# Deformations and abelian extensions of compatible pre-Lie algebras

Shanshan Liu  School of Mathematics and Statistics, Northeast Normal
University, Changchun 130024, Jilin, China<EMAIL_ADDRESS>and  Liangyun
Chen  School of Mathematics and Statistics, Northeast Normal University,
Changchun 130024, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first give the notation of a compatible pre-Lie algebra and
its representation. We study the relation between compatible Lie algebras and
compatible pre-Lie algebras. We also construct a new bidifferential graded Lie
algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We
give the bidifferential graded Lie algebra which controls deformations of a
compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible
pre-Lie algebra with coefficients in itself. We study infinitesimal
deformations of compatible pre-Lie algebras and show that equivalent
infinitesimal deformations are in the same second cohomology group. We further
give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We
study formal deformations of compatible pre-Lie algebras. If the second
cohomology group $\mathcal{H}^{2}(\mathfrak{g};\mathfrak{g})$ is trivial, then
the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a
compatible pre-Lie algebra with coefficients in arbitrary representation and
study abelian extensions of compatible pre-Lie algebras using this cohomology.
We show that abelian extensions are classified by the second cohomology group.

###### Key words and phrases:

compatible pre-Lie algebra, Maurer-Cartan element, cohomology, deformation,
abelian extension

000* Corresponding author000MSC 2020: 17A36, 17A40, 17B10, 17B40, 17B60,
17B63, 17D25

###### Contents

  1. 1 Introduction
  2. 2 Maurer-Cartan characterizations of compatible pre-Lie algebras
  3. 3 Formal deformations of compatible pre-Lie algebras
    1. 3.1 Cohomologies of compatible pre-Lie algebras
    2. 3.2 Infinitesimal deformations of compatible pre-Lie algebras
    3. 3.3 Formal deformations of compatible pre-Lie algebras
  4. 4 Abelian extensions of compatible pre-Lie algebras
    1. 4.1 Cohomologies of compatible pre-Lie algebras with coefficients in arbitrary representation
    2. 4.2 Abelian extensions of compatible pre-Lie algebras

##  1\. Introduction

The notion of a pre-Lie algebra (also called left-symmetric algebras, quasi-
associative algebras, Vinberg algebras and so on) has been introduced
independently by M. Gerstenhaber in deformation theory of rings and algebras
[9]. Pre-Lie algebra arose from the study of affine manifolds and affine
structures on Lie group [14], homogeneous convex cones [22]. Its defining
identity is weaker than associativity. This algebraic structure describes some
properties of cochains space in Hochschild cohomology of an associative
algebra, rooted trees and vector fields on affine spaces. Moreover, it is
playing an increasing role in algebra, geometry and physics due to their
applications in nonassociative algebras, combinatorics, numerical Analysis and
quantum field theory, see also in [1, 2, 3, 6]. There is a close relationship
between pre-Lie algebras and Lie algebras: a pre-Lie algebra
$(\mathfrak{g},\cdot)$ gives rise to a Lie algebra
$(\mathfrak{g},[\cdot,\cdot]_{C})$ via the commutator bracket, which is called
the subadjacent Lie algebra and denoted by ${\mathfrak{g}}^{C}$. Furthermore,
the map $L:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$, defined by
$L_{x}y=x\cdot y$ for all $x,y\in\mathfrak{g}$, gives rise to a representation
of the subadjacent Lie algebra ${\mathfrak{g}}^{C}$ on $\mathfrak{g}$.

Compatible algebraic structures refer to two algebraic structures of the same
kind in a linear category such that any linear combination of multiplications
corresponding to these two algebraic structures still defines the same kind of
algebraic structure. Compatible algebraic structures have been widely studied
in mathematics and mathematical physics. Golubchik and Sokolov studied
compatible Lie algebras with the background of integrable equations [10],
classical Yang-Baxter equations [11], loop algebras over Lie algebras [12].
Compatible Lie algebras are also related to elliptic theta functions [19].
Classification, operads and bialgebra theory of compatible Lie algebras were
also studied in [20, 21, 24]. Recently, in [17], deformation theory and the
cohomology theory of compatible Lie algebras were established by using the
theory of bidifferential graded Lie algebras. Using similar ideas, compatible
$L_{\infty}$-algebras were studied in [8], compatible associative algebras
were studied in [5] and compatible $3$-Lie algebras were studied in [13].

The purpose of the paper is to study the cohomology of a compatible pre-Lie
algebra and its applications. We construct a bidifferential graded Lie algebra
whose Maurer-Cartan elements are compatible pre-Lie structures. Then, we
introduce a cohomology of a compatible pre-Lie algebra with coefficients in
itself. As applications, we study formal deformations and infinitesimal
deformations of compatible pre-Lie algebras and give the notion of a Nijenhuis
operator on a compatible pre-Lie algebra. We give a compatible pre-Lie algebra
$(\mathfrak{g},\pi_{1},\pi_{2})$ and its representation
$(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ and construct a bidifferential graded
Lie algebra whose Maurer-Cartan elements is
$(\pi_{1}+\rho+\mu,\pi_{2}+\tilde{\rho}+\tilde{\mu})$. Then we give a
cohomology of a compatible pre-Lie algebra with coefficients in arbitrary
representation. As applications, we study abelian extensions of compatible
pre-Lie algebras.

The paper is organized as follows. In Section 2, we give the notation of a
compatible pre-Lie algebra and its representation. We study the relation
between compatible Lie algebras and compatible pre-Lie algebras. We recall the
notion of bidifferential graded Lie algebra and construct a new bidifferential
graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie
structures. Furthermore, we give the bidifferential graded Lie algebra which
controls deformations of a compatible pre-Lie algebra. In Section 3, we
introduce a cohomology of a compatible pre-Lie algebra with coefficients in
itself. Using this cohomology, we study infinitesimal deformations of
compatible pre-Lie algebras and show that equivalent infinitesimal
deformations are in the same second cohomology group. We give the notion of a
Nijenhuis operator on a compatible pre-Lie algebra. We show that a trivial
deformation gives rise to a Nijenhuis operator. Conversely, a Nijenhuis
operator gives rise to a trivial deformation. We study formal deformations of
compatible pre-Lie algebras. If the second cohomology group
$\mathcal{H}^{2}(\mathfrak{g};\mathfrak{g})$ is trivial, then the compatible
pre-Lie algebra is rigid. In Section 4, we give a compatible pre-Lie algebra
$(\mathfrak{g},\pi_{1},\pi_{2})$ and its representation
$(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ and construct a bidifferential graded
Lie algebra whose Maurer-Cartan elements is
$(\pi_{1}+\rho+\mu,\pi_{2}+\tilde{\rho}+\tilde{\mu})$. Using this
bidifferential graded Lie algebra, we give a cohomology of a compatible pre-
Lie algebra with coefficients in arbitrary representation. We study abelian
extensions of compatible pre-Lie algebras using this cohomological approach
and show that abelian extensions are classified by the second cohomology
group.

##  2\. Maurer-Cartan characterizations of compatible pre-Lie algebras

In this section, first, we give the notation of a compatible pre-Lie algebra
and its representation. Then, we study the relation between compatible Lie
algebras and compatible pre-Lie algebras. Finally, we construct a new
bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible
pre-Lie structures. We give the bidifferential graded Lie algebra which
controls deformations of a compatible pre-Lie algebra.

######  Definition 2.1.

([1]) A pre-Lie algebra $(\mathfrak{g},\cdot)$ is a vector space
$\mathfrak{g}$ equipped with a bilinear product
$\cdot:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$, such that
for all $x,y,z\in\mathfrak{g}$, the following equality is satisfied:

$(x\cdot y)\cdot z-x\cdot(y\cdot z)=(y\cdot x)\cdot z-y\cdot(x\cdot z).$

Let $(\mathfrak{g},\cdot)$ be a pre-Lie algebra. The commutator
$[x,y]_{C}=x\cdot y-y\cdot x$ gives a Lie algebra
$(\mathfrak{g},[\cdot,\cdot]_{C})$, which is denoted by ${\mathfrak{g}}^{C}$
and called the sub-adjacent Lie algebra of $(\mathfrak{g},\cdot)$.

######  Definition 2.2.

([2]) A representation of a pre-Lie algebra $(\mathfrak{g},\cdot)$ on a vector
space $V$ consists of a pair $(\rho,\mu)$, where
$\rho:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ is a representation of the
sub-adjacent Lie algebra ${\mathfrak{g}}^{C}$ on $V$, and
$\mu:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ is a linear map, such that
for all $x,y\in\mathfrak{g}$:

$\mu(y)\circ\mu(x)-\mu(x\cdot y)=\mu(y)\circ\rho(x)-\rho(x)\circ\mu(y).$

We denote a representation of a pre-Lie algebra $(\mathfrak{g},\cdot)$ by
$(V,\rho,\mu)$. Furthermore, let
$L,R:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$ be linear maps,
where $L_{x}y=x\cdot y,R_{x}y=y\cdot x$. Then $(\mathfrak{g},L,R)$ is also a
representation, which is called the regular representation.

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

Let $\mathfrak{g}$ be a vector space. We denote
$C^{n}(\mathfrak{g};\mathfrak{g})=\mathrm{Hom}(\wedge^{n-1}\mathfrak{g}\otimes\mathfrak{g},\mathfrak{g})$
and consider the graded vector space
$C^{*}(\mathfrak{g};\mathfrak{g})=\oplus_{n=1}^{+\infty}C^{n}(\mathfrak{g};\mathfrak{g})=\oplus_{n=1}^{+\infty}\mathrm{Hom}(\wedge^{n-1}\mathfrak{g}\otimes\mathfrak{g},\mathfrak{g})$.
It was shown in [4, 18, 23] that $C^{*}(\mathfrak{g};\mathfrak{g})$ equipped
with the Matsushima-Nijenhuis bracket

$[P,Q]^{MN}=P\circ Q-(-1)^{pq}Q\circ P,\quad\forall P\in
C^{p+1}(\mathfrak{g};\mathfrak{g}),Q\in C^{q+1}(\mathfrak{g};\mathfrak{g})$

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

$\displaystyle P\circ Q(x_{1},\dots,x_{p+q+1})$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}(q,1,p-1)}\mathrm{sgn}(\sigma)P(Q(x_{\sigma(1)},\dots,x_{\sigma(q)},x_{\sigma(q+1)}),x_{\sigma(q+2)},\dots,x_{\sigma(p+q)},x_{p+q+1})$
$\displaystyle+(-1)^{pq}\sum_{\sigma\in\mathbb{S}(p,q)}\mathrm{sgn}(\sigma)P(x_{\sigma(1)},\dots,x_{\sigma(p)},Q(x_{\sigma(p+1)},\dots,x_{\sigma(p+q)},x_{p+q+1})).$

In particular, $\pi\in\mathrm{Hom}(\otimes^{2}\mathfrak{g},\mathfrak{g})$
defines a pre-Lie algebra if and only if $[\pi,\pi]^{MN}=0$. If $\pi$ is a
pre-Lie algebra structure, then $d_{\pi}:=[\pi,\cdot]^{MN}$ is a graded
derivation of the graded Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN})$ satisfying
$d_{\pi}\circ d_{\pi}=0$, so that
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},d_{\pi})$ becomes a
differential graded Lie algebra.

######  Definition 2.3.

A compatible pre-Lie algebra is a triple $(\mathfrak{g},\cdot,\ast)$, where
$\mathfrak{g}$ is a vector space, $``\cdot$” and $``\ast$” are pre-Lie
structures on $\mathfrak{g}$, such that for all $x,y,z\in\mathfrak{g}$, the
following equality is satisfied:

(1) $(x\ast y)\cdot z+(x\cdot y)\ast z-x\cdot(y\ast z)-x\ast(y\cdot z)-(y\ast
x)\cdot z-(y\cdot x)\ast z+y\cdot(x\ast z)+y\ast(x\cdot z)=0.$

######  Proposition 2.4.

A triple $(\mathfrak{g},\cdot,\ast)$ is a compatible pre-Lie algebra if and
only if $``\cdot$” and $``\ast$” are pre-Lie structures on $\mathfrak{g}$,
such that for all $k_{1},k_{2}\in\mathbb{K}$, the following bilinear operation

(2) $x\diamond y=k_{1}x\cdot y+k_{2}x\ast y,\quad\forall x,y\in\mathfrak{g}.$

defines a pre-Lie algebra structure on $\mathfrak{g}$.

###### Proof.

It is straightforward. ∎

######  Definition 2.5.

Let $(\mathfrak{g},\cdot,\ast)$ and
$(\mathfrak{g}^{\prime},\cdot^{\prime},\ast^{\prime})$ be two compatible pre-
Lie algebras. A homomorphism
$\varphi:(\mathfrak{g},\cdot,\ast)\longrightarrow(\mathfrak{g}^{\prime},\cdot^{\prime},\ast^{\prime})$
is both a pre-Lie homomorphism from $(\mathfrak{g},\cdot)$ to
$(\mathfrak{g}^{\prime},\cdot^{\prime})$ and a pre-Lie homomorphism from
$(\mathfrak{g},\ast)$ to $(\mathfrak{g}^{\prime},\ast^{\prime})$.

######  Definition 2.6.

A representation of a compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$
on a vector space $V$ consists of a quadruple
$(\rho,\mu,\tilde{\rho},\tilde{\mu})$, where $(V,\rho,\mu)$ is a
representation of the pre Lie algebra $(\mathfrak{g},\cdot)$ and
$(V,\tilde{\rho},\tilde{\mu})$ is a representation of the pre Lie algebra
$(\mathfrak{g},\ast)$, such that for all $x,y\in\mathfrak{g}$:

(3) $\displaystyle\rho(x\ast y)+\tilde{\rho}(x\cdot
y)-\rho(x)\tilde{\rho}(y)-\tilde{\rho}(x)\rho(y)$ $\displaystyle=$
$\displaystyle\rho(y\ast x)+\tilde{\rho}(y\cdot
x)-\rho(y)\tilde{\rho}(x)-\tilde{\rho}(y)\rho(x),$ (4)
$\displaystyle\mu(y)\tilde{\rho}(x)-\rho(x)\tilde{\mu}(y)-\mu(y)\tilde{\mu}(x)+\mu(x\ast
y)$ $\displaystyle=$
$\displaystyle-\tilde{\mu}(y)\rho(x)+\tilde{\rho}(x)\mu(y)+\tilde{\mu}(y)\mu(x)-\tilde{\mu}(x\cdot
y).$

We denote a representation of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ by $(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$.
Furthermore, let
$L,R,\tilde{L},\tilde{R}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$
be linear maps, where $L_{x}y=x\cdot y,R_{x}y=y\cdot x,\tilde{L}_{x}y=x\ast
y,\tilde{R}_{x}y=y\ast x$. Then $(\mathfrak{g},L,R,\tilde{L},\tilde{R})$ is
also a representation, which is called the regular representation.

