# On BiHom-associative dialgebras

Ahmed Zahari
Université de Haute Alsace,
IRIMAS-Département de Mathématiques,
6, rue des Frères Lumière F-68093 Mulhouse, France.
Ibrahima BAKAYOKO
Département de Mathématiques, Université de N’Zérékoré,
BP 50 N’Zérékoré, Guinée.  e-mail address<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
Abstract. The aim of this paper is to introduce and study BiHom-associative
dialgebras. We give various constructions and study its connections with
BiHom-Poisson dialgebras and BiHom-Leibniz algebras. Next we discuss the
central extensions of BiHom-diassociative and we describe the classification
of $n$-dimensional BiHom-diassociative algebras for $n\leq 4$. Finally, we
discuss their derivations.

AMS Subject Classification:  .

Keywords: BiHom-associative dialgebra, BiHom-Poisson dialgebra, centroid,
averaging operator, Nijenhuis operator, Rota-Baxter operator, Extension,
Derivation, Classification.

##  1 Introduction

The associative dialgebras (also known as diassociative algebras) has been
introduced by Loday in 1990 (see [6] and references therein) as a
generalization of associative algebras. They are a generalization of
associative algebras in the sens that they possess two associative
multiplications and obey to three other conditions; when the two associative
low are equal we recover associative algebra. One of his motivation were to
find an algebra whose commutator give rises to Leibniz algebra as it is the
case in the relation between Lie and associative algebra. Another motivation
come from the research of an obstruction to the periodicity in algebraic
K-theory. Now, these algebras found their applications in classical geometry,
non-commutative geometry and physics.

The centroid plays an important role in understanding forms of an algebra. It
is an element in the classification of associative and diassociative algebras.
They occurs naturally is in the study of derivations of an algebra. The
centroid and averaging operators are used in the deformation of algebra in
order to generate another algebraic structure. The Nijenhuis operator on an
associative algebra was introduced in [16] to study quantum bi-Hamiltonian
systems while the notion Nijenhuis operator on a Lie algebra originated from
the concept of Nijenhuis tensor that was introduced by Nijenhuis in the study
of pseudo-complex manifolds and was related to the well known concepts of
Schouten-Nijenhuis bracket , the Frolicher-Nijenhuis bracket [1], and the
Nijenhuis-Richardson bracket. The associative analog of the Nijenhuis relation
may be regaded as the homogeneous version of Rota-Baxter relation[23].

BiHom-algebraic structures were introduced in 2015 by G. Graziani, A.
Makhlouf, C. Menini and F. Panaite in [7] from a categorical approach as an
extension of the class of Hom-algebras. Since then, other interesting BiHom-
type algebraic structures of many Hom-algebraic structures has been
intensively studied as BiHom-Lie colour algebras structures [17],
Representations of BiHom-Lie algebras [30], BiHom-Lie superalgebra structures
[28], $\\{\sigma,\tau\\}$-Rota-Baxter operators, infinitesimal Hom-bialgebras
and the associative (Bi)Hom-Yang-Baxter equation [22], The construction and
deformation of BiHom-Novikov algebras [27], On n-ary Generalization of BiHom-
Lie algebras and BiHom-Associative Algebras [3], Rota-Baxter operators on
BiHom-associative algebras and related structures [19].

The goal of this paper is to introduce, classify and study structures, central
extensions and derivations of BiHom-associative algebras. The paper is
organized as follows. In section 2, we define BiHom-associative dialgebras,
give some constructions using twisting, direct sum, elements of centroid,
averaging operator, Nijenhuis operator and Rota-Baxter relation. We give a
connection between BiHom-associative dialgebras and BiHom-Leibniz algebras. We
introduce action of a BiHom-Leibniz algebra onto another and give a Leibniz
structure on the semidirect structure. Then, we show that the semidirect sum
of BiHom-Leibniz algebras associated to BiHom-associative dialgebras is the
same that the BiHom-Leibniz algebra associated to the semidirect of BiHom-
associative dialgebras. Finally, we introduce BiHom-associative dialgebras and
show that any BiHom-associative dialgebra carries a structure of BiHom-Poisson
dialgebra. In section 3, we introduce the notion of central extension of
BiHom-associative dialgebras and define $2$-cocycles and $2$-coboundaries of
BiHom-associative dialgebras with coefficients in a trivial BiHom-module. Then
we establish relationship between $2$-cocycles and central extensions. Section
4, is devoted to the classification of $n$-dimensional BiHom-associative
dialgebras for $n\leq 4$. We dedicated Section 5 to the derivations of BiHom-
associative dialgebras.

##  2 Structure of BiHom-associative dialgebras

######  Definition 2.1.

A BiHom-associative dialgebras is a $5$-truple
$(A,\dashv,\vdash,\alpha,\beta)$ consisting of a linear space $A$ linear maps
$\dashv,\vdash,:A\times A\longrightarrow A$ and $\alpha,\beta:A\longrightarrow
A$ satisfying, for all $x,y,z\in A$ the following conditions :

$\displaystyle\alpha\circ\beta$ $\displaystyle=$
$\displaystyle\beta\circ\alpha,$ (2.1) $\displaystyle(x\dashv
y)\dashv\beta(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\dashv(y\dashv z),$
(2.2) $\displaystyle(x\dashv y)\dashv\beta(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\dashv(y\vdash z),$ (2.3) $\displaystyle(x\vdash
y)\dashv\beta(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\vdash(y\dashv z),$
(2.4) $\displaystyle(x\dashv y)\vdash\beta(z)$ $\displaystyle=$
$\displaystyle\alpha(x)\vdash(y\vdash z),$ (2.5) $\displaystyle(x\vdash
y)\vdash\beta(z)$ $\displaystyle=$ $\displaystyle\alpha(x)\vdash(y\vdash z).$
(2.6)

We called $\alpha$ and $\beta$ ( in this order ) the structure maps of A.

######  Example 2.2.

Any Hom-associative dialgebra [10] or any associative dialgebra is a BiHom-
associative dialgebra by setting $\beta=\alpha$ or $\alpha=\beta=id$.

######  Example 2.3.

Let $(A,\dashv,\vdash,\alpha,\beta)$ a BiHom-associative dialgebra. Consider
the module of $n\times n$-matrices
$\mathcal{M}_{n}(D)=\mathcal{M}_{n}(\mathbb{K})\otimes D$ with the linear maps
${\bf\alpha}(A)=(\alpha(a_{ij}))$, ${\bf\beta}(A)=(\beta(a_{ij})$ for all
$A\in\mathcal{M}_{n}(D)$ and the products $(a\triangleleft
b)_{ij}=\sum_{k}a_{ik}\dashv b_{kj}$ and $(a\triangleright
b)_{ij}=\sum_{k}a_{ik}\vdash b_{kj}$. Then,
$(\mathcal{M}_{n}(D),\triangleleft,\triangleright,{\bf\alpha},{\bf\beta})$ is
a BiHom-associative dialgebra.

######  Definition 2.4.

A morphism $f:({D},\dashv,\vdash,\alpha,\beta)$ and
$({D}^{\prime},\dashv^{\prime},\vdash^{\prime},\alpha^{\prime},\beta^{\prime})$
be a BiHom-associative dialgebras is a linear map
$f:{D}\rightarrow{D}^{\prime}$ such that $\alpha^{\prime}\circ
f=f\circ\alpha,\,\beta^{\prime}\circ f=f\circ\beta$ and $f(x\dashv
y)=f(x)\dashv^{\prime}f(y),\quad f(x\vdash y)=f(x)\vdash^{\prime}f(y)$, for
all $x,y\in{D}.$

######  Definition 2.5.

A BiHom-associative dialgebra $(A,\dashv,\vdash,\alpha,\beta)$ in which
$\alpha$ and $\beta$ are morphism is said to be a multiplicative BiHom-
associative dialgebra.
If moreover, $\alpha$ and $\beta$ are bijective (i.e. automorphisms), then
$(A,\dashv,\vdash,\alpha,\beta)$ is said to be a regular BiHom-associative
dialgebra.

We prove in the following proposition that any BiHom-associative dialgebra
turn to another one via morphisms.

######  Theorem 2.6.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$\alpha^{\prime},\beta^{\prime}:D\rightarrow D$ two morphisms of BiHom-
associative dialgebras such that the maps
$\alpha,\alpha^{\prime},\beta,\beta^{\prime}$ commute pairewise. Then

$D_{(\alpha^{\prime},\beta^{\prime})}=(D,\triangleleft:=\dashv(\alpha^{\prime}\otimes\beta^{\prime}),\triangleright:=\vdash(\alpha^{\prime}\otimes\beta^{\prime}),\alpha\alpha^{\prime},\beta\beta^{\prime})$

is a BiHom-associative dialgebra.

###### Proof.

We prove only one axiom and leave the rest to the reader. For any $x,y,z\in
D$,

$\displaystyle(x\triangleleft
y)\triangleleft\beta\beta^{\prime}(z)-\alpha\alpha^{\prime}(x)\triangleleft(y\triangleright
z)$ $\displaystyle=$
$\displaystyle\alpha^{\prime}(\alpha^{\prime}(x)\dashv\beta^{\prime}(y))\dashv\beta^{\prime}\beta\beta^{\prime}(z)-\alpha^{\prime}\alpha\alpha^{\prime}(x)\dashv\beta^{\prime}(\alpha^{\prime}(y)\vdash\beta^{\prime}(z))$
$\displaystyle=$
$\displaystyle(\alpha^{\prime}\alpha^{\prime}(x)\dashv\alpha^{\prime}\beta^{\prime}(y))\dashv\beta\beta^{\prime}\beta^{\prime}(z)-\alpha\alpha^{\prime}\alpha^{\prime}(x)\dashv(\alpha^{\prime}\beta^{\prime}(y)\vdash\beta^{\prime}\beta^{\prime}(z)).$

The left hand side vanishes by (2.3). And, this ends the proof. ∎

######  Corollary 2.7.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a multiplicative BiHom-associative
dialgebra. Then

$(D,\dashv\circ(\alpha^{n}\otimes\beta^{n}),\vdash\circ(\alpha^{n}\otimes\beta^{n}),\alpha^{n+1},\beta^{n+1})$

is also a multiplicative BiHom-associative dialgebra.

###### Proof.

It suffises to take $\alpha^{\prime}=\alpha^{n}$ and
$\beta^{\prime}=\beta^{n}$ in Theorem 2.6. ∎

######  Corollary 2.8.

Let $(D,\dashv,\vdash,\alpha)$ be a multiplicative Hom-associative dialgebra
and $\beta:D\rightarrow D$ an endomorphism of $D$. Then

$(D,\dashv\circ(\alpha\otimes\beta),\vdash\circ(\alpha\otimes\beta),\alpha^{2},\beta)$

is also a Hom-associative dialgebra.

###### Proof.

It suffises to take $\alpha^{\prime}=\alpha$ and replace $\beta$ by $Id_{D}$,
and $\beta^{\prime}$ by $\beta$ in Theorem 2.6. ∎

Any regular Hom-associative dialgebra give rises to associative dialgebra as
stated in the next corollary.

######  Corollary 2.9.

If $(D,\dashv,\vdash,\alpha,\beta)$ is a regular BiHom-associative dialgebra,
then

$(D,\dashv\circ(\alpha^{-1}\otimes\beta^{-1}),\vdash\circ(\alpha^{-1}\otimes\beta^{-1}))$

is an associative dialgebra.

###### Proof.

We have to take $\alpha^{\prime}=\alpha^{-1}$ and $\beta^{\prime}=\beta^{-1}$
in Theorem 2.6. ∎

######  Corollary 2.10.

Let $(D,\dashv,\vdash)$ be an associative dialgebra and $\alpha:D\rightarrow
D$ and $\beta:D\rightarrow D$ a pair of commuting endomorphisms of $D$. Then

$(D,\dashv\circ(\alpha\otimes\beta),\vdash\circ(\alpha\otimes\beta),\alpha,\beta)$

is a BiHom-associative dialgebra.

###### Proof.

We have to take $\alpha=\beta=Id_{D}$ and replace $\alpha^{\prime}$ by
$\alpha$, and $\beta^{\prime}$ by $\beta$ in Theorem 2.6. ∎

######  Definition 2.11.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra. For any
integers $k,l$, an even linear map $\theta:D\rightarrow D$ is called an
element of $(\alpha^{k},\beta^{l})$-centroid on $D$ if

$\displaystyle\alpha\circ\theta$ $\displaystyle=$
$\displaystyle\theta\circ\alpha,\quad\beta\circ\theta=\theta\circ\beta,$ (2.7)
$\displaystyle\theta(x)\dashv\alpha^{k}\beta^{l}(y)$ $\displaystyle=$
$\displaystyle\theta(x)\dashv\theta(y)=\alpha^{k}\beta^{l}(x)\dashv\theta(y),$
(2.8) $\displaystyle\theta(x)\vdash\alpha^{k}\beta^{l}(y)$ $\displaystyle=$
$\displaystyle\theta(x)\vdash\theta(y)=\alpha^{k}\beta^{l}(x)\vdash\theta(y),$
(2.9)

for all $x,y\in D$.

The set of elements of centroid is called centroid.

######  Proposition 2.12.

Let $(A,\dashv,\vdash,\alpha,\beta)$ a BiHom-associative dialgebra and
$\phi:A\rightarrow A$ and $\psi:A\rightarrow A$ be a paire of commuting
elements of cenroid. Let us defined

$x\triangleleft y:=\phi(x)\dashv y\quad\mbox{and}\quad x\triangleright
y:=\psi(x)\vdash y.$

Then, $(A,\triangleleft,\triangleright,\alpha,\beta)$ is a BiHom-associative
dialgebra if and only if
$Im(\phi-\psi)\in Z_{\dashv}(A):=\\{x\in A/x\dashv y=0,\forall y\in
A\\}\;\mbox{and}\;Im(\phi-\psi)\in Z_{\vdash}(A):=\\{x\in A/y\vdash
x=0,\forall y\in A\\}$.

