111Corresponding author

#  Cohomology and Deformation theory of crossed homomorphisms on Leibniz
algebras

Yizheng Li  School of Mathematical Sciences, Qufu Normal University, Qufu
273165, P. R. of China<EMAIL_ADDRESS>and  $\mbox{Dingguo\ Wang}^{1}$
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, P. R. of
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we construct a differential graded Lie algebra whose Maurer-
Cartan elements are given by crossed homomorphisms on Leibniz algebras. This
allows us to define cohomology for a crossed homomorphism. Finally, we study
linear deformations, formal deformations and extendibility of finite order
deformations of a crossed homomorphism in terms of the cohomology theory.

###### Key words and phrases:

crossed homomorphism, cohomology, Maurer-Cartan elements, deformation

###### 2010 Mathematics Subject Classification:

16E40 16S80 17A30

###### Contents

  1. 1 Leibniz algebras, representations and their cohomology theory
  2. 2 Crossed homomorphisms on Leibniz algebras
  3. 3 Cohomology theory of crossed homomorphisms on Leibniz algebras
  4. 4 Deformations of crossed homomorphisms on Leibniz algebras
    1. 4.1 Linear deformations
    2. 4.2 Formal deformations
    3. 4.3 Extensions of finite order deformations

## Introduction

In 1960, Baxter [3] introduced the notion of Rota-Baxter operators on
associative algebras in his study of fluctuation theory in probability. Rota-
Baxter operators have been found many applications, including in Connes-
Kreimer’s algebraic approach to the renormalization in perturbative quantum
field theory [4]. For more details on the Rota-Baxter operator, see [9].

The notion of crossed homomorphisms of Lie algebras was first introduced by
Lue [12] in the study of non-abelian extensions of Lie algebras. A crossed
homomorphism is nothing but a differential operator of weight 1. The authors
showed that the category of weak representations (resp. admissible
representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left
module category over the monoidal category of representations of Lie algebras
using crossed homomorphisms [13]. Recently, the author considerd crossed
homomorphisms between associative algebras [6].

The concept of Leibniz algebras was introduced by Loday [10, 11] in the study
of the algebraic $K$-theory. Relative Rota-Baxter operators on Leibniz
algebras were studied in [15], which is the main ingredient in the study of
the twisting theory and the bialgebra theory for Leibniz algebras [14].
Recently, the author has considered weighted relative Rota-Baxter operators on
Leibniz algebras in [7]. Our aim in this paper is to consider crossed
homomorphisms between Leibniz algebras using Leibniz representation introduced
by [7], More precisely, we construct a differential graded Lie algebra whose
Maurer-Cartan elements are given by crossed homomorphisms on Leibniz algebras.
This allows us to define cohomology for a crossed homomorphism. Finally, we
study linear deformations, formal deformations and extendibility of finite
order deformations of a crossed homomorphism in terms of the cohomology
theory.

The paper is organized as follows. In Section 1, we recall some basic
definitions about Leibniz algebras and their cohomology. In Section 2, we
consider crossed homomorphisms between Leibniz algebras. In Section 3, we
construct a differential graded Lie algebra whose Maurer-Cartan elements are
given by crossed homomorphisms on Leibniz algebras and define cohomology for a
crossed homomorphism. In Section 4, we study linear deformations, formal
deformations and extendibility of finite order deformations of a crossed
homomorphism in terms of the cohomology theory.

Throughout this paper, let ${\bf k}$ be a field of characteristic $0$. Except
specially stated, vector spaces are ${\bf k}$-vector spaces and all tensor
products are taken over ${\bf k}$.

##  1\. Leibniz algebras, representations and their cohomology theory

We start with the background of Leibniz algebras and their cohomology that we
refer the reader to [5, 11, 7] for more details.

######  Definition 1.1.

A Leibniz algebra is a vector space ${\mathfrak{g}}$ together with a bilinear
operation (called bracket)
$[\cdot,\cdot]_{\mathfrak{g}}:{\mathfrak{g}}\otimes{\mathfrak{g}}\rightarrow{\mathfrak{g}}$
satisfying

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

A Leibniz algebra as above may be denoted by the pair
$({\mathfrak{g}},[\cdot,\cdot])$ or simply by ${\mathfrak{g}}$ when no
confusion arises. A Leibniz algebra whose bilinear bracket is skewsymmetric is
nothing but a Lie algebra. Thus, Leibniz algebras are the non-skewsymmetric
analogue of Lie algebras.

######  Definition 1.2.

A representation of a Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ consists of a triple
$(V,\rho^{L},\rho^{R})$ in which $V$ is a vector space and
$\rho^{L},\rho^{R}:{\mathfrak{g}}\rightarrow gl(V)$ are linear maps satisfying
for $x,y\in{\mathfrak{g}}$,

$\displaystyle\begin{cases}\rho^{L}([x,y]_{\mathfrak{g}})=\rho^{L}(x)\rho^{L}(y)-\rho^{L}(y)\rho^{L}(x),\\\
\rho^{R}([x,y]_{\mathfrak{g}})=\rho^{L}(x)\rho^{R}(y)-\rho^{R}(y)\rho^{L}(x),\\\
\rho^{R}([x,y]_{\mathfrak{g}})=\rho^{L}(x)\rho^{R}(y)+\rho^{R}(y)\rho^{R}(x).\end{cases}$

It follows that any Leibniz algebra ${\mathfrak{g}}$ is a representation of
itself with

$\displaystyle\rho^{L}(x)=L_{x}=[x,~{}]_{\mathfrak{g}}~{}~{}\text{ and
}~{}~{}\rho^{R}(x)=R_{x}=[~{},x]_{\mathfrak{g}},\text{ for
}x\in{\mathfrak{g}}.$

Here $L_{x}$ and $R_{x}$ denotes the left and right multiplications by $x$,
respectively. This is called the regular representation.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ be a Leibniz algebra and
$(V,\rho^{L},\rho^{R})$ be a representation of it. The cohomology of the
Leibniz algebra ${\mathfrak{g}}$ with coefficients in $V$ is the cohomology of
the cochain complex $\\{C^{\ast}({\mathfrak{g}},V),\delta\\}$, where
$C^{n}(\mathfrak{g},V)=\mathrm{Hom}({\mathfrak{g}}^{\otimes n},V)$ for $n\geq
0$, and the coboundary operator $\delta:C^{n}({\mathfrak{g}},V)\rightarrow
C^{n+1}({\mathfrak{g}},V)$ given by

$\displaystyle(\delta f)(x_{1},\ldots,x_{n+1})$
$\displaystyle=\sum^{n}_{i=1}(-1)^{i+1}\rho^{L}(x_{i})f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})+(-1)^{n+1}\rho^{R}(x_{n+1})f(x_{1},\ldots,x_{n})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{g}},x_{j+1},\ldots,x_{n+1}),$

for $x_{1},\ldots,x_{n+1}\in{\mathfrak{g}}$. The corresponding cohomology
groups are denoted by $H^{\ast}({\mathfrak{g}},V).$ This cohomology has been
first appeared in [5] and rediscover by Loday and Pirashvili [11]. This
cohomology is also the Loday-Pirashvili cohomology.

