathrm{Hom}(V^{\otimes 2},V)$
defined by

$\displaystyle\theta((x,u),(y,v))=([x,y]_{\mathfrak{g}},~{}\rho^{L}(x,v)+\rho^{R}(u,y)),~{}\text{
for }(x,y),(y,v)\in V$

satisfies $[\theta,\theta]_{B}=0$. Hence $\theta$ induces a degree $1$
differential $d_{\theta}=[\theta,-]_{B}$ on the graded vector space $L$.
Moreover, the graded subspace $\mathfrak{a}=\oplus_{n\geq
0}\mathrm{Hom}(\mathfrak{h}^{\otimes n+1},\mathfrak{g})\subset L$ is an
abelian Lie subalgebra. Hence, by the derived bracket construction [17], the
shifted space $\mathfrak{a}[-1]=\oplus_{n\geq
1}\mathrm{Hom}(\mathfrak{h}^{\otimes n},\mathfrak{g})$ carries a graded Lie
bracket given by

$\displaystyle\llbracket
P,Q\rrbracket:=(-1)^{m}[d_{\theta}(P),Q]_{B}=(-1)^{m}[[\theta,P]_{B},Q]_{B},$

for $P\in\mathrm{Hom}(\mathfrak{h}^{\otimes
m},\mathfrak{g}),Q\in\mathrm{Hom}(\mathfrak{h}^{\otimes n},\mathfrak{g}).$ The
explicit description of the bracket is as follows (see [12])

(10) $\displaystyle\llbracket P,Q\rrbracket(u_{1},\ldots,u_{m+n})$
$\displaystyle=\sum_{i=1}^{m}\sum_{\sigma\in
Sh(i-1,n)}(-1)^{(i-1)n}(-1)^{\sigma}P\big{(}u_{\sigma(1)},\ldots,u_{\sigma(i-1)},\rho^{L}(Q(u_{\sigma(i)},\ldots,u_{\sigma(i+n-1)}),u_{i+n}),\ldots,u_{m+n}\big{)}$
$\displaystyle+\sum_{i=1}^{m}\sum_{{\sigma\in
Sh(i-1,1,n-1)}}(-1)^{(i-1)n}(-1)^{\sigma}P\big{(}u_{\sigma(1)},\ldots,\rho^{R}(u_{\sigma(i)},Q(u_{\sigma(i+1)},\ldots,u_{\sigma(i+n-1)},u_{i+n})),\ldots,u_{m+n}\big{)}$
$\displaystyle+\sum_{\sigma\in
Sh(m,n-1)}(-1)^{mn}(-1)^{\sigma}[P(u_{\sigma(1)},\ldots,u_{\sigma(m)}),Q(u_{\sigma(m+1)},\ldots,u_{\sigma(m+n-1)},u_{m+n})]_{\mathfrak{g}}$
$\displaystyle-(-1)^{mn}\big{\\{}\sum_{i=1}^{n}\sum_{\sigma\in
Sh(i-1,m)}(-1)^{(i-1)m}(-1)^{\sigma}Q\big{(}u_{\sigma(1)},\ldots,\rho^{L}(P(u_{\sigma(i)},\ldots,u_{\sigma(i+m-1)}),u_{i+m}),\ldots,u_{m+n}\big{)}$
$\displaystyle+\sum_{i=1}^{n}\sum_{\sigma\in
Sh(i-1,1,m-1)}(-1)^{(i-1)m}(-1)^{\sigma}Q\big{(}u_{\sigma(1)},\ldots,\rho^{R}(u_{\sigma(i)},P(u_{\sigma(i+1)},\ldots,u_{\sigma(i+m-1)},u_{i+m})),\ldots,u_{m+n}\big{)}$
$\displaystyle+\sum_{\sigma\in
Sh(n,m-1)}(-1)^{mn}(-1)^{\sigma}[Q(u_{\sigma(1)},\ldots,u_{\sigma(n)}),P(u_{\sigma(n+1)},\ldots,u_{\sigma(m+n-1)},u_{m+n})]_{\mathfrak{g}}\big{\\}}.$

On the other hand, for any $\lambda\in{\bf k}$, the element
$\theta^{\prime}=-\lambda\mu_{\mathfrak{h}}\in\mathrm{Hom}(V^{\otimes 2},V)$
defined by

$\displaystyle\theta^{\prime}((x,u),(y,v))=(0,-\lambda[u,v]_{\mathfrak{h}}),~{}\text{for
}(x,u),(y,v)\in V$

satisfies $[\theta^{\prime},\theta^{\prime}]_{B}=0$. Hence the element
$\theta^{\prime}$ also induces a differential
$d_{\theta^{\prime}}=[\theta^{\prime},-]_{B}$ on the graded vector space $L$.
It is then easy to observe that the graded subspace
$\mathfrak{a}[-1]=\oplus_{n\geq 1}(\mathfrak{h}^{\otimes n},\mathfrak{g})$ is
closed under the differential $d_{\theta^{\prime}}$. Let us denote the
restriction of $d_{\theta^{\prime}}$ to this subspace by $d$. This is
explicitly given by

(11) $\displaystyle(dP)(u_{1},\ldots,u_{n+1})=(-1)^{n}\sum_{1\leq i<j\leq
n+1}(-1)^{i}P(u_{1},\ldots,\widehat{u_{i}},\ldots,u_{j-1},\lambda[u_{i},u_{j}]_{\mathfrak{h}},u_{j+1},\ldots,u_{n+1}).$

Finally, we observe that the elements $\theta$ and $\theta^{\prime}$
additionally satisfies the compatibility $[\theta,\theta^{\prime}]_{B}=0$.
This implies that

$\displaystyle d\llbracket P,Q\rrbracket=$
$\displaystyle(-1)^{m}[\theta^{\prime},[[\theta,P]_{B},Q]_{B}]_{B}$
$\displaystyle=$
$\displaystyle(-1)^{m}[[\theta^{\prime},[\theta,P]_{B}]_{B},Q]_{B}+(-1)^{m}(-1)^{m}[[\theta,P]_{B},[\theta^{\prime},Q]_{B}]_{B}$
$\displaystyle=$
$\displaystyle(-1)^{m+1}[[[\theta,[\theta^{\prime},P]_{B}]_{B},Q]_{B}+[[\theta,P]_{B},dQ]_{B}$
$\displaystyle=$
$\displaystyle(-1)^{m+1}[[\theta,dP]_{B},Q]_{B}+[[\theta,P]_{B},dQ]_{B}=\llbracket
dP,Q\rrbracket+(-1)^{m}\llbracket P,dQ\rrbracket.$

Thus, we get the following.