###### Proof.

We only prove axioms (2.3) and (2.5), the three other comes from BiHom-
associativity. So for any $x,y,z\in A$,

$\displaystyle(x\triangleleft
y)\triangleleft\beta(z)-\alpha(x)\triangleleft(y\triangleright z)$
$\displaystyle=$ $\displaystyle(\phi(x)\dashv
y)\dashv\phi\beta(z)-\phi\alpha(x)\dashv(y\vdash\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv
y)\dashv\beta\phi(z)-\alpha\phi(x)\dashv(y\vdash\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv y)\vdash\beta\phi(z)-(\phi(x)\dashv
y)\vdash\beta\psi(z))$ $\displaystyle=$ $\displaystyle(\phi(x)\dashv
y)\vdash\beta(\phi-\psi)(z)$ $\displaystyle=$
$\displaystyle\alpha\phi(x)\dashv(y\vdash(\phi-\psi)(z)).$

and

$\displaystyle(x\triangleleft
y)\triangleright\beta(z)-\alpha(x)\triangleright(y\triangleright z)$
$\displaystyle=$ $\displaystyle(\phi(x)\dashv
y)\vdash\beta\psi(z)-\psi\alpha(x)\vdash(y\vdash\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv
y)\vdash\beta\psi(z)-\alpha\psi(x)\vdash(y\psi(z))$ $\displaystyle=$
$\displaystyle(\phi(x)\dashv y)\vdash\beta\psi(z)-(\psi(x)\dashv
y)\vdash\beta\psi(z)$ $\displaystyle=$ $\displaystyle[(\phi(x)-\psi(x))\dashv
y]\vdash\beta\psi(z).$

A study of cancellation of the two equalities allows to conclude. ∎

######  Proposition 2.13.

Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra, and
$\phi:A\rightarrow A$ and $\psi:A\rightarrow A$ be a paire of commuting
elements of cenroid. Let us defined

$x\dashv y:=\phi(x)\cdot y\quad\mbox{and}\quad x\vdash y:=\psi(x)\cdot y.$

Then, $(A,\dashv,\vdash,\alpha,\beta)$ is a BiHom-associative dialgebra if and
only if $Im(\phi-\psi)$ is contained in the set of isotropic vectors.

###### Proof.

We only prove axioms (2.3) and (2.5), the three other comes from BiHom-
associativity. So for any $x,y,z\in A$,

$\displaystyle(x\dashv y)\dashv\beta(z)-\alpha(x)\dashv(y\vdash z)$
$\displaystyle=$ $\displaystyle(\phi(x)y)\phi\beta(z)-\phi\alpha(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\phi(z)-\alpha\phi(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\phi(z)-(\phi(x)y)\beta\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta(\phi-\psi)(z)$ $\displaystyle=$
$\displaystyle\alpha\phi(x)(y(\phi-\psi)(z)).$

and

$\displaystyle(x\dashv y)\vdash\beta(z)-\alpha(x)\vdash(y\vdash z)$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\psi(z)-\psi\alpha(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\psi(z)-\alpha\psi(x)(y\psi(z))$
$\displaystyle=$ $\displaystyle(\phi(x)y)\beta\psi(z)-(\psi(x)y)\beta\psi(z)$
$\displaystyle=$ $\displaystyle[(\phi(x)-\psi(x))y]\beta\psi(z).$

A study of cancellation of the two equalities allow to conclude. ∎

######  Remark 2.14.

Proposition 2.13 may be seen as a consequence of Proposition 2.12.

######  Proposition 2.15.

Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and
$(M,\ast_{L},\ast_{R},\alpha_{M},\beta_{M})$ an $A$-BiHom-bimodule i.e. $M$ is
a vector space, $\alpha_{M}:M\rightarrow M$ and $\beta_{M}:M\rightarrow M$ are
two linear maps, and $\ast_{L}:A\rightarrow M$ and $\ast_{R}:M\rightarrow A$
two bilinear maps such that

$\displaystyle\alpha(x)\ast_{L}(y\ast_{L}m)$ $\displaystyle=$
$\displaystyle(x\cdot y)\ast_{L}\beta_{M}(m)$ (2.10)
$\displaystyle\alpha(x)\ast_{L}(m\ast_{R}y)$ $\displaystyle=$
$\displaystyle(x\ast_{L}m)\ast_{R}\beta(y)$ (2.11)
$\displaystyle\alpha_{M}(m)\ast_{R}(x\cdot y)$ $\displaystyle=$
$\displaystyle(m\ast_{R}x)\ast_{R}\beta(y).$ (2.12)

Suppose that $f:M\rightarrow A$ is a morphism of $A$-BiHom-bimodule i.e. $f$
is linear such that $\alpha\circ f=f\circ\alpha_{M}$, $\beta\circ
f=f\circ\beta_{M}$ and

$\displaystyle f(x\ast_{L}m)$ $\displaystyle=$ $\displaystyle x\cdot f(m)$
(2.13) $\displaystyle f(m\ast_{R}x)$ $\displaystyle=$ $\displaystyle f(m)\cdot
x.$ (2.14)

Then, $(M,\triangleleft,\triangleright,\alpha_{M},\beta_{M})$ is a BiHom-
associative dialgebra with

$m\triangleleft n=f(m)\ast_{R}n\quad\mbox{and}\quad m\triangleright
n=m\ast_{R}f(n),$

for all $m,n\in M$.

###### Proof.

We only prove axiom (2.6), the other being proved similarly. For any $x,y,z\in
A$,

$\displaystyle(m\triangleleft n)\triangleright\beta_{M}(p)$ $\displaystyle=$
$\displaystyle(f(m)\ast_{L}n)\ast_{R}f\beta_{M}(p)$ $\displaystyle=$
$\displaystyle(f(m)\ast_{L}n)\ast_{R}\beta f(p).$

By (2.11),

$\displaystyle(m\triangleleft n)\triangleright\beta_{M}(p)$ $\displaystyle=$
$\displaystyle\alpha f(m)\ast_{L}(n\ast_{R}f(p))$ $\displaystyle=$
$\displaystyle f\alpha_{M}(m)\ast_{L}(n\triangleright p)$ $\displaystyle=$
$\displaystyle\alpha_{M}(m)\triangleleft(n\triangleright p).$

This completes the proof. ∎

######  Remark 2.16.

Any $(\alpha^{0},\beta^{0})$-element of centroid of a BiHom-associative
algebra is a morphism of BiHom-bimodule.

Thanks to the above remark, we have what follows :

######  Corollary 2.17.

Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and let $\theta$
be an element of cenroid on $A$. Then,
$(A,\triangleleft,\triangleright,\alpha,\beta)$ is a BiHom-associative
dialgebra with

$x\triangleleft y=\theta(x)\cdot y\quad\mbox{and}\quad x\triangleright
y=x\cdot\theta(y),$

for any $x,y\in A.$

######  Proposition 2.18.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$R:D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$ i.e. $R$ is
linear and $\alpha\circ R=R\circ\alpha$ , $\beta\circ R=R\circ\beta$, and

$\displaystyle R(x)\dashv R(y)$ $\displaystyle=$ $\displaystyle R(R(x)\dashv
y+x\dashv R(y))$ (2.16) $\displaystyle R(x)\vdash R(y)$ $\displaystyle=$
$\displaystyle R(R(x)\vdash y+x\vdash R(y))$ (2.17)

Then, $(D,\triangleleft,\triangleright,\alpha,\beta)$ is also a BiHom-
associative algebra with

$\displaystyle x\triangleleft y=R(x)\dashv y+x\dashv R(y),$ (2.18)
$\displaystyle x\triangleright y=R(x)\vdash y+x\vdash R(y),$ (2.19)

for all $x,y\in D$.

###### Proof.

We only prove axiom (2.6), the other being proved in a similar way. Thus, For
any $x,y,z\in A$,

$\displaystyle\qquad(x\triangleleft
y)\triangleleft\beta(z)-\alpha(x)\triangleleft(y\triangleright z)=$
$\displaystyle=(x\dashv R(y)+R(x)\dashv y)\dashv R\beta(z)+R(R(x)\dashv
y+x\dashv R(y))\dashv\beta(z)$ $\displaystyle\quad-\alpha(x)\dashv
R(R(y)\dashv z+y\dashv R(z))-R\alpha(x)\dashv(R(y)\vdash z+y\vdash R(z))$
$\displaystyle=(x\dashv R(y))\dashv\beta R(z)+(R(x)\dashv y)\dashv\beta
R(z)+(R(x)\dashv R(y))\dashv\beta(z)$
$\displaystyle\quad-\alpha(x)\dashv(R(y)\vdash R(z))-\alpha R(x)\dashv(y\vdash
R(z))\alpha R(x)\dashv(R(y)\vdash z).$

The left hand side vanishes by axiom (2.6). This ends the proof. ∎

######  Corollary 2.19.

Let $(D,\dashv,\vdash,\alpha,\beta)$ BiHom-associative dialgebra and
$R:D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$. Then,
$(D,\ast,\alpha,\beta)$ is a BiHom-associative algebra with $x\ast
y=x\triangleleft y+x\triangleright y$.

######  Corollary 2.20.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$R:D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$. Then,
$(D,[-,-],\alpha,\beta)$ is a BiHom-Lie algebra with

$[x,y]=x\ast y-\alpha^{-1}\beta(y)\ast\alpha\beta^{-1}(x),$

with $x\ast y=x\triangleleft y+x\triangleright y$.

As in the previous proposition, it is well known that a Nijenhuis operator on
an associative algebra allows to define another associative algebra. In the
next result, we try to establish an analoq of this result for BiHom-
associative dialgebras.

######  Proposition 2.21.

Let $(D,\dashv,\vdash,\alpha,\beta)$ BiHom-associative dialgebra and
$N:D\rightarrow D$ a Nijenhuis operator on $D$ i.e. $N$ is linear and
$\alpha\circ N=N\circ\alpha$ , $\beta\circ N=N\circ\beta$, and

$\displaystyle N(x)\dashv N(y)$ $\displaystyle=$ $\displaystyle N(N(x)\dashv
y+x\dashv N(y)-N(x\cdot y))$ (2.20) $\displaystyle N(x)\vdash N(y)$
$\displaystyle=$ $\displaystyle N(N(x)\vdash y+x\vdash N(y)-N(x\cdot y))$
(2.21)

Then, $(D,\triangleleft,\triangleright,\alpha,\beta)$ is also a BiHom-
associative algebra with

$\displaystyle x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),$
(2.22) $\displaystyle x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash
y),$ (2.23)

for all $x,y\in D$.

###### Proof.

We only prove axiom (2.4) for the products $\triangleleft$ and
$\triangleright$. The others are leave to the reader.

$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=N\Big{(}N(x)\vdash y+x\dashv y-N(x\dashv
y)\Big{)}\dashv\beta(z)+\Big{(}N(x)\dashv y+x\vdash N(y)-N(x\vdash
y)\Big{)}\dashv N\beta(z)$ $\displaystyle\quad-N\Big{(}(N(x)\vdash y+x\vdash
N(y)-N(x\vdash y))\dashv\beta(z)\Big{)}-N\alpha(x)\vdash\Big{(}N(y)\dashv
z+y\dashv N(z)-N(y\dashv z)\Big{)}$ $\displaystyle-\alpha(x)\vdash
N\Big{(}N(y)\dashv z+y\dashv N(z)-N(y\dashv
z)\Big{)}+N\Big{(}\alpha(x)\vdash(N(y)\dashv z+y\dashv N(z)-N(y\dashv
z))\Big{)}.$

By (2.20) and (2.21), we have

$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=(N(x)\vdash N(y))\dashv\beta(z)+(N(x)\dashv y)\dashv\beta
N(z)+(x\vdash N(y))\dashv\beta N(z)-N(x\vdash y))\dashv N\beta(z)$
$\displaystyle\quad-N\Big{(}(N(x)\vdash
y)\dashv\beta(z)\Big{)}-N\Big{(}(x\vdash
N(y))\dashv\beta(z)\Big{)}-N\Big{(}N(x\vdash y)\dashv\beta(z)\Big{)}$
$\displaystyle-\alpha N(x)\vdash(N(y)\dashv z)-\alpha N(x)\vdash(y\dashv
N(z))+N\alpha(x)\vdash N(y\dashv z))$
$\displaystyle-\alpha(x)\vdash(N(y)\dashv
N(z))+N\Big{(}\alpha(x)\vdash(N(y)\dashv
z\Big{)}+N\Big{(}\alpha(x)\vdash(y\dashv N(z))\Big{)}-N\Big{(}\alpha(x)\vdash
N(y\dashv z))\Big{)}.$

By (2.4), we have

$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=-N(x\vdash y))\dashv N\beta(z)-N\Big{(}(N(x)\vdash
y)\dashv\beta(z)\Big{)}-N\Big{(}(x\vdash
N(y))\dashv\beta(z)\Big{)}-N\Big{(}N(x\vdash y)\dashv\beta(z)\Big{)}$
$\displaystyle\quad+N\alpha(x)\vdash N(y\dashv
z))+N\Big{(}\alpha(x)\vdash(N(y)\dashv
z\Big{)}+N\Big{(}\alpha(x)\vdash(y\dashv N(z))\Big{)}-N\Big{(}\alpha(x)\vdash
N(y\dashv z))\Big{)}.$

Using again (2.20) and (2.21), it comes

$\displaystyle\qquad(x\triangleright
y)\triangleleft\beta(z)-\alpha(x)\triangleright(y\triangleleft z)=$
$\displaystyle=-N\Big{(}N(x\vdash y)\dashv\beta(z)+(x\vdash y)\dashv\beta
N(z)-N((x\vdash y)\dashv\beta(z))\Big{)}$ $\displaystyle-N\Big{(}(N(x)\vdash
y)\dashv\beta(z)\Big{)}-N\Big{(}(x\vdash
N(y))\dashv\beta(z)\Big{)}-N\Big{(}N(x\vdash y)\dashv\beta(z)\Big{)}$
$\displaystyle\quad+N\Big{(}\alpha(x)\vdash(N(y)\dashv z)+\alpha(x)\vdash
N(y\dashv z)-N(\alpha(x)\vdash(y\dashv z))\Big{)}$
$\displaystyle+N\Big{(}\alpha(x)\vdash(N(y)\dashv
z\Big{)}+N\Big{(}\alpha(x)\vdash(y\dashv N(z))\Big{)}-N\Big{(}\alpha(x)\vdash
N(y\dashv z))\Big{)}.$

The left hand side vanishes by (2.4). ∎

######  Corollary 2.22.

If $(D,\dashv,\vdash,\alpha)$ is a Hom-associative dialgebra and
$N:D\rightarrow D$ a Nijenhuis operator on $D$, then
$(D,\triangleleft,\triangleright,\alpha)$ is also a Hom-associative algebra
with

$\displaystyle x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),$
$\displaystyle x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash y),$

for all $x,y\in D$.

