Double derivations of $n-$Hom-Lie color algebras

Valiollah Khalili111Department of mathematics, Faculty of sciences, Arak
University, Arak 385156-8-8349, Po.Box: 879, Iran<EMAIL_ADDRESS>
ppp2000 Mathematics Subject Classification(s): 17B40, 17B61, 17B75.
pppKeywords: :n-Hom-Lie color algebras, derivations, inner derivations, double
derivations

ABSTRACT: : We study the double derivation algebra $\mathcal{D}(\mathcal{L})$
of $n-$Hom Lie color algebra $\mathcal{L}$ and describe the relation between
$\mathcal{D}(\mathcal{L})$ and the usual derivation Hom-Lie color algebra
$Der(\mathcal{L}).$ We prove that the inner derivation algebra
$Inn(\mathcal{L})$ is an ideal of the double derivation algebra
$\mathcal{D}(\mathcal{L}).$ We also show that if $\mathcal{L}$ is a perfect
$n-$Hom Lie color algebra with certain constraints on the base field, then the
centralizer of $Inn(\mathcal{L})$ in $\mathcal{D}(\mathcal{L})$ is trivial. In
addition, we obtain that for every centerless perfect $n-$Hom Lie color
algebra $\mathcal{L},$ the triple derivations of the derivation algebra
$Der(\mathcal{L})$ are exactly the derivations of $Der(\mathcal{L}).$

##  1\. Introduction

The first motivation to study $n-$ary algebras appeared in Physics when Nambu
suggested in 1973 a generalization of Hamiltonian Mechanics with more than one
hamiltonian [26]. The mathematical algebraic foundations of Nambu mechanics
have been developed by Takhtajan in [28, 29]. The abstract definition of
$n-$Lie algebra is due to Filippov in 1985 [10], then Kasymov [16] deeper
investigated their properties. This approach uses the interpretation of Jacobi
identity expressing the fact that the adjoint map is a derivation. Nambu
mechanics, $n-$Lie algebras revealed to have many applications in physics.

The theory of Hom-algebras originated from Hom-Lie algebras introduced by J.
T. Hartwig, D. Larsson and S. D. Silvestrov in [15] in the study of quasi-
deformations of Lie algebras of vector fields, including q-deformations of
Witt algebras and Virasoro algebras. Hom-type generalization of $n-$ary
algebras, such as $n-$Hom-Lie algebras and other $n-$ary Hom algebras of Lie
type and associative type, were introduced in [1], by twisting the identities
defining them using an algebra morphism. Further properties, construction
methods, examples, cohomology and central extensions of n-ary Hom-algebras
have been considered in [2, 3, 4, 32].

The concept of derivations appear in different mathematical fields with many
different forms. In algebra systems, derivations are linear maps that satisfy
the Leibniz relation. There are several kinds of derivations in the theory of
Lie algebras, such as Leibniz derivations [19], Jordan derivations [14] and
$\delta-$derivations [11, 12, 13, 17]. The notion of a generalized derivation
of a Lie algebra and their subalgebras is a generalization of
$\delta-$derivation was due to Leger and Luks [22]. Their results were
generalized by many authors. For example, in the case of color Lie algebras
and Hom-Lie color algebras [9, 37], Hom–Poisson color algebras [18] Lie triple
systems and Hom-Lie triple systems [34, 35], color $n-$ary $\Omega-$algebras
and multiplicative $n-$ary Hom-$\Omega-$algebras [20, 8] and many other works.
Another generalization of derivations of Lie algebras are Lie triple
derivations and generalized Lie triple derivations. It was first introduced
independently by Muller where it was called prederivation [24]. It can be
easily checked that, for any Lie algebra, every derivation is a Lie triple
derivation, but the converse does not always hold [33]. The author proved that
every triple derivation of a perfect Lie algebra with zero center is a
derivation and every derivation of the derivation algebra is an inner
derivation. In recent years, much progress has been obtained on the Lie triple
derivations and $n-$Lie drivations of Lie algebras (see [5, 21, 23, 25, 30,
31, 33, 36, 39]).

The main purpose of our work is to generalize the results of R. Bai, Y. Gao,
Y. Zhang and Z. Li [7] and B. Sun, L. Chen and X. Zhou [27] to the case of
multiplicative $n-$Hom- Lie color algebras. Double derivations are similar to
the triple derivations of Lie algebras to some extent [33]. However, the
structure of the algebra of double derivations of $n-$Hom-Lie algebras are
different from that of triple derivations of Lie algebras. The double
derivations provide a useful tool to study $n-$Lie algebras by making use of
linear mappings.

The paper is organized as follows. Section 2 contains some necessary basic
definitions and notions. In this section, we show that the double derivation
algebra of an $n-$Hom-Lie color algebra $\mathcal{L}$ is a color Hom-
subalgebra of the Hom-Lie color algebra $End(\mathcal{L}),$ and for a perfect
$n-$Hom-Lie color algebra $\mathcal{L}$ its inner derivation algebra is an
ideal of the double derivation algebra. In Section 3, we construct a new
double derivation of a perfect centerless multiplicative $n-$Hom-Lie color
algebra $\mathcal{L}$ and obtain a Hom-Lie color algebra homomorphism.
Moreover, we study triple derivations of the derivation algebra
$Der(\mathcal{L})$ and the inner derivation algebra $Inn(\mathcal{L}).$

##  2\. Double Derivations of $n-$Hom-Lie color algebras

Let us begin with some necessary important basic definitions and notations on
graded spaces, algebras and $n-$Hom-Lie color algebras used in other sections.
For a detailed discussion of this subject, we refer the reader to the
literature [6].

Let $G$ be any abelian group with unit element $1,$ a vector space $V$ is said
to be $G-$graded, if there is a family $\\{V_{g}\\}_{g\in G}$ vector subspaces
such that $V=\bigoplus_{g\in G}V_{g}.$ An element $v\in V$ is said to be
homogeneous of the degree $g$ if $v\in V_{g},~{}g\in G,$ and in this case, $g$
is called the color of $v.$ We denote $\mathcal{H}(V)$ the set of all
homogeneous elements in $V.$

Let $V=\bigoplus_{g\in G}V_{g}$ and $W=\bigoplus_{h\in G}W_{h}$ be two
$G-$graded vector spaces. A linear mapping $f:V\longrightarrow W$ is said to
be homogeneous of degree $\theta\in G$ if

$f(V_{g})\subset W_{g+\theta},~{}~{}~{}\forall g\in G.$

If in addition, $f$ is homogeneous of degree zero, namely, $f(V_{g})\subset
W_{g}$ holds for any $g\in G,$ then we call $f$ is even.

An algebra $A$ is said to be $G-$graded if its underlying vector space is
$G-$graded, i.e., $A=\bigoplus_{g\in G}A_{g},$ and if $A_{g}A_{h}\subset
A_{g+h},$ for $g,h\in G.$ A subalgebra of $A$ is said to be graded if it is a
graded as a subspace of $A.$

Let $B$ be another $G-$graded algebra. A morphism $\varphi:A\longrightarrow B$
of $G-$graded algebras is a homomorphism of the algebra $A$ into the algebra
$B,$ which is an even mapping.

######  Definition 2.1.

Let $G$ be an abelian group. A map $\epsilon:G\times
G\longrightarrow{\mathbb{K}}\setminus\\{0\\}$ is called a skew-symmetric bi-
character on $G$ if for all $g,h,k\in G,$

  * (i)

$\epsilon(g,h)\epsilon(h,g)=1,$

  * (ii)

$\epsilon(g+h,k)=\epsilon(g,k)\epsilon(h,k),$

  * (iii)

$\epsilon(g,h+k)=\epsilon(g,h)\epsilon(g,k).$

The definition above implies that in particular, the following relations hold

$\epsilon(g,0)=1=\epsilon(0,g),~{}~{}~{}~{}\epsilon(g,g)=1(\hbox{or}~{}-1),~{}~{}~{}\forall
g\in G.$

Throughout this paper, if $x$ and $y$ are homogeneous elements of a $G-$graded
vector space and $|x|$ and $|y|$ which are in $G$ denote their degrees
respectively, then for convenience, we write $\epsilon(x,y)$ instead of
$\epsilon(|x|,|y|).$ It is worth mentioning that, We unless otherwise stated,
in the sequel all the graded spaces are over the same abelian group $G$ and
the bi-character will be the same for all structures.

######  Definition 2.2.

[10] An $n-$Lie algebra is a linear spaces $L$ equipped with $n-$ary operation
which is skew-symmetric for any pair of variables and satisfies the following
identity:

$[x_{1},...,x_{n-1},[y_{1},...,y_{n}]]=\sum_{i=1}^{n}[y_{1},...,y_{i-1},[x_{1},...,x_{n-1},y_{i}],y_{i+1},...y_{n}].$

######  Definition 2.3.

[6] An $n-$Hom-Lie color algebra is a graded linear space
$\mathcal{L}=\bigoplus_{g\in G}\mathcal{L}_{g}$ with an $n-$linear map
$[.,...,.]~{}:~{}\mathcal{L}\times...\times\mathcal{L}\longrightarrow\mathcal{L},$
a bicharacter $\epsilon:G\times G\longrightarrow{\mathbb{K}}\setminus\\{0\\}$
and an even linear map $\alpha:\mathcal{L}\longrightarrow\mathcal{L}$ such
that

  * (i)

$[x_{1},...x_{i},x_{i+1},...,x_{n}]=-\epsilon(x_{i},x_{i+1})[x_{1},...x_{i+1},x_{i},...,x_{n}],~{}~{}i=1,...,n,$

  * (ii)

$\displaystyle[\alpha(x_{1}),...,\alpha(x_{n-1}),[y_{1},y_{2},...,y_{n}]]=$
(2.4)
$\displaystyle\sum_{i=1}^{n}\epsilon(X,Y_{i})[\alpha(y_{1}),...,\alpha(y_{i-1}),[x_{1},...,x_{n-1},y_{i}],\alpha(y_{i+1}),...,\alpha(y_{n})],$

where
$x_{i},y_{j}\in\mathcal{H}(\mathcal{L}),X=\sum_{i=1}^{n-1}x_{i},Y_{i}=\sum_{j=1}^{i}y_{j-1}$
and $y_{0}=1.$

Whenever $n=2$ (resp. $n=3$) we recover Hom-Lie color algebras (resp. ternairy
Hom-Lie color algebras).

An $n-$Hom-Lie color algebra $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ is said
to be multiplicative if $\alpha$ is an endomorphism, i.e. a linear map on
$\mathcal{L}$ which is also a homomorphism with respect to multiplication
$[.,...,.].$ We also call it a regular Hom-Lie color algebra if $\alpha$ is an
automorphism. We recover $n-$Lie color algebra when we have $\alpha=id,$
$n-$Lie color algebra is a generalization of Lie algebra and Lie super algebra
(if $G=\\{0\\},$ we have $\mathcal{L}=\mathcal{L}_{0}$ is a Lie algebra and if
$G={\mathbb{Z}}_{2}$ and $\epsilon(\bar{1},\bar{1})=-1,$ then $\mathcal{L}$ is
a Lie superalgebra). Clearly, $n-$Hom-Lie algebras and Lie color algebras are
examples of $n-$Hom-Lie color algebras.

######  Definition 2.5.

(1) A graded subspace $S$ of an $n-$Hom-Lie color algebra $\mathcal{L}$ is a
color Hom-subalgebra of $\mathcal{L}$ if $\alpha(S)\subseteq S$ and
$[S,S,...,S]\subseteq S.$

(2) A Hom-ideal $I$ of an $n-$Hom-Lie color algebra $\mathcal{L}$ is a graded
subspace of $\mathcal{L}$ such that $\alpha(I)\subseteq I$ and
$[I,\mathcal{L},...,\mathcal{L}]\subseteq I.$

Let $S_{1},S_{2},...,S_{n}$ be Hom-subalgebras of an $n-$Hom-Lie color algebra
$\mathcal{L}.$ Denote by $[S_{1},S_{2},...,S_{n}]$ the Hom-subalgebra of
$\mathcal{L}$ generated by all elements $[x_{1},x_{2},...,x_{n}],$ where
$x_{i}\in S_{i},i=1,2,...,n.$ The algebra $[\mathcal{L},...,\mathcal{L}]$ is
called the derived algebra of $\mathcal{L},$ and is denoted by
$\mathcal{L}^{1}.$ If $\mathcal{L}^{1}=\mathcal{L},$ then $\mathcal{L}$ is
called a perfect $n-$Hom-Lie color algebra. The center of an $n-$Hom-Lie color
algebra $\mathcal{L}$ is denoted by
$Z(\mathcal{L})=\\{x\in\mathcal{L}~{}:~{}[x,\mathcal{L},...,\mathcal{L}]=0\\}$
which is a Hom-ideal of $\mathcal{L}$ (see Theorem 2.16 in [6]). For a subset
$S$ of $\mathcal{L},$ the centralizer of $S$ in $\mathcal{L}$ is defined by
$C_{\mathcal{L}}(S)=\\{x\in\mathcal{L}~{}:~{}[x,S,\mathcal{L},...,\mathcal{L}]=0\\}.$
Note that $C_{\mathcal{L}}(\mathcal{L})=Z(\mathcal{L}).$

From now on, we let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be a
multiplicative $n-$Hom-Lie color algebra, which we will denote by
$\mathcal{L},$ for short. For any nonnegative integer $k,$ denote by
$\alpha^{k}$ the $k-$times composition of $\alpha.$ In particular
$\alpha^{0}=id,~{}\alpha^{1}=\alpha.$

Here we give a more general definition of derivation and related objects.

