# Admissible Hom-Novikov-Poisson and Hom-Gelfand-Dorfman color Hom-algebras

Ismail Laraiedh _Corresponding author_ : Ismail Laraiedh,
<EMAIL_ADDRESS>Departement of Mathematics, Faculty of Sciences,
Sfax University, Box 1171, 3000 Sfax, Tunisia
<EMAIL_ADDRESS>Departement of Mathematics,
College of Sciences and Humanities Al Quwaiiyah,
Shaqra University, Kingdom of Saudi Arabia
<EMAIL_ADDRESS>Sergei Silvestrov  Division of Mathematics and
Physics,
School of Education, Culture and Communication,
Mälardalen University, Box 883, 72123 Västerås, Sweden
<EMAIL_ADDRESS>
###### Abstract

The main feature of color Hom-algebras is that the identities defining the
structures are twisted by even linear maps. The purpose of this paper is to
introduce and give some constructions of admissible Hom-Novikov-Poisson color
Hom-algebras and Hom-Gelfand-Dorfman color Hom-algebras. Their bimodules and
matched pairs are defined and the relevant properties and theorems are given.
Also, the connections between Hom-Novikov-Poisson color Hom-algebras and Hom-
Gelfand-Dorfman color Hom-algebras is proved. Furthermore, we show that the
class of admissible Hom-Novikov-Poisson color Hom-algebras is closed under
tensor product.

##  1 Introduction

A Novikov algebra has a binary operation such that the associator is left-
symmetric and that the right multiplication operators commute. Novikov
algebras play a major role in the studies of Hamiltonian operators and Poisson
brackets of hydrodynamic type [15, 28, 29, 32, 33, 34]. The left-symmetry of
the associator implies that every Novikov algebra is Lie admissible, i.e., the
commutator bracket $[x,y]=xy-yx$ gives it a Lie algebra structure.

Poisson algebras are used in many fields in mathematics and physics. In
mathematics, Poisson algebras play a fundamental role in Poisson geometry
[76], quantum groups [21, 27], and deformation of commutative associative
algebras [35]. In physics, Poisson algebras are a major part of deformation
quantization [43], Hamiltonian mechanics [4], and topological field theories
[63]. Poisson-like structures are also used in the study of vertex operator
algebras [31].

The theory of Hom-algebras has been initiated in [37, 54, 55] motivated by
quasi-deformations of Lie algebras of vector fields, in particular
q-deformations of Witt and Virasoro algebras. Hom-Lie algebras and more
general quasi-Hom-Lie algebras were introduced first by Hartwig, Larsson and
Silvestrov in [37] where a general approach to discretization of Lie algebras
of vector fields using general twisted derivations ($\sigma$-derivations) and
a general method for construction of deformations of Witt and Virasoro type
algebras based on twisted derivations have been developed. The general quasi-
Lie algebras, containing the quasi-Hom-Lie algebras and Hom-Lie algebras as
subclasses, as well their graded color generalization, the color quasi-Lie
algebras including color quasi-Hom-Lie algebras, color Hom-Lie algebras and
their special subclasses the quasi-Hom-Lie superalgebras and Hom-Lie
superalgebras, have been first introduced in [37, 54, 55, 56, 53, 69].
Subsequently, various classes of Hom-Lie admissible algebras have been
considered in [45]. In particular, in [45], the Hom-associative algebras have
been introduced and shown to be Hom-Lie admissible, that is leading to Hom-Lie
algebras using commutator map as new product, and in this sense constituting a
natural generalization of associative algebras as Lie admissible algebras
leading to Lie algebras using commutator map. Furthermore, in [45], more
general $G$-Hom-associative algebras including Hom-associative algebras, Hom-
Vinberg algebras (Hom-left symmetric algebras), Hom-pre-Lie algebras (Hom-
right symmetric algebras), and some other Hom-algebra structures, generalizing
$G$-associative algebras, Vinberg and pre-Lie algebras respectively, have been
introduced and shown to be Hom-Lie admissible, meaning that for these classes
of Hom-algebras, the operation of taking commutator leads to Hom-Lie algebras
as well. Also, flexible Hom-algebras have been introduced, connections to Hom-
algebra generalizations of derivations and of adjoint maps have been noticed,
and some low-dimensional Hom-Lie algebras have been described. In Hom-algebra
structures, defining algebra identities are twisted by linear maps. Since the
pioneering works [37, 54, 55, 56, 53, 45], Hom-algebra structures have
developed in a popular broad area with increasing number of publications in
various directions. Hom-algebra structures include their classical
counterparts and open new broad possibilities for deformations, extensions to
Hom-algebra structures of representations, homology, cohomology and formal
deformations, Hom-modules and Hom-bimodules, Hom-Lie admissible Hom-
coalgebras, Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras, $L$-modules,
$L$-comodules and Hom-Lie quasi-bialgebras, $n$-ary generalizations of BiHom-
Lie algebras and BiHom-associative algebras and generalized derivations, Rota-
Baxter operators, Hom-dendriform color Hom-algebras, Rota-Baxter bisystems and
covariant bialgebras, Rota-Baxter cosystems, coquasitriangular mixed
bialgebras, coassociative Yang-Baxter pairs, coassociative Yang-Baxter
equation and generalizations of Rota-Baxter systems and algebras, curved
$\mathcal{O}$-operator systems and their connections with tridendriform
systems and pre-Lie algebras, BiHom-algebras, BiHom-Frobenius algebras and
double constructions, infinitesimal BiHom-bialgebras and Hom-dendriform
$D$-bialgebras, Hom-algebras have been considered [2, 3, 6, 9, 10, 12, 14, 13,
18, 16, 17, 20, 25, 26, 30, 36, 38, 39, 40, 41, 42, 51, 52, 54, 53, 57, 49,
44, 46, 58, 47, 48, 60, 61, 62, 68, 67, 70, 72, 73, 74, 75, 81, 83, 86, 82,
87].

In [84] the author initiated the study of a twisted generalization of Novikov
algebras, called Hom-Novikov algebras. A Hom-Novikov algebra $A$ has a binary
operation $\cdot$ and a linear self-map $\alpha$, and it satisfies some
$\alpha$-twisted versions of the defining identities of a Novikov algebra. In
[84] several constructions of Hom-Novikov algebras were given and some low
dimensional Hom-Novikov algebras were classified. Further, Hom-Poisson
algebras were defined in [46] by Makhlouf and Silvestrov. It is shown in [46]
that Hom-Poisson algebras play the same role in the deformation of commutative
Hom-associative algebras as Poisson algebras do in the deformation of
commutative associative algebras.

In this paper, we introduce and obtain some results on construction of
admissible Hom-Novikov-Poisson color Hom-algebras and Hom-Gelfand-Dorfman
color Hom-algebras. Their bimodules and matched pairs are defined and the
relevant properties and theorems are obtained. We also show that the class of
admissible Hom-Novikov-Poisson color Hom-algebras are closed under tensor
product. In Section 2, we introduce the notions of bimodules and matched pairs
of Hom-associative color Hom-algebras, Hom-Novikov color Hom-algebras and Hom-
Lie color Hom-algebras in which we give some results and some examples. In
Section 3, we establish the notions of admissible Hom-Novikov-Poisson color
Hom-algebras and we give some explicit constructions. Their bimodule and
matched pair are defined and their related relevant properties are also given.
Finally, we show that the much larger class of admissible Hom-Novikov-Poisson
color Hom-algebras is also closed under tensor products. In Section 4, we
introduce the notions of Hom-Gelfand-Dorfman color Hom-algebras and we discuss
some basic properties and examples of these objects. Moreover, we characterize
the representations and matched pairs of Hom-Gelfand-Dorfman color Hom-
algebras and provide some key constructions.

##  2 Preliminaries and some results

Throughout the article, we assume that all linear spaces are over an
algebraically closed field $\mathbb{K}$ of characteristic $0$, and denote by
$\mathbb{K^{\ast}}=\mathbb{K}\backslash\\{0\\}$ the group of invertible
elements of $\mathbb{K}$ with respect to the multiplication in $\mathbb{K}$.

In this section, we introduce the notions of bimodules and matched pairs of
Hom-associative color Hom-algebras, Hom-Novikov color Hom-algebras and Hom-Lie
color Hom-algebras in which we give some results and examples.

Let $\Gamma$ be an abelian group. A linear space $V$ is said to be
$\Gamma$-graded, if there is a family $(V_{\gamma})_{\gamma\in\Gamma}$ of
vector subspace of $V$ such that

$V=\bigoplus_{\gamma\in\Gamma}V_{\gamma}.$

An element $x\in V$ is said to be homogeneous of degree $\gamma\in\Gamma$ if
$x\in V_{\gamma},\gamma\in\Gamma$, and in this case, $\gamma$ is called the
degree of $x$. In the sequel, we will denote the set of all the homogeneous
elements of $V$ by $\mathcal{H}(V)$. As usual, we denote by $\overline{x}$ the
degree of an element $x\in V$. Thus each homogeneous element $x\in V$
determines a unique group element $\overline{x}\in\Gamma$ by $x\in
V_{\overline{x}}$. Thus, when no confusion occur, we can drop $"-"$ in
notation of degree for convenience of exposition.

Let $V=\bigoplus_{\gamma\in\Gamma}V_{\gamma}$ and
$V^{{}^{\prime}}=\bigoplus_{\gamma\in\Gamma}V^{{}^{\prime}}_{\gamma}$ be two
$\Gamma$-graded linear spaces. A linear mapping $f:V\longrightarrow
V^{{}^{\prime}}$ is said to be homogeneous of degree $\upsilon\in\Gamma$ if
$f(V_{\gamma})\subseteq V^{{}^{\prime}}_{\gamma+\upsilon}$ for all
$\gamma\in\Gamma.$ If in addition $f$ is homogeneous of degree zero, i.e.
$f(V_{\gamma})\subseteq V^{{}^{\prime}}_{\gamma}$ holds for any
$\gamma\in\Gamma$, then $f$ is said to be even.

An algebra $\mathcal{A}$ is said to be $\Gamma$-graded if its underlying
linear space is $\Gamma$-graded, i.e.
$\mathcal{A}=\bigoplus_{\gamma\in\Gamma}\mathcal{A}_{\gamma}$, and if,
furthermore
$\mathcal{A}_{\gamma}\mathcal{A}_{\gamma^{\prime}}\subseteq\mathcal{A}_{\gamma+\gamma^{\prime}}$,
for all $\gamma,\gamma^{\prime}\in\Gamma$. It is easy to see that if
$\mathcal{A}$ has a unit element $e$, it follows that $e\in\mathcal{A}_{0}$. A
subalgebra of $\mathcal{A}$ is said to be $\Gamma$-graded if it is
$\Gamma$-graded as a subspace of $\mathcal{A}$. Let
$\mathcal{A}^{{}^{\prime}}$ be another $\Gamma$-graded algebra. A homomorphism
$f:\mathcal{A}\longrightarrow\mathcal{A}^{{}^{\prime}}$ of $\Gamma$-graded
algebras is by definition a homomorphism of the algebra $\mathcal{A}$ into the
algebra $\mathcal{A}^{{}^{\prime}}$, which is, in addition an even mapping.

######  Definition 2.1 ([7, 22, 23, 50, 59, 64, 65, 66, 71]).

Let $\mathbb{K}$ be a field and $\Gamma$ be an abelian group. A map
$\varepsilon:\Gamma\times\Gamma\rightarrow\mathbb{K^{\ast}}$ is called a
commutation factor on ${\Gamma}$ if the following identities hold, for all
$a,b,c$ in $\Gamma$

  1. (i)

$\varepsilon(a,b)~{}\varepsilon(b,a)=1,$

  2. (ii)

$\varepsilon(a,b+c)=\varepsilon(a,b)~{}\varepsilon(a,c),$

  3. (iii)

$\varepsilon(a+b,c)=\varepsilon(a,c)~{}\varepsilon(b,c).$

The definition above implies, in particular, the following relations

$\varepsilon(a,0)=\varepsilon(0,a)=1,\ \varepsilon(a,a)=\pm 1,\ \textrm{for\
all}\ a\in\Gamma.$

If $x$ and $x^{\prime}$ are two homogeneous elements of degree $\gamma$ and
$\gamma^{\prime}$ respectively and $\varepsilon$ is a skewsymmetric
bicharacter, then we shorten the notation by writing
$\varepsilon(x,x^{\prime})$ instead of $\varepsilon(\gamma,\gamma^{\prime})$
since the degree of every homogeneous element is unique.

######  Remark 2.2.

Let $A$ and $V$ be two $\Gamma$-graded linear spaces such that

$A\oplus V=\bigoplus_{\gamma\in\Gamma}(A\oplus
V)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus V_{\gamma}),$

then, for all $X_{1}=x_{1}+v_{1}\in A_{\gamma_{1}}\oplus
V_{\gamma_{1}},X_{2}=x_{2}+v_{2}\in A_{\gamma_{2}}\oplus V_{\gamma_{2}}$ we
have

$\varepsilon(x_{1},x_{2})=\varepsilon(x_{1},v_{2})=\varepsilon(v_{1},x_{2})=\varepsilon(v_{1},v_{2})=\varepsilon(X_{1},X_{2}).$

######  Example 2.3.

Some standard examples of skew-symmetric bicharacters are:

  1. 1)

$\Gamma=\mathbb{Z}_{2},\quad\varepsilon(i,j)=(-1)^{ij}$,

  2. 2)

$\Gamma=\mathbb{Z}_{2}^{n},\quad\varepsilon((\alpha_{1},\dots,\alpha_{n}),(\beta_{1},\dots,\beta_{n})):=(-1)^{\alpha_{1}\beta_{1}+\dots+\alpha_{n}\beta_{n}}.$

  3. 3)

$\Gamma=\mathbb{Z}_{2}\times\mathbb{Z}_{2},\quad\varepsilon((i_{1},i_{2}),(j_{1},j_{2}))=(-1)^{i_{1}j_{2}-i_{2}j_{1}}$,

  4. 4)

$\Gamma=\mathbb{Z}\times\mathbb{Z},\quad\varepsilon((i_{1},i_{2}),(j_{1},j_{2}))=(-1)^{(i_{1}+i_{2})(j_{1}+j_{2})}$.

######  Definition 2.4.

A color Hom-algebra or a Hom-color algebra, $(A,\cdot,\varepsilon,\alpha)$ is
a $\Gamma$-graded linear space $A$ equipped with even bilinear multiplication
$\cdot$, even twisting map $\alpha$ and commutation factor $\varepsilon$.

######  Definition 2.5.

A derivation of degree $d\in\Gamma$ on a color Hom-algebra
$(A,\cdot,\varepsilon,\alpha)$ is a linear map $D:A\rightarrow A$ such that
for any $x,y\in\mathcal{H}(A)$,

$D(x\cdot y)=D(x)\cdot y+\varepsilon(d,x)x\cdot D(y).$

In particular, an even derivation $D:A\rightarrow A$ is a derivation of degree
zero, i.e., for all $x,y\in\mathcal{H}(A)$,

$D(x\cdot y)=D(x)\cdot y+x\cdot D(y).$

###  2.1 $\boldsymbol{\varepsilon}$-Commutative Hom-associative color Hom-
algebras

######  Definition 2.6 ([80]).

A Hom-associative color Hom-algebra is a color Hom-algebra
$(A,\cdot,\varepsilon,\alpha)$ satisfying for $x,y,z\in\mathcal{H}(A)$,

$\displaystyle as_{A}(x,y,z):=\alpha(x)\cdot(y\cdot z)-(x\cdot
y)\cdot\alpha(z)=0.$ (Hom-associativity) (2.1)

If in addition, for any $x,y\in\mathcal{H}(A)$,

$x\cdot y=\varepsilon(x,y)y\cdot x,$ (2.2)

then $(A,\cdot,\varepsilon,\alpha)$ is said to be a $\varepsilon$-commutative
Hom-associative color Hom-algebra.

######  Example 2.7.

Let $A=A_{0}\oplus A_{1}=<e_{1}>\oplus<e_{2},e_{3}>$ be a $3$-dimensional
superspace. Then $A$ is a $\varepsilon$-commutative Hom-associative color Hom-
algebra with

$\displaystyle\text{the bicharacter:}\quad\varepsilon(i,j)=(-1)^{ij},$
$\displaystyle\text{the multiplication:}\quad e_{1}\cdot e_{2}=e_{2}\cdot
e_{1}=-2e_{3},$ $\displaystyle\text{the even linear map $\alpha:A\rightarrow
A$ defined
by}\begin{array}[t]{l}\alpha(e_{1})=\sqrt{2}e_{1},\quad\alpha(e_{2})=e_{3}-e_{2},\\\
\quad\alpha(e_{3})=e_{3}.\end{array}$

In the following, we introduce the notion of bimodule of
$\varepsilon$-commutative Hom-associative color Hom-algebra.

######  Definition 2.8.

Let $(A,\cdot,\varepsilon,\alpha)$ be an $\varepsilon$-commutative Hom-
associative color Hom-algebra, $(V,\beta)$ be a pair consisting of
$\Gamma$-graded linear space $V$ and an even linear map $\beta:V\rightarrow
V$, and $s:A\rightarrow End(V)$ be an even linear map. The triple
$(s,\beta,V)$ is called a bimodule of $(A,\cdot,\varepsilon,\alpha)$ if for
all $x,y\in\mathcal{H}(A),v\in\mathcal{H}(V)$,

$\displaystyle s(x\cdot y)\beta(v)=s(\alpha(x))s(y)v.$ (2.3)

######  Proposition 2.9.

Let $(s,\beta,V)$ is a bimodule of a $\varepsilon$-commutative Hom-associative
color Hom-algebra $(A,\cdot,\varepsilon,\alpha)$. Then the direct sum of
$\Gamma$-graded linear spaces,

$A\oplus V=\bigoplus_{\gamma\in\Gamma}(A\oplus
V)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus V_{\gamma}),$

is turned into a $\varepsilon$-commutative Hom-associative color Hom-algebra
by defining multiplication and the twisting map in $A\oplus V$ for all
$X_{1}=x_{1}+v_{1}\in A_{\gamma_{1}}\oplus V_{\gamma_{1}},X_{2}=x_{2}+v_{2}\in
A_{\gamma_{2}}\oplus V_{\gamma_{2}}$ by

$\displaystyle(x_{1}+v_{1})\ast(x_{2}+v_{2})$ $\displaystyle=$ $\displaystyle
x_{1}\cdot
x_{2}+\big{(}s(x_{1})v_{2}+\varepsilon(v_{1},x_{2})s(x_{2})v_{1}\big{)},$
(2.4) $\displaystyle(\alpha\oplus\beta)(x_{1}+v_{1})$ $\displaystyle=$
$\displaystyle\alpha(x_{1})+\beta(v_{1}).$ (2.5)

###### Proof.

We prove the commutativity and Hom-associativity in $A\oplus V$. For all
elements $X_{i}=x_{i}+v_{i}\in A_{\gamma_{i}}\oplus V_{\gamma_{i}},\ i=1,2,3$,

$\displaystyle X_{1}\ast X_{2}=(x_{1}+v_{1})\ast(x_{2}+v_{2})$
$\displaystyle\quad=x_{1}\cdot
x_{2}+\big{(}s(x_{1})v_{2}+\varepsilon(v_{1},x_{2})s(x_{2})v_{1}\big{)}$
$\displaystyle\quad=\varepsilon(x_{1},x_{2})x_{2}\cdot
x_{1}+\big{(}\varepsilon(v_{1},x_{2})s(x_{2})v_{1}+s(x_{1})v_{2}\big{)}$
$\displaystyle\quad=\varepsilon(X_{1},X_{2})x_{2}\cdot
x_{1}+\big{(}\varepsilon(X_{1},X_{2})s(x_{2})v_{1}+s(x_{1})v_{2}\big{)}$
$\displaystyle\quad=\varepsilon(X_{1},X_{2})\big{(}x_{2}\cdot
x_{1}+s(x_{2})v_{1}+\varepsilon(X_{2},X_{1})s(x_{1})v_{2}\big{)}$
$\displaystyle\quad=\varepsilon(X_{1},X_{2})\big{(}x_{2}\cdot
x_{1}+s(x_{2})v_{1}+\varepsilon(v_{2},x_{1})s(x_{1})v_{2}\big{)}$
$\displaystyle\quad=\varepsilon(X_{1},X_{2})(x_{2}+v_{2})\ast(x_{1}+v_{1})$
$\displaystyle\quad=\varepsilon(X_{1},X_{2})X_{2}\ast X_{1},$
$\displaystyle(X_{1}\ast
X_{2})\ast(\alpha+\beta)X_{3}-(\alpha+\beta)X_{1}\ast(X_{2}\ast X_{3})$
$\displaystyle\quad=\big{(}(x_{1}+v_{1})\ast(x_{2}+v_{2})\big{)}\ast(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle\quad\quad-(\alpha+\beta)(x_{1}+v_{1})\ast\big{(}(x_{2}+v_{2})\ast(x_{3}+v_{3})\big{)}$
$\displaystyle\quad=\big{(}x_{1}\cdot
x_{2}+s(x_{1})v_{2}+\varepsilon(v_{1},x_{2})s(x_{2})v_{1}\big{)}\ast(\alpha(x_{3})+\beta(v_{3}))$
$\displaystyle\quad\quad-(\alpha(x_{1})+\beta(v_{1}))\ast\big{(}x_{2}\cdot
x_{3}+s(x_{2})v_{3}+\varepsilon(v_{2},x_{3})s(x_{3})v_{2}\big{)}$
$\displaystyle\quad=(x_{1}\cdot x_{2})\cdot\alpha(x_{3})+s(x_{1}\cdot
x_{2})\beta(v_{3})+\varepsilon(x_{1}+x_{2},x_{3})s(\alpha(x_{3}))s(x_{1})v_{2}$
$\displaystyle\quad\quad+\varepsilon(x_{1}+x_{2},x_{3})\varepsilon(x_{1},x_{2})s(\alpha(x_{3}))s(x_{2})v_{1}-\alpha(x_{1})\cdot(x_{2}\cdot
x_{3})$
$\displaystyle\quad\quad-s(\alpha(x_{1}))s(x_{2})v_{3}-\varepsilon(x_{2},x_{3})s(\alpha(x_{1}))s(x_{3})v_{2}$
$\displaystyle\quad\quad-\varepsilon(v_{1},x_{2}+x_{3})s(x_{2}\cdot
x_{3})\beta(v_{1})$ $\displaystyle\quad=\underbrace{\Big{(}(x_{1}\cdot
x_{2})\cdot\alpha(x_{3})-\alpha(x_{1})\cdot(x_{2}\cdot
x_{3})\Big{)}}_{\text{$=0$ by \eqref{Homass:homassociator}}}$
$\displaystyle\quad\quad+\underbrace{\Big{(}s(x_{1}\cdot
x_{2})\beta(v_{3})-s(\alpha(x_{1}))s(x_{2})v_{3}\Big{)}}_{\text{$=0$ by
\eqref{bimod:comm.hom.ass.color.algeb.lara.silvest}}}$
$\displaystyle\quad\quad+\underbrace{\varepsilon(x_{2},x_{3})\Big{(}\varepsilon(x_{1},x_{3})s(\alpha(x_{3}))s(x_{1})v_{2}-s(\alpha(x_{1}))s(x_{3})v_{2}\Big{)}}_{\text{$=0$
by \eqref{commutative:homcoloralgeb} and
\eqref{bimod:comm.hom.ass.color.algeb.lara.silvest}}}$
$\displaystyle\quad\quad+\underbrace{\varepsilon(x_{1},x_{2}+x_{3})\Big{(}\varepsilon(x_{2},x_{3})s(\alpha(x_{3}))s(x_{2})v_{1}-s(x_{2}\cdot
x_{3})\beta(v_{1})\Big{)}}_{\text{$=0$ by \eqref{commutative:homcoloralgeb}
and \eqref{bimod:comm.hom.ass.color.algeb.lara.silvest}}}=0.\qed$

The $\varepsilon$-commutative Hom-associative color Hom-algebra constructed in
Proposition 2.9 is denoted by $(A\oplus V,\ast,\varepsilon,\alpha+\beta)$ or
$A\ltimes_{s,\alpha,\beta}V.$

######  Example 2.10.

Let $(A,\cdot,\varepsilon,\alpha)$ be a $\varepsilon$-commutative Hom-
associative color Hom-algebra. Then, $(S,\alpha,A)$ with $S(x)y=x\cdot y$ for
all $x,y\in\mathcal{H}(A)$, is a bimodule of $(A,\cdot,\varepsilon,\alpha)$
called the regular bimodule of $(A,\cdot,\varepsilon,\alpha)$.

