# Deformation cohomology of morphisms of Lie-Yamaguti algebras

Bibhash Mondal1, Ripan Saha2111 Corresponding author (Ripan Saha), Email:
<EMAIL_ADDRESS>
1\. Department of Mathematics, Behala College
Behala, 700060, Kolkata, India
Email<EMAIL_ADDRESS>
2\. Department of Mathematics, Raiganj University
Raiganj 733134, West Bengal, India

ABSTRACT

We study cohomology of morphisms of Lie-Yamaguti algebras. As an application,
we establish that this cohomology ‘controls’ the formal deformations.
Additionally, we demonstrate its connection to the abelian extension of
morphisms of Lie-Yamaguti algebras.

Key words: Lie-Yamaguti algebra, morphism, cohomohology, deformation,
rigidity.

2020 MSC: 17A30, 17A40, 17B56, 17D99.

##  1 Introduction

Lie-Yamaguti algebras, a pivotal concept in non-associative algebra theory,
find their roots in the pioneering work of Jacobson [10], who formally
introduced the notion of a Lie triple system. This algebraic structure emerged
from the context of quantum mechanics [2], demonstrating its profound
relevance in this domain. Nomizu [16] further extended this theory,
establishing a compelling connection between affine connections with parallel
torsion and curvature, and invariant connections on reductive homogeneous
spaces. Notably, each such space possesses a canonical connection that aligns
geodesic parallel translation with the natural group action.

Building on these foundational ideas, K. Yamaguti [17] introduced the concept
of a general Lie triple system: a vector space equipped with both bilinear and
trilinear operations, satisfying intriguing relations. This innovation was
crucial in characterizing the torsion and curvature tensors associated with
Nomizu’s canonical connection. The subsequent work of M. Kikkawa [11] added a
crucial insight, recognizing the intimate relationship between this problem,
the canonical connection, and the general Lie triple system defined on the
tangent space.

In a transformative contribution, Kikkawa renamed the notion of a general Lie
system as a Lie triple algebra, establishing a nomenclature that resonates
within the mathematical community. Kinyon and Weinstein [12] expanded this
framework, identifying that Lie triple algebras, which they later termed Lie-
Yamaguti algebras, could be constructed from Leibniz algebras. This insight
demonstrated the broad applicability of Lie-Yamaguti algebras, which arose
initially from Cartan’s investigations into Riemannian geometry.

In [20], the author studied infinitesimal deformations of a Lie triple system
using the cohomology groups as introduced by K. Yamaguti in [19]. In [21], the
authors studied $(2,3)$-cohomology groups and related deformation theory of
Lie-Yamaguti algebras. In [13], the authors discussed the one-paramater formal
deformation theory of Lie-Yamaguti algebras, and recently, in [9] the author
studied correct obstruction theory to integrate a finite order deformation to
a full-blown deformation.

In 1983, Gerstenhaber and Schack [8] initiated the study of deformation of
morphism of associative algebras. In [15], the author studied deformations of
Leibniz algebra morphisms. Recently, Das [1] studied cohomology and
deformation theory of Lie algebra morphisms. As Lie-Yamaguti algebra is a
generalization (see Example 2.2 and 2.3) of both Lie and Leibniz algebra [14],
it is very natural to study morphism of Lie-Yamaguti algebra from
cohomological point of view. The aim of this paper is to study an algebraic
deformation theory of Lie-Yamaguti algebra morphisms. We study the deformation
theory of morphism of Lie-Yamaguti algebra following the idea of
Gerstenhaber’s [3, 4, 5, 6, 7] algebraic deformation theory. We define a
suitable deformation cohomology for such formal deformation. We also show that
isomorphism classes of abelian extension of morphism of Lie-Yamaguti algebra
has one to one correspondence with the ‘simple’ $(2,3)$\- cohomology group.

The paper is organised as follows: In Section 2, we recall the definition and
cohomology of Lie-Yamaguti algebras. In Section 3, we discuss representation
of morphisms of Lie-Yamaguti algebras. In Section 4, we introduce cohomology
of morphism of Lie-Yamaguti algebra which involves both Lie-Yamaguti part and
the morphism part. In Section 5, we discuss one-parameter formal deformation
theory of morphism of Lie-Yamaguti algebra and show that the deformation is
‘controlled’ by our cohomology defined in Section 4. In Section 6, we study
abelian extension of morphism of Lie-Yamaguti algebra and its relationship
with the cohomology. Finally, in Section 7, we conclude our paper by pointing
out that the work of this paper is a special case of a more general problem of
diagram of Lie-Yamaguti algebras.

##  2 Lie-Yamaguti algebra and its cohomology

In this section, we recall the cohomology of Lie-Yamaguti algebra as in [17,
18, 19], and [9]. Let $(L,[~{},~{}],\\{~{},~{},~{}\\})$ be a Lie-Yamaguti
algebra with representation $V$.

######  Definition 2.1.

A Lie-Yamaguti algebra is a triple $(L,[~{},~{}],\\{~{},~{},~{}\\})$, where
$L$ is a $k$-vector space, $[~{},~{}]$ is a binary operation and
$\\{~{},~{},~{}\\}$ is a ternary operation on $L$ such that

$\displaystyle(1)~{}~{}[x,y]=-[y,x],$
$\displaystyle(2)~{}~{}\\{x,y,z\\}=-\\{y,x,z\\},$
$\displaystyle(3)~{}~{}\circlearrowleft_{(x,y,z)}([[x,y],z]+\\{x,y,z\\})=0,$
$\displaystyle(4)~{}~{}\circlearrowleft_{(x,y,z)}(\\{[x,y],z,u\\})=0,$
$\displaystyle(5)~{}~{}\\{x,y,[u,v]\\}=[\\{x,y,u\\},v]+([u,\\{x,y,v\\}],$
$\displaystyle(6)~{}~{}\\{x,y,\\{u,v,w\\}\\}=\\{\\{x,y,u\\},v,w\\}+\\{u,\\{x,y,v\\},w\\}+\\{u,v,\\{x,y,w\\}\\},$

for all $x,y,z,u,v,w\in L$ and where $\circlearrowleft_{(x,y,z)}$ denotes the
sum over cyclic permutation of $x,y,z$.

######  Example 2.2.

Let $(\mathfrak{g},[~{},~{}])$ be a Lie algebra over $k$. Then, $\mathfrak{g}$
has a Lie-Yamaguti algebra structure induced by the given Lie bracket, the
trilinear operation being:

$\\{a,b,c\\}=[[a,b],c]$

for all $a,b,c\in\mathfrak{g}$.

######  Example 2.3.

Let $(\mathfrak{g},\cdot)$ be a Leibniz algebra. Consider a bilinear operation
and a trilinear operation as follows:

$[~{},~{}]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g},\quad[a,b]:=a\cdot
b-b\cdot a,~{}~{}a,b\in\mathfrak{g};$
$\\{~{},~{},~{}\\}:\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g},\quad\\{a,b,c\\}:=-(a\cdot
b)\cdot c,~{}~{}a,b,c\in\mathfrak{g}.$

Then, $(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$ is a Lie-Yamaguti algebra.

######  Definition 2.4.

Let $(L,[~{},~{}],\\{~{},~{},~{}\\})$ be a Lie-Yamaguti algebra and $V$ be a
vector space. A representation of $L$ on $V$ consists of a linear map
$\rho:L\rightarrow$ End($V$) and two bilinear maps $D,\theta:L\times
L\rightarrow$ End($V$) satisfying the following conditions

$\displaystyle(1)~{}D(x,y)-\theta(y,x)+\theta(x,y)+\rho([x,y])-\rho(x)\rho(y)+\rho(y)\rho(x)=0,$
$\displaystyle(2)~{}D([x,y],z)+D([y,z],x)+D([z,x],y)=0,$
$\displaystyle(3)~{}\theta([x,y],z)=\theta(x,z)\rho(y)-\theta(y,z)\rho(x),$
$\displaystyle(4)~{}D(x,y)\rho(z)=\rho(z)D(x,y)+\rho(\\{x,y,z\\}),$
$\displaystyle(5)~{}\theta(x,[y,z])=\rho(y)\theta(x,z)-\rho(z)\theta(x,y),$
$\displaystyle(6)~{}D(x,y)\theta(u,v)=~{}\theta(u,v)D(x,y)+\theta(\\{x,y,u\\},v)+\theta(u,\\{x,y,v\\}),$
$\displaystyle(7)~{}\theta(x,\\{y,z,u\\})=\theta(z,u)\theta(x,y)-\theta(y,u)\theta(x,z)+D(y,z)\theta(x,u),$

for any $x,y,z,u,v\in L$. Here we say the $(\rho,D,\theta)$ is a
representation of $L$ on $V.$

Let us define the cohomology groups of $L$ with coefficients in $V$. Let
$f:L\times L\times\cdot\cdot\cdot\times L$ be an $n$-linear map of $L$ into
$V$ such that the following conditions are satisfied:

$\displaystyle
f(x_{1},\ldots,x_{2i-1},x_{2i},\ldots,x_{n})=0,~{}~{}~{}~{}\mbox{if}~{}~{}~{}x_{2i-1}=x_{2i}.$

The vector space spanned by such linear maps is called an $n$-cochain of $L$,
which is denoted by $C^{n}(L,V)$ for $n\geq 1$.

######  Definition 2.5.

For any $(f,g)\in C^{2n}(L,V)\times C^{2n+1}(L,V)$, $n\geq 1$ the coboundary
operator $\delta:(f,g)\rightarrow(\delta_{I}f,\delta_{II}g)$ is a mapping from
$C^{2n}(L,V)\times C^{2n+1}(L,V)$ into $C^{2n+2}(L,V)\times C^{2n+3}(L,V)$
defined as follows:

$\displaystyle(\delta_{I}f)(x_{1},x_{2},\ldots,x_{2n+2})$ $\displaystyle=$
$\displaystyle\rho(x_{2n+1})g(x_{1},x_{2},\ldots,x_{2n},x_{2n+2})-\rho(x_{2n+2})g(x_{1},x_{2},\ldots,x_{2n},x_{2n+1})$
$\displaystyle-g(x_{1},x_{2},\ldots,x_{2n},[x_{2n+1},x_{2n+2}])$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k+1}D(x_{2k-1},x_{2k})f(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,x_{2n+2})$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+2}(-1)^{n+k}f(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,\\{x_{2k-1},x_{2k},x_{j}\\},\ldots,x_{2n+2}),$
$\displaystyle(\delta_{II}g)(x_{1},x_{2},\ldots,x_{2n+3})$ $\displaystyle=$
$\displaystyle\theta(x_{2n+2},x_{2n+3})g(x_{1},\ldots,x_{2n+1})-\theta(x_{2n+1},x_{2n+3})g(x_{1},\ldots,x_{2n},x_{2n+1})$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{n+k+1}D(x_{2k-1},x_{2k})g(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,x_{2n+3})$
$\displaystyle+\sum_{k=1}^{n+1}\sum_{j=2k+1}^{2n+3}(-1)^{n+k}g(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,\\{x_{2k-1},x_{2k},x_{j}\\},\ldots,x_{2n+3}).$

We define $C^{(2n,2n+1)}(L,V):=C^{2n}(L,V)\times C^{2n+1}(L,V)$. Then
$\\{C^{(\bullet,\bullet)}(L,V),\delta\\}$ is a cochain complex.
Let $C^{1}(L,V)$ denote the space of all linear maps from $L$ into $V$. We
define the subspace $C(L,V)$ by the diagonal elements $(f,f)\in
C^{1}(L,V)\times C^{1}(L,V.)$. The map $\delta:C(L,V)\rightarrow
C^{(2,3)}(L,V)$ is defined by $\delta(f,f)=(\delta_{I}f,\delta_{II}f)$, where

$\displaystyle\delta_{I}f(a,b)=\rho(a)f(b)-\rho(b)f(a)-f([a,b]),$
$\displaystyle\delta_{II}(a,b,c)=\theta(b,c)f(a)-\theta(a,c)f(b)+D(a,b)f(c)-f(\\{a,b,c\\})$

for all $a,b,c\in L.$ From [19], we have for any $f\in C^{1}(L,V)$ ,
$\delta_{I}\delta_{I}f=\delta_{II}\delta_{II}f=0$ and In general for any
$(f,g)\in C^{(2p,2p+1)}(L,V)~{}\mbox{we
have}~{}~{}(\delta\circ\delta)(f,g)=(\delta_{I}\circ\delta(f),\delta_{II}\circ\delta_{II}(g))=0.$

##  3 Morphism of Lie-Yamaguti algebra and its representation

In this section, we discuss momorphism between Lie-Yamaguti algebras and
representation of Lie-Yamaguti algebra as defined in [17, 19].