######  Corollary 2.23.

If $(D,\dashv,\vdash,\alpha,\beta)$ is an associative dialgebra and
$N:D\rightarrow D$ a Nijenhuis operator on $D$, then
$(D,\triangleleft,\triangleright,\alpha,\beta)$ is also an associative algebra
with

$\displaystyle x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),$
$\displaystyle x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash y),$

for all $x,y\in D$.

The next proposition asserts that the twist of the products of any BiHom-
associative dialgebra by an averaging operator gives rise to another BiHom-
associative dialgebra.

######  Proposition 2.24.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$\theta:D\rightarrow D$ an injective averaging operator on $D$ i.e. $\theta$
is an injective linear map such that $\alpha\circ\theta=\theta\circ\alpha$ ,
$\beta\circ\theta=\theta\circ\beta$, and

$\displaystyle\theta(x)\dashv\theta(y)$ $\displaystyle=$
$\displaystyle\theta(\alpha^{k}\beta^{l}(x)\dashv\theta(y))=\theta(\theta(x)\dashv\alpha^{k}\beta^{l}(y)),$
(2.24) $\displaystyle\theta(x)\vdash\theta(y)$ $\displaystyle=$
$\displaystyle\theta(\alpha^{k}\beta^{l}(x)\vdash\theta(y))=\theta(\theta(x)\vdash\alpha^{k}\beta^{l}(y)),$
(2.25)

for any $x,y\in D$. Then, $(D,\triangleleft,\triangleright,\alpha,\beta)$ is
also a BiHom-associative algebra with

$\displaystyle x\triangleleft y=\theta(x)\dashv\alpha^{k}\beta^{l}(y))$ (2.26)
$\displaystyle x\triangleright y=\alpha^{k}\beta^{l}(x)\vdash\theta(y),$
(2.27)

for all $x,y\in D$.

###### Proof.

We only prove one identity, the others have a similar proof. For any $x,y,z\in
D$, one has :

$\displaystyle\qquad\theta[(x\triangleleft
y)\triangleright\beta(z)-\alpha(x)\triangleright(y\triangleleft z)]=$
$\displaystyle=\theta[\theta(\theta(x)\dashv\alpha^{k}\beta^{l}(y)))\vdash\alpha^{k}\beta^{l+1}(z))-\theta\alpha(x)\vdash(\theta(y)\dashv\alpha^{k}\beta^{l}(z)))]$
$\displaystyle=\theta[(\theta(x)\dashv\theta(y)\vdash\alpha^{k}\beta^{l+1}(z)]-\theta\alpha(x)\vdash\theta(\theta(y)\dashv\alpha^{k}\beta^{l}(z)))]$
$\displaystyle=(\theta(x)\vdash\theta(y))\dashv\beta\theta(z)-\alpha\theta(x)\vdash(\theta(y)\dashv\theta(z)).$

Which vanishes by axiom (2.4), and the conclusion holds by injectivity. ∎

At this moment, we introduce ideals for BiHom-associative dialgebra in order
to give another construction of BiHom-associative dialgebras.

######  Definition 2.25.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$D_{o}$ a subset of $D$. We say that $D_{o}$ is a BiHom-subalgebra of $D$ if
$D_{o}$ is stable under $\alpha$ and $\beta$, and $x\dashv y,x\vdash y\in
D_{o}$, for any $x,y\in D_{o}$.

######  Example 2.26.

If $\varphi:D_{1}\rightarrow D_{2}$ is a homomorphism of BiHom-associative
dialgebras, the image $Im\varphi$ is a BiHom-subalgebra of $D_{2}$.

######  Definition 2.27.

A two side BiHom-ideal of a BiHom-associative dialgebra
$(D,\dashv,\vdash,\alpha,\beta)$ is subspace $I$ such that $\alpha(I)\subset
I,x\ast y,y\ast x\in I$ for all $x\in D,y\in I$ with $\ast=\dashv$ and
$\vdash$. Note that $I$ is called the left and right BiHom-ideal if $x\dashv
y,x\vdash y$ and $y\dashv x,y\vdash x$ are in $I$, respectively, for all $x\in
D.y\in I$.

######  Example 2.28.

i) Obviously $I=\\{0\\}$ and $I=D$ are two-sided ideals.
ii) If $\varphi:D_{1}\rightarrow D_{2}$ is a homomorphism of BiHom-associative
dialgebras, the kernel $Ker\varphi$ is a two sided ideal in $D_{1}$.
iii) If $I_{1}$ and $I_{2}$ are two sided ideals of $D$, then so is
$I_{1}+I_{2}$.

In the below proposition, we prove that BiHom-associative dialgebras are
closed under direct summation, and give a condition for which a linear map
becomes a morphism.

######  Proposition 2.29.

Let $({A},\dashv_{A},\vdash_{A},\alpha_{A},\beta_{A})$ and
$({B},\dashv_{B},\vdash_{B},\alpha_{B},\beta_{B})$ be two BiHom-associative
dialgebras. Then there exists a BiHom-associative dialgebra structure on
${A}\oplus{B}$ with the bilinear maps
$\triangleleft,\triangleright:({A}\oplus{B})^{\otimes
2}\rightarrow{A}\oplus{B}$ given by

$(a_{1}+b_{2})\dashv(a_{2}+b_{2})=a_{1}\dashv_{A}a_{2}+b_{2}\dashv_{B}b_{2},$
$(a_{1}+b_{1})\vdash(a_{2}+b_{2})=a_{1}\vdash_{A}a_{2}+b_{A}\vdash_{B}b_{2}$

and the linear maps
$\alpha=\alpha_{A}+\alpha_{B},\,\beta=\beta_{A}+\beta_{B}:{A}\oplus{B}\rightarrow{A}\oplus{B}$
given by

$(\alpha_{A}+\alpha_{B})(a+b)=\alpha_{A}(a)+\alpha_{B}(b),\,(\beta_{A}+\beta_{B})(a+b)=\beta_{A}(a)+\beta_{B}(b),\,\forall(a,b)\in({A}\times{B}).$

Moreover, if $\xi:{A}\rightarrow{B}$ is a linear map. Then
$\xi:({A},\dashv_{A},\vdash_{A},\alpha_{A},\beta_{A})$ to
$({B},\dashv_{B},\vdash_{B},\alpha_{B},\beta_{B})$ is a morphism if and only
if its graph $\Gamma_{\xi}=\\{(x,\xi(x)),x\in A\\}$ is a BiHom-subalgebra of
$({A}\oplus{B},\triangleleft,\triangleright,\alpha,\beta)$.

###### Proof.

The proof of the first part of the proposition comes from a simple
computation.
Let us suppose that
$\xi:({A},\dashv_{A},\vdash_{A},\alpha_{A},\beta_{A})\rightarrow({B},\dashv_{B},\vdash_{B},\alpha_{B},\beta_{B})$
is a morphism of BiHom-associative dialgebras. Then

$(u+\xi(u))\dashv(v+\xi(v))=(u\dashv_{A}v+\xi(u)\dashv_{B}\xi(v))=(u\dashv_{A}v+\xi(u\dashv_{A}v)$
$(u+\xi(u))\vdash(v+\xi(v))=(u\vdash_{A}v+\xi(u)\vdash_{B}\xi(v))=(u\vdash_{A}v+\xi(u\vdash_{A}v).$

Thus the graph $\Gamma_{\xi}$ is closed under the operations $\dashv$ and
$\vdash$.
Furthermore since $\xi\circ\alpha_{A}=\alpha_{B}\circ\xi,$ and
$\xi\circ\beta_{A}=\beta_{B}\circ\xi,$ we have

$(\alpha_{A}\oplus\alpha_{B})(u,\xi(u))=(\alpha_{A}(u),\alpha_{B}\circ\xi(u))=(\alpha_{A}(u),\xi\circ\alpha_{A}(u)).$

and

$(\beta_{A}\oplus\beta_{B})(u,\xi(u))=(\beta_{A}(u),\beta_{B}\circ\xi(u))=(\beta_{A}(u),\xi\circ\beta_{A}(u)),$

implies that $\Gamma_{\xi}$ is closed $\alpha_{A}\oplus\alpha_{B}$ and
$\beta_{A}\oplus\beta_{B}.$ Thus, $\Gamma_{\xi}$ is a BiHom-subalgebra of
$({A}\otimes{B},\dashv,\vdash,\alpha,\beta).$
Conversely, if the graph $\Gamma_{\xi}\subset{A}\oplus{B}$ is a BiHom-
subalgebra of $({A}\oplus{B},\dashv,\vdash,\alpha,\beta)$ then we

$(u+\xi(u))\dashv(v+\xi(v))=(u\dashv_{A}v+\xi(u)\dashv_{B}\xi(v))\in\Gamma_{\xi}$
$(u+\xi(u))\vdash(v+\xi(v))=(u\vdash_{A}v+\xi(u)\vdash_{B}\xi(v))\in\Gamma_{\xi}.$

Furthermore,
$(\alpha_{A}\oplus\alpha_{B})(\Gamma_{\xi})\subset\Gamma_{\xi},\,(\beta_{A}\oplus\beta_{B})(\Gamma_{\xi})\subset\Gamma_{\xi},$
implies

$(\alpha_{A}\oplus\alpha_{B})(u,\xi(u))=(\alpha_{A}(u),\alpha_{B}\circ\xi(u))\in\Gamma_{\xi},\,(\beta_{A}\oplus\beta_{B})(u,\xi(u))=(\beta_{A}(u),\beta_{B}\circ\xi(u))\in\Gamma_{\xi},$

which is equivalent to the condition
$\alpha_{B}\circ\xi(u)=\xi\circ\alpha_{A}(u),$ i.e
$\alpha_{B}\circ\xi=\xi\circ\alpha_{A}.$ Similary,
$\beta_{B}\circ\xi=\xi\circ\beta_{A}$. Therefore, $\xi$ is a morphism BiHom-
associative dialgebras. ∎

######  Proposition 2.30.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and $I$
be a two sided BiHom-ideal of $(D,\dashv,\vdash,\alpha,\beta)$. Then,
$(D/I,\left[\cdot,\cdot\right],\overline{\dashv},\overline{\vdash},\overline{\alpha},\overline{\beta})$
is a BiHom-associative dialgebra where

$\overline{x}\;\overline{\dashv}\;\overline{y}:=\overline{x\dashv
y},\;\;\overline{x}\;\overline{\vdash}\;\overline{y}:=\overline{x\vdash
y},\;\;\overline{\alpha}(\overline{x}):=\overline{\alpha(x)},\;\;\overline{\beta}(\overline{x}):=\overline{\beta(x)},$

for all $\overline{x},\overline{y}\in A/I.$

###### Proof.

We only prove left associativity, the other being proved similarly. For all
$\overline{x},\overline{y},\overline{z}\in D/I$, we have

$\displaystyle(\overline{x}\overline{\vdash}\overline{y})\overline{\vdash}\overline{\beta}(\overline{z})-\overline{\alpha}(\overline{x})\overline{\vdash}(\overline{y}\overline{\vdash}\overline{z})=\overline{(x\vdash
y)\vdash\beta(z)-\alpha(x)\vdash(y\vdash z)}=0.$

Then,
$(D/I,\overline{\dashv},\overline{\vdash},\overline{\alpha},\overline{\beta})$
is BiHom-associative dialgebra. ∎

Now, let us recall the definition of BiHom-Lie algebra.

######  Definition 2.31.

[7] $A$ BiHom-Lie algebra $(L,\left[\cdot,\cdot\right],\alpha,\beta)$ is a
$4$-tuple in where L is linear space, $\alpha,\beta:A\rightarrow A$,are linear
maps and $\left[\cdot,\cdot\right]:L\otimes L\rightarrow L$ is a bilinear
maps, such that, for all $x,y,z\in L$ :

$\alpha\circ\beta=\beta\circ\alpha,$ (2.28)
$\alpha(\left[x,y\right])=\left[\alpha(x),\alpha(y)\right],\,\text{and},\,\beta(\left[x,y\right])=\left[\beta(x),\beta(y)\right],$
(2.29)
$\left[\beta(x),\alpha(y)\right])=-\left[\beta(y),\alpha(x)\right],\,(\text{BiHom-
skew-symetry}),$ (2.30)
$\left[\beta^{2}(x),\left[\beta(y),\alpha(z)\right]\right]+\left[\beta^{2}(y),\left[\beta(z),\alpha(x)\right]\right]+\left[\beta^{2}(z),\left[\beta(x),\alpha(y)\right]\right]=0,$
(2.31)

(BiHom-Jacobi identity).

