# Cohomology of morphism Lie algebras and some applications

Apurba Das††Email<EMAIL_ADDRESS>
###### Abstract

A morphism Lie algebra is a triple $(\mathfrak{g},\mathfrak{h},\phi)$
consisting of two Lie algebras $\mathfrak{g},\mathfrak{h}$ and a Lie algebra
homomorphism $\phi:\mathfrak{g}\rightarrow\mathfrak{h}$. We define
representations and cohomology of morphism Lie algebras. As applications of
our cohomology, we study some aspects of deformations, abelian extensions of
morphism Lie algebras and classify skeletal morphism sh Lie algebras. Finally,
we consider the cohomology of morphism Lie groups and find a relation with the
cohomology of morphism Lie algebras.

2010 MSC classification: 17B55, 17B56.

Keywords: Morphism Lie algebras, Cohomology, Deformations, Morphism sh Lie
algebras, Morphism Lie groups.

###### Contents

  1. 1 Introduction
  2. 2 Morphism Lie algebras and their representations
  3. 3 Cohomology of morphism Lie algebras
  4. 4 Deformation theories
  5. 5 Abelian extensions of morphism Lie algebras
  6. 6 Classification of skeletal morphism sh Lie algebras
  7. 7 Morphism Lie groups

##  1 Introduction

Cohomology of some algebraic structures (e.g. associative, Lie, commutative
algebras) plays a key role in the study of deformations and extensions of
algebras [11, 13]. They are also useful in representation theory [3], category
theory [15], differential geometry of Lie groups [4], homotopy algebras [2]
and quantization of Poisson manifolds in mathematical physics [14]. In the
last twenty years, cohomology theory has been formulated for various algebras
including algebras over binary quadratic operads [1] and combinatorial Loday-
type algebras [7]. Our main objective in this paper is the notion that
consists of a triple $(\mathfrak{g},\mathfrak{h},\phi)$ of two Lie algebras
$\mathfrak{g},\mathfrak{h}$ and a Lie algebra homomorphism
$\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ between them. We call such a triple
a morphism Lie algebra. Thus, a morphism Lie algebra captures information of
two Lie algebras as well as a Lie algebra homomorphism between them. Morphism
Lie algebras arise from Rota-Baxter operators, Nijenhuis operators and
differentiating Lie group homomorphisms. The aim of this paper is to define
and study the cohomology theory of morphism Lie algebras. We provide
applications of our cohomology in some aspects of deformation problems,
abelian extensions and homotopy algebras. We also consider morphism Lie
groups, define their cohomology and find a relation with morphism Lie
algebras.

Deformation theory is not only limited to algebraic structures. In [17]
Nijenhuis and Richardson first considered deformations of algebra
homomorphisms by keeping the underlying algebras intact. Later, simultaneous
deformations of algebras and homomorphisms were considered by Gerstenhaber and
Schack [12]. Simultaneous deformations of two Lie algebras
$\mathfrak{g},\mathfrak{h}$ and a Lie algebra homomorphism
$\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ were studied by Frégier [10] by
introducing a suitable cohomology. As mentioned before, we call the whole
triple $(\mathfrak{g},\mathfrak{h},\phi)$ a morphism Lie algebra. We define
representations of a morphism Lie algebra $(\mathfrak{g},\mathfrak{h},\phi)$
and introduce cohomology with coefficients in a representation. When
considering the cohomology with coefficients in the adjoint representation,
our cohomology coincides with the one introduced by Frégier. We also find a
sufficient condition for the vanishing of our cohomology in terms of the
vanishing of some Lie algebra cohomology groups.

The cohomology of some type of algebras with coefficients in the adjoint
representation only allows the study of deformations of algebras. However, to
study some other deformation problems (e.g. deformations of homomorphisms
between algebras and deformations of subalgebras), one needs to know the
cohomology with coefficients in a representation [17, 18]. Therefore, our
cohomology of morphism Lie algebras with coefficients in a representation gave
us the freedom to consider the above-mentioned deformation problems for
morphism Lie algebras. First, we study deformations of homomorphisms between
morphism Lie algebras. We also find a sufficient condition for the rigidity of
a homomorphism between morphism Lie algebras. Second, we study deformations of
morphism Lie subalgebras in terms of the cohomology of a suitable morphism Lie
algebra.

In the next part, we study abelian extensions of morphism Lie algebras in
terms of cohomology. More precisely, we introduce a notion of ‘simple’
cohomology of morphism Lie algebras and show that isomorphism classes of
abelian extensions are classified by the simple second cohomology group.
Another justification of our cohomology can be given in terms of some suitable
morphism sh Lie algebras. Recall that, sh Lie algebras (strongly homotopy Lie
algebras) are the homotopy analogue of Lie algebras where the Jacobi identity
holds only up to a homotopy. In [2] Baez and Crans showed that skeletal sh Lie
algebras are classified by the third cohomology groups of Lie algebras. We
generalize their result in the context of morphism Lie algebras. More
precisely, we introduce skeletal morphism sh Lie algebras and show that such
structures can be classified by the third cohomology groups of morphism Lie
algebras introduced in the paper.

Finally, we consider morphism Lie groups, the global object for morphism Lie
algebras. We introduce the cohomology of morphism Lie groups with coefficients
in a module. This cohomology is obtained as byproducts of known group
cohomologies. In the end, we define the differential cohomology of a morphism
Lie group and find its relation with the cohomology of morphism Lie algebra.

The paper is organized as follows. In Sections 2 and 3, we respectively study
representations and cohomology of morphism Lie algebras. In Section 4, we
study deformations of homomorphisms between morphism Lie algebras and
deformations of morphism Lie subalgebras in terms of cohomology. Abelian
extensions of morphism Lie algebras and classification of skeletal morphism sh
Lie algebras are given in Sections 5 and 6, respectively. Finally, morphism
Lie groups and their cohomology theoretic results are considered in Section 7.

All vector spaces, (multi)linear maps, tensor products and wedge products are
over a field k of characteristic $0$.

##  2 Morphism Lie algebras and their representations

In this section, we introduce morphism Lie algebras and define their
representations.

Let $\mathfrak{g}$ be a Lie algebra. We denote the Lie bracket on
$\mathfrak{g}$ by $[~{},~{}]_{\mathfrak{g}}$. A representation of the Lie
algebra $\mathfrak{g}$ is a vector space $V$ together with a linear map
(called the action map) $\rho:\mathfrak{g}\rightarrow\mathrm{End}(V)$
satisfying

$\displaystyle\rho([x,y]_{\mathfrak{g}})=\rho(x)\circ\rho(y)-\rho(y)\circ\rho(x),~{}\text{
for }x,y\in\mathfrak{g}.$

We denote a representation as above simply by $V$ when the action map is clear
from the context. Note that any Lie algebra $\mathfrak{g}$ is a representation
of itself with the action map
$\rho:\mathfrak{g}\rightarrow\mathrm{End}(\mathfrak{g})$ given by
$\rho(x)(y)=[x,y]_{\mathfrak{g}}$, for $x,y\in\mathfrak{g}$. This is called
the adjoint representation.

Given a Lie algebra $\mathfrak{g}$ and a representation $V$, there is a
cochain complex (called the Chevalley-Eilenberg cochain complex)
$\\{C^{\ast}_{\mathsf{CE}}(\mathfrak{g},V),\delta_{\mathsf{CE}}\\}$, where
$C^{n}_{\mathsf{CE}}(\mathfrak{g},V)=\mathrm{Hom}(\wedge^{n}\mathfrak{g},V))$
for $n\geq 0$, and the differential
$\delta_{\mathsf{CE}}:C^{n}_{\mathsf{CE}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathsf{CE}}(\mathfrak{g},V)$ given by

$\displaystyle(\delta_{\mathsf{CE}}f)(x_{1},\ldots,x_{n+1})=~{}$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}~{}\rho(x_{i})f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}~{}f([x_{i},x_{j}],x_{1},\ldots,\widehat{x_{i}},\ldots,\widehat{x_{j}},\ldots,x_{n+1}),$

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

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras. A linear map
$\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ is said to be a homomorphism of Lie
algebras if $\phi$ satisfies
$\phi([x,y]_{\mathfrak{g}})=[\phi(x),\phi(y)]_{\mathfrak{h}}$, for
$x,y\in\mathfrak{g}$.

######  2.1 Definition.

A morphism Lie algebra is a triple $(\mathfrak{g},\mathfrak{h},\phi)$ consist
of two Lie algebras $\mathfrak{g},\mathfrak{h}$ and a homomorphism
$\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ of Lie algebras.

######  2.2 Definition.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ and
$(\mathfrak{g}^{\prime},\mathfrak{h}^{\prime},\phi^{\prime})$ be two morphism
Lie algebras. A homomorphism between them is a pair $(\alpha,\beta)$ consists
of Lie algebra homomorphisms
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ and
$\beta:\mathfrak{h}\rightarrow\mathfrak{h}^{\prime}$ satisfying
$\phi^{\prime}\circ\alpha=\beta\circ\phi$. It is called an isomorphism if
$\alpha$ and $\beta$ are both isomorphisms.

######  2.3 Definition.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra. A
representation of it consists of a triple $(V,W,\psi)$ in which $V$ is a
representation of $\mathfrak{g}$, $W$ is a representation of $\mathfrak{h}$
and $\psi:V\rightarrow W$ is a linear map satisfying

$\displaystyle\psi(\rho_{V}(x)v)=\rho_{W}(\phi(x))\psi(v),~{}\text{ for
}x\in\mathfrak{g},v\in V.$

######  2.4 Example.

Any morphism Lie algebra $(\mathfrak{g},\mathfrak{h},\phi)$ is a
representation of itself, where $\mathfrak{g}$ and $\mathfrak{h}$ are equipped
with respective adjoint representations. We call this the adjoint
representation of the morphism Lie algebra $(\mathfrak{g},\mathfrak{h},\phi)$.

######  2.5 Example.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ and
$(\mathfrak{g}^{\prime},\mathfrak{h}^{\prime},\phi^{\prime})$ be two morphism
Lie algebras, and $(\alpha,\beta)$ be a homomorphism between them. Then the
triple $(\mathfrak{g}^{\prime},\mathfrak{h}^{\prime},\phi^{\prime})$ can be
regarded as a representation of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$, where $\mathfrak{g}^{\prime}$ is a
representation of $\mathfrak{g}$ via the action map
$\rho_{\mathfrak{g}^{\prime}}:\mathfrak{g}\rightarrow\mathrm{End}(\mathfrak{g}^{\prime})$,
$\rho_{\mathfrak{g}^{\prime}}(x)(x^{\prime})=[\alpha(x),x^{\prime}]_{\mathfrak{g}^{\prime}}$
and $\mathfrak{h}^{\prime}$ is a representation of $\mathfrak{h}$ via the
action map
$\rho_{\mathfrak{h}^{\prime}}:\mathfrak{h}\rightarrow\mathrm{End}(\mathfrak{h}^{\prime})$,
$\rho_{\mathfrak{h}^{\prime}}(h)(h^{\prime})=[\beta(h),h^{\prime}]_{\mathfrak{h}^{\prime}}$.

######  2.6 Example.

Let $\mathfrak{g}$ be a Lie algebra. A Rota-Baxter operator of weight
$\lambda\in{\bf k}$ on $\mathfrak{g}$ is a linear map
$R:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying

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

Let $R$ be a Rota-Baxter operator of weight $\lambda$. Then $R$ induces a new
Lie algebra structure on the underlying vector space $\mathfrak{g}$ with
bracket given by
$[x,y]_{R}:=[R(x),y]_{\mathfrak{g}}+[x,R(y)]_{\mathfrak{g}}+\lambda[x,y]_{\mathfrak{g}},~{}\text{
for }x,y\in\mathfrak{g}.$ We denote this Lie algebra by $\mathfrak{g}_{R}$. It
follows from (1) that $(\mathfrak{g}_{R},\mathfrak{g},R)$ is a morphism Lie
algebra.

A pair $(\mathfrak{g},R)$ consisting of a Lie algebra $\mathfrak{g}$ and a
Rota-Baxter operator of weight $\lambda$ on it is called a Rota-Baxter Lie
algebra (of weight $\lambda$). A representation of a Rota-Baxter Lie algebra
$(\mathfrak{g},R)$ consists of a pair $(V,R_{V})$ in which $V$ is a
representation of $\mathfrak{g}$ and $R_{V}:V\rightarrow V$ is a linear map
satisfying

$\displaystyle\rho(R(x))R_{V}(v)=R_{V}\big{(}\rho(R(x))v+\rho(x)R_{V}(v)+\lambda\rho(x)v\big{)},\text{
for }x\in\mathfrak{g},v\in V.$

In this case, $V$ van be equipped with a representation of the Lie algebra
$\mathfrak{g}_{R}$ with the action map
$\rho_{R_{V}}:\mathfrak{g}_{R}\rightarrow\mathrm{End}(V)$ given by
$\rho_{R_{V}}(x)v:=\rho(R(x))v+\rho(x)R_{V}(v)+\lambda\rho(x)v$, for
$x\in\mathfrak{g}_{R}$ and $v\in V$ [8]. We denote this representation by
$V_{R_{V}}.$ Then it is easy to see that $(V_{R_{V}},V,R_{V})$ is a
representation of the morphism Lie algebra
$(\mathfrak{g}_{R},\mathfrak{g},R)$.

######  2.7 Example.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra. Suppose $U$
is a representation of $\mathfrak{h}$ with the action map
$\rho_{U}^{\mathfrak{h}}:\mathfrak{h}\rightarrow\mathrm{End}(U)$. Then $\phi$
induces a representation of $\mathfrak{g}$ on the vector space $U$ with action
map
$\rho_{U}^{\mathfrak{g}}:\mathfrak{g}\rightarrow\mathrm{End}(U),~{}\rho_{U}^{\mathfrak{g}}(x)u=\rho^{\mathfrak{h}}_{U}(\phi(x))u$,
for $x\in\mathfrak{g}$, $u\in U$.