######  Definition 2.6.

For any $k\geq 1,$ we call $D\in End(\mathcal{L})$ an $\alpha^{k}-$derivation
of degree $d$ of the multiplicative $n-$Hom-Lie color algebra $\mathcal{L}$ if

(1)- $D\circ\alpha=\alpha\circ D.$

(2)-For all $x_{1},x_{2},...,x_{n}\in\mathcal{H}(\mathcal{L}),$

$D([x_{1},...,x_{n}])=\sum_{i=1}^{n}\epsilon(d,X_{i})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{i-1}),D(x_{i}),\alpha^{k}(x_{i+1}),...,\alpha^{k}(x_{n})],$

where $X_{i}=\sum_{j=1}^{i}x_{j}.$

We denote the set of $\alpha^{k}-$derivations of the multiplicative $n-$Hom-
Lie color algebras $\mathcal{L}$ by $Der_{\alpha^{k}}(\mathcal{L}).$ For any
$D\in Der_{\alpha^{k}}(\mathcal{L})$ and $D^{\prime}\in
Der_{\alpha^{s}}(\mathcal{L}),$ let us define their color commutator
$[D,D^{\prime}]=D\circ D^{\prime}-\epsilon(d,d^{\prime})D^{\prime}\circ D.$
Then $[D,D^{\prime}]\in Der_{\alpha^{k+s}}$ (see Lemma 5.6 in [6]). Denote by
$Der(\mathcal{L})=\bigoplus_{k\geq-1}Der_{\alpha^{k}}(\mathcal{L}).$ We also
have that $(Der(\mathcal{L}),[.,.],\tilde{\alpha}=D\circ\alpha,\epsilon)$ is a
Hom-Lie color algebra (see Proposition 5.7 in [6]).

For $x_{1},...,x_{n-1}\in\mathcal{L}$ satisfying
$\alpha(x_{i})=x_{i},i=1,2,...,n-1,$ we define the map
$ad_{k}(x_{1},...,x_{n-1})~{}:~{}\mathcal{L}\longrightarrow\mathcal{L}$ by

$ad_{k}(x_{1},...,x_{n-1})(y):=[x_{1},...,x_{n-1},\alpha^{k}(y)],~{}\hbox{~{}and~{}}~{}k\geq
1,$

for all $y\in\mathcal{H}(\mathcal{L}).$ Then $ad_{k}(x_{1},...,x_{n-1}),$ is
an $\alpha^{k+1}-$derivation of $\mathcal{L}$ (see Lemma 2.2 in [4]) We call
$ad_{k}(x_{1},...,x_{n-1})$ an inner $\alpha^{k+1}-$derivation. Denote by
$Inn_{\alpha^{k}}(\mathcal{L})$ the space generate by all the inner
$\alpha^{k+1}-$derivations. Set $Inn(\mathcal{L})=\bigoplus_{k\geq
0}Inn_{\alpha^{k}}(\mathcal{L}).$ It is not difficult to show that
$(Inn(\mathcal{L}),[.,.],\tilde{\alpha}=D\circ\alpha,\epsilon)$ is a Hom-Lie
color algebra. In addition, $Inn(\mathcal{L})$ is a Hom-ideal of
$Der(\mathcal{L})$ (see Proposition 2.4 in [4]).

######  Definition 2.7.

Let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be a multiplicative $n-$Hom-Lie
color algebra with $n\geq 3.$ A linear map
$D~{}:~{}\mathcal{L}\longrightarrow\mathcal{L}$ is called a double
$\alpha^{k}-$derivation of degree $d$ of $\mathcal{L}$ if

(1)- $D\circ\alpha=\alpha\circ D.$

(2)-For all $x_{1},x_{2},...,x_{n-1},y_{1},...,y_{n}\in\mathcal{L},$

$\displaystyle D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$ (2.8)
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d,X_{i})[\alpha^{k}(x_{1}),...,D(x_{i}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d,X+Y_{j})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{j}),...,\alpha^{k}(y_{n})]].$

Let us denote the set of double $\alpha^{k}-$derivations of the multiplicative
$n-$Hom-Lie color algebras $\mathcal{L}$ by
$\mathcal{D}_{\alpha^{k}}(\mathcal{L})$ and denote by
$\mathcal{D}(\mathcal{L})=\bigoplus_{k\geq-1}\mathcal{D}_{\alpha^{k}}(\mathcal{L}),$
the vector space spanned by double derivations of $\mathcal{L}.$ We call it
double derivation algebra of $\mathcal{L}.$

It is obvious that the derivations of an $n-$Hom-Lie color algebra
$\mathcal{L}$ are double derivations, and hence we have
$Der(\mathcal{L})\subset\mathcal{D}(\mathcal{L}).$ Note that there exists an
$n-$Lie superalgebra $\mathcal{L}$ such that
$Der(\mathcal{L})\neq\mathcal{D}(\mathcal{L})$ (see Remark 2.5 in [27]).

######  Theorem 2.9.

Let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be a multiplicative $n-$Hom-Lie
color algebra. Then $\mathcal{D}(\mathcal{L})$ is a color Hom-subalgebra of
the general linear Hom-Lie color algebra
$(End(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon)$ with
$\tilde{\alpha}(D)=D\circ\alpha$ for all $D\in\mathcal{D}(\mathcal{L}).$

Proof. First, let us prove that
$\tilde{\alpha}(\mathcal{D}(\mathcal{L}))\subseteq\mathcal{D}(\mathcal{L}).$
Suppose that $D\in\mathcal{D}_{\alpha^{k}}(\mathcal{L}),$ for any
$x_{1},x_{2},...,x_{n-1},y_{1},...,y_{n}\in\mathcal{L},$ we have

$\displaystyle\tilde{\alpha}(D)([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=$
$\displaystyle(D\circ\alpha)([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=$ $\displaystyle
D([\alpha(x_{1}),...,\alpha(x_{n-1}),[\alpha(y_{1}),...,\alpha(y_{n})]])$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d,X_{i})[\alpha^{k+1}(x_{1}),...,D(\alpha(x_{i})),...,\alpha^{k+1}(x_{n-1}),[\alpha^{k+1}(y_{1}),...,$
$\displaystyle\alpha^{k+1}(y_{n})]]$ $\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d,X+Y_{j})[\alpha^{k+1}(x_{1}),...,\alpha^{k+1}(x_{n-1}),[\alpha^{k+1}(y_{1}),...,D(\alpha(y_{j})),...,$
$\displaystyle\alpha^{k+1}(y_{n})]]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d,X_{i})[\alpha^{k+1}(x_{1}),...,(D\circ\alpha)(x_{i}),...,\alpha^{k+1}(x_{n-1}),[\alpha^{k+1}(y_{1}),...,$
$\displaystyle\alpha^{k+1}(y_{n})]]$ $\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d,X+Y_{j})[\alpha^{k+1}(x_{1}),...,\alpha^{k+1}(x_{n-1}),[\alpha^{k+1}(y_{1}),...,(D\circ\alpha)(y_{j}),...,$
$\displaystyle\alpha^{k+1}(y_{n})]]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d,X_{i})[\alpha^{k+1}(x_{1}),...,(\tilde{\alpha}(D))(x_{i}),...,\alpha^{k+1}(x_{n-1}),[\alpha^{k+1}(y_{1}),...,$
$\displaystyle\alpha^{k+1}(y_{n})]]$ $\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d,X+Y_{j})[\alpha^{k+1}(x_{1}),...,\alpha^{k+1}(x_{n-1}),[\alpha^{k+1}(y_{1}),...,(\tilde{\alpha}(D))(y_{j}),...,$
$\displaystyle\alpha^{k+1}(y_{n})]].$

This means that $\tilde{\alpha}(D)$ is a duble $\alpha^{k+1}-$derivation i.e.
$\tilde{\alpha}(D)\in\mathcal{D}(\mathcal{L}).$

Secondly, assum that $D_{1}\in\mathcal{D}_{\alpha^{k}}(\mathcal{L})$ of degree
$d_{1}$ and $D_{2}\in\mathcal{D}_{\alpha^{s}}(\mathcal{L})$ of degree $d_{2}.$
It is clear that $[D_{1},D_{2}]\circ\alpha=\alpha\circ[D_{1},D_{2}].$ We also
have

$\displaystyle D_{1}D_{2}([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=$ $\displaystyle
D_{1}(\sum_{i=1}^{n-1}\epsilon(d_{2},X_{i})[\alpha^{s}(x_{1}),...,D_{2}(x_{i}),...,\alpha^{s}(x_{n-1}),[\alpha^{s}(y_{1}),...,\alpha^{s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d_{2},X+Y_{j})[\alpha^{s}(x_{1}),...,\alpha^{s}(x_{n-1}),[\alpha^{s}(y_{1}),...,D_{2}(y_{j}),...,\alpha^{s}(y_{n})]])$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d_{1}+d_{2},X_{i})[\alpha^{k+s}(x_{1}),...,D_{1}D_{2}(x_{i}),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,$
$\displaystyle\alpha^{k+s}(y_{n})]]$ $\displaystyle+$
$\displaystyle\sum_{i=1}^{n-1}\sum_{r<i}\epsilon(d_{1},X_{r})\epsilon(d_{2},X_{i})[\alpha^{k+s}(x_{1}),...,D_{1}(\alpha^{s}(x_{r})),...,D_{2}(\alpha^{k}(x_{i})),...,$
$\displaystyle\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n-1}\sum_{i<r}\epsilon(d_{1},X_{r})\epsilon(d_{2},X_{i})\epsilon(d_{1},d_{2})[\alpha^{k+s}(x_{1}),...,D_{2}(\alpha^{k}(x_{i})),...,$
$\displaystyle
D_{1}(\alpha^{s}(x_{r})),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n-1}\sum_{t=1}^{n-1}\epsilon(d_{2},X_{i})\epsilon(d_{1},X)\epsilon(d_{1},d_{2})[\alpha^{k+s}(x_{1}),...,D_{2}(\alpha^{k}(x_{i})),...,$
$\displaystyle\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,D_{1}(\alpha^{s}(y_{t})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\sum_{m=1}^{n-1}\epsilon(d_{1},X_{m})\epsilon(d_{2},X+Y_{j})[\alpha^{k+s}(x_{1}),...,D_{1}(\alpha^{s}(x_{j})),...,$
$\displaystyle\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,D_{2}(\alpha^{k}(y_{m})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\sum_{l<j}\epsilon(d_{1},X+Y_{l})\epsilon(d_{2},X+Y_{j})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),$
$\displaystyle[\alpha^{k+s}(y_{1}),...,D_{1}(\alpha^{s}(y_{l})),...,D_{2}(\alpha^{k}(y_{j})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\sum_{j<l}\epsilon(d_{1},X+Y_{l})\epsilon(d_{2},X+Y_{j})\epsilon(d_{1},d_{2})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),$
$\displaystyle[\alpha^{k+s}(y_{1}),...,D_{2}(\alpha^{k}(y_{j})),...,D_{1}(\alpha^{s}(y_{l})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X+Y_{j})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,D_{1}D_{2}(x_{j}),$
$\displaystyle...,\alpha^{k+s}(y_{n})]].$