######  Definition 1.3.

([2, 8]) The graded vector space $C^{\ast}(\mathfrak{g},\mathfrak{g})$
equipped with Balavoine bracket

$\displaystyle\llbracket
P,Q\rrbracket:=P\overline{\diamond}Q-(-1)^{pq}Q\overline{\diamond}P~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
P\in C^{p+1}(\mathfrak{g},\mathfrak{g}),Q\in
C^{q+1}(\mathfrak{g},\mathfrak{g})$

is a graded Lie algebra, where $P\overline{\diamond}Q\in
C^{p+q+1}(\mathfrak{g},\mathfrak{g})$ is defined by

$\displaystyle
P\overline{\diamond}Q=\sum_{k=1}^{p+1}(-1)^{(k-1)q}P\diamond_{\mathrm{{\bf
k}}}Q$

and $\diamond_{\mathrm{{\bf k}}}$ is defined by

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

for all $x_{1},\cdots,x_{p+q+1}\in\mathfrak{g}$.

Moreover,
$\mu_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ is
a Leibniz bracket if and only if
$\llbracket\mu_{\mathfrak{g}},\mu_{\mathfrak{g}}\rrbracket=0$, i. e.
$\mu_{\mathfrak{g}}$is a Maurer
-Cartan element of the graded Lie algebra $(C^{\ast}(\mathfrak{g},\mathfrak{g}),\llbracket-,-\rrbracket)$.

######  Definition 1.4.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ and
$({\mathfrak{h}},[\cdot,\cdot]_{\mathfrak{h}})$ be two Leibniz algebras. We
say that ${\mathfrak{h}}$ is a Leibniz ${\mathfrak{g}}$-representation if
there are bilinear maps $\rho^{L},\rho^{R}:{\mathfrak{g}}\rightarrow$
End$({\mathfrak{h}})$ that make $({\mathfrak{h}},\rho^{L},\rho^{R})$ into a
representation of the Leibniz algebra ${\mathfrak{g}}$ satisfying additionally

$\displaystyle(La)\qquad\rho^{L}(x)[h,k]_{\mathfrak{h}}=[\rho^{L}(x)h,k]_{\mathfrak{h}}+[h,\rho^{L}(x)k]_{\mathfrak{h}},$
$\displaystyle(Lb)\qquad[h,\rho^{R}(x)k]_{\mathfrak{h}}=\rho^{R}(x)[h,k]_{\mathfrak{h}}+[k,\rho^{R}(x)h]_{\mathfrak{h}},$
$\displaystyle(Lc)\qquad[h,\rho^{L}(x)k]_{\mathfrak{h}}=[\rho^{R}(x)h,k]_{\mathfrak{h}}+\rho^{L}(x)[h,k]_{\mathfrak{h}},$

for $h,k\in{\mathfrak{h}},x\in{\mathfrak{g}}$.

Note that, for any Leibniz algebra
$({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$, the regular representation is
a Leibniz ${\mathfrak{g}}$-representation.

##  2\. Crossed homomorphisms on Leibniz algebras

In this section, we study crossed homomorphisms between Leibniz algebras.

######  Definition 2.1.

[7] Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ and
$({\mathfrak{h}},[\cdot,\cdot]_{\mathfrak{h}})$ be Leibniz algebras and
$({\mathfrak{h}},\rho^{L},\rho^{R})$ be a Leibniz
${\mathfrak{g}}$-representation. If a linear map
$H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ is said to be a crossed
homomorphism from ${\mathfrak{g}}$ to ${\mathfrak{h}}$ such that the following
equation

(1) $\displaystyle
H([x,y]_{\mathfrak{g}})=\rho^{L}(x)H(y)+\rho^{R}(y)H(x)+[H(x),H(y)]_{\mathfrak{h}}$

holds for any $x,y\in{\mathfrak{g}}$.

######  Remark 2.2.

A crossed homomorphism from ${\mathfrak{g}}$ to ${\mathfrak{g}}$ with respect
to the regular representation is also called a differential operator of weight
1.

######  Example 2.3.

If the action $\rho^{L},\rho^{R}$ of ${\mathfrak{g}}$ on ${\mathfrak{h}}$ is
zero, then any crossed homomorphism from ${\mathfrak{g}}$ to ${\mathfrak{h}}$
is nothing but a Leibniz algebra homomorphism.

######  Definition 2.4.

Let $H,H^{\prime}:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ be two crossed
homomorphisms from ${\mathfrak{g}}$ to ${\mathfrak{h}}$. A morphism from $H$
to $H^{\prime}$ consists of two Leibniz algebra morphisms
$\phi_{\mathfrak{g}}:{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ and
$\phi_{\mathfrak{h}}:{\mathfrak{h}}\rightarrow{\mathfrak{h}}$ satisfying
$\phi_{\mathfrak{h}}\circ H=H^{\prime}\circ\phi_{\mathfrak{g}}$,
$\phi_{\mathfrak{h}}(\rho^{L}(x)h)=\rho^{L}(\phi_{\mathfrak{g}}(x))\phi_{\mathfrak{h}}(h)$
and
$\phi_{\mathfrak{h}}(\rho^{R}(x)h)=\rho^{R}(\phi_{\mathfrak{g}}(x))\phi_{\mathfrak{h}}(h)$,
for all $x\in{\mathfrak{g}}$ and $h\in{\mathfrak{h}}$.

One can construct a new Leibniz ${\mathfrak{g}}$-representation.

######  Lemma 2.5.

Let $H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ be a crossed homomorphism.
Define maps $\rho_{H}^{L},\rho_{H}^{R}:{\mathfrak{g}}\rightarrow$
End$({\mathfrak{h}})$ by

(2) $\displaystyle\rho^{L}_{H}(x)h:=\rho^{L}(x)h+[H(x),h]_{\mathfrak{h}},$
$\displaystyle\rho^{R}_{H}(x)h:=\rho^{R}(x)h+[h,H(x)]_{\mathfrak{h}},$

for $x\in{\mathfrak{g}}$ and $h\in{\mathfrak{h}}$. Then
$({\mathfrak{h}},\rho^{L}_{H},\rho^{R}_{H})$ is a Leibniz
${\mathfrak{g}}$-representation.