We define two bilinear operations $\cdot_{\mathfrak{g}\oplus
V}:\otimes^{2}(\mathfrak{g}\oplus V)\,\rightarrow\,(\mathfrak{g}\oplus V)$ and
$\ast_{\mathfrak{g}\oplus V}:\otimes^{2}(\mathfrak{g}\oplus
V)\,\rightarrow\,(\mathfrak{g}\oplus V)$ respectively by

$\displaystyle(x+u)\cdot_{\mathfrak{g}\oplus V}(y+v)$ $\displaystyle=$
$\displaystyle x\cdot y+\rho(x)(v)+\mu(y)(u),\quad\forall
x,y\in\mathfrak{g},u,v\in V,$ $\displaystyle(x+u)\ast_{\mathfrak{g}\oplus
V}(y+v)$ $\displaystyle=$ $\displaystyle x\ast
y+\tilde{\rho}(x)(v)+\tilde{\mu}(y)(u),\quad\forall x,y\in\mathfrak{g},u,v\in
V.$

######  Proposition 2.7.

With the above notation, $(\mathfrak{g}\oplus V,\cdot_{\mathfrak{g}\oplus
V},\ast_{\mathfrak{g}\oplus V})$ is a compatible pre-Lie algebra, which is
denoted by $\mathfrak{g}\ltimes_{(\rho,\mu,\tilde{\rho},\tilde{\mu})}V$ and
called the semi-direct product of the compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ and the representation
$(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$.

###### Proof.

Obviously, $(\mathfrak{g}\oplus V,\cdot_{\mathfrak{g}\oplus V})$ and
$(\mathfrak{g}\oplus V,\ast_{\mathfrak{g}\oplus V})$ are pre-Lie algebras. For
all $x,y,z\in\mathfrak{g},u,v,w\in V$, by (1), (3) and (4), we have

$\displaystyle((x+u)\ast_{\mathfrak{g}\oplus V}(y+v))\cdot_{\mathfrak{g}\oplus
V}(z+w)+((x+u)\cdot_{\mathfrak{g}\oplus V}(y+v))\ast_{\mathfrak{g}\oplus
V}(z+w)$ $\displaystyle-(x+u)\cdot_{\mathfrak{g}\oplus
V}((y+v)\ast_{\mathfrak{g}\oplus V}(z+w))-(x+u)\ast_{\mathfrak{g}\oplus
V}((y+v)\cdot_{\mathfrak{g}\oplus V}(z+w))$
$\displaystyle-((y+v)\ast_{\mathfrak{g}\oplus
V}(x+u))\cdot_{\mathfrak{g}\oplus V}(z+w)-((y+v)\cdot_{\mathfrak{g}\oplus
V}(x+u))\ast_{\mathfrak{g}\oplus V}(z+w)$
$\displaystyle+(y+v)\cdot_{\mathfrak{g}\oplus V}((x+u)\ast_{\mathfrak{g}\oplus
V}(z+w))+(y+v)\ast_{\mathfrak{g}\oplus V}((x+u)\cdot_{\mathfrak{g}\oplus
V}(z+w))$ $\displaystyle=$ $\displaystyle(x\ast y)\cdot z+\rho(x\ast
y)w+\mu(z)\tilde{\rho}(x)v+\mu(z)\tilde{\mu}(y)u+(x\cdot y)\ast
z+\tilde{\rho}(x\cdot y)w+\tilde{\mu}(z)\rho(x)v+\tilde{\mu}(z)\mu(y)u$
$\displaystyle-x\cdot(y\ast
z)-\rho(x)\tilde{\rho}(y)w-\rho(x)\tilde{\mu}(z)v-\mu(y\ast z)u-x\ast(y\cdot
z)-\tilde{\rho}(x)\rho(y)w-\tilde{\rho}(x)\mu(z)v-\tilde{\mu}(y\cdot z)u$
$\displaystyle-(y\ast x)\cdot z-\rho(y\ast
x)w-\mu(z)\tilde{\rho}(y)u-\mu(z)\tilde{\mu}(x)v-(y\cdot x)\ast
z-\tilde{\rho}(y\cdot x)w-\tilde{\mu}(z)\rho(y)u-\tilde{\mu}(z)\mu(x)v$
$\displaystyle+y\cdot(x\ast
z)+\rho(y)\tilde{\rho}(x)w+\rho(y)\tilde{\mu}(z)u+\mu(x\ast z)v+y\ast(x\cdot
z)+\tilde{\rho}(y)\rho(x)w+\tilde{\rho}(y)\mu(z)u+\tilde{\mu}(x\cdot z)v$
$\displaystyle=$ $\displaystyle 0.$

This finishes the proof. ∎

Now, we will give the relation between compatible Lie algebras and compatible
pre-Lie algebras. First, we will recall the notation of a compatible Lie
algebra and its representation.

######  Definition 2.8.

([10, 11, 19]) A compatible Lie algebra is a triple
$(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$, where $\mathfrak{g}$ is a
vector space, $[\cdot,\cdot]$ and $\\{\cdot,\cdot\\}$ are Lie algebra
structures on $\mathfrak{g}$, such that for all $x,y,z\in\mathfrak{g}$, the
following equality is satisfied:

$[\\{x,y\\},z]+[\\{y,z\\},x]+[\\{z,x\\},y]+\\{[x,y],z\\}+\\{[y,z],x\\}+\\{[z,x],y\\}=0.$

######  Definition 2.9.

([24]) A representation of a compatible Lie algebra
$(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$ on a vector space $V$
consists of a pair $(\rho,\mu)$, where $(V,\rho)$ is a representation of the
Lie algebra $(\mathfrak{g},[\cdot,\cdot])$ and $(V,\mu)$ is a representation
of the Lie algebra $(\mathfrak{g},\\{\cdot,\cdot\\})$ such that for all
$x,y\in\mathfrak{g}$:

$\rho(\\{x,y\\})+\mu([x,y])=[\rho(x),\mu(y)]-[\rho(y),\mu(x)].$

We denote a representation of a compatible Lie algebra
$(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$ by $(V,\rho,\mu)$.

######  Proposition 2.10.

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra. Define two
brackets $[\cdot,\cdot]$ and $\\{\cdot,\cdot\\}$ respectively by

$[x,y]=x\cdot y-y\cdot x,\quad\\{x,y\\}=x\ast y-y\ast x.$

Then, $(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$ is a compatible Lie
algebra, which is denoted by ${\mathfrak{g}}^{C}$ and called the sub-adjacent
compatible Lie algebra of $(\mathfrak{g},\cdot,\ast)$. Moreover, let
$L,\tilde{L}$ be linear maps, where $L_{x}y=x\cdot y,\tilde{L}_{x}y=x\ast y$.
Then $(V,L,\tilde{L})$ is a representation of the sub-adjacent compatible Lie
algebra $(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$.

###### Proof.

Obviously, $(\mathfrak{g},[\cdot,\cdot])$ and
$(\mathfrak{g},\\{\cdot,\cdot\\})$ are Lie algebras. For all
$x,y,z\in\mathfrak{g}$, by (1), we have

$\displaystyle[\\{x,y\\},z]+[\\{y,z\\},x]+[\\{z,x\\},y]+\\{[x,y],z\\}+\\{[y,z],x\\}+\\{[z,x],y\\}$
$\displaystyle=$ $\displaystyle[x\ast y-y\ast x,z]+[y\ast z-z\ast y,x]+[z\ast
x-x\ast z,y]$ $\displaystyle+\\{x\cdot y-y\cdot x,z\\}+\\{y\cdot z-z\cdot
y,x\\}+\\{z\cdot x-x\cdot z,y\\}$ $\displaystyle=$ $\displaystyle(x\ast
y)\cdot z-(y\ast x)\cdot z-z\cdot(x\ast y)+z\cdot(y\ast x)+(y\ast z)\cdot
x-(z\ast y)\cdot x$ $\displaystyle-x\cdot(y\ast z)+x\cdot(z\ast y)+(z\ast
x)\cdot y-(x\ast z)\cdot y-y\cdot(z\ast x)+y\cdot(x\ast z)$
$\displaystyle+(x\cdot y)\ast z-(y\cdot x)\ast z-z\ast(x\cdot y)+z\ast(y\cdot
x)+(y\cdot z)\ast x-(z\cdot y)\ast x$ $\displaystyle-x\ast(y\cdot
z)+x\ast(z\cdot y)+(z\cdot x)\ast y-(x\cdot z)\ast y-y\ast(z\cdot
x)+y\ast(x\cdot z)$ $\displaystyle=$ $\displaystyle 0.$

Thus, $(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$ is a compatible Lie
algebra.

Obviously, $(V,L)$ is a representation of $(\mathfrak{g},[\cdot,\cdot])$ and
$(V,\tilde{L})$ is a representation of $(\mathfrak{g},\\{\cdot,\cdot\\})$. For
all $x,y,z\in\mathfrak{g}$, by (1), we have

$\displaystyle
L_{\\{x,y\\}}z+\tilde{L}_{[x,y]}z-[L_{x},\tilde{L}_{y}]z+[L_{y},\tilde{L}_{x}]z$
$\displaystyle=$ $\displaystyle\\{x,y\\}\cdot z+[x,y]\ast
z-L_{x}\tilde{L}_{y}z+\tilde{L}_{y}L_{x}z+L_{y}\tilde{L}_{x}z-\tilde{L}_{x}L_{y}z$
$\displaystyle=$ $\displaystyle(x\ast y)\cdot z-(y\ast x)\cdot z+(x\cdot
y)\ast z-(y\cdot x)\ast z-x\cdot(y\ast z)+y\ast(x\cdot z)+y\cdot(x\ast
z)-x\ast(y\cdot z)$ $\displaystyle=$ $\displaystyle 0,$

which implies that

$L_{\\{x,y\\}}+\tilde{L}_{[x,y]}=[L_{x},\tilde{L}_{y}]-[L_{y},\tilde{L}_{x}].$

Thus, $(V,L,\tilde{L})$ is a representation of the sub-adjacent compatible Lie
algebra $(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$. ∎

######  Proposition 2.11.

Let $(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ be a representation of a
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$. Then
$(V,\rho-\mu,\tilde{\rho}-\tilde{\mu})$ is a representation of the sub-
adjacent compatible Lie algebra
$(\mathfrak{g},[\cdot,\cdot],\\{\cdot,\cdot\\})$.

###### Proof.

Obviously, $(V,\rho-\mu)$ is a representation of the Lie algebra
$(\mathfrak{g},[\cdot,\cdot])$ and $(V,\tilde{\rho}-\tilde{\mu})$ is a
representation of the Lie algebra $(\mathfrak{g},\\{\cdot,\cdot\\})$. For all
$x,y\in\mathfrak{g},u\in V$, by (3) and (4), we have

$\displaystyle(\rho-\mu)(\\{x,y\\})+(\tilde{\rho}-\tilde{\mu})([x,y])-[(\rho-\mu)(x),(\tilde{\rho}-\tilde{\mu})(y)]+[(\rho-\mu)(y),(\tilde{\rho}-\tilde{\mu})(x)]$
$\displaystyle=$
$\displaystyle\rho\\{x,y\\}-\mu\\{x,y\\}+\tilde{\rho}[x,y]-\tilde{\mu}[x,y]-[\rho(x),\tilde{\rho}(y)]+[\rho(x),\tilde{\mu}(y)]+[\mu(x),\tilde{\rho}(y)]$
$\displaystyle-[\mu(x),\tilde{\mu}(y)]+[\rho(y),\tilde{\rho}(x)]-[\rho(y),\tilde{\mu}(x)]-[\mu(y),\tilde{\rho}(x)]+[\mu(y),\tilde{\mu}(x)]$
$\displaystyle=$ $\displaystyle\rho(x\ast y)-\rho(y\ast x)-\mu(x\ast
y)+\mu(y\ast x)+\tilde{\rho}(x\cdot y)-\tilde{\rho}(y\cdot
x)-\tilde{\mu}(x\cdot y)+\tilde{\mu}(y\cdot x)$
$\displaystyle-\rho(x)\tilde{\rho}(y)+\tilde{\rho}(y)\rho(x)+\rho(x)\tilde{\mu}(y)-\tilde{\mu}(y)\rho(x)+\mu(x)\tilde{\rho}(y)-\tilde{\rho}(y)\mu(x)-\mu(x)\tilde{\mu}(y)+\tilde{\mu}(y)\mu(x)$
$\displaystyle+\rho(y)\tilde{\rho}(x)-\tilde{\rho}(x)\rho(y)-\rho(y)\tilde{\mu}(x)+\tilde{\mu}(x)\rho(y)-\mu(y)\tilde{\rho}(x)+\tilde{\rho}(x)\mu(y)+\mu(y)\tilde{\mu}(x)-\tilde{\mu}(x)\mu(y)$
$\displaystyle=$ $\displaystyle 0,$

which implies that

$(\rho-\mu)(\\{x,y\\})+(\tilde{\rho}-\tilde{\mu})([x,y])=[(\rho-\mu)(x),(\tilde{\rho}-\tilde{\mu})(y)]-[(\rho-\mu)(y),(\tilde{\rho}-\tilde{\mu})(x)].$

This finishes the proof. ∎

######  Definition 2.12.

([17]) Let $(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{1})$ and
$(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{2})$ be two differential graded Lie
algebras, where $\mathcal{G}=\oplus_{i=0}^{\infty}{\mathfrak{g}}_{i}$. We call
the quadruple $(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{1},\mathrm{d}_{2})$ a
bidifferential graded Lie algebra if $\mathrm{d}_{1}$ and $\mathrm{d}_{2}$
satisfy

(5) $\mathrm{d}_{1}\circ\mathrm{d}_{2}+\mathrm{d}_{2}\circ\mathrm{d}_{1}=0.$

######  Definition 2.13.

([17]) Let $(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{1},\mathrm{d}_{2})$ be a
bidifferential graded Lie algebra. A pair
$(P_{1},P_{2})\in{\mathfrak{g}}_{1}\oplus{\mathfrak{g}}_{1}$ is called a
Maurer-Cartan element of
$(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{1},\mathrm{d}_{2})$ if $P_{1}$ and
$P_{2}$ are Maurer-Cartan elements of the differential graded Lie algebra
$(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{1})$ and
$(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{2})$ respectively, such that

(6) $\mathrm{d}_{1}P_{2}+\mathrm{d}_{2}P_{1}+[P_{1},P_{2}]=0.$

Let $(\mathcal{G},[\cdot,\cdot])$ be a graded Lie algebra. It is obviously
that $(\mathcal{G},[\cdot,\cdot],\mathrm{d}_{1}=0,\mathrm{d}_{2}=0)$ is a
bidifferential graded Lie algebra. Consider the graded Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN})$, we obtain the
following main result.