######  4.1 Theorem.

Let $\mathfrak{g},\mathfrak{h}$ be two Leibniz algebras and $\mathfrak{h}$ be
a Leibniz $\mathfrak{g}$-representation. Then the triple $(\oplus_{n\geq
1}\mathrm{Hom}(\mathfrak{h}^{\otimes
n},\mathfrak{g}),\llbracket~{},~{}\rrbracket,d)$ is a differential graded Lie
algebra. Moreover, a linear map $T:\mathfrak{h}\rightarrow\mathfrak{g}$ is a
$\lambda$-weighted relative Rota-Baxter operator if and only if $T$ is a
Maurer-Cartan element in the differential graded Lie algebra $(\oplus_{n\geq
1}\mathrm{Hom}(\mathfrak{h}^{\otimes
n},\mathfrak{g}),\llbracket~{},~{}\rrbracket,d)$.

###### Proof.

The first part follows from previous discussions. For the second part, we
observe that for any linear map $T:\mathfrak{h}\rightarrow\mathfrak{g}$, we
have

$\displaystyle\llbracket T,T\rrbracket(u,v)=~{}$ $\displaystyle
2\big{(}T(\rho^{L}(Tu,v))+T(\rho^{R}(u,Tv)-[Tu,Tv]_{\mathfrak{g}}\big{)}\qquad(\text{from
}(\ref{b-formula}))$ $\displaystyle(dT)(u,v)=~{}$ $\displaystyle\lambda
T([u,v]_{\mathfrak{h}})\qquad(\text{from }\ref{d-formula})).$

Therefore,

$\displaystyle(dT+\frac{1}{2}\llbracket
T,T\rrbracket)(u,v)=2\big{(}T(\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}})-[Tu,Tv]_{\mathfrak{g}}\big{)}.$

This shows that $T$ is a Maurer-Cartan element (i.e. $dT+\frac{1}{2}\llbracket
T,T\rrbracket=0$) if and only if $T$ is a $\lambda$-weighted relative Rota-
Baxter operator. ∎

Let $T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a $\lambda$-weighted relative
Rota-Baxter operator. That is, $T$ can be equivalently seen as a Maurer-Cartan
element in the differential graded Lie algebra constructed in Theorem 4.1.
Therefore, $T$ induces a differential

$\displaystyle d_{T}=d+\llbracket
T,-\rrbracket:\mathrm{Hom}(\mathfrak{h}^{\otimes
n},\mathfrak{g})\rightarrow\mathrm{Hom}(\mathfrak{h}^{\otimes
n+1},\mathfrak{g}),~{}\text{ for }n\geq 1.$

Explicitly, we have

(12) $\displaystyle(d_{T}f)(u_{1},\ldots,u_{n+1})$
$\displaystyle=(-1)^{n}\sum_{i=1}^{n}(-1)^{i+1}\big{(}[Tu_{i},f(u_{1},\ldots,\widehat{u_{i}},\ldots,u_{n+1})]_{\mathfrak{g}}-T(\rho^{R}(u_{i},f(u_{1},\ldots,\widehat{u_{i}},\ldots,u_{n+1})))\big{)}$
$\displaystyle-\big{(}[f(u_{1},\ldots,u_{n}),T(u_{n+1})]_{\mathfrak{g}}-T(\rho^{L}(f(u_{1},\ldots,u_{n}),u_{n+1}))\big{)}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+n}f(u_{1},\ldots,\widehat{u_{i}},\ldots,u_{j-1},\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}},u_{j+1},\ldots,u_{n+1}).$

In the following, we show that the differential (12) can be seen as the
differential of the Leibniz algebra cohomology of $\mathfrak{h}_{T}$ (see
Remark 3.7) with coefficients in a suitable representation. We first observe
the following.

######  4.2 Proposition.

Let $T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a $\lambda$-weighted relative
Rota-Baxter operator. Then $(\mathfrak{g},\varrho^{L},\varrho^{R})$ is a
representation of the induced Leibniz algebra $\mathfrak{h}_{T}$ with left and
right $\mathfrak{h}_{T}$-actions given by

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

###### Proof.

For any $u,v\in\mathfrak{h}$ and $x\in\mathfrak{g}$,

$\displaystyle\varrho^{L}(u,\varrho^{L}(v,x))-\varrho^{L}([u,v]_{T},x)-\varrho^{L}(v,\varrho^{L}(u,x))$
$\displaystyle=\varrho^{L}\big{(}u,[Tv,x]_{\mathfrak{g}}-T\rho^{R}(v,x)\big{)}-[T[u,v]_{T},x]_{\mathfrak{g}}+T\rho^{R}([u,v]_{T},x)-\varrho^{L}(v,[Tu,x]_{\mathfrak{g}}-T\rho^{R}(u,x))$
$\displaystyle=\cancel{[Tu,[Tv,x]_{\mathfrak{g}}]_{\mathfrak{g}}}-[Tu,T\rho^{R}(v,x)]_{\mathfrak{g}}-T\rho^{R}(u,[Tv,x]_{\mathfrak{g}})+T\rho^{R}(u,T\rho^{R}(v,x))$
$\displaystyle-\cancel{[[Tu,Tv]_{\mathfrak{g}},x]_{\mathfrak{g}}}+T\rho^{R}\big{(}\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}},x\big{)}$
$\displaystyle-\cancel{[Tv,[Tu,x]_{\mathfrak{g}}]_{\mathfrak{g}}}+[Tv,T\rho^{R}(u,x)]_{\mathfrak{g}}+T\rho^{R}(v,[Tu,x]_{\mathfrak{g}})-T\rho^{L}(v,T\rho^{R}(u,x))$
$\displaystyle=-T\big{(}\rho^{L}(Tu,\rho^{R}(v,x))+\rho^{R}(u,T\rho^{R}(v,x))+\lambda[u,\rho^{R}(v,x)]_{\mathfrak{h}}\big{)}+T\rho^{R}\big{(}\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}},x\big{)}$
$\displaystyle+T\big{(}\rho^{L}(Tv,\rho^{R}(u,x))+\rho^{R}(v,T\rho^{R}(u,x))+\lambda[v,\rho^{R}(u,x)]_{\mathfrak{h}}\big{)}+T(\rho^{R}(v,[Tu,x]_{\mathfrak{g}}))-T(\rho^{R}(v,T\rho^{R}(u,x)))$
$\displaystyle=0.$

