# Isoclinic relative central extensions of a pair of regular Hom-Lie algebras

Rudra Narayan Padhan  Institute of Technical Education and Research
Siksha ‘O’ Anusandhan (A Deemed to be University)
Bhubaneswar-751030
Odisha, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and
Tofan Kumar Khuntia  Department of Mathematics, National Institute of
Technology
Rourkela, Odisha-769008
India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we show the relation among the relative central extensions in
an isoclinism family of a particular relative central extension of Hom-Lie
algebras. We define the notion of isoclinism on the central relative
extensions of a pair of Hom-Lie algebras. Then, we figure out the concept of
isomorphism in the equivalence class of isoclinisms on the central relative
extensions of a pair of Hom-Lie algebras.

###### Key words and phrases:

Hom-Lie algebra; relative central extension; isoclinism

###### 2020 Mathematics Subject Classification:

Primary 17B61.

##  1\. Introduction and preliminaries

The concept of Hom-Lie algebras was first introduced by Hartwig et al. [5], as
a part of a study of deformations of the Witt and the Virasoro algebras. A
Hom-Lie algebra is basically a non-associative algebra satisfying the skew-
symmetry and the Hom-Jacobi identity twisted by a linear map. If we take this
linear map as the identity map, then the Hom-Lie algebra becomes a Lie
algebra. Therefore, the generalizations of the known theories from Lie
algebras to the Hom-Lie play a crucial role as these generalizations are not
straight forward. Also, because of the close relation to discrete and deformed
vector fields and differential calculus, Hom-Lie algebras are widely studied
recently [2, 3, 13].

The notion of isoclinism for Lie algebras was first introduced by Moneyhun [9]
in 1994. Eshrati et al. [4] studied various properties of isoclinism in
$n$-Lie algebras. Isoclinism of various algebraic structures have been
exploid, for more details see [6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 11].
Recently in 2020, Arabyani and Sheikh-mohseni [1] introduced the concept of
isoclinism on the relative central extensions of pairs of Lie algebras.
Lately, isoclinism has been defined and studied for Hom-Lie algebra by Padhan
et al. [13]. In this paper, we study the relative central extensions of pairs
of Hom-Lie algebras.

Firstly,x we define the notions such as isoclinism between a pair of Hom-Lie
algebras, Hom-action and relative central extensions. In Section 2, we
introduce the concept of isoclinism on the relative central extensions of a
pair of Hom-Lie algebras which is a generalization of the work in [1]. We also
give some equivalent conditions under which two relative central extensions
are isoclinic. In Section 3, we define the concept of factor sets in the
context of relative central extensions of a pair of Hom-Lie algebras. Then we
show that under certain conditions, two concepts of isoclinism and isomorphism
between the relative central extensions of a pair of finite dimensional Hom-
Lie algebras are same. In fact, we show that under some conditions two finite
dimensional relative central extensions are isoclinic if and only if they are
isomorphic.

######  Definition 1.1.

A Hom-Lie algebra $(L,\alpha_{L})$ is a non-associative algebra $L$ together
with a bilinear map $[.,.]:L\times L\rightarrow L$ and a linear map
$\alpha_{L}:L\rightarrow L$ satisfying

$[x,y]=-[y,x]$ (skew-symmetry)

$[\alpha_{L}(x),[y,z]]+[\alpha_{L}(y),[z,x]]+[\alpha_{L}(z),[x,y]]=0$ (Hom-
Jacobi identity)

for all $x,y,z\in L$.

A Hom-Lie subalgebra $(H,\alpha_{H})$ of a Hom-Lie algebra $(L,\alpha_{L})$ is
a vector subspace $H$ of $L$ closed under the product, i.e., $[x,y]\in
H~{}\forall~{}x,y\in H$, together with the endomorphism
$\alpha_{H}:H\rightarrow H$ being the restriction of $\alpha_{L}$ on $H$. A
Hom-Lie subalgebra $(H,\alpha_{H})$ of $(L,\alpha_{L})$ is said to be an ideal
if $[x,y]\in H~{}\forall~{}x\in H,~{}y\in L$. The center of a Hom-Lie algebra
$(L,\alpha_{L})$ is the vector subspace

$Z(L)=\\{x\in L~{}|~{}[x,y]=0~{}\forall~{}y\in L\\}.$

A Hom-Lie algebra $(L,\alpha_{L})$ is said to be $multiplicative$ if
$\varphi([l_{1},l_{2}])=[\alpha(l_{1}),\alpha(l_{2})]$ for all $l_{1},l_{2}\in
L$. A multiplicative Hom-Lie algebra $(L,\alpha_{L})$ is said to be $regular$
if $\alpha$ is bijective.

######  Definition 1.2.

Let $(L_{1},\alpha_{1})$ and $(L_{2},\alpha_{2})$ be two Hom-Lie algebras.
Then a morphism from $L_{1}$ to $L_{2}$ is a linear map $f:L_{1}\rightarrow
L_{2}$ such that $f([l,m]_{L_{1}})=[f(l),f(m)]_{L_{2}}$ for all $l,m\in L_{1}$
and $f\circ\alpha_{1}=\alpha_{2}\circ f$ i.e., the following diagram commutes;

$L_{1}$$L_{1}$$L_{2}$$L_{2}$f$\alpha_{1}$f$\alpha_{2}$

######  Definition 1.3.

Let $(M,\alpha_{M})$ be an ideal of a Hom-Lie algebra $(L,\alpha_{L})$. Then
$(M,L)$ is said to be a pair of Hom-Lie algebras and we may define the
commutator as

$[M,L]=\langle[m,l]:m\in M,~{}l\in L\rangle$

and the centre of $(M,L)$ can be defined as

$Z(M,L)=\\{m\in M:[m,l]=0,\forall~{}l\in L\\}.$

If $L$ is regular Hom-Lie algebra, then $[M,L]$ and $Z(M,L)$ are ideals of
$L$. Now onwards we always assume that the the given Hom-Lie algebra is
regular.

######  Definition 1.4.

Let $(M_{1},L_{1})$ and $(M_{2},L_{2})$ be two pairs of Hom-Lie algebras.
These two pairs are said to be isoclinic, denoted by
$(M_{1},L_{1})\sim(M_{2},L_{2})$, if there exists a pair of Hom-Lie
isomorphisms $(\varphi,\theta)$ where,
$\varphi:L_{1}/Z(M_{1},L_{1})\rightarrow L_{2}/Z(M_{2},L_{2})$ is such that
$\varphi(\bar{M_{1}})=\bar{M_{2}}$,
$\bar{M_{j}}=M_{j}/Z(M_{j},L_{j}),~{}j=1,2$ and
$\theta:[M_{1},L_{1}]\rightarrow[M_{2},L_{2}]$ satisfies
$\theta([m_{1},l_{1}])=[m_{2},l_{2}],$ for all $m_{1}\in M_{1},l_{1}\in
L_{1},\bar{m_{2}}\in\varphi(\bar{m_{1}}),\bar{l_{2}}\in\varphi(\bar{l_{1}})$,
and the following diagram is commutative:

$M_{1}/Z(M_{1},L_{1})\times L_{1}/Z(M_{1},L_{1})$$M_{2}/Z(M_{2},L_{2})\times
L_{2}/Z(M_{2},L_{2})$$[M_{1},L_{1}]$$[M_{2},L_{2}]$$\mu$$\varphi^{2}$$\rho$$\theta$

Observe that when $M_{i}=L_{i}$ for $i=1,2$ on the above definition, we have
the definition for isoclinism of Hom-Lie algebras [13].