######  Definition 3.1.

A momorphism between two Lie-Yamaguti algebras
$(L,[~{},~{}],\\{~{},~{},~{}\\})$ and
$(L^{\prime},[~{},~{}]^{\prime},\\{~{},~{},~{}\\}^{\prime})$ is a linear map
$\varphi:L\rightarrow L^{\prime}$ satisfying

$\displaystyle\varphi([x,y])=[\varphi(x),\varphi(y)]^{\prime},~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi(\\{x,y,z\\})=\\{\varphi(x),\varphi(y),\varphi(z)\\}^{\prime}$

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

######  Definition 3.2.

Let $\phi:L_{1}\rightarrow L_{2}$ and
$\phi^{{}^{\prime}}:L_{1}^{{}^{\prime}}\rightarrow L_{2}^{{}^{\prime}}$ be two
morphism of Lie-Yamaguti algebras. A homomorphism between these two morphism
of Lie-Yamaguti algebra is a pair $(\alpha,\beta)$, where
$\alpha:L_{1}\rightarrow L_{1}^{{}^{\prime}}$ and $\beta:L_{2}\rightarrow
L_{2}^{{}^{\prime}}$ are two Lie-Yamaguti algebra homomorphism satisfying the
condition $\phi^{{}^{\prime}}\circ\alpha=\beta\circ\phi$.

######  Definition 3.3.

Let $\phi:L_{1}\rightarrow L_{2}$ be a morphism of Lie-Yamaguti algebra. A
representation of it is a ordered triple $(V,W,\psi)$, where $V$, $W$ are
representations of $L_{1}$ and $L_{2}$ respectively, and $\psi:V\rightarrow W$
is a linear operator satisfying the following conditions

$\displaystyle\psi(\rho_{V}(x)v)=\rho_{W}(\phi(x))\psi(v)$
$\displaystyle\psi(D_{V}(x,y)v)=D_{W}(\phi(x),\phi(y))\psi(v)$
$\displaystyle\psi(\theta_{V}(x,y)v)=\theta_{W}(\phi(x),\phi(y))\psi(v)$

for all $x,y\in L_{1}$ and $v\in V.$

The following proposition gives an example of representation of a Lie-Yamaguti
algebra induced by representation of morphism of Lie-Yamaguti algebra.

######  Proposition 3.4.

Let $\phi:L_{1}\rightarrow L_{2}$ be a morphism of Lie-Yamaguti algebra and
$(V,W,\psi)$ be a representation of it. Now the maps

$\displaystyle\rho_{W}^{L_{1}}:L_{1}\rightarrow End(W)$ $\displaystyle
D_{W}^{L_{1}}:L_{1}\times L_{1}\rightarrow End(W)$
$\displaystyle\theta_{W}^{L_{1}}:L_{1}\times L_{1}\rightarrow End(W)$

defined by

$\displaystyle\rho_{W}^{L_{1}}(x)v=\rho_{W}(\phi(x))v$
$\displaystyle\theta_{W}^{L_{1}}(x,y)v=\theta_{W}(\phi(x),\phi(y))v$
$\displaystyle D_{W}^{L_{1}}(x,y)v=D_{W}(\phi(x),\phi(y))v$

respectively for all $x,y\in L_{1}$ and $v\in V$ makes
$(\rho_{W}^{L_{1}},D_{W}^{L_{1}},\theta_{W}^{L_{1}})$ is representation of
$L_{1}$ on $W.$

###### Proof.

Let $x,y,z\in L_{1}$ and $w\in W.$ Observe that

$\displaystyle
D_{W}^{L_{1}}(x,y)w-\theta_{W}^{L_{1}}(y,x)w+\theta_{W}^{L_{1}}(x,y)w+\rho_{W}^{L_{1}}([x,y])w$
$\displaystyle-\rho_{W}^{L_{1}}(x)\rho_{W}^{L_{1}}(y)w+\rho_{W}^{L_{1}}(y)\rho_{W}^{L_{1}}(x)w$
$\displaystyle=D_{W}(\phi(x),\phi(y))w-\theta_{W}(\phi(y),\phi(x))w+\theta_{W}(\phi(x),\phi(y))w+\rho_{W}([\phi(x),\phi(y)])w$
$\displaystyle-\rho_{W}^{L_{1}}(x)\rho_{W}(\phi(y))w+\rho_{W}^{L_{1}}(y)\rho_{W}(\phi(x))w$
$\displaystyle=D_{W}(\phi(x),\phi(y))w-\theta_{W}(\phi(y),\phi(x))w+\theta_{W}(\phi(x),\phi(y))w+\rho_{W}([\phi(x),\phi(y)])w$
$\displaystyle-\rho_{W}(\phi(x))\rho_{W}(\phi(y))w+\rho(\phi(y))\rho_{W}(\phi(x))w$
$\displaystyle=0,~{}~{}~{}\mbox{as}~{}W~{}\mbox{is a representation
of}~{}L_{2}.$

On the other hand, we have

$\displaystyle
D_{W}^{L_{1}}([x,y],z)w+D_{W}^{L_{1}}([y,z],x)w+D_{W}^{L_{1}}([z,x],y)$
$\displaystyle=D_{W}(\phi([x,y]),\phi(z))w+D_{W}(\phi([y,z]),\phi(x))w+D_{W}(\phi([z,x]),\phi(y))w$
$\displaystyle=D_{W}([\phi(x),\phi(y)],\phi(z))w+D_{W}([\phi(y),\phi(z)],\phi(x))w+D_{W}([\phi(z),\phi(x)],\phi(y))w$
$\displaystyle=0,~{}~{}~{}\mbox{as}~{}W~{}\mbox{is a representation
of}~{}L_{2}.$

The other axioms of representation of a Lie-Yamaguti algebra can be proved
similarly as above. Hence,
$(\rho_{W}^{L_{1}},D_{W}^{L_{1}},\theta_{W}^{L_{1}})$ is a representation of
$L_{1}$ on $W.$ ∎

We denote the above representation by $W_{\phi}.$ The following proposition
shows that given a homomorphism between two morphism of Lie-Yamaguti algebra
induces a representation of a morphism of Lie-Yamaguti algebra.

######  Proposition 3.5.

Let $\phi:L_{1}\rightarrow L_{2}$,
$\phi^{{}^{\prime}}:L_{1}^{{}^{\prime}}\rightarrow L_{2}^{{}^{\prime}}$ be two
morphism of Lie Yamaguti algebras and $(\alpha,\beta)$ be a homomorphism
between them. Then,
$(L_{1}^{{}^{\prime}},L_{2}^{{}^{\prime}},\phi^{{}^{\prime}})$ is a
representation of Lie-Yamaguti algebra morphism $\phi:L_{1}\rightarrow L_{2}$,
where $\rho_{L_{1}^{{}^{\prime}}}(x):L_{1}\rightarrow
End(L_{1}^{{}^{\prime}}),~{}D_{L_{1}^{{}^{\prime}}}(x,y):L_{1}\times
L_{1}\rightarrow
End(L_{1}^{{}^{\prime}}),~{}\theta_{L_{1}^{{}^{\prime}}}(x,y):L_{1}\times
L_{1}\rightarrow End(L_{1}^{{}^{\prime}})$ are defined by

$\displaystyle\rho(x)x^{{}^{\prime}}=[\alpha(x),x^{{}^{\prime}}]$
$\displaystyle
D(x,y)x^{{}^{\prime}}=\\{\alpha(x),\alpha(y),x^{{}^{\prime}}\\}$
$\displaystyle\theta(x,y)x^{{}^{\prime}}=\\{x^{{}^{\prime}},\alpha(x),\alpha(y)\\}$

respectively for all $x,y\in L_{1}$ and $x^{{}^{\prime}}\in
L_{1}^{{}^{\prime}}$. Also, $\rho_{L_{2}^{{}^{\prime}}}(x):L_{2}\rightarrow
End(L_{2}^{{}^{\prime}}),~{}D_{L_{2}^{{}^{\prime}}}(x,y):L_{2}\times
L_{2}\rightarrow
End(L_{2}^{{}^{\prime}}),~{}\theta_{L_{2}^{{}^{\prime}}}(x,y):L_{2}\times
L_{2}\rightarrow End(L_{2}^{{}^{\prime}})$ are defined by

$\displaystyle\rho(x)x^{{}^{\prime}}=[\beta(x),x^{{}^{\prime}}]$
$\displaystyle D(x,y)x^{{}^{\prime}}=\\{\beta(x),\beta(y),x^{{}^{\prime}}\\}$
$\displaystyle\theta(x,y)x^{{}^{\prime}}=\\{x^{{}^{\prime}},\beta(x),\beta(y)\\}$

respectively for all $x,y\in L_{2}$ and $x^{{}^{\prime}}\in
L_{2}^{{}^{\prime}}$.

###### Proof.

For any $x,y,z\in L_{1}$ and $x^{{}^{\prime}}\in L_{1}^{{}^{\prime}}$, we have

$\displaystyle
D(x,y)x^{{}^{\prime}}-\theta(y,x)x^{{}^{\prime}}+\theta(x,y)x^{{}^{\prime}}+\rho([x,y])x^{{}^{\prime}}-\rho(x)\rho(y)x^{{}^{\prime}}+\rho(y)\rho(x)x^{{}^{\prime}}$
$\displaystyle=\\{\alpha(x),\alpha(y),x^{{}^{\prime}}\\}-\\{x^{{}^{\prime}},\alpha(y),\alpha(x)\\}+\\{x^{{}^{\prime}},\alpha(x),\alpha(y)\\}$
$\displaystyle+[[\alpha(x),\alpha(y)],x^{{}^{\prime}}]-[\alpha(x),[\alpha(y),x^{{}^{\prime}}]]+[\alpha(y),[\alpha(x),x^{{}^{\prime}}]]$
$\displaystyle=\\{\alpha(x),\alpha(y),x^{{}^{\prime}}\\}+\\{\alpha(y),x^{{}^{\prime}},\alpha(x)\\}+\\{x^{{}^{\prime}},\alpha(x),\alpha(y)\\}$
$\displaystyle+[[\alpha(x),\alpha(y)],x^{{}^{\prime}}]+[[\alpha(y),x^{{}^{\prime}}],\alpha(x)]+[[x^{{}^{\prime}},\alpha(x)],\alpha(y)]$
$\displaystyle=0.$

Again,

$\displaystyle
D([x,y],z)x^{{}^{\prime}}+D([y,z],x)x^{{}^{\prime}}+D([z,x],y)x^{{}^{\prime}}$
$\displaystyle=\\{\alpha([x,y]),\alpha(z),x^{{}^{\prime}}\\}+\\{\alpha([y,z]),\alpha(x),x^{{}^{\prime}}\\}+\\{\alpha([z,x]),\alpha(y),x^{{}^{\prime}}\\}$
$\displaystyle=\\{[\alpha(x),\alpha(y)],\alpha(z),x^{{}^{\prime}}\\}+\\{[\alpha(y),\alpha(z)],\alpha(x),x^{{}^{\prime}}\\}+\\{[\alpha(z),\alpha(x)],\alpha(y),x^{{}^{\prime}}\\}$
$\displaystyle=0.$

In a similar way other axioms of representation of Lie-Yamaguti algebra can be
shown. Therefore, $L_{1}^{{}^{\prime}}$ is a representation of $L_{1}$. In a
similar way as above $L_{2}^{{}^{\prime}}$ is a representation of $L_{2}.$
Now,

$\phi^{{}^{\prime}}(\rho_{L_{1}^{{}^{\prime}}}(x)x^{{}^{\prime}})=\phi^{{}^{\prime}}([\alpha(x),x^{{}^{\prime}}])=[\phi^{{}^{\prime}}\circ\alpha(x),\phi^{{}^{\prime}}(x^{{}^{\prime}})]\\\
=[\beta\circ\phi^{{}^{\prime}}(x^{{}^{\prime}}),\phi^{{}^{\prime}}(x^{{}^{\prime}})]=\rho_{L_{2}^{{}^{\prime}}}(\phi(x))\phi^{{}^{\prime}}(x^{{}^{\prime}})$

The other two conditions for representation for morphism of Lie-Yamaguti
algebra can be shown same way as above. Hence,
$(L_{1}^{{}^{\prime}},L_{2}^{{}^{\prime}},\phi)$ is a representation of Lie-
Yamaguti algebra morphism $\phi:L_{1}\rightarrow L_{2}.$ ∎

##  4 Cohomology of morphism of Lie-Yamaguti algebras

In this section, we define the cohomology of morphism of Lie-Yamaguti algebra
using cohomolgy theory of Lie-Yamaguti algebra motivated by the cohomolgoy
theory of morphism of different algebras in [8, 15, 1].
Let $\phi:L_{1}\rightarrow L_{2}$ be a morphism of Lie-Yamaguti algebra and
$(V,W,\psi)$ be a representation of it. Now we have the following three
cochain complexes

  * •

$\\{C^{(\bullet,\bullet)}(L_{1},V),\delta^{{}^{\prime}}\\}$, the cochain
complex of $L_{1}$ with coefficients in $V.$