Similarly,

$\displaystyle\varrho^{L}(u,\varrho^{R}(x,v))-\varrho^{R}(\varrho^{L}(u,x),v)-\varrho^{R}(x,[u,v]_{T})$
$\displaystyle=\varrho^{L}(u,[x,Tv]_{\mathfrak{g}}-T\rho^{L}(x,v))-\varrho^{R}([Tu,x]_{\mathfrak{g}}-T\rho^{R}(u,x),v)-[x,T[u,v]_{T}]_{\mathfrak{g}}+T\rho^{L}(x,[u,v]_{T})$
$\displaystyle=\cancel{[Tu,[x,Tv]_{\mathfrak{g}}]_{\mathfrak{g}}}-[Tu,T\rho^{L}(x,v)]_{\mathfrak{g}}-T\rho^{R}(u,[x,Tv]_{\mathfrak{g}})+T\rho^{R}(u,T\rho^{L}(x,v))$
$\displaystyle-\cancel{[[Tu,x]_{\mathfrak{g}},Tv]_{\mathfrak{g}}}+T\rho^{L}([Tu,x]_{\mathfrak{g}},v)+[T\rho^{R}(u,x),Tv]_{\mathfrak{g}}-T\rho^{L}(T\rho^{R}(u,x),v)$
$\displaystyle-\cancel{[x,[Tu,Tv]_{\mathfrak{g}}]_{\mathfrak{g}}}+T\rho^{L}\big{(}x,\rho^{L}(Tu,v))+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}\big{)}$
$\displaystyle=-T\big{(}\rho^{L}(Tu,\rho^{L}(x,v))+\rho^{R}(u,T\rho^{L}(x,v))+\lambda[u,\rho^{L}(x,v)]_{\mathfrak{h}}\big{)}-T\rho^{R}(u,[x,Tv]_{\mathfrak{g}})+T\rho^{R}(u,T\rho^{L}(x,v))$
$\displaystyle+T\rho^{L}([Tu,x]_{\mathfrak{g}},v)+T\big{(}\rho^{L}(T\rho^{R}(u,x),v)+\rho^{R}(\rho^{R}(u,x),Tv)+\lambda[\rho^{R}(u,x),v]_{\mathfrak{h}}\big{)}-T\rho^{L}(T\rho^{R}(u,x),v)$
$\displaystyle+T\rho^{L}\big{(}x,\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}\big{)}$
$\displaystyle=0$

and

$\displaystyle\varrho^{L}(x,[u,v]_{T})-\varrho^{R}(\varrho^{R}(x,u),v)-\varrho^{L}(u,\varrho^{R}(x,v))$
$\displaystyle=[x,T[u,v]_{T}]_{\mathfrak{g}}-T\rho^{R}(x,[u,v]_{T})-\varrho^{R}([x,Tu]_{\mathfrak{g}}-T\rho^{L}(x,u),v)-\varrho^{L}(u,[x,Tv]_{\mathfrak{g}}-T\rho^{L}(x,v))$
$\displaystyle=\cancel{[x,[Tu,Tv]_{\mathfrak{g}}]_{\mathfrak{g}}}-T\rho^{R}\big{(}x,\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}\big{)}$
$\displaystyle-\cancel{[[x,Tu]_{\mathfrak{g}},Tv]_{\mathfrak{g}}}+T\rho^{L}([x,Tu]_{\mathfrak{g}},v)+[T\rho^{L}(x,u),Tv]_{\mathfrak{g}}-T\rho^{L}(T\rho^{L}(x,u),v)$
$\displaystyle-\cancel{[Tu,[x,Tv]_{\mathfrak{g}}]_{\mathfrak{g}}}+T\rho^{R}(u,[x,Tv]_{\mathfrak{g}})+[Tu,T\rho^{L}(x,v)]_{\mathfrak{g}}-T\rho^{R}(u,T\rho^{L}(x,v))$
$\displaystyle=-T\rho^{R}\big{(}x,\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}\big{)}$
$\displaystyle+T\rho^{L}([x,Tu]_{\mathfrak{g}},v)+T\big{(}\rho^{L}(T\rho^{L}(x,u),v)+\rho^{R}(\rho^{L}(x,u),TV)+\lambda[\rho^{L}(x,u),v]_{\mathfrak{h}}\big{)}-T\rho^{L}(T\rho^{L}(x,u),v)$
$\displaystyle+T\rho^{R}(u,[x,Tv]_{\mathfrak{g}})+T\big{(}\rho^{L}(Tu,\rho^{L}(x,v))+\rho^{R}(u,T\rho^{L}(x,v))+\lambda[u,\rho^{L}(x,v)]_{\mathfrak{h}}\big{)}-T\rho^{R}(u,T\rho^{L}(x,v))$
$\displaystyle=0.$

This proves that $(\mathfrak{g},\varrho^{L},\varrho^{R})$ is a representation
of the Leibniz algebra $\mathfrak{h}_{T}$. ∎

Therefore, by the previous proposition, we can consider the cochain complex
$\\{C^{\ast}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g}),\delta_{\mathrm{Leib}}\\}$
of the Leibniz algebra $\mathfrak{h}_{T}$ with coefficients in the
representation $(\mathfrak{g},\varrho^{L},\varrho^{R})$. More precisely,
$C^{n}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})=\mathrm{Hom}(\mathfrak{h}^{\otimes
n},\mathfrak{g})$, for $n\geq 0$, and the differential
$\delta_{\mathrm{Leib}}:C^{n}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})\rightarrow
C^{n+1}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})$ given by

(13) $\displaystyle(\delta_{\mathrm{Leib}}f)(u_{1},\ldots,u_{n+1})$
$\displaystyle=\sum_{i=1}^{n}(-1)^{i+1}\big{(}[Tu_{i},f(u_{1},\ldots,\widehat{u_{i}},\ldots,u_{n+1})]_{\mathfrak{g}}-T(\rho^{R}(u_{i},f(u_{1},\ldots,\widehat{u_{i}},\ldots,u_{n+1})))\big{)}$
$\displaystyle+(-1)^{n+1}\big{(}[f(u_{1},\ldots,u_{n}),T(u_{n+1})]_{\mathfrak{g}}-T(\rho^{L}(f(u_{1},\ldots,u_{n}),u_{n+1}))\big{)}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(u_{1},\ldots,\widehat{u_{i}},\ldots,u_{j-1},\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}},u_{j+1},\ldots,u_{n+1}).$

Let $Z^{n}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})$ and
$B^{n}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})$ be the space of
$n$-cocycles and $n$-coboundaries, respectively. Then we have
$B^{n}_{\mathrm{Leib}}(\mathfrak{h},\mathfrak{g})\subset
Z^{n}_{\mathrm{Leib}}(\mathfrak{h},\mathfrak{g})$. The corresponding quotients

$\displaystyle
H^{n}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g}):=\frac{Z^{n}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})}{B^{n}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})},~{}\text{
for }n\geq 0$

are called the cohomology groups of the $\lambda$-weighted relative Rota-
Baxter operator $T$.