######  Theorem 2.14.

Let $\mathfrak{g}$ be a vector space and
$\pi_{1},\pi_{2}\in\mathrm{Hom}(\otimes^{2}\mathfrak{g},\mathfrak{g})$. Then
$(\mathfrak{g},\pi_{1},\pi_{2})$ is a compatible pre-Lie algebra if and only
if $(\pi_{1},\pi_{2})$ is a Maurer-Cartan element of the bidifferential graded
Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{1}=0,\mathrm{d}_{2}=0)$.

###### Proof.

Obviously, $(\mathfrak{g},\pi_{1})$ and $(\mathfrak{g},\pi_{2})$ are pre-Lie
algebra if and only if

$[\pi_{1},\pi_{1}]^{MN}=0,\quad[\pi_{2},\pi_{2}]^{MN}=0.$

For all $x,y,z\in\mathfrak{g}$, we have

$\displaystyle[\pi_{1},\pi_{2}]^{MN}(x,y,z)$ $\displaystyle=$
$\displaystyle\pi_{1}(\pi_{2}(x,y),z)-\pi_{1}(\pi_{2}(y,x),z)-\pi_{1}(x,\pi_{2}(y,z))+\pi_{1}(y,\pi_{2}(x,z))$
$\displaystyle+\pi_{2}(\pi_{1}(x,y),z)-\pi_{2}(\pi_{1}(y,x),z)-\pi_{2}(x,\pi_{1}(y,z))+\pi_{2}(y,\pi_{1}(x,z)),$

which implies that equation (1) is equivalent to $[\pi_{1},\pi_{2}]^{MN}=0$.
Thus, $(\mathfrak{g},\pi_{1},\pi_{2})$ is a compatible pre-Lie algebra if and
only if $(\pi_{1},\pi_{2})$ is a Maurer-Cartan element of the bidifferential
graded Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{1}=0,\mathrm{d}_{2}=0)$.
∎

Now, we give a new bidifferential graded Lie algebra that controls
deformations of a compatible pre-Lie algebra.

######  Proposition 2.15.

Let $(\mathfrak{g},\pi_{1},\pi_{2})$ be a compatible pre-Lie algebra. Then
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{\pi_{1}},\mathrm{d}_{\pi_{2}})$
is a bidifferential graded Lie algebra. Moreover,
$(\mathfrak{g},\pi_{1}+\pi_{1}^{\prime},\pi_{2}+\pi_{2}^{\prime})$ is a
compatible pre-Lie algebra for all
$\pi^{\prime}_{1},\pi^{\prime}_{2}\in\mathrm{Hom}(\otimes^{2}\mathfrak{g},\mathfrak{g})$
if and only if $(\pi^{\prime}_{1},\pi^{\prime}_{2})$ is a Maurer-Cartan
element of the bidifferential graded Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{\pi_{1}},\mathrm{d}_{\pi_{2}})$.

###### Proof.

Since $(\mathfrak{g},\pi_{1},\pi_{2})$ is a compatible pre-Lie algebra, by
Theorem 2.14, $(\pi_{1},\pi_{2})$ is a Maurer-Cartan element of the
bidifferential graded Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{1}=0,\mathrm{d}_{2}=0)$.
Thus, we have

$[\pi_{1},\pi_{1}]^{MN}=0,\quad[\pi_{2},\pi_{2}]^{MN}=0,\quad[\pi_{1},\pi_{2}]^{MN}=0.$

Thus,
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{\pi_{1}})$
and
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{\pi_{2}})$
are differential graded Lie algebras. For all $P\in
C^{p+1}(\mathfrak{g},\mathfrak{g})$, by graded Jacobi identity, we have

$\mathrm{d}_{\pi_{1}}(\mathrm{d}_{\pi_{2}}P)+\mathrm{d}_{\pi_{2}}(\mathrm{d}_{\pi_{1}}P)=[\pi_{1},[\pi_{2},P]^{MN}]^{MN}+[\pi_{2},[\pi_{1},P]^{MN}]^{MN}=[[\pi_{1},\pi_{2}]^{MN},p]^{MN}=0,$

which implies that
$\mathrm{d}_{\pi_{1}}\circ\mathrm{d}_{\pi_{2}}+\mathrm{d}_{\pi_{2}}\circ\mathrm{d}_{\pi_{1}}=0$.
Thus,
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{\pi_{1}},\mathrm{d}_{\pi_{2}})$
is a bidifferential graded Lie algebra.

If $(\mathfrak{g},\pi_{1}+\pi_{1}^{\prime},\pi_{2}+\pi_{2}^{\prime})$ is a
compatible pre-Lie algebra, by Theorem 2.14,
$(\pi_{1}+\pi_{1}^{\prime},\pi_{2}+\pi_{2}^{\prime})$ is a Maurer-Cartan
element of the bidifferential graded Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{1}=0,\mathrm{d}_{2}=0)$.
Thus, we have

(7) $\displaystyle[\pi_{1}+\pi_{1}^{\prime},\pi_{1}+\pi_{1}^{\prime}]^{MN}$
$\displaystyle=$ $\displaystyle 0,$ (8)
$\displaystyle[\pi_{2}+\pi_{2}^{\prime},\pi_{2}+\pi_{2}^{\prime}]^{MN}$
$\displaystyle=$ $\displaystyle 0,$ (9)
$\displaystyle[\pi_{1}+\pi_{1}^{\prime},\pi_{2}+\pi_{2}^{\prime}]^{MN}$
$\displaystyle=$ $\displaystyle 0,$

By (7), (8) and (9), we have

$\displaystyle\mathrm{d}_{\pi_{1}}\pi_{1}^{\prime}+\frac{1}{2}[\pi_{1}^{\prime},\pi_{1}^{\prime}]^{MN}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\mathrm{d}_{\pi_{2}}\pi_{2}^{\prime}+\frac{1}{2}[\pi_{2}^{\prime},\pi_{2}^{\prime}]^{MN}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\mathrm{d}_{\pi_{1}}\pi_{2}^{\prime}+\mathrm{d}_{\pi_{2}}\pi_{1}^{\prime}+[\pi_{1}^{\prime},\pi_{2}^{\prime}]^{MN}$
$\displaystyle=$ $\displaystyle 0.$

Thus, $(\pi^{\prime}_{1},\pi^{\prime}_{2})$ is a Maurer-Cartan element of
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{\pi_{1}},\mathrm{d}_{\pi_{2}})$.
The converse part can be proved similarly. We omit details. The proof is
finished. ∎

##  3\. Formal deformations of compatible pre-Lie algebras

In this section, first, we introduce a cohomology of a compatible pre-Lie
algebra with coefficients in itself. Then, we study infinitesimal deformations
of compatible pre-Lie algebras using this cohomology, we show that equivalent
infinitesimal deformations are in the same second cohomology group. We give
the notion of a Nijenhuis operator on a compatible pre-Lie algebra and show
that a Nijenhuis operator gives rise to a trivial deformation. Finally, we
study formal deformations of compatible pre-Lie algebras. If the second
cohomology group $\mathcal{H}^{2}(\mathfrak{g};\mathfrak{g})$ is trivial, then
the compatible pre-Lie algebra is rigid.

###  3.1. Cohomologies of compatible pre-Lie algebras

Let $(\mathfrak{g},\pi)$ be a pre-Lie algebra, where $\pi(x,y)=x\cdot y$.
Because of the graded Jacobi identity, we define a coboundary operator
$\delta_{\pi}:C^{n}(\mathfrak{g},\mathfrak{g})\longrightarrow
C^{n+1}(\mathfrak{g},\mathfrak{g})$ by

$\delta_{\pi}f=(-1)^{n-1}[\pi,f]^{MN},\quad\forall f\in
C^{n}(\mathfrak{g};\mathfrak{g}).$

Thus, we obtain a cochain complex
$(\oplus_{n=1}^{+\infty}C^{n}(\mathfrak{g};\mathfrak{g}),\delta_{\pi})$. More
precisely, for all $x_{1},\dots,x_{n+1}\in\mathfrak{g}$, we have

$\displaystyle(\delta_{\pi}f)(x_{1},\dots,x_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}x_{i}\cdot
f(x_{1},\dots,\hat{x_{i}},\dots,x_{n+1})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i+1}f(x_{1},\dots,\hat{x_{i}},\dots,x_{n},x_{i})\cdot
x_{n+1}$
$\displaystyle-\sum_{i=1}^{n}(-1)^{i+1}f(x_{1},\dots,\hat{x_{i}}\dots,x_{n},x_{i}\cdot
x_{n+1})$ $\displaystyle+\sum_{1\leq i<j\leq
n}(-1)^{i+j}f([x_{i},x_{j}]_{C},x_{1},\dots,\hat{x_{i}},\dots,\hat{x_{j}},\dots,x_{n+1}),$

which is a coboundary operator of pre-Lie algebra $(\mathfrak{g},\pi)$ with
coefficients in the regular representation $(V,L,R)$. We can see more details
in [8].

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra with
$\pi_{1}(x,y)=x\cdot y$ , $\pi_{2}(x,y)=x\ast y$. We define the set of
$n$-cochains $(n\geq 1)$ by

$\mathfrak{C}^{n}(\mathfrak{g};\mathfrak{g})=\underbrace{C^{n}(\mathfrak{g};\mathfrak{g})\oplus
C^{n}(\mathfrak{g};\mathfrak{g})\oplus\cdots\oplus
C^{n}(\mathfrak{g};\mathfrak{g})}_{n~{}copies}.$

Define the operator
$\delta:\mathfrak{C}^{n}(\mathfrak{g};\mathfrak{g})\longrightarrow\mathfrak{C}^{n+1}(\mathfrak{g};\mathfrak{g})$
by

$\displaystyle\delta^{1}f$ $\displaystyle=$
$\displaystyle(\delta_{\pi_{1}}f,\delta_{\pi_{2}}f),\quad\forall
f\in\mathrm{Hom}(\mathfrak{g},\mathfrak{g}),n=1,$
$\displaystyle\delta^{n}(f_{1},\dots,f_{n})$ $\displaystyle=$
$\displaystyle(\delta_{\pi_{1}}f_{1},\dots,\underbrace{\delta_{\pi_{2}}f_{i-1}+\delta_{\pi_{1}}f_{i}}_{i},\dots,\delta_{\pi_{2}}f_{n}),\quad\forall(f_{1},\dots,f_{n})\in\mathfrak{C}^{n}(\mathfrak{g},\mathfrak{g}),2\leq
i\leq n.$

######  Theorem 3.1.

The operator
$\delta:\mathfrak{C}^{n}(\mathfrak{g};\mathfrak{g})\longrightarrow\mathfrak{C}^{n+1}(\mathfrak{g};\mathfrak{g})$
defined as above satisfies $\delta\circ\delta=0$.

###### Proof.

By Theorem 2.14, we obtain that $(\pi_{1},\pi_{2})$ is a Maurer-Cartan element
of the bidifferential graded Lie algebra
$(C^{*}(\mathfrak{g};\mathfrak{g}),[\cdot,\cdot]^{MN},\mathrm{d}_{1}=0,\mathrm{d}_{2}=0)$.
Thus, by the fact that
$[\pi_{1},\pi_{1}]^{MN}=[\pi_{2},\pi_{2}]^{MN}=[\pi_{1},\pi_{2}]^{MN}=0$ and
the graded Jacobi identity, for all
$f\in\mathrm{Hom}(\mathfrak{g},\mathfrak{g})$, we have

$\displaystyle\delta^{2}(\delta^{1}f)$ $\displaystyle=$
$\displaystyle\delta^{2}([\pi_{1},f]^{MN},[\pi_{2},f]^{MN})$ $\displaystyle=$
$\displaystyle-([\pi_{1},[\pi_{1},f]^{MN}]^{MN},[\pi_{2},[\pi_{1},f]^{MN}]^{MN}+[\pi_{1},[\pi_{2},f]^{MN}]^{MN},[\pi_{2},[\pi_{2},f]^{MN}]^{MN})$
$\displaystyle=$
$\displaystyle-(\frac{1}{2}[[\pi_{1},\pi_{1}]^{MN},f]^{MN},[[\pi_{1},\pi_{2}]^{MN},f]^{MN},\frac{1}{2}[[\pi_{2},\pi_{2}]^{MN},f]^{MN})$
$\displaystyle=$ $\displaystyle(0,0,0).$

By $[\pi_{1},\pi_{1}]^{MN}=[\pi_{2},\pi_{2}]^{MN}=[\pi_{1},\pi_{2}]^{MN}=0$
and the graded Jacobi identity, for all
$(f_{1},\dots,f_{n})\in\mathfrak{C}^{n}(\mathfrak{g},\mathfrak{g}),2\leq i\leq
n$, we have

$\displaystyle\delta^{n+1}\delta^{n}(f_{1},\dots,f_{n})$ $\displaystyle=$
$\displaystyle(-1)^{n-1}\delta^{n+1}([\pi_{1},f_{1}]^{MN},\dots,\underbrace{[\pi_{2},f_{i-1}]^{MN}+[\pi_{1},f_{i}]^{MN}}_{i},\dots,[\pi_{2},f_{n}]^{MN})$
$\displaystyle=$
$\displaystyle-([\pi_{1},[\pi_{1},f_{1}]^{MN}]^{MN},[\pi_{2},[\pi_{1},f_{1}]^{MN}]^{MN}+[\pi_{1},[\pi_{2},f_{1}]^{MN}]^{MN}+[\pi_{1},[\pi_{1},f_{2}]^{MN}]^{MN},\dots,$
$\displaystyle\underbrace{[\pi_{2},[\pi_{2},f_{i-2}]^{MN}]^{MN}+[\pi_{2},[\pi_{1},f_{i-1}]^{MN}]^{MN}+[\pi_{1},[\pi_{2},f_{i-1}]^{MN}]^{MN}+[\pi_{1},[\pi_{1},f_{i}]^{MN}]^{MN}}_{3\leq
i\leq n-1},\dots,$
$\displaystyle[\pi_{2},[\pi_{2},f_{n-1}]^{MN}]^{MN}+[\pi_{2},[\pi_{1},f_{n}]^{MN}]^{MN}+[\pi_{1},[\pi_{2},f_{n}]^{MN}]^{MN},[\pi_{2},[\pi_{2},f_{n}]^{MN}]^{MN})$
$\displaystyle=$
$\displaystyle-(\frac{1}{2}[[\pi_{1},\pi_{1}]^{MN},f_{1}]^{MN},[[\pi_{1},\pi_{2}]^{MN},f_{1}]^{MN}+\frac{1}{2}[[\pi_{1},\pi_{1}]^{MN},f_{2}]^{MN},\dots,$
$\displaystyle\underbrace{\frac{1}{2}[[\pi_{2},\pi_{2}]^{MN},f_{i-2}]^{MN}+[[\pi_{1},\pi_{2}]^{MN},f_{i-1}]^{MN}+\frac{1}{2}[[\pi_{1},\pi_{1}]^{MN},f_{i}]^{MN}}_{3\leq
i\leq n-1},\dots,$
$\displaystyle\frac{1}{2}[[\pi_{2},\pi_{2}]^{MN},f_{n-1}]^{MN}+[[\pi_{1},\pi_{2}]^{MN},f_{n}]^{MN},\frac{1}{2}[[\pi_{2},\pi_{2}]^{MN},f_{n}]^{MN})$
$\displaystyle=$ $\displaystyle(0,0,\dots,0).$

Thus, we have $\delta\circ\delta=0$. ∎

######  Definition 3.2.

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra. The
cohomology of the cochain complex
$(\oplus_{n=1}^{+\infty}\mathfrak{C}^{n}(\mathfrak{g};\mathfrak{g}),\delta)$
is called the cohomology of $(\mathfrak{g},\cdot,\ast)$. The corresponding
$n$-th cohomology group is denoted by
$\mathcal{H}^{n}(\mathfrak{g};\mathfrak{g})$.