  * •

$\\{C^{(\bullet,\bullet)}(L_{2},W),\delta^{{}^{\prime\prime}}\\}$, the cochain
complex of $L_{2}$ with coefficients in $W.$

  * •

$\\{C^{(\bullet,\bullet)}(L_{1},W_{\phi}),\delta^{{}^{\prime\prime\prime}}\\}$,
the cochain complex of $L_{1}$ with coefficients in $W_{\phi}.$

Now, using the above three cochain complex we define the cohomology of the
morphism of Lie Yamaguti algebra $\phi:L_{1}\rightarrow L_{2}$ with
coefficients in $(V,W,\psi).$

For each $n\geq 1$, we define the $(2n,2n+1)$-th cochain group
$C^{(2n,2n+1)}_{mLYA}(\phi,\psi)$ by

$C^{(2n,2n+1)}_{mLYA}(\phi,\psi)=\begin{cases}&C^{(2,3)}(L_{1},V)\oplus
C^{(2,3)}(L_{2},W)\oplus C(L_{1},W_{\phi})~{}~{}\mbox{if}~{}~{}n=1\\\
&C^{(2n,2n+1)}(L_{1},V)\oplus C^{(2n,2n+1)}(L_{2},W)\oplus
C^{(2n-2,2n-1)}(L_{1},W_{\phi}),~{}~{}~{}\mbox{if}~{}~{}n\geq 2.\end{cases}$

We define the coboundary map
$\delta_{mLYA}:C^{(2n,2n+1)}_{mLYA}(\phi,\psi)\rightarrow
C^{(2n+2,2n+3)}_{mLYA}(\phi,\psi)$ by

$\delta_{mLYA}(\alpha,\beta,\gamma)=\begin{cases}(\delta(\alpha),\delta(\beta),\phi\circ\alpha-\beta\phi-\delta(\gamma))&\text{for
}(\alpha,\beta,\gamma)\in C^{(2,3)}_{mLYA}(\phi,\psi),n=1\\\
(\delta^{{}^{\prime}}(\alpha),\delta^{{}^{\prime\prime}}(\beta),\psi\circ\alpha-\beta\circ\wedge^{n}\phi-\delta^{{}^{\prime\prime\prime}}(\gamma))&\text{for
}(\alpha,\beta,\gamma)\in C^{(2n,2n+1)}_{mLYA}(\phi,\psi),n\geq 2,\end{cases}$

where $\delta:C(L_{1},W_{\phi})\rightarrow C^{(2,3)}(L_{1},W_{\phi})$ is
defined by $\delta(\gamma)=(\delta_{I}\gamma,\delta_{II}\gamma)$ such that

$\displaystyle\delta_{I}\gamma(x,y)=[\phi(x),\gamma(y)]+[\gamma(x),\phi(y)]-\gamma([x,y])$
and
$\displaystyle\delta_{II}\gamma(x,y,z)=\\{\gamma(x),\phi(y),\phi(z)\\}+\\{\phi(x),\gamma(y),\phi(z)\\}+\\{\phi(x),\phi(y),\gamma(z)\\}-\gamma(\\{x,y,z\\})$

for all $x,y,z\in L_{1}.$ Also, note that $\beta\phi:L_{1}^{\otimes
n}\rightarrow W_{\phi}$ is a linear function defined by

$\beta\phi(x_{1},x_{2},\ldots,x_{n})=\beta(\phi(x_{1}),\phi(x_{2}),\ldots,\phi(x_{n}))$

for all $x_{1},x_{2},\ldots,x_{n}\in L_{1}.$

######  Proposition 4.1.

The map $\delta_{mLYA}$ defined above satisfies $\delta_{mLYA}^{2}=0.$

###### Proof.

Case-I:
For $n=1,$ and $(\alpha,\beta,\gamma)\in C^{(2,3)}_{mLYA}(\phi,\psi)$, we have

$\displaystyle d^{2}(\alpha,\beta,\gamma)$
$\displaystyle=d(\delta(\alpha),\delta(\beta),\phi\circ\alpha-\beta\phi-\delta(\gamma))$
$\displaystyle=(\delta^{2}(\alpha),\delta^{2}(\beta),\phi(\delta(\alpha))-\delta(\beta)\phi-\delta(\phi\circ\alpha-\beta\phi-\delta(\gamma)))$
$\displaystyle=(\delta^{2}(\alpha),\delta^{2}(\beta),\phi(\delta(\alpha))-\delta(\beta)\phi-\delta(\phi\circ\alpha)-\delta(\beta\phi))$
$\displaystyle=0,~{}\mbox{since}~{}\delta^{2}=0~{}by~{}\cite[cite]{[\@@bibref{}{KY}{}{}]}.$

Case-II: For $n\geq 2.$
For $(\alpha,\beta,\gamma)\in C^{(2n,2n+1)}_{mLYA}(\phi,\psi)$, we have

$\displaystyle\delta_{mLYA}^{2}(\alpha,\beta,\gamma)$
$\displaystyle=(\delta^{{}^{\prime}}(\alpha),\delta^{{}^{\prime\prime}}(\beta),\psi\circ\alpha-\beta\circ\wedge^{n}\phi-\delta^{{}^{\prime\prime\prime}}(\gamma))$
$\displaystyle=((\delta^{{}^{\prime}})^{2}(\alpha),(\delta^{{}^{\prime\prime}})^{2}(\beta),\psi\circ\delta^{{}^{\prime}}(\alpha)-\delta^{{}^{\prime\prime}}(\beta)\circ\wedge^{n+1}\phi-\delta^{{}^{\prime\prime\prime}}(\psi\circ\alpha)+\delta^{{}^{\prime\prime\prime}}(\beta\circ\wedge^{n}\phi)+(\delta^{{}^{\prime\prime\prime}})^{2}(\gamma))$
$\displaystyle=(0,0,\psi\circ\delta^{{}^{\prime}}(\alpha)-\delta^{{}^{\prime\prime}}(\beta)\circ\wedge^{n+1}\phi-\delta^{{}^{\prime\prime\prime}}(\psi\circ\alpha)+\delta^{{}^{\prime\prime\prime}}(\beta\circ\wedge^{n}\phi))\
$ (1)

For $x_{1},x_{2},\ldots,x_{2n+2}\in L_{1}$, observe that

$\displaystyle\delta^{{}^{\prime\prime\prime}}(\psi\circ\alpha)(x_{1},x_{2},\ldots,x_{2n+2})$
$\displaystyle=\delta^{{}^{\prime\prime\prime}}(\psi\circ\alpha_{I},\psi\circ\alpha_{II})(x_{1},x_{2},\ldots,x_{2n+2})$
$\displaystyle=(\delta^{{}^{\prime\prime\prime}}_{I}(\psi\circ\alpha_{I}),\delta^{{}^{\prime\prime\prime}}_{II}(\psi\circ\alpha_{II}))(x_{1},x_{2},\ldots,x_{2n+2}).$

Again,

$\displaystyle\delta^{{}^{\prime\prime\prime}}_{I}(\psi\circ\alpha_{I})(x_{1},x_{2},\ldots,x_{2n+2})$
$\displaystyle=\rho_{W}^{L_{1}}(x_{2n+1})(\psi\circ\alpha_{II})(x_{1},x_{2},\ldots,x_{2n},x_{2n+2})-\rho_{W}^{L_{1}}(x_{2n+2})(\psi\circ\alpha_{II})(x_{1},x_{2},\ldots,x_{2n},x_{2n+1})$
$\displaystyle-(\psi\circ\alpha_{II})(x_{1},x_{2},\ldots,x_{2n},[x_{2n+1},x_{2n+2}])$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k+1}D_{W}^{L_{1}}(x_{2k-1},x_{2k})(\psi\circ\alpha_{I})(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,x_{2n+2})$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+2}(-1)^{n+k}(\psi\circ\alpha_{I})(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,\\{x_{2k-1},x_{2k},x_{j}\\},\ldots,x_{2n+2}),$
$\displaystyle=\psi(\rho_{V}(x_{2n+1})\alpha_{II}(x_{1},x_{2},\ldots,x_{2n},x_{2n+2}))-\psi(\rho_{V}(x_{2n+2})\alpha_{II}(x_{1},x_{2},\ldots,x_{2n},x_{2n+1}))$
$\displaystyle-\psi(\alpha_{II}(x_{1},x_{2},\ldots,x_{2n},[x_{2n+1},x_{2n+2}]))$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k+1}\psi_{V}(D_{V}(x_{2k-1},x_{2k})\alpha_{I}(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,x_{2n+2}))$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+2}(-1)^{n+k}\psi(\alpha_{I}(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,\\{x_{2k-1},x_{2k},x_{j}\\},\ldots,x_{2n+2})),$
$\displaystyle=(\psi\circ\delta^{{}^{\prime}}(\alpha_{I}))(x_{1},x_{2},\ldots,x_{2n+2}).$

Thus, we have

$\delta^{{}^{\prime\prime\prime}}_{I}(\psi\circ\alpha_{I})=(\psi\circ\delta^{{}^{\prime}}(\alpha_{I}))$

Similarly, it can be proved that

$\delta^{{}^{\prime\prime\prime}}_{II}(\psi\circ\alpha_{II})=(\psi\circ\delta^{{}^{\prime}}(\alpha_{II})).$

Now from the Equation (4), we have

$\displaystyle\delta^{{}^{\prime\prime\prime}}(\psi\circ\alpha)(x_{1},x_{2},\ldots,x_{2n+2})$
$\displaystyle=(\delta^{{}^{\prime\prime\prime}}_{I}(\psi\circ\alpha_{I}),\delta^{{}^{\prime\prime\prime}}_{II}(\psi\circ\alpha_{II}))(x_{1},x_{2},\ldots,x_{2n+2})$
$\displaystyle=(\psi\circ\delta^{{}^{\prime}}(\alpha_{I}),\psi\circ\delta^{{}^{\prime}}(\alpha_{II}))(x_{1},x_{2},\ldots,x_{2n+2})$
$\displaystyle=(\psi\circ\delta^{{}^{\prime}}(\alpha))(x_{1},x_{2},\ldots,x_{2n+2}).$

Hence,
$\delta^{{}^{\prime\prime\prime}}(\psi\circ\alpha)=(\psi\circ\delta^{{}^{\prime}}(\alpha)).$

Now, we want to show that

$\delta^{{}^{\prime\prime\prime}}(\beta\circ\wedge^{n}\phi)=\delta^{{}^{\prime\prime}}(\beta)\circ\wedge^{n+1}(\phi)$

Note that

$\displaystyle\delta^{{}^{\prime\prime\prime}}(\beta\circ\wedge^{n}\phi)(x_{1},x_{2},\dots,x_{2n+2})$
$\displaystyle=\delta^{{}^{\prime\prime\prime}}(\beta_{I}\circ\wedge^{n}\phi,\beta_{II}\circ\wedge^{n}\phi)(x_{1},x_{2},\dots,x_{2n+2})$
$\displaystyle=(\delta^{{}^{\prime\prime\prime}}_{I}(\beta_{I}\circ\wedge^{n}\phi),\delta_{II}^{{}^{\prime\prime\prime}}(\beta_{II}\circ\wedge^{n}\phi))(x_{1},x_{2},\dots,x_{2n+2}).$
(2)

Again, we have

$\displaystyle\delta^{{}^{\prime\prime\prime}}_{I}(\beta_{I}\circ\wedge^{n}\phi)(x_{1},x_{2},\dots,x_{2n+2})$
$\displaystyle=\rho_{W}^{L_{1}}(x_{2n+1})(\beta_{II}\circ\wedge^{n}\phi)(x_{1},x_{2},\ldots,x_{2n},x_{2n+2})-\rho_{W}^{L_{1}}(x_{2n+2})(\beta_{II}\circ\wedge^{n}\phi)(x_{1},x_{2},\ldots,x_{2n},x_{2n+1})$
$\displaystyle-(\beta_{II}\circ\wedge^{n}\phi)(x_{1},x_{2},\ldots,x_{2n},[x_{2n+1},x_{2n+2}])$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k+1}D_{W}^{L_{1}}(x_{2k-1},x_{2k})(\beta_{I}\circ\wedge^{n}\phi)(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,x_{2n+2})$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+2}(-1)^{n+k}(\beta_{I}\circ\wedge^{n}\phi)(x_{1},\ldots,\widehat{x}_{2k-1},\widehat{x}_{2k},\ldots,\\{x_{2k-1},x_{2k},x_{j}\\},\ldots,x_{2n+2}),$
$\displaystyle=\rho_{W}(\phi(x_{2n+1}))\beta_{II}(\phi(x_{1}),\phi(x_{2}),\cdots,\phi(x_{2n}),\phi(x_{2n+2}))$
$\displaystyle-\rho_{W}(\phi(x_{2n+2}))\beta_{II}(\phi(x_{1}),\phi(x_{2}),\cdots,\phi(x_{2n}),\phi(x_{2n+1}))$
$\displaystyle-\beta_{II}(\phi(x_{1}),\phi(x_{2}),\cdots,\phi(x_{2n}),[\phi(x_{2n+1}),\phi(x_{2n+2})])$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k+1}D_{W}(\phi(x_{2k-1}),\phi(x_{2k}))\beta_{I}(\phi(x_{1}),\cdots,\widehat{\phi(x_{2k-1})},\widehat{\phi(x_{2k})},\ldots,\phi(x_{2n+2}))$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+2}(-1)^{n+k}\beta_{I}(\phi(x_{1}),\ldots,\widehat{\phi(x_{2k-1})},\widehat{\phi(x_{2k})},\ldots,\\{\phi(x_{2k-1}),\phi(x_{2k}),\phi(x_{j})\\},\ldots,\phi(x_{2n+2}))$
$\displaystyle=\delta_{I}^{{}^{\prime\prime}}(\beta_{I})\circ\wedge^{n+1}(\phi)(x_{1},x_{2},\dots,x_{2n+2}).$