Similarly,

$\displaystyle\epsilon(d_{1},d_{2})D_{2}D_{1}([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d_{1}+d_{2},X_{i})[\alpha^{k+s}(x_{1}),...,D_{2}D_{1}(x_{i}),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,$
$\displaystyle\alpha^{k+s}(y_{n})]]$ $\displaystyle+$
$\displaystyle\sum_{i=1}^{n-1}\sum_{r<i}\epsilon(d_{2},X_{r})\epsilon(d_{1},X_{i})[\alpha^{k+s}(x_{1}),...,D_{2}(\alpha^{k}(x_{r})),...,D_{1}(\alpha^{s}(x_{i})),...,$
$\displaystyle\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n-1}\sum_{i<r}\epsilon(d_{2},X_{r})\epsilon(d_{1},X_{i})\epsilon(d_{2},d_{1})[\alpha^{k+s}(x_{1}),...,D_{1}(\alpha^{s}(x_{i})),...,$
$\displaystyle
D_{2}(\alpha^{k}(x_{r})),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n-1}\sum_{t=1}^{n-1}\epsilon(d_{1},X_{i})\epsilon(d_{2},X)\epsilon(d_{2},d_{1})[\alpha^{k+s}(x_{1}),...,D_{1}(\alpha^{s}(x_{i})),...,$
$\displaystyle\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,D_{2}(\alpha^{k}(y_{t})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\sum_{m=1}^{n-1}\epsilon(d_{2},X_{m})\epsilon(d_{1},X+Y_{j})[\alpha^{k+s}(x_{1}),...,D_{2}(\alpha^{k}(x_{j})),...,$
$\displaystyle\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,D_{1}(\alpha^{k}(y_{m})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\sum_{l<j}\epsilon(d_{2},X+Y_{l})\epsilon(d_{1},X+Y_{j})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),$
$\displaystyle[\alpha^{k+s}(y_{1}),...,D_{2}(\alpha^{k}(y_{l})),...,D_{1}(\alpha^{s}(y_{j})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\sum_{j<l}\epsilon(d_{2},X+Y_{l})\epsilon(d_{1},X+Y_{j})\epsilon(d_{2},d_{1})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),$
$\displaystyle[\alpha^{k+s}(y_{1}),...,D_{1}(\alpha^{k}(y_{j})),...,D_{2}(\alpha^{k}(y_{l})),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X+Y_{j})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,$
$\displaystyle D_{2}D_{1}(x_{j}),...,\alpha^{k+s}(y_{n})]].$

Then from easy computation we have

$\displaystyle[D_{1},D_{2}]([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=$
$\displaystyle(D_{1}D_{2}-\epsilon(d_{1},d_{2})D_{2}D_{1})([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}[\alpha^{s}(x_{1}),...,(D_{1}D_{2}-\epsilon(d_{1},d_{2})D_{2}D_{1})(x_{i}),...,\alpha^{s}(x_{n-1}),[\alpha^{s}(y_{1}),...,\alpha^{s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X+Y_{j})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,$
$\displaystyle(D_{1}D_{2}-\epsilon(d_{1},d_{2})D_{2}D_{1})(y_{j}),...,\alpha^{k+s}(y_{n})]]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}[\alpha^{s}(x_{1}),...,[D_{1},D_{2}](x_{i}),...,\alpha^{s}(x_{n-1}),[\alpha^{s}(y_{1}),...,\alpha^{s}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X+Y_{j})[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),[\alpha^{k+s}(y_{1}),...,$
$\displaystyle[D_{1},D_{2}](x_{j}),...,\alpha^{k+s}(y_{n})]]$

This implies that, $[D_{1},D_{2}]$ is a double $\alpha^{k+s}-$derivation of
degree $d_{1}+d_{2}$ and so $[D_{1},D_{2}]\in\mathcal{D}(\mathcal{L})$ as
required. $\Box$

######  Theorem 2.10.

If $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ is a perfect multiplicative
$n-$Hom-Lie color algebra, then $Inn(\mathcal{L})$ is a Hom-ideal of the Hom-
Lie color algebra $(\mathcal{D}(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon).$

Proof. Let us prove that $\tilde{\alpha}(Inn(\mathcal{L}))\subseteq
Inn(\mathcal{L}).$

For $x_{1},x_{2},...,x_{n-1}\in\mathcal{L}$ and $y\in\mathcal{L},$ we have

$\displaystyle\tilde{\alpha}(ad_{k}(x_{1},...,x_{n-1}))(y)$ $\displaystyle=$
$\displaystyle ad_{k}(x_{1},...,x_{n-1})(\alpha(y))$ $\displaystyle=$
$\displaystyle[x_{1},...,x_{n-1},\alpha^{k+1}(y)]$ $\displaystyle=$
$\displaystyle ad_{k+1}(x_{1},...,x_{n-1})(y).$

This means that $\tilde{\alpha}(ad_{k}(x_{1},...,x_{n-1}))$ is an inner
$\alpha^{k+2}-$derivation i.e. $\tilde{\alpha}(Inn(\mathcal{L}))\subseteq
Inn(\mathcal{L}).$

Next, let $D\in\mathcal{D}_{\alpha^{s}}(\mathcal{L})$ and
$ad_{k}(x_{1},x_{2},...,x_{n-1})$ be an inner $\alpha^{k+1}-$derivation. Then

$[D,ad_{k}(x_{1},x_{2},...,x_{n-1})]\in Der_{\alpha^{k+s+1}}(\mathcal{L}).$

Since $\mathcal{L}$ is perfect, there exists a finite index set $I$ and
$x_{i_{j}}\in\mathcal{L},i\in I,j=1,...,n,$ such that

$x_{1}=\sum_{i\in I}[x_{i_{1}},x_{i_{2}},...,x_{i_{n}}].$

For any $y\in\mathcal{L}$ we have

$\displaystyle[D,ad_{k}(x_{1},...,x_{n-1})](y)$ $\displaystyle=$
$\displaystyle D\circ
ad_{k}(x_{1},...,x_{n-1})(y)-\epsilon(d,X)ad_{k}(x_{1},...,x_{n-1})\circ D(y)$
$\displaystyle=$ $\displaystyle
D([\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),\alpha^{k}(y)])-\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),$
$\displaystyle\alpha^{k}(D(y))]$ $\displaystyle=$ $\displaystyle\sum_{i\in
I}(D([[\alpha^{k}(x_{i_{1}}),\alpha^{k}(x_{i_{2}}),...,\alpha^{k}(x_{i_{n}})],\alpha^{k}(x_{2}),...,\alpha^{k}(x_{n-1}),$
$\displaystyle\alpha^{k}(y)])$ $\displaystyle-$
$\displaystyle\epsilon(d,X)[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),D(\alpha^{k}(y))])$
$\displaystyle=$ $\displaystyle\sum_{i\in
I}((-1)^{n-1}\epsilon(x_{1},x_{2}+...+x_{n-1}+y)D([\alpha^{k}(x_{2}),...,\alpha^{k}(x_{n-1}),$
$\displaystyle\alpha^{k}(y),[\alpha^{k}(x_{i_{1}}),\alpha^{k}(x_{i_{2}}),...,\alpha^{k}(x_{i_{n}})]]))$
$\displaystyle-$
$\displaystyle\epsilon(d,X)[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),D(\alpha^{k}(y))]$
$\displaystyle=$ $\displaystyle\sum_{i\in
I}(-1)^{n-1}\epsilon(x_{1},x_{2}+...+x_{n-1}+y)(\sum_{j=2}^{n-2}\epsilon(d,x_{2}+...+x_{j-1})$
$\displaystyle[\alpha^{k+s}(x_{2}),...,D(x_{j}),...,\alpha^{k+s}(x_{n-1}),\alpha^{k+s}(y),[\alpha^{k+s}(x_{i_{1}}),$
$\displaystyle...,\alpha^{k+s}(x_{i_{n}})]]$ $\displaystyle+$
$\displaystyle\epsilon(d,x_{2}+...x_{n-1})[\alpha^{k+s}(x_{2}),...,\alpha^{k+s}(x_{n-1}),D(\alpha^{k}(y)),[\alpha^{k+s}(x_{i_{1}}),$
$\displaystyle...,\alpha^{k+s}(x_{i_{n}})]]$ $\displaystyle+$
$\displaystyle\sum_{t=1}^{n}\epsilon(d,x_{2}+...+x_{n-1}+y+X_{i_{t}})[\alpha^{k+s}(x_{2}),...,\alpha^{k+s}(x_{n-1}),$
$\displaystyle\alpha^{k+s}(y),[\alpha^{k+s}(x_{i_{1}}),...,D(x_{i_{t}}),...,\alpha^{k+s}(x_{i_{n}})]])$
$\displaystyle-$
$\displaystyle\epsilon(d,X)[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),D(\alpha^{k}(y))]$
$\displaystyle=$ $\displaystyle\sum_{i\in
I}\sum_{j=2}^{n-1}\epsilon(d,X_{j})[[\alpha^{k+s}(x_{i_{1}}),...,\alpha^{k+s}(x_{i_{n}})],\alpha^{k+s}(x_{2}),...,D(x_{j}),$
$\displaystyle...,\alpha^{k+s}(x_{n-1}),\alpha^{k+s}(y)]$ $\displaystyle+$
$\displaystyle\sum_{i\in
I}\epsilon(d,X)[[\alpha^{k+s}(x_{i_{1}}),...,\alpha^{k+s}(x_{i_{n}})],\alpha^{k+s}(x_{2}),...,\alpha^{k+s}(x_{n-1}),$
$\displaystyle D(\alpha^{k}(y))]$ $\displaystyle+$ $\displaystyle\sum_{i\in
I}\sum_{t=1}^{n}\epsilon(d,X_{i_{t}})[[\alpha^{k+s}(x_{i_{1}}),...,\alpha^{k+s}(x_{i_{n}})],\alpha^{k+s}(x_{2}),...,\alpha^{k+s}(x_{n-1}),$
$\displaystyle\alpha^{k+s}(y)])$ $\displaystyle-$
$\displaystyle\epsilon(d,X)[\alpha^{k+s}(x_{1}),...,\alpha^{k+s}(x_{n-1}),D(\alpha^{k}(y))]$
$\displaystyle=$
$\displaystyle\sum_{j=2}^{n-1}\epsilon(d,X_{j})[\alpha^{k+s}(x_{1}),...,D(x_{j}),...,\alpha^{k+s}(x_{n-1}),\alpha^{k+s}(y)]$
$\displaystyle+$ $\displaystyle\sum_{i\in
I}(\sum_{t=1}^{n}\epsilon(d,X_{i_{t}})[[\alpha^{k+s}(x_{i_{1}}),...,D(x_{i_{t}}),...,\alpha^{k+s}(x_{i_{n}})],$
$\displaystyle\alpha^{k+s}(x_{2}),...,\alpha^{k+s}(x_{n-1}),\alpha^{k+s}(y)])$
$\displaystyle=$
$\displaystyle\sum_{j=2}^{n-1}\epsilon(d,X_{j})ad_{k+s}(x_{i_{1}},...,D(x_{j}),...,x_{n-1})(y)$
$\displaystyle+$ $\displaystyle\sum_{i\in
I}(\sum_{t=1}^{n}\epsilon(d,X_{i_{t}})ad_{k+s}([x_{i_{1}},...,D(x_{i_{t}}),...,x_{i_{n}}],x_{2},...,x_{n-1})(y).$

Consequently, $[D,ad_{k}(x_{1},...,x_{n-1}]\in
Inn_{\alpha^{k+s+1}}(\mathcal{L}),$ which implies that $Inn(\mathcal{L})$ is
an ideal of Hom-Lie color algebra
$(\mathcal{D}(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon).$ $\Box$

Note that, if $\mathcal{L}$ is not a perfect multiplicative $n-$Hom-Lie color
algebra, then $Inn(\mathcal{L})$ may not be an ideal of
$\mathcal{D}(\mathcal{L})$ (see Rimark 2.8 in [27]).

##  3\. Double Derivations of Perfect multiplicative $n-$Hom-Lie color
algebras

In this section, we focus on the Perfectnes of multiplicative $n-$Hom-Lie
color algebras by centering our attention in those of centerless.

Let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be an arbitrary centerless
multiplicative $n-$Hom-Lie color algebra. For all
$D\in\mathcal{D}_{\alpha^{k}}(\mathcal{L}),$ we define a linear map
$\delta_{D}~{}:~{}\mathcal{L}\longrightarrow\mathcal{L},$ with
$\delta_{D}\circ\alpha=\alpha\circ\delta_{D}$ by

$\delta_{D}(x)=\left\\{\begin{array}[]{ll}D(x)&;~{}x\in\mathcal{L}\setminus\mathcal{L}^{1}\\\
\sum_{i\in
I}\sum_{t=1}^{n}\epsilon(d,X_{i_{t}})[\alpha^{k}(x_{i_{1}}),...,D(x_{i_{t}}),...,\alpha^{k}(x_{i_{n}})]&;~{}x=\sum_{i\in
I}[x_{i_{1}},\\\ &...,x_{i_{n}}]\in\mathcal{L}^{1},\end{array}\right.$ (3.1)

for all $x\in\mathcal{L}.$

######  Lemma 3.2.

The linear map $\delta_{D}$ defined as Eq. (3.1) is well defined.

