# Classification, $\alpha^{k}$-Derivations and Centroids of 4-dimensional
Complex Hom-associative Dialgebras

Ahmed Zahari
Université de Haute Alsace,
IRIMAS-Département de Mathématiques,
18, rue des Frères Lumière F-68093 Mulhouse, France.  e-mail address:
<EMAIL_ADDRESS>
Abstract. The basic objective of the current research paper is to investigate
the structure and the algebraic varieties of Hom-associative dialgebras. We
elaborate a classification of $n$-dimensional Hom-associative dialgebras for
$n\leq 4$. Additionally, using the classification result of Hom-associative
dialgebras, we characterize the $\alpha$-derivations and centroids of low-
dimensional Hom-associative dialgebras. Furthermore, we equally tackle certain
features of derivations and centroids in the light of associative dialgebras
and compute the centroids of low-dimensional associative dialgebras.

AMS Subject Classification:  .

Keywords: Hom-associative dialgebra, Classification, Derivation, Triple
system.

__

_Dedicated to the memory of professor Marie-Hélène TUILIER, Professor of
physics and Responsible at the University of Haute-Alsace of doctoral school
who has helped me morally and financially during my doctoral years._

##  1 Introduction

A Hom-diassociative algebra $(\mathcal{A},\dashv,\vdash,\alpha)$ consists of a
vector space, two multiplications, and a linear self map. It may be regarded
as a deformation of an associative algebra, where the associativity condition
is twisted by a linear map $\alpha$ , such that when $\alpha=id$, the Hom-
associative dialgebra degenerates to exactly a associative dialgebra. The
central focus of this work is to explore the structure of Hom-associative
dialgebras. Let $\mathcal{A}$ be an $n$-dimensional $\mathbb{K}$-linear vector
space and let $\left\\{e_{1},e_{2},\cdots,e_{n}\right\\}$ be a basis of
$\mathcal{A}$, where $\mathbb{K}$ will always be an algebraically closed field
of characteristic $0$. A Hom-dialgebra structure on $\mathcal{A}$ with
products $\mu$ and $\lambda$ is determined by $2n^{3}$ structure constants
$\mathcal{\gamma}_{ij}^{k}$ and $\mathcal{\delta}_{ij}^{k}$, were $e_{i}\dashv
e_{j}=\sum_{k=1}^{n}\gamma_{ij}^{k}e_{k},\quad e_{i}\vdash
e_{j}=\sum_{k=1}^{n}\delta_{ij}^{k}e_{k}$ and by $\alpha$ which is given by
${n^{2}}$ structure constants $a_{ij}$, where
$\alpha(e_{i})=\sum_{j=1}^{n}a_{ji}e_{j}$. Requiring the algebra structure to
be Hom-diassociative and unital gives rise to a sub-variety $\mathcal{H}d_{n}$
of $k^{2n^{3}+n^{2}}$. Base changes in $A$ result in the natural transport of
structure action of $GL_{n}(k)$ on $\mathcal{H}d_{n}$. Thus isomorphism
classes of $n$-dimensional Hom-dialgebras are in one-to-one correspondence
with the orbits of the action of $GL_{n}(k)$ on $\mathcal{H}d_{n}$.

In this paper, we tackle the problem of classification. We set forward an
algorithm to compute classification. We apply the algorithm in low-dimensional
cases. We obtain the classification results of two and three-dimensional
complex associative dialgebras from Rikhsiboev [10] and revise a list of four-
dimensional nilpotent diassociative algebras from Rakhimov and Fiidov [13].
Within this framwork, A. Zahari and I. Bakayoko studied the classification of
BiHom-associative dialgebras [2]. The classification of two and three-
dimensional Hom-associative dialgebras was undertaken by A. Zahari and A.
Makhlouf [1] and the classification of $3$-dimensional BiHom-associative and
BiHom-bialgebras [16] was performed by A. Zahari. Furthermore, we shall
consider the class of Hom-associative dialgebras. We shall also establish a
classification of these algebras up to isomorphism in low dimension $n\leq 4$.

The paper is laid out as follows. In the first section, we identify the basics
about Hom-associative dialgebras and provide several new properties.

In section $2$, we address the structure of Hom-associative dialgebras.

Section $3$ is devoted to the description of the algebraic varieties of Hom-
diassociative algebras, and classifications, up to isomorphism, of two-
dimensional, three-dimensional and four-Hom-associative dialgebras are
introduced.

In section $4$, we determine certain new properties of derivations and we
focus upon the classification of the derivations.

Eventually, in Section $5$, we handle the classification of the centroids. In
this case, the concept of derivations and centroids is notably inspired from
that of finite-dimensional algebras. The algebra of centroids plays a key role
in terms of the classification problems as well as in different applications
of algebras. As far as our work is concerned, we elaborate classification
results of two, three and four-dimensional Hom-associative dialgebras. All
considered algebras and vectors spaces are supposed to be over a field
$\mathbb{K}$ of characteristic zero.

##  2 Structure of Hom-associative dialgebras

######  Definition 2.1.

A Hom-associative dialgebra is a $4$-truple
$(\mathcal{A},\dashv,\vdash,\alpha)$ consisting of a linear space
$\mathcal{A}$ linear maps
$\dashv,\vdash,:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$ and
$\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ satisfying, for all
$x,y,z\in\mathcal{A}$, the following conditions :

$\displaystyle(x\dashv y)\dashv\alpha(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\dashv(y\dashv z),$ (2.1) $\displaystyle(x\dashv
y)\dashv\alpha(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\dashv(y\vdash z),$
(2.2) $\displaystyle(x\vdash y)\dashv\alpha(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\vdash(y\dashv z),$ (2.3) $\displaystyle(x\dashv
y)\vdash\alpha(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\vdash(y\vdash z),$
(2.4) $\displaystyle(x\vdash y)\vdash\alpha(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\vdash(y\vdash z).$ (2.5)

We call $\alpha$ ( in this order ) the structure maps of $\mathcal{A}$.

If in addition, $\alpha$ is an endomorphism with respect to $\dashv$ and
$\vdash$, then $\mathcal{A}$ is said to be a multiplicative Hom-dialgebra :

$\alpha(x\dashv y)=\alpha(x)\dashv\alpha(y)\quad\text{and}\quad\alpha(x\vdash
y)=\alpha(x)\vdash\alpha(y)$ (2.6)

for $x,y,z$ in $\mathcal{A}.$

As we are dealing only with multiplicative Hom-associative dialgebras, we
shall call them Hom-diassociative algebras for simplicity. We denote the set
of all Hom-associative dialgebras by $\mathcal{H}d$. The kernel and the image
of homomorphism is defined naturally. One of the basic problems in structure
theory of algebras resides in the problem of classification. The
classification implies the description of the orbits under base change linear
transformations and list representatives of the orbits.

######  Definition 2.2.

Let $(\mathcal{A}_{1},\dashv_{1},\vdash_{1},\alpha_{1})$ and
$(\mathcal{A}_{2},\dashv_{2},\vdash_{2},\alpha_{2})$ be Hom-associative
dialgebras over a field $\mathbb{K}$. Then, a homomorphism of Hom-associative
dialgebras $\mathcal{A}_{1}$ to $\mathcal{A}_{2}$ is a $\mathbb{K}$-linear
mapping $\phi:\mathcal{A}_{1}\longrightarrow\mathcal{A}_{2}$ such that
$\phi:\mathcal{A}_{1}\longrightarrow\mathcal{}_{2}$ such that

$\phi(x\dashv_{1}y)=\phi(x)\dashv_{2}\phi(y),\,\phi(x\vdash_{1}y)=\phi(x)\vdash_{2}\phi(y),\,\phi\circ\alpha_{1}(x)=\alpha_{2}\circ\phi(x),$
(2.7)

for all $x,y\in\mathcal{A}.$ A bijective homomorphism is said to be an
isomorphism.

######  Definition 2.3.

A bar unit in $\mathcal{A}$ is an element $e\in\mathcal{A}$ such that $x\dashv
e=e\vdash x=\alpha(x)=x.$

######  Definition 2.4.

A Hom-dendrifom algebra is a quadriple $(\mathcal{A},\prec,\succ,\alpha)$
consisting of a vector pace $\mathcal{A}$ on which the operations
$\prec,\succ:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$ and
$\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ are linear maps satisfying :

$\displaystyle(x\prec y)\prec\alpha(z)$ $\displaystyle=\alpha(x)\prec(y\prec
z+y\succ z),$ (2.8) $\displaystyle\alpha(x)\succ(y\succ z)$
$\displaystyle=(x\prec y+x\succ y)\prec\alpha(z),$ $\displaystyle(x\succ
y)\prec\alpha(z)$ $\displaystyle=\alpha(x)\succ(y\prec z),$

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

######  Example 2.5.

In fact, a Hom-dendrifom algebra structure on an n-dimensional vector space
$V$ with a basis $\left\\{e_{1},e_{2},\dots,e_{n}\right\\}$ can be obtained
through defining the products and maps of the vectors
$\left\\{e_{1},e_{2},\dots,e_{n}\right\\}$. In 2 dimensions, we have :

$e_{2}\prec e_{1}=e_{1},\,e_{2}\prec e_{2}=e_{1},\,e_{1}\succ
e_{2}=e_{1},\,e_{2}\succ e_{1}=e_{1},\,e_{2}\succ
e_{2}=e_{1},\,\alpha(e_{2})=e_{1}.$

On a 3-dimensional vector space, define the following Hom-dendrifom algebra
structure respectively :

$e_{1}\prec e_{1}=e_{1},\ ,e_{3}\prec e_{2}=e_{3},\,e_{3}\prec
e_{3}=e_{3},\,e_{2}\succ e_{3}=e_{3},\,e_{3}\succ
e_{3}=e_{3},\,\alpha(e_{1})=e_{1}.$

######  Definition 2.6.

A Hom-Zinbiel algebra is a triple $(R,\circ,\alpha)$ consisting of vector
space A on which $\ast:A\otimes A\longrightarrow A$ and
$\alpha:A\longrightarrow A$ are linear maps satisfying

$(x\circ y)\circ\alpha(z)=\alpha(x)\circ(y\circ z)+\alpha(x)\circ(z\circ y)$
(2.9)

for $x,y,y\in\mathcal{A}.$

######  Example 2.7.

Let $\left\\{e_{1},e_{2}\right\\}$ be a basis of $2$-dimensional
multiplicative linear space $\mathcal{A}$ over $\mathbb{K}$. The following
multiplications and linear map $\alpha$ on $\mathcal{A}$ are defined by

  1. 1.

$e_{1}\circ e_{2}=e_{1},\quad e_{2}\circ
e_{2}=e_{1},\quad\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=e_{2}.$

  2. 2.

$e_{1}\circ e_{1}=e_{1},\quad e_{2}\circ
e_{2}=e_{1},\quad\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=e_{1}+e_{2}.$

  3. 3.

$e_{1}\circ e_{1}=e_{2},\quad e_{2}\circ
e_{1}=-e_{2},\quad\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=e_{1}+e_{2}.$

######  Proposition 2.8.

Let R be a Hom-Zinbiel algebra and put $x\prec y=x\circ y,\quad x\succ
y=y\circ x,\,x,y\in R.$ Then $(R,\prec,\succ,\alpha)$ is a Hom-dendriform
algebra. Conversely, a commutative Hom-dendriform algebra $(i.e$ a Hom-
dendriform algebra for which $x\prec y=y\succ x)$ is a Hom-dendriform algebra.

###### Proof.

Indeed, $(i)$ $(x\prec y)\prec\alpha(z)=(x\circ y)\circ\alpha(z)$.
But $\alpha(x)\prec(y\prec z)+\alpha(x)\prec(y\succ y)=\alpha(x)\circ(y\circ
z)+\alpha(x)\circ(z\circ y).$ Therefore, (i) holds.
$(ii)$ $(x\succ y)\prec\alpha(z)=(y\circ x)\circ\alpha(z)$ and
$\alpha(x)\succ(y\prec z)=(y\circ z)\circ\alpha(x).$ However, these two
expressions are the same according to the axioms of Hom-Zinbiel algebras.
$(iii)$ $(x\prec y)\succ\alpha(z)+(x\succ
y)\succ\alpha(z)=\alpha(z)\circ(x\circ y)+\alpha(z)\circ(y\circ x)$, which is
equal to $(z\circ y)\circ\alpha(z)=\alpha(x)\succ(y\succ z).$ As a matter of
fact, $(iii)$ also holds. ∎

######  Proposition 2.9.

Let $(R,\cdot,\alpha)$ be a Hom-Zinbiel algebra. Then, the symmetrized product
$x\cdot y=x\circ y+y\circ x$ is Hom-associative (,i.e., under the symmetrized
product, R becomes a Hom-associative and commutative algebra.

###### Proof.

Indeed,

$\begin{aligned} (x\cdot y)\alpha(z)&=(x\circ y+y\circ x)\alpha(z)\\\
&=(x\circ y+y\circ x)\circ\alpha(z)+\alpha(z)\circ(x\circ y+y\circ
x)\circ\alpha(z)\\\ &=(x\circ y)\circ\alpha(z)+(y\circ
x)\circ\alpha(z)+\alpha(z)\circ(x\circ y)+(y\circ
x)\circ\alpha(z)\end{aligned}$

and

$\begin{aligned} \alpha(x)(y\cdot z)&=\alpha(x)(y\circ z+z\circ y)\\\
&=\alpha(x)\circ(y\circ z+z\circ y)+(y\circ z+z\circ y)\alpha(x)\\\
&=\alpha(x)\circ(y\circ z)+\alpha(x)\circ(z\circ y)+(y\circ
z)\circ\alpha(x)+(z\circ y)\circ(x).\end{aligned}$

Now, if we take into account Hom-Zinbiel indentity and its consequence
$(y\circ x)\circ\alpha(z)=(y\circ z)\alpha(z)$, then we get the following
required equality $(x\cdot y)\alpha(z)=(x\cdot y)\alpha(z).$ ∎

######  Definition 2.10.

A Hom-dipterous algebra is a quadruple $(Z,\prec,\ast,\alpha)$ consisting of a
vector space $\mathcal{A}$ with the operations
$\prec,\ast:\mathcal{A}\otimes\mathcal{A}\longrightarrow\mathcal{A}$ and
$\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ which are linear maps
satisfying

$(x\ast y)\succ\alpha(z)=\alpha(x)\succ(y\succ z),\quad(x\ast
y)\ast\alpha(z)=\alpha(x)\ast(y\ast z)$ (2.10)

for all $x,y,z\in\mathcal{A}.$ Similarly, a right Hom-dipterous algebra is
defined by the following relations

$(x\prec y)\prec\alpha(z)=\alpha(x)\prec(y\ast z),\quad(x\ast
y)\ast\alpha(z)=\alpha(x)\ast(y\ast z),$ (2.11)

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

######  Example 2.11.

Let $\left\\{e_{1},e_{2},e_{3}\right\\}$ be a basis of $3$-dimensional
multiplicative linear space $A$ over $\mathbb{K}$. The following
multiplications and linear map $\alpha$ on $\mathcal{A}$ define the structure
of a Hom-dipterous algebra:

  1. 1.