###  3.2. Infinitesimal deformations of compatible pre-Lie algebras

######  Definition 3.3.

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra and
$(\omega_{1},\omega_{2})\in\mathfrak{C}^{2}(\mathfrak{g},\mathfrak{g})$.
Define

$x\cdot_{t}y=x\cdot y+t\omega_{1}(x,y),\quad x\ast_{t}y=x\ast
y+t\omega_{2}(x,y),\quad\forall x,y\in\mathfrak{g}.$

If for all $t\in\mathbb{K}$, $(\mathfrak{g},\cdot_{t},\ast_{t})$ is still a
compatible pre-Lie algebra, then we say that $(\omega_{1},\omega_{2})$
generates an infinitesimal deformation of $(\mathfrak{g},\cdot,\ast)$.

It is straightforward to verify that $(\omega_{1},\omega_{2})$ generates an
infinitesimal deformation of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ if and only if for all $k_{1},k_{2}\in\mathbb{K}$,
$k_{1}\omega_{1}+k_{2}\omega_{2}$ generates an infinitesimal deformation of
the pre-Lie algebra $(\mathfrak{g},\diamond)$, where $``\diamond$” is given by
(2).

By Theorem 2.14, $(\omega_{1},\omega_{2})$ generates an infinitesimal
deformation of a compatible pre-Lie algebra $(\mathfrak{g},\pi_{1},\pi_{2})$,
where $x\cdot y=\pi_{1}(x,y)$ and $x\ast y=\pi_{2}(x,y)$ if and only if

(10)
$\displaystyle[\pi_{1},\omega_{1}]^{MN}=0,\quad[\pi_{2},\omega_{2}]^{MN}=0,\quad[\pi_{1},\omega_{2}]^{MN}+[\pi_{2},\omega_{1}]^{MN}=0,$
(11)
$\displaystyle[\omega_{1},\omega_{1}]^{MN}=0,\quad[\omega_{2},\omega_{2}]^{MN}=0,\quad[\omega_{1},\omega_{2}]^{MN}=0.$

Obviously, (10) means that $(\omega_{1},\omega_{2})$ is a $2$-cocycle of the
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$, i.e.
$\delta(\omega_{1},\omega_{2})=0$. (11) means that
$(\mathfrak{g},\omega_{1},\omega_{2})$ is a compatible pre-Lie algebra.

######  Theorem 3.4.

With the above notation, $(\omega_{1},\omega_{2})$ is a $2$-cocycle of the
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$.

######  Definition 3.5.

Two infinitesimal deformations $(\mathfrak{g},\cdot_{t},\ast_{t})$ and
$(\mathfrak{g}^{\prime},\cdot^{\prime}_{t},\ast^{\prime}_{t})$ generated by
$(\omega_{1},\omega_{2})$ and $(\omega^{\prime}_{1},\omega^{\prime}_{2})$
respectively are said to be equivalent if there exists a linear operator
$N\in\mathfrak{gl}(\mathfrak{g})$ such that $\rm{Id}+tN$ is a compatible pre-
Lie algebra homomorphism from
$(\mathfrak{g}^{\prime},\cdot^{\prime}_{t},\ast^{\prime}_{t})$ to
$(\mathfrak{g},\cdot_{t},\ast_{t})$.

Two infinitesimal deformations $(\mathfrak{g},\cdot_{t},\ast_{t})$ and
$(\mathfrak{g}^{\prime},\cdot^{\prime}_{t},\ast^{\prime}_{t})$ generated by
$(\omega_{1},\omega_{2})$ and $(\omega^{\prime}_{1},\omega^{\prime}_{2})$
respectively are equivalent if and only if for all $x,y\in\mathfrak{g}$, the
following equalities hold:

(12) $\displaystyle\omega^{\prime}_{1}(x,y)-\omega_{1}(x,y)$ $\displaystyle=$
$\displaystyle N(x)\cdot y+x\cdot N(y)-N(x\cdot y),$ (13) $\displaystyle
N(\omega^{\prime}_{1}(x,y))$ $\displaystyle=$
$\displaystyle\omega_{1}(x,N(y))+\omega_{1}(N(x),y)+N(x)\cdot N(y),$ (14)
$\displaystyle\omega_{1}(N(x),N(y))$ $\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle\omega^{\prime}_{2}(x,y)-\omega_{2}(x,y)$ $\displaystyle=$
$\displaystyle N(x)\ast y+x\ast N(y)-N(x\ast y),$ (16) $\displaystyle
N(\omega^{\prime}_{2}(x,y))$ $\displaystyle=$
$\displaystyle\omega_{2}(x,N(y))+\omega_{2}(N(x),y)+N(x)\ast N(y),$ (17)
$\displaystyle\omega_{2}(N(x),N(y))$ $\displaystyle=$ $\displaystyle 0.$

Note that (12) and (15) means that
$(\omega^{\prime}_{1}-\omega_{1},\omega^{\prime}_{2}-\omega_{2})=\delta
N=([\pi_{1},N]^{MN},[\pi_{2},N]^{MN})$. Thus, we have

######  Theorem 3.6.

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra. If two
infinitesimal deformations $(\mathfrak{g},\cdot_{t},\ast_{t})$ and
$(\mathfrak{g}^{\prime},\cdot^{\prime}_{t},\ast^{\prime}_{t})$ generated by
$(\omega_{1},\omega_{2})$ and $(\omega^{\prime}_{1},\omega^{\prime}_{2})$
respectively are equivalent, then $(\omega_{1},\omega_{2})$ and
$(\omega^{\prime}_{1},\omega^{\prime}_{2})$ are in the same cohomology class
of $\mathcal{H}^{2}(\mathfrak{g};\mathfrak{g}).$

######  Definition 3.7.

([23]) Let $(\mathfrak{g},\cdot)$ be a pre-Lie algebra. A linear operator
$N\in\mathfrak{gl}(\mathfrak{g})$ is called a Nijenhuis operator on
$(\mathfrak{g},\cdot)$ if for all $x,y\in\mathfrak{g}$

$N(x)\cdot N(y)=N(x\cdot_{N}y),$

where the product $``\cdot_{N}$” is defined by

$x\cdot_{N}y\triangleq N(x)\cdot y+x\cdot N(y)-N(x\cdot y).$

######  Proposition 3.8.

([23]) Let $N$ be a Nijenhuis operator on a pre-Lie algebra
$(\mathfrak{g},\cdot)$, then $(\mathfrak{g},\cdot_{N})$ is a pre-Lie algebra,
and $N$ is a homomorphism from $(\mathfrak{g},\cdot_{N})$ to
$(\mathfrak{g},\cdot)$.

######  Definition 3.9.

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra. A linear
operator $N\in\mathfrak{gl}(\mathfrak{g})$ is called a Nijenhuis operator on
$(\mathfrak{g},\cdot,\ast)$ if $N$ is both a Nijenhuis operator on the pre-Lie
algebra $(\mathfrak{g},\cdot)$ and a Nijenhuis operator on the pre-Lie algebra
$(\mathfrak{g},\ast)$.

######  Proposition 3.10.

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra and
$N\in\mathfrak{gl}(\mathfrak{g})$ a linear map. Then $N$ is a Nijenhuis
operator on the compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$ if and
only if $N$ is a Nijenhuis operator on the pre-Lie algebra
$(\mathfrak{g},\diamond)$, where $``\diamond$” is given by (2).

###### Proof.

If $N$ is a Nijenhuis operator on the compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$, for all $x,y\in\mathfrak{g}$, we have

$\displaystyle N(x)\diamond N(y)-N(N(x)\diamond y+x\diamond N(y)-N(x\diamond
y))$ $\displaystyle=$ $\displaystyle k_{1}N(x)\cdot N(y)+k_{2}N(x)\ast
N(y)-N(k_{1}N(x)\cdot y+k_{2}N(x)\ast y$ $\displaystyle+k_{1}x\cdot
N(y)+k_{2}x\ast N(y)-N(k_{1}x\cdot y+k_{2}x\ast y))$ $\displaystyle=$
$\displaystyle k_{1}(N(x)\cdot N(y)-N(N(x)\cdot y+x\cdot N(y)-N(x\cdot y)))$
$\displaystyle+k_{2}(N(x)\ast N(y)-N(N(x)\ast y+x\ast N(y)-N(x\ast y)))$
$\displaystyle=$ $\displaystyle 0,$

which implies that $N$ is a Nijenhuis operator on the pre-Lie algebra
$(\mathfrak{g},\diamond)$. The converse part can be proved similarly. We omit
details. The proof is finished. ∎

######  Proposition 3.11.

Let $N\in\mathfrak{gl}(\mathfrak{g})$ be a Nijenhuis operator on the
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$. Then
$(\mathfrak{g},\cdot_{N},\ast_{N})$ is a compatible pre-Lie algebra and $N$ is
a homomorphism from $(\mathfrak{g},\cdot_{N},\ast_{N})$ to
$(\mathfrak{g},\cdot,\ast)$.

###### Proof.

By Proposition 3.10, $N$ is a Nijenhuis operator on the pre-Lie algebra
$(\mathfrak{g},\diamond)$. For all $x,y\in\mathfrak{g}$, we have

$\displaystyle x\diamond_{N}y$ $\displaystyle=$ $\displaystyle N(x)\diamond
y+x\diamond N(y)-N(x\diamond y)$ $\displaystyle=$ $\displaystyle
k_{1}N(x)\cdot y+k_{2}N(x)\ast y+k_{1}x\cdot N(y)+k_{2}x\ast
N(y)-k_{1}N(x\cdot y)-k_{2}N(x\ast y)$ $\displaystyle=$ $\displaystyle
k_{1}(x\cdot_{N}y)+k_{2}(x\ast_{N}y).$

By Proposition 3.8, $(\mathfrak{g},\diamond_{N})$ is a pre-Lie algebra. By
Proposition 2.4, $(\mathfrak{g},\cdot_{N},\ast_{N})$ is a compatible pre-Lie
algebra. By Proposition 3.8, $N$ is a homomorphism from
$(\mathfrak{g},\cdot_{N})$ to $(\mathfrak{g},\cdot)$ and a homomorphism from
$(\mathfrak{g},\ast_{N})$ to $(\mathfrak{g},\ast)$. Thus, $N$ is a
homomorphism from $(\mathfrak{g},\cdot_{N},\ast_{N})$ to
$(\mathfrak{g},\cdot,\ast)$. ∎

######  Definition 3.12.

An infinitesimal deformation $(\mathfrak{g},\cdot_{t},\ast_{t})$ of a
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$ generated by
$(\omega_{1},\omega_{2})$ is trivial if there exists a linear operator
$N\in\mathfrak{gl}(\mathfrak{g})$ such that $\rm{Id}+tN$ is a compatible pre-
Lie algebra homomorphism from $(\mathfrak{g},\cdot_{t},\ast_{t})$ to
$(\mathfrak{g},\cdot,\ast)$.

$(\mathfrak{g},\cdot_{t},\ast_{t})$ is a trivial infinitesimal deformation if
and only if for all $x,y\in\mathfrak{g}$, the following equalities hold:

(18) $\displaystyle\omega_{1}(x,y)$ $\displaystyle=$ $\displaystyle N(x)\cdot
y+x\cdot N(y)-N(x\cdot y),$ (19) $\displaystyle N(\omega_{1}(x,y))$
$\displaystyle=$ $\displaystyle N(x)\cdot N(y),$ (20)
$\displaystyle\omega_{2}(x,y)$ $\displaystyle=$ $\displaystyle N(x)\ast
y+x\ast N(y)-N(x\ast y),$ (21) $\displaystyle N(\omega_{2}(x,y))$
$\displaystyle=$ $\displaystyle N(x)\ast N(y).$

By (18) and (19), we obtain that $N$ is a Nijenhuis operator on the pre-Lie
algebra $(\mathfrak{g},\cdot)$. By (20) and (21), we obtain that $N$ is a
Nijenhuis operator on the pre-Lie algebra $(\mathfrak{g},\ast)$. Thus, by
(18), (19), (20) and (21), we obtain that a trivial infinitesimal deformation
of a compatible pre-Lie algebra gives rise to a Nijenhuis operator $N$ on the
compatible pre-Lie algebra. Conversely, a Nijenhuis operator can also generate
a trivial infinitesimal deformation as the following theorem shows.

######  Theorem 3.13.

Let N be a Nijenhuis operator on a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$. Then a infinitesimal deformation
$(\mathfrak{g},\cdot_{t},\ast_{t})$ of the compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ can be obtained by putting

(22) $\displaystyle\omega_{1}(x,y)$ $\displaystyle=$ $\displaystyle N(x)\cdot
y+x\cdot N(y)-N(x\cdot y),$ (23) $\displaystyle\omega_{2}(x,y)$
$\displaystyle=$ $\displaystyle N(x)\ast y+x\ast N(y)-N(x\ast y).$

Furthermore, this infinitesimal deformation
$(\mathfrak{g},\cdot_{t},\ast_{t})$ is trivial.