Proof. Suppose that

$x=\sum_{i\in I}[x_{i_{1}},...,x_{i_{n}}]=\sum_{j\in
J}[x_{j_{1}},...,x_{j_{n}}]=y,$

we set $\delta_{D}(x)=A$ and $\delta_{D}(y)=B.$ For
$D\in\mathcal{D}_{\alpha^{k}}(\mathcal{L})$ of degree $d$ and
$z_{1},...,z_{n-1}\in\mathcal{L},$ we have

$\displaystyle[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),A]$ $\displaystyle=$
$\displaystyle[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),\sum_{i\in
I}\sum_{t=1}^{n}\epsilon(d,X_{i_{t}})[\alpha^{k}(x_{i_{1}}),...,D(x_{i_{t}}),...,\alpha^{k}(x_{i_{n}})]]$
$\displaystyle=$
$\displaystyle\epsilon(d,Z)(D([z_{1},...,z_{n-1},x])-\sum_{s=1}^{n-1}\epsilon(d,Z_{s})[\alpha^{k}(z_{1}),...,D(z_{s}),...,\alpha^{k}(z_{n-1}),$
$\displaystyle\alpha^{k}(x)])$ $\displaystyle=$
$\displaystyle\epsilon(d,Z)(D([z_{1},...,z_{n-1},y])-\sum_{s=1}^{n-1}\epsilon(d,Z_{s})[\alpha^{k}(z_{1}),...,D(z_{s}),...,\alpha^{k}(z_{n-1}),$
$\displaystyle\alpha^{k}(y)])$ $\displaystyle=$
$\displaystyle\epsilon(d,Z)(D([z_{1},...,z_{n-1},\sum_{j\in
J}[x_{j_{1}},...,x_{j_{n}}]])-\sum_{s=1}^{n-1}\epsilon(d,Z_{s})[\alpha^{k}(z_{1}),...,D(z_{s}),$
$\displaystyle...,\alpha^{k}(z_{n-1}),\sum_{j\in
J}[\alpha^{k}(x_{j_{1}}),...,\alpha^{k}(x_{j_{n}})]])$ $\displaystyle=$
$\displaystyle[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),\sum_{j\in
J}\sum_{s=1}^{n}\epsilon(d,Y_{j_{s}})[\alpha^{k}(y_{j_{1}}),...,D(y_{j_{s}}),...,\alpha^{k}(y_{j_{n}})]]$
$\displaystyle=$ $\displaystyle[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),B].$

Therefore, we get that $[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),A-B]=0.$
This implies $A-B\in Z(\mathcal{L})=0,$ we obtain
$\delta_{D}(x)=\delta_{D}(y).$ Thus, $\delta_{D}$ is well-defined. $\Box$

Consequently, we obtain a linear map
$\delta~{}:~{}\mathcal{D}(\mathcal{L})\longrightarrow End(\mathcal{L})$
defined by

$\delta(D)=\delta_{D},~{}~{}\forall D\in\mathcal{D}(\mathcal{L}).$

######  Proposition 3.3.

Let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be a centerless perfect
multiplicative $n-$Hom-Lie color algebra. Suppose that
$D\in\mathcal{D}(\mathcal{L}),$ then the following assertions hold.

  * (1)

$\delta_{D}$ is a double derivation of $\mathcal{L}.$

  * (2)

For all $x_{1},...,x_{n}\in\mathcal{L},$ we have

$(D-\delta_{D})([x_{1},...,x_{n}])=\epsilon(d,X_{i})[\alpha^{k}(x_{1}),...,(D-\delta_{D})(x_{i}),...,\alpha^{k}(x_{n})],~{}i=1,...,n.$

  * (3)

$\delta_{D-\delta_{D}}=-n(D-\delta_{D}).$

Proof. (1) Suppose that $D\in\mathcal{D}_{\alpha^{k}}(\mathcal{L}).$ Since
$\mathcal{L}$ is perfect, we can write

$x_{j}=\sum_{i\in{I_{j}}}[x_{i_{j_{1}}},...,x_{i_{j_{n}}}],~{}~{}~{}j=1,...,n-1~{}~{}\hbox{~{}~{}and~{}~{}}~{}~{}y_{s}=\sum_{p\in{P_{s}}}[y_{p_{s_{1}}},...,y_{p_{s_{n}}}],~{}~{}s=1,...,n.$

From Eq. (3.1), we have

$\delta_{D}(x_{j})=\sum_{l=1}^{n}\sum_{i\in{I_{j}}}\epsilon(d,X_{I_{j_{l}}})([\alpha^{k}(x_{i_{j_{1}}}),...,D(x_{i_{j_{l}}}),...,\alpha^{k}(x_{i_{j_{n}}})]),$

and similarly for $\delta_{D}(y_{s}).$ Next, we obtain

$\displaystyle\sum_{j=1}^{n-1}\epsilon(d,X_{j})[\alpha^{k}(x_{1}),...,\delta_{D}(x_{j}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{s=1}^{n}\epsilon(d,X+Y_{s})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\delta_{D}(y_{s}),...,\alpha^{k}(y_{n})]]$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}\epsilon(d,X_{j})[\alpha^{k}(x_{1}),...,\sum_{l=1}^{n}\sum_{i\in{I_{j}}}\epsilon(d,X_{I_{j_{l}}})([\alpha^{k}(x_{i_{j_{1}}}),...,D(x_{i_{j_{l}}})$
$\displaystyle,...,\alpha^{k}(x_{i_{j_{n}}})]),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{s=1}^{n}\epsilon(d,X+Y_{s})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\sum_{l=1}^{n}\epsilon(d,Y_{p_{s_{l}}})$
$\displaystyle[\alpha^{k}(y_{p_{s_{1}}}),...,D(y_{p_{s_{l}}}),...,\alpha^{k}(y_{n})]]$
$\displaystyle=$
$\displaystyle\sum_{i\in{I_{j}}}\sum_{j=1}^{n-1}(-1)^{(n-1)(n-j)}\epsilon(d,X_{j})\epsilon(Y,d+X)\epsilon(\sum_{t=j+1}^{n-1}(x_{t},\hat{X_{t}}+Y+d))$
$\displaystyle[\alpha^{k}(x_{j+1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})],\alpha^{k}(x_{1}),...,\alpha^{k}(x_{j-1}),$
$\displaystyle\sum_{l=1}^{n}\epsilon(d,X_{i_{j_{l}}})[\alpha^{k}(x_{i_{j_{1}}}),...,D(x_{i_{j_{l}}}),...,\alpha^{k}(x_{i_{j_{n}}})]]$
$\displaystyle+$
$\displaystyle\sum_{p\in{P_{s}}}\sum_{s=1}^{n}(-1)^{(n-1)(n-j)}\epsilon(d,X+Y_{s})\epsilon(\sum_{t=s+1}^{n}(y_{t},\hat{Y_{t}}+d))$
$\displaystyle[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{s+1}),...,\alpha^{k}(y_{n}),\alpha^{k}(y_{1}),...,\alpha^{k}(y_{s_{t}}),$
$\displaystyle\sum_{l=1}^{n}\epsilon(d,Y_{p_{s_{l}}})[\alpha^{k}(y_{p_{s_{1}}}),...,D(y_{p_{s_{l}}}),...,\alpha^{k}(y_{p_{s_{n}}})]]]$
$\displaystyle=$
$\displaystyle\sum_{i\in{I_{j}}}\sum_{j=1}^{n}(-1)^{(n-1)(n-j)}\epsilon(d,X_{j})\epsilon(Y,d+X)\epsilon(\sum_{t=j+1}^{n-1}(x_{t},\hat{X_{t}}+Y+d))$
$\displaystyle\epsilon(d,\sum_{t=j+1}^{n-1}(\hat{X_{t}}+Y))(D([x_{j+1},...,x_{n-1},[y_{1},...,y_{n}],x_{1},...,x_{j-1},$
$\displaystyle[x_{i_{j_{1}}},...,x_{i_{j_{n}}}]])$ $\displaystyle-$
$\displaystyle\sum_{m=j+1}^{n-1}\epsilon(d,x_{j+1}+...+x_{m-1})[\alpha^{k}(x_{j+1}),...,D(x_{m}),...,\alpha^{k}(x_{n-1}),$
$\displaystyle[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})],\alpha^{k}(x_{1}),...,\alpha^{k}(x_{j-1}),[x_{i_{j_{1}}},...,x_{i_{j_{n}}}]]$
$\displaystyle-$
$\displaystyle\sum_{m=1}^{j-1}\epsilon(d,x_{j+1}+...+x_{n-1}+Y+X_{m})[\alpha^{k}(x_{j+1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),$
$\displaystyle...,\alpha^{k}(y_{n})],\alpha^{k}(x_{1}),...,D(x_{m}),...,\alpha^{k}(x_{j-1}),[x_{i_{j_{1}}},...,x_{i_{j_{n}}}]]$
$\displaystyle-$
$\displaystyle\epsilon(d,x_{j+1}+...+x_{n-1})[\alpha^{k}(x_{j+1}),...,\alpha^{k}(x_{n-1}),D([y_{1},...,y_{n}]),\alpha^{k}(x_{1}),...,$
$\displaystyle\alpha^{k}(x_{j-1}),[x_{i_{j_{1}}},...,x_{i_{j_{n}}}]]$
$\displaystyle+$
$\displaystyle\sum_{p\in{P_{s}}}\sum_{s=1}^{n}(-1)^{(n-1)(n-j)}\epsilon(d,X+Y_{s})\epsilon(y_{s},\sum_{t=s+1}^{n}\hat{Y_{t}}+d)\epsilon(d,\sum_{t=s+1}^{n}\hat{Y_{t}})$
$\displaystyle[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),D([y_{s+1},...,y_{n},y_{1},...,y_{s-1},[y_{p_{s_{1}}},...,y_{p_{s_{n}}}]])]$
$\displaystyle-$
$\displaystyle\sum_{m=s+1}^{n}\epsilon(d,y_{s+1}+...+y_{m-1})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{s+1}),...,D(y_{m}),$
$\displaystyle...,\alpha^{k}(y_{n}),\alpha^{k}(y_{1}),...,\alpha^{k}(y_{s-1}),[\alpha^{k}(y_{p_{s_{1}}}),...,\alpha^{k}(y_{p_{s_{n}}})]]]$
$\displaystyle-$
$\displaystyle\sum_{m=1}^{s-1}\epsilon(d,y_{s+1}+...+y_{n}+Y_{m})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{s+1}),...,\alpha^{k}(y_{n}),$
$\displaystyle\alpha^{k}(y_{1}),...,D(y_{m}),...,\alpha^{k}(y_{s-1}),[\alpha^{k}(y_{p_{s_{1}}}),...,\alpha^{k}(y_{p_{s_{n}}})]]])$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}(-1)^{(n-1)(n-j)}\epsilon(d,X_{j})\epsilon(Y,d+X)\epsilon(\sum_{t=j+1}^{n-1}(x_{t},\hat{X_{t}}+Y+d))$
$\displaystyle((-1)^{(n-1)(n-j)}\epsilon(\sum_{t=j+1}^{n-1}(x_{t},\hat{X_{t}}+Y+d))\epsilon(Y,X)D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle-$
$\displaystyle\sum_{m=j+1}^{n-1}\epsilon(d,x_{j+1}+...+x_{m-1})\epsilon(d+x_{m},\hat{X_{m}}+Y)\epsilon(\sum_{t=j+1}^{n-1}(x_{t},\hat{X_{t}}+Y+d))$
$\displaystyle\epsilon(Y,d+X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle-$
$\displaystyle\sum_{m=1}^{j-1}(-1)^{(n-1)(n-j)}\epsilon(d,x_{j+1}+...+x_{n-1}+Y+X_{m})\epsilon(\sum_{t=j+1}^{n-1}(x_{t},$
$\displaystyle\hat{X_{t}}+Y+d))\epsilon(Y,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle-$
$\displaystyle(-1)^{(n-1)(n-j)}\epsilon(d,x_{j-1}+...+x_{n-1})\epsilon(\sum_{t=j+1}^{n-1}(x_{t},\hat{X_{t}}+Y+d))\epsilon(d+Y,X)$
$\displaystyle[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n}),D([y_{1},...,y_{n}])])$
$\displaystyle+$
$\displaystyle\sum_{s=1}^{n}(-1)^{(n-1)(n-j)}\epsilon(d,X+Y_{s})\epsilon(\sum_{t=j+1}^{n-1}(y_{t},\hat{Y_{t}})$
$\displaystyle((-1)^{(n-1)(n-j)}\epsilon(\sum_{t=j+1}^{n-1}(y_{t},\hat{Y_{t}}))[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n}),D([y_{1},...,y_{s},...,y_{n}])]$
$\displaystyle-$
$\displaystyle\sum_{m=s+1}^{n}\epsilon(d,y_{s+1}+...+y_{m-1})\epsilon(\sum_{\begin{subarray}{c}t=s+1\\\
t\neq m\end{subarray}}^{n}(y_{t},d+\hat{Y_{t}}))\epsilon(d+y_{m},\hat{Y_{m}})$
$\displaystyle[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n}),[\alpha^{k}(y_{1}),...,D(y_{m}),...,\alpha^{k}(y_{n})]]$
$\displaystyle-$
$\displaystyle\sum_{m=1}^{s-1}(-1)^{(n-1)(s-1)}\epsilon(d,y_{s+1}+...+y_{n}+Y_{m})\epsilon(\sum_{t=s+1}^{n}(y_{t},d+\hat{Y_{t}}))$
$\displaystyle[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{m}),...,\alpha^{k}(y_{n})]])$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}D([x_{1},...,x_{j},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle-$
$\displaystyle\sum_{j=1}^{n-1}\sum_{m=j+1}^{n-1}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{j}),...,D(x_{m}),...,\alpha^{k}(x_{n-1}),$
$\displaystyle[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$ $\displaystyle-$
$\displaystyle\sum_{j=1}^{n-1}\sum_{m=1}^{j-1}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,D(x_{m}),...,\alpha^{k}(x_{j}),...,\alpha^{k}(x_{n-1}),$
$\displaystyle[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$ $\displaystyle-$
$\displaystyle\sum_{j=1}^{n-1}\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{j}),...,\alpha^{k}(x_{n-1}),D([y_{1},...,y_{n}])]$
$\displaystyle+$
$\displaystyle\sum_{s=1}^{n}\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),D([y_{1},...,y_{s},...,y_{n}])]$
$\displaystyle-$
$\displaystyle\sum_{s=1}^{n}\sum_{m=s+1}^{n}\epsilon(d,X+Y_{m})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{s}),...,$
$\displaystyle D(y_{m}),...,\alpha^{k}(y_{n})]]$ $\displaystyle-$
$\displaystyle\sum_{s=1}^{n}\sum_{m=1}^{s-1}\epsilon(d,X+Y_{m})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{m}),...,$
$\displaystyle\alpha^{k}(y_{s}),...,\alpha^{k}(y_{n})]]$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n-1}D([x_{1},...,x_{j},...,x_{n-1},[y_{1},...,y_{n}]]])$
$\displaystyle-$ $\displaystyle\sum_{j=1}^{n-1}\sum_{\begin{subarray}{c}m=1\\\
m\neq
j\end{subarray}}^{n-1}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,D(x_{m}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),D([y_{1},...,y_{n}])]$
$\displaystyle-$
$\displaystyle\sum_{s=1}^{n-1}\sum_{m=1}^{n}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{m}),...,\alpha^{k}(y_{n})]]$
$\displaystyle=$
$\displaystyle(n-1)D([x_{1},...,x_{j},...,x_{n-1},[y_{1},...,y_{n}]]])$
$\displaystyle-$
$\displaystyle(n-2)\sum_{m=1}^{n}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,D(x_{m}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),D([y_{1},...,y_{n}])]$
$\displaystyle-$
$\displaystyle(n-1)(\sum_{m=1}^{n-1}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,D(x_{m}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{m=1}^{n}\epsilon(d,X+Y_{m})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{m}),...,\alpha^{k}(y_{n})]])$
$\displaystyle-$
$\displaystyle(n-2)\sum_{m=1}^{n-1}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,D(x_{m}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),D([y_{1},...,y_{n}])]$
$\displaystyle-$
$\displaystyle(n-1)\sum_{m=1}^{n}\epsilon(d,X+Y_{m})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{m}),...,$
$\displaystyle\alpha^{k}(y_{n})]])$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{n-1}\epsilon(d,X_{m})[\alpha^{k}(x_{1}),...,D(x_{m}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),D([y_{1},...,y_{n}])]$
$\displaystyle=$
$\displaystyle\delta_{D}([x_{1},...,x_{n-1},[y_{1},...,y_{n}]]),$