$e_{1}\ast e_{2}=e_{3},\,e_{3}\ast e_{2}=e_{3},\,e_{3}\ast e_{3}=e_{3},\quad
e_{1}\prec e_{2}=e_{3},\,e_{2}\prec e_{3}=e_{3},\,\alpha(e_{1})=e_{1}.$

  2. 2.

$e_{1}\ast e_{1}=e_{1},e_{1}\ast e_{2}=e_{3},e_{3}\ast e_{2}=e_{3},e_{3}\ast
e_{3}=e_{3},e_{1}\prec e_{2}=e_{3},e_{2}\prec
e_{3}=e_{3},\,\alpha(e_{1})=e_{1}.$

######  Proposition 2.12.

Let $(\mathcal{A},\ast,\succ)$ be a dipterous algebra and let
$\alpha:\mathcal{A}\longrightarrow\mathcal{A}$ be a dipterous algebra
endomorphism. Hence,
$\mathcal{A}_{\alpha}=(\mathcal{A},\ast_{\alpha},\succ_{\alpha},\alpha)$,
where $\ast_{\alpha}=\alpha\circ\ast$ and $\succ_{\alpha}=\alpha\circ\succ$,
is a Hom-dipterous algebra.

###### Proof.

Notice that

$\displaystyle(x\ast_{\alpha}y)\succ_{\alpha}\alpha(z)$
$\displaystyle=\alpha^{2}((x\ast y)\succ z)$
$\displaystyle\alpha(x)\succ_{\alpha}(y\succ_{\alpha}z)$
$\displaystyle=\alpha^{2}(x\succ(y\succ z))$
$\displaystyle(x\ast_{\alpha}y)\ast_{\alpha}\alpha(z)$
$\displaystyle=\alpha^{2}((x\ast y)\ast z)$
$\displaystyle\alpha(x)\ast_{\alpha}(y\ast_{\alpha}z)$
$\displaystyle=\alpha^{2}(x\ast(y\ast z)).$

Similarly,

$\displaystyle(x\prec_{\alpha}y)\prec_{\alpha}\alpha(z)$
$\displaystyle=\alpha^{2}((x\prec y)\prec z)$
$\displaystyle\alpha(x)\prec_{\alpha}(y\ast_{\alpha}z)$
$\displaystyle=\alpha^{2}(x\prec(y\ast z))$
$\displaystyle(x\ast_{\alpha}y)\ast_{\alpha}\alpha(z)$
$\displaystyle=\alpha^{2}((x\ast y)\ast z)$
$\displaystyle\alpha(x)\ast_{\alpha}(y\ast_{\alpha}z)$
$\displaystyle=\alpha^{2}(x\ast(y\ast z)).$

Thus, the identities (2.10) and (2.11) obviously follow from the identities
satisfied by $(\mathcal{A},\ast,\succ)$. ∎

######  Definition 2.13.

Let $(\mathcal{A},\dashv,\vdash,\alpha)$ be a Hom-associative dialgebra. If
there is an associative dialgebra $(A,\dashv^{\prime},\vdash^{\prime})$ such
that $\dashv^{\prime}=\alpha\circ\dashv$ and
$\vdash^{\prime}=\alpha\circ\vdash$, we state that
$(\mathcal{A},\dashv^{\prime},\vdash^{\prime})$ is the untwist of
$(\mathcal{A},\dashv,\vdash,\alpha)$.

######  Proposition 2.14.

Let $(\mathcal{A},\dashv,\vdash,\alpha)$ be an $n$-dimensional Hom-associative
dialgebra and let $\Phi:\mathcal{A}\rightarrow\mathcal{A}$ be an invertible
linear map. Then, there is an isomorphism with an n-dimensional Hom-
associative dialgebra
$(\mathcal{A},\dashv^{\prime},\vdash^{\prime},\Phi\alpha\Phi^{-1})$ where
$\dashv^{\prime}=\Phi\circ\dashv\circ(\Phi^{-1}\otimes\Phi^{-1}$) and
$\vdash^{\prime}=\Phi\circ\vdash\circ(\Phi^{-1}\otimes\Phi^{-1}$).

###### Proof.

We prove that for any invertible linear map
$\Phi:\mathcal{A}\rightarrow\mathcal{A},\,(\mathcal{A},\dashv^{\prime},\vdash^{\prime},\Phi\alpha\Phi^{-1})$
is a Hom-associative dialgebra.

$\begin{array}[]{ll}(x\dashv^{\prime}y)\dashv^{\prime}\Phi\alpha\Phi^{-1}(z)&=(\Phi\circ(\Phi^{-1}(x)\dashv\Phi^{-1}(y)))\dashv^{\prime}\Phi\circ\alpha\circ\Phi^{-1}(z)\\\
&=\Phi\circ(\Phi^{-1}(x)\dashv\Phi^{-1}(y))\dashv\alpha\circ\Phi^{-1}(z)\\\
&=\Phi\circ\Phi^{-1}((x)\dashv(y))\dashv\alpha(z)\\\
&=\Phi\circ\Phi^{-1}(\alpha(x)\dashv(y))\vdash z)\\\
&=\Phi\circ(\alpha\Phi^{-1}(x)\dashv\Phi^{-1}(y))\vdash\Phi^{-1}(z))\\\
&=\Phi\circ((\Phi^{-1}\otimes\Phi^{-1})(\Phi\otimes\Phi)\alpha\Phi^{-1}(x)\dashv\Phi^{-1}(y))\vdash\Phi^{-1}(z))\\\
&=\Phi\circ(\Phi\circ\alpha\Phi^{-1}(x))\dashv\Phi\circ(\Phi^{-1}(y)\vdash\Phi^{-1}(z))\\\
&=\Phi\circ\alpha\Phi^{-1}(x)\dashv^{\prime}(y\vdash^{\prime}z).\end{array}$

$\begin{array}[]{ll}(x\vdash^{\prime}y)\dashv^{\prime}\Phi\alpha\Phi^{-1}(z)&=\Phi\circ(\Phi^{-1}(x)\vdash\Phi^{-1}(y))\dashv^{\prime}\Phi\circ\alpha\circ\Phi^{-1}(z)\\\
&=\Phi\circ(\Phi^{-1}(x)\vdash\Phi^{-1}(y))\dashv\alpha\circ\Phi^{-1}(z)\\\
&=\Phi\circ(\Phi^{-1}((x\vdash y)\dashv\alpha(z)))\\\
&=\Phi\circ\Phi^{-1}((\alpha(x)\vdash(y\vdash z)))\\\
&=\Phi\circ(\alpha\Phi^{-1}(x)\vdash(\Phi^{-1}(y)\dashv\Phi^{-1}(z)))\\\
&=\Phi\circ((\Phi^{-1}\otimes\Phi^{-1})(\Phi\otimes\Phi)\alpha\Phi^{-1}(x)\vdash(\Phi^{-1}(y)\dashv\Phi^{-1}(z)))\\\
&=\Phi\circ((\Phi^{-1}\otimes\Phi^{-1})\circ\Phi\alpha\Phi^{-1}(x))\vdash\Phi\circ(\Phi^{-1}(y)\dashv\Phi^{-1}(z)))\\\
&=\Phi\circ\alpha\Phi^{-1}(x)\vdash^{\prime}(y\dashv^{\prime}z).\end{array}$

$\begin{array}[]{ll}(x\vdash^{\prime}y)\vdash^{\prime}\Phi\alpha\Phi^{-1}(z)&=\Phi\circ(\Phi^{-1}(x)\vdash\Phi^{-1}(y))\vdash^{\prime}\Phi\circ\alpha\circ\Phi^{-1}(z)\\\
&=\Phi\circ(\Phi^{-1}(x)\vdash\Phi^{-1}(y))\vdash\alpha\circ\Phi^{-1}(z)\\\
&=\Phi\circ(\Phi^{-1}((x\vdash y)\vdash\alpha(z)))\\\
&=\Phi\circ\Phi^{-1}((x\dashv y)\vdash z)))\\\
&=\Phi\circ(\Phi^{-1}(x)\dashv(\Phi^{-1}y))\vdash\Phi^{-1}(z)))\\\
&=\Phi\circ((\Phi^{-1}\otimes\Phi^{-1})(\Phi\otimes\Phi)\circ(\Phi^{-1}(x)\dashv\Phi^{-1}(y))\vdash\Phi^{-1}(z)))\\\
&=\Phi\circ((\Phi^{-1}\otimes\Phi^{-1})\circ\Phi(\Phi^{-1}(x)\dashv\Phi^{-1}(y))\vdash\Phi\circ\alpha\circ\Phi^{-1}(z)))\\\
&=(x\dashv^{\prime}y)\vdash\Phi\circ\alpha\Phi^{-1}(z).\end{array}$

From this perspective,
$(\mathcal{A},\dashv^{\prime},\vdash^{\prime},\Phi\alpha\Phi^{-1})$ is a Hom-
associative dialgebra.It is also multiplicative.
Indeed, for $\alpha$

$\begin{array}[]{ll}\Phi\alpha\Phi^{-1}\circ(x\dashv^{\prime}y)&=\Phi\alpha\Phi^{-1}\Phi\circ
x\dashv(\Phi^{-1}\otimes\Phi^{-1})(y)\\\ &=\Phi\alpha\circ
x\dashv(\Phi^{-1}\otimes\Phi^{-1})(y)\\\
&=\Phi\alpha\circ\Phi^{-1}(x)\dashv\Phi^{-1}(y)\\\
&=\Phi\circ(\alpha\Phi^{-1}(x)\dashv\alpha\Phi^{-1}(y))\\\
&=\Phi\circ(\Phi^{-1}\otimes\Phi^{-1})(\Phi\otimes\Phi)(\alpha\Phi^{-1}(x)\dashv\alpha\Phi^{-1}(y))\\\
&=\Phi\circ(\Phi^{-1}\otimes\Phi^{-1})(\Phi\alpha\Phi^{-1}(x)\dashv\alpha\Phi\Phi^{-1}(y))\\\
&=\Phi\alpha\Phi^{-1}(x)\dashv^{\prime}\Phi\alpha\Phi^{-1}(y).\end{array}$

It follows that

$\begin{array}[]{ll}\Phi\alpha\Phi^{-1}\circ(x\vdash^{\prime}y)&=\Phi\alpha\Phi^{-1}\Phi\circ
x\vdash(\Phi^{-1}\otimes\Phi^{-1})(y)\\\ &=\Phi\alpha\circ
x\vdash(\Phi^{-1}\otimes\Phi^{-1})(y)\\\
&=\Phi\alpha\circ\Phi^{-1}(x)\vdash\Phi^{-1}(y)\\\
&=\Phi\circ(\alpha\Phi^{-1}(x)\vdash\alpha\Phi^{-1}(y))\\\
&=\Phi\circ(\Phi^{-1}\otimes\Phi^{-1})(\Phi\otimes\Phi)(\alpha\Phi^{-1}(x)\vdash\alpha\Phi^{-1}(y))\\\
&=\Phi\circ(\Phi^{-1}\otimes\Phi^{-1})(\Phi\alpha\Phi^{-1}(x)\vdash\alpha\Phi\Phi^{-1}(y))\\\
&=\Phi\alpha\Phi^{-1}(x)\vdash^{\prime}\Phi\alpha\Phi^{-1}(y).\end{array}$

Therefore,
$\Lambda:(\mathcal{A},\dashv,\vdash,\alpha)\rightarrow(\mathcal{A},\dashv^{\prime},\vdash^{\prime},\Phi\alpha\Phi^{-1})$
is a Hom-associative dialgebras morphism, since
$\Phi\circ\dashv=\Phi\circ\dashv\circ(\Phi^{-1}\otimes\Phi^{-1})\circ(\Phi\otimes\Phi)=\dashv^{\prime}\circ(\Phi\otimes\Phi)$
and $(\Phi\alpha\Phi^{-1})\circ\Phi=\Phi\circ\alpha.$ ∎

######  Proposition 2.15.

Let $(\mathcal{A},\dashv,\vdash,\alpha)$ be a Hom-associative dialgebra over
$\mathbb{K}$.
Let $(\mathcal{A},\dashv^{\prime},\vdash^{\prime},\Phi\alpha\Phi^{-1})$ be its
isomorphic Hom-associative dialgebra described in Proposition 2.14. If $\psi$
is an automorphism of $(\mathcal{A},\dashv,\vdash,\alpha)$, then
$\Phi\psi\Phi^{-1}$ is an automorphism of
$(\mathcal{A},\dashv,\vdash,\Phi\alpha\Phi^{-1})$.