###### Proof.

By (22) and (23), we obtain that $(\omega_{1},\omega_{2})=\delta N$. Thus,
$(\omega_{1},\omega_{2})$ is a $2$-cocycle of the compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$. Since N is a Nijenhuis operator on a compatible
pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$, by Proposition 3.11,
$(\mathfrak{g},\omega_{1},\omega_{2})$ is a compatible pre-Lie algebra. Thus,
$(\mathfrak{g},\cdot_{t},\ast_{t})$ is a infinitesimal deformation of
$(\mathfrak{g},\cdot,\ast)$. It is straightforward to deduce that $\rm{Id}+tN$
is a compatible pre-Lie algebra homomorphism from
$(\mathfrak{g},\cdot_{t},\ast_{t})$ to $(\mathfrak{g},\cdot,\ast)$. Thus, this
infinitesimal deformation is trivial. ∎

###  3.3. Formal deformations of compatible pre-Lie algebras

######  Definition 3.14.

Let $(\mathfrak{g},\pi_{1},\pi_{2})$ be a compatible pre-Lie algebra,
$\pi_{1}^{t}=\pi_{1}+\sum_{i=1}^{+\infty}\pi_{1}^{i}t^{i},\pi_{2}^{t}=\pi_{2}+\sum_{i=1}^{+\infty}\pi_{2}^{i}t^{i}:\mathfrak{g}[[t]]\otimes\mathfrak{g}[[t]]\longrightarrow\mathfrak{g}[[t]]$
be $\mathbb{K}[[t]]$-bilinear maps, where
$\pi_{1}^{i},\pi_{2}^{i}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$
are linear maps. If $(\mathfrak{g}[[t]],\pi_{1}^{t},\pi_{2}^{t})$ is still a
compatible pre-Lie algebra, we say that $\\{\pi_{1}^{i},\pi_{2}^{i}\\}_{i\geq
1}$ generates a $1$-parameter formal deformation of a compatible pre-Lie
algebra $(\mathfrak{g},\pi_{1},\pi_{2})$.

If $\\{\pi_{1}^{i},\pi_{2}^{i}\\}_{i\geq 1}$ generates a $1$-parameter formal
deformation of a compatible pre-Lie algebra $(\mathfrak{g},\pi_{1},\pi_{2})$,
for all $x,y,z\in\mathfrak{g}$ and $n=1,2,\dots$, we have

$\sum_{i+j=n\atop i,j\geq
0}\pi_{1}^{i}(\pi_{1}^{j}(x,y),z)-\pi_{1}^{i}(x,\pi_{1}^{j}(y,z))-\pi_{1}^{i}(\pi_{1}^{j}(y,x),z)+\pi_{1}^{i}(y,\pi_{1}^{j}(x,z))=0.$

Moreover, we have

(24) $\displaystyle\sum_{i+j=n\atop 0<i,j\leq
n-1}\pi_{1}^{i}(\pi_{1}^{j}(x,y),z)-\pi_{1}^{i}(x,\pi_{1}^{j}(y,z))-\pi_{1}^{i}(\pi_{1}^{j}(y,x),z)+\pi_{1}^{i}(y,\pi_{1}^{j}(x,z))=-[\pi_{1},\pi_{1}^{n}]^{MN}(x,y,z).$

Similarly, we have

(25) $\displaystyle\sum_{i+j=n\atop 0<i,j\leq
n-1}\pi_{2}^{i}(\pi_{2}^{j}(x,y),z)-\pi_{2}^{i}(x,\pi_{2}^{j}(y,z))-\pi_{2}^{i}(\pi_{2}^{j}(y,x),z)+\pi_{2}^{i}(y,\pi_{2}^{j}(x,z))=-[\pi_{2},\pi_{2}^{n}]^{MN}(x,y,z).$

For all $x,y,z\in\mathfrak{g}$ and $n=1,2,\dots$, we have

$\displaystyle\sum_{i+j=n\atop i,j\geq
0}\pi_{1}^{i}(\pi_{2}^{j}(x,y),z)+\pi_{2}^{j}(\pi_{1}^{i}(x,y),z)-\pi_{1}^{i}(x,\pi_{2}^{j}(y,z))-\pi_{2}^{j}(x,\pi_{1}^{i}(y,z))$
$\displaystyle-\pi_{1}^{i}(\pi_{2}^{j}(y,x),z)-\pi_{2}^{j}(\pi_{1}^{i}(y,x),z)+\pi_{1}^{i}(y,\pi_{2}^{j}(x,z))+\pi_{2}^{j}(y,\pi_{1}^{i}(x,z))=0.$

Moreover, we have

$\displaystyle\sum_{i+j=n\atop 0<i,j\leq
n-1}\pi_{1}^{i}(\pi_{2}^{j}(x,y),z)+\pi_{2}^{j}(\pi_{1}^{i}(x,y),z)-\pi_{1}^{i}(x,\pi_{2}^{j}(y,z))-\pi_{2}^{j}(x,\pi_{1}^{i}(y,z))$
$\displaystyle-\pi_{1}^{i}(\pi_{2}^{j}(y,x),z)-\pi_{2}^{j}(\pi_{1}^{i}(y,x),z)+\pi_{1}^{i}(y,\pi_{2}^{j}(x,z))+\pi_{2}^{j}(y,\pi_{1}^{i}(x,z))$
$\displaystyle=$
$\displaystyle-([\pi_{1},\pi_{2}^{n}]^{MN}+[\pi_{2},\pi_{1}^{n}]^{MN})(x,y,z).$

######  Definition 3.15.

Let
$({\pi_{1}^{t}}^{\prime}=\pi_{1}+\sum_{i=1}^{+\infty}{\pi_{1}^{i}}^{\prime}t^{i},{\pi_{2}^{t}}^{\prime}=\pi_{2}+\sum_{i=1}^{+\infty}{\pi_{2}^{i}}^{\prime}t^{i})$
and
$(\pi_{1}^{t}=\pi_{1}+\sum_{i=1}^{+\infty}\pi_{1}^{i}t^{i},\pi_{2}^{t}=\pi_{2}+\sum_{i=1}^{+\infty}\pi_{2}^{i}t^{i})$
be two $1$-parameter formal deformations of a compatible pre-Lie algebra
$(\mathfrak{g},\pi_{1},\pi_{2})$. A formal isomorphism from
$(\mathfrak{g}[[t]],{\pi_{1}^{t}}^{\prime},{\pi_{2}^{t}}^{\prime})$ to
$(\mathfrak{g}[[t]],\pi_{1}^{t},\pi_{2}^{t})$ is a power series
$\Phi_{t}=\sum_{i=0}^{+\infty}\varphi_{i}t^{i}$, where
$\varphi_{i}:A\longrightarrow A$ are linear maps with $\varphi_{0}={\rm{Id}}$,
such that

$\displaystyle\Phi_{t}\circ{\pi_{1}^{t}}^{\prime}$ $\displaystyle=$
$\displaystyle\pi_{1}^{t}\circ(\Phi_{t}\times\Phi_{t}),$
$\displaystyle\Phi_{t}\circ{\pi_{2}^{t}}^{\prime}$ $\displaystyle=$
$\displaystyle\pi_{2}^{t}\circ(\Phi_{t}\times\Phi_{t}).$

Two $1$-parameter formal deformations
$(\mathfrak{g}[[t]],{\pi_{1}^{t}}^{\prime},{\pi_{2}^{t}}^{\prime})$ and
$(\mathfrak{g}[[t]],\pi_{1},\pi_{2})$ are said to be equivalent if there
exists a formal isomorphism $\Phi_{t}=\sum_{i=0}^{+\infty}\varphi_{i}t^{i}$
from $(\mathfrak{g}[[t]],{\pi_{1}^{t}}^{\prime},{\pi_{2}^{t}}^{\prime})$ to
$(\mathfrak{g}[[t]],\pi_{1},\pi_{2})$.

######  Definition 3.16.

A $1$-parameter formal deformation
$(\mathfrak{g}[[t]],\pi_{1}^{t},\pi_{2}^{t})$ of a compatible pre-Lie algebra
$(\mathfrak{g},\pi_{1},\pi_{2})$ is said to be trivial if it is equivalent to
$(\mathfrak{g},\pi_{1},\pi_{2})$, i.e. there exists
$\Phi_{t}=\sum_{i=0}^{+\infty}\varphi_{i}t^{i}$, where
$\varphi_{i}:A\longrightarrow A$ are linear maps with $\varphi_{0}={\rm{Id}}$,
such that

$\displaystyle\Phi_{t}\circ\pi_{1}^{t}$ $\displaystyle=$
$\displaystyle\pi_{1}\circ(\Phi_{t}\times\Phi_{t}),$
$\displaystyle\Phi_{t}\circ\pi_{2}^{t}$ $\displaystyle=$
$\displaystyle\pi_{2}\circ(\Phi_{t}\times\Phi_{t}).$

######  Definition 3.17.

Let $(\mathfrak{g},\pi_{1},\pi_{2})$ be a compatible pre-Lie algebra. If all
$1$-parameter formal deformations are trivial, we say that
$(\mathfrak{g},\pi_{1},\pi_{2})$ is rigid.

######  Theorem 3.18.

Let $(\mathfrak{g},\pi_{1},\pi_{2})$ be a compatible pre-Lie algebra. If
$\mathcal{H}^{2}(\mathfrak{g};\mathfrak{g})=0$, then
$(\mathfrak{g},\pi_{1},\pi_{2})$ is rigid.

###### Proof.

Let
$(\pi_{1}^{t}=\pi_{1}+\sum_{i=1}^{+\infty}\pi_{1}^{i}t^{i},\pi_{2}^{t}=\pi_{2}+\sum_{i=1}^{+\infty}\pi_{2}^{i}t^{i})$
be a $1$-parameter formal deformation and assume that $n\geq 1$ is the minimal
number such that $(\pi_{1}^{n},\pi_{2}^{n})$ is not zero. By (24), (25), (3.3)
and $\mathcal{H}^{2}(\mathfrak{g};\mathfrak{g})=0$, we have
$(\pi_{1}^{n},\pi_{2}^{n})\in B^{2}(A;A)$. Thus, there exists
$\varphi_{n}\in\mathfrak{C}^{1}(\mathfrak{g};\mathfrak{g})$ such that
$(\pi_{1}^{n},\pi_{2}^{n})=\delta(-\varphi_{n})=(\delta_{\pi_{1}}(-\varphi_{n}),\delta_{\pi_{2}}(-\varphi_{n}))$.
Let $\Phi_{t}={\rm{Id}}+\varphi_{n}t^{n}$ and define a new formal deformation
$({\pi_{1}^{t}}^{\prime},{\pi_{2}^{t}}^{\prime})$ by
${\pi_{1}^{t}}^{\prime}(x,y)=\Phi_{t}^{-1}\circ\pi_{1}^{t}(\Phi_{t}(x),\Phi_{t}(y))$,
${\pi_{2}^{t}}^{\prime}(x,y)=\Phi_{t}^{-1}\circ\pi_{2}^{t}(\Phi_{t}(x),\Phi_{t}(y))$.
Then $({\pi_{1}^{t}}^{\prime},{\pi_{2}^{t}}^{\prime})$ and
$(\pi_{1}^{t},\pi_{2}^{t})$ are equivalent. By straightforward computation,
for all $x,y\in\mathfrak{g}$, we have

$\displaystyle{\pi_{1}^{t}}^{\prime}(x,y)$ $\displaystyle=$
$\displaystyle\Phi_{t}^{-1}\circ\pi_{1}^{t}(\Phi_{t}(x),\Phi_{t}(y))$
$\displaystyle=$
$\displaystyle({\rm{Id}}-\varphi_{n}t^{n}+\dots)\pi_{1}^{t}\big{(}x+\varphi_{n}(x)t^{n},y+\varphi_{n}(y)t^{n}\big{)}$
$\displaystyle=$ $\displaystyle({\rm{Id}}-\varphi_{n}t^{n}+\dots)\Big{(}x\cdot
y+\big{(}x\cdot\varphi_{n}(y)+\varphi_{n}(x)\cdot
y+\pi_{1}^{n}(x,y)\big{)}t^{n}+\dots\Big{)}$ $\displaystyle=$ $\displaystyle
x\cdot y+\Big{(}x\cdot\varphi_{n}(y)+\varphi_{n}(x)\cdot
y+\pi_{1}^{n}(x,y)-\varphi_{n}(x\cdot y)\Big{)}t^{n}+\dots.$

Thus, we have
${\pi_{1}^{1}}^{\prime}={\pi_{1}^{2}}^{\prime}=\dots={\pi_{1}^{n-1}}^{\prime}=0$.
Moreover, we have

$\displaystyle{\pi_{1}^{n}}^{\prime}(x,y)$ $\displaystyle=$ $\displaystyle
x\cdot\varphi_{n}(y)+\varphi_{n}(x)\cdot y+\pi_{1}^{n}(x,y)-\varphi_{n}(x\cdot
y)$ $\displaystyle=$
$\displaystyle\delta_{\pi_{1}}\varphi_{n}(x,y)+\pi_{1}^{n}(x,y)$
$\displaystyle=$ $\displaystyle 0.$

Similarly, we have ${\pi_{2}^{n}}^{\prime}(x,y)=0$. Keep repeating the
process, we obtain that $(\mathfrak{g}[[t]],\pi_{1}^{t},\pi_{2}^{t})$ is
equivalent to $(\mathfrak{g},\pi_{1},\pi_{2})$. The proof is finished. ∎

##  4\. Abelian extensions of compatible pre-Lie algebras

In this section, first, we give a compatible pre-Lie algebra
$(\mathfrak{g},\pi_{1},\pi_{2})$ and its representation
$(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$. We construct a bidifferential graded
Lie algebra whose Maurer-Cartan elements is
$(\pi_{1}+\rho+\mu,\pi_{2}+\tilde{\rho}+\tilde{\mu})$. Then, we give a
cohomology of a compatible pre-Lie algebra with coefficients in arbitrary
representation. Finally, we study abelian extensions of compatible pre-Lie
algebras using this cohomological approach. We show that abelian extensions
are classified by the second cohomology group.