where $\hat{X_{t}}=\sum_{j=1}^{n-1}y_{j}.$ Thus,
$\delta_{D}\in\mathcal{D}_{\alpha^{k}}(\mathcal{L}),$ and so is a double
derivation of $\mathcal{L}.$

(2) For any $D\in\mathcal{D}_{\alpha^{k}}(\mathcal{L})$ and
$x_{i},y_{j},z_{i}\in\mathcal{L}$ with $1\leq i\leq n-1,1\leq j\leq n,$ it
follows from Eqs. (2.8) and (3.1) that

$\displaystyle[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d,X_{i})[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),[\alpha^{k}(x_{1}),...,D(x_{i}),...\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,$
$\displaystyle\alpha^{k}(y_{n})]]]$ $\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d,X+Y_{j})[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),[\alpha^{k}(x_{1}),...\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{j}),$
$\displaystyle...,\alpha^{k}(y_{n})]]]$ $\displaystyle=$
$\displaystyle[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),\delta_{D}([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])]$
$\displaystyle-$
$\displaystyle\epsilon(d,X))[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),[\alpha^{k}(x_{1}),...\alpha^{k}(x_{n-1}),D([y_{1},...,y_{n}])]]$
$\displaystyle+$
$\displaystyle\epsilon(d,X))[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),[\alpha^{k}(x_{1}),...\alpha^{k}(x_{n-1}),\delta_{D}([y_{1},...,y_{n}])]].$

Hence,

$\displaystyle[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),(D-\delta_{D})([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])]$
$\displaystyle=$
$\displaystyle-[\alpha^{k}(z_{1}),...,\alpha^{k}(z_{n-1}),[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),\epsilon(d,X)(D-\delta_{D})[y_{1},...,y_{n}]])]$

Taking into account $[\mathcal{L},\mathcal{L},...,\mathcal{L}]=\mathcal{L}$
and $Z(\mathcal{L})=0$ we have that

$\displaystyle(D-\delta_{D})([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=-\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),$
$\displaystyle(D-\delta_{D})([y_{1},...,y_{n}])],$

and

$(D-\delta_{D})([x_{1},...,x_{n}])=\epsilon(d,X_{i})[\alpha^{k}(x_{1}),...,(D-\delta_{D})(x_{i}),...,\alpha^{k}(x_{n})],~{}i=1,...,n,$

for all $x_{1},...,x_{n}\in\mathcal{L}.$

(3) By part (1), if $D$ is a double derivation of $\mathcal{L},$ then
$\delta_{D}$ is a double derivation of $\mathcal{L}.$ Hence, $D-\delta_{D}$ is
a double derivation of $\mathcal{L}.$ Now, by part (2) we find that

$\displaystyle\delta_{D-\delta_{D}}([x_{1},...,x_{n}])$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\epsilon(d,X_{i})[\alpha^{k}(x_{1}),...,(D-\delta_{D})(x_{i}),...,\alpha^{k}(x_{n})]$
$\displaystyle=$ $\displaystyle-n(D-\delta_{D})([x_{1},...,x_{n}]),$

which gives us the result. $\Box$

######  Proposition 3.4.

If $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ is a centerless multiplicative
perfect $n-$Hom-Lie color algebra. Then the following assertions hold.

  * (1)

If $D$ is a double derivation of $\mathcal{L},$ then $\delta_{D}$ is a
derivation of $\mathcal{L}$ if and only if $D$ is a derivation of
$\mathcal{L}.$ In particular, $\delta_{D}=D$ if $D$ is a derivation of
$\mathcal{L}.$

  * (2)

For all $x_{1},...,x_{n}\in\mathcal{L}$ and any $\alpha^{k}-$double derivation
$D$ of $\mathcal{L},$

$\displaystyle[D,ad_{s}(x_{1},...,x_{n})]$ $\displaystyle=$
$\displaystyle\sum_{j=2}^{n-1}\epsilon(d,X_{j})ad_{k+s}(x_{1},...,D(x_{j}),...,x_{n-1})$
$\displaystyle+$ $\displaystyle
ad_{k+s}(\delta_{D}(x_{1}),x_{2},...,x_{n-1}).$

Proof. (1) Let $D$ be a derivation of $\mathcal{L}.$ Then by Eq. (3.1),

$\delta_{D}([x_{1},...,x_{n}])=D([x_{1},...,x_{n}])$

Taking into account $[\mathcal{L},\mathcal{L},...,\mathcal{L}]=\mathcal{L},$
we get $\delta_{D}=D.$ Next, suppose that
$D\in\mathcal{D}_{\alpha^{k}}(\mathcal{L})$ but not contained in
$Der(\mathcal{L})$ and so $\delta_{D}\neq D.$ Then there exist
$x_{1},...,x_{n}\in\mathcal{L}$ such that
$(D-\delta_{D})([x_{1},...,x_{n}])\neq 0.$ Suppose that

$x_{j}=\sum_{i\in{I_{j}}}[x_{i_{j_{1}}},...,x_{i_{j_{n}}}],~{}~{}~{}j=1,...,n.$

We have

$\displaystyle\sum_{j=1}^{n}\epsilon(d,X_{j})[\alpha^{k}(x_{1}),...,\delta_{D}(x_{j}),...\alpha^{k}(x_{n})]$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\epsilon(d,X_{j})[\alpha^{k}(x_{1}),...,\sum_{i\in{I_{j}}}\sum_{l=1}^{n}\epsilon(d,X_{i_{j_{l}}})[\alpha^{k}(x_{i_{j_{1}}}),...,D(x_{i_{j_{l}}}),...,$
$\displaystyle\alpha^{k}(x_{i_{j_{n}}})],...\alpha^{k}(x_{n})]$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\sum_{i\in{I_{j}}}(-1)^{(n-1)(n-j)}\epsilon(d,X_{j})\epsilon(\sum_{m=j+1}^{n}(x_{m},d+\hat{X_{m}}))[\alpha^{k}(x_{j+1}),...,$
$\displaystyle\alpha^{k}(x_{n}),\alpha^{k}(x_{1}),...,\sum_{l=1}^{n}\epsilon(x,X_{i_{j_{l}}})[\alpha^{k}(x_{i_{j_{1}}}),...,D(x_{i_{j_{l}}}),...,\alpha^{k}(x_{i_{j_{n}}})]]$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\sum_{i\in{I_{j}}}(-1)^{(n-1)(n-j)}\epsilon(d,X_{j})\epsilon(\sum_{m=j+1}^{n}(x_{m},d+\hat{X_{m}}))\epsilon(d,\hat{X_{j}})$
$\displaystyle(D([x_{j+1},...,x_{n},x_{1},...,x_{j-1},[x_{i_{j_{1}}},...,x_{i_{j_{n}}}]])-\sum_{s=j+1}^{n}\epsilon(d,x_{j+1}+...+x_{s-1})$
$\displaystyle[\alpha^{k}(x_{j+1}),...,D(x_{s}),...,\alpha^{k}(x_{n}),\alpha^{k}(x_{1}),...,\alpha^{k}(x_{j-1}),[\alpha^{k}(x_{i_{j_{1}}}),...,\alpha^{k}(x_{i_{j_{n}}})]]$
$\displaystyle-$
$\displaystyle\sum_{s=1}^{j-1}\epsilon(d,x_{j+1}+...+x_{s-1})[\alpha^{k}(x_{j+1}),...,\alpha^{k}(x_{n}),\alpha^{k}(x_{1}),...,D(x_{s}),...,$
$\displaystyle\alpha^{k}(x_{j-1}),[\alpha^{k}(x_{i_{j_{1}}}),...,\alpha^{k}(x_{i_{j_{n}}})]])$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}(-1)^{(n-1)(n-j)}\epsilon(d,X_{j})\epsilon(\sum_{m=j+1}^{n}(x_{m},\hat{X_{m}}))\epsilon(d,\hat{X_{j}})$
$\displaystyle((-1)^{(n-1)(n-j)}\epsilon(\sum_{m=j+1}^{n}(x_{m},\hat{X_{m}}))D([x_{1},...,x_{n}])$
$\displaystyle-$
$\displaystyle\sum_{s=j+1}^{n}\epsilon(d,x_{j+1}+...+x_{s-1})\epsilon(\sum_{m=j+1}^{n}(x_{m},d+\hat{X_{m}}))\epsilon(d+x_{s},\hat{X_{s}})$
$\displaystyle[\alpha^{k}(x_{1}),...,D(x_{s}),...,\alpha^{k}(x_{n})]$
$\displaystyle-\sum_{s=1}^{j-1}(-1)^{(n-1)(n-j)}\epsilon(d,x_{j+1}+...+x_{s-1})\epsilon(\sum_{m=j+1}^{n}(x_{m},d+\hat{X_{m}}))$
$\displaystyle[\alpha^{k}(x_{1}),...,D(x_{s}),...,\alpha^{k}(x_{n})])$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}D([x_{1},...,x_{n}])-\sum_{j=1}^{n}\sum_{s=j+1}^{n}\epsilon(d,X_{s})[\alpha^{k}(x_{1}),...,D(x_{s}),...,\alpha^{k}(x_{n})]$
$\displaystyle-$
$\displaystyle\sum_{j=1}^{n}\sum_{s=1}^{j-1}\epsilon(d,X_{s})[\alpha^{k}(x_{1}),...,D(x_{s}),...,\alpha^{k}(x_{n})]$
$\displaystyle=$ $\displaystyle
nD([x_{1},...,x_{n}])-\sum_{j=1}^{n}\sum_{{\begin{subarray}{c}s=1\\\ s\neq
j\end{subarray}}}^{n}\epsilon(d,X_{s})[\alpha^{k}(x_{1}),...,D(x_{s}),...,\alpha^{k}(x_{n})]$
$\displaystyle=$ $\displaystyle
nD([x_{1},...,x_{n}])-(n-1)\sum_{s=1}^{n}\epsilon(d,X_{s})[\alpha^{k}(x_{1}),...,D(x_{s}),...,\alpha^{k}(x_{n})]$
$\displaystyle=$ $\displaystyle
nD([x_{1},...,x_{n}])-(n-1)\delta_{D}([x_{1},...,x_{n}])$ $\displaystyle=$
$\displaystyle
n(D-\delta_{D})([x_{1},...,x_{n}])+\delta_{D}([x_{1},...,x_{n}])$
$\displaystyle\neq$ $\displaystyle\delta_{D}([x_{1},...,x_{n}]).$

