# LIE-YAMAGUTI ALGEBRA BUNDLE

,  Saikat Goswami<EMAIL_ADDRESS>RKMVERI, Belur, Howrah-711202,
India.  TCG Centres for Research and Education in Science and Technology,
Institute for Advancing Intelligence, Sector V, Salt lake, Kolkata 700091,
INDIA.  and  Goutam Mukherjee<EMAIL_ADDRESS><EMAIL_ADDRESS>TCG Centres for Research and Education in Science and
Technology, Institute for Advancing Intelligence, Sector V, Salt lake, Kolkata
700091, INDIA.

###### Abstract.

We introduce the notion of Lie-Yamaguti algebra bundle, define its cohomology
groups with coefficients in a representation and show that such bundles
appeared naturally from geometric considerations in the work of M. Kikkawa,
which motivates us to introduce this object in the proper mathematical
framework. We also study abelian extensions of Lie-Yamaguti algebra bundles
and investigate their relationship with suitable cohomology group.

###### Key words and phrases:

Vector bundle, Lie-Yamaguti algebra, Non-associative algebra, Cohomology.

###### 2020 Mathematics Subject Classification:

53B05, 58A05, 16E99, 17A30, 17A40

##  1\. Introduction

Triple systems in algebra may be traced back to the works of P. Jordan, J. v.
Neumann and E. Wigner [9] in quantum mechanics, and N. Kemmer [10, 11] in
particle physics. The notion of Lie triple system was formally introduced as
an algebraic object by Jacobson [8] in connection with problems which arose
from quantum mechanics.

Nomizu [18] proved that affine connections with parallel torsion and curvature
are locally equivalent to invariant connections on reductive homogeneous
spaces, and that each such space has a canonical connection for which parallel
translation along geodesics agrees with the natural action of the group.

Let $M$ be a smooth manifold equipped with a linear connection $\nabla.$ Let
$e\in M$ be a given fixed point. Then there is a local multiplication $\mu$ at
$e$ compatible with $\nabla$, which is given by

$\mu(x,y)=exp_{x}\circ\tau_{e,x}\circ exp_{e}^{-1}(y),$

where $exp_{x}$ denotes the exponential mapping at $x$ and $\tau_{e,x}$
denotes the parallel displacement of tangent vectors along the geodesic
joining $e$ to $x$ in a normal neighbourhood of $e$ [13].

If $M$ is a reductive homogeneous space $A/K$ with the canonical connection,
due to K. Nomizu, then the local multiplication $\mu$ given above satisfies
some special property (cf. [18]). In particular, if M is a Lie group $A$
itself, then the canonical connection is reduced to the connection of [3] and
the local multiplication $\mu$ coincides with the multiplication of $A$ in
local.

Motivated by this fact, M. Kikkawa [13] investigated the problem of the
existence of a global differentiable binary system on a reductive homogeneous
space $A/K,$ which coincides locally with the above geodesic local
multiplication $\mu$ and observed that the problem is related to the canonical
connection and to the general Lie triple system defined on the tangent space
$T_{e}M.$ In this paper, Kikkawa renamed the notion of general Lie triple
system as Lie triple algebra. Kinyon and Weinstein [15] observed that Lie
triple algebras, which they called Lie-Yamaguti algebras in their paper, can
be constructed from Leibniz algebras. Leibniz algebras are non anti-symmetric
analogue of Lie algebras introduced by J. L. Loday [16].

Organization of the paper: In §2, we set up notations, recall some known
definitions and results. In §3, we introduce the main object of study of the
present paper, namely, the notion of a Lie-Yamaguti algebra bundle, illustrate
examples of such bundles and describe a general method of constructing such
bundles. In §4, we introduce the concept of representation of Lie-Yamaguti
algebra bundles which is required to introduce cohomology groups of Lie-
Yamaguti algebra bundles. In §5, we define cohomology groups of a Lie-Yamaguti
algebra bundle with coefficients in a given representation. Finally, in §6, we
study (abelian) extensions of Lie-Yamaguti algebra bundles and establish its
connection to cohomology.

##  2\. Preliminaries

The aim of this section is to recall some basic definitions and set up
notations to be followed throughout the paper. Let $\mathbb{K}$ be a given
field.

######  2.1 Definition.

A Lie algebra is a vector space $\mathfrak{g}$ over $\mathbb{K}$ equipped with
a $\mathbb{K}$-bilinear operation
$[~{},~{}]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying

  1. (1)

(Anti-symmetry): $[x,y]=-[y,x]$ for all $x,y\in\mathfrak{g}$;

  2. (2)

(Jacobi identity): $[[x,y],z]+[[y,z],x]+[[z,x],y]=0$ for all
$x,y,z\in\mathfrak{g}.$

######  2.2 Definition.

A Leibniz algebra is a vector space $\mathfrak{g}$ over $\mathbb{K}$ equipped
with a $\mathbb{K}$-bilinear operation
$\cdot:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying the
Leibniz identity

$x\cdot(y\cdot z)=(x\cdot y)\cdot z+y\cdot(x\cdot z)$

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

It is easy to see that in presence of the anti-symmetric condition the Leibniz
identity reduces to Jacobi identity. Thus, Lie algebras are examples of
Leibniz algebras. See [16] for many other non-trivial examples of Leibniz
algebras.

######  2.3 Definition.

A Lie triple system is a vector space $\mathfrak{g}$ over $\mathbb{K}$
equipped with a $\mathbb{K}$-trilinear operation

$\\{~{},~{},~{}\\}:\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$

satisfying

  1. (1)

$\\{x,y,z\\}=-\\{y,x,z\\}$ for all $x,y,z\in\mathfrak{g}$;

  2. (2)

$\\{x,y,z\\}+\\{y,z,x\\}+\\{z,x,y\\}=0$ for all $x,y,z\in\mathfrak{g}$;

  3. (3)

$\\{x,y,\\{u,v,w\\}\\}=\\{\\{x,y,u\\},v,w\\}+\\{u,\\{x,y,v\\},w\\}+\\{u,v,\\{x,y,w\\}\\}$
for all $x,y,u,v,w\in\mathfrak{g}.$

The following is an interesting example of a Lie triple system which arose
from Physics [8].

######  2.4 Example.

We denote by $M_{n+1}(\mathbb{R})$, the set of all $(n+1)\times(n+1)$ matrices
over the field $\mathbb{R}$, which is an associative algebra with respect to
matrix multiplication. Let $\delta_{ij}$ denote the Kronecker delta symbol

$\delta_{ij}=\left\\{\begin{array}[]{ll}0&i\neq j\\\ 1&i=j\end{array}\right.$

and $e_{i,j}$ denote the elementary matrix which has $1$ in the $(i,j)$-entry
as its only non-zero entry. Let $\mathfrak{m}$ be the subspace of
$M_{n+1}(\mathbb{R})$ spanned by the matrices $G_{i}$ for $i=1,2,\cdots,n$,
where $G_{i}=e_{i,n+1}-e_{n+1,i}.$ As an example, for $n=3,$ the matrix
$G_{2}\in M_{4}(\mathbb{R})$ is given by

$G_{2}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\ 0&-1&0&0\\\
\end{pmatrix}.$

Then, the subspace $\mathfrak{m}$ is closed under the ternary product

$\\{A,B,C\\}:=[[A,B],C],~{}~{}A,B,C\in\mathfrak{g}$

where $[A,B]:=AB-BA$ is the commutator bracket. Explicitly, the trilinear
product of the basis elements are given by

$[[G_{i},G_{j}],G_{k}]=\delta_{ki}G_{j}-\delta_{kj}G_{i}.$

It turns out that $(\mathfrak{m},\\{~{},~{},~{}\\})$ is a Lie triple system,
first used in [6] to provide a significant and elegant algebraic formalism of
Meson equations. It was introduced formally as a Lie triple system by Jacobson
[8] and hence known as Meson field.

######  2.5 Remark.

Note that any Lie algebra $(\mathfrak{g},[~{},~{}])$ can be viewed as a Lie
triple system with the trilinear operation

$\\{x,y,z\\}:=[[x,y],z]$

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

######  2.6 Definition.

A Lie-Yamaguti Algebra $(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$ is a
vector space $\mathfrak{g}$ equipped with a $\mathbb{K}$-bilinear operation

$[~{},~{}]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$

and a $\mathbb{K}$-trilinear operation

$\\{~{},~{},~{}\\}:\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$

such that for all $x,y,z,u,v,w\in\mathfrak{g}$ the following relations hold:

(LY1) $[x,y]=-[y,x];$ (LY2) $\\{x,y,z\\}=-\\{y,x,z\\};$ (LY3)
$\Sigma_{\circlearrowleft(x,y,z)}([[x,y],z]+\\{x,y,z\\})=0;$ (LY4)
$\Sigma_{\circlearrowleft(x,y,z)}\\{[x,y],z,u\\}=0;$ (LY5)
$\\{x,y,[u,v]\\}=[\\{x,y,u\\},v]+[u,\\{x,y,v\\}];$ (LY6)
$\\{x,y,\\{u,v,w\\}\\}\\\
=\\{\\{x,y,u\\},v,w\\}+\\{u,\\{x,y,v\\},w\\}+\\{u,v,\\{x,y,w\\}\\}.$

Here, $\Sigma_{\circlearrowleft(x,y,z)}$ denotes the sum over cyclic
permutations of x, y, and z.

######  2.7 Remark.

Notice that if the trilinear product in a Lie-Yamaguti algebra is trivial,
that is, if $\\{~{},~{},~{}\\}=0,$ then (LY$2$), (LY$4$), (LY$5$), and (LY$6$)
are trivial, and (LY$1$) and (LY$3$) define a Lie algebra structure on
$\mathfrak{g}$. On the other hand, if the binary product is trivial, that is,
$[~{},~{}]=0,$ then (LY$1$), (LY$4$), and (LY$5$) are trivial, and (LY$2$),
(LY$3$), together with (LY$6$) define a Lie triple system on $\mathfrak{g}.$

The following result is well-known.

######  2.8 Lemma.

Let $(\mathfrak{g},[~{},~{}])$ be a Lie algebra over $\mathbb{K}$. 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 $a,b,c\in\mathfrak{g}$.

######  2.9 Example.

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

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

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

Let $(\mathfrak{g},\langle~{},~{}\rangle)$ be a Lie algebra. Recall that a
reductive decomposition of $\mathfrak{g}$ is a vector space direct sum
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ satisfying
$\langle\mathfrak{h},\mathfrak{h}\rangle\subseteq\mathfrak{h}$ and
$\langle\mathfrak{h},\mathfrak{m}\rangle\subseteq\mathfrak{m}.$ In this case,
we call $(\mathfrak{h},\mathfrak{m})$ a reductive pair.

######  2.10 Example.

Let $(\mathfrak{g},\langle~{},~{}\rangle)$ be a Lie algebra with a reductive
decomposition $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}.$ Then, there exist
a natural binary and a ternary product on $\mathfrak{m}$ defined by

$[a,b]:=\pi_{\mathfrak{m}}(\langle
a,b\rangle),~{}~{}\\{a,b,c\\}:=\langle\pi_{\mathfrak{h}}(\langle
a,b\rangle),c\rangle,$

where $\pi_{\mathfrak{m}}$ and $\pi_{\mathfrak{h}}$ are the projections on
$\mathfrak{m}$ and $\mathfrak{h}$, respectively. These products endow
$\mathfrak{m}$ with the structure of a Lie-Yamaguti algebra [2].

######  2.11 Example.

Consider the vector space $\mathfrak{g}$ over $\mathbb{K}$ generated by
$\\{e_{1},e_{2},e_{3}\\}.$ Define a bilinear operation $[~{},~{}]$ and a
trilinear operation $\\{~{},~{},~{}\\}$ on $\mathfrak{g}$ as follows.

$[e_{1},e_{2}]=e_{3};~{}~{}\\{e_{1},e_{2},e_{1}\\}=e_{3}.$

All other brackets of the basis elements are either determined by the
definition of Lie-Yamaguti algebra or else are zero. Then, $\mathfrak{g}$ with
the above operations is a Lie-Yamaguti algebra.