###### Proof.

First we prove that $\rho_{H}^{L},\rho_{H}^{R}$ satisfy the conditions
L(a)-L(c) as follows.

For any $x\in{\mathfrak{g}}$ and $h,k\in{\mathfrak{h}}$, we have

$\displaystyle(La^{\prime})\qquad\qquad$
$\displaystyle[\rho^{L}_{H}(x)h,k]_{\mathfrak{h}}+[h,\rho^{L}_{H}(x)k]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle[\rho^{L}(x)h,k]_{\mathfrak{h}}+[[H(x),h]_{\mathfrak{h}},k]_{\mathfrak{h}}+[h,\rho^{L}(x)k]_{\mathfrak{h}}+[h,[H(x),k]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle\rho^{L}(x)[h,k]_{\mathfrak{h}}+[H(x),[h,k]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle=$ $\displaystyle\rho^{L}_{H}(x)[h,k]_{\mathfrak{h}}.$

Similar to prove that

$\displaystyle(Lb^{\prime})\qquad\qquad[h,\rho_{H}^{R}(x)k]_{\mathfrak{h}}=\rho_{H}^{R}(x)[h,k]_{\mathfrak{h}}+[k,\rho_{H}^{R}(x)h]_{\mathfrak{h}}.$

Furthermore, we have

$\displaystyle(Lc^{\prime})\qquad\qquad$
$\displaystyle[\rho_{H}^{R}(x)h,k]_{\mathfrak{h}}+\rho_{H}^{L}(x)[h,k]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle[\rho^{R}(x)h,k]_{\mathfrak{h}}+[[h,H(x)]_{\mathfrak{h}},k]_{\mathfrak{h}}+\rho^{L}(x)[h,k]_{\mathfrak{h}}+[H(x),[h,k]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle[h,\rho^{L}(x)k]_{\mathfrak{h}}+[h,[H(x),k]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle=$ $\displaystyle[h,\rho^{L}_{H}(x)k]_{\mathfrak{h}}.$

Next we prove that $({\mathfrak{h}},\rho_{H}^{L},\rho_{H}^{R})$ is a
representation over $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$.

For any $x,y\in{\mathfrak{g}}$ and $h\in{\mathfrak{h}}$, we have

$\displaystyle\rho_{H}^{L}(x)\rho_{H}^{L}(y)h-\rho_{H}^{L}(y)\rho_{H}^{L}(x)h$
$\displaystyle=$
$\displaystyle\rho_{H}^{L}(x)(\rho^{L}(y)h+[H(y),h]_{\mathfrak{h}})-\rho_{H}^{L}(y)(\rho^{L}(x)h+[H(x),h]_{\mathfrak{h}})$
$\displaystyle=$
$\displaystyle\rho^{L}(x)\rho^{L}(y)h+[H(x),\rho^{L}(y)h]_{\mathfrak{h}}+\rho^{L}(x)[H(y),h]_{\mathfrak{h}}+[H(x),[H(y),h]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle-\rho^{L}(y)\rho^{L}(x)h-[H(y),\rho^{L}(x)h]_{\mathfrak{h}}-\rho^{L}(y)[H(x),h]_{\mathfrak{h}}-[H(y),[H(x),h]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle\stackrel{{\scriptstyle(La,Lc)}}{{=}}$
$\displaystyle\rho^{L}([x,y]_{\mathfrak{g}})h+[\rho^{L}(y)H(x),h]_{\mathfrak{h}}+[\rho^{R}(x)H(y),h]_{\mathfrak{h}}+[H(x),H(y)]_{\mathfrak{h}},h]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle\rho^{L}([x,y]_{\mathfrak{g}})h+[H([x,y]_{\mathfrak{g}}),h]_{\mathfrak{h}}$
$\displaystyle=$ $\displaystyle\rho^{L}_{H}([x,y]_{\mathfrak{g}})h.$

Similarly, we have

$\displaystyle\rho_{H}^{R}([x,y]_{\mathfrak{g}})=\rho_{H}^{L}(x)\rho_{H}^{R}(y)-\rho_{H}^{R}(y)\rho_{H}^{L}(x).$

Furthermore, we have

$\displaystyle\rho_{H}^{R}(y)\rho_{H}^{L}(x)h+\rho_{H}^{R}(y)\rho_{H}^{R}(x)h$
$\displaystyle=$
$\displaystyle\rho_{H}^{R}(y)(\rho^{L}(x)h+[H(x),h]_{\mathfrak{h}})+\rho_{H}^{R}(y)(\rho^{R}(x)h+[h,H(x)]_{\mathfrak{h}})$
$\displaystyle=$
$\displaystyle\rho^{R}(y)\rho^{L}(x)h+[\rho^{L}(x)h,H(y)]_{\mathfrak{h}}+\rho^{R}_{H}(y)[H(x),h]_{\mathfrak{h}}$
$\displaystyle+\rho^{R}(y)\rho^{R}(x)h+[\rho^{R}(x)h,H(y)]_{\mathfrak{h}}+\rho^{R}_{H}(y)[h,H(x)]_{\mathfrak{h}}$
$\displaystyle\stackrel{{\scriptstyle(La,Lb^{\prime},Lc)}}{{=}}$
$\displaystyle 0.$

Hence, $({\mathfrak{h}},\rho^{L}_{H},\rho^{R}_{H})$ is a Leibniz
${\mathfrak{g}}$-representation. ∎

Since $({\mathfrak{h}},[\cdot,\cdot]_{\mathfrak{h}})$ is a Leibniz
${\mathfrak{g}}$-representation, there is a semi-direct product Leibniz
algebra structure on ${\mathfrak{g}}\oplus{\mathfrak{h}}$ given by

$\displaystyle[(x,h),(y,k)]=[x,y]_{\mathfrak{g}}+\rho^{L}(x)k+\rho^{R}(y)h+[h,k]_{\mathfrak{h}},$

for $(x,h),(y,k)\in{\mathfrak{g}}\oplus{\mathfrak{h}}$. We denote this semi-
direct product algebra by ${\mathfrak{g}}\ltimes{\mathfrak{h}}$. Moreover, it
follows from Lemma 2.5 that the direct sum
${\mathfrak{g}}\ltimes{\mathfrak{h}}$ carries another semi-direct product
Leibniz algebra structure given by

$\displaystyle[(x,h),(y,k)]_{H}=[x,y]_{\mathfrak{g}}+\rho^{L}_{H}(x)k+\rho^{R}_{H}(y)h+[h,k]_{\mathfrak{h}}.$

We denote this semi-direct product algebra by
${\mathfrak{g}}\ltimes_{H}{\mathfrak{h}}$.