It follows that $\delta_{D}\notin Der_{\alpha^{k}}(\mathcal{L})$ and so
$\delta_{D}$ is not a derivation of $\mathcal{L}.$

(2) For all $x_{1},...,x_{n}\in\mathcal{L}$ and any
$D\in\mathcal{D}_{\alpha^{k}}(\mathcal{L}),$ by Theorem 2.10 and Eq. (3.1), we
have

$\displaystyle[D,ad_{s}(x_{1},...,x_{n})]$ $\displaystyle=$
$\displaystyle\sum_{j=2}^{n-1}\epsilon(d,X_{j})ad_{k+s}(x_{1},...,D(x_{j}),...,x_{n-1})$
$\displaystyle+$ $\displaystyle\sum_{i\in
I}\sum_{t=1}^{n}\epsilon(d,X_{i_{t}})ad_{k+s}([x_{i_{1}},...,D(x_{i_{t}}),...,x_{i_{n}}],x_{2},...,x_{n-1})$
$\displaystyle=$
$\displaystyle\sum_{j=2}^{n-1}\epsilon(d,X_{j})ad_{k+s}(x_{1},...,D(x_{j}),...,x_{n-1})$
$\displaystyle+$ $\displaystyle\sum_{i\in
I}\sum_{t=1}^{n}\epsilon(d,X_{i_{t}})ad_{k+s}([\alpha^{k}(x_{i_{1}}),...,D(x_{i_{t}}),...,\alpha^{k}(x_{i_{n}})],x_{2},...,x_{n-1})$
$\displaystyle=$
$\displaystyle\sum_{j=2}^{n-1}\epsilon(d,X_{j})ad_{k+s}(x_{1},...,D(x_{j}),...,x_{n-1})$
$\displaystyle+$ $\displaystyle
ad_{k+s}(\delta_{D}(x_{1}),x_{2},...,x_{n-1}).$

Therefore, the result holds. $\Box$

######  Proposition 3.5.

If $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ is a centerless perfect $n-$Hom-
Lie color algebra. Then the map
$\delta~{}:~{}\mathcal{D}(\mathcal{L})\longrightarrow End(\mathcal{L})$ is a
Hom-Lie color algebra homomorphism, that is,
$\delta_{[D_{1},D_{2}]}=[\delta_{D_{1}},\delta_{D_{2}}].$

Proof. Suppose that $D_{1}\in\mathcal{D}_{\alpha^{k}}(\mathcal{L})$ and
$D_{2}\in\mathcal{D}_{\alpha^{s}}(\mathcal{L}).$ For any $x=\sum_{i\in
I}[x_{i_{1}},...,x_{i_{n}}]\in\mathcal{L}$ by Eq. (3.1) we have

$\displaystyle[\delta_{D_{1}},\delta_{D_{2}}](x)=(\delta_{D_{1}}\delta_{D_{2}}-\epsilon(d_{1},d_{2})\delta_{D_{2}}\delta_{D_{1}})(x)$
$\displaystyle=$ $\displaystyle\delta_{D_{1}}(\sum_{i\in
I}\delta_{D_{2}}([x_{i_{1}},...,x_{i_{n}}]))-\epsilon(d_{1},d_{2})\delta_{D_{2}}(\sum_{i\in
I}\delta_{D_{1}}([x_{i_{1}},...,x_{i_{n}}]))$ $\displaystyle=$
$\displaystyle\delta_{D_{1}}(\sum_{i\in
I}\sum_{j=1}^{n}\epsilon(d_{2},X_{i_{j}})[\alpha^{s}(x_{i_{1}}),...,D_{2}(x_{i_{j}}),...,\alpha^{s}(x_{i_{n}})])$
$\displaystyle-$ $\displaystyle\epsilon(d_{1},d_{2})\delta_{D_{2}}(\sum_{i\in
I}\sum_{j=1}^{n}\epsilon(d_{1},X_{i_{j}})[\alpha^{k}(x_{i_{1}}),...,D_{1}(x_{i_{j}}),...,\alpha^{k}(x_{i_{n}})])$
$\displaystyle=$ $\displaystyle\sum_{i\in
I}\sum_{j=1}^{n}(\sum_{l=1}^{j-1}\epsilon(d_{2},X_{i_{j}})\epsilon(d_{1},X_{i_{l}})[\alpha^{k+s}(x_{i_{1}}),...,D_{1}(\alpha^{k}(x_{i_{l}})),...,$
$\displaystyle D_{2}(x_{i_{j}}),...,\alpha^{k+s}(x_{i_{n}})]$
$\displaystyle+\sum_{l=j+1}^{n}\epsilon(d_{2},X_{i_{j}})\epsilon(d_{1},X_{i_{l}})[\alpha^{k+s}(x_{i_{1}}),...,D_{2}(\alpha^{s}(x_{i_{l}})),...,$
$\displaystyle D_{1}(x_{i_{j}}),...,\alpha^{k+s}(x_{i_{n}})])$
$\displaystyle-$ $\displaystyle\epsilon(d_{1},d_{2})\sum_{i\in
I}\sum_{j=1}^{n}(\sum_{l=1}^{j-1}\epsilon(d_{1},X_{i_{j}})\epsilon(d_{2},X_{i_{l}})[\alpha^{k+s}(x_{i_{1}}),...,$
$\displaystyle
D_{2}(\alpha^{s}(x_{i_{l}})),...,D_{1}(x_{i_{j}}),...,\alpha^{k+s}(x_{i_{n}})]$
$\displaystyle+$
$\displaystyle\sum_{l=j+1}^{n}\epsilon(d_{2},X_{i_{j}})\epsilon(d_{1},X_{i_{l}})[\alpha^{k+s}(x_{i_{1}}),...,D_{1}(\alpha^{k}(x_{i_{l}})),...,$
$\displaystyle D_{2}(x_{i_{j}}),...,\alpha^{k+s}(x_{i_{n}})])$
$\displaystyle=$ $\displaystyle\sum_{i\in
I}\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X_{i_{j}})[\alpha^{k+s}(x_{i_{1}}),...,D_{1}D_{2}(x_{i_{j}}),...,\alpha^{k+s}(x_{i_{n}})]$
$\displaystyle-$ $\displaystyle\sum_{i\in
I}\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X_{i_{j}})[\alpha^{k+s}(x_{i_{1}}),...,\epsilon(d_{1},d_{2})D_{2}D_{1}(x_{i_{j}}),...,$
$\displaystyle\alpha^{k+s}(x_{i_{n}})]$ $\displaystyle=$
$\displaystyle\sum_{i\in
I}\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X_{i_{j}})[\alpha^{k+s}(x_{i_{1}}),...,(D_{1}D_{2}-\epsilon(d_{1},d_{2})D_{2}D_{1})(x_{i_{j}}),...,$
$\displaystyle\alpha^{k+s}(x_{i_{n}})]$ $\displaystyle=$
$\displaystyle\sum_{i\in
I}\sum_{j=1}^{n}\epsilon(d_{1}+d_{2},X_{i_{j}})[\alpha^{k+s}(x_{i_{1}}),...,[D_{1},D_{2}](x_{i_{j}}),...,\alpha^{k+s}(x_{i_{n}})]$
$\displaystyle=$ $\displaystyle\delta_{[D_{1},D_{2}]}(x).$

$\Box$

######  Theorem 3.6.

If $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ is a perfect $n-$Hom-Lie color
algebra. Then the centralizer of $Inn(\mathcal{L})$ in
$\mathcal{D}(\mathcal{L})$ is trivial. In particular, the center of the Hom-
Lie color algebra $\mathcal{D}(\mathcal{L})$ is zero.

Proof. Let $D\in\mathcal{D}_{\alpha^{s}}(\mathcal{L})$ of degree $d$ and $D\in
C_{\mathcal{D}(\mathcal{L})}(Inn(\mathcal{L})).$ Then

$[D,ad_{k}(x_{1},...,x_{n-1})]=0,~{}~{}\forall
x_{1},x_{2},...,x_{n-1}\in\mathcal{L}.$

For all $z\in\mathcal{L},$ we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle[D,ad_{k}(x_{1},...,x_{n-1})](z)$ $\displaystyle=$
$\displaystyle D\circ
ad_{k}(x_{1},...,x_{n-1})(z)-\epsilon(d,X)ad_{k}(x_{1},...,x_{n-1})\circ D(z)$
$\displaystyle=$ $\displaystyle
D([x_{1},...,x_{n-1},\alpha^{k}(z)])-\epsilon(d,X)[x_{1},...,x_{n-1},\alpha^{k}(D(z))].$

Consequently, for arbitrary $x_{1},x_{2},...,x_{n-1},z\in\mathcal{L},$ we
obtain

$\displaystyle D([x_{1},...,x_{n-1},\alpha^{k}(z)])$ $\displaystyle=$
$\displaystyle\epsilon(d,X)[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),\alpha^{k}(D(z))]$
$\displaystyle=$
$\displaystyle\epsilon(d,X_{i})[\alpha^{k}(x_{1}),...,D(x_{i}),...,\alpha^{k}(x_{n-1}),\alpha^{k}(z)],$

for $1\leq i\leq n-1.$ Now, let
$x_{1},...,x_{n-1},y_{1},...,y_{n}\in\mathcal{L}.$ Taking into account Eq.
(3), we obtain

$\displaystyle D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\epsilon(d,X_{i})[\alpha^{k}(x_{1}),...,D(x_{i}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,\alpha^{k}(y_{n})]]$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}\epsilon(d,X+Y_{j})[\alpha^{k}(x_{1}),...,\alpha^{k}(x_{n-1}),[\alpha^{k}(y_{1}),...,D(y_{j}),...,\alpha^{k}(y_{n})]]$
$\displaystyle=$
$\displaystyle(n-1)D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])+nD([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])$
$\displaystyle=$
$\displaystyle(2n-1)D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])+nD([x_{1},...,x_{n-1},[y_{1},...,y_{n}]]),$

so we get

$(2n-2)D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])=0.$

That is,

$D([x_{1},...,x_{n-1},[y_{1},...,y_{n}]])=0.$

Taking into account $[\mathcal{L},\mathcal{L},...,\mathcal{L}]=\mathcal{L},$
we conclude that $D=0.$ $\Box$

In the rest of this section, we are particularly interested in the Lie triple
derivations of the Hom-Lie color algebra
$(\mathcal{D}(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon)$ consisting of
double derivations algebra as well as in the triple derivations of the Hom-Lie
color algebra $(Inn(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon)$ consisting of
inner derivations algebra of a perfect $n-$Hom-Lie color algebra
$\mathcal{L}.$

Recall that, a linear map $D$ of a multiplicative Hom-Lie color algebra
$(L,[.,.],\alpha,\epsilon)$ is called an $\alpha^{k}-$Lie triple derivation if
it satisfies

(1)- $D\circ\alpha=\alpha\circ D.$

(2)-For all $x,y,z\in L,$

$\displaystyle D([x,[y,z]])$ $\displaystyle=$
$\displaystyle[D(x),[\alpha^{k}(y),\alpha^{k}(z)]]+\epsilon(d,x)[\alpha^{k}(x),[D(y),\alpha^{k}(z)]]$
$\displaystyle+$
$\displaystyle\epsilon(d,x+y)[\alpha^{k}(x),[\alpha^{k}(y),D(z)]].$

Let us denote the set of $\alpha^{k}-$Lie triple derivations of $L$ by
$TDer_{\alpha^{k}}(L)$ and denote $TDer(L)=\bigoplus_{k\geq
0}TDer_{\alpha^{k}}(L),$ which is a Hom-Lie color algebra and it is called the
Lie triple derivation algebra of $L.$

In the following we suppose that $\mathcal{L}$ is a centerless perfect
$n-$Hom-Lie color algebra and
$L=(Inn(\mathcal{L}),[.,.],\tilde{\alpha}=D\circ\alpha,\epsilon)$ is a perfect
Hom-Lie color algebra. Then we have the following results.