######  4.3 Remark.

It follows from the expressions of (12) and (13) that
$d_{T}f=(-1)^{n}\delta_{\mathrm{Leib}}f$, for
$f\in\mathrm{Hom}(\mathfrak{h}^{\otimes n},\mathfrak{g})$ with $n\geq 1$.
Hence the cohomology groups induced by the differential $d_{T}$ are isomorphic
to the cohomology groups of $T$.

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

In this section, we studey formal and finite order deformations of a
$\lambda$-weighted relative Rota-Baxter operator in terms of the cohomology
theory introduced in the previous section. In particular, we find a sufficient
condition for the rigidity of a $\lambda$-weighted relative Rota-Baxter
operator, and find a obstruction class for the extensibility of a finite order
deformation.

Formal deformations. Let $\mathfrak{g},\mathfrak{h}$ be two Leibniz algebras
and $\mathfrak{h}$ be a Leibniz $\mathfrak{g}$-representation. We consider the
spaces $\mathfrak{g}[[t]]$ and $\mathfrak{h}[[t]]$ of formal power series in
$t$ with coefficients from $\mathfrak{g}$ and $\mathfrak{h}$, respectively.
Note that $\mathfrak{g}[[t]]$ and $\mathfrak{h}[[t]]$ are both ${\bf
k}[[t]]$-modules. It is easy to see that the Leibniz algebra structures on
$\mathfrak{g}$ and $\mathfrak{h}$ can be extended by ${\bf
k}[[t]]$-bilinearity to Leibniz algebra structures on $\mathfrak{g}[[t]]$ and
$\mathfrak{h}[[t]]$, respectively. Moreover, the Leibniz
$\mathfrak{g}$-representation on $\mathfrak{h}$ also induces a Leibniz
$\mathfrak{g}[[t]]$-representation on $\mathfrak{h}[[t]]$.

######  5.1 Definition.

Let $T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a $\lambda$-weighted relative
Rota-Baxter operator. A formal deformation of $T$ consists of a formal sum
$T_{t}=\sum_{i=0}^{\infty}t^{i}T_{i}\in\mathrm{Hom}(\mathfrak{h},\mathfrak{g})[[t]]$
with $T_{0}=T$, such that the ${\bf k}[[t]]$-linear map
$T_{t}:\mathfrak{h}[[t]]\rightarrow\mathfrak{g}[[t]]$ is a $\lambda$-weighted
relative Rota-Baxter operator (on $\mathfrak{h}[[t]]$ over the Leibniz algebra
$\mathfrak{g}[[t]]$).

It follows from the above definition that $T_{t}$ is a formal deformation of
$T$ if and only if

(14)
$\displaystyle\sum_{i+j=n}[T_{i}(u),T_{j}(v)]_{\mathfrak{g}}=\sum_{i+j=n}T_{i}\big{(}\rho^{L}(T_{j}(u),v)+\rho^{R}(u,T_{j}(v))\big{)}+\lambda
T_{n}([u,v]_{\mathfrak{h}}),$

for all $u,v\in\mathfrak{h}$ and $n\geq 0$. The above system of equations are
called deformation equations. Note that (14) holds for $n=0$ as $T_{0}=T$ is a
$\lambda$-weighted relative Rota-Baxter operator. However, for $n=1$, we get
that

$\displaystyle[Tu,T_{1}v]_{\mathfrak{g}}+[T_{1}u,Tv]_{\mathfrak{g}}=T(\rho^{L}(T_{1}u,v)+\rho^{R}(u,T_{1}v))+T_{1}\big{(}\rho^{L}(Tu,v)+\rho^{R}(u,Tv)+\lambda[u,v]_{\mathfrak{h}}\big{)}.$

This implies that $\delta_{\mathrm{Leib}}(T_{1})=0$ (follows from (13)). In
other words, $T_{1}$ is a $1$-cocycle in the cohomology complex of $T$. This
is called the infinitesimal of the deformation.

######  5.2 Definition.

Two deformations $T_{t}$ and $T_{t}^{\prime}$ of a $\lambda$-weighted relative
Rota-Baxter operator $T$ are said to be equivalent if there exists an element
$x_{0}\in\mathfrak{g}$ and linear maps
$\phi_{i}\in\mathrm{Hom}(\mathfrak{g},\mathfrak{g})$,
$\psi_{i}\in\mathrm{Hom}(\mathfrak{h},\mathfrak{h})$ for $i\geq 2$, such that
the pair of maps

$\displaystyle\big{(}\Phi_{t}=\mathrm{id}_{\mathfrak{g}}+t[x_{0},-]_{\mathfrak{g}}+\sum_{i\geq
2}t^{i}\phi_{i}:\mathfrak{g}[[t]]\rightarrow\mathfrak{g}[[t]]~{},~{}\Psi_{t}=\mathrm{id}_{\mathfrak{h}}+t\rho^{L}(x_{0},-)+\sum_{i\geq
2}t^{i}\psi_{i}:\mathfrak{h}[[t]]\rightarrow\mathfrak{h}[[t]]\big{)}$

is a morphism of $\lambda$-weighted relative Rota-Baxter operators from
$T_{t}$ to $T_{t}^{\prime}$.

Therefore, if $T_{t}$ and $T_{t}^{\prime}$ are equivalent, then we must have

$\displaystyle(\Phi_{t}\circ
T_{t})(u)=(T_{t}^{\prime}\circ\Psi_{t})(u),\quad\Psi_{t}(\rho^{L}(x,u))=\rho^{L}(\Phi_{t}(x),\Psi_{t}(u))~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}\Psi_{t}(\rho^{R}(u,x))=\rho^{R}(\Psi_{t}(u),\Phi_{t}(x)),$

for $x\in\mathfrak{g}$, $u\in\mathfrak{h}$. By expanding the identity
$(\Phi_{t}\circ T_{t})(u)=(T_{t}^{\prime}\circ\Psi_{t})(u)$ and equating
coefficients of $t$ from both sides, we get

$\displaystyle
T_{1}(u)-T_{1}^{\prime}(u)=T\rho^{L}(x_{0},u)-[x_{0},Tu]_{\mathfrak{g}}=\delta_{\mathrm{Leib}}(x_{0})(u).$

As a summary of the above discussions, we get the following.

######  5.3 Theorem.

Let $T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a $\lambda$-weighted relative
Rota-Baxter operator. If $T_{t}=\sum_{i=0}^{\infty}t^{i}T_{i}$ is a formal
deformation of $T$, then $T_{1}$ is a $1$-cocycle in the cohomology complex of
$T$. Moreover, the corresponding cohomology class depends only on the
equivalence class of the deformation $T_{t}$.

