# Cohomology and deformations of compatible Hom-Lie algebras

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

In this paper, we consider compatible Hom-Lie algebras as a twisted version of
compatible Lie algebras. Compatible Hom-Lie algebras are characterized as
Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We
also define a cohomology theory for compatible Hom-Lie algebras generalizing
the recent work of Liu, Sheng and Bai. As applications of cohomology, we study
abelian extensions and deformations of compatible Hom-Lie algebras.

2020 Mathematics Subject Classification: 17B61, 17A30, 17B56, 16S80.

Keywords: Hom-Lie algebras, Compatible structures, Cohomology, Extensions,
Deformations.

###### Contents

  1. 1 Introduction
  2. 2 Hom-Lie algebras and bidifferential graded Lie algebras
  3. 3 Compatible Hom-Lie algebras
  4. 4 Cohomology of compatible Hom-Lie algebras
  5. 5 Deformations of compatible Hom-Lie algebras

##  1 Introduction

The notion of Hom-algebras first appeared in the $q$-deformations of the Witt
and Virasoro algebra by the work of Hartwig, Larsson and Silvestrov [11]. More
precisely, they introduced Hom-Lie algebras as a twisted version of Lie
algebras, where the usual Jacobi identity is twisted by a linear homomorphism
(see Definition 2.1). Later, various others algebras (e.g. associative,
Leibniz, Poisson, …) twisted by homomorphisms are also studied [14, 3, 23].
These Hom-algebras are widely explored in the last 15 years. In [1, 15] the
authors study cohomology and deformations of Hom-associative and Hom-Lie
algebras. In particular, they generalize the classical Gerstenhaber bracket
and Nijenhuis-Richardson bracket on the cochain complex of Hom-associative and
Hom-Lie algebras. See also [20, 23, 24] and references therein for more on
Hom-algebras.

Two algebraic structures of the same kind are said to be compatible if their
sum also defines a same type of algebraic structure. They appeared in many
contexts of mathematics and mathematical physics. For instance, the notion of
compatibility of two Poisson structures on a manifold was first appeared in
the mathematical study of biHamiltonian mechanics [12, 16]. Using the
correspondence between Lie algebra structures on a vector space $\mathfrak{g}$
and linear Poisson structures on $\mathfrak{g}^{\ast}$, one lead to a notion
of compatible Lie algebras [12]. See [9, 19] for more study on compatible Lie
algebras. Some other compatible structures include compatible associative
algebras [18], compatible Lie bialgebras [22], compatible Lie algebroids and
Lie bialgebroids [5]. See also [7, 21] for the operadic study of compatible
algebraic structures.

Recently, a cohomology theory for compatible Lie algebras has been introduced
by Liu, Sheng and Bai [13]. This cohomology is based on the characterization
of a compatible Lie algebra as Maurer-Cartan element in a bidifferential
graded Lie algebra. It is also seen that this cohomology is related to
extensions and deformations of compatible Lie algebras. These results have
been extended to compatible associative algebras in [4]. The present paper
aims to define and study compatible Hom-Lie algebras. We observe that
compatible Hom-Lie algebras are related to compatible Hom-Poisson manifolds.
We first define representations and cohomology of a compatible Hom-Lie
algebra. We relate our cohomology of a compatible Hom-Lie algebra with the
cohomology of Hom-Lie algebra. As applications of our cohomology, we study
abelian extensions and linear, finite order deformations of a compatible Hom-
Lie algebra.

The paper is organized as follows. In Section 2 (preliminary section), we
recall Hom-Lie algebras and some basics on bidifferential graded Lie algebras.
In Section 3, we introduce compatible Hom-Lie algebras and their Maurer-Cartan
characterizations in a suitably constructed bidifferential graded Lie algebra.
We also define the notion of representation of a compatible Hom-Lie algebra
and construct the semidirect product. The cohomology of a compatible Hom-Lie
algebra with coefficients in a representation is given in Section 4. We also
introduce abelian extensions of a compatible Hom-Lie algebra and characterize
equivalence classes of abelian extensions by the second cohomology group. In
Section 5, we first define linear deformations of a compatible Hom-Lie algebra
and introduce Nijenhuis operators that induce trivial linear deformations. We
show that the equivalence classes of infinitesimal deformations of a
compatible Hom-Lie algebra are in one-to-one correspondence with the second
cohomology group with coefficients in itself. Finally, we consider finite
order deformations of compatible Hom-Lie algebra and study their extensions.

All vector spaces, (multi)linear maps, tensor products are over a field
$\mathbb{K}$ of characteristic zero.

##  2 Hom-Lie algebras and bidifferential graded Lie algebras

In this preliminary section, we recall some basics on Hom-Lie algebras [14, 1,
20] and bidifferential graded Lie algebras [13].

######  2.1 Definition.

A Hom-Lie algebra is a vector space $\mathfrak{g}$ together with a bilinear
skew-symmetric bracket
$[~{},~{}]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ and a
linear map $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying the
followings:

$\displaystyle\alpha[x,y]=[\alpha(x),\alpha(y)]\qquad(\text{multiplicativity}),$
$\displaystyle[[x,y],\alpha(z)]+[[y,z],\alpha(x)]+[[z,x],\alpha(y)]=0\qquad(\text{Hom-
Jacobi identity}),$

for all $x,y,z\in\mathfrak{g}$.

A Hom-Lie algebra as above may be denoted by the triple
$(\mathfrak{g},[~{},~{}],\alpha)$ or simply by $\mathfrak{g}$ when no
confusion arises. The bracket $[~{},~{}]$ is said to be the Hom-Lie bracket on
$\mathfrak{g}$ when the twisting map $\alpha$ is clear from the context.

It follows from the above definition that Hom-Lie algebras are a twisted
version of Lie algebras. More specifically, a Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$ with $\alpha=\mathrm{id}$ is nothing but a
Lie algebra.

######  2.2 Example.

Let $(\mathfrak{g},[~{},~{}])$ be a Lie algebra and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Lie algebra homomorphism.
Then the triple
$(\mathfrak{g},[~{},~{}]_{\alpha}=\alpha\circ[~{},~{}],\alpha)$ is a Hom-Lie
algebra, called induced by composition.

######  2.3 Example.

Let $(A,\mu,\alpha)$ be a Hom-associative algebra, i.e., $A$ is a vector
space, $\mu:A\otimes A\rightarrow A,~{}(a,b)\mapsto a\cdot b$ is a bilinear
map and $\alpha:A\rightarrow A$ a linear map satisfying $\alpha(a\cdot
b)=\alpha(a)\cdot\alpha(b)$ and the following Hom-associativity $(a\cdot
b)\cdot\alpha(c)=\alpha(a)\cdot(b\cdot c),$ for $a,b,c\in A.$ If
$(A,\mu,\alpha)$ is a Hom-associative algebra, then $(A,[~{},~{}],\alpha)$ is
a Hom-Lie algebra, where $[a,b]=a\cdot b-b\cdot a$, for $a,b\in A$.

######  2.4 Example.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. Then for any
$n\geq 0$, the triple
$(\mathfrak{g},[~{},~{}]^{(n)}=\alpha^{n}\circ[~{},~{}],\alpha^{n+1})$ is a
Hom-Lie algebra, called the $n$-th derived Hom-Lie algebra.

######  2.5 Definition.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ and
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime},\alpha^{\prime})$ be two Hom-Lie
algebras. A linear map $\phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ is
said to be a Hom-Lie algebra morphism if
$\alpha^{\prime}\circ\phi=\phi\circ\alpha$ and
$\phi[x,y]=[\phi(x),\phi(y)]^{\prime},$ for $x,y\in\mathfrak{g}.$

Next we recall the graded Lie bracket (called the Nijenhuis-Richardson
bracket) whose Maurer-Cartan elements are given by Hom-Lie algebra structures
[1]. This generalizes the classical Nijenhuis-Richardson bracket in the
context of Lie algebras [17]. Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map. For each $n\geq
0$, consider the spaces $C^{n}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ by

$\displaystyle
C^{0}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})=\\{x\in\mathfrak{g}|~{}\alpha(x)=x\\}~{}~{}\text{
and
}~{}~{}C^{n}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})=\\{f:\wedge^{n}\mathfrak{g}\rightarrow\mathfrak{g}|~{}\alpha\circ
f=f\circ\alpha^{\wedge n}\\},~{}n\geq 1.$

Then the shifted graded vector space
$C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq
0}C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ carries a graded Lie
bracket defined as follows. For $P\in
C^{m+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ and $Q\in
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$, the Nijenhuis-Richardson
bracket $[P,Q]_{\mathsf{NR}}\in
C^{m+n+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ given by

$\displaystyle[P,Q]_{\mathsf{NR}}=P\diamond Q-(-1)^{mn}~{}Q\diamond
P,~{}\text{ where }$ $\displaystyle(P\diamond
Q)(x_{1},\ldots,x_{m+n+1})=\sum_{\sigma\in
Sh(n+1,m)}(-1)^{\sigma}~{}P\big{(}Q(x_{\sigma(1)},\ldots,x_{\sigma(n+1)}),\alpha^{n}(x_{\sigma(n+2)}),\ldots,\alpha^{n}(x_{\sigma(m+n+1)}\big{)}.$

With this notation, we have the following.