Let $V,W$ be two vector spaces and a $\psi:V\rightarrow W$ be a linear map. We
define a representation of $\mathfrak{g}$ on the space $\mathrm{End}(U,V)$ by
$\rho_{U,V}:\mathfrak{g}\rightarrow\mathrm{End}(\mathrm{End}(U,V))$,
$(\rho_{U,V}(x)f)(u)=-f(\rho_{U}^{\mathfrak{g}}(x)u)$, for $x\in\mathfrak{g}$,
$f\in\mathrm{End}(U,V)$ and $u\in U$. Similarly, there is a representation of
$\mathfrak{h}$ on the space $\mathrm{End}(U,W)$ by
$\rho_{U,W}:\mathfrak{h}\rightarrow\mathrm{End}(\mathrm{End}(U,W))$,
$(\rho_{U,W}(h)g)(u)=-g(\rho_{U}^{\mathfrak{h}}(h)u)$, for $h\in\mathfrak{h}$,
$g\in\mathrm{End}(U,W)$ and $u\in U$. We also define a map
$\Psi:\mathrm{End}(U,V)\rightarrow\mathrm{End}(U,W)$ by $\Psi(f)=\psi\circ f$.
For $x\in\mathfrak{g}$, $f\in\mathrm{End}(U,V)$ and $u\in U$, we observe that

$\displaystyle(\Psi(\rho_{U,V}(x)f))u=\psi((\rho_{U,V}(x)f)u)=-(\psi\circ
f)(\rho_{U}^{\mathfrak{g}}(x)u)=-\Psi(f)(\rho_{U}^{\mathfrak{h}}(\phi(x))u)=(\rho_{U,W}(\phi(x))\Psi(f))u.$

This shows that $(\mathrm{End}(U,V),\mathrm{End}(U,W),\Psi)$ is a
representation of the morphism Lie algebra $(\mathfrak{g},\mathfrak{h},\phi)$.

It is known that a representation of a Lie algebra $\mathfrak{g}$ can be
equivalently described as a left module over the universal enveloping algebra
$U(\mathfrak{g})$. Note that $U(\mathfrak{g})$ is obtained from the tensor
algebra $T(\mathfrak{g})$ quotient by the two-sided ideal $I_{\mathfrak{g}}$
generated by elements of the form $x\otimes y-y\otimes
x-[x,y]_{\mathfrak{g}}$, for $x,y\in\mathfrak{g}$. More precisely, if $V$ is a
left $U(\mathfrak{g})$-module, then the representation of the Lie algebra
$\mathfrak{g}$ on $V$ is given by the action map
$\rho_{V}:\mathfrak{g}\rightarrow\mathrm{End}(V)$, $\rho_{V}(x)v=x\cdot v$,
for $x\in\mathfrak{g},v\in V$. We will generalize this result to
representations of morphism Lie algebras.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra. Then $\phi$
induces an algebra homomorphism $T(\phi):T(\mathfrak{g})\rightarrow
T(\mathfrak{h})$ between tensor algebras given by

$\displaystyle T(\phi)(x_{1}\otimes\cdots\otimes
x_{n})=\phi(x_{1})\otimes\cdots\otimes\phi(x_{n}),~{}\text{ for
}x_{1}\otimes\cdots\otimes x_{n}\in\mathfrak{g}^{\otimes n}\subset
T(\mathfrak{g}).$

By composing with the projection $T(\mathfrak{h})\rightarrow
U(\mathfrak{h})=T(\mathfrak{h})/I_{\mathfrak{h}}$, we obtain a map (denoted by
the same notation) $T(\phi):T(\mathfrak{g})\rightarrow U(\mathfrak{h})$. Since

$\displaystyle T(\phi)(x\otimes y-y\otimes
x-[x,y]_{\mathfrak{g}})=\phi(x)\otimes\phi(y)-\phi(y)\otimes\phi(x)-[\phi(x),\phi(y)]_{\mathfrak{h}}=0~{}~{}~{}(\text{mod
}I_{\mathfrak{h}}),$

the map $T(\phi)$ induces an algebra homomorphism
$U(\phi):U(\mathfrak{g})\rightarrow U(\mathfrak{h})$. In other words, the
triple $(U(\mathfrak{g}),U(\mathfrak{h}),U(\phi))$ is a morphism associative
algebra.

A left module over the associative algebra
$(U(\mathfrak{g}),U(\mathfrak{h}),U(\phi))$ is a triple $(V,W,\psi)$ in which
$V$ is a left $U(\mathfrak{g})$-module, $W$ is a left $U(\mathfrak{h})$-module
and $\phi:V\rightarrow W$ is a linear map satisfying $\psi(x\cdot
v)=U(\phi)(x)\cdot\psi(v)$, for $x\in U(\mathfrak{g}),v\in V$. Then we have
the following result.

######  2.8 Proposition.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra. A
representation of $(\mathfrak{g},\mathfrak{h},\phi)$ is equivalent to a left
module over the morphism associative algebra
$(U(\mathfrak{g}),U(\mathfrak{h}),U(\phi))$.

##  3 Cohomology of morphism Lie algebras

In this section, we introduce the cohomology of a morphism Lie algebra with
coefficients in a representation. Our cohomology generalizes the cohomology of
Lie algebra homomorphisms defined in [10].

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(V,W,\psi)$ be a representation of it. Then there are two obvious cochain
complexes, namely

  * •

$\\{C^{\ast}_{\mathsf{CE}}(\mathfrak{g},V),\delta^{\prime}_{\mathsf{CE}}\\}$,
the Chevalley-Eilenberg complex of $\mathfrak{g}$ with coefficients in $V$,

  * •

$\\{C^{\ast}_{\mathsf{CE}}(\mathfrak{h},W),\delta^{\prime\prime}_{\mathsf{CE}}\\}$,
the Chevalley-Eilenberg complex of $\mathfrak{h}$ with coefficients in $W$.

We define a new map
$\rho_{W}^{\mathfrak{g}}:\mathfrak{g}\rightarrow\mathrm{End}(W)$ by
$\rho_{W}^{\mathfrak{g}}(x)w=\rho_{W}(\phi(x))w$, for $x\in\mathfrak{g}$ and
$w\in W$. For any $x,y\in\mathfrak{g}$ and $w\in W$, we have

$\displaystyle\rho_{W}^{\mathfrak{g}}([x,y]_{\mathfrak{g}})w=\rho_{W}([\phi(x),\phi(y)]_{\mathfrak{h}})w=~{}$
$\displaystyle\rho_{W}(\phi(x))\rho_{W}(\phi(y))w~{}-~{}\rho_{W}(\phi(y))\rho_{W}(\phi(x))w$
$\displaystyle=~{}$
$\displaystyle\rho_{W}^{\mathfrak{g}}(x)\rho_{W}^{\mathfrak{g}}(y)w~{}-~{}\rho_{W}^{\mathfrak{g}}(y)\rho_{W}^{\mathfrak{g}}(x)w.$

This shows that $\rho_{W}^{\mathfrak{g}}$ defines a representation of the Lie
algebra $\mathfrak{g}$ on the vector space $W$. We denote this representation
by $W_{\phi}$. Hence we may consider another cochain complex, namely

  * •

$\\{C^{\ast}_{\mathsf{CE}}(\mathfrak{g},W_{\phi}),\delta^{\prime\prime\prime}_{\mathsf{CE}}\\}$,
the Chevalley-Eilenberg complex of $\mathfrak{g}$ with coefficients in
$W_{\phi}$.

We are now in a position to define the cohomology of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$ with coefficients in $(V,W,\psi)$. For each
$n\geq 0$, we define the $n$-th cochain group
$C^{n}_{\mathsf{mLA}}(\phi,\psi)$ by

$\displaystyle C^{n}_{\mathsf{mLA}}(\phi,\psi)=\begin{cases}V&\quad\text{ if
}n=0,\\\ C^{n}_{\mathsf{CE}}(\mathfrak{g},V)\oplus
C^{n}_{\mathsf{CE}}(\mathfrak{h},W)\oplus
C^{n-1}_{\mathsf{CE}}(\mathfrak{g},W_{\phi})\\\
=\mathrm{Hom}(\wedge^{n}\mathfrak{g},V)\oplus\mathrm{Hom}(\wedge^{n}\mathfrak{h},W)\oplus\mathrm{Hom}(\wedge^{n-1}\mathfrak{g},W)&\quad\text{
if }n\geq 1\end{cases}$

and a map $\delta_{\mathsf{mLA}}:C^{n}_{\mathsf{mLA}}(\phi,\psi)\rightarrow
C^{n+1}_{\mathsf{mLA}}(\phi,\psi)$ by

$\displaystyle\delta_{\mathsf{mLA}}(v)=~{}$
$\displaystyle(\delta^{\prime}_{\mathsf{CE}}(v),~{}\delta^{\prime\prime}_{\mathsf{CE}}(\psi(v)),~{}0),~{}\text{
for }v\in C^{0}_{\mathsf{mLA}}(\phi,\psi),$
$\displaystyle\delta_{\mathsf{mLA}}(\theta,\gamma,\eta)=~{}$
$\displaystyle(\delta^{\prime}_{\mathsf{CE}}(\theta),~{}\delta^{\prime\prime}_{\mathsf{CE}}(\gamma),~{}\psi\circ\theta-\gamma\circ{\wedge^{n}\phi}-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\eta)),~{}\text{
for }(\theta,\gamma,\eta)\in C^{n\geq 1}_{\mathsf{mLA}}(\phi,\psi).$

######  3.1 Proposition.

With the above notations, we have $(\delta_{\mathsf{mLA}})^{2}=0.$

###### Proof.

For $v\in C^{0}_{\mathsf{mLA}}(\phi,\psi)$,

$\displaystyle(\delta_{\mathsf{mLA}})^{2}(v)=~{}$
$\displaystyle\delta_{\mathsf{mLA}}\big{(}\delta^{\prime}_{\mathsf{CE}}(v),~{}\delta^{\prime\prime}_{\mathsf{CE}}(\psi(v)),~{}0\big{)}$
$\displaystyle=~{}$
$\displaystyle\big{(}(\delta^{\prime}_{\mathsf{CE}})^{2}(v),~{}(\delta^{\prime\prime}_{\mathsf{CE}})^{2}(\psi(v)),~{}\psi\circ\delta^{\prime}_{\mathsf{CE}}(v)-\delta^{\prime\prime}_{\mathsf{CE}}(\psi(v))\circ\phi\big{)}=0.$

The lase equality holds as
$(\psi\circ\delta^{\prime}_{\mathsf{CE}}(v))(x)=\psi(\rho_{V}(x)v)=\rho_{W}(\phi(x))\psi(v)=(\delta^{\prime\prime}_{\mathsf{CE}}(\psi(v))\circ\phi)(x)$,
for all $x\in\mathfrak{g}$. Similarly, for $(\theta,\gamma,\eta)\in C^{n\geq
1}_{\mathsf{mLA}}(\phi,\psi),$ we have

$\displaystyle(\delta_{\mathsf{mLA}})^{2}(\theta,\gamma,\eta)$
$\displaystyle=\delta_{\mathsf{mLA}}(\delta^{\prime}_{\mathsf{CE}}(\theta),~{}\delta^{\prime\prime}_{\mathsf{CE}}(\gamma),~{}\psi\circ\theta-\gamma\circ{\wedge^{n}\phi}-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\eta))$
$\displaystyle=((\delta^{\prime}_{\mathsf{CE}})^{2}(\theta),~{}(\delta^{\prime\prime}_{\mathsf{CE}})^{2}(\gamma),~{}\psi\circ\delta^{\prime}_{\mathsf{CE}}(\theta)-\delta^{\prime\prime}_{\mathsf{CE}}(\gamma)\circ\wedge^{n+1}\phi-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\psi\circ\theta)+\delta^{\prime\prime\prime}_{\mathsf{CE}}(\gamma\circ\wedge^{n}\phi)+(\delta^{\prime\prime\prime}_{\mathsf{CE}})^{2}(\eta)).$
(2)

We observe that

$\displaystyle\delta^{\prime\prime\prime}_{\mathsf{CE}}(\psi\circ\theta)(x_{1},\ldots,x_{n+1})=~{}$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}~{}\rho_{W}^{\mathfrak{g}}(x_{i})(\psi\circ\theta)(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}~{}(\psi\circ\theta)([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\ldots,\widehat{x_{i}},\ldots,\widehat{x_{j}},\ldots,x_{n+1})$
$\displaystyle=~{}$
$\displaystyle\psi\big{(}\sum_{i=1}^{n+1}(-1)^{i+1}~{}\rho_{V}(x_{i})\theta(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}~{}(\psi\circ\theta)([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\ldots,\widehat{x_{i}},\ldots,\widehat{x_{j}},\ldots,x_{n+1})\big{)}$
$\displaystyle=~{}$
$\displaystyle(\psi\circ\delta^{\prime}_{\mathsf{CE}}(\theta))(x_{1},\ldots,x_{n+1})$

and

$\displaystyle\delta^{\prime\prime\prime}_{\mathsf{CE}}(\gamma\circ\wedge^{n}\phi)(x_{1},\ldots,x_{n+1})=~{}$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}~{}\rho_{W}^{\mathfrak{g}}(x_{i})(\gamma\circ\wedge^{n}\phi)(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}~{}(\gamma\circ\wedge^{n}\phi)([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\ldots,\widehat{x_{i}},\ldots,\widehat{x_{j}},\ldots,x_{n+1})$
$\displaystyle=~{}$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}~{}\rho_{W}(\phi(x_{i}))\gamma(\phi(x_{1}),\ldots,\widehat{\phi(x_{i})},\ldots,\phi(x_{n+1}))$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}~{}\gamma([\phi(x_{i}),\phi(x_{j})]_{\mathfrak{h}},\phi(x_{1}),\ldots,\widehat{\phi(x_{i})},\ldots,\widehat{\phi(x_{j})},\ldots)$
$\displaystyle=~{}$
$\displaystyle(\delta^{\prime\prime}_{\mathsf{CE}}(\gamma)\circ\wedge^{n+1}\phi)(x_{1},\ldots,x_{n+1}).$

Hence it follows from (3) that $(\delta_{\mathsf{mLA}})^{2}=0$. This completes
the proof. ∎

It follows from the above proposition that
$\\{C^{\ast}_{\mathsf{mLA}}(\phi,\psi),\delta_{\mathsf{mLA}}\\}$ is a cochain
complex. Let $Z^{n}_{\mathsf{mLA}}(\phi,\psi)$ denote the space of
$n$-cocycles and $B^{n}_{\mathsf{mLA}}(\phi,\psi)$ denote the space of
$n$-coboundaries. Then we have $B^{n}_{\mathsf{mLA}}(\phi,\psi)\subset
Z^{n}_{\mathsf{mLA}}(\phi,\psi)$. The corresponding quotients

$\displaystyle
H^{n}_{\mathsf{mLA}}(\phi,\psi):=\frac{Z^{n}_{\mathsf{mLA}}(\phi,\psi)}{B^{n}_{\mathsf{mLA}}(\phi,\psi)},~{}\text{
for }n\geq 0$ (3)

are called the cohomology groups of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$ with coefficients in the representation
$(V,W,\psi)$.