See [1] for classification of some low dimensional Lie-Yamaguti algebras.

######  2.12 Definition.

Let $(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$,
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime},\\{~{},~{},~{}\\}^{\prime})$ be two
Lie-Yamaguti algebras. A homomorphism

$\phi:(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})\rightarrow(\mathfrak{g}^{\prime},[~{},~{}]^{\prime},\\{~{},~{},~{}\\}^{\prime})$

of Lie-Yamaguti algebras is a $\mathbb{K}$-linear map
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ satisfying

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

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

A homomorphism

$\phi:(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})\rightarrow(\mathfrak{g}^{\prime},[~{},~{}]^{\prime},\\{~{},~{},~{}\\}^{\prime})$

of Lie-Yamaguti algebras is an isomorphism if there exists a homomorphism

$\phi^{\prime}:(\mathfrak{g}^{\prime},[~{},~{}]^{\prime},\\{~{},~{},~{}\\}^{\prime})\rightarrow(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$

such that $\phi^{\prime}\circ\phi=id_{\mathfrak{g}}$ and
$\phi\circ\phi^{\prime}=id_{\mathfrak{g}^{\prime}}.$ The set of all self-
isomorphisms of a Lie-Yamaguti algebra
$(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$ is obviously a group under
composition of maps and is denoted by ${Aut}_{LY}(\mathfrak{g}).$

The notion of Lie algebra bundle was introduced in [5]. For smooth Lie algebra
bundle we refer [17]. Other notions of algebra bundles are available in the
literature and appeared in various context.

Let $M$ be a smooth manifold (Hausdorff and second countable, hence,
paracompact). Let $C^{\infty}(M)$ be the algebra of smooth functions on $M$.
Let $TM$ be the tangent bundle of $M$. Recall that a vector field on $M$ is a
smooth section of the tangent bundle $TM.$ Let us denote the space of vector
fields on $M$ by $\chi(M).$ It is well-known that $\chi(M)$ is a
$C^{\infty}(M)$-module. Moreover, $\chi(M)$ is a Lie algebra with the
commutator bracket:

$[\alpha,\beta]:=\alpha\beta-\beta\alpha$

for $\alpha,\beta\in\chi(M).$ Here, for $\alpha,\beta\in\chi(M)$ and $p\in M,$
the action of $\alpha\beta(p)$ on a smooth function $f\in C^{\infty}(M)$ is
given by

$\alpha\beta(p)(f)=\alpha_{p}(\beta f),$

where $\beta f\in C^{\infty}(M)$ is given by $\beta
f(m)=\beta_{m}(f),~{}~{}m\in M.$

For a (smooth) vector bundle $p:L\rightarrow M,$ often denoted by
$\xi=(L,p,M),$ we denote the space of smooth sections of $L$ by $\Gamma L.$ It
is well-known that $\Gamma L$ is a $C^{\infty}(M)$-module. For any $m\in M,$
we denote the fibre of the vector bundle $\xi$ over $m$ by $L_{m}$ or
sometimes by $\xi_{m}.$

Henceforth, we will work in the smooth category and with
$\mathbb{K}=\mathbb{R}.$

######  2.13 Definition.

Let $(L,p,M)$ be a vector bundle and let $[~{},~{}]$ be a section of the
bundle $Alt^{2}(L)$ such that for each $m\in M,$

$[~{},~{}]_{m}:L_{m}\times L_{m}\rightarrow L_{m}$

is a Lie algebra bracket on $L_{m}.$ We call such a section a field of Lie
algebra brackets in $L.$

######  2.14 Definition.

A Lie algebra bundle is a vector bundle $(L,p,M)$ together with a field of Lie
algebra brackets

$m\mapsto[~{},~{}]_{m},~{}m\in M.$

Thus, for a Lie algebra bundle $(L,p,M),$ each fibre $L_{m}$ is a Lie algebra
which varies smoothly as $m\in M$ varies over $M.$ In other words, the
assignment $m\mapsto[~{},~{}]_{m},~{}~{}m\in M$ is smooth.

######  2.15 Definition.

Let $\mathfrak{g}$ be a given Lie algebra. A locally trivial Lie algebra
bundle with fibre $\mathfrak{g}$ is a vector bundle $(L,p,M)$ together with a
field of Lie algebra brackets

$m\mapsto[~{},~{}]_{m},~{}~{}m\in M$

such that $M$ admits an open covering $\\{U_{i}\\}$ equipped with local
trivializations $\\{\psi_{i}:U_{i}\times\mathfrak{g}\rightarrow
p^{(-1)}(U_{i})\\}$ for which each $\psi_{i,m},~{}~{}m\in M$ ($\psi_{i}$
restricted to each fibre $L_{m}$) is a Lie algebra isomorphism.

A homomorphism $\phi:(L,p,M)\rightarrow(L^{\prime},p^{\prime},M^{\prime})$ of
Lie algebra bundles is a vector bundle morphism $(\phi,\phi_{0}),$ where
$\phi_{0}:M\rightarrow M^{\prime}$ such that $\phi|_{L_{m}}:L_{m}\rightarrow
L^{\prime}_{\phi_{0}(m)},~{}m\in M$ is a Lie algebra homomorphism.

##  3\. Lie-Yamaguti Algebra Bundle

In this section we introduce the notion of Lie-Yamuguti algebra bundle and
related results. All vector bundles and vector bundle maps are assumed to be
smooth and $\mathbb{K}=\mathbb{R}.$

######  3.1 Definition.

Let $\xi=(L,p,M)$ be a (real) vector bundle. Let
$\langle~{},\cdots,~{}\rangle$ be a section of the bundle $Hom(\xi^{\otimes
k},\xi).$ We call such a section a $k$-field of ($\mathbb{K}$-multinear)
brackets in $\xi.$ Thus, a $k$-field of brackets in $\xi$ is a smooth
assignment

$m\mapsto(\langle~{},\cdots,~{}\rangle_{m}:\xi_{m}\times\cdots\times\xi_{m}\rightarrow\xi_{m})$

of multilinear operation on $\xi_{m}$, $m\in M.$

######  3.2 Definition.

A Lie-Yamaguti algebra bundle is a vector bundle $\xi=(L,p,M)$ together with a
$2$-field of brackets

$m\mapsto[~{},~{}]_{m},~{}~{}m\in M$

and a $3$-field of brackets

$m\mapsto\\{~{},~{},~{}\\}_{m},~{}~{}m\in M$

which make each fibre $\xi_{m},$ $m\in M$ a Lie-Yamaguti algebra.

######  3.3 Definition.

Let $(\mathfrak{g},[~{},~{}]_{\mathfrak{g}},\\{~{},~{},~{}\\}_{\mathfrak{g}})$
be a given Lie-Yamaguti algebra. A locally trivial Lie-Yamaguti algebra bundle
is a vector bundle $\xi=(L,p,M)$ together with a $2$-field of brackets

$m\mapsto[~{},~{}]_{m},~{}~{}m\in M$

and a $3$-field of brackets

$m\mapsto\\{~{},~{},~{}\\}_{m},~{}~{}m\in M$

such that $M$ admits an open covering $\\{U_{i}\\}$ equipped with local
trivializations $\\{\psi_{i}:U_{i}\times\mathfrak{g}\rightarrow
p^{(-1)}(U_{i})\\}$ for which each $\psi_{i,m},~{}~{}m\in M$ ($\psi_{i}$
restricted to each fibre $\xi_{m}$) is a Lie-Yamaguti algebra isomorphism.

######  3.4 Remark.

Thus, for a Lie-Yamagutil algebra bundle as defined above each fibre
$\xi_{m}=p^{-1}(m),~{}~{}m\in M,$ together with the binary operation
$[~{},~{}]_{m}$ and the ternary operation $\\{~{},~{},~{}\\}_{m}$ is a Lie-
Yamaguti algebra isomorphic to $\mathfrak{g},$ and the assignments

$m\mapsto[~{},~{}]_{m},~{}~{}m\mapsto\\{~{},~{},~{}\\}_{m}$

varies smoothly over $M.$

In other words, a Lie-Yamaguti algebra bundle over $M$ is a vector bundle over
$M$ such that each fibre of the bundle has a Lie-Yamaguti algebra structure
isomorphic to $\mathfrak{g}.$

An obvious example of a Lie-Yamaguti algebra bundle is the trivial bundle over
a smooth manifold $M$ with fibres a Lie-Yamaguti algebra.

######  3.5 Example.

Let $(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$ be a given Lie-Yamaguti
algebra and $M$ be any smooth manifold. Then the trivial vector bundle
$\xi=M\times\mathfrak{g}$ with the projection onto the first factor
$\pi_{1}:M\times\mathfrak{g}\to M$ is a Lie-Yamaguti algebra bundle, called
the product Lie-yamaguti algebra bundle.

We have the following example from the Lemma 2.8.

######  3.6 Example.

Any Lie algebra bundle $(L,p,M,[~{},~{}])$ is a Lie-Yamaguti algebra bundle,
where the $3$-field of brackets on $M$ induced by the $2$-field of Lie
brackets

$m\mapsto[~{},~{}]_{m},~{}~{}m\in M$

is defined by

$\\{a,b,c\\}_{m}:=[[a,b]_{m},c]_{m},~{}~{}m\in M,$

for $a,b,c\in L_{m},~{}m\in M.$

######  3.7 Definition.

Let $\xi=(L,p,M)$ be a Lie algebra bundle with the field of Lie algebra
bracket $m\mapsto[~{},~{}]_{m},~{}~{}m\in M.$ A reductive decomposition of
$\xi$ is a pair $(L^{1},L^{2})$ of sub-bundles of $L$ such that $L$ is a
Whitney sum $L=L^{1}\oplus L^{2}$ satisfying
$[L^{1}_{m},L^{1}_{m}]_{m}\subseteq L^{1}_{m}$ and
$[L^{1}_{m},L^{2}_{m}]_{m}\subseteq L^{2}_{m}.$ In this case, we call
$(L^{1},L^{2})$ a reductive pair.

For a reductive pair as above, let $\pi^{i}:L\to L^{i},~{}~{}i=1,2$ denote the
vector bundle projection maps.

######  3.8 Example.

Let $(L^{1},L^{2})$ be a reductive decomposition of a Lie algebra bundle
$\xi=(L,p,M)$ as described in the above definition. Then, define a $2$-field
of brackets and a $3$-field of brackets

$m\mapsto\langle~{},~{}\rangle_{m},~{}~{}m\mapsto\\{~{},~{},~{}\\}_{m},~{}m\in
M$

on the vector bundle $(L^{2},p|_{L^{2}},M)$ as follows. Let $a,b,c\in
L^{2}_{m},~{}~{}m\in M.$

$[a,b]_{m}:=\pi^{1}(\langle
a,b\rangle_{m}),~{}~{}\\{a,b,c\\}:=\langle\pi^{2}(\langle
a,b\rangle_{m}),c]_{m}.$

Then, as in the case of Example 2.10, the vector bundle $(L^{2},p|_{L^{2}},M)$
is a Lie-Yamaguti algebra bundle equipped with the $2$-field of brackets and
the $3$-field of brackets as defined above.

Next, we discuss an interesting example of a Lie-Yamaguti algebra bundle that
arose from the work of M. Kikkawa [12, 13, 14] to characterize some local
geometric properties. We recall some definitions which are necessary to
describe our next example.

Recall that a linear connection on a smooth manifold $M$ is an
$\mathbb{R}$-bilinear map

$\nabla:\chi(M)\times\chi(M)\to\chi(M)$

written $\nabla_{X}Y$ for $\nabla(X,Y)$, satisfying two properties stated
below: For all $X,Y\in\chi(M)$

  * •

$\nabla_{X}Y$ is a $C^{\infty}(M)$-linear in $X$.