######  5.4 Definition.

A $\lambda$-weighted relative Rota-Baxter operator $T$ is said to be rigid if
any formal deformation $T_{t}$ is equivalent to the undeformed one
$T_{t}^{\prime}=T.$

Next, we introduce certain specific elements associated to $T$ whose
importance will be clear in the next theorem.

######  5.5 Definition.

An element $x_{0}\in\mathfrak{g}$ is said to be a Nijenhuis element associated
with $T$ if

$\displaystyle[[x_{0},x]_{\mathfrak{g}},[x_{0},y]_{\mathfrak{g}}]_{\mathfrak{g}}=0,\quad[\rho^{L}(x_{0},u),\rho^{L}(x_{0},v)]_{\mathfrak{h}}=0,$
$\displaystyle\rho^{L}([x_{0},x]_{\mathfrak{g}},\rho^{L}(x_{0},u))=0,\quad\rho^{R}(\rho^{L}(x_{0},u),[x_{0},x]_{\mathfrak{g}})=0,$

for $x,y\in\mathfrak{g}$ and $u,v\in\mathfrak{h}$. We denote the set of all
Nijenhuis elements by $\mathrm{Nij}(T).$

Note that any element lying in the intersection

$\displaystyle\\{x_{0}\in\mathfrak{g}~{}|~{}[x_{0},x]_{\mathfrak{g}}=0,\forall
x\in\mathfrak{g}\\}\cap\\{x_{0}\in\mathfrak{g}~{}|~{}\rho^{L}(x_{0},u)=0,\forall
u\in\mathfrak{h}\\}$

belongs to $\mathrm{Nij}(T).$ We now have the following interesting result.

######  5.6 Theorem.

Let $T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a $\lambda$-weighted relative
Rota-Baxter operator. If
$Z^{1}(\mathfrak{h}_{T},\mathfrak{g})=\delta_{\mathrm{Leib}}(\mathrm{Nij}(T))$
then $T$ is rigid.

###### Proof.

Let $T_{t}=\sum_{i=0}^{\infty}t^{i}T_{i}$ be a formal deformation of $T$. Then
by Theorem 5.3, we get that $T_{1}\in
Z^{1}_{\mathrm{Leib}}(\mathfrak{h}_{T},\mathfrak{g})$ is a $1$-cocycle in the
cohomology complex of $T$. Hence by hypothesis, there exists an element
$x_{0}\in\mathrm{Nij}(T)$ such that $T_{1}=\delta_{\mathrm{Leib}}(x_{0})$
(i.e. $T_{1}(u)=-[x_{0},Tu]_{\mathfrak{g}}+T\rho^{L}(x_{0},u)$ for all
$u\in\mathfrak{h}$). Sett
$\Phi_{t}=\mathrm{id}_{\mathfrak{g}}+t[x_{0},-]_{\mathfrak{g}}$ and
$\Psi_{t}=\mathrm{id}_{\mathfrak{h}}+t\rho^{L}(x_{0},-)$, and define
$T_{t}^{\prime}=\Phi_{t}\circ T_{t}\circ\Psi_{t}^{-1}$. Then $T_{t}^{\prime}$
is a formal deformation of $T$. Since $x_{0}\in\mathrm{Nij}(T)$, it follows
that $\Phi_{t}$ and $\Psi_{t}$ are Leibniz algebra morphisms, also
$\Psi_{t}(\rho^{L}(x,u))=\rho^{L}(\Phi_{t}(x),\Psi_{t}(u))$ and
$\Psi_{t}(\rho^{R}(u,x))=\rho^{R}(\Psi_{t}(u),\Phi_{t}(x))$ hold. Hence
$T_{t}^{\prime}$ is equivalent to $T_{t}$. We also have

$\displaystyle T_{t}^{\prime}(u)=~{}$
$\displaystyle(\mathrm{id}_{\mathfrak{g}}+t[x_{0},-]_{\mathfrak{g}})\circ(\sum_{i=0}^{\infty}t^{i}T_{i})(u-t\rho^{L}(x_{0},u)+\text{powers
of }t^{\geq 2})$ $\displaystyle=~{}$
$\displaystyle(\mathrm{id}_{\mathfrak{g}}+t[x_{0},-]_{\mathfrak{g}})\big{(}Tu-t~{}T\rho^{L}(x_{0},u)+tT_{1}(u)+\text{powers
of }t^{\geq 2}\big{)}$ $\displaystyle=~{}$ $\displaystyle
Tu+t\big{(}T\rho^{L}(x_{0},u)+T_{1}(u)+[x_{0},Tu]_{\mathfrak{g}}\big{)}+\text{powers
of }t^{\geq 2}.$

It follows from the above expression that the coefficients of $t$ vanishes. By
repeating the above argument, we get that $T_{t}$ is equivalent to $T$. ∎

Finite order deformations. Let $T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a
$\lambda$-weighted relative Rota-Baxter operator. For a fixed
$N\in\mathbb{N}$, consider the spaces $\mathfrak{g}[[t]]/(t^{N+1})$ and
$\mathfrak{h}[[t]]/(t^{N+1})$. Both of them are $\mathbf{k}[[t]]/(t^{N+1})$.
Moreover, by $\mathbf{k}[[t]]/(t^{N+1})$-bilinearity, the Leibniz structures
on $\mathfrak{g}$ and $\mathfrak{h}$ can be extended to
$\mathfrak{g}[[t]]/(t^{N+1})$ and $\mathfrak{h}[[t]]/(t^{N+1})$, respectively.
Further, $\mathfrak{h}[[t]]/(t^{N+1})$ is a Leibniz
$\mathfrak{g}[[t]]/(t^{N+1})$-representation.

######  5.7 Definition.

An order $N$ deformation of $T$ is given by a finite polynomial of the form
$T_{t}^{N}=\sum_{i=0}^{N}t^{i}T_{i}\in\mathrm{Hom}(\mathfrak{h},\mathfrak{g})[[t]]/(t^{N+1})$
with $T_{0}=T$, such that the $\mathbf{k}[[t]]/(t^{N+1})$-linear map
$T_{t}^{N}:\mathfrak{h}[[t]]/(t^{N+1})\rightarrow\mathfrak{g}[[t]]/(t^{N+1})$
is a $\lambda$-weighted relative Rota-Baxter operator.