###  4.1. Cohomologies of compatible pre-Lie algebras with coefficients in
arbitrary representation

Let $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be vector spaces and elements in
$\mathfrak{g}_{1}$ will be denoted by $x,y,x_{i}$ and elements in
$\mathfrak{g}_{2}$ will be denoted by $u,v,v_{i}$. Let
$c:\wedge^{n-1}\mathfrak{g}_{1}\otimes\mathfrak{g}_{1}\longrightarrow\mathfrak{g}_{2}$
be a linear map. We can construct a linear map $\hat{c}\in
C^{n}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
by

$\hat{c}(x_{1}+v_{1},\dots,x_{n}+v_{n}):=c(x_{1},\dots,x_{n}).$

In general, for a given linear map
$f:\wedge^{k-1}\mathfrak{g}_{1}\otimes\wedge^{l}\mathfrak{g}_{2}\otimes\mathfrak{g}_{1}\longrightarrow\mathfrak{g}_{j}$
for $j\in\\{1,2\\}$, we define a linear map $\hat{f}\in
C^{k+l}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
by

$\hat{f}(x_{1}+v_{1},\dots,x_{k+l}+v_{k+l})=\sum_{\sigma\in\mathbb{S}(k-1,l)}\mathrm{sgn}(\sigma)f(x_{\sigma(1)},\dots,x_{\sigma(k-l)},v_{\sigma(k)},\dots,v_{\sigma(k+l-1)},x_{k+l}).$

Similarly, for
$f:\wedge^{k}\mathfrak{g}_{1}\otimes\wedge^{l-1}\mathfrak{g}_{2}\otimes\mathfrak{g}_{2}\longrightarrow\mathfrak{g}_{j}$
for $j\in\\{1,2\\}$, we define a linear map $\hat{f}\in
C^{k+l}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
by

$\hat{f}(x_{1}+v_{1},\dots,x_{k+l}+v_{k+l})=\sum_{\sigma\in\mathbb{S}(k,l-1)}\mathrm{sgn}(\sigma)f(x_{\sigma(1)},\dots,x_{\sigma(k)},v_{\sigma(k+1)},\dots,v_{\sigma(k+l-1)},v_{k+l}).$

We call the linear map $\hat{f}$ a horizontal lift of $f$, or simply a lift.
We define
$\mathcal{G}^{k,l}=\wedge^{k-1}\mathfrak{g}_{1}\otimes\wedge^{l}\mathfrak{g}_{2}\otimes\mathfrak{g}_{1}+\wedge^{k}\mathfrak{g}_{1}\otimes\wedge^{l-1}\mathfrak{g}_{2}\otimes\mathfrak{g}_{2}$.
The vector space
$\wedge^{n-1}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})\otimes(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
is isomorphic to the direct sum of $\mathcal{G}^{k,l},k+l=n$. In the sequel,
we will omit the notation $\hat{\cdot}$.

######  Definition 4.1.

([15]) A linear map
$f\in\mathrm{Hom}(\wedge^{n-1}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})\otimes(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}),\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
has a bidegree $k|l$ if the following four conditions hold:

  * $\rm(i)$

$k+l+1=n$;

  * $\rm(ii)$

If X is an element in $\mathcal{G}^{k+1,l}$, then $f(X)\in\mathfrak{g}_{1}$;

  * $\rm(iii)$

If X is an element in $\mathcal{G}^{k,l+1}$, then $f(X)\in\mathfrak{g}_{2}$;

  * $\rm(iv)$

All the other case, $f(X)=0$.

We denote a linear map $f$ with bidegree $k|l$ by $||f||=k|l$.

We call a linear map $f$ homogeneous if $f$ has a bidegree. We denote the set
of homogeneous linear maps of bidegree $k|l$ by
$C^{k|l}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$.
We have $k+l\geq 0,k,l\geq-1$ because $n\geq 1$ and $k+1,l+1\geq 0$.

By the above lift, we have the following isomorphisms:

$\displaystyle
C^{k|0}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
$\displaystyle\cong$
$\displaystyle\mathrm{Hom}(\wedge^{k}\mathfrak{g}_{1}\otimes\mathfrak{g}_{1},\mathfrak{g}_{1})\oplus\mathrm{Hom}(\wedge^{k}\mathfrak{g}_{1}\otimes\mathfrak{g}_{2},\mathfrak{g}_{2})\oplus\mathrm{Hom}(\wedge^{k-1}\mathfrak{g}_{1}\otimes\mathfrak{g}_{2}\otimes\mathfrak{g}_{1},\mathfrak{g}_{2});$
$\displaystyle
C^{l|-1}(\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\mathfrak{g}_{1}\oplus\mathfrak{g}_{2})$
$\displaystyle\cong$
$\displaystyle\mathrm{Hom}(\wedge^{l-1}\mathfrak{g}_{1}\otimes\mathfrak{g}_{1},\mathfrak{g}_{2}).$

######  Lemma 4.2.

([15]) If $||f||=k_{f}|l_{f}$ and $||g||=k_{g}|l_{g}$, then $[f,g]^{MN}$ has
the bidegree $k_{f}+k_{g}|l_{f}+l_{g}$.

######  Proposition 4.3.

([16]) Let $(V,\rho,\mu)$ be a representation of the pre-Lie algebra
$(\mathfrak{g},\pi)$. Then we have

$[\pi+\rho+\mu,\pi+\rho+\mu]^{MN}=0.$

Let $(V,\rho,\mu)$ be a representation of the pre-Lie algebra
$(\mathfrak{g},\pi)$, where $\pi(x,y)=x\cdot y$. Denote the set of
$n$-cochains by

$C^{n}(\mathfrak{g};V)=\mathrm{Hom}(\wedge^{n-1}\mathfrak{g}\otimes\mathfrak{g},V),\quad
n\geq 1.$

By Proposition 4.3 and graded Jacobi identity, we define a coboundary operator
$\partial_{\pi+\rho+\mu}:C^{n}(\mathfrak{g},V)\longrightarrow
C^{n+1}(\mathfrak{g},V)$ by

$\partial_{\pi+\rho+\mu}f=(-1)^{n-1}[\pi+\rho+\mu,f]^{MN},\quad\forall f\in
C^{n}(\mathfrak{g},V).$

In facet, since $\pi+\rho+\mu\in C^{1|0}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)$ and $f\in C^{n|-1}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)$, by Lemma 4.2, we obtain that
$[\pi+\rho+\mu,f]^{MN}\in C^{n+1|-1}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus
V)$. Thus, $[\pi+\rho+\mu,f]^{MN}\in C^{n+1}(\mathfrak{g},V)$, we obtain a
well-defined cochain complex
$(\oplus_{n=1}^{+\infty}C^{n}(\mathfrak{g};V),\partial_{\pi+\rho+\mu})$. More
precisely, for all $x_{1},\dots,x_{n+1}\in\mathfrak{g}$, we have

$\displaystyle(\partial_{\pi+\rho+\mu}f)(x_{1},\dots,x_{n+1})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}\rho(x_{i})f(x_{1},\dots,\hat{x_{i}},\dots,x_{n+1})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i+1}\mu(x_{n+1})f(x_{1},\dots,\hat{x_{i}},\dots,x_{n},x_{i})$
$\displaystyle-\sum_{i=1}^{n}(-1)^{i+1}f(x_{1},\dots,\hat{x_{i}}\dots,x_{n},x_{i}\cdot
x_{n+1})$ $\displaystyle+\sum_{1\leq i<j\leq
n}(-1)^{i+j}f([x_{i},x_{j}]_{C},x_{1},\dots,\hat{x_{i}},\dots,\hat{x_{j}},\dots,x_{n+1}),$

which is a coboundary operator of pre-Lie algebra $(\mathfrak{g},\pi)$ with
coefficients in the representation $(V,\rho,\mu)$. We can see more details in
[8].

Let $(\mathfrak{g},\cdot,\ast)$ be a compatible pre-Lie algebra with
$\pi_{1}(x,y)=x\cdot y$ , $\pi_{2}(x,y)=x\ast y$ and
$(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ a representation of
$(\mathfrak{g},\cdot,\ast)$.

######  Proposition 4.4.

With the above notation, $(\pi_{1}+\rho+\mu,\pi_{2}+\tilde{\rho}+\tilde{\mu})$
is a Maurer-Cartan element of the bidifferential graded Lie algebra
$(C^{*}(\mathfrak{g}\oplus V;\mathfrak{g}\oplus
V),[\cdot,\cdot]^{MN},\mathrm{d}_{1}=0,\mathrm{d}_{2}=0)$, i.e.

$[\pi_{1}+\rho+\mu,\pi_{1}+\rho+\mu]^{MN}=0,\quad[\pi_{2}+\tilde{\rho}+\tilde{\mu},\pi_{2}+\tilde{\rho}+\tilde{\mu}]^{MN}=0,\quad[\pi_{1}+\rho+\mu,\pi_{2}+\tilde{\rho}+\tilde{\mu}]^{MN}=0.$

###### Proof.

Since $(V,\rho,\mu)$ is a representation of the pre-Lie algebra
$(\mathfrak{g},\cdot)$, by Proposition 4.3, we have

$[\pi_{1}+\rho+\mu,\pi_{1}+\rho+\mu]^{MN}=0.$

Similarly, since $(V,\tilde{\rho},\tilde{\mu})$ is a representation of the
pre-Lie algebra $(\mathfrak{g},\ast)$, by Proposition 4.3, we have

$[\pi_{2}+\tilde{\rho}+\tilde{\mu},\pi_{2}+\tilde{\rho}+\tilde{\mu}]^{MN}=0.$

For all $x,y,z\in\mathfrak{g},u,v,w\in V$, by (1), (3) and (4), we have

$\displaystyle[\pi_{1}+\rho+\mu,\pi_{2}+\tilde{\rho}+\tilde{\mu}]^{MN}(x+u,y+v,z+w)$
$\displaystyle=$
$\displaystyle(\pi_{1}+\rho+\mu)(\pi_{2}(x,y)+\tilde{\rho}(x)v+\tilde{\mu}(y)u,z+w)-(\pi_{1}+\rho+\mu)(\pi_{2}(y,x)+\tilde{\rho}(y)u+\tilde{\mu}(x)v,z+w)$
$\displaystyle-(\pi_{1}+\rho+\mu)(x+u,\pi_{2}(y,z)+\tilde{\rho}(y)w+\tilde{\mu}(z)v)+(\pi_{1}+\rho+\mu)(y+v,\pi_{2}(x,z)+\tilde{\rho}(x)w+\tilde{\mu}(z)u)$
$\displaystyle+(\pi_{2}+\tilde{\rho}+\tilde{\mu})(\pi_{1}(x,y)+\rho(x)v+\mu(y)u,z+w)-(\pi_{2}+\tilde{\rho}+\tilde{\mu})(\pi_{1}(y,x)+\rho(y)u+\mu(x)v,z+w)$
$\displaystyle-(\pi_{2}+\tilde{\rho}+\tilde{\mu})(x+u,\pi_{1}(y,z)+\rho(y)w+\mu(z)v)+(\pi_{2}+\tilde{\rho}+\tilde{\mu})(y+v,\pi_{1}(x,z)+\rho(x)w+\mu(z)u)$
$\displaystyle=$
$\displaystyle\pi_{1}(\pi_{2}(x,y),z)+\rho(\pi_{2}(x,y))w+\mu(z)\tilde{\rho}(x)v+\mu(z)\tilde{\mu}(y)u$
$\displaystyle-\pi_{1}(\pi_{2}(y,x),z)-\rho(\pi_{2}(y,x))w-\mu(z)\tilde{\rho}(y)u-\mu(z)\tilde{\mu}(x)v$
$\displaystyle-\pi_{1}(x,\pi_{2}(y,z))-\rho(x)\tilde{\rho}(y)w-\rho(x)\tilde{\mu}(z)v-\mu(\pi_{2}(y,z))u$
$\displaystyle+\pi_{1}(y,\pi_{2}(x,z))+\rho(y)\tilde{\rho}(x)w+\rho(y)\tilde{\mu}(z)u+\mu(\pi_{2}(x,z))v$
$\displaystyle+\pi_{2}(\pi_{1}(x,y),z)+\tilde{\rho}(\pi_{1}(x,y))w+\tilde{\mu}(z)\rho(x)v+\tilde{\mu}(z)\mu(y)u$
$\displaystyle-\pi_{2}(\pi_{1}(y,x),z)-\tilde{\rho}(\pi_{1}(y,x))w-\tilde{\mu}(z)\rho(y)u-\tilde{\mu}(z)\mu(x)v$
$\displaystyle-\pi_{2}(x,\pi_{1}(y,z))-\tilde{\rho}(x)\rho(y)w-\tilde{\rho}(x)\mu(z)v-\tilde{\mu}(\pi_{1}(y,z))u$
$\displaystyle+\pi_{2}(y,\pi_{1}(x,z))+\tilde{\rho}(y)\rho(x)w+\tilde{\rho}(y)\mu(z)u+\tilde{\mu}(\pi_{1}(x,z))v$
$\displaystyle=$ $\displaystyle 0,$

which implies that

$[\pi_{1}+\rho+\mu,\pi_{2}+\tilde{\rho}+\tilde{\mu}]^{MN}=0.$

This finishes the proof. ∎

We define the set of $n$-cochains $(n\geq 1)$ by

$\mathfrak{C}^{n}(\mathfrak{g};V)=\underbrace{C^{n}(\mathfrak{g};V)\oplus
C^{n}(\mathfrak{g};V)\oplus\cdots\oplus C^{n}(\mathfrak{g};V)}_{n~{}copies}.$

Define the operator
$\partial:\mathfrak{C}^{n}(\mathfrak{g};V)\longrightarrow\mathfrak{C}^{n+1}(\mathfrak{g};V)$
by

$\partial^{1}f=(\partial_{\pi_{1}+\rho+\mu}f,\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}f),\quad\forall
f\in\mathrm{Hom}(\mathfrak{g},V),n=1.$

And for all $(f_{1},\dots,f_{n})\in\mathfrak{C}^{n}(\mathfrak{g},V),2\leq
i\leq n$, we have

$\partial^{n}(f_{1},\dots,f_{n})=(\partial_{\pi_{1}+\rho+\mu}f_{1},\dots,\underbrace{\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}f_{i-1}+\partial_{\pi_{1}+\rho+\mu}f_{i}}_{i},\dots,\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}f_{n}).$

######  Theorem 4.5.

The operator
$\partial:\mathfrak{C}^{n}(\mathfrak{g};V)\longrightarrow\mathfrak{C}^{n+1}(\mathfrak{g};V)$
defined as above satisfies $\partial\circ\partial=0$.

###### Proof.

By Proposition 4.4 and the graded Jacobi identity, similarly to the proof of
Theorem 3.1, we have $\partial\circ\partial=0$. ∎

######  Definition 4.6.

Let $(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ be a representation of the
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$. The cohomology of the
cochain complex
$(\oplus_{n=1}^{+\infty}\mathfrak{C}^{n}(\mathfrak{g};V),\partial)$ is called
the cohomology of $(\mathfrak{g},\cdot,\ast)$ with coefficients in the
representation $(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$. The corresponding
$n$-th cohomology group is denoted by $\mathcal{H}^{n}(\mathfrak{g};V)$.

###  4.2. Abelian extensions of compatible pre-Lie algebras

######  Definition 4.7.

Let $(\mathfrak{g},\cdot,\ast)$ and $(V,\cdot_{V},\ast_{V})$ be two compatible
pre-Lie algebras. An extension of $(\mathfrak{g},\cdot,\ast)$ by
$(V,\cdot_{V},\ast_{V})$ is a short exact sequence of compatible pre-Lie
algebras morphisms:

$0\longrightarrow
V\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0,$

where
$(\hat{\mathfrak{g}},\cdot_{\hat{\mathfrak{g}}},\ast_{\hat{\mathfrak{g}}})$ is
a compatible pre-Lie algebra.

It is called an abelian extension if $(V,\cdot_{V},\ast_{V})$ is an abelian
compatible pre-Lie algebra, i.e. for all $u,v\in V,u\cdot_{V}v=u\ast_{V}v=0$ .