The maps $\alpha$ and $\beta$ (in this order) are called the structure maps of
L.

######  Definition 2.32.

A morphism between two BiHom-Lie algebras
$f:(L,[-,-],\alpha,\beta)\rightarrow(L^{\prime},[-,-]^{\prime},\alpha^{\prime},\beta^{\prime})$
is a linear map $f:L\rightarrow L^{\prime}$ such that $\alpha^{\prime}\circ
f=f\circ\alpha,\,\beta^{\prime}\circ f=f\circ\beta$ and
$f(\left[x,y\right])=\left[f(x),f(y)\right]^{\prime}$, for all $x,y\in L.$

The following lemma asserts that the commutator of any BiHom-associative
algebra gives rise to BiHom-Lie.

######  Lemma 2.33.

[7] Let $(A,\cdot,\alpha,\beta)$ be a regular BiHom-associative algebra. Then

$L(A)=(A,[-,-],\alpha,\beta)$

is a regular BiHom-Lie algebra, where

$[x,y]=x\cdot y-\alpha^{-1}\beta(y)\cdot\alpha\beta^{-1}(x),$

for any $x,y\in A.$

######  Proposition 2.34.

Let $(L,[-,-],\alpha,\beta)$ be a BiHom-Lie algebra and $N:L\rightarrow L$ be
a Nijenhuis operator on $L$ i.e. $\alpha\circ N=N\circ\alpha$, $\beta\circ
N=N\circ\beta$ and

$\displaystyle[N(x),N(y)]=N([N(x),y]+[x,N(y)]-N([x,y]))$

for any $x,y\in L$. Then, $(L,[-,-]_{N},\alpha,\beta)$ is a BiHom-Lie algebra
with

$\displaystyle[x,y]_{N}=[N(x),y]+[x,N(y)]-N([x,y])$

for all $x,y\in L$.

###### Proof.

It follows from direct computation. ∎

######  Corollary 2.35.

Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and
$N:A\rightarrow A$ be a Nijenhuis operator on $A$ i.e. $\alpha\circ
N=N\circ\alpha$, $\beta\circ N=N\circ\beta$ and

$\displaystyle N(x)\cdot N(y)=N(N(x)\cdot y+x\cdot N(y)-N(x\cdot y))$

for any $x,y\in A$. Let us denote by $L(A)$ the BiHom-Lie algebra associated
with $A$ as in Proposition 2.33. Then, $(A,[-,-]_{N},\alpha,\beta)$ is a
BiHom-Lie algebra.

######  Corollary 2.36.

Let $(A,\cdot,\alpha,\beta)$ be a BiHom-associative algebra and
$N:A\rightarrow A$ be a Nijenhuis operator on $A$ i.e. $\alpha\circ
N=N\circ\alpha$, $\beta\circ N=N\circ\beta$ and

$\displaystyle N(x)\cdot N(y)=N(N(x)\cdot y+x\cdot N(y)-N(x\cdot y))$

for any $x,y\in A$. Then, $(A,\\{-,-\\},\alpha,\beta)$ is a BiHom-Lie algebra
with

$\displaystyle\\{x,y\\}=x\ast_{N}y-\alpha^{-1}\beta(y)\ast_{N}\alpha\beta^{-1}(x)$

and

$\displaystyle x\ast_{N}y=N(x)\cdot y+x\cdot N(y)-N(x\cdot y)$

for all $x,y\in A$.

###### Proof.

It is similar to the one of Proposition 2.21. And the Lemma 2.33 will end the
proof. ∎

######  Remark 2.37.

The BiHom-Lie algebra generated by Corollary 2.35 and Corollary 2.36 are
equal.

######  Proposition 2.38.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra.
Then,for all $x,y\in D$, the bracket

$[x,y]=[x,y]_{L}+[x,y]_{R},$

where

$\displaystyle[x,y]_{L}$ $\displaystyle=$ $\displaystyle x\dashv
y-\alpha^{-1}\beta(y)\dashv\alpha\beta^{-1}(x),$ $\displaystyle{[x,y]_{R}}$
$\displaystyle=$ $\displaystyle x\vdash
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x),$

is a BiHom-Lie bracket if and only if

$\displaystyle\alpha(x)\dashv(y\vdash z)=(x\dashv y)\vdash\beta(z),$ (2.32)
$\displaystyle\alpha(x)\dashv(y\dashv z)=(x\vdash y)\vdash\beta(z).$ (2.33)

###### Proof.

It is essentialy based on Lemma 2.33. That is, an expansion of BiHom-Jacobi
identity leads to $48$ terms including $8$ terms which cancel pairewise by
axiom (2.2), $4$ terms cancel pairewise by axiom (2.3), $12$ terms cancel
pairewise by axiom (2.4), $6$ terms cancel pairewise by axiom (2.5) and $6$
terms cancel pairewise by axiom (2.6).
For the of the $12$ terms, $8$ terms cancel pairewise by axiom (2.32) and $4$
terms cancel pairewise by axiom (2.33). ∎

######  Definition 2.39.

A (right ) BiHom-Leibniz algebra is a $4$-tuple
$(L,\left[\cdot,\cdot\right],\alpha,\beta)$, where L is a linear space,
$\left[\cdot,\cdot\right]:L\times L\rightarrow L$ is a bilinear map and
$\alpha,\beta:L\rightarrow L$ are linear maps satisfying

$\left[\left[x,y\right],\alpha\beta(z),\right]=\left[\left[x,\beta(z)\right],\alpha(y)\right]+\left[\alpha(x),\left[y,\alpha(z)\right]\right],$
(2.34)

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

######  Example 2.40.

Let $L$ be a two-dimensional vector space and $\left\\{e_{1},e_{2}\right\\}$
be a basis of $L$. Then, $(L,[-,-],\alpha,\beta)$ is a BiHom-Leibniz algebra
with

$\left[e_{1},e_{2}\right]=ae_{1},\left[e_{2},e_{2}\right]=be_{1},\;\alpha(e_{2})=\beta(e_{2})=e_{1},a,b\in\mathbb{R}.$

Now, we introduce BiHom-Poisson dialgebras and study its connection with
BiHom-associative dialgebras.

######  Definition 2.41.

A BiHom-Poisson dialgebra is a BiHom-associative dialgebra
$({P},\dashv,\vdash,\alpha,\beta)$ and a BiHom-Leibniz algebra
$({P},[-,-],\alpha,\beta)$ such that

$\displaystyle{[x\dashv y,\alpha\beta(z)]}$ $\displaystyle=$
$\displaystyle\alpha(x)\dashv[y,\alpha(z)]+[x,\beta(z)]\dashv\alpha(y),$
$\displaystyle{[x\vdash y,\alpha\beta(z)]}$ $\displaystyle=$
$\displaystyle\alpha(x)\vdash[y,\alpha(z)]+[x,\beta(z)]\vdash\alpha(y),$
$\displaystyle\\{\alpha\beta(x),y\dashv z\\}$ $\displaystyle=$
$\displaystyle\beta(y)\vdash[\alpha(x),z]+[\beta(x),y]\dashv\beta(z)=[\alpha\beta(x),y\vdash
z],$

are satisfied for $x,y,z\in{P}$.

######  Theorem 2.42.

Let $({D},\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra. Then,

$P(D)=(D,[-,-],\dashv,\vdash,\alpha,\beta)$

is a BiHom-Poisson dialgebra, where $[x,y]=x\dashv y-y\vdash x$, for any
$x,y\in{D}$.

###### Proof.

By Theorem 2.46 $P(D)$ is a BiHom-Leibniz algebra. Moreover, for any $x,y,z\in
D$,

$\displaystyle[x\dashv
y,\alpha\beta(z)]-\alpha(x)\dashv[y,\alpha(z)]-[x,\beta(z)]\dashv\alpha(y)=$
$\displaystyle=(x\dashv
y)\dashv\alpha\beta(z)-\alpha^{-1}\beta\alpha\beta(z)\vdash\alpha\beta^{-1}(x\dashv
y)-\alpha(x)\dashv(y\dashv\alpha(z)-\alpha^{-1}\beta\alpha(z)\vdash\alpha\beta^{-1}(y))$
$\displaystyle-(x\dashv\beta(z)-\alpha^{-1}\beta\beta(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y)$
$\displaystyle=(x\dashv
y)\dashv\alpha\beta(z)-\beta^{2}(z)\vdash(\alpha\beta^{-1}(x)\dashv\alpha\beta^{-1}(y))-\alpha(x)\dashv(y\dashv\alpha(z)$
$\displaystyle\quad+\alpha(x)\dashv(\beta(z)\vdash\alpha\beta^{-1}(y))-(x\dashv\beta(z))\dashv\alpha(y)+(\alpha^{-1}\beta^{2}(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y).$

The last three axioms are proved analagously. This completes the proof. ∎

######  Theorem 2.43.

Let $(P,\dashv,\vdash,[-,-],\alpha,\beta)$ be a BiHom-Poisson dialgebra and
$\alpha^{\prime},\beta^{\prime}:D\rightarrow D$ two morphisms of BiHom-Poisson
dialgebras such that the maps $\alpha,\alpha^{\prime},\beta,\beta^{\prime}$
commute pairewise. Then

$P_{(\alpha^{\prime},\beta^{\prime})}=(D,\,\triangleleft:=\dashv(\alpha^{\prime}\otimes\beta^{\prime}),\;\triangleright:=\vdash(\alpha^{\prime}\otimes\beta^{\prime}),\;\\{-,-\\}:=[-,-](\alpha^{\prime}\otimes\beta^{\prime}),\;\alpha\alpha^{\prime},\;\beta\beta^{\prime}),$

is a BiHom-Poisson dialgebra.

###### Proof.

It is essentialy based on the one of Theorem 2.6. ∎

Now, we introduce action of BiHom-Leibniz algebra on another one.

######  Definition 2.44.

Let $D$ and $L$ be two BiHom-Leibniz algebras. An action of $D$ on $L$
consists of a pair of bilinear maps, $D\times L\rightarrow
L,(x,a)\mapsto[x,a]$ and $L\times D\rightarrow[x,a]$, such that

$\displaystyle\left[\alpha(x),\left[a,\alpha(b)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[x,a\right],\alpha\beta(b),\right]-\left[\left[x,\beta(b)\right],\alpha(a)\right]$
(2.35) $\displaystyle\left[\alpha(a),\left[x,\alpha(b)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[a,x\right],\alpha\beta(b),\right]-\left[\left[a,\beta(b)\right],\alpha(x)\right]$
(2.36) $\displaystyle\left[\alpha(a),\left[b,\alpha(x)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[a,b\right],\alpha\beta(x),\right]-\left[\left[a,\beta(x)\right],\alpha(b)\right]$
(2.37) $\displaystyle\left[\alpha(a),\left[x,\alpha(y)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[a,x\right],\alpha\beta(y),\right]-\left[\left[a,\beta(y)\right],\alpha(x)\right]$
(2.38) $\displaystyle\left[\alpha(x),\left[a,\alpha(y)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[x,a\right],\alpha\beta(y),\right]-\left[\left[x,\beta(y)\right],\alpha(a)\right]$
(2.39) $\displaystyle\left[\alpha(x),\left[y,\alpha(a)\right]\right]$
$\displaystyle=$
$\displaystyle\left[\left[x,y\right],\alpha\beta(a),\right]-\left[\left[x,\beta(a)\right],\alpha(y)\right]$
(2.40)

for all $x,y\in D,a,b\in L$.

######  Lemma 2.45.

Given a BiHom-Leibniz action of $D$ on $L$, we can consider the semidirect
product Leibniz algebra $L\rJoin D$, which consists of vector space $D\oplus
L$ together with the Leibniz bracket given by

$\displaystyle[(x,a),(y,b)]=([x,y]+[x,b]+[a,y],[a,b])$ (2.41)

for all $(x,a),(x,b)\in D\times L$.

###### Proof.

$\displaystyle[\alpha(x,a),[(y,b),\alpha(z,c)]]$ $\displaystyle=$
$\displaystyle[(\alpha(x),\alpha(a)),([y,\alpha(z)]+[y,\alpha(c)]+[b,\alpha(z)],[b,\alpha(c)])]$
$\displaystyle=$
$\displaystyle\Big{(}[\alpha(x),[y,\alpha(z)]]+[\alpha(x),[y,\alpha(c)]]+[\alpha(x),[b,\alpha(z)]]+[\alpha(x),[b,\alpha(c)]]$
$\displaystyle+[\alpha(a),[y,\alpha(z)]]+[\alpha(a),[y,\alpha(z)]]+[\alpha(a),[y,\alpha(c)]]+[\alpha(a),[b,\alpha(z)],$
$\displaystyle[\alpha(a),[b,\alpha(c)]]\Big{)}.$
$\displaystyle{[[(x,a),(y,b)],\alpha\beta(z,c)]}$ $\displaystyle=$
$\displaystyle[([x,y]+[x,b]+[a,y]),[a,b]),(\alpha\beta(z),\alpha\beta(c))]$
$\displaystyle=$
$\displaystyle([[x,y],\alpha\beta(z)]+[[x,b],\alpha\beta(z)]+[[a,y],\alpha\beta(z)]+[[x,y],\alpha\beta(c)]$
$\displaystyle+[[x,b],\alpha\beta(c)]+[[a,y],\alpha\beta(c)]+[[a,b],\alpha\beta(c)],[[a,b],\alpha\beta(c)].$
$\displaystyle{[[(x,a),\beta(z,c)],\alpha(y,b)]}$ $\displaystyle=$
$\displaystyle[([x,\beta(z)]+[x,\beta(c)]+[a,\beta(z)],[a,\beta(c)]),(\alpha(y),\alpha(b))]$
$\displaystyle=$
$\displaystyle([[x,\beta(z)],\alpha(y)]+[[x,\beta(c)],\alpha(y)]+[[a,\beta(z)],\alpha(y)]+[[x,\beta(z)],\alpha(b)]$
$\displaystyle+[[x,\beta(c)],\alpha(b)]+[[a,\beta(z)],\alpha(b)]+[[a,\beta(c)],\alpha(y)],[[a,\beta(c)],\alpha(b)]).$

Using axioms in Definition 2.35, it follows that

${[[(x,a),(y,b)],\alpha\beta(z,c)]}={[[(x,a),\beta(z,c)],\alpha(y,b)]}+[\alpha(x,a),[(y,b),\alpha(z,c)]].$

Which proves the proposition. ∎

######  Theorem 2.46.

Let $({D},\dashv,\vdash,\alpha,\beta)$ be a regular BiHom-associative
dialgebra. Then the bracket defined by $\left[x,y\right]=x\dashv
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x)$, defines a structure of BiHom-
Leibniz algebra on ${D}$, and denoted ${\bf Lb}(D)$.