###### Proof.

Note that $\gamma=\Phi\alpha\Phi^{-1}$. We have

$\Phi\psi\Phi^{-1}\gamma=\Phi\psi\Phi^{-1}\Phi\alpha\Phi^{-1}=\Phi\psi\alpha\Phi^{-1}=\Phi\alpha\psi\Phi^{-1}=\Phi\alpha\Phi^{-1}\Phi\psi\Phi^{-1}=\gamma\Phi\psi\Phi^{-1}.$

For any $x,y\in\mathcal{A},$

$\begin{array}[]{ll}\Phi\psi\Phi^{-1}\circ(\Phi(x)\dashv^{\prime}\Phi(y))&=\Phi\psi\Phi^{-1}\circ\Phi\circ(x\dashv
y)\\\ &=\Phi\circ\psi\circ(x\dashv y)\\\ &=\Phi\circ(\psi(x)\dashv\psi(y))\\\
&=\Phi\circ\psi(x)\dashv^{\prime}\Phi\circ\psi(y))\\\
&=\Phi\psi\Phi^{-1}\Phi(x)\dashv^{\prime}\Phi\psi\Phi^{-1}\Phi(y)\\\
\end{array}$

This entails,

$\begin{array}[]{ll}\Phi\psi\Phi^{-1}\circ(\Phi(x)\vdash^{\prime}\Phi(y))&=\Phi\psi\Phi^{-1}\circ\Phi\circ(x\vdash
y)\\\ &=\Phi\circ\psi\circ(x\vdash y)\\\ &=\Phi\circ(\psi(x)\vdash\psi(y))\\\
&=\Phi\circ\psi(x)\vdash^{\prime}\Phi\circ\psi(y))\\\
&=\Phi\psi\Phi^{-1}\Phi(x)\vdash^{\prime}\Phi\psi\Phi^{-1}\Phi(y)\\\
\end{array}$

By Definition, $\Phi\psi\Phi^{-1}$ is an automorphism of
$(\mathcal{A},\dashv^{\prime},\vdash^{\prime},\Phi\alpha\Phi^{-1})$. ∎

##  3 Classification in low dimensions

Let $\mathcal{H}d_{n}(K)$ denote the variety of $n$ dimensional Hom-
associative dialgebras over a field $\mathcal{A}$. If $\mathcal{A}$ is an
$n$-dimensional algebra, then the product of any two elements $x$ and $y$ can
be expressed by the product of basis elements
$\left\\{e_{1},e_{2},e_{3},\cdots,e_{n}\right\\}$. Recall that a Hom-
diassociative structure on $\mathcal{A}$ can then be defined by two bilinear
mappings : $\dashv:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$
representing the left product,
$\dashv:\mathcal{A}\times\mathcal{A}\longrightarrow\mathcal{A}$ representing
the left product $\vdash$ and $\alpha:\mathcal{A}\longrightarrow\mathcal{A}$
representing a linear map, all satisfying the above-mentionend identities when
a Hom-associative dialgebra $\mathcal{D}$ can be regarded as a quadruplet
$\mathcal{D}=(\mathcal{A},\dashv,\vdash,\alpha)$ where $\dashv,\vdash$ and
$\alpha$ are Hom-associative dialgebra laws on $\mathcal{A}.$

Let us denote by $\gamma_{ij}^{k}$, $\delta_{st}^{q}$ and $a_{ij}$, where
$i,j,k,s,t,q=1,2,3,\dots,n$, the structure constants of a Hom-associative
dialgebra with respect to the basis $e_{1},e_{2},\dots,e_{n}$ of
$\mathcal{A}$.

As a result, $\mathcal{H}d$ can be considered as a closed subset of
$2n^{3}+n^{2}$-dimensional affine space specified by the following system of
polynomial equations with respect to the structure constants
$\gamma_{ij}^{k}$, $\delta_{st}^{q}$ and $a_{ji}$ :

Let $\left\\{e_{1},e_{2},e_{3},\cdots,e_{n}\right\\}$ be a basis of an
$n$-dimensional Hom-associative dialgebra $\mathcal{A}.$ The product of basis
is denoted by

$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}\gamma_{ij}^{p}a_{qk}\gamma_{pq}^{r}$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}a_{pi}\gamma_{jk}^{q}\gamma_{pq}^{r},$
(3.1)
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}\gamma_{ij}^{p}a_{qk}\gamma_{pq}^{r}$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}a_{pi}\delta_{jk}^{q}\gamma_{pq}^{r}=,$
(3.2)
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}\delta_{ij}^{p}a_{qk}\gamma_{pq}^{r}$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}a_{pi}\gamma_{jk}^{q}\delta_{pq}^{r},$
(3.3)
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}\gamma_{ij}^{p}a_{qk}\delta_{pq}^{r}$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}a_{pi}\gamma_{jk}^{q}\delta_{pq}^{r},$
(3.4)
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}\delta_{ij}^{p}a_{qk}\delta_{pq}^{r}$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}\sum_{q=1}^{n}a_{pi}\delta_{jk}^{q}\delta_{pq}^{r}.$
(3.5)

We seek for all $2$-dimensional Hom-associative dialgebras, we consider two
classes of morphisms which are given by their Jordan forms. This implies that
they are represented by the matrices

$\left(\begin{array}[]{ccc}a&0\\\
0&b\end{array}\right)\quad\text{and}\quad\left(\begin{array}[]{ccc}a&1\\\
0&a\end{array}\right).$

Using similar calculations as depicted in the previous section, we obtain the
following classification.

######  Theorem 3.1.

Every $2$-dimensional complex Hom-associative dialgebra is isomorphic to one
of the following pairwise non-isomorphic Hom-associative dialgebra
$(\mathcal{A},\dashv,\vdash,\alpha)$ where $\dashv,\vdash$ are the left
product and right product and $\alpha$ of the structure map.

$\mathcal{H}d_{2}^{1}$ : $\begin{array}[]{ll}e_{1}\dashv e_{1}=e_{1},\\\
e_{1}\dashv e_{2}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{2}\dashv
e_{1}=e_{2},\\\ e_{2}\dashv e_{2}=e_{2},\\\ \end{array}$
$\begin{array}[]{ll}e_{1}\vdash e_{1}=e_{1},\\\ e_{1}\vdash e_{2}=e_{2},\\\
\end{array}$ $\begin{array}[]{ll}e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash
e_{2}=e_{2},\\\ \end{array}$ $\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\
\alpha(e_{2})=e_{2}.\end{array}$

$\mathcal{H}d_{2}^{2}$ : $\begin{array}[]{ll}e_{1}\dashv
e_{2}=e_{2},\,e_{2}\dashv e_{1}=e_{1},\,e_{1}\vdash e_{1}=e_{1},\,e_{2}\vdash
e_{1}=e_{1},\end{array}$ $\begin{array}[]{ll}\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{2}^{3}$ : $\begin{array}[]{ll}e_{2}\dashv
e_{2}=-\frac{e_{1}}{2}-e_{2},\\\ e_{2}\vdash e_{1}=e_{1},\quad e_{2}\vdash
e_{2}=-e_{1}-e_{2},\end{array}$
$\begin{array}[]{ll}\alpha(e_{1})=-e_{1},\quad\alpha(e_{2})=e_{1}+e_{2}.\end{array}$

$\mathcal{H}d_{2}^{4}$ : $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{2}=e_{1}+e_{2},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash
e_{2}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1}+e_{2}.\end{array}$
$\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\
\alpha(e_{2})=e_{1}+e_{2}.\end{array}$

$\mathcal{H}d_{2}^{5}$ : $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{1}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{2}\dashv
e_{2}=e_{1},\quad e_{2}\vdash e_{1}=e_{1},\end{array}$
$\begin{array}[]{ll}\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{2}^{6}$ : $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{2}=e_{2},\\\ \end{array}$ $\begin{array}[]{ll}e_{2}\vdash
e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\ \end{array}$
$\begin{array}[]{ll}\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=e_{2}.\end{array}$

$\mathcal{H}d_{2}^{7}$ : $\begin{array}[]{ll}e_{1}\dashv
e_{1}=e_{1}+e_{2},\,e_{1}\vdash e_{1}=e_{1},\end{array}$
$\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ \end{array}$
$\begin{array}[]{ll}\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=e_{2}.\end{array}$

$\mathcal{H}d_{2}^{8}$ : $\begin{array}[]{ll}e_{1}\dashv
e_{2}=e_{1},\,e_{2}\dashv e_{1}=e_{1},\,e_{2}\dashv e_{2}=e_{1},\,e_{2}\vdash
e_{2}=e_{1},\end{array}$ $\begin{array}[]{ll}\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{2}^{9}$ : $\begin{array}[]{ll}e_{1}\dashv e_{2}=ae_{1},\\\
e_{2}\dashv e_{2}=be_{1}+ce_{2},\\\ \end{array}$
$\begin{array}[]{ll}e_{2}\vdash e_{1}=fe_{1},\\\ e_{2}\vdash
e_{2}=ge_{1}+ke_{2},\end{array}$ $\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\
\alpha(e_{2})=e_{1}+e_{2}.\end{array}$

###### Proof.

Let $\mathcal{A}$ be a two-dimensional vector space. To determine a Hom-
associative dialgebra structure on $\mathcal{A}$, we consider $\mathcal{A}$
with respect to Hom-diassociative operation.

Let $\mathcal{H}_{2}^{{}^{\prime}}=(\mathcal{A},\vdash,\alpha)$ be the algebra

$e_{1}\vdash e_{1}=e_{1},\quad e_{1}\vdash e_{2}=e_{2},\,\,e_{2}\vdash
e_{1}=e_{1},\,\,e_{2}\vdash
e_{2}=e_{2},\quad\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=e_{2}.$

The second multiplication operation $\dashv$ in $\mathcal{A}$, is indicated as
follows :

$e_{1}\dashv e_{1}=a_{1}e_{1}+a_{2},\,e_{1}\dashv
e_{2}=a_{3}e_{1}+a_{4}e_{2},\,e_{2}\dashv
e_{1}=a_{5}e_{1}+a_{6}e_{2},\,e_{2}\vdash e_{2}=a_{7}e_{1}+a_{8}e_{2}.$

Now, verifying Hom-associative dialgebras axioms, we get several constants for
the coefficients $a_{i}$ where, $i=1,2,\dots,8.$

Applying $(e_{1}\dashv
e_{1})\vdash\alpha(e_{1})=\alpha(e_{1})\vdash(e_{1}\vdash e_{1})$, we have
$(a_{1}e_{1}+a_{2}e_{2})\vdash e_{1}=e_{1}\vdash e_{1}$ . Then,
$a_{1}e_{1}=e_{1}.$ Therefore $a_{1}=1.$

The verification, $(e_{1}\vdash
e_{1})\dashv\alpha(e_{1})=\alpha(e_{1})\vdash(e_{1}\dashv e_{1})$ yields
$(e_{1}+a_{2}e_{2})\vdash e_{1}=e_{1}\vdash e_{1}$. We have
$e_{1}+a_{2}e_{2}=e_{1}$. Hence we get $a_{2}=0.$

Consider $(e_{1}\dashv
e_{2})\vdash\alpha(e_{1})=\alpha(e_{1})\vdash(e_{2}\vdash e_{1})$. It implies
that $(a_{3}e_{1}+a_{4}e_{2})\dashv e_{1}=e_{1}\dashv e_{1}$. Hence, $a_{3}=1$
and $a_{4}=0.$

The next relation to consider is $(e_{2}\vdash
e_{1})\vdash\alpha(e_{2})=\alpha(e_{2})\vdash(e_{1}\dashv e_{2})$. Hence,
$a_{3}=0$ and $a_{4}=1.$

Consider $(e_{2}\dashv
e_{1})\vdash\alpha(e_{2})=\alpha(e_{2})\vdash(e_{1}\vdash e_{2})$. It implies
that $(a_{5}e_{1}+a_{6}e_{2})\vdash e_{2})=e_{2}\vdash e_{2}$, Therefore,
$a_{5}=0$ and $a_{6}=1.$

Finally, we apply $(e_{2}\dashv
e_{2})\vdash\alpha(e_{2})=\alpha(e_{2})\vdash(e_{2}\vdash e_{2})$. We obtain
$(a_{7}e_{1}+a_{8}e_{2})\vdash e_{2}=e_{2}\vdash e_{2}$ and we get $a_{7}=0$
and $a_{8}=1.$
The verification of all other cases leads to the obtained constraints.

If $a_{3}=1$ and $a_{4}=0$, then the right and left product coincide and we
get the Hom-associative dialgebra.

If $a_{3}=0$ and $a_{4}=1$, we obtain the Hom-associative dialgebras
$\mathcal{H}d_{2}^{1}$. The other Hom-associative dialgebras of the list of
Theorem 3.1 can be obtained by minor modification of the above observation. ∎

We seek for all $3$-dimensional Hom-associative dialgebras, we consider two
classes of morphisms that are provided by their Jordan forms. This implies
that they are represented by the matrices

$\left(\begin{array}[]{ccc}a&0&0\\\ 0&b&0\\\
0&0&c\end{array}\right),\quad\left(\begin{array}[]{ccc}a&1&0\\\ 0&a&0\\\
0&0&b\end{array}\right),\quad\left(\begin{array}[]{ccc}a&1&0\\\ 0&a&0\\\
0&0&a\end{array}\right),\quad\left(\begin{array}[]{ccc}a&1&0\\\ 0&a&1\\\
0&0&a\end{array}\right).$

Using similar calculations as in the previous Section, we obtain the following
classification.

######  Theorem 3.2.

Every $3$-dimensional multiplicative Hom-associative dialgebra is isomorphic
to one of the following pairwise non-isomorphic Hom-associative dialgebras
$(\mathcal{A},\dashv,\vdash,\alpha)$, where $\dashv,\vdash$ are left product
and right product and $\alpha$ of the structure map.