  * •

(Leibniz rule) $\nabla_{X}Y$ satisfies the Leibniz rule in $Y$: For all $f\in
C^{\infty}(M)$,

$\nabla_{X}(fY)=(Xf)Y+f(\nabla_{X}Y).$

Now, let $M$ be a smooth manifold along with linear connection $\nabla$.

Recall that

  * •

a torsion tensor of the connection $\nabla$ is a $C^{\infty}(M)$-bilinear map
$S:\chi(M)\times\chi(M)\to\chi(M)$ defined as

$S(X,Y):=\nabla_{X}Y-\nabla_{Y}X-[X,Y],~{}~{}~{}X,Y\in\chi(M),$

where $[X,Y]$ is the Lie bracket of $\chi(M)$ and

  * •

a curvature tensor of the connection $\nabla$ is a $C^{\infty}(M)$-trilinear
map $R:\chi(M)\times\chi(M)\times\chi(M)\to\chi(M)$ defined as

$R(X,Y)Z:=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,~{}~{}~{}X,Y,Z\in\chi(M).$

Recall the following definitions [13].

######  3.9 Definition.

Let $M$ be a smooth manifold with a connection $\nabla$. Let $S$ and $R$
denote the torsion and curvature tensors of $\nabla,$ respectively. Then,
$(M,\nabla)$ is said to be a locally reductive space if $~{}\nabla
S=0~{}~{}\&~{}~{}\nabla R=0;$ that is,

  * •

for all $X,Y,Z\in\chi(M)$; $\nabla_{X}S(Y,Z)=0;$

  * •

for all $X,U,V,W\in\chi(M)$; $\nabla_{X}R(U,V)W=0.$

######  3.10 Definition.

Let $G$ be a connected Lie group and $H$ be a closed subgroup of $G.$ Then the
homogeneous space $M=G/H$ is said to be reductive if and only if $G$ acts
effectively on $M$ and the Lie algebra $\mathfrak{g}$ of $G$ admits a direct
sum decomposition as

$\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h},$

where $\mathfrak{h}$ is the Lie algebra of $H$ and $\mathfrak{m}$ is a
subspace of $\mathfrak{g}.$

Next, we recall the notion of homogeneous Lie loops.

######  3.11 Definition.

Let $G=(G,\mu)$ be a binary system with the binary operation

$\mu:G\times G\to G$

$G$ is a loop if there is a (two-sided) identity $e\in G$, $xe=ex=x~{}~{}(x\in
G)$, and the left and right translations of $G$ by any element $x\in G$,
denoted by

$L_{x},R_{x}:G\to G;~{}L_{x}(y)=xy,~{}R_{x}(y)=yx~{}~{}~{}~{}(y\in G),$

are permutations of $G$.

######  3.12 Definition.

A loop $G$ is said to have the left inverse property, if for any $x\in G$
there exists an element $x^{-1}\in G$ such that

$x^{-1}(xy)=y~{}~{}~{}~{}(y\in G)$

######  3.13 Definition.

Let $L_{0}(G)$ be the group generated by all left inner mappings, i.e.,

$L_{x,y}=L_{xy}^{-1}\circ L_{x}\circ L_{y}~{}~{}~{}~{}(x,y\in G)$

A loop $G$ is called a left A-loop, if the left inner mapping group $L_{0}(G)$
is a subgroup of the automorphism group $AUT(G)$ of $G$.

######  3.14 Definition.

A Loop $(G,\mu)$ is said to be a homogeneous loop, if it is a left A-loop with
the left inverse property.

######  3.15 Definition.

A homogeneous Lie loop $G$ is a homogeneous loop, and is also a smooth
manifold such that the loop multiplication $\mu:G\times G\to G$ is smooth.

Here are some examples of locally reductive spaces.

  * •

Let $G$ be a connected homogeneous Lie loop equipped with the canonical
connection.

  * •

Define $K(G):=$ the closure of $L_{0}(G)$ in the smooth automorphism group
$Aut(G)$ of $G$, and consider the semi-direct product $A(G)=G\times K(G)$.
Since $G$ is connected, $L_{0}(G)$ is connected, and consequently $K(G)$ is
also connected. $A(G)$ is also a connected Lie group with the product manifold
structure. Further $A(G)$ contains $K(G)$ as a closed subgroup.

  * •

The homogeneous space $A(G)/K(G)$ is reductive.

Consider the reductive homogeneous space $A(G)/K(G)$ equipped with the
canonical connection. Then, we have the following results from [13].

######  3.16 Theorem.

For a connected homogeneous Lie loop $G$, the map

$i:G\to A(G)/K(G),~{}~{}~{}i(x)=x\times K(G)$

is a connection preserving loop isomorphism onto $A(G)/K(G)$ with
multiplication

$(x\times K(G)).(y\times K(G))=(xy)\times K(G)~{}~{}~{}~{}~{}(x,y\in G)$

with respect to the canonical connections on $G$ and $A(G)/K(G)$.

As a result, any connected homogeneous Lie loop with canonical connection can
be identified with a reductive homogeneous space with canonical connection.
The following result of M. Kikkawa tells us that any reductive homogeneous
space with canonical connection is locally reductive.

######  3.17 Theorem.

Let $S$ and $R$ denote the torsion and curvature tensors of the canonical
connection $\nabla$ of a reductive homogeneous space $M=G/H$, respectively.
Then $\nabla$ is locally reductive, i.e., $\nabla S=0$ and $\nabla R=0$

######  3.18 Corollary.

Any connected homogeneous Lie loop with the canonical connection is a locally
reductive space.

Below is a list of some examples of homogeneous Lie loops.

######  3.19 Example.

Any Lie group is a homogeneous Lie loop.

######  3.20 Example.

The set of all positive definite real symmetric matrices, denoted by $P_{n}$,
is a homogeneous Lie loop. Loop multiplication $\mu$ being

$\mu(X,Y)=X^{\frac{1}{2}}YX^{\frac{1}{2}},~{}~{}X,Y\in P_{n}.$

We are now in a position to describe a Lie-Yamaguti algebra bundle which arose
from the work of M. Kikkawa.

Since any connected homogeneous Lie loop with canonical connection is a
locally reductive space, we obtain the following example (cf. Theorem $7.2$
[13]).

######  3.21 Example.

Let $M$ be a connected homogeneous Lie loop with the canonical connection. Let
the associated torsion and curvature tensors be $S$ and $R$, respectively. Let
$\xi=(TM,p,M)$ be the tangent bundle of $M.$ Define a $2$-field of brackets
and a $3$-field of brackets on $M$ as follows:

$m\mapsto[a,b]_{m}=S_{m}(a,b);~{}~{}m\mapsto\\{a,b,c\\}=R_{m}(a,b)c~{}~{}~{}~{}~{}(a,b,c\in
T_{m}G).$

Then $\xi$ is a Lie-Yamaguti algebra bundle.

Next, we discuss a general existence theorem for locally trivial Lie-Yamaguti
algebra bundle.

######  3.22 Definition.

Let $(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$ be a Lie-Yamaguti algebra and
$G$ be a Lie group. We say that $G$ acts on $\mathfrak{g}$ if there exists a
smooth homomorphism

$\phi:G\to{Aut}_{LY}(\mathfrak{g}),~{}~{}g\mapsto\phi_{g}.$

Given such an action $\phi,$ we simply write $ga=:\phi_{g}(a),~{}g\in
G,~{}a\in\mathfrak{g}.$

Note that any closed subgroup of ${Aut}_{LY}(\mathfrak{g})$ acts smoothly on
$\mathfrak{g}$ and is a closed subgroup of the general linear group
$GL_{n}(\mathbb{R}).$

######  3.23 Definition.

Let $G$ be a Lie group and $M$ a smooth manifold. A family of smooth
transition maps in $M$ with values in $G$ is an atlas $\\{U_{i}:i\in I\\}$ of
$M$ together with a collection of smooth maps

$g_{ij}:U_{i}\cap U_{j}\to G,~{}~{}i,j\in I,$

where $I$ is any index set which we may assume to be countable satisfying the
following condition. For $i,j,k\in I,$ with $U_{i}\cup U_{j}\cup
U_{k}\neq\emptyset,$

$g_{ij}(m)\cdot g_{jk}(m)=g_{ik}(m),~{}m\in U_{i}\cap U_{j}\cap U_{k}.$

It follows from the above condition by taking $i=j=k$ that for any $i\in I,$
$g_{ii}(m),~{}m\in M$ is the identity of $G.$ The above condition is known as
the cocycle condition.

We have the following existence result of locally trivial Lie-Yamguti algebra
bundles whose proof is parallel to the proof of clutching construction in the
theory of fibre bundles [19]. We outline the sketch of the proof.

######  3.24 Theorem.

Let $(\mathfrak{g},[~{},~{}],\\{~{},~{},~{}\\})$ be a Lie-Yamaguti algebra
equipped with a smooth action of a Lie group $G.$ Let $M$ be a smooth manifold
with a given countable atlas $\\{U_{i}:\ i\in I\\}$ together with a family of
smooth transition maps

$g_{ij}:U_{i}\cap U_{j}\to G,~{}~{}i,j\in I,$

in $M$ with values in $G.$ Then, there exists a locally trivial Lie-Yamuguti
algebra bundle over $M,$ with $\mathfrak{g}$ as the fibre, $G$ as the
structure group of the bundle and with $\\{g_{ij}\\}$ as the associated
transition maps.

###### Proof.

Consider the following space where $I$ has the discrete topology

$\tilde{L}:=\bigcup_{i\in I}\\{(u,a,i)|u\in U_{i},~{}a\in\mathfrak{g},~{}i\in
I\\}.$

Define an equivalence relation on $\tilde{L}$ by $(u,a,i)\sim(v,b,j)$ if and
only if $u=v,~{}b=g_{ij}(u)a.$ Let $L=\tilde{L}/\sim.$ Let us denote the
equivalence class of $(u,a,i)$ by $[u,a,i].$ Let $q:\tilde{L}\to
L,(u,a,i)\mapsto[u,a,i]$ be the quotient map and $p:L\to
M,~{}~{}[u,a,i]\mapsto u$ be the natural projection map.

If $q_{i}=q|_{(U_{i}\times\mathfrak{g}\times\\{i\\})},$ then it is readily
seen that $q_{i}$ is injective,
$(q_{i}(U_{i}\times\mathfrak{g}\times\\{i\\}),q_{i}^{-1})$ is a smooth chart
on $L$ and $p:L\to M$ is a smooth vector bundle.

We now show that $\xi=(L,p,M)$ is a Lie-Yamaguti algebra bundle. Let $m\in M$
and $\xi_{m}$ be the fibre over $m.$ Define a $2$-field of brackets
$m\mapsto[~{},~{}]_{m}$ and a $3$-filed of brackets
$m\mapsto\\{~{},~{},~{}\\}_{m}$ as follows. Note that for $i\in I,$ the map

$\\{\psi_{i}:U_{i}\times\mathfrak{g}\rightarrow p^{(-1)}(U_{i})\\}$

defined by

$\psi(u,a)=q(u,a,i),~{}u\in U_{i},~{}a\in\mathfrak{g}$

gives the local trivialization of the vector bundle $\xi.$ Let
$\psi_{i,m},~{}~{}m\in U_{i}\subset M$ denotes the restriction of $\psi_{i}$
to $\\{m\\}\times\mathfrak{g}.$

Let $a,b,c\in\xi_{m},~{}m\in M.$ Choose $i\in I$ such that $m\in U_{i}.$
Define

$[a,b]_{m}:=\psi_{i,m}([\psi_{i,m}^{-1}(a),\psi_{i,m}^{-1}(b)])$

and

$\\{a,b,c\\}_{m}:=\psi_{i,m}(\\{\psi_{i,m}^{-1}(a),\psi_{i,m}^{-1}(b),\psi_{i,m}^{-1}(c)\\}).$

Then, it is routine to verify that $\xi$ is a locally trivial Lie-Yamaguti
algebra bundle with fibre $\mathfrak{g}.$ ∎

######  3.25 Remark.

The above theorem provides a general method of constructing a locally trivial
Lie-Yamaguti algebra bundle from any Lie group of symmetry of a given Lie-
Yamaguti algebra on a manifold, equipped with a family of smooth transition
maps taking values in the group of symmetry. In particular, we may apply the
above method for any Lie group of symmetry of the Lie-Yamaguti algebras
discussed in the previous section to construct examples of Lie-Yamaguti
algebra bundles.