It follows from the above definition that

$\displaystyle H^{0}_{\mathsf{mLA}}(\phi,\psi)=\\{v\in
V~{}|~{}\rho_{V}(x)v=0\text{ and }\rho_{W}(h)\psi(v)=0,\text{ for all
}x\in\mathfrak{g},h\in\mathfrak{h}\\}.$

A triple
$(d,\delta,w)\in\mathrm{Hom}(\mathfrak{g},V)\oplus\mathrm{Hom}(\mathfrak{h},W)\oplus
W$ is said to be a derivation for the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$ with coefficients in $(V,W,\psi)$ if

$\displaystyle d([x,y]_{\mathfrak{g}})=~{}$
$\displaystyle\rho_{V}(x)d(y)-\rho_{V}(y)d(x),~{}\text{ for
}x,y\in\mathfrak{g},$ $\displaystyle\delta([h,k]_{\mathfrak{h}})=~{}$
$\displaystyle\rho_{W}(h)\delta(k)-\rho_{W}(k)\delta(h),~{}\text{ for
}h,k\in\mathfrak{h},$ $\displaystyle\rho_{W}(\phi(x))w=~{}$
$\displaystyle\psi(d(x))-\delta(\phi(x)),\text{ for }x\in\mathfrak{g}.$

In other words, $d$ and $\delta$ are both derivations on Lie algebras
$\mathfrak{g}$ and $\mathfrak{h}$ respectively, and the presence of $w$
obstructs the triviality of $\psi\circ d-\delta\circ\phi$. We denote the set
of all derivations by $\mathrm{Der}(\phi,\psi)$. An inner derivation is a
derivation of the form
$(\delta^{\prime}_{\mathsf{CE}}(v),~{}\delta^{\prime\prime}_{\mathsf{CE}}(\psi(v)),~{}0)$,
for some $v\in V$. The set of all inner derivations are denoted by
$\mathrm{InnDer}(\phi,\psi)$. It follows from (3) that
$H^{1}_{\mathsf{mLA}}(\phi,\psi)=\frac{\mathrm{Der}(\phi,\psi)}{\mathrm{InnDer}(\phi,\psi)}$,
called the space of outer derivations.

######  3.2 Remark.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra. In [10]
Frégier introduced a cohomology that governs deformation of the morphism Lie
algebra $(\mathfrak{g},\mathfrak{h},\phi)$. The cohomology of [10] can be seen
as our cohomology of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$ with coefficients in the adjoint
representation.

In the following, we give a sufficient condition for the vanishing of the
cohomology groups $H^{\ast}_{\mathsf{mLA}}(\phi,\psi)$ in terms of the
vanishing of some Chevalley-Eilenberg cohomology groups.

######  3.3 Proposition.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(V,W,\psi)$ be a representation of it. If
$H^{n}_{\mathsf{CE}}(\mathfrak{g},V)$, $H^{n}_{\mathsf{CE}}(\mathfrak{h},W)$
and $H^{n-1}_{\mathsf{CE}}(\mathfrak{g},W_{\phi})$ are trivial, so is
$H^{n}_{\mathsf{mLA}}(\phi,\psi)$.

###### Proof.

Let $(\theta,\gamma,\eta)\in Z^{n}_{\mathsf{mLA}}(\phi,\psi)$ be an
$n$-cocycle. It follows that $\theta\in Z^{n}_{\mathsf{CE}}(\mathfrak{g},V)$
and $\gamma\in Z^{n}_{\mathsf{CE}}(\mathfrak{h},W)$ are $n$-cocycles in the
respective Chevalley-Eilenberg cochain complexes and
$\psi\circ\theta-\gamma\circ\wedge^{n}\phi-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\eta)=0$.
Hence by the hypothesis, there exist $(n-1)$-cochains $\theta_{0}\in
C^{n-1}_{\mathsf{CE}}(\mathfrak{g},V)$ and $\gamma_{0}\in
C^{n-1}_{\mathsf{CE}}(\mathfrak{h},W)$ such that
$\theta=\delta^{\prime}_{\mathsf{CE}}(\theta_{0})$ and
$\gamma=\delta^{\prime\prime}_{\mathsf{CE}}({\gamma}_{0})$. This implies that

$\displaystyle\delta^{\prime\prime\prime}_{\mathsf{CE}}(\psi\circ\theta_{0}-\gamma_{0}\circ\wedge^{n-1}\phi-\eta)=~{}$
$\displaystyle\psi\circ\delta^{\prime}_{\mathsf{CE}}(\theta_{0})-\delta^{\prime\prime}_{\mathsf{CE}}(\gamma_{0})\circ\wedge^{n}\phi-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\eta)$
$\displaystyle=~{}$
$\displaystyle\phi\circ\theta-\gamma\circ\wedge^{n}\phi-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\eta)=0.$

In other words, $(\psi\circ\theta_{0}-\gamma_{0}\circ\wedge^{n-1}\phi-\eta)\in
Z^{n-1}_{\mathsf{CE}}(\mathfrak{g},W_{\phi})$ is a $(n-1)$-cocycle. Hence by
the hypothesis, there exists an element $\eta_{0}\in
C^{n-1}_{\mathsf{CE}}(\mathfrak{g},W_{\phi})$ such that
$(\psi\circ\theta_{0}-\gamma_{0}\circ\wedge^{n-1}\phi-\eta)=\delta^{\prime\prime\prime}_{\mathsf{CE}}(\eta_{0}).$
This implies that
$(\theta,\gamma,\eta)=\delta_{\mathsf{mLA}}(\theta_{0},\gamma_{0},\eta_{0})\in
B^{n}_{\mathsf{mLA}}(\phi,\psi)$ is a $n$-coboundary. Hence the result
follows. ∎

######  3.4 Corollary.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(V,W,\psi)$ be a representation of it. Suppose $\mathfrak{g},\mathfrak{h}$
are both semisimple Lie algebras and $V,W$ are nontrivial irreducible
representations of $\mathfrak{g}$ and $\mathfrak{h}$, respectively.
Additionally, we assume that the representation $W_{\phi}$ of the Lie algebra
$\mathfrak{g}$ is nontrivial and irreducible. Then by the Whitehead’s theorem
[19] and Proposition 3.3, we get that $H^{\ast}_{\mathsf{mLA}}(\phi,\psi)=0$.

##  4 Deformation theories

Deformations of morphism Lie algebras was considered by Frégier in [10]. In a
deformation of a morphism Lie algebra $(\mathfrak{g},\mathfrak{h},\phi)$, we
simultaneously deform the underlying Lie algebras $\mathfrak{g},\mathfrak{h}$
and the Lie algebra homomorphism $\phi$. The main result of [10] says that the
cohomology (with coefficients in the adjoint representation) of the morphism
Lie algebra $(\mathfrak{g},\mathfrak{h},\phi)$ govern such deformations. In
the following, we study some other aspects of deformation problems in the
context of morphism Lie algebras.

### Deformations of homomorphisms between morphism Lie algebras

Let $(\mathfrak{g},\mathfrak{h},\phi)$ and
$(\mathfrak{g}^{\prime},\mathfrak{h}^{\prime},\phi^{\prime})$ be two morphism
Lie algebras and $(\alpha,\beta)$ be a homomorphism between them. We have seen
in Example 2.5 that the triple
$(\mathfrak{g}^{\prime},\mathfrak{h}^{\prime},\phi^{\prime})$ can be
considered as a representation of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$, where the representation of the Lie
algebra $\mathfrak{g}$ on the vector space $\mathfrak{g}^{\prime}$ is given by
$\rho_{\mathfrak{g}^{\prime}}:\mathfrak{g}\rightarrow\mathrm{End}(\mathfrak{g}^{\prime}),~{}\rho_{\mathfrak{g}^{\prime}}(x)(x^{\prime})=[\alpha(x),x^{\prime}]_{\mathfrak{g}^{\prime}}$,
and the representation of $\mathfrak{h}$ on the space $\mathfrak{h}^{\prime}$
is given by
$\rho_{\mathfrak{h}^{\prime}}:\mathfrak{h}\rightarrow\mathrm{End}(\mathfrak{h}^{\prime}),~{}\rho_{\mathfrak{h}^{\prime}}(h)(h^{\prime})=[\beta(h),h^{\prime}]_{\mathfrak{h}^{\prime}}$.
Hence we may consider the cochain complex
$\\{C^{\ast}_{\mathsf{mLA}}(\phi,\phi^{\prime}),\delta_{\mathsf{mLA}}\\}$,
where

$\displaystyle
C^{0}_{\mathsf{mLA}}(\phi,\phi^{\prime})=\mathfrak{g}^{\prime}\quad\text{ and
}\quad C^{n\geq
1}_{\mathsf{mLA}}(\phi,\phi^{\prime})=\mathrm{Hom}(\wedge^{n}\mathfrak{g},\mathfrak{g}^{\prime})\oplus\mathrm{Hom}(\wedge^{n}\mathfrak{h},\mathfrak{h}^{\prime})\oplus\mathrm{Hom}(\wedge^{n-1}\mathfrak{g},\mathfrak{h}^{\prime}).$

We will use the corresponding cohomology
$H^{\ast}_{\mathrm{mLA}}(\phi,\phi^{\prime})$ to study deformations of the
homomorphism $(\alpha,\beta)$.

######  4.1 Definition.

A deformation of $(\alpha,\beta)$ consists of a pair $(\alpha_{t},\beta_{t})$
of two formal power series

$\displaystyle\alpha_{t}=\sum_{i\geq 0}\alpha_{i}t^{i}~{}\text{ and
}~{}\beta_{t}=\sum_{i\geq 0}\beta_{i}t^{i}~{}~{}~{}(\text{with
}\alpha_{i}\in\mathrm{Hom}(\mathfrak{g},\mathfrak{g}^{\prime}),~{}\beta_{i}\in\mathrm{Hom}(\mathfrak{h},\mathfrak{h}^{\prime})\text{
and }\alpha_{0}=\alpha,~{}\beta_{0}=\beta)$

satisfying for $x,y\in\mathfrak{g}$ and $h,k\in\mathfrak{h},$

$\displaystyle\alpha_{t}([x,y]_{\mathfrak{g}})=[\alpha_{t}(x),\alpha_{t}(y)]_{\mathfrak{g}^{\prime}},\quad\beta_{t}([h,k]_{\mathfrak{h}})=[\beta_{t}(h),\beta_{t}(k)]_{\mathfrak{h}^{\prime}}~{}~{}~{}\text{
and }~{}~{}~{}\phi^{\prime}\circ\alpha_{t}=\beta_{t}\circ\phi.$

By equating coefficients of $t$ in each of the above identities, we get

$\displaystyle\alpha_{1}([x,y]_{\mathfrak{g}})=~{}$
$\displaystyle[\alpha(x),\alpha_{1}(y)]_{\mathfrak{g}^{\prime}}+[\alpha_{1}(x),\alpha(y)]_{\mathfrak{g}^{\prime}},$
$\displaystyle\beta_{1}([h,k]_{\mathfrak{h}})=~{}$
$\displaystyle[\beta_{1}(h),\beta_{1}(k)]_{\mathfrak{h}^{\prime}}+[\beta_{1}(h),\beta(k)]_{\mathfrak{h}^{\prime}},$
$\displaystyle\phi^{\prime}\circ\alpha_{1}=~{}$
$\displaystyle\beta_{1}\circ\phi.$

The first condition is equivalent to
$\delta^{\prime}_{\mathsf{CE}}(\alpha_{1})(x,y)=0$, for $x,y\in\mathfrak{g}$,
and the second condition is equivalent to
$\delta^{\prime\prime}_{\mathsf{CE}}(\beta_{1})(h,k)=0$. Therefore, we have

$\displaystyle\delta_{\mathsf{mLA}}(\alpha_{1},\beta_{1},0)=(\delta^{\prime}_{\mathsf{CE}}(\alpha_{1}),~{}\delta^{\prime\prime}_{\mathsf{CE}}(\beta_{1}),~{}\phi^{\prime}\circ\alpha_{1}-\beta_{1}\circ\phi)=0.$

This shows that $(\alpha_{1},\beta_{1},0)\in
C^{1}_{\mathsf{mLA}}(\phi,\phi^{\prime})$ is a $1$-cocycle (that is, lies in
$Z^{1}_{\mathsf{mLA}}(\phi,\phi^{\prime})$). This is called the infinitesimal
(or linear component) of the deformation. In particular, if
$(\alpha_{1},\beta_{1},0)=\cdots=(\alpha_{n-1},\beta_{n-1},0)=0$ and
$(\alpha_{n},\beta_{n},0)$ is nonzero, then $(\alpha_{n},\beta_{n},0)\in
Z^{1}_{\mathsf{mLA}}(\phi,\phi^{\prime})$ is a $1$-cocycle, called the $n$-th
infinitesimal.