In the following, we introduce the notion of matched pair of
$\varepsilon$-commutative Hom-associative color Hom-algebras.

######  Proposition 2.11.

Let $(A,\cdot_{A},\varepsilon,\alpha)$ and $(B,\cdot_{B},\varepsilon,\beta)$
be $\varepsilon$-commutative Hom-associative color Hom-algebras. Suppose that
there are even linear maps $s_{A}:A\rightarrow End(B)$ and $s_{B}:B\rightarrow
End(A)$ such that $(s_{A},\beta,B)$ is a bimodule of $A$, and
$(s_{B},\alpha,A)$ is a bimodule of $B$, satisfying, for any
$x,y\in\mathcal{H}(A),a,b\in\mathcal{H}(B)$, the following conditions:

$\displaystyle\varepsilon(b,x)\beta(a)\cdot_{B}(s_{A}(x)b)+\varepsilon(a,b+x)s_{A}(s_{B}(b)x)\beta(a)$
$\displaystyle=\varepsilon(a+b,x)s_{A}(\alpha(x))(a\cdot_{B}b),$ (2.6)
$\displaystyle\beta(a)\cdot_{B}(s_{A}(x)b)+\varepsilon(a,x+b)\varepsilon(x,b)s_{A}(s_{B}(b)x)\beta(a)$
$\displaystyle=\varepsilon(a,x)s_{A}(x)a\cdot_{B}\beta(b)+s_{A}(s_{B}(a)x)\beta(b),$
(2.7)
$\displaystyle\varepsilon(y,a)\alpha(x)\cdot_{A}(s_{B}(a)y)+\varepsilon(x,y+a)s_{B}(s_{A}(y)a)\alpha(x)$
$\displaystyle=\varepsilon(x+y,a)s_{B}(\beta(a))(x\cdot_{A}y),$ (2.8)
$\displaystyle\alpha(x)\cdot_{A}(s_{B}(a)y)+\varepsilon(x,a+y)\varepsilon(a,y)s_{B}(s_{A}(y)a)\alpha(x)$
$\displaystyle=\varepsilon(x,a)s_{B}(a)x\cdot_{A}\alpha(y)+s_{B}(s_{A}(x)a)\alpha(y).$
(2.9)

Then, $(A,B,l_{A},r_{A},\beta,l_{B},r_{B},\alpha)$ is called a matched pair of
$\varepsilon$-commutative Hom-associative color Hom-algebras. In this case,
there is a $\varepsilon$-commutative Hom-associative color Hom-algebra
structure on the direct sum of the underlying $\Gamma$-graded linear spaces of
$A$ and $B$,

$A\oplus B=\bigoplus_{\gamma\in\Gamma}(A\oplus
B)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus B_{\gamma}),$

given for all $x+a\in A_{\gamma_{1}}\oplus B_{\gamma_{1}},y+b\in
A_{\gamma_{2}}\oplus B_{\gamma_{2}}$ by

$\displaystyle(x+a)\cdot(y+b)$ $\displaystyle=$
$\displaystyle\big{(}x\cdot_{A}y+s_{B}(a)y+\varepsilon(x,b)s_{B}(b)x\big{)}$
(2.10) $\displaystyle+$
$\displaystyle\big{(}a\cdot_{B}b+s_{A}(x)b+\varepsilon(a,y)s_{A}(y)a\big{)},$
$\displaystyle(\alpha\oplus\beta)(x+a)$ $\displaystyle=$
$\displaystyle\alpha(x)+\beta(a).$ (2.11)

###### Proof.

Let $X=x+a\in A_{\gamma_{1}}\oplus B_{\gamma_{1}},Y=y+b\in
A_{\gamma_{2}}\oplus B_{\gamma_{2}},Z=z+c\in A_{\gamma_{3}}\oplus
B_{\gamma_{3}}$. First, we prove the commutativity condition:

$\displaystyle X\cdot Y-\varepsilon(X,Y)Y\cdot
X=(x+a)\cdot(y+b)-\varepsilon(X,Y)(y+b)\cdot(x+a)$
$\displaystyle\quad=x\cdot_{A}y+s_{B}(a)y+\varepsilon(x,b)s_{B}(b)x+a\cdot_{B}b+s_{A}(x)b+\varepsilon(a,y)s_{A}(y)a$
$\displaystyle\quad\quad-\varepsilon(X,Y)\big{(}y\cdot_{A}x+s_{B}(b)x+\varepsilon(y,a)s_{B}(a)y+b\cdot_{B}a+s_{A}(y)a+\varepsilon(b,x)s_{A}(x)b\big{)}$
$\displaystyle\quad=\big{(}x\cdot_{A}y-\varepsilon(X,Y)y\cdot_{A}x\big{)}+\big{(}a\cdot_{B}b-\varepsilon(X,Y)b\cdot_{B}a\big{)}$
$\displaystyle\quad\quad+\big{(}s_{B}(a)y-\varepsilon(X,Y)\varepsilon(y,a)s_{B}(a)y\big{)}+\big{(}\varepsilon(x,b)s_{B}(b)x-\varepsilon(X,Y)s_{B}(b)x\big{)}$
$\displaystyle\quad\quad+\big{(}s_{A}(x)b-\varepsilon(X,Y)\varepsilon(b,x)s_{A}(x)b\big{)}+\big{(}\varepsilon(a,y)s_{A}(y)a-\varepsilon(X,Y)s_{A}(y)a\big{)}=0.$
(using Remark 2.2 and (2.2))

Next, we prove the Hom-associativity condition:

$\displaystyle(X\cdot
Y)\cdot(\alpha+\beta)Z=\big{(}(x+a)\cdot(y+b)\big{)}\cdot(\alpha+\beta)(z+c)$
(using Remark 2.2)
$\displaystyle\quad=\Big{(}\big{(}x\cdot_{A}y+s_{B}(a)y+\varepsilon(x,y)s_{B}(b)x\big{)}$
$\displaystyle\quad\quad+\big{(}a\cdot_{B}b+s_{A}(x)b+\varepsilon(x,y)s_{A}(y)a\big{)}\Big{)}\cdot(\alpha(z)+\beta(c))$
$\displaystyle\quad=(x\cdot_{A}y)\cdot_{A}\alpha(z)+s_{B}(a)y\cdot_{A}\alpha(z)+\varepsilon(x,y)s_{B}(b)x\cdot_{A}\alpha(z)$
$\displaystyle\quad\quad+s_{B}(a\cdot_{B}b)\alpha(z)+s_{B}(s_{A}(x)b)\alpha(z)+\varepsilon(x,y)s_{B}(s_{A}(y)a)\alpha(z)$
$\displaystyle\quad\quad+\varepsilon(x+y,z)\Big{(}s_{B}(\beta(c))(x\cdot_{A}y)+s_{B}(\beta(c))s_{B}(a)y$
$\displaystyle\quad\quad+\varepsilon(x,y)s_{B}(\beta(c))s_{B}(b)x\Big{)}$
$\displaystyle\quad\quad+(a\cdot_{B}b)\cdot_{B}\beta(c)+s_{A}(x)b\cdot_{B}\beta(c)+\varepsilon(x,y)s_{A}(y)a\cdot_{B}\beta(c)$
$\displaystyle\quad\quad+s_{A}(x\cdot_{A}y)\beta(c)+s_{A}(s_{B}(a)y)\beta(c)+\varepsilon(x,y)s_{A}(s_{B}(b)x)\beta(c)$
$\displaystyle\quad\quad+\varepsilon(x+y,z)\Big{(}s_{A}(\alpha(z))(a\cdot_{B}b)+s_{A}(\alpha(z))s_{A}(x)b+\varepsilon(x,y)s_{A}(\alpha(z))s_{A}(y)a\Big{)}$
(using Remark 2.2)
$\displaystyle\quad=\Big{(}(x\cdot_{A}y)\cdot_{A}\alpha(z)+s_{B}(a)y\cdot_{A}\alpha(z)+\varepsilon(x,y)s_{B}(b)x\cdot_{A}\alpha(z)$
$\displaystyle\quad\quad+s_{B}(a\cdot_{B}b)\alpha(z)+s_{B}(s_{A}(x)b)\alpha(z)+\varepsilon(x,y)s_{B}(s_{A}(y)a)\alpha(z)$
$\displaystyle\quad\quad+(a\cdot_{B}b)\cdot_{B}\beta(c)+s_{A}(x)b\cdot_{B}\beta(c)+\varepsilon(x,y)s_{A}(y)a\cdot_{B}\beta(c)$
$\displaystyle\quad\quad+s_{A}(x\cdot_{A}y)\beta(c)+s_{A}(s_{B}(a)y)\beta(c)+\varepsilon(x,y)s_{A}(s_{B}(b)x)\beta(c)\Big{)}$
$\displaystyle\quad\quad+\varepsilon(x+y,z)\Big{(}s_{B}(\beta(c))(x\cdot_{A}y)+s_{B}(\beta(c))s_{B}(a)y$
$\displaystyle\quad\quad+\varepsilon(x,y)s_{B}(\beta(c))s_{B}(b)x+s_{A}(\alpha(z))(a\cdot_{B}b)+s_{A}(\alpha(z))s_{A}(x)b$
$\displaystyle\quad\quad+\varepsilon(x,y)s_{A}(\alpha(z))s_{A}(y)a\Big{)},$
$\displaystyle(\alpha+\beta)X\cdot(Y\cdot Z)$
$\displaystyle\quad=(\alpha+\beta)(x+a)\cdot\big{(}(y+b)\cdot(z+c)\big{)}$
$\displaystyle\quad=(\alpha(x)+\beta(a))\cdot\Big{(}\big{(}y\cdot_{A}z+s_{B}(b)z+\varepsilon(y,c)s_{B}(c)y\big{)}$
$\displaystyle\quad\quad+\big{(}b\cdot_{B}c+s_{B}(y)c+\varepsilon(b,z)s_{A}(z)b\big{)}\Big{)}$
(using Remark 2.2)
$\displaystyle\quad=\alpha(x)\cdot_{A}(y\cdot_{A}z)+\alpha(x)\cdot_{A}s_{B}(b)z+\varepsilon(y,z)\alpha(x)\cdot_{A}s_{B}(c)y$
$\displaystyle\quad\quad+s_{B}(\beta(a))(y\cdot_{A}z)+s_{B}(\beta(a))s_{B}(b)z+\varepsilon(y,c)s_{B}(\beta(a))s_{B}(c)y$
$\displaystyle\quad\quad+\varepsilon(x,y+z)\Big{(}s_{B}(b\cdot_{B}c)\alpha(x)+s_{B}(s_{A}(y)c)\alpha(x)$
$\displaystyle\quad\quad+\varepsilon(b,z)s_{B}(s_{A}(z)b)\alpha(x)\Big{)}+\beta(a)\cdot_{B}(b\cdot_{B}c)+\beta(a)\cdot_{B}s_{A}(y)c$
$\displaystyle\quad\quad+\varepsilon(b,z)\beta(a)\cdot_{B}s_{A}(z)b+s_{A}(\alpha(x))(b\cdot_{B}c)+s_{A}(\alpha(x))s_{A}(y)c$
$\displaystyle\quad\quad+\varepsilon(y,z)s_{A}(\alpha(x))s_{A}(z)b+\varepsilon(x,y+z)\Big{(}s_{A}(y\cdot_{A}z)\beta(a)$
$\displaystyle\quad\quad+s_{A}(s_{B}(b)z)\beta(a)+\varepsilon(y,z)s_{A}(s_{B}(c)y)\beta(a)\Big{)}.$

Using (2.11)-(2.11) and that $(s_{A},\beta,B)$ and $(s_{B},\alpha,A)$ are
bimodules of $(A,\cdot_{A},\varepsilon,\alpha)$ and
$(B,\cdot_{B},\varepsilon,\beta)$, respectively, we derive that $(A\oplus
B,\cdot,\varepsilon,\alpha+\beta)$ is $\varepsilon$-commutative Hom-
associative color Hom-algebra. This completes the proof. ∎

This $\varepsilon$-commutative Hom-associative color Hom-algebra, constructed
in Proposition 2.11, is denoted by $(A\bowtie
B,\cdot,\varepsilon,\alpha+\beta)$ or
$A\bowtie^{s_{A},\beta}_{s_{B},\alpha}B.$

###  2.2 On Hom-Novikov color Hom-algebras

######  Definition 2.12 ([8]).

A color Hom-algebra $(A,\cdot,\varepsilon,\alpha)$ is called a Hom-Novikov
color Hom-algebra if the following identities are satisfied for all
$x,y,z\in\mathcal{H}(A)$:

$\displaystyle(x\cdot y)\cdot\alpha(z)-\alpha(x)\cdot(y\cdot
z)=\varepsilon(x,y)\big{(}(y\cdot x)\cdot\alpha(z)-\alpha(y)\cdot(x\cdot
z)\big{)},$ (2.12) $\displaystyle(x\cdot
y)\cdot\alpha(z)=\varepsilon(y,z)(x\cdot z)\cdot\alpha(y).$ (2.13)

######  Remark 2.13.

If $\alpha=id_{A}$ in Definition 2.12, we recover a Novikov color Hom-algebra.
So, Novikov color Hom-algebras are a special case of Hom-Novikov color Hom-
algebras when the twisting linear map is the identity map.

######  Examples 2.14 (Hom-Novikov color Hom-algebras).

Here there are some examples of Hom-Novikov color Hom-algebras.

  1. (i)

Any $\varepsilon$-commutative Hom-associative color Hom-algebra is a Hom-
Novikov color Hom-algebra.

  2. (ii)

Let $A=A_{0}\oplus A_{1}=<e_{1},e_{2}>\oplus<e_{3},e_{4}>$ be a
$4$-dimensional superspace. Then $A$ is a Hom-Novikov color Hom-algebra with

$\displaystyle\text{the bicharacter}\quad\varepsilon(i,j)=(-1)^{ij},$
$\displaystyle\text{the multiplication:}\quad\begin{array}[t]{ll}e_{1}\cdot
e_{1}=\lambda_{1}e_{2},&e_{1}\cdot e_{3}=\lambda_{2}e_{4},\\\ e_{3}\cdot
e_{3}=\lambda_{3}e_{2},&e_{3}\cdot
e_{1}=\lambda_{4}e_{4},\quad\lambda_{i}\in\mathbb{K}\end{array}$
$\displaystyle\text{the even linear map $\alpha:A\rightarrow A$ defined
by}\begin{array}[t]{lll}&\alpha(e_{1})=-e_{1},&\alpha(e_{2})=e_{1}-e_{2},\\\
&\alpha(e_{3})=e_{4},&\alpha(e_{4})=e_{3}+2e_{4}.\end{array}$

######  Proposition 2.15 ([8]).

Let $\mathcal{A}=(A,\cdot,\varepsilon)$ be a Novikov color Hom-algebra and
$\alpha:\mathcal{A}\rightarrow\mathcal{A}$ be a Novikov color Hom-algebras
morphism. Define $\cdot_{\alpha}:A\times A\rightarrow A$ for all
$x,y\in\mathcal{H}(A)$, by $x\cdot_{\alpha}y=\alpha(x\cdot y)$. Then,
$\mathcal{A}_{\alpha}=(A,\cdot_{\alpha},\varepsilon,\alpha)$ is a Hom-Novikov
color Hom-algebra called the $\alpha$-twist or Yau twist of
$(A,\cdot,\varepsilon)$.

In the following we introduce the notions of bimodule and matched pair of
Novikov color Hom-algebras.

######  Definition 2.16.

Let $(A,\cdot,\varepsilon,\alpha)$ be a Hom-Novikov color Hom-algebra,
$(V,\beta)$ is a pair of $\Gamma$-graded linear space $V$ and an even linear
map $\beta:V\rightarrow V$. Let $l,r:A\rightarrow End(V)$ be two even linear
maps. The quadruple $(l,r,\beta,V)$ is called a bimodule of
$(A,\cdot,\varepsilon,\alpha)$ if for all
$x,y\in\mathcal{H}(A),v\in\mathcal{H}(V)$,

$\displaystyle l(x\cdot
y)\beta(v)-l(\alpha(x))l(y)v=\varepsilon(x,y)\big{(}l(y\cdot
x)\beta(v)-l(\alpha(y))l(x)v\big{)},$ (2.14) $\displaystyle
r(\alpha(y))l(x)v-l(\alpha(x))r(y)v=\varepsilon(x,v)\big{(}r(\alpha(y))r(x)v-r(x\cdot
y)\beta(v)\big{)},$ (2.15) $\displaystyle r(\alpha(y))r(x)v-r(x\cdot
y)\beta(v)=\varepsilon(v,x)\big{(}r(\alpha(y))l(x)v-l(\alpha(x))r(y)v\big{)},$
(2.16) $\displaystyle l(x\cdot y)\beta(v)=\varepsilon(y,v)r(\alpha(y))l(x)v,$
(2.17) $\displaystyle r(\alpha(y))l(x)v=\varepsilon(v,y)l(x\cdot y)\beta(v),$
(2.18) $\displaystyle r(\alpha(y))r(x)v=\varepsilon(x,y)r(\alpha(x))r(y)v.$
(2.19)

######  Proposition 2.17.

Let $(l,r,\beta,V)$ is a bimodule of a Hom-Novikov color Hom-algebra
$(A,\cdot,\varepsilon,\alpha)$. Then the direct sum of $\Gamma$-graded linear
spaces,

$A\oplus V=\bigoplus_{\gamma\in\Gamma}(A\oplus
V)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus V_{\gamma}),$

is turned into a Hom-Novikov color Hom-algebra by defining multiplication in
$A\oplus V$ for all $X_{1}=x_{1}+v_{1}\in A_{\gamma_{1}}\oplus
V_{\gamma_{1}},X_{2}=x_{2}+v_{2}\in A_{\gamma_{2}}\oplus V_{\gamma_{2}}$ by

$\displaystyle(x_{1}+v_{1})\ast(x_{2}+v_{2})$ $\displaystyle=$ $\displaystyle
x_{1}\cdot x_{2}+(l(x_{1})v_{2}+r(x_{2})v_{1}),$ (2.20)
$\displaystyle(\alpha\oplus\beta)(x_{1}+v_{1})$ $\displaystyle=$
$\displaystyle\alpha(x_{1})+\beta(v_{1}).$ (2.21)

###### Proof.

We prove the axioms (2.12) and (2.13) in $A\oplus V$. For all
$X_{i}=x_{i}+v_{i}\in A_{\gamma_{i}}\oplus V_{\gamma_{i}},\ i\in\\{1;2;3\\}$,

$\displaystyle(X_{1}\ast
X_{2})\ast(\alpha+\beta)X_{3}-(\alpha+\beta)X_{1}\ast(X_{2}\ast X_{3})$
$\displaystyle\quad-\varepsilon(X_{1},X_{2})\big{(}(X_{2}\ast
X_{1})\ast(\alpha+\beta)X_{3}-(\alpha+\beta)X_{2}\ast(X_{1}\ast X_{3})\big{)}$
$\displaystyle=\big{(}(x_{1}+v_{1})\ast(x_{2}+v_{2})\big{)}\ast(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle\quad-(\alpha+\beta)(x_{1}+v_{1})\ast((x_{2}+v_{2})\ast(x_{3}+v_{3}))$
$\displaystyle\quad-\varepsilon(X_{1},X_{2})\Big{(}\big{(}(x_{2}+v_{2})\ast(x_{1}+v_{1})\big{)}\ast(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle\quad-(\alpha+\beta)(x_{2}+v_{2})\ast\big{(}(x_{1}+v_{1})\ast(x_{3}+v_{3})\big{)}\Big{)}$
$\displaystyle=\big{(}x_{1}\cdot
x_{2}+l(x_{1})v_{2}+r(x_{2})v_{1}\big{)}\ast(\alpha(x_{2})+\beta(v_{3}))$
$\displaystyle\quad-(\alpha(x_{1})+\beta(v_{1}))\big{(}x_{2}\cdot
x_{3}+l(x_{2})v_{3}+r(x_{3})v_{2}\big{)}$
$\displaystyle\quad-\varepsilon(x_{1},x_{2})\Big{(}\big{(}x_{2}\cdot
x_{1}+l(x_{2})v_{1}+r(x_{1})v_{2}\big{)}\ast(\alpha(x_{3})+\beta(v_{3}))$
$\displaystyle\quad-(\alpha(x_{2})+\beta(v_{2}))\ast\big{(}x_{1}\cdot
x_{3}+l(x_{1})v_{3}+r(x_{3})v_{1}\big{)}\Big{)}$ $\displaystyle=(x_{1}\cdot
x_{2})\cdot\alpha(x_{3})+l(x_{1}\cdot
x_{2})\beta(v_{3})+r(\alpha(x_{3}))l(x_{1})v_{2}+r(\alpha(x_{3}))r(x_{2})v_{1}$
$\displaystyle\quad-\alpha(x_{1})(x_{2}\cdot
x_{3})-l(\alpha(x_{1}))l(x_{2})v_{3}-l(\alpha(x_{1}))r(x_{3})v_{2}-r(x_{2}\cdot
x_{3})\beta(v_{1})$
$\displaystyle\quad-\varepsilon(x_{1},x_{2})\Big{(}(x_{2}\cdot
x_{1})\cdot\alpha(x_{3})+l(x_{2}\cdot
x_{1})\beta(v_{3})+r(\alpha(x_{3}))l(x_{2})v_{1}$
$\displaystyle\quad+r(\alpha(x_{3}))r(x_{1})v_{2}-\alpha(x_{2})(x_{1}\cdot
x_{3})-l(\alpha(x_{2}))l(x_{1})v_{3}-l(\alpha(x_{2}))r(x_{3})v_{1}$
$\displaystyle\hskip 142.26378pt-r(x_{1}\cdot x_{3})\beta(v_{2})\Big{)}$
$\displaystyle=\underbrace{\Big{(}(x_{1}\cdot
x_{2})\cdot\alpha(x_{3})-\alpha(x_{1})(x_{2}\cdot
x_{3})-\varepsilon(x_{1},x_{2})((x_{2}\cdot
x_{1})\cdot\alpha(x_{3})-\alpha(x_{2})\cdot(x_{1}\cdot
x_{3}))\Big{)}}_{\text{$=0$ by \eqref{eqhomNovikov1} in $A$}}$
$\displaystyle+\underbrace{\Big{(}l(x_{1}\cdot
x_{2})\beta(v_{3})-l(\alpha(x_{1}))l(x_{2})v_{3}-\varepsilon(x_{1},x_{2})(l(x_{2}\cdot
x_{1})\beta(v_{3})-l(\alpha(x_{2}))l(x_{1})v_{3})\Big{)}}_{\text{$=0$ by
\eqref{Cond1}}}$
$\displaystyle+\underbrace{\Big{(}r(\alpha(x_{3}))l(x_{1})v_{2}-l(\alpha(x_{1}))r(x_{3})v_{2}-\varepsilon(x_{1},v_{2})(r(\alpha(x_{3}))r(x_{1})v_{2}-r(x_{1}\cdot
x_{3})\beta(v_{2}))\Big{)}}_{\text{$=0$ by \eqref{Cond2}}}$
$\displaystyle+\underbrace{\Big{(}r(\alpha(x_{3}))r(x_{2})v_{1}-r(x_{2}\cdot
x_{3})\beta(v_{1})-\varepsilon(v_{1},x_{2})(r(\alpha(x_{3}))l(x_{2})v_{1}-l(\alpha(x_{2}))r(x_{3})v_{1})\Big{)}}_{\text{$=0$
by \eqref{Cond3}}}$ $\displaystyle\quad=0,$ $\displaystyle(X_{1}\ast
X_{2})\ast(\alpha+\beta)X_{3}-\varepsilon(X_{2},X_{3})(X_{1}\ast
X_{3})\ast(\alpha+\beta)X_{2}$
$\displaystyle=\big{(}(x_{1}+v_{1})\ast(x_{2}+v_{2})\big{)}\ast(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle\quad-\varepsilon(X_{2},X_{3})\big{(}(x_{1}+v_{1})\ast(x_{3}+v_{3})\big{)}\ast(\alpha+\beta)(x_{2}+v_{2})$
$\displaystyle=\big{(}x_{1}\cdot
x_{2}+l(x_{1})v_{2}+r(x_{2})v_{1}\big{)}\ast(\alpha(x_{3})+\beta(v_{3}))$
$\displaystyle\quad-\varepsilon(x_{2},x_{3})\big{(}x_{1}\cdot
x_{3}+l(x_{1})v_{3}+r(x_{3})v_{1}\big{)}\ast(\alpha(x_{2})+\beta(v_{2}))$
$\displaystyle=(x_{1}\cdot x_{2})\cdot\alpha(x_{3})+l(x_{1}\cdot
x_{2})\beta(v_{3})+r(\alpha(x_{3}))l(x_{1})v_{2}+r(\alpha(x_{3}))r(x_{2})v_{1}$
$\displaystyle\quad-\varepsilon(x_{2},x_{3})((x_{1}\cdot
x_{3})\cdot\alpha(x_{2})+l(x_{1}\cdot
x_{3})\beta(v_{2})+r(\alpha(x_{2}))l(x_{1})v_{3}+r(\alpha(x_{2}))r(x_{3})v_{1})$
$\displaystyle=\underbrace{\Big{(}(x_{1}\cdot
x_{2})\cdot\alpha(x_{3})-(x_{1}\cdot
x_{3})\cdot\alpha(x_{2})\Big{)}}_{\text{$=0$ by \eqref{eqhomNovikov2} in
$A$}}+\underbrace{\Big{(}l(x_{1}\cdot
x_{2})\beta(v_{3})-\varepsilon(x_{2},x_{3})r(\alpha(x_{2}))l(x_{1})v_{3}\Big{)}}_{\text{$=0$
by \eqref{Cond4}}}$
$\displaystyle\quad+\underbrace{\Big{(}r(\alpha(x_{2}))l(x_{1})v_{3}-\varepsilon(v_{2},x_{3})l(x_{1}\cdot
x_{3})\beta(v_{2})\Big{)}}_{\text{$=0$ by \eqref{Cond5}}}$
$\displaystyle\hskip
85.35826pt+\underbrace{\Big{(}r(\alpha(x_{3}))r(x_{2})v_{1}-\varepsilon(x_{2},x_{3})r(\alpha(x_{2}))r(x_{3})v_{1}\Big{)}}_{\text{$=0$
by \eqref{Cond6}}}=0.\qed$