$\mathcal{H}d_{3}^{1}:\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{2}=e_{1},\\\ e_{3}\dashv e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash
e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\vdash e_{3}=e_{1},\\\
\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{2}:\begin{array}[]{ll}e_{2}\dashv e_{1}=e_{1},\\\
e_{2}\dashv e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{2}=e_{1},\\\ e_{3}\dashv e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash
e_{3}=e_{2},\\\ \end{array}\begin{array}[]{ll}e_{3}\vdash e_{3}=e_{1},\\\
\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{3}:\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{2}=e_{1},\\\ e_{3}\dashv e_{3}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash
e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{4}:\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{2}=e_{1},\\\ e_{3}\dashv e_{3}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash
e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\vdash e_{3}=e_{1},\\\
\alpha(e_{1})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{5}:\begin{array}[]{ll}e_{1}\dashv e_{1}=e_{1},\\\
e_{2}\dashv e_{2}=e_{2},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{2}=e_{2},\\\ e_{1}\vdash e_{1}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash
e_{2}=e_{2},\quad\alpha(e_{1})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{6}:\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\dashv e_{3}=e_{1},\\\ e_{3}\dashv
e_{2}=e_{3},\\\ e_{1}\vdash e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash
e_{2}=e_{1},\\\ e_{2}\vdash e_{3}=e_{3},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{2}=e_{3},\\\ e_{3}\vdash
e_{3}=e_{1},\\\ \alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{7}:\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{2},\\\
e_{2}\dashv e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{3}=e_{3},\\\ e_{2}\vdash e_{2}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1},\\\ e_{3}\vdash
e_{2}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{3}\vdash e_{3}=e_{1},\\\
\alpha(e_{1})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{8}:\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{2},\\\
e_{2}\dashv e_{3}=e_{2},\\\ \end{array}\begin{array}[]{ll}e_{2}\vdash
e_{2}=e_{2},\\\ e_{2}\vdash e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash
e_{2}=e_{1},\quad\alpha(e_{1})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{9}:\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{2},\\\
e_{2}\dashv e_{3}=e_{2},\\\ \end{array}\begin{array}[]{ll}e_{2}\vdash
e_{2}=e_{2},\\\ e_{2}\vdash e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{2}=e_{1},\\\ e_{3}\vdash
e_{3}=e_{3},\\\ \end{array}\begin{array}[]{ll}\alpha(e_{1})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{10}:\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{3}\dashv e_{2}=e_{3},\\\ e_{1}\vdash
e_{2}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{3},\\\ e_{3}\vdash
e_{2}=e_{1},\\\ \end{array}\begin{array}[]{ll}\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{11}:\begin{array}[]{ll}e_{1}\dashv
e_{2}=(-1)^{2/3}e_{1},\\\ e_{2}\dashv e_{1}=ae_{1},\\\ e_{2}\dashv
e_{2}=be_{1}+ce_{3},\end{array}\begin{array}[]{ll}e_{2}\dashv
e_{3}=e_{1}+de_{3},\\\ e_{3}\dashv e_{2}=e_{1}+(-1)^{2/3}e_{3},\\\ e_{3}\dashv
e_{3}=\frac{i}{\sqrt{3}}e_{1},\end{array}\begin{array}[]{ll}e_{1}\vdash
e_{2}=(-1)^{2/3}e_{1},\\\ e_{2}\vdash e_{1}=fe_{1},\\\ e_{2}\vdash
e_{2}=e_{1}+g(-1)^{2/3}e_{3},\end{array}\begin{array}[]{ll}e_{3}\vdash
e_{2}=e_{1}+e_{3},\\\ e_{3}\vdash e_{3}=\frac{i}{\sqrt{3}}e_{1},\\\
\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{3}^{12}:\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1}+e_{3},\\\
\end{array}\begin{array}[]{ll}e_{2}\dashv e_{3}=e_{1}+e_{3},\\\ e_{3}\dashv
e_{2}=e_{1}+e_{3},\\\ e_{3}\dashv e_{3}=e_{3},\\\
\end{array}\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash
e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1}+e_{3},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1}+e_{3},\\\ e_{3}\vdash
e_{2}=e_{1}+e_{3},\\\ \alpha(e_{2})=e_{1}.\end{array}$
$\mathcal{H}d_{3}^{13}:\begin{array}[]{ll}e_{1}\dashv
e_{1}=(-1)^{1/3}e_{2},\\\ e_{1}\dashv e_{3}=ae_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\dashv e_{1}=be_{2},\\\ e_{3}\dashv
e_{3}=\frac{(-1)^{1/3}}{\sqrt{3}}ie_{2},\\\
\end{array}\begin{array}[]{ll}e_{1}\vdash e_{1}=(-1)^{1/3}e_{2},\\\
e_{1}\vdash e_{3}=ce_{1},\\\ e_{3}\vdash
e_{1}=\frac{(-1)^{1/3}}{\sqrt{3}}ie_{2},\\\
\end{array}\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\ \alpha(e_{2})=e_{2},\\\
\alpha(e_{3})=e_{3}.\end{array}$
$\mathcal{H}d_{3}^{14}:\begin{array}[]{ll}e_{1}\dashv
e_{1}=e_{1}+a(-1)^{2/3}e_{3},\\\ e_{1}\dashv e_{3}=be_{3},\\\ e_{3}\dashv
e_{1}=\frac{i}{\sqrt{3}}e_{1}+c(-1+(-1)^{2/3})e_{3},\\\
\end{array}\begin{array}[]{ll}e_{1}\vdash e_{1}=e_{1}+d(-1)^{2/3}e_{3},\\\
e_{1}\vdash e_{3}=e_{1},\\\ e_{2}\vdash e_{2}=fe_{1}+ge_{3},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{1}=e_{1}+h(-1)^{2/3}e_{3},\\\
e_{3}\vdash e_{3}=e_{1}+\frac{i}{\sqrt{3}}e_{3},\\\
\alpha(e_{2})=e_{2}.\end{array}$

###### Proof.

The proof is similar to Theorem 3.1 ∎

We seek for all $4$-dimensional Hom-associative dialgebras, we consider two
classes of morphisms which are given by their Jordan forms. This implies that
they are represented by the matrices

$\left(\begin{array}[]{cccc}a&0&0&0\\\ 0&b&0&0\\\ 0&0&c&0\\\
0&0&0&d\end{array}\right),\quad\left(\begin{array}[]{cccc}a&1&0&0\\\
0&a&1&0\\\ 0&0&a&1\\\
0&0&0&a\end{array}\right),\quad\left(\begin{array}[]{cccc}a&1&0&0\\\
0&a&0&0\\\ 0&0&a&1\\\
0&0&0&a\end{array}\right),\quad\left(\begin{array}[]{cccc}a&1&0&0\\\
0&a&1&0\\\ 0&0&a&0\\\
0&0&0&a\end{array}\right),\quad\left(\begin{array}[]{cccc}a&1&0&0\\\
0&a&0&0\\\ 0&0&a&0\\\ 0&0&0&a\end{array}\right).$

Using similar calculations as displayed in the previous Section, we obtain the
following classification.

######  Theorem 3.3.

Every $4$-dimensional multiplicative real Hom-associative dialgebra is
isomorphic to one of the following pairwise non-isomorphic Hom-associative
dialgebras $(\mathcal{A},\dashv,\vdash,\alpha)$, where $\dashv,\vdash$ are
left product and right product and $\alpha$ the structure map.

$\mathcal{H}d_{4}^{1}:\\\ \begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{1}\dashv e_{4}=e_{3},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv
e_{3}=ae_{3},\\\ \end{array}\begin{array}[]{ll}e_{2}\dashv e_{4}=e_{1},\\\
e_{3}\dashv e_{2}=e_{1},\\\ e_{3}\dashv e_{4}=e_{1},\\\ e_{4}\dashv
e_{1}=e_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{1},\\\
e_{1}\vdash e_{2}=e_{1},\\\ e_{1}\vdash e_{4}=e_{3},\\\ e_{2}\vdash
e_{2}=e_{3},\\\ \end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1},\\\
e_{2}\vdash e_{4}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\\\ e_{3}\vdash
e_{4}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{4}\vdash e_{2}=e_{1},\\\
e_{4}\vdash e_{3}=e_{1},\\\ e_{4}\vdash e_{4}=e_{1},\\\
\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{4}^{2}:\\\ \begin{array}[]{ll}e_{1}\dashv e_{4}=e_{3},\\\
e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{3},\\\ e_{2}\dashv
e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{2}\dashv e_{4}=e_{1},\\\
e_{3}\dashv e_{2}=e_{1},\\\ e_{3}\dashv e_{4}=e_{1},\\\ e_{4}\dashv
e_{1}=e_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{1},\\\
e_{1}\vdash e_{2}=e_{1},\\\ e_{1}\vdash e_{4}=e_{3},\\\ e_{2}\vdash
e_{2}=e_{3},\\\ \end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1},\\\
e_{2}\vdash e_{4}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\\\ e_{3}\vdash
e_{4}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{4}\vdash e_{2}=e_{1},\\\
e_{4}\vdash e_{3}=e_{1},\\\ e_{4}\vdash e_{4}=e_{1},\\\
\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{4}^{3}:\\\ \begin{array}[]{ll}e_{1}\dashv e_{4}=e_{3},\\\
e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1}+e_{3},\\\ e_{2}\dashv
e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{2}\dashv e_{4}=e_{1},\\\
e_{3}\dashv e_{2}=e_{1},\\\ e_{3}\dashv e_{4}=e_{1},\\\ e_{4}\dashv
e_{1}=e_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{1},\\\
e_{1}\vdash e_{2}=e_{1},\\\ e_{1}\vdash e_{4}=e_{3},\\\ e_{2}\vdash
e_{2}=e_{3},\\\ \end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1},\\\
e_{2}\vdash e_{4}=e_{3},\\\ e_{3}\vdash e_{2}=e_{1},\\\ e_{3}\vdash
e_{4}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{4}\vdash e_{2}=e_{1},\\\
e_{4}\vdash e_{3}=e_{1},\\\ e_{4}\vdash e_{4}=e_{1},\\\
\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{4}^{4}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{3}=e_{1}+e_{3},\\\
e_{3}\dashv e_{2}=e_{1}+e_{3},\\\ e_{3}\dashv e_{4}=e_{1}+e_{3},\\\
e_{4}\dashv e_{1}=e_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\dashv
e_{3}=e_{1}+e_{3},\\\ e_{4}\dashv e_{4}=e_{1}+e_{3},\\\ e_{1}\vdash
e_{2}=e_{1},\\\ e_{1}\vdash e_{4}=e_{3},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1}+e_{3},\\\ e_{2}\vdash
e_{4}=e_{1}+e_{3},\\\ e_{3}\vdash e_{2}=e_{1}+e_{3},\\\ e_{3}\vdash
e_{4}=e_{1}+e_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\vdash
e_{3}=e_{1}+e_{3},\\\ e_{4}\vdash e_{4}=e_{1}+e_{3},\\\
\alpha(e_{2})=e_{1},\\\ \alpha(e_{4})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{5}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{3}=e_{3},\\\
e_{2}\dashv e_{4}=e_{1}+e_{3},\\\ e_{3}\dashv e_{2}=e_{1}+e_{3},\\\
e_{3}\dashv e_{4}=e_{1}+e_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\dashv
e_{1}=e_{3},\\\ e_{4}\dashv e_{3}=e_{1}+e_{3},\\\ e_{4}\dashv
e_{4}=e_{1}+e_{3},\\\ e_{2}\vdash e_{1}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1}+e_{3},\\\ e_{2}\vdash
e_{4}=e_{1}+e_{3},\\\ e_{3}\vdash e_{2}=e_{1}+e_{3},\\\ e_{3}\vdash
e_{4}=e_{1}+e_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\vdash
e_{3}=e_{1}+e_{3},\\\ e_{4}\vdash e_{4}=e_{1}+e_{3},\\\
\alpha(e_{2})=e_{1},\\\ \alpha(e_{4})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{6}:\\\ \begin{array}[]{ll}e_{1}\dashv e_{2}=e_{3},\\\
e_{2}\dashv e_{3}=e_{1}+e_{3},\\\ e_{2}\dashv e_{4}=e_{1}+e_{3},\\\
e_{3}\dashv e_{2}=e_{1}+e_{3},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{4}=e_{1},\\\ e_{4}\dashv e_{2}=e_{3},\\\ e_{4}\dashv e_{3}=e_{1}+e_{3},\\\
e_{4}\dashv e_{4}=e_{1}+e_{3},\\\ \end{array}\begin{array}[]{ll}e_{2}\vdash
e_{1}=e_{1},\\\ e_{2}\vdash e_{3}=e_{1}+e_{3},\\\ e_{2}\vdash
e_{4}=e_{1}+e_{3},\\\ e_{3}\vdash e_{2}=e_{1}+e_{3},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{4}=e_{1}+e_{3},\\\ e_{4}\vdash
e_{3}=e_{1}+e_{3},\\\ \alpha(e_{2})=e_{1},\\\ \alpha(e_{4})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{7}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{3}=e_{4},\\\
e_{3}\dashv e_{2}=e_{4},\\\ e_{3}\dashv e_{4}=e_{4},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{3}=e_{4},\\\ e_{4}\dashv
e_{4}=e_{2}+e_{4},\\\ e_{3}\vdash e_{2}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{4}=e_{2}+e_{4},\\\ e_{4}\vdash
e_{3}=e_{2}+e_{4},\\\ e_{4}\vdash e_{4}=e_{2}+e_{4},\\\
\end{array}\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\
\alpha(e_{2})=e_{2}.\end{array}$

$\mathcal{H}d_{4}^{8}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{3}=e_{4},\\\
e_{3}\dashv e_{2}=e_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\
\end{array}\begin{array}[]{ll}e_{3}\dashv e_{4}=e_{4},\\\ e_{4}\dashv
e_{3}=e_{4},\\\ e_{4}\dashv e_{4}=e_{4},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{2}=e_{2},\\\ e_{3}\vdash
e_{4}=e_{2}+e_{4},\\\ e_{4}\vdash e_{3}=e_{2}+e_{4},\\\
\end{array}\begin{array}[]{ll}e_{4}\vdash e_{4}=e_{2}+e_{4},\\\
\alpha(e_{1})=e_{1},\\\ \alpha(e_{2})=e_{2}.\end{array}$