Next we define an equivalence relation on the space of space of all
deformations. Let $G^{\prime}$ and $H^{\prime}$ be the unique simply connected
Lie groups integrating the Lie algebras $\mathfrak{g}^{\prime}$ and
$\mathfrak{h}^{\prime}$, respectively. Then the Lie algebra homomorphism
$\phi^{\prime}:\mathfrak{g}^{\prime}\rightarrow\mathfrak{h}^{\prime}$
integrates to a Lie group homomorphism $\Phi^{\prime}:G^{\prime}\rightarrow
H^{\prime}$.

######  4.2 Definition.

Two deformations $(\alpha_{t},\beta_{t})$ and
$(\alpha^{\prime}_{t},\beta^{\prime}_{t})$ are said to be equivalent if there
exists a smooth curve $g_{t}^{\prime}$ in $G^{\prime}$ starting at the
identity (i.e. $g_{0}^{\prime}=1_{G^{\prime}}$) such that

$\displaystyle\alpha_{t}^{\prime}(x)=\mathrm{Ad}^{G^{\prime}}_{g^{\prime}_{t}}\circ\alpha_{t}(x)~{}~{}~{}\text{
and
}~{}~{}~{}\beta_{t}^{\prime}(h)=\mathrm{Ad}^{H^{\prime}}_{\Phi^{\prime}(g_{t}^{\prime})}\circ\beta_{t}(h),~{}\text{
for }x\in\mathfrak{g},h\in\mathfrak{h}.$ (4)

By differentiating both the identities of (4) at $t=0$, we get

$\displaystyle(\alpha_{1}-\alpha_{1}^{\prime})(x)=\delta^{\prime}_{\mathsf{CE}}(\frac{d}{dt}|_{t=0}~{}g_{t}^{\prime})(x)~{}~{}~{}\text{
and
}~{}~{}~{}(\beta_{1}-\beta_{1}^{\prime})(h)=\delta^{\prime\prime}_{\mathsf{CE}}(\frac{d}{dt}|_{t=0}~{}\Phi^{\prime}(g_{t}^{\prime}))(h).$

In other words,
$(\alpha_{1},\beta_{1},0)-(\alpha_{1}^{\prime},\beta_{1}^{\prime},0)=\delta_{\mathsf{mLA}}(\frac{d}{dt}|_{t=0}~{}g_{t}^{\prime})$,
where the derivative
$\frac{d}{dt}|_{t=0}~{}g_{t}^{\prime}\in\mathfrak{g}^{\prime}=C^{0}_{\mathsf{mLA}}(\phi,\phi^{\prime})$.
As a summary of the above discussions, we get the following.

######  4.3 Proposition.

Let $(\alpha_{t},\beta_{t})$ be a deformation of the homomorphism
$(\alpha,\beta)$ of morphism Lie algebras. Then the infinitesimal is
$1$-cocycle in $Z^{1}_{\mathsf{mLA}}(\phi,\phi^{\prime})$. Moreover, the
corresponding cohomology class in $H^{1}_{\mathsf{mLA}}(\phi,\phi^{\prime})$
depends only on the equivalence class of the deformation
$(\alpha_{t},\beta_{t})$.

Next, we consider the rigidity of a homomorphism between morphism Lie algebras
and find a sufficient condition for the rigidity.

######  4.4 Definition.

A homomorphism $(\alpha,\beta)$ is said to be rigid if any deformation
$(\alpha_{t},\beta_{t})$ of $(\alpha,\beta)$ is equivalent to the undeformed
one $(\overline{\alpha}_{t}=\alpha,\overline{\beta}_{t}=\beta)$.

######  4.5 Theorem.

Let $(\alpha,\beta)$ be a homomorphism of morphism Lie algebras from
$(\mathfrak{g},\mathfrak{h},\phi)$ to
$(\mathfrak{g}^{\prime},\mathfrak{h}^{\prime},\phi^{\prime})$. If
$H^{1}_{\mathsf{mLA}}(\phi,\phi^{\prime})=0$ then the homomorphism
$(\alpha,\beta)$ is rigid.

###### Proof.

Let $(\alpha_{t},\beta_{t})$ be any deformation of $(\alpha,\beta)$. Then we
know from Proposition 4.3 that $(\alpha_{1},\beta_{1},0)\in
Z^{1}_{\mathsf{mLA}}(\phi,\phi^{\prime})$ is a $1$-cocycle. Thus, from the
hypothesis, there exists an element $x^{\prime}\in\mathfrak{g}^{\prime}$ such
that $\alpha_{1}=\delta^{\prime}_{\mathsf{CE}}(x^{\prime})$ and
$\beta_{1}=\delta^{\prime\prime}_{\mathsf{CE}}(\phi^{\prime}(x^{\prime}))$.
Let $g^{\prime}_{t}$ be a smooth curve in $G^{\prime}$ starting at the
identity $1_{G^{\prime}}$ such that
$\frac{d}{dt}|_{t=0}g^{\prime}_{t}=x^{\prime}$ and
$\frac{d}{dt}|_{t=0}\Phi^{\prime}(g_{t}^{\prime})=\phi^{\prime}(x^{\prime})$.
We define a deformation $(\alpha_{t}^{\prime},\beta_{t}^{\prime})$ by
$\alpha_{t}^{\prime}=\mathrm{Ad}^{G^{\prime}}_{g_{t}^{\prime}}\circ\alpha_{t}$
and
$\beta_{t}^{\prime}=\mathrm{Ad}^{H^{\prime}}_{\Phi^{\prime}(g_{t}^{\prime})}\circ\beta_{t}$.
Then $(\alpha_{t}^{\prime},\beta^{\prime}_{t})$ is equivalent to
$(\alpha_{t},\beta_{t})$. Moreover, we have

$\displaystyle\alpha_{1}^{\prime}=\frac{d}{dt}|_{t=0}\alpha_{t}^{\prime}=0~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}\beta_{1}^{\prime}=\frac{d}{dt}|_{t=0}\beta_{t}^{\prime}=0.$

Therefore, the linear component $(\alpha_{1}^{\prime},\beta_{1}^{\prime},0)$
associated to the deformation $(\alpha_{t}^{\prime},\beta_{t}^{\prime})$ is
null. By repeating this process, we can show that $(\alpha_{t},\beta_{t})$ is
equivalent to $(\overline{\alpha}_{t}=\alpha,\overline{\beta}_{t}=\beta)$.
This completes the proof. ∎

### Deformations of morphism Lie subalgebras

Here we study infinitesimal deformation theory of morphism Lie subalgebras in
terms of cohomology.

######  4.6 Definition.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra. A morphism
Lie subalgebra of $(\mathfrak{g},\mathfrak{h},\phi)$ is a triple
$(\mathfrak{p},\mathfrak{q},\varphi)$ in which
$\mathfrak{p}\hookrightarrow\mathfrak{g}$ and
$\mathfrak{q}\hookrightarrow\mathfrak{h}$ are Lie subalgebras and
$\varphi:\mathfrak{p}\rightarrow\mathfrak{q}$ is a linear map satisfying
$\varphi=\phi|_{\mathfrak{p}}$. We often write
$(\mathfrak{p},\mathfrak{q},\varphi)\hookrightarrow(\mathfrak{g},\mathfrak{h},\phi)$
to denote a morphism Lie subalgebra.

It follows that $(\mathfrak{p},\mathfrak{q},\varphi)$ is a morphism Lie
algebra in its own right and the pair of inclusion maps defines a homomorphism
of morphism Lie algebras from $(\mathfrak{p},\mathfrak{q},\varphi)$ to
$(\mathfrak{g},\mathfrak{h},\phi)$.

Let
$(\mathfrak{p},\mathfrak{q},\varphi)\hookrightarrow(\mathfrak{g},\mathfrak{h},\phi)$
be a morphism Lie subalgebra. Note that the composition
$\mathfrak{g}\xrightarrow{\phi}\mathfrak{h}\rightarrow\mathfrak{h}/\mathfrak{q}$
induces a map (denoted by $\phi/\varphi$)
$\mathfrak{g}/\mathfrak{p}\rightarrow\mathfrak{h}/\mathfrak{q}$. Moreover,
there is a representation of the Lie algebra $\mathfrak{p}$ on the space
$\mathfrak{g}/\mathfrak{p}$ given by the action map
$\rho_{\mathfrak{g}/\mathfrak{p}}(p)(x\text{ mod
}\mathfrak{p})=[p,x]_{\mathfrak{g}}\text{ mod }\mathfrak{p}$, for
$p\in\mathfrak{p}$ and $x\in\mathfrak{g}$. Similarly, there is a
representation of the Lie algebra $\mathfrak{q}$ on the space
$\mathfrak{h}/\mathfrak{q}$ with the action map
$\rho_{\mathfrak{h}/\mathfrak{q}}(q)(h\text{ mod
}\mathfrak{q})=[q,h]_{\mathfrak{h}}\text{ mod }\mathfrak{q}$, for
$q\in\mathfrak{q}$ and $h\in\mathfrak{h}$. It is further easy to see that the
triple
$(\mathfrak{g}/\mathfrak{p},~{}\mathfrak{h}/\mathfrak{q},~{}\phi/\varphi)$ is
a representation of the morphism Lie algebra
$(\mathfrak{p},\mathfrak{q},\varphi)$. Therefore, we may consider the cochain
complex
$\\{C^{\ast}_{\mathsf{mLA}}(\varphi,\phi/\varphi),\delta_{\mathsf{mLA}}\\}$
and the corresponding cohomology groups
$H^{\ast}_{\mathsf{mLA}}(\varphi,\phi/\varphi)$.

Let $\mathrm{dim}(\mathfrak{p})=d$ and $\mathrm{dim}(\mathfrak{q})=l$. We
consider the following Grassmannians of $d$-dimensional subspaces of
$\mathfrak{g}$ and $l$-dimensional subspaces of $\mathfrak{h}$, i.e.

$\displaystyle\mathrm{Gr}_{d}(\mathfrak{g})=~{}$
$\displaystyle\\{P\subset\mathfrak{g}~{}|~{}P\text{ is a }d\text{-dimensional
subspace of }\mathfrak{g}\\},$
$\displaystyle\mathrm{Gr}_{l}(\mathfrak{h})=~{}$
$\displaystyle\\{Q\subset\mathfrak{h}~{}|~{}Q\text{ is a }l\text{-dimensional
subspace of }\mathfrak{h}\\}.$

Observation. [6, Remark 3.13] Note that
$\mathfrak{p}\in\mathrm{Gr}_{d}(\mathfrak{g})$. Moreover, the tangent space
$T_{\mathfrak{p}}(\mathrm{Gr}_{d}(\mathfrak{g}))$ can be canonically
identified with
$\mathfrak{p}^{*}\otimes\mathfrak{g}/\mathfrak{p}=\mathrm{Hom}(\mathfrak{p},\mathfrak{g}/\mathfrak{p})$
as follows. If $\mathfrak{p}_{t}$ is a smooth curve in
$\mathrm{Gr}_{d}(\mathfrak{g})$ starting at $\mathfrak{p}$, we can find a
curve $a_{t}$ in $\mathrm{GL}(\mathfrak{g})$ starting at the identity and such
that $\mathfrak{p}_{t}=a_{t}(\mathfrak{p})$. Then the tangent vector
$\dot{\mathfrak{p}}_{0}=\frac{d}{dt}|_{t=0}\mathfrak{p}_{t}\in
T_{\mathfrak{p}}(\mathrm{Gr}_{d}(\mathfrak{g}))$ can be considered as an
element in
$\mathfrak{p}^{*}\otimes\mathfrak{g}/\mathfrak{p}=\mathrm{Hom}(\mathfrak{p},\mathfrak{g}/\mathfrak{p})$
by

$\displaystyle\dot{\mathfrak{p}}_{0}(p)=\frac{d}{dt}|_{t=0}a_{t}(p)\text{ mod
}\mathfrak{p},~{}\text{ for }p\in\mathfrak{p}.$ (5)

Similarly, the tangent space $T_{\mathfrak{q}}(\mathrm{Gr}_{l}(\mathfrak{h}))$
can be identified with the space
$\mathfrak{q}^{*}\otimes\mathfrak{h}/\mathfrak{q}=\mathrm{Hom}(\mathfrak{q},\mathfrak{h}/\mathfrak{q}).$

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(\mathfrak{p},\mathfrak{q},\varphi)\hookrightarrow(\mathfrak{g},\mathfrak{h},\phi)$
be a morphism Lie subalgebra. Since the map $\varphi$ is the restriction of
$\phi$ to the subspace $\mathfrak{p}$, the above morphism Lie subalgebra can
be written as $(\mathfrak{p},\mathfrak{q},\phi|_{\mathfrak{p}})$.