######  2.6 Proposition.

Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map. Then the Hom-Lie
brackets on $\mathfrak{g}$ are precisely the Maurer-Cartan elements in the
graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}}).$

In the following, we recall the Chevalley-Eilenberg cohomology of a Hom-Lie
algebra $(\mathfrak{g},[~{},~{}],\alpha)$ with coefficients in a
representation.

######  2.7 Definition.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. A representation
of it consists of a vector space $V$ together with a bilinear operation
(called the action) $\bullet:\mathfrak{g}\otimes V\rightarrow V$,
$(x,v)\mapsto x\bullet v$ and a linear map $\beta:V\rightarrow V$ satisfying

$\displaystyle\beta(x\bullet v)=\alpha(x)\bullet\beta(v),$
$\displaystyle[x,y]\bullet\beta(v)=\alpha(x)\bullet(y\bullet
v)-\alpha(y)\bullet(x\bullet v),~{}\text{ for }x,y\in\mathfrak{g},v\in V.$

We denote a representation as above by $(V,\bullet,\beta)$ or simply by $V$.
It follows that any Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$ is a
representation of itself with the action given by the bracket $[~{},~{}].$
This is called the adjoint representation.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra and
$(V,\bullet,\beta)$ be a representation. For each $n\geq 0$, we define the
$n$-th cochain group $C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$ as

$\displaystyle C^{0}_{\mathrm{Hom}}(\mathfrak{g},V)=\\{v\in
V|~{}\beta(v)=v\\}~{}~{}\text{ and
}~{}~{}C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)=\\{f:\wedge^{n}\mathfrak{g}\rightarrow
V|~{}\beta\circ f=f\circ\alpha^{\wedge n}\\},~{}n\geq 1.$

The coboundary operator
$\delta_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},V)$, for $n\geq 0$, given by

(1) $\displaystyle(\delta_{\mathrm{Hom}}v)(x)=~{}$ $\displaystyle x\bullet
v,~{}\text{ for }v\in C^{0}_{\mathrm{Hom}}(\mathfrak{g},V),x\in\mathfrak{g},$
(2) $\displaystyle(\delta_{\mathrm{Hom}}f)(x_{1},\ldots,x_{n+1})=$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}~{}\alpha^{n-1}(x_{i})\bullet
f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$ $\displaystyle+\sum_{1\leq
i<j\leq
n+1}(-1)^{i+j}~{}f([x_{i},x_{j}],\alpha(x_{1}),\ldots,\widehat{\alpha(x_{i})},\ldots,\widehat{\alpha(x_{j})},\ldots,\alpha(x_{n+1})),$

for $f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$ and
$x_{1},\ldots,x_{n+1}\in\mathfrak{g}$. The cohomology groups of the cochain
complex $\\{C^{\ast}_{\mathrm{Hom}}(\mathfrak{g},V),\delta_{\mathrm{Hom}}\\}$
are called the Chevalley-Eilenberg cohomology groups, denoted by
$H^{\ast}_{\mathrm{Hom}}(\mathfrak{g},V).$

It is important to note that the coboundary operator for the Chevalley-
Eilenberg cohomology of the Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$
with coefficients in itself is simply given by

$\displaystyle\delta_{\mathrm{Hom}}f=(-1)^{n-1}[\mu,f]_{\mathsf{NR}},~{}\text{for
}f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),$

where $\mu\in C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ corresponds to
the Hom-Lie bracket $[~{},~{}]$.

Bidifferential graded Lie algebras. Next, we recall bidifferential graded Lie
algebras [13]. Before that, let us first give the definition of a differential
graded Lie algebra.

######  2.8 Definition.

A differential graded Lie algebra is a triple $(L=\oplus L^{i},[~{},~{}],d)$
consisting of a graded Lie algebra together with a degree $+1$ differential
$d:L\rightarrow L$ which is a derivation for the bracket $[~{},~{}]$.

An element $\theta\in L^{1}$ is said to be a Maurer-Cartan element in the
differential graded Lie algebra $(L,[~{},~{}],d)$ if $\theta$ satisfies

$\displaystyle d\theta+\frac{1}{2}[\theta,\theta]=0.$

######  2.9 Definition.

A bidifferential graded Lie algebra is a quadruple $(L=\oplus
L^{i},[~{},~{}],d_{1},d_{2})$ in which the triples $(L,[~{},~{}],d_{1})$ and
$(L,[~{},~{}],d_{2})$ are differential graded Lie algebras additionally
satisfying $d_{1}\circ d_{2}+d_{2}\circ d_{1}=0.$

######  2.10 Remark.

Any graded Lie algebra can be considered as a bidifferential graded Lie
algebra with both the differentials $d_{1}$ and $d_{2}$ to be trivial.

######  2.11 Definition.

Let $(L,[~{},~{}],d_{1},d_{2})$ be a bidifferential graded Lie algebra. A pair
of elements $(\theta_{1},\theta_{2})\in L^{1}\oplus L^{1}$ is said to be a
Maurer-Cartan element if

  * (i)

$\theta_{1}$ is a Maurer-Cartan element in the differential graded Lie algebra
$(L,[~{},~{}],d_{1})$;

  * (ii)

$\theta_{2}$ is a Maurer-Cartan element in the differential graded Lie algebra
$(L,[~{},~{}],d_{2})$;

  * (iii)

the following compatibility condition holds

$\displaystyle d_{1}\theta_{2}+d_{2}\theta_{1}+[\theta_{1},\theta_{2}]=0.$

Like a differential graded Lie algebra can be twisted by a Maurer-Cartan
element, the same result holds for bidifferential graded Lie algebras.

######  2.12 Proposition.

Let $(L,[~{},~{}],d_{1},d_{2})$ be a bidifferential graded Lie algebra and let
$(\theta_{1},\theta_{2})$ be a Maurer-Cartan element. Then the quadruple
$(L,[~{},~{}],d_{1}^{\theta_{1}},d_{2}^{\theta_{2}})$ is a bidifferential
graded Lie algebra, where

$\displaystyle d_{1}^{\theta_{1}}=d_{1}+[\theta_{1},-]~{}~{}~{}\text{ and
}~{}~{}~{}d_{2}^{\theta_{2}}=d_{2}+[\theta_{2},-].$

For any $\vartheta_{1},\vartheta_{2}\in L^{1}$, the pair
$(\theta_{1}+\vartheta_{1},\theta_{2}+\vartheta_{2})$ is a Maurer-Cartan
element in the bidifferential graded Lie algebra $(L,[~{},~{}],d_{1},d_{2})$
if and only if $(\vartheta_{1},\vartheta_{2})$ is a Maurer-Cartan element in
the bidifferential graded Lie algebra
$(L,[~{},~{}],d_{1}^{\theta_{1}},d_{2}^{\theta_{2}}).$

##  3 Compatible Hom-Lie algebras

In this section, we introduce compatible Hom-Lie algebras and give a Maurer-
Cartan characterization. We end this section by defining representations of
compatible Hom-Lie algebras.

Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map.

######  3.1 Definition.

Two Hom-Lie algebras $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$ are said to be compatible if for all
$\lambda,\eta\in\mathbb{K}$, the triple
$(\mathfrak{g},\lambda[~{},~{}]_{1}+\eta[~{},~{}]_{2},\alpha)$ is a Hom-Lie
algebra.

The condition in the above definition is equivalent to the following

(3)
$\displaystyle[[x,y]_{1},\alpha(z)]_{2}+[[y,z]_{1},\alpha(x)]_{2}+[[z,x]_{1},\alpha(y)]_{2}+[[x,y]_{2},\alpha(z)]_{1}+[[y,z]_{2},\alpha(x)]_{1}+[[z,x]_{2},\alpha(y)]_{1}=0,$

for all $x,y,z\in\mathfrak{g}.$

######  3.2 Definition.

A compatible Hom-Lie algebra is a quadruple
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ in which
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$ are both Hom-Lie algebras and are
compatible.

In this case, we say that the pair $([~{},~{}]_{1},[~{},~{}]_{2})$ is a
compatible Hom-Lie algebra structure on $\mathfrak{g}$ when the twisting map
$\alpha$ is clear from the context.

Compatible Hom-Lie algebras are twisted version of compatible Lie algebras
[12]. Recall that a compatible Lie algebra is a triple
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2})$ in which
$(\mathfrak{g},[~{},~{}]_{1})$ and $(\mathfrak{g},[~{},~{}]_{2})$ are Lie
algebras and are compatible in the sense that
$\lambda[~{},~{}]_{1}+\eta[~{},~{}]_{2}$ is a Lie bracket on $\mathfrak{g}$,
for all $\lambda,\eta\in\mathbb{K}$. Thus, a compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ with $\alpha=\mathrm{id}$
is nothing but a compatible Lie algebra.