######  Lemma 3.9.

If $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ is a perfect $n-$Hom-Lie color
algebra. Then every Lie triple derivation $D$ of
$(\mathcal{D}(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon)$ keeps
$L=Inn(\mathcal{L})$ invariant. Furthermore, if $D(L)=0,$ then $D=0.$

Proof. First, we show that $D(L)\subseteq L,$ for $D\in
TDer_{\tilde{\alpha}^{k}}(\mathcal{D}(\mathcal{L})).$ Since
$L=\\{ad_{s}(x_{1},...,x_{n-1})~{}:~{}\alpha(x_{i})=x_{i}\in\mathcal{L},i=1,...,n-1\\}$
is a perfect Hom-Lie color algebra, for all $x_{1},...,x_{n-1}\in\mathcal{L}$
there exist
$x_{1_{i}},...,x_{{n-1}_{i}},y_{1_{i_{j}}},...,y_{{n-1}_{i_{j}}}\in\mathcal{L}$
and $z_{1_{i_{j}}},...,z_{{n-1}_{i_{j}}}\in\mathcal{L}$ such that

$\displaystyle ad_{s}(x_{1},...,x_{n-1})$ $\displaystyle=\sum_{i\in
I}[ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}}),ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}})]$
$\displaystyle=\sum_{i\in I}[ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}}),\sum_{j\in
J}[ad_{s}(y_{1_{i_{j}}},...,y_{{n-1}_{i_{j}}}),ad_{s}(z_{1_{i_{j}}},...,z_{{n-1}_{i_{j}}})]],$

for some finite index sets $I$ and $J.$ Now, by definition of Lie triple
derivation we have

$\displaystyle D(ad_{s}(x_{1},...,x_{n-1}))$ $\displaystyle\sum_{i\in
I}\sum_{j\in
J}D([ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}}),[ad_{s}(y_{1_{i_{j}}},...,y_{{n-1}_{i_{j}}}),ad_{s}(z_{1_{i_{j}}},...,z_{{n-1}_{i_{j}}})]])$
$\displaystyle=$ $\displaystyle\sum_{i\in I}\sum_{j\in
J}([D(ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}})),[ad_{s}(\tilde{\alpha}^{k}(y_{1_{i_{j}}}),...,\tilde{\alpha}^{k}(y_{{n-1}_{i_{j}}})),$
$\displaystyle
ad_{s}(\tilde{\alpha}^{k}(z_{1_{i_{j}}}),...,\tilde{\alpha}^{k}(z_{{n-1}_{i_{j}}}))]]$
$\displaystyle+$
$\displaystyle\epsilon(d,X_{i})[ad_{s}(\tilde{\alpha}^{k}(x_{1_{i}}),...,\tilde{\alpha}^{k}(x_{{n-1}_{i}})),[D(ad_{s}(\tilde{\alpha}^{k}(y_{1_{i_{j}}}),...,\tilde{\alpha}^{k}(y_{{n-1}_{i_{j}}}))),$
$\displaystyle
ad_{s}(\tilde{\alpha}^{k}(z_{1_{i_{j}}}),...,\tilde{\alpha}^{k}(z_{{n-1}_{i_{j}}}))]]$
$\displaystyle+$
$\displaystyle\epsilon(d,X_{i}+Y_{i_{j}})[ad_{s}(\tilde{\alpha}^{k}(x_{1_{i}}),...,\tilde{\alpha}^{k}(x_{{n-1}_{i}})),[ad_{s}(\tilde{\alpha}^{k}(y_{1_{i_{j}}}),...,\tilde{\alpha}^{k}(y_{{n-1}_{i_{j}}})),$
$\displaystyle D(ad_{s}(z_{1_{i_{j}}},...,z_{{n-1}_{i_{j}}}))]]).$

Thanks to Theorem 2.10, we obtain $D(ad_{s}(x_{1},...,x_{n-1}))$ is an inner
$\tilde{\alpha}^{k+s}-$derivation and so $D(L)\subseteq L.$

Next, let $D$ be an $\tilde{\alpha}^{k}-$triple derivation of
$\mathcal{D}(\mathcal{L})$ which satisfies $D(L)=0.$ Then for each double
derivation $E$ of $\mathcal{L},$ again by perfectness of $L$ we have

$\displaystyle[D(E),ad_{s}(x_{1},...,x_{n-1})]$ $\displaystyle=$
$\displaystyle\sum_{i\in
I}[D(E),[ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}}),ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}})]]$
$\displaystyle=$ $\displaystyle\sum_{i\in
I}D([E,[ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}}),ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}})]]$
$\displaystyle-\epsilon(d,e)[\tilde{\alpha}^{k}(E),[D(ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}})),\tilde{\alpha}^{k}(ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}}))]]$
$\displaystyle-\epsilon(d,e+X_{i})[\tilde{\alpha}^{k}(E),[\tilde{\alpha}^{k}(ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}})),D(ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}}))]].$

Since $[E,L]\subseteq L$ (tanks to Theorem 2.10), we get
$D([E,ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}})])=0.$ Thus, we can obtain
$[D(E),ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}})]=0.$ Therefore, $D(E)$ is in the
centralizer of $L$ in $\mathcal{D}(\mathcal{L}),$ and by Proposition 3.6,
$D(E)=0,~{}\forall E\in\mathcal{D}(\mathcal{L}).$ Hence, $D=0.$ The proof is
completed. $\Box$

######  Lemma 3.10.

Let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be a centerles perfect $n-$Hom-
Lie color algebra. If $D\in TDer(Der(\mathcal{L})),$ then there exists
$\Delta\in Der(\mathcal{L})$ such that $D(ad_{s}(x))=ad_{s}(\Delta(x))$ for
all $x\in\mathcal{L}.$

Proof. For any $D\in TDer_{\tilde{\alpha}^{k}}(Der(\mathcal{L})),$ by Lemma
3.9 $D(ad_{s}(x))\in Inn_{\tilde{\alpha}^{k+s}}(\mathcal{L})$ for
$x\in\mathcal{L}.$ Now, let $y\in\mathcal{L}$ such that
$D(ad_{s}(x))=ad_{k+s}(y).$ Note that $y$ is unique, thanks to
$Z(\mathcal{L})=0.$ Define the map
$\Delta~{}:~{}\mathcal{L}\longrightarrow\mathcal{L}$ by $\Delta(x)=y$ which is
an endomorphism. Let $x_{1},x_{2}\in\mathcal{H}(\mathcal{L})$ and
$x_{3}\in\mathcal{L}.$ We have

$\displaystyle
ad_{k+s}(\Delta([[x_{1},x_{2}],x_{3}]))=D(ad_{s}([[x_{1},x_{2}],x_{3}]))$
$\displaystyle=$ $\displaystyle
D([[ad_{s}(x_{1}),ad_{(}x_{2})],ad_{s}(x_{3})])$ $\displaystyle=$
$\displaystyle[[D(ad_{s}(x_{1})),\tilde{\alpha}^{k}(ad_{(}x_{2}))],\tilde{\alpha}^{k}(ad_{s}(x_{3}))]+\epsilon(d,x_{1})[[\tilde{\alpha}^{k}(ad_{s}(x_{1}))$
$\displaystyle,D(ad_{(}x_{2}))],\tilde{\alpha}^{k}(ad_{s}(x_{3}))]+\epsilon(d,x_{1}+x_{2})[[\tilde{\alpha}^{k}(ad_{s}(x_{1})),\tilde{\alpha}^{k}(ad_{(}x_{2}))],D(ad_{s}(x_{3}))]$
$\displaystyle=$
$\displaystyle[[ad_{k+s}(\Delta(x_{1})),\tilde{\alpha}^{k}(ad_{(}x_{2}))],\tilde{\alpha}^{k}(ad_{s}(x_{3}))]$
$\displaystyle+\epsilon(d,x_{1})[[\tilde{\alpha}^{k}(ad_{s}(x_{1})),ad_{k+s}(\Delta(x_{2}))],\tilde{\alpha}^{k}(ad_{s}(x_{3}))]$
$\displaystyle+\epsilon(d,x_{1}+x_{2})[[\tilde{\alpha}^{k}(ad_{s}(x_{1})),\tilde{\alpha}^{k}(ad_{(}x_{2}))],ad_{k+s}(\Delta(x_{3}))]$
$\displaystyle=$ $\displaystyle
ad_{k+s}([d(x_{1}),[\alpha^{k}(x_{2}),\alpha^{k}(x_{3})]]+\epsilon(d,x_{1})[\alpha^{k}(x_{1}),[\Delta(y),\alpha^{k}(x_{3})]]$
$\displaystyle+\epsilon(d,x_{1}+x_{2})[\alpha^{k}(x_{1}),[\alpha^{k}(x_{2}),\Delta(x_{3})]]).$

Since $Z(\mathcal{L})=0,$ we obtain

$\displaystyle\Delta([[x_{1},x_{2}],x_{3}])$ $\displaystyle=$
$\displaystyle[[\Delta(x_{1}),\alpha^{k}(x_{2})],\alpha^{k}(x_{3})]+\epsilon(d,x_{1})[[\alpha^{k}(x_{1}),\Delta(x_{2})],\alpha^{k}(x_{3})]$
$\displaystyle+$
$\displaystyle\epsilon(d,x_{1}+x_{2})[[\alpha^{k}(x_{1}),\alpha^{k}(x_{2})],\Delta(x_{3})].$

That is, $\Delta\in TDer_{\tilde{\alpha}^{k}}(\mathcal{L}).$ By Proposition
3.4, $\Delta\in Der(\mathcal{L}).$ $\Box$

######  Theorem 3.11.

Let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be a centerles perfect $n-$Hom-
Lie color algebra. Then the triple derivation algebra of $Inn(\mathcal{L})$
coincides with the derivation algebra of $Inn(\mathcal{L}).$

Proof. We have $L=(Inn(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon)$ is a
perfect Hom-Lie color algebra with zero center. For any $D\in
TDer_{\tilde{\alpha}^{k}}(L),$ by Lemma 3.10 there exists $\Delta\in
Der(\mathcal{L})$ such that $D(ad_{s}(x))=ad_{s}(\Delta(x))$ for all
$x\in\mathcal{L}.$ As well known that $ad_{s}(\Delta(x))=[\Delta,ad_{s}(x)].$
We also have

$D(ad_{s}(x))=ad_{s}(\Delta(x))=[\Delta,ad_{s}(x)]=ad_{s}(\Delta)(ad_{s}(x))$

Hence, $(D-ad_{s}(\Delta(x)))(ad_{s}(x))=0.$ By Lemma 3.9,
$D=ad_{s}(\Delta(x)).$ Therefore, $TDer(L)$ equal to derivation algebra of
$Inn(\mathcal{L}).$ $\Box$

######  Theorem 3.12.

Let $(\mathcal{L},[.,...,.],\alpha,\epsilon)$ be a perfect $n-$Hom-Lie color
algebra. Then the triple derivation algebra of Hom-Lie color algebra
$(Der(\mathcal{L}),[.,.],\tilde{\alpha},\epsilon)$ is equal to the derivation
algebra $Der(\mathcal{L}).$