###### Proof.

For any $x,y,z\in{D}$, we have

$\displaystyle[[x,y],\alpha\beta(z)]$ $\displaystyle=$ $\displaystyle(x\dashv
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(z)$
$\displaystyle-\alpha^{-1}\beta\alpha\beta(z)\vdash(x\dashv
y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1})$ $\displaystyle=$
$\displaystyle(x\dashv
y)\dashv\alpha\beta(z)-(\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(z)$
$\displaystyle-\beta^{2}(z)\vdash(\alpha\beta^{-1}(x)\dashv\alpha\beta^{-1}(y)+\beta^{2}(z)\vdash(y\vdash\alpha^{2}\beta^{-2}(x)).$
$\displaystyle{[[x,\beta(z)],\alpha(y)]}$ $\displaystyle=$
$\displaystyle(x\dashv\beta(z)-\alpha^{-1}\beta^{2}(z)\vdash\alpha\beta^{-1}(x)\dashv\alpha(y)$
$\displaystyle-\alpha^{-1}\beta\alpha(z)\vdash(\alpha\beta^{-1}(x\dashv
y+\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1})$ $\displaystyle=$
$\displaystyle(x\dashv\beta(z))\dashv\alpha(y)-(\alpha^{-1}\beta^{2}(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y)$
$\displaystyle-\beta(y)\vdash(\alpha\beta^{-1}(x)\dashv\alpha(z))+\beta(y)\vdash(\beta(z)\vdash\alpha^{2}\beta^{-2}(x)).$
$\displaystyle{[\alpha(x),[y,\alpha(z)]]}$ $\displaystyle=$
$\displaystyle\alpha(x)\dashv(y\dashv\alpha(z)-\alpha^{-1}\beta\alpha(z)\vdash\alpha\beta^{-1}(y))-\alpha^{-1}\beta(y\dashv\alpha(z)$
$\displaystyle-\beta(z)\vdash\alpha\beta^{-1}(y))\vdash\alpha\beta^{-1}\alpha(x)$
$\displaystyle=$
$\displaystyle\alpha(x)\dashv(y\dashv\alpha(z))-\alpha(x)\dashv(\beta(z)\vdash\alpha\beta^{-1}(y))$
$\displaystyle-(\alpha^{-1}\beta(y)\dashv\beta(z))\vdash\alpha^{2}\beta^{-1}(x)-(\alpha^{-1}\beta^{2}(z)\vdash
y)\vdash\alpha^{2}\beta^{-1}(x).$

By axioms in Definition 2.1, the conclusion holds. ∎

In the relations contained in the below definition, we omitted the subsript
for simplifying the typography.

######  Definition 2.47.

Let $D$ and $L$ be dialgebras. An action of $D$ on $L$ consists of four linear
maps, two of them denoted by the symbol $\dashv$ and other two by $\vdash$,

$\displaystyle\dashv:D\otimes L\rightarrow L,$ $\displaystyle\dashv:L\otimes
D\rightarrow L,$ $\displaystyle\dashv:D\otimes L\rightarrow L,$
$\displaystyle\dashv:L\otimes D\rightarrow L$

such that the following $30$ equalities hold :

${(01)}\quad(x\dashv a)\dashv\beta(b)=\alpha(x)\dashv(a\dashv b)$, |  $\qquad\qquad{(16)}\quad(a\dashv x)\dashv\beta(y)=\alpha(a)\dashv(x\dashv y)$,
---|---
${(02)}\quad(x\dashv a)\dashv\beta(b)=\alpha(x)\dashv(a\vdash b)$, |  $\qquad\qquad{(17)}\quad(a\dashv x)\dashv\beta(y)=\alpha(a)\dashv(x\vdash y)$,
${(03)}\quad(x\vdash a)\dashv\beta(b)=\alpha(x)\vdash(a\dashv b)$, |  $\qquad\qquad{(18)}\quad(a\vdash x)\dashv\beta(y)=\alpha(a)\vdash(x\dashv y)$,
${(04)}\quad(x\dashv a)\vdash\beta(b)=\alpha(x)\vdash(a\vdash b)$, |  $\qquad\qquad{(19)}\quad(a\dashv x)\vdash\beta(y)=\alpha(a)\vdash(x\vdash y)$,
${(05)}\quad(x\vdash a)\vdash\beta(b)=\alpha(x)\vdash(a\vdash b)$, |  $\qquad\qquad{(20)}\quad(a\vdash x)\vdash\beta(y)=\alpha(a)\vdash(x\vdash y)$,
${(06)}\quad(a\dashv x)\dashv\beta(b)=\alpha(a)\dashv(x\dashv b)$, |  $\qquad\qquad{(21)}\quad(x\dashv a)\dashv\beta(y)=\alpha(x)\dashv(a\dashv y)$,
${(07)}\quad(a\dashv x)\dashv\beta(b)=\alpha(a)\dashv(x\vdash b)$, |  $\qquad\qquad{(22)}\quad(x\dashv a)\dashv\beta(y)=\alpha(x)\dashv(a\vdash y)$,
${(08)}\quad(a\vdash x)\dashv\beta(b)=\alpha(a)\vdash(x\dashv b)$, |  $\qquad\qquad{(23)}\quad(x\vdash a)\dashv\beta(y)=\alpha(x)\vdash(a\dashv y)$,
${(09)}\quad(a\dashv x)\vdash\beta(b)=\alpha(a)\vdash(x\vdash b)$, |  $\qquad\qquad{(24)}\quad(x\dashv a)\vdash\beta(y)=\alpha(x)\vdash(a\vdash y)$,
${(10)}\quad(a\vdash x)\vdash\beta(b)=\alpha(a)\vdash(x\vdash b)$, |  $\qquad\qquad{(25)}\quad(x\vdash a)\vdash\beta(y)=\alpha(x)\vdash(a\vdash y)$,

${(11)}\quad(a\dashv b)\dashv\beta(x)=\alpha(a)\dashv(b\dashv x)$, |  $\qquad\qquad{(26)}\quad(x\dashv y)\dashv\beta(a)=\alpha(x)\dashv(y\dashv a)$,
---|---
${(12)}\quad(a\dashv b)\dashv\beta(x)=\alpha(a)\dashv(b\vdash x)$, |  $\qquad\qquad{(27)}\quad(x\dashv y)\dashv\beta(a)=\alpha(x)\dashv(y\vdash a)$,
${(13)}\quad(a\vdash b)\dashv\beta(x)=\alpha(a)\vdash(b\dashv x)$, |  $\qquad\qquad{(28)}\quad(x\vdash y)\dashv\beta(a)=\alpha(x)\vdash(y\dashv a)$,
${(14)}\quad(a\dashv b)\vdash\beta(x)=\alpha(a)\vdash(b\vdash x)$, |  $\qquad\qquad{(29)}\quad(x\dashv y)\vdash\beta(a)=\alpha(x)\vdash(y\vdash a)$,
${(15)}\quad(a\vdash b)\vdash\beta(x)=\alpha(a)\vdash(b\vdash x)$, |  $\qquad\qquad{(30)}\quad(x\vdash y)\vdash\beta(a)=\alpha(x)\vdash(y\vdash a)$,

for all $x,y\in D,a,b\in L$. The action is called trivial if these four maps
are trivial.

######  Example 2.48.

i) Any BiHom-associative dialgebra may be seen as acting on itself ii)Given a
homomorphism $\varphi:D\rightarrow L$ of BiHom-associative dialgebras, then
there is an action of $D$ on $L$ via the maps $x\triangleleft
a:=\varphi(x)\dashv a,\;x\triangleright
a:=\varphi(x)\triangleright,a\;a\triangleleft
x:=a\vdash\varphi(x)\;\;\mbox{and}\;\;a\triangleright x:=a\vdash\varphi(x)$.
iii)If $\psi:L\rightarrow D$ is an isomorphism of BiHom-associative
dialgebras, then there is an action of $D$ on $L$ via the maps $x\triangleleft
a:=\psi^{-1}(x)\dashv a,\;x\triangleright a:=\psi^{-1}(x)\triangleright
a,\;a\triangleleft x:=a\vdash\psi^{-1}(x)\;\;\mbox{and}\;\;a\triangleright
x:=a\vdash\psi^{-1}(x)$.
iv) If $I$ is an ideal of $D$, then the left and the right product yield an
action of $D$ on I.

######  Lemma 2.49.

Given two regular BiHom-associative dialgebras $D$ and $L$ together with an
action of $D$ on $L$, there is an action an action ${\bf Lb}(D)$ on ${\bf
Lb}(L)$ given by

$\displaystyle[x,a]$ $\displaystyle=$ $\displaystyle x\dashv
a-\alpha^{-1}\beta(a)\dashv\alpha\beta^{-1}(x),$ $\displaystyle{[a,x]}$
$\displaystyle=$ $\displaystyle a\vdash
x-\alpha^{-1}\beta(x)\vdash\alpha\beta^{-1}(a),$

for all $x\in{\bf Lb}(D)$, $a\in{\bf Lb}(L)$.

###### Proof.

For all $x\in{\bf Lb}(D)$, $a\in{\bf Lb}(L)$,

$\displaystyle[[x,a],\alpha\beta(b)]$ $\displaystyle=$ $\displaystyle(x\dashv
a-\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(b)$
$\displaystyle-\alpha^{-1}\beta\alpha\beta(b)\vdash\alpha\beta^{-1}(x\dashv
a-\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))$ $\displaystyle=$
$\displaystyle(x\dashv
a)\dashv\beta\alpha(b)-(\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))\dashv\beta\alpha(b)$
$\displaystyle-\beta^{2}(b)\vdash(\alpha\beta^{-1}(x)\dashv\beta^{-1}\alpha(a))+\beta^{2}(b)\vdash(a\vdash\alpha^{2}\beta^{-2}(x)).$

On the other hand,

$\displaystyle\qquad[[x,\beta(b)],\alpha(a)]+[\alpha(x),[a,\alpha(b)]]=$
$\displaystyle=\Big{(}x\dashv\beta(b)-\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x)\Big{)}\dashv\alpha(a)-\alpha^{-1}\beta\alpha(a)\vdash\alpha\beta^{-1}\Big{(}x\dashv\beta(b)-\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x)\Big{)}$
$\displaystyle\quad+\alpha(x)\dashv\Big{(}a\dashv\alpha(b)-\alpha^{-1}\beta\alpha(b)\vdash\alpha\beta^{-1}(a)\Big{)}-\alpha^{-1}\beta\Big{(}a\dashv\alpha(b)-\alpha^{-1}\beta\alpha(b)\vdash\alpha\beta^{-1}(a)\Big{)}\vdash\alpha\beta^{-1}\alpha(x)$
$\displaystyle=(x\dashv\beta(b))\dashv\alpha(a)-(\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x))\dashv\alpha(a)-\beta(a)\vdash(\alpha\beta^{-1}(x)\dashv\alpha(b))$
$\displaystyle\quad+\beta(a)\vdash(\beta(b)\vdash\alpha^{2}\beta^{-2}(x))+\alpha(x)\dashv(a\dashv\alpha(b))-\alpha(x)\dashv(\beta(b)\vdash\alpha\beta^{-1}(a))$
$\displaystyle\quad-(\alpha^{-1}\beta(a)\dashv\beta(b))\vdash\beta^{-1}\alpha^{2}(x)+(\alpha^{-1}\beta^{2}(b)\vdash
a)\vdash\beta^{-1}\alpha^{2}(x).$

Using axioms (2.3), (2.5), it comes

$\displaystyle\qquad[[x,\beta(b)],\alpha(a)]+[\alpha(x),[a,\alpha(b)]]=$
$\displaystyle=-(\alpha^{-1}\beta^{2}(b)\vdash\alpha\beta^{-1}(x))\dashv\alpha(a)-\beta(a)\vdash(\alpha\beta^{-1}(x)\dashv\alpha(b))$
$\displaystyle\quad+\alpha(x)\dashv(a\dashv\alpha(b))+(\alpha^{-1}\beta^{2}(b)\vdash
a)\vdash\alpha^{2}\beta^{-1}(x).$

By comparing, we get the attended result. The five other axioms are proved in
the same way. ∎

######  Lemma 2.50.

Let $D$ and $L$ be two regular BiHom-associative dialgebras together with an
action of $D$ on $L$. There is a BiHom-associative dialgebra structure on
$L\rJoin D$ which consists with vector space $L\oplus D$ and