The Hom-Novikov color Hom-algebra constructed in Proposition 2.17 is denoted
by $(A\oplus V,\ast,\varepsilon,\alpha+\beta)$ or
$A\times_{l,r,\alpha,\beta}V.$

######  Example 2.18.

Let $(A,\cdot,\varepsilon,\alpha)$ be a Hom-Novikov color Hom-algebra. Then
$(L,R,\alpha,A)$ is a bimodule of $(A,\cdot,\varepsilon,\alpha)$, where
$L(x)y=x\cdot y$ and $R(x)y=y\cdot x$ for all $x,y\in\mathcal{H}(A)$, called
the regular bimodule of $(A,\cdot,\varepsilon,\alpha)$.

######  Proposition 2.19.

Let $(A,\cdot_{A},\varepsilon,\alpha)$ and $(B,\cdot_{B},\varepsilon,\beta)$
be two Hom-Novikov color Hom-algebras. Suppose there are even linear maps
$l_{A},r_{A}:A\rightarrow End(B)$ and $l_{B},r_{B}:B\rightarrow End(A)$ such
that the quadruple $(l_{A},r_{A},\beta,B)$ is a bimodule of $A,$ and
$(l_{B},r_{B},\alpha,A)$ is a bimodule of $B,$ satisfying, for any
$x,y\in\mathcal{H}(A),a,b\in\mathcal{H}(B)$, the following conditions:

$\displaystyle\begin{array}[]{l}r_{A}(\alpha(x))(a\cdot_{B}b)-\beta(a)\cdot_{B}(r_{A}(x)b)-r_{A}(l_{B}(b)x)\beta(a)\\\
\quad=\varepsilon(a,b)\big{(}r_{A}(\alpha(x))(b\cdot_{B}a)-\beta(b)\cdot_{B}(r_{A}(x)a)-r_{A}(l_{B}(a)x)\beta(b)\big{)},\end{array}$
(2.24)
$\displaystyle\begin{array}[]{l}(r_{A}(x)a)\cdot_{B}\beta(b)+l_{A}(l_{B}(a)x)\beta(b)-\beta(a)\cdot_{B}(l_{A}(x)b)-r_{A}(r_{B}(b)x)\beta(a)\\\
\quad=\varepsilon(a,x)\big{(}(l_{A}(x)a)\cdot_{B}\beta(b)+l_{A}(r_{B}(a)x)\beta(b)-l_{A}(\alpha(x))(a\cdot_{B}b)\big{)},\end{array}$
(2.27)
$\displaystyle\begin{array}[]{l}(l_{A}(x)a)\cdot_{B}\beta(b)-l_{A}(r_{B}(a)x)\beta(b)-l_{A}(\alpha(x))(a\cdot_{B}b)\\\
\quad=\varepsilon(x,a)\big{(}(r_{A}(x)a)\cdot_{B}\beta(b)+l_{A}(l_{B}(a)x)\beta(b)-\beta(a)\cdot_{B}(l_{A}(x)b)\\\
\hskip 227.62204pt\big{.}-r_{A}(r_{B}(b)x)\beta(a)\big{)},\end{array}$ (2.31)
$\displaystyle\begin{array}[]{l}r_{B}(\beta(a))(x\cdot_{A}y)-\alpha(x)\cdot_{A}(r_{B}(a)y)-r_{B}(l_{A}(y)a)\alpha(x)\\\
\quad=\varepsilon(x,y)\big{(}r_{B}(\beta(a))(y\cdot_{A}x)-\alpha(y)\cdot_{A}(r_{B}(a)x)-r_{B}(l_{A}(x)a)\alpha(y)\big{)},\end{array}$
(2.34)
$\displaystyle\begin{array}[]{l}(r_{B}(a)x)\cdot_{A}\alpha(y)+l_{B}(l_{A}(x)a)\alpha(y)-\alpha(x)\cdot_{A}(l_{B}(a)y)-r_{B}(r_{A}(y)a)\alpha(x)\\\
\quad=\varepsilon(x,a)\big{(}(l_{B}(a)x)\cdot_{A}\alpha(y)+l_{B}(r_{A}(x)a)\alpha(y)-l_{B}(\beta(a))(x\cdot_{A}y)\big{)},\end{array}$
(2.37)
$\displaystyle\begin{array}[]{l}(l_{B}(a)x)\cdot_{A}\alpha(y)-l_{B}(r_{A}(x)a)\alpha(y)-l_{B}(\beta(a))(x\cdot_{A}y)\\\
\quad=\varepsilon(a,x)\big{(}(r_{B}(a)x)\cdot_{A}\alpha(y)+l_{B}(l_{A}(x)a)\alpha(y)-\alpha(x)\cdot_{A}(l_{B}(a)y)\big{.}\\\
\hskip 227.62204pt\big{.}-r_{B}(r_{A}(y)a)\alpha(x)\big{)}.\end{array}$ (2.41)

Then, $(A,B,l_{A},r_{A},\beta,l_{B},r_{B},\alpha)$ is called a matched pair of
Hom-Novikov color Hom-algebras. In this case, there is a Hom-Novikov color
Hom-algebra structure on the direct sum $A\oplus
B=\bigoplus_{\gamma\in\Gamma}(A\oplus
B)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus B_{\gamma}),$ of the
underlying $\Gamma$-graded linear spaces of $A$ and $B$ given for all $x+a\in
A_{\gamma_{1}}\oplus B_{\gamma_{1}},y+b\in A_{\gamma_{2}}\oplus
B_{\gamma_{2}}$ by

$\begin{array}[]{l}(x+a)\cdot(y+b)=\big{(}x\cdot_{A}y+l_{B}(a)y+r_{B}(b)x\big{)}+\big{(}a\cdot_{B}b+l_{A}(x)b+r_{A}(y)a\big{)},\\\
(\alpha\oplus\beta)(x+a)=\alpha(x)+\beta(a).\end{array}$ (2.42)

We denote this Hom-Novikov color Hom-algebra by $(A\bowtie
B,\cdot,\varepsilon,\alpha+\beta)$ or
$A\bowtie^{l_{A},r_{A},\beta}_{l_{B},r_{B},\alpha}B.$

###  2.3 On Hom-Lie color Hom-algebras

######  Definition 2.20 ([1, 24, 19, 55, 56, 69, 80]).

A Hom-Lie color Hom-algebra is a quadruple
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ consisting of a $\Gamma$-graded vector
space $A$, a bi-character $\varepsilon$, an even bilinear mapping
$[\cdot,\cdot]:A\times A\rightarrow A$, (i.e. $[A_{a},A_{b}]\subseteq
A_{a+b}$, for all $a,b\in\Gamma$) and an even homomorphism
$\alpha:A\rightarrow A$ such that for homogeneous elements $x,y,z\in A$ we
have

$\displaystyle[x,y]=-\varepsilon(x,y)[y,x],$
$\displaystyle\quad\text{$\varepsilon$-skew symmetry},$ (2.43)
$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)[\alpha(x),[y,z]]=0,$
$\varepsilon$-Hom-Jacobi identity (2.44)

where $\circlearrowleft_{x,y,z}$ denotes the cyclic sum over $(x,y,z)$.

######  Remark 2.21.

Hom-Lie color Hom-algebras contain ordinary Lie color algebras, Lie
superalgebras and Lie algebras, as well as Hom-Lie superalgebras and Hom-Lie
algebras for specific choices of the twisting map, grading group and
commutation factor.

  1. (i)

When $\alpha=id_{A}$, one recovers Lie color algebras, and in particular if
the grading group is $\mathbb{Z}_{2}$ and the commutation factor is defined as
$\varepsilon(i,j)=(-1)^{ij}$ for all $i,j\in\mathbb{Z}$, then one gets Lie
superalgebras. [7, 22, 23, 50, 59, 64, 65, 66].

  2. (ii)

When $\alpha=id_{A}$ and $A$ is trivially graded, by the group with one
element, we recover Lie algebras.

  3. (iii)

When $A$ is trivially graded, while $\alpha$ is an arbitrary linear map, we
recover Hom-Lie algebras [37, 54, 56, 55, 45], and if $A$ is graded by the
group of two elements $\mathbb{Z}_{2}$, while $\alpha$ is an arbitrary even
linear map, and the commutation factor is defined as
$\varepsilon(i,j)=(-1)^{ij}$ for all $i,j\in\mathbb{Z}_{2}$, then we get Hom-
Lie superalgebras.

######  Proposition 2.22 ( [8]).

Let $(A,\cdot,\varepsilon,\alpha)$ be a Hom-Novikov color Hom-algebra. Then,
there exists a Hom-Lie color algebra structure on $A$ given for all
$x,y\in\mathcal{H}(A)$ by

$[x,y]=x\cdot y-\varepsilon(x,y)y\cdot x.$ (2.45)

######  Example 2.23.

Let $A=A_{0}\oplus A_{1}=<e_{1}>\oplus<e_{2},e_{3}>$ be a $3$-dimensional
superspace. The quintuple $(A,\cdot,\varepsilon,\alpha)$ is a Hom-Novikov
color Hom-algebra with

$\displaystyle\text{the multiplication:}\quad e_{1}\cdot e_{2}=e_{3},\quad
e_{2}\cdot e_{1}=-e_{3},$ $\displaystyle\text{the
bicharacter:}\quad\varepsilon(i,j)=(-1)^{ij},$ $\displaystyle\text{the even
linear map $\alpha:A\rightarrow A$ defined
by}\quad\begin{array}[t]{l}\alpha(e_{1})=-2e_{1},\quad\alpha(e_{2})=e_{3},\\\
\alpha(e_{3})=e_{2}-e_{3}.\end{array}$

Therefore, by Proposition 2.22, $(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a
Hom-Lie color Hom-algebra with

$[e_{1},e_{2}]=-[e_{2},e_{1}]=2e_{3}.$

######  Proposition 2.24 ([1]).

Let $\mathcal{A}=(A,[\cdot,\cdot],\varepsilon)$ be a Hom-Lie color Hom-algebra
and $\alpha:\mathcal{A}\rightarrow\mathcal{A}$ be a Hom-Lie color Hom-algebras
morphism. Define $[\cdot,\cdot]_{\alpha}:A\times A\rightarrow A$ for all
$x,y\in A$, by $[x,y]_{\alpha}=\alpha([x,y])$. Then,
$\mathcal{A}_{\alpha}=(A,[\cdot,\cdot]_{\alpha},\varepsilon,\alpha)$ is a Hom-
Lie color Hom-algebra called the $\alpha$-twist or Yau twist of
$(A,[\cdot,\cdot],\varepsilon)$.

######  Definition 2.25.

Let $(A,[\cdot,\cdot],\varepsilon,\alpha)$ be a Hom-Lie color Hom-algebra,
$(V,\beta)$ is a pair of $\Gamma$-graded linear space $V$ and an even linear
map $\beta:V\rightarrow V$. Let $\rho:A\rightarrow End(V)$ an even linear map.
The triple $(\rho,\beta,V)$ is called a representation of
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ if for all
$x,y\in\mathcal{H}(A),v\in\mathcal{H}(V)$,

$\displaystyle\rho([x,y])\beta(v)=\rho(\alpha(x))\rho(y)v-\varepsilon(x,y)\rho(\alpha(y))\rho(x)v.$
(2.46)

######  Proposition 2.26.

Let $(\rho,\beta,V)$ is a representation of a Hom-Lie color Hom-algebra
$(A,[\cdot,\cdot],\varepsilon,\alpha)$. Then the direct sum of $\Gamma$-graded
linear spaces,

$A\oplus V=\bigoplus_{\gamma\in\Gamma}(A\oplus
V)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus V_{\gamma}),$

is turned into a Hom-Lie color Hom-algebra by defining multiplication in
$A\oplus V$ for all $X_{1}=x_{1}+v_{1}\in A_{\gamma_{1}}\oplus
V_{\gamma_{1}},X_{2}=x_{2}+v_{2}\in A_{\gamma_{2}}\oplus V_{\gamma_{2}}$ by

$\displaystyle[x_{1}+v_{1},x_{2}+v_{2}]^{\prime}$ $\displaystyle=$
$\displaystyle[x_{1},x_{2}]+\rho(x_{1})v_{2}-\varepsilon(v_{1},x_{2})\rho(x_{2})v_{1},$
(2.47) $\displaystyle(\alpha\oplus\beta)(x_{1}+v_{1})$ $\displaystyle=$
$\displaystyle\alpha(x_{1})+\beta(v_{1}).$ (2.48)

The Hom-Lie color Hom-algebra constructed in previous Proposition is denoted
by $(A\oplus V,[\cdot,\cdot]^{\prime},\varepsilon,\alpha+\beta)$ or
$A\ltimes_{\rho,\alpha,\beta}V.$

######  Example 2.27.

Let $(A,[\cdot,\cdot],\varepsilon,\alpha)$ be a Hom-Lie algebra. Then
$(ad,\alpha,A)$ is a representation of $(A,[\cdot,\cdot],\varepsilon,\alpha)$,
where $ad(x)y=[x,y]$ for all $x,y\in\mathcal{H}(A)$, called the adjoint
representation of $(A,[\cdot,\cdot],\varepsilon,\alpha)$.

Now, we introduce the notion of matched pair of Hom-Lie color Hom-algebra

######  Proposition 2.28.

Suppose that $(A,[\cdot,\cdot]_{A},\varepsilon,\alpha)$ and
$(B,[\cdot,\cdot]_{B},\varepsilon,\beta)$ are Hom-Lie color Hom-algebras, and
there are even linear maps $\rho_{A}:A\rightarrow End(B)$ and
$\rho_{B}:B\rightarrow End(A)$ such that $(\rho_{A},\beta,B)$ is a
representation of $A$ and $(\rho_{B},\alpha,A)$ is a representation of $B$
satisfying for any $x,y\in\mathcal{H}(A),~{}a,b\in\mathcal{H}(B)$,

$\displaystyle\begin{array}[]{r}\varepsilon(x,a)\big{(}\rho_{A}(\rho_{B}(a)x)\beta(b)-[\beta(a),\rho_{A}(x)b]_{B}\big{)}+\varepsilon(a+x,b)\big{(}[\beta(b),\rho_{A}(x)a]_{B}\\\
-\rho_{A}(\rho_{B}(b)x)\beta(a)\big{)}+\rho_{A}(\alpha(x))([a,b]_{B})=0,\end{array}$
(2.51)
$\displaystyle\begin{array}[]{r}\varepsilon(a,x)\big{(}\rho_{B}(\rho_{A}(x)a)\alpha(y)-[\alpha(x),\rho_{B}(a)y]_{A}\big{)}+\varepsilon(x+a,y)\big{(}[\alpha(y),\rho_{B}(a)x]_{A}\\\
-\rho_{B}(\rho_{A}(y)a)\alpha(x)\big{)}+\rho_{B}(\beta(a))([x,y]_{A})=0.\end{array}$
(2.53)

Then, $(A,B,\rho_{A},\beta,\rho_{B},\alpha)$ is called a matched pair of Hom-
Lie color Hom-algebras. In this case, there is a Hom-Lie color Hom-algebra
structure on the linear space of the underlying $\Gamma$-graded linear spaces
of $A$ and $B$,

$A\oplus B=\bigoplus_{\gamma\in\Gamma}(A\oplus
B)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus B_{\gamma}),$

given for all $x+a\in A_{\gamma_{1}}\oplus B_{\gamma_{1}},y+b\in
A_{\gamma_{2}}\oplus B_{\gamma_{2}}$ by

$\displaystyle\begin{array}[]{ll}[x+a,y+b]=&[x,y]_{A}+\rho_{A}(x)b-\varepsilon(a,y)\rho_{A}(y)a\\\
&\hskip
28.45274pt+[a,b]_{B}+\rho_{B}(a)y-\varepsilon(x,b)\rho_{B}(b)x,\end{array}$
(2.56) $\displaystyle(\alpha\oplus\beta)(x+a)=\alpha(x)+\beta(a).$ (2.57)

##  3 Admissible Hom-Novikov-Poisson color Hom-algebras

In this Section, we recall the main result of Hom-Novikov-Poisson color Hom-
algebras in [5] and we introduce their notions of bimodules and matched pairs.
Next, we introduce the dfinition of admissible Hom-Novikov-Poisson color Hom-
algebras and we give some explicit constructions. Finally, we show that the
much larger class of admissible Hom-Novikov-Poisson color Hom-algebras is also
closed under tensor products

###  3.1 Constructions and bimodules of (admissible) Hom-Novikov-Poisson color
Hom-algebras

######  Definition 3.1 ([5]).

Hom-Novikov-Poisson color Hom-algebras are defined as quintuples
$(A,\cdot,\diamond,\varepsilon,\alpha)$ consisting of an
$\varepsilon$-commutative Hom-associative color Hom-algebra
$(A,\cdot,\varepsilon,\alpha)$ and a Hom-Novikov color Hom-algebra
$(A,\diamond,\varepsilon,\alpha)$ obeying for $x,y,z\in\mathcal{H}(A)$,

$\displaystyle(x\cdot y)\diamond\alpha(z)$
$\displaystyle=\varepsilon(y,z)(x\diamond z)\cdot\alpha(y),$ (3.1)
$\displaystyle(x\diamond y)\cdot\alpha(z)-\alpha(x)\diamond(y\cdot z)$
$\displaystyle=\varepsilon(x,y)\big{(}(y\diamond
x)\cdot\alpha(z)-\alpha(y)\diamond(x\cdot z)\big{)}.$ (3.2)

A Hom-Novikov-Poisson color Hom-algebra is called multiplicative if the linear
map $\alpha:A\rightarrow A$ is multiplicative with respect to $\cdot$ and
$\diamond$, that is, for all $x,y\in\mathcal{H}(A)$,

$\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y),\quad\alpha(x\diamond
y)=\alpha(x)\diamond\alpha(y).$

######  Remark 3.2.

Hom-Novikov-Poisson color Hom-algebras contain Novikov-Poisson color Hom-
algebras, Hom-Novikov-Poisson algebras and Novikov-Poisson algebras for
special choices of the twising map and grading group.

  1. (i)

When $\alpha=id$, we get Novikov-Poisson color Hom-algebra.

  2. (ii)

When $\Gamma=\\{e\\}$ and $\alpha\neq id$, we get Hom-Novikov-Poisson algebra
[85].

  3. (iii)

When $\Gamma=\\{e\\}$ and $\alpha=id$, we get Novikov-Poisson algebra [77,
78].

######  Example 3.3.

Let $A=A_{0}\oplus A_{1}=<e_{1},e_{2}>\oplus<e_{3},e_{4}>$ be a
$4$-dimensional superspace. Then $(A,\cdot,\diamond,\varepsilon,\alpha)$ is a
Hom-Novikov-Poisson color Hom-algebra with

$\displaystyle\text{the bicharacter}\quad\varepsilon(i,j)=(-1)^{ij},$
$\displaystyle\text{the multiplications:}\begin{array}[t]{lllll}e_{2}\cdot
e_{2}=\lambda_{1}e_{1},\quad e_{2}\cdot e_{4}=e_{4}\cdot
e_{2}=\lambda_{2}e_{3},\quad\lambda_{i}\in\mathbb{K},\\\ e_{2}\diamond
e_{4}=\mu_{2}e_{3},\quad e_{4}\diamond e_{2}=\mu_{3}e_{3},\quad e_{4}\diamond
e_{4}=\mu_{4}e_{1},\quad\mu_{i}\in\mathbb{K},\end{array}$
$\displaystyle\text{the even linear map $\alpha:A\rightarrow A$ defined
by}\begin{array}[t]{l}\alpha(e_{1})=2e_{1},\quad\alpha(e_{2})=e_{2}-e_{1},\\\
\alpha(e_{3})=-e_{4},\quad\alpha(e_{4})=e_{3}.\end{array}$

######  Definition 3.4.

Let $(A,\cdot,\diamond,\alpha)$ and
$(A^{\prime},\cdot^{\prime},\diamond^{\prime},\alpha^{\prime})$ be Hom-
Novikov-Poisson color Hom-algebras. A linear map of degree zero
$f:A\rightarrow A^{\prime}$ is a Hom-Novikov-Poisson color Hom-algebra
morphism if

$\displaystyle\cdot^{\prime}\circ(f\otimes
f)=f\circ\cdot,\quad\diamond^{\prime}\circ(f\otimes f)=f\circ\diamond\mbox{
and }f\circ\alpha=\alpha^{\prime}\circ f.$

######  Theorem 3.5 ([5]).

Let $\mathcal{A}=(A,\cdot,\diamond,\varepsilon)$ be a Hom-Novikov-Poisson
color Hom-algebra and $\alpha:\mathcal{A}\rightarrow\mathcal{A}$ be a Hom-
Novikov-Poisson color Hom-algebras morphism. Define $\cdot_{\alpha},\
\diamond_{\alpha}:A\times A\rightarrow A$ for all $x,y\in\mathcal{H}(A)$, by
$x\cdot_{\alpha}y=\alpha(x\cdot y)$ and $x\diamond_{\alpha}y=\alpha(x\diamond
y)$. Then,
$\mathcal{A}_{\alpha}=(A_{\alpha}=A,\cdot_{\alpha},\diamond_{\alpha},\varepsilon,\alpha)$
is a Hom-Novikov-Poisson color Hom-algebra called the $\alpha$-twist or Yau
twist of $(A,\cdot,\diamond,\varepsilon)$.

