# Nearly associative and nearly Hom-associative algebras and bialgebras

Mafoya Landry Dassoundo1 Sergei Silvestrov2
1Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071,
China
e-mail<EMAIL_ADDRESS>
2 Division of Mathematics and Physics, School of Education, Culture and
Communication,
Mälardalen University, Box 883, 72123 Västeras, Sweden.
e-mail<EMAIL_ADDRESS>
###### Abstract

Basic definitions and properties of nearly associative algebras are described.
Nearly associative algebras are proved to be Lie-admissible algebras. Two-
dimensional nearly associative algebras are classified, and its main classes
are derived. The bimodules, matched pairs and Manin triple of a nearly
associative algebras are derived and their equivalence with nearly associative
bialgebras is proved. Basic definitions and properties of nearly Hom-
associative algebras are described. Related bimodules and matched pairs are
given, and associated identities are established.

000Corresponding author: Sergei Silvestrov<EMAIL_ADDRESS>
##  1 Introduction

An algebra $A$ with a bilinear product $\cdot:A\times A\rightarrow A$ is not
necessarily associative or possibly non-associative if possibly there exist
$x,y,z\in A$ such that $(x\cdot y)\cdot z-x\cdot(y\cdot z)\neq 0.$ If such
$x,y,z\in A$ exist, then algebra is not associative. The term non-associative
algebras is used often to mean all possibly non-associative algebras,
including also the associative algebras. Associative algebras, Lie algebras,
and Jordan algebras are well-known sub-classes of non-associative algebras in
the sense of possibly not associative algebras [60].

Hom-algebraic structures originated from quasi-deformations of Lie algebras of
vector fields which gave rise to quasi-Lie algebras, defined as generalized
Lie structures in which the skew-symmetry and Jacobi conditions are twisted.
Hom-Lie algebras and more general quasi-Hom-Lie algebras where introduced
first by Silvestrov and his students Hartwig and Larsson in [27], where the
general quasi-deformations and discretizations of Lie algebras of vector
fields using general twisted derivations, $\sigma$-derivations, and a general
method for construction of deformations of Witt and Virasoro type algebras
based on twisted derivations have been developed. The initial motivation came
from examples of $q$-deformed Jacobi identities discovered in $q$-deformed
versions and other discrete modifications of differential calculi and
homological algebra, $q$-deformed Lie algebras and other algebras important in
string theory, vertex models in conformal field theory, quantum mechanics and
quantum field theory, such as the $q$-deformed Heisenberg algebras,
$q$-deformed oscillator algebras, $q$-deformed Witt, $q$-deformed Virasoro
algebras and related $q$-deformations of infinite-dimensional algebras [1, 16,
17, 18, 19, 20, 21, 22, 33, 34, 41, 42, 43].

Possibility of studying, within the same framework, $q$-deformations of Lie
algebras and such well-known generalizations of Lie algebras as the color and
super Lie algebras provided further general motivation for development of
quasi-Lie algebras and subclasses of quasi-Hom-Lie algebras and Hom-Lie
algebras. The general abstract quasi-Lie algebras and the subclasses of quasi-
Hom-Lie algebras and Hom-Lie algebras, as well as their color (graded)
counterparts, color (graded) quasi-Lie algebras, color (graded) quasi-Hom-Lie
algebras and color (graded) Hom-Lie algebras, including in particular the
super quasi-Lie algebras, super quasi-Hom-Lie algebras, and super Hom-Lie
algebras, have been introduced in [27, 37, 38, 39, 63, 64]. In [48], Hom-
associative algebras were introduced, generalizing associative algebras by
twisting the associativity law by a linear map. Hom-associative algebra is a
triple $(A,\cdot,\alpha)$ consisting of a linear space $A$, a bilinear product
$\cdot:A\times A\rightarrow A$ and a linear map $\alpha:A\rightarrow A$,
satisfying $a_{\alpha,\cdot}(x,y,z)=(x\cdot
y)\cdot\alpha(z)-\alpha(x)\cdot(y\cdot z)=0,$ for any $x,y,z\in A$. In [48],
alongside Hom-associative algebras, the Hom-Lie admissible algebras
generalizing Lie-admissible algebras, were introduced as Hom-algebras such
that the commutator product, defined using the multiplication in a Hom-
algebra, yields a Hom-Lie algebra, and also Hom-associative algebras were
shown to be Hom-Lie admissible. Moreover, in [48], more general $G$-Hom-
associative algebras including Hom-associative algebras, Hom-Vinberg algebras
(Hom-left symmetric algebras), Hom-pre-Lie algebras (Hom-right symmetric
algebras), and some other Hom-algebra structures, generalizing $G$-associative
algebras, Vinberg and pre-Lie algebras respectively, have been introduced and
shown to be Hom-Lie admissible, meaning that for these classes of Hom-
algebras, the operation of taking commutator leads to Hom-Lie algebras as
well. Also, flexible Hom-algebras have been introduced, connections to Hom-
algebra generalizations of derivations and of adjoint maps have been noticed,
and some low-dimensional Hom-Lie algebras have been described. The enveloping
algebras of Hom-Lie algebras were considered in [67] using combinatorial
objects of weighted binary trees. In [29], for Hom-associative algebras and
Hom-Lie algebras, the envelopment problem, operads, and the Diamond Lemma and
Hilbert series for the Hom-associative operad and free algebra have been
studied. Strong Hom-associativity yielding a confluent rewrite system and a
basis for the free strongly hom-associative algebra has been considered in
[28]. An explicit constructive way, based on free Hom-associative algebras
with involutive twisting, was developed in [25] to obtain the universal
enveloping algebras and Poincaré-Birkhoff-Witt type theorem for Hom-Lie
algebras with involutive twisting map. Free Hom-associative color algebra on a
Hom-module and enveloping algebra of color Hom-Lie algebras with involutive
twisting and also with more general conditions on the powers of twisting map
was constructed, and Poincaré-Birkhoff-Witt type theorem was obtained in [4,
5]. It is worth noticing here that, in the subclass of Hom-Lie algebras, the
skew-symmetry is untwisted, whereas the Jacobi identity is twisted by a single
linear map and contains three terms as in Lie algebras, reducing to ordinary
Lie algebras when the twisting linear map is the identity map.

Hom-algebra structures include their classical counterparts and open new broad
possibilities for deformations, extensions to Hom-algebra structures of
representations, homology, cohomology and formal deformations, Hom-modules and
hom-bimodules, Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-
bialgebras, Hom-Hopf algebras, $L$-modules, $L$-comodules and Hom-Lie quasi-
bialgebras, $n$-ary generalizations of biHom-Lie algebras and biHom-
associative algebras and generalized derivations, Rota-Baxter operators, Hom-
dendriform color algebras, Rota-Baxter bisystems and covariant bialgebras,
Rota-Baxter cosystems, coquasitriangular mixed bialgebras, coassociative Yang-
Baxter pairs, coassociative Yang-Baxter equation and generalizations of Rota-
Baxter systems and algebras, curved $\mathcal{O}$-operator systems and their
connections with tridendriform systems and pre-Lie algebras, BiHom-algebras,
BiHom-Frobenius algebras and double constructions, infinitesimal biHom-
bialgebras and Hom-dendriform $D$-bialgebras, Hom-algebras has been considered
from a category theory point of view [3, 8, 9, 10, 12, 11, 13, 14, 15, 24, 26,
30, 31, 35, 36, 37, 40, 44, 45, 46, 49, 50, 51, 52, 56, 57, 62, 61, 65, 66,
67, 68, 69, 70].

This paper is organized as follows. In Section 2, basic definitions and
fundamental identities and some elementary examples of nearly associative
algebras are given. In Section 3, we derive the classification of the two-
dimensional nearly associative algebras and main classes are provided. In
Section 4, bimodules, duals bimodules and matched pair of nearly associative
algebras are established and related identities are derived and proved. In
Section 5, Manin triple of nearly associative algebras is given and its
equivalence to the nearly associative bialgebras is derived. In Section 6,
Hom-Lie-admissible, $G$-Hom-associative, flexible Hom-algebras, the result on
Lie-admissibility of $G$-Hom-admissible algebras and subclasses of $G$-Hom-
admissible algebras are reviewed. In Section 7, main definitions and
fundamental identities of Hom-nearly associative algebras are given.
Furthermore, the bimodules, and matched pair of the Hom-nearly associative
algebras are derived and related properties are obtained.

##  2 Nearly associative algebras: basic definitions and properties

Throughout this paper, for simplicity of exposition, all linear spaces are
assumed to be over field $\mathbb{K}$ of characteristic is $0$, even though
many results hold in general for other fields as well unchanged or with minor
modifications. An algebra is a couple $(A,\mu)$ consisting of a linear space
$A$ and a bilinear product $\mu:A\times A\rightarrow A$.

######  Definition 2.1.

An algebra $(A,\cdot)$ is called nearly associative if, for all $x,y,z\in A$,

$\displaystyle x\cdot(y\cdot z)=(z\cdot x)\cdot y.$ (1)

######  Example 2.2.

Consider a two-dimensional linear space $A$ with basis $\\{e_{1},e_{2}\\}$.

  * •

Then, $(A,\cdot)$ is a nearly associative algebra, where $e_{1}\cdot
e_{1}=e_{1}+e_{2}$ and for all $(i,j)\neq(1,1)$ with $i,j\in\\{1,2\\}$,
$e_{i}\cdot e_{j}=0$.

  * •

The linear product defined on $A$ by: $e_{1}\cdot e_{1}=e_{2}$, $e_{1}\cdot
e_{2}=e_{1}=e_{2}\cdot e_{1}$ and $e_{2}\cdot e_{2}=e_{2}$, is such that
$(A,\cdot)$ is a nearly associative algebra.

######  Example 2.3.

Consider a three-dimensional linear space $A$ with basis
$\\{e_{1},e_{2},e_{3}\\}$.

  * •

The linear space $A$ equipped with the linear product defined on $A$ by:
$e_{1}\cdot e_{1}=e_{2}+e_{3}$, $e_{2}\cdot e_{2}=e_{1}+e_{2}-e_{3}$,
$e_{3}\cdot e_{3}=-e_{1}+e_{2}$ and for all $i\neq j,e_{i}\cdot e_{j}=0$,
where $i,j\in\\{1,2,3\\}$, is a nearly associative algebra.

  * •

The linear space $A$ equipped with the linear product defined on $A$ by:
$e_{1}\cdot e_{1}=e_{2}-e_{3}$, $e_{2}\cdot e_{2}=e_{2}+e_{3}$, $e_{3}\cdot
e_{3}=e_{1}-e_{2}+e_{3}$ and for all $i\neq j,e_{i}\cdot e_{j}=0$, where
$i,j\in\\{1,2,3\\}$, is a nearly associative algebra.

  * •

The linear space $A$ equipped with the linear product defined on $A$ by:
$e_{2}\cdot e_{2}=e_{1}+e_{3}$, $e_{1}\cdot e_{1}=e_{1}+e_{2}+e_{3}$,
$e_{3}\cdot e_{3}=e_{1}+e_{2}$ and for all $i\neq j,e_{i}\cdot e_{j}=0$, where
$i,j\in\\{1,2,3\\}$, is a nearly associative algebra.

######  Definition 2.4 ([2, 23, 53, 54, 55, 58, 59]).

An algebra $(A,\cdot)$ is called Lie admissible if $(A,[.,.])$ is a Lie
algebra, where $[x,y]=x\cdot y-y\cdot x$ for all $x,y\in A$.

For a Lie admissible algebra $(A,\cdot)$, the Lie algebra
$\mathcal{G}(A)=(A,[.,.])$ is called an underlying Lie algebra of $(A,\cdot)$.

It is known that associative algebras, left-symmetric algebras and anti-
flexible algebras (center-symmetric algebras) are Lie-admissible [6, 7, 30].

######  Proposition 2.5.

Any nearly associative algebra is Lie-admissible.

###### Proof.

For $[.,.]:(v,w)\mapsto v\cdot w-w\cdot v$ and $x,y,z$ in a nearly associative
algebra $(A,\cdot)$,

$\displaystyle[x,[y,z]]+[y,[z,x]]+[z,[x,y]]$ $\displaystyle=[x,y\cdot z-z\cdot
y]+[y,z\cdot x-x\cdot z]+[z,x\cdot y-y\cdot x]$ $\displaystyle=x\cdot(y\cdot
z)-x\cdot(z\cdot y)-(y\cdot z)\cdot x+(z\cdot y)\cdot x$
$\displaystyle\quad+y\cdot(z\cdot x)-y\cdot(x\cdot z)-(z\cdot x)\cdot
y+(x\cdot z)\cdot y$ $\displaystyle\quad+z\cdot(x\cdot y)-z\cdot(y\cdot
x)-(x\cdot y)\cdot z+(y\cdot x)\cdot z$ $\displaystyle=\\{x\cdot(y\cdot
z)-(z\cdot x)\cdot y\\}+\\{(y\cdot x)\cdot z-x\cdot(z\cdot y)\\}$
$\displaystyle\quad+\\{y\cdot(z\cdot x)-(x\cdot y)\cdot z\\}+\\{z\ast(x\cdot
y)-(y\cdot z)\cdot x\\}$ $\displaystyle\quad+\\{(z\cdot y)\cdot
x-y\cdot(x\cdot z)\\}+\\{(x\cdot z)\cdot y-z\cdot(y\cdot x)\\}=0.$

Therefore, $(A,[.,.])$ is a Lie algebra. ∎

######  Remark 2.6.

In a nearly associative algebra $(A,\cdot)$, for $x,y\in A$,

$\displaystyle L(x)L(y)$ $\displaystyle=$ $\displaystyle R(y)R(x),$ (2a)
$\displaystyle L(x)R(y)$ $\displaystyle=$ $\displaystyle L({y\cdot x}),$ (2b)
$\displaystyle R(x)L(y)$ $\displaystyle=$ $\displaystyle R(x\cdot y),$ (2c)

where $L,R:A\rightarrow{\rm End}(A)$ are the operators of left and right
multiplications.