$\mathcal{H}d_{4}^{9}:\\\ \begin{array}[]{ll}e_{1}\dashv e_{1}=e_{1},\\\
e_{2}\dashv e_{4}=e_{1},\\\ e_{3}\dashv e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\dashv e_{4}=e_{1},\\\ e_{4}\dashv
e_{1}=e_{1},\\\ e_{4}\dashv e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{3}=e_{1},\\\ e_{4}\dashv
e_{4}=e_{1},\\\ e_{2}\vdash e_{4}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{3}=e_{2},\\\ e_{3}\vdash
e_{4}=e_{1}+e_{2},\\\ e_{4}\vdash e_{3}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}e_{4}\vdash e_{4}=e_{2},\\\
\alpha(e_{2})=e_{2},\\\ \alpha(e_{3})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{10}:\\\ \begin{array}[]{ll}e_{2}\dashv
e_{2}=e_{2}+e_{4},\\\ e_{2}\dashv e_{4}=e_{2}+e_{4},\\\ e_{3}\dashv
e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{4},\\\
e_{2}\vdash e_{2}=e_{2}+e_{4},\\\ e_{2}\dashv e_{4}=e_{2}+e_{4},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{3}=e_{2}+e_{4},\\\ e_{4}\vdash
e_{2}=e_{4},\\\ e_{4}\vdash e_{4}=e_{2}+e_{4},\\\
\end{array}\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\
\alpha(e_{3})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{11}:\\\ \begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\
e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\dashv e_{3}=e_{1},\\\ e_{3}\dashv
e_{2}=e_{1},\\\ e_{4}\dashv e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{3},\\\ e_{1}\vdash
e_{2}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{2}\vdash e_{3}=e_{1},\\\ e_{2}\vdash
e_{4}=e_{1}+e_{3},\\\ e_{4}\vdash e_{2}=e_{1}+e_{3},\\\
\end{array}\begin{array}[]{ll}e_{4}\vdash e_{4}=e_{1},\\\
\alpha(e_{3})=e_{3},\\\ \alpha(e_{4})=e_{4}.\end{array}$

$\mathcal{H}d_{4}^{12}:\\\ \begin{array}[]{ll}e_{2}\dashv
e_{2}=e_{2}+e_{4},\\\ e_{2}\dashv e_{4}=e_{2}+e_{4},\\\ e_{3}\dashv
e_{3}=e_{1},\\\ \end{array}\begin{array}[]{ll}e_{4}\dashv e_{2}=e_{2},\\\
e_{4}\dashv e_{4}=e_{4},\\\ e_{2}\vdash e_{2}=e_{2}+e_{4},\\\
\end{array}\begin{array}[]{ll}e_{2}\dashv e_{4}=e_{2}+e_{4},\\\ e_{3}\vdash
e_{3}=e_{2}+e_{4},\\\ e_{4}\vdash e_{2}=e_{4},\\\
\end{array}\begin{array}[]{ll}e_{4}\vdash e_{4}=e_{2}+e_{4},\\\
\alpha(e_{1})=e_{1},\\\ \alpha(e_{3})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{13}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{2}=e_{2},\\\
e_{2}\dashv e_{3}=e_{2}+e_{3},\\\ e_{3}\dashv e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{1},\\\ e_{2}\vdash
e_{2}=e_{1}+e_{3},\\\ e_{2}\vdash e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\dashv e_{2}=e_{1}+e_{3},\\\ e_{3}\vdash
e_{3}=e_{1}+e_{3},\\\ e_{4}\vdash e_{4}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\
\alpha(e_{4})=e_{4}.\end{array}$

$\mathcal{H}d_{4}^{14}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{2}=e_{2},\\\
e_{2}\dashv e_{3}=e_{2}+e_{3},\\\ e_{3}\dashv e_{3}=e_{2}+e_{3},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{1},\\\ e_{2}\vdash
e_{2}=e_{1}+e_{3},\\\ e_{2}\vdash e_{3}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\dashv e_{2}=e_{1}+e_{3},\\\ e_{3}\vdash
e_{3}=e_{1}+e_{3},\\\ e_{4}\vdash e_{4}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}\alpha(e_{1})=e_{1},\\\
\alpha(e_{4})=e_{4}.\end{array}$

$\mathcal{H}d_{4}^{15}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{4}=e_{1},\\\
e_{3}\dashv e_{3}=e_{2},\\\ e_{3}\dashv e_{4}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{1}=e_{1},\\\ e_{4}\dashv
e_{2}=e_{1},\\\ e_{4}\dashv e_{3}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{1},\\\ e_{2}\vdash
e_{4}=e_{2},\\\ e_{3}\vdash e_{3}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{4}=e_{1}+e_{2},\\\ e_{4}\vdash
e_{1}=e_{1}+e_{2},\\\ e_{4}\vdash e_{2}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}e_{4}\vdash e_{3}=e_{1}+e_{2},\\\
\alpha(e_{2})=e_{2},\\\ \alpha(e_{3})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{16}:\\\ \begin{array}[]{ll}e_{2}\dashv e_{4}=e_{1},\\\
e_{3}\dashv e_{3}=e_{2},\\\ e_{3}\dashv e_{4}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{1}=e_{1},\\\ e_{4}\dashv
e_{2}=e_{1},\\\ e_{4}\dashv e_{3}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{4}=e_{1},\\\ e_{2}\vdash
e_{4}=e_{1}+e_{2},\\\ e_{3}\vdash e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{4}=e_{1}+e_{2},\\\ e_{4}\vdash
e_{1}=e_{1}+e_{2},\\\ e_{4}\vdash e_{2}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}e_{4}\vdash e_{4}=e_{2},\\\
\alpha(e_{2})=e_{2},\\\ \alpha(e_{3})=e_{3}.\end{array}$
$\mathcal{H}d_{4}^{17}:\\\ \begin{array}[]{ll}e_{1}\dashv e_{4}=e_{1},\\\
e_{2}\dashv e_{4}=e_{1},\\\ e_{3}\dashv e_{3}=e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\dashv e_{4}=e_{1},\\\ e_{4}\dashv
e_{1}=e_{1},\\\ e_{4}\dashv e_{2}=e_{1},\\\
\end{array}\begin{array}[]{ll}e_{4}\dashv e_{3}=e_{1},\\\ e_{4}\dashv
e_{4}=e_{1},\\\ e_{2}\vdash e_{4}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}e_{3}\vdash e_{3}=e_{1}+e_{2},\\\ e_{4}\vdash
e_{2}=e_{2},\\\ e_{4}\vdash e_{3}=e_{1}+e_{2},\\\
\end{array}\begin{array}[]{ll}e_{4}\vdash e_{4}=e_{2},\\\
\alpha(e_{2})=e_{2},\\\ \alpha(e_{3})=e_{3}.\end{array}$

$\mathcal{H}d_{4}^{18}:\\\ \begin{array}[]{ll}e_{1}\dashv
e_{2}=e_{3}+e_{4},\\\ e_{2}\dashv e_{1}=e_{3}+e_{4},\\\ e_{2}\dashv
e_{2}=e_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{2}\dashv
e_{3}=e_{3}+e_{4},\\\ e_{2}\dashv e_{4}=e_{3}+e_{4},\\\ e_{3}\dashv
e_{2}=e_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{3}=e_{3}+e_{4},\\\ e_{3}\dashv e_{4}=e_{3}+e_{4},\\\ e_{4}\dashv
e_{2}=e_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{1}\vdash
e_{1}=e_{3}+e_{4},\\\ e_{2}\vdash e_{2}=e_{3}+e_{4},\\\ e_{2}\vdash
e_{3}=e_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{3}\vdash
e_{2}=e_{3}+e_{4},\\\ e_{4}\vdash e_{4}=e_{3}+e_{4},\\\
\alpha(e_{2})=e_{1}.\end{array}$

$\mathcal{H}d_{4}^{19}:\\\ \begin{array}[]{ll}e_{1}\dashv
e_{2}=ae_{3}+be_{4},\\\ e_{2}\dashv e_{1}=e_{3}+ce_{4},\\\ e_{2}\dashv
e_{2}=e_{3}+de_{4},\\\ \end{array}\begin{array}[]{ll}e_{2}\dashv
fe_{3}=e_{3}+e_{4},\\\ e_{2}\dashv e_{4}=he_{3}+e_{4},\\\ e_{3}\dashv
e_{2}=le_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{3}=e_{3}+e_{4},\\\ e_{3}\dashv e_{4}=e_{3}+e_{4},\\\ e_{4}\dashv
e_{2}=e_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{1}\vdash
e_{1}=me_{3}+ne_{4},\\\ e_{2}\vdash e_{2}=oe_{3}+pe_{4},\\\ e_{2}\vdash
e_{3}=ce_{3}+de_{4},\\\ \end{array}\begin{array}[]{ll}e_{3}\vdash
e_{2}=e_{3},\\\ e_{4}\vdash e_{4}=e_{4},\\\ \alpha(e_{2})=e_{1}.\end{array}$
$\mathcal{H}d_{4}^{20}:\\\ \begin{array}[]{ll}e_{2}\dashv
e_{1}=e_{3}+ae_{4},\\\ e_{2}\dashv e_{2}=e_{3}+be_{4},\\\ e_{2}\dashv
e_{3}=ce_{3}+de_{4},\\\ \end{array}\begin{array}[]{ll}e_{2}\dashv
e_{4}=e_{3}+e_{4},\\\ e_{3}\dashv e_{2}=fe_{3}+ge_{4},\\\ e_{3}\dashv
e_{3}=e_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{3}\dashv
e_{4}=e_{3}+e_{4},\\\ e_{4}\dashv e_{2}=e_{3}+e_{4},\\\ e_{1}\vdash
e_{1}=e_{3}+e_{4},\\\ \end{array}\begin{array}[]{ll}e_{2}\vdash
e_{2}=he_{3}+e_{4},\\\ e_{2}\vdash e_{3}=ke_{3}+e_{4},\\\ e_{3}\vdash
e_{2}=le_{3},\\\ \end{array}\begin{array}[]{ll}e_{4}\vdash e_{3}=e_{3},\\\
e_{4}\vdash e_{4}=e_{4},\\\ \alpha(e_{2})=e_{1}.\end{array}$

###### Proof.

The proof is similar to Theorem 3.1 ∎

##  4 Derivations of Complex Hom-associative dialgebras

This section is notably devoted to the description of derivations of two,
three and four-dimensional Complex Hom-associative dialgebras.

###  4.1 Derivations of Complex Hom-associative dialgebras

Let $(\mathcal{A},\dashv,\vdash,\alpha)$ be a multiplicative Hom-associative
dialgebra. For any nonnegative integer $k$, we denote by $\alpha^{k}$ the
$k$-fold composition of $\alpha$ with itself,i.e.,
$\alpha^{k}=\alpha\circ\cdots\circ\alpha$ ($k$-times). In particular,
$\alpha^{0}=id$ and $\alpha^{1}=\alpha$.

######  Definition 4.1.

For any non-negative integer $k$, a linear map
$D:\mathcal{A}\longrightarrow\mathcal{A}$ is called an $\alpha^{k}-$derivation
of a Hom-associative dialgebra $(\mathcal{A},\dashv,\vdash,\alpha)$, if

$D\circ\alpha=\alpha\circ D$ (4.1) $D\circ(f\dashv
g)=(D(f)\dashv\alpha^{k}(g))+(\alpha^{k}(f)\dashv D(g)).$ (4.2)

$D\circ(f\vdash g)=(D(f)\vdash\alpha^{k}(g))+(\alpha^{k}(f)\vdash D(g)).$
(4.3)

The map $D(f)$ is an $\alpha^{k+1}$-derivation, which we will call an inner
$\alpha^{k+1}$-derivation. In fact, we have
$D_{k}(f)(\alpha(g))=\alpha^{k+1}(g)\dashv f=\alpha(\alpha^{k}(g)\dashv
f)=\alpha\circ D_{k}(f)(g)$,
$D_{k}(f)(\alpha(g))=\alpha^{k+1}(g)\vdash f=\alpha(\alpha^{k}(g)\vdash
f)=\alpha\circ D_{k}(f)(g)$,

which implies that identity (4.1) in Definition 4.1 is satisfied. On the other
side, we have

$\begin{array}[]{ll}D_{k}(f)(g\dashv h)&=\alpha^{k}((g\dashv h)\dashv
f)=(\alpha^{k}(g)\dashv\alpha^{k}(h))\dashv\alpha(f))\\\
&=\alpha^{k+1}(g)\dashv\alpha^{k}(h))\dashv f+(\alpha^{k}(g)\dashv
f)\dashv\alpha^{k+1}(h)\\\ &=(\alpha^{k+1}(g)\dashv
D_{k}(f)(h))+D_{k}(f)(g)\dashv\alpha^{k+1}(h).\end{array}$

Therefore, $D_{k}(f)$ is an $\alpha^{k+1}$-derivation. The set of
$\alpha^{k}$-derivations denoted by Inner${}_{\alpha^{k}}(A)$ are expressed in
terms of

$\textbf{Inner}_{\alpha^{k}}(A)=\left\\{\alpha^{k-1}(\bullet)\dashv f|f\in
A,\alpha(f)=f\right\\}.$ (4.4)
$\textbf{Inner}_{\alpha^{k}}(A)=\left\\{\alpha^{k-1}(\bullet)\vdash f|f\in
A,\alpha(f)=f\right\\}.$ (4.5)

For any $D\in Der_{\alpha^{k}}(A)$ and $D^{\prime}\in Der_{\alpha^{s}}(A)$, we
define their commutator $\left[D,D^{\prime}\right]$ as usual :
$\left[D,D^{\prime}\right]=D\circ D^{\prime}-D^{\prime}\circ D$.