######  Definition 3.6.

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be a Hom-Novikov-Poisson color
Hom-algebra. A bimodule of $(A,\cdot,\diamond,\varepsilon,\alpha)$ is a
quintuple $(s,l,r,\beta,V)$ such that $(s,\beta,V)$ is a bimodule of the
$\varepsilon$-commutative Hom-associative color Hom-algebra
$(A,\cdot,\varepsilon,\alpha)$ and $(l,r,\beta,V)$ is a bimodule of the Hom-
Novikov color Hom-algebra $(A,\diamond,\varepsilon,\alpha)$ and satisfying,
for all $x,y\in\mathcal{H}(A),v\in\mathcal{H}(V)$,

$\displaystyle l(x\cdot y)\beta(v)=\varepsilon(x,y)s(\alpha(y))l(x)v,$ (3.3)
$\displaystyle r(\alpha(y))s(x)v=\varepsilon(v,y)s(x\diamond y)\beta(v),$
(3.4) $\displaystyle r(\alpha(y))s(x)v=s(\alpha(x))r(y)v,$ (3.5)
$\displaystyle s(x\diamond
y)\beta(v)-l(\alpha(x))s(y)v=\varepsilon(x,y)\big{(}s(y\diamond
x)\beta(v)-l(\alpha(y))s(x)v\big{)},$ (3.6)
$\displaystyle\begin{array}[]{l}\varepsilon(x+v,y)\big{(}s(\alpha(y))l(x)v-\varepsilon(x,v)s(\alpha(y))r(x)v\big{)}=\varepsilon(v,y)l(\alpha(x))s(y)v\\\
\hskip 256.0748pt-\varepsilon(x,v)r(x\cdot y)\beta(v).\end{array}$ (3.9)

######  Proposition 3.7.

Let $A\oplus V=\bigoplus_{\gamma\in\Gamma}(A\oplus
V)_{\gamma}=\bigoplus_{\gamma\in\Gamma}(A_{\gamma}\oplus V_{\gamma})$, the
direct sum of $\Gamma$-graded linear spaces. Then, $(A\oplus
V,\cdot^{\prime},\diamond^{\prime},\varepsilon,\alpha+\beta)$ is a Hom-
Novikov-Poisson color Hom-algebra, where $(A\oplus
V,\cdot^{\prime},\varepsilon,\alpha+\beta)$ is the semi-direct product
$\varepsilon$-commutative Hom-associative color Hom-algebra
$A\ltimes_{s,\alpha,\beta}V$ and $(A\oplus
V,\diamond^{\prime},\varepsilon,\alpha+\beta)$ is the semi-direct product Hom-
Novikov color Hom-algebra $A\ltimes_{l,r,\alpha,\beta}V$.

###### Proof.

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be a Hom-Novikov-Poisson color
Hom-algebra, and $(s,l,r,\beta,V)$ be a bimodule. By Proposition 2.9 and
Proposition 2.17, $(A\oplus V,\cdot^{\prime},\varepsilon,\alpha+\beta)$ is a
$\varepsilon$-commutative Hom-associative color Hom-algebra, and $(A\oplus
V,\diamond^{\prime},\varepsilon,\alpha+\beta)$ is a Hom-Novikov color Hom-
algebra respectively. Now, we show that the compatibility conditions
(3.1)-(3.2) are satisfied. For all $X_{i}=x_{i}+v_{i}\in A_{\gamma_{i}}\oplus
V_{\gamma_{i}},\ i=1,2,3$ we have

$\displaystyle(X_{1}\cdot^{\prime}X_{2}\big{)}\diamond^{\prime}(\alpha+\beta)X_{3}-\varepsilon(X_{2},X_{3})(X_{1}\diamond^{\prime}X_{3})\cdot^{\prime}(\alpha+\beta)X_{2}$
$\displaystyle\quad=\big{(}(x_{1}+v_{1})\cdot^{\prime}(x_{2}+v_{2})\big{)}\diamond^{\prime}(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle\quad\quad-\varepsilon(X_{2},X_{3})\big{(}(x_{1}+v_{1})\diamond^{\prime}(x_{3}+v_{3})\big{)}\cdot^{\prime}(\alpha+\beta)(x_{2}+v_{2})$
$\displaystyle\quad=\big{(}x_{1}\cdot
x_{2}+s(x_{1})v_{2}+\varepsilon(v_{1},x_{2})s(x_{2})v_{1}\big{)}\diamond^{\prime}(\alpha(x_{3})+\beta(v_{3}))$
$\displaystyle\quad\quad-\varepsilon(x_{2},x_{3})\big{(}(x_{1}\diamond
x_{3}+l(x_{1})v_{3}+r(x_{3})v_{1}\big{)}\cdot^{\prime}(\alpha(x_{2})+\beta(v_{2}))$
$\displaystyle\quad=(x_{1}\cdot x_{3})\diamond\alpha(x_{3})+l(x_{1}\cdot
x_{2})\beta(v_{3})+r(\alpha(x_{3}))s(x_{1})v_{2}$
$\displaystyle\quad\quad+\varepsilon(v_{1},x_{2})r(\alpha(x_{3}))s(x_{2})v_{1}-\varepsilon(x_{2},x_{3})\big{(}x_{1}\diamond
x_{3})\cdot\alpha(x_{2})+s(x_{1}\diamond x_{2})\beta(v_{2})\big{)}$
$\displaystyle\quad\quad-\varepsilon(x_{1},x_{2})\big{(}s(\alpha(x_{2}))l(x_{1})v_{3}+s(\alpha(x_{2}))r(x_{3})v_{1}\big{)}$
$\displaystyle\quad=\underbrace{\Big{(}(x_{1}\cdot
x_{2})\diamond\alpha(x_{3})-\varepsilon(x_{2},x_{3})(x_{1}\diamond
x_{3})\cdot\alpha(x_{2})\Big{)}}_{\text{$=0$ by
\eqref{HomNovPoisscoloralg1}}}$
$\displaystyle\quad\quad+\underbrace{\Big{(}l(x_{1}\cdot
x_{2})\beta(v_{3})-\varepsilon(x_{1},x_{2})s(\alpha(x_{2}))l(x_{1})v_{3}\Big{)}}_{\text{$=0$
by \eqref{Condit1}}}$
$\displaystyle\quad\quad+\underbrace{\Big{(}r(\alpha(x_{3}))s(x_{1})v_{2}-\varepsilon(x_{2},x_{3})s(x_{1}\diamond
x_{3})\beta(v_{2})\Big{)}}_{\text{$=0$ by \eqref{Condit2} and Remark
\ref{remarkepsilon}}}$
$\displaystyle\quad\quad+\underbrace{\varepsilon(x_{1},x_{2})\Big{(}r(\alpha(x_{3}))s(x_{2})v_{1}-s(\alpha(x_{2}))r(x_{3})v_{1}\Big{)}}_{\text{$=0$
by \eqref{Condit3}}}=0,$
$\displaystyle(X_{1}\diamond^{\prime}X_{2})\cdot^{\prime}(\alpha+\beta)X_{3}-(\alpha+\beta)X_{1}\diamond^{\prime}(X_{2}\cdot^{\prime}X_{3})$
$\displaystyle-\varepsilon(X_{1},X_{2})\big{(}(X_{2}\diamond^{\prime}X_{1})\cdot^{\prime}(\alpha+\beta)X_{3}-(\alpha+\beta)X_{2}\diamond^{\prime}(X_{1}\cdot^{\prime}X_{3})\big{)}$
$\displaystyle\quad=\big{(}(x_{1}+v_{1})\diamond^{\prime}(x_{2}+v_{2})\big{)}\cdot^{\prime}(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle\quad\quad-(\alpha+\beta)(x_{1}+v_{1})\diamond^{\prime}\big{(}(x_{2}+v_{2})\cdot^{\prime}(x_{3}+v_{3})\big{)}$
$\displaystyle\quad\quad-\varepsilon(X_{1},X_{2})\big{(}(x_{2}+v_{2})\diamond^{\prime}(x_{1}+v_{1})\big{)}\cdot^{\prime}(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle\quad\quad-(\alpha+\beta)(x_{2}+v_{2})\diamond^{\prime}\big{(}(x_{1}+v_{1})\cdot^{\prime}(x_{3}+v_{3})\big{)}$
$\displaystyle\quad=(x_{1}\diamond x_{2})\alpha(x_{3})+s(x_{1}\diamond
x_{2})\beta(v_{3})+\varepsilon(x_{1}+x_{2},v_{3})s(\alpha(x_{3}))l(x_{1})v_{2}$
$\displaystyle\quad\quad+\varepsilon(x_{1}+x_{2},v_{3})s(\alpha(x_{3}))r(x_{2})v_{1}-\alpha(x_{1})\diamond(x_{2}\cdot
x_{3})$
$\displaystyle\quad\quad-l(\alpha(x_{1}))s(x_{2})v_{3}+\varepsilon(v_{2},x_{3})l(\alpha(x_{1}))s(x_{3})v_{2}+r(x_{2}\cdot
x_{3})\beta(v_{1})$
$\displaystyle\quad\quad-\varepsilon(x_{1},x_{2})\Big{(}x_{2}\diamond
x_{1})\cdot\alpha(x_{3})+s(x_{2}\diamond
x_{1})\beta(v_{3})+\varepsilon(x_{1}+x_{2},v_{3})s(\alpha(x_{3}))l(x_{2})v_{1}$
$\displaystyle\quad\quad+\varepsilon(x_{1}+x_{2},v_{3})s(\alpha(x_{3}))r(x_{1})v_{2}-\alpha(x_{2})\diamond(x_{1}\cdot
x_{3})-l(\alpha(x_{2}))s(x_{1})v_{3}$
$\displaystyle\quad\quad+\varepsilon(v_{1},x_{3})l(\alpha(x_{2}))s(x_{3})v_{1}+r(x_{1}\cdot
x_{3})\beta(v_{2})\Big{)}$
$\displaystyle\quad=\underbrace{\Big{(}(x_{1}\diamond
x_{2})\cdot\alpha(x_{3})-\alpha(x_{2})\diamond(x_{2}\cdot x_{3})}$
$\displaystyle\hskip
113.81102pt-\underbrace{\varepsilon(x_{1},x_{2})\big{(}(x_{2}\diamond
x_{1})\cdot\alpha(x_{3})-\alpha(x_{2})\diamond(x_{1}\cdot
x_{3}))\big{)}\Big{)}}_{\text{$=0$ by \eqref{HomNovPoisscoloralg2}}}$
$\displaystyle\quad\quad+\underbrace{\Big{(}s(x_{1}\diamond
x_{2})\beta(v_{3})-l(\alpha(x_{1}))s(x_{2})v_{3}}$ $\displaystyle\hskip
113.81102pt-\underbrace{\varepsilon(x_{1},x_{2})(s(x_{2}\diamond
x_{1})\beta(v_{3})-l(\alpha(x_{2}))s(x_{1})v_{3}\Big{)}}_{\text{$=0$ by
\eqref{Condit4}}}$
$\displaystyle\quad\quad+\underbrace{\Big{(}\varepsilon(x_{1}+x_{2},v_{3})\big{(}s(\alpha(x_{3}))l(x_{1})v_{2}-\varepsilon(x_{1},x_{2})s(\alpha(x_{3}))r(x_{1})v_{2}\big{)}}$
$\displaystyle\hskip
142.26378pt-\underbrace{\varepsilon(x_{1},x_{2})r(x_{1}\cdot
x_{3})\beta(v_{2})+\varepsilon(x_{2},v_{3})l(\alpha(x_{1}))s(x_{3})v_{2}\Big{)}}_{\text{$=0$
by \eqref{Condit5} and Remark \ref{remarkepsilon}}}$
$\displaystyle\quad\quad-\underbrace{\varepsilon(x_{1},x_{2})\Big{(}\varepsilon(x_{1}+x_{2},v_{3})\big{(}s(\alpha(x_{3}))l(\alpha(x_{2}))v_{1}-\varepsilon(x_{2},x_{1})s(\alpha(x_{3}))r(x_{2})v_{1}\big{)}}$
$\displaystyle\hskip
113.81102pt-\underbrace{\varepsilon(x_{1},x_{3})l(\alpha(x_{2}))s(x_{3})v_{1}+\varepsilon(x_{2},x_{1})r(x_{2}\cdot
x_{3})\beta(v_{1})\Big{)}}_{\text{$=0$ by \eqref{Condit5} and Remark
\ref{remarkepsilon}}}=0.$

Hence, $(A\oplus
V,\cdot^{\prime},[\cdot,\cdot]^{\prime},\varepsilon,\alpha+\beta)$ is a Hom-
Novikov-Poisson color Hom-algebra. ∎

The Hom-Novikov-Poisson color Hom-algebra constructed in previous Proposition
is denoted by $A\ltimes_{s,l,r,\alpha,\beta}V.$

######  Examples 3.8.

  1. (i)

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be a Hom-Novikov-Poisson color
Hom-algebra, and let $S(x)y=x\cdot y=\varepsilon(x,y)y\cdot x$,
$L(x)y=x\diamond y$ and $R(x,y)=y\diamond x$, for all $x,y\in\mathcal{H}(A)$.
Then, $(S,L,R,\alpha,A)$ is a bimodule of
$(A,\cdot,\diamond,\varepsilon,\alpha)$, called the regular bimodule of
$(A,\cdot,\diamond,\varepsilon,\alpha)$.

  2. (ii)

If
$f:\mathcal{A}=(A,\cdot_{1},\diamond_{1},\varepsilon,\alpha)\longrightarrow(A^{\prime},\cdot_{2},\diamond_{2},\varepsilon,\beta)$
is a Hom-Novikov-Poisson color Hom-algebras morphism, then
$(s,l,r,\beta,A^{\prime})$ becomes a bimodule of $\mathcal{A}$ via $f$, that
is, for all $(x,y)\in\mathcal{H}(A)\times\mathcal{H}(A^{\prime})$,
$s(x)y=f(x)\cdot_{2}y,$ $l(x)y=f(x)\diamond_{2}y,$ $r(x)y=y\diamond_{2}f(x).$

######  Theorem 3.9.

Suppose that $\mathcal{A}=(A,\cdot_{A},\diamond_{A},\varepsilon,\alpha)$ and
$\mathcal{B}=(B,\cdot_{B},\diamond_{B},\varepsilon,\beta)$ are two Hom-
Novikov-Poisson color Hom-algebras, and there are even linear maps
$s_{A},l_{A},r_{A}:A\rightarrow End(B)$ and $s_{B},l_{B},r_{B}:B\rightarrow
End(A)$ such that $A\bowtie^{s_{A},\beta}_{s_{B},\alpha}B$ is a matched pair
of $\varepsilon$-commutative Hom-associative color Hom-algebras,
$A\bowtie^{l_{A},r_{A},\beta}_{l_{B},r_{B},\alpha}B$ is a matched pair of Hom-
Novikov color Hom-algebras, and for all
$x,y\in\mathcal{H}(A),~{}a,b\in\mathcal{H}(B)$, the following equalities hold:

$\displaystyle\
r_{A}(\alpha(x))(a\cdot_{B}b)=\varepsilon(b,x)\big{(}r_{A}(x)a\cdot_{B}\beta(b)+s_{A}(l_{B}(a)x)\beta(b),$
(3.10) $\displaystyle\
l_{A}(s_{B}(a)x)\beta(b)+\varepsilon(a,x)s(x)a\diamond_{B}\beta(b)=\varepsilon(x,b)\big{(}\varepsilon(a+b,x)s(\alpha(x))(a\diamond_{B}b)\big{)},$
(3.11)
$\displaystyle\begin{array}[]{l}\varepsilon(a,x)l_{A}(s_{B}(a)x)\beta(b)+s_{A}(x)a\diamond_{B}\beta(b)\\\
\hskip
113.81102pt=\varepsilon(a,b)\big{(}l_{A}(x)b\cdot_{B}\beta(a)+s_{A}(r_{B}(b)x)\beta(a)\big{)},\end{array}$
(3.14)
$\displaystyle\begin{array}[]{l}\varepsilon(a+b,x)s(\alpha(x))(a\diamond_{B}b)-\varepsilon(b,x)\beta(a)\diamond_{B}s_{A}(x)b-r_{A}(s_{B}(b)x)\beta(a)\\\
\quad=\varepsilon(a,b)\big{(}\varepsilon(a+b,x)s(\alpha(x))(b\diamond_{B}a)-\varepsilon(a,x)\beta(b)\diamond_{B}(s_{A}(x)a)\big{.}\\\
\hskip 227.62204pt\big{.}-r_{A}(s_{B}(a)x)\beta(b)\big{)},\end{array}$ (3.18)
$\displaystyle\begin{array}[]{l}r_{A}(x)b\cdot_{B}\beta(b)+s_{A}(l_{B}(a)x)\beta(b)-\beta(a)\diamond_{B}(s_{A}(x)b)-\varepsilon(x,b)r_{A}(s_{B}(b)x)\beta(a)\\\
\quad=\varepsilon(a,x)\big{(}l_{A}(x)b\cdot_{B}\beta(b)-s_{A}(r_{B}(a)x)\beta(b)-l_{A}(\alpha(x))(a\cdot_{B}b)\big{)},\end{array}$
(3.21)
$\displaystyle\begin{array}[]{l}l_{A}(x)a\cdot_{B}\beta(b)+s_{A}(r_{B}(a)x)\beta(b)-l_{A}(\alpha(x))(a\cdot_{B}b)\\\
\hskip
85.35826pt=\varepsilon(x,a)\big{(}r_{A}(x)a\cdot_{B}\beta(b)+s_{A}(l_{B}(a)x)\beta(b)\big{.}\\\
\hskip
113.81102pt\big{.}-\beta(a)\diamond_{B}(s_{A}(x)b)-\varepsilon(x,b)r_{A}(s_{B}(b)x)\beta(a)\big{)},\end{array}$
(3.25) $\displaystyle\
r_{B}(\beta(a))(x\cdot_{A}y)=\varepsilon(y,a)\big{(}r_{B}(a)x\cdot_{A}\alpha(y)+s_{B}(l_{A}(x)a)\alpha(y),$
(3.26) $\displaystyle\
l_{B}(s_{A}(x)a)\alpha(y)+\varepsilon(x,a)s(a)x\diamond_{A}\alpha(y)=\varepsilon(a,y)\big{(}\varepsilon(x+y,a)s(\beta(a))(x\diamond_{A}y)\big{)},$
(3.27)
$\displaystyle\begin{array}[]{l}\varepsilon(x,a)l_{B}(s_{A}(x)a)\alpha(y)+s_{B}(a)x\diamond_{A}\alpha(y)\\\
\hskip
113.81102pt=\varepsilon(x,y)\big{(}l_{B}(a)y\cdot_{A}\alpha(x)+s_{B}(r_{A}(y)a)\alpha(x)\big{)},\end{array}$
(3.30)
$\displaystyle\begin{array}[]{l}\varepsilon(x+y,a)s(\beta(a))(x\diamond_{A}y)-\varepsilon(y,a)\alpha(x)\diamond_{A}s_{B}(a)y-r_{B}(s_{A}(y)a)\alpha(x)\\\
\quad=\varepsilon(x,y)\big{(}\varepsilon(x+y,a)s(\beta(a))(y\diamond_{A}x)-\varepsilon(x,a)\alpha(y)\diamond_{A}(s_{B}(a)x)\big{.}\\\
\hskip 227.62204pt\big{.}-r_{B}(s_{A}(x)a)\alpha(y)\big{)},\end{array}$ (3.34)
$\displaystyle\begin{array}[]{l}r_{B}(a)y\cdot_{A}\alpha(y)+s_{B}(l_{A}(x)a)\alpha(y)-\alpha(x)\diamond_{A}(s_{B}(a)y)-\varepsilon(a,y)r_{B}(s_{A}(y)a)\alpha(x)\\\
\quad=\varepsilon(x,a)\big{(}l_{B}(a)y\cdot_{A}\alpha(y)-s_{B}(r_{A}(x)a)\alpha(y)-l_{B}(\beta(a))(x\cdot_{A}y)\big{)},\end{array}$
(3.36)
$\displaystyle\begin{array}[]{l}l_{B}(a)x\cdot_{A}\alpha(y)+s_{B}(r_{A}(x)a)\alpha(y)-l_{B}(\beta(a))(x\cdot_{A}y)\\\
\hskip
85.35826pt=\varepsilon(a,x)\big{(}r_{B}(a)x\cdot_{A}\alpha(y)+s_{B}(l_{A}(x)a)\alpha(y)\big{.}\\\
\hskip
113.81102pt\big{.}-\alpha(x)\diamond_{A}(s_{B}(a)y)-\varepsilon(a,y)r_{B}(s_{A}(y)a)\alpha(x)\big{)}.\end{array}$
(3.40)

Then, $(A,B,s_{A},l_{A},r_{A},\beta,s_{B},l_{B},r_{B},\alpha)$ is called a
matched pair of the Hom-Novikov-Poisson color Hom-algebras. In this case, on
the direct sum $A\oplus B$ of the underlying linear spaces of $\mathcal{A}$
and $\mathcal{B}$, there is a Hom-Novikov-Poisson color Hom-algebra structure
which is given for any $x+a\in A_{\gamma_{1}}\oplus B_{\gamma_{1}},y+b\in
A_{\gamma_{2}}\oplus B_{\gamma_{2}}$ by

$\displaystyle\begin{array}[]{lcl}(x+a)\cdot(y+b)&=&x\cdot_{A}y+(s_{A}(x)b+\varepsilon(a,y)s_{A}(y)a)\\\
&&+a\cdot_{B}b+(s_{B}(a)y+\varepsilon(x,b)s_{B}(b)x),\end{array}$ (3.43)
$\displaystyle\begin{array}[]{lcl}(x+a)\diamond(y+b)&=&x\diamond_{A}y+(l_{A}(x)b+r_{A}(y)a)\\\
&&+a\diamond_{B}b+(l_{B}(a)y+r_{B}(b)x).\end{array}$ (3.46)

###### Proof.

By Proposition 2.11 and Proposition 2.19, we deduce that $(A\oplus
B,\cdot,\varepsilon,\alpha+\beta)$ is a $\varepsilon$-commutative Hom-
associative color Hom-algebra and $(A\oplus B,\diamond,\alpha+\beta)$ is a
Hom-Novikov color Hom-algebra. Now, the rest, it is easy (in a similar way as
for Proposition 2.11) to verify the compatibility conditions are satisfied. ∎

######  Definition 3.10.

A transposed Hom-Poisson color Hom-algebra is defined as a quintuple
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$, where
$(A,\cdot,\varepsilon,\alpha)$ is a $\varepsilon$-commutative Hom-associative
color Hom-algebra and $(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie
color Hom-algebra, satisfying the transposed Hom-$\varepsilon$-Leibniz
identity for $x,y,z\in\mathcal{H}(A)$,

$2\alpha(z)\cdot[x,y]=[z\cdot x,\alpha(y)]+\varepsilon(z,x)[\alpha(x),z\cdot
y].$ (3.47)

######  Proposition 3.11.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ be a multiplicative
transposed Hom-Posson color Hom-algebra. Then the following identities hold
for all $h,x,y,z\in\mathcal{H}(A)$,

$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)\alpha(x)\cdot[y,z]=0,$
(3.48)
$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)[\alpha(h)\cdot[x,y],\alpha^{2}(z)]=0,$
(3.49)
$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)[\alpha(h)\cdot\alpha(x),[\alpha(y),\alpha(z)]]=0,$
(3.50)
$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)[\alpha(h),\alpha(x)]\cdot[\alpha(y),\alpha(z)]=0.$
(3.51)

###### Proof.

Proof of (3.48): Let $x,y,z\in\mathcal{H}(A)$. By the transposed
Hom-$\varepsilon$-Leibniz identity,

$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}2\alpha(x)\cdot[y,z]\Big{)}=\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}[x\cdot
y,\alpha(z)]+\varepsilon(x,y)[\alpha(y),x\cdot z]\Big{)}$
$\displaystyle\quad=\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}[x\cdot
y,\alpha(z)]+\varepsilon(x+y,z)[z\cdot x,\alpha(y)]\Big{)}$
$\displaystyle\quad=\circlearrowleft_{x,y,z}\Big{(}\varepsilon(z,x)[x\cdot
y,\alpha(z)]-\varepsilon(y,z)[z\cdot x,\alpha(y)]\Big{)}=0.$

Then we have (3.48).