######  Definition 2.7.

An anti-flexible algebra is a couple $(A,\cdot)$ where $A$ is a linear space,
and $\cdot:A\times A\rightarrow A$ is a bilinear product such that for all
$x,y,z\in A$,

$\displaystyle(x\cdot y)\cdot z-(z\cdot y)\cdot x=x\cdot(y\cdot
z)-z\cdot(y\cdot x).$ (3)

Using associator $a(x,y,z)=(x\cdot y)\cdot z-x\cdot(y\cdot z)$, the equality
(3) is equivalent to

$\displaystyle a(x,y,z)=a(z,y,x).$ (4)

In view of (4), anti-flexible algebras were called center-symmetric algebras
in [30].

######  Proposition 2.8.

Any commutative nearly associative algebra is anti-flexible.

###### Proof.

For all $x,y,z\in A$ in a commutative nearly associative algebra $(A,\cdot)$,
by using nearly associativity, commutativity and again nearly associativity,

$\displaystyle a(x,y,z)=(x\cdot y)\cdot z-x\cdot(y\cdot z)=y\cdot(z\cdot
x)-(z\cdot x)\cdot y=[y,z\cdot x]=[y,x\cdot z]=$ $\displaystyle y\cdot(x\cdot
z)-(x\cdot z)\cdot y=(z\cdot y)\cdot x-z\cdot(y\cdot x)=a(z,y,x)$

proves (4) meaning that $(A,\cdot)$ is anti-flexible. ∎

##  3 Classification of the two-dimensional nearly associative algebras

######  Theorem 3.1.

Any two-dimensional algebra $(A,\cdot)$ is nearly associative if and only if

$\displaystyle e_{1}\cdot(e_{1}\cdot e_{1})=(e_{1}\cdot e_{1})\cdot
e_{1},\qquad e_{1}\cdot(e_{1}\cdot e_{2})=(e_{2}\cdot e_{1})\cdot e_{1},$
$\displaystyle e_{1}\cdot(e_{2}\cdot e_{1})=(e_{1}\cdot e_{1})\cdot
e_{2},\qquad e_{2}\cdot(e_{1}\cdot e_{1})=(e_{1}\cdot e_{2})\cdot e_{1},$
$\displaystyle e_{1}\cdot(e_{2}\cdot e_{2})=(e_{2}\cdot e_{1})\cdot
e_{2},\qquad e_{2}\cdot(e_{1}\cdot e_{2})=(e_{2}\cdot e_{2})\cdot e_{1},$
$\displaystyle e_{2}\cdot(e_{2}\cdot e_{1})=(e_{1}\cdot e_{2})\cdot
e_{2},\qquad e_{2}\cdot(e_{2}\cdot e_{2})=(e_{2}\cdot e_{2})\cdot e_{2},$

where $\\{e_{1},e_{2}\\}$ is a basis of $A.$

######  Theorem 3.2.

Any two-dimensional nearly associative algebra is isomorphic to one of the
following nearly associative algebras:

  * •

For all $(\alpha,\beta)\in\mathbb{K}^{2}\backslash\\{(0,0)\\}$, $e_{1}\cdot
e_{1}=\alpha e_{2},e_{1}\cdot e_{2}=\beta e_{1}=e_{2}\cdot e_{1},e_{2}\cdot
e_{2}=\beta e_{2}$.

  * •

For all $(\alpha,\beta)\in\mathbb{K}^{2}\backslash\\{(0,0)\\}$, $e_{1}\cdot
e_{1}=\alpha e_{1}+\beta e_{2},e_{1}\cdot e_{2}=\beta e_{1}+\alpha
e_{2}=e_{2}\cdot e_{1},\\\ e_{2}\cdot e_{2}=\alpha e_{1}+\beta e_{2}$.

  * •

For all $(\alpha,\beta,\gamma)\in\mathbb{K}^{3}$, such that
$\gamma^{2}+4\alpha\beta\geq 0,$ $e_{1}\cdot e_{1}=\alpha e_{1},e_{2}\cdot
e_{2}=\beta e_{1}+\gamma e_{2},\\\ e_{1}\cdot
e_{2}=\frac{1}{2}\left(\gamma+\sqrt{\gamma^{2}+4\alpha\beta}\right)e_{1}=e_{2}\cdot
e_{1}.$

###### Proof.

Equip the linear space $A$ with the basis $\\{e_{1},e_{2}\\}$, and for all
$i;j\in\\{1;2\\}$, set $e_{i}\cdot e_{j}=a_{ij}e_{1}+b_{ij}e_{2}$, where
$a_{ij}\in\mathbb{K}$ and $b_{ij}\in\mathbb{K}$. In addition, for all
$i,j,k\in\\{1,2\\}$,
$a_{jk}a_{i1}+b_{jk}a_{i2}=a_{ki}a_{1j}+b_{ki}a_{2j},a_{jk}b_{i1}+b_{jk}b_{i2}=a_{ki}b_{1j}+b_{ki}b_{2j}.$
By Theorem 3.1,

$\displaystyle\left\\{\begin{array}[]{lllllllllllllllll}e_{1}\cdot(e_{1}\cdot
e_{1})=(e_{1}\cdot e_{1})\cdot e_{1}\\\ e_{1}\cdot(e_{1}\cdot
e_{2})=(e_{2}\cdot e_{1})\cdot e_{1}\\\ e_{1}\cdot(e_{2}\cdot
e_{1})=(e_{1}\cdot e_{1})\cdot e_{2}\\\ e_{2}\cdot(e_{1}\cdot
e_{1})=(e_{1}\cdot e_{2})\cdot e_{1}\\\ e_{1}\cdot(e_{2}\cdot
e_{2})=(e_{2}\cdot e_{1})\cdot e_{2}\\\ e_{2}\cdot(e_{1}\cdot
e_{2})=(e_{2}\cdot e_{2})\cdot e_{1}\\\ e_{2}\cdot(e_{2}\cdot
e_{1})=(e_{1}\cdot e_{2})\cdot e_{2}\\\ e_{2}\cdot(e_{2}\cdot
e_{2})=(e_{2}\cdot e_{2})\cdot e_{2}\end{array}\right.$
$\displaystyle\Longleftrightarrow\left\\{\begin{array}[]{lllllllllllllllll}a_{11}a_{11}+b_{11}a_{12}=a_{11}a_{11}+b_{11}a_{21},\\\
a_{11}b_{11}+b_{11}b_{12}=a_{11}b_{11}+b_{11}b_{21},\\\
a_{12}a_{11}+b_{12}a_{12}=a_{21}a_{11}+b_{21}a_{21},\\\
a_{12}b_{11}+b_{12}b_{12}=a_{21}b_{11}+b_{21}b_{21},\\\
a_{21}a_{11}+b_{21}a_{12}=a_{11}a_{12}+b_{11}a_{22},\\\
a_{21}b_{11}+b_{21}b_{12}=a_{11}b_{12}+b_{11}b_{22},\\\
a_{11}a_{21}+b_{11}a_{22}=a_{12}a_{11}+b_{12}a_{21},\\\
a_{11}b_{21}+b_{11}b_{22}=a_{12}b_{11}+b_{12}b_{21},\\\
a_{22}a_{11}+b_{22}a_{12}=a_{21}a_{12}+b_{21}a_{22},\\\
a_{22}b_{11}+b_{22}b_{12}=a_{21}b_{12}+b_{21}b_{22},\\\
a_{12}a_{21}+b_{12}a_{22}=a_{22}a_{11}+b_{22}a_{21},\\\
a_{12}b_{21}+b_{12}b_{22}=a_{22}b_{11}+b_{22}b_{21},\\\
a_{21}a_{21}+b_{21}a_{22}=a_{12}a_{12}+b_{12}a_{22},\\\
a_{21}b_{21}+b_{21}b_{22}=a_{12}b_{12}+b_{12}b_{22},\\\
a_{22}a_{21}+b_{22}a_{22}=a_{22}a_{12}+b_{22}a_{22},\\\
a_{22}b_{21}+b_{22}b_{22}=a_{22}b_{12}+b_{22}b_{22}\end{array}\right.$
$\displaystyle\Longleftrightarrow\left\\{\begin{array}[]{lllllllllllllllll}e(b-c)=0,e(f-g)=0,\\\
h(b-c)=0,d(b-c)=0,\\\ d(f-g)=0,a(f-g)=0\\\ (b-c)(b+c)=0,\\\ (f-g)(f+g)=0\\\
e(b-c)+f(g-a)=0\\\ d(a-g)+b(h-c)=0\\\ a(b-c)=0,h(f-g)=0\\\ (bf-
cg)=0,bg=de=fc\end{array}\right.$
$\displaystyle\Longleftrightarrow\left\\{\begin{array}[]{cccccccccccccc}\left\\{\begin{array}[]{llllllllllll}a=r_{1},b=r_{2},c=r_{2},d=r_{1},\\\
e=r_{2},f=r_{1},g=r_{1},h=r_{2}\\\ \end{array}\right.\mbox{or
}\left\\{\begin{array}[]{llllllllllll}a=r_{1},b=r_{2},c=r_{2},d=r_{2},\\\
e=r_{1},f=r_{1},g=r_{1},h=r_{2}\\\ \end{array}\right.\\\ \mbox{or }\\\
\left\\{\begin{array}[]{lllllllllll}a=r_{5},b=0,c=0,d=0,\\\
e=r_{6},f=r_{5},g=r_{5},h=r_{8}\\\ \end{array}\right.\mbox{or
}\left\\{\begin{array}[]{lllllllllll}a=r_{5},b=0,c=0,d=r_{6},\\\
e=0,f=r_{5},g=r_{5},h=r_{8}\\\ \end{array}\right.\\\ \mbox{or }\\\
\left\\{\begin{array}[]{llllllllll}a=r_{9},b=\frac{|r_{12}|+r_{12}}{2},\\\
c=\frac{|r_{12}|+r_{12}}{2},d=0,\\\ e=r_{11},f=0,g=0,h=r_{12}\\\
\end{array}\right.\mbox{or
}\left\\{\begin{array}[]{llllllllll}a=r_{9},b=\frac{\sqrt{4r_{10}\,r_{9}+{{r_{12}}^{2}}}+r_{12}}{2},\\\
c=\frac{\sqrt{4r_{10}\,r_{9}+{{r_{12}}^{2}}}+r_{12}}{2},d=r_{10},\\\
e=0,f=0,g=0,h=r_{12}\end{array}\right.\\\ \mbox{or }\\\
\left\\{\begin{array}[]{lllllllllllllll}a=r_{13},b=\frac{r_{16}-\sqrt{{{r_{16}}^{2}}}}{2},\\\
c=\frac{r_{16}-\sqrt{{{r_{16}}^{2}}}}{2},d=0,\\\ e=r_{14},f=0\\\
,g=0,h=r_{16}\\\ \end{array}\right.\mbox{or
}\left\\{\begin{array}[]{lllllllllllllll}a=r_{13},b=\frac{r_{16}-\sqrt{{{r_{16}}^{2}}+4r_{13}\,r_{14}}}{2},\\\
c=\frac{r_{16}-\sqrt{{{r_{16}}^{2}}+4r_{13}\,r_{14}}}{2},d=r_{14},\\\
e=0,f=0,g=0,h=r_{16}\\\ \end{array}\right.\\\ \mbox{or }\\\
\left\\{\begin{array}[]{llllllllllllllll}a=r_{17},b=0,c=0,d=0,\\\
e=r_{18},f=0,g=0,h=0\\\ \end{array}\right.\mbox{or
}\left\\{\begin{array}[]{llllllllllllllll}a=r_{17},b=\sqrt{r_{17}\,r_{18}},c=\sqrt{r_{17}\,r_{18}},\\\
d=r_{18},e=0,f=0,g=0,h=0\\\ \end{array}\right.\\\ \mbox{or }\\\
\left\\{\begin{array}[]{lllllllllllllll}a=r_{20},b=0,c=0,d=0,\\\
e=r_{21},f=0,g=0,h=0\\\ \end{array}\right.\mbox{or
}\left\\{\begin{array}[]{lllllllllllllll}a=r_{20},b=-\sqrt{r_{20}\,r_{21}},\\\
c=-\sqrt{r_{20}\,r_{21}},d=r_{21},\\\ e=0,f=0,g=0,h=0\\\ \end{array}\right.\\\
\mbox{or }\\\ \left\\{\begin{array}[]{lllllllllllll}a=0,b=0,c=0,d=0,\\\
e=r_{24},f=0,g=0,h=r_{25}\\\ \end{array}\right.\mbox{or
}\left\\{\begin{array}[]{lllllllllllll}a=0,b=0,c=0,d=r_{23},\\\
e=0,f=0,g=0,h=r_{25}\\\ \end{array}\right.\\\ \mbox{or }\\\
\left\\{\begin{array}[]{lllllllllllllll}a=0,b=r_{26},c=r_{26},d=0,\\\
e=r_{28},f=0,g=0,h=r_{26}\\\ \end{array}\right.\mbox{or
}\left\\{\begin{array}[]{lllllllllllllll}a=0,b=r_{26},c=r_{26},d=r_{27},\\\
e=0,f=0,g=0,h=r_{26}\\\ \end{array}\right.\\\ \mbox{or }\\\
\left\\{\begin{array}[]{llllllllllllllll}a=r_{29},b=0,c=0,d=0,\\\
e=0,f=0,g=0,h=0\end{array}\right.\end{array}\right.$

with
$a_{11}=a,a_{12}=b,a_{21}=c,a_{22}=d,b_{11}=e,b_{12}=f,b_{21}=g,b_{22}=h.$

Therefore, the non-isomorphic algebras generated by these constants structures
are:

  * •

For all $(\alpha,\beta)\in\mathbb{K}^{2}\backslash\\{(0,0)\\}$,$e_{1}\cdot
e_{1}=\alpha e_{2},e_{1}\cdot e_{2}=\beta e_{1}=e_{2}\cdot e_{1},e_{2}\cdot
e_{2}=\beta e_{2}$.