######  Definition 1.5.

Let $(L,\alpha_{L})$ and $(K,\alpha_{K})$ be two Hom-Lie algebras. A Hom-
action of $L$ on $K$ is a linear map $L\otimes K\rightarrow K,~{}x\otimes
k\mapsto~{}^{x}k$, satisfying the following properties:

  1. (1)

${}^{[x,y]}\alpha_{K}(k)=~{}^{\alpha_{L}{(x)}}(^{y}k)-~{}^{\alpha_{L}{(y)}}(^{x}k)$

  2. (2)

${}^{\alpha_{L}{(x)}}[k,k^{\prime}]=[^{x}k,\alpha_{K}(k^{\prime})]+[\alpha_{K}(k),~{}^{x}(k^{\prime})]$

  3. (3)

$\alpha_{K}(^{x}k)=~{}^{\alpha_{L}{(x)}}\alpha_{K}(k)$

for all $x,y\in L,$ and $k,k^{\prime}\in K$.

It is easy to see that if $K$ is an ideal of $L$, then by Lie multiplication
we have an action of $L$ on $K$.

######  Definition 1.6.

A relative central extension of a pair of Hom-Lie algebras $(M,L)$ is a
homomorphism of Hom-Lie algebras
$\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ together with an action
of $L$ on $M^{*}$ satisfying the following conditions:

  1. (1)

$\sigma(M^{*})=M,$

  2. (2)

$\sigma(^{l}{\alpha_{*}(m)})=[\alpha_{L}(l),\sigma(\alpha_{*}(m))],~{}$ for
all $l\in L,~{}m\in M^{*}$,

  3. (3)

${}^{\sigma(m^{\prime})}\alpha_{*}(m)=[m^{\prime},m],~{}$ for all
$m,m^{\prime}\in M^{*}$,

For a given relative central extension $\sigma:M^{*}\rightarrow L$, the
$L$-commutator and the $L$-central subalgebras of $M^{*}$ are defined
respectively, as follows:

$[M^{*},L]=\\{~{}^{l}\alpha_{*}(m)|~{}l\in L,m\in M^{*}\\}$

and

$Z(M^{*},L)=\\{m\in M^{*}|~{}^{l}\alpha_{*}(m)=0,\forall~{}l\in L\\}.$

Observe that $\textnormal{Ker}~{}\sigma\subseteq Z(M^{*},L)$ and
$\textnormal{Ker}~{}\sigma\subseteq Z(M^{*})$.

##  2\. Isoclinic relative extensions

In this section, we introduce the concept of isoclinism on the relative
central extensions of pairs of Hom-Lie algebras.

######  Definition 2.1.

Let $(M_{1},L_{1})$ and $(M_{2},L_{2})$ be two pairs of Hom-Lie algebras. Let
there be two Hom-Lie homomorphisms
$\gamma:(L_{1},\alpha_{1})\rightarrow(L_{2},\alpha_{2})$ and
$\beta:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(M_{2}^{*},\alpha_{2}^{*})$ with
$\gamma(M_{1})=M_{2}$ and $\beta([M_{1}^{*},L_{1}])=[M_{2}^{*},L_{2}]$ such
that $\gamma\sigma_{1}(m_{1})=\sigma_{2}\beta(m_{1})$, where
$\sigma_{i}:(M_{i}^{*},\alpha_{i}^{*})\rightarrow(L_{i},\alpha_{i}),~{}(i=1,2)$
are relative central extensions of $(M_{i},L_{i}),~{}(i=1,2)$ respectively.
Then the pair $(\gamma,\beta):\sigma_{1}\rightarrow\sigma_{2}$ is called a
morphism from $\sigma_{1}$ to $\sigma_{2}$. If $\gamma$ and $\beta$ are
isomorphic then the pair $(\gamma,\beta)$ is called an isomorphism. The
isomorphism between $\sigma_{1}$ and $\sigma_{2}$ is denoted by
$\sigma_{1}\cong\sigma_{2}$.

######  Definition 2.2.

The relative central extensions
$\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$ and
$\sigma_{2}:(M_{2}^{*},\alpha_{2}^{*})\rightarrow(L_{2},\alpha_{2})$ are said
to be isoclinic if there exist Hom-Lie isomorphisms
$\gamma:(L_{1},\alpha_{1})\rightarrow(L_{2},\alpha_{2})$ with
$\gamma(M_{1})=M_{2}$ and
$\beta^{\prime}:[M_{1}^{*},L_{1}]\rightarrow[M_{2}^{*},L_{2}]$ such that for
all $m_{1}\in M_{1}^{*},l_{1}\in L_{1}$ we have
$\beta^{\prime}(^{l_{1}}\alpha_{1}^{*}(m_{1}))=~{}^{l_{2}}\alpha_{2}^{*}(m_{2})$,
where $m_{2}\in M_{2}^{*},l_{2}\in L_{2}$ and
$\gamma\sigma_{1}(\alpha_{1}^{*}(m_{1}))=\sigma_{2}(\alpha_{2}^{*}(m_{2})),\gamma\alpha_{1}(l_{1})=\alpha_{2}(l_{2}).$
In this case, the pair $(\gamma,\beta^{\prime})$ is called an isoclinism from
$\sigma_{1}$ to $\sigma_{2}$ and we write
$(\gamma,\beta^{\prime}):\sigma_{1}\cong\sigma_{2}$. A morphism
$(\gamma,\beta):\sigma_{1}\rightarrow\sigma_{2}$ is called isoclinic if the
pair $(\gamma,\beta|_{[M_{1}^{*},L_{1}]})$ is an isoclinism from $\sigma_{1}$
to $\sigma_{2}$.

Suppose that $(M_{1},L_{1})$ and $(M_{2},L_{2})$ be two pairs of Hom-Lie
algebras and let for
$i=1,2,~{}\overline{M_{i}}=M_{i}/Z(M_{i},L_{i}),~{}\overline{L_{i}}=L_{i}/Z(M_{i},L_{i})$.
If $(\gamma,\beta^{\prime})$ be an isoclinism between relative central
extensions $\overline{\sigma_{1}}:\overline{M_{1}}\rightarrow\overline{L_{1}}$
and $\overline{\sigma_{2}}:\overline{M_{2}}\rightarrow\overline{L_{2}}$, then
$\beta^{\prime}([\overline{m_{1}},\overline{l_{1}}])=[\overline{m_{2}},\overline{l_{2}}]$,
where $\gamma(\overline{l_{1}})=\overline{l_{2}}$ and
$\gamma(\overline{m_{1}})=\overline{m_{2}}$. Now, if we consider
$\beta:[M_{1},L_{1}]\rightarrow[M_{2},L_{2}]$, a Hom-Lie isomorphism defined
by $\beta([m_{1},l_{1}])=[m_{2},l_{2}]$, then the pair $(\gamma,\beta)$ will
be an isoclinism from $(M_{1},L_{1})$ to $(M_{2},L_{2})$. Therefore, two pairs
of Hom-Lie algebras $(M_{1},L_{1})$ and $(M_{2},L_{2})$ are isoclinic if the
relative central extensions $\overline{\sigma_{1}}$ and
$\overline{\sigma_{2}}$ are isoclinic.