It is known that compatible Lie algebras are closely related with compatible
Poisson structures [12]. Hence compatible Poisson structures can be thought of
as a motivation to study compatible Lie algebras. In the following, we study
compatible Hom-Poisson structures and relate them with compatible Hom-Lie
algebras. Recall that a Hom-Poisson manifold is a triple
$(M,\\{~{},~{}\\},\triangle)$ consisting of a smooth manifold $M$, a bilinear
skew-symmetric operation $\\{~{},~{}\\}:C^{\infty}(M)\times
C^{\infty}(M)\rightarrow C^{\infty}(M)$ and a linear map
$\triangle:C^{\infty}(M)\rightarrow C^{\infty}(M)$ satisfying for $f,g,h\in
C^{\infty}(M)$,

$\displaystyle\triangle\\{f,g\\}=\\{\triangle f,\triangle
g\\},\qquad\triangle(fg)=(\triangle f)(\triangle g),$
$\displaystyle\\{f,gh\\}=(\triangle g)\\{f,h\\}+\\{f,g\\}(\triangle h),$
$\displaystyle\\{\\{f,g\\},\triangle h\\}+\\{\\{g,h\\},\triangle
f\\}+\\{\\{h,f\\},\triangle g\\}=0.$

See [2] for more details. If $(M,\\{~{},~{}\\})$ is a Poisson manifold and
$\varphi:M\rightarrow M$ is a Poisson morphism then
$(M,\\{~{},~{}\\}_{\varphi}=\varphi^{*}\circ\\{~{},~{}\\},\triangle=\varphi^{*})$
is a Hom-Poisson manifold.

Let $(M,\\{~{},~{}\\}_{1},\triangle)$ and $(M,\\{~{},~{}\\}_{1},\triangle)$ be
two Hom-Poisson manifolds with same underlying manifold $M$ and same twisting
map $\triangle$. These two Hom-Poisson manifolds are said to be compatible if
for any $\lambda,\eta\in\mathbb{K}$, the triple
$(M,\lambda\\{~{},~{}\\}_{1}+\eta\\{~{},~{}\\}_{2},\triangle)$ is also a Hom-
Poisson manifold. In this case, we say that
$(M,\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is a compatible Hom-
Poisson manifold. A compatible Hom-Poisson manifold
$(M,\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is said to be ‘linear’ if
$M$ is a vector space and $\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle$
takes linear maps to linear maps.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. For any $f\in C^{\infty}(\mathfrak{g}^{*})$ and
$\xi\in\mathfrak{g}^{*}$, we consider the tangent map of $f$ at the point
$\alpha^{*}\xi$,

$\displaystyle\mathfrak{g}^{*}~{}\cong~{}T_{\alpha^{*}\xi}\mathfrak{g}^{*}\xrightarrow{T_{\alpha^{*}\xi}f}T_{f(\alpha^{*}\xi)}\mathbb{K}~{}\cong~{}\mathbb{K}.$

Since $T_{\alpha^{*}\xi}f$ is a linear map, it corresponds to an element of
$\mathfrak{g}$. Using this notation, we will now define brackets
$\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{1}:C^{\infty}(\mathfrak{g}^{*})\times
C^{\infty}(\mathfrak{g}^{*})\rightarrow C^{\infty}(\mathfrak{g}^{*})$ as
follows:

$\displaystyle\\{f,g\\}_{1}(\xi):=\langle[T_{\alpha^{*}\xi}f,T_{\alpha^{*}\xi}g]_{1},\xi\rangle~{}~{}~{}~{}\text{
and
}~{}~{}~{}~{}\\{f,g\\}_{2}(\xi):=\langle[T_{\alpha^{*}\xi}f,T_{\alpha^{*}\xi}g]_{2},\xi\rangle,~{}\text{
for }f,g\in C^{\infty}(\mathfrak{g}^{*}).$

By using the fact that $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ is a Hom-Lie
algebra, it is easy to verify that the triple
$(\mathfrak{g}^{*},\\{~{},~{}\\}_{1},\triangle)$ is a Hom-Poisson manifold,
where $\triangle:C^{\infty}(\mathfrak{g}^{*})\rightarrow
C^{\infty}(\mathfrak{g}^{*})$ is the map $\triangle(f)=f\circ\alpha^{*}$.
Similarly, $(\mathfrak{g},[~{},~{}]_{2},\alpha)$ is a Hom-Lie algebra implies
that $(\mathfrak{g}^{*},\\{~{},~{}\\}_{2},\triangle)$ is a Hom-Poisson
manifold. Finally, the compatibility condition of the compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ ensures that
$(\mathfrak{g}^{*},\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is a
compatible Hom-Poisson manifold. Finally, if $f,g$ are two linear functions on
$\mathfrak{g}^{*}$, then $\\{f,g\\}_{1},\\{f,g\\}_{2}$ and $\triangle(f)$ are
all linear functions on $\mathfrak{g}^{*}$. Hence it follows that
$(\mathfrak{g}^{*},\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is a linear
compatible Hom-Poisson manifold.

######  3.3 Definition.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ and
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime}_{1},[~{},~{}]^{\prime}_{2},\alpha^{\prime})$
be two compatible Hom-Lie algebras. A morphism between them is a linear map
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ which is a Hom-Lie algebra
morphism from $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ to
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime}_{1},\alpha^{\prime})$, and a Hom-
Lie algebra morphism from $(\mathfrak{g},[~{},~{}]_{2},\alpha)$ to
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime}_{2},\alpha^{\prime}).$

In the following, we give some examples of compatible Hom-Lie algebras.

######  3.4 Example.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2})$ be a compatible Lie algebra
and $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a compatible Hom-Lie
algebra homomorphism, i.e., $\alpha$ is a Lie algebra homomorphism for both
the Lie algebras $(\mathfrak{g},[~{},~{}]_{1})$ and
$(\mathfrak{g},[~{},~{}]_{2})$. Then the quadruple
$(\mathfrak{g},\alpha\circ[~{},~{}]_{1},\alpha\circ[~{},~{}]_{2},\alpha)$ is a
compatible Hom-Lie algebra.

######  3.5 Example.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. Then for each $n\geq 0$, the quadruple
$(\mathfrak{g},[~{},~{}]_{1}^{(n)}=\alpha^{n}\circ[~{},~{}]_{1},[~{},~{}]_{2}^{(n)}=\alpha^{n}\circ[~{},~{}]_{2},\alpha^{n+1})$
is a compatible Hom-Lie algebra. This is the $n$-th derived compatible Hom-Lie
algebra.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. A Nijenhuis
operator on this Hom-Lie algebra is a linear map
$N:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying $\alpha\circ
N=N\circ\alpha$ and

$\displaystyle[Nx,Ny]=N([Nx,y]+[x,Ny]-N[x,y]),~{}\text{ for
}x,y\in\mathfrak{g}.$

Nijenhuis operators are useful to study linear deformations of a Hom-Lie
algebra [6]. Consider the $4$-dimensional Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$, where $\mathfrak{g}=\langle
e_{1},e_{2},e_{3},e_{4}\rangle$ and structure maps are given by

$\displaystyle[e_{1},e_{2}]=ae_{1}+ae_{2},$
$\displaystyle\alpha(e_{1})=e_{2},~{}~{}~{}~{}\alpha(e_{2})=e_{1},~{}~{}~{}~{}\alpha(e_{3})=0~{}~{}~{}\text{
and }~{}~{}~{}\alpha(e_{4})=e_{3}.$

Then it is easy to verify that the map $N:\mathfrak{g}\rightarrow\mathfrak{g}$
defined by

$\displaystyle
N(e_{1})=e_{2},~{}~{}~{}~{}N(e_{2})=e_{1},~{}~{}~{}~{}N(e_{3})=e_{3},~{}~{}~{}~{}N(e_{4})=e_{4}$

is a Nijenhuis operator on the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$.

######  3.6 Example.

Let $N$ be a Nijenhuis operator on a Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$. Then there is a deformed Hom-Lie bracket on
$\mathfrak{g}$ given by

$\displaystyle[x,y]_{N}:=[Nx,y]+[x,Ny]-N[x,y],~{}\text{ for
}x,y\in\mathfrak{g}.$

In other words $(\mathfrak{g},[~{},~{}]_{N},\alpha)$ is a Hom-Lie algebra. It
is easy to see that the quadruple
$(\mathfrak{g},[~{},~{}],[~{},~{}]_{N},\alpha)$ is a compatible Hom-Lie
algebra.

######  3.7 Example.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra and
$(V,\bullet,\beta)$ be a representation of it. Suppose $f\in
C^{2}_{\mathrm{Hom}}(\mathfrak{g},V)$ is a $2$-cocycle in the Chevalley-
Eilenberg cohomology complex of the Hom-Lie algebra $\mathfrak{g}$ with
coefficients in $V$. Then the direct sum vector space $\mathfrak{g}\oplus V$
inherits a Hom-Lie algebra structure (called the $f$-twisted semidirect
product) whose bracket is given by

$\displaystyle[(x,u),(y,v)]_{\ltimes_{f}}:=([x,y],x\bullet v-y\bullet
u+f(x,y)),~{}\text{ for }(x,u),(y,v)\in\mathfrak{g}\oplus V$

and the linear twisting map on $\mathfrak{g}\oplus V$ is given by
$\alpha\oplus\beta$. It is easy to verify that the quadruple
$(\mathfrak{g}\oplus
V,[~{},~{}]_{\ltimes_{0}},[~{},~{}]_{\ltimes_{f}},\alpha\oplus\beta)$ is a
compatible Hom-Lie algebra.