Hence, we have

$\delta^{{}^{\prime\prime\prime}}_{I}(\beta_{I}\circ\wedge^{n}\phi)=\delta_{I}^{{}^{\prime\prime}}(\beta_{I})\circ\wedge^{n+1}(\phi).$

Similarly, we have

$\delta^{{}^{\prime\prime\prime}}_{II}(\beta_{II}\circ\wedge^{n}\phi)=\delta_{II}^{{}^{\prime\prime}}(\beta_{II})\circ\wedge^{n+1}(\phi).$

Now from the Equation (2), we have

$\displaystyle\delta^{{}^{\prime\prime\prime}}(\beta\circ\wedge^{n}\phi)(x_{1},x_{2},\dots,x_{2n+2})$
$\displaystyle=(\delta^{{}^{\prime\prime\prime}}_{I}(\beta_{I}\circ\wedge^{n}\phi),\delta_{II}^{{}^{\prime\prime\prime}}(\beta_{II}\circ\wedge^{n}\phi))(x_{1},x_{2},\dots,x_{2n+2})$
$\displaystyle=(\delta_{I}^{{}^{\prime\prime}}(\beta_{I})\circ\wedge^{n+1}(\phi),\delta_{II}^{{}^{\prime\prime}}(\beta_{II})\circ\wedge^{n+1}(\phi))(x_{1},x_{2},\dots,x_{2n+2})$
$\displaystyle=\delta^{{}^{\prime\prime}}(\beta)\circ\wedge^{n+1}(\phi)(x_{1},x_{2},\dots,x_{2n+2}).$

Hence, we have

$\delta^{{}^{\prime\prime\prime}}(\beta\circ\wedge^{n}\phi)=\delta^{{}^{\prime\prime}}(\beta)\circ\wedge^{n+1}(\phi).$

Therefore, from the Equation (1), we have $\delta_{mLYA}^{2}=0.$ ∎

Thus, it follows from the above proposition that
$\\{C^{(\bullet,\bullet)}_{mLYA},\delta_{mLYA}\\}$ is a cochain complex. Let
the space of $(2n,2n+1)$-th cocyles are denoted by
$Z^{(2n,2n+1)}_{mLYA}(\phi,\psi)$ and the space of $(2n,2n+1)$-th coboundaries
are denoted by $B^{(2n,2n+1)}_{mLYA}(\phi,\psi)$. Observe that, we have
$B^{(2n,2n+1)}_{mLYA}(\phi,\psi)\subset Z^{(2n,2n+1)}_{mLYA}(\phi,\psi).$ The
corresponding quotient groups

$H^{(2n,2n+1)}_{mLYA}(\phi,\psi):=\frac{Z^{(2n,2n+1)}_{mLYA}(\phi,\psi)}{B^{(2n,2n+1)}_{mLYA}(\phi,\psi)}~{}~{},~{}\mbox{for}~{}~{}n\geq
1,$

are called the cohomology groups of the morphism of Lie-Yamaguti algebra
$\phi:L_{1}\rightarrow L_{2}$ with coefficients in the representation
$(V,W,\psi).$

##  5 Deformation of morphisms of Lie-Yamaguti algebra

In this section we develop one-parameter formal deformation theory of
morphisms of Lie-Yamaguti algebra over a field $k$ of $Char(k)=0.$ Let
$K=k[[t]]$ be the formal power series ring with coefficient in $k$. For a Lie-
Yamaguti algebra $L$ define $L[[t]]:=L\otimes_{k}K$, then $L[[t]]$ is a module
over $K.$

######  Definition 5.1.

Let $\phi:L_{1}\rightarrow L_{2}$ be a morphism of Lie Yamaguti algebra. A
formal one parameter deformation of it is a order triple
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
, with the formal power series of the forms

$\displaystyle f_{t}=[~{},~{}]+\sum_{i\geqslant
1}f_{i}t^{i};f^{{}^{\prime}}_{t}=[~{},~{}]+\sum_{i\geqslant
1}f^{{}^{\prime}}_{i}t^{i}$ $\displaystyle
g_{t}=\\{~{},~{},~{}\\}+\sum_{i\geqslant
1}g_{i}t^{i};g^{{}^{\prime}}_{t}=\\{~{},~{},~{}\\}+\sum_{i\geqslant
1}g^{{}^{\prime}}_{i}t^{i}$
$\displaystyle\phi_{t}=\sum_{i=0}\phi_{i}t^{{}^{\prime}},\phi_{0}=\phi,$

where for $i\geq 0$, $f_{i}:L_{1}\times L_{1}\rightarrow L_{1}$ and
$f^{{}^{\prime}}_{i}:L_{2}\times L_{2}\rightarrow L_{2}$ are $k$\- bilinear
map (extended to be $K$\- bilinear) and also, for $i\geq 0$,
$g_{i}:L_{1}\times\times L_{1}\rightarrow L_{1}$ and
$g^{{}^{\prime}}_{i}:L_{2}\times L_{2}\times L_{2}\rightarrow L_{2}$ are $k$\-
trilinear map (extended to be $K$\- trilinear) such that the following
conditions are satisfied

  1. 1.

$(L_{1}[[t]],f_{t},g_{t})$ is a Lie-Yamaguti algebra.

  2. 2.

$(L_{2}[[t]],f^{{}^{\prime}}_{t},g^{{}^{\prime}}_{t})$ is a Lie-Yamaguti
algebra.

  3. 3.

$\phi_{t}:L_{1}[[t]]\rightarrow L_{2}[[t]]$ is a morphism of Lie-Yamaguti
algebra.

######  Definition 5.2.

The $(2,3)$\- cochain
$((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))$
is called the infinitesimal of the deformation
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$.
More generally
$((f_{i},g_{i}),(f_{i}^{{}^{\prime}},g_{i}^{{}^{\prime}}),(\phi_{i},\phi_{i}))=0$
for $1\leq i\leq n-1$ and
$((f_{n},g_{n}),(f_{n}^{{}^{\prime}},g_{n}^{{}^{\prime}}),(\phi_{n},\phi_{n}))$
is non zero, then
$((f_{n},g_{n}),(f_{n}^{{}^{\prime}},g_{n}^{{}^{\prime}}),(\phi_{n},\phi_{n}))$
is called the $n$\- infinitesimal of the deformation
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$.

Let
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
be a formal deformation of a morphism of Lie-Yamaguti algebra
$\phi:L_{1}\rightarrow L_{2}.$ Since $(L_{1}[[t]],f_{t},g_{t})$ is a Lie
Yamaguti algebra, we have

$\displaystyle f_{t}(x,x)=0,g_{t}(x,x,y)=0,$ (3)
$\displaystyle\sum_{\circlearrowleft(x,y,z)}(f_{t}(f_{t}(x,y),z)+g_{t}(x,y,z))=0$
(4) $\displaystyle\sum_{\circlearrowleft(x,y,z,w)}g_{t}(f_{t}(x,y),z,w)=0$ (5)
$\displaystyle
g_{t}(x,y,f_{t}(z,w))=f_{t}(g_{t}(x,y,z),w)+f_{t}(z,g_{t}(x,y,z))$ (6)
$\displaystyle
g_{t}(x,y,g_{t}(z,w,u))=g_{t}(g_{t}(x,y,z),w,u)+g_{t}(z,g_{t}(x,y,w),u)+g_{t}(z,w,g_{t}(x,y,u))$
(7)

for all $x,y,z,w,u\in L_{1}.$
Now since $f_{t}=\sum_{i\geq 0}f_{i}t^{i}$ and $g_{t}=\sum_{i\geq
0}g_{i}t^{i}$, where $f_{0}=[~{},~{}],~{}g_{0}=\\{~{},~{},~{}\\}$. Thus,
equating the coefficients of $t^{n}$, $n\geqslant 0$ from both sides of the
Equations $(\ref{LYAD 1})-(\ref{LYAD 2})$ we get,

$\displaystyle f_{n}(x,x)=0$ $\displaystyle g_{n}(x,x,y)=0$
$\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geqslant
0\end{subarray}}\sum_{\circlearrowleft(x,y,z)}(f_{i}(f_{i}(x,y),z)+g_{n}(x,y,z))=0$
(8) $\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geqslant
0\end{subarray}}\sum_{\circlearrowleft(x,y,z)}(g_{i}(f_{i}(x,y),z,w))=0$ (9)
$\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geqslant
0\end{subarray}}(g_{i}(x,y,f_{i}(z,w))-f_{i}(g_{i}(x,y,z),w)-f_{i}(z,g_{j}(x,y,w)))=0$
(10) $\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geqslant
0\end{subarray}}(g_{i}(x,y,g_{j}(z,w,u))-g_{i}(g_{j}(x,y,z),w,u)-g_{i}(z,g_{i}(x,y,w),u)-g_{i}(z,w,g_{j}(x,y,u)))=0$
(11)

where $x,y,z,w,u\in L_{1}.$
Now putting $n=1$ in the Equations (10)-(11)we get

$\displaystyle\\{x,y,f_{1}(u,v)\\}+g_{1}(x,y,[u,v])-[g_{1}(x,y,u),v]$
$\displaystyle-f_{1}(\\{x,y,u\\},v)-(z,g_{1}(x,y,z))-f_{1}(z,\\{x,y,u\\})=0;$
and
$\displaystyle\\{x,y,g_{1}(u,v,w)\\}+g_{1}(x,y,\\{u,v,w\\})-\\{g_{1}(x,y,u),v,w\\}$
$\displaystyle-
g_{1}(\\{x,y,u\\},v,w)-\\{u,g_{1}(x,y,v),w\\}-g_{1}(u,\\{x,y,v\\},w)$
$\displaystyle-\\{u,v,g_{1}(x,y,w)\\}-g_{1}(u,v,\\{x,y,w\\})-\\{u,v,g_{1}(x,y,w)\\}=0.$

Therefore, we have $(\delta_{I},\delta_{II})(f_{1},g_{1})=(0,0)$, i.e.,
$\delta^{{}^{\prime}}(f_{1},g_{1})=0$

Now using the fact $(L_{2}[[t]],f^{{}^{\prime}}_{t},g^{{}^{\prime}}_{t})$ is a
Lie-Yamaguti algebra we get similarly as above,
$(\delta_{I},\delta_{II})(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}})=(0,0)$,
i.e., $\delta^{{}^{\prime\prime}}(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}})=0.$

Now, since $\phi_{t}:L_{1}[[t]]\rightarrow L_{2}[[t]]$ is morphism of Lie-
Yamaguti algebra, we have

$\displaystyle\phi_{t}(f_{t}(x,y))=f_{t}^{{}^{\prime}}(\phi_{t}(x),\phi_{t}(y))$
(12)
$\displaystyle\phi_{t}(g_{t}(x,y,z))=g_{t}^{{}^{\prime}}(\phi_{t}(x),\phi_{t}(y),\phi_{t}(z))$
(13)

for all $x,y,z\in L_{1}.$
Now the Equations (12) and (13) reduces to

$\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geqslant
0\end{subarray}}\phi_{i}(f_{j}(x,y))=\sum_{\begin{subarray}{c}i+j+k=n\\\
i,j,k\geq 0\end{subarray}}f_{i}^{{}^{\prime}}(\phi_{j}(x),\phi_{k}(y))$
$\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geqslant
0\end{subarray}}\phi_{i}(g_{j}(x,y,z))=\sum_{\begin{subarray}{c}i+j+p+q=n\\\
i,j,p,q\geq
0\end{subarray}}g_{i}^{{}^{\prime}}(\phi_{j}(x),\phi_{p}(y),\phi_{q}(z))$