######  Proposition 4.2.

For any $D\in Der_{\alpha^{k}}(A)$ and $D^{\prime}\in Der_{\alpha^{s}}(A)$, we
have $\left[D,D^{\prime}\right]\in Der_{\alpha^{k+s}}(A).$

###### Proof.

For any $f,g\in\mathcal{A},$ we have

$\begin{array}[]{ll}&\left[D,D^{\prime}\right](f\dashv g)=D\circ
D^{\prime}(f\dashv g)-D^{\prime}\circ D(f\dashv g)\\\
&=D(D^{\prime}(f)\dashv\alpha^{s}(g))+\alpha^{s}(f)\dashv
D^{\prime}(g)))-D^{\prime}(\mu(D(f),\alpha^{k}(g))+\mu(\alpha^{k}(f),D(g)))\\\
&=D\circ D^{\prime}(f)\dashv\alpha^{k+s}(g))+\alpha^{k}\circ
D^{\prime}(f)\dashv
D\circ\alpha^{s}(g)+D\circ\alpha^{s}(f)\dashv\alpha^{k}\circ D^{\prime}(g)\\\
&+\alpha^{k+s}(f)\dashv D\circ D^{\prime}(g))-D^{\prime}\circ
D(f)\dashv\alpha^{k+s}(g))-\alpha^{s}\circ D(f)\dashv
D^{\prime}\circ\alpha^{k}(g)\\\
&-D^{\prime}\circ\alpha^{k}(f)\dashv\alpha^{s}\circ D(g)-\alpha^{k+s}(f)\dashv
D^{\prime}\circ D(g).\end{array}$

Since $D$ and $D^{\prime}$ satisfy $D\circ\alpha=\alpha\circ D,\quad
D^{\prime}\circ\alpha=\alpha\circ D^{\prime}$, we obtain
$\alpha^{k}\circ D^{\prime}=D^{\prime}\circ\alpha^{k},\quad
D\circ\alpha^{s}=\alpha^{s}\circ D$. Therefore, we get

$\left[D,D^{\prime}\right](f\dashv
g)=\alpha^{k+s}(f)\dashv\left[D,D^{\prime}\right](g))+\left[D,D^{\prime}\right](f)\dashv\alpha^{k+s}(g)).$

Furthermore, it is straightforward to infer that

$\left[D,D^{\prime}\right]\circ\alpha=D\circ
D^{\prime}\circ\alpha-D^{\prime}\circ D\circ\alpha=\alpha\circ D\circ
D^{\prime}-\alpha\circ D^{\prime}\circ
D=\alpha\circ\left[D,D^{\prime}\right],$

which yields that $\left[D,D^{\prime}\right]\in
Der_{\alpha^{k+s}}(\mathcal{A})$. ∎

######  Definition 4.3.

A Hom-associative triple system is a vector space $A$ over field $\mathbb{K}$
with a trilinear multiplications satisfying

$\begin{array}[]{ll}(((x\dashv
y)\dashv\alpha(z))\dashv\alpha(u))\dashv\alpha(w)&=((\alpha(x)\dashv(y\dashv
z))\dashv\alpha(u))\dashv\alpha(w))\\\
&=\alpha(x)\dashv(\alpha(y)\dashv(z\dashv u)\dashv\alpha(w))),\end{array}$
$\begin{array}[]{ll}(((x\vdash
y)\vdash\alpha(z))\vdash\alpha(u))\vdash\alpha(w)&=((\alpha(x)\vdash(y\vdash
z))\vdash\alpha(u))\vdash\alpha(w))\\\
&=\alpha(x)\vdash(\alpha(y)\vdash(z\vdash u)\vdash\alpha(w))),\end{array}$

for any $x,y,z,u,w\in\mathcal{A}$.

######  Definition 4.4.

An associative triple derivation of Hom-associative dialgebra
$(A,\dashv,\vdash,\alpha)$ is a linear transformation
$D:\mathcal{A}\longrightarrow\mathcal{A}$ such that

$D\circ\alpha=\alpha\circ D$ $\begin{array}[]{ll}D\circ((x\dashv y)\dashv
z))&=(D(x)\dashv\alpha^{k}(y))\dashv\alpha^{k}(z)\\\ &+(\alpha^{k}(x)\dashv
D(y))\dashv\alpha^{k}(z)+(\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv
D(z)\end{array}$ $\begin{array}[]{ll}D\circ((x\vdash y)\vdash
z))&=(D(x)\vdash\alpha^{k}(y))\vdash\alpha^{k}(z)\\\ &+(\alpha^{k}(x)\vdash
D(y))\vdash\alpha^{k}(z)+(\alpha^{k}(x)\vdash\alpha^{k}(y))\vdash
D(z)\end{array}$

for $x,y,z\in\mathcal{A}.$

######  Definition 4.5.

A Jordan associative triple derivation of Hom-associative
$(\mathcal{A},\dashv,\vdash\alpha)$ is a linear transformation
$D^{\prime}:\mathcal{A}\longrightarrow A$ such that

$D^{\prime}\circ\alpha=\alpha\circ D^{\prime}$

and

$\begin{array}[]{ll}D^{\prime}\circ((x\dashv y)\dashv
x))&=(D^{\prime}(x)\dashv\alpha^{k}(y))\dashv\alpha^{k}(x)\\\
&+(\alpha^{k}(x)\dashv
D^{\prime}(y))\dashv\alpha^{k}(x)+(\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv
D^{\prime}(x)\end{array}$ $\begin{array}[]{ll}D^{\prime}\circ((x\vdash
y)\vdash x))&=(D^{\prime}(x)\vdash\alpha^{k}(y))\vdash\alpha^{k}(x)\\\
&+(\alpha^{k}(x)\vdash
D^{\prime}(y))\vdash\alpha^{k}(x)+(\alpha^{k}(x)\vdash\alpha^{k}(y))\vdash
D^{\prime}(x)\end{array}$

for $x,y\in\mathcal{A}.$

######  Proposition 4.6.

D is an associative triple derivation of $(\mathcal{A},\dashv,\vdash,\alpha)$
if and only if D is a Jordan triple derivation of
$(\mathcal{A},\dashv,\vdash\alpha)$ such that $A(x,y,z)+A(y,z,x)+A(z,x,y)=0$
with $A(x,y,z)=(\alpha\circ D)\circ(x\dashv y)\dashv z)$.

###### Proof.

If D is a Jordan triple derivation of $(\mathcal{A},\dashv,\vdash,\alpha)$,
then $1$ following immediately. $2$ holds because

$\begin{array}[]{ll}&A(x,y,z)+A(y,z,x)+A(z,x,y)=\\\
&=\alpha((D(x)\dashv\alpha^{k}(y))\dashv\alpha^{k}(z))+(\alpha^{k}(x)\dashv
D(y))\dashv\alpha^{k}(z)+(\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv D(z))\\\
&+\alpha((D(y)\dashv\alpha^{k}(z))\dashv\alpha^{k}(x)+(\alpha^{k}(y))\dashv
D(z))\dashv\alpha^{k}(x)+(\alpha^{k}(y)\dashv\alpha^{k}(z))\dashv D(x))\\\
&+\alpha((D(z)\dashv\alpha^{k}(x))\dashv\alpha^{k}(y))+(\alpha^{k}(z)\dashv
D(x))\dashv\alpha^{k}(y)+(\alpha^{k}(z)\dashv\alpha^{k}(x))\dashv D(y))\\\
&=\alpha((D(x)\dashv\alpha^{k}(y))\dashv\alpha^{k}(z))+(\alpha^{k}(y)\dashv\alpha^{k}(z))\dashv
D(x)+(\alpha^{k}(z)\dashv D(x))\dashv\alpha^{k}(y)\\\
&+\alpha((\alpha^{k}(x)\dashv
D(y))\dashv\alpha^{k}(z))+(D(y)\dashv\alpha^{k}(y))\dashv\alpha^{k}(x)+(\alpha^{k}(z)\dashv\alpha^{k}(x))\dashv
D(y)\\\ &+\alpha((\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv
D(z))+(\alpha^{k}(y)\dashv
D(z))\dashv\alpha^{k}(x)+(D(z)\dashv\alpha^{k}(x))\dashv\alpha^{k}(y)\\\
&=0.\end{array}$

Therefore, D is a generalized associative triple derivation of Hom-associative
$(\mathcal{A},\dashv,\vdash,\alpha)$. ∎

######  Proposition 4.7.

D is an associative triple derivation of $(\mathcal{A},\dashv,\vdash,\alpha)$
with respect to associative derivation $\delta$ if and only if $D$ is Jordan
triple derivation of $(A,\dashv,\vdash,\alpha)$ with respect to a Jordan
triple derivation $\delta$ such that

$(\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv(D-\delta)(z))+(\alpha^{k}(y)\dashv\alpha^{k}(z))\dashv(D-\delta)(x))+(\alpha^{k}(z)\dashv\alpha^{k}(x))\dashv(D-\delta)(y))=0,$

with
$B(x,y,z)=\alpha((\delta(x)\dashv\alpha^{k}(y)\dashv\alpha^{k}(z))+(\alpha^{k}(x)\dashv\delta(y))\dashv\alpha^{k}(z))+((\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv\delta(z)).$

###### Proof.

If D is a Jordan triple derivation of $(\mathcal{A},\dashv,\vdash,\alpha)$,
then $1$ follows immediately. $2$ holds because

$\begin{array}[]{ll}&B(x,y,z)+B(y,z,x)+B(z,x,y)=\\\
&=\alpha((\delta(x)\dashv\alpha^{k}(y))\dashv\alpha^{k}(z))+(\alpha^{k}(x)\dashv\delta(y))\dashv\alpha^{k}(z))+(\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv\delta(z)))\\\
&+\alpha((\delta(y)\dashv\alpha^{k}(z))\dashv\alpha^{k}(x))+(\alpha^{k}(y)\dashv\delta(z))\dashv\alpha^{k}(x))+(\alpha^{k}(y)\dashv\alpha^{k}(z))\dashv\delta(x)))\\\
&+\alpha((\delta(z)\dashv\alpha^{k}(x))\dashv\alpha^{k}(y))+(\alpha^{k}(z)\dashv\delta(x))\dashv\alpha^{k}(y))+(\alpha^{k}(z)\dashv\alpha^{k}(x))\dashv\delta(y)))\\\
&=\alpha((\delta(x)\dashv\alpha^{k}(y))\dashv\alpha^{k}(z))+(\alpha^{k}(x)\dashv\delta(y))\dashv\alpha^{k}(z))+(\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv\delta(z)))\\\
&+\alpha((\delta(y)\dashv\alpha^{k}(z))\dashv\alpha^{k}(x))+(\alpha^{k}(y)\dashv\delta(z))\dashv\alpha^{k}(x))+(\alpha^{k}(y)\dashv\alpha^{k}(z))\dashv\delta(x)))\\\
&+\alpha((\delta(z)\dashv\alpha^{k}(x))\dashv\alpha^{k}(y))+(\alpha^{k}(z)\dashv\delta(x))\dashv\alpha^{k}(y))+(\alpha^{k}(z)\dashv\alpha^{k}(x))\dashv\delta(y)))\\\
&=\alpha((\delta(x)\dashv\alpha^{k}(y))\dashv\alpha^{k}(z))+(\alpha^{k}(y)\dashv\alpha^{k}(z))\dashv\delta(x))+(\alpha^{k}(z)\dashv\delta(x))\dashv\alpha^{k}(y)))\\\
&+\alpha((\alpha^{k}(x)\dashv\delta(y))\dashv\alpha^{k}(z))+(\delta(y)\dashv\alpha^{k}(z))\dashv\alpha^{k}(x))+(\alpha^{k}(z)\dashv\alpha^{k}(x))\dashv\delta(y)))\\\
&+\alpha((\alpha^{k}(x)\dashv\alpha^{k}(y))\dashv\delta(z))+(\alpha^{k}(y)\dashv\delta(z))\dashv\alpha^{k}(x))+(\delta(z)\dashv\alpha^{k}(x))\dashv\alpha^{k}(y)))\\\
&=0.\end{array}$

Note that $\delta$ is an associative derivation referring to Proposition 4.6.
Therefore, D is a generalized associative triple derivation of
$(\mathcal{A},\dashv,\vdash,\alpha)$ with respect to an associative derivation
$\delta$ according to Proposition 4.6. ∎

######  Definition 4.8.

An $\alpha$-derivation of the BiHom-associative trialgebra $\mathcal{A}$ is a
linear transformation $d:\mathcal{A}\rightarrow\mathcal{A}$ satisfying

$\displaystyle\alpha\circ d=d\circ\alpha$ , (4.6) $\displaystyle d(x\dashv y)$
$\displaystyle=$ $\displaystyle d(x)\dashv\alpha(y)+\alpha(x)\dashv d(y)$
(4.7) $\displaystyle d(x\vdash y)$ $\displaystyle=$ $\displaystyle
d(x)\vdash\alpha(y)+\alpha(x)\vdash d(y),$ (4.8)

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

###  4.2 Derivations of complex Hom-associative dialgebras

This section illustrates in depth, $\alpha$-derivation of Hom-associative
dialgebras in dimension two and three over the field $\mathbb{K}.$ Let
$\left\\{e_{1},e_{2},e_{3},\cdots,e_{n}\right\\}$ be a basis of an
$n$-dimensional Hom-associative dialgebra $\mathcal{A}.$ The product of basis
is denoted by

$\displaystyle d(e_{p})=\sum_{q=1}^{n}d_{qp}e_{q}.$

We have

$\displaystyle\sum_{p=1}^{n}d_{pk}a_{qp}=\sum_{p=1}^{n}a_{pk}d_{qp}$ , (4.9)
$\displaystyle\sum_{k=1}^{n}\gamma_{ij}^{p}d_{rp}=\sum_{k=1}^{n}\sum_{p=1}^{n}d_{ki}a_{pj}\gamma_{kp}^{q}$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{n}\sum_{p=1}a_{ki}d_{pj}\gamma_{kp}^{q},$ (4.10)
$\displaystyle\sum_{k=1}^{n}\delta_{ij}^{p}d_{rp}=\sum_{k=1}^{n}\sum_{p=1}^{n}d_{ki}a_{pj}\delta_{kp}^{q}$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{n}\sum_{p=1}a_{ki}d_{pj}\delta_{kp}^{q}.$ (4.11)

######  Theorem 4.9.

The derivations of 2-dimensional Hom-associative dialgebras have the following
form

IC | Der$(d)$ | $Dim(d)$ | IC | Der$(d)$ | $Dim(d)$
---|---|---|---|---|---
$\mathcal{TH}_{2}^{4}$ | $\left(\begin{array}[]{cccc}0&0\\\ d_{21}&0\end{array}\right)$ | 1 | $\mathcal{TH}_{2}^{5}$ | $\left(\begin{array}[]{cccc}0&0\\\ d_{21}&0\end{array}\right)$ | 1
$\mathcal{TH}_{2}^{6}$ | $\left(\begin{array}[]{cccc}0&0\\\ d_{21}&0\end{array}\right)$ | 1 | $\mathcal{TH}_{2}^{8}$ | $\left(\begin{array}[]{cccc}0&0\\\ d_{21}&0\end{array}\right)$ | 1

######  Theorem 4.10.

The derivations of $3$-dimensional Hom-associative dialgebras have the
following form