Another example of a compatible Hom-Lie algebra arises from compatible Rota-
Baxter operators on a Hom-Lie algebra. Rota-Baxter operators are an algebraic
abstraction of the integral operator and operator analogue of Poisson
structures [10].

######  3.8 Definition.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. A linear map
$R:\mathfrak{g}\rightarrow\mathfrak{g}$ is said to be a Rota-Baxter operator
of weight $\lambda\in\mathbb{K}$ on the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$ if $R$ satisfies $\alpha\circ
R=R\circ\alpha$ and

$\displaystyle[Rx,Ry]=R([Rx,y]+[x,Ry]+\lambda[x,y]),~{}\text{ for
}x,y\in\mathfrak{g}.$

A Rota-Baxter operator $R$ induces a new Hom-Lie algebra structure on
$\mathfrak{g}$ with the Hom-Lie bracket

$\displaystyle[x,y]_{R}:=[Rx,y]+[x,Ry]+\lambda[x,y],~{}\text{ for
}x,y\in\mathfrak{g}.$

Compatible Poisson structures first appeared in the context of bihamiltonian
mechanics [12, 16]. The operator version of compatible Poisson structures in
the Hom-Lie algebra context is given by the following.

######  3.9 Definition.

Two Rota-Baxter operators $R$ and $S$ of same weight $\lambda\in\mathbb{K}$ on
a Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$ are said to be compatible
if

$\displaystyle[Rx,Sy]+[Sx,Ry]=R([Sx,y]+[x,Sy])+S([Rx,y]+[x,Ry]),~{}\text{ for
}x,y\in\mathfrak{g}.$

######  3.10 Example.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be the Hom-Lie algebra given by
$\mathfrak{g}=\langle e_{1},e_{2}\rangle$ and the structure maps are given by

$\displaystyle[e_{1},e_{2}]=ae_{1}+ae_{2},~{}~{}~{}\alpha(e_{1})=e_{2},~{}~{}~{}\alpha(e_{2})=e_{1}.$

Then the map $R=\alpha$ is a Rota-Baxter operator of weight $\lambda=-1$.

Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra and let
$R:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Rota-Baxter operator of weight
$\lambda$. Then $-\lambda\mathrm{id}-R:\mathfrak{g}\rightarrow\mathfrak{g}$ is
also a Rota-Baxter operator of the same weight. Moreover, $R$ and
$-\lambda\mathrm{id}-R$ are compatible.

The proof of the following proposition is straightforward.

######  3.11 Proposition.

Let $R$ and $S$ be two compatible Rota-Baxter operators of weight
$\lambda\in\mathbb{K}$ on a Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha).$
Then $(\mathfrak{g},[~{},~{}]_{R},[~{},~{}]_{S},\alpha)$ is a compatible Hom-
Lie algebra.

Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map. Consider the
graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}})$
given in Section 2. Hence by Remark 2.10, the quadruple

$\displaystyle(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=0,d_{2}=0)$

is a bidifferential graded Lie algebra. Then we have the following Maurer-
Cartan characterization of compatible Hom-Lie algebras.

######  3.12 Theorem.

There is a one-to-one correspondence between compatible Hom-Lie algebra
structures on $\mathfrak{g}$ and Maurer-Cartan elements in the bidifferential
graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=0,d_{2}=0)$.

###### Proof.

Let $[~{},~{}]_{1}$ and $[~{},~{}]_{2}$ be two multiplicative skew-symmetric
bilinear brackets on $\mathfrak{g}$. Then the brackets $[~{},~{}]_{1}$ and
$[~{},~{}]_{2}$ correspond to elements (say, $\mu_{1}$ and $\mu_{2}$,
respectively) in $C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$. Then

$\displaystyle[~{},~{}]_{1}\text{ is a Hom-Lie bracket }\Leftrightarrow~{}$
$\displaystyle~{}~{}[\mu_{1},\mu_{1}]_{\mathsf{NR}}=0;$
$\displaystyle[~{},~{}]_{2}\text{ is a Hom-Lie bracket }\Leftrightarrow~{}$
$\displaystyle~{}~{}[\mu_{2},\mu_{2}]_{\mathsf{NR}}=0;$ $\displaystyle\text{
compatibility condition }(\ref{comp-cond-e})~{}\Leftrightarrow~{}$
$\displaystyle~{}~{}[\mu_{1},\mu_{2}]_{\mathsf{NR}}=0.$

Hence $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a compatible Hom-
Lie algebra if and only if $(\mu_{1},\mu_{2})$ is a Maurer-Cartan element in
the bidifferential graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=0,d_{2}=0)$.
∎

Hence from Proposition 2.12, we get the following.

######  3.13 Proposition.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. Then for any multiplicative skew-symmetric bilinear operations
$[~{},~{}]_{1}^{\prime}$ and $[~{},~{}]_{2}^{\prime}$ on $\mathfrak{g}$, the
quadruple

$\displaystyle(\mathfrak{g},[~{},~{}]_{1}+[~{},~{}]_{1}^{\prime},[~{},~{}]_{2}+[~{},~{}]_{2}^{\prime},\alpha)$

is a compatible Hom-Lie algebra if and only if
$(\mu_{1}^{\prime},\mu_{2}^{\prime})$ is a Maurer-Cartan element in the
bidifferential graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=[\mu_{1},-],d_{2}=[\mu_{2},-])$.
Here $\mu_{1}^{\prime},\mu_{2}^{\prime}\in
C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ denote the elements
corresponding to the brackets $[~{},~{}]_{1}^{\prime}$ and
$[~{},~{}]_{2}^{\prime}$, respectively.

In the following, we define representations of a compatible Hom-Lie algebra.

######  3.14 Definition.

A representation of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ consists of a quadruple
$(V,\bullet_{1},\bullet_{2},\beta)$ such that

  * (i)

$(V,\bullet_{1},\beta)$ is a representation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$;

  * (ii)

$(V,\bullet_{2},\beta)$ is a representation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$;

  * (iii)

the following compatibility condition holds

$\displaystyle[x,y]_{1}\bullet_{2}\beta(v)+[x,y]_{2}\bullet_{1}\beta(v)=\alpha(x)\bullet_{1}(y\bullet_{2}v)-\alpha(y)\bullet_{2}(x\bullet_{1}v)+\alpha(x)\bullet_{2}(y\bullet_{1}v)-\alpha(y)\bullet_{1}(x\bullet_{2}v),$

for all $x,y\in\mathfrak{g}$ and $v\in V$.

It follows that any compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a representation of
itself, where $\bullet_{1}=[~{},~{}]_{1}$ and $\bullet_{2}=[~{},~{}]_{2}$.
This is called the adjoint representation.

######  3.15 Remark.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
Then for any $\lambda,\eta\in\mathbb{K}$, the triple
$(\mathfrak{g},\lambda[~{},~{}]_{1}+\eta[~{},~{}]_{2},\alpha)$ is a Hom-Lie
algebra and $(V,\lambda~{}\bullet_{1}+\eta~{}\bullet_{2},\beta)$ is a
representation of it.

The proof of the following proposition is similar to the standard case.

######  3.16 Proposition.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
Then the direct sum $\mathfrak{g}\oplus V$ carries a compatible Hom-Lie
algebra structure with the linear homomorphism $\alpha\oplus\beta$, and Hom-
Lie brackets

$\displaystyle[(x,u),(y,v)]_{i}^{\ltimes}:=([x,y]_{i},x\bullet_{i}v-y\bullet_{i}u),~{}\text{
for }i=1,2\text{ and }(x,u),(y,v)\in\mathfrak{g}\oplus V.$

This is called the semidirect product.

##  4 Cohomology of compatible Hom-Lie algebras

In this section, we introduce the cohomology of a compatible Hom-Lie algebra
with coefficients in a representation. We also define abelian extensions of a
compatible Hom-Lie algebra and prove that isomorphism classes of abelian
extensions are classified by the second cohomology group.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
Let
${}^{1}\delta_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},V)$ (resp.
${}^{2}\delta_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},V)$ ), for $n\geq 0$, be the coboundary
operator for the Chevalley-Eilenberg cohomology of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$ with coefficients in the representation
$(V,\bullet_{1},\beta)$ (resp. of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$ with coefficients in the representation
$(V,\bullet_{2},\beta)$ ). Then we have

$\displaystyle({}^{1}\delta_{\mathrm{Hom}})^{2}=0~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}({}^{2}\delta_{\mathrm{Hom}})^{2}=0.$

Moreover, we have the following.