  * •

For all $(\alpha,\beta)\in\mathbb{K}^{2}\backslash\\{(0,0)\\}$,$e_{1}\cdot
e_{1}=\alpha e_{1}+\beta e_{2},e_{1}\cdot e_{2}=\beta e_{1}+\alpha
e_{2}=e_{2}\cdot e_{1},\\\ e_{2}\cdot e_{2}=\alpha e_{1}+\beta e_{2}$.

  * •

For all $(\alpha,\beta,\gamma)\in\mathbb{K}^{3}$, such that
$\gamma^{2}+4\alpha\beta\geq 0,$ $e_{1}\cdot e_{1}=\alpha e_{1},e_{2}\cdot
e_{2}=\beta e_{1}+\gamma e_{2},\\\ e_{1}\cdot
e_{2}=\frac{1}{2}\left(\gamma+\sqrt{\gamma^{2}+4\alpha\beta}\right)e_{1}=e_{2}\cdot
e_{1}.$

∎

##  4 Bimodules and matched pairs nearly associative algebras

######  Definition 4.1.

Let $(A,\cdot)$ be a nearly associative algebra. Consider the linear maps
$l;r:A\rightarrow{\rm End}(V)$, where $V$ is a linear space. A triple
$(l,r,V)$ is a bimodule of $(A,\cdot)$ if for all $x,y\in A$, the following
relations

$\displaystyle l(x)l(y)$ $\displaystyle=$ $\displaystyle r(y)r(x),$ (18a)
$\displaystyle l(x)r(y)$ $\displaystyle=$ $\displaystyle l({y\cdot x}),$ (18b)
$\displaystyle r(x)l(y)$ $\displaystyle=$ $\displaystyle r({x\cdot y})$ (18c)

are satisfied.

######  Example 4.2.

Let $(A,\cdot)$ be a nearly associative algebra. The triple $(L,R,A)$ is a
bimodule of $(A,\cdot)$, where for any $x,y\in A$, $L(x)y=x\cdot y=R(y)x$.

######  Proposition 4.3.

Let $(l,r,V)$ be a bimodule of a nearly associative algebra $(A,\cdot)$, where
$l;r:A\rightarrow{\rm End}(V)$ are two linear maps and $V$ a linear space.
There is a nearly associative algebra defined on $A\oplus V$ by, for any
$x,y\in A$ and any $u,v\in V,$

$\displaystyle(x+u)\ast(y+v)=x\cdot y+l(x)v+r(y)u.$ (19)

###### Proof.

Consider the bimodule $(l,r,V)$ of the nearly associative algebra $(A,\cdot)$.
For all $x,y,z\in A$ and $u,v,w\in V$ we have:

$\displaystyle(x+u)\ast((y+v)\ast(z+w))=x\cdot(y\cdot
z)+l(x)l(y)w+l(x)r(z)v+r({y\cdot z})u$ (20a)
$\displaystyle((z+w)\ast(x+u))\ast(y+v)=(z\cdot x)\cdot y+l({z\cdot
x})v+r(y)l(z)u+r(y)r(x)w$ (20b)

Using (18a) - (18c) in (20a) and (20b) we easily deduce that $(A\oplus
V,\ast)$ is a nearly associative algebra. ∎

######  Corollary 4.4.

Let $(l,r,V)$ be a bimodule of a nearly associative algebra $(A,\cdot)$, where
$l,r:\rm A\rightarrow{\rm End}(V)$ are two linear maps and $V$ a linear space.
Then there is a Lie algebra product on $A\oplus V$ given by

$\displaystyle[x+u,y+v]=[x,y]_{{}_{\cdot}}+(l(x)-r(x))v-(l(y)-r(y))u$ (21)

for all $x,y\in A$ and for any $u,v\in V$.

###### Proof.

It is simple to remark that the commutator of the product defined in (19) is
the product defined in (21). By taking into account Proposition 2.5, the
Jacobi identity of the product given in (21) is satisfied. ∎

######  Definition 4.5.

Let $(\mathcal{G},[.,.]_{{}_{\mathcal{G}}})$ be a Lie algebra. A
representation of $(\mathcal{G},[.,.]_{{}_{\mathcal{G}}})$ over the linear
space $V$ is a linear map $\rho:\mathcal{G}\rightarrow{\rm End}(V)$ satisfying

$\displaystyle\rho([x,y]_{{}_{\mathcal{G}}})=\rho(x)\circ\rho(y)-\rho(y)\circ\rho(x)$
(22)

for all $x,y\in\mathcal{G}$.

######  Proposition 4.6.

Let $(A,\cdot)$ be a nearly associative algebra and let $V$ be a finite-
dimensional linear space over the field $\mathbb{K}$ such that $(l,r,V)$ is a
bimodule of $(A,\cdot)$, where $l,r:A\rightarrow{\rm End}(V)$ are two linear
maps. Then, the linear map

$l-r:A\rightarrow{\rm End}(V),\quad x\mapsto l(x)-r(x)$

is a representation of the underlying Lie algebra $\mathcal{G}(A)$ underlying
$(A,\cdot)$.

###### Proof.

Let $(l,r,V)$ be a bimodule of the nearly associative algebra $(A,\cdot)$. For
$x,y\in A$,

$(l(x)-r(x))(l(y)-r(y))-(l(y)-r(y))(l(x)-r(x))=\cr
l(x)l(y)-l(x)r(y)-r(x)l(y)+r(x)r(y)-l(y)l(x)+l(y)r(x)+r(y)l(x)-r(y)r(x)\cr=-l(x)r(y)-r(x)l(y)+l(y)r(x)+r(y)l(x)=-l({y\cdot
x})-r({x\cdot y})+l({x\cdot x})+r({y\cdot x})\cr=(l-r)(x\cdot y-y\cdot
x)=(l-r)([x,y]).$

Therefore, (22) is satisfied for $l-r=\rho$. ∎

######  Definition 4.7.

Let $(A,\cdot)$ be a nearly associative algebra and $(l,r,V)$ its associated a
bimodule, where $V$ is a finite-dimensional linear space. The dual maps
$l^{*},r^{*}$ of linear maps $l,r,$ respectively are defined as $\displaystyle
l^{*},r^{*}:A\rightarrow{\rm End}(V^{*})$ such that for any $x\in A,u^{*}\in
V^{*},v\in V,$

$\displaystyle\left<l^{*}(x)u^{*},v\right>=\left<u^{*},l(x)v\right>,$ (23a)
$\displaystyle\left<r^{*}(x)u^{*},v\right>=\left<u^{*},r(x)v\right>.$ (23b)

######  Proposition 4.8.

Let $(A,\cdot)$ be a nearly associative algebra and $(l,r,V)$ be its bimodule.
The following relations are equivalent:

  1. (i)

$(r^{*},l^{*},V^{*})$ is a bimodule of $(A,\cdot)$,

  2. (ii)

$l(x)r(y)=r(y)l(x)$, for all $x,y\in A$,

  3. (iii)

$(l^{*},r^{*},V^{*})$ is a bimodule of $(A,\cdot)$.

###### Proof.

Let $(A,\cdot)$ be a nearly associative algebra and $(l,r,V)$ be its
associated bimodule i.e. the linear maps $l,r:A\rightarrow{\rm End}(V)$
satisfying (18a) - (18c) and $V$ is a finite-dimensional linear space.

  * •

Suppose that $(r^{*},l^{*},V^{*})$ is a bimodule of $(A,\cdot)$, i.e. with
correspondences $l\rightarrow r^{*}$ and $r\rightarrow l^{*}$, (18a) - (18c)
are satisfied. For any $x,y\in A$, $v\in V$, $u^{*}\in V^{*}$:

$\langle l(x)r(y)v,u^{*}\rangle=\langle v,r^{*}(y)l^{*}(x)u^{*}\rangle=\langle
v,r^{*}({y\cdot x})u^{*}\rangle\\\ =\langle r({y\cdot
x})v,u^{*}\rangle=\langle r(y)l(x)v,u^{*}\rangle.$

Therefore, the relation $l(x)r(y)=r(y)l(x)$ is satisfied.

  * •

Suppose $l(x)r(y)=r(y)l(x)$ for any $x,y\in A$. For any $x,y\in A$, $v\in V$,
$u^{*}\in V^{*}$:

$\left<l^{*}(x)l^{*}(y)u^{*},v\right>=\left<u^{*},l(y)l(x)v\right>=\left<u^{*},r(x)r(y)v\right>=\left<r^{*}(y)r^{*}(x)u^{*},v\right>,$

yields $l^{*}(x)l^{*}(y)=r^{*}(y)r^{*}(x);$

$\left<l^{*}(x)r^{*}(y)u^{*},v\right>=\left<u^{*},r(y)l(x)v\right>=\left<u^{*},l(x)r(y)v\right>=\\\
\left<u^{*},l({y\cdot x})v\right>=\left<l^{*}({y\cdot x})u^{*},v\right>,$

yields $l^{*}(x)r^{*}(y)=l^{*}({y\cdot x})$;

$\left<r^{*}(y)l^{*}(x)u^{*},v\right>=\left<u^{*},l(x)r(y)v\right>=\left<u^{*},r(y)l(x)v\right>=\\\
\left<u^{*},r({y\cdot x})v\right>=\left<r^{*}({y\cdot x})u^{*},v\right>,$

yields $r^{*}(y)l^{*}(x)=r^{*}({y\cdot x}).$ Thus, with correspondences
$r^{*}\rightarrow l$ and $l^{*}\rightarrow r$, (18a) - (18c) are satisfied.

Similarly, one obtains the equivalence between $l(x)r(y)=r(y)l(x)$, for any
$x,y\in A$, and $(l^{*},r^{*},V^{*})$ being a bimodule of $(A,\cdot)$. ∎

######  Remark 4.9.

It is clear that $(L_{\cdot}^{*},R_{\cdot}^{*},A^{*})$ and
$(R_{\cdot}^{*},L_{\cdot}^{*},A^{*})$ are bimodules of the nearly associative
algebra $(A,\cdot)$ if and only if $L$ and $R$ commute.

######  Theorem 4.10.

Let $(A,\cdot)$ and $(B,\circ)$ be two nearly associative algebras. Suppose
that $(l_{A},r_{A},B)$ and $(l_{B},r_{B},A)$ are bimodules of $(A,\cdot)$ and
$(B,\circ)$, respectively, where $l_{A},r_{A}:A\rightarrow{\rm End}(B)$,
$l_{B},r_{B}:B\rightarrow{\rm End}(A)$ are four linear maps satisfying for all
$x,y\in A$, $a,b\in B$ the following relations

$\displaystyle r_{B}(l_{A}(x)a)y+y\cdot(r_{B}(a)x)-(l_{B}(a)y)\cdot
x-l_{B}(r_{A}(y)a)x=0,$ (24a) $\displaystyle r_{B}(a)(x\cdot
y)-y\cdot(l_{B}(a)x)-r_{B}(r_{A}(x)a)y=0,$ (24b) $\displaystyle
l_{B}(a)(x\cdot y)-(r_{B}(a)y)\cdot x-l_{B}(l_{A}(y)a)x=0,$ (24c)
$\displaystyle r_{A}(l_{B}(a)x)b+b\circ(r_{A}(x)a)-(l_{A}(x)b)\circ
a-l_{A}(r_{B}(b)x)a=0,$ (24d) $\displaystyle r_{A}(x)(a\circ
b)-b\circ(l_{A}(x)a)-r_{A}(r_{B}(a)x)b=0,$ (24e) $\displaystyle
l_{A}(x)(a\circ b)-(r_{A}(x)b)\circ a-l_{A}(l_{B}(b)x)a=0.$ (24f)

Then, $(A\oplus B,\ast)$ is a nearly associative algebra, where

$\displaystyle(x+a)\ast(y+b)=(x\cdot y+l_{B}(a)y+r_{B}(b)x)+(a\circ
b+l_{A}(x)b+r_{A}(y)a).$ (25)

for all $x,y\in A,a,b\in B$.