$\displaystyle(a,x)\triangleleft(b,y)$ $\displaystyle=$ $\displaystyle(a\dashv
b+a\dashv y+x\dashv b,x\dashv y),$ $\displaystyle(a,x)\triangleright(b,y)$
$\displaystyle=$ $\displaystyle(a\vdash b+a\vdash y+x\vdash b,x\vdash y),$

for any $(a,x),(b,y)\in L\times D$

###### Proof.

For any $a,b,c\in L,x,y,z\in D$, one has

$\displaystyle\qquad\Big{(}(a,x)\triangleleft(b,y)\Big{)}\triangleleft\beta(c,z)-\alpha(a,x)\triangleleft\Big{(}(b,y)\triangleright(c,z)\Big{)}=$
$\displaystyle=(a\dashv b+a\dashv y+x\dashv b,x\dashv
y)\triangleleft(\beta(c),\beta(z))-(\alpha(a),\alpha(x))\triangleleft(b\vdash
c+b\vdash z+y\vdash c,y\vdash z)$ $\displaystyle=\Big{(}(a\dashv b+a\dashv
y+x\dashv b)\dashv\beta(c)+(a\dashv b+a\dashv y+x\dashv
b)\dashv\beta(z)+(x\dashv y)\dashv\beta(c),\;\;(x\dashv
y)\dashv\beta(z)\Big{)}$ $\displaystyle\quad-\Big{(}\alpha(a)\dashv(b\vdash
c+b\vdash z+y\vdash c)+\alpha(a)\dashv(y\vdash z)+\alpha(x)\dashv(b\vdash
c+b\vdash z+y\vdash c),\;\;\alpha(x)\dashv(y\vdash z)\Big{)}$
$\displaystyle=\Big{(}(a\dashv b)\dashv\beta(c)+(a\dashv
y)\dashv\beta(c)+(x\dashv b)\dashv\beta(c)+(a\dashv b)\dashv\beta(z)+(a\dashv
y)\dashv\beta(z)+(x\dashv b)\dashv\beta(z)$ $\displaystyle\quad+(x\dashv
y)\dashv\beta(c)-\alpha(a)\dashv(b\vdash c)-\alpha(a)\dashv(b\vdash
z)-\alpha(a)\dashv(y\vdash c)-\alpha(a)\dashv(y\vdash
z)-\alpha(x)\dashv(b\vdash c)$ $\displaystyle\quad-\alpha(x)\dashv(b\vdash
z)-\alpha(x)\dashv(y\vdash c),\;\;(x\dashv
y)\dashv\beta(z)-\alpha(x)\dashv(y\vdash z)\Big{)}.$

The left hand side vanishes by axiom (2.3) and axioms
$(02),(07),(12),(17),(22),(27)$ in Definition 2.47. The other axioms are
proved in the same way. ∎

######  Theorem 2.51.

Let $D$ and $L$ be two regular BiHom-associative dialgebras together with an
action of $D$ on $L$. Then, ${\bf Lb}(L\rJoin D)={\bf Lb}(L)\rJoin{\bf
Lb}(D)$.

###### Proof.

By lemma 2.49, ${\bf Lb}(D)$ acts on ${\bf Lb}(L)$, so it makes sense to
consider the semidirect product Leibniz algebra ${\bf Lb}(L)\rJoin{\bf
Lb}(D)$. It is clear that ${\bf Lb}(L\rJoin D)$ and ${\bf Lb}(L)\rJoin{\bf
Lb}(D)$ are egal as vector space, so we only need to verify that they share
the same bracket. Let $(a,x),(b,y)\in L\times D$. If we use the bracket in
${\bf Lb}(L)\rJoin{\bf Lb}(D)$, we get :

$\displaystyle[(a,x),(b,y)]$ $\displaystyle=$
$\displaystyle([a,b]+[x,b]+[a,y],[x,y])$ $\displaystyle=$
$\displaystyle(a\dashv y-b\vdash+x\dashv b-b\vdash x+a\dashv y-y\vdash
a,x\dashv y-y\vdash x).$

On the other hand, if we use the Leibniz bracket in ${\bf Lb}(L\rJoin D)$
(Lemma 2.50), we get

$\displaystyle\\{(a,x),(b,y)\\}$ $\displaystyle=$
$\displaystyle(a,x)\triangleleft(b,y)-(b,y)\triangleright(a,x)$
$\displaystyle=$ $\displaystyle(a\dashv b+x\dashv b+a\dashv y,x\dashv
y)-(b\vdash a+y\vdash a+b\vdash x,y\vdash x),$

So the brackets are equal. ∎

##  3 Central extensions

This section concerns the central extension of BiHom-associative dialgebras in
relation with cocycles.

######  Definition 3.1.

Let $(D_{i},\dashv_{i},\vdash_{i},\alpha_{i},\beta_{i}),i=1,2,3$ be three
BiHom-associative dialgebras. The BiHom-associative dialgebra $D_{2}$ is
called the extension of $D_{3}$ by $D_{1}$ if there are homomorphisms
$\phi:D_{1}\rightarrow D_{2}$ and $\psi:D_{2}\rightarrow D_{3}$ such that the
following sequence

$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}D_{3}\rightarrow
0$

is exact.

######  Definition 3.2.

An extension is called trivial if there exists a BiHom-ideal $I$ of $D_{2}$
complementary to $Ker\psi$ i.e.

$D_{2}=Ker\psi\oplus I$

It may happen that there exist several extensions of $D_{3}$ by $D_{1}$. To
classify extensions the notion of equivalent extensions is defined.

######  Definition 3.3.

Two sequences

$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}D_{3}\rightarrow
0$

and

$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi^{\prime}}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi^{\prime}}}{{\rightarrow}}D_{3}\rightarrow
0$

are equivalent extensions if there exists a associative dialgebra isomorphism
$f:D_{2}\rightarrow D^{\prime}_{2}$ such that

$f\circ\phi=\phi^{\prime}\quad\mbox{and}\quad\psi^{\prime}\circ f=\psi.$

######  Definition 3.4.

An extension

$0\rightarrow
D_{1}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}D_{2}\stackrel{{\scriptstyle\psi}}{{\rightarrow}}D_{3}\rightarrow
0$

is called central if the kernel of $\psi$ is contained in the center
$Z(D_{2})$ of $D_{2}$, i.e. $Ker\psi\subset Z(D)$.

Now, we introduce $2$-cocycle on BiHom-associative dialgebra with values in a
BiHom-module.

######  Definition 3.5.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra and
$(M,\alpha_{M},\beta_{M})$ a BiHom-module over the same field that $D$. A pair
$\Theta=(\theta_{1},\theta_{2})$ of bilinear maps $\theta_{1}:D\times
D\rightarrow V$ and $\theta_{2}:D\times D\rightarrow V$ is called a
$2$-cocycle on $D$ with values in $V$ if $\theta_{1}$ and $\theta_{2}$ satisfy

$\displaystyle\theta_{1}(x\dashv y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{1}(\alpha(x),y\dashv z),$ (3.1)
$\displaystyle\theta_{1}(x\dashv y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{1}(\alpha(x),y\vdash z),$ (3.2)
$\displaystyle\theta_{2}(x\vdash y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{2}(\alpha(x),y\vdash z),$ (3.3)
$\displaystyle\theta_{2}(x\dashv y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{2}(\alpha(x),y\vdash z),$ (3.4)
$\displaystyle\theta_{1}(x\vdash y,\beta(z))$ $\displaystyle=$
$\displaystyle\theta_{2}(\alpha(x),y\dashv z),$ (3.5)

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

The set of all $2$-cocycles on $D$ with values in $M$ is denoted $Z^{2}(D,M)$,
which a vector space.
In the below lemma, we give a special type of $2$-cocycles which are called
$2$-coboundaries.

######  Lemma 3.6.

Let $\nu:D\rightarrow V$ be a linear map, and define
$\varphi_{1}(x,y)=\nu(x\dashv y)$ and $\varphi_{2}(x,y)=\nu(x\vdash y)$. Then,
$\Phi=(\varphi_{1},\varphi_{2})$ is a $2$-cocycle on $D$.

###### Proof.

We will prove one equality, the others being proved in the same way. For any
$x,y,z\in D$, one has

$\displaystyle\varphi_{1}(\alpha(x),y\dashv z)$ $\displaystyle=$
$\displaystyle\nu(\alpha(x)\dashv(y\dashv z))=\nu((x\dashv y)\dashv\beta(z))$
$\displaystyle=$ $\displaystyle\nu(\alpha(x)\dashv(y\vdash
z))=\varphi_{1}(\alpha(x),y\vdash z).$

This finishes the proof. ∎

The set of all $2$-coboundaries is denoted by $B^{2}(D,M)$ and it is a
subgroup of $Z^{2}(D,M)$. The group $H^{2}(D,M)=Z^{2}(D,M)/B^{2}(D,M)$ is said
to be a second cohomology group of $D$ with values in $M$. Two cocycles
$\Theta_{1}$ and $\Theta_{2}$ are said to be cohomologous cocycles if
$\Theta_{1}-\Theta_{2}$ is a coboundary.

######  Theorem 3.7.

Let $(D,\dashv,\vdash,\alpha_{D},\beta_{D})$ be a BiHom-associative dialgebra,
$(M,\alpha_{M},\beta_{M})$ a BiHom-module,

$\theta_{1}:D\times D\rightarrow M\quad\mbox{and}\quad\theta_{2}:D\times
D\rightarrow M$

be bilinear maps. Let us set $D_{\Theta}=D\oplus M$, where
$\Theta=(\theta_{1},\theta_{2})$. For any $x,y\in D$, $v,w\in M$, let us
define

$\displaystyle(x+u)\triangleleft(y+v)=x\dashv
y+\theta_{1}(x,y)\quad\mbox{and}\quad(x+u)\triangleright(y+v)=x\vdash
y+\theta_{2}(x,y).$

Then,
$(D_{\Theta},\triangleleft,\triangleright,\alpha_{A}\otimes\alpha_{M},\beta_{A}\otimes\beta_{M})$
is a BiHom-associative dialgebra if and only if $\Theta$ is a $2$-cocycle.

###### Proof.

For any $x,y,z\in D,u,v,w\in M$, we have

$\displaystyle((x+v)\triangleleft(y+w))\triangleleft(\beta(z)+w)-(\alpha(x)+v)\triangleleft((y+w)\triangleleft(z+w))=$
$\displaystyle=$
$\displaystyle((x+v)\triangleleft(y+w))\triangleleft(\beta(z)+w)-(\alpha(x)+v)\triangleleft((y\dashv
z)+\theta_{1}(y,z))$ $\displaystyle=$ $\displaystyle((x\dashv
y)\dashv\beta(z))+\theta_{1}(x\dashv y,\beta(z))-(\alpha(x)\dashv(y\dashv
z))-\theta_{1}(\alpha(x),y\dashv z).$

The left hand vanishes by axioms (2.2) and (3.1). The other axioms are proved
analagously. ∎

######  Lemma 3.8.

Let $\Theta$ be a $2$-cocycle and $\Phi$ a $2$-coboundary. Then,
$D_{\Theta+\Phi}$ is a BiHom-associative dialgebra with

$\displaystyle(x+u)\unlhd(y+v)=x\dashv y+\varphi_{1}(x,y)+\theta_{1}(x,y),$
$\displaystyle(x+u)\unrhd(y+v)=x\vdash y+\varphi_{2}(x,y)+\theta_{2}(x,y).$

Moreover, $D_{\Theta}\cong D_{\Theta+\Phi}$.

###### Proof.

First, we have to shown that $D_{\Theta+\Phi}$ is a BiHom-associative
dialgebra. So, for any $x+u,\\\ y+v,z+w\in D\oplus M$,

$\displaystyle\qquad((x+u)\unlhd(y+v))\unlhd\beta(z+w)-\alpha(x+u)\unlhd((y+v))\unlhd(z+w))=$
$\displaystyle=(x\dashv
y+\varphi_{1}(x,y)+\theta_{1}(x,y))\unlhd(\beta(z)+\beta(w))-(\alpha(x)+\alpha(u))\unlhd(y\dashv
z+\varphi_{1}(y,z)+\theta_{1}(y,z))$ $\displaystyle=(x\dashv
y)\dashv\beta(z)+\varphi_{1}(x\dashv y,\beta(z))+\theta_{1}(x\dashv
y,\beta(z))-\alpha(x)\dashv y\dashv z-\varphi_{1}(\alpha(x),y\dashv
z)+\theta_{1}(\alpha(x),y\dashv z)$

The left hand side vanishes by (2.2) and (3.1). The proofs of the rest of
axioms are leaved to the reader.
Next, the isomorphism $f:D_{\Theta}\rightarrow D_{\Theta+\Phi}$ is given by
$f(x+v)=x+\nu(x)+v$. In fact, it is clear that $f$ is a bijective linear map
and

$\displaystyle f(\alpha_{D}+\alpha_{M})(x+v)$ $\displaystyle=$ $\displaystyle
f(\alpha_{D}(x)+\alpha_{M}(v))$ $\displaystyle=$
$\displaystyle\alpha_{D}(x)+\nu\alpha_{D}(x)+\alpha_{M}(v)$ $\displaystyle=$
$\displaystyle\alpha_{D}(x)+\alpha_{M}\nu(x)+\alpha_{M}(v)$ $\displaystyle=$
$\displaystyle(\alpha_{D}+\alpha_{M})(x+\nu(x)+v)$ $\displaystyle=$
$\displaystyle(\alpha_{D}+\alpha_{M})\circ f(x+v).$

Thus, $f$ commutes $\alpha_{D}+\alpha_{M}$, and similarly with
$\beta_{D}+\beta_{M}$.
Then,

$\displaystyle f((x+v)\triangleleft(y+w))$ $\displaystyle=$ $\displaystyle
f(x\dashv y+\theta_{1}(x,y))$ $\displaystyle=$ $\displaystyle f(x\dashv
y)+f(\theta_{1}(x,y))$ $\displaystyle=$ $\displaystyle x\dashv y+\nu(x\dashv
y)+\theta_{1}(x,y)$ $\displaystyle=$ $\displaystyle x\dashv
y+\varphi_{1}(x,y)+\theta_{1}(x,y).$

and

$\displaystyle f(x+v)\unlhd f(y+w)$ $\displaystyle=$
$\displaystyle(x+\nu(x)+v)\unlhd(y+\nu(y)+w)$ $\displaystyle=$
$\displaystyle(x\dashv y)+\varphi_{1}(x,y)+\theta_{1}(x,y).$

∎

######  Corollary 3.9.

Let $\Theta_{1},\Theta_{2}$ be two cohomologous $2$-cocycles on a BiHom-
associative dialgebra $D$, and $D_{1},D_{2}$ be the central extensions
constructed with these $2$-cocycles, respectively. The the central extensions
$D_{1}$ and $D_{2}$ are equivalent extensions. In particular a central
extension defined by a coboundary is equivalent with a trivial central
extension.