######  Definition 4.8.

A section of an extension
$(\hat{\mathfrak{g}},\cdot_{\hat{\mathfrak{g}}},\ast_{\hat{\mathfrak{g}}})$ of
a compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$ by
$(V,\cdot_{V},\ast_{V})$ is a linear map
$s:\mathfrak{g}\longrightarrow\hat{\mathfrak{g}}$ such that $p\circ
s=\rm{Id}_{\mathfrak{g}}$.

Let
$(\hat{\mathfrak{g}},\cdot_{\hat{\mathfrak{g}}},\ast_{\hat{\mathfrak{g}}})$ be
an abelian extension of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ by $(V,\cdot_{V},\ast_{V})$ and
$s:\mathfrak{g}\longrightarrow\hat{\mathfrak{g}}$ a section. For all
$x,y\in\mathfrak{g}$, define linear maps
$\theta,\tilde{\theta}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow V$
respectively by

(26) $\displaystyle\theta(x,y)$ $\displaystyle=$ $\displaystyle
s(x)\cdot_{\hat{\mathfrak{g}}}s(y)-s(x\cdot y),$ (27)
$\displaystyle\tilde{\theta}(x,y)$ $\displaystyle=$ $\displaystyle
s(x)\ast_{\hat{\mathfrak{g}}}s(y)-s(x\ast y).$

And for all $x,y\in\mathfrak{g},u\in V$, define
$\rho,\mu,\tilde{\rho},\tilde{\mu}:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$
respectively by

(28)
$\displaystyle\rho(x)(u)=s(x)\cdot_{\hat{\mathfrak{g}}}u,\quad\mu(x)(u)=u\cdot_{\hat{\mathfrak{g}}}s(x),$
(29)
$\displaystyle\tilde{\rho}(x)(u)=s(x)\ast_{\hat{\mathfrak{g}}}u,\quad\tilde{\mu}(x)(u)=u\ast_{\hat{\mathfrak{g}}}s(x).$

Obviously, $\hat{\mathfrak{g}}$ is isomorphic to $\mathfrak{g}\oplus V$ as
vector spaces. Transfer the compatible pre-Lie algebra structure on
$\hat{\mathfrak{g}}$ to that on $\mathfrak{g}\oplus V$, we obtain a compatible
pre-Lie algebra $(\mathfrak{g}\oplus
V,\cdot_{(\theta,\rho,\mu)},\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})})$,
where $``\cdot_{(\theta,\rho,\mu)}$” and
$``\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}$” are given by

(30) $\displaystyle(x+u)\cdot_{(\theta,\rho,\mu)}(y+v)$ $\displaystyle=$
$\displaystyle x\cdot
y+\theta(x,y)+\rho(x)(v)+\mu(y)(u),\quad\forall~{}x,y\in\mathfrak{g},u,v\in
V,$ (31)
$\displaystyle(x+u)\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(y+v)$
$\displaystyle=$ $\displaystyle x\ast
y+\tilde{\theta}(x,y)+\tilde{\rho}(x)(v)+\tilde{\mu}(y)(u),\quad\forall~{}x,y\in\mathfrak{g},u,v\in
V.$

######  Theorem 4.9.

With the above notation, $(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ is a
representation of the compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$.
Moreover, this representation is independent of the choice of sections.

###### Proof.

For all $x,y\in\mathfrak{g}$, $u\in V$, by the definition of a pre-Lie
algebra, we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(x\cdot_{(\theta,\rho,\mu)}y)\cdot_{(\theta,\rho,\mu)}u-x\cdot_{(\theta,\rho,\mu)}(y\cdot_{(\theta,\rho,\mu)}u)-(y\cdot_{(\theta,\rho,\mu)}x)\cdot_{(\theta,\rho,\mu)}u+y\cdot_{(\theta,\rho,\mu)}(x\cdot_{(\theta,\rho,\mu)}u)$
$\displaystyle=$ $\displaystyle(x\cdot
y+\theta(x,y))\cdot_{(\theta,\rho,\mu)}u-x\cdot_{(\theta,\rho,\mu)}\rho(y)u-(y\cdot
x+\theta(y,x))\cdot_{(\theta,\rho,\mu)}u+y\cdot_{(\theta,\rho,\mu)}\rho(x)u$
$\displaystyle=$ $\displaystyle\rho(x\cdot y)u-\rho(x)\rho(y)u-\rho(y\cdot
x)u+\rho(y)\rho(x)u,$

and

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(u\cdot_{(\theta,\rho,\mu)}x)\cdot_{(\theta,\rho,\mu)}y-u\cdot_{(\theta,\rho,\mu)}(x\cdot_{(\theta,\rho,\mu)}y)-(x\cdot_{(\theta,\rho,\mu)}u)\cdot_{(\theta,\rho,\mu)}y+x\cdot_{(\theta,\rho,\mu)}(u\cdot_{(\theta,\rho,\mu)}y)$
$\displaystyle=$
$\displaystyle\mu(x)u\cdot_{(\theta,\rho,\mu)}y-u\cdot_{(\theta,\rho,\mu)}(x\cdot
y+\theta(x,y))-\rho(x)u\cdot_{(\theta,\rho,\mu)}y+x\cdot_{(\theta,\rho,\mu)}\mu(y)u$
$\displaystyle=$ $\displaystyle\mu(y)\mu(x)u-\mu(x\cdot
y)u-\mu(y)\rho(x)u+\rho(x)\mu(y)u,$

which implies that

$\displaystyle\rho([x,y]_{C}$ $\displaystyle=$
$\displaystyle\rho(x)\circ\rho(y)-\rho(y)\circ\rho(x),$
$\displaystyle\mu(y)\circ\mu(x)-\mu(x\cdot y)$ $\displaystyle=$
$\displaystyle\mu(y)\circ\rho(x)-\rho(x)\circ\mu(y).$

Thus, $(V,\rho,\mu)$ is a representation of the pre-Lie algebra
$(\mathfrak{g},\cdot)$. Similarly, $(V,\tilde{\rho},\tilde{\mu})$ is a
representation of the pre-Lie algebra $(\mathfrak{g},\ast)$.

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}y)\cdot_{(\theta,\rho,\mu)}u+(x\cdot_{(\theta,\rho,\mu)}y)\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}u-x\cdot_{(\theta,\rho,\mu)}(y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}u)-x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(y\cdot_{(\theta,\rho,\mu)}u)$
$\displaystyle-(y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}x)\cdot_{(\theta,\rho,\mu)}u-(y\cdot_{(\theta,\rho,\mu)}x)\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}u+y\cdot_{(\theta,\rho,\mu)}(x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}u)+y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(x\cdot_{(\theta,\rho,\mu)}u)$
$\displaystyle=$ $\displaystyle(x\ast
y+\tilde{\theta}(x,y))\cdot_{(\theta,\rho,\mu)}u+(x\cdot
y+\theta(x,y))\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}u-x\cdot_{(\theta,\rho,\mu)}\tilde{\rho}(y)u-x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}\rho(y)u$
$\displaystyle-(y\ast x+\tilde{\theta}(y,x))\cdot_{(\theta,\rho,\mu)}u-(y\cdot
x+\theta(y,x))\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}u+y\cdot_{(\theta,\rho,\mu)}\tilde{\rho}(x)u+y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}\rho(x)u$
$\displaystyle=$ $\displaystyle\rho(x\ast y)u+\tilde{\rho}(x\cdot
y)u-\rho(x)\tilde{\rho}(y)u-\tilde{\rho}(x)\rho(y)u-\rho(y\ast
x)u-\tilde{\rho}(y\cdot x)u+\rho(y)\tilde{\rho}(x)u+\tilde{\rho}(y)\rho(x)u,$

which implies that

$\rho(x\ast y)+\tilde{\rho}(x\cdot
y)-\rho(x)\tilde{\rho}(y)-\tilde{\rho}(x)\rho(y)=\rho(y\ast
x)+\tilde{\rho}(y\cdot x)-\rho(y)\tilde{\rho}(x)-\tilde{\rho}(y)\rho(x).$

Similarly, by

$\displaystyle(x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}u)\cdot_{(\theta,\rho,\mu)}y+(x\cdot_{(\theta,\rho,\mu)}u)\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}y-x\cdot_{(\theta,\rho,\mu)}(u\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}y)-x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(u\cdot_{(\theta,\rho,\mu)}y)$
$\displaystyle-(u\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}x)\cdot_{(\theta,\rho,\mu)}y-(u\cdot_{(\theta,\rho,\mu)}x)\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}y+u\cdot_{(\theta,\rho,\mu)}(x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}y)+u\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(x\cdot_{(\theta,\rho,\mu)}y)$
$\displaystyle=$ $\displaystyle 0,$

we have

$\mu(y)\tilde{\rho}(x)-\rho(x)\tilde{\mu}(y)-\mu(y)\tilde{\mu}(x)+\mu(x\ast
y)=-\tilde{\mu}(y)\rho(x)+\tilde{\rho}(x)\mu(y)+\tilde{\mu}(y)\mu(x)-\tilde{\mu}(x\cdot
y).$

Thus, $(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ is a representation of the
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$. Let $s^{\prime}$ be
another section and
$(V,\rho^{\prime},\mu^{\prime},\tilde{\rho}^{\prime},\tilde{\mu}^{\prime})$ be
the corresponding representation of the compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$. Since $s(x)-s^{\prime}(x)\in V$, then we have

$\displaystyle\rho(x)u-\rho^{\prime}(x)u=(s(x)-s^{\prime}(x))\cdot_{\hat{\mathfrak{g}}}u=0,$
$\displaystyle\mu(x)u-\mu^{\prime}(x)u=u\cdot_{\hat{\mathfrak{g}}}(s(x)-s^{\prime}(x))=0,$

which implies that $\rho=\rho^{\prime}$ and $\mu=\mu^{\prime}$. Similarly, we
have $\tilde{\rho}=\tilde{\rho}^{\prime}$ and
$\tilde{\mu}=\tilde{\mu}^{\prime}$. Thus, this representation is independent
of the choice of sections. ∎

######  Theorem 4.10.

With the above notation, $(\theta,\tilde{\theta})$ is a $2$-cocycle of the
compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$ with coefficients in
the representation $(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$.

###### Proof.

For all $x,y,z\in\mathfrak{g}$, by the definition of a pre-Lie algebra, we
have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(x\cdot_{(\theta,\rho,\mu)}y)\cdot_{(\theta,\rho,\mu)}z-x\cdot_{(\theta,\rho,\mu)}(y\cdot_{(\theta,\rho,\mu)}z)-(y\cdot_{(\theta,\rho,\mu)}x)\cdot_{(\theta,\rho,\mu)}z+y\cdot_{(\theta,\rho,\mu)}(x\cdot_{(\theta,\rho,\mu)}z)$
$\displaystyle=$ $\displaystyle(x\cdot
y+\theta(x,y)\big{)}\cdot_{(\theta,\rho,\mu)}z-x\cdot_{(\theta,\rho,\mu)}\big{(}y\cdot
z+\theta(y,z)\big{)}$ $\displaystyle-\big{(}y\cdot
x+\theta(y,x)\big{)}\cdot_{(\theta,\rho,\mu)}z+y\cdot_{(\theta,\rho,\mu)}\big{(}x\cdot
z+\theta(x,z))$ $\displaystyle=$ $\displaystyle\theta(x\cdot
y,z)+\mu(z)\theta(x,y)-\theta(x,y\cdot z)-\rho(x)\theta(y,z)$
$\displaystyle-\theta(y\cdot x,z)-\mu(z)\theta(y,x)+\theta(y,x\cdot
z)+\rho(y)\theta(x,z)$ $\displaystyle=$
$\displaystyle-\partial_{\pi_{1}+\rho+\mu}\theta(x,y)$

which implies that $\partial_{\pi_{1}+\rho+\mu}\theta=0.$ Similarly, we have
$\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}\tilde{\theta}=0.$

For all $x,y,z\in\mathfrak{g}$, by (1), we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}y)\cdot_{(\theta,\rho,\mu)}z+(x\cdot_{(\theta,\rho,\mu)}y)\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}z-x\cdot_{(\theta,\rho,\mu)}(y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}z)-x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(y\cdot_{(\theta,\rho,\mu)}z)$
$\displaystyle-(y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}x)\cdot_{(\theta,\rho,\mu)}z-(y\cdot_{(\theta,\rho,\mu)}x)\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}z+y\cdot_{(\theta,\rho,\mu)}(x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}z)+y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(x\cdot_{(\theta,\rho,\mu)}z)$
$\displaystyle=$ $\displaystyle(x\ast
y+\tilde{\theta}(x,y)\big{)}\cdot_{(\theta,\rho,\mu)}z+(x\cdot
y+\theta(x,y))\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}z$
$\displaystyle-x\cdot_{(\theta,\rho,\mu)}(y\ast
z+\tilde{\theta}(y,z))-x\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(y\cdot
z+\theta(y,z))$ $\displaystyle-(y\ast
x+\tilde{\theta}(y,x)\big{)}\cdot_{(\theta,\rho,\mu)}z-(y\cdot
x+\theta(y,x))\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}z$
$\displaystyle+y\cdot_{(\theta,\rho,\mu)}(x\ast
z+\tilde{\theta}(x,z))+y\ast_{(\tilde{\theta},\tilde{\rho},\tilde{\mu})}(x\cdot
z+\theta(x,z))$ $\displaystyle=$ $\displaystyle\theta(x\ast
y,z)+\mu(z)\tilde{\theta}(x,y)+\tilde{\theta}(x\cdot
y,z)+\tilde{\mu}(z)\theta(x,y)$ $\displaystyle-\theta(x,y\ast
z)-\rho(x)\tilde{\theta}(y,z)-\tilde{\theta}(x,y\cdot
z)-\tilde{\rho}(x)\theta(y,z)$ $\displaystyle-\theta(y\ast
x,z)-\mu(z)\tilde{\theta}(y,x)-\tilde{\theta}(y\cdot
x,z)-\tilde{\mu}(z)\theta(y,x)$ $\displaystyle+\theta(y,x\ast
z)+\rho(y)\tilde{\theta}(x,z)+\tilde{\theta}(y,x\cdot
z)+\tilde{\rho}(y)\theta(x,z)$ $\displaystyle=$
$\displaystyle-(\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}\theta+\partial_{\pi_{1}+\rho+\mu}\tilde{\theta})(x,y,z),$

which implies that
$\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}\theta+\partial_{\pi_{1}+\rho+\mu}\tilde{\theta}=0$.
Thus, we have

$\partial(\theta,\tilde{\theta})=(\partial_{\pi_{1}+\rho+\mu}\theta,\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}\theta+\partial_{\pi_{1}+\rho+\mu}\tilde{\theta},\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}\tilde{\theta})=0.$

This finishes the proof. ∎

######  Definition 4.11.

Let
$(\hat{{\mathfrak{g}}_{1}},\cdot_{\hat{{\mathfrak{g}}_{1}}},\ast_{\hat{{\mathfrak{g}}_{1}}})$
and
$(\hat{{\mathfrak{g}}_{2}},\cdot_{\hat{{\mathfrak{g}}_{2}}},\ast_{\hat{{\mathfrak{g}}_{2}}})$
be two abelian extensions of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ by $(V,\cdot_{V},\ast_{V})$. They are said to be
isomorphic if there exists a compatible pre-Lie algebra isomorphism
$\zeta:(\hat{{\mathfrak{g}}_{1}},\cdot_{\hat{{\mathfrak{g}}_{1}}},\ast_{\hat{{\mathfrak{g}}_{1}}})\longrightarrow(\hat{{\mathfrak{g}}_{2}},\cdot_{\hat{{\mathfrak{g}}_{2}}},\ast_{\hat{{\mathfrak{g}}_{2}}})$
such that the following diagram is commutative:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{1}}$$\textstyle{\hat{{\mathfrak{g}}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\scriptstyle{p_{1}}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$$\textstyle{\hat{{\mathfrak{g}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

######  Lemma 4.12.

Let
$(\hat{\mathfrak{g}}_{1},\cdot_{\hat{\mathfrak{g}}_{1}},\ast_{\hat{\mathfrak{g}}_{1}})$
and
$(\hat{\mathfrak{g}}_{2},\cdot_{\hat{\mathfrak{g}}_{2}},\ast_{\hat{\mathfrak{g}}_{2}})$
be two isomorphic abelian extensions of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ by $(V,\cdot_{V},\ast_{V})$. Then they are give
rise to the same representation of $(\mathfrak{g},\cdot,\ast)$.