######  4.1 Proposition.

The coboundary operators ${}^{1}\delta_{\mathrm{Hom}}$ and
${}^{2}\delta_{\mathrm{Hom}}$ satisfy the following compatibility

$\displaystyle{}^{1}\delta_{\mathrm{Hom}}\circ{}^{2}\delta_{\mathrm{Hom}}+{}^{2}\delta_{\mathrm{Hom}}\circ{}^{1}\delta_{\mathrm{Hom}}=0.$

Before we prove the above proposition, we first observe the followings. For a
compatible Hom-Lie algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$
and a representation $(V,\bullet_{1},\bullet_{2},\beta)$, we consider the
semidirect product compatible Hom-Lie algebra structure on $\mathfrak{g}\oplus
V$ given in Proposition 3.16. We denote by $\pi_{1},\pi_{2}\in
C^{2}_{\mathrm{Hom}}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus V)$ the elements
corresponding to the Hom-Lie brackets $[~{},~{}]_{1}^{\ltimes}$ and
$[~{},~{}]_{2}^{\ltimes}$ on $\mathfrak{g}\oplus V$, respectively. Let

$\displaystyle\delta^{1}_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)\rightarrow C^{n+1}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V),~{}~{}\text{ for }n\geq 0,$
$\displaystyle\delta^{2}_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)\rightarrow C^{n+1}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V),~{}~{}\text{ for }n\geq 0,$

denote respectively the coboundary operator for the Chevalley-Eilenberg
cohomology of the Hom-Lie algebra $(\mathfrak{g}\oplus
V,[~{},~{}]_{1}^{\ltimes},\alpha\oplus\beta)$ (resp. of the Hom-Lie algebra
$(\mathfrak{g}\oplus V,[~{},~{}]_{2}^{\ltimes},\alpha\oplus\beta)$ ) with
coefficients in itself. Note that any map $f\in
C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus V)$ can be lifted to a map
$\widetilde{f}\in C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus
V)$ by

$\displaystyle\widetilde{f}\big{(}(x_{1},v_{1}),\ldots,(x_{n},v_{n})\big{)}=\big{(}0,f(x_{1},\ldots,x_{n})\big{)}.$

Then $f=0$ if and only if $\widetilde{f}=0$.

With these notations, for any $f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$, we
have

$\displaystyle\widetilde{({}^{1}\delta_{\mathrm{Hom}}f)}=\delta^{1}_{\mathrm{Hom}}(\widetilde{f})=(-1)^{n-1}[\pi_{1},\widetilde{f}]_{\mathsf{NR}}~{}~{}~{}~{}\text{
and
}~{}~{}~{}~{}\widetilde{({}^{2}\delta_{\mathrm{Hom}}f)}=\delta^{2}_{\mathrm{Hom}}(\widetilde{f})=(-1)^{n-1}[\pi_{2},\widetilde{f}]_{\mathsf{NR}}.$

###### Proof.

(of Proposition 4.1) For any $f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$, we
have

$\displaystyle\widetilde{({}^{1}\delta_{\mathrm{Hom}}\circ{}^{2}\delta_{\mathrm{Hom}}+{}^{2}\delta_{\mathrm{Hom}}\circ{}^{1}\delta_{\mathrm{Hom}})(f)}$
$\displaystyle=\widetilde{{}^{1}\delta_{\mathrm{Hom}}({}^{2}\delta_{\mathrm{Hom}}{f})}~{}+~{}\widetilde{{}^{2}\delta_{\mathrm{Hom}}({}^{1}\delta_{\mathrm{Hom}}{f})}$
$\displaystyle=(-1)^{n}~{}[\pi_{1},\widetilde{{}^{2}\delta_{\mathrm{Hom}}{f}}]_{\mathsf{NR}}~{}+~{}(-1)^{n}~{}[\pi_{2},\widetilde{{}^{1}\delta_{\mathrm{Hom}}{f}}]_{\mathsf{NR}}$
$\displaystyle=-[\pi_{1},[\pi_{2},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}-[\pi_{2},[\pi_{1},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=-[[\pi_{1},\pi_{2}]_{\mathsf{NR}},\widetilde{f}]_{\mathsf{NR}}~{}+~{}[\pi_{2},[\pi_{1},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}-[\pi_{2},[\pi_{1},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=0~{}~{}~{}~{}\qquad(\because~{}[\pi_{1},\pi_{2}]_{\mathsf{NR}}=0).$

Therefore, it follows that
$({}^{1}\delta_{\mathrm{Hom}}\circ{}^{2}\delta_{\mathrm{Hom}}+{}^{2}\delta_{\mathrm{Hom}}\circ{}^{1}\delta_{\mathrm{Hom}})(f)=0$.
Hence the result follows. ∎

We are now in a position to define the cohomology of a compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ with coefficients
in a representation $(V,\bullet_{1},\bullet_{2},\beta)$. For each $n\geq 0$,
we define an abelian group $C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)$ as follows:

$\displaystyle C^{0}_{\mathrm{cHom}}(\mathfrak{g},V):=~{}$ $\displaystyle
C^{0}_{\mathrm{Hom}}(\mathfrak{g},V)\cap\\{v\in
V|~{}x\bullet_{1}v=x\bullet_{2}v,\forall x\in\mathfrak{g}\\}$
$\displaystyle=~{}$ $\displaystyle\\{v\in V|~{}\beta(v)=v\text{ and
}x\bullet_{1}v=x\bullet_{2}v,\forall x\in\mathfrak{g}\\},$ $\displaystyle
C^{n}_{\mathrm{cHom}}(\mathfrak{g},V):=\underbrace{C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\oplus\cdots\oplus
C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)}_{n\text{ copies }},~{}\text{ for }n\geq
1.$

Define a map
$\delta_{\mathrm{cHom}}:C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{cHom}}(\mathfrak{g},V)$, for $n\geq 0$ by

$\displaystyle\delta_{\mathrm{cHom}}(v)(x):=x\bullet_{1}v=x\bullet_{2}v,~{}\text{
for }v\in C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)~{}\text{ and
}x\in\mathfrak{g},$
$\displaystyle\delta_{\mathrm{cHom}}(f_{1},\ldots,f_{n}):=\big{(}{}^{1}\delta_{\mathrm{Hom}}f_{1},\ldots,\underbrace{{}^{1}\delta_{\mathrm{Hom}}f_{i}+{}^{2}\delta_{\mathrm{Hom}}f_{i-1}}_{i\text{-th
position}},\ldots,{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)},$

for $(f_{1},\ldots,f_{n})\in C^{n}_{\mathrm{cHom}}(\mathfrak{g},V).$

Then we have the following.

######  4.2 Proposition.

The map $\delta_{\mathrm{cHom}}$ is a coboundary map, i.e.,
$(\delta_{\mathrm{cHom}})^{2}=0$.

###### Proof.

For any $v\in C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$, we have

$\displaystyle(\delta_{\mathrm{cHom}})^{2}(v)=\delta_{\mathrm{cHom}}(\delta_{\mathrm{cHom}}v)=~{}$
$\displaystyle({}^{1}\delta_{\mathrm{Hom}}\delta_{\mathrm{cHom}}v~{},{}^{2}\delta_{\mathrm{Hom}}\delta_{\mathrm{cHom}}v)$
$\displaystyle=~{}$
$\displaystyle({}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}v~{},{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}v)=0.$

Moreover, for any $(f_{1},\ldots,f_{n})\in
C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)$, $n\geq 1$, we have

$\displaystyle(\delta_{\mathrm{cHom}})^{2}(f_{1},\ldots,f_{n})$
$\displaystyle=\delta_{\mathrm{cHom}}\big{(}{}^{1}\delta_{\mathrm{Hom}}f_{1},\ldots,{}^{1}\delta_{\mathrm{Hom}}f_{i}+{}^{2}\delta_{\mathrm{Hom}}f_{i-1},\ldots,{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)}$
$\displaystyle=\big{(}{}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{1}~{},{}^{2}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{1}+{}^{1}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{1}+{}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{2}~{},\ldots,$
$\displaystyle\qquad\underbrace{{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{i-2}+{}^{2}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{i-1}+{}^{1}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{i-1}+{}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{i}}_{3\leq
i\leq n-1},\ldots,$
$\displaystyle\qquad{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{n-1}+{}^{2}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{n}+{}^{1}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{n}~{},{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)}$
$\displaystyle=0~{}~{}~{}\quad(\text{from Proposition }\ref{d-comp}).$

This proves that $(\delta_{\mathrm{cHom}})^{2}=0$. ∎

It follows from the above proposition that
$\\{C^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V),\delta_{\mathrm{cHom}}\\}$ is a
cochain complex. The corresponding cohomology groups

$\displaystyle
H^{n}_{\mathrm{cHom}}(\mathfrak{g},V):=\frac{Z^{n}_{\mathrm{cHom}}(\mathfrak{g},V)}{B^{n}_{\mathrm{cHom}}(\mathfrak{g},V)}=\frac{\mathrm{Ker~{}}\delta_{\mathrm{cHom}}:C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{cHom}}(\mathfrak{g},V)}{\mathrm{Im~{}}\delta_{\mathrm{cHom}}:C^{n-1}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)},\text{ for }n\geq 0$

are called the cohomology of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ with coefficients in the
representation $(V,\bullet_{1},\bullet_{2},\beta)$.