The following theorem is proved Mutatis Mutandis as ([25], Theorem 4.1). So we
omitted the proof.

######  Theorem 3.10.

There exists one to one correspondence between elements of $H^{2}(D,M)$ and
nonequivalents central extensions of associative dialgebra $D$ by $M$.

##  4 Classification

In this section, we give classification of BiHom-associative dialgebras in low
dimension.

Let $(D,\dashv,\vdash,\alpha,\beta)$ be an $n$-dimensional BiHom-associative
dialgebra, $\\{e_{i}\\}$ be a basis of $D$. For any $i,j\in\mathbb{N},1\leq
i,j\leq n$, let us put

$e_{i}\dashv e_{j}=\sum_{k=1}^{n}\gamma_{ij}^{k}e_{k},\quad e_{i}\vdash
e_{j}\sum_{k=1}^{n}\delta_{ij}^{k}e_{k},\quad\alpha(e_{j})=\sum_{k=1}^{n}\alpha_{kj}e_{k},\quad\beta(e_{j})=\sum_{k=1}^{n}\beta_{kj}e_{k}.$

The axioms in Definition 2.1 are respectively equivalent to

$\displaystyle\beta_{kj}\alpha_{pk}-\alpha_{ji}\beta_{pj}$ $\displaystyle=$
$\displaystyle 0,$ (4.1)
$\displaystyle\gamma_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\gamma_{jk}^{q}\gamma_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.2)
$\displaystyle\gamma_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\delta_{jk}^{q}\gamma_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.3)
$\displaystyle\gamma_{ij}^{p}\beta_{qk}\delta_{pq}^{r}-\alpha_{pi}\delta_{jk}^{q}\delta_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.4)
$\displaystyle\delta_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\gamma_{jk}^{q}\delta_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0,$ (4.5)
$\displaystyle\delta_{ij}^{p}\beta_{qk}\gamma_{pq}^{r}-\alpha_{pi}\delta_{jk}^{q}\delta_{pq}^{r}$
$\displaystyle=$ $\displaystyle 0.$ (4.6)

###  4.1 One dimensional

There is only one $1$-dimensional BiHom-associative dialgebra ; the nul (or
trivial) BiHom-associative dialgebra.

###  4.2 Two dimensional

$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{1}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=ae_{1},\\\ e_{2}\dashv e_{1}=be_{1},\\\ e_{1}\vdash e_{2}=ce_{1},\\\ e_{2}\vdash e_{1}=de_{1},\\\ e_{2}\vdash e_{2}=fe_{1}.\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$
$\mathcal{A}lg_{2}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=ae_{1},\\\ e_{2}\dashv e_{1}=ae_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$
$\mathcal{A}lg_{3}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=ae_{1},\\\ e_{1}\vdash e_{2}=be_{1},\\\ e_{2}\vdash e_{1}=ce_{1},\\\ e_{2}\vdash e_{2}=de_{1}\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$
$\mathcal{A}lg_{4}$ | $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=ae_{1},\\\ e_{1}\vdash e_{2}=be_{1},\\\ e_{2}\vdash e_{1}=ce_{1},\\\ e_{2}\vdash e_{2}=de_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1},\\\ \beta(e_{2})=e_{1}\end{array}$

######  Remark 4.1.

In two dimensional, all of the BiHom-associative dialgebras are Hom-
associative dialgebras i.e. $\alpha=\beta$.

###  4.3 Three dimensional

$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{1}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=ae_{1},\\\ e_{2}\dashv e_{3}=be_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{2}=ce_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=de_{1},\\\ e_{3}\vdash e_{2}=fe_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$

$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{2}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ e_{3}\dashv e_{2}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
$\mathcal{A}lg_{3}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{,}\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ e_{3}\dashv e_{2}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{3}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\\\ \end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
$\mathcal{A}lg_{4}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{3}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$
$\mathcal{A}lg_{5}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{1}=e_{1},\\\ e_{2}\dashv e_{2}=e_{1},\\\ e_{2}\dashv e_{3}=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{1},\\\ e_{2}\vdash e_{1}=e_{1},\\\ e_{2}\vdash e_{3}=e_{1},\\\ e_{3}\vdash e_{2}=e_{1},\end{array}$ | $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \beta(e_{2})=e_{1},\\\ \beta(e_{3})=be_{3}\end{array}$

###  4.4 Four dimensional

$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{1}$ |  $\begin{array}[]{ll}e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{1}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ e_{4}\dashv e_{4}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=ce_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{4}=de_{3},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=be_{2}\\\ \beta(e_{2})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{2}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{1}=ae_{4},\\\ e_{2}\dashv e_{3}=be_{4},\\\ e_{3}\dashv e_{1}=-ce_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=de_{4},\\\ e_{3}\vdash e_{3}=fe_{4},\\\ e_{3}\vdash e_{4}=e_{4},\\\ e_{4}\vdash e_{4}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \beta(e_{2})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{3}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=be_{4},\\\ e_{3}\dashv e_{1}=ce_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{2}=ce_{4},\\\ e_{3}\vdash e_{3}=de_{4},\\\ e_{4}\vdash e_{4}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{3})=e_{3}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{4}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=ae_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{1}=e_{4},\\\ e_{3}\dashv e_{2}=ce_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{4}\vdash e_{4}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{3})=e_{3}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$

$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{5}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{4}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{4}=e_{4},\\\ e_{1}\vdash e_{3}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{6}$ |  $\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{4}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{3}=e_{4},\\\ e_{1}\vdash e_{4}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{1}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{4})=e_{4}\end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{7}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{4}=fe_{4},\\\ e_{3}\dashv e_{3}=-ge_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{4}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{1}=e_{4},\\\ e_{3}\vdash e_{2}=-he_{4},\\\ e_{3}\vdash e_{3}=ke_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{3})=e_{3},\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{8}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{3}=e_{4},\\\ e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{4}=e_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{1}=e_{4},\\\ e_{3}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{3})=e_{3}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1},\\\ \beta(e_{3})=e_{2},\\\ \beta(e_{4})=e_{3},\\\ \end{array}$
$\mathcal{A}lg_{9}$ |  $\begin{array}[]{ll}e_{2}\dashv e_{2}=e_{1}+e_{4},\\\ e_{2}\dashv e_{3}=e_{1}+e_{4},\\\ e_{3}\dashv e_{2}=e_{1}+e_{4},\\\ e_{4}\dashv e_{2}=e_{1}+e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=-e_{1}+e_{4},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{3}\vdash e_{3}=e_{1}+e_{4},\\\ e_{4}\vdash e_{2}=e_{1}+e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \alpha(e_{3})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{3},\\\ \beta(e_{4})=e_{4},\\\ \end{array}$
$\mathcal{A}lg_{10}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{2}=e_{1}+e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{2}=e_{1},\\\ e_{3}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{4}\dashv e_{2}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{1},\\\ e_{3}\vdash e_{3}=e_{1}+e_{4},\\\ e_{4}\vdash e_{2}=e_{1}+e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \alpha(e_{3})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{3},\\\ \beta(e_{4})=e_{4},\\\ \end{array}$
$\mathcal{A}lg_{11}$ |  $\begin{array}[]{ll}e_{2}\dashv e_{2}=fe_{1}+ge_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{2}=e_{1}+e_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ e_{4}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=he_{1}-ke_{4},\\\ e_{3}\vdash e_{3}=e_{1}+e_{4},\\\ e_{4}\vdash e_{2}=e_{1}+e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{1}\\\ \alpha(e_{3})=e_{2}\\\ \alpha(e_{4})=e_{4}\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{3})=e_{3},\\\ \beta(e_{4})=e_{4},\\\ \end{array}$
$\mathcal{A}lg_{12}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{4}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=ae_{4},\\\ e_{2}\dashv e_{4}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=-be_{4},\\\ e_{3}\vdash e_{2}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{3})=e_{3}\\\ \alpha(e_{4})=e_{4}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
$\mathcal{A}lg_{13}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{1}\dashv e_{3}=e_{4},\\\ e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{1}=e_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \alpha(e_{3})=e_{3}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
$\mathcal{A}lg_{14}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{1}=e_{4},\\\ e_{1}\dashv e_{3}=-ce_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=e_{4},\\\ e_{3}\dashv e_{1}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{3}\dashv e_{3}=-2ae_{4},\\\ e_{1}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{2}=e_{4},\\\ e_{3}\vdash e_{3}=be_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{1})=e_{1}\\\ \alpha(e_{2})=e_{2}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$

$Algebras$ | Multiplications | Morphisms $\alpha,\beta$.
---|---|---
$\mathcal{A}lg_{15}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{1}=-e_{4},\\\ e_{1}\dashv e_{2}=ae_{4},\\\ e_{2}\dashv e_{3}=be_{4},\\\ e_{3}\dashv e_{1}=ce_{4},\\\ e_{3}\dashv e_{2}=de_{4},\\\ e_{3}\dashv e_{3}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=fe_{4},\\\ e_{1}\vdash e_{4}=e_{4},\\\ e_{2}\vdash e_{2}=e_{4},\\\ e_{2}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{2}=ge_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{4}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{2})=e_{2}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$
$\mathcal{A}lg_{16}$ |  $\begin{array}[]{ll}e_{1}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{1}=e_{4},\\\ e_{2}\dashv e_{2}=e_{4},\\\ e_{2}\dashv e_{3}=ae_{4},\\\ e_{2}\dashv e_{4}=e_{4},\\\ e_{3}\dashv e_{2}=e_{4},\\\ \end{array}$ $\begin{array}[]{ll}e_{1}\vdash e_{2}=be_{4},\\\ e_{2}\vdash e_{2}=ce_{4},\\\ e_{3}\vdash e_{2}=de_{4},\\\ e_{3}\vdash e_{3}=e_{4},\\\ e_{3}\vdash e_{4}=e_{4},\\\ \end{array}$ |  $\begin{array}[]{ll}\alpha(e_{1})=ae_{1}\\\ \beta(e_{1})=e_{1},\\\ \end{array}$ $\begin{array}[]{ll}\beta(e_{2})=e_{1}+e_{2},\\\ \beta(e_{3})=e_{2}+e_{3},\\\ \beta(e_{4})=e_{3}+e_{4},\\\ \end{array}$

##  5 Derivation of BiHom-associative dialgebras

In this section, we introduce and study derivations of BiHom-dendrifom, BiHom-
dialgebras.

######  Definition 5.1.

Let $(A,\mu,\alpha,\beta)$ be a BiHom-associative algebra. A linear map
${D}:A\longrightarrow A$ is called an $(\alpha^{s},\beta^{r})$-derivation of
$(A,\mu,\alpha,\beta)$, if it satisfies

${D}\circ\alpha=\alpha\circ{D}\quad\text{and}\quad{D}\circ\beta=\beta\circ{D}$
${D}\circ\mu(x,y)=\mu({D}(x),\alpha^{s}\beta^{r}(y))+\mu(\alpha^{s}\beta^{r}(x),{D}(y))$

######  Example 5.2.

We consider the $2$-dimensional BiHom-associative with a basis
$\left\\{e_{1},e_{2}\right\\}$. For
$\mu(e_{1},e_{1})=-e_{1},\quad\mu(e_{1},e_{2})=-e_{2},\quad\mu(e_{2},e_{1})=0,\quad\mu(e_{2},e_{2})=e_{2}$
and
$\alpha(e_{1})=e_{1},\quad\alpha(e_{2})=-e_{2},\quad\beta(e_{1})=e_{1},\quad\beta(e_{2})=e_{2}.$
A direct computation gives that :
${D}(e_{1})=d_{22}e_{1},\quad{D}(e_{2})=d_{22}e_{2},$
$\alpha^{s}(e_{1})=\frac{\alpha_{21}\beta_{22}}{\beta_{21}}e_{1}+\frac{e_{2}}{2\beta_{21}},\quad\alpha^{s}(e_{2})=\alpha_{21}e_{1},\quad\beta^{r}(e_{1})=\frac{e_{2}}{2\beta_{21}}e_{2},\quad\beta^{r}(e_{2})=\beta_{21}e_{1}+\beta_{22}e_{2}$.

######  Definition 5.3.

Let $({D},\dashv,\vdash,\alpha,\beta)$ be a BiHom-associative dialgebra. A
linear map ${D}:{D}\rightarrow{D}$ is called an
$(\alpha^{k},\beta^{l})$-derivation of ${D}$ if it satisfies

  1. $1.$

${D}\circ\alpha=\alpha\circ{D},\,{D}\circ\beta=\beta\circ{D}$;

  2. $2.$

${D}(x\dashv
y)=\alpha^{k}\beta^{l}(x)\dashv{D}(y)+{D}(x)\dashv\alpha^{k}\beta^{l}(y);$

  3. $3.$

${D}(x\vdash
y)=\alpha^{k}\beta^{l}(x)\vdash{D}(y)+{D}(x)\vdash\alpha^{k}\beta^{l}(y),$

for $x,y\in{D}.$

We denote by $Der({D}):=\displaystyle\bigoplus_{k\geq
0}\displaystyle\bigoplus_{l\geq 0}Der_{(\alpha^{k},\beta^{l})}({D})$, where
$Der_{(\alpha^{k},\beta^{l})}({D})$ is the set of all
$(\alpha^{k},\beta^{l})$-derivations of ${D}$.