######  Theorem 2.6.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ and
$({\mathfrak{h}},[\cdot,\cdot]_{\mathfrak{h}})$ be Leibniz algebras with
respect to the Leibniz ${\mathfrak{g}}$-representation
$({\mathfrak{h}},\rho^{L},\rho^{R})$ and
$H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ be a linear map.
(Ca) Suppose that $({\mathfrak{h}},\rho^{L}_{H},\rho^{R}_{H})$ is a Leibniz
${\mathfrak{g}}$-representation given by (2). Then the linear map
$\hat{H}:{\mathfrak{g}}\ltimes_{H}{\mathfrak{h}}\rightarrow{\mathfrak{g}}\ltimes{\mathfrak{h}}$
defined by

$\displaystyle\hat{H}(x,h)=(x,H(x)+h),\forall
x\in{\mathfrak{g}},h\in{\mathfrak{h}},$

is a Leibniz algebra isomorphism if and only if $H$ is a crossed homomorphism
from ${\mathfrak{g}}$ to ${\mathfrak{h}}$ with respect to the Leibniz
${\mathfrak{g}}$-representation $({\mathfrak{h}},\rho^{L},\rho^{R})$.
(Cb) $H$ is a crossed homomorphism from ${\mathfrak{g}}$ to ${\mathfrak{h}}$
with respect to the Leibniz ${\mathfrak{g}}$-representation
$({\mathfrak{h}},\rho^{L},\rho^{R})$ if and only if the map
$\iota_{H}:{\mathfrak{g}}\rightarrow{\mathfrak{g}}\ltimes_{H}{\mathfrak{h}}$
defined by

$\displaystyle\iota_{H}(x)=(x,H(x)),\forall x\in{\mathfrak{g}}$

is a Leibniz algebra homomorphism.

###### Proof.

(Ca) Clearly $\hat{H}$ is an invertible linear map. For all
$x,y\in{\mathfrak{g}},h,k\in{\mathfrak{h}}$, we have

$\displaystyle[\hat{H}(x,h),\hat{H}(y,k)]$ $\displaystyle=$
$\displaystyle[(x,H(x)+h),(y,H(y)+k)]$ $\displaystyle=$
$\displaystyle([x,y]_{\mathfrak{g}},\rho^{L}(x)(H(y)+k)+\rho^{R}(y)(H(x)+h)+[H(x)+h,H(y)+k]_{\mathfrak{h}})$
$\displaystyle=$
$\displaystyle([x,y]_{\mathfrak{g}},\rho^{L}(x)k+\rho^{R}(y)h+[H(x),k]_{\mathfrak{h}}+[h,H(y)]_{\mathfrak{h}}+[h,k]_{\mathfrak{h}}$
$\displaystyle+[H(x),H(y)]_{\mathfrak{g}}+\rho^{L}(x)H(y)+\rho^{R}(y)H(x))$
$\displaystyle\hat{H}[(x,h),(y,k)]_{H}$ $\displaystyle=$
$\displaystyle([x,y]_{\mathfrak{g}},H([x,y]_{\mathfrak{g}})+\rho_{H}^{L}(x)k+\rho_{H}^{R}(y)h+[h,k]_{\mathfrak{h}})$
$\displaystyle=$
$\displaystyle([x,y]_{\mathfrak{g}},H([x,y]_{\mathfrak{g}})+\rho^{L}(x)k+[H(x),k]_{\mathfrak{h}}+\rho^{R}(y)h+[h,H(y)]_{\mathfrak{g}}$
$\displaystyle+[h,k]_{\mathfrak{h}}).$

Thus $[\hat{H}(x,h),\hat{H}(y,k)]=\hat{H}[(x,h),(y,k)]_{H}$ if and only if (1)
holds for $H$, which is equivalent to that $H$ is a crossed homomorphism from
from ${\mathfrak{g}}$ to ${\mathfrak{h}}$ with respect to the Leibniz
${\mathfrak{g}}$-representation $({\mathfrak{h}},\rho^{L},\rho^{R})$.
(Cb) follows from the proof of (Ca) by taking $h=k=0$. ∎

##  3\. Cohomology theory of crossed homomorphisms on Leibniz algebras

In this section, we consider a differential graded Lie algebra (dgLa) whose
Maurer-Cartan elements are given by crossed homomorphisms on Leibniz algebras.
This characterizations of a crossed homomorphism allow us to define cohomology
for a crossed homomorphism.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ and
$({\mathfrak{h}},[\cdot,\cdot]_{\mathfrak{h}})$ be two Leibniz algebras and
$({\mathfrak{h}},\rho^{L},\rho^{R})$ be a Leibniz
${\mathfrak{g}}$-representation. We denote the Leibniz products on
${\mathfrak{g}}$ and ${\mathfrak{h}}$ respectively by $\mu_{\mathfrak{g}}$ and
$\mu_{\mathfrak{h}}$. Consider the semidirect product
${\mathfrak{g}}\oplus{\mathfrak{h}}$. Note that $\mu_{\mathfrak{h}}$ is a
Maurer-Cartan element in the graded Lie algebra
$\oplus_{n}\mathrm{Hom}(({\mathfrak{g}}\oplus{\mathfrak{h}})^{\otimes
n},{\mathfrak{g}}\oplus{\mathfrak{h}})$. Therefore, we can define a
differential
$d_{\mu_{\mathfrak{h}}}=\llbracket\mu_{\mathfrak{h}},~{}\rrbracket$ and the
derived bracket on the graded space
$\oplus_{n}\mathrm{Hom}({\mathfrak{g}}^{\otimes n},{\mathfrak{h}})$ by

$\displaystyle\widehat{\llbracket
f,g\rrbracket}:=(-1)^{m}\llbracket\llbracket\mu_{\mathfrak{h}},f\rrbracket,g\rrbracket,$

for any $f\in\mathrm{Hom}({\mathfrak{g}}^{\otimes m},{\mathfrak{h}})$ and
$g\in\mathrm{Hom}({\mathfrak{g}}^{\otimes n},{\mathfrak{h}})$.