Thus, $T_{t}^{N}=\sum_{i=0}^{N}t^{i}T_{i}$ is an order $N$ deformation of $T$
if and only if the identity (14) holds for $n=0,1,\ldots,N$. These can be
equivalently written as

(15) $\displaystyle d_{T}(T_{n})=-\frac{1}{2}\sum_{\begin{subarray}{c}i+j=n\\\
i,j\geq 1\end{subarray}}\llbracket T_{i},T_{j}\rrbracket,~{}\text{for
}n=0,1,\ldots,N.$

######  5.8 Definition.

(Obstruction cochain) Let $T_{t}^{N}=\sum_{i=0}^{N}t^{i}T_{i}$ be an order $N$
deformation of $T$. We define a $2$-cochain $Ob_{T_{t}^{N}}\in
C^{2}_{T}(\mathfrak{h},\mathfrak{g})$, called the obstruction cochain, by

(16) $\displaystyle
Ob_{T_{t}^{N}}=-\frac{1}{2}\sum_{\begin{subarray}{c}i+j=N+1\\\ i,j\geq
1\end{subarray}}\llbracket T_{i},T_{j}\rrbracket.$

######  5.9 Proposition.

The obstruction cochain $Ob_{T^{N}_{t}}$ is a $2$-cocycle in the cohomology
complex of $T$, i.e. $\delta_{\mathrm{Leib}}(Ob_{T^{N}_{t}})=0$ (equivalently,
$d_{T}(Ob_{T^{N}_{t}})=0$).

###### Proof.

We have

$\displaystyle d_{T}(-\frac{1}{2}\sum_{\begin{subarray}{c}i+j=N+1\\\ i,j\geq
1\end{subarray}}\llbracket T_{i},T_{j}\rrbracket)$
$\displaystyle=-\frac{1}{2}\sum_{\begin{subarray}{c}i+j=N+1\\\ i,j\geq
1\end{subarray}}(d~{}\llbracket T_{i},T_{j}\rrbracket+\llbracket T,\llbracket
T_{i},T_{j}\rrbracket\rrbracket)$
$\displaystyle=-\frac{1}{2}\sum_{\begin{subarray}{c}i+j=N+1\\\ i,j\geq
1\end{subarray}}\big{(}\llbracket dT_{i},T_{j}\rrbracket-\llbracket
T_{i},dT_{j}\rrbracket+\llbracket\llbracket
T,T_{i}\rrbracket,T_{j}\rrbracket-\llbracket T_{i},\llbracket
T,T_{j}\rrbracket\rrbracket\big{)}$
$\displaystyle=-\frac{1}{2}\sum_{\begin{subarray}{c}i+j=N+1\\\ i,j\geq
1\end{subarray}}\big{(}\llbracket d_{T}T_{i},T_{j}\rrbracket-\llbracket
T_{i},d_{T}T_{j}\rrbracket\big{)}$
$\displaystyle=\frac{1}{4}\sum_{\begin{subarray}{c}i_{1}+i_{2}+j=N+1\\\
i_{1},i_{2},j\geq 1\end{subarray}}\llbracket\llbracket
T_{i_{1}},T_{i_{2}}\rrbracket,T_{j}\rrbracket-\frac{1}{4}\sum_{\begin{subarray}{c}i+j_{1}+j_{2}=N+1\\\
i,j_{1},j_{2}\geq 1\end{subarray}}\llbracket T_{i},\llbracket
T_{j_{1}},T_{j_{2}}\rrbracket\rrbracket\quad(\text{from }(\ref{fin-eq}))$
$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}i+j+k=N+1\\\ i,j,k\geq
1\end{subarray}}\llbracket\llbracket T_{i},T_{j}\rrbracket,T_{k}\rrbracket=0.$

This completes the proof. This proves the result. ∎

It follows from the above result that we obtain a cohomology class
$[Ob_{T^{N}_{t}}]\in H^{2}_{T}(\mathfrak{h},\mathfrak{g})$ associated to the
order $N$ deformation $T^{N}_{t}.$ This cohomology class is called the
obstruction class.

######  5.10 Definition.

An order $N$ deformation $T^{N}_{t}=\sum_{i=0}^{N}t^{i}T_{i}$ is said to be
extensible if there exists a linear map
$T_{N+1}:\mathfrak{h}\rightarrow\mathfrak{g}$ which makes
$T_{t}^{N+1}=T^{N}_{t}+t^{N+1}T_{N+1}$ into an order $N+1$ deformation.

The following result gives a necessary and sufficient condition for the
extensibility of a finite order deformation in terms of its obstruction class.

######  5.11 Theorem.

An order $N$ deformation $T^{N}_{t}$ is extensible if and only if the
corresponding obstruction class $[Ob_{T^{N}_{t}}]$ vanishes.

###### Proof.

Let $T^{N}_{t}=\sum_{i=0}^{N}t^{i}T_{i}$ be an extensible order $N$
deformation. Since there exists a linear map $T_{N+1}$ which makes
$T^{N+1}_{t}=\sum_{i=0}^{N}t^{i}T_{i}$ into an order $N+1$ deformation, we
have that $Ob_{T^{N}_{t}}=d_{T}(T_{N+1}).$ In other words, $Ob_{T^{N}_{t}}$ is
a coboundary. Hence the cohomology class $[Ob_{T^{N}_{t}}]$ vanishes. The
converse part follows by similar argument. ∎

##  6\. Post-Leibniz algebras

In this section, we introduce post-Leibniz algebras as a non-skewsymmetric
analogue of post-Lie algebras. We show that post-Leibniz algebras arise
naturally from $\lambda$-weighted relative Rota-Baxter operators. Finally, we
study some properties of post-Leibniz algebras.

######  6.1 Definition.

A post-Leibniz algebra is a quadruple
$(\mathfrak{a},\triangleleft,\triangleright,[~{},~{}]_{\mathfrak{a}})$
consisting of a vector space $\mathfrak{a}$ together with three bilinear
operations
$\triangleleft,\triangleright,[~{},~{}]_{\mathfrak{a}}:\mathfrak{a}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$
satisfying for $u,v,w\in\mathfrak{a}$,

(17) $\displaystyle u\triangleleft[v,w]_{\star}=~{}$
$\displaystyle(u\triangleleft v)\triangleleft w+v\triangleright(u\triangleleft
w),$ (18) $\displaystyle u\triangleright(v\triangleleft w)=~{}$
$\displaystyle(u\triangleright v)\triangleleft w+v\triangleleft[u,w]_{\star},$
(19) $\displaystyle u\triangleright(v\triangleright w)=~{}$
$\displaystyle[u,v]_{\star}\triangleright w+v\triangleright(u\triangleright
w),$ (20) $\displaystyle u\triangleright[v,w]_{\mathfrak{a}}=~{}$
$\displaystyle[u\triangleright v,w]_{\mathfrak{a}}+[v,u\triangleright
w]_{\mathfrak{a}},$ (21) $\displaystyle[u.v\triangleright
w]_{\mathfrak{a}}=~{}$ $\displaystyle[u\triangleleft
v,w]_{\mathfrak{a}}+v\triangleright[u,w]_{\mathfrak{a}},$ (22)
$\displaystyle[u,v\triangleleft w]_{\mathfrak{a}}=~{}$
$\displaystyle[u,v]_{\mathfrak{a}}\triangleleft w+[v,u\triangleleft
w]_{\mathfrak{a}},$ (23)
$\displaystyle[u,[v,w]_{\mathfrak{a}}]_{\mathfrak{a}}=~{}$
$\displaystyle[[u,v]_{\mathfrak{a}},w]_{\mathfrak{a}}+[v,[u,w]_{\mathfrak{a}}]_{\mathfrak{a}},$

where $[u,v]_{\star}=u\triangleleft v+u\triangleright v+[u,v]_{\mathfrak{a}}$.