######  4.7 Definition.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(\mathfrak{p},\mathfrak{q},\phi|_{\mathfrak{p}})\hookrightarrow(\mathfrak{g},\mathfrak{h},\phi)$
be a morphism Lie subalgebra. A deformation of
$(\mathfrak{p},\mathfrak{q},\phi|_{\mathfrak{p}})$ consists of a pair
$(\mathfrak{p}_{t},\mathfrak{q}_{t})$ of smooth curves

$\displaystyle\mathfrak{p}_{t}\in
C^{\infty}([0,1],\mathrm{Gr}_{d}(\mathfrak{g}))~{}~{}~{}\text{ and
}~{}~{}~{}\mathfrak{q}_{t}\in C^{\infty}([0,1],\mathrm{Gr}_{l}(\mathfrak{h}))$

such that for all $t$,

  * (i)

$\mathfrak{p}_{t}$ is a Lie subalgebra of $\mathfrak{g}$ with
$\mathfrak{p}_{0}=\mathfrak{p}$,

  * (ii)

$\mathfrak{q}_{t}$ is a Lie subalgebra of $\mathfrak{h}$ with
$\mathfrak{q}_{0}=\mathfrak{q}$,

  * (iii)

$\phi(\mathfrak{p}_{t})\subset\mathfrak{q}_{t}$ and the map
$\phi|_{\mathfrak{p}_{t}}:\mathfrak{p}_{t}\rightarrow\mathfrak{q}_{t}$ is a
Lie algebra homomorphism.

Let $G$ and $H$ be the unique simply connected Lie groups integrating the Lie
algebras $\mathfrak{g}$ and $\mathfrak{h}$, respectively. The Lie algebra
homomorphism $\phi$ then integrates to a Lie group homomorphism
$\Phi:G\rightarrow H$.

######  4.8 Definition.

Two deformations $(\mathfrak{p}_{t},\mathfrak{q}_{t})$ and
$(\mathfrak{p}^{\prime}_{t},\mathfrak{q}^{\prime}_{t})$ are said to be
equivalent if there exists a smooth curve $g_{t}$ in $G$ starting at the
identity $1_{G}\in G$ such that
$\mathfrak{p}_{t}^{\prime}=\mathrm{Ad}^{G}_{g_{t}}\mathfrak{p}_{t}$ and
$\mathfrak{q}_{t}^{\prime}=\mathrm{Ad}^{G}_{\Phi(g_{t})}\mathfrak{q}_{t}$.

######  4.9 Theorem.

Let $(\mathfrak{p}_{t},\mathfrak{q}_{t})$ be a deformation of a morphism Lie
subalgebra $(\mathfrak{p},\mathfrak{q},\phi|_{\mathfrak{p}})$. Then

$\displaystyle(\dot{\mathfrak{p}}_{0},\dot{\mathfrak{q}}_{0},0)\in\mathrm{Hom}(\mathfrak{p},\mathfrak{g}/\mathfrak{p})\oplus\mathrm{Hom}(\mathfrak{q},\mathfrak{h}/\mathfrak{q})\oplus\mathfrak{h}/\mathfrak{q}=C^{1}_{\mathsf{mLA}}(\varphi,\phi/\varphi)$

is a $1$-cocycle, i.e. lies in $Z^{1}_{\mathsf{mLA}}(\varphi,\phi/\varphi)$.
Moreover, the corresponding cohomology class in
$H^{1}_{\mathsf{mLA}}(\varphi,\phi/\varphi)$ depends only on the equivalence
class of the deformation.

###### Proof.

Let $a_{t}$ be a smooth curve in $\mathrm{GL}(\mathfrak{g})$ starting at the
identity such that $a_{t}(\mathfrak{p})=\mathfrak{p}_{t}$, and $b_{t}$ be a
smooth curve in $\mathrm{GL}(\mathfrak{h})$ such that
$b_{t}(\mathfrak{q})=\mathfrak{q}_{t}$. We may choose $a_{t}$ and $b_{t}$ in
such a way that $\phi(a_{t}(p))=b_{t}(\phi(p))$, for $p\in\mathfrak{p}$ and
for all $t$. Let
$\overline{a_{t}}:\mathfrak{g}/\mathfrak{p}\rightarrow\mathfrak{g}/\mathfrak{p}_{t}$
and
$\overline{b_{t}}:\mathfrak{h}/\mathfrak{q}\rightarrow\mathfrak{h}/\mathfrak{q}_{t}$
be the induced isomorphisms. For each $t$, we define a map
$\sigma_{t}\in\mathrm{Hom}(\wedge^{2}\mathfrak{p},\mathfrak{g}/\mathfrak{p})$
by

$\displaystyle\sigma_{t}(p,r):=\overline{a_{t}}^{-1}([a_{t}(p),a_{t}(r)]_{\mathfrak{g}}~{}\text{mod
}\mathfrak{p}_{t})=(a_{t}^{-1}\circ[a_{t}(p),a_{t}(r)]_{\mathfrak{g}})\text{
mod }\mathfrak{p},~{}\text{ for }p,r\in\mathfrak{p}.$

Since $\mathfrak{p}_{t}$ is a Lie subalgebra of $\mathfrak{g}$, we have
$\sigma_{t}=0$, for all $t$. By differentiating $\sigma_{t}=0$ at $t=0$, we
obtain

$\displaystyle(-\dot{a}_{0}[p,r]_{\mathfrak{g}}+[\dot{a}_{0}(p),r]_{\mathfrak{g}}+[p,\dot{a}_{0}(r)]_{\mathfrak{g}})\text{
mod }\mathfrak{p}=0.$

In view of (5), the above identity is same as
$\delta^{\prime}_{\mathsf{CE}}(\dot{\mathfrak{p}}_{0})=0$. Similarly, one can
show that $\delta^{\prime\prime}_{\mathsf{CE}}(\dot{\mathfrak{q}}_{0})=0$.
Moreover, for any $p\in\mathfrak{p}$, we have

$\displaystyle((\phi/\varphi)\circ\dot{\mathfrak{p}}_{0}-\dot{\mathfrak{q}}_{0}\circ\varphi)(p)=~{}$
$\displaystyle(\phi/\varphi)(\frac{d}{dt}|_{t=0}a_{t}(p)~{}\text{ mod
}\mathfrak{p})-\frac{d}{dt}|_{t=0}b_{t}(\phi(p))~{}\text{ mod }\mathfrak{q}$
$\displaystyle=~{}$
$\displaystyle\frac{d}{dt}|_{t=0}\big{(}\phi(a_{t}(p))-b_{t}(\phi(p))\big{)}~{}\text{
mod }\mathfrak{q}~{}=0.$

Hence
$\delta_{\mathsf{mLA}}(\dot{\mathfrak{p}}_{0},\dot{\mathfrak{q}}_{0},0)=(\delta^{\prime}_{\mathsf{CE}}(\dot{\mathfrak{p}}_{0}),~{}\delta^{\prime\prime}_{\mathsf{CE}}(\dot{\mathfrak{q}}_{0}),~{}(\phi/\varphi)\circ\dot{\mathfrak{p}}_{0}-\dot{\mathfrak{q}}_{0}\circ\varphi)=0$.

Next, let $(\mathfrak{p}_{t},\mathfrak{q}_{t})$ and
$(\mathfrak{p}^{\prime}_{t},\mathfrak{q}^{\prime}_{t})$ be two equivalent
deformations as of Definition 4.8. Then
$a_{t}^{\prime}=\mathrm{Ad}^{G}_{g_{t}}\circ a_{t}$ is a curve in
$\mathrm{GL}(\mathfrak{g})$ and
$b_{t}^{\prime}=\mathrm{Ad}^{H}_{\Phi(g_{t})}\circ b_{t}$ is a curve in
$\mathrm{GL}(\mathfrak{h})$ satisfying
$a_{t}^{\prime}(\mathfrak{p})=\mathfrak{p}^{\prime}_{t}$ and
$b_{t}^{\prime}(\mathfrak{q})=\mathfrak{q}^{\prime}_{t}$. By differentiating
$a_{t}^{\prime}=\mathrm{Ad}^{G}_{g_{t}}\circ a_{t}$ and
$b_{t}^{\prime}=\mathrm{Ad}^{H}_{\Phi(g_{t})}\circ b_{t}$ at $t=0$ and taking
quotient by $\mathfrak{p}$ and $\mathfrak{q}$, respectively, we get

$\displaystyle\dot{\mathfrak{p}}_{0}^{\prime}(p)=\dot{\mathfrak{p}}_{0}(p)+[\dot{g}_{0},p]_{\mathfrak{g}}~{}\text{
mod }\mathfrak{p}~{}~{}~{}\text{ and
}~{}~{}~{}\dot{\mathfrak{q}}_{0}^{\prime}(q)=\dot{\mathfrak{q}}_{0}(q)+[\phi(\dot{g}_{0}),q]_{\mathfrak{h}}~{}\text{
mod }\mathfrak{q},~{}\text{ for }p\in\mathfrak{p},q\in\mathfrak{q}.$

This shows that
$(\dot{\mathfrak{p}}_{0},\dot{\mathfrak{q}}_{0},0)-(\dot{\mathfrak{p}}^{\prime}_{0},\dot{\mathfrak{q}}^{\prime}_{0},0)=\delta_{\mathsf{mLA}}(\dot{g}_{0})$
is a coboundary. Hence the proof. ∎

##  5 Abelian extensions of morphism Lie algebras

In this section, we study abelian extensions of morphism Lie algebras. Our
main result says that isomorphism classes of abelian extensions are classified
by the ‘simple’ second cohomology group of morphism Lie algebras.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(V,W,\psi)$ be a representation of it. A $n$-coboundary in
$B^{n}_{\mathsf{mLA}}(\phi,\psi)$ is said to be ‘simple’ if it is of the form
$\delta_{\mathsf{mLA}}(d_{0},\delta_{0},0)$, for some $(d_{0},\delta_{0},0)\in
C^{n-1}_{\mathsf{mLA}}(\phi,\psi)$. The space of all simple $n$-coboundaries
are denoted by $B^{n}_{\mathsf{mLA},s}(\phi,\psi)$. The quotients
$H^{n}_{\mathsf{mLA},s}:=\frac{Z^{n}_{\mathsf{mLA}}(\phi,\psi)}{B^{n}_{\mathsf{mLA},s}(\phi,\psi)},$
for $n\geq 0$, are called the simple cohomology groups of the morphism Lie
algebra $(\mathfrak{g},\mathfrak{h},\phi)$ with coefficients in $(V,W,\psi).$

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra. Let
$(V,W,\psi)$ be a triple consisting of two vector spaces $V,W$ and a linear
map $\psi:V\rightarrow W$. Note that the triple $(V,W,\psi)$ can be regarded
as a morphism Lie algebra with abelian Lie brackets on $V$ and $W$.

######  5.1 Definition.

(i) An abelian extension of a morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$ by a triple $(V,W,\psi)$ is a short exact
sequence of morphism Lie algebras

(10)

(ii) Two abelian extensions
$(\widehat{\mathfrak{g}},\widehat{\mathfrak{h}},\widehat{\phi})$ and
$(\widehat{\mathfrak{g}}^{\prime},\widehat{\mathfrak{h}}^{\prime},\widehat{\phi}^{\prime})$
are said to be isomorphic if there is an isomorphism $(\alpha,\beta)$ of
morphism Lie algebras from
$(\widehat{\mathfrak{g}},\widehat{\mathfrak{h}},\widehat{\phi})$ to
$(\widehat{\mathfrak{g}}^{\prime},\widehat{\mathfrak{h}}^{\prime},\widehat{\phi}^{\prime})$
making the following diagram commutative

(19)

Let (10) be an abelian extension of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$ by the triple $(V,W,\psi)$. A section of
this abelian extension is a pair $(s,\overline{s})$ of linear maps
$s:\mathfrak{g}\rightarrow\widehat{\mathfrak{g}}$ and
$\overline{s}:\mathfrak{h}\rightarrow\widehat{\mathfrak{h}}$ such that $p\circ
s=\mathrm{id}_{\mathfrak{g}}$ and
$\overline{p}\circ\overline{s}=\mathrm{id}_{\mathfrak{h}}$. Note that, a
section always exists.

Let $(s,\overline{s})$ be a section. Then there is a representation of
$\mathfrak{g}$ on the vector space $V$ given by
$\rho_{V}:\mathfrak{g}\rightarrow\mathrm{End}(V)$,
$\rho_{V}(x)v:=[s(x),i(v)]_{\widehat{\mathfrak{g}}}$, for $x\in\mathfrak{g}$
and $v\in V$. Note that the right hand side of the above equality lies in
$\mathrm{ker}~{}p=\mathrm{im}~{}i$. Hence it can be considered as an element
in $V$ via the embedding $i:V\rightarrow\widehat{\mathfrak{g}}$. Similarly,
there is a representation of $\mathfrak{h}$ on the vector space $W$ given by
$\rho_{W}:\mathfrak{h}\rightarrow\mathrm{End}(W)$,
$\rho_{W}(h)w:=[\overline{s}(h),\overline{i}(w)]_{\widehat{\mathfrak{h}}}$,
for $h\in\mathfrak{g}$ and $w\in W$. Finally, we observe that

$\displaystyle\psi(\rho_{V}(x)v)-\rho_{W}(\phi(x))\psi(v)$
$\displaystyle=\widehat{\phi}[s(x),i(v)]_{\widehat{\mathfrak{g}}}-[\overline{s}\circ\phi(x),\overline{i}\circ\psi(v)]_{\widehat{\mathfrak{h}}}$
$\displaystyle=[\widehat{\phi}\circ
s(x)-\overline{s}\circ\phi(x),~{}\overline{i}\circ\psi(v)]_{\widehat{\mathfrak{h}}}=0~{}~{}~{}(\text{as
}\widehat{\phi}\circ
s(x)-\overline{s}\circ\phi(x)\in\mathrm{ker}~{}\overline{p}=\mathrm{im}~{}\overline{i}).$

This shows that $(V,W,\psi)$ is a representation of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$. This representation is infact independent
of the choice of the section. To prove this, take another section
$(s^{\prime},\overline{s}^{\prime})$ of the abelian extension (10). We observe
that

$\displaystyle
s(x)-s^{\prime}(x)\in\mathrm{ker}~{}p=\mathrm{im}~{}i~{}~{}~{}\text{ and
}~{}~{}~{}\overline{s}(h)-\overline{s}^{\prime}(h)\in\mathrm{ker}~{}\overline{p}=\mathrm{im}~{}\overline{i}.$

Hence
$\rho_{V}(x)v-\rho^{\prime}_{V}(x)v=[s(x)-s^{\prime}(x),i(v)]_{\widehat{\mathfrak{g}}}=0$
and
$\rho_{W}(h)w-\rho^{\prime}_{W}(h)w:=[\overline{s}(h)-\overline{s}^{\prime}(h),\overline{i}(w)]_{\widehat{\mathfrak{h}}}=0$.
Here $\rho^{\prime}_{V}$ (resp. $\rho^{\prime}_{W}$) is the representation of
the Lie algebra $\mathfrak{g}$ (resp. $\mathfrak{h}$) on the vector space $V$
(resp. $W$) induced by the section $(s^{\prime},\overline{s}^{\prime}).$

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(V,W,\psi)$ be a fixed representation of it. Let $\mathrm{Ext}(\phi,\psi)$
denote the isomorphism classes of abelian extensions of
$(\mathfrak{g},\mathfrak{h},\phi)$ by the triple $(V,W,\psi)$ so that the
induced representation is the prescribed one.