IC | Der$(d)$ | $Dim(d)$ | IC | Der$(d)$ | $Dim(d)$
---|---|---|---|---|---
$\mathcal{TH}_{3}^{1}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&0\\\ d_{31}&0&d_{33}\end{array}\right)$ | 2 | $\mathcal{TH}_{3}^{2}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&0\\\ d_{31}&0&d_{33}\end{array}\right)$ | 2
$\mathcal{TH}_{3}^{3}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ d_{21}&0&d_{23}\\\ d_{31}&0&d_{33}\end{array}\right)$ | 4 | $\mathcal{TH}_{3}^{4}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&d_{22}&d_{23}\\\ 0&d_{32}&d_{33}\end{array}\right)$ | 4
$\mathcal{TH}_{3}^{5}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&0\\\ 0&d_{32}&d_{33}\end{array}\right)$ | 2 | $\mathcal{TH}_{3}^{8}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&0\\\ 0&d_{32}&d_{33}\end{array}\right)$ | 2
$\mathcal{TH}_{3}^{10}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ d_{21}&0&d_{23}\\\ 0&0&0\end{array}\right)$ | 2 | $\mathcal{TH}_{3}^{11}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ d_{21}&0&d_{23}\\\ 0&0&0\end{array}\right)$ | 2
$\mathcal{TH}_{3}^{12}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&d_{23}\\\ 0&0&0\end{array}\right)$ | 1 | $\mathcal{TH}_{3}^{13}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&0\\\ 0&d_{32}&0\end{array}\right)$ | 1

###### Proof.

Departing from Theorem 4.10, we provide the proof only for one case to
illustrate the used approach, the other cases can be addressed similarly with
or without modification(s). Let’s consider $\mathcal{TH}_{3}^{3}$. Applying
the systems of equations (4.9), (4.10) and (4.11), we get
$d_{11}=d_{12}=d_{13}=d_{22}=d_{32}=0$. Hence, the derivations of
$\mathcal{TH}_{3}^{3}$ are indicated as follows
$d(e_{1})=\left(\begin{array}[]{cccc}0&0&0\\\ 1&0&0\\\
0&0&0\end{array}\right)$, $d(e_{2})=\left(\begin{array}[]{cccc}0&0&0\\\
0&0&0\\\ 1&0&0\end{array}\right)$,
$d(e_{3})=\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&1\\\
0&0&0\end{array}\right)$, and $d(e_{4})=\left(\begin{array}[]{cccc}0&0&0\\\
0&0&0\\\ 0&0&1\end{array}\right)$ is the basis of $Der(Cent(\mathcal{A}))$ and
Dim$Der(Cent(\mathcal{A}))=4.$ The derivations of the remaining parts of
dimension two associative dialgebras can be handled in a similar manner as
illustrated above. ∎

######  Theorem 4.11.

The derivations of $4$-dimensional Hom-associative dialgebras have the
following form

IC | Der$(d)$ | $Dim(d)$ | IC | Der$(d)$ | $Dim(d)$
---|---|---|---|---|---
$\mathcal{TH}_{4}^{1}$ | $\left(\begin{array}[]{ccccc}0&0&0&0\\\ d_{21}&0&d_{23}&0\\\ 0&0&0&0\\\ d_{41}&0&d_{43}&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{2}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ d_{21}&0&d_{23}&0\\\ 0&0&0&0\\\ d_{41}&0&d_{43}&0\\\ \end{array}\right)$ | 2
$\mathcal{TH}_{4}^{3}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ d_{21}&0&d_{23}&0\\\ 0&0&0&0\\\ d_{41}&0&d_{43}&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{4}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ d_{21}&0&d_{23}&0\\\ 0&0&0&0\\\ d_{41}&0&d_{43}&0\\\ \end{array}\right)$ | 4
$\mathcal{TH}_{4}^{5}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ d_{21}&0&d_{23}&0\\\ 0&0&0&0\\\ d_{41}&0&d_{43}&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{6}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ d_{21}&0&d_{23}&0\\\ 0&0&0&0\\\ d_{41}&0&d_{43}&0\\\ \end{array}\right)$ | 4
$\mathcal{TH}_{4}^{7}$ | $\left(\begin{array}[]{cccc}d_{11}&d_{12}&0&0\\\ 0&0&0&0\\\ 0&0&0&d_{34}\\\ 0&0&0&0\\\ \end{array}\right)$ | 3 | $\mathcal{TH}_{4}^{8}$ | $\left(\begin{array}[]{cccc}d_{11}&d_{12}&0&0\\\ 0&0&0&0\\\ 0&0&0&d_{34}\\\ 0&0&0&0\\\ \end{array}\right)$ | 3
$\mathcal{TH}_{4}^{9}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ d_{41}&0&0&0\\\ \end{array}\right)$ | 1 | $\mathcal{TH}_{4}^{10}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ d_{31}&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 1
$\mathcal{TH}_{4}^{11}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ d_{21}&0&0&0\\\ 0&0&0&0\\\ 0&0&d_{43}&0\\\ \end{array}\right)$ | 1 | $\mathcal{TH}_{4}^{12}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ d_{31}&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 1
$\mathcal{TH}_{4}^{13}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ d_{41}&0&0&0\\\ \end{array}\right)$ | 1 | $\mathcal{TH}_{4}^{14}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ d_{41}&0&0&0\\\ \end{array}\right)$ | 1
$\mathcal{TH}_{4}^{15}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ d_{41}&0&0&0\\\ \end{array}\right)$ | 1 | $\mathcal{TH}_{4}^{16}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ d_{41}&0&0&0\\\ \end{array}\right)$ | 1
$\mathcal{TH}_{4}^{17}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&d_{23}&0\\\ d_{41}&0&0&0\\\ \end{array}\right)$ | 2 | $\mathcal{TH}_{4}^{18}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&d_{23}&d_{24}\\\ 0&0&-d_{23}&-d_{24}\\\ 0&0&d_{23}&d_{24}\\\ \end{array}\right)$ | 2

IC | Der$(d)$ | $Dim(d)$ | IC | Der$(d)$ | $Dim(d)$
---|---|---|---|---|---
$\mathcal{TH}_{4}^{19}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&d_{23}&d_{24}\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 2 | $\mathcal{TH}_{4}^{20}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&d_{23}&d_{24}\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 2

###### Proof.

Departing from Theorem 4.11, we provide the proof only for one case to
illustrate the used approach, the other cases can be addressed similarly with
or without modification(s). Let’s consider $\mathcal{TH}_{4}^{1}$. Applying
the systems of equations (4.9), (4.10) and (4.11), we get
$d_{11}=d_{12}=d_{13}=d_{14}=d_{22}=d_{24}=d_{31}=d_{32}=d_{33}=d_{34}=d_{42}=d_{44}=0$.
Hence, the derivations of $\mathcal{TH}_{4}^{1}$ are indicated as follows
$d(e_{1})=\left(\begin{array}[]{ccccc}0&0&0&0\\\ 1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right)$, $d(e_{2})=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&0\end{array}\right)$,
$d(e_{3})=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right)$ and $d(e_{4})=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\end{array}\right)$ is the basis of
$Der(Cent(\mathcal{A}))$ and Dim$Der(Cent(\mathcal{A}))=4.$ The derivations of
the remaining parts of dimension two associative dialgebras can be handled in
a similar manner as illustrated above. ∎

######  Corollary 4.12.

  * •

The dimensions of the derivations of two-dimensional associative trialgebras
range between zero and two.

  * •

The dimensions of the derivations of three-dimensional associative trialgebras
range between zero and three.

  * •

The dimensions of the derivations of four-dimensional associative trialgebras
range between one and four.

##  5 Centroids of Complex Hom-associative dialgebras

###  5.1 Properties of centroids Hom-associative dialgebras

In this section, we set forward the following results on properties of
centroids of Hom-associative dialgebras $\mathcal{A}$.

######  Definition 5.1.

Let $(\mathcal{A},\dashv,\vdash,\alpha)$ be a Hom-associative dialgebra. A
linear map $\psi:\mathcal{A}\rightarrow\mathcal{A}$ is called an element of
$(\alpha)$-element of centroids on $\mathcal{A}$ if, for all
$x,y\in\mathcal{A}$,

$\displaystyle\alpha\circ\psi$ $\displaystyle=$
$\displaystyle\psi\circ\alpha,$ (5.1) $\displaystyle\psi(x)\dashv\alpha(y)$
$\displaystyle=$ $\displaystyle\psi(x)\dashv\psi(y)=\alpha(x)\dashv\psi(y),$
(5.2) $\displaystyle\psi(x)\vdash\alpha(y)$ $\displaystyle=$
$\displaystyle\psi(x)\vdash\psi(y)=\alpha(x)\vdash\psi(y).$ (5.3)

The set of all elements of $(\alpha)$-centroid of $\mathcal{A}$ is denoted
$Cent_{(\alpha)}(\mathcal{A})$. The centroid of $\mathcal{A}$ is denoted
$Cent(\mathcal{A})$.

######  Definition 5.2.

Let $\mathcal{H}$ be a nonempty subset of $\mathcal{A}$. The subset

$Z_{\mathcal{A}}(\mathcal{H})=\left\\{x\in\mathcal{H}|\alpha(x)\bullet\mathcal{H}=\mathcal{H}\bullet\alpha(x)=0\right\\},$
(5.4)

is said to be centralizer of $\mathcal{H}$ in $\mathcal{A}$, where $\bullet$
is $\dashv$ and $\vdash$, respectively.

######  Definition 5.3.

Let $\psi\in End(\mathcal{A})$. If $\psi(\mathcal{A})\subseteq Z(\mathcal{A})$
and $\psi(\mathcal{A}^{2})=0$, then $\psi$ is called a central derivation. The
set of all central derivations of $\mathcal{A}$ is denoted by
$\mathcal{C}(\mathcal{A})$.

######  Proposition 5.4.

Consider $(\mathcal{A},\dashv,\vdash,\alpha)$ a Hom-associative dialgebra.
Then,

  1. i)

$\Gamma(\mathcal{A})Der(\mathcal{T})\subseteq Der(\mathcal{A})$.

  2. ii)

$\left[\Gamma(\mathcal{A}),Dr(\mathcal{A})\right]\subseteq\Gamma(\mathcal{A}).$

  3. iii)

$\left[\Gamma(\mathcal{A}),\Gamma(\mathcal{A})\right](\mathcal{A})\subseteq\Gamma(\mathcal{A})$
and $\left[\Gamma(\mathcal{A}),\Gamma(\mathcal{A})\right](\mathcal{A}^{2})=0.$

###### Proof.

The proof of parts $i)-iii)$ is straightforward by using definitions of
derivations and centroids. ∎

######  Proposition 5.5.

Let $(\mathcal{A},\dashv,\vdash\alpha)$ be a Hom-associative dialgebra and
$\varphi\in Cent(\mathcal{A}),\,d\in Der(\mathcal{A}).$ Then, $\varphi\circ d$
is an $\alpha$-derivation of $\mathcal{A}.$

###### Proof.

Indeed, if $x,y\in\mathcal{A}$, then

$\begin{array}[]{ll}(\varphi\circ d)(x\bullet
y)&=\varphi(d(x)\bullet\alpha(y)+\alpha(x)\bullet d(y))\\\
&=\varphi(d(x)\bullet y)+\varphi(x\bullet d(y))=(\varphi\circ
d)(x)\bullet\alpha(y)+\alpha(x)\bullet(\varphi\circ d)(y),\end{array}$

where $\bullet$ is $\dashv$ and $\vdash$, respectively. ∎

######  Proposition 5.6.

Let $(\mathcal{A},\dashv,\vdash,\alpha)$ be a Hom-associative dialgebra over a
field $\mathbb{F}$. Hence, $\mathcal{C}(\mathcal{A})=Cent(\mathcal{A})\cap
Der(\mathcal{A}).$

###### Proof.

If $\psi\in Cent(\mathcal{A})\cap Der(\mathcal{A})$, then grounded on
definition of $Cent(\mathcal{A}$ and $Der(\mathcal{A})$, we have

$\psi(x\bullet y)=\psi(x)\bullet\alpha(y)+\alpha(x)\bullet\psi(y)$ and
$\psi(x\bullet y)=\psi(x)\circ\alpha(y)=\alpha(x)\circ\psi(y)$ for
$x,y\in\mathcal{A}.$ The yields $\psi(\mathcal{A}\mathcal{A})=0$ and
$\psi(\mathcal{A})\subseteq Z(\mathcal{A})$ i.e $Cent(\mathcal{A})\cap
Der(\mathcal{A})\subseteq Cent(\mathcal{A}).$ The inverse is obvious since
$\mathcal{C}(\mathcal{A})$ is in both $Cent(\mathcal{A})$ and
$Der(\mathcal{A}),$ where $\bullet$ is $\dashv$ and $\vdash$, respectively. ∎

######  Proposition 5.7.

Let $(\mathcal{A},\dashv,\vdash,\alpha)$ be a Hom-associative dialgebra.
Therefore, for any $d\in Der(\mathcal{A})$ and $\varphi\in Cent(\mathcal{A})$,
we have

  1. (i)

The composition $d\circ\varphi$ is in $Cent(\mathcal{A})$, if and only if
$\varphi\circ d$ is a central $\alpha$-derivation of $\mathcal{A}.$

  2. (ii)

The composition $d\circ\varphi$ is a $\alpha$-derivation of $\mathcal{A}$, if
and only if $\left[d,\varphi\right]$ is a central $\alpha$-derivation of
$\mathcal{A}.$

###### Proof.

  1. i)

For any $\varphi\in Cent(\mathcal{A}),\,d\in
Der(\mathcal{A}),\,\forall\,x,y\in\mathcal{A}$, we have

$\begin{array}[]{ll}d\circ\varphi(x\bullet y)=d\circ\varphi(x)\bullet
y&=d\circ\varphi(x)\bullet y+\varphi(x)\bullet d(y)\\\
&=d\circ\varphi(x)\bullet y+\varphi\circ d(x\bullet y)-\varphi\circ
d(x)\bullet y.\end{array}$

Thus, $(d\circ\varphi-\varphi\circ d)(x\bullet
y)=(d\bullet\varphi-\varphi\circ d)(x)\bullet y.$