Proof of (3.49): Let $x,y,z,h\in\mathcal{H}(A)$. First, by (3.47), we have

$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}2\alpha^{2}(h)[[x,y],\alpha(z)]\Big{)}$
$\displaystyle=\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}[\alpha(h),[x,y],\alpha^{2}(z)]$
$\displaystyle+\varepsilon(h,x+y)[\alpha([x,y]),\alpha(h\cdot z)]\Big{)}.$

Applying the Hom-Jacobi identity of the above equality, we obtain

$\begin{array}[]{l}\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}[\alpha(h),[x,y],\alpha^{2}(z)]\Big{)}\\\
\quad\quad+\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}\varepsilon(h,x+y)[\alpha([x,y]),\alpha(h\cdot
z)]\Big{)}=0.\end{array}$ (3.52)

Next, by the Hom-Jacobi identity, we have

$\displaystyle\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot z)]$
$\displaystyle+\varepsilon(x+y,z)[[h\cdot z,\alpha(x)],\alpha^{2}(y)]$
$\displaystyle+\varepsilon(h,y)\varepsilon(x,y+z)[[\alpha(y),h\cdot
z],\alpha^{2}(x)]=0,$

and by (3.47) we have

$\displaystyle\varepsilon(h,y)\varepsilon(x,y+z)[[\alpha(y),h\cdot
z],\alpha^{2}(x)]=$ $\displaystyle
2\varepsilon(x,y+z)[\alpha(h)\cdot[y,z],\alpha^{2}(x)]$
$\displaystyle-\varepsilon(x,y+z)[[h\cdot y,\alpha(z)],\alpha^{2}(x)]=0.$

Thus we obtain

$\displaystyle\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot
z)]+\varepsilon(x+y,z)[[h\cdot z,\alpha(x)],\alpha^{2}(y)]$
$\displaystyle+\varepsilon(x,y+z)\big{(}2[\alpha(h)\cdot[y,z],\alpha^{2}(x)]-\varepsilon(x,y+z)[[h\cdot
y,\alpha(z)],\alpha^{2}(x)]\big{)}=0.$

Similarly, we have

$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot
z)]+\varepsilon(x+y,z)[[h\cdot z,\alpha(x)],\alpha^{2}(y)]$
$\displaystyle+\varepsilon(x,y+z)\big{(}2[\alpha(h)\cdot[y,z],\alpha^{2}(x)]-\varepsilon(x,y+z)[[h\cdot
y,\alpha(z)],\alpha^{2}(x)]\big{)}\Big{)}=0.$

Taking the above sum, we obtain

$\begin{array}[]{l}\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot
z)]\Big{)}\\\
\quad\quad+\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}2[\alpha(h)\cdot[x,y],\alpha^{2}(x)]\Big{)}=0.\end{array}$
(3.53)

Finally, taking the difference between the two equations (3.52) and (3.53) we
obtain (3.49).

Proof of (3.50): Taking the difference between the two equations (3.49) and
(3.52) we obtain (3.50)

Proof of (3.51): Let $x,y,z,h\in\mathcal{H}(A)$. By (3.47) we have

$\displaystyle\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}2\varepsilon(h,x+y)[\alpha(x),\alpha(y)]\cdot[\alpha(h),\alpha(z)]\Big{)}$
$\displaystyle\quad=\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big{(}[\alpha(h)\cdot[x,y],\alpha^{2}(z)]+\varepsilon(x+y,z)[\alpha^{2}(h),\alpha(z)\cdot[x,y].]\Big{)}.$

Applying equations (3.48) and (3.49) to above equality, we obtain (3.51). ∎

######  Proposition 3.12.

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be a Hom-Novikov-Poisson color
Hom-algebra. Define

$[x,y]=x\diamond y-\varepsilon(x,y)y\diamond x,\;\;\forall
x,y\in\mathcal{H}(A).$ (3.54)

Then $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-transposed-Poisson
color Hom-algebra.

###### Proof.

By definition, we have $(A,\cdot,\varepsilon,\alpha)$ is a
$\varepsilon$-commutative Hom-associative color Hom-algebra and by Proposition
2.22, $(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie color Hom-algebra.
Now, we show that the $\varepsilon$-transposed Hom-Leibniz identity is
satisfied. For any $x,y,z\in\mathcal{H}(A)$ we have

$\displaystyle\varepsilon(z,x)[x\cdot
z,\alpha(y)]+\varepsilon(z,x+y)[\alpha(x),y\cdot z]-2\alpha(z)\cdot[x,y]$
$\displaystyle=\varepsilon(z,x)\Big{(}(x\cdot
z)\diamond\alpha(y)-\varepsilon(x+z,y)\alpha(y)\diamond(x\cdot z)\Big{)}$
$\displaystyle+\varepsilon(z,x+y)\Big{(}\alpha(x)\diamond(y\cdot
z)-\varepsilon(x,y+z)(y\cdot
z)\diamond\alpha(x)\Big{)}-2\alpha(z)\cdot\Big{(}x\cdot
y-\varepsilon(x,y)y\cdot x\Big{)}$ (using Eqs. (3.1) and (3.2))
$\displaystyle=\varepsilon(z,x)(x\cdot z)\diamond\alpha(y)-\alpha(z)(x\diamond
y)+\varepsilon(x,y)\alpha(z)\cdot(y\diamond x)$
$\displaystyle\quad-\varepsilon(x+z,y)(y\cdot
z)\diamond\alpha(x)-\Big{(}\varepsilon(z,x+y)(x\diamond y)\cdot\alpha(z)$
$\displaystyle\quad-\varepsilon(x,y)(y\diamond
x)\cdot\alpha(z)-\varepsilon(z,x+y)\alpha(x)\diamond(y\cdot
z)+\varepsilon(x,y)\alpha(y)\diamond(x\cdot z)\Big{)}=0.$

Hence the conclusion holds. ∎

######  Example 3.13.

Let $A=A_{0}\oplus A_{1}=<e_{1},e_{2}>\oplus<e_{3},e_{4}>$ be a
$4$-dimensional superspace. The quintuple
$(A,\cdot,\diamond,\varepsilon,\alpha)$ is a Hom-Novikov-Poisson color Hom-
algebra with

$\displaystyle\text{the bicharacter:}\quad\varepsilon(i,j)=(-1)^{ij},$
$\displaystyle\text{the multiplications:}\quad\begin{array}[t]{l}e_{2}\cdot
e_{2}=e_{1},\quad e_{2}\cdot e_{4}=e_{4}\cdot e_{2}=e_{3}\\\ e_{2}\diamond
e_{4}=-e_{3},\quad e_{4}\diamond e_{2}=e_{3},\quad e_{4}\diamond
e_{4}=2e_{1},\end{array}$ $\displaystyle\text{the even linear map
$\alpha:A\rightarrow A$ defined
by}\begin{array}[t]{ll}\alpha(e_{1})=2e_{1},&\quad\alpha(e_{2})=-e_{2},\\\
\alpha(e_{3})=-e_{4},&\quad\alpha(e_{4})=e_{3}.\end{array}$

Therefore, using the Proposition 3.12,
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a transposed Hom-Poisson color
Hom-algebra with

$[e_{2},e_{4}]=[e_{4},e_{2}]=-2e_{3},\quad[e_{4},e_{4}]=4e_{1}.$

######  Definition 3.14 ([11]).

A _Hom-Poisson color Hom-algebra_ is defined as a quintuple
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ such that
$(A,\cdot,\varepsilon,\alpha)$ is a $\varepsilon$-commutative Hom-associative
color Hom-algebra, and $(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie
color Hom-algebra, satisfying for all $x,y,z$ in $\mathcal{H}(A)$, the
Hom-$\varepsilon$-Leibniz identity,

$[\alpha(x),y\cdot
z]=\varepsilon(x,y)\alpha(y)\cdot[x,z]+\varepsilon(x+y,z)\alpha(z)\cdot[x,y].$
(3.55)

Condition (3.55), expressing the compatibility between the multiplication and
the Poisson bracket, can be reformulated equivalently as

$[x\cdot
y,\alpha(z)]=\varepsilon(y,z)[x,z]\cdot\alpha(y)+\alpha(x)\cdot[y,z].$ (3.56)

######  Definition 3.15.

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be Hom-Novikov-Poisson color Hom-
algebra. Then $A$ is called admissible if
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Poisson color Hom-
algebra, where

$[x,y]=x\diamond y-\varepsilon(x,y)y\diamond x.$ (3.57)

######  Lemma 3.16.

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be a Hom-Novikov-Poisson color
Hom-algebra. Then for any $x,y,z\in\mathcal{H}(A)$,

$(x\cdot y)\diamond\alpha(z)=\alpha(x)\diamond(y\cdot z).$ (3.58)

###### Proof.

For all $x,y,z\in\mathcal{H}(A)$, we have

($(A,\cdot,\varepsilon,\alpha)$ is $\varepsilon$-commutative)
$\displaystyle(x\cdot y)\circ\alpha(z)=\varepsilon(x,y)(y\cdot
x)\circ\alpha(z)$ (by (3.1)) $\displaystyle=\varepsilon(x,y+z)(y\circ
z)\cdot\alpha(x)$ ($(A,\cdot,\varepsilon,\alpha)$ is
$\varepsilon$-commutative) $\displaystyle=\alpha(x)\cdot(y\circ z).\qed$

The following result gives a necessary and sufficient condition under which a
Hom-Novikov-Poisson color algebra is admissible.

######  Theorem 3.17.

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be Hom-Novikov-Poisson color Hom-
algebra. Then $A$ is an admissible if and only if

$as^{l}_{A}(x,y,z)=(x\cdot y)\diamond\alpha(z)-\alpha(x)\diamond(y\cdot z)=0.$
(3.59)

###### Proof.

By definition, $(A,\cdot,\varepsilon,\alpha)$ is a $\varepsilon$-commutative
Hom-associative color Hom-algebra and by Proposition 2.22,
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie color Hom-algebra.
Therefore,

(using (2.45)) $\displaystyle[\alpha(x),y\cdot z]=\alpha(x)\diamond(y\cdot
z)-\varepsilon(x+y,z)(y\cdot z)\diamond\alpha(x)$ (by (3.2) and Lemma 3.16)
$\displaystyle=(x\diamond y)\cdot\alpha(z)-\varepsilon(x,y)(y\diamond
x)\cdot\alpha(z)+\varepsilon(x,y)\alpha(y)\diamond(x\cdot
z)-\varepsilon(x,y+z)\alpha(y)\cdot(z\diamond x),$
$\displaystyle[x,y]\cdot\alpha(x)+\varepsilon(x,y)\alpha(y)[x,z]$ (using
(2.45)) $\displaystyle=(x\diamond y)\cdot\alpha(z)-\varepsilon(x,y)(y\diamond
x)\cdot\alpha(z)+\varepsilon(x,y)\alpha(y)\cdot(x\diamond
z)-\varepsilon(x,y+z)\alpha(y)\cdot(z\diamond x)$ (by Lemma 3.16)
$\displaystyle=(x\diamond y)\cdot\alpha(z)-\varepsilon(x,y)(y\diamond
x)\cdot\alpha(z)+\varepsilon(x,y)(y\cdot
x)\diamond\alpha(z)-\varepsilon(x,y+z)\alpha(y)\cdot(z\diamond x).$

Then $A$ satisfies the Hom-$\varepsilon$-Leibniz-identity if and only if
$\alpha(y)\diamond(x\cdot z)=(y\cdot x)\diamond\alpha(z).$ ∎

######  Example 3.18.

Let $A=A_{0}\oplus A_{1}=<e_{1},e_{2}>\oplus<e_{3},e_{4}>$ be a
$4$-dimensional superspace. The quintuple
$(A,\cdot,\diamond,\varepsilon,\alpha)$ is a Hom-Novikov-Poisson color Hom-
algebra with

$\displaystyle\text{the bicharacter:}\quad\varepsilon(i,j)=(-1)^{ij},$
$\displaystyle\text{the
multiplications:}\quad\begin{array}[t]{rclrcll}e_{2}\cdot
e_{2}&=&\lambda_{1}e_{1},&\quad e_{2}\cdot e_{4}&=&e_{4}\cdot
e_{2}=\lambda_{2}e_{3},\quad\lambda_{i}\in\mathbb{K}\\\ e_{2}\diamond
e_{2}&=&\mu_{1}e_{1},&\quad e_{4}\diamond e_{2}&=&\mu_{2}e_{3},\\\ \quad
e_{4}\diamond e_{4}&=&\mu_{3}e_{1},&\mu_{i}\in\mathbb{K}\end{array}$
$\displaystyle\text{the even linear map $\alpha:A\rightarrow A$ defined
by}\begin{array}[t]{l}\alpha(e_{1})=2e_{1}-e_{2},\quad\alpha(e_{2})=e_{1},\\\
\alpha(e_{3})=-e_{4},\quad\alpha(e_{4})=e_{3}-e_{4}.\end{array}$

Then by Theorem 3.17, the Hom-Novikov-Poisson color Hom-algebra
$(A,\cdot,\diamond,\varepsilon,\alpha)$ is admissible.

######  Theorem 3.19.

Let $\mathcal{A}=(A,\cdot,\diamond,\varepsilon)$ be an admissible Novikov-
Poisson color Hom-algebra and $\alpha:\mathcal{A}\rightarrow\mathcal{A}$ be an
admissible Novikov-Poisson color Hom-algebras morphism. Define
$\cdot_{\alpha},\ \diamond_{\alpha}:A\times A\rightarrow A$ for all
$x,y\in\mathcal{H}(A)$, by $x\cdot_{\alpha}y=\alpha(x\cdot y)$ and
$x\diamond_{\alpha}y=\alpha(x\diamond y)$. Then,
$\mathcal{A}_{\alpha}=(A,\cdot_{\alpha},\diamond_{\alpha},\varepsilon,\alpha)$
is an admissible Hom-Novikov-Poisson color Hom-algebra called the
$\alpha$-twist or Yau twist of $(A,\cdot,\diamond,\varepsilon)$.

###### Proof.

By Theorem 3.4, $\mathcal{A}_{\alpha}$ is a Hom-Novikov-Poisson color Hom-
algebra. Moreover, the left Hom-associators in $\mathcal{A}$ and
$\mathcal{A}_{\alpha}$ are related as

$as_{\mathcal{A}_{\alpha}}^{l}(x,y,z)=\alpha^{2}as_{\mathcal{A}}^{l}(x,y,z)\quad
for~{}all~{}x,y,z\in\mathcal{H}(A).$

Since $\mathcal{A}$ is left Hom-associative by Theorem 3.17, it follows that
so is $\mathcal{A}_{\alpha}$. Therefore, by Theorem 3.17 again
$\mathcal{A}_{\alpha}$ is admissible. ∎

######  Corollary 3.20.

If $\mathcal{A}=(A,\cdot,\diamond,\varepsilon,\alpha)$ is a multiplicative
admissible Novikov-Poisson color algebra, then for any
$n\in\mathbb{N}^{\ast}$,

  1. (i)

The $n{\rm th}$ derived admissible Hom-Novikov-Poisson color Hom-algebra of
type $1$ of $\mathcal{A}$ is defined by

$\mathcal{A}_{1}^{n}=(A,\cdot^{(n)}=\alpha^{n}\circ\cdot,\diamond^{(n)}=\alpha^{n}\circ\diamond,\varepsilon,\alpha^{n+1}).$

  2. (ii)

The $n{\rm th}$ derived admissible Hom-Novikov-Poisson color Hom-algebra of
type $2$ of $A$ is defined by

$\mathcal{A}_{2}^{n}=(A,\cdot^{(2^{n}-1)}=\alpha^{2^{n}-1}\circ\cdot,\diamond^{(2^{n}-1)}=\alpha^{2^{n}-1}\circ\diamond,\varepsilon,\alpha^{2^{n}}).$

######  Example 3.21.

Let $A=A_{0}\oplus A_{1}=<e_{1},e_{2}>\oplus<e_{3},e_{4}>$ be a
$4$-dimensional superspace. There is a multiplicative admissible Hom-Novikov-
Poisson color Hom-algebra $(A,\cdot,\diamond,\varepsilon,\alpha)$ with the
bicharacter, $\varepsilon(i,j)=(-1)^{ij},$ and the multiplications tables for
a basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$:

$\cdot$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$
---|---|---|---|---
$e_{1}$ | $0$ | $0$ | $0$ | $0$
$e_{2}$ | $0$ | $e_{1}$ | $0$ | $4e_{3}$
$e_{3}$ | $0$ | $0$ | $0$ | $0$
$e_{4}$ | $0$ | $4e_{3}$ | $0$ | $0$
$\diamond$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$
---|---|---|---|---
$e_{1}$ | $0$ | $0$ | $0$ | $0$
$e_{2}$ | $0$ | $4e_{3}$ | $0$ | $4e_{3}$
$e_{3}$ | $0$ | $0$ | $0$ | $0$
$e_{4}$ | $0$ | $0$ | $0$ | $e_{1}$
$\begin{array}[]{ll}\alpha(e_{1})=4e_{1},&\alpha(e_{2})=-2e_{2},\\\
\alpha(e_{3})=e_{3},&\alpha(e_{4})=-2e_{4}.\end{array}$

Then there are admissible Hom-Novikov-Poisson color Hom-algebras
$\mathcal{A}_{1}^{n}$ and $\mathcal{A}_{2}^{n}$ with multiplications tables
respectively:

$\cdot^{(n)}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$
---|---|---|---|---
$e_{1}$ | $0$ | $0$ | $0$ | $0$
$e_{2}$ | $0$ | $2^{2n}e_{1}$ | $0$ | $4e_{3}$
$e_{3}$ | $0$ | $0$ | $0$ | $0$
$e_{4}$ | $0$ | $4e_{3}$ | $0$ | $0$
$\diamond^{(n)}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$
---|---|---|---|---
$e_{1}$ | $0$ | $0$ | $0$ | $0$
$e_{2}$ | $0$ | $4e_{3}$ | $0$ | $4e_{3}$
$e_{3}$ | $0$ | $0$ | $0$ | $0$
$e_{4}$ | $0$ | $0$ | $0$ | $2^{2n}e_{1}$
$\begin{array}[]{ll}\alpha^{n+1}(e_{1})=4^{n+1}e_{1},&\alpha^{n+1}(e_{2})=(-2)^{n+1}e_{2},\\\ \alpha^{n+1}(e_{3})=e_{3},&\alpha^{n+1}(e_{4})=(-2)^{n+1}e_{4},\end{array}$ $\cdot^{(2^{n}-1)}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$
---|---|---|---|---
$e_{1}$ | $0$ | $0$ | $0$ | $0$
$e_{2}$ | $0$ | $2^{2(2^{n}-1)}e_{1}$ | $0$ | $4e_{3}$
$e_{3}$ | $0$ | $0$ | $0$ | $0$
$e_{4}$ | $0$ | $4e_{3}$ | $0$ | $0$
$\diamond^{(2^{n}-1)}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$
---|---|---|---|---
$e_{1}$ | $0$ | $0$ | $0$ | $0$
$e_{2}$ | $0$ | $4e_{3}$ | $0$ | $4e_{3}$
$e_{3}$ | $0$ | $0$ | $0$ | $0$
$e_{4}$ | $0$ | $0$ | $0$ | $2^{2(2^{n}-1)}e_{1}$
$\begin{array}[]{ll}\alpha^{2^{n}}(e_{1})=4^{2^{n}}e_{1},&\alpha^{2^{n}}(e_{2})=2^{2^{n}}e_{2},\\\
\alpha^{2^{n}}(e_{3})=e_{3},&\alpha^{2^{n}}(e_{4})=2^{2^{n}}e_{4}.\end{array}$

###  3.2 Tensor products of admissible Hom-Novikov-Poisson color Hom-algebras

Now, we show that the much larger class of admissible Hom-Novikov-Poisson
color Hom-algebras is also closed under tensor products.

######  Theorem 3.22.

Let $(A_{1},\cdot_{1},\diamond_{1},\varepsilon,\alpha_{1})$ and
$(A_{2},\cdot_{2},\diamond_{2},\varepsilon,\alpha_{2})$ be admissible Hom-
Novikov-Poisson color Hom-algebras and let $A=A_{1}\otimes A_{2}$. Define the
operations $\alpha:A\rightarrow A$ and $\cdot,\diamond:A\otimes A\rightarrow
A$ by the following formulae for all
$x_{i},y_{i}\in\mathcal{H}(A_{i}),i\in\\{1;2\\}$,

$\displaystyle\alpha=\alpha_{1}\otimes\alpha_{2},$ $\displaystyle(x_{1}\otimes
x_{2})\cdot(y_{1}\otimes
y_{2})=\varepsilon(x_{2},y_{1})(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2}),$
$\displaystyle(x_{1}\otimes x_{2})\diamond(y_{1}\otimes
y_{2})=\varepsilon(x_{2},y_{1})\big{(}(x_{1}\diamond_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})+(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\diamond_{2}y_{2})\big{)},$

Then $(A,\cdot,\diamond,\varepsilon,\alpha)$ is an admissible Hom-Novikov-
Poisson color Hom-algebra.

###### Proof.

Pick $x=x_{1}\otimes x_{2},y=y_{1}\otimes y_{2}$ and $z=z_{1}\otimes z_{2}$
homogeneous elements in $A$.