######  5.2 Theorem.

Let $(\mathfrak{g},\mathfrak{h},\phi)$ be a morphism Lie algebra and
$(V,W,\psi)$ be a representation of it. Then there is a one-to-one
correspondence between $\mathrm{Ext}(\phi,\psi)$ and the simple second
cohomology group $H^{2}_{\mathsf{mLA},s}(\phi,\psi).$

###### Proof.

Let (10) be an abelian extension of the morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$ by the triple $(V,W,\psi)$. Let
$(s,\overline{s})$ be a section of it. We define maps

$\displaystyle\theta\in\mathrm{Hom}(\wedge^{2}\mathfrak{g},V),~{}\theta(x,y):=[s(x),s(y)]_{\widehat{\mathfrak{g}}}-s[x,y]_{\mathfrak{g}},$
$\displaystyle\gamma\in\mathrm{Hom}(\wedge^{2}\mathfrak{h},W),~{}\theta(h,k):=[\overline{s}(h),\overline{s}(k)]_{\widehat{\mathfrak{h}}}-\overline{s}[h,k]_{\mathfrak{h}},$
$\displaystyle\eta\in\mathrm{Hom}(\mathfrak{g},W),~{}\eta(x):=\widehat{\phi}(s(x))-\overline{s}(\phi(x)).$

Thus, we have $(\theta,\gamma,\eta)\in C^{2}_{\mathsf{mLA}}(\phi,\psi).$ It is
known from the classical theory of abelian extensions of Lie algebras [19]
that $\delta^{\prime}_{\mathsf{CE}}(\theta)=0$ and
$\delta^{\prime\prime}_{\mathsf{CE}}(\gamma)=0$. Moreover, by a
straightforward computation we have
$(\psi\circ\theta-\gamma\circ\wedge^{2}\phi-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\eta))=0$.
Hence we have $(\theta,\gamma,\eta)\in Z^{2}_{\mathsf{mLA}}(\phi,\psi)$.
Therefore $(\theta,\gamma,\eta)$ corresponds to a cohomology class in
$H^{2}_{\mathsf{mLA},s}(\phi,\psi)$. It is easy to verify that this cohomology
class does not depend on the choice of section. Next, we let
$(\widehat{\mathfrak{g}},\widehat{\mathfrak{h}},\widehat{\phi})$ and
$(\widehat{\mathfrak{g}}^{\prime},\widehat{\mathfrak{h}}^{\prime},\widehat{\phi}^{\prime})$
be two isomorphic abelian extensions and the isomorphism is given by
$(\alpha,\beta)$ (see Definition 5.1 (ii)). Let $(s,\overline{s})$ be a
section of the first abelian extension. We have $p^{\prime}\circ(\alpha\circ
s)=p\circ s=\mathrm{id}_{\mathfrak{g}}$ and
$\overline{p}^{\prime}\circ(\beta\circ\overline{s})=\overline{p}\circ\overline{s}=\mathrm{id}_{\mathfrak{h}}$,
which shows that $(\alpha\circ s,\beta\circ\overline{s})$ is a section of the
second abelian extension. If
$(\theta^{\prime},\gamma^{\prime},\eta^{\prime})\in
Z^{2}_{\mathsf{mLA}}(\phi,\psi)$ is the $2$-cocycle corresponding to the
second abelian extension and its section $(\alpha\circ
s,\beta\circ\overline{s})$ then

$\displaystyle\theta^{\prime}(x,y)=~{}$ $\displaystyle[(\alpha\circ
s)(x),(\alpha\circ s)(y)]_{\widehat{\mathfrak{g}}^{\prime}}-(\alpha\circ
s)([x,y]_{\mathfrak{g}})$ $\displaystyle=~{}$
$\displaystyle\alpha([s(x),s(y)]_{\widehat{\mathfrak{g}}}-s[x,y]_{\mathfrak{g}})=\theta(x,y)\quad(\text{as
}\alpha|_{V}=\mathrm{id}_{V}).$

Similarly, one can show that $\gamma^{\prime}=\gamma$ and
$\eta^{\prime}=\eta$. Hence
$(\theta,\gamma,\eta)=(\theta^{\prime},\gamma^{\prime},\eta^{\prime})$. This
shows that there is a well-defined map
$\Theta_{1}:\mathrm{Ext}(\phi,\psi)\rightarrow
H^{2}_{\mathsf{mLA},s}(\phi,\psi)$.

Conversely, let $(\theta,\gamma,\eta)\in Z^{2}_{\mathsf{mLA}}(\phi,\psi)$ be a
$2$-cocycle. Take $\widehat{\mathfrak{g}}=\mathfrak{g}\oplus V$ and
$\widehat{\mathfrak{h}}=\mathfrak{h}\oplus W$, and define a map
$\widehat{\phi}:\widehat{\mathfrak{g}}\rightarrow\widehat{\mathfrak{h}}$ by
$\widehat{\phi}(x,v)=(\phi(x),~{}\psi(v)+\eta(x))$, for
$(x,v)\in\widehat{\mathfrak{g}}$. We also define bilinear skew-symmetric
brackets on $\widehat{\mathfrak{g}}$ and $\widehat{\mathfrak{h}}$ by

$\displaystyle[(x,v),(x^{\prime},v^{\prime})]_{\widehat{\mathfrak{g}}}:=~{}$
$\displaystyle([x,x^{\prime}]_{\mathfrak{g}},~{}\rho_{V}(x)v^{\prime}-\rho_{V}(x^{\prime})v+\theta(x,x^{\prime})),$
$\displaystyle[(h,w),(h^{\prime},w^{\prime})]_{\widehat{\mathfrak{h}}}:=~{}$
$\displaystyle([h,h^{\prime}]_{\mathfrak{h}},~{}\rho_{W}(h)w^{\prime}-\rho_{W}(h^{\prime})w+\gamma(h,h^{\prime})),$

for $(x,v),(x^{\prime},v^{\prime})\in\widehat{\mathfrak{g}}$ and
$(h,w),(h^{\prime},w^{\prime})\in\widehat{\mathfrak{h}}$. Since
$(\theta,\gamma,\eta)$ is a $2$-cocycle, it follows that
$(\widehat{\mathfrak{g}},\widehat{\mathfrak{h}},\widehat{\phi})$ is a morphism
Lie algebra with the above (Lie) brackets on $\widehat{\mathfrak{g}}$ and
$\widehat{\mathfrak{h}}$, respectively. Moreover, this is an abelian extension
of the morphism Lie algebra $(\mathfrak{g},\mathfrak{h},\phi)$ by the triple
$(V,W,\psi)$. Finally, let $(\theta,\gamma,\eta)$ and
$(\theta^{\prime},\gamma^{\prime},\eta^{\prime})$ be two $2$-cocycles such
that
$(\theta,\gamma,\eta)-(\theta^{\prime},\gamma^{\prime},\eta^{\prime})=\delta_{\mathsf{mLA}}(d_{0},\delta_{0},0)$,
for some $(d_{0},\delta_{0},0)\in C^{1}_{\mathsf{mLA}}(\phi,\psi)$. Then the
abelian extensions
$(\widehat{\mathfrak{g}},\widehat{\mathfrak{h}},\widehat{\phi})$ and
$(\widehat{\mathfrak{g}}^{\prime},\widehat{\mathfrak{h}}^{\prime},\widehat{\phi}^{\prime})$
are isomorphic via the map $(\alpha,\beta)$, where
$\alpha:\widehat{\mathfrak{g}}\rightarrow\widehat{\mathfrak{g}}^{\prime}$ and
$\beta:\widehat{\mathfrak{h}}\rightarrow\widehat{\mathfrak{h}}^{\prime}$ are
the maps

$\displaystyle\alpha(x,v)=(x,v+d_{0}(x))\quad\text{ and
}\quad\beta(h,w)=(h,w+\delta_{0}(h)).$

In other words, there is a well-defined map
$\Theta_{2}:H^{2}_{\mathsf{mLA},s}(\phi,\psi)\rightarrow\mathrm{Ext}(\phi,\psi)$.
Finally, the maps $\Theta_{1}$ and $\Theta_{2}$ are inverses to each other.
This completes the proof. ∎

##  6 Classification of skeletal morphism sh Lie algebras

A morphism sh Lie algebra is a triple $(\mathcal{G},\mathcal{H},\Phi)$
consisting of two sh Lie algebras $\mathcal{G},\mathcal{H}$ and a homomorphism
$\Phi:\mathcal{G}\rightarrow\mathcal{H}$ of sh Lie algebras. In this section,
we focus on morphism sh Lie algebras whose underlying sh Lie algebras are
concentrated in degrees $0$ and $1$. We first recall some definitions from
[2].

######  6.1 Definition.

A $2$-term sh Lie algebra $\mathcal{G}$ consists of a chain complex
$\mathfrak{g}_{1}\xrightarrow{d}\mathfrak{g}_{0}$ together with a skew-
symmetric bilinear map
$l_{2}:\mathfrak{g}_{i}\otimes\mathfrak{g}_{j}\rightarrow\mathfrak{g}_{i+j}$
$(0\leq i+j\leq 1)$ and a skew-symmetric trilinear map
$l_{3}:\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\rightarrow\mathfrak{g}_{1}$
satisfying the following identities

  * (i)

$dl_{2}(x,p)=l_{2}(x,dp),$

  * (ii)

$l_{2}(dp,q)=l_{2}(p,dq),$

  * (iii)

$dl_{3}(x,y,z)=l_{2}(x,l_{2}(y,z))+l_{2}(y,l_{2}(z,x))+l_{2}(z,l_{2}(x,y)),$

  * (iv)

$l_{3}(x,y,dp)=l_{2}(x,l_{2}(y,p))+l_{2}(y,l_{2}(p,x))+l_{2}(p,l_{2}(x,y)),$

  * (v)

$l_{2}(x,l_{3}(y,z,t))-l_{2}(y,l_{3}(x,z,t))+l_{2}(z,l_{3}(x,y,t))-l_{2}(t,l_{3}(x,y,z))-l_{3}(l_{2}(x,y),z,t)\\\
+l_{3}(l_{2}(x,z),y,t)-l_{3}(l_{2}(x,t),y,z)-l_{3}(l_{2}(y,z),x,t)+l_{3}(l_{2}(y,t),x,z)-l_{3}(l_{2}(z,t),x,y)=0,$

for $x,y,z,t\in\mathfrak{g}_{0}$ and $p,q\in\mathfrak{g}_{1}$. We denote a
$2$-term sh Lie algebra by the tuple
$\mathcal{G}=(\mathfrak{g}_{1}\xrightarrow{d}\mathfrak{g}_{0},l_{2},l_{3}).$

######  6.2 Definition.

Let
$\mathcal{G}=(\mathfrak{g}_{1}\xrightarrow{d}\mathfrak{g}_{0},l_{2},l_{3})$
and
$\mathcal{H}=(\mathfrak{h}_{1}\xrightarrow{d^{\prime}}\mathfrak{h}_{0},l^{\prime}_{2},l^{\prime}_{3})$
be $2$-term sh Lie algebras. A homomorphism
$\Phi:\mathcal{G}\rightarrow\mathcal{H}$ is a tuple
$\Phi=(\phi_{0},\phi_{1},\phi_{2})$ of linear maps
$\phi_{0}:\mathfrak{g}_{0}\rightarrow\mathfrak{h}_{0}$,
$\phi_{1}:\mathfrak{g}_{1}\rightarrow\mathfrak{h}_{1}$ and a skew-symmetric
bilinear map
$\phi_{2}:\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\rightarrow\mathfrak{h}_{1}$
satisfying for $x,y,z\in\mathfrak{g}_{0}$ and $p\in\mathfrak{g}_{1}$,

  * (i)

$\phi_{0}\circ d=d^{\prime}\circ\phi_{1}$,

  * (ii)

$d\phi_{2}(x,y)=\phi_{0}(l_{2}(x,y))-l_{2}^{\prime}(\phi_{0}(x),\phi_{0}(y)),$

  * (iii)

$\phi_{2}(x,dp)=\phi_{1}(l_{2}(x,p))-l_{2}^{\prime}(\phi_{0}(x),\phi_{1}(p)),$

  * (iv)

$l_{2}^{\prime}(\phi_{0}(x),\phi_{2}(y,z))+c.p.+\phi_{2}(x,l_{2}(y,z))+c.p.=\phi_{1}(l_{3}(x,y,z))-l_{3}^{\prime}(\phi_{0}(x),\phi_{0}(y),\phi_{0}(z)).$

######  6.3 Definition.

A $2$-term morphism sh Lie algebra is a triple
$(\mathcal{G},\mathcal{H},\Phi)$ in which
$\mathcal{G}=(\mathfrak{g}_{1}\xrightarrow{d}\mathfrak{g}_{0},l_{2},l_{3})$
and
$\mathcal{H}=(\mathfrak{h}_{1}\xrightarrow{d^{\prime}}\mathfrak{h}_{0},l^{\prime}_{2},l^{\prime}_{3})$
are $2$-term sh Lie algebras and
$\Phi=(\phi_{0},\phi_{1},\phi_{2}):\mathcal{G}\rightarrow\mathcal{H}$ is a
homomorphism between them.