Moreover, we know that $\mu_{\mathfrak{g}}+\rho^{L}+\rho^{R}$ is a Maurer-
Cartan element in the graded Lie algebra
$\oplus_{n}\mathrm{Hom}(({\mathfrak{g}}\oplus{\mathfrak{h}})^{\otimes
n},{\mathfrak{g}}\oplus{\mathfrak{h}})$ from [14], Thus it induces a
differential
$d_{\mu_{\mathfrak{g}}+\rho^{L}+\rho^{R}}=\llbracket\mu_{\mathfrak{g}}+\rho^{L}+\rho^{R},~{}\rrbracket$,
and the graded space $\oplus_{n}\mathrm{Hom}({\mathfrak{g}}^{\otimes
n},{\mathfrak{h}})$ is closed under the differential
$d=d_{\mu_{\mathfrak{g}}+\rho^{L}+\rho^{R}}$ and is given by

$\displaystyle(df)(x_{1},\ldots,x_{n+1})$
$\displaystyle=(-1)^{n+1}\sum^{n}_{i=1}(-1)^{i+1}\rho^{L}(x_{i})f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})+\rho^{R}(x_{n+1})f(x_{1},\ldots,x_{n})$
$\displaystyle+(-1)^{n+1}\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{g}},x_{j+1},\ldots,x_{n+1}).$

Finally, we have
$\llbracket\mu_{\mathfrak{g}}+\rho^{L}+\rho^{R},\mu_{\mathfrak{h}}\rrbracket$=0.
Hence, $(\oplus_{n}\mathrm{Hom}({\mathfrak{g}}^{\otimes
n},{\mathfrak{h}}),\widehat{\llbracket~{},~{}\rrbracket},d)$ is a dgLa.

######  Proposition 3.1.

A linear map $H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ is a crossed
homomorphism from ${\mathfrak{g}}$ to ${\mathfrak{h}}$ if and only if $H\in
C^{1}({\mathfrak{g}},{\mathfrak{h}})$ is a Maurer-Cartan element in the dgLa
$(\oplus_{n}\mathrm{Hom}({\mathfrak{g}}^{\otimes
n},{\mathfrak{h}}),\widehat{\llbracket~{},~{}\rrbracket},d)$.

###### Proof.

For any linear map $H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ and
$x,y\in{\mathfrak{g}}$, we have

$\displaystyle(dH+\frac{1}{2}\widehat{\llbracket
H,H\rrbracket})(x,y)=\rho^{L}(x)H(y)+\rho^{R}(y)H(x)-H([x,y]_{\mathfrak{g}})+[H(x),H(y)]_{\mathfrak{h}}.$

Hence $H$ is a crossed homomorphism if and only if $H$ is a Maurer-Cartan
element. ∎

It follows from the above proposition that a crossed homomorphism $H$ induces
a differential $d_{H}=d+\widehat{\llbracket H,~{}\rrbracket}$ on the graded
Lie algebra $(\oplus_{n}\mathrm{Hom}({\mathfrak{g}}^{\otimes
n},{\mathfrak{h}}),\widehat{\llbracket~{},~{}\rrbracket})$. Define
$C^{n}({\mathfrak{g}},{\mathfrak{h}})=\mathrm{Hom}({\mathfrak{g}}^{\otimes
n},{\mathfrak{h}})$ and
$C^{\ast}({\mathfrak{g}},{\mathfrak{h}})=\oplus_{n}C^{n}({\mathfrak{g}},{\mathfrak{h}})$.
Thus a crossed homomorphism induces a dgLa
$(C^{\ast}({\mathfrak{g}},{\mathfrak{h}}),\widehat{\llbracket~{},~{}\rrbracket},d_{H})$.

######  Theorem 3.2.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ and
$({\mathfrak{h}},[\cdot,\cdot]_{\mathfrak{h}})$ be two Leibniz algebras and
$({\mathfrak{h}},\rho^{L},\rho^{R})$ be a Leibniz
${\mathfrak{g}}$-representation. Suppose $H$ is a crossed homomorphism. Then
for any linear map $H^{\prime}:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$, the
sum $H+H^{\prime}$ is also a crossed homomorphism if and only if $H^{\prime}$
is a Maurer-Cartan element in the dgLa
$(C^{\ast}({\mathfrak{g}},{\mathfrak{h}}),\widehat{\llbracket~{},~{}\rrbracket},d_{H})$.

###### Proof.

$\displaystyle(d(H+H^{\prime})+\frac{1}{2}\widehat{\llbracket
H+H^{\prime},H+H^{\prime}\rrbracket})$
$\displaystyle=dH+dH^{\prime}+\frac{1}{2}(\widehat{\llbracket
H,H\rrbracket}+\widehat{\llbracket H,H^{\prime}\rrbracket}+\widehat{\llbracket
H^{\prime},H\rrbracket}+\widehat{\llbracket H^{\prime},H^{\prime}\rrbracket})$
$\displaystyle=dH^{\prime}+\widehat{\llbracket
H,H^{\prime}\rrbracket}+\frac{1}{2}\widehat{\llbracket
H^{\prime},H^{\prime}\rrbracket}$
$\displaystyle=d_{H}(H^{\prime})+\frac{1}{2}\widehat{\llbracket
H^{\prime},H^{\prime}\rrbracket}.$

And the proof is finished. ∎

The cohomology of the cochain complex
$(C^{\ast}({\mathfrak{g}},{\mathfrak{h}}),d_{H})$ is called the cohomology of
the crossed homomorphism $H$, if
$\mathcal{Z}_{H}^{k}({\mathfrak{g}},{\mathfrak{h}})=\\{f\in
C^{k}({\mathfrak{g}},{\mathfrak{h}})|d_{H}(f)=0\\}$ is the space of
$k$-cocycles and
$\mathcal{B}^{k}_{H}({\mathfrak{g}},{\mathfrak{h}})=\\{d_{H}(f)\in
C^{k}({\mathfrak{g}},{\mathfrak{h}})|f\in
C^{k-1}({\mathfrak{g}},{\mathfrak{h}})\\}$ is the space of $k$-coboundaries
then
$\mathcal{B}^{k}_{H}({\mathfrak{g}},{\mathfrak{h}})\subset\mathcal{Z}^{k}_{H}({\mathfrak{g}},{\mathfrak{h}})$,
for $k\geq 0$. The corresponding cohomology groups

$\displaystyle\mathcal{H}^{k}_{H}({\mathfrak{g}},{\mathfrak{h}})=\frac{\mathcal{Z}^{k}_{H}({\mathfrak{g}},{\mathfrak{h}})}{\mathcal{B}^{k}_{H}({\mathfrak{g}},{\mathfrak{h}})},k\geq
0.$

We denote the corresponding cohomology groups simply by
$\mathcal{H}^{\ast}({\mathfrak{g}},{\mathfrak{h}})$.