Step 1: We show that $(A,\cdot,\varepsilon,\alpha)$ is
$\varepsilon$-commutative Hom-associative-color Hom-algebra:

$\displaystyle x\cdot y$ $\displaystyle=(x_{1}\otimes x_{2})\cdot(y_{1}\otimes
y_{2})$
$\displaystyle=\varepsilon(x_{2},y_{1})(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})$
$\displaystyle=\varepsilon(x_{2},y_{1})\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})(y_{1}\cdot_{1}x_{1})\otimes(y_{2}\cdot_{2}x_{2})$
$\displaystyle=\varepsilon(x_{1}+x_{2},y_{1}+y_{2})\big{(}\varepsilon(y_{2},x_{1})(y_{1}\cdot_{1}x_{1})\otimes(y_{2}\cdot_{2}x_{2})\big{)}$
$\displaystyle=\varepsilon(x_{1}+x_{2},y_{1}+y_{2})(y_{1}\otimes
y_{2})\cdot(x_{1}\otimes x_{2})$ $\displaystyle=\varepsilon(x,y)y\cdot x,$
$\displaystyle(x\cdot y)\cdot\alpha(z)$ $\displaystyle=((x_{1}\otimes
x_{2})\cdot(y_{1}\otimes
y_{2}))\cdot(\alpha_{1}\otimes\alpha_{2})(z_{1}\otimes z_{2})$
$\displaystyle=\Big{(}\varepsilon(x_{2},y_{1})\cdot(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\otimes_{2}y_{2})\Big{)}\cdot(\alpha_{1}\otimes\alpha_{2})(z_{1}\otimes
z_{2})$
$\displaystyle=\varepsilon(x_{2}+y_{2},z_{1})\varepsilon(x_{2},y_{1})\Big{(}(x_{1}\cdot_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})\Big{)}$
$\displaystyle=\varepsilon(x_{2},y_{1}+z_{1})\varepsilon(y_{2},z_{1})\Big{(}\alpha_{1}(x_{1})\cdot_{1}(y_{1}\cdot_{1}z_{1})\otimes\alpha_{2}(x_{2})\cdot_{2}(y_{2}\cdot_{2}z_{2})\Big{)}$
$\displaystyle=(\alpha_{1}\otimes\alpha_{2})(x_{1}\otimes
x_{2})\cdot((y_{1}\otimes y_{2})\cdot(z_{1}\otimes z_{2}))$
$\displaystyle=\alpha(x)\cdot(y\cdot z).$

Hence, $(A_{1}\otimes A_{2},\cdot,\varepsilon,\alpha)$ is a
$\varepsilon$-commutative Hom-associative color Hom-algebra.

Step 2: We show that $(A,\diamond,\varepsilon,\alpha)$ is Hom-Novikov-color
Hom-algebra.

$\displaystyle(x\diamond y)\diamond\alpha(z)-\alpha(x)\diamond(y\diamond
z)-\varepsilon(x,y)\big{(}(y\diamond
x)\diamond\alpha(z)-\alpha(y)\diamond(x\diamond z))$
$\displaystyle=\big{(}(x_{1}\otimes x_{2})\diamond(y_{1}\otimes
y_{2})\big{)}\diamond\alpha(z_{1}\otimes
z_{2})-(\alpha_{1}\otimes\alpha_{2})(x_{1}\otimes
x_{2})\diamond\big{(}(y_{1}\otimes y_{2})\diamond(z_{1}\otimes z_{2})\big{)}$
$\displaystyle\quad-\varepsilon(x_{1}+x_{2},y_{1}+y_{2})\Big{(}\big{(}(y_{1}\otimes
y_{2})\diamond(x_{1}\otimes
x_{2})\big{)}\diamond(\alpha_{1}\otimes\alpha_{2})(z_{1}\otimes z_{2})$
$\displaystyle\quad-(\alpha_{1}\otimes\alpha_{2})(y_{1}\otimes
y_{2})\diamond\big{(}(x_{1}\otimes x_{2})\diamond(z_{1}\otimes
z_{2})\big{)}\Big{)}$
$\displaystyle=\varepsilon(x_{2},y_{1})\big{(}(x_{1}\diamond_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})+(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\diamond_{2}y_{2})\big{)}\diamond(\alpha_{1}\otimes\alpha_{2})(z_{1}\otimes
z_{2})$
$\displaystyle\quad-\varepsilon(y_{2},z_{1})(\alpha_{1}\otimes\alpha_{2})(x_{1}\otimes
x_{2})\diamond\big{(}(y_{1}\diamond_{1}z_{1})\otimes(y_{2}\cdot_{2}z_{2})+(y_{1}\cdot_{1}z_{1})\otimes(y_{2}\diamond_{2}z_{2})\big{)}$
$\displaystyle\quad-\varepsilon(x_{1}+x_{2},y_{1}+y_{2})\Big{(}\varepsilon(y_{2},x_{1})\big{(}(y_{1}\diamond_{1}x_{1})\otimes(y_{2}\cdot_{2}x_{2})+(y_{1}\cdot_{1}x_{1})\otimes(y_{2}\diamond_{2}x_{2})\big{)}\diamond$
$\displaystyle\quad\quad\quad(\alpha_{1}\otimes\alpha_{2})(z_{1}\otimes
z_{2})-\varepsilon(x_{2},z_{1})(\alpha_{1}\otimes\alpha_{2})(y_{1}\otimes
y_{2})\diamond\big{(}(x_{1}\diamond_{1}z_{1})\otimes(x_{2}\cdot_{2}z_{2})$
$\displaystyle\quad+(x_{1}\cdot_{1}z_{1})\otimes(x_{2}\diamond_{2}z_{2})\big{)}\Big{)}$
$\displaystyle=\varepsilon(x_{1}+x_{2},y_{1}+y_{2})\times$
$\displaystyle\Big{[}\underbrace{(x_{1}\diamond_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})}_{A_{1}}\underbrace{+(x_{1}\diamond_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})}_{A_{2}}\Big{.}$
$\displaystyle\underbrace{+(x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})}_{A_{3}}\underbrace{+(x_{1}\cdot_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})}_{A_{4}}$
$\displaystyle\underbrace{-\alpha_{1}(x_{1})\diamond_{1}(y_{1}\diamond_{1}z_{1})\otimes\alpha_{2}(x_{2})\cdot_{2}(y_{2}\cdot_{2}z_{2})}_{A_{5}}\underbrace{-\alpha_{1}(x_{1})\cdot_{1}(y_{1}\diamond_{1}z_{1})\otimes\alpha_{2}(x_{2})\diamond_{2}(y_{2}\cdot_{2}z_{2})}_{A_{6}}$
$\displaystyle\underbrace{-\alpha_{1}(x_{1})\diamond_{1}(y_{1}\cdot_{1}z_{1})\otimes\alpha_{2}(x_{2})\cdot_{2}(y_{2}\diamond_{2}z_{2})}_{A_{7}}\underbrace{-\alpha_{1}(x_{1})\cdot_{1}(y_{1}\cdot_{1}z_{1})\otimes\alpha_{2}(x_{2})\diamond_{2}(y_{2}\diamond_{2}z_{2})}_{A_{8}}$
$\displaystyle\quad-\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})\times$
$\displaystyle\Big{(}\underbrace{(y_{1}\diamond_{1}x_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}x_{2})\cdot_{2}\alpha_{2}(z_{2})}_{B_{1}}\underbrace{+(x_{1}\diamond_{1}x_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}x_{2})\diamond_{2}\alpha_{2}(z_{2})}_{B_{2}}$
$\displaystyle\underbrace{+(x_{1}\cdot_{1}x_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}x_{2})\cdot_{2}\alpha_{2}(z_{2})}_{B_{3}}\underbrace{+(x_{1}\cdot_{1}x_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}x_{2})\diamond_{2}\alpha_{2}(z_{2})}_{B_{4}}$
$\displaystyle\underbrace{-\alpha_{1}(x_{1})\diamond_{1}(x_{1}\diamond_{1}z_{1})\otimes\alpha_{2}(x_{2})\cdot_{2}(x_{2}\cdot_{2}z_{2})}_{B_{5}}\underbrace{-\alpha_{1}(x_{1})\cdot_{1}(x_{1}\diamond_{1}z_{1})\otimes\alpha_{2}(x_{2})\diamond_{2}(x_{2}\cdot_{2}z_{2})}_{B_{6}}$
$\displaystyle\underbrace{-\alpha_{1}(x_{1})\diamond_{1}(x_{1}\cdot_{1}z_{1})\otimes\alpha_{2}(x_{2})\cdot_{2}(x_{2}\diamond_{2}z_{2})}_{B_{7}}$
$\displaystyle\hskip
113.81102pt\underbrace{\Big{.}\Big{.}-\alpha_{1}(x_{1})\cdot_{1}(x_{1}\cdot_{1}z_{1})\otimes\beta(x_{2})\diamond_{2}(x_{2}\diamond_{2}z_{2})\Big{)}\Big{]}}_{B_{8}}.$

Furthermore, we have

$\displaystyle(A_{1}+A_{5})-\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})(B_{1}+B_{5})$
(by (2.1) and (2.2))
$\displaystyle\quad=\Big{[}\big{(}(x_{1}\diamond_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})-\alpha_{1}(x_{1})\diamond_{1}(y_{1}\diamond_{1}z_{1})\big{)}$
$\displaystyle\quad\quad-\varepsilon(x_{1},y_{1})\big{(}(y_{1}\diamond_{1}x_{1})\diamond_{1}\alpha_{1}(z_{1})-\alpha_{1}(y_{1})\diamond_{1}(x_{1}\diamond_{1}z_{1})\big{)}\Big{]}\otimes(x_{2}\cdot_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})$
$\displaystyle\text{\scriptsize(by \eqref{eqhomNovikov1})}\quad=0,$
$\displaystyle(A_{4}+A_{8})-\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})(B_{4}+B_{8})$
(by (2.1) and (2.2))
$\displaystyle\quad=(x_{1}\cdot_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes\Big{[}\big{(}(x_{2}\diamond_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})$
$\displaystyle\quad\quad-\alpha_{2}(x_{2})\diamond_{2}(y_{2}\diamond_{2}z_{2})\big{)}-\varepsilon(x_{2},y_{2})\big{(}(y_{2}\diamond_{2}x_{2})\diamond_{2}\alpha_{2}(z_{2})-\alpha_{2}(y_{2})\diamond_{2}(x_{2}\diamond_{2}z_{2})\big{)}\Big{]}$
$\displaystyle\text{\scriptsize(by \eqref{eqhomNovikov1})}\quad=0,$
$\displaystyle(A_{2}+A_{7})-\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})(B_{2}+B_{7})$
(by (2.1), (3.1) and (2.2))
$\displaystyle\quad=\Big{[}\big{(}(x_{1}\diamond_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})-\alpha_{1}(x_{1})\diamond_{1}(y_{1}\cdot_{1}z_{1})\big{)}$
$\displaystyle\quad\quad-\varepsilon(x_{1},y_{1})\big{(}(y_{1}\diamond_{1}x_{1})\cdot_{1}\alpha_{1}(z_{1})-\alpha_{1}(y_{1})\diamond_{1}(x_{1}\cdot_{1}z_{1})\big{)}\Big{]}\otimes(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})$
$\displaystyle\text{\scriptsize(by \eqref{HomNovPoisscoloralg2})}\quad=0,$
$\displaystyle(A_{3}+A_{6})-\varepsilon(x_{1},y_{1})\varepsilon(x_{2},y_{2})(B_{3}+B_{6})$
(by (2.1), (3.1) and (2.2))
$\displaystyle\quad=(x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes\Big{[}\big{(}(x_{2}\diamond_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})$
$\displaystyle\quad\quad-\alpha_{2}(x_{2})\diamond_{2}(y_{2}\diamond_{2}z_{2})\big{)}-\varepsilon(x_{2},y_{2})\big{(}(y_{2}\diamond_{2}x_{2})\cdot_{2}\alpha_{2}(z_{2})-\alpha_{2}(y_{2})\diamond_{2}(x_{2}\cdot_{2}z_{2})\big{)}\Big{]}$
$\displaystyle\text{\scriptsize(by \eqref{HomNovPoisscoloralg2})}\quad=0.$

Then, we obtain

$\displaystyle(x\diamond y)\diamond\alpha(z)-\alpha(x)\diamond(y\diamond
z)-\varepsilon(x,y)\big{(}(y\diamond
x)\diamond\alpha(z)-\alpha(y)\diamond(x\diamond z)\big{)}=0.$
$\displaystyle(x\diamond y)\diamond\alpha(z)-\varepsilon(y,z)\big{(}(x\diamond
z)\diamond\alpha(y)$ $\displaystyle\quad=\big{(}(x_{1}\otimes
x_{2})\diamond(y_{1}\otimes y_{2})\big{)}\diamond\alpha(z_{1}\otimes z_{2})$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\Big{(}\big{(}(x_{1}\otimes
x_{2})\diamond(z_{1}\otimes
z_{2})\big{)}\diamond(\alpha_{1}\otimes\alpha_{2})(y_{1}\otimes y_{2})\Big{)}$
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\Big{(}(x_{1}\diamond_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})+(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\diamond_{2}y_{2})\Big{)}\diamond(\alpha_{1}\otimes\alpha_{2})(z_{1}\otimes
z_{2})$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\Big{(}\varepsilon(x_{2},z_{1})\big{(}(x_{1}\diamond_{1}z_{1})\otimes(x_{2}\cdot_{2}z_{2})+(x_{1}\cdot_{1}z_{1})\otimes(x_{2}\diamond_{2}z_{2})\big{)}\diamond$
$\displaystyle\quad\quad\quad(\alpha_{1}\otimes\alpha_{2})(y_{1}\otimes
y_{2})\Big{)}$
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\times$
$\displaystyle\Big{[}\underbrace{(x_{1}\diamond_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})}_{C_{1}}\underbrace{+(x_{1}\diamond_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})}_{C_{2}}$
$\displaystyle\underbrace{+(x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})}_{C_{3}}\underbrace{+(x_{1}\cdot_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})}_{C_{4}}\Big{]}$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})\times$
$\displaystyle\Big{[}\underbrace{(x_{1}\diamond_{1}z_{1})\diamond_{1}\alpha_{1}(y_{1})\otimes(x_{2}\cdot_{2}z_{2})\cdot_{2}\alpha_{2}(y_{2})}_{D_{1}}\underbrace{+(x_{1}\diamond_{1}z_{1})\cdot_{1}\alpha_{1}(y_{1})\otimes(x_{2}\cdot_{2}z_{2})\diamond_{2}\alpha_{2}(y_{2})}_{D_{2}}$
$\displaystyle\underbrace{+(x_{1}\cdot_{1}z_{1})\diamond_{1}\alpha_{1}(y_{1})\otimes(x_{2}\diamond_{2}z_{2})\cdot_{2}\alpha_{2}(y_{2})}_{D_{3}}\underbrace{+(x_{1}\cdot_{1}z_{1})\cdot_{1}\alpha_{1}(y_{1})\otimes(x_{2}\diamond_{2}z_{2})\diamond_{2}\alpha_{2}(y_{2})}_{D_{4}}\Big{]}.$

Furthermore, we have

$\displaystyle\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})C_{1}-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})D_{1}$
(by (2.2))
$\displaystyle\quad=\varepsilon(x_{2},y_{1}+y_{2})\varepsilon(x_{2}+y_{2},z_{1})\times$
$\displaystyle\quad\quad\big{(}(x_{1}\diamond_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\big{)}\otimes(y_{2}\cdot_{2}x_{2})\cdot_{2}\alpha_{2}(z_{2})$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2}+z_{2},y_{1}+y_{2})\varepsilon(x_{2},z_{1})\times$
$\displaystyle\quad\quad\big{(}(x_{1}\diamond_{1}z_{1})\diamond_{1}\alpha_{1}(y_{1})\big{)}\otimes\alpha_{2}(y_{2})\cdot_{2}(x_{2}\cdot_{2}z_{2})$
(by (2.1))
$\displaystyle\quad=\varepsilon(x_{2},y_{1}+y_{2})\varepsilon(x_{2}+y_{2},z_{1})\times$
$\displaystyle\quad\quad\Big{[}(x_{1}\diamond_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})-\varepsilon(y_{1},z_{1})(x_{1}\diamond
z_{1})\diamond_{1}\alpha_{1}(y_{1})\Big{]}\otimes\alpha_{2}(y_{2})\cdot_{2}(x_{2}\cdot_{2}z_{2})$
$\displaystyle\text{\scriptsize(by \eqref{eqhomNovikov2})}\quad=0.$
$\displaystyle\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})C_{4}-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})D_{4}$
(by (2.2))
$\displaystyle\quad=\varepsilon(x_{1}+x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\times$
$\displaystyle\quad\quad(y_{1}\cdot_{1}x_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes\big{(}(x_{2}\diamond_{2}y_{1})\diamond_{1}\alpha_{1}(z_{2})\big{)}$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2}+z_{2},y_{1})\varepsilon(x_{1}+y_{1},z_{1})\times$
$\displaystyle\quad\quad\alpha_{1}(y_{1})\cdot_{1}(x_{1}\cdot_{2}1_{2})\otimes\big{(}(x_{2}\diamond_{2}z_{2})\diamond_{2}\alpha_{2}(y_{2})\big{)}$
(by (2.1))
$\displaystyle\quad=\varepsilon(x_{1}+x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\times$
$\displaystyle\quad\quad(y_{1}\cdot_{1}x_{1})\cdot_{1}\alpha_{1}(z_{1})\Big{[}(x_{2}\diamond_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})-\varepsilon(y_{2},z_{2})(x_{2}\diamond_{2}z_{2})\diamond_{2}\alpha_{2}(y_{2})\Big{]}$
$\displaystyle\text{\scriptsize(by \eqref{eqhomNovikov2})}\quad=0.$
$\displaystyle\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})C_{2}-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})D_{3}$
(by (3.1))
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\Big{[}x_{1}\diamond_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})$
$\displaystyle\quad\quad-(x_{1}\diamond_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\otimes(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})\Big{]}=0,$
$\displaystyle\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})C_{3}-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})D_{2}$
(by (3.1))
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\Big{[}x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})$
$\displaystyle\quad\quad-(x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\otimes(x_{2}\diamond_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})\Big{]}=0.$

Then, we obtain

$(x\diamond y)\diamond\alpha(z)-\varepsilon(y,z)\big{(}(x\diamond
z)\diamond\alpha(y)=0.$

Hence, $(A_{1}\otimes A_{2},\diamond,\varepsilon,\alpha)$ is a Hom-Novikov
color Hom-algebra.

Step 3: We show that the compatibility conditions of Hom-Novikov-Poisson color
Hom-algebras are satisfied

$\displaystyle(x\cdot y)\diamond\alpha(z)-\varepsilon(y,z)(x\diamond
z)\cdot\alpha(y)$ $\displaystyle\quad=\big{(}(x_{1}\otimes
x_{2})\cdot(y_{1}\otimes
y_{2})\big{)}\diamond(\alpha_{1}\otimes\alpha_{2})(z_{1}\otimes z_{2})$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\big{(}(x_{1}\otimes
x_{2})\diamond(z_{1}\otimes
z_{2})\big{)}\cdot(\alpha_{1}\otimes\alpha_{2})(y_{1}\otimes y_{2})$
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\big{(}(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})\big{)}\diamond(\alpha_{1}(z_{1})\otimes\alpha_{2}(z_{2}))$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1}\Big{(}(x_{1}\diamond_{1}z_{1})\otimes(x_{2}\diamond_{2}z_{2})$
$\displaystyle\quad\quad+(x_{1}\cdot_{1}z_{1})\otimes(x_{2}\cdot_{2}z_{2})\Big{)}\cdot(\alpha_{1}\otimes\alpha_{2})(y_{1}\otimes
y_{2})$
$\displaystyle\quad\quad=\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\times$
$\displaystyle\quad\quad\Big{(}\underbrace{\big{(}(x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\big{)}\otimes\big{(}(x_{2}\cdot_{2}y_{2})\cdot_{2}\alpha_{2}(z_{2})\big{)}}_{E_{1}}$
$\displaystyle\quad\quad\underbrace{\big{(}(x_{1}\cdot_{1}y_{1})\cdot_{1}\alpha_{1}(z_{1})\big{)}\otimes\big{(}(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})\big{)}}_{E_{2}}\Big{)}$
$\displaystyle\quad\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1}\varepsilon(x_{2}+z_{2},y_{1})\times$
$\displaystyle\quad\quad\Big{(}\underbrace{\big{(}x_{1}\diamond_{1}z_{1})\cdot_{1}\alpha_{1}(y_{1})\big{)}\otimes\big{(}(x_{2}\cdot_{2}z_{2})\cdot_{2}\alpha_{2}(y_{2})\big{)}}_{F_{1}}$
$\displaystyle\quad\quad+\underbrace{\big{(}(x_{1}\cdot_{1}z_{1})\cdot_{1}\alpha_{1}(y_{1})\big{)}\otimes\big{(}(x_{2}\diamond_{2}z_{2})\cdot_{2}\alpha_{2}(y_{2})\big{)}}_{F_{2}}\Big{)}.$

Furthermore, we have

$\displaystyle\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})E_{1}-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})F_{1}$
(by (2.2))
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\varepsilon(x_{2},y_{2})\big{(}(x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})\big{)}\otimes(y_{2}\cdot_{2}x_{2})\cdot_{2}\alpha_{2}(z_{2})$
$\displaystyle\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1}+y_{2})\times$
$\displaystyle\quad\quad\big{(}(x_{1}\cdot_{1}z_{1})\cdot\alpha_{1}(y_{1})\big{)}\otimes\big{(}\alpha_{2}(y_{2})\cdot_{2}(x_{2}\cdot_{2}z_{2})\big{)}$
(by (2.1))
$\displaystyle\quad=\varepsilon(x_{2},y_{1}+y_{2})\varepsilon(x_{2}+y_{2},z_{1})\times$
$\displaystyle\quad\quad\Big{(}(x_{1}\cdot_{1}y_{1})\diamond_{1}\alpha_{1}(z_{1})-\varepsilon(y_{1},z_{1})(x_{1}\diamond
z_{1})\cdot_{1}\alpha_{1}(y_{1})\Big{)}\otimes\alpha_{2}(y_{2})\cdot_{2}(x_{2}\cdot_{2}z_{2})$
$\displaystyle\text{\scriptsize(by \eqref{HomNovPoisscoloralg1})}\quad=0,$
$\displaystyle\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})E_{2}-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})F_{2}$
(by (2.2))
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\varepsilon(x_{2}+y_{2},z_{1})\varepsilon(x_{1}+y_{1},z_{1})\times$
$\displaystyle\quad\quad\big{(}\alpha_{1}(z_{1})\cdot(x_{1}\cdot
y_{1})\big{)}\otimes\big{(}(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})\big{)}$
$\displaystyle\quad-\varepsilon(y_{1}+y_{2},z_{1}+z_{2})\varepsilon(x_{1}+x_{2},z_{1})\varepsilon(x_{2}+z_{2},y_{1})\times$
$\displaystyle\quad\quad\big{(}(z_{1}\cdot_{1}x_{1})\cdot_{1}\alpha_{1}(y_{1})\big{)}\otimes\big{(}(x_{2}\diamond_{2}z_{2})\diamond_{2}\alpha_{2}(y_{2})\big{)}$
(by (2.1))
$\displaystyle\quad=\varepsilon(x_{2},y_{1})\varepsilon(x_{1}+x_{2}+y_{1}+y_{2},z_{1})\times$
$\displaystyle\quad\alpha_{1}(z_{1})\cdot_{1}(x_{1}\cdot_{1}y_{1})\otimes\Big{(}(x_{2}\cdot_{2}y_{2})\diamond_{2}\alpha_{2}(z_{2})-\varepsilon(y_{2},z_{2})(x_{2}\diamond_{2}z_{2})\cdot_{2}\alpha_{2}(y_{2})\Big{)}$
$\displaystyle\text{\scriptsize(by \eqref{HomNovPoisscoloralg1})}\quad=0.$

Then, $(x\cdot y)\diamond\alpha(z)-\varepsilon(y,z)(x\diamond
z)\cdot\alpha(y)=0.$ Similarly,

$(x\diamond y)\cdot\alpha(z)-\alpha(x)\diamond(y\cdot
z)=\varepsilon(x,y)\big{(}(y\diamond x)\cdot\alpha(z)-\alpha(y)\diamond(x\cdot
z)\big{)}.$

Hence, $(A_{1}\otimes A_{2},\cdot,\diamond,\varepsilon,\alpha)$ is a Hom-
Novikov-Poisson color Hom-algebra.