######  3.26 Definition.

Let $\xi=(L,p,M)$ and $\xi^{\prime}=(L^{\prime},p^{\prime},M^{\prime})$ be two
Lie-Yamaguti algebra bundles. A homomorphism
$\phi:(L,p,M)\rightarrow(L^{\prime},p^{\prime},M^{\prime})$ from $\xi$ to
$\xi^{\prime}$ is a vector bundle morphism $(\tilde{\phi},\phi),$ where
$\tilde{\phi}:L\rightarrow L^{\prime},$ is the map between total spaces and
$\phi:M\rightarrow M^{\prime}$ is the maps of the base spaces such that
$\tilde{\phi}|_{L_{m}}:L_{m}\rightarrow L^{\prime}_{\phi(m)}$ is a Lie-
Yamaguti algebra homomorphism, where $m\in M.$

A homomorphism $\phi:\xi\rightarrow\xi^{\prime}$ of two Lie-Yamaguti algebra
bundles over the same base space $M$ is a vector bundle morphism
$\phi:\xi\rightarrow\xi^{\prime}$ such that
$\phi|_{\xi_{m}}:\xi_{m}\rightarrow\xi^{\prime}_{m}$ is a Lie-Yamaguti algebra
homomorphism for all $m\in M.$ Moreover, if $\phi|_{\xi_{m}}$ is a linear
bijection then $\xi=(L,p,M)$ is said to be isomorphic to
$\xi^{\prime}=(L^{\prime},p^{\prime},M).$

######  3.27 Definition.

A Lie-Yamaguti algebra bundle $\xi$ is said to be trivial if it is isomorphic
to a product Lie-Yamaguti algebra bundle.

##  4\. Representation of Lie-Yamaguti Algebra Bundles

The aim of this section is to introduce the notion of representation of Lie-
Yamaguti algebra bundles.

Our definition of representation of a Lie-Yamaguti algebra bundle is based on
the definition of representation of a Lie-Yamaguti algebra [20].

######  4.1 Definition.

Let $\xi=(L,p,M)$ be a Lie-Yamaguti algebra bundle and $\eta=(E,q,M)$ be a
vector bundle. For any point $m\in M,$ let $\eta_{m}$ denote the fibre
$\eta_{m}=q^{-1}(m)$ of the bundle $\eta$ over $m.$

A representation of the Lie-Yamaguti algebra bundle $\xi$ on the vector bundle
$\eta$ consists of vector bundle morphisms

$\rho:\xi\to\textrm{End}(\eta),~{}~{}D,~{}\theta:\xi\otimes\xi\to\textrm{End}(\eta)$

such that these maps restricted to each fibre satisfy the conditions (RLYB$1$)
- (RLYB$6$) as described below, where the bilinear maps

$D|_{\xi_{m}},~{}\theta|_{\xi_{m}}:\xi_{m}\times\xi_{m}\to\textrm{End}(\eta_{m}),$

obtained by restricting $D,~{}\theta$ to a fibre $\xi_{m}$ are denoted by
$D_{m}$ and $\theta_{m},$ respectively and similarly, $\rho_{m}$ is the linear
map

$\rho|_{\xi_{m}}:\xi_{m}\to\textrm{End}(\eta_{m}).$

For any $m\in M$ and $a,b,c,d\in\xi_{m},$

(RLYB1) $\displaystyle
D_{m}(a,b)+\theta_{m}(a,b)-\theta_{m}(b,a)=[\rho_{m}(a),\rho_{m}(b)]_{m}-\rho_{m}([a,b]);$
(RLYB2)
$\displaystyle\theta_{m}(a,[b,c]_{m})-\rho_{m}(b)\theta_{m}(a,c)+\rho_{m}(c)\theta_{m}(a,b)=0;$
(RLYB3)
$\displaystyle\theta_{m}([a,b]_{m},c)-\theta_{m}(a,c)\rho_{m}(b)+\theta_{m}(b,c)\rho_{m}(a)=0;$
(RLYB4)
$\displaystyle\theta_{m}(c,d)\theta_{m}(a,b)-\theta_{m}(b,d)\theta_{m}(a,c)-\theta_{m}(a,\\{b,c,d\\}_{m})+D_{m}(b,c)\theta_{m}(a,d)=0;$
(RLYB5) $\displaystyle[D_{m}(a,b),\rho_{m}(c)]_{m}=\rho_{m}(\\{a,b,c\\}_{m});$
(RLYB6)
$\displaystyle[D_{m}(a,b),\theta_{m}(c,d)]_{m}=\theta_{m}(\\{a,b,c\\}_{m},d)+\theta_{m}(c,\\{a,b,d\\}_{m}).$

We shall denote a representation of a Lie-Yamaguti algebra bundle $\xi$ on a
vector bundle $\eta$ as described above by $(\eta;~{}\rho,~{}D,~{}\theta).$ A
representation $(\eta;~{}\rho,~{}D,~{}\theta)$ of a Lie-Yamaguti algebra
bundle $\xi$ is also called a $\xi$-module.

######  4.2 Remark.

Like a representation of a Lie-Yamaguti algebra [20], given a representation
$(\eta;~{}\rho,~{}D,~{}\theta)$ of a Lie-Yamaguti algebra bundle $\xi,$ we
have for every $m\in M$

(RLYB7) $D_{m}([a,b]_{m},c)+D_{m}([b,c]_{m},a)+D_{m}([c,a]_{m},b)=0,$

for any $a,~{}b,~{}c\in\xi_{m}.$

######  4.3 Example.

Given a Lie-Yamaguti algebra bundle $\xi$ over $M$, we may consider $\xi$ as a
$\xi$-module which gives us the adjoint representation of $\xi$ on itself.
Explicitly, for each $m\in M,$ $\rho_{m},~{}D_{m},~{}\theta_{m}$ are given by

$\rho_{m}(a):b\mapsto[a,b]_{m};~{}~{}D_{m}(a,b):c\mapsto\\{a,b,c\\}_{m};~{}~{}\theta_{m}(a,b):c\mapsto\\{c,a,b\\}_{m},$

for any $a,~{}b,~{}c\in\xi_{m}.$

######  4.4 Remark.

Observe that for a $0$-dimensional manifold $M=\\{pt\\},$ a Lie-Yamaguti
algebra bundle $\xi$ over $M$ is simply a Lie-Yamaguti algebra and a
representation $\eta$ of $\xi$ in this case, reduces to a representation of
the Lie-Yamaguti algebra $\xi.$ More generally, given any representation
$(\eta;\rho,D,\theta)$ of a Lie-Yamaguti algebra bundle $\xi=(L,p,M)$ on the
vector bundle $\eta=(E,q,M)$ over a smooth manifold $M,$
$(\eta_{m};\rho_{m},D_{m},\theta_{m})$ may be viewed as a representation of
the Lie-Yamaguti algebra $\xi_{m}$ for any $m\in M.$

##  5\. Cohomology of Lie-Yamaguti Algebra Bundle

In this section we introduce cohomology of Lie-Yamaguti algebra bundle with
coefficients in a representation. The definition is motivated by the
definition of cohomology of a Lie-Yamaguti algebra as introduced in [21]. We
use the Remark 4.4 to introduce our definition.

######  5.1 Definition.

Let $\xi=(L,p,M)$ be a Lie-Yamaguti algebra bundle and $(\eta;\rho,D,\theta)$
be a $\xi$-module. Let us denote the $2$-field and the $3$-field of brackets
which make the vector bundle $\xi$ a Lie-Yamaguti algebra bundle by

$m\mapsto[~{},~{}]_{m},~{}~{}m\mapsto\\{~{},~{},~{}\\}_{m},~{}~{}m\in M.$

Let

$C^{1}(\xi;\eta)=\mbox{Hom}~{}(\xi;\eta).$

Let $C^{0}(\xi;\eta)$ be the subspace spanned by the diagonal elements
$(f,f)\in C^{1}(\xi;\eta)\times C^{1}(\xi,\eta)$. For $n\geq 2,$ let
$C^{n}(\xi;\eta)$ be the space of all vector bundle maps $f:\xi^{\otimes
n}\to\eta,$ that is, $f\in\mbox{Hom}~{}(\xi^{\otimes n};\eta)$ such that the
resulting $n$-linear maps $f_{m}=f|_{\xi^{\otimes
n}_{m}}:\xi_{m}\times\cdots\times\xi_{m}\to\eta_{m}$ satisfy

$f_{m}(x_{1},\ldots,x_{2i-1},x_{2i},\ldots,x_{n})=0,$

if $x_{2i-1}=x_{2i},~{}~{}x_{i}\in\xi_{m},~{}i=1,\ldots,[n/2].$

For $p\geq 1,$ set

$C^{(2p,2p+1)}(\xi;\eta):=C^{2p}(\xi;\eta)\times C^{2p+1}(\xi;\eta).$

Any element $(f,g)\in C^{(2p,2p+1}(\xi;\eta)$ will be referred to as a
$(2p,2p+1)$-cochain.

For $p\geq 1,$ we define a coboundary operator

$\delta=(\delta_{I},\delta_{II}):C^{(2p,2p+1)}(\xi;\eta)\to
C^{(2p+2,2p+3)}(\xi;\eta),$
$(f,g)\mapsto\delta(f,g)=(\delta_{I}f,\delta_{II}g)$

by defining it fibre-wise using the formula introduced by K. Yamaguti [21]. In
other words, for any $m\in M,$

$\delta(f,g)_{m}=((\delta_{I})_{m}f_{m},(\delta_{II})_{m}g_{m}).$

Explicitly, for $m\in M$ and $x_{1},\ldots,x_{2p+2}\in\xi_{m},$

$\displaystyle(\delta_{I})_{m}f_{m}(x_{1},\ldots,x_{2p+2})$ $\displaystyle=$
$\displaystyle(-1)^{p}[\rho_{m}(x_{2p+1})g_{m}(x_{1},\ldots,x_{2p},x_{2p+2})-\rho_{m}(x_{2p+2})g_{m}(x_{1},\ldots,x_{2p},x_{2p+1})$
$\displaystyle-$ $\displaystyle
g_{m}(x_{1},\ldots,x_{2p},[x_{2p+1},x_{2p+2}]_{m})]$ $\displaystyle+$
$\displaystyle\sum_{k=1}^{p}(-1)^{k+1}D_{m}(x_{2k-1},x_{2k})f_{m}(x_{1},\ldots,\hat{x}_{2k-1},\hat{x}_{2k},\ldots,x_{2p+2})$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{p+1}\sum_{j=2k+1}^{2p+2}(-1)^{k}f_{m}(x_{1},\ldots,\hat{x}_{2k-1},\hat{x}_{2k},\ldots,\\{x_{2k-1},x_{2k},x_{j}\\}_{m},\ldots,x_{2p+2}).$

Let $x_{1},\ldots,x_{2p+3}\in\xi_{m}.$ Then,

$\displaystyle(\delta_{II})_{m}g_{m}(x_{1},\ldots,x_{2p+3})$ $\displaystyle=$
$\displaystyle(-1)^{p}[\theta_{m}(x_{2p+2},x_{2p+3})g_{m}(x_{1},\ldots,x_{2p+1})$
$\displaystyle-$
$\displaystyle\theta_{m}(x_{2p+1},x_{2p+3})g_{m}(x_{1},\ldots,x_{2p},x_{2p+2})]$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{p+1}(-1)^{k+1}D_{m}(x_{2k-1},x_{2k})g_{m}(x_{1},\ldots,\hat{x}_{2k-1},\hat{x}_{2k},\ldots,x_{2p+3})$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{p+1}\sum_{j=2k+1}^{2p+3}(-1)^{k}g_{m}(x_{1},\ldots,\hat{x}_{2k-1},\hat{x}_{2k},\ldots,\\{x_{2k-1},x_{2k},x_{j}\\}_{m},\ldots,x_{2p+3}).$

Now observe that for any $m\in M,$ the coboundary operator $\delta_{m}$ is
precisely the coboundary operator for the Lie-Yamaguti algebra $\xi_{m}$ with
coefficient in $\eta_{m}$ (cf. Remark 4.4) and since
$\delta_{m}\circ\delta_{m}=0$ [21] we obtain the following result.