Now putting $n=1$, the above equations reduces to

$\displaystyle\phi_{1}(f_{0}(x,y))+\phi(f_{1}(x,y))=f_{1}^{{}^{\prime}}(\phi(x),\phi(y))+f_{0}^{{}^{\prime}}(\phi_{1}(x),\phi(x))+f_{0}^{{}^{\prime}}(\phi(x),\phi_{1}(y)).$

and

$\displaystyle\phi_{1}(g_{0}(x,y,z))+\phi(g_{1}(x,y,z))$
$\displaystyle=g_{1}^{{}^{\prime}}(\phi(x),\phi(y),\phi(z))+g_{0}^{{}^{\prime}}(\phi_{1}(x),\phi(y),\phi(z))$
$\displaystyle+g_{0}^{{}^{\prime}}(\phi(x),\phi_{1}(y),\phi(z))+g_{0}^{{}^{\prime}}(\phi(x),\phi(y),\phi_{1}(z)).$

Now, the above equation reduces to
$\phi\circ f_{1}-f_{1}^{{}^{\prime}}\phi-\delta_{I}(\phi_{1})=0$
$~{}\mbox{and}~{}\\\ \phi\circ
g_{1}-g_{1}^{{}^{\prime}}\phi-\delta_{II}(\phi_{1})=0.$
Hence,
$\phi\circ(f_{1},g_{1})-(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}})\phi-\delta(\phi_{1})=0$.
Thus, we have $\delta(f_{1},g_{1})=0$,
$\delta(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}})=0$ and
$\phi\circ(f_{1},g_{1})-(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}})\phi-\delta(\phi_{1})=0.$
Hence,
$d((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))=0.$
Therefore,
$((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))\in
Z^{(2,3)}_{mLYA}(\phi,\phi).$

######  Definition 5.3.

Let
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
and
$((F_{t},G_{t}),(F_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}}),(\Phi_{t},\Phi_{t}))$
be two one parameter formal deformation of a morphism of Lie-Yamaguti algebra
$\phi:L_{1}\rightarrow L_{2}$. Then this two one-parameter deformation is
called equivalent if there exists a formal isomorphisms
$(\psi_{t},\psi^{{}^{\prime}}_{t})$ such that
$\psi_{t}(x):(L_{1}[[t]],f_{t},g_{t})\rightarrow(L_{1}[[t]],F_{t},G_{t})$ and
$\psi^{{}^{\prime}}_{t}(x):(L_{2}[[t]],f^{{}^{\prime}}_{t},g^{{}^{\prime}}_{t})\rightarrow(L_{2}[[t]],F^{{}^{\prime}}_{t},G^{{}^{\prime}}_{t})$,
where $\psi_{i}:L_{1}\rightarrow L_{1}$ and
$\psi^{{}^{\prime}}:L_{2}\rightarrow L_{2}$ are two $k-$ linear map (extended
to be $k[[t]]$ linear ) such that
$\Phi_{t}=\psi_{t}^{{}^{\prime}}\circ\phi_{t}\circ\psi_{t}^{-1}.$

######  Definition 5.4.

Any deformation of $\phi:L_{1}\rightarrow L_{2}$, which is equivalent to
$((f_{0},g_{0}),(f_{0}^{{}^{\prime}},g_{0}^{{}^{\prime}}),(\phi,\phi))$ is
called a trivial deformation, where $f_{0},g_{0}$ are the brackets of $L_{1}$
and $f_{0}^{{}^{\prime}},g_{0}^{{}^{\prime}}$ are the brackets of $L_{2}.$

######  Theorem 5.5.

Let
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
and
$((F_{t},G_{t}),(F_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}}),(\Phi_{t},\Phi_{t}))$
be two equivalent one parameter formal deformation of morphism of Lie-Yamaguti
algebra $\phi:L_{1}\rightarrow L_{2}$. Then
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
and
$((F_{t},G_{t}),(F_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}}),(\Phi_{t},\Phi_{t}))$
are belongs to the same cohomology class in $H^{(2,3)}_{mLYA}(\phi,\phi).$

###### Proof.

Since $(\hat{f_{t}},\hat{g_{t}},\phi_{t})$ and
$(\hat{F_{t}},\hat{G_{t}},\Phi_{t})$ are two equivalent deformation, there
exists isomorphisms say
$\psi_{t}:(L_{1}[[t]],f_{t},g_{t})\rightarrow(L_{1}[[t]],F_{t},G_{t})$ and
$\psi^{\prime}_{t}:(L_{2}[[t]],f_{t}^{\prime},g_{t}^{\prime})\rightarrow(L_{2}[[t]],F_{t}^{\prime},G_{t}^{\prime})$.

Therefore,

$\displaystyle\psi_{t}\circ
f_{t}(x,y)=F_{t}(\psi_{t}(x),\psi_{t}(y)),\psi_{t}\circ
g_{t}(x,y,z)=G_{t}(\psi_{t}(x),\psi_{t}(y),\psi_{t}(z))$ (14)

for all $x,y,z\in L_{1}$ with $\psi_{0}=id_{L_{1}}$ and

$\displaystyle\psi^{{}^{\prime}}_{t}\circ
f^{{}^{\prime}}_{t}(x,y)=F^{{}^{\prime}}_{t}(\psi^{{}^{\prime}}_{t}(x),\psi^{{}^{\prime}}_{t}(y)),\psi^{{}^{\prime}}_{t}\circ
g^{{}^{\prime}}_{t}(x,y,z)=G^{{}^{\prime}}_{t}(\psi^{{}^{\prime}}_{t}(x),\psi^{{}^{\prime}}_{t}(y),\psi^{{}^{\prime}}_{t}(z))$
(15)

for all $x,y,z\in L_{2}$ with $\psi^{{}^{\prime}}_{0}=id_{L_{2}}.$

From the Equation (14), we have

$\sum_{i\geq 0}\psi_{i}(\sum_{j\geq 0}f_{j}(x,y)t^{j})t^{i}=\sum_{i\geq
0}F_{i}(\sum_{p\geq 0}\psi_{p}(x)t^{p},\sum_{q\geq 0}\psi_{q}(y)t^{q})t^{i}$

and

$\sum_{i\geq 0}\psi_{i}(\sum_{j\geq 0}g_{j}(x,y,z)t^{j})t^{i}=\sum_{i\geq
0}G_{i}(\sum_{p\geq 0}\psi_{(}x)t^{,}\sum_{q\geq
0}\psi_{q}(y)t^{q},\sum_{l\geq 0}\psi_{l}(z)t^{l})t^{i}$

Comparing the coefficients of $t$ from both sides of the above equation we get

$f_{1}(x,y)+\psi_{1}([x,y])=F_{1}(x,y)+[\psi_{1}(x),y]+[x,\psi_{1}(y)]$

i.e.,

$f_{1}(x,y)-F_{1}(x,y)=\delta_{I}\psi_{1}(x,y)$

and

$g_{1}(x,y,z)+\psi_{1}(\\{x,y,z\\})=G_{1}(x,y,z)+\\{\psi_{1}(x),y,z\\}+\\{x,\psi_{1}(y),z\\}+\\{x,y,\psi_{1}(z)\\}$

i.e.,

$g_{1}(x,y,z)-G_{1}(x,y,z)=\delta_{II}\psi_{1}(x,y,z).$

Therefore, $(f_{1}-F_{1},g_{1}-G_{1})=\delta(\psi_{1},\psi_{1}).$ Similarly,
as above from the Equation (15), we get
$(f_{1}^{{}^{\prime}}-F_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}-G_{1}^{{}^{\prime}})=\delta(\psi_{1}^{{}^{\prime}},\psi_{1}^{{}^{\prime}}).$

Now, we have $\Phi_{t}\circ\psi_{t}=\psi_{t}^{{}^{\prime}}\circ\phi_{t}.$
Expanding and equating the coefficients of $t^{n}$ from both sides of the
above equation we get,

$\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geqslant
0\end{subarray}}\Phi_{i}\circ\psi_{j}=\sum_{\begin{subarray}{c}i+j=n\\\
i,j\geqslant 0\end{subarray}}\psi_{i}^{{}^{\prime}}\circ\phi_{j}.$

Now putting $n=1$ in the above equation we get

$\displaystyle\Phi_{0}\circ\psi_{1}+\Phi_{1}\circ\psi_{0}=\psi_{0}^{{}^{\prime}}\circ\phi_{1}+\psi_{1}^{{}^{\prime}}\circ\phi_{0}$
$\displaystyle\phi_{1}-\Phi_{1}=\Phi\circ\psi_{1}-\psi_{1}^{{}^{\prime}}\circ\phi$

Now,

$\displaystyle((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))-((F_{1},G_{1}),(F_{1}^{{}^{\prime}},G_{1}^{{}^{\prime}}),(\Phi_{1},\Phi_{1}))$
$\displaystyle=((f_{1}-G_{1},g_{1}-F_{1}),(f_{1}^{{}^{\prime}}-F_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}-G_{1}^{{}^{\prime}}),(\phi_{1}-\Phi_{1},\phi_{1}-\Phi_{1}))$
$\displaystyle=(\delta(\psi_{1},\psi_{1}),\delta(\psi_{1}^{{}^{\prime}},\psi_{1}^{{}^{\prime}}),(\Phi\circ\psi_{1}-\psi_{1}^{{}^{\prime}}\circ\phi,\Phi\circ\psi_{1}-\psi_{1}^{{}^{\prime}}\circ\phi))$
$\displaystyle=d((\psi_{1},\psi_{1}),(\psi_{1}^{{}^{\prime}},\psi_{1}^{{}^{\prime}}),(0,0))$

Hence,
$(f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))$
and
$((F_{1},G_{1}),(F_{1}^{{}^{\prime}},G_{1}^{{}^{\prime}}),(\Phi_{1},\Phi_{1})$
are cohomologous and belongs to the same cohomology class
$B^{(2,3)}_{mLYA}(\phi,\phi).$

∎

######  Theorem 5.6.

Let $((f_{t},g_{t}),(f_{t}^{\prime},g_{t}^{\prime}),(\phi_{t},\phi_{t}))$ be a
formal one parameter deformation of a morphism of Lie-Yamaguti algebra
$\phi:L_{1}\rightarrow L_{2}$, with infinitesimal
$((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))$.
Let
$((F_{1},G_{1}),(F_{1}^{{}^{\prime}},G_{1}^{{}^{\prime}}),(\Phi_{1},\Phi_{1}))$
be a $(2,3)-$ cocyle cohomologous to
$((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))$,
then there exists a deformation equivalent to
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
with infinitesimal
$((F_{1},G_{1}),(F_{1}^{{}^{\prime}},G_{1}^{{}^{\prime}}),(\Phi_{1},\Phi_{1}))$.

###### Proof.

Since
$((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))$
and
$((F_{1},G_{1}),(F_{1}^{{}^{\prime}},G_{1}^{{}^{\prime}}),(\Phi_{1},\Phi_{1}))$
are cohomologous, we have

$\displaystyle((F_{1},G_{1}),(F_{1}^{{}^{\prime}},G_{1}^{{}^{\prime}}),(\Phi_{1},\Phi_{1}))=((f_{1},g_{1}),(f_{1}^{{}^{\prime}},g_{1}^{{}^{\prime}}),(\phi_{1},\phi_{1}))+(\delta(\lambda_{1},\lambda_{1})+\delta(\lambda_{2},\lambda_{2}),(m,m))$

where $\lambda_{1}\in C^{1}(L_{1},L_{1})$, $\lambda_{2}\in C^{1}(L_{2},L_{2})$
and $(m,m)\in Z(L_{1},L_{2})$.
Let $\psi_{t}:L_{1}[[t]]\rightarrow L_{1}[[t]]$ and
$\psi_{t}^{{}^{\prime}}:L_{2}[[t]]\rightarrow L_{2}[[t]]$ be the formal
isomorphisms obtained by extending linearly the map
$\psi_{t}(a)=a+\lambda_{1}(a)t$, $a\in L_{1}$ to $L_{1}[[t]]$ and
$\psi_{t}^{{}^{\prime}}(a)=b+\lambda_{2}(b)t$, $b\in L_{2}$ to $L_{2}[[t]]$
respectively. Then, the inverse maps are given by

$\displaystyle\psi_{t}^{-1}(a)=a+\sum_{i\geq
0}(-1)^{i}\lambda_{1}^{i}(a)t^{i},~{}a\in L_{1}$
$\displaystyle{\psi_{t}^{{}^{\prime}}}^{-1}(b)=b+\sum_{i\geq
0}(-1)^{i}\lambda_{2}^{i}(b)t^{i},~{}b\in L_{2}.$

Now define the maps $(F_{t},G_{t})$,
$(f_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}})$ and $\Phi_{t}$ by

$\displaystyle
F_{t}(x,y)=\psi_{t}^{-1}(f_{t}(\psi_{t}(x),\psi_{t}(y))),~{}x,y\in L_{1}$ (16)
$\displaystyle
G_{t}(x,y,z)=\psi_{t}^{-1}(g_{t}(\psi_{t}(x),\psi_{t}(y),\psi_{t}(z))),~{}x,y,z\in
L_{1}$ (17) $\displaystyle
F_{t}^{{}^{\prime}}(x,y)={\psi_{t}^{{}^{\prime}}}^{-1}(f_{t}^{{}^{\prime}}(\psi_{t}^{{}^{\prime}}(x),\psi_{t}^{{}^{\prime}}(y))),~{}x,y\in
L_{2}$ (18) $\displaystyle
G_{t}^{{}^{\prime}}(x,y,z)={\psi_{t}^{{}^{\prime}}}^{-1}(g_{t}^{{}^{\prime}}(\psi_{t}^{{}^{\prime}}(x),\psi_{t}^{{}^{\prime}}(y),\psi_{t}^{{}^{\prime}}(z))),~{}x,y,z\in
L_{2}$ (19)
$\displaystyle\Phi_{t}=\psi_{t}^{{}^{\prime}}\circ\phi_{t}\circ\psi_{t}^{-1}.$
(20)

Note that
$((F_{t},G_{t}),(F_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}}),(\Phi_{t},\Phi_{t}))$
is equivalent to
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$.