###### Proof.

For any $x,y,z\in A$, and for any $a,b,c\in B$, we have

$(x+a)\ast((y+b)\ast(z+c))=x\cdot(y\cdot
z)+\\{x\cdot(l_{B}(b)z)+r_{B}(r_{A}(z)b)x\\}+l_{B}(a)(y\cdot
z)\cr+\\{x\cdot(r_{B}(c)y)+r_{B}(l_{A}(y)c)x\\}+l_{B}(a)(l_{B}(b)z)+r_{B}(b\circ
c)x\cr+l_{B}(a)(r_{B}(c)y)+a\circ(b\circ
c)+\\{a\circ(l_{A}(y)c)+r_{B}(r_{B}(c)y)a\\}\cr+\\{a\circ(r_{A}(z)b)+r_{B}(l_{B}(b)z)a\\}+l_{A}(x)(l_{A}(y)c)+l_{A}(x)(b\circ
c)+l_{A}(x)(r_{A}(z)b)+r_{A}(y\cdot z)a;\\\\[8.5359pt]
((z+c)\ast(x+a))\ast(y+b)=(z\cdot x)\cdot y+\\{(l_{B}(c)x)\cdot
y+l_{B}(r_{A}(x)c)y\\}+r_{B}(x)(z\cdot x)\cr+\\{(r_{A}(a)z)\cdot
y+l_{B}(l_{A}(z)a)y\\}+l_{B}(c\circ
a)y+r_{B}(b)(l_{B}(c)x)\cr+r_{B}(b)(r_{B}(a)z)(c\circ a)\circ
y+\\{(l_{A}(z)a)\circ b+l_{A}(r_{B}(a)z)b\\}\cr+\\{(r_{A}(x)c)\circ
b+l_{A}(l_{B}(c)x)b\\}+r_{A}(y)(r_{A}(x)c)+r_{A}(y)(c\circ
a)+r_{A}(y)(l_{A}(z)a)+l_{A}(z\cdot x)b.$

Using (24a) - (24f) and that $(l_{A},r_{A},B)$ and $(l_{B},r_{B},A)$ are
bimodules of $(A,\cdot)$ and $(B,\circ)$, respectively, we derive that
$(A\oplus B,\ast)$ is a nearly associative algebra. ∎

######  Definition 4.11 ([47]).

Let $(\mathcal{G},[.,.]_{{}_{\mathcal{G}}})$ and
$({\mathcal{H},[.,.]_{{}_{\mathcal{H}}}})$ be two Lie algebras such that
$\rho:\mathcal{G}\rightarrow{\rm End}(\mathcal{H})$ and
$\mu:\mathcal{H}\rightarrow{\rm End}(\mathcal{G})$ are representations of
$\mathcal{G}$ and $\mathcal{H}$, respectively. A matched pair of Lie algebras
$\mathcal{G}$ and $\mathcal{H}$ is $(\mathcal{G},\mathcal{H},\rho,\mu)$ such
that $\rho$ and $\mu$ are satisfying the following relations, for all
$x,y\in\mathcal{G}$ and $a,b\in\mathcal{H}$,

$\displaystyle\rho(x)[a,b]_{{}_{\mathcal{G}}}-[\rho(x)a,b]_{{}_{\mathcal{H}}}-[a,\rho(x)b]_{{}_{\mathcal{H}}}+\rho(\mu(a)x)b-\rho(\mu(b)x)a=0,$
(26a)
$\displaystyle\mu(a)[x,y]_{{}_{\mathcal{G}}}-[\mu(a)x,y]_{{}_{\mathcal{G}}}-[x,\mu(a)y]_{{}_{\mathcal{G}}}+\mu(\rho(x)a)y-\mu(\rho(y)a)x=0.$
(26b)

######  Corollary 4.12.

Let $(A,B,l_{A},r_{A},l_{B},r_{B})$ be a matched pair of the nearly
associative algebras $(A,\cdot)$ and $(B,\circ)$. Then
$(\mathcal{G}(A),\mathcal{G}(B),l_{A}-r_{A},l_{B}-r_{B})$ is a matched pair of
Lie algebras $\mathcal{G}(A)$ and $\mathcal{G}(B)$.

###### Proof.

Let $(A,B,l_{A},r_{A},l_{B},r_{B})$ be a matched pair of the nearly
associative algebras $(A,\cdot)$ and $(B,\circ)$. In view of Proposition 4.6,
the linear maps $l_{A}-r_{A}:A\longrightarrow{\rm End}(B)$ and
$l_{B}-r_{B}:B\longrightarrow{\rm End}(A)$ are representations of the
underlying Lie algebras $\mathcal{G}(A)$ and $\mathcal{G}(B)$, respectively.
Therefore, by direct calculation we have (26a) is equivalent to (24a) - (24c)
and similarly, (26b) is equivalent to (24d) - (24f). ∎

######  Proposition 4.13.

Let $(A,\cdot)$ be a nearly associative algebra. Suppose that there is a
nearly associative algebra structure $\circ$ on its the dual space $A^{*}$. If
in addition, the linear maps $L$ and $R$ commute then
$(A,A^{*},R_{\cdot}^{*},L_{\cdot}^{*},R_{\circ}^{*},L_{\circ}^{*})$ is a
matched pair of the nearly associative algebras $(A,\cdot)$ and
$(A^{*},\circ)$ if and only if the following relations are satisfied for any
$x,y\in A$ and $a\in A^{*}$

$\displaystyle
L_{\circ}^{*}(R_{\cdot}^{*}(x)a)y-y\cdot(L_{\circ}^{*}(a)x)-(R_{\circ}^{*}(a)y)\cdot
x-R_{\circ}^{*}(L_{\cdot}^{*}(y)a)x=0,$ (27a) $\displaystyle
L_{\circ}^{*}(a)(x\cdot
y)-y\cdot(R_{\circ}^{*}(a)x)-L_{\circ}^{*}(L_{\cdot}^{*}(x)a)y=0,$ (27b)
$\displaystyle R_{\circ}^{*}(a)(x\cdot y)-(L_{\circ}^{*}(a)y)\cdot
x-R_{\circ}^{*}(R_{\cdot}^{*}(y)a)x=0.$ (27c)

###### Proof.

Since $L$ and $R$ commute, according to Remark 4.9 and Proposition 4.8, both
$(R_{\cdot}^{*},L_{\cdot}^{*},A^{*})$ and
$(L_{\cdot}^{*},R_{\cdot}^{*},A^{*})$ are bimodules of $(A,\cdot)$. Setting
$l_{A}=R_{\cdot}^{*}$, $r_{A}=L_{\cdot}^{*},l_{B}=R_{\circ}^{*}$ and
$r_{B}=L_{\circ}^{*}$ in Theorem 4.10 the equivalences among Eq (24a) and
(27a), (24b) and (27b), and finally (24c) and (27c) are straightforward.
Besides, for any $x,y\in A$ and any $a,b\in A^{*}$, we have

$\displaystyle\langle L_{\circ}^{*}(R_{\cdot}^{*}(x)a)y,b\rangle=\langle
y,L_{\circ}(R_{\cdot}^{*}(x)a)b\rangle=\langle y,(R_{\cdot}^{*}(x)a)\circ
b\rangle,$ $\displaystyle\langle y\cdot(L_{\circ}^{*}(a)x),b\rangle=\langle
R_{\cdot}(L_{\circ}^{*}(a)x)y,b\rangle=\langle
y,R_{\cdot}^{*}(L_{\circ}^{*}(a)x)b\rangle,$
$\displaystyle\langle(R_{\circ}^{*}(a)y)\cdot x,b\rangle=\langle
R_{\circ}^{*}(a)y,R_{\cdot}^{*}(x)b\rangle=\langle y,(R_{\cdot}^{*}(x)b)\circ
a\rangle,$ $\displaystyle\langle
R_{\circ}^{*}(L_{\cdot}^{*}(y)a)x,b\rangle=\langle
L_{\circ}^{*}(b)x,L_{\cdot}^{*}(y)a\rangle=\langle
y\cdot(L_{\circ}^{*}(b)x),a\rangle=\langle
y,R_{\cdot}^{*}(L_{\circ}^{*}(b)x)a\rangle,$ $\displaystyle\langle
L_{\circ}^{*}(a)(x\cdot y),b\rangle=\langle R_{\cdot}(y)x,a\circ
b\rangle=\langle x,R_{\cdot}^{*}(y)(a\circ b)\rangle,$ $\displaystyle\langle
y\cdot(R_{\circ}^{*}(a)x),b\rangle=\langle
R_{\circ}^{*}(a)x,L_{\cdot}^{*}(y)b\rangle=\langle x,(L_{\cdot}^{*}(y)b)\circ
a\rangle,$ $\displaystyle\langle
L_{\circ}^{*}(L_{\cdot}^{*}(x)a)y,b\rangle=\langle
R_{\circ}^{*}(b)y,L_{\circ}^{*}(x)a\rangle=\langle
x\cdot(R_{\circ}^{*}(b)y),a\rangle=\langle
x,R_{\cdot}^{*}(R_{\circ}^{*}(b)y)a\rangle,$ $\displaystyle\langle
R_{\circ}^{*}(a)(x\cdot y),b\rangle=\langle L_{\cdot}(x)y,b\circ
a\rangle=\langle y,L_{\cdot}(x)^{*}(b\circ a)\rangle,$
$\displaystyle\langle(L_{\circ}^{*}(a)y)\cdot x,b\rangle=\langle
L_{\circ}^{*}(a)y,R_{\cdot}^{*}(b)\rangle=\langle
y,a\circ(R_{\cdot}^{*}(x)b)\rangle,$ $\displaystyle\langle
R_{\circ}^{*}(R_{\cdot}^{*}(y)a)x,b\rangle=\langle
L_{\circ}^{*}(b)x,R_{\cdot}^{*}(y)a\rangle=\langle(L_{\circ}^{*}(b)x)\cdot
y,a\rangle=\langle y,L_{\cdot}^{*}(L_{\circ}^{*}(b)x)a\rangle$

Then, Eq (24a) holds if and only if (24d) holds, Eq (24b) holds if and only if
(24e) holds, and finally Eq (24c) holds if and only if (24f) holds. ∎

##  5 Manin triple and bialgebra of nearly associative algebras

######  Definition 5.1.

A bilinear form $\mathfrak{B}$ on a nearly associative algebra $(A,\cdot)$ is
called left-invariant if $forallx,y,z\in A$,

$\mathfrak{B}(x\cdot y,z)=\mathfrak{B}(x,y\cdot z).$ (28)

######  Proposition 5.2.

Let $(A,\cdot)$ be a nearly associative algebra. If there is a nondegenerate
symmetric invariant bilinear form $\mathfrak{B}$ defined on $A$, then as
bimodules of the nearly associative algebra $(A,\cdot)$, $(L,R,A)$ and
$(R^{*},L^{*},A^{*})$ are equivalent. Conversely, if $(L,R,A)$ and
$(R^{*},L^{*},A^{*})$ are equivalent bimodules of a nearly associative algebra
$(A,\cdot)$, then there exists a nondegenerate invariant bilinear form
$\mathfrak{B}$ on $A$.

######  Definition 5.3.

A Manin triple of nearly associative algebras is a triple of nearly
associative algebras $(A,A_{1},A_{2})$ together with a nondegenerate symmetric
invariant bilinear form $\mathfrak{B}$ on $A$ such that the following
conditions are satisfied.

  1. (i)

$A_{1}$ and $A_{2}$ nearly associative subalgebras of $A$;

  2. (ii)

as linear spaces, $A=A_{1}\oplus A_{2}$;

  3. (iii)

$A_{1}$ and $A_{2}$ are isotropic with respect to $\mathfrak{B}$, i.e. for any
$x_{1},y_{1}\in A_{1}$ and any $x_{2},y_{2}\in A_{2}$,
$\mathfrak{B}(x_{1},y_{1})=0=\mathfrak{B}(x_{2},y_{2})=0.$

######  Definition 5.4.

Let $(A,\cdot)$ be a nearly associative algebra. Suppose that $\circ$ is a
nearly associative algebra structure on the dual space $A^{*}$ of $A$ and
there is a nearly associative algebra structure on the direct sum $A\oplus
A^{*}$ of the underlying linear spaces of $A$ and $A^{*}$ such that
$(A,\cdot)$ and $(A^{*},\circ)$ are subalgebras and the natural symmetric
bilinear form on $A\oplus A^{*}$ given by $\forall x,y\in A;\forall
a^{*},b^{*}\in A^{*},$

$\mathfrak{B}_{d}(x+a^{*},y+b^{*}):=\langle a^{*},y\rangle+\langle
x,b^{*}\rangle,\;$ (29)

is left-invariant, then $(A\oplus A^{*},A,A^{*})$ is called a standard Manin
triple of nearly associative algebras associated to $\mathfrak{B}_{d}$.

Obviously, a standard Manin triple of nearly associative algebras is a Manin
triple of nearly associative algebras. By symmetric role of $A$ and $A^{*}$,
we have

######  Proposition 5.5.

Every Manin triple of nearly associative algebras is isomorphic to a standard
one.

######  Proposition 5.6.

Let $(A,\cdot)$ be a nearly associative algebra. Suppose that there is a
nearly associative algebra structure $\circ$ on the dual space $A^{*}$. There
exists a nearly associative algebra structure on the linear space $A\oplus
A^{*}$ such that $(A\oplus A^{*},A,A^{*})$ is a standard Manin triple of
nearly associative algebras associated to $\mathfrak{B}_{d}$ defined by (29)
if and only if
$(A,A^{*},R_{\cdot}^{*},L_{\cdot}^{*},R_{\circ}^{*},L_{\circ}^{*})$ is a
matched pair of nearly associative algebras.