First recall from Lemma (3) that a crossed homomorphism $H$ induces a Leibniz
${\mathfrak{g}}$-representation given by

$\displaystyle\rho^{L}_{H}(x)h:=\rho^{L}(x)h+[H(x),h]_{\mathfrak{h}},~{}~{}~{}~{}~{}~{}~{}~{}\rho^{R}_{H}(x)h:=\rho^{R}(x)h+[h,H(x)]_{\mathfrak{h}}.$

The corresponding cochain groups are given by
$C^{n}_{\mathrm{Leib}}({\mathfrak{g}},{\mathfrak{h}})=\mathrm{Hom}({\mathfrak{g}}^{\otimes
n},{\mathfrak{h}})$, for $n\geq 0$, and the coboundary operator
$\delta_{\mathrm{Leib}}:C^{n}_{\mathrm{Leib}}({\mathfrak{g}},{\mathfrak{h}})\rightarrow
C^{n+1}_{\mathrm{Leib}}({\mathfrak{g}},{\mathfrak{h}})$ is given by

$\displaystyle(\delta_{\mathrm{Leib}}f)(x_{1},\ldots,x_{n+1})$
$\displaystyle=$
$\displaystyle\sum^{n}_{i=1}(-1)^{i+1}\rho^{L}(x_{i})f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})+[H(x_{i}),f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})]_{\mathfrak{h}}$
$\displaystyle+(-1)^{n+1}\rho^{R}(x_{n+1})f(x_{1},\ldots,x_{n})+(-1)^{n+1}[f(x_{1},\ldots,x_{n}),H(x)]_{\mathfrak{h}}$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{g}},x_{j+1},\ldots,x_{n+1}),$

for $x_{1},\ldots,x_{n+1}\in{\mathfrak{g}}$.

######  Proposition 3.3.

The coboundary operators $d_{H}$ and $\delta_{\mathrm{Leib}}$ are related by

$\displaystyle d_{H}(f)=(-1)^{n-1}\delta_{\mathrm{Leib}}(f),\forall f\in
C^{n}({\mathfrak{g}},{\mathfrak{h}}).$

###### Proof.

For any $f\in C^{n}({\mathfrak{g}},{\mathfrak{h}})$, we have

$\displaystyle(-1)^{n-1}(d_{H}(f))(x_{1},\ldots,x_{n+1})$ $\displaystyle=$
$\displaystyle(-1)^{n-1}(df+\widehat{\llbracket
H,f\rrbracket})(x_{1},\ldots,x_{n+1})$ $\displaystyle=$
$\displaystyle\sum^{n}_{i=1}(-1)^{i+1}\rho^{L}(x_{i})f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})+(-1)^{n+1}\rho^{R}(x_{n+1})f(x_{1},\ldots,x_{n})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i}f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{g}},x_{j+1},\ldots,x_{n+1})$
$\displaystyle+\\{[H(x_{i}),f(x_{1},\ldots,\hat{x}_{i},\ldots,x_{n+1})]_{\mathfrak{h}}+(-1)^{n+1}[f(x_{1},\ldots,x_{n}),H(x)]_{\mathfrak{h}}\\}$
$\displaystyle=$ $\displaystyle\delta_{\mathrm{Leib}}(f).$

And the proof is finished. ∎

##  4\. Deformations of crossed homomorphisms on Leibniz algebras

In this section, we study deformations of a crossed homomorphism on Leibniz
algebras.

###  4.1. Linear deformations

Let $H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ be a crossed homomorphism. A
linear deformation of $H$ consists of a sum $H_{t}=H+tH_{1}$ such that $H_{t}$
is a crossed homomorphism, for all values of $t$. In such a case, we say that
$H_{1}$ generates a one-parameter linear deformation of $H$. Thus, if $H_{1}$
generates a linear deformation of $H$ then $H_{t}=H+tH_{1}$ satisfies

$\displaystyle
H_{t}([x,y]_{\mathfrak{g}})=\rho^{L}(x)H_{t}(y)+\rho^{L}(y)H_{t}(x)+[H_{t}(x),H_{t}(y)]_{\mathfrak{g}},\forall
x,y\in{\mathfrak{g}}.$

That is

(3) $\displaystyle
H_{1}([x,y]_{\mathfrak{g}})=\rho^{L}(x)H_{1}(y)+\rho^{L}(y)H_{1}(x)+[H_{1}(x),H(y)]_{\mathfrak{g}}+[H(x),H_{1}(y)]_{\mathfrak{g}},$
(4) $\displaystyle[H_{1}(x),H_{1}(y)]_{\mathfrak{g}}=0.$

It follows from (3) that $H_{1}$ is a 1-cocycle in the cohomology of the
crossed homomorphism $H$.

######  Definition 4.1.

Two linear deformations $H_{t}=H+tH_{1}$ and
$H^{\prime}_{t}=H+tH^{\prime}_{1}$ are said to be equivalent if there exists
$x\in{\mathfrak{g}}$ such that

$\displaystyle\phi_{t}=\mathrm{id}_{\mathfrak{g}}+tL_{x},\quad\psi_{t}=\mathrm{id}_{\mathfrak{h}}+t\rho^{L}(x)$

is a homomorphism from $H_{t}$ to $H^{\prime}_{t}$.

Thus, if $H_{t}$ and $H^{\prime}_{t}$ are equivalent linear deformations, then
the following conditions hold:

$\displaystyle(i)~{}~{}[\phi_{t}(y),\phi_{t}(z)]_{\mathfrak{g}}=\phi_{t}([y,z]_{\mathfrak{g}}),~{}~{}~{}[\psi_{t}(h),\psi_{t}(k)]_{\mathfrak{h}}=\psi_{t}([h,k]_{\mathfrak{h}}),$
$\displaystyle(ii)~{}~{}\phi_{t}(\rho^{L}(y)h)=\rho^{L}(\phi_{t}(y))\psi_{t}(h),$
$\displaystyle(iii)~{}~{}\phi_{t}(\rho^{R}(y)h)=\rho^{R}(\phi_{t}(y))\psi_{t}(h),$
$\displaystyle(iv)~{}~{}H^{\prime}_{t}\circ\phi_{t}(y)=\psi_{t}\circ
H_{t}(y),$

for all $y,z\in{\mathfrak{g}}$ and $h,k\in{\mathfrak{h}}$.

Note that (i) is equivalent

(5)
$\displaystyle[[x,y]_{\mathfrak{g}},[x,z]_{\mathfrak{g}}]_{\mathfrak{g}}=0,~{}~{}~{}~{}~{}[\rho^{L}(x)h,\rho^{L}(x)k]_{\mathfrak{h}}=0.$

Further, (ii)and (iii) imply that

(6) $\displaystyle\rho^{L}([x,y]_{\mathfrak{g}})\rho^{L}(x)=0,$ (7)
$\displaystyle\rho^{R}([x,y]_{\mathfrak{g}})\rho^{L}(x)=0,~{}~{}~{}~{}\forall
y\in{\mathfrak{g}}.$

Finally, the condition (iv) is equivalent to

(8) $\displaystyle
H_{1}(y)-H^{\prime}_{1}(y)=\rho^{R}(y)H(x)+[H(x),H(y)]_{\mathfrak{h}},$ (9)
$\displaystyle\rho^{L}(x)H_{1}(y)=H_{1}^{\prime}([x,y]_{\mathfrak{g}}).$

It follows from (8) that
$H_{1}(y)-H^{\prime}_{1}(y)=\delta_{\mathrm{Leib}}(-H(x))y.$ Hence, we obtain
the following

######  Theorem 4.2.

Let $H_{t}=H+tH_{1}$ be a linear deformation of $H$. Then the linear term
$H_{1}$ is a 1-cocycle in the cohomology of $H$. Its cohomology class depends
only on the equivalence class of $H_{t}$.