Now we give some examples of isoclinic morphisms.

Example 1. Let $(L,\alpha_{L})$ be a Hom-Lie algebra and $A$ be an abelian
Hom-Lie subalgebra of $L$, then the maps $\pi_{L}:L\times A\rightarrow L$ and
$i_{L}:L\rightarrow L\times A$ denote the projective and canonical Hom-Lie
homomorphisms, respectively. Let
$\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ be a relative central
extension of the pair $(M,L)$. Then the morphism
$\sigma\pi_{M^{*}}:M^{*}\times A\rightarrow L$ is a relative central extension
of $(M,L)$ together with an action of $L$ on $M^{*}\times A$ defined by
${}^{l}(m,a)=(^{l}m,a)$. It can be easily verified that
$(1_{L},\pi_{M^{*}}):\sigma\pi_{M^{*}}\rightarrow\sigma$ and
$(1_{L},i_{M^{*}}):\sigma\rightarrow\sigma\pi_{M^{*}}$ are isoclinic
epimorphism and isoclinic monomorphism, respectively.

Example 2. Let $\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ be a
relative central extension of the pair $(M,L)$ and $N$ be an ideal of $M^{*}$
such that $N\subseteq\textnormal{Ker}{\sigma}$. Let
$\bar{\sigma}:M^{*}/N\rightarrow L$ be the morphism induced by $\sigma$. This
$\bar{\sigma}$ along with an action of $L$ on $M^{*}/N$ given by
${}^{l}(m+N)=(~{}^{l}m+N)$ forms a relative central extension of $(M,L)$. Let
$\rho_{M^{*}}:M^{*}\rightarrow M^{*}/N$ denote the natural epimorphism. Then
we find an isoclinic epimorphism
$(1_{L},\rho_{M^{*}}):\sigma\rightarrow\bar{\sigma}$ from $\sigma$ to
$\bar{\sigma}$.

######  Lemma 2.3.

Let $\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$ and
$\sigma_{2}:(M_{2}^{*},\alpha_{2}^{*})\rightarrow(L_{2},\alpha_{2})$ be
relative central extensions of the pairs of Hom-Lie algebras $(M_{1},L_{1})$
and $(M_{2},L_{2})$, respectively. If
$(\gamma,\beta^{\prime}):\sigma_{1}\rightarrow\sigma_{2}$ is an isoclinism,
then

  1. (1)

$\gamma\sigma_{1}(x)=\sigma_{2}\beta^{\prime}(x)$, for all
$x\in[M_{1}^{*},L_{1}]$.

  2. (2)

$\beta^{\prime}(\textnormal{Ker}\sigma_{1}\cap[M_{1}^{*},L_{1}])=\textnormal{Ker}\sigma_{2}\cap[M_{2}^{*},L_{2}]$.

  3. (3)

$\beta^{\prime}(^{l_{1}}x)=~{}^{l_{2}}\beta^{\prime}(x)$, for all
$x\in[M_{1}^{*},L_{1}],l_{i}\in L_{i},i=1,2$ with
$\gamma\alpha_{1}(l_{1})=\alpha_{2}(l_{2})$.

###### Proof.

  1. (1)

Let $x\in[M_{1}^{*},L_{1}]$ such that $x=~{}^{l_{1}}\alpha_{1}^{*}(m_{1})$,
for some $l_{1}\in L_{1}$ and $m_{1}\in M_{1}^{*}$. Then,

$\gamma\sigma_{1}(x)=\gamma\sigma_{1}(^{l_{1}}\alpha_{1}^{*}(m_{1}))=\gamma[\alpha_{1}(l_{1}),\sigma_{1}(\alpha_{1}^{*}(m_{1}))]=[\gamma\alpha_{1}(l_{1}),\gamma\sigma_{1}(\alpha_{1}^{*}(m_{1}))].$

Now if $\gamma\alpha_{1}(l_{1})=\alpha_{2}(l_{2})$ and
$\gamma\sigma_{1}(\alpha_{1}^{*}(m_{1}))=\sigma_{2}(\alpha_{2}^{*}(m_{2}))$
where $l_{2}\in L_{2}$ and $m_{2}\in M_{2}^{*}$, then

$\gamma\sigma_{1}(x)=[\alpha_{2}(l_{2}),\sigma_{2}(\alpha_{2}^{*}(m_{2}))]=\sigma_{2}(^{l_{2}}\alpha_{2}^{*}(m_{2}))=\sigma_{2}(\beta^{\prime}(x)).$

  2. (2)

Let $x\in\textnormal{Ker}\sigma_{1}\cap[M_{1}^{*},L_{1}]$, then
$\sigma_{1}(x)=0$ and $x=~{}^{l_{1}}\alpha_{1}^{*}(m_{1})$, for some $l_{1}\in
L_{1},~{}m_{1}\in M_{1}^{*}$. Suppose that
$\gamma\alpha_{1}(l_{1})=\alpha_{2}(l_{2})$ and
$\gamma\sigma_{1}(\alpha_{1}^{*}(m_{1}))=\sigma_{2}(\alpha_{2}^{*}(m_{2}))$,
where $l_{2}\in L_{2}$ and $m_{2}\in M_{2}^{*}$. Now from the definition of
isoclinism we have
$\beta^{\prime}(^{l_{1}}\alpha_{1}^{*}(m_{1}))=~{}^{l_{2}}\alpha_{2}^{*}(m_{2})$.
Also we have,

$\sigma_{2}(^{l_{2}}\alpha_{2}^{*}(m_{2}))=[\alpha_{2}(l_{2}),\sigma_{2}(\alpha_{2}^{*}(m_{2}))]=\gamma[\alpha_{1}(l_{1}),\sigma_{1}(\alpha_{1}^{*}(m_{1}))]=\gamma\sigma_{1}(x)=0.$

Therefore,
$\beta^{\prime}(x)\in\textnormal{Ker}\sigma_{2}\cap[M_{2}^{*},L_{2}]$ and the
converse is obvious.

  3. (3)

From (1), we have $\gamma\sigma_{1}(x)=\sigma_{2}(\beta^{\prime}(x))$. So by
the isoclinic properties of the relative central extensions $\sigma_{1}$,
$\sigma_{2}$ and for all $x\in[M_{1}^{*},L_{1}],l_{1}\in
L_{1},\gamma\alpha_{1}(l_{1})=\alpha_{2}(l_{2})$, we can clearly have
$\beta^{\prime}(^{l_{1}}x)=~{}^{l_{2}}\beta^{\prime}(x)$.