######  6.2 Remark.

(i) The notion of post-Lie algebras was first introduced by Vallette [16] in
the operadic study of generalized partitioned posets. A post-Lie algebra is a
triple $(\mathfrak{a},\circ,[~{},~{}])$ consisting of a vector space
$\mathfrak{a}$ together with bilinear operations
$\circ,[~{},~{}]:\mathfrak{a}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$ in
which $[~{},~{}]$ is skewsymmetric and satisfying the following identities

$\displaystyle(u\circ v)\circ w-u\circ(v\circ w)-(v\circ u)\circ
w+v\circ(u\circ w)+[u,v]\circ w=0,$ $\displaystyle u\circ[v,w]=[u\circ
v,w]+[v,u\circ w],$
$\displaystyle[u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0~{}~{}~{}~{}~{}(\text{Jacobi
identity}).$

Next, let
$(\mathfrak{a},\triangleleft,\triangleright,[~{},~{}]_{\mathfrak{a}})$ be a
post-Leibniz algebra with the property that $u\triangleleft v=-v\triangleright
u$ and $[u,v]_{\mathfrak{a}}=-[v,u]_{\mathfrak{a}}$ for all
$u,v\in\mathfrak{a}$ (such post-Leibniz algebras are called ‘skewsymmetric’).
In this case, it can be checked that
$(\mathfrak{a},\triangleright,[~{},~{}]_{\mathfrak{a}})$ is a post-Lie
algebra. Therefore, ordinary post-Leibniz algebras can be thought of as a non-
skewsymmetric analogue of post-Lie algebras.

(ii) The notion of pre-Leibniz algebras was recently introduced in the study
of weight zero relative Rota-Baxter operators on Leibniz algebras. More
precisely, a pre-Leibniz algebra is a triple
$(\mathfrak{a},\triangleleft,\triangleright)$ consisting of a vector space
$\mathfrak{a}$ together with bilinear operations
$\triangleleft,\triangleright:\mathfrak{a}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$
satisfying for $u,v,w\in\mathfrak{a}$,

$\displaystyle u\triangleleft(v\triangleleft w+v\triangleright w)=~{}$
$\displaystyle(u\triangleleft v)\triangleleft w+v\triangleright(u\triangleleft
w),$ $\displaystyle u\triangleright(v\triangleleft w)=~{}$
$\displaystyle(u\triangleright v)\triangleleft w+v\triangleleft(u\triangleleft
w+u\triangleright w),$ $\displaystyle u\triangleright(v\triangleright w)=~{}$
$\displaystyle(u\triangleleft v+u\triangleright v)\triangleright
w+v\triangleright(u\triangleright w).$

Therefore, it follows that any pre-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright)$ is a post-Leibniz algebra
$(\mathfrak{a},\triangleleft,\triangleright,[~{},~{}]_{\mathfrak{a}}=0)$.

######  6.3 Proposition.

Let $(\mathfrak{a},\triangleleft,\triangleright,[~{},~{}]_{\mathfrak{a}})$ be
a post-Leibniz algebra. Then $(\mathfrak{a},[~{},~{}]_{\star})$ is a Leibniz
algebra. This is called the total Leibniz algebra, denoted by
$\mathfrak{a}_{\mathrm{Tot}}$.

###### Proof.

By adding the left hand sides of the identities (17)-(23), we simply get that
$[u,[v,w]_{\star}]_{\star}$. On the other hand, by adding the right hand sides
of (17)-(23), we get $[[u,v]_{\star},w]_{\star}+[v,[u,w]_{\star}]_{\star}$.
Therefore, the Leibniz identity (1) holds for the bracket $[~{},~{}]_{\star}$.
∎

######  6.4 Proposition.

Let $\mathfrak{g},\mathfrak{h}$ be two Leibniz algebras and $\mathfrak{h}$ be
a Leibniz $\mathfrak{g}$-representation. Let
$T:\mathfrak{h}\rightarrow\mathfrak{g}$ be a $\lambda$-weighted relative Rota-
Baxter operator. Then
$(\mathfrak{h},\triangleleft,\triangleright,[~{},~{}]^{\lambda}_{\mathfrak{h}})$
is a post-Leibniz algebra, where

$\displaystyle u\triangleleft v=\rho^{R}(u,Tv),\quad u\triangleright
v=\rho^{L}(Tu,v)~{}~{}~{}~{}\text{~{}~{} and ~{}~{}
}~{}~{}~{}~{}[u,v]^{\lambda}_{\mathfrak{h}}=\lambda[u,v]_{\mathfrak{h}},~{}\text{
for }u,v\in\mathfrak{h}.$

###### Proof.

For any $u,v,w\in\mathfrak{h}$, we have

$\displaystyle\rho^{R}(u,[Tv,Tw]_{\mathfrak{g}})=\rho^{R}(u,T[v,w]_{\star})=u\triangleleft[v,w]_{\star}.$

On the other hand,

$\displaystyle\rho^{R}(\rho^{R}(u,Tv),Tw)+\rho^{L}(Tv,\rho^{R}(u,Tw))=(u\triangleleft
v)\triangleleft w+v\triangleright(u\triangleleft w).$

Since $(\rho^{L},\rho^{R})$ satisfies (4), it follows that the right hand
sides of the above equations are same. Hence (17) holds. Similarly, we observe
that

$\displaystyle\rho^{L}(Tu,\rho^{R}(v,Tw))=u\triangleright(v\triangleleft w),$
$\displaystyle\rho^{R}(\rho^{L}(Tu,v),Tw)+\rho^{R}(v,[Tu,Tw]_{\mathfrak{g}})=(u\triangleright
v)\triangleleft w+v\triangleleft[u,w]_{\star}.$