  2. ii)

Let $d\circ\varphi\in Der(\mathcal{A})$. Using $\left[d,\varphi\right]\in
Cent(\mathcal{A})$, we get

$\left[d,\varphi\right](x\bullet
y)=(\left[d,\varphi\right](x))\bullet\alpha(y)=\alpha(x)\bullet(\left[d,\varphi\right](y))$
(5.5)

On the other side, $\left[d,\varphi\right]d\circ\varphi-\varphi\circ d$ and
$d\circ\varphi,\varphi\circ d\in Der(\mathcal{A}).$ Therefore,

$\left[d,\varphi\right](x\bullet
y)=(d(\varphi\circ(x))\bullet\alpha(y)+\alpha(x)\bullet(d\circ\varphi(y))-(\varphi\circ
d(x))\bullet\alpha(y)-\alpha(x)\bullet(\varphi\circ d(y)).$ (5.6)

Referring to (5.5) and (5.6), we get
$\alpha(x)\bullet(\left[d,\varphi\right])(y)=(\left[d,\varphi\right])(x)\bullet\alpha(y)=0.$

At this stage of analysis,let $\left[d,\varphi\right]$ be a central
$\alpha$-derivation of $\mathcal{A}$. Then,

$\begin{array}[]{ll}d\circ\varphi(x\bullet
y)&=\left[d\circ\varphi\right](x\bullet y)+(\varphi\circ d)(x\bullet y)\\\
&=\varphi(\circ d(x)\bullet\alpha(y))+\varphi(\alpha(x)\bullet d(y))\\\
&=(\varphi\circ d)(x)\bullet\alpha(y)+\alpha(x)\bullet(\varphi\circ
d)(y),\end{array}$

where $\bullet$ represents the products $\dashv$ and $\vdash$, respectively. ∎

###  5.2 Centroids of complex Hom-associative dialgebras

This section elaborates the details of $\alpha$-centroids of Hom-associative
dialgebras in dimension two and three over the field $\mathbb{K}.$ Let
$\left\\{e_{1},e_{2},e_{3},\cdots,e_{n}\right\\}$ be a basis of an
$n$-dimensional Hom-associative dialgebra $\mathcal{A}.$ The product of basis

$\displaystyle\psi(e_{p})=\sum_{q=1}^{n}c_{qp}e_{q}.$

$\displaystyle\sum_{p=1}^{n}c_{pi}a_{qp}$ $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}a_{pi}c_{qp},$ (5.7)
$\displaystyle\sum_{k=1}^{n}\sum_{p=1}^{n}c_{ki}a_{pj}\gamma_{kp}^{q}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\sum_{p=1}^{n}c_{ki}c_{pj}\gamma_{kp}^{q}=\sum_{k=1}^{n}\sum_{p=1}^{n}a_{ki}c_{pj}\gamma_{kp}^{q},$
(5.8) $\displaystyle\sum_{k=1}^{n}\sum_{p=1}^{n}c_{ki}a_{pj}\delta_{kp}^{q}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\sum_{p=1}^{n}c_{ki}c_{pj}\delta_{kp}^{q}=\sum_{k=1}^{n}\sum_{p=1}^{n}a_{ki}c_{pj}\delta_{kp}^{q}.$
(5.9)

######  Theorem 5.8.

The centroids of $2$-dimensional Hom-associative dialgebras have the following
form

IC | $Cent(\mathcal{A})$ | $Dim(Cent(\mathcal{A}))$ | IC | $Cent(\mathcal{A})$ | $Dim(Cent(\mathcal{A}))$
---|---|---|---|---|---
$\mathcal{TH}_{2}^{1}$ | $\left(\begin{array}[]{cccc}c_{11}&0\\\ 0&c_{22}\end{array}\right)$ | 2 | $\mathcal{TH}_{2}^{2}$ | $\left(\begin{array}[]{cccc}0&0\\\ c_{21}&0\end{array}\right)$ | 1
$\mathcal{TH}_{2}^{3}$ | $\left(\begin{array}[]{cccc}-c_{11}&0\\\ c_{21}&c_{22}\end{array}\right)$ | 3 | $\mathcal{TH}_{2}^{4}$ | $\left(\begin{array}[]{cccc}c_{11}&0\\\ c_{21}&c_{22}\end{array}\right)$ | 3
$\mathcal{TH}_{2}^{5}$ | $\left(\begin{array}[]{cccc}0&0\\\ c_{21}&0\end{array}\right)$ | 1 | $\mathcal{TH}_{2}^{6}$ | $\left(\begin{array}[]{cccc}c_{11}&0\\\ 0&c_{22}\end{array}\right)$ | 2
$\mathcal{TH}_{2}^{7}$ | $\left(\begin{array}[]{cccc}c_{11}&0\\\ 0&c_{22}\end{array}\right)$ | 2 | $\mathcal{TH}_{2}^{8}$ | $\left(\begin{array}[]{cccc}0&0\\\ c_{21}&0\end{array}\right)$ | 1
$\mathcal{TH}_{2}^{9}$ | $\left(\begin{array}[]{cccc}c_{11}&0\\\ c_{21}&c_{22}\end{array}\right)$ | 3 |  |  |

###### Proof.

Departing from Theorem 5.8, we provide the proof only for one case to
illustrate the used approach, the other cases can be addressed similarly with
or without modification(s). Let’s consider $\mathcal{TH}_{2}^{1}$. Applying
the systems of equations (5.7), (5.8) and (5.9), we get $c_{12}=c_{21}=0$.
Hence, the centroids of $\mathcal{TH}_{2}^{1}$ are indicated as follows
$c(e_{1})=\left(\begin{array}[]{cccc}1&0\\\ 0&0\end{array}\right)$ and
$c(e_{2})=\left(\begin{array}[]{cccc}0&0\\\ 0&1\end{array}\right)$ is the
basis of $Der(Cent(\mathcal{A}))$ and Dim$Der(Cent(\mathcal{A}))=2.$ The
centroids of the remaining parts of dimension two associative dialgebras can
be handled in a similar manner as illustrated above. ∎

######  Theorem 5.9.

The centroids of $3$-dimensional Hom-associative dialgebras have the following
form

IC | $Cent(\mathcal{A})$ | $Dim(Cent(\mathcal{A}))$ | IC | $Cent(\mathcal{A})$ | $Dim(Cent(\mathcal{A}))$
---|---|---|---|---|---
$\mathcal{TH}_{3}^{1}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ c_{21}&0&0\\\ c_{31}&0&0\end{array}\right)$ | 2 | $\mathcal{TH}_{3}^{2}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ c_{21}&0&0\\\ c_{31}&0&0\end{array}\right)$ | 2
$\mathcal{TH}_{3}^{3}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0\\\ c_{21}&c_{11}&c_{11}\\\ c_{31}&0&0\end{array}\right)$ | 3 | $\mathcal{TH}_{3}^{4}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | 1
$\mathcal{TH}_{3}^{5}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&c_{23}\\\ 0&0&c_{33}\end{array}\right)$ | 2 | $\mathcal{TH}_{3}^{6}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ \frac{1}{2}c_{21}&0&0\\\ c_{31}&0&0\end{array}\right)$ | 2
$\mathcal{TH}_{3}^{7}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | 1 | $\mathcal{TH}_{3}^{8}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0\\\ 0&0&c_{23}\\\ 0&0&c_{33}\end{array}\right)$ | 3
$\mathcal{TH}_{3}^{9}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)$ | 1 | $\mathcal{TH}_{3}^{10}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ c_{21}&0&c_{23}\\\ c_{31}&0&c_{33}\end{array}\right)$ | 4
$\mathcal{TH}_{3}^{11}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ c_{21}&0&0\\\ c_{31}&0&0\end{array}\right)$ | 2 | $\mathcal{TH}_{3}^{12}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ c_{21}&0&0\\\ c_{31}&0&0\end{array}\right)$ | 2
$\mathcal{TH}_{3}^{13}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0\\\ 0&c_{22}&0\\\ 0&0&c_{33}\end{array}\right)$ | 3 | $\mathcal{TH}_{3}^{14}$ | $\left(\begin{array}[]{cccc}0&0&0\\\ 0&c_{22}&0\\\ 0&0&0\end{array}\right)$ | 1

###### Proof.

Departing from Theorem 5.9, we provide the proof only for one case to
illustrate the used approach, the other cases can be addressed similarly with
or without modification(s). Let’s consider $\mathcal{TH}_{3}^{3}$. Applying
the systems of equations (5.7), (5.8) and (5.9), we get
$c_{12}=c_{13}=c_{32}=c_{33}=0$ and $c_{22}=c_{23}=c_{11}$. Hence, the
centroids of $\mathcal{TH}_{3}^{3}$ are indicated as follows
$c(e_{1})=\left(\begin{array}[]{cccc}1&0&0\\\ 0&1&1\\\
0&0&0\end{array}\right)$, $c(e_{2})=\left(\begin{array}[]{cccc}0&0&0\\\
1&0&0\\\ 0&0&0\end{array}\right)$ and
$c(e_{3})=\left(\begin{array}[]{cccc}0&0&0\\\ 0&0&0\\\
1&0&0\end{array}\right)$ is the basis of $Der(c)$ and Dim$Der(c)=3.$ The
centroids of the remaining parts of dimension three associative dialgebras can
be handled in a similar manner as illustrated above. ∎

######  Theorem 5.10.

The centroids of $4$-dimensional Hom-associative dialgebras have the following
form

IC | $Cent(\mathcal{A})$ | $Dim$ | IC | $Cent(\mathcal{A})$ | $Dim$
---|---|---|---|---|---
$\mathcal{TH}_{4}^{1}$ | $\left(\begin{array}[]{ccccc}0&0&0&0\\\ c_{21}&0&c_{23}&0\\\ 0&0&0&0\\\ c_{41}&0&c_{43}&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{2}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ c_{21}&0&c_{23}&0\\\ c_{31}&0&c_{33}&0\\\ c_{41}&0&c_{43}&0\\\ \end{array}\right)$ | 6
$\mathcal{TH}_{4}^{3}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ d_{21}&0&d_{23}&0\\\ c_{31}&0&c_{33}&0\\\ d_{41}&0&d_{43}&0\\\ \end{array}\right)$ | 6 | $\mathcal{TH}_{4}^{4}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ c_{21}&0&0&0\\\ 0&0&0&0\\\ 0&0&c_{43}&0\\\ \end{array}\right)$ | 2

IC | $Cent(\mathcal{A})$ | $Dim$ | IC | $Cent(\mathcal{A})$ | $Dim$
---|---|---|---|---|---
$\mathcal{TH}_{4}^{5}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ c_{21}&0&c_{23}&0\\\ 0&0&0&0\\\ c_{41}&0&c_{43}&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{6}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ c_{21}&0&c_{23}&0\\\ 0&0&0&0\\\ c_{41}&0&c_{43}&0\\\ \end{array}\right)$ | 4
$\mathcal{TH}_{4}^{7}$ | $\left(\begin{array}[]{cccc}c_{11}&c_{12}&0&0\\\ c_{21}&c_{22}&0&0\\\ 0&0&0&0\\\ c_{41}&0&c_{43}&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{8}$ | $\left(\begin{array}[]{cccc}0&c_{12}&0&0\\\ 0&c_{22}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 2
$\mathcal{TH}_{4}^{9}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&c_{22}&0&0\\\ 0&0&c_{33}&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 2 | $\mathcal{TH}_{4}^{10}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0&0\\\ 0&0&0&0\\\ c_{31}&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 2
$\mathcal{TH}_{4}^{11}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0&0\\\ 0&0&0&0\\\ c_{31}&0&c_{33}&0\\\ 0&0&c_{43}&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{12}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0&0\\\ 0&0&0&0\\\ c_{31}&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 2
$\mathcal{TH}_{4}^{13}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ c_{41}&0&0&c_{44}\\\ \end{array}\right)$ | 3 | $\mathcal{TH}_{4}^{14}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ c_{41}&0&0&0\\\ \end{array}\right)$ | 2
$\mathcal{TH}_{4}^{15}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&c_{22}&0&0\\\ 0&c_{32}&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ | 2 | $\mathcal{TH}_{4}^{16}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0&0\\\ 0&c_{22}&0&0\\\ 0&c_{32}&0&0\\\ c_{41}&0&0&0\\\ \end{array}\right)$ | 4
$\mathcal{TH}_{4}^{17}$ | $\left(\begin{array}[]{cccc}c_{11}&0&0&0\\\ 0&c_{22}&0&0\\\ 0&c_{32}&0&0\\\ c_{41}&0&0&0\\\ \end{array}\right)$ | 4 | $\mathcal{TH}_{4}^{18}$ | $\left(\begin{array}[]{cccc}0&0&0&0\\\ \frac{1-\sqrt{1-4c_{21}^{2}}}{2}&0&0&c_{24}\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right)$ | 2

###### Proof.

Departing from Theorem 5.10, we provide the proof only for one case to
illustrate the used approach, the other cases can be addressed similarly with
or without modification(s). Let’s consider $\mathcal{TH}_{4}^{1}$. Applying
the systems of equations (5.7), (5.8) and (5.9), we get
$c_{11}=c_{12}=c_{13}=c_{14}=0=c_{22}=c_{24}=c_{31}=c_{32}=c_{33}=c_{34}=c_{42}=c_{44}=0$.
Hence, the centroids of $\mathcal{TH}_{4}^{1}$ are indicated as follows
$c(e_{1})=\left(\begin{array}[]{ccccc}0&0&0&0\\\ 1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right)$, $c(e_{2})=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&0\end{array}\right)$,
$c(e_{3})=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right)$ and $c(e_{4})=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\end{array}\right)$ is the basis of
$Der(Cent(\mathcal{A}))$ and Dim$Der(Cent(\mathcal{A}))=4.$ The derivations of
the remaining parts of dimension two associative dialgebras can be handled in
a similar manner as illustrated above. ∎

######  Corollary 5.11.

  * •

The dimensions of the centroids of two-dimensional associative trialgebras
range between one and three.

  * •

The dimensions of the centroids of three-dimensional associative trialgebras
range between one and five.

  * •

The dimensions of the centroids of four-dimensional associative trialgebras
range between zero and six.

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