∎

Let $(M_{1},L_{1})$ and $(M_{2},L_{2})$ be two pairs of Hom-Lie algebras and
suppose that
$\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$ and
$\sigma_{2}:(M_{2}^{*},\alpha_{2}^{*})\rightarrow(L_{2},\alpha_{2})$ are
relative central extensions of these pairs, respectively. Let
$\gamma:(L_{1},\alpha_{1})\rightarrow(L_{2},\alpha_{2})$ be an isomorphism
such that $\gamma(M_{1})=M_{2}$ and

$\overline{M^{*}}=\\{(m_{1},m_{2})|m_{i}\in
M_{i}^{*},i=1,2,~{}\gamma\sigma_{1}(\alpha_{1}^{*}(m_{1}))=\sigma_{2}(\alpha_{2}^{*}(m_{2}))\\}.$

Then $\overline{M^{*}}$ is a subalgebra of $M_{1}^{*}\times M_{2}^{*}$. Also,
the homomorphism

$\displaystyle\hat{\sigma}:\overline{M^{*}}$ $\displaystyle\longrightarrow
L_{1}$ $\displaystyle(m_{1},m_{2})$
$\displaystyle\longmapsto\sigma_{1}(m_{1})$

together with an action of $L_{1}$ on $\overline{M^{*}}$ defined by
${}^{l_{1}}(m_{1},m_{2})=(^{l_{1}}m_{1},~{}^{l_{2}}m_{2}),$ where
$\gamma\alpha_{1}(l_{1})=\alpha_{2}(l_{2})$ is a relative central extension of
the pair $(M_{1},L_{1})$.

######  Proposition 2.4.

The isomorphism $\gamma:(L_{1},\alpha_{1})\rightarrow(L_{2},\alpha_{2})$
induces an isoclinism from $\sigma_{1}$ to $\sigma_{2}$ if and only if there
exist isoclinic epimorphisms from $\hat{\sigma}$ onto $\sigma_{1}$ and
$\sigma_{2}$.

###### Proof.

The necessary condition holds true obviously. To prove the converse part, let
$\gamma$ induces an isoclinism from $\sigma_{1}$ to $\sigma_{2}$. Then there
exists an isomorphism
$\beta^{\prime}:[M_{1}^{*},L_{1}]\rightarrow[M_{2}^{*},L_{2}]$ induced by
$\gamma$ such that the pair $(\gamma,\beta^{\prime})$ is an isoclinism from
$\sigma_{1}$ to $\sigma_{2}$. Now according to the hypothesis and by Lemma 2.3
(1) we have

$[\overline{M^{*}},L_{1}]=\\{(m_{1},\beta^{\prime}(m_{1}))|m_{1}\in[M_{1}^{*},L_{1}]\\}.$

Let $\beta_{i}:\overline{M^{*}}\rightarrow M_{i}^{*}$ be the projective
homomorphisms and let $\gamma_{1}=1_{L_{1}},\gamma_{2}=\gamma$. Now it can be
easily checked that $(\beta_{1},\gamma_{1}):\hat{\sigma}\rightarrow\sigma_{1}$
and $(\beta_{2},\gamma_{2}):\hat{\sigma}\rightarrow\sigma_{2}$ are isoclinic
epimorphisms and the result follows. ∎

Let $A=\overline{M^{*}}/[\overline{M^{*}},L_{1}]$. Then $A$ is an abelian Hom-
Lie algebra. Thus by example 1, for any central relative extension
$\sigma:M^{*}\rightarrow L$ of the pair $(M,L)$,
$\sigma{\pi_{M^{*}}}:M^{*}\times A\rightarrow L$ is also a relative central
extension. The following proposition explains more of this.

######  Proposition 2.5.

Let $(\gamma,\beta^{\prime})$ be an isoclinism from $\sigma_{1}$ to
$\sigma_{2}$. Then there exist isoclinic monomorphisms from $\hat{\sigma}$
into $\sigma_{1}\pi_{M_{1}^{*}}$ and $\sigma_{2}\pi_{M_{2}^{*}}$.

###### Proof.

Let’s consider a map $\overline{\beta_{i}}:\overline{M^{*}}\rightarrow
M_{i}^{*}\oplus A$ given by

$\overline{\beta_{i}}(x)=(\beta_{i}(x),x+[\overline{M^{*}},L_{1}]),$

for $i=1,2$ and $\beta_{i}$’s are as defined in Proposition 2.4. Clearly,
$\overline{\beta_{i}}$ is a Hom-Lie homomorphism. Also we have,
$[\overline{M^{*}},L_{1}]=\\{(m_{1},\beta^{\prime}(m_{1}))|m_{1}\in[M_{1}^{*},L_{1}]\\}.$
Now if $x=(m_{1},m_{2})\in\textnormal{Ker}\overline{\beta_{i}}$, then
$m_{i}=0$ and $m_{2}=\beta^{\prime}(m_{1})$, which implies that
$\textnormal{Ker}\overline{\beta_{i}}=0$. Hence $\overline{\beta_{i}}$ is a
Hom-Lie monomorphism and it can be easily seen that the morphisms
$(\overline{\beta_{1}},1_{L_{1}}):\hat{\sigma}\rightarrow\sigma_{1}{\pi_{M_{1}^{*}}}$
and
$(\overline{\beta_{2}},\gamma):\hat{\sigma}\rightarrow\sigma_{2}{\pi_{M_{2}^{*}}}$
are isoclinic monomorphisms. ∎

######  Proposition 2.6.

The isomorphism $\gamma:(L_{1},\alpha_{1})\rightarrow(L_{2},\alpha_{2})$
induces an isoclinism from $\sigma_{1}$ to $\sigma_{2}$ if and only if for
some ideal $T$ of $M_{1}^{*}\oplus A$ with
$T\subseteq\textnormal{Ker}\sigma_{1}{\pi_{M_{1}^{*}}}$, there exist isoclinic
monomorphisms from $\sigma_{1}$ and $\sigma_{2}$ into the relative central
extension $\overline{\sigma_{1}{\pi_{M_{1}^{*}}}}:(M_{1}^{*}\times
A)/T\rightarrow L_{1}$.

###### Proof.

The necessary condition is obvious. To prove the converse part, let’s suppose
that $\gamma$ induces an isoclinism $(\gamma,\beta^{\prime})$ from
$\sigma_{1}$ to $\sigma_{2}$. Let
$T=\\{(x,(x,0)+[\overline{M^{*}},L_{1}])|x\in\textnormal{Ker}\sigma_{1}\\}$.
Then clearly $T$ is an ideal of $M_{1}^{*}\times A$ with
$T\subseteq\textnormal{Ker}\sigma_{1}{\pi_{M_{1}^{*}}}$.
Now, we define two Hom-Lie homomorphisms
$\delta_{1}:M_{1}^{*}\rightarrow(M_{1}^{*}\times A)/T$ and
$\delta_{2}:M_{2}^{*}\rightarrow(M_{1}^{*}\times A)/T$ as follows:

$\delta_{1}(m_{1})=(m_{1},[\overline{M^{*}},L_{1}])+T$

and

$\delta_{2}(m_{2})=(m_{1},(m_{1},m_{2})+[\overline{M^{*}},L_{1}])+T,$

where $m_{i}\in M_{i}^{*}~{}(i=1,2)$ and
$\gamma\sigma_{1}(\alpha_{1}^{*}(m_{1}))=\sigma_{2}(\alpha_{2}^{*}(m_{2}))$.
We claim that $\delta_{1}$ and $\delta_{2}$ are monomorphisms. If
$m_{1}\in\textnormal{Ker}\delta_{1}$, then
$m_{1}\in\textnormal{Ker}\sigma_{1}$ and $\beta^{\prime}(m_{1})=0$ and thereby
$m_{1}=0$. Similarly, $\textnormal{Ker}\delta_{2}=0$. Now it can be easily
checked that the morphisms
$(\delta_{1},1_{L_{1}}):\sigma_{1}\rightarrow\overline{\sigma_{1}{\pi_{M_{1}^{*}}}}$
and
$(\delta_{2},\gamma^{-1}):\sigma_{2}\rightarrow\overline{\sigma_{1}{\pi_{M_{1}^{*}}}}$
are isoclinic monomorphisms, as required. ∎

##  3\. Factor set on an isoclinic relative extension

######  Definition 3.1.

Let $(M,L)$ be a pair of Hom-Lie algebras. A relative central extension
$\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ is said to be finite
dimensional , if $(M^{*},\alpha_{*})$ is a finite dimensional Hom-Lie algebra.