Step 4: We show that the Equation (3.59) is satisfied:

$\displaystyle\alpha(x)\cdot(y\diamond z)=\alpha(x_{1}\otimes
x_{2})\cdot\varepsilon(y_{2},z_{1})\Big{(}(y_{1}\diamond_{1}z_{1})\otimes(y_{2}\cdot_{2}z_{2})+(y_{1}\cdot_{1}z_{1})\otimes(y_{2}\diamond_{2}z_{2})\Big{)}$
$\displaystyle\quad=\varepsilon(y_{2},z_{1})\varepsilon(x_{2},y_{1}+z_{1})\Big{(}\big{(}\alpha(x_{1})\cdot_{1}(y_{1}\diamond_{1}z_{1})\big{)}\otimes\big{(}\alpha_{2}(x_{2})\cdot_{2}(y_{2}\cdot_{2}z_{2})\big{)}$
$\displaystyle\quad\quad+\big{(}\alpha(x_{1})\cdot_{1}(y_{1}\cdot_{1}z_{1})\big{)}\otimes\big{(}\alpha_{2}(x_{2})\cdot_{2}(y_{2}\diamond
z_{2})\big{)}\Big{)},$ $\displaystyle\alpha(x)\diamond(y\cdot
z)=(\alpha(x_{1})\otimes\alpha_{2}(x_{2}))\diamond\big{(}(y_{1}\otimes
y_{2})\cdot(z_{1}\otimes z_{2})\big{)}$
$\displaystyle\quad=(\alpha_{1}(x_{1})\otimes\alpha_{2}(x_{2}))\diamond\Big{(}\varepsilon(y_{2},z_{1})(y_{1}\diamond_{1}z_{1})\otimes(y_{2}\cdot_{2}z_{2})\Big{)}$
$\displaystyle\quad=\varepsilon(y_{2},z_{1})\varepsilon(x_{2},y_{1}+z_{1})\Big{(}\big{(}\alpha(x_{1})\diamond_{1}(y_{1}\cdot_{1}z_{1})\big{)}\otimes\big{(}\alpha_{2}(x_{2})\cdot_{2}(y_{2}\cdot_{2}z_{2})\big{)}$
$\displaystyle\quad\quad+\big{(}\alpha(x_{1})\cdot_{1}(y_{1}\cdot_{1}z_{1})\big{)}\otimes\big{(}\alpha_{2}(x_{2})\diamond_{2}(y_{2}\cdot_{2}z_{2})\big{)}\Big{)}.$

Now, using the Theorem 3.17 we conclude that $\alpha(x)\cdot(y\diamond
z)=\alpha(x)\diamond(y\cdot z)$. Therefore $as_{A}^{l}(x,y,z)=0$ and hence,
$(A=A_{1}\otimes A_{2},\cdot,\diamond,\varepsilon,\alpha)$ is an admissible
Hom-Novikov-Poisson color Hom-algebra. ∎

By taking in Theorem 3.22, $\alpha_{1}=id_{A_{1}}$ and
$\alpha_{2}=id_{A_{2}}$, we have the following result

######  Corollary 3.23.

Let $(A_{1},\cdot_{1},\diamond_{1},\varepsilon)$ and
$(A_{2},\cdot_{2},\diamond_{2},\varepsilon)$ be admissible Novikov-Poisson
color Hom-algebras and let $A=A_{1}\otimes A_{2}$. Define the operations
$\cdot,\diamond:A\otimes A\rightarrow A$ by the following formulae for
$x_{i},y_{i}\in\mathcal{H}(A_{i}),\ i\in\\{1;2\\}$,

$\displaystyle(x_{1}\otimes x_{2})\cdot(y_{1}\otimes
y_{2})=\varepsilon(x_{2},y_{1})(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2}),$
$\displaystyle(x_{1}\otimes x_{2})\diamond(y_{1}\otimes
y_{2})=\varepsilon(x_{2},y_{1})\big{(}(x_{1}\diamond_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})+(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\diamond_{2}y_{2})\big{)}.$

Then $(A,\cdot,\diamond,\varepsilon)$ is an admissible Novikov-Poisson color
Hom-algebra.

By taking in Theorem 3.22, $\Gamma=\\{e\\}$, we recover the following result

######  Corollary 3.24 ([85]).

Let $(A_{1},\cdot_{1},\diamond_{1},\alpha_{1})$ and
$(A_{2},\cdot_{2},\diamond_{2},\alpha_{2})$ be admissible Hom-Novikov-Poisson
algebras and let $A=A_{1}\otimes A_{2}$. Define the operations
$\alpha:A\rightarrow A$ and $\cdot,\diamond:A\otimes A\rightarrow A$ by the
following formulae for $x_{i},y_{i}\in A_{i},\ i\in\\{1;2\\}$,

$\displaystyle\alpha=\alpha_{1}\otimes\alpha_{2},$ $\displaystyle(x_{1}\otimes
x_{2})\cdot(y_{1}\otimes
y_{2})=(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2}),$
$\displaystyle(x_{1}\otimes x_{2})\diamond(y_{1}\otimes
y_{2})=\big{(}(x_{1}\diamond_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})+(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\diamond_{2}y_{2})\big{)}.$

Then $(A,\cdot,\diamond,\alpha)$ is an admissible Hom-Novikov-Poisson algebra.

By taking in Theorem 3.22, $\alpha_{1}=id_{A_{1}},\alpha_{2}=id_{A_{2}}$ and
$\Gamma=\\{e\\}$, we have the following result

######  Corollary 3.25 ([77]).

Let $(A_{1},\cdot_{1},\diamond_{1})$ and $(A_{2},\cdot_{2},\diamond_{2})$ be
admissible Novikov-Poisson algebras and let $A=A_{1}\otimes A_{2}$. Define the
operations $\cdot,\diamond:A\otimes A\rightarrow A$ by the following formulae
for $x_{i},y_{i}\in A_{i},\ i\in\\{1;2\\}$,

$\displaystyle(x_{1}\otimes x_{2})\cdot(y_{1}\otimes
y_{2})=(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2}),$
$\displaystyle(x_{1}\otimes x_{2})\diamond(y_{1}\otimes
y_{2})=\big{(}(x_{1}\diamond_{1}y_{1})\otimes(x_{2}\cdot_{2}y_{2})+(x_{1}\cdot_{1}y_{1})\otimes(x_{2}\diamond_{2}y_{2})\big{)}.$

Then $(A,\cdot,\diamond)$ is an admissible Novikov-Poisson algebra.

##  4 Hom-Gelfand-Dorfman color Hom-algebras

In this section, our goals are to introduce Hom-Gelfand-Dorfman color Hom-
algebras and to discuss some basic properties and examples of these objects.
Moreover we characterize the representation of Hom-Gelfand-Dorfman color Hom-
algebras and provide some key constructions.

######  Definition 4.1.

A Gelfand-Dorfman color Hom-algebra is a quadruple
$(A,\cdot,[\cdot,\cdot],\varepsilon)$ such that $(A,\cdot,\varepsilon)$ is a
Novikov color Hom-algebra and $(A,[\cdot,\cdot],\varepsilon)$ is a Lie color
Hom-algebra satisfying for all $x,y,z\in\mathcal{H}(A)$, the following
compatibility condition:

$y\cdot[x,z]=\varepsilon(y,x)[x,y\cdot z]-\varepsilon(x+y,z)[z,y\cdot
x]+[y,x]\cdot z-\varepsilon(x,z)[y,z]\cdot x.$ (4.1)

######  Definition 4.2.

A Hom-Gelfand-Dorfman color Hom-algebra is defined as a quintuple
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ such that
$(A,\cdot,\varepsilon,\alpha)$ is a Hom-Novikov color Hom-algebra and
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie color Hom-algebra
satisfying for all $x,y,z\in\mathcal{H}(A)$, the following compatibility
condition:

$\begin{array}[]{rcl}\alpha(y)\cdot[x,z]&=&\varepsilon(y,x)[\alpha(x),y\cdot
z]-\varepsilon(x+y,z)[\alpha(z),y\cdot x]+[y,x]\cdot\alpha(z)\\\ &&\hskip
170.71652pt-\varepsilon(x,z)[y,z]\cdot\alpha(x).\end{array}$ (4.2)

A Hom-Gelfand-Dorfman color Hom-algebra is called multiplicative if the even
linear map $\alpha:A\rightarrow A$ is multiplicative with respect to $\cdot$
and $[\cdot,\cdot]$, that is, for all $x,y\in\mathcal{H}(A)$,

$\alpha(x\cdot
y)=\alpha(x)\cdot\alpha(y),\quad\alpha([x,y])=[\alpha(x),\alpha(y)].$

######  Remark 4.3.

Hom-Gelfand-Dorfman color Hom-algebras contain the Gelfand-Dorfman algebras
and the Hom-Gelfand-Dorfman Hom-algebras for special choices of grading group
and the twisting map.

  1. (i)

When $\Gamma=\\{e\\}$ and $\alpha=id$, we get Gelfand-Dorfman algebra [34,
79].

  2. (ii)

When $\Gamma=\\{e\\}$ and $\alpha\neq id$, we get Hom-Gelfand-Dorfman Hom-
algebra [88].

######  Example 4.4.

Let $\Gamma=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ be an abelian group and $A$ be
a $4$-dimensional $\Gamma$-graded linear space with one-dimensional
homogeneous subspaces

$A_{(0,0)}=<e_{1}>,\quad A_{(0,1)}=<e_{2}>,\quad A_{(1,0)}=<e_{3}>,\quad
A_{(1,1)}=<e_{4}>.$

Then $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Gelfand-Dorfman
color Hom-algebra with

$\displaystyle\text{the
bicharacter:}\quad\varepsilon\big{(}(i_{1},i_{2}),(j_{1},j_{2})\big{)}=(-1)^{i_{1}j_{1}+i_{2}j_{2}},$
$\displaystyle\text{the multiplication:}\quad e_{2}\cdot
e_{3}=\lambda_{1}e_{4},\quad e_{3}\cdot e_{2}=\lambda_{2}e_{4},\quad
e_{3}\cdot e_{3}=\lambda_{3}e_{1},\quad\lambda_{i}\in\mathbb{K},$
$\displaystyle\text{the
bracket:}\quad[e_{2},e_{2}]=\mu_{1}e_{1},\quad[e_{3},e_{2}]=\mu_{2}e_{4},\quad\mu_{i}\in\mathbb{K},$
$\displaystyle\text{the even linear map $\alpha:A\rightarrow A$ given
by}\quad\begin{array}[t]{l}\alpha(e_{1})=-e_{1},\quad\alpha(e_{2})=2e_{2},\\\
\alpha(e_{3})=-2e_{3},\quad\alpha(e_{4})=e_{4}.\\\ \end{array}$

######  Definition 4.5.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ and
$(A^{\prime},\cdot^{\prime},[\cdot,\cdot]^{\prime},\varepsilon^{\prime},\alpha^{\prime})$
be Hom-Gelfand-Dorfman color Hom-algebras. A linear map of degree zero
$f:A\rightarrow A^{\prime}$ is a Hom-Gelfand-Dorfman color Hom-algebra
morphism if

$\displaystyle\cdot^{\prime}\circ(f\otimes
f)=f\circ\cdot,\quad[\cdot,\cdot]^{\prime}\circ(f\otimes
f)=f\circ[\cdot,\cdot],\quad f\circ\alpha=\alpha^{\prime}\circ f.$

######  Proposition 4.6.

Let $(A,\cdot,\varepsilon,\alpha)$ be a Hom-Novikov color Hom-algebra. For all
$x,y\in\mathcal{H}(A)$, let

$[x,y]=x\cdot y-\varepsilon(x,y)y\cdot x.$ (4.3)

Then $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Gelfand-Dorfman
color Hom-algebra.

###### Proof.

Let $(A,\cdot,\varepsilon,\alpha)$ be a Hom-Novikov color Hom-algebra. By
Proposition 2.22 $(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie color
Hom-algebra. Now, we show that the compatibility condition (4.2) is satisfied.
For any $x,y,z\in\mathcal{H}(A)$ we have

$\displaystyle\alpha(y)\cdot[x,z]-\varepsilon(y,x)[\alpha(x),y\cdot
z]+\varepsilon(x+y,z)[\alpha(z),y\cdot x]-[y,x]\cdot\alpha(z)$
$\displaystyle\quad\quad+\varepsilon(x,z)[y,z]\cdot\alpha(x)$
$\displaystyle\quad=\alpha(y)\big{(}x\cdot z-\varepsilon(x,y+z)(y\cdot
z)\cdot\alpha(x)\big{)}+\varepsilon(x+y,z)\big{(}\alpha(z)\cdot(y\cdot x)$
$\displaystyle\quad\quad-\varepsilon(z,y+x)(y\cdot
x)\cdot\alpha(z)\big{)}-\big{(}y\cdot x-\varepsilon(y,x)x\cdot
y)\big{)}\cdot\alpha(z)+\varepsilon(x,z)\big{(}y\cdot z$
$\displaystyle\quad\quad-\varepsilon(y,z)z\cdot y\big{)}\cdot\alpha(x)$
$\displaystyle\quad=\alpha(y)\cdot(x\cdot
z)-\varepsilon(x,z)\alpha(y)\cdot(z\cdot
x)-\varepsilon(y,x)\alpha(x)\cdot(y\cdot z)+\varepsilon(x,z)(y\cdot
z)\cdot\alpha(x)$
$\displaystyle\quad\quad+\varepsilon(x+y,z)\alpha(z)\cdot(y\cdot x)-(y\cdot
x)\cdot\alpha(z)-(y\cdot x)\cdot\alpha(z)+\varepsilon(y,x)(x\cdot
y)\cdot\alpha(z)$ $\displaystyle\quad\quad+\varepsilon(x,z)(y\cdot
z)\cdot\alpha(x)-\varepsilon(x+y,z)(x\cdot
y)\cdot\alpha(x)+\varepsilon(y,x)(x\cdot y)\cdot\alpha(z)$
$\displaystyle\quad=\underbrace{\Big{(}\alpha(y)\cdot(x\cdot z)-(y\cdot
x)\cdot\alpha(z)-\varepsilon(y,x)\big{(}\alpha(x)\cdot(y\cdot z)-(x\cdot
y)\cdot\alpha(z)\big{)}\Big{)}}_{\text{$=0$ by \eqref{eqhomNovikov1}}}$
$\displaystyle\quad\quad+\varepsilon(x,z)\underbrace{\Big{(}(y\cdot
z)\cdot\alpha(x)-\alpha(y)\cdot(z\cdot
x)-\varepsilon(y,z)\big{(}\alpha(z)\cdot(y\cdot x)-(z\cdot
y)\cdot\alpha(x)\Big{)}}_{\text{$=0$ by \eqref{eqhomNovikov1}}}$
$\displaystyle\quad\quad-\underbrace{\Big{(}(y\cdot
x)\cdot\alpha(z)-\varepsilon(x,z)(y\cdot z)\cdot\alpha(x)\Big{)}}_{\text{$=0$
by \eqref{eqhomNovikov2}}}=0.$

Hence, $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Gelfand-Dorfman
color Hom-algebra. ∎

######  Definition 4.7.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ be a Hom-Gelfand-Dorfman
color Hom-algebra. A $\Gamma$-graded subspace $H$ of $A$ is called

  1. (i)

color Hom-subalgebra of $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ if

$\alpha(H)\subseteq H,\ H\cdot H\subseteq H,\ [H,H]\subseteq H,$

  2. (ii)

color Hom-ideal of $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ if

$\alpha(H)\subseteq H,\ A\cdot H\subseteq H,\ H\cdot A\subseteq H,\
[A,H]\subseteq H.$

######  Proposition 4.8.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ a Hom-Gelfand-Dorfman color
Hom-algebra and $I$ a color Hom-ideal of
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$. Then
$(A/I,\overline{\cdot},\\{\cdot,\cdot\\},\varepsilon,\alpha)$ is a Hom-
Gelfand-Dorfman color Hom-algebra where
$\overline{x}~{}\overline{\cdot}~{}\overline{y}=\overline{x\cdot y}$,
$\\{\overline{x},\overline{y}\\}=\overline{[x,y]},\overline{\alpha}(\overline{x})=\overline{\alpha(x)}$
and $\varepsilon(\overline{x},\overline{y})=\varepsilon(x,y)$, for all
$\overline{x},\overline{y}\in\mathcal{H}(A/I)$.

###### Proof.

It follows from a straightforward computation. ∎

######  Proposition 4.9.

Any transposed Hom-Poisson color Hom-algebra is a Hom-Gelfand-Dorfman color
Hom-algebra.

###### Proof.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ be a transposed Hom-Poisson
color Hom-algebra. By definition $(A,\cdot,\varepsilon,\alpha)$ be a
$\varepsilon$-commutative Hom-associative color Hom-algebra, then
$(A,\cdot,\varepsilon,\alpha)$ is a Hom-Novikov color Hom-algebra and
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie color Hom-algebra. Now, we
show that the compatibility condition (4.2) is satisfied. For any
$x,y,z\in\mathcal{H}(A)$ we have

$\displaystyle\alpha(y)\cdot[x,z]-\varepsilon(y,x)[\alpha(x),y\cdot
z]+\varepsilon(x+y,z)[\alpha(z),y\cdot x]$
$\displaystyle-[y,x]\cdot\alpha(z)+\varepsilon(x,z)[y,z]\cdot\alpha(x)$
$\displaystyle\quad=\alpha(y)\cdot\big{(}x\cdot z-\varepsilon(x,z)z\cdot
x\big{)}-\varepsilon(y,x)\big{(}\alpha(x)\cdot(y\cdot
z)-\varepsilon(x,y+z)(y\cdot z)\cdot\alpha(x)\big{)}$
$\displaystyle\quad\quad-\varepsilon(x+y,z)\big{(}\alpha(z)\cdot(y\cdot
x)-\varepsilon(z,y+x)(y\cdot x)\cdot\alpha(z)\big{)}$
$\displaystyle\quad\quad+\big{(}y\cdot x-\varepsilon(y,x)x\cdot
y\big{)}\cdot\alpha(z)-\varepsilon(x,z)\big{(}y\cdot z-\varepsilon(y,z)z\cdot
y\big{)}\cdot\alpha(x)$ $\displaystyle\quad=\alpha(y)\cdot(x\cdot
z)-\varepsilon(x,z)\alpha(y)\cdot(z\cdot
x)-\varepsilon(y,x)\alpha(x)\cdot(y\cdot z)+\varepsilon(x,z)(y\cdot
z)\cdot\alpha(x)$
$\displaystyle\quad\quad+\varepsilon(x+y,z)\alpha(z)\cdot(y\cdot x)-(y\cdot
x)\cdot\alpha(z)-(y\cdot x)\cdot\alpha(z)+\varepsilon(y,x)(x\cdot
y)\cdot\alpha(z)$ $\displaystyle\quad\quad+\varepsilon(x,z)(y\cdot
z)\cdot\alpha(x)-\varepsilon(x+y,z)(z\cdot y)\cdot\alpha(x)$
$\displaystyle\quad=\underbrace{\big{(}\alpha(y)\cdot(x\cdot z)-(y\cdot
x)\cdot\alpha(z)\big{)}}_{\text{$=0$ by
\eqref{Homass:homassociator}}}-\varepsilon(x,z)\underbrace{\big{(}\alpha(y)\cdot(z\cdot
x)-(y\cdot z)\cdot\alpha(x)\big{)}}_{\text{$=0$ by
\eqref{Homass:homassociator}}}$
$\displaystyle\quad\quad-\varepsilon(y,x)\underbrace{\big{(}\alpha(x)\cdot(y\cdot
z)-(x\cdot y)\cdot\alpha(z)\big{)}}_{\text{$=0$ by
\eqref{Homass:homassociator}}}+\varepsilon(x+y,z)\underbrace{\big{(}\alpha(z)\cdot(y\cdot
x)-(z\cdot y)\cdot\alpha(x)\big{)}}_{\text{$=0$ by
\eqref{Homass:homassociator}}}$
$\displaystyle\quad\quad+\underbrace{\big{(}\varepsilon(x,z)(y\cdot
z)\cdot\alpha(x)-(y\cdot x)\cdot\alpha(z)\big{)}}_{\text{$=0$ by
\eqref{Homass:homassociator} and \eqref{commutative:homcoloralgeb}}}=0,$

which completes the proof. ∎

######  Lemma 4.10.

[8] Let $(A,\cdot,\varepsilon,\alpha)$ be a $\varepsilon$-commutative Hom-
associative color Hom-algebra with an even derivation $D$ such that
$\alpha\circ D=D\circ\alpha$. Define

$x\diamond y=x\cdot D(y),\quad for~{}all~{}x,y\in\mathcal{H}(A).$ (4.4)

Then $(A,\diamond,\varepsilon,\alpha)$ is a Hom-Novikov color Hom-algebra.

######  Theorem 4.11.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ be a Hom-Poisson color Hom-
algebra with an even derivation $D$ relative to the both products. Define a
new operation $\diamond$ on $A$ in the following way:

$x\diamond y=x\cdot D(y).$ (4.5)

Then $(A,\diamond,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Gelfand-Dorfman
color Hom-algebra.

###### Proof.

By Lemma 4.10, $(A,\diamond,\varepsilon,\alpha)$ is a Hom-Novikov color Hom-
algebra and by definition of Hom-Poisson color Hom-algebra,
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie color Hom-algebra. Now, we
show that the compatibility condition (4.2) is satisfied. For any
$x,y,z\in\mathcal{H}(A)$,

$\displaystyle\alpha(y)\diamond[x,z]-\varepsilon(y,x)[\alpha(x),y\diamond
z]+\varepsilon(x+y,z)[\alpha(z),y\diamond x]$
$\displaystyle\quad\quad-[y,x]\diamond\alpha(z)+\varepsilon(x,z)[y,z]\diamond\alpha(x)$
(using (4.4)) $\displaystyle\quad=\alpha(y)\cdot
D([x,z])-\varepsilon(y,x)[\alpha(x),y\cdot
D(z))]+\varepsilon(x+y,z)[\alpha(z),y\cdot D(x)]$
$\displaystyle\quad\quad-[y,x]\cdot D(\alpha(z))+\varepsilon(x,z)[y,z]\cdot
D(\alpha(x))$ ($D$ is derivation)
$\displaystyle\quad=\alpha(y)\cdot[D(x),z]+\alpha(y)\cdot[x,D(z)]-\varepsilon(y,x)[\alpha(x),y\cdot
D(z)]$ $\displaystyle\quad\quad+\varepsilon(x+y,z)[\alpha(z),y\cdot
D(x)]-[y,x]\cdot\alpha(D(x))$
$\displaystyle\quad\quad+\varepsilon(x,z)[y,z]\cdot\alpha(D(x))$
$\displaystyle\quad=-\varepsilon(y,x)\underbrace{\Big{(}[\alpha(x),y\cdot
D(z)]-\varepsilon(x,y)\alpha(y)\cdot[x,D(z)]-\varepsilon(y+x,z)\alpha(D(x))\cdot[x,y]\Big{)}}_{\text{$=0$
by \eqref{CompatibiltyPoissonLeibform}}}$
$\displaystyle\quad\quad+\varepsilon(x+y,z)\underbrace{\Big{(}[\alpha(z),y\cdot
D(x)]-\varepsilon(z,y)\alpha(y)\cdot[z,D(x)]-\alpha(D(x))\cdot[z,y]\Big{)}}_{\text{$=0$
by \eqref{CompatibiltyPoissonLeibform}}}=0.\qed$

Let us call a Hom-Gelfand-Dorfman color Hom-algebra
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is special if it can be embedded
into a differential Hom-Poisson color Hom-algebra with operations
$[\cdot,\cdot]$ and $\diamond$ given by (4.5).