A skeletal morphism sh Lie algebra is a $2$-term morphism sh Lie algebra
$(\mathcal{G},\mathcal{H},\Phi)$ in which $d=0$ and $d^{\prime}=0$. In other
words, the $2$-term sh Lie algebras $\mathcal{G}$ and $\mathcal{H}$ are
skeletal in the sense of [2].

Let
$(\mathcal{G}=(\mathfrak{g}_{1}\xrightarrow{0}\mathfrak{g}_{0},l_{2},l_{3}),~{}\mathcal{H}=(\mathfrak{h}_{1}\xrightarrow{0}\mathfrak{h}_{0},l^{\prime}_{2},l^{\prime}_{3}),~{}\Phi=(\phi_{0},\phi_{1},\phi_{2}))$
be a skeletal morphism sh Lie algebra. Suppose there are skew-symmetric
bilinear maps
$\sigma:\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\rightarrow\mathfrak{g}_{1}$
and
$\sigma^{\prime}:\mathfrak{h}_{0}\otimes\mathfrak{h}_{0}\rightarrow\mathfrak{h}_{1}$,
and a linear map $\phi:\mathfrak{g}_{0}\rightarrow\mathfrak{h}_{1}$. We define
two skew-symmetric trilinear maps and a skew-symmetric bilinear map

$\underline{l_{3}}:\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\rightarrow\mathfrak{g}_{1},\qquad\underline{l^{\prime}_{3}}:\mathfrak{h}_{0}\otimes\mathfrak{h}_{0}\otimes\mathfrak{h}_{0}\rightarrow\mathfrak{h}_{1}\quad~{}~{}~{}\text{
and
}~{}~{}~{}\underline{\phi_{2}}:\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\rightarrow\mathfrak{h}_{1}$

by

$\displaystyle\underline{l_{3}}(x,y,z):=l_{3}(x,y,z)+\\{l_{2}(x,\sigma(y,z))+c.p.\\}+\\{\sigma(x,l_{2}(y,z))+c.p.\\},$
(20)
$\displaystyle\underline{l^{\prime}_{3}}(h,j,k):=l^{\prime}_{3}(h,j,k)+\\{l^{\prime}_{2}(h,\sigma^{\prime}(j,k))+c.p.\\}+\\{\sigma^{\prime}(h,l^{\prime}_{2}(j,k))+c.p.\\},$
(21)
$\displaystyle\underline{\phi_{2}}(x,y):=\phi_{2}(x,y)+\phi_{1}(\sigma(x,y))-\sigma^{\prime}(\phi_{0}(x),\phi_{0}(y))-l_{2}^{\prime}(\phi_{0}(x),\phi(y))-l_{2}^{\prime}(\phi(x),\phi_{0}(y))+\phi(l_{2}(x,y)),$
(22)

for $x,y,z\in\mathfrak{g}_{0}$ and $h,j,k\in\mathfrak{h}_{0}$. Then it is easy
to see that

$(\underline{\mathcal{G}}=(\mathfrak{g}_{1}\xrightarrow{0}\mathfrak{g}_{0},l_{2},\underline{l_{3}}),~{}\underline{\mathcal{H}}=(\mathfrak{h}_{1}\xrightarrow{0}\mathfrak{h}_{0},l^{\prime}_{2},\underline{l^{\prime}_{3}}),~{}\underline{\Phi}=(\phi_{0},\phi_{1},\underline{\phi_{2}}))$

is a skeletal morphism sh Lie algebra. In this case, we say that the skeletal
morphism sh Lie algebras $(\mathcal{G},\mathcal{H},\Phi)$ and
$(\underline{\mathcal{G}},\underline{\mathcal{H}},\underline{\Phi})$ are
equivalent. In the following result, we show that (equivalence classes of)
skeletal morphism sh Lie algebras are closely related to the cohomology of
morphism Lie algebras introduced in Section 3.

######  6.4 Theorem.

There is a one-to-one correspondence between skeletal morphism sh Lie algebras
and triples of the form
$((\mathfrak{g},\mathfrak{h},\phi),(V,W,\psi),(\theta,\gamma,\eta))$ in which
$(\mathfrak{g},\mathfrak{h},\phi)$ is a morphism Lie algebra, $(V,W,\psi)$ is
a representation and $(\theta,\gamma,\eta)\in Z^{3}_{\mathsf{mLA}}(\phi,\psi)$
is a $3$-cocycle.

Moreover, it extends to a one-to-one correspondence between equivalence
classes of skeletal morphism sh Lie algebras and triples
$((\mathfrak{g},\mathfrak{h},\phi),(V,W,\psi),[(\theta,\gamma,\eta)])$, where
$[(\theta,\gamma,\eta)]\in H^{3}_{\mathsf{mLA}}(\phi,\psi)$.

###### Proof.

Let $(\mathcal{G},\mathcal{H},\Phi)$ be a skeletal morphism sh Lie algebra,
where
$\mathcal{G}=(\mathfrak{g}_{1}\xrightarrow{0}\mathfrak{g}_{0},l_{2},l_{3})$,
$\mathcal{H}=(\mathfrak{h}_{1}\xrightarrow{0}\mathfrak{h}_{0},l^{\prime}_{2},l^{\prime}_{3})$
and $\Phi=(\phi_{0},\phi_{1},\phi_{2})$. Since
$\mathcal{G}=(\mathfrak{g}_{1}\xrightarrow{0}\mathfrak{g}_{0},l_{2},l_{3})$ is
a skeletal $2$-term sh Lie algebra, it follows from [2, Theorem 6.7] that
$\mathfrak{g}_{0}$ is a Lie algebra with the Lie bracket
$[x,y]_{\mathfrak{g}_{0}}:=l_{2}(x,y)$, for $x,y\in\mathfrak{g}_{0}$, and
$\mathfrak{g}_{1}$ is a representation of the Lie algebra $\mathfrak{g}_{0}$
with the action map
$\rho_{\mathfrak{g}_{1}}:\mathfrak{g}_{0}\rightarrow\mathrm{End}(\mathfrak{g}_{1})$
given by $\rho_{\mathfrak{g}_{1}}(x)(p):=l_{2}(x,p)$, for
$x\in\mathfrak{g}_{0}$, $p\in\mathfrak{g}_{1}$. Moreover, the skew-symmetric
trilinear map
$l_{3}\in\mathrm{Hom}(\wedge^{3}\mathfrak{g}_{0},\mathfrak{g}_{1})$ is a
$3$-cocycle in the Chevalley-Eilenberg cohomology complex of
$\mathfrak{g}_{0}$ with coefficients in the representation $\mathfrak{g}_{1}$.
Similarly, since
$\mathcal{H}=(\mathfrak{h}_{1}\xrightarrow{0}\mathfrak{h}_{0},l^{\prime}_{2},l^{\prime}_{3})$
is a skeletal $2$-term sh Lie algebra, we have that $\mathfrak{h}_{0}$ is a
Lie algebra with the bracket $[h,k]_{\mathfrak{h}_{0}}:=l^{\prime}_{2}(h,k)$,
for $h,k\in\mathfrak{h}_{0}$; the space $\mathfrak{h}_{1}$ is a representation
of the Lie algebra $\mathfrak{h}_{0}$ with the action map
$\rho_{\mathfrak{h}_{1}}:\mathfrak{h}_{0}\rightarrow\mathrm{End}(\mathfrak{h}_{1})$,
$\rho_{\mathfrak{h}_{1}}(h)(r):=l^{\prime}_{2}(h,r)$, for
$h\in\mathfrak{h}_{0},r\in\mathfrak{h}_{1}$; and
$l^{\prime}_{3}\in\mathrm{Hom}(\wedge^{3}\mathfrak{h}_{0},\mathfrak{h}_{1})$
is a $3$-cocycle in the Chevalley-Eilenberg complex of $\mathfrak{h}_{0}$ with
coefficients in $\mathfrak{h}_{1}$. Finally, since
$\Phi=(\phi_{0},\phi_{1},\phi_{2}):\mathcal{G}\rightarrow\mathcal{H}$ is a
homomorphism of $2$-term sh Lie algebras, we have from Definition 6.2 that

$\displaystyle\phi_{0}([x,y]_{\mathfrak{g}_{0}})=[\phi_{0}(x),\phi_{0}(y)]_{\mathfrak{h}_{0}},$
(23)
$\displaystyle\phi_{1}(\rho_{\mathfrak{g}_{1}}(x)(p))=\rho_{\mathfrak{h}_{1}}(\phi_{0}(x))\phi_{1}(p),$
(24)
$\displaystyle\rho_{\mathfrak{h}_{1}}(\phi_{0}(x))\phi_{2}(y,z)+c.p.+\phi_{2}(x,[y,z]_{\mathfrak{g}_{0}})+c.p.=(\phi_{1}\circ
l_{3}-l^{\prime}_{3}\circ\wedge^{3}\phi_{0})(x,y,z),$ (25)

for $x,y,z\in\mathfrak{g}_{0}$ and $p\in\mathfrak{g}_{1}$. The condition (23)
says that $\phi_{0}:\mathfrak{g}_{0}\rightarrow\mathfrak{h}_{0}$ is a Lie
algebra homomorphism. In other words,
$(\mathfrak{g}_{0},\mathfrak{h}_{0},\phi_{0})$ is a morphism Lie algebra. The
condition (24) says that the triple
$(\mathfrak{g}_{1},\mathfrak{h}_{1},\phi_{1})$ is a representation of the
morphism Lie algebra $(\mathfrak{g}_{0},\mathfrak{h}_{0},\phi_{0})$.

It follows that we may consider the representation of the Lie algebra
$\mathfrak{g}_{0}$ on the vector space $\mathfrak{h}_{1}$ with the action map
$\rho_{\mathfrak{h}_{1}}^{\mathfrak{g}_{0}}:\mathfrak{g}_{0}\rightarrow\mathrm{End}(\mathfrak{h}_{1})$,
$\rho_{\mathfrak{h}_{1}}^{\mathfrak{g}_{0}}(x)(r):=\rho_{\mathfrak{h}_{1}}(\phi_{0}(x))r=l^{\prime}_{2}(\phi_{0}(x),r)$,
for $x\in\mathfrak{g}_{0},r\in\mathfrak{h}_{1}$. Denote this representation by
$(\mathfrak{h}_{1})_{\phi_{0}}$. Let
$\\{C^{\ast}_{\mathsf{CE}}(\mathfrak{g}_{0},(\mathfrak{h}_{1})_{\phi_{0}}),\delta^{\prime\prime\prime}_{\mathsf{CE}}\\}$
be the corresponding Chevalley-Eilenberg cochain complex. Note that the
condition (25) is equivalent to $(\phi_{1}\circ
l_{3}-l^{\prime}_{3}\circ\wedge^{3}\phi_{0}-\delta^{\prime\prime\prime}_{\mathsf{CE}}(\phi_{2}))=0$.
Thus, combining all these facts, we get
$\delta_{\mathsf{mLA}}(l_{3},l_{3}^{\prime},\phi_{2})=0.$ Hence we get a
required triple
$((\mathfrak{g}_{0},\mathfrak{h}_{0},\phi_{0}),(\mathfrak{g}_{1},\mathfrak{h}_{1},\phi_{1}),(l_{3},l_{3}^{\prime},\phi_{2}))$.

Finally, if $(\mathcal{G},\mathcal{H},\Phi)$ and
$(\underline{\mathcal{G}},\underline{\mathcal{H}},\underline{\Phi})$ are
equivalent skeletal morphism sh Lie algebras, then we have from (20), (21) and
(22) that

$\displaystyle(\underline{l_{3}},\underline{l_{3}^{\prime}},\underline{\phi_{2}})-(l_{3},l_{3}^{\prime},\phi_{2})=\delta_{\mathsf{mLA}}(\sigma,\sigma^{\prime},\phi).$

Hence
$[(l_{3},l_{3}^{\prime},\phi_{2})]=[(\underline{l_{3}},\underline{l_{3}^{\prime}},\underline{\phi_{2}})]$
in $H^{3}_{\mathsf{mLA}}(\phi_{0},\phi_{1})$.

Conversely, let
$((\mathfrak{g},\mathfrak{h},\phi),(V,W,\psi),(\theta,\gamma,\eta))$ be a
triple consisting of a morphism Lie algebra
$(\mathfrak{g},\mathfrak{h},\phi)$, a representation $(V,W,\psi)$ and a
$3$-cocycle $(\theta,\gamma,\eta)\in Z^{3}_{\mathsf{mLA}}(\phi,\psi)$. Then by
a straightforward observation one can show that
$(\mathcal{G},\mathcal{H},\Phi)$ is a skeletal morphism sh Lie algebra, where

$\displaystyle\mathcal{G}=(V\xrightarrow{0}\mathfrak{g},l_{2},\theta),~{}\text{
where }l_{2}(x,v)=\rho_{V}(x)v,\text{ for }x\in\mathfrak{g},v\in V,$
$\displaystyle\mathcal{H}=(W\xrightarrow{0}\mathfrak{h},l_{2}^{\prime},\gamma),~{}\text{
where }l_{2}^{\prime}(h,w)=\rho_{W}(h)w,\text{ for }h\in\mathfrak{h},w\in W,$
$\displaystyle\Phi=(\phi,\psi,\eta).$

Moreover, if $(\theta,\gamma,\eta)$ and
$(\underline{\theta},\underline{\gamma},\underline{\eta})$ are two
cohomologous $3$-cocycles, then the corresponding skeletal morphism sh Lie
algebras are obviously equivalent. This completes the proof. ∎

##  7 Morphism Lie groups

In this section, we consider morphism Lie groups and define their cohomology
with coefficients in a module. We also define differentiable cohomology of
morphism Lie groups and construct a map from the differentiable cohomology of
a morphism Lie group to the cohomology of the corresponding morphism Lie
algebra.