######  5.2 Lemma.

For $p\geq 1,$ the coboundary operator

$\delta=(\delta_{I},\delta_{II}):C^{2p}(\xi;\eta)\times C^{2p+1}(\xi;\eta)\to
C^{2p+2}(\xi;\eta)\times C^{2p+3}(\xi;\eta)$

satisfy $\delta\circ\delta=0.$

######  5.3 Definition.

For the case $p\geq 2,$ let $Z^{(2p,2p+1)}(\xi;\eta)$ be the subspace of
$C^{(2p,2p+1)}(\xi;\eta)$ spanned by $(f,g)$ such that $\delta(f,g)=0$ and
$B^{(2p,2p+1)}(\xi;\eta)$ be the subspace $\delta(C^{(2p-2,2p-1)}(\xi;\eta)).$
Then, the $(2p,2p+1)$-cohomology group of the Lie-Yamaguti algebra bundle
$\xi$ with coefficients in $\eta$ is defined by

$H^{(2p,2p+1)}(\xi;\eta):=\frac{Z^{(2p,2p+1)}(\xi;\eta)}{B^{(2p,2p+1)}(\xi;\eta)}.$

We next consider the case $p=1,$ and define the cohomology group
$H^{(2,3)}(\xi;\eta).$ Define a coboundary operator

$\delta=(\delta_{I},\delta_{II}):C^{0}(\xi;\eta)\to
C^{(2,3)}(\xi;\eta),~{}~{}(f,f)\mapsto(\delta_{I}f,\delta_{II}f),$

where for $x_{1},~{}x_{2},~{}x_{3}\in\xi_{m},~{}m\in M,$

$\displaystyle(\delta_{I})_{m}f_{m}(x_{1},x_{2})$
$\displaystyle=\rho_{m}(x_{1})f_{m}(x_{2})-\rho_{m}(x_{2})f_{m}(x_{1})-f_{m}([x_{1},x_{2}]_{m})$
$\displaystyle(\delta_{II})_{m}f_{m}(x_{1},x_{2},x_{3})$
$\displaystyle=\theta_{m}(x_{2},x_{3})f_{m}(x_{1})-\theta_{m}(x_{1},x_{3})f_{m}(x_{2})$
$\displaystyle+D_{m}(x_{1},x_{2})f_{m}(x_{3})-f_{m}(\\{x_{1},x_{2},x_{3}\\}_{m}).$

Furthermore, we define another coboundary operator

$\delta^{*}=(\delta^{*}_{I},\delta^{*}_{II}):C^{(2,3)}(\xi;\eta)\to
C^{(3,4)}(\xi;\eta)$

as follows.

Let $m\in M$ and $x_{1},~{}x_{2},~{}x_{3}~{}x_{4}\in\xi_{m}.$ Then for
$(f,g)\in C^{(2,3)}(\xi;\eta),$

$\displaystyle(\delta^{*}_{I})_{m}f_{m}(x_{1},x_{2},x_{3})$ $\displaystyle=$
$\displaystyle\rho_{m}(x_{1})f_{m}(x_{2},x_{3}\rho_{m}(x_{2})f_{m}(x_{3},x_{1})-\rho_{m}(x_{3})f_{m}(x_{1},x_{2})$
$\displaystyle+$ $\displaystyle
f_{m}([x_{1},x_{2}]_{m},x_{3})+f_{m}([x_{2},x_{3}]_{m},x_{1})+f_{m}([x_{3},x_{1}]_{m},x_{2})$
$\displaystyle+$ $\displaystyle
g_{m}(x_{1},x_{2},x_{3})+g_{m}(x_{2},x_{3},x_{3})+g_{m}(x_{3},x_{1},x_{2}),$

and

$\displaystyle(\delta^{*}_{II})_{m}g_{m}(x_{1},x_{2},x_{3},x_{4})$
$\displaystyle=\theta_{m}(x_{1},x_{4})f_{m}(x_{2},x_{3})+\theta_{m}(x_{2},x_{4})f_{m}(x_{3},x_{1})+\theta_{m}(x_{3},x_{4})f_{m}(x_{1},x_{2})$
$\displaystyle+g_{m}([x_{1},x_{2}]_{m},x_{3},x_{4})+g_{m}([x_{2},x_{3}]_{m},x_{1},x_{4})+g_{m}([x_{3},x_{1}]_{m},x_{2},x_{4}).$

Following [21], we have for each $f\in C^{1}(\xi;\eta)$

$\delta_{I}\delta_{I}f=\delta^{*}_{I}\delta_{I}f=0~{}~{}\mbox{and}~{}~{}\delta_{II}\delta_{II}f=\delta^{*}_{II}\delta_{II}f=0.$

In general, for $(f,g)\in C^{(2p,2p+1)}(\xi;\eta)$

$(\delta\circ\delta)(f,g)=(\delta_{I}\circ\delta_{I}(f),\delta_{II}\circ\delta_{II}(g))=0.$

We define

######  5.4 Definition.

$H^{1}(\xi;\eta):=\\{f\in C^{1}(\xi;\eta)|\delta_{I}f=0,~{}\delta_{II}f=0\\}.$

For $p=1$, we define the cohomology $H^{(2,3)}(\xi;\eta)$ as follows.

######  5.5 Definition.

Let $Z^{(2,3)}(\xi;\eta)$ be the subspace of $C^{(2,3)}(\xi;\eta)$ spanned by
$(f,g)$ such that $\delta_{I}f=\delta^{*}_{I}f=0,$ and
$\delta_{II}g=\delta^{*}_{II}g=0.$ Let

$B^{(2,3)}(\xi;\eta)=\\{\delta(f,f)|f\in C^{1}(\xi;\eta)\\}.$

Then, the $(2,3)$-cohomology group of the Lie-Yamaguti algebra bundle $\xi$
with coefficients in $\eta$ is defined by

$H^{(2,3)}(\xi;\eta)=\frac{Z^{(2,3)}(\xi;\eta)}{B^{(2,3)}(\xi;\eta)}.$

##  6\. Extensions of Lie-Yamaguti algebra bundles

Chevally and Eilenberg [4] showed that extensions of algebras can be
interpreted in terms of certain Hochschild cohomology group. Later, D. K.
Harrison [7] showed that certain Harrison cohomology group of commutative
algebras can be related to extensions of commutative algebras. In the same
spirit, Yamaguti showed that that $(2,3)$-cohomology of a Lie-Yamaguti algebra
with coefficients in a representation may be interpreted in terms of
isomorphism classes of extensions of the Lie-Yamaguti algebra. The aim of this
section is to introduce the notion of extension of Lie-Yamaguti algebra
bundles and relate isomorphism classes of extensions in terms of
$(2,3)$-cohomology of such bundle as introduced in the previous section. We
begin with some definitions.

######  6.1 Definition.

Let $\xi=(L,p,M)$ be a Lie-Yamaguti algebra bundle. Let us denote the
associated $2$-field and $3$-field of brackets by

$m\to[~{},~{}]_{m}~{}~{}\mbox{and}~{}~{}m\to\\{~{},~{},~{}\\}_{m},~{}m\in M,$

respectively. An ideal of $\xi$ is a sub-bundle $\eta$ of the vector bundle
$\xi$ such that for all $m\in M,$ $v\in\eta_{m},$ $a,~{}b\in\xi_{m}$

$[v,a]_{m}\in\eta_{m}~{}~{}\text{and}~{}~{}\\{v,a,b\\}_{m}\in\eta_{m},~{}\\{a,b,v\\}_{m}\in\eta_{m}.$

An ideal $\eta$ of $\xi$ is said to be abelian if for all $m\in M,$
$u,v\in\eta_{m}$ and $a\in\xi_{m},$

$[u,v]_{m}=0,~{}~{}\\{u,v,a\\}_{m}=\\{u,a,v\\}_{m}=\\{a,u,v\\}_{m}=0.$

######  6.2 Definition.

Let $\tilde{\xi}=(\tilde{L},\tilde{p},M)$ and $\eta=(E,q,M)$ be Lie-Yamaguti
algebra bundles. An extension of Lie-Yamaguti algebra bundle over $M$ is a
short exact sequence in the category of Lie-Yamaguti algebra bundles over $M$
(that is, restricted to each fibre yields a short exact sequence of vector
spaces where the maps involved are Lie-Yamaguti algebra homomorphisms)

${0}$${\eta}$${\tilde{\xi}}$${\xi}$${0.}$$\scriptstyle{i}$$\scriptstyle{j}$

We call $\tilde{\xi}$ an extension of $\xi$ by $\eta$, and denote it by
${Ext}_{\tilde{\xi}}$.

An extension as above is said to be abelian if $\eta$ is an abelian ideal of
$\tilde{\xi}.$

Throughout in this section, we will consider only abelian extensions.

A section of the vector bundle map $j:\tilde{\xi}\to\xi$ is a vector bundle
map $\sigma:\xi\to\tilde{\xi}$ such that $j\circ\sigma=id_{\xi}$. Note that if
$\sigma$ is a splitting as above then $\tilde{\xi}$ may be viewed as as a
Whitney sum of $\xi$ and $\eta.$ Here, we identify $\eta$ with its image
$i(\eta).$

In other words, $\tilde{\xi}=\xi\oplus\eta$. Since we are concerned with
smooth vector bundles over smooth manifolds (paracompact), any short exact
sequence splits.

We now introduce the notion of equivalence of two extensions.

######  6.3 Definition.

Two extensions of $\xi$ by $\eta$, say
${Ext}_{\tilde{\xi}}:0\to\eta\xrightarrow[]{i}\tilde{\xi}\xrightarrow[]{j}\xi\to
0$ and
${Ext}_{\hat{\xi}}:0\to\eta\xrightarrow[]{\hat{i}}\hat{\xi}\xrightarrow[]{\hat{j}}\xi\to
0$, are said to be equivalent, if $\exists$ a Lie-Yamaguti algebra bundle
isomorphism $f:\tilde{\xi}\to\hat{\xi}$ such that the following diagram is
commutative.

${0}$${\eta}$${\tilde{\xi}}$${\xi}$${0}$${0}$${\eta}$${\hat{\xi}}$${\xi}$${0}$$\scriptstyle{i}$$\scriptstyle{\hat{j}}$$\scriptstyle{f}$$\scriptstyle{id}$$\scriptstyle{id}$$\scriptstyle{\hat{i}}$$\scriptstyle{j}$

Note that, any extension $\tilde{\xi}$ of $\xi$ by $\eta$, is isomorphic to
$\xi\oplus\eta$ as vector bundles.

Next, we show that any given extension
${Ext}_{\tilde{\xi}}:0\to\eta\xrightarrow[]{i}\tilde{\xi}\xrightarrow[]{j}\xi\to
0$ induces a representation of the Lie-Yamaguti algebra bundle on the vector
bundle $\eta.$ It turns out that equivalent extensions induce the same
representation on $\eta$.

Let
${Ext}_{\tilde{\xi}}:0\to\eta\xrightarrow[]{i}\tilde{\xi}\xrightarrow[]{j}\xi\to
0$ be a given extension of $\xi$ by $\eta$, that is,

${0}$${\eta}$${{\tilde{\xi}}}$${\xi}$${0}$$\scriptstyle{i}$$\scriptstyle{j}$

and $\sigma:\xi\to\tilde{\xi}$ be a section of $j:\tilde{\xi}\to\xi$. Let us
denote the associated $2$-field and $3$-field of brackets of the Lie-Yamaguti
algebra bundle $\tilde{\xi}$ by

$m\to[~{},~{}]^{\sim}_{m}~{}~{}\mbox{and}~{}~{}m\to\\{~{},~{},~{}\\}^{\sim}_{m},~{}m\in
M,$

respectively.