######  Definition 3.2.

Let $\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ be a relative central
extension of a pair of Hom-Lie algebras $(M,L)$. A factor set on $\sigma$ is
defined as a skew-bilinear map

$f:(L,\alpha_{L})\times(L,\alpha_{L})\rightarrow\textnormal{Ker}(\sigma)$

such that the following conditions are satisfied:

  1. (1)

$f(l_{1},l_{1})=0$

  2. (2)

$f([{l_{1}},{l_{2}}],\alpha_{L}({l_{3}}))+f([{l_{2}},{l_{3}}],\alpha_{L}({l_{1}}))+f([{l_{3}},{l_{1}}],\alpha_{L}({l_{2}}))=0$

for all ${l_{1}},{l_{2}},{l_{3}}\in L$.

######  Lemma 3.3.

Let $f$ be a factor set on the relative central extension $\sigma$ of the pair
$(M,L)$. Then the product $\textnormal{Ker}\sigma\times M$ is a Hom-Lie
algebra with the bracket defined as,

$[(x_{1},m_{1}),(x_{2},m_{2})]=(f(m_{1},m_{2}),[m_{1},m_{2}]),$

along with a linear map ${\alpha_{f}}:\textnormal{Ker}\sigma\times
M\rightarrow\textnormal{Ker}\sigma\times M$ such that
${\alpha_{f}}(x,m)=(\alpha_{*}(x),\alpha_{L}(m)),$ where $\alpha_{*}$ and
$\alpha_{L}$ are as defined in Definition 3.2.

###### Proof.

First we will check the anti-symmetry

$\displaystyle[(x_{1},m_{1}),(x_{2},m_{2})]$
$\displaystyle=(f(m_{1},m_{2}),[m_{1},m_{2}])$
$\displaystyle=(-f(m_{2},m_{1}),-[m_{2},m_{1}])$
$\displaystyle=-[(x_{2},m_{2}),(x_{1},m_{1})].$

Now to check the Hom-Jacobi identity, we have

$\displaystyle[\alpha_{f}(x_{1},m_{1}),[(x_{2},m_{2}),(x_{3},m_{3})]]$
$\displaystyle=[(\alpha_{*}(x_{1}),\alpha_{L}(m_{1})),(f(m_{2},m_{3}),[m_{2},m_{3}])]$
$\displaystyle=(f(\alpha_{L}(m_{1}),[m_{2},m_{3}]),[\alpha_{L}(m_{1}),[m_{2},m_{3}]]),$

by performing similar calculations we can find that

$\displaystyle[\alpha_{f}(x_{1},m_{1}),[(x_{2},m_{2}),(x_{3},m_{3})]]$
$\displaystyle+[\alpha_{f}(x_{2},m_{2}),[(x_{3},m_{3}),(x_{1},m_{1})]]$
$\displaystyle+[\alpha_{f}(x_{3},m_{3}),[(x_{1},m_{1}),(x_{2},m_{2})]]=0.$

Therefore $\textnormal{Ker}\sigma\times M$ is a Hom-Lie algebra. ∎

We denote the above Hom-Lie algebra by $(\textnormal{Ker}\sigma\times M)_{f}$.
Then the projective map,

$\displaystyle\sigma_{f}:(\textnormal{Ker}\sigma\times M)_{f}$
$\displaystyle~{}\rightarrow~{}L$ $\displaystyle(x,m)$
$\displaystyle~{}\mapsto~{}m$

together with an action of $L$ on $(\textnormal{Ker}\sigma\times M)_{f}$
defined by ${}^{l}(x,m)=(f(m,l),[m,l])$is a relative central extension of the
pair $(M,L)$.

######  Definition 3.4.

Let $\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ be a relative central
extension of a pair of Hom-Lie algebras $(M,L)$. If
$\textnormal{Ker}\sigma\subseteq[M^{*},L]$, then $\sigma$ is called as a stem
relative central extension.

######  Lemma 3.5.

Let $\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$ and
$\sigma_{2}:(M_{2}^{*},\alpha_{2}^{*})\rightarrow(L_{2},\alpha_{2})$ be two
relative central extensions of the pairs $(M_{1},L_{1})$ and $(M_{2},L_{2})$
of finite dimensional Hom-Lie algebras, respectively. Then,

  1. (1)

$\sigma_{1}$ is a stem relative central extension if and only if
$\textnormal{Ker}\sigma_{1}$ contains no non-zero ideal $N$ satisfying
$N\cap[M_{1}^{*},L_{1}]=0$.

  2. (2)

If $\sigma_{1}$ and $\sigma_{2}$ are isoclinic stem relative central
extensions, then $\textnormal{Ker}\sigma_{1}\cong\textnormal{Ker}\sigma_{2}$.
In particular, if the Hom-Lie algebras $M_{1}^{*}$ and $M_{2}^{*}$ are finite
dimensional, then $\textnormal{dim}M_{1}^{*}=\textnormal{dim}M_{2}^{*}$.

###### Proof.

  1. (1)

Let’s suppose that $\sigma_{1}$ is stem and $N$ is an ideal of $M_{1}^{*}$
with $N\subseteq\textnormal{Ker}\sigma_{1}$ and $N\cap[M_{1}^{*},L_{1}]=0$.
But we know that $\textnormal{Ker}\sigma_{1}\subseteq Z(M_{1}^{*},L_{1})$.
Also, since $\sigma_{1}$ is stem, so
$\textnormal{Ker}\sigma_{1}\subseteq[M_{1}^{*},L_{1}]$. Therefore, we have

$N\subseteq N\cap\textnormal{Ker}\sigma_{1}\subseteq N\cap
Z(M_{1}^{*},L_{1})\cap[M_{1}^{*},L_{1}]=N\cap[M_{1}^{*},L_{1}]=0,$

and the result follows.

Conversely, let $\textnormal{Ker}\sigma_{1}\subsetneq
Z(M_{1}^{*},L_{1})\cap[M_{1}^{*},L_{1}].$ Then there exists a non-zero element
$x\in\textnormal{Ker}\sigma_{1}\setminus(Z(M_{1}^{*},L_{1})\cap[M_{1}^{*},L_{1}])$.
Now $\langle x\rangle$ is clearly a central ideal of $M_{1}^{*}$ and
consequently, by hypothesis, we have $\langle
x\rangle\cap[M_{1}^{*},L_{1}]\neq 0$. So, for some non-zero element $a\in
F,~{}ax\in[M_{1}^{*},L_{1}]$, which is a contradiction. Therefore,
$\textnormal{Ker}\sigma_{1}\subseteq Z(M_{1}^{*},L_{1})\cap[M_{1}^{*},L_{1}]$
and $\sigma_{1}$ is a stem relative central extension.