Hence it follows from (3) that the identity (18) holds. We also have

$\displaystyle\rho^{L}(Tu,\rho^{L}(Tv,w))=u\triangleright(v\triangleright w),$
$\displaystyle\rho^{L}([Tu,Tv]_{\mathfrak{g}},w)+\rho^{L}(Tv,\rho^{L}(Tu,w))=[u,w]_{\star}\triangleright
w+v\triangleright(u\triangleright w).$

Therefore, the identity (19) also holds. By the same way, we observe

$\displaystyle\rho^{L}(Tu,\lambda[v,w]_{\mathfrak{h}})=u\triangleright[v,w]^{\lambda}_{\mathfrak{h}},$
$\displaystyle\lambda[\rho^{L}(Tu,v),w]_{\mathfrak{h}}+\lambda[v,\rho^{L}(Tu,w)]_{\mathfrak{h}}=[u\triangleright
v,w]_{\mathfrak{h}}^{\lambda}+[v,u\triangleright w]_{\mathfrak{h}}^{\lambda}.$

It follows from (6) that the identity (20) also holds. Similarly, we get the
identities (21), (23) as we have (7), (8). Finally, the identity (23)
automatically holds for the bracket $[~{},~{}]_{\mathfrak{h}}^{\lambda}$ as
the bracket $[~{},~{}]_{\mathfrak{h}}$ satisfies the same. This completes the
proof. ∎

In the previous proposition, we show that a $\lambda$-weighted relative Rota-
Baxter operator induces a post-Leibniz algebra structure. Next, we prove the
converse: any post-Leibniz algebra is always induced by a $1$-weighted
relative Rota-Baxter operator.

Let $(\mathfrak{a},\triangleleft,\triangleright,[~{},~{}]_{\mathfrak{a}})$ be
a post-Leibniz algebra. Consider the total Leibniz algebra
$\mathfrak{a}_{\mathrm{Tot}}=(\mathfrak{a},[~{},~{}]_{\star})$. We define maps
$\varrho^{L}:\mathfrak{a}_{\mathrm{Tot}}\otimes\mathfrak{a}\rightarrow\mathfrak{a}$
and
$\varrho^{R}:\mathfrak{a}\otimes\mathfrak{a}_{\mathrm{Tot}}\rightarrow\mathfrak{a}$
by

$\displaystyle\varrho^{L}(u,v)=u\triangleright v~{}~{}~{}\text{ and
}~{}~{}~{}\varrho^{R}(v,u)=v\triangleleft u,~{}\text{ for
}u\in\mathfrak{a}_{\mathrm{Tot}},v\in\mathfrak{a}.$

Then it can be checked that $\varrho^{L},\varrho^{R}$ makes the Leibniz
algebra $\mathfrak{a}=(\mathfrak{a},[~{},~{}]_{\mathfrak{a}})$ into a Leibniz
$\mathfrak{a}_{\mathrm{Tot}}$-representation. With this notation, the identity
map $\mathrm{id}:\mathfrak{a}\rightarrow\mathfrak{a}_{\mathrm{Tot}}$ is a
$1$-weighted relative Rota-Baxter operator. Moreover, the induced post-Leibniz
algebra structure on the vector space $\mathfrak{a}$ coincides with the given
one.

Given a Leibniz algebra, the following result gives a necessary and sufficient
condition to have a compatible post-Leibniz algebra structure.

######  6.5 Proposition.

Let $(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$ be any Leibniz algebra. Then
there is a compatible post-Leibniz algebra structure if and only if there
exists a Leibniz $\mathfrak{g}$-representation $\mathfrak{h}$ and an
invertible $1$-weighted relative Rota-Baxter operator
$T:\mathfrak{h}\rightarrow\mathfrak{g}$.

###### Proof.

Suppose $(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$ has a compatible post-
Leibniz algebra structure given by
$(\mathfrak{g},\triangleleft,\triangleright,[~{},~{}]^{\prime}_{\mathfrak{g}})$,
i.e. the quadruple
$(\mathfrak{g},\triangleleft,\triangleright,[~{},~{}]^{\prime}_{\mathfrak{g}})$
is a post-Leibniz algebra and
$[~{},~{}]_{\mathfrak{g}}=\triangleleft+\triangleright+[~{},~{}]^{\prime}_{\mathfrak{g}}$.
As discussed above, the Leibniz algebra
$(\mathfrak{g},[~{},~{}]^{\prime}_{\mathfrak{g}})$ is a Leibniz
$\mathfrak{g}$-representation with left and right actions given by

$\displaystyle\varrho^{L}(x,y)=x\triangleright y~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}\varrho^{R}(x,y)=x\triangleleft y,~{}\text{ for
}x,y\in\mathfrak{g}.$

Then the identity map $\mathrm{id}:\mathfrak{g}\rightarrow\mathfrak{g}$ (which
is invertible) is a $1$-weighted relative Rota-Baxter operator. The induced
Leibniz algebra structure on $\mathfrak{g}$ is given by
$(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$.

Conversely, let $\mathfrak{h}$ be a Leibniz $\mathfrak{g}$-representation and
$T:\mathfrak{h}\rightarrow\mathfrak{g}$ be an invertible $1$-weighted relative
Rota-Baxter operator. We know from Proposition 6.4 that $\mathfrak{h}$ carries
a post-Leibniz algebra structure. Using the invertibility of $T$, we get a
post-Leibniz algebra structure on $\mathfrak{g}$ which is given by

$\displaystyle x\triangleleft
y=T(\rho^{R}(T^{-1}x,y)),~{}~{}~{}~{}x\triangleright
y=T(\rho^{L}(x,T^{-1}y))~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}[x,y]^{\prime}_{\mathfrak{g}}=T[T^{-1}x,T^{-1}y]_{\mathfrak{h}},~{}\text{
for }x,y\in\mathfrak{g}.$

Moreover, we have

$\displaystyle x\triangleleft y+x\triangleright
y+[x,y]^{\prime}_{\mathfrak{g}}=~{}$ $\displaystyle
T\big{(}\rho^{R}(T^{-1}x,y)+\rho^{L}(x,T^{-1}y)+[T^{-1}x,T^{-1}y]_{\mathfrak{h}}\big{)}$
$\displaystyle=~{}$
$\displaystyle[TT^{-1}x,TT^{-1}y]_{\mathfrak{g}}=[x,y]_{\mathfrak{g}}.$

Hence
$(\mathfrak{g},\triangleleft,\triangleright,[~{},~{}]^{\prime}_{\mathfrak{g}})$
is a compatible post-Leibniz algebra structure for the Leibniz algebra
$(\mathfrak{g},[~{},~{}]_{\mathfrak{g}})$. ∎

Acknowledgements. The author would like to thank IIT Kharagpur (India) for
providing the beautiful academic atmosphere where the research has been
carried out.

Data availability statement. Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

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