Define vector bundle morphisms $\rho:\xi\to\textrm{End}(\eta)$ and
$D,\theta:\xi\otimes\xi\to\textrm{End}(\eta)$ fibre-wise as follows. Let $m\in
M.$ For any $a,b\in\xi_{m}$ and $v\in\eta_{m}$

(1) $\displaystyle\rho_{m}(a)(v)$
$\displaystyle:=[\sigma_{m}(a),v]^{\sim}_{m}$ (2) $\displaystyle
D_{m}(a,b)(v)$ $\displaystyle:=\\{\sigma_{m}(a),\sigma_{m}(b),v\\}^{\sim}_{m}$
(3) $\displaystyle\theta_{m}(a,b)(v)$
$\displaystyle:=\\{v,\sigma_{m}(a),\sigma_{m}(b)\\}^{\sim}_{m}.$

######  6.4 Proposition.

The above data yield a representation $(\eta;~{}\rho,~{}D,~{}\theta)$ of
$\xi$. Furthermore,

  1. (1)

The definition of $\rho,D$, and $\theta$ does not depend on the choice of the
section $\sigma$, that is, given any two sections of $j$, say $\sigma^{1}$,
and $\sigma^{2}$, we have

$[\sigma^{1}_{m}(a),v]^{\sim}=[\sigma^{2}_{m}(a),v]^{\sim}~{},~{}~{}\\{\sigma^{1}_{m}(a),\sigma^{1}_{m}(b),v\\}^{\sim}=\\{\sigma^{2}_{m}(a),\sigma^{2}_{m}(b),v\\}^{\sim}~{},$
$\text{and}~{}~{}\\{v,\sigma^{1}_{m}(a),\sigma^{1}(b)\\}^{\sim}=\\{v,\sigma^{2}_{m}(a),\sigma^{2}_{m}(b)\\}^{\sim}.$

  2. (2)

Equivalent extensions induce the same representation on $\eta$, that is, given
any two equivalent extensions, say ${Ext}_{\tilde{\xi}}$ and
${Ext}_{\hat{\xi}}$ with induced representations being
$(\eta;~{}\rho,~{}D,~{}\theta)$ and
$(\eta;~{}\rho^{\prime},~{}D^{\prime},~{}\theta^{\prime})$ respectively. Then
$\rho=\rho^{\prime},~{}D=D^{\prime},~{}\theta=\theta^{\prime}$.

###### Proof.

Let $m\in M,$ $a,b,c,d\in\xi_{m}$ and $v\in eta_{m}.$ Let
$\sigma:\xi\to\tilde{\xi}$ be a given section of $j:\tilde{\xi}\to\xi.$ Then,
we have the following equalities.

From the condition (LY3) of $\tilde{\xi}_{m}$ we get

$\sum_{\circlearrowleft(\sigma_{m}(a),\sigma_{m}(b),v)}[[\sigma_{m}(a),\sigma_{m}(b)]^{\sim},v]^{\sim}+\sum_{\circlearrowleft(\sigma_{m}(a),\sigma_{m}(b),v)}\\{\sigma_{m}(a),\sigma_{m}(b),v\\}^{\sim}=0$

which reduces to (RLYB1):

$D_{m}(a,b)+\theta_{m}(a,b)-\theta_{m}(b,a)=[\rho_{m}(a),\rho_{m}(b)]_{m}-\rho_{m}([a,b]_{m}).$

By (LY5) of $\tilde{\xi}$ we get

$\displaystyle\\{\sigma_{m}(a),v,[\sigma_{m}(b),\sigma_{m}(c)]_{m}^{\sim}$
$\displaystyle=[\\{\sigma_{m}(a),v,\sigma_{m}(b)\\}_{m}^{\sim},\sigma_{m}(c)]_{m}^{\sim}+[\sigma_{m}(b),\\{\sigma_{m}(a),v,\sigma_{m}(c)\\}_{m}^{\sim}]_{m}^{\sim}$

which reduces to (RLYB2):

$\theta_{m}(a,[b,c]_{m})=\rho_{m}(b)\theta_{m}(a,c)-\rho_{m}(c)\theta_{m}(a,b).$

By (LY4) of $\tilde{\xi}_{m}$ we get

$\sum_{\circlearrowleft(\sigma_{m}(a),\sigma_{m}(b),v)}\\{[\sigma_{m}(a),\sigma_{m}(b)]_{m}^{\sim},v,\sigma_{m}(c)\\}_{m}^{\sim}=0$

which reduces to (RLYB3):

$\theta_{m}([a,b]_{m},c)=\theta_{m}(a,c)\rho_{m}(b)-\theta_{m}(b,c)\rho_{m}(a).$

By (LY6) of $\tilde{\xi}_{m}$ we get

$\displaystyle\\{v,\sigma_{m}(a),\\{\sigma_{m}(b),\sigma_{m}(c),\sigma_{m}(d)\\}_{m}^{\sim}\\}_{m}^{\sim}$
$\displaystyle=\\{\\{v,\sigma_{m}(a),\sigma_{m}(b)\\}_{m}^{\sim},\sigma_{m}(c),\sigma_{m}(d)\\}_{m}^{\sim}+\\{\sigma_{m}(b),\\{v,\sigma_{m}(a),\sigma_{m}(c)\\_m^{\sim},\sigma_{m}(d)\\}_{m}^{\sim}$
$\displaystyle+\\{\sigma_{m}(b),\sigma_{m}(c),\\{v,\sigma_{m}(a),\sigma_{m}(d)\\}_{m}^{\sim}\\}_{m}^{\sim}$

which reduces to (RLYB4):

$\theta_{m}(a,\\{b,c,d\\}_{m})=\theta_{m}(c,d)\theta_{m}(a,b)-\theta_{m}(b,d)\theta_{m}(a,c)+D_{m}(b,c)\theta_{m}(a,d).$

By (LY5) of $\tilde{\xi}_{m}$ we get

$\displaystyle\\{\sigma_{m}(a),\sigma_{m}(b),[\sigma_{m}(c),v]_{m}^{\sim}\\}_{m}^{\sim}$
$\displaystyle=[\\{\sigma_{m}(a),\sigma_{m}(b),\sigma_{m}(c)\\}_{m}^{\sim},v]_{m}^{\sim}+[\sigma_{m}(c),\\{\sigma_{m}(a),\sigma_{m}(b),v\\}_{m}^{\sim}]_{m}^{\sim}$

which reduces to (RLYB5):

$D_{m}(a,b)\rho_{m}(c)=\rho_{m}(c)D_{m}(a,b)+\rho_{m}(\\{a,b,c\\}_{m}).$

By (LY6) of $\tilde{\xi}_{m}$ we get

$\displaystyle\\{\sigma_{m}(a),\sigma_{m}(b),\\{v,\sigma_{m}(c),\sigma_{m}(d)\\}_{m}^{\sim}\\}_{m}^{\sim}$
$\displaystyle=\\{\\{\sigma_{m}(a),\sigma_{m}(b),v\\}_{m}^{\sim},\sigma_{m}(c),\sigma_{m}(d)\\}_{m}^{\sim}+\\{v,\\{\sigma_{m}(a),\sigma_{m}(b),\sigma_{m}(c)\\}_{m}^{\sim},\sigma_{m}(d)\\}_{m}^{\sim}$
$\displaystyle+\\{v,\sigma_{m}(c),\\{\sigma_{m}(a),\sigma_{m}(b),\sigma_{m}(d)\\}_{m}^{\sim}\\}_{m}^{\sim}$

which reduces to (RLYB6):

$D_{m}(a,b)\theta_{m}(c,d)=\theta_{m}(c,d)D_{m}(a,b)+\theta_{m}(\\{a,b,c\\}_{m},d)+\theta_{m}(c,\\{a,b,d\\}_{m}).$

Therefore, $(\eta;\rho,~{}D,~{}\theta)$ is a representation of $\xi$. Hence
any extension of $\xi$ by $\eta$ gives a representation of $\xi$ on $\eta$.

Next we show that the definition of $\theta$ is independent of the choice of
the section. The proofs that the definitions of $\rho$ and $D$ do not depend
on the choice of the section $\sigma$ are similar, hence, we omit the details.

Let $\sigma,~{}\sigma^{\prime}:\xi\to\tilde{\xi}$ be two sections of
$j:\tilde{\xi}\to\xi.$ Let $m\in M.$ Then, for any $a\in\xi_{m}$

$j(\sigma_{m}(a)-\sigma_{m}^{\prime}(a))=0.$

Therefore,
$\sigma_{m}(a)-\sigma_{m}^{\prime}(a)\in\operatorname*{Ker}(j)=\eta_{m}$, so
that $\sigma_{m}(a)=\sigma_{m}^{\prime}(a)+v_{a}$ for some $v_{a}\in\eta_{m}$.
Since we are considering abelian extension, for any $v\in\eta_{m},$
$a,~{}b\in\xi_{m}$ we have

$\displaystyle\\{v,\sigma_{m}(a),\sigma_{m}(b)\\}_{m}^{\sim}$
$\displaystyle=\\{v,\sigma_{m}^{\prime}(a)+v_{a},\sigma_{m}^{\prime}(b)+v_{b}\\}_{m}^{\sim}$
$\displaystyle=\\{v,\sigma_{m}^{\prime}(a),\sigma_{m}^{\prime}(b)+v_{b}\\}_{m}^{\sim}+\\{v,v_{a},\sigma_{m}^{\prime}(b)+v_{b}\\}_{m}^{\sim}$
$\displaystyle=\\{v,\sigma_{m}^{\prime}(a),\sigma_{m}^{\prime}(b)\\}_{m}^{\sim}+\\{v,\sigma_{m}^{\prime}(a),v_{b}\\}_{m}^{\sim}$
$\displaystyle=\\{v,\sigma_{m}^{\prime}(a),\sigma_{m}^{\prime}(b)\\}_{m}^{\sim}.$

Finally, we show that two equivalent extensions of $\xi$ by $\eta$ induce the
same representation. Suppose that ${Ext}_{\tilde{\xi}}$ and
${Ext}_{\hat{\xi}}$ are two equivalent extensions of $\xi.$ Let us denote the
associated $2$-field and $3$-field of brackets of the Lie-Yamaguti algebra
bundle $\hat{\xi}$ by

$m\to[~{},~{}]^{\wedge}_{m}~{}~{}\mbox{and}~{}~{}m\to\\{~{},~{},~{}\\}^{\wedge}_{m},~{}m\in
M,$

respectively.

Let $f:\tilde{\xi}\to\hat{\xi}$ be a Lie-Yamaguti algebra isomorphism
satisfying $f\circ i=\hat{i}$ and $\hat{j}\circ f=j.$ Thus, the following
diagram is commutative.