######  Theorem 5.7.

Let $(A,\cdot)$ be a nearly associative algebra such that the left and right
multiplication operators commute. Suppose that there is a nearly associative
algebra structure $\circ$ on its the dual space $A^{*}$ given by
$\Delta^{*}:A^{*}\otimes A^{*}\rightarrow A^{*}$. Then,
$(A,A^{*},R_{\cdot}^{*},L_{\cdot}^{*},R_{\circ}^{*},L_{\circ}^{*})$ is a
matched pair of the nearly associative algebras $(A,\cdot)$ and
$(A^{*},\circ)$ if and only if $\Delta:A\rightarrow A\otimes A$ satisfies the
following relations

$\displaystyle(R_{\cdot}(x)\otimes{\rm id}-\sigma(R_{\cdot}(x)\otimes{\rm
id}))\Delta(y)+({\rm id}\otimes L_{\cdot}(y)-\sigma({\rm id}\otimes
L_{\cdot}(y)))\Delta(x)=0,$ (30a)
$\displaystyle\begin{array}[]{r}(L_{\cdot}(x)\otimes{\rm
id})\Delta(y)+\sigma(L_{\cdot}(y)\otimes{\rm id})\Delta(x)=\Delta(x\cdot y)\\\
=\sigma({\rm id}\otimes R_{\cdot}(x))\Delta(y)+({\rm id}\otimes
R_{\cdot}(y))\Delta(x).\end{array}$ (30d)

###### Proof.

For any $a,b\in A^{*}$ and any $x,y\in A$ we have

$\displaystyle\langle(R_{\cdot}(x)\otimes{\rm id})\Delta(y),a\otimes
b\rangle=\langle y,(R_{\cdot}^{*}(x)a)\circ b\rangle=\langle
L_{\circ}^{*}(R_{\cdot}^{*}(x)a)y,b\rangle,$
$\displaystyle\langle\sigma(R_{\cdot}(x)\otimes{\rm id})\Delta(y),a\otimes
b\rangle=\langle y,(R_{\cdot}^{*}(x)b)\circ a\rangle=\langle
R_{\circ}^{*}(a)y,R_{\cdot}^{*}(x)b\rangle=\langle(R_{\circ}^{*}(a)y)\cdot
x,b\rangle,$ $\displaystyle\langle({\rm id}\otimes
L_{\cdot}(y))\Delta(x),a\otimes b\rangle=\langle
x,a\circ(L_{\cdot}^{*}(y)b)\rangle=\langle
y\cdot(L_{\circ}^{*}(a)x),b\rangle,$ $\displaystyle\langle\sigma({\rm
id}\otimes L_{\cdot}(y))\Delta(x),a\otimes b\rangle=\langle
x,b\circ(L_{\cdot}^{*}(y)a)\rangle=\langle
R_{\circ}^{*}(L_{\cdot}^{*}(y)a)x,b\rangle.$

Hence (27a) is equivalent to (30a).

Similarly, we have for any $x,y\in A$ and any $a,b\in A^{*}$

$\displaystyle\langle\Delta(x\cdot y),a\otimes b\rangle=\langle x\cdot
y,a\circ b\rangle=\langle L_{\circ}^{*}(a)(x\cdot y),b\rangle=\langle
R_{\circ}^{*}(b)(x\cdot y),a\rangle,$
$\displaystyle\langle(L_{\cdot}(x)\otimes{\rm id})\Delta(y),a\otimes
b\rangle=\langle y,(L_{\cdot}^{*}(x)a)\circ b\rangle=\langle
L_{\circ}^{*}(L_{\cdot}^{*}(x)a)y,b\rangle,$
$\displaystyle\langle\sigma(L_{\cdot}(y)\otimes{\rm id})\Delta(x),a\otimes
b\rangle=\langle x,(L_{\cdot}^{*}(y)b)\circ a\rangle=\langle
y\cdot(R_{\circ}^{*}(a)x),b\rangle,$ $\displaystyle\langle\sigma({\rm
id}\otimes R_{\cdot}(x))\Delta(y),a\otimes b\rangle=\langle
y,b\circ(R_{\cdot}^{*}(x)a)\rangle=\langle
R_{\circ}^{*}(R_{\cdot}^{*}(x)a)y,b\rangle,$ $\displaystyle\langle({\rm
id}\otimes R_{\cdot}(y))\Delta(x),a\otimes b\rangle=\langle
x,a\circ(R_{\cdot}^{*}(y)b)\rangle=\langle(L_{\circ}^{*}(a)x)\cdot
y,b\rangle,$

Therefore, (27b) and (27c) and is equivalent to (30d). ∎

######  Remark 5.8.

Obviously, if $L$ and $R$ commute, then $L^{*}$ and $R^{*}$ commute too and if
in addition $\gamma:A^{*}\rightarrow A^{*}\otimes A^{*}$ is a linear maps such
that its dual $\gamma^{*}:A\otimes A\rightarrow A$ defines a nearly
associative algebra structure $\cdot$ on $A$, then $\Delta$ satisfies (30a)
and (30d) if and only if $\gamma$ satisfies for all $a,b\in A^{*}$,

$\displaystyle(R_{\circ}(a)\otimes{\rm id}-\sigma(R_{\circ}(a)\otimes{\rm
id}))\gamma(b)+({\rm id}\otimes L_{\circ}(b)-\sigma({\rm id}\otimes
L_{\circ}(b)))\gamma(a)=0,$ $\displaystyle(L_{\circ}(x)\otimes{\rm
id})\gamma(b)+\sigma(L_{\circ}(b)\otimes{\rm id})\gamma(a)=$
$\displaystyle\qquad\qquad\qquad\gamma(a\circ b)=\sigma({\rm id}\otimes
R_{\circ}(a))\gamma(b)+({\rm id}\otimes R_{\circ}(b))\gamma(a).$

######  Definition 5.9.

Let $(A,\cdot)$ be a nearly associative algebra in which the left ($L$) and
right ($R$) multiplication operators commute. A nearly anti-flexible bialgebra
structure is a linear map $\Delta:A\rightarrow A\otimes A$ such that

  * •

$\Delta^{*}:A^{*}\otimes A^{*}\rightarrow A^{*}$ defines a nearly associative
algebra structure on $A,$

  * •

$\Delta$ satisfies (30d) and (30d).

######  Theorem 5.10.

Let $(A,\cdot)$ be a nearly associative algebra in which the left and right
multiplication operators commute. Suppose that there is a nearly associative
algebra structure on $A^{*}$ denoted by $\circ$ which defined a linear map
$\Delta:A\rightarrow A\otimes A$. Then the following conditions are
equivalent:

  1. (i)

$(A\oplus A^{*},A,A^{*})$ is a standard Manin triple of nearly associative
algebras $(A,\cdot)$ and $(A^{*},\circ)$ such that its associated symmetric
bilinear form $\mathfrak{B}_{d}$ is defined by (29).

  2. (ii)

$(A,A^{*},R_{\cdot}^{*},L_{\cdot}^{*},R_{\circ}^{*},L_{\circ}^{*})$ is a
matched pair of nearly associative algebras $(A,\cdot)$ and $(A^{*},\circ)$.

  3. (iii)

$(A,A^{*})$ is a nearly associative bialgebra.

##  6 Hom-Lie admissible, G-Hom-associative, flexible and anti-flexible Hom-
algebras

Hom-Lie admissible algebras along with Hom-associative algebras and more
general $G$-Hom-associative algebras were first introduced, and Hom-
associative algebras and $G$-Hom-associative algebras were shown to be Hom-Lie
admissible in [48].

Hom-algebra is a triple $(A,\mu,\alpha)$ consisting of a linear space $A$ over
a field $\mathbb{K}$, a bilinear product $\mu:A\times A\rightarrow A$ and a
linear map $\alpha:A\rightarrow A$.

######  Definition 6.1 ([48]).

Hom-Lie, Hom-Lie admissible, Hom-associative and $G$-Hom-associative Hom-
algebras (over a field $\mathbb{K}$) are defined as follows:

  1. 1)

Hom-Lie algebras are triples $(A,[.,.],\alpha)$, consisting of a linear space
$A$ over a field $\mathbb{K}$, bilinear map (bilinear product) $[.,.]:A\times
A\rightarrow A$ and a linear map $\alpha:A\rightarrow A$ satisfying, for all
$x,y,z\in A$,

$\displaystyle[x,y]=-[y,x],$ (Skew-symmetry) (31)
$\displaystyle[\alpha(x),[y,z]]+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]=0.$ (Hom-
Jacobi identity) (32)

  2. 2)

Hom-Lie admissible algebras are Hom-algebras $(A,\mu,\alpha)$ consisting of
possibly non-associative algebra $(A,\mu)$ and a linear map
$\alpha:A\rightarrow A$, such that $(A,[.,.],\alpha)$ is a Hom-Lie algebra,
where $[x,y]=\mu(x,y)-\mu(y,x)$ for all $x,y\in A$.

  3. 3)

Hom-associative algebras are triples $(A,\cdot,\alpha)$ consisting of a linear
space $A$ over a field $\mathbb{K}$, a bilinear product $\mu:A\times
A\rightarrow A$ and a linear map $\alpha:A\rightarrow A$, satisfying for all
$x,y,z\in A$,

$\mu(\mu(x,y),\alpha(z))=\mu(\alpha(x),\mu(y,z)).\quad\quad\text{\rm(Hom-
associativity)}$ (33)

  4. 4)

Let $G$ be a subgroup of the permutations group $\mathcal{S}_{3}$. Hom-algebra
$(A,\mu,\alpha)$ is said to be $G$-Hom-associative if

$\sum_{\sigma\in
G}{(-1)^{\varepsilon({\sigma})}(\mu(\mu(x_{\sigma(1)},x_{\sigma(2)}),\alpha(x_{\sigma(3)}))}-\mu(\alpha(x_{\sigma(1)}),\mu(x_{\sigma(2)},x_{\sigma(3)}))=0,$
(34)

where $x_{i}\in A,i=1,2,3$ and $(-1)^{\varepsilon({\sigma})}$ is the signature
of the permutation $\sigma$.

For any Hom-algebra $(A,\mu,\alpha)$, the Hom-associator, called also
$\alpha$-associator of $\mu$, is a trilinear map (ternary product)
$a_{\alpha,\mu}:A\times A\times A\rightarrow A$ defined by

$a_{\alpha,\mu}(x_{1},x_{2},x_{3})=\mu(\mu(x_{1},x_{2}),\alpha(x_{3}))-\mu(\alpha(x_{1}),\mu(x_{2},x_{3}))$

for all $x_{1},x_{2},x_{3}\in A$. The ordinary associator

$a_{\mu}(x_{1},x_{2},x_{3})=a_{{\rm
id},\mu}(x_{1},x_{2},x_{3})=\mu((x_{1},x_{2}),(x_{3}))-\mu((x_{1}),\mu(x_{2},x_{3}))$

on an algebra $(A,\mu)$ is $\alpha$-associator for the Hom-algebra
$(A,\mu,\alpha)=(A,\mu,{\rm id})$ with $\alpha={\rm id}:A\rightarrow A$, the
identity map on $A$.

Using Hom-associator $a_{\alpha,\mu}$ and notation
$\sigma(x_{1},x_{2},x_{3})=(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)})$, the
Hom-associativity (33) can be written as

$a_{\alpha,\mu}(x,y,z)=\mu(\mu(x,y),\alpha(z))-\mu(\alpha(x),\mu(y,z))=0,\quad\quad\text{\rm(Hom-
associativity)}$ (35)

or as $a_{\alpha,\mu}=0$, and the $G$-Hom-associativity (34) as

$\sum_{\sigma\in G}{(-1)^{\varepsilon({\sigma})}a_{\alpha,\mu}\circ\sigma}=0.$
(36)

If $\mu$ is the multiplication of a Hom-Lie admissible Lie algebra, then (34)
is equivalent to $[x,y]=\mu(x,y)-\mu(y,x)$ satisfying the Hom-Jacobi identity,
or equivalently,

$\sum_{\sigma\in\mathcal{S}_{3}}{(-1)^{\varepsilon({\sigma})}(\mu(\mu(x_{\sigma(1)},x_{\sigma(2)}),\alpha(x_{\sigma(3)}))}-\mu(\alpha(x_{\sigma(1)}),\mu(x_{\sigma(2)},x_{\sigma(3)})))=0,$
(37)

which may be written as

$\sum_{\sigma\in\mathcal{S}_{3}}{(-1)^{\varepsilon({\sigma})}a_{\alpha,\mu}\circ\sigma}=0.$
(38)

Thus, Hom-Lie admissible Hom-algebras are $\mathcal{S}_{3}$-associative Hom-
algebras. In general, for all subgroups $G$ of the permutations group
$\mathcal{S}_{3}$, all $G$-Hom-associative Hom-algebras are Hom-Lie
admissible, or in other words, all Hom-algebras from the six classes of
$G$-Hom-associative Hom-algebras, corresponding to the six subgroups of the
symmetric group $\mathcal{S}_{3}$, are Hom-Lie admissible [48, Proposition
3.4]. All six subgroups of $\mathcal{S}_{3}$ are

$\displaystyle G_{1}=\mathcal{S}_{3}({\rm id})=\\{{\rm
id}\\},G_{2}=\mathcal{S}_{3}(\tau_{12})=\\{{\rm
id},\tau_{12}\\},G_{3}=\mathcal{S}_{3}(\tau_{23})=\\{{\rm id},\tau_{23}\\},$
$\displaystyle G_{4}=\mathcal{S}_{3}(\tau_{13})=\\{{\rm
id},\tau_{13}\\},G_{5}=\mathcal{A}_{3},G_{6}=\mathcal{S}_{3}$

where $\mathcal{A}_{3}$ is the alternating group and $\tau_{ij}$ is the
transposition of $i$ and $j$.