######  Definition 4.12.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ be a Hom-Gelfand-Dorfman
color Hom-algebra. A representation of
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a quintuple
$(l,r,\rho,\beta,V)$ such that $(l,r,\beta,V)$ is a bimodule of the Hom-
Novikov color Hom-algebra $(A,\cdot,\varepsilon,\alpha)$ and $(\rho,\beta,V)$
is a representation of the Hom-Lie color Hom-algebra
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ satisfying, for all
$x,y\in\mathcal{H}(A),v\in\mathcal{H}(V)$,

$\displaystyle l(\alpha(y))\rho(x)v=\rho(y\cdot x)\beta(v)$ $\displaystyle+$
$\displaystyle\varepsilon(y,x)\rho(\alpha(x))l(y)v$ (4.6) $\displaystyle-$
$\displaystyle\varepsilon(x,v)r(\alpha(x))\rho(y)v+l([y,x])\beta(v),$
$\displaystyle r([x,y])\beta(v)$ $\displaystyle=$
$\displaystyle\varepsilon(v,x)(\rho(\alpha(x))r(y)v-r(\alpha(y))\rho(x)v)$
(4.7) $\displaystyle+$
$\displaystyle\varepsilon(x+v,y)(r(\alpha(x))\rho(y)v-\rho(\alpha(y))r(x)v).$

######  Proposition 4.13.

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ be a Hom-Gelfand-Dorfman
color Hom-algebra, and let $(l,r,\rho,\beta,V)$ its representation. Then,
$(A\oplus V,\cdot^{\prime},[\cdot,\cdot]^{\prime},\varepsilon,\alpha+\beta)$
is a Hom-Gelfand-Dorfman color Hom-algebra, where $(A\oplus
V,\cdot^{\prime},\varepsilon,\alpha+\beta)$ is the semi-direct product Hom-
Novikov color Hom-algebra $A\ltimes_{l,r,\alpha,\beta}V$, and $(A\oplus
V,[\cdot,\cdot]^{\prime},\varepsilon,\alpha+\beta)$ is the semi-direct product
Hom-Lie color Hom-algebra $A\ltimes_{\rho,\alpha,\beta}V$.

###### Proof.

Let $(l,r,\rho,\beta,V)$ be a representation of a Hom-Gelfand-Dorfman color
Hom-algebra $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$. By Proposition 2.17
and Proposition 2.26, $(A\oplus V,\cdot^{\prime},\varepsilon,\alpha+\beta)$ is
a Hom-Novikov color Hom-algebra, and $(A\oplus
V,[\cdot,\cdot]^{\prime},\varepsilon,\alpha+\beta)$ is a Hom-Lie color Hom-
algebra respectively. Now, we show that the compatibility condition (4.2) is
satisfied. For all $X_{i}=x_{i}+v_{i}\in A_{\gamma_{i}}\oplus
V_{\gamma_{i}},\quad i=1,2,3$ we have

$\displaystyle(\alpha+\beta)(x_{2}+v_{2})\ast[x_{1}+v_{1},x_{3}+v_{3}]^{\prime}$
$\displaystyle-\varepsilon(x_{2}+v_{2},x_{1}+v_{1})[(\alpha+\beta)(x_{1}+v_{1}),(x_{2}+v_{2})\cdot^{\prime}(x_{3}+v_{3})]^{\prime}$
$\displaystyle+\varepsilon(x_{1}+x_{2},x_{3})[(\alpha+\beta)(x_{3}+v_{3}),(x_{2}+v_{2})\cdot^{\prime}(x_{1}+v_{1})]^{\prime}$
$\displaystyle-[(x_{2}+v_{2}),(x_{1}+v_{1})]^{\prime}\cdot^{\prime}(\alpha+\beta)(x_{3}+v_{3})$
$\displaystyle+\varepsilon(x_{1},x_{3})[(x_{2}+v_{2}),(x_{3}+v_{3})]^{\prime}\cdot^{\prime}(\alpha+\beta)(x_{1}+v_{1})$
$\displaystyle\quad=(\alpha(x_{2})+\beta(v_{2}))\cdot^{\prime}([x_{1},x_{3}]+\rho(x_{1})v_{3}-\varepsilon(x_{1},x_{3})\rho(x_{3})v_{1})$
$\displaystyle\quad\quad-\varepsilon(x_{2},x_{1})[\alpha(x_{1})+\beta(v_{1}),x_{2}\cdot
x_{3}+l(x_{2})v_{3}+r(x_{3})v_{2}]^{\prime}$
$\displaystyle\quad\quad+\varepsilon(x_{1}+x_{2},x_{3})[\alpha(x_{3})+\beta(v_{3}),x_{2}\cdot
x_{1}+l(x_{2})v_{1}+r(x_{1})v_{2}]^{\prime}$
$\displaystyle\quad\quad-([x_{2},x_{1}]+\rho(x_{2})v_{1}-\varepsilon(x_{2},x_{1})\rho(x_{1})v_{2})\cdot^{\prime}(\alpha(x_{3})+\beta(v_{3}))$
$\displaystyle\quad\quad+\varepsilon(x_{1},x_{3})([x_{2},x_{3}]+\rho(x_{2})v_{3}-\varepsilon(v_{2},x_{3})\rho(x_{3})v_{2}$
$\displaystyle\quad=\alpha(x_{2})\cdot[x_{1},x_{3}]+l(\alpha(x_{2}))\rho(x_{1})v_{3}$
$\displaystyle\quad\quad-\varepsilon(x_{1},x_{3})l(\alpha(x_{2}))\rho(x_{3})v_{1}+r([x_{1},x_{3}])\beta(v_{2})$
$\displaystyle\quad\quad-\varepsilon(x_{2},x_{1})\Big{(}[\alpha(x_{1}),x_{2}\cdot
x_{3}]+\rho(\alpha(x_{1}))l(x_{2})v_{3}$
$\displaystyle\quad\quad+\rho(\alpha(x_{1}))r(x_{3})v_{2}-\varepsilon(x_{1},x_{2}+x_{3})\rho(x_{2}\cdot
x_{3})\beta(v_{1})\Big{)}$
$\displaystyle\quad\quad+\varepsilon(x_{1}+x_{2},x_{3})\Big{(}[\alpha(x_{3}),x_{2}\cdot
x_{1}]+\rho(\alpha(x_{3}))l(x_{2})v_{1}$
$\displaystyle\quad\quad+\rho(\alpha(x_{3}))r(x_{1})v_{2}-\varepsilon(v_{3},x_{1}+x_{2})\rho(x_{2}\cdot
x_{1})\beta(v_{3})\Big{)}$
$\displaystyle\quad\quad-\Big{(}[x_{2},x_{1}]\cdot\alpha(x_{3})+l([x_{2},x_{1}])\beta(v_{3})$
$\displaystyle\quad\quad+r(\alpha(x_{3}))\rho(x_{2})v_{1}-\varepsilon(x_{2},x_{1})r(\alpha(x_{3}))\rho(x_{1})v_{2}$
$\displaystyle\quad\quad+\varepsilon(x_{1},x_{3})\Big{(}[x_{2},x_{3}]\cdot\alpha(x_{1})+l([x_{2},x_{3}])\beta(v_{1})$
$\displaystyle\quad\quad+r(\alpha(x_{1}))\rho(x_{2})v_{3}-\varepsilon(x_{2},x_{3})r(\alpha(x_{1}))\rho(x_{3})v_{2}$
$\displaystyle\quad=\alpha(x_{2})\cdot[x_{1},x_{3}]+l(\alpha(x_{2}))\rho(x_{1})v_{3}$
$\displaystyle\quad\quad-\varepsilon(x_{1},x_{3})l(\alpha(x_{2}))\rho(x_{3})v_{1}+r([x_{1},x_{3}])\beta(v_{2})$
$\displaystyle\quad\quad-\varepsilon(x_{2},x_{1})[\alpha(x_{1}),x_{2}\cdot
x_{3}]-\varepsilon(x_{2},x_{1})\rho(\alpha(x_{1}))l(x_{2})v_{3}$
$\displaystyle\quad\quad-[x_{2},x_{1}]\cdot\alpha(x_{3})-l([x_{2},x_{1}])\beta(v_{3})$
$\displaystyle\quad\quad-r(\alpha(x_{3}))\rho(x_{2})v_{1}+\varepsilon(x_{2},x_{1})r(\alpha(x_{3}))\rho(x_{1})v_{2}$
$\displaystyle\quad\quad+\varepsilon(x_{1},x_{3})[x_{2},x_{3}]\cdot\alpha(x_{1})+\varepsilon(x_{1},x_{3})l([x_{2},x_{3}])\beta(v_{1})$
$\displaystyle\quad\quad\varepsilon(x_{1},x_{3})r(\alpha(x_{1}))\rho(x_{2})v_{3}-\varepsilon(x_{1}+x_{2},x_{3})r(\alpha(x_{1}))\rho(x_{3})v_{2}$
$\displaystyle\quad=\Big{(}\alpha(x_{2})\cdot[x_{1},x_{3}]-\varepsilon(x_{2},x_{1})[\alpha(x_{1}),x_{2}\cdot
x_{3}]$
$\displaystyle\quad\quad+\varepsilon(x_{1}+x_{2},x_{3})[\alpha(x_{3}),x_{2}\cdot
x_{1}]-[x_{2},x_{1}]\cdot\alpha(x_{3})$
$\displaystyle\quad\quad+\varepsilon(x_{1},x_{3})[x_{2},x_{3}]\cdot\alpha(x_{1})\Big{)}$
$\displaystyle\quad\quad+\Big{(}l(\alpha(x_{2}))\rho(x_{1})v_{3}-\rho(x_{2}\cdot
x_{1})\beta(v_{3})-\varepsilon(x_{2},x_{1})\rho(\alpha(x_{1}))l(x_{2})v_{3}$
$\displaystyle\quad\quad+\varepsilon(x_{1},x_{3})r(\alpha(x_{1}))\rho(x_{2})v_{3}-l([x_{2},x_{1}]\beta(v_{3})\Big{)}$
$\displaystyle\quad\quad-\varepsilon(x_{1},x_{3})\Big{(}l(\alpha(x_{2}))\rho(x_{3})v_{1}-\rho(x_{2}\cdot
x_{3})\beta(v_{1})$
$\displaystyle\quad\quad-\varepsilon(x_{2},x_{3})\rho(\alpha(x_{3}))l(x_{2})v_{1}+\varepsilon(x_{3},x_{1})r(\alpha(x_{3}))\rho(x_{2})v_{1}-l([x_{2},x_{3}])\beta(v_{1})\Big{)}$
$\displaystyle\quad\quad+\Big{(}r([x_{1},x_{3}])\beta(v_{2})-\varepsilon(x_{2},x_{1})(\rho(\alpha(x_{1}))r(x_{3})v_{2}-r(\alpha(x_{3}))\rho(x_{1})v_{2})$
$\displaystyle\quad\quad-\varepsilon(x_{1}+x_{2},x_{3})(r(\alpha(x_{1}))\rho(x_{3})v-\rho(\alpha(x_{3}))r(x_{1})v_{2})\Big{)}=0.$

Hence, $(A\oplus
V,\cdot^{\prime},[\cdot,\cdot]^{\prime},\varepsilon,\alpha+\beta)$ is a Hom-
Gelfand-Dorfman color Hom-algebra. ∎

######  Examples 4.14.

Some important examples of representations of Hom-Gelfand-Dorfman color Hom-
algebras can be constructed as follows.

  1. (i)

Let $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ be a Hom-Gelfand-Dorfman
color Hom-algebra. If

$L(a)b=a\cdot b,\quad R(a)b=b\cdot a,\quad
ad(a)b=[a,b]=-\varepsilon(a,b)[b,a],\quad\forall a,b\in\mathcal{H}(A),$

then $(L,R,ad,\alpha,A)$ is a representation of
$(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$.

  2. (ii)

If
$f:\mathcal{A}=(A,\cdot_{1},[\cdot,\cdot]_{1},\varepsilon,\alpha)\rightarrow(A^{\prime},\cdot_{2},[\cdot,\cdot]_{2},\varepsilon,\beta)$
is a Hom-Gelfand-Dorfman color Hom-algebras morphism, then
$(l,r,\rho,\beta,A^{\prime})$ becomes a representation of $\mathcal{A}$ via
$f$, that is, for all $(x,y)\in\mathcal{H}(A)\times\mathcal{H}(A^{\prime})$,

$l(x)y=f(x)\cdot_{2}y,\quad r(x)y=y\cdot_{2}f(x),\quad\rho(x)y=[f(x),y]_{2}.$

######  Theorem 4.15.

Let $\mathcal{A}=(A,\cdot_{A},[\cdot,\cdot]_{A},\varepsilon,\alpha)$ and
$\mathcal{B}=(B,\cdot_{B},[\cdot,\cdot]_{B},\varepsilon,\beta)$ be Hom-
Gelfand-Dorfman color Hom-algebras. Suppose that there are such even linear
maps $l_{A},r_{A},\rho_{A}:A\rightarrow End(B)$ and
$l_{B},r_{B},\rho_{B}:B\rightarrow End(A)$ that
$A\bowtie^{\rho_{A},\beta}_{\rho_{B},\alpha}B$ is a matched pair of Hom-Lie
color Hom-algebras, and $A\bowtie^{l_{A},r_{A},\beta}_{l_{B},r_{B},\alpha}B$
is a matched pair of Hom-Novikov color Hom-algebras, and for all
$x,y\in\mathcal{H}(A),~{}a,b\in\mathcal{H}(B)$, the following equalities hold:

$\displaystyle\begin{array}[]{l}r_{A}(\rho_{B}(a)x)\beta(b)-r_{A}(\alpha(x))([b,a])\\\
=\varepsilon(a,x)\big{(}\beta(b)\cdot_{B}\rho_{A}(x)a-\rho_{A}(l_{B}(b)x)\beta(a)\\\
\quad-
r_{A}(\rho_{B}(b)x)\beta(a)\big{)}-\varepsilon(a+b,x)\big{(}\rho_{A}(\alpha(x))(b\cdot_{B}a)\\\
\quad-\rho_{A}(x)b\cdot_{B}\beta(a)+\varepsilon(b,a)[\beta(a),r_{A}(x)b]_{B},\end{array}$
(4.12)
$\displaystyle\begin{array}[]{l}l_{A}(\alpha(x))([a,b]_{B})-\rho_{A}(x)a\cdot_{B}\beta(b)-\rho_{A}(r_{B}(a)x)\beta(b)\\\
=\varepsilon(x,a)\big{(}[\beta(a),l_{A}(x)b]_{B}-r_{A}(\rho_{B}(a)x)\beta(b)\big{)}\\\
\quad+\varepsilon(a,b)\big{(}\rho_{A}(r_{B}(b)x)\beta(a)-\rho_{A}(x)b\cdot_{B}\beta(a)\big{)}\\\
\quad+\varepsilon(a+x,b)\big{(}l_{A}(\rho_{B}(b)x)\beta(a)-[\beta(b),l_{A}(x)a]_{B}\big{)},\end{array}$
(4.17)
$\displaystyle\begin{array}[]{l}r_{B}(\rho_{A}(x)a)\alpha(y)-r_{B}(\beta(a))([y,x])\\\
=\varepsilon(x,a)\big{(}\alpha(y)\cdot_{A}\rho_{B}(a)x-\rho_{B}(l_{A}(y)a)\alpha(x)\\\
\quad-
r_{B}(\rho_{A}(y)a)\alpha(x)\big{)}-\varepsilon(x+y,a)\big{(}\rho_{B}(\beta(a))(y\cdot_{A}x)\\\
\quad-\rho_{B}(a)y\cdot_{A}\alpha(x)+\varepsilon(y,x)[\alpha(x),r_{B}(a)y]_{A},\end{array}$
(4.22)
$\displaystyle\begin{array}[]{l}l_{B}(\beta(a))([x,y]_{A})-\rho_{B}(a)x\cdot_{A}\alpha(y)-\rho_{B}(r_{A}(x)a)\alpha(y)\\\
=\varepsilon(a,x)\big{(}[\alpha(x),l_{B}(a)y]_{A}-r_{B}(\rho_{A}(x)a)\beta(y)\big{)}\\\
\quad+\varepsilon(x,y)\big{(}\rho_{B}(r_{A}(y)a)\alpha(x)-\rho_{B}(a)y\cdot_{A}\alpha(x)\big{)}\\\
\quad+\varepsilon(a+x,y)\big{(}l_{B}(\rho_{A}(y)a)\alpha(x)-[\alpha(y),l_{B}(a)x]_{A}\big{)},\end{array}$
(4.27)

Then, $(A,B,l_{A},r_{A},\rho_{A},\beta,l_{B},r_{B},\rho_{B},\alpha)$ is called
a matched pair of the Hom-Gelfand-Dorfman color Hom-algebras. In this case, on
the direct sum $A\oplus B$ of the underlying linear spaces of $\mathcal{A}$
and $\mathcal{B}$, there is a Hom-Gelfand-Dorfman color Hom-algebra structure
which is given for any $x+a\in A_{\gamma_{1}}\oplus B_{\gamma_{1}},y+b\in
A_{\gamma_{2}}\oplus B_{\gamma_{2}}$ by

$\displaystyle(x+a)\cdot(y+b)$ $\displaystyle=$ $\displaystyle
x\cdot_{A}y+(s_{A}(x)b+\varepsilon(a,y)s_{A}(y)a)$ (4.28) $\displaystyle+$
$\displaystyle a\cdot_{B}b+(s_{B}(a)y+\varepsilon(x,b)s_{B}(b)x),$ (4.29)
$\displaystyle~{}[x+a,y+b]$ $\displaystyle=$
$\displaystyle[x,y]_{A}+(\rho_{A}(x)b-\rho_{A}(y)a)$ (4.30) $\displaystyle+$
$\displaystyle[a,b]_{B}+(\rho_{B}(a)y-\rho_{B}(b)x).$ (4.31)

###### Proof.

By Proposition 2.19 and Proposition 2.28, we deduce that $(A\oplus
B,\cdot,\varepsilon,\alpha+\beta)$ is a Hom-Novikov color Hom-algebra and
$(A\oplus B,[\cdot,\cdot],\alpha+\beta)$ is a Hom-Lie color Hom-algebra. Now,
the rest, it is easy (in a similar way as for Proposition 2.11) to verify the
compatibility condition satisfied. ∎

Taking the color $\varepsilon$-commutator in a Hom-Novikov-Poisson color Hom-
algebra, we obtain the following result.

######  Theorem 4.16.

Let $(A,\cdot,\diamond,\varepsilon,\alpha)$ be a Hom-Novikov-Poisson color
Hom-algebra, and

$[x,y]=x\diamond y-\varepsilon(x,y)y\diamond x,\quad\forall
x,y\in\mathcal{H}(A).$ (4.32)

Then $(A,\cdot,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Gelfand-Dorfman
color Hom-algebra.

###### Proof.

By definition $(A,\cdot,\varepsilon,\alpha)$ is a $\varepsilon$-commutative
Hom-associative color Hom-algebra. Then $(A,\cdot,\varepsilon,\alpha)$ is a
Hom-Novikov color Hom-algebra. Moreover, by Proposition 2.22,
$(A,[\cdot,\cdot],\varepsilon,\alpha)$ is a Hom-Lie color Hom-algebra. Now, we
show that the compatibility condition (4.2) is satisfied. For any
$x,y,z\in\mathcal{H}(A)$,

$\displaystyle\alpha(y)\cdot[x,z]-\varepsilon(y,x)[\alpha(x),y\cdot
z]+\varepsilon(x+y,z)[\alpha(z),y\cdot x]$
$\displaystyle\quad\quad-[y,x]\cdot\alpha(z)+\varepsilon(x,z)[y,z]\cdot\alpha(x)$
(using (4.32)) $\displaystyle=\alpha(y)\cdot\big{(}x\diamond
z-\varepsilon(x,z)z\diamond
x\big{)}-\varepsilon(y,x)\big{(}\alpha(x)\diamond(y\cdot z)$
$\displaystyle\quad\quad-\varepsilon(x,y+z)(y\cdot
z)\diamond\alpha(x)\big{)}+\varepsilon(x+y,z)\big{(}\alpha(z)\diamond(y\cdot
x)$ $\displaystyle\quad\quad-\varepsilon(z,y+x)(y\cdot
x^{)}\diamond\alpha(z)\big{)}-\big{(}y\diamond x-\varepsilon(y,x)x\diamond
y\big{)}\cdot\alpha(z)$
$\displaystyle\quad\quad+\varepsilon(x,z)\big{(}(y\diamond
z-\varepsilon(y,z)z\diamond y)\cdot\alpha(x)\big{)}$
$\displaystyle=\alpha(y)\cdot(x\diamond
z)-\varepsilon(x,z)\alpha(y)\cdot(z\diamond
x)-\varepsilon(y,x)\alpha(x)\diamond(y\cdot z)$
$\displaystyle\quad\quad+\varepsilon(x,z)(y\cdot
z)\diamond\alpha(x)+\varepsilon(x+y,z)\alpha(z)\diamond(y\cdot x)$
$\displaystyle\quad\quad-(y\cdot x)\diamond\alpha(z)-(y\diamond
x)\cdot\alpha(z)+\varepsilon(y,x)(x\diamond y)\cdot\alpha(z)$
$\displaystyle\quad\quad+\varepsilon(x,z)(y\diamond
z)\cdot\alpha(x)-\varepsilon(x+y,z)(z\circ y)\cdot\alpha(x)$
$\displaystyle=\varepsilon(y,x)\underbrace{\Big{(}(x\diamond
y)\cdot\alpha(z)-\alpha(x)\diamond(y\cdot z)-\varepsilon(x,y)\big{(}(y\diamond
x)\cdot\alpha(z)-\alpha(y)\cdot(x\diamond z)\big{)}\Big{)}}_{\text{$=0$ by
\eqref{HomNovPoisscoloralg2}}}$
$\displaystyle-\varepsilon(x+y,z)\underbrace{\Big{(}(z\diamond
y)\cdot\alpha(x)-\alpha(z)\diamond(y\cdot x)-\varepsilon(z,y)\big{(}(y\diamond
z)\cdot\alpha(x)-\alpha(y)\diamond(z\cdot x)\big{)}\Big{)}}_{\text{$=0$ by
\eqref{HomNovPoisscoloralg2}}}$
$\displaystyle\quad\quad-\underbrace{\Big{(}(y\cdot
x)\diamond\alpha(z)-\varepsilon(x,z)(y\diamond
z)\cdot\alpha(x)\Big{)}}_{\text{$=0$ by \eqref{HomNovPoisscoloralg1}}}=0.\qed$

######  Example 4.17.

Let $\Gamma=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ be an abelian group and $A$ be
a $4$-dimensional $\Gamma$-graded linear space defined by
$A_{(0,0)}=<e_{1}>,\quad A_{(0,1)}=<e_{2}>,\quad A_{(1,0)}=<e_{3}>$ and
$A_{(1,1)}=<e_{4}>.$ The quintuple $(A,\cdot,\diamond,\varepsilon,\alpha)$ is
a Hom-Novikov-Poisson color Hom-algebra with

$\displaystyle\text{the
bicharacter:}\quad\varepsilon\big{(}(i_{1},i_{2}),(j_{1},j_{2})\big{)}=(-1)^{i_{1}j_{1}+i_{2}j_{2}},$
$\displaystyle\text{the multiplication $"\cdot"$:}\quad e_{2}\cdot
e_{3}=e_{3}\cdot e_{2}=\mu e_{4},\quad\mu\in\mathbb{K},$