######  Definition 4.3.

A linear deformation $H_{t}$ of a crossed homomorphism $H$ is said to be
trivial if $H_{t}$ is equivalent to $H^{\prime}_{t}=H$.

######  Definition 4.4.

Let $H$ be a crossed homomorphism from ${\mathfrak{g}}$ to ${\mathfrak{h}}$.
An element $x\in{\mathfrak{g}}$ is called a Nijenhuis element associated to
$H$ if $x$ satisfies (5), (6), (7) and

$\displaystyle[x,\rho^{R}(y)H(x)+[H(x),H(y)]_{\mathfrak{h}}]_{\mathfrak{g}}=0,~{}~{}~{}~{}\forall
y\in{\mathfrak{g}}.$

Denote by $Nij(H)$ the set of Nijenhuis elements associated to $H$. Then we
have

######  Theorem 4.5.

Let $H$ be a crossed homomorphism from ${\mathfrak{g}}$ to ${\mathfrak{h}}$..
Then for any $x\in Nij(H)$, the sum $H_{t}=H+tH_{1}$ with
$H_{1}=-\delta_{\mathrm{Leib}}(H(x))$ is a trivial deformation of $H$.

###  4.2. Formal deformations

Let $H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ be a crossed homomorphism. Let
$\mathbf{k}[[t]]$ be the ring of power series in one variable $t$. For any
$\mathbf{k}$-linear space ${\mathfrak{g}}$, let ${\mathfrak{g}}[[t]]$ denotes
the vector space of formal power series in $t$ with coefficients from
${\mathfrak{g}}$. Then ${\mathfrak{g}}[[t]]$ is Leibniz algebra structure over
$\mathbf{K}[[t]]$. If ${\mathfrak{h}}$ is a Leibniz algebra which is also a
Leibniz ${\mathfrak{g}}$-representation, then ${\mathfrak{h}}[[t]]$ is a
Leibniz algebra over $\mathbf{k}[[t]]$ and a Leibniz
${\mathfrak{g}}[[t]]$-representation.

######  Definition 4.6.

A formal one-parameter deformation of a crossed homomorphism
$H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ consists of a formal sum

(10) $\displaystyle
H_{t}=H_{0}+tH_{1}+t^{2}H_{2}+\cdots\in\mathrm{Hom}({\mathfrak{g}},{\mathfrak{h}})[[t]]$

with $H_{0}=H$ such that
$H_{t}:{\mathfrak{g}}[[t]]\rightarrow{\mathfrak{h}}[[t]]$ is a crossed
homomorphism from ${\mathfrak{g}}[[t]]$ to ${\mathfrak{h}}[[t]]$.

Note that (10) is equivalent to: for each $n\geq 0$

$\displaystyle
H_{n}([x,y]_{\mathfrak{g}})=\rho^{L}(x)H_{n}(y)+\rho^{R}(y)H_{n}(x)+\sum_{i+j=n}[H_{i}(x),H_{j}(y)]_{\mathfrak{h}}.$

That is

$\displaystyle d_{H}(H_{n})=-\frac{1}{2}\sum_{i+j=n,i,j\geq
1}\widehat{\llbracket H_{i},H_{j}\rrbracket},n\geq 0.$

The identity holds for $n=0$ as $H$ is a crossed homomorphism. For $n=1$, we
get $d_{H}(H_{1})=0$. Hence $H_{1}$ is a 1-cocycle in the cohomology complex
of $H$. This is called the infinitesimal of the deformation $H_{t}$.

######  Definition 4.7.

Two o formal deformations $H_{t}$ and $H^{\prime}_{t}$ of a crossed
homomorphism $H$ are said to be equivalent if there exists
$x\in{\mathfrak{g}}$ such that

$\displaystyle\phi_{t}=\mathrm{id}_{\mathfrak{g}}+tL_{x},\quad\psi_{t}=\mathrm{id}_{\mathfrak{h}}+t\rho^{L}(x)$

is a homomorphism from $H_{t}$ to $H^{\prime}_{t}$.

######  Proposition 4.8.

Let $H_{t}$ be a formal deformation of a crossed homomorphism $H$. Then the
linear term $H_{1}$ is a 1-cocycle in the cohomology of $H$. (It is called the
infinitesimal of the deformation.) Moreover, the corresponding cohomology
class depends only on the equivalence class of $H_{t}$.

######  Definition 4.9.

Let $({\mathfrak{g}},[\cdot,\cdot]_{\mathfrak{g}})$ and
$({\mathfrak{h}},[\cdot,\cdot]_{\mathfrak{h}})$ be two Leibniz algebras and
$({\mathfrak{h}},\rho^{L},\rho^{R})$ a Leibniz
${\mathfrak{g}}$-representation. A crossed homomorphism
$H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ is said to be rigid if any formal
deformation $H_{t}$ of $H$ is equivalent to $H^{\prime}_{t}=H$.

######  Theorem 4.10.

Let $H$ be a crossed homomorphism. If
$\mathcal{Z}^{1}_{H}({\mathfrak{g}},{\mathfrak{h}})=\delta_{\mathrm{Leib}}(H(Nij(H)))$,
then $H$ is rigid.