Table 1: $G$-Hom-associative algebras Subgroup of $\mathcal{S}_{3}$ |  Hom-algebras class names |  Defining Identity (Notation: $\mu(a,b)=ab$)
---|---|---
$G_{1}=$ $\mathcal{S}_{3}({\rm id})$ |  Hom-associative | $\alpha(x)(yz)=(xy)\alpha(z)$
$G_{2}=$ $\mathcal{S}_{3}(\tau_{12})$ |  Hom-left symmetric Hom-Vinberg | $\alpha(x)(yz)-\alpha(y)(xz)=(xy)\alpha(z)-(yx)\alpha(z)$
$G_{3}=$ $\mathcal{S}_{3}(\tau_{23})$ |  $\mathcal{S}_{3}(\tau_{23})$-Hom-associative Hom-right symmetric Hom-pre-Lie | $\alpha(x)(yz)-\alpha(x)(zy)=(xy)\alpha(z)-(xz)\alpha(y)$
$G_{4}=$ $\mathcal{S}_{3}(\tau_{13})$ |  $\mathcal{S}_{3}(\tau_{13})$-Hom-associative Hom-anti-flexible Hom-center symmetric | $\alpha(x)(yz)-\alpha(z)(yx)=(xy)\alpha(z)-(zy)\alpha(x)$
$G_{5}=$ $\mathcal{A}_{3}$ |  $\mathcal{A}_{3}$-Hom-associative | $\begin{array}[]{l}\alpha(x)(yz)+\alpha(y)(zx)+\alpha(z)(xy)=\\\ (xy)\alpha(z)+(yz)\alpha(x)+(zx)\alpha(y)\end{array}$
$G_{6}=$ $\mathcal{S}_{3}$ |  Hom-Lie admissible | $\begin{array}[]{r}\displaystyle{\sum_{\sigma\in\mathcal{S}_{3}}(-1)^{\varepsilon({\sigma})}}\left((x_{\sigma(1)}x_{\sigma(2)})\alpha(x_{\sigma(3)})\right.\\\ \left.-\alpha(x_{\sigma(1)})(x_{\sigma(2)}x_{\sigma(3)})\right)=0\end{array}$

The skew-symmetric $G_{5}$-Hom-associative Hom-algebras and Hom-Lie algebras
form the same class of Hom-algebras for linear spaces over fields of
characteristic different from $2$, since then the defining identity of
$G_{5}$-Hom-associative algebras is equivalent to the Hom-Jacobi identity of
Hom-Lie algebras when the product $\mu$ is skew-symmetric.

A Hom-right symmetric (Hom-pre-Lie) algebra is the opposite algebra of a Hom-
left-symmetric algebra.

Hom-flexible algebras introduced in [48] is a generalization to Hom-algebra
context of flexible algebras [2, 53, 55].

######  Definition 6.2 ([48]).

A Hom-algebra $(A,\mu,\alpha)$ is called flexible if

$\mu(\mu(x,y),\alpha(x))=\mu(\alpha(x),\mu(y,x)))$ (39)

for any $x,y$ in $A$.

Using the $\alpha$-associator
$a_{\alpha,\mu}(x,y,z)=\mu(\mu(x,y),\alpha(z))-\mu(\alpha(x),\mu(y,z)),$ the
condition (39) may be written as

$a_{\alpha,\mu}(x,y,x)=0.$ (40)

Since Hom-associator map $a_{\alpha,\mu}$ is a trilinear map,

$a_{\alpha,\mu}(z-x,y,z-x)=a_{\alpha,\mu}(z,y,z)+a_{\alpha,\mu}(x,y,x)-a_{\alpha,\mu}(x,y,z)-a_{\alpha,\mu}(z,y,x),$

and hence (40) yields

$a_{\alpha,\mu}(x,y,z)=-a_{\alpha,\mu}(z,y,x)$ (41)

in linear spaces over any field, whereas setting $x=z$ in (41) gives
$2a_{\alpha,\mu}(x,y,x)=0$, implying that (40) and (41) are equivalent in
linear spaces over fields of characteristic different from $2$. The equality
(41) written in terms of the Hom-algebra producs $\mu$ is

$\mu(\mu(x,y),\alpha(z))-\mu(\alpha(x),\mu(y,z))=\mu(\alpha(z),\mu(y,x))-\mu(\mu(z,y),\alpha(x)).$
(42)

######  Definition 6.3.

A Hom-algebra $(A,\mu,\alpha)$ is called anti-flexible if

$\displaystyle\mu(\mu(x,y),\alpha(z))-\mu(\mu(z,y),\alpha(x))=\mu(\mu(\alpha(x),\mu(y,z))-\mu(\mu(\alpha(z),\mu(y,x))$
(43)

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

The equality (43) can be written as

$\displaystyle a_{\alpha,\mu}(x,y,z)=a_{\alpha,\mu}(z,y,x),$ (44)

in terms of the Hom-associator $a_{\alpha,\mu}(x,y,z)$.

Hom-anti-flexible algebras were first introduced in [48] as
$\mathcal{S}_{3}(\tau_{13})$-Hom-associative algebras, the subclass of
$G$-Hom-associative algebras corresponding to the subgroup
$G=\mathcal{S}_{3}(\tau_{13})\subset\mathcal{S}_{3}$ (see Table 1). In view of
(44), anti-flexible algebras have been called Hom-center symmetric in [32].

Note that (44) differs from (41) by absence of the minus sign on the right
hand side, meaning that for any $y$, the bilinear map $a_{\alpha,\mu}(.,y,.)$
is symmetric on Hom-anti-flexible algebras and skew-symmetric on Hom-flexible
algebras. Unlike (39) and (41) in Hom-flexible algebras, in Hom-anti-flexible
algebras, (44) is generally not equivalent to the restriction of (44) to $z=x$
trivially identically satisfied for any $x$ and $y$. In view of (44), Hom-
anti-flexible algebras are called Hom-center-symmetric algebras in [32].

##  7 Nearly Hom-associative algebras, bimodules and matched pairs

######  Definition 7.1.

A nearly Hom-associative algebra is a triple $(A,\ast,\alpha)$, where $A$ is a
linear space endowed to the bilinear product $\ast:A\times A\rightarrow A$ and
$\alpha:A\rightarrow A$ is a linear map such that for all $x,y,z\in A$,

$\displaystyle\alpha(x)\ast(y\ast z)=(z\ast x)\ast\alpha(y).$ (45)

Nearly Hom-associative algebras are Hom-Lie admissible.

######  Proposition 7.2.

Any nearly Hom-associative algebra $(A,\ast,\alpha)$ is Hom-Lie admissible,
that is $(A,[.,.],\alpha)$ is a Hom-Lie algebra, where $[x,y]=x\ast y-y\ast x$
for all $x,y\in A$.

###### Proof.

Let $(A,\ast,\alpha)$ be a nearly Hom-associative algebra. The commutator is
skew-symmetric since $[x,y]=x\ast y-y\ast x=-(y\ast x-x\ast y)=-[y,x].$ For
all $x,y,z\in A$,

$\displaystyle[\alpha(x),[y,z]]+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]$
$\displaystyle=[\alpha(x),y\ast z-z\ast y]+[\alpha(y),z\ast x-x\ast
z]+[\alpha(z),x\ast y-y\ast x]$ $\displaystyle=\alpha(x)\ast(y\ast
z)-\alpha(x)\ast(z\ast y)-(y\ast z)\ast\alpha(x)$ $\displaystyle+(z\ast
y)\ast\alpha(x)+\alpha(y)\ast(z\ast x)-\alpha(y)\ast(x\ast z)$
$\displaystyle-(z\ast x)\ast\alpha(y)+(x\ast
z)\ast\alpha(y)+\alpha(z)\ast(x\ast y)$ $\displaystyle-\alpha(z)\ast(y\ast
x)-(x\ast y)\ast\alpha(z)+(y\ast x)\ast\alpha(z)$
$\displaystyle=\\{\alpha(x)\ast(y\ast z)-(z\ast x)\ast\alpha(y)\\}$
$\displaystyle+\\{(y\ast x)\ast\alpha(z)-\alpha(x)\ast(z\ast y)\\}$
$\displaystyle+\\{\alpha(y)\ast(z\ast x)-(x\ast y)\ast\alpha(z)\\}$
$\displaystyle+\\{\alpha(z)\ast(x\ast y)-(y\ast z)\ast\alpha(x)\\}$
$\displaystyle+\\{(z\ast y)\ast\alpha(x)-\alpha(y)\ast(x\ast z)\\}$
$\displaystyle+\\{(x\ast z)\ast\alpha(y)-\alpha(z)\ast(y\ast x)\\}=0.$

Therefore, $(A,[.,.],\alpha)$ is a Hom-Lie algebra. ∎

Commutative nearly Hom-associative algebras are Hom-anti-flexible.

######  Proposition 7.3.

If $(A,\ast,\alpha)$ is a commutative nearly Hom-associative algebra, then
$(A,\ast,\alpha)$ is a Hom-anti-flexible algebra.

###### Proof.

In a commutative nearly Hom-associative algebra $(A,\ast,\alpha)$.

$\displaystyle a_{\alpha,\ast}(x,y,z)$ $\displaystyle=(x\ast
y)\ast\alpha(z)-\alpha(x)\ast(y\ast z)$ $\displaystyle=\alpha(y)\ast(z\ast
x)-(z\ast x)\ast\alpha(y)$ (nearly Hom-associativity)
$\displaystyle=\alpha(y)\ast(x\ast z)-(x\ast z)\ast\alpha(y)$ (commutativity)
$\displaystyle=(z\ast y)\ast\alpha(x)-\alpha(z)\ast(y\ast x)$ (nearly Hom-
associativity) $\displaystyle=a_{\alpha,\ast}(z,y,x).$

So any commutative nearly Hom-associative algebra is a Hom-anti-flexible
algebra. ∎

######  Definition 7.4.

A bimodule of a nearly Hom-associative algebra $(A,\ast,\alpha)$ is a
quadruple $(l,r,V,\varphi)$, where $V$ is a linear space,
$l,r:A\rightarrow{\rm End}(V)$ are two linear maps and $\varphi\in{\rm
End}(V)$ satisfying the relations, for all $x,y\in A$,

$\displaystyle\varphi\circ l(x)=l({\alpha(x)})\circ\varphi,$
$\displaystyle\varphi\circ r(x)=r({\alpha(x)})\circ\varphi,$ (46a)
$\displaystyle l({\alpha(x)})\circ l(y)$ $\displaystyle=$ $\displaystyle
r({\alpha(y)})\circ r(x),$ (46b) $\displaystyle l({\alpha(x)})\circ r(y)$
$\displaystyle=$ $\displaystyle l({y\ast x})\circ\varphi,$ (46c)
$\displaystyle r({\alpha(x)})\circ l(y)$ $\displaystyle=$ $\displaystyle
r({x\ast y})\circ\varphi.$ (46d)

######  Proposition 7.5.

Consider a nearly Hom-associative $(A,\ast,\alpha)$. Let $l,r:A\rightarrow{\rm
End}(V)$ be two linear maps such that $V$ is a linear space and
$\varphi\in{\rm End}(V)$. The quadruple $(l,r,V,\varphi)$ is a bimodule of
$(A,\ast,\alpha)$ if and only if there is a structure of a nearly Hom-
associative algebra $\star$ on $A\oplus V$ given by, for all $x,y\in A$ and
all $u,v\in V$,

$\displaystyle(\alpha\oplus\varphi)(x+u)$ $\displaystyle=$
$\displaystyle\alpha(x)+\varphi(u),$ (47) $\displaystyle(x+u)\star(y+v)$
$\displaystyle=$ $\displaystyle(x\ast y)+(l(x)v+r(y)u).$ (48)

######  Definition 7.6.

A representation of a Hom-Lie algebra
$(\mathcal{G},[.,.]_{{}_{\mathcal{G}}},\alpha_{{}_{\mathcal{G}}})$ on a linear
space $V$ with respect to $\psi\in{\rm End}(V)$ is a linear map
$\rho_{{}_{\mathcal{G}}}:\mathcal{G}\rightarrow{\rm End}(V)$ obeying for all
$x,y\in\mathcal{G}$,

$\displaystyle\rho_{{}_{\mathcal{G}}}(\alpha_{{}_{\mathcal{G}}}(x))\circ\psi$
$\displaystyle=$ $\displaystyle\psi\circ\rho_{{}_{\mathcal{G}}}(x),$ (49)
$\displaystyle\rho_{{}_{\mathcal{G}}}([x,y]_{{}_{\mathcal{G}}})\circ\psi$
$\displaystyle=$
$\displaystyle\rho_{{}_{\mathcal{G}}}(\alpha_{{}_{\mathcal{G}}}(x))\circ\rho_{{}_{\mathcal{G}}}(y)-\rho_{{}_{\mathcal{G}}}(\alpha_{{}_{\mathcal{G}}}(y))\circ\rho_{{}_{\mathcal{G}}}(x).$
(50)

######  Proposition 7.7.

Let $(A,\cdot,\alpha)$ be a nearly Hom-associative algebra and $V$ be a
finite-dimensional linear space over the field $\mathbb{K}$ such that
$(l,r,\varphi,V)$ is a bimodule of $(A,\cdot,\alpha)$, where
$l,r:A\rightarrow{\rm End}(V)$ are two linear maps and $\varphi\in{\rm
End}(V)$. Then the linear map $l-r:A\rightarrow{\rm End}(V),x\mapsto
l(x)-r(x)$ is a representation of the underlying Hom-Lie algebra
$(\mathcal{G}(A),\alpha)$ associated to the nearly Hom-associative algebra
$(A,\cdot,\alpha)$.