  2. (2)

Let $\sigma_{1}$ and $\sigma_{2}$ be isoclinic stem relative central
extensions, then there exists an isomorphism
$\gamma:(L_{1},\alpha_{1})\rightarrow(L_{2},\alpha_{2})$ with
$\gamma(M_{1})=M_{2}$. Therefore,
$\textnormal{dim}M_{1}=\textnormal{dim}M_{2}$. Also, by Lemma 2.3 (2),
$\textnormal{Ker}\sigma_{1}\cong\textnormal{Ker}\sigma_{2}$. If $M_{1}^{*}$
and $M_{2}^{*}$ are finite dimensional, then by the isomorphism theorems, we
have $\textnormal{dim}M_{1}^{*}=\textnormal{dim}M_{2}^{*}$.

∎

The next result shows that every isoclinic family of the relative central
extensions of pairs of Hom-Lie algebras contains a stem relative central
extension.

######  Corollary 3.6.

Any relative central extension is isoclinic to a stem relative central
extension.

###### Proof.

Let
$\mathcal{A}=\\{N|N\textnormal{~{}is~{}an~{}ideal~{}of~{}Ker}\sigma~{}\textnormal{and}~{}N\cap[M^{*},L]=0\\}$,
where $\sigma:M^{*}\rightarrow L$ is a relative central extension of $(M,L)$.
Then the set $\mathcal{A}$ is non-empty, as it contains the zero ideal. Now if
we define a partial ordering on $\mathcal{A}$ by inclusion then by Zorn’s
lemma, we can find a maximal ideal $N$ in $\mathcal{A}$. Since,
$N\subset\textnormal{Ker}\sigma$, by example 2,
$\overline{\sigma}:M^{*}/n\rightarrow L$ is a relative central extension and
$\sigma\sim\overline{\sigma}$.

Now, let $K/N$ is an ideal of $M^{*}/N$ such that
$K/N\subseteq\textnormal{Ker}\overline{\sigma}$ and $K/N\cap[M^{*}/N,L]=0$.
Then $(K\cap[M^{*},L]+N)/N=0$ and hence $K\cap[M^{*},L]\subseteq N$. Since
$N\cap[M^{*},L]=0$, we conclude that $K\in\mathcal{A}$. Also, we have
$N\subseteq K$. So by the maximality of $N$, it follows that $K=N$. Therefore,
$K/N$ is zero ideal and hence $\overline{\sigma}$ is a stem relative central
extension by Lemma 3.5 (1). ∎

######  Lemma 3.7.

Let $\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$ and
$\sigma_{2}:(M_{2}^{*},\alpha_{2}^{*})\rightarrow(L_{2},\alpha_{2})$ be
relative central extensions of the pairs of Hom-Lie algebras $(M_{1},L_{1})$
and $(M_{2},L_{2})$, respectively. Then

  1. (1)

There exists a factor set
$f:(L_{1},\alpha_{1})\times(L_{1},\alpha_{1})\rightarrow\textnormal{Ker}(\sigma_{1})$
such that the relative central extensions $\sigma_{1}$ and $\sigma_{1f}$ are
isomorphic.

  2. (2)

If $\sigma_{1}$ and $\sigma_{2}$ are stem and
$(\gamma,\beta^{\prime}):\sigma_{1}\rightarrow\sigma_{2}$ is an isoclinism,
then there exists a factor set
$g:(L_{1},\alpha_{1})\times(L_{1},\alpha_{1})\rightarrow\textnormal{Ker}(\sigma_{1})$
such that the relative central extensions $\sigma_{1}$ and $\sigma_{1g}$ are
isomorphic.

###### Proof.

  1. (1)

Let $T_{1}$ be the vector space complement of $\textnormal{Ker}(\sigma_{1})$
in $M_{1}^{*}$. Then for any $m_{1}\in M_{1}$, there is a unique element
$t_{m_{1}}\in T_{1}$ with $\sigma_{1}(t_{m_{1}})=m_{1}$. Therefore we have

$\displaystyle\sigma_{1}(t_{[m_{1},m_{2}]})=[m_{1},m_{2}]=-[m_{2},m_{1}]$
$\displaystyle=-\sigma_{1}(t_{[m_{2},m_{1}]})$
$\displaystyle=\sigma_{1}(-t_{[m_{2},m_{1}]}),$

which implies that $t_{[m_{1},m_{2}]}=-t_{[m_{2},m_{1}]}$ for all
$m_{1},m_{2}\in M_{1}.$

Again since $\sigma_{1}$ is a homomorphism, so we have

$\displaystyle\sigma_{1}([t_{[m_{1},m_{2}]},t_{\alpha_{1}(m_{3})}])=[\sigma_{1}({t_{[m_{1},m_{2}]}}),\sigma_{1}({t_{\alpha_{1}(m_{3})}})]=[[m_{1},m_{2}],\alpha_{1}(m_{3})]=\sigma_{1}(t_{[[m_{1},m_{2}],\alpha_{1}(m_{3})]})$
$\implies[t_{[m_{1},m_{2}]},t_{\alpha_{1}(m_{3})}]=t_{[[m_{1},m_{2}],\alpha_{1}(m_{3})]}.$

Let’s define a map
$f:(L_{1},\alpha_{1})\times(L_{1},\alpha_{1})\rightarrow\textnormal{Ker}(\sigma_{1})$
given by $f(m_{1},m_{2})=[t_{m_{1}},t_{m_{2}}]-t_{[m_{1},m_{2}]}$ for all
$m_{1},m_{2}\in M_{1}$ and $f(x,y)=0$, if $x,y\in L_{1}\setminus M_{1}$. We
will show that $f$ is a factor set.

From the definition it is clear that $f$ satisfies all the conditions of a
factor set on $L_{1}\setminus M_{1}$. So it only remains to show that $f$ is
also a factor set on $M_{1}$. It is obvious that
$f(m_{1},m_{2})\in\textnormal{Ker}(\sigma_{1})$. Now to check the skew-
symmetric property of $f$, we take for all $m_{1},m_{2}\in M_{1}$

$\displaystyle f(m_{1},m_{2})$
$\displaystyle=[t_{m_{1}},t_{m_{2}}]-t_{[m_{1},m_{2}]}$
$\displaystyle=-[t_{m_{2}},t_{m_{1}}]+t_{[m_{2},m_{1}]}$
$\displaystyle=-\\{[t_{m_{2}},t_{m_{1}}]-t_{[m_{2},m_{1}]}\\}$
$\displaystyle=-f(m_{2},m_{1}).$