###### Proof.

Let $H_{t}=\sum_{i\geq 0}t^{i}H_{i}$ be any formal deformation of $H$. By
Proposition 4.8, we deduce
$H_{1}\in\mathcal{Z}^{1}({\mathfrak{g}},{\mathfrak{h}})$. By the assumption
$\mathcal{Z}^{1}_{H}({\mathfrak{g}},{\mathfrak{h}})=\delta_{\mathrm{Leib}}(H(Nij(H)))$,
we obtain $H_{1}=-\delta_{\mathrm{Leib}}(H(x))$ for some $x\in Nij(H)$. Then
setting
$\phi_{t}=\mathrm{id}_{\mathfrak{g}}+tL_{x},\quad\psi_{t}=\mathrm{id}_{\mathfrak{h}}+t\rho^{L}(x)$,
we get a formal deformation $\overline{H}_{t}:=\psi_{t}\circ
H_{t}\phi_{t}^{-1}$ Thus, $H_{t}$ is equivalent to $\overline{H}_{t}$.
Moreover, we have

$\displaystyle\overline{H}_{t}(y))(\mathrm{mod}~{}t^{2})$ $\displaystyle=$
$\displaystyle(\mathrm{id}_{\mathfrak{g}}+t\rho^{L}(x))\circ(H+tH_{1})(y-t[x,y]_{\mathfrak{g}})(\mathrm{mod}~{}t^{2})$
$\displaystyle=$
$\displaystyle(\mathrm{id}_{\mathfrak{g}}+t\rho^{L}(x))(H(y)+tH_{1}(y)-tH([x,y]_{\mathfrak{g}}))(\mathrm{mod}~{}t^{2})$
$\displaystyle=$ $\displaystyle
H(y)+t(\rho^{L}(x)H(y)-H([x,y]_{\mathfrak{g}}+H_{1}(y)).$

Since

$\displaystyle H_{1}(y)$ $\displaystyle=$
$\displaystyle-\delta_{\mathrm{Leib}}(H(x))(y)$ $\displaystyle=$
$\displaystyle\rho^{R}(y)H(x)+[H(x),H(y)]_{\mathfrak{h}}$ $\displaystyle=$
$\displaystyle-(\rho^{L}(x)H(y)-H([x,y]_{\mathfrak{g}}).$

The coefficient of $t$ in the expression of $\overline{H}_{t}$ is zero. Then
by repeating this argument, one get the equivalence between $H_{t}$ and $H$.
Hence the proof. ∎

###  4.3. Extensions of finite order deformations

In this subsection, we introduce a cohomology class associated to any order
$N$ deformation of a crossed homomorphism, and show that an order $n$
deformation is extensible if and only if this cohomology class is trivial.
Thus, we call this cohomology class the obstruction class of the order $N$
deformation being extensible.

Let $H:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$ be a crossed homomorphism.
Consider the space ${\mathfrak{g}}[[t]]/(t^{N+1})$ which inherits a Leibniz
algebra structure over $\mathbf{k}[[t]]/(t^{N+1})$, similarly,
${\mathfrak{h}}[[t]]/(t^{N+1})$ is a Leibniz algebra and a Leibniz
${\mathfrak{g}}[[t]]/(t^{N+1})$-representation.

######  Definition 4.11.

A deformation of $H$ of order $N$ consists of a finite sum
$H_{t}=\sum_{i=0}^{N}t^{i}H_{i}$ with $H_{0}=H$ such that $H_{t}$ is a crossed
homomorphism from ${\mathfrak{g}}[[t]]/(t^{N+1})$ to
${\mathfrak{h}}[[t]]/(t^{N+1})$.

######  Definition 4.12.

If there exists a linear map $H_{N+1}:{\mathfrak{g}}\rightarrow{\mathfrak{h}}$
such that $\hat{H}_{t}=H_{t}+t^{N+1}H_{N+1}$ is a deformation of order $N+1$,
we say that $H_{t}$ is extensible.

Thus, if a deformation $H_{t}$ of order $N$ is extensible then one more
deformation equation need to be satisfied, namely,

$\displaystyle d_{H}(H_{N+1})=-\frac{1}{2}\sum_{i+j=N+1,i,j\geq
1}\widehat{\llbracket H_{i},H_{j}\rrbracket},n\geq 0.$

for some $H_{N+1}$. Note that the right hand side of the above equation does
not involve $H_{N+1}$, we denote it by $Ob_{H_{t}}$. Obviously $Ob_{H_{t}}$ is
a 2-cochain in the cohomology of $H$, we have the following.

######  Proposition 4.13.

The $2$-cochain $Ob_{H_{t}}$ is a 2-cocycle, that is, $d_{H}(Ob_{H_{t}})=0$.

###### Proof.

We have

$\displaystyle d_{H}(Ob_{H_{t}})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i+j=N+1,i,j\geq 1}(d\widehat{\llbracket
H_{i},H_{j}\rrbracket}+\widehat{\llbracket H,\widehat{\llbracket
H_{i},H_{j}\rrbracket}\rrbracket})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i+j=N+1,i,j\geq 1}(\widehat{\llbracket
d(H_{i}),H_{j}\rrbracket}-\widehat{\llbracket
H_{i},d(H_{j})\rrbracket}+\widehat{\llbracket\widehat{\llbracket
H,H_{i}\rrbracket},H_{j}\rrbracket}-\widehat{\llbracket
H_{i},\widehat{\llbracket H,H_{j}\rrbracket}\rrbracket})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i+j=N+1,i,j\geq 1}(\widehat{\llbracket
d_{H}(H_{i}),H_{j}\rrbracket}-\widehat{\llbracket
H_{i},d_{H}(H_{j})\rrbracket})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{i_{1}+i_{2}+j=N+1,i_{1},i_{2},j\geq
1}\widehat{\llbracket\widehat{\llbracket
H_{i_{1}},H_{i_{2}}\rrbracket},H_{j}\rrbracket}-\frac{1}{4}\sum_{i+j_{1}+j_{2}=N+1,i_{1},i_{2},j\geq
1}\widehat{\llbracket H_{i},\widehat{\llbracket
H_{j_{1}},H_{j_{2}}\rrbracket}\rrbracket}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i+j+k=N+1,i,j,k\geq
1}\widehat{\llbracket\widehat{\llbracket
H_{i},H_{j}\rrbracket},H_{k}\rrbracket}$ $\displaystyle=$ $\displaystyle 0.$

∎

The cohomology class
$[Ob_{H_{t}}]\in\mathcal{H}_{H}^{2}({\mathfrak{g}},{\mathfrak{h}})$ is called
the obstruction class to extend the deformation $H_{t}$. we have the
following.

######  Theorem 4.14.

A finite order deformation $H_{t}$ of a crossed homomorphism $H$ extends to a
deformation of next order if and only if the obstruction class
$[Ob_{H_{t}}]\in\mathcal{H}_{H}^{2}({\mathfrak{g}},{\mathfrak{h}})$ vanishes.

######  Corollary 4.15.

If $\mathcal{H}_{H}^{2}({\mathfrak{g}},{\mathfrak{h}})=0$, then any finite
order deformation of $H$ extends to a deformation of next order.

######  Theorem 4.16.

If $\mathcal{H}_{H}^{2}({\mathfrak{g}},{\mathfrak{h}})=0$, then every
1-cocycle in the cohomology of $H$ is the infinitesimal of some formal
deformation of $H$.

Acknowledgments. The work is supported by Natural Science Foundation of China
(Grant Nos. 11871301, 12271292).

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