It follows from the above definition that

$\displaystyle H^{0}_{\mathrm{cHom}}(\mathfrak{g},V)=\\{v\in
V|~{}\beta(v)=v\text{ and }x\bullet_{1}v=x\bullet_{2}v=0,~{}\forall
x\in\mathfrak{g}\\}.$

A linear map $D:\mathfrak{g}\rightarrow V$ is said to be a derivation if
$\beta\circ D=D\circ\alpha$ and

$\displaystyle D[x,y]_{1}=x\bullet_{1}Dy-y\bullet_{1}Dx,\qquad
D[x,y]_{2}=x\bullet_{2}Dy-y\bullet_{2}Dx,\text{ for }x,y\in\mathfrak{g}.$

We denote the space of derivations by $\mathrm{Der}(\mathfrak{g},V)$. A
derivation $D$ is said to be inner if it is of the form
$D=-\bullet_{1}v=-\bullet_{2}v$, for some $v\in
C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$. The space of inner derivations are
denoted by $\mathrm{InnDer}(\mathfrak{g},V)$. Then we have

$\displaystyle
H^{1}_{\mathrm{cHom}}(\mathfrak{g},V)=\frac{\mathrm{Der}(\mathfrak{g},V)}{\mathrm{InnDer}(\mathfrak{g},V)}=\mathrm{OutDer}(\mathfrak{g},V),\text{
the space of outer derivations.}$

Let $\mathfrak{g}=(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a
compatible Hom-Lie algebra and $V=(V,\bullet_{1},\bullet_{2},\beta)$ be a
representation of it. Then we know from Remark 3.15 that
$\mathfrak{g}_{+}=(\mathfrak{g},[~{},~{}]_{1}+[~{},~{}]_{2},\alpha)$ is a Hom-
Lie algebra and $V_{+}=(V,\bullet_{1}+\bullet_{2},\beta)$ is a representation
of it. Consider the cochain complex
$\\{C^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V),\delta_{\mathrm{cHom}}\\}$ of the
compatible Hom-Lie algebra $\mathfrak{g}$ with coefficients in the
representation $V$, and the cochain complex
$\\{C^{\ast}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+}),\delta_{\mathrm{Hom}}\\}$
of the Hom-Lie algebra $\mathfrak{g}_{+}$ with coefficients in $V_{+}$.

For each $n\geq 0$, we define a map

$\displaystyle\triangle_{n}:C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+})~{}~{}~{}\text{ by
}~{}~{}~{}\begin{cases}\triangle_{0}(v)=\frac{1}{2}v,\\\
\triangle_{n}((f_{1},\ldots,f_{n}))=f_{1}+\cdots+f_{n},\end{cases}$

for $v\in C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$ and $(f_{1},\ldots,f_{n})\in
C^{n\geq 1}_{\mathrm{cHom}}(\mathfrak{g},V)$. If $v\in
C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$, then

$\displaystyle(\delta_{\mathrm{Hom}}\circ\triangle_{0}(v))(x)=\frac{1}{2}(\delta_{\mathrm{Hom}}(v))(x)=\frac{1}{2}(x\bullet_{1}v+x\bullet_{2}v)=~{}$
$\displaystyle x\bullet_{1}v=x\bullet_{2}v$ $\displaystyle=~{}$
$\displaystyle\delta_{\mathrm{cHom}}(v)(x)=(\triangle_{1}\circ\delta_{\mathrm{cHom}}(v))(x).$

Moreover, if $(f_{1},\ldots,f_{n})\in C^{n\geq
1}_{\mathrm{cHom}}(\mathfrak{g},V)$, then

$\displaystyle(\delta_{\mathrm{Hom}}\circ\triangle_{n})((f_{1},\ldots,f_{n}))=~{}$
$\displaystyle\delta_{\mathrm{Hom}}(f_{1}+\cdots+f_{n})$ $\displaystyle=~{}$
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}(f_{1},\ldots,f_{n})+{}^{2}\delta_{\mathrm{Hom}}(f_{1},\ldots,f_{n})$
$\displaystyle=~{}$
$\displaystyle\triangle_{n+1}\big{(}{}^{1}\delta_{\mathrm{Hom}}f_{1},\ldots,{}^{1}\delta_{\mathrm{Hom}}f_{i}+{}^{2}\delta_{\mathrm{Hom}}f_{i-1},\ldots,{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)}$
$\displaystyle=~{}$
$\displaystyle(\triangle_{n+1}\circ\delta_{\mathrm{cHom}})((f_{1},\ldots,f_{n})).$

Therefore, we get the following.

######  4.3 Theorem.

The collection $\\{\triangle_{n}\\}_{n\geq 0}$ defines a morphism of cochain
complexes from
$\\{C^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V),\delta_{\mathrm{cHom}}\\}$ to
$\\{C^{\ast}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+}),\delta_{\mathrm{Hom}}\\}$.
Hence, it induces a morphism
$H^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
H^{\ast}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+})$ between corresponding
cohomologies.

Abelian extensions of compatible Hom-Lie algebras. Here, we study abelian
extensions of compatible Hom-Lie algebras and give a classification of
equivalence classes of abelian extensions.

Let $\mathfrak{g}=(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a
compatible Hom-Lie algebra and $(V,\beta)$ be a pair of a vector space with a
linear map. Note that $(V,\beta)$ can be considered as a compatible Hom-Lie
algebra with both the Hom-Lie brackets on $V$ are trivial.

######  4.4 Definition.

An abelian extension of $\mathfrak{g}$ by $V$ is an exact sequence of
compatible Hom-Lie algebras

(6)

together with a $\mathbb{K}$-splitting (given by $s$) satisfying

(7) $\displaystyle\alpha^{\mathfrak{h}}\circ s=s\circ\alpha.$

We denote an abelian extension as above simply by $\mathfrak{h}$ when all the
structures of the exact sequence (6) are understood. Note that an abelian
extension induces a representation of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ on $(V,\beta)$ with the
actions

$\displaystyle x\bullet_{1}v=[s(x),i(v)]_{1}^{\mathfrak{h}}~{}~{}~{}\text{ and
}~{}~{}~{}x\bullet_{2}v=[s(x),i(v)]_{2}^{\mathfrak{h}},\text{ for
}x\in\mathfrak{g},v\in V.$

It is easy to see that the above action is independent of the choice of $s$.

######  4.5 Remark.

Let $(\mathfrak{h},\alpha^{\mathfrak{h}})$ and $(\mathfrak{g},\alpha)$ be two
pairs of vector spaces endowed with linear maps. Suppose
$j:\mathfrak{h}\rightarrow\mathfrak{g}$ is a linear map satisfying
$\alpha\circ j=j\circ\alpha^{\mathfrak{h}}$. Then there might not be a linear
map $s:\mathfrak{g}\rightarrow\mathfrak{h}$ that satisfies
$\alpha^{\mathfrak{h}}\circ s=s\circ\alpha$. Take $\mathfrak{h}=\langle
e_{1},e_{2}\rangle$ and $\alpha^{\mathfrak{h}}(e_{1})=e_{2}$,
$\alpha^{\mathfrak{h}}(e_{2})=0$. Also take $\mathfrak{g}=\langle f\rangle$
and $\alpha(f)=0$. Let $j:\mathfrak{h}\rightarrow\mathfrak{g}$ be the map
defined by $j(e_{1})=f$ and $j(e_{2})=0$. Let
$s:\mathfrak{g}\rightarrow\mathfrak{h}$ be a map satisfying
$\alpha^{\mathfrak{h}}\circ s=s\circ\alpha$. For $s(f)=\lambda e_{1}+\eta
e_{2}$, we have $f=(j\circ s)(f)=\lambda f$ which implies that $\lambda=1$.
Thus,

$\displaystyle 0=(s\circ\alpha)(f)=(\alpha^{\mathfrak{h}}\circ
s)(f)=\alpha^{\mathfrak{h}}(e_{1}+\eta e_{2})=e_{2}$

leads to a contradiction.

######  4.6 Definition.

Two abelian extensions $\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ are said to
be equivalent if there is a compatible Hom-Lie algebra homomorphism
$\phi:\mathfrak{h}\rightarrow\mathfrak{h}^{\prime}$ making the following
diagram commutative

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V,0,0,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{(\mathfrak{h},[~{},~{}]_{1}^{\mathfrak{h}},[~{},~{}]_{2}^{\mathfrak{h}},\alpha^{\mathfrak{h}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{j}$$\textstyle{(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V,0,0,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{(\mathfrak{h}^{\prime},[~{},~{}]_{1}^{\mathfrak{h}^{\prime}},[~{},~{}]_{2}^{\mathfrak{h}^{\prime}},\alpha^{\mathfrak{h}^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\textstyle{(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{\prime}}$$\textstyle{0.}$

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
We denote $\mathrm{Ext}(\mathfrak{g},V)$ by the set of equivalence classes of
abelian extensions of $\mathfrak{g}$ by $V$ for which the induced
representation on $V$ is the given one.

With the above notation, we have the following.

######  4.7 Theorem.

There is a bijection
$H^{2}_{\mathrm{cHom}}(\mathfrak{g},V)\cong\mathrm{Ext}(\mathfrak{g},V).$

###### Proof.

Let $(f_{1},f_{2})\in Z^{2}_{\mathrm{cHom}}(\mathfrak{g},V)$ be any
$2$-cocycle. Then it induces a compatible Hom-Lie algebra structure on
$\mathfrak{h}=\mathfrak{g}\oplus V$ with structure maps

$\displaystyle[(x,u),(y,v)]_{i}^{\ltimes}:=~{}$
$\displaystyle([x,y],x\bullet_{i}v-y\bullet_{i}u+f_{i}(x,y)),~{}\text{ for
}i=1,2,$ $\displaystyle\alpha^{\mathfrak{h}}(x,u):=~{}$
$\displaystyle(\alpha(x),\beta(u)),~{}\text{ for }(x,u),(y,v)\in\mathfrak{h}.$

Moreover, this compatible Hom-Lie algebra makes

into an abelian extension with the obvious splitting $s$. It is easy to see
that any other cohomologous $2$-cocycle gives rise to an equivalent abelian
extension. Hence the map
$H^{2}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow\mathrm{Ext}(\mathfrak{g},V)$
is well defined.