To check the Jacobi identity, we take for all $m_{1},m_{2},m_{3}\in M_{1}$

$\displaystyle
f([m_{1},m_{2}],\alpha_{1}(m_{3}))=[t_{[m_{1}m_{2}]},t_{\alpha_{1}(m_{3})}]-t_{[[m_{1},m_{2}],\alpha_{1}(m_{3})]}=0,$

and similarly,
$f([{m_{2}},{m_{3}}],\alpha_{1}({m_{1}}))=0=f([{m_{3}},{m_{1}}],\alpha_{1}({m_{2}}))$.
i.e., $f$ also satisfies the Jacobi identity and hence is a factor set. Now
consider the mapping $\beta:(\textnormal{Ker}\sigma_{1}\times
M_{1})_{f}\rightarrow M_{1}^{*}$ given by
$\beta(x_{1},m_{1})=x_{1}+t_{m_{1}}$. Then we have,
$(x_{1},m_{1})=(x_{2},m_{2})\Leftrightarrow
x_{1}+t_{m_{1}}=x_{2}+t_{m_{2}}\Leftrightarrow\beta{(x_{1},m_{1})}=\beta{(x_{2},m_{2})}$
i.e., $\beta$ is well defined and obviously bijective. Also,

$\displaystyle\beta([(x_{1},m_{1}),(x_{2},m_{2})])=\beta(f(m_{1},m_{2}),[m_{1},m_{2}])$
$\displaystyle=f(m_{1},m_{2})+t_{[m_{1},m_{2}]}$
$\displaystyle=[t_{m_{1}},t_{m_{2}}]$
$\displaystyle=[x_{1}+t_{m_{1}},x_{2}+t_{m_{2}}]$
$\displaystyle=[\beta{(x_{1},m_{1})},\beta{(x_{2},m_{2})}].$

i.e., $\beta$ preserves the bracket operation. Again, we know that the
relative central extension
$\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$ being a
Hom-Lie homomorphism satisfies
$\sigma_{1}\alpha_{1}^{*}=\alpha_{1}\sigma_{1}$. So, we have
$\sigma_{1}\alpha_{1}^{*}(t_{m})=\alpha_{1}\sigma_{1}(t_{m})=\alpha_{1}(m)$,
which implies that $\alpha_{1}^{*}(t_{m})=t_{\alpha_{1}(m)}$. Now for some
$(x,m)\in(\textnormal{Ker}\sigma_{1}\times M_{1})_{f}$, we get

$\displaystyle\beta{\alpha_{f}}(x,m)=\beta(\alpha_{1}^{*}(x),\alpha_{1}(m))$
$\displaystyle=\alpha_{1}^{*}(x)+t_{\alpha_{1}(m)}$
$\displaystyle=\alpha_{1}^{*}(x)+\alpha_{1}^{*}(t_{m})$
$\displaystyle=\alpha_{1}^{*}(x+t_{m})$
$\displaystyle=\alpha_{1}^{*}\beta(x,m),$

i.e., $\beta{\alpha_{f}}=\alpha_{1}^{*}\beta$, which confirms that $\beta$ is
a Hom-Lie isomorphism. Therefore,
$(1_{L_{1}},\beta):\sigma_{1f}\rightarrow\sigma_{1}$ is an isomorphism between
the central extensions $\sigma_{1f}$ and $\sigma_{1}$.

  2. (2)

By Lemma 2.3(2), we have
$\beta^{\prime}(\textnormal{Ker}\sigma_{1})=\textnormal{Ker}\sigma_{2}$. Since
$\sigma_{2}$ is a stem relative extension, so from the above proof, we can
conclude that there exists a factor set
$h:(L_{2},\alpha_{2})\times(L_{2},\alpha_{2})\rightarrow\textnormal{Ker}(\sigma_{2})$
such that the relative central extensions $\sigma_{2}$ and $\sigma_{2h}$ are
isomorphic. Define a map
$g:(L_{1},\alpha_{1})\times(L_{1},\alpha_{1})\rightarrow\textnormal{Ker}(\sigma_{1})$
given by $g(m_{1},m_{2})=\beta^{\prime-1}(h(\gamma(m_{1}),\gamma(m_{2})))$ for
all $m_{1},m_{2}\in M_{1}$ and otherwise $g(x,y)=0.$ It can be easily shown
that $g$ is a factor set.
Now consider a map $\theta:(\textnormal{Ker}\sigma_{1}\times
M_{1})_{g}\rightarrow(\textnormal{Ker}\sigma_{2}\times M_{2})_{h}$ defined by
$\theta(x_{1},m_{1})=(\beta^{\prime}(x_{1}),\gamma(m_{1}))$. It is obvious
that $\theta$ is well defined and bijective. Also $\theta$ is a Hom-Lie
homomorphism as,

$\displaystyle\theta[(x_{1},m_{1}),(x_{2},m_{2})]_{g}$
$\displaystyle=\theta(g(m_{1},m_{2}),[m_{1},m_{2}])$
$\displaystyle=\theta(\beta^{\prime-1}(h(\gamma(m_{1}),\gamma(m_{2}))),[m_{1},m_{2}])$
$\displaystyle=(\beta^{\prime}(\beta^{\prime-1}(h(\gamma(m_{1}),\gamma(m_{2})))),\gamma([m_{1},m_{2}]))$
$\displaystyle=(h(\gamma(m_{1}),\gamma(m_{2})),[\gamma(m_{1}),\gamma(m_{2})])$
$\displaystyle=[(\beta^{\prime}(x_{1}),\gamma(m_{1})),(\beta^{\prime}(x_{2}),\gamma(m_{2}))]_{h}$
$\displaystyle=[\theta(x_{1},m_{1}),\theta(x_{2},m_{2})]_{h},$

and if $\alpha_{g}$ and $\alpha_{h}$ are the linear maps in the Hom-Lie
algebras $(\textnormal{Ker}\sigma_{1}\times M_{1})_{g}$ and
$(\textnormal{Ker}\sigma_{2}\times M_{2})_{h}$ respectively, then

$\displaystyle\theta\alpha_{g}(x_{1},m_{1})=\theta(\alpha_{1}^{*}(x_{1}),\alpha_{1}(m_{1}))$
$\displaystyle=(\beta^{\prime}\alpha_{1}^{*}(x_{1}),\gamma\alpha_{1}(m_{1}))$
$\displaystyle=(\alpha_{2}^{*}\beta^{\prime}(x_{1}),\alpha_{2}\gamma(m_{1}))$
$\displaystyle=\alpha_{h}(\beta^{\prime}(x_{1}),\gamma(m_{1}))$
$\displaystyle=\alpha_{h}\theta(x_{1},m_{1}).$

i.e., $\theta$ is a Hom-Lie isomorphism and hence $(\gamma,\theta)$ is an
isomorphism between $\sigma_{1g}$ and $\sigma_{2h}$. Therefore,
$\sigma_{1g}\cong\sigma_{2h}\cong\sigma_{2}$.

∎

######  Proposition 3.8.

Let $\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ be a finite
dimensional relative central extension and
$f,g:(L,\alpha_{L})\times(L,\alpha_{L})\rightarrow\textnormal{Ker}\sigma$ be
factor sets on $\sigma$ such that
$\textnormal{dim}(\textnormal{Ker}\sigma\times
M)_{f}=\textnormal{dim}(\textnormal{Ker}\sigma\times M)_{g}$, $\sigma_{f}$ is
a stem relative extension and the extensions $\sigma_{f}$ and $\sigma_{g}$ are
isoclinic. Then $\sigma_{f}$ and $\sigma_{g}$ are isomorphic.