${0}$${\eta}$${\tilde{\xi}}$${\xi}$${0}$${0}$${\eta}$${\hat{\xi}}$${\xi}$${0}$$\scriptstyle{i}$$\scriptstyle{\hat{j}}$$\scriptstyle{f}$$\scriptstyle{id}$$\scriptstyle{id}$$\scriptstyle{\hat{i}}$$\scriptstyle{j}$

Let $\sigma:\xi\to\tilde{\xi}$ and $\sigma^{\prime}:\xi\to\hat{\xi}$ be
sections of $j$ and $\hat{j}$ respectively. Then, for any $a\in\xi_{m},~{}m\in
M$ we have

$\hat{j}\circ
f(\sigma_{m}(a))=j\circ(\sigma_{m}(a))=a=\hat{j}\circ(\sigma_{m}^{\prime}(a))$
$\Rightarrow\hat{j}(f(\sigma_{m}(a))-\sigma_{m}^{\prime}(a))=0.$

This implies
$f(\sigma_{m}(a))-\sigma_{m}^{\prime}(a)\in\operatorname*{Ker}(\hat{j}_{m})=\eta_{m}$,
that is, $f(\sigma_{m}(a))=\sigma_{m}^{\prime}(a)+v_{a}$ for some
$v_{a}\in\eta_{m}$. Thus, we have for any $a,b\in\xi_{m}$ and $v\in\eta_{m}$

$f\big{(}\\{v,\sigma_{m}(a),\sigma_{m}(b)\\}_{m}^{\sim}\big{)}=\\{f(v),f(\sigma_{m}(a)),f(\sigma_{m}(b))\\}_{m}^{\wedge}=\\{v,\sigma_{m}^{\prime}(a),\sigma_{m}^{\prime}(b)\\}_{m}^{\wedge}.$

Note that $f(v)=v$ follows from the commutativity of first box in the diagram.
Therefore, equivalent extensions induce the same $\theta$. Similarly one can
show that equivalent extensions induce the same $D$ and $\rho$. ∎

As of now, we have seen that any extension ${Ext}_{\tilde{\xi}}$

${0}$${\eta}$${{\tilde{\xi}}}$${\xi}$${0}$$\scriptstyle{i}$$\scriptstyle{j}$

of $\xi$ by $\eta$, induces a representation $(\eta;\rho,D,\theta)$ of $\xi$,
where the vector bundle maps $\rho,D$, and $\theta$ are defined by (1)- (3) in
terms of a section $\sigma:\xi\to\tilde{\xi}$ of $j:\tilde{\xi}\to\xi$.
Therefore, as discussed in the previous section, we have the following cochain
complex of the Lie-Yamaguti algebra bundle $\xi$ with coefficients in the
induced representation $(\eta;\rho,D,\theta)$ of $\xi.$

${{C^{0}(\xi;\eta)}}$${{C^{(2,3)}(\xi;\eta)}}$${{C^{(4,5)}(\xi;\eta)}}$${\cdots}$${{C^{(3,4)}(\xi;\eta)}}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta^{*}}$$\scriptstyle{\delta}$

Our next goal is to attach a $(2,3)$-cocycle of the above cochain complex to
${Ext}_{\tilde{\xi}}.$

Fix a section $\sigma:\xi\to\tilde{\xi}$ of $j:\tilde{\xi}\to\xi$. Define two
maps; $f:\xi\otimes\xi\to\eta$ and $g:\xi\otimes\xi\otimes\xi\to\eta$ in the
following way. Let $m\in M.$ Denote by $f_{m}$ and $g_{m}$ the resulting
bilinear and trilinear maps obtained by restricting $f$ and $g$ to the fibres
$(\xi\otimes\xi)_{m}$ and $(\xi\otimes\xi\otimes\xi)_{m}$ respectively. For
all $a_{1},~{}a_{2},~{}a_{3}\in\xi_{m},$ define

(4) $\displaystyle f_{m}(a_{1},a_{2})$
$\displaystyle:=[\sigma_{m}(a_{1}),\sigma_{m}(a_{2})]_{m}^{\sim}-\sigma_{m}([a_{1},a_{2}]_{m})$
(5) $\displaystyle g_{m}(a_{1},a_{2},a_{3})$
$\displaystyle:=\\{\sigma_{m}(a_{1}),\sigma_{m}(a_{2}),\sigma_{m}(a_{3})\\}_{m}^{\sim}-\sigma_{m}(\\{a_{1},a_{2},a_{3}\\}_{m})$

Note that $(f,g)\in C^{(2,3)}(\xi;\eta).$

######  6.5 Proposition.

For any given abelian extension ${Ext}_{\tilde{\xi}}$ of $\xi$ by $\eta$, the
cochain $(f,g)\in C^{(2,3)}(\xi;\eta)$ as defined above is a $(2,3)$-cocycle.

###### Proof.

To show that $(f,g)$ is a $(2,3)$-cocycle, we need to show

$\delta(f,g)=0~{}~{}\text{and}~{}~{}\delta^{*}(f,g)=0,$

that is,

$\delta_{I}f=0,~{}\delta_{II}g=0~{}~{}\text{and}~{}~{}\delta_{I}^{*}f=0,~{}\delta_{II}^{*}g=0.$

Recall that the representation induced by the given extension are given by the
vector bundle morphisms $\rho,D,$ and $\theta,$ where for $a,b\in\xi_{m}$ and
$v\in\eta_{m}$, $m\in M,$

$\displaystyle\rho_{m}(a)(v)$ $\displaystyle=[\sigma_{m}(a),v]_{m}^{\sim}$
$\displaystyle D_{m}(a,b)(v)$
$\displaystyle=\\{\sigma_{m}(a),\sigma_{m}(b),v\\}_{m}^{\sim}$
$\displaystyle\theta_{m}(a,b)(v)$
$\displaystyle=\\{v,\sigma_{m}(a),\sigma_{m}(b)\\}_{m}^{\sim}.$

Let $a_{i}\in\xi_{m},$ $1\leq i\leq 5.$

By the definitions of $\delta$ and $\delta^{*}$ we get

$\displaystyle(\delta_{I})_{m}f_{m}(a_{1},a_{2},a_{3},a_{4})$
$\displaystyle=-\rho_{m}(a_{3})g_{m}(a_{1},a_{2},a_{4})+\rho_{m}(a_{4})g_{m}(a_{1},a_{2},a_{3})+g_{m}(a_{1},a_{2},[a_{3},a_{4}]_{m})$
$\displaystyle+D_{m}(a_{1},a_{2})f_{m}(a_{3},a_{4})-f_{m}(\\{a_{1},a_{2},a_{3}\\}_{m},a_{4})-f(a_{3},\\{a_{1},a_{2},a_{4}\\}_{m})$
$\displaystyle=0,$

where we have used the definition of representation as given above and (LY5).
Similarly, we obtain using (LY6)

$\displaystyle(\delta_{II})_{m}g_{m}(a_{1},a_{2},a_{3},a_{4},a_{5})$
$\displaystyle=-\theta_{m}(a_{4},a_{5})g_{m}(a_{1},a_{2},a_{3})+\theta_{m}(a_{3},a_{5})g_{m}(a_{1},a_{2},a_{4})$
$\displaystyle+D_{m}(a_{1},a_{2})g_{m}(a_{3},a_{4},a_{5})-D_{m}(a_{3},a_{4})g_{m}(a_{1},a_{2},a_{5})$
$\displaystyle-
g_{m}(\\{a_{1},a_{2},a_{3}\\}_{m},a_{4},a_{5})-g_{m}(a_{3},\\{a_{1},a_{2},a_{4}\\}_{m},a_{5})$
$\displaystyle-
g_{m}(a_{3},a_{4},\\{a_{1},a_{2},a_{5}\\}_{m})+g_{m}(a_{1},a_{2},\\{a_{3},a_{4},a_{5}\\}_{m})$
$\displaystyle=0.$

Moreover, from the above definition of representation and (LY3) we get

$\displaystyle(\delta_{I}^{*})_{m}f_{m}(a_{1},a_{2},a_{3})$
$\displaystyle=-\sum_{\circlearrowleft(a_{1},a_{2},a_{3})}\rho_{m}(a_{1})f_{m}(a_{2},a_{3})+\sum_{\circlearrowleft(a_{1},a_{2},a_{3})}f_{m}([a_{1},a_{2}]_{m},a_{3})$
$\displaystyle+\sum_{\circlearrowleft(a_{1},a_{2},a_{3})}g_{m}(a_{1},a_{2},a_{3})$
$\displaystyle=0,$

and from (LY4) we obtain

$\displaystyle(\delta_{II}^{*})_{m}g_{m}(a_{1},a_{2},a_{3},a_{4})$
$\displaystyle=\theta_{m}(a_{1},a_{4})f_{m}(a_{2},a_{3})+\theta_{m}(a_{2},a_{4})f_{m}(a_{3},a_{4})+\theta_{m}(a_{3},a_{4})f_{m}(a_{1},a_{2})$
$\displaystyle+g_{m}([a_{1},a_{2}]_{m},a_{3},a_{4})+g_{m}([a_{2},a_{3}]_{m},a_{1},a_{4})+g_{m}([a_{3},a_{1}]_{m},a_{2},a_{4})$
$\displaystyle=0.$

Thus, $(f,g)\in C^{(2,3)}(\xi;\eta)$ is a $(2,3)$-cocycle. ∎

By a routine calculation we obtain the following result.

######  6.6 Corollary.

If $\sigma,\sigma^{\prime}:\xi\to\tilde{\xi}$ are any two chosen sections of
$j:\tilde{\xi}\to\xi$ and $(f,g),~{}(f^{\prime},g^{\prime})$ are the
corresponding cocycles as obtained in Proposition 6.5, then $(f,g)$ and
$(f^{\prime},g^{\prime})$ are cohomologous. Hence, the extension
${Ext}_{\tilde{\xi}}$ of $\xi$ by $\eta$ determines uniquely an element of
$H^{(2,3)}(\xi;\eta).$

On the other hand, given a Lie-Yamaguti algebra bundle $\xi$ equipped with a
representation $(\eta;\rho,D,\theta),$ any $(2,3)$-cocycle in
$Z^{(2,3)}(\xi;\eta)$ determines an abelian extension of $\xi$ by $\eta$ which
is unique up to equivalence.

Let $\xi$ be a given Lie-Yamaguti algebra bundle over $M$ and
$(\eta;\rho,D,\theta)$ be a given representation of $\xi.$ Let $(f,g)\in
Z^{(2,3)}(\xi;\eta).$ Then, we have the following result.

######  6.7 Lemma.

Given $(f,g)\in Z^{(2,3)}(\xi;\eta),$ the vector bundle
$\tilde{\xi}=\xi\oplus\eta$ becomes a Lie-Yamaguti algebra bundle, where the
associated $2$-field and $3$-field

$m\mapsto[~{},~{}]_{m}^{\sim},~{}~{}m\mapsto\\{~{},~{},~{}\\}_{m}^{\sim},~{}~{}m\in
M$

are given by

$\displaystyle[a_{1}+w_{1},a_{2}+w_{2}]_{m}^{\sim}$
$\displaystyle:=[a_{1},a_{2}]_{m}+f_{m}(a_{1},a_{2})+\rho_{m}(a_{1})(w_{2})-\rho_{m}(a_{2})(w_{1})$
$\displaystyle\\{a_{1}+w_{1},a_{2}+w_{2},a_{3}+w_{3}\\}_{m}^{\sim}$
$\displaystyle:=\\{a_{1},a_{2},a_{3}\\}_{m}+g_{m}(a_{1},a_{2},a_{3})+D_{m}(a_{1},a_{2})(w_{3})$
$\displaystyle-\theta_{m}(a_{1},a_{3})(w_{2})+\theta_{m}(a_{2},a_{3})(w_{1})$

where $a_{1},a_{2},a_{3}\in\xi_{m}$ and $w_{1},w_{2},w_{3}\in\eta_{m}$. It is
convenient to denote this Lie-Yamaguti algebra bundle by
$\xi\oplus_{(f,g)}\eta$ to emphasize that it is induced by the given cocycle.

###### Proof.

Clearly the assignments

$m\mapsto[~{},~{}]_{m}^{\sim},~{}~{}m\mapsto\\{~{},~{},~{}\\}_{m}^{\sim},~{}~{}m\in
M$

as defined in the statement are smooth. So, it is enough to show that for any
$m\in M,~{}\tilde{\xi}_{m}$ is a Lie-Yamaguti algebra. Let $m\in M$. It is
easy to see that (LY1) and (LY2) holds for $[~{},~{}]_{m}^{\sim}$ and
$\\{~{},~{},~{}\\}_{m}^{\sim}$ defined above. To verify (LY6) proceed as
follows.