######  Proposition 5.4.

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

###### Proof.

For $x,y\in A$, we have

$\begin{array}[]{ll}\left[{D},{D^{\prime}}\right]\circ\mu(x,y)&={D}\circ{D^{\prime}}\circ\mu(x,y)-{D^{\prime}}\circ{D}\circ\mu(x,y)\\\
&={D}(\mu({D}^{\prime}(x),\alpha^{s}\beta^{r}(y))+\mu(\alpha^{s}\beta^{r}(x),{D}^{\prime}(y)))\\\
&-{D}^{\prime}(\mu({D}(x),\alpha^{s}\beta^{r}(y))+\mu(\alpha^{s}\beta^{r}(x),{D}(y)))\\\
&=\mu({D}\circ{D^{\prime}}(x),\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(y))+\mu(\alpha^{s}\beta^{r}\circ{D^{\prime}}(x),{D}\circ\alpha^{s}\beta^{r}(y))\\\
&+\mu({D}\circ\alpha^{s}\beta^{r}(x),\alpha^{s}\beta^{r}\circ{D}^{\prime}(y))+\mu(\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(x),{D}\circ{D^{\prime}}(y))\\\
&-\mu({D^{\prime}}\circ{D}(x),\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(y))-\mu(\alpha^{s}\beta^{r}\circ{D}(x),{D^{\prime}}\circ\alpha^{s}\beta^{r}(y))\\\
&-\mu({D^{\prime}}\circ\alpha^{s}\beta^{r}(x),\alpha^{s}\beta^{r}{D}(y))-\mu(\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(x),{D^{\prime}}\circ{D}(y)).\end{array}$

Since ${D}$ and ${D}^{\prime}$ satisfy
${D}\circ\alpha=\alpha\circ{D},\,{D}^{\prime}\circ\alpha=\alpha\circ{D}^{\prime}$,
${D}\circ\beta=\beta\circ{D},\,{D}^{\prime}\circ\beta=\beta\circ{D}^{\prime}$.
We obtain
$\alpha^{s}\beta^{r}\circ{D^{\prime}}={D^{\prime}}\circ\alpha^{s}\beta^{r},\,{D}\circ\alpha^{s^{\prime}}\beta^{r^{\prime}}=\alpha^{s^{\prime}}\beta^{r^{\prime}}\circ{D}.$
Therefore, we arrive at
$\left[{D},{D^{\prime}}\right]\circ\mu(x,y)=\mu(\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(x),\left[{D},{D^{\prime}}\right](y))+\mu(\left[{D},{D^{\prime}}\right](x),\alpha^{s+s^{\prime}}\beta^{r+r^{\prime}}(y)).$

Furthermore, it is straightforward to see that

$\begin{array}[]{ll}\left[{D},{D^{\prime}}\right]\circ\alpha&={D}\circ{D^{\prime}}\circ\alpha-{D^{\prime}}\circ{D}\circ\alpha\\\
&=\alpha\circ{D}\circ{D}^{\prime}-\alpha\circ{D}^{\prime}\circ{D}=\alpha\circ\left[{D},{D^{\prime}}\right].\end{array}$
$\begin{array}[]{ll}\left[{D},{D^{\prime}}\right]\circ\beta&={D}\circ{D^{\prime}}\circ\beta-{D^{\prime}}\circ{D}\circ\beta\\\
&=\beta\circ{D}\circ{D}^{\prime}-\beta\circ{D}^{\prime}\circ{D}=\beta\circ\left[{D},{D^{\prime}}\right]\end{array}$

which yields that $\left[{D},{D^{\prime}}\right]\in
Der_{(\alpha^{s+s^{\prime}},\beta^{r+r^{\prime}})}(A)$ with
$\mu=\dashv=\vdash.$ ∎

######  Proposition 5.5.

The space $Der_{(\alpha^{s},\beta^{r})}(A)$ is an invariant of the triple
BiHom-associative algebra A.

###### Proof.

Let
$\sigma:(A,\dashv_{A},\vdash_{A},\alpha^{s},\beta^{r})\longrightarrow(B,\dashv_{B},\vdash_{B},\alpha^{s},\beta^{r})$
be a triple BiHom-associative algebra isomorphism and let ${D}$ be a
$(\alpha^{s},\beta^{r})$-derivation of A. Then for any $x,y,z\in B$. We have :

$\begin{array}[]{ll}\sigma{D}\sigma^{-1}\circ(((x)\dashv_{B}(y))\dashv_{B}(z))&=\sigma{D}\circ((\sigma^{-1}(x)\dashv_{A}\sigma^{-1}(y))\dashv_{A}\sigma^{-1}(z))\\\
&=\sigma({D}\circ\sigma^{-1}(x)\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(y))\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(z))\\\
&+\sigma(\sigma^{-1}\circ\alpha^{s}\beta^{r}(x)\vdash_{A}{D}\circ\sigma^{-1}(y))\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(z))\\\
&+\sigma(\sigma^{-1}\circ\alpha^{s}\beta^{r}(x)\vdash_{A}\sigma^{-1}\circ\alpha^{s}\beta^{r}(y)\vdash_{A}{D}\circ\sigma^{-1}(z))\\\
&=({D}\circ\sigma^{-1}(x)\dashv_{B}\alpha^{s}\beta^{r}(y))\dashv_{B}\alpha^{s}\beta^{r}(z))\\\
&+(\alpha^{s}\beta^{r}(x)\dashv_{B}\sigma\circ{D}\circ\sigma^{-1}(y))\dashv_{B}\alpha^{s}\beta^{r}(z))\\\
&+(\alpha^{s}\beta^{r}(x)\dashv_{B}\alpha^{s}\beta^{r}(y))\dashv_{B}{D}\circ\sigma^{-1}(z)).\end{array}$

Thus $\sigma\circ{D}\circ\sigma^{-1}$ is a $(\alpha^{s},\beta^{r})$-derivation
of $B$, hence the mapping.
$\psi:Der_{(\alpha^{s},\beta^{r})}(A)\longrightarrow
Der_{(\alpha^{s},\beta^{r})}(B)$,
${D}\longmapsto\sigma{D}\sigma^{-1}$ is an isomorphism of triple BiHom-
associative algebras.

In fact, it is easy to see that $\psi$ is linear. Moreover let
${D}_{1},{D}_{2},{D}_{3}$ be derivations of A :

$\begin{array}[]{ll}&\alpha^{s}\beta^{r}\circ\psi({D}_{1}\dashv_{tr}{D}_{2})\dashv_{tr}{D}_{3})=\\\
&=\alpha^{s}\beta^{r}\psi(tr({D}_{1})({D}_{2}\dashv{D}_{3}))+\alpha^{s}\beta^{r}\psi(tr({D}_{3})({D}_{1}{D}_{2}))+\alpha^{s}\beta^{r}\psi(tr({D}_{2})({D}_{3}\dashv{D}_{1}))\\\
&=\alpha^{s}\beta^{r}tr({D}_{1})\psi({D}_{2}\dashv{D}_{3})+\alpha^{s}\beta^{r}tr({D}_{3})\psi({D}_{1}\dashv{D}_{2})+\alpha^{s}\beta^{r}tr({D}_{2})\psi({D}_{3}\dashv{D}_{1})\\\
&=\alpha^{s}\beta^{r}tr(\psi({D}_{1}))(\psi({D}_{2})\dashv\psi({D}_{3}))+\alpha^{s}\beta^{r}tr(\psi({D}_{3}))(\psi({D}_{1})\dashv\psi({D}_{2}))\\\
&+\alpha^{s}\beta^{r}tr(\psi({D}_{2}))\psi((\psi({D}_{3})\dashv\psi({D}_{1})),\end{array}$

since $\psi$ is a morphism of the $Der_{(\alpha^{s},\beta^{r})}(A)$ and
$Der_{(\alpha^{s},\beta^{r})}(B)$, and
$tr({D})=tr(\sigma\circ{D}\circ\sigma^{-1}).$
Then
$\alpha^{s}\beta^{r}\psi(({D}_{1}\dashv_{tr}{D}_{2})\dashv_{tr}{D}_{3}))=\alpha^{s}\beta^{r}((\psi({D}_{1})\dashv_{tr}\psi({D}_{2}))\dashv_{tr}\psi({D}_{3})).$
∎

## References

  * [1] A. Frolicher, A. Nijenhuis, Theory of vector valued differential forms, Part I. Indag Math, 1956, 18: 338-360
  * [2] A. Zahari and A. Makhlouf, Structure and Classification of Hom-Associative Algebras, Acta et commentationes universitis Tartuensis de mathematica, vol 24 (1) 2020.
  * [3] A. Kitouni, A. Makhlouf, S. Silvestrov, On n-ary Generalization of BiHom-Lie algebras and BiHom-Associative Algebras, arXiv:1812.00094, 2018.
  * [4] A. Majumdar, and G. Mukherjee, (2002). Deformation theory of dialgebras. K-theory, 27(1):33-60.
  * [5] A. P. Pozhidaev, (2008). Dialgebras and related triple systems. Siberian Mathematical Journal, 49(4):696-708.
  * [6] Basri, W., Rakhimov, I., Rikhsiboev, I., et al. (2015). Four-dimensional nilpotent dias- sociative algebras. Journal of Generalized Lie Theory and Applications, 9(1):1-7.
  * [7] G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras, Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 086, 34 pages.
  * [8] H. Adimi, T. Chtioui, S. Mabrouk, S. Massoud, Construction of BiHom-post-Lie algebras, arXiv:math.RA/2001.02308.
  * [9] I. BAKAYOKO and M. BANGOURA, Bimodules and Rota-Baxter relations, J. Appl. Mech. Eng 4:178, doi:10.4172/2168-9873.1000178, 2015.
  * [10] I. BAKAYOKO and M. BANGOURA, Left-Hom-symmetric and Hom-Poisson dialgebras, Konuralp Journal of Mathematics, $\bf 3$ No.2, 42-53, 2015.
  * [11] I.Rikhsiboev, I.Rakhimov and W. Basri, (2014). Diassociative algebras and their derivations. In Journal of Physics: Conference Series, volume 553, pages 1?9. IOP Publishing.
  * [12] I. M.Rikhsiboev, I. Rakhimov, and W. Basri (2010). Classification of 3-dimensional complex diassociative algebras. Malaysian Journal of Mathematical Sciences, 4(2):241-254.
  * [13] J. Li, L. Chen, B. Sun, Bihom-Nijienhuis operators and extensions of Bihom-Lie superalgebras.
  * [14] J. M. Casas, R. F. Casado, E. Khmaladze and M. Ladra, More on crossed modules of Lie, Leibniz, associative and diassociative algebras, available as arXiv:1508.01147 (05.08.2015).
  * [15] J.-L. Loday, Dialgebras. Dialgebras and related operads, pp. 7-66, Lecture Notes in Math., 1763, Springer, Berlin, 2001.
  * [16] J. Carinena, J. Grabowski, G. Marmo, Quantum bi-Hamiltonian systems, Internat J. Modern Phys A, 2000, 15: 4797-4810.
  * [17] K. Abdaoui, B. H. Abdelkader and A. Makhlouf, BiHom-Lie colour algebras structures, arXiv 1706.02188v1[math. RT] 6 Juin 2017.
  * [18] L. Lin and Y. Zhang, F [x, y] as a dialgebra and a Leibniz Algebra, Comm. Algebra 38(9) (2010), 3417-3447.
  * [19] L. Liu, A. Makhlouf, C. Menini, F. Panaite, Rota-Baxter operators on BiHom-associative algebras and related structures, arXiv:math.QA/1703.07275.
  * [20] L. Ling, A. Makhlouf, Claudia M. and Florin P., BiHom-Novikov algebras and infinitesimal BiHom-bialgebras, arXiv 1903.08145v1[Math. QA] 18 Mars 2019.
  * [21] L. Ling, A. Makhlouf, Claudia M. and Florin P., BiHom-pre-Lie algebras, BiHom-Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras, arXiv 1706.00457v2[Math. QA] 2 Fevrier 2020.
  * [22] L. Liu, A. Makhlouf, C. Menini, F. Panaite, $\\{\sigma,\tau\\}$-ota-Baxter operators, infinitesimal Hom-bialgebras and the associative (Bi)Hom-Yang-Baxter equation, Canad. Math. Bull., DOI: 10.4153/CMB-2018-028-8.
  * [23] P. Leroux, Contruction of Nijenhuis operators and Dendriform trialgebras, February 2004.
  * [24] P. Kolesnikov, and V. Y. Voronin, (2013). On special identities for dialgebras. Linear and Multilinear Algebra, 61(3):377-391.
  * [25] S. Isamiddin and Rakhimov, On central Extensions of Associative Dialgebras, J. Physics : conf. Ser. 697 (2016).
  * [26] Salazar-Diaz, O. Velasquez, R., and Wills-Toro, L. A. (2016). Construction of dialgebras through bimodules over algebras. Linear and Multilinear Algebra, pages 1-22.
  * [27] S. Guo, X. Zhang, S. Wang,, The construction and deformation of BiHom-Novikov algebras, J. Geom. Phys. 132 (2018), 460-472.
  * [28] S. Wang, S. Guo, BiHom-Lie superalgebra structures, arXiv:1610.02290v1 (2016).
  * [29] X. LI, BiHom-Poisson algebra and its application, International Journal of Algebra, Vol 13, 2019, no 2, 73-81.
  * [30] Y. Cheng, H. Qi, Representations of BiHom-Lie algebras, arXiv:1610.04302v1(2016).