Now from the Equation (16) we get the expression of $F_{t}(x,y)$ as

$\psi_{t}^{-1}(f_{t}(\psi_{t}(x),\psi_{t}(y)))=\psi_{t}^{-1}(\sum_{i\geq
0}(f_{i}(x+\lambda_{1}(x)t,y+\lambda_{1}(y)t)t^{i}),~{}x,y\in L_{1}.$

Expanding the above equation we get the coefficient of $t$ in $F_{t}(x,y)$ as

$f_{1}(x,y)+[\lambda_{1}(x),y]+[x,\lambda_{1}(y)]-\lambda_{1}([x,y])=f_{1}(x,y)+\delta_{I}\lambda_{1}(x,y)=F_{1}(x,y)$

i.e., $F_{1}-f_{1}=\delta_{I}\lambda_{1}.$

Again, from the Equation (17) we get the expression of $G_{t}(x,y,z)$ as

$\psi_{t}^{-1}(g_{t}(\psi_{t}(x),\psi_{t}(y),\psi_{t}(z)))=\psi_{t}^{-1}(\sum_{i\geq
0}(g_{i}(x+\lambda_{1}(x)t,y+\lambda_{1}(y)t,z+\lambda_{1}(z))t^{i}),~{}x,y,z\in
L_{2}.$

Expanding the above equation we get the coefficient of $t$ in $G_{t}(x,y,z)$
as

$\displaystyle
g_{1}(x,y,z)+\\{\lambda_{1}(x),y,z\\}+\\{x,\lambda(y),z\\}+\\{x,y,\lambda_{1}(z)\\}-\lambda_{1}(\\{x,y,z\\})$
$\displaystyle=g_{1}(x,y,z)+\delta_{II}\lambda_{1}(x,y,z)=G_{1}(x,y,z)$

i.e., $G_{1}-g_{1}=\delta_{II}\lambda_{1}.$
Similarly from the Equations (18), (19) we get,
$F_{1}^{{}^{\prime}}-f_{1}^{{}^{\prime}}=\delta_{I}\lambda_{2},$ and
$G_{1}^{{}^{\prime}}-g_{1}^{{}^{\prime}}=\delta_{II}\lambda_{2}.$

Again, from the Equation (20) we get,

$\displaystyle\Phi_{t}(a)=\psi_{t}^{{}^{\prime}}(\phi_{t}(a+\sum_{i\geq
0}(-1)^{i}\lambda_{1}^{i}(a)t^{i}))$ $\displaystyle=\phi_{t}(a+\sum_{i\geq
0}(-1)^{i}\lambda_{1}^{i}(a)t^{i})+\lambda_{2}(a+\sum_{i\geq
0}(-1)^{i}\lambda_{1}^{i}(a)t^{i})t,~{}~{}a\in L_{1}.$

From the above equation we get the coefficient of $t$ in $\Phi_{t}$ as
$\phi_{1}-\phi\circ\lambda_{1}+\lambda_{2}\phi=\Phi_{1}$ i.e.,
$\Phi_{1}-\phi_{1}=\lambda_{2}\phi-\phi\circ\lambda_{1}.$
Now as
$d((\delta_{I}\lambda_{1},\delta_{II}\lambda_{1}),(\delta_{I}\lambda_{2},\delta_{II}\lambda_{2}),(\lambda_{2}\phi-\phi\circ\lambda_{1},\lambda_{2}\phi-\phi\circ\lambda_{1}))=0$.
Hence,
$((F_{1},G_{1}),(F_{1}^{{}^{\prime}},G_{1}^{{}^{\prime}}),(\Phi_{1},\Phi_{1}))$
is the infinitesimal of
$((F_{t},G_{t}),(F_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}}),(\Phi_{t},\Phi_{t}))$.

∎

######  Definition 5.7.

A Lie-Yamaguti algebra morphism $\phi:L_{1}\rightarrow L_{2}$ is said to be
rigid if and only if every deformation of $\phi$ is trivial.

######  Theorem 5.8.

A non-trivial deformation of a morphism of Lie-Yamaguti algebra is equivalent
to a deformation whose $n$-infinitesimal is not a coboundary for some $n\geq
1.$

###### Proof.

Let
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
be a deformation of $\phi:L_{1}\rightarrow L_{2}$ with $n$-infinitesimal
$((f_{n},g_{n}),(f_{n}^{{}^{\prime}},g_{n}^{{}^{\prime}}),(\phi_{n},\phi_{n}))$
for some $n\geq 1.$ Assume that there exists a $(2,3)$\- cochain
$((\psi,\psi),(\psi^{{}^{\prime}},\psi^{{}^{\prime}}),(m,m))\in
C^{(2,3)}_{mLYA}(\phi,\phi)$ with
$d((\psi,\psi),(\psi^{{}^{\prime}},\psi^{{}^{\prime}}),(m,m))=((f_{n},g_{n}),(f_{n}^{{}^{\prime}},g_{n}^{{}^{\prime}}),(\phi_{n},\phi_{n})).$

Now we may assume that $(m,m)=(0,0)$ as
$d((\psi,\psi),(\psi^{{}^{\prime}},\psi^{{}^{\prime}}),(m,m))=d((\psi,\psi),(\psi^{{}^{\prime}},\psi^{{}^{\prime}})+\delta(m,m),(0,0)).$

Therefore, we have

$\displaystyle\delta_{I}(\psi)=f_{n}~{};~{}\delta_{II}(\psi)=g_{n},$
$\displaystyle\delta_{I}(\psi^{{}^{\prime}})=f_{n}^{{}^{\prime}}~{};~{}\delta_{II}(\psi^{{}^{\prime}})=g_{n}^{{}^{\prime}}~{}\mbox{and}~{}$
$\displaystyle\phi_{n}=\phi\circ\psi-\psi^{{}^{\prime}}\phi.$

let $\psi_{t}:L_{1}[[t]]\rightarrow L_{1}[[t]]$ be formal isomorphism defined
by $\psi_{t}=id_{L_{1}}+\psi t^{n}$
and
$\psi_{t}^{{}^{\prime}}:L_{2}[[t]]\rightarrow L_{2}[[t]]$ be formal
isomorphism defined by
$\psi_{t}^{{}^{\prime}}=id_{L_{2}}+\psi^{{}^{\prime}}t^{n}.$ Now, we have a
deformation
$((F_{t},G_{t}),(F_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}}),(\Phi_{t},\Phi_{t}))$
where

$\displaystyle
F_{t}(x,y)=\psi_{t}^{-1}(f_{t}(\psi_{t}(x),\psi_{t}(y))),~{}x,y\in L_{1}$ (21)
$\displaystyle
G_{t}(x,y,z)=\psi_{t}^{-1}(g_{t}(\psi_{t}(x),\psi_{t}(y),\psi_{t}(z))),~{}x,y,z\in
L_{1}$ (22) $\displaystyle
F_{t}^{{}^{\prime}}(x,y)={\psi_{t}^{{}^{\prime}}}^{-1}(f_{t}^{{}^{\prime}}(\psi_{t}^{{}^{\prime}}(x),\psi_{t}^{{}^{\prime}}(y))),~{}x,y\in
L_{2}$ (23) $\displaystyle
G_{t}^{{}^{\prime}}(x,y,z)={\psi_{t}^{{}^{\prime}}}^{-1}(g_{t}^{{}^{\prime}}(\psi_{t}^{{}^{\prime}}(x),\psi_{t}^{{}^{\prime}}(y),\psi_{t}^{{}^{\prime}}(z))),~{}x,y,z\in
L_{2}$ (24)
$\displaystyle\Phi_{t}=\psi_{t}^{{}^{\prime}}\circ\phi_{t}\circ\psi_{t}^{-1}.$
(25)

Expanding both sides of the Equation (21) and equating the coefficients of
$t^{i},~{}i\leq n$ we get $F_{i}=0,~{}1\leq i\leq n-1$ and
$F_{n}=f_{n}-\delta_{I}(\psi)=0.$
Similarly using the Equations (22)-(25) we can show that
$G_{i}=0,~{}F_{i}^{{}^{\prime}}=0,~{}G_{i}^{{}^{\prime}}=0,~{}\Phi_{i}=0$ for
all $1\leq i\leq n.$ Therefore, the given deformation
$((f_{t},g_{t}),(f_{t}^{{}^{\prime}},g_{t}^{{}^{\prime}}),(\phi_{t},\phi_{t}))$
is equivalent to
$((F_{t},G_{t}),(F_{t}^{{}^{\prime}},G_{t}^{{}^{\prime}}),(\Phi_{t},\Phi_{t}))$
with
$((F_{i},G_{i}),(F_{i}^{{}^{\prime}},G_{i}^{{}^{\prime}}),(\Phi_{i},\Phi_{i}))=0$
for $1\leq i\leq n.$ Hence, we can repeat the argument to kill off any
infinitesimal which is coboundary, so the process must stop if the deformation
is non trivial.

∎

######  Corollary 5.9.

If $H^{(2,3)}_{mLYA}(\phi,\phi)=0$, then $\phi:L_{1}\rightarrow L_{2}$ is
rigid.

##  6 Abelian extension of morphisms of Lie-Yamaguti algebras

In this section we discuss the abelian extensions of morphism of Lie-Yamaguti
algebras using the theory of abelian extension of Lie-Yamaguti algebra ([21])
and motivated by the abelian extensions of morphisms of Lie algebra ([1]). We
show that isomorphism classes of abelian extension of morphism of Lie-Yamaguti
algebra has one to one correspondence with the ‘simple’ $(2,3)$\- cohomology
group.

Note that any vector space $V$ can be regarded as a Lie-Yamaguti algebra where
$[a,b]=0,~{}\\{a,b,c\\}=0$ for all $a,b,c\in V.$ Thus, any linear operator
$\phi:V\rightarrow W$, where $V,~{}W$ are vector spaces , have a structure of
morphism of Lie-Yamaguti algebra.

Let $\phi:L_{1}\rightarrow L_{2}$ be a morphism of Lie-Yamaguti algebra with a
representation $(V,W,\psi).$ A $(2n,2n+1)$-coboundary in
$B_{mLYA}^{(2n,2n+1)}(\phi,\psi)$ is called simple if it has the form
$d_{mLYA}(\alpha,\beta,0)$ for some $(\alpha,\beta,0)\in
C_{mLYA}^{(2n-2,2n-1)}(\phi,\psi).$ Then the space of all simple
$(2n,2n+1)$-coboundaries are denoted by $B_{mLYA,s}^{(2n,2n+1)}(\phi,\psi)$
and the quotients
$H_{mLYA,s}^{(2n,2n+1)}(\phi,\psi):=\frac{Z_{mLYA}^{(2n,2n+1)}(\phi,\psi)}{B_{mLYA,s}^{(2n,2n+1)}(\phi,\psi)}$,
for $n\geq 1$ are called the simple cohomology groups of the morphism of Lie-
Yamaguti algebra $\phi:L_{1}\rightarrow L_{2}$ with coefficients in
$(V,W,\psi).$

######  Definition 6.1.

An abelian extension of a morphism of Lie-Yamaguti algebra
$\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi)$ is a linear map between the
vectors spaces $\psi:V\rightarrow W$ and is a short exact sequence of
morphisms of Lie-Yamaguti algebra

${0}$${V}$${{\hat{L_{1}}}}$${{L_{1}}}$${0}$${0}$${W}$${{\hat{L_{2}}}}$${{L_{2}}}$${0}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{\psi}$$\scriptstyle{\hat{\phi}}$$\scriptstyle{\phi}$$\scriptstyle{\bar{i}}$$\scriptstyle{\bar{p}}$

In this case we say that $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ is an
abelian extension of the morphism of Lie-Yamaguti algebra
$\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi).$

######  Definition 6.2.

Two abelian extension $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ and
$\hat{\phi^{{}^{\prime}}}:\hat{L_{1}^{{}^{\prime}}}\rightarrow\hat{L_{2}^{{}^{\prime}}}$
are said to be isomorphic if there exists an isomorphism $(\alpha,\beta)$ of
morphism Lie-Yamaguti algebra from
$\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ to
$\hat{\phi^{{}^{\prime}}}:\hat{L_{1}^{{}^{\prime}}}\rightarrow\hat{L_{2}^{{}^{\prime}}}$
so that the following diagram is commutative