$\displaystyle\big{\\{}a_{1}$
$\displaystyle+w_{1},a_{2}+w_{2},\\{b_{1}+v_{1},b_{2}+v_{2},b_{3}+v_{3}\\}_{m}^{\sim}\big{\\}}_{m}^{\sim}$
$\displaystyle=\big{\\{}a_{1}+w_{1},~{}a_{2}+w_{2},~{}\\{b_{1},b_{2},b_{3}\\}_{m}+g_{m}(b_{1},b_{2},b_{3})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+D_{m}(b_{1},b_{2})(v_{3})-\theta_{m}(b_{1},b_{3})(v_{2})+\theta_{m}(b_{2},b_{3})(v_{1})\big{\\}}_{m}^{\sim}$
$\displaystyle=\\{a_{1},a_{2},\\{b_{1},b_{2},b_{3}\\}_{m}\\}_{m}+g_{m}(a_{1},a_{2},\\{b_{1},b_{2},b_{3}\\})+D_{m}(a_{1},a_{2})g_{m}(b_{1},b_{2},b_{3})$
$\displaystyle~{}~{}~{}~{}~{}~{}+D_{m}(a_{1},a_{2})D_{m}(b_{1},b_{2})(v_{3})-D_{m}(a_{1},a_{2})\theta_{m}(b_{1},b_{3})(v_{2})+D_{m}(a_{1},a_{2})\theta_{m}(b_{2},b_{3})(v_{1})$
$\displaystyle~{}~{}~{}~{}~{}~{}-\theta_{m}(a_{1},\\{b_{1},b_{2},b_{3}\\}_{m})(w_{2})+\theta_{m}(a_{2},\\{b_{1},b_{2},b_{3}\\}_{m})(w_{1})$
$\displaystyle\big{\\{}\\{a_{1}$
$\displaystyle+w_{1},a_{2}+w_{2},b_{1}+v_{1}\\}_{m}^{\sim},b_{2}+v_{2},b_{3}+v_{3}\big{\\}}_{m}^{\sim}$
$\displaystyle=\big{\\{}\\{a_{1},a_{2},b_{1}\\}_{m}+g_{m}(a_{1},a_{2},b_{1})+D_{m}(a_{1},a_{2})(v_{1})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\theta_{m}(a_{1},b_{1})(w_{2})+\theta_{m}(a_{2},b_{1})(w_{1}),~{}b_{2}+v_{2},~{}b_{3}+v_{3}\big{\\}}_{m}^{\sim}$
$\displaystyle=\\{\\{a_{1},a_{2},b_{1}\\}_{m},b_{2},b_{3}\\}_{m}+g_{m}(\\{a_{1},a_{2},b_{1}\\}_{m},b_{2},b_{3})+D_{m}(\\{a_{1},a_{2},b_{1}\\}_{m},b_{2})(v_{3})$
$\displaystyle~{}~{}~{}~{}~{}-\theta_{m}(\\{a_{1},a_{2},b_{1}\\}_{m},b_{3})(v_{2})+\theta_{m}(b_{2},b_{3})g_{m}(a_{1},a_{2},b_{1})+\theta_{m}(b_{2},b_{3})D_{m}(a_{1},a_{2})(v_{1})$
$\displaystyle~{}~{}~{}~{}~{}-\theta_{m}(b_{2},b_{3})\theta_{m}(a_{1},b_{1})(w_{2})+\theta_{m}(b_{2},b_{3})\theta_{m}(a_{2},b_{1})(w_{1})$
$\displaystyle\big{\\{}b_{1}$
$\displaystyle+v_{1},\\{a_{1}+w_{1},a_{2}+w_{2},b_{2}+v_{2}\\}_{m}^{\sim},b_{3}+v_{3}\big{\\}}_{m}^{\sim}$
$\displaystyle=\big{\\{}b_{1}+v_{1},~{}\\{a_{1},a_{2},b_{2}\\}_{m}+g_{m}(a_{1},a_{2},b_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+D_{m}(a_{1},a_{2})(v_{2})-\theta_{m}(a_{1},b_{2})(w_{2})+\theta_{m}(a_{2},b_{2})(w_{1}),~{}b_{3}+v_{3}\big{\\}}_{m}^{\sim}$
$\displaystyle=\\{b_{1},\\{a_{1},a_{2},b_{2}\\}_{m},b_{3}\\}_{m}+g_{m}(b_{1},\\{a_{1},a_{2},b_{2}\\}_{m},b_{3})+D_{m}(b_{1},\\{a_{1},a_{2},b_{2}\\}_{m})(v_{3})$
$\displaystyle~{}~{}~{}~{}~{}+\theta_{m}(\\{a_{1},a_{2},b_{2}\\}_{m},b_{3})(v_{1})-\theta_{m}(b_{1},b_{3})g_{m}(a_{1},a_{2},b_{2})-\theta_{m}(b_{1},b_{3})D_{m}(a_{1},a_{2})(v_{2})$
$\displaystyle~{}~{}~{}~{}~{}+\theta_{m}(b_{1},b_{3})\theta_{m}(a_{1},b_{2})(w_{2})-\theta_{m}(b_{1},b_{3})\theta_{m}(a_{2},b_{2})(w_{1})$
$\displaystyle\big{\\{}b_{1}$
$\displaystyle+v_{1},b_{2}+v_{2},\\{a_{1}+w_{1},a_{2}+w_{2},b_{3}+v_{3}\\}_{m}^{\sim}\big{\\}}_{m}^{\sim}$
$\displaystyle=\big{\\{}b_{1}+v_{1},~{}b_{2}+v_{2},~{}\\{a_{1},a_{2},b_{3}\\}_{m}+g_{m}(a_{1},a_{2},b_{3})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+D_{m}(a_{1},a_{2})(v_{3})-\theta_{m}(a_{1},a_{3})(w_{2})+\theta_{m}(a_{2},a_{3})(w_{1})\big{\\}}_{m}^{\sim}$
$\displaystyle=\\{b_{1},b_{2},\\{a_{1},a_{2},b_{3}\\}_{m}\\}_{m}+g_{m}(b_{1},b_{2},\\{a_{1},a_{2},b_{3}\\}_{m})+D_{m}(b_{1},b_{2})g_{m}(a_{1},a_{2},b_{3})$
$\displaystyle~{}~{}~{}~{}~{}+D_{m}(b_{1},b_{2})D_{m}(a_{1},a_{2})(v_{3})-D_{m}(b_{1},b_{2})\theta_{m}(a_{1},a_{3})(w_{2})+D_{m}(b_{1},b_{2})\theta_{m}(a_{2},a_{3})(w_{1})$
$\displaystyle~{}~{}~{}~{}~{}-\theta_{m}(b_{1},\\{a_{1},a_{2},b_{3}\\})(v_{2})+\theta_{m}(b_{2},\\{a_{1},a_{2},b_{3}\\})(v_{1})$

Using (RLYB6), (RLYB4) and the definition of coboundary maps we can show

$\displaystyle\big{\\{}a_{1}+w_{1},$ $\displaystyle
a_{2}+w_{2},\\{b_{1}+v_{1},b_{2}+v_{2},b_{3}+v_{3}\\}_{m}^{\sim}\big{\\}}_{m}^{\sim}$
$\displaystyle=\big{\\{}\\{a_{1}+w_{1},a_{2}+w_{2},b_{1}+v_{1}\\}_{m}^{\sim},b_{2}+v_{2},b_{3}+v_{3}\big{\\}}_{m}^{\sim}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\big{\\{}b_{1}+v_{1},\\{a_{1}+w_{1},a_{2}+w_{2},b_{2}+v_{2}\\}_{m}^{\sim},b_{3}+v_{3}\big{\\}}_{m}^{\sim}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\big{\\{}b_{1}+v_{1},b_{2}+v_{2},\\{a_{1}+w_{1},a_{2}+w_{2},b_{3}+v_{3}\\}_{m}^{\sim}\big{\\}}_{m}^{\sim}$

giving us (LY6). Other relations, (LY3), (LY4), (LY5) can also be obtained in
the same way. Thus making $\xi\oplus_{(f,g)}\eta$ a Lie-Yamaguti algebra
bundle.

∎

Observe that the Lie-Yamaguti algebra brackets of the fibres of
$\xi\oplus_{(f,g)}\eta$ makes $\eta$ an abelian ideal in
$\xi\oplus_{(f,g)}\eta$ and we have the following extension of $\xi$ by
$\eta$:

${0}$${\eta}$${{\xi\oplus_{(f,g)}\eta}}$${\xi}$${0}$$\scriptstyle{i}$$\scriptstyle{j}$

where $i$ is the inclusion map and $j$ is the projection map.

Let $(h,k)\in Z^{(2,3)}(\xi;\eta)$ be another cocycle. Then, we have the
following result.

######  6.8 Lemma.

Two extensions $0\to\eta\to\xi\oplus_{(f,g)}\eta\to\xi\to 0$ and
$0\to\eta\to\xi\oplus_{(h,k)}\to\xi\to 0$ are equivalent iff
$(f,g),~{}(h,k)\in Z^{(2,3)}(\xi;\eta)$ are cohomologous.

###### Proof.

Let the two extensions
$0\to\eta\xrightarrow[]{i}\xi\oplus_{(f,g)}\eta\xrightarrow[]{p}\xi\to 0$ and
$0\to\eta\xrightarrow[]{i}\xi\oplus_{(h,k)}\eta\xrightarrow[]{p}\xi\to 0$ be
equivalent through a Lie-Yamaguti algebra isomorphism

$\gamma:\xi\oplus_{(f,g)}\eta\to\xi\oplus_{(h,k)}\eta$

Then for each $m\in M$ we have the following equivalence of abelian extension.

${0}$${{\eta_{m}}}$${{\xi_{m}\oplus_{(f_{m},g_{m})}\eta_{m}}}$${{\xi_{m}}}$${0}$${0}$${{\eta_{m}}}$${{\xi_{m}\oplus_{(h_{m},k_{m})}\eta_{m}}}$${{\xi_{m}}}$${0}$$\scriptstyle{id}$$\scriptstyle{\gamma_{m}}$$\scriptstyle{id}$

To show that $(f,g)$ and $(h,k)$ are cohomologous it is enough to show for
each $m\in M$, $(f_{m},g_{m})$ and $(h_{m},k_{m})$ are cohomologous, that is,

$(f_{m},g_{m})-(h_{m},k_{m})\in B^{(2,3)}(\xi_{m};\eta_{m})$

We define a map $\lambda_{m}:\xi_{m}\to\eta_{m}$ by
$\lambda_{m}(a)=\gamma_{m}(a)-a$ by which one can show

$f_{m}-h_{m}=(\delta_{I})_{m}(\lambda_{m})~{}~{}\text{and}~{}~{}g_{m}-k_{m}=(\delta_{II})_{m}(\lambda_{m})$

Conversely, assume that for each $m\in M,~{}(f_{m},g_{m})$ and $(h_{m},k_{m})$
are in the same cohomology class, that is,
$(f_{m},g_{m})-(h_{m},k_{m})=(\delta)_{m}(\lambda_{m})$. Then,
$\gamma_{m}:\xi_{m}\oplus_{(f,g)}\eta_{m}\to\xi_{m}\oplus_{(h,k)}\eta_{m}$
defined by

$\gamma_{m}(a+v)=a+\lambda_{m}(a)+v$

gives the required isomorphism. ∎

By summarizing the above observations we have the following theorem.

######  6.9 Theorem.

To each equivalence class of abelian extensions of $\xi$ by $\eta$ there
corresponds an element of $H^{(2,3)}(\xi;\eta)$.
Suppose $\xi$ is a given Lie-Yamaguti algebra bundle over $M$ equipped with a
representation $(\eta;\rho,~{}D,\theta).$ To each cohomology class $[(f,g)]\in
H^{(2,3)}(\xi;\eta)$, there is an extension of $\xi$ by $\eta$

${0}$${\eta}$${{\xi\oplus_{(f,g)}\eta}}$${\xi}$${0}$$\scriptstyle{i}$$\scriptstyle{j}$

which is unique up to equivalence of extensions.

We conclude addressing the following questions.

Question: Does there exist a notion of Lie-Yamaguti algebroid which appears as
the infinitesimal version of some groupoid with relevant structure?

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