Conversely, let (6) be an abelian extension with splitting $s$. Then we may
consider $\mathfrak{h}=\mathfrak{g}\oplus V$ and $s$ is the map $s(x)=(x,0)$,
for $x\in\mathfrak{g}.$ The map $i$ and $j$ are the obvious ones. Moreover, it
follows from (7) that $\alpha^{\mathfrak{h}}=(\alpha,\beta)$. Finally, since
$j$ is a compatible Hom-Lie algebra map, we have

$\displaystyle j[(x,0),(y,0)]_{i}^{\mathfrak{h}}=[x,y]_{i},~{}\text{ for
}x,y\in\mathfrak{g}.$

This implies that $[(x,0),(y,0)]_{i}^{\mathfrak{h}}=([x,y]_{i},f_{i}(x,y)),$
for $i=1,2$ and some $f_{1},f_{2}\in C^{2}_{\mathrm{Hom}}(\mathfrak{g},V)$. As
$([~{},~{}]_{1}^{\mathfrak{h}},[~{},~{}]_{2}^{\mathfrak{h}})$ is a compatible
Hom-Lie algebra structure on $\mathfrak{h}$, we get that the pair
$(f_{1},f_{2})\in Z^{2}_{\mathrm{cHom}}(\mathfrak{g},V)$ is a $2$-cocycle. It
is left to the reader to check that equivalent abelian extensions give rise to
cohomologous $2$-cocycles. Hence the map
$\mathrm{Ext}(\mathfrak{g},V)\rightarrow
H^{2}_{\mathrm{cHom}}(\mathfrak{g},V)$ is also well defined. Finally, these
two maps are inverses to each other. Hence the proof. ∎

##  5 Deformations of compatible Hom-Lie algebras

In this section, we study linear deformations and finite order deformations of
a compatible Hom-Lie algebra generalizing the classical deformation theory of
Gerstenhaber [8]. We introduce Nijenhuis operators that generate trivial
linear deformations. We also define infinitesimal deformations of a compatible
Hom-Lie algebra and characterize equivalence classes of infinitesimal
deformations in terms of cohomology. Finally, we consider extensibility of
finite order deformations.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(\omega_{1},\omega_{2})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ be a $2$-cochain. Define two
bilinear operations on $\mathfrak{g}$ depending on the parameter $t$ as
follows:

$\displaystyle[x,y]_{1}^{t}:=[x,y]_{1}+t\omega_{1}(x,y)~{}~{}\text{ and
}~{}~{}[x,y]_{2}^{t}:=[x,y]_{2}+t\omega_{2}(x,y),\text{ for
}x,y\in\mathfrak{g}.$

######  5.1 Definition.

We say that $(\omega_{1},\omega_{2})$ generates a linear deformation of the
compatible Hom-Lie algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$
if for all $t$, the quadruple
$(\mathfrak{g},[~{},~{}]_{1}^{t},[~{},~{}]_{2}^{t},\alpha)$ is a compatible
Hom-Lie algebra.

If $\mu_{1},\mu_{2}\in C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ denote
the elements corresponding to the Hom-Lie brackets $[~{},~{}]_{1}$ and
$[~{},~{}]_{2}$, respectively, then the above definition is equivalent to
saying that

$\displaystyle[\mu_{1}+t\omega_{1},\mu_{1}+t\omega_{1}]_{\mathsf{NR}}=0,\qquad[\mu_{2}+t\omega_{2},\mu_{2}+t\omega_{2}]_{\mathsf{NR}}=0~{}~{}\text{
and }~{}~{}[\mu_{1}+t\omega_{1},\mu_{2}+t\omega_{2}]_{\mathsf{NR}}=0.$

In other words, the followings are hold

$\displaystyle[\mu_{1},\omega_{1}]_{\mathsf{NR}}=0,\qquad[\mu_{2},\omega_{2}]_{\mathsf{NR}}=0,\qquad[\mu_{1},\omega_{2}]_{\mathsf{NR}}+[\mu_{2},\omega_{1}]_{\mathsf{NR}}=0,$
$\displaystyle[\omega_{1},\omega_{1}]_{\mathsf{NR}}=0,\qquad[\omega_{2},\omega_{2}]_{\mathsf{NR}}=0,\qquad[\omega_{1},\omega_{2}]_{\mathsf{NR}}=0.$

The first three condition implies that $(\omega_{1},\omega_{2})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is a $2$-cocycle in the
cohomology of the compatible Hom-Lie algebra $\mathfrak{g}$ with coefficients
in the adjoint representation. Moreover, the last three conditions implies
that $(\mathfrak{g},\omega_{1},\omega_{2},\alpha)$ is a compatible Hom-Lie
algebra.

######  5.2 Definition.

Let $(\omega_{1},\omega_{2})$ and $(\omega_{1}^{\prime},\omega_{2}^{\prime})$
generate linear deformations
$(\mathfrak{g},[~{},~{}]_{1}^{t},[~{},~{}]_{2}^{t},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{1}^{{}^{\prime}t},[~{},~{}]_{2}^{{}^{\prime}t},\alpha)$
of a compatible Hom-Lie algebra $\mathfrak{g}.$ They are said to be equivalent
if there exists a linear map $N:\mathfrak{g}\rightarrow\mathfrak{g}$
satisfying $\alpha\circ N=N\circ\alpha$ and such that

$\displaystyle\mathrm{id}+tN:(\mathfrak{g},[~{},~{}]_{1}^{t},[~{},~{}]_{2}^{t},\alpha)\rightarrow(\mathfrak{g},[~{},~{}]_{1}^{{}^{\prime}t},[~{},~{}]_{2}^{{}^{\prime}t},\alpha)$

is a morphism of compatible Hom-Lie algebras.

One can equivalently write the explicit identities as follows:

$\displaystyle\omega_{i}(x,y)-\omega_{i}^{\prime}(x,y)=~{}$
$\displaystyle[x,Ny]_{i}+[Nx,y]_{i}-N[x,y]_{i},$ $\displaystyle
N\omega_{i}(x,y)=~{}$
$\displaystyle\omega_{i}^{\prime}(x,Ny)+\omega_{i}^{\prime}(Nx,y)+[Nx,Ny]_{i},$
$\displaystyle\omega_{i}^{\prime}(Nx,Ny)=~{}$ $\displaystyle 0,$

for $x,y\in\mathfrak{g}$ and $i=1,2$. Note that from the first identity, we
get

$\displaystyle(\omega_{1},\omega_{2})-(\omega_{1}^{\prime},\omega_{2}^{\prime})=\delta_{\mathrm{cHom}}N,$

where $N$ is considered as an element in
$C^{1}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$. Hence, summarizing the
above discussions, we get the following.

######  5.3 Theorem.

Let $\mathfrak{g}$ be a compatible Hom-Lie algebra. Then there is a map from
the set of equivalence classes of linear deformations of $\mathfrak{g}$ to the
second cohomology group $H^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$.

Next, we introduce trivial linear deformations of a compatible Hom-Lie algebra
and introduce Nijenhuis operators that generate trivial linear deformations.

######  5.4 Definition.

A linear deformation $([~{},~{}]_{1}+t\omega_{1},[~{},~{}]_{2}+t\omega_{2})$
of compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is said to be trivial if
the deformation is equivalent to the undeformed one
$([~{},~{}]_{1},[~{},~{}]_{2}).$

Thus, a linear deformation
$([~{},~{}]_{1}+t\omega_{1},[~{},~{}]_{2}+t\omega_{2})$ is trivial if and only
if there exists a linear map $N:\mathfrak{g}\rightarrow\mathfrak{g}$
satisfying $\alpha\circ N=N\circ\alpha$ and

$\displaystyle\omega_{i}(x,y)=~{}$
$\displaystyle[x,Ny]_{i}+[Nx,y]_{i}-N[x,y]_{i},$ $\displaystyle
N\omega_{i}(x,y)=~{}$ $\displaystyle[Nx,Ny]_{i},~{}\text{ for }i=1,2\text{ and
}x,y\in\mathfrak{g}.$

This motivates us to introduce Nijenhuis operators on a compatible Hom-Lie
algebra.

######  5.5 Definition.

A Nijenhuis operator on a compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a linear map
$N:\mathfrak{g}\rightarrow\mathfrak{g}$ which is a Nijenhuis operator for both
the Hom-Lie algebras $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$, i.e., $N$ satisfies $\alpha\circ
N=N\circ\alpha$ and

$\displaystyle[Nx,Ny]_{i}=N([Nx,y]_{i}+[x,Ny]_{i}-N[x,y]_{i}),~{}\text{ for
}i=1,2\text{ and }x,y\in\mathfrak{g}.$

It follows that any trivial linear deformation of a compatible Hom-Lie algebra
induces a Nijenhuis operator. The converse is given by the next result whose
proof is straightforward.