${0}$${V}$${{\hat{L_{1}}}}$${{L_{1}}}$${0}$${0}$${V}$${{\hat{L_{1}^{{}^{\prime}}}}}$${{L_{1}}}$${0}$${0}$${W}$${{\hat{L_{2}}}}$${{L_{2}}}$${0}$${0}$${W}$${{\hat{L_{2}^{{}^{\prime}}}}}$${{L_{2}}}$${0}$$\scriptstyle{\beta}$$\scriptstyle{\bar{p}}$$\scriptstyle{\bar{i}}$$\scriptstyle{i}$$\scriptstyle{\alpha}$$\scriptstyle{p}$

######  Definition 6.3.

A section is a pair $(s,\bar{s})$ of linear maps
$s:L_{1}\rightarrow\hat{L_{1}}$ and $\bar{s}:L_{2}\rightarrow\hat{L_{2}}$ such
that $p\circ s=id_{L_{1}},~{}\bar{p}\circ\bar{s}=id_{L_{2}}.$

Let $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ be an abelian extension of
the morphism of Lie-Yamaguti algebra $\phi:L_{1}\rightarrow L_{2}$ by
$(V,W,\psi).$ Let $(s,\bar{s})$ be a section of this abelian extension. Now
define a linear map $\rho_{V}:L_{1}\rightarrow End(V)$ and two bilinear maps
$D_{V}:L_{1}\times L_{1}\rightarrow End(V),~{}\theta_{V}:L_{1}\times
L_{1}\rightarrow End(V)$ by

$\displaystyle\rho_{V}(x)v=[s(x),i(v)]_{\hat{L_{1}}},~{}D_{V}(x,y)v=\\{s(x),s(y),i(v)\\}_{\hat{L_{1}}}$
$\displaystyle\theta_{V}(x,y)v=\\{i(v),s(x),s(y)\\}_{\hat{L_{1}}}~{}~{}\mbox{for
all}~{}x,y\in L_{1},~{}v\in V.$

and also define a linear map $\rho_{W}:L_{2}\rightarrow End(W)$ and two
bilinear maps $D_{W}:L_{2}\times L_{2}\rightarrow
End(W),~{}\theta_{W}:L_{2}\times L_{2}\rightarrow End(W)$ by

$\displaystyle\rho_{W}(a)w=[\bar{s}(a),\bar{i}(w)]_{\hat{L_{2}}},~{}D_{W}(a,b)w=\\{\bar{s}(a),\bar{s}(b),\bar{i}(w)\\}_{\hat{L_{2}}}$
$\displaystyle\theta_{W}(a,b)w=\\{\bar{i}(w),\bar{s}(a),\bar{s}(b)\\}_{\hat{L_{2}}}~{}~{}\mbox{for
all}~{}a,b\in L_{2},~{}w\in W.$

Now with the above definitions we have the following proposition.

######  Proposition 6.4.

If $(s,\bar{s})$ is a section then $(\rho_{V},D_{V},\theta_{V})$ is a
representation of $L_{1}$ on $V$ and $(\rho_{W},D_{W},\theta_{W})$ is a
representation of $L_{2}$ on $W.$

###### Proof.

Now for any $x,y,z\in L_{1}$ and $v\in V$, we have
1.

$\displaystyle
D_{V}(x,y)v-\theta_{V}(y,x)v+\theta_{V}(x,y)v+\rho_{V}([x,y])v-\rho_{V}(x)\rho_{V}(y)v+\rho_{V}(y)\rho_{V}(x)v$
$\displaystyle=\\{s(x),s(y),i(v)\\}_{\hat{L_{1}}}-\\{i(v),s(y),s(x)\\}_{\hat{L_{1}}}+\\{i(v),s(x),s(y)\\}_{\hat{L_{1}}}$
$\displaystyle+[[s(x),s(y)]_{\hat{L_{1}}},i(v)]_{\hat{L_{1}}}-[s(x),[s(y),i(v)]_{\hat{L_{1}}}]_{\hat{L_{1}}}+[s(y),[s(x),i(v)]_{\hat{L_{1}}}]_{\hat{L_{1}}}$
$\displaystyle=\\{s(x),s(y),i(v)\\}_{\hat{L_{1}}}+\\{s(y),i(v),s(x)\\}_{\hat{L_{1}}}+\\{i(v),s(x),s(y)\\}_{\hat{L_{1}}}$
$\displaystyle+[[s(x),s(y)]_{\hat{L_{1}}},i(v)]_{\hat{L_{1}}}+[[s(y),i(v)]_{\hat{L_{1}}},s(x)]_{\hat{L_{1}}}+[[i(v),s(x)]_{\hat{L_{1}}},s(y)]_{\hat{L_{1}}}$
$\displaystyle=0~{}~{}~{}~{}~{}~{}~{}(\mbox{since}~{}\hat{L_{1}}~{}\mbox{is a
Lie-Yamaguti alebra.})$

2.

$\displaystyle D([x,y],z)v+D([y,z],x)v+D([z,x],y)v$
$\displaystyle=\\{s([x,y]_{\hat{L_{1}}}),s(z),i(v)\\}_{\hat{L_{1}}}+\\{s([y,z]_{\hat{L_{1}}}),s(x),i(v)\\}_{\hat{L_{1}}}+\\{s([z,x]_{\hat{L_{1}}}),s(y),i(v)\\}_{\hat{L_{1}}}$
$\displaystyle=\\{[s(x),s(y)]_{\hat{L_{1}}},s(z),i(v)\\}_{\hat{L_{1}}}+\\{[s(y),s(z)]_{\hat{L_{1}}},s(x),i(v)\\}_{\hat{L_{1}}}+\\{[s(z),s(x)]_{\hat{L_{1}}},s(y),i(v)\\}_{\hat{L_{1}}}$
$\displaystyle=0~{}~{}~{}~{}~{}~{}~{}(\mbox{since}~{}\hat{L_{1}}~{}\mbox{is a
Lie-Yamaguti alebra.})$

The other axioms of representation of a Lie-Yamaguti algebra $L_{1}$ can be
proved similarly as above.

Now for any $x,y\in L_{1}$ and $v\in V$ we have

$\displaystyle\psi(\rho_{V}(x)v)-\rho_{W}(\phi(x))\psi(v)$
$\displaystyle=\hat{\phi}[s(x),i(v)]_{\hat{L_{1}}}-[\bar{s}\circ\phi(x),\bar{i}\circ\psi(v)]_{\hat{L_{1}}}$
$\displaystyle=[\hat{\phi}\circ s(x)-\bar{s}\circ\phi(x),\hat{\phi}\circ
i(v)-\bar{i}\circ\psi(v)]_{\hat{L_{2}}}$
$\displaystyle=0,~{}\mbox{as}~{}\hat{\phi}\circ
s(x)-\bar{s}\circ\phi(x),~{}\hat{\phi}\circ
i(v)-\bar{i}\circ\psi(v)\in\mbox{ker}~{}(\bar{p})=\mbox{im}~{}(\bar{i}).$

Again we have,

$\displaystyle\psi(D_{V}(x,y)v)-D_{W}(\phi(x),\phi(y))\psi(v)$
$\displaystyle=\hat{\phi}\\{s(x),s(y),i(v)\\}_{\hat{L_{1}}}-\\{\bar{s}\circ\phi(x),\bar{s}\circ\phi(y),\bar{i}\circ\psi(v)\\}_{\hat{L_{2}}}$
$\displaystyle=\\{\hat{\phi}\circ s(x)-\bar{s}\circ\phi(x),\hat{\phi}\circ
s(y)-\bar{s}\circ\phi(y),\hat{\phi}\circ i(v)-\bar{i}\circ\psi(v)\\}$
$\displaystyle=0$

Similar as above we get
$\psi(\theta_{V}(x,y)v)-\theta_{W}(\phi(x),\phi(y))\psi(v)=0.$

Therefore, $(\rho_{V},D_{V},\theta_{V})$ is a representation of $L_{1}$ on
$V.$ Also by the same argument as above we get $(\rho_{W},D_{W},\theta_{W})$
is a representation of $L_{2}$ on $W.$

∎

######  Proposition 6.5.

The representation of $L_{1}$ on $V$ and the representation of $L_{2}$ on $W$
induced from a section of an abelian extension is independent of the choice of
section.

###### Proof.

Let $(s,\bar{s})$ and $(s^{{}^{\prime}},\bar{s^{{}^{\prime}}})$ be two
section. Now
$\rho_{V}(x)-\rho_{V}^{{}^{\prime}}(x)=[s(x)-s^{{}^{\prime}}(x),i(v)]_{\hat{L_{1}}}=0$
and
$\rho_{W}(a)-\rho_{W}^{{}^{\prime}}(a)=[s(a)-s^{{}^{\prime}}(a),i(w)]_{\hat{L_{2}}}=0,$
since $s(x)-s^{{}^{\prime}}(x)\in\mbox{ker}~{}p=\mbox{im}~{}i$ and
$\bar{s}(a)-\bar{s^{{}^{\prime}}}(a)\in\mbox{ker}~{}\bar{p}=\mbox{im}~{}\bar{i}$
for all $x\in L_{1},~{}a\in L_{2},~{}v\in V,~{}w\in W.$

Similarly, $D_{V}(x,y)v-D_{V}^{{}^{\prime}}(x,y)v=0$,
$D_{W}(a,b)w-D_{W}^{{}^{\prime}}(a,b)w=0$ and
$\theta(x,y)v-\theta^{{}^{\prime}}(x,y)v=0$,
$\theta_{W}(a,b)w-\theta_{W}^{{}^{\prime}}(a,b)w=0$ for all $x,y\in
L_{1},~{}a,b\in L_{2},~{}v\in V,~{}w\in W.$ Therefore, the representation
induced by the sections $(s,\bar{s})$ and
$(s^{{}^{\prime}},\bar{s^{{}^{\prime}}})$ are same. ∎

Let $\phi:L_{1}\rightarrow L_{2}$ be a morphism of Lie-Yamaguti algebra with a
representation $(V,W,\psi)$. Now define $Ext(\phi,\psi)$ be the isomorphism
classes of the abelian extensions of $\phi:L_{1}\rightarrow L_{2}$ by the
representation $(V,W,\psi)$. It is clear from the Proposition (6.4)
$Ext(\phi,\psi)$ is non empty.

######  Lemma 6.6.

Every abelian extension of $\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi)$ gives
a cohomology class in $H^{(2,3)}_{mLYA,s}(\phi,\psi).$

###### Proof.

Let $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ be an abelian extension of
$\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi)$. Let $(s,\bar{s})$ be a section.
Now define the maps

$\alpha_{I}:L_{1}\times L_{1}\rightarrow V,~{}\beta:L_{2}\times
L_{2}\rightarrow W,~{}\gamma:L_{1}\rightarrow W$ by

$\displaystyle\alpha_{I}(x,y):=[s(x),s(y)]_{\hat{L_{1}}}-s([x,y]_{L_{1}}),$
$\displaystyle\beta_{I}(a,b):=[\bar{s}(a),\bar{s}(b)]_{\hat{L_{2}}}-\bar{s}([a,b]_{L_{2}})~{}~{}~{}\mbox{and}$
$\displaystyle\gamma(x):=\hat{\phi}(s(x))-\bar{s}(\phi(s)),~{}~{}~{}\mbox{for
all }~{}x,y\in L_{1},~{}a,b\in L_{2}.$

and $\alpha_{II}:L_{1}\times L_{1}\times L_{1}\rightarrow
V,~{}\beta:L_{2}\times L_{2}\times L_{2}\rightarrow W$ by

$\displaystyle\alpha_{II}(x,y,z):=\\{s(x),s(y),s(z)\\}_{\hat{L_{1}}}-s(\\{x,y,z\\}_{L_{1}})$
$\displaystyle\beta_{II}(a,b,c):=\\{\bar{s}(a),\bar{s}(b),\bar{s}(c)\\}_{\hat{L_{2}}}-\bar{s}(\\{a,b,c\\}_{L_{2}})~{}~{}~{}$

Therefore,
$((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{II}),(\gamma,\gamma))\in
C^{(2,3)}_{mLYA}(\phi,\psi).$ By the theory of abelian extension of Lie
Yamaguti algebra ([21]) we have
$\delta(\alpha_{I},\alpha_{II})=0,\delta(\beta_{I},\beta_{II})=0$ and a direct
computation shows that
$\phi\circ(\alpha_{I},\alpha_{II})-(\beta_{I},\beta_{II})\phi-\delta(\gamma,\gamma)=0.$
Hence, $((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{II}),(\gamma,\gamma))\in
Z^{(2,3)}_{mLYA}(\phi,\psi)$. Therefore,
$((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{II}),(\gamma,\gamma))$
corresponds to a cohomology class in $H^{(2,3)}_{mLYA,s}(\phi,\psi).$

∎

######  Lemma 6.7.

The cohomology class induced from an abelian extension of
$\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi)$ does not depend on the choice of
section.