###### Proof.

Let $s_{1}:{\mathfrak{g}}_{1}\longrightarrow\hat{\mathfrak{g}}_{1}$ and
$s_{2}:{\mathfrak{g}}_{2}\longrightarrow\hat{\mathfrak{g}}_{2}$ be two
sections of
$(\hat{\mathfrak{g}}_{1},\cdot_{\hat{\mathfrak{g}}_{1}},\ast_{\hat{\mathfrak{g}}_{1}})$
and
$(\hat{\mathfrak{g}}_{2},\cdot_{\hat{\mathfrak{g}}_{2}},\ast_{\hat{\mathfrak{g}}_{2}})$
respectively. By Theorem 4.9, we obtain that
$(V,\rho_{1},\mu_{1},\tilde{\rho}_{1},\tilde{\mu}_{1})$ and
$(V,\rho_{2},\mu_{2},\tilde{\rho}_{2},\tilde{\mu}_{2})$ are their
representations respectively. Define
$s^{\prime}_{1}:{\mathfrak{g}}_{1}\longrightarrow\hat{\mathfrak{g}}_{1}$ by
$s^{\prime}_{1}=\zeta^{-1}\circ s_{2}$. Since
$\zeta:(\hat{\mathfrak{g}}_{1},\cdot_{\hat{\mathfrak{g}}_{1}},\ast_{\hat{\mathfrak{g}}_{1}})\longrightarrow(\hat{\mathfrak{g}}_{2},\cdot_{\hat{\mathfrak{g}}_{2}},\ast_{\hat{\mathfrak{g}}_{2}})$
is a compatible pre-Lie algebra isomorphism satisfying the commutative diagram
in Definition 4.11, by $p_{2}\circ\zeta=p_{1}$, we have

$p_{1}\circ s^{\prime}_{1}=p_{2}\circ\zeta\circ\zeta^{-1}\circ
s_{2}=\rm{Id}_{\mathfrak{g}}.$

Thus, we obtain that $s^{\prime}_{1}$ is a section of
$(\hat{\mathfrak{g}}_{1},\cdot_{\hat{\mathfrak{g}}_{1}},\ast_{\hat{\mathfrak{g}}_{1}})$.
For all $x\in\mathfrak{g},u\in V$, by $\zeta\mid_{V}=\rm{Id}_{V}$, we have

$\displaystyle\rho_{1}(x)(u)=s^{\prime}_{1}(x)\cdot_{\hat{\mathfrak{g}}_{1}}u=(\zeta^{-1}\circ
s_{2})(x)\cdot_{\hat{\mathfrak{g}}_{1}}u=\zeta^{-1}(s_{2}(x)\cdot_{\hat{\mathfrak{g}}_{2}}u)=\rho_{2}(x)(u),$
$\displaystyle\mu_{1}(x)(u)=u\cdot_{\hat{\mathfrak{g}}_{1}}s^{\prime}_{1}(x)=u\cdot_{\hat{\mathfrak{g}}_{1}}(\zeta^{-1}\circ
s_{2})(x)=\zeta^{-1}(u\cdot_{\hat{\mathfrak{g}}_{2}}s_{2}(x))=\mu_{2}(x)(u),$

which implies that $\rho_{1}=\rho_{2}$ and $\mu_{1}=\mu_{2}$. Similarly, we
have $\tilde{\rho}_{1}=\tilde{\rho}_{2}$ and
$\tilde{\mu}_{1}=\tilde{\mu}_{2}$. This finishes the proof. ∎

So in the sequel, we fixed a representation
$(V,\rho,\mu,\tilde{\rho},\tilde{\mu})$ of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ and consider abelian extensions that induce the
given representation.

######  Theorem 4.13.

Abelian extensions of a compatible pre-Lie algebra $(\mathfrak{g},\cdot,\ast)$
by $(V,\cdot_{V},\ast_{V})$ are classified by
$\mathcal{H}^{2}(\mathfrak{g};V)$.

###### Proof.

Let
$(\hat{\mathfrak{g}},\cdot_{\hat{\mathfrak{g}}},\ast_{\hat{\mathfrak{g}}})$ be
an abelian extension of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ by $(V,\cdot_{V},\ast_{V})$. Choosing a section
$s:\mathfrak{g}\longrightarrow\hat{\mathfrak{g}}$, by Theorem 4.10, we obtain
that $(\theta,\tilde{\theta})$ is a $2$-cocycle. Now we show that the
cohomological class of $(\theta,\tilde{\theta})$ does not depend on the choice
of sections. In fact, let $s$ and $s^{\prime}$ be two different sections.
Define $\varphi:\mathfrak{g}\longrightarrow V$ by
$\varphi(x)=s(x)-s^{\prime}(x)$. Then for all $x,y\in\mathfrak{g}$, we have

$\displaystyle\theta(x,y)$ $\displaystyle=$ $\displaystyle
s(x)\cdot_{\hat{\mathfrak{g}}}s(y)-s(x\cdot y)$ $\displaystyle=$
$\displaystyle\big{(}s^{\prime}(x)+\varphi(x)\big{)}\cdot_{\hat{\mathfrak{g}}}\big{(}s^{\prime}(y)+\varphi(y)\big{)}-s^{\prime}(x\cdot
y)-\varphi(x\cdot y)$ $\displaystyle=$ $\displaystyle
s^{\prime}(x)\cdot_{\hat{\mathfrak{g}}}s^{\prime}(y)+\rho(x)\varphi(y)+\mu(y)\varphi(x)-s^{\prime}(x\cdot
y)-\varphi(x\cdot y)$ $\displaystyle=$
$\displaystyle\theta^{\prime}(x,y)+\partial_{\pi_{1}+\rho+\mu}\varphi(x,y),$

which implies that
$\theta-\theta^{\prime}=\partial_{\pi_{1}+\rho+\mu}\varphi$. Similarly, we
have
$\tilde{\theta}-\tilde{\theta}^{\prime}=\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}\varphi$.

Therefore, we obtain that
$(\theta-\theta^{\prime},\tilde{\theta}-\tilde{\theta}^{\prime})=\partial\varphi$,
$(\theta,\theta^{\prime})$ and $(\tilde{\theta},\tilde{\theta}^{\prime})$ are
in the same cohomological class.

Now we prove that isomorphic abelian extensions give rise to the same element
in $\mathcal{H}^{2}(\mathfrak{g};V)$. Assume that
$(\hat{{\mathfrak{g}}_{1}},\cdot_{\hat{{\mathfrak{g}}_{1}}},\ast_{\hat{{\mathfrak{g}}_{1}}})$
and
$(\hat{{\mathfrak{g}}_{2}},\cdot_{\hat{{\mathfrak{g}}_{2}}},\ast_{\hat{{\mathfrak{g}}_{2}}})$
are two isomorphic abelian extensions of a compatible pre-Lie algebra
$(\mathfrak{g},\cdot,\ast)$ by $(V,\cdot_{V},\ast_{V})$, and
$\zeta:(\hat{{\mathfrak{g}}_{1}},\cdot_{\hat{{\mathfrak{g}}_{1}}},\ast_{\hat{{\mathfrak{g}}_{1}}})\longrightarrow(\hat{{\mathfrak{g}}_{2}},\cdot_{\hat{{\mathfrak{g}}_{2}}},\ast_{\hat{{\mathfrak{g}}_{2}}})$
is a compatible pre-Lie algebra isomorphism satisfying the commutative diagram
in Definition 4.11. Assume that
$s_{1}:\mathfrak{g}\longrightarrow\hat{{\mathfrak{g}}_{1}}$ is a section of
$\hat{{\mathfrak{g}}_{1}}$. By $p_{2}\circ\zeta=p_{1}$, we have

$p_{2}\circ(\zeta\circ s_{1})=p_{1}\circ s_{1}=\rm{Id}_{\mathfrak{g}}.$

Thus, we obtain that $\zeta\circ s_{1}$ is a section of
$\hat{{\mathfrak{g}}_{2}}$. Define $s_{2}=\zeta\circ s_{1}$. Since $\zeta$ is
an isomorphism of compatible pre-Lie algebras and $\zeta\mid_{V}=\rm{Id}_{V}$,
for all $x,y\in\mathfrak{g}$, we have

$\displaystyle\theta_{2}(x,y)$ $\displaystyle=$ $\displaystyle
s_{2}(x)\cdot_{\hat{{\mathfrak{g}}_{2}}}s_{2}(y)-s_{2}(x\cdot y)$
$\displaystyle=$ $\displaystyle(\zeta\circ
s_{1})(x)\cdot_{\hat{{\mathfrak{g}}_{2}}}(\zeta\circ s_{1})(y)-(\zeta\circ
s_{1})(x\cdot y)$ $\displaystyle=$
$\displaystyle\zeta\big{(}s_{1}(x)\cdot_{\hat{{\mathfrak{g}}_{1}}}s_{1}(y)-s_{1}(x\cdot
y)\big{)}$ $\displaystyle=$ $\displaystyle\theta_{1}(x,y),$

Similarly, we have $\tilde{\theta}_{1}=\tilde{\theta}_{2}$. Thus, isomorphic
abelian extensions gives rise to the same element in
$\mathcal{H}^{2}(\mathfrak{g};V)$.

Conversely, given two 2-cocycles $(\theta_{1},\tilde{\theta}_{1})$ and
$(\theta_{2},\tilde{\theta}_{2})$, we can construct two abelian extensions
$(\mathfrak{g}\oplus
V,\cdot_{(\theta_{1},\rho,\mu)},\ast_{(\tilde{\theta}_{1},\tilde{\rho},\tilde{\mu})})$
and $(\mathfrak{g}\oplus
V,\cdot_{(\theta_{2},\rho,\mu)},\ast_{(\tilde{\theta}_{2},\tilde{\rho},\tilde{\mu})})$.
If
$(\theta_{1},\tilde{\theta}_{1}),(\theta_{2},\tilde{\theta}_{2})\in\mathcal{H}^{2}(\mathfrak{g};V)$,
then there exists $\varphi:\mathfrak{g}\longrightarrow V$, such that
$\theta_{1}=\theta_{2}+\partial_{\pi_{1}+\rho+\mu}\varphi$ and
$\tilde{\theta}_{1}=\tilde{\theta}_{2}+\partial_{\pi_{2}+\tilde{\rho}+\tilde{\mu}}\varphi$.
We define $\zeta:\mathfrak{g}\oplus V\longrightarrow\mathfrak{g}\oplus V$ by

$\zeta(x+u)=x+u+\varphi(x),\quad\forall~{}x\in\mathfrak{g},u\in V.$

For all $x,y\in\mathfrak{g},u,v\in V$, by
$\theta_{1}=\theta_{2}+\partial_{\pi_{1}+\rho+\mu}\varphi$, we have

$\displaystyle\zeta\big{(}(x+u)\cdot_{(\theta_{1},\rho,\mu)}(y+v)\big{)}-\zeta(x+u)\cdot_{(\theta_{2},\rho,\mu)}\zeta(y+v)$
$\displaystyle=$ $\displaystyle\zeta\big{(}x\cdot
y+\theta_{1}(x,y)+\rho(x)v+\mu(y)u\big{)}-\big{(}x+u+\varphi(x)\big{)}\cdot_{(\theta_{2},\rho,\mu)}\big{(}y+v+\varphi(y)\big{)}$
$\displaystyle=$ $\displaystyle\theta_{1}(x,y)+\varphi(x\cdot
y)-\theta_{2}(x,y)-\rho(x)\varphi(y)-\mu(y)\varphi(x)$ $\displaystyle=$
$\displaystyle\theta_{1}(x,y)-\theta_{2}(x,y)-\partial_{\pi_{1}+\rho+\mu}\varphi$
$\displaystyle=$ $\displaystyle 0,$

which implies that

(32)
$\zeta\big{(}(x+u)\cdot_{(\theta_{1},\rho,\mu)}(y+v)\big{)}=\zeta(x+u)\cdot_{(\theta_{2},\rho,\mu)}\zeta(y+v).$

Similarly, we have

(33)
$\zeta\big{(}(x+u)\ast_{(\tilde{\theta}_{1},\tilde{\rho},\tilde{\mu})}(y+v)\big{)}=\zeta(x+u)\ast_{(\tilde{\theta}_{2},\tilde{\rho},\tilde{\mu})}\zeta(y+v).$

Thus, by (32) and (33), $\zeta$ is a compatible pre-Lie algebra isomorphism
from $(\mathfrak{g}\oplus
V,\cdot_{(\theta_{1},\rho,\mu)},\ast_{(\tilde{\theta}_{1},\tilde{\rho},\tilde{\mu})})$
to $(\mathfrak{g}\oplus
V,\cdot_{(\theta_{2},\rho,\mu)},\ast_{(\tilde{\theta}_{2},\tilde{\rho},\tilde{\mu})})$.
Moreover, it is obvious that the diagram in Definition 4.11 is commutative.
This finishes the proof. ∎

Acknowledgement: This work is supported by NSF of Jilin Province (No.
YDZJ202201ZYTS589), NNSF of China (Nos. 12271085, 12071405) and the
Fundamental Research Funds for the Central Universities.

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