###### Proof.

Let $(\gamma,\beta^{\prime})$ be an isoclinism between $\sigma_{f}$ and
$\sigma_{g}$. Then from Lemma 2.3(2), it follows that
$\beta^{\prime}(\textnormal{Ker}\sigma_{f})=\textnormal{Ker}\sigma_{g}$. Now
for all $m\in M,l\in L$, we have

$\beta^{\prime}(^{l}(0,m))=~{}^{\gamma(l)}(0,\gamma(m))=(g(\gamma(m),\gamma(l)),\gamma([m,l])),$

and

$\beta^{\prime}(^{l}(0,m))=\beta^{\prime}(f(m,l),[m,l])=\beta^{\prime}(f(m,l),0)+\beta^{\prime}(0,[m,l]).$

Let $d([m,l])$ be the first component of $\beta^{\prime}(^{l}(0,m))$. Then we
can define a linear map $d:[M,L]\rightarrow\textnormal{Ker}\sigma$ satisfying
$\rho\beta^{\prime}(f(m,l),0)+d([m,l])=g(\gamma(m),\gamma(l))$, where
$\rho:\textnormal{Ker}\sigma_{g}\rightarrow\textnormal{Ker}\sigma$ is the
projective map. If we take $d$ to be zero on the vector space complement of
$[M,L]$ in $M$ then $d$ can be defined on whole $M$. Now it is easy to see
that the map $\lambda:(\textnormal{Ker}\sigma\times
N)_{f}\rightarrow(\textnormal{Ker}\sigma\times N)_{g}$ given by
$\lambda(x,l)=\beta^{\prime}(x,0)+(d(l),\gamma(l))$, is an isomorphism. As
$\lambda|_{\textnormal{Ker}\sigma_{f}}=\beta^{\prime}|_{\textnormal{Ker}\sigma_{f}}$
and $(\lambda,\gamma):\sigma_{f}\rightarrow\sigma_{g}$ is a morphism, we
conclude that $\sigma_{f}$ and $\sigma_{g}$ are isomorphic. ∎

######  Corollary 3.9.

Let $\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$ and
$\sigma_{2}:(M_{2}^{*},\alpha_{2}^{*})\rightarrow(L_{2},\alpha_{2})$ be finite
dimensional stem relative central extensions of the pairs $(M_{1},L_{1})$ and
$(M_{2},L_{2})$ respectively. Then $\sigma_{1}$ and $\sigma_{2}$ are isoclinic
if and only if they are isomorphic.

###### Proof.

Let $\sigma_{1}$ and $\sigma_{2}$ are isoclinic. Then by lemma 3.7, there
exist factor sets
$f,g:(L_{1},\alpha_{1})\times(L_{1},\alpha_{1})\rightarrow\textnormal{Ker}\sigma_{1}$
such that $\sigma_{1}\cong\sigma_{1f}$ and $\sigma_{2}\cong\sigma_{1g}$.
Therefore, $\sigma_{1f}$ and $\sigma_{1g}$ are isoclinic and since
$(\textnormal{Ker}\sigma_{1}\times M_{1})_{f}$ and
$(\textnormal{Ker}\sigma_{1}\times M_{1})_{g}$ are finite dimensional, so we
can have $\textnormal{dim}(\textnormal{Ker}\sigma_{1}\times
M_{1})_{f}=\textnormal{dim}(\textnormal{Ker}\sigma_{1}\times M_{1})_{g}$.
Hence by Proposition 3.8, $\sigma_{1f}\cong\sigma_{1g}.$ Therefore,
$\sigma_{1}\cong\sigma_{1f}\cong\sigma_{1g}\cong\sigma_{2}$ and the result
follows. ∎

Example 3. Let $\sigma:(M^{*},\alpha_{*})\rightarrow(L,\alpha_{L})$ be a
relative central extension of $(M,L)$. Let $T$ be a Hom-Lie subalgebra of
$M^{*}$ such that $M^{*}=T+\textnormal{Ker}\sigma$ and also $\sigma(T)=M$.
Then the map $\sigma_{T}:T\rightarrow L$ defined by $\sigma_{T}(t)=\sigma(t)$
for all $t\in T$ is a relative central extension of $(M,L)$ and
$(1_{L},i):\sigma_{T}\rightarrow\sigma$ is an isoclinic monomorphism, where
$i:T\rightarrow M^{*}$ is an inclusion map.

The next theorem helps to determine the structure of all the relative central
extensions in an isoclinism family of a given relative central extension.

######  Theorem 3.10.

Let $\mathcal{C}$ be an isoclinism family of finite dimensional relative
central extensions. Then $\mathcal{C}$ contains a stem relative central
extension $\sigma_{1}:(M_{1}^{*},\alpha_{1}^{*})\rightarrow(L_{1},\alpha_{1})$
and relative central extension in $\mathcal{C}$ is the form of
$\sigma_{1}\pi_{M_{1}^{*}}:M_{1}^{*}\times A\rightarrow L_{1}$, in which $A$
is a finite dimensional abelian Hom-Lie algebra.

###### Proof.

From Corollary 3.6, it is clear that $\mathcal{C}$ contains a stem relative
central extension $\sigma_{1}$. We know that for any finite dimensional
abelian Hom-Lie algebra $A$, the relative central extensions
$\sigma_{1}:M_{1}^{*}\rightarrow L_{1}$ and
$\sigma_{1}\pi_{M_{1}^{*}}:M_{1}^{*}\times A\rightarrow L_{1}$ are isoclinic
and consequently $\sigma_{1}\pi_{M_{1}^{*}}\in\mathcal{C}$.

Now, let $\sigma:M^{*}\rightarrow L$ be an arbitrary relative central
extension in $\mathcal{C}$. Then there exists an ideal $N$ of
$\textnormal{Ker}\sigma$ such that
$\textnormal{Ker}\sigma=N\times(\textnormal{Ker}\sigma\cap(M^{*})^{2})$, where
$(M^{*})^{2}$ is the derived Hom-Lie subalgebra of $M^{*}$ and by the proof of
Corollary 3.6, and example 2, the map $\bar{\sigma}:M^{*}/N\rightarrow L$ is a
stem relative central extension in $\mathcal{C}$. If we choose the subspace
$T$ of $M^{*}$ such that $(M^{*})^{2}\subseteq T$ and $T\times N=M^{*}$, then
$T$ is a Hom-Lie subalgebra of $M^{*}$ with $\sigma(T)=M$ and, $\sigma_{1}$
and $\bar{\sigma}$ are isoclinic. By Corollary 3.9, we get
$\sigma_{1}\cong\sigma_{T}$. Therefore,
$\sigma\cong\sigma_{T}\pi_{T}\cong\sigma_{1}\pi{M_{1}^{*}}$, where
$\sigma_{T}\pi_{T}:T\times N\rightarrow L$ and
$\sigma_{1}\pi{M_{1}^{*}}:M_{1}^{*}\times N\rightarrow L$ are relative central
extensions of the pair $(M,L)$ and $N$ is a finite dimensional Hom-Lie
algebra. ∎

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