######  5.6 Proposition.

Let $N:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Nijenhuis operator on a
compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$. Then
$(\omega_{1},\omega_{2})$ generates a trivial linear deformation of the
compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$, where

$\displaystyle\omega_{i}(x,y)=[Nx,y]_{i}+[x,Ny]_{i}-N[x,y]_{i},~{}\text{ for
}i=1,2\text{ and }x,y\in\mathfrak{g}.$

In the following, we introduce infinitesimal deformations of a compatible Hom-
Lie algebra as a generalization of linear deformations.

######  5.7 Definition.

An infinitesimal deformation of a compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a linear deformation
over $\mathbb{K}[[t]]/(t^{2})$.

One can similarly define equivalences between two infinitesimal deformations.
It is easy to see that any $2$-cocycle $(\omega_{1},\omega_{2})\in
Z^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ induces an infinitesimal
deformation and cohomologous $2$-cocycles give rise to equivalent
infinitesimal deformations. Summarizing this fact with Theorem 5.3, we get the
following.

######  5.8 Theorem.

Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. Then the equivalence classes of infinitesimal deformations are in
one-to-one correspondence with
$H^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g}).$

Finite order deformations. Next, we consider finite order deformations of a
compatible Hom-Lie algebra and study their extensions. Let
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-Lie
algebra. For any natural number $p\in\mathbb{N}$, consider the ring
$\mathbb{K}[[t]]/(t^{p+1})$. Then $\mathfrak{g}[[t]]/(t^{p+1})$ is a module
over $\mathbb{K}[[t]]/(t^{p+1})$. Note that the linear map
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ extends to a
$\mathbb{K}[[t]]/(t^{p+1})$-linear map (denoted by the same notation) on the
space $\mathfrak{g}[[t]]/(t^{p+1})$.

######  5.9 Definition.

An order $p$ deformation of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a pair
$(\mu_{1,t},\mu_{2,t})$, where $\mu_{1,t}=\sum_{i=0}^{p}t^{i}\mu_{1,i}$ and
$\mu_{2,t}=\sum_{i=0}^{p}t^{i}\mu_{2,i}$ with $\mu_{1,i},\mu_{2,i}\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$, and $\mu_{1,0}=\mu_{1}$,
$\mu_{2,0}=\mu_{2}$ which makes the quadruple
$(\mathfrak{g}[[t]]/(t^{p+1}),\mu_{1,t},\mu_{2,t},\alpha)$ into a compatible
Hom-Lie algebra over $\mathbb{K}[[t]]/(t^{p+1})$.

Therefore, in an order $p$ deformation, we have

$\displaystyle[\mu_{1,t},\mu_{1,t}]_{\mathsf{NR}}=0,\quad[\mu_{2,t},\mu_{2,t}]_{\mathsf{NR}}=0~{}~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}~{}[\mu_{1,t},\mu_{2,t}]_{\mathsf{NR}}=0.$

These are equivalent to the following system of identities

$\displaystyle{}^{1}\delta_{\mathrm{Hom}}(\mu_{1,n})=~{}$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},$
$\displaystyle{}^{2}\delta_{\mathrm{Hom}}(\mu_{2,n})=~{}$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geq
1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}},$
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}(\mu_{2,n})+{}^{2}\delta_{\mathrm{Hom}}(\mu_{1,n})=~{}$
$\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}},$

for $n=0,1,\ldots,p$.

######  5.10 Definition.

An order $p$ deformation $(\mu_{1,t},\mu_{2,t})$ of the compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is said to be
extensible if there exists an element $(\mu_{1,p+1},\mu_{2,p+1})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ which makes the pair

$\displaystyle\big{(}\overline{\mu_{1,t}}=\mu_{1,t}+t^{p+1}\mu_{1,p+1},~{}\overline{\mu_{2,t}}=\mu_{2,t}+t^{p+1}\mu_{2,p+1}\big{)}$

into a deformation of order $p+1$.

Let $(\mu_{1,t},\mu_{2,t})$ be an order $p$ deformation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$. Define an element
$Ob_{(\mu_{1,t},\mu_{2,t})}\in
C^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ by

$\displaystyle
Ob_{(\mu_{1,t},\mu_{2,t})}=\big{(}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},~{}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}},~{}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}.$

Since $\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}}$ is the obstruction
cochain to extend the order $p$ deformation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$, we have
${}^{1}\delta_{\mathrm{Hom}}(\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}})=0$. Similarly,
${}^{2}\delta_{\mathrm{Hom}}(\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}})=0$. Moreover, we
have

$\displaystyle{}^{1}\delta_{\mathrm{Hom}}\big{(}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}~{}+~{}{}^{2}\delta_{\mathrm{Hom}}\big{(}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}}\big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}[\mu_{1},[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}}]_{\mathsf{NR}}~{}+~{}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{2},[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}\big{(}[[\mu_{1},\mu_{1,i}]_{\mathsf{NR}},\mu_{2,j}]_{\mathsf{NR}}-[\mu_{1,i},[\mu_{1},\mu_{2,j}]_{\mathsf{NR}}]_{\mathsf{NR}}\big{)}$
$\displaystyle\quad+\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}\big{(}[[\mu_{2},\mu_{1,i}]_{\mathsf{NR}},\mu_{1,j}]_{\mathsf{NR}}-[\mu_{1,i},[\mu_{2},\mu_{1,j}]_{\mathsf{NR}}]_{\mathsf{NR}}\big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}[[\mu_{1},\mu_{1,i}]_{\mathsf{NR}},\mu_{2,j}]_{\mathsf{NR}}-\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},[\mu_{1},\mu_{2,j}]_{\mathsf{NR}}+[\mu_{2},\mu_{1,j}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=-\sum_{\begin{subarray}{c}i_{1}+i_{2}+j=p+1\\\
i_{1},i_{2},j\geq
1\end{subarray}}\frac{1}{2}[[\mu_{1,i_{1}},\mu_{1,i_{2}}]_{\mathsf{NR}},\mu_{2,j}]_{\mathsf{NR}}+\sum_{\begin{subarray}{c}i+j_{1}+j_{2}=p+1\\\
i,j_{1},j_{2}\geq
1\end{subarray}}[\mu_{1,i},[\mu_{1,j_{1}},\mu_{2,j_{2}}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=-\sum_{\begin{subarray}{c}i+j+k=p+1\\\ i,j,k\geq
1\end{subarray}}\frac{1}{2}[[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},\mu_{2,k}]_{\mathsf{NR}}+\sum_{\begin{subarray}{c}i+j+k=p+1\\\
i,j,k\geq
1\end{subarray}}\frac{1}{2}[[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},\mu_{2,k}]_{\mathsf{NR}}=0.$

Similarly, we can show that

$\displaystyle{}^{1}\delta_{\mathrm{Hom}}\big{(}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}~{}+~{}{}^{2}\delta_{\mathrm{Hom}}\big{(}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}=0.$

Therefore, we have $\delta_{\mathrm{cHom}}(Ob_{(\mu_{1,t},\mu_{2,t})})=0$. The
corresponding cohomology class $[Ob_{(\mu_{1,t},\mu_{2,t})}]\in
H^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is called the obstruction
class to extend the deformation $(\mu_{1,t},\mu_{2,t}).$

######  5.11 Theorem.

An order $p$ deformation $(\mu_{1,p},\mu_{2,p})$ of the compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is extensible if
and only if the corresponding obstruction class
$[Ob_{(\mu_{1,t},\mu_{2,t})}]\in
H^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is trivial.

###### Proof.

Suppose $(\mu_{1,t},\mu_{2,t})$ is extensible. Let
$(\mu_{1,p+1},\mu_{2,p+1})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ be an element which makes
the pair
$\big{(}\overline{\mu_{1,t}}=\mu_{1,t}+t^{p+1}\mu_{1,p+1},~{}\overline{\mu_{2,t}}=\mu_{2,t}+t^{p+1}\mu_{2,p+1}\big{)}$
into a deformation of order $p+1$. Then it follows from the expression of
$Ob_{(\mu_{1,t},\mu_{2,t})}$ that

$\displaystyle
Ob_{(\mu_{1,t},\mu_{2,t})}=\delta_{\mathrm{cHom}}\big{(}(\mu_{1,p+1},\mu_{2,p+1})\big{)}.$

This shows that the cohomology class $[Ob_{(\mu_{1,t},\mu_{2,t})}]$ is
trivial.

Conversely, suppose $(\mu_{1,t},\mu_{2,t})$ is an order $p$ deformation for
which $[Ob_{(\mu_{1,t},\mu_{2,t})}]\in
H^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is trivial. Then we have
$Ob_{(\mu_{1,t},\mu_{2,t})}=\delta_{\mathrm{cHom}}\big{(}(\mu_{1,p+1},\mu_{2,p+1})\big{)}$,
for some $(\mu_{1,p+1},\mu_{2,p+1})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$. Then it is easy to see that
$\big{(}\overline{\mu_{1,t}}=\mu_{1,t}+t^{p+1}\mu_{1,p+1},~{}\overline{\mu_{2,t}}=\mu_{2,t}+t^{p+1}\mu_{2,p+1}\big{)}$
is a deformation of order $p+1$. In other words, $(\mu_{1,t},\mu_{2,t})$ is
extensible. ∎

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

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