###### Proof.

Let $(A,\cdot,\alpha)$ be a nearly Hom-associative algebra and $V$ a finite-
dimensional linear space over the field $\mathbb{K}$ such that
$(l,r,\varphi,V)$ is a bimodule of $(A,\cdot,\alpha)$, where
$l,r:A\rightarrow{\rm End}(V)$ are two linear maps and $\varphi\in{\rm
End}(V)$. For all $x,y\in A$,

$\displaystyle(l-r)({\alpha(x)})\circ\varphi=l({\alpha(x)})\circ\varphi-r({\alpha(x)})\circ\varphi=\varphi\circ
l(x)-\varphi\circ r(x)=\varphi\circ(l-r)(x),$
$\displaystyle(l-r)({(\alpha(x))})\circ(l-r)(y)-(l-r)({(\alpha(y))})\circ(l-r)(x)$
$\displaystyle\quad=l({\alpha(x)})\circ l(y)-l({\alpha(x)})\circ
r(y)-r({\alpha(x)})\circ l(y)+r({\alpha(x)})\circ r(y)$
$\displaystyle\quad\quad-l({\alpha(y)})\circ l(x)+l({\alpha(y)})\circ
r(x)+r({\alpha(y)})\circ l(x)-r({\alpha(y)})\circ r(x)$
$\displaystyle\quad=\\{l({\alpha(x)})\circ l(y)-r({\alpha(y)})\circ
r(x)\\}-l({\alpha(x)})\circ r(y)-r({\alpha(x)})\circ l(y)$
$\displaystyle\quad\quad+\\{r({\alpha(x)})\circ r(y)-l({\alpha(y)})\circ
l(x)\\}+r({\alpha(y)})\circ l(x)+l({\alpha(y)})\circ r(x)$
$\displaystyle\quad=r({\alpha(y)})\circ l(x)-l({\alpha(x)})\circ
r(y)+l({\alpha(y)})\circ r(x)-r({\alpha(x)})\circ l(y)$
$\displaystyle\quad=r({y\cdot x})\circ\varphi-l({y\cdot
x})\circ\varphi+l({x\cdot y})\circ\varphi-r({x\cdot
y})\circ\varphi=(l-r)({[x,y]})\circ\varphi.$

Therefore, (49) and (50) are satisfied. ∎

######  Definition 7.8.

Let
$\displaystyle(\mathcal{G},[.,.]_{{}_{\mathcal{G}}},\alpha_{{}_{\mathcal{G}}})$
and
$\displaystyle(\mathcal{H},[.,.]_{{}_{\mathcal{H}}},\alpha_{{}_{\mathcal{H}}})$
be two Hom-Lie algebras. Let
$\displaystyle\rho_{{}_{\mathcal{H}}}:\mathcal{H}\rightarrow{\rm
End}(\mathcal{G})$ and
$\displaystyle\mu_{{}_{\mathcal{G}}}:\mathcal{G}\rightarrow{\rm
End}(\mathcal{H})$ be two Hom-Lie algebra representations, and
$\alpha_{{}_{\mathcal{G}}}:\mathcal{G}\rightarrow\mathcal{G}$ and
$\alpha_{{}_{\mathcal{H}}}:\mathcal{H}\rightarrow\mathcal{H}$ two linear maps
such that for all $x,y\in\mathcal{G},a,b\in\mathcal{H},$

$\displaystyle\begin{array}[]{lll}\mu_{{}_{\mathcal{G}}}(\alpha_{{}_{\mathcal{G}}}(x))\left[a,b\right]_{{}_{\mathcal{H}}}&=&\left[\mu_{{}_{\mathcal{G}}}(x)a,\alpha_{{}_{\mathcal{H}}}(b)\right]_{{}_{\mathcal{H}}}+\left[\alpha_{{}_{\mathcal{H}}}(a),\mu_{{}_{\mathcal{G}}}(x)b\right]_{{}_{\mathcal{H}}}\\\
&&-\mu_{{}_{\mathcal{G}}}(\rho_{{}_{\mathcal{H}}}(a)x)(\alpha_{{}_{\mathcal{H}}}(b))+\mu_{{}_{\mathcal{G}}}(\rho_{{}_{\mathcal{H}}}(b)x)(\alpha_{{}_{\mathcal{H}}}(a)),\end{array}$
(51c)
$\displaystyle\begin{array}[]{lll}\rho_{{}_{\mathcal{H}}}(\alpha_{{}_{\mathcal{H}}}(a))\left[x,y\right]_{{}_{\mathcal{G}}}&=&\left[\rho_{{}_{\mathcal{H}}}(a)x,\alpha_{{}_{\mathcal{G}}}(y)\right]_{{}_{\mathcal{G}}}+\left[\alpha_{{}_{\mathcal{G}}}(x),\rho_{{}_{\mathcal{H}}}(a)y\right]_{{}_{\mathcal{G}}}\\\
&&-\rho_{{}_{\mathcal{H}}}(\mu_{{}_{\mathcal{G}}}(x)a)(\alpha_{{}_{\mathcal{G}}}(y))+\rho_{{}_{\mathcal{H}}}(\mu_{{}_{\mathcal{G}}}(y)a)(\alpha_{{}_{\mathcal{G}}}(x)).\end{array}$
(51f)

Then,
$\displaystyle(\mathcal{G},\mathcal{H},\mu,\rho,\alpha_{{}_{\mathcal{G}}},\alpha_{{}_{\mathcal{H}}})$
is called a matched pair of the Hom-Lie algebras $\displaystyle{\mathcal{G}}$
and $\displaystyle\mathcal{H}$, and denoted by
$\displaystyle\mathcal{H}\bowtie_{\mu_{{}_{\mathcal{G}}}}^{\rho_{{}_{\mathcal{H}}}}{\mathcal{G}}.$
In this case,
$\displaystyle(\mathcal{G}\oplus\mathcal{H},[.,.]_{{}_{\mathcal{G}\oplus\mathcal{H}}},\alpha_{{}_{\mathcal{G}}}\oplus\alpha_{{}_{\mathcal{H}}})$
defines a Hom-Lie algebra, where

$\displaystyle[(x+a),(y+b)]_{{}_{\mathcal{G}\oplus\mathcal{H}}}=[x,y]_{{}_{\mathcal{G}}}+\rho_{{}_{\mathcal{H}}}(a)y-\rho_{{}_{\mathcal{H}}}(b)x+[a,b]_{{}_{\mathcal{H}}}+\mu_{{}_{\mathcal{G}}}(x)b-\mu_{{}_{\mathcal{G}}}(y)a.$
(52)

######  Theorem 7.9.

Let $(A,\cdot,\alpha_{A})$ and $(B,\circ,\alpha_{B})$ be two nearly Hom-
associative algebras. Suppose there are linear maps
$l_{A},r_{A}:A\rightarrow{\rm End}(B)$ and $l_{B},r_{B}:B\rightarrow{\rm
End}(A)$ such that $(l_{A},r_{A},B,\alpha_{B})$ and
$(l_{B},r_{B},A,\alpha_{A})$ are bimodules of the nearly Hom-associative
algebras $(A,\cdot,\alpha_{A})$ and $(B,\circ,\alpha_{B})$, respectively and
satisfying the following conditions for all $x,y\in A$ and $a,b\in B:$

$\displaystyle\alpha_{A}(x)\cdot(r_{B}(a)y)+(r_{B}(l_{A}(y)a)\alpha_{A}(x)-(l_{B}(a)x)\cdot\alpha_{A}(y)-l_{B}(r_{A}(x)a)\alpha_{A}(y)=0,$
(53a)
$\displaystyle\alpha_{A}(x)\cdot(l_{B}(a)y)+r_{B}(r_{A}(y)a)\alpha_{A}(x)-r_{B}(\alpha_{B}(a))(y\cdot
x)=0,$ (53b) $\displaystyle l_{B}(\alpha_{B}(a))(x\cdot
y)-(r_{B}(a)y)\cdot\alpha_{A}(x)-l_{B}(l_{A}(y)a)\alpha_{A}(x)=0,$ (53c)
$\displaystyle\alpha_{B}(a)\circ(r_{A}(x)b)+r_{A}(l_{B}(b)x)\alpha_{B}(a)-(l_{A}(x)a)\circ\alpha_{B}(b)-l_{A}(r_{B}(a)x)\alpha_{B}(b)=0,$
(53d)
$\displaystyle\alpha_{B}(a)\circ(l_{A}(x)b)+r_{A}(r_{B}(b)x)\alpha_{B}(a)-r_{A}(\alpha_{A}(x))(b\circ
a)=0,$ (53e) $\displaystyle l_{A}(\alpha_{A}(x))(b\circ
a)-(r_{A}(x)a)\circ\alpha_{B}(b)-l_{A}(l_{B}(a)x)\alpha_{B}(b)=0.$ (53f)

Then, there is a bilinear product defined on $A\oplus B$ for all $x,y\in A$,
and all $a,b\in B$, by

$\displaystyle(x+a)\ast(y+b)=(x\cdot y+l_{B}(a)y+r_{B}(b)x)+(a\circ
b+l_{A}(x)b+r_{A}(y)a)$ (54)

such that $(A\oplus B,\ast,\alpha_{A}\oplus\alpha_{B})$ is a nearly Hom-
associative algebra.

###### Proof.

Let $(A,\cdot,\alpha_{A})$, $(B,\circ,\alpha_{B})$ be two nearly Hom-
associative algebras, $(l_{A},r_{A},B,\alpha_{B})$ a bimodule of
$(A,\cdot,\alpha_{A})$ and $(l_{B},r_{B},A,\alpha_{A})$ a bimodule of
$(B,\circ,\alpha_{B})$. For all $x,y\in A$ and all $a,b\in B$,

$\displaystyle(\alpha_{A}(x)+\alpha_{B}(a))\ast((y+b)\ast(z+c))$
$\displaystyle\quad=\\{(\alpha_{A}(x))\cdot(l_{B}(b)z)+r_{B}(r_{A}(z)b)\cdot(\alpha_{A}(x))\\}$
$\displaystyle\quad\quad+\\{(\alpha_{A}(x))\cdot(r_{B}(c)y)+r_{B}(l_{A}(y)c)\alpha_{A}(x)\\}$
$\displaystyle\quad\quad+(\alpha_{A}(x))\cdot(y\cdot
z)+l_{B}(\alpha_{B}(a))(y\cdot z)+l_{B}(\alpha_{B}(a))(l_{B}(b)z)$
$\displaystyle\quad\quad+l_{B}(\alpha_{B}(a))(r_{B}(c)y)+r_{B}(b\circ
c)(\alpha_{A}(x))$
$\displaystyle\quad\quad+\\{(\alpha_{B}(a))\circ(l_{A}(y)c)+r_{A}(r_{B}(c)y)\alpha_{B}(a)\\}$
$\displaystyle\quad\quad+\\{(\alpha_{B}(a))\circ(r_{A}(z)b)+r_{A}(l_{B}(b)z)\alpha_{B}(a)\\}$
$\displaystyle\quad\quad+(\alpha_{B}(a))\circ(b\circ c)+r_{A}(y\cdot
z)\alpha_{B}(a)+l_{A}(\alpha_{A}(x))(b\circ c)$
$\displaystyle\quad\quad+l_{A}(\alpha_{A}(x))(l_{A}(y)c)+l_{A}(\alpha_{A}(x))(r_{A}(z)b);$
$\displaystyle((z+c)\ast(x+a))\ast(\alpha_{A}(y)+\alpha_{B}(b))$
$\displaystyle\quad=\\{(l_{B}(c)x)\cdot(\alpha_{A}(y))+l_{B}(r_{A}(x)c)\alpha_{A}(y)\\}$
$\displaystyle\quad\quad+\\{l_{B}(l_{A}(z)a)\alpha_{A}(y)+(l_{B}(c)x)\cdot\alpha_{A}(y)\\}$
$\displaystyle\quad\quad+(z\cdot x)\cdot(\alpha_{A}(y))+l_{B}(c\circ
a)(\alpha_{A}(y))+r_{B}(\alpha_{B}(b)(z\cdot x)$
$\displaystyle\quad\quad+r_{B}(\alpha_{B}(b)(l_{B}(c)x)+r_{B}(\alpha_{B}(b)(r_{B}(a)z)$
$\displaystyle\quad\quad+\\{(l_{A}(z)a)\circ(\alpha_{B}(b))+l_{A}(r_{B}(a)z)\alpha_{B}\\}$
$\displaystyle\quad\quad+\\{(r_{A}(x)c)\circ(\alpha_{B}(b))+l_{A}(l_{B}(c)x)\alpha_{B}(b)\\}$
$\displaystyle\quad\quad+(c\circ a)\circ(\alpha_{B}(b))+l_{A}(z\cdot
x)(\alpha_{B}(b))+r_{A}(\alpha_{A}(y))(c\circ a)$
$\displaystyle\quad\quad+r_{A}(\alpha_{A}(y))(l_{A}(z)a)+r_{A}(\alpha_{A}(y))(r_{A}(x)c)$

Using (53a) - (53f) and the fact that $(l_{A},r_{A},B,\alpha_{B})$ and
$(l_{B},r_{B},A,\alpha_{A})$ are bimodules of the nearly Hom-associative
algebras $(A,\cdot,\alpha_{A})$ and $(B,\circ,\alpha_{B})$, respectively, we
obtain that $(A\oplus B,\ast,\alpha_{A}\oplus\alpha_{B})$ is a nearly
associative algebra. ∎

######  Definition 7.10.

A matched pair of the nearly Hom-associative algebras $(A,\cdot,\alpha_{A})$
and $(B,\circ,\alpha_{B})$ is the high-tuple
$(A,B,l_{A},r_{A},\alpha_{B},l_{B},r_{B},\alpha_{A})$, where
$l_{A},r_{A}:A\rightarrow{\rm End}(B)$ and $l_{B},r_{B}:B\rightarrow{\rm
End}(A)$ are linear maps such that $(l_{A},r_{A},B,\alpha_{B})$ and
$(l_{B},r_{B},A,\alpha_{A})$ are bimodules of the nearly Hom-associative
algebras $(A,\cdot,\alpha_{A})$ and $(B,\circ,\alpha_{B})$, respectively, and
satisfying (53a) - (53f).