A morphism Lie group is a triple $(G,H,\Phi)$ consisting of two Lie groups
$G,H$ and a homomorphism $\Phi:G\rightarrow H$ of Lie groups. Let $(G,H,\Phi)$
be a morphism Lie group. Let $\mathrm{Lie}(G)$ and $\mathrm{Lie}(H)$ be the
Lie algebras corresponding to the Lie groups $G$ and $H$, respectively. The
Lie group homomorphism $\Phi:G\rightarrow H$ induces a Lie algebra
homomorphism $\mathrm{Lie}(\Phi):\mathrm{Lie}(G)\rightarrow\mathrm{Lie}(H)$ by

$\displaystyle(\mathrm{Lie}(\Phi))(X)=(T_{1_{G}}\Phi)(X),~{}\text{ for
}X\in\mathrm{Lie}(G)=T_{1_{G}}(G).$

In other words, the triple
$(\mathrm{Lie}(G),\mathrm{Lie}(H),\mathrm{Lie}(\Phi))$ is a morphism Lie
algebra, called the ‘infinitesimal’ of the morphism Lie group $(G,H,\Phi)$.

Let $(G,H,\Phi)$ be a morphism Lie group. A module over it consists of a
triple $(V,W,\psi)$ in which $V,W$ are vector spaces with Lie group
homomorphisms $\varrho_{V}:G\rightarrow\mathrm{Aut}(V)$ and
$\varrho_{W}:H\rightarrow\mathrm{Aut}(W)$, and $\psi:V\rightarrow W$ is a
linear map satisfying $\psi(\varrho_{V}(g)v)=\varrho_{W}(\Phi(g))\psi(v)$, for
$g\in G$, $v\in V$. If $(V,W,\psi)$ is a module over the morphism Lie group
$(G,H,\Phi)$, then it can be considered as a representation of the
infinitesimal morphism Lie algebra
$(\mathrm{Lie}(G),\mathrm{Lie}(H),\mathrm{Lie}(\Phi))$. Here the
representation of the Lie algebra $\mathrm{Lie}(G)$ on $V$ is given by

$\displaystyle\rho_{V}:\mathrm{Lie}(G)\rightarrow\mathrm{End}(V),~{}\rho_{V}(X)v=\frac{d}{dt}|_{t=0}\varrho_{V}(\mathrm{exp}(tX))v,~{}\text{
for }X\in\mathrm{Lie}(G),v\in V,$

where $\mathrm{exp}:\mathfrak{g}\rightarrow G$ is the exponential map. The
representation of the Lie algebra $\mathrm{Lie}(H)$ on $W$ is defined
similarly.

In the following, we first recall the group cohomology [19] and using it we
define cohomology of morphism Lie groups. Let $G$ be a Lie group and
$\varrho_{V}:G\rightarrow\mathrm{Aut}(V)$ be a Lie group homomorphism (in this
case we say that $V$ is a $G$-module). Then there is a cochain complex
$\\{C^{\ast}_{\mathsf{gp}}(G,V),\delta_{\mathsf{gp}}\\}$, where $C^{n\geq
0}_{\mathsf{gp}}(G,V)=\mathrm{Maps}(G^{\times n},V)$ and the coboundary
operator $\delta_{\mathsf{gp}}:C^{n}_{\mathsf{gp}}(G,V)\rightarrow
C^{n+1}_{\mathsf{gp}}(G,V)$ is given by

$\displaystyle(\delta_{\mathsf{gp}}f)(g_{1},\ldots,g_{n+1})=$
$\displaystyle\varrho_{V}(g_{1})(f(g_{2},\ldots,g_{n+1}))+(-1)^{i}f(g_{1},\ldots,g_{i}\cdot
g_{i+1},\ldots,g_{n+1})+(-1)^{n+1}f(g_{1},\ldots,g_{n}),$

for $f\in C^{n}_{\mathsf{gp}}(G,V)$ and $g_{1},\ldots,g_{n+1}\in G$. The
corresponding cohomology groups are called the group cohomology of $G$ with
coefficients in the $G$-module $V$. The differential cohomology of $G$ with
coefficients in $V$ is the cohomology of the subcomplex
$\\{C^{\ast}_{\mathsf{gp},d}(G,V),\delta_{\mathsf{gp}}\\}$, where

$\displaystyle C^{n}_{\mathsf{gp},d}(G,V):=\\{f\in
C^{n}_{\mathsf{gp}}(G,V)~{}|~{}f(g_{1},\ldots,g_{n})=0\text{ if
}g_{i}=1_{G}\\}.$

The van Est map [15] is a map between cochain complexes
$\mathrm{vE}:C^{\ast}_{\mathsf{gp},d}(G,V)\rightarrow
C^{\ast}_{\mathsf{CE}}(\mathrm{Lie}(G),V)$ from
$\\{C^{\ast}_{\mathsf{gp},d}(G,V),\delta_{\mathsf{gp}}\\}$ to the Chevalley-
Eilenberg cochain complex
$\\{C^{\ast}_{\mathsf{CE}}(\mathrm{Lie}(G),V),\delta_{\mathsf{CE}}\\}.$ It is
explicitly given by

$\displaystyle(\mathrm{vE})(f)(X_{1},\ldots,X_{n})=\sum_{\sigma\in
S_{n}}\mathrm{sgn}(\sigma)R_{X_{\sigma(1)}}\cdots R_{X_{\sigma(n)}}(f),$ (26)

where $R_{X}:C^{n}_{\mathsf{gp},d}(G,V)\rightarrow
C^{n-1}_{\mathsf{gp},d}(G,V)$ is the map which differentiates
$f(-,g_{2},\ldots,g_{n})$ at the point $1_{G}$ with respect to the right
invariant vector field corresponding to $X$.

Let $(G,H,\Phi)$ be a morphism Lie group and $(V,W,\psi)$ be a module over it.
Then there is a $G$-module structure on $W$ given by the Lie group
homomorphism
$\varrho_{W}^{G}:G\rightarrow\mathrm{Aut}(W),\varrho_{W}^{G}(g)(w)=\varrho_{W}(\Phi(g))w,$
for $g\in G,w\in W$. We denote this $G$-module by $W_{\Phi}$. Let
$\delta_{\mathsf{gp}}^{\prime}$ (resp. $\delta_{\mathsf{gp}}^{\prime\prime}$)
be the coboundary operator for the group cohomology complex of $G$ with
coefficients in $V$ (resp. of $H$ with coefficients in $W$), and
$\delta^{\prime\prime\prime}_{\mathsf{gp}}$ be the coboundary operator for the
group cohomology complex of $G$ with coefficients in $W_{\Phi}$. For each
$n\geq 0$, we define an abelian group $C^{n}_{\mathsf{mLG}}(\Phi,\psi)$ by

$\displaystyle C^{0}_{\mathsf{mLG}}(\Phi,\psi)=V\quad\text{ and }\quad
C^{n\geq 1}_{\mathsf{mLG}}(\Phi,\psi)=C^{n}_{\mathsf{gp}}(G,V)\oplus
C^{n}_{\mathsf{gp}}(H,W)\oplus C^{n-1}_{\mathsf{gp}}(G,W_{\Phi}).$

We define a map
$\delta_{\mathsf{mLG}}:C^{n}_{\mathsf{mLG}}(\Phi,\psi)\rightarrow
C^{n+1}_{\mathsf{mLG}}(\Phi,\psi)$, for $n\geq 0$, by

$\displaystyle\delta_{\mathsf{mLG}}(v)=(\delta^{\prime}_{\mathsf{gp}}(v),~{}\delta^{\prime\prime}_{\mathsf{gp}}(\psi(v)),~{}0)~{}~{}\text{
and
}~{}~{}\delta_{\mathsf{mLG}}(\Theta,\Gamma,\Lambda)=(\delta^{\prime}_{\mathsf{gp}}(\Theta),~{}\delta^{\prime\prime}_{\mathsf{gp}}(\Gamma),~{}\psi\circ\Theta-\Gamma\circ\Phi^{\times
n}-\delta^{\prime\prime\prime}_{\mathsf{gp}}(\Lambda)),$

for $v\in V$ and $(\Theta,\Gamma,\Lambda)\in C^{n\geq
1}_{\mathsf{mLG}}(\Phi,\psi).$ Then it is easy to see that
$(\delta_{\mathsf{mLG}})^{2}=0$. The cohomology groups of the cochain complex
$\\{C^{\ast}_{\mathsf{mLG}}(\Phi,\psi),\delta_{\mathsf{mLG}}\\}$ are called
the cohomology of the morphism Lie group $(G,H,\Phi)$ with coefficients in the
module $(V,W,\psi)$, and denoted by $H^{\ast}_{\mathsf{mLG}}(\Phi,\psi).$

######  7.1 Remark.

For any Lie group $G$, the corresponding Lie algebra $\mathrm{Lie}(G)$ can be
given a $G$-module structure with the Lie group homomorphism
$\rho_{\mathrm{Lie}(G)}:G\rightarrow\mathrm{Aut}(\mathrm{Lie}(G))$,
$(\rho_{\mathrm{Lie}(G)}(g))X:=\mathrm{Ad}^{G}_{g}(X)$, for $g\in G$ and
$X\in\mathrm{Lie}(G)$. This is called the adjoint $G$-module. It has been
shown in [5] that the group cohomology of $G$ with coefficients in the adjoint
$G$-module $\mathrm{Lie}(G)$ controls the deformations of $G$. Similar to the
definition of adjoint module of a Lie group, we may define the adjoint module
of a morphism Lie group $(G,H,\Phi)$ on the triple
$(\mathrm{Lie}(G),\mathrm{Lie}(H),\mathrm{Lie}(\Phi))$. By following [5] one
may show that the cohomology of the morphism Lie group $(G,H,\Phi)$ with
coefficients in the adjoint module governs the deformations. We will address
deformation related questions for morphism Lie groups in a separate article.

In the following, we consider the differentiable subcomplex
$\\{C^{\ast}_{\mathsf{mLG},d}(\Phi,\psi),\delta_{\mathsf{mLG}}\\}$ of the
cochain complex
$\\{C^{\ast}_{\mathsf{mLG}}(\Phi,\psi),\delta_{\mathsf{mLG}}\\}$, where

$\displaystyle C^{0}_{\mathsf{mLG},d}(\Phi,\psi)=V~{}~{}~{}\text{ and
}~{}~{}~{}C^{n}_{\mathsf{mLG},d}(\Phi,\psi):=C^{n}_{\mathsf{gp},d}(G,V)\oplus
C^{n}_{\mathsf{gp},d}(H,W)\oplus C^{n-1}_{\mathsf{gp},d}(G,W_{\Phi})\subset
C^{n}_{\mathsf{mLG}}(\Phi,\psi).$

We call the corresponding cohomology groups as the differentiable cohomology
of the morphism Lie group $(G,H,\Phi)$ with coefficients in the module
$(V,W,\psi)$. We denote cohomology groups by
$H^{\ast}_{\mathsf{mLG},d}(\Phi,\psi)$.

Let $(G,H,\Phi)$ be a morphism Lie group and $(V,W,\psi)$ be a bimodule over
it. Therefore, we can consider the differentiable complex
$\\{C^{\ast}_{\mathsf{mLG},d}(\Phi,\psi),\delta_{\mathsf{mLG}}\\}$. On the
other hand, the triple $(V,W,\psi)$ can be regarded as a representation of the
morphism Lie algebra $(\mathrm{Lie}(G),\mathrm{Lie}(H),\mathrm{Lie}(\Phi))$.
Thus, we can consider the cochain complex
$\\{C^{\ast}_{\mathsf{mLA}}(\mathrm{Lie}(\Phi),\psi),\delta_{\mathsf{mLA}}\\}$.
Following the van Est map (26), we are now able to find a connection between
these two cochain complexes and induced cohomologies.

######  7.2 Theorem.

With the above notations, the collection of maps
$\mathrm{vE}:C^{n}_{\mathsf{mLG},d}(\Phi,\psi)\rightarrow
C^{n}_{\mathsf{mLA}}(\mathrm{Lie}(\Phi),\psi)$ given by

$\displaystyle(\mathrm{vE})(\Theta,\Gamma,\Lambda)=\big{(}(\mathrm{vE})(\Theta),~{}(\mathrm{vE})(\Gamma),~{}(\mathrm{vE})(\Lambda)\big{)}$

defines a homomorphism of cochain complexes from
$\\{C^{\ast}_{\mathsf{mLG},d}(\Phi,\psi),\delta_{\mathsf{mLG}}\\}$ to
$\\{C^{\ast}_{\mathsf{mLA}}(\mathrm{Lie}(\Phi),\psi),\delta_{\mathsf{mLA}}\\}$.
Hence there is a induced map
$\mathrm{vE}:H^{\ast}_{\mathsf{mLG},d}(\Phi,\psi)\rightarrow
H^{\ast}_{\mathsf{mLA}}(\mathrm{Lie}(\Phi),\psi)$ on the level of cohomology.

Acknowledgements. The author would like to thank IIT Kanpur, India where some
parts of the work have been carried out.

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