###### Proof.

Let $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ be an abelian extension of
a morphism of Lie-Yamaguti algebra $\phi:L_{1}\rightarrow L_{2}$ by
$(V,W,\psi).$ Let $(s,\bar{s})$ and $(s^{{}^{\prime}},\bar{s^{{}^{\prime}}})$
be two section of it. Now define $\eta:L_{1}\rightarrow V$ by
$\eta(x)=s(x)-s^{{}^{\prime}}(x)$ for all $x\in L_{1}$ and
$\eta^{{}^{\prime}}:L_{2}\rightarrow W$ by
$\eta^{{}^{\prime}}(a)=\bar{s}(a)-\bar{s^{{}^{\prime}}}(a)$ for all $a\in
L_{2}.$

Now,

$\displaystyle\alpha_{I}(x,y)$
$\displaystyle=[s(x),s(y)]_{\hat{L_{1}}}-s([x,y]_{L_{1}})$
$\displaystyle=[s^{{}^{\prime}}(x)+\eta(x),s^{{}^{\prime}}(y)+\eta(y)]_{\hat{L_{1}}}-(s^{{}^{\prime}}([x,y])+\eta([x,y]))$
$\displaystyle=[s^{{}^{\prime}}(x),s^{{}^{\prime}}(y)]_{\hat{L_{1}}}-s^{{}^{\prime}}([x,y]_{L_{1}})+[\eta(x),s^{{}^{\prime}}(y)]_{\hat{L_{1}}}+[s^{{}^{\prime}}(x),\eta(y)]_{\hat{L_{1}}}-\eta([x,y]_{L_{1}})~{}(\mbox{as}~{}[\eta(x),\eta(y)]=0)$
$\displaystyle=\alpha_{I}^{{}^{\prime}}(x,y)+\delta_{I}(\eta)(x,y)$

Similarly we have,
$\alpha_{II}(x,y,z)=\alpha^{{}^{\prime}}(x,y,z)+\eta_{II}(\eta)(x,y,z)$
Therefore,
$(\alpha_{I},\alpha_{II})-(\alpha^{{}^{\prime}}_{I},\alpha^{{}^{\prime}}_{II})=(\delta_{I}(\eta),\delta_{II}(\eta))\in
Z^{(2,3)}(L_{1},V).$
Similarly using $\eta^{{}^{\prime}}$ we will get
$(\beta_{I},\beta_{II})-(\beta_{I}^{{}^{\prime}}\beta_{II}^{{}^{\prime}})\in
Z^{(2,3)}(L_{2},W)$ and
$(\gamma,\gamma)-(\gamma^{{}^{\prime}},\gamma^{{}^{\prime}})\in Z(L_{1},W).$
Therefore, $((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{I}I),(\gamma,\gamma))$
and
$((\alpha_{I}^{{}^{\prime}},\alpha_{II}^{{}^{\prime}}),(\beta_{I}^{{}^{\prime}},\beta_{I}I^{{}^{\prime}}),(\gamma^{{}^{\prime}},\gamma^{{}^{\prime}}))$
are in the same cohomology class in $H_{mLYA,s}^{(2,3)}(\phi,\psi).$

∎

######  Theorem 6.8.

Isomorphic abelian extensions of $\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi)$
gives rise to the same element in $H^{(2,3)}_{mLYA,s}(\phi,\psi)$.

###### Proof.

Let $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ and
$\hat{\phi^{{}^{\prime}}}:\hat{L_{1}^{{}^{\prime}}}\rightarrow\hat{L_{2}^{{}^{\prime}}}$
be two isomorphic abelian extensions and let $(\alpha,\beta)$ be the
isomorphism between them. Now suppose $(s,\bar{s})$ be a section of of the
abelian extension $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}.$ Therefore,
$p^{{}^{\prime}}\circ(\alpha\circ s)=p\circ s=id_{L_{1}}$ and
$\bar{p^{{}^{\prime}}}\circ(\beta\circ\bar{s})=\bar{p}\circ\bar{s}=id_{L_{2}}.$
Thus, $(\alpha\circ s,\beta\circ\bar{s})$ is a section of
$\hat{\phi^{{}^{\prime}}}:\hat{L_{1}^{{}^{\prime}}}\rightarrow\hat{L_{2}^{{}^{\prime}}}$.
Now let
$((\alpha_{I}^{{}^{\prime}},\alpha_{II}^{{}^{\prime}}),(\beta_{I}^{{}^{\prime}},\beta_{I}I^{{}^{\prime}}),(\gamma^{{}^{\prime}},\gamma^{{}^{\prime}}))\in
Z^{(2,3)}_{mLYA}$ be the $(2,3)$\- cocycle corresponding to the abelian
extension
$\hat{\phi^{{}^{\prime}}}:\hat{L_{1}^{{}^{\prime}}}\rightarrow\hat{L_{2}^{{}^{\prime}}}$
with section $(\alpha\circ s,\beta\circ\bar{s}).$ Then we have

$\displaystyle\alpha_{I}^{{}^{\prime}}(x,y)$ $\displaystyle=[\alpha\circ
s(x),\alpha\circ s(y)]_{\hat{L_{1}}}-\alpha\circ s([x,y]_{L_{1}})$
$\displaystyle=\alpha([s(x),s(y)]_{\hat{L_{1}}}-s([x,y]_{L_{1}}))$
$\displaystyle=\alpha_{I}(x,y)~{}~{}(\mbox{as}~{}\alpha_{I}|_{V}=id_{V}),~{}\mbox{for
all}~{}x,y\in L_{1}.$

Similarly, we will get $\alpha_{II}^{{}^{\prime}}(x,y,z)=\alpha_{II}(x,y,z)$
for all $x,y,z\in L_{1}$;
$\beta_{I}^{{}^{\prime}}(a,b)=\beta_{I}(a,b)$ and
$\beta_{II}^{{}^{\prime}}(a,b,c)=\beta_{II}(a,b,c)$ for all $a,b,c\in L_{2}$
and $\gamma^{{}^{\prime}}(x)=\gamma(x)$ for all $x\in L_{1}.$ Hence,
$((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{I}I),(\gamma,\gamma))=((\alpha_{I}^{{}^{\prime}},\alpha_{II}^{{}^{\prime}}),(\beta_{I}^{{}^{\prime}},\beta_{I}I^{{}^{\prime}}),(\gamma^{{}^{\prime}},\gamma^{{}^{\prime}})).$

Thus, isomorphic abelian extension gives rise to the same element in
$H^{(2,3)}_{mLYA,s}(\phi,\psi)$.

∎

######  Theorem 6.9.

Every element in $H^{(2,3)}_{mLYA,s}(\phi,\psi)$ induces an abelian extension
of $\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi)$.

###### Proof.

Let $((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{I}I),(\gamma,\gamma))\in
Z^{(2,3)}_{mLYA}(\phi,\psi)$ be a $(2,3)$\- cocycle. Now define
$\hat{L_{1}}:=L_{1}\oplus V,~{}\hat{L_{2}}=L_{2}\oplus W$ and define
$\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ by
$\hat{\phi}(x,v)=(\phi(x),\psi(v)+\gamma(x))$, for all $(x,v)\in\hat{L_{1}}.$
Now define the bilinear and trilinear brackets on $\hat{L_{1}}$ and
$\hat{L_{2}}$ by

$\displaystyle[(x,v_{1}),(y,v_{2})]_{\hat{L_{1}}}:=([x,y]_{L_{1}},\rho_{V}(x)v_{2}-\rho_{V}(y)v_{1}+\delta_{I}(x,y))$
$\displaystyle\\{(x,v_{1}),(y,v_{2}),(z,v_{3})\\}_{\hat{L_{1}}}:=(\\{x,y,z\\}_{L_{1}},\theta_{V}(y,z)v_{1}-\theta_{V}(x,z)v_{2}+D_{V}(x,y)v_{3}+\delta_{II}(x,y,z))$
$\displaystyle\mbox{for all}~{}x,y,z\in L_{1},~{}v_{1},v_{2},v_{3}\in V.$ and
$\displaystyle[(a,w_{1}),(b,w_{2})]_{\hat{L_{2}}}:=([a,b]_{L_{2}},\rho_{V}(a)w_{2}-\rho_{V}(b)w_{1}+\delta_{I}(a,b))$
$\displaystyle\\{(a,w_{1}),(b,w_{2}),(c,w_{3})\\}_{\hat{L_{2}}}:=(\\{a,b,c\\}_{L_{2}},\theta_{V}(b,c)w_{1}-\theta_{V}(a,c)w_{2}+D_{V}(a,b)w_{3}+\delta_{II}(a,b,c))$
$\displaystyle\mbox{for all}~{}a,b,c\in L_{2},~{}w_{1},w_{2},w_{3}\in W.$

Then $\hat{L_{1}}$, $\hat{L_{2}}$ are Lie-Yamaguti algebra with the above
defined brackets (for proof see [21]). It is easy to show that
$\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ is a morphisms of Lie-Yamaguti
algebra and $\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ is an abelian
extension of $(L_{1},L_{2},\phi)$ by $(V,W,\psi)$ (For details see [21]). ∎

######  Lemma 6.10.

Cohomologous elements in $H^{(2,3)}_{mLYA,s}(\phi,\psi)$ gives isomorphic
abelian extensions of $\phi:L_{1}\rightarrow L_{2}$ by $(V,W,\psi)$.

###### Proof.

Let $((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{I}I),(\gamma,\gamma))$,
$((\alpha_{I}^{{}^{\prime}},\alpha_{II}^{{}^{\prime}}),(\beta_{I}^{{}^{\prime}},\beta_{I}I^{{}^{\prime}}),(\gamma^{{}^{\prime}},\gamma^{{}^{\prime}}))\in
Z^{(2,3)}_{mLYA}(\phi,\psi)$ such that

$\displaystyle((\alpha_{I},\alpha_{II}),(\beta_{I},\beta_{I}I),(\gamma,\gamma))-((\alpha_{I}^{{}^{\prime}},\alpha_{II}^{{}^{\prime}}),(\beta_{I}^{{}^{\prime}},\beta_{I}I^{{}^{\prime}}),(\gamma^{{}^{\prime}},\gamma^{{}^{\prime}}))$
$\displaystyle=d_{mLYA}((\xi,\xi),(\xi^{{}^{\prime}},\xi^{{}^{\prime}}),(0,0))~{}\mbox{for
some}~{}\xi\in C^{1}(L_{1},V),~{}\xi^{{}^{\prime}}\in C^{2}(L_{2},W).$

Then the corresponding induced abelian extensions
$\hat{\phi}:\hat{L_{1}}\rightarrow\hat{L_{2}}$ and
$\hat{\phi^{{}^{\prime}}}:\hat{L_{1}^{{}^{\prime}}}\rightarrow\hat{L_{2}^{{}^{\prime}}}$
respectively are isomorphic via the map $(\alpha,\beta)$ where
$\alpha:\hat{L_{1}}\rightarrow\hat{L_{1}{{}^{\prime}}}$ and
$\beta:\hat{L_{2}}\rightarrow\hat{L_{2}^{{}^{\prime}}}$ defined by
$\alpha(x,v)=(x,v+\xi(x))$, $\beta(a,w)=(a,w+\xi^{{}^{\prime}}(a))$, $x\in
L_{1},~{}a\in L_{2},~{}v\in V,~{}w\in W$ respectively.

∎

From the above Lemmas and Theorems, we have the following main result of this
section:

######  Theorem 6.11.

Let $\phi:L_{1}\rightarrow L_{2}$ be a morphism of Lie-Yamaguti algebra with a
representation $(V,W,\psi)$. Then there exists a one-to-one correspondence
between $Ext(\phi,\psi)$ and the $(2,3)$-cohomology group
$H^{(2,3)}_{mLYA,s}(\phi,\psi).$

##  7 Discussion about the study of a more general problem

Suppose $\bf{LYA}$ is the category of all Lie-Yamaguti algebras over a field
$k$. Let $I=\\{i,j,k,\ldots\\}$ be a partially ordered set. One can think of
$I$ as a category whose set of objects is $I$ itself and there is a unique
morphism $i\to j$, whenever $i\leq j$. A diagram of Lie-Yamaguti algebras over
$I$ is a contravariant functor $\mathfrak{L}:I\to\bf{LYA}$. The diagram is
called finite if $I$ is a finite set. In this paper, we have considered a
special case of this general form with $I=\\{i,j\\}$. Therefore, in this
paper, our diagram is nothing but a single morphism of Lie-Yamaguti algebras.
One can study the cohomology and deformation theory of a diagram of Lie-
Yamaguti algebras generalizing our case of single morphism and following [8].

Acknowledgements: The second/corresponding author is supported by the Science
and Engineering Research Board (SERB), Department of Science and Technology
(DST), Govt. of India (Grant Number- CRG/2022/005332).

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