######  Corollary 7.11.

Let $(A,B,l_{A},r_{A},\alpha_{B},l_{B},r_{B},\alpha_{A})$ be a matched pair of
the nearly Hom-associative algebras $(A,\cdot,\alpha_{A})$ and
$(B,\circ,\alpha_{B}).$ Then,
$(\mathcal{G}(A),\mathcal{G}(B),l_{A}-r_{A},l_{B}-r_{B},\alpha_{A},\alpha_{B})$
is a matched pair of the underlying Hom-Lie algebras $\mathcal{G}(A)$ and
$\mathcal{G}(B)$ of the nearly Hom-associative algebras $(A,\cdot,\alpha_{A})$
and $(B,\circ,\alpha_{B})$.

###### Proof.

Let $(A,B,l_{A},r_{A},\alpha_{B},l_{B},r_{B},\alpha_{A})$ be a matched pair of
nearly Hom-associative algebras $(A,\cdot,\alpha_{A})$ and
$(B,\circ,\alpha_{B})$. In view of Proposition 7.7, the linear maps
$l_{A}-r_{A}:A\longrightarrow{\rm End}(B)$ and
$l_{B}-r_{B}:B\longrightarrow{\rm End}(A)$ are representations of the
underlying Hom-Lie algebras $(\mathcal{G}(A),\alpha_{A})$ and
$(\mathcal{G}(B),\alpha_{B})$, respectively. Therefore, (51c) is equivalent to
(53a) - (53c) and similarly, (51f) is equivalent to (53d) - (53f). ∎

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# Time Series Anomaly Detection with CNN for Environmental Sensors in
Healthcare-IoT
††thanks: This publication has emanated from research conducted with the
financial support of Science Foundation Ireland under Grant number
18/CRT/6049.

Mirza Akhi Khatun
Ciarán Eising  Dept. Electronic & Computer Engineering,
SFI CRT Foundations in Data Science
University of Limerick
Limerick, V94 T9PX, Ireland
<EMAIL_ADDRESS>Dept. Electronic & Computer Engineering
University of Limerick
Limerick, V94 T9PX, Ireland
<EMAIL_ADDRESS>Mangolika Bhattacharya
Lubna Luxmi Dhirani  School of Information Technology
Illinois State University
Illinois, USA
<EMAIL_ADDRESS>Dept. Electronic & Computer Engineering
University of Limerick
Limerick, V94 T9PX, Ireland
<EMAIL_ADDRESS>
###### Abstract

This research develops a new method to detect anomalies in time series data
using Convolutional Neural Networks (CNNs) in healthcare-IoT. The proposed
method creates a Distributed Denial of Service (DDoS) attack using an IoT
network simulator, Cooja, which emulates environmental sensors such as
temperature and humidity. CNNs detect anomalies in time series data, resulting
in a 92% accuracy in identifying possible attacks.

###### Index Terms:

Healthcare-IoT (H-IoT), Anomaly Detection, DDoS Attack, Cyberattack, Wireless
Sensor Network (WSN), Cooja Simulator, Contiki OS.

##  I Introduction

The Internet of Things (IoT) is a network of physical devices that communicate
with each other through sensors, software, and connectivity [1],[2],[3].

Considering that Wireless Sensor Networks (WSNs) are used to monitor the
environment in health facilities, the necessity for precise and reliable data
is significant. Aligning an organization’s cybersecurity measures is crucial
for protecting patient privacy and safety and ensuring the continuous and
efficient provision of high-quality care [4]. The massive influx of data makes
it challenging to discern the presence of a malfunctioning sensor,
environmental changes, or abrupt temperature fluctuations. For example,
patients’ rooms with high humidity can create a breeding ground for bacteria,
posing a heightened risk of infection, especially in patients with weakened
immune systems [5]. As another example, fluctuations in temperature within an
operating room can be detrimental to the patient’s well-being [6]. Thus,
providing integrity to the transmitted data is essential in the healthcare-IoT
(H-IoT) environment. Any disruption can adversely affect patient care,
including delayed or prematurely accelerated data caused by cyber threats [7],
reducing potential disruptions that could adversely affect clinical results.

This research investigates the accuracy of convolutional neural network (CNN)
models in detecting anomalies in time series data and their adaptability to
real-world constraints within healthcare-IoT, as simulated within the Cooja
environment. CNN is widely used for various applications, including the
analysis of time series data, for which it can achieve competitive and better
results than traditional time series analysis models like Autoregressive
Integrated Moving Average (ARIMA) [8].

The primary goal is to detect anomalous readings on environmental sensors
within the hospital’s IoT ecosystem. Moreover, our dataset improves abnormal
reading detection, reduces risks, enhances efficiency, and elevates healthcare
quality in H-IoT environments.

##  II Methodology

This paper proposes a novel CNN-based method for identifying time-series
healthcare-IoT data anomalies. The methodology is explicitly tailored to the
healthcare-IoT environment. Models and data are analyzed using Python 3.9,
integrated with a neural network framework. To ensure robust and efficient
deep learning processing, all experiments are conducted on a MacBook Pro
equipped with the M1 chip. The performance of the algorithms has been measured
using our developed dataset “WSN_DDoS_Attack_H-IoT2023” in this research.

###  II-A Data Acquisition

To detect time series anomalies, we divide the healthcare environment into
specific zones such as patient wards, waiting rooms, medical equipment areas,
and high-traffic areas.

###  II-B Data Preprocessing

The first step is establishing a baseline profile for generating datasets from
healthcare devices’ raw data packets. The baseline profile consists of three
features, namely Time, Mote, and Message.

Feature Extraction: Several features are extracted from the raw data
descriptions through pattern matching, including ‘Previous Send Time,’
‘Interval,’ ‘Previous Send Interval (Same/current Node),’ etc. The goal is to
increase accuracy by experimenting with five selected features [9].

  * •

IDs: In the Cooja simulation, IDs serve as unique identifiers for nodes and
servers, facilitating efficient tracking and management of client-server
interactions. Furthermore, a unique identifier for a device, node, mote, or
client.

  * •

Interval: A period between two or more data transmissions. The equation for
finding the interval is shown in equation (1).

$\text{Interval}=\text{TimeStamp}-\text{PreviousSendTime}$ (1)

  * •

Previous Send Interval (Same Node): Approximately one minute has passed since
the last transmission of data from the current node. In equation (2), “PSI”
stands for “PreviousSendInterval,” “TS” stands for “TimeStamp,” and “PST”
stands for “PreviousSendTime”, calculated as,

$\displaystyle\text{PSI(same node)}=\text{TS}-\text{PST(same node)}$ (2)

  * •

Average Previous Send Interval (Same Node): The Average Previous send interval
(same node) is calculated based on the Previous send interval (same node). In
this equation (3), “Avg. PSI” stands for “Average Previous Send Interval,”
“Sum of PSI” represents the sum of all previous send intervals, and “Total
Intervals” signifies the total number of intervals for the same node (SM). The
calculation is as follows:

$\text{Avg. PSI (SM)}=\frac{\sum\text{PSI(SM)}}{\text{Total Intervals (SM)}}$
(3)

  * •

Previous Sender: Before the current sensor node, a node transmits data, for
example, sensor node ID:8 and the current node ID:2.

Data Analysis: To capture average behaviors across different nodes, features
such as ‘Average Previous Send Interval (Same Node), ‘Average Previous Send
Interval (Normal Nodes), and ‘Average Previous Send Interval (Malicious
Nodes)’ are computed.

Time Series Data Splitting: When handling time series data, paying attention
to the temporal structure is crucial. A time series dataset differs from a
standard dataset in that data points are arranged sequentially instead of
independently. A data sequence is essential for identifying underlying
patterns and relationships. When splitting, a specialized method is used to
preserve the temporal structure. This approach divides the dataset based on
sequential order rather than random splits. In particular, the first part of
the set (e.g., 70%) is used for training, while the remaining portion is used
for testing. As a result, any potential data leakage is eliminated since the
training dataset always precedes the test dataset sequentially. It is,
therefore, possible to predict tasks that depend on past events by maintaining
the natural order of the data.

###  II-C CNN Model Architecture

A time series CNN architecture is explicitly developed for one-dimensional
time series data. A 3D array is created by reshaping an input into samples,
timesteps, and individual features. Following this, a convolutional layer is
created using 32 filters of size three and a Rectified Linear Unit (ReLU)
activation function, which excels at detecting temporal patterns. Following
that, a max-pooling layer with a pool size of 2 reduces the dimensionality of
the data. In binary classification, the dense layer is initially flattened.
The input is passed through a Sigmoid activation function in the next dense
layer.

The sigmoid activation for the dense layer is shown below:

$\sigma(z)=\frac{1}{1+e^{-z}}$

As compared to Softmax, ReLU, and Tanh, Sigmoid performed better in our binary
classification experiment. Table I indicates this. In binary classification,
Sigmoid is preferred due to its effective outcome mapping. Softmax, Tanh, and
ReLU comparisons also highlight the importance of carefully selecting
activation functions based on data and task characteristics.

Activation Function | Accuracy (%) | Estimated Loss (%)
---|---|---
Sigmoid | 92% | 8%
Softmax | 47% | 53%
ReLU | 68% | 32%
Tanh | 76% | 24%
TABLE I: Scores with Different Activation Functions

###  II-D Experimental Setup

The simulations are conducted using the Cooja simulator on Contiki 3.0 OS.
This version contains several improvements, including enhancements to Cooja,
improvements to MSPSim, new regression tests, new platforms, enhancements to
the radio API, enhancements to the IPv6 stack, implementation of the message
queuing telemetry transport (MQTT) protocol, updates to the constrained
application protocol (CoAP), and added hypertext transfer protocol (HTTP)
sockets [10]. The latest version of Contiki comes with all the tools and
compilers necessary for emulating hardware devices and debugging programs. A
free, open-source platform is available for download and testing anytime.
Cooja has two emulator software packages: Avrora and MSPSim [11]. As a result,
it emulates MSP430-based devices through MSPSim. Most platforms are equipped
with the MSP430 microcontroller [12]. Consequently, MSPSim is the most widely
used software package for simulating WSNs. Cooja can emulate multiple
platforms, including TelosB/SkyMote, Zolertia Z1 mote, Wismote, ESB, and MicaZ
mote. In this work, we rely on the SkyMote platform. This allowed us to
evaluate code and systems before their deployment on real hardware. Also,
efficient node power estimation and radio transmission display are ensured.

The technical specifications of the system used during the simulation stage
are as follows:

  * •

CPU: 12th Gen Intel(R) Core(TM) i5-12500H 2.50 GHz

  * •

Random Access Memory: 16 GB

  * •

Operating System: Windows 11

  * •

GPU: NVIDIA GeForce RTX

  * •

Emulated Environment: Oracle VM VirtualBox 7.0 Platform

  * •

Simulation Spotlight: Contiki 3.0 OS

Integrating these hardware characteristics with Oracle VM VirtualBox 7.0
creates a flexible and efficient platform. This setup makes detecting
abnormalities and significant events from IoT devices in healthcare easier.

###  II-E Metrics for Performance Evaluation

This section illustrates the level of precision of the suggested
classification method and compares its performance to that of established
methods, which employ a CNN classifier. The accuracy is calculated as,

$\text{Accuracy}=\frac{Correctly\ classified\ nodes}{Total\ nodes\ in\ test\
data}*100\%$ (4)

Thus, the error rate is calculated in equation

$\text{Error\ Rate}=\big{(}100\ -\ Accuracy)*100\%$ (5)

###  II-F Convolutional Neural Network Classifier

This paper relies on CNN as the basis for classification. The proposed model
extracts features from input data using a 1D convolutional layer, particularly
suited for time-series analysis. The initial layer consists of 32 filters,
each using a three-unit kernel, optimizing the model’s performance in
isolating time-related attributes. A ‘relu’ activation function is
incorporated into the convolutional framework to enhance the model’s ability
to recognize complex relationships in data. As a result of this particular
function, the model can distinguish complex features and anomalies more
easily. Following the convolution stage, the data is strategically streamlined
using MaxPooling1D.

Figure 1: Comparative Performance: Accuracy vs. Error Rate

In our experiments, all algorithms displayed in the graph 1 are accurate to a
reasonable degree. Nevertheless, the decision leaned towards the CNN model for
various reasons. CNN not only demonstrated faster training times but also
required fewer computational resources. It outperforms other algorithms by
extracting relevant features from our dataset. In addition, CNN consistently
performs better than other methods, such as support vector machine (SVM) and
ensemble learning, regarding accuracy and loss metrics. The following list
summarizes the execution times for CNN and SVM:

  * •

CNN (10 epochs): 1 minute 40 seconds

  * •

SVM: 2 minutes 17 seconds

Based on these aspects, we conclude that the CNN model is the most effective
and efficient.

##  III Conclusion

This research presents an innovative method for identifying anomalies in time
series data using CNN for environmental IoT sensors in healthcare. This paper
describes a unique dataset called “WSN_DDoS_Attack_H-IoT2023”. As a result of
methodically assembling data on room temperature and humidity, this dataset
reflects the commitment to cater to the unique requirements of healthcare
environments. The CNN model achieved a 92% accuracy rate on this dataset,
performing better than other methods such as SVM, ensemble learning, long
short-term memory network (LSTM), etc.

## Acknowledgment

This publication has emanated from research conducted with the financial
support of Science Foundation Ireland under Grant number 18/CRT/6049.

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