Proof. Suppose that $D$ is a $\tilde{\alpha}^{k}-$triple derivation of
$Der(\mathcal{L}),$ then $D$ is an $\tilde{\alpha}^{k}-$triple derivation of
$L=Inn(\mathcal{L})$ by Lemma 3.10 and $D$ is an
$\tilde{\alpha}^{k}-$derivation of $L=Inn(\mathcal{L})$ by Theorem 3.11. Since
$L$ is a perfect Hom-Lie color algebra, for any
$x_{1},...,x_{n-1}\in\mathcal{L},$ there exist
$x_{1_{i}},...,x_{{n-1}_{i}},y_{1_{i}},...,y_{{n-1}_{i}}\in\mathcal{L}$ such
that

$ad_{s}(x_{1},...,x_{n-1})=\sum_{i\in
I}[ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}}),ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}})],$

for some finite index set $I.$ Then for all $d_{1},d_{2}\in Der(\mathcal{L})$
of degrees $\lambda_{1},\lambda_{2}$ respectively,

$\displaystyle D([[d_{1},d_{2}],ad_{s}(x_{1},...,x_{n-1})])$ $\displaystyle=$
$\displaystyle D([[d_{1},d_{2}],\sum_{i\in
I}[ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}}),ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}})]])$
$\displaystyle=$ $\displaystyle[D([d_{1},d_{2}]),\sum_{i\in
I}[ad_{s}(\tilde{\alpha}^{k}(x_{1_{i}}),...,\tilde{\alpha}^{k}(x_{{n-1}_{i}})),ad_{s}(\tilde{\alpha}^{k}(y_{1_{i}}),...,\tilde{\alpha}^{k}(y_{{n-1}_{i}}))]]$
$\displaystyle-$
$\displaystyle\epsilon(d,\lambda_{1}+\lambda_{2})[\tilde{\alpha}^{k}([d_{1},d_{2}]),\sum_{i\in
I}[D(ad_{s}(x_{1_{i}},...,x_{{n-1}_{i}})),ad_{s}(\tilde{\alpha}^{k}(y_{1_{i}}),...,$
$\displaystyle\tilde{\alpha}^{k}(y_{{n-1}_{i}}))]$ $\displaystyle-$
$\displaystyle\epsilon(d,\lambda_{1}+\lambda_{2}+X_{i})[\tilde{\alpha}^{k}([d_{1},d_{2}]),\sum_{i\in
I}[ad_{s}(\tilde{\alpha}^{k}(x_{1_{i}}),...,\tilde{\alpha}^{k}(x_{{n-1}_{i}}))),$
$\displaystyle D(ad_{s}(y_{1_{i}},...,y_{{n-1}_{i}}))]]$ $\displaystyle=$
$\displaystyle[D([d_{1},d_{2}]),\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1}))]+\epsilon(d,\lambda_{1}+\lambda_{2})[\tilde{\alpha}^{k}([d_{1},d_{2}]),$
$\displaystyle D(ad_{s}(x_{1},...,x_{n-1}))]$ $\displaystyle=$
$\displaystyle[D([d_{1},d_{2}]),\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1}))]-\epsilon(\lambda_{1}+\lambda_{2},X)D([ad_{s}(x_{1},...,x_{n-1}),[d_{1},d_{2}]])$
$\displaystyle+$
$\displaystyle\epsilon(d,X)\epsilon(\lambda_{1}+\lambda_{2},X)[\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1})),[D(d_{1}),\tilde{\alpha}^{k}(d_{2})]]$
$\displaystyle+$
$\displaystyle\epsilon(d,\lambda_{1}+X)\epsilon(\lambda_{1}+\lambda_{2},X)[[\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1})),[\tilde{\alpha}^{k}(d_{1}),D(d_{2})]]$
$\displaystyle=$
$\displaystyle[D([d_{1},d_{2}]),\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1}))]+D([[d_{1},d_{2}],ad_{s}(x_{1},...,x_{n-1})])$
$\displaystyle=$
$\displaystyle[[D(d_{1}),\tilde{\alpha}^{k}(d_{2})],\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1}))]-e(d,\lambda_{1})[[\tilde{\alpha}^{k}(d_{1}),D(d_{2})],$
$\displaystyle\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1}))].$

Thus,

$D([d_{1},d_{2}])-[D(d_{1}),\tilde{\alpha}^{k}(d_{2})]-\epsilon(d,\lambda_{1})[\tilde{\alpha}^{k}(d_{1}),D(d_{2})],\tilde{\alpha}^{k}(ad_{s}(x_{1},...,x_{n-1}))]=0.$

Now, Proposition 3.6 gives us

$D([d_{1},d_{2}])=[D(d_{1}),\tilde{\alpha}^{k}(d_{2})]-\epsilon(d,\lambda_{1})[\tilde{\alpha}^{k}(d_{1}),D(d_{2})].$

That is $D$ is an $\tilde{\alpha}^{k}-$derivation of Hom-Lie color algebra
$Der(\mathcal{L}).$ $\Box$

## References

  * [1] Ataguema A., Makhlouf A., Silvestrov S., Generalization of $n-$ary Nambu algebras and beyond, J. Math. Phys. 50, 083501, (2009). (DOI:10.1063/1.3167801).
  * [2] Ammar F., Makhlouf A. and Silvestrov S., Ternary $q-$Virasoro-Witt Hom-Nambu-Lie algebras, Journal of Physics A - Mathematical and Theoretical, 43,no. 26, 265204, (2010).
  * [3] Arnlind J., Makhlouf A. and Silvestrov S., Construction of $n-$Lie algebras and $n-$ary Hom-Nambu-Lie algebras, J. Math. Phys. 52, 123502, 13 pp. (2011).
  * [4] Ammar F., Mabrouk S., and Makhlouf A., Representations and cohomology of $n-$ary multiplicative Hom-Nambu-Lie algebras, J. Geom. Physics , 61, no. 10, 1898-1913, (2011).
  * [5] Asef S. and Wu Z., Generalized Lie Triple Derivations of Lie Color Algebras and Their Subalgebras, Symmetry, 13(7), 1280, (2021). https://doi.org/10.3390/sym13071280
  * [6] Bakayoko I. and Silvestrov S., Multiplicative $n-$Hom-Lie color algebras, arXiv:1912.10216v2 [math.QA].
  * [7] Bai R., Gao Y., Zhang Y. and Li Z., Double derivations of $n-$Lie algebras, Bull. Iranian Math. Soc. 3. 897–910, (2017).
  * [8] Beites, P. D., Kaygorodov, I. and Popov, Y., Generalized derivations of multiplicative $n-$ary Hom-$\Omega$ color algebras, Bull. Malays. Math. Sci. Soc. 41, 315–335, (2019).
  * [9] Chen, L., Ma, Y. and Ni, L., Generalized Derivations of Lie color algebras, Results Math. 63(3–4), 923–936 (2013).
  * [10] Filippov, V., $n-$Lie algebras, Sib. Math. J. 26, 879–891 (1985). (Transl. from Russian: Sib. Mat. Zh. 26:126-140 (1985)).
  * [11] Filippov V., $\delta-$Derivations of Lie algebras, Siberian Mathematical Journal, 39, 6, 1218–1230, (1998).
  * [12] Filippov V., $\delta-$Derivations of prime Lie algebras, Siberian Mathematical Journal, 40, 1, 174–184, (1999).
  * [13] Filippov V., $\delta-$Derivations of prime alternative and Malcev algebras, Algebra and Logic, 39, 5, 354–358, (2000).
  * [14] Herstein I., Jordan derivations of prime rings, Proceedings of the American Mathematical Society, 8, 1104–1110, (1957).
  * [15] Hartwig J. T., Larsson D. and Silvestrov S., Deformations of Lie algebras using $\sigma-$derivations, J. Algebra 295, 314–361, (2006).
  * [16] Kasymov Sh. M., On a theory of n-Lie algebras, Algebra and Logic, 26, 155–166 (1987).
  * [17] Kaygorodov I., $\delta-$derivations of $n-$ary algebras, Izv. Math. 76(5), 1150–1162, (2012).
  * [18] Khalili V., Generalized derivations of Hom–Poisson color algebras, São Paulo Journal of Mathematical Sciences https://doi.org/10.1007/s40863-021-00233-4
  * [19] Kaygorodov I. and Popov Yu., A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations, Journal of Algebra, 456, 323–347, (2016).
  * [20] Kaygorodov I. and Popov Yu., Generalized derivations of (color) $n-$ary algebras, Linear Multilinear Algebra, 64(6), 1086-1106, (2016).
  * [21] Lian H. and Chen C., $N-$Derivations for Finitely Generated Graded Lie Algebras, Algebra ColloquiumVol. 23, No. 02, pp. 205-212, (2016).
  * [22] Leger, G. and Luks, E., Generalized derivations of Lie algebras, J. Algebra 228, 165–203 (2000).
  * [23] Li, H. and Wang, Y., Generalized Lie triple derivations, Linear Multilinear Algebra, 59, 237–247, (2011).
  * [24] Müller D., Isometries of bi-invariant pseudo-Riemannian metries on Lie groups, Geom. Dedicata 29, 65–96, (1989).
  * [25] Miers C.R., Lie triple derivations of Von Neumann algebras, Proc. Amer. Math. Soc. 71, 57–61, (1978).
  * [26] Nambu Y., Generalized Hamiltonian mechanics, Phys. Rev. D7, 2405–2412, (1973).
  * [27] Sun B., Chen L. and Zhou X., Double Derivations of n-Lie Superalgebras, Algebra Colloquium 25 : 1, 161–180, (2018). DOI: 10.1142/S1005386718000111
  * [28] Takhtajan L., On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160, 295–315, (1994).
  * [29] Takhtajan, L., Higher order analog of Chevalley-Eilenberg complex and deformation theory of $n-$gebras, St. Petersburg Math. J. 6 no. 2, 429-438 (1995).
  * [30] Wang, D.N. and Xiao Z.K., Lie triple derivations of incidence algebras, Comm. Algebra, 47, 1841–1852, (2019).
  * [31] Xiaoa Z., Wei F., Lie triple derivations of triangular algebras, Linear Algebra and its Applicatiions, 437:5, 1234-1249, (2012). Doi: 10.1016/j.laa.2012.04.015
  * [32] Yau, D., On n-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys. 62, 506-522 (2012).
  * [33] Zhou J., Triple derivations of perfect Lie algebras, Comm. Algebra 41, 1647–1654, (2013).
  * [34] Zhou J., Chen L. and Ma Y., Generalized Derivations of Lie triple systems, Open Mathematics, 14, 260–271, (2016).
  * [35] Zhou J., Chen L. and Ma Y., Generalized Derivations of Hom-Lie triple systems, Bulletin of the Malaysian Mathematical Sciences Society, 39 (2016), DOI 10.1007/s40840-016-0334-2
  * [36] Zhou, J.; Chen, L. and Max, Y., Triple derivations and triple homomorphisms of perfect Lie superalgebras, Indag. Math. 28, 436–445, (2015).
  * [37] Zhou J. and Fan G., Generalized Derivations of Hom-Lie Color Algebra (Chinese), Pure Mathematics, 6, 3, 182-189, (2016).
  * [38] Zhou J. and Fan G., Generalized Derivations of $n-$Hom Lie superalgebras, Mathematica Aeterna, Vol. 6, no. 4, 533 - 550, (2016).
  * [39] Zhang J.Z., Wu B.W. and Cao H.X., Lie triple derivations of nest algebras, Linear Algebra Appl. 416, 559–567, (2006)..
  * [40] Zhang R. and Zhang Y., Generalized derivations of Lie superalgebras, Communications in Algebra, 38, 10, 3737–3751, (2018).


# Understanding human mobility patterns in Chicago: an analysis of taxi data
using clustering techniques††thanks: Project done as part of the course on
Mathematical Tools for Data Science at NYU Center for Data Science.

Harish Chauhan, Nikunj Gupta22footnotemark: 2, Zoe Haskell-
Craig22footnotemark: 2 All authors contributed equally to this work.

## Motivation

Understanding human mobility patterns is important in applications as diverse
as urban planning, public health and political organizing. One rich source of
data on human mobility is taxi ride data. Using the city of Chicago as a case
study, we examine data from taxi rides in 2016 with the goal of understanding
how neighborhoods are interconnected. This analysis will provide a sense of
which neighborhoods individuals are using taxis to travel between, suggesting
regions to focus new public transit development efforts. Additionally, this
analysis will map traffic circulation patterns and provide an understanding of
where in the city people are travelling from and where they are heading to -
perhaps informing traffic or road pollution mitigation efforts. For the first
application, representing the data as an undirected graph will suffice.
Transit lines run in both directions so simply a knowledge of which
neighborhoods have high rates of taxi travel between them provides an argument
for placing public transit along those routes. However, in order to understand
the flow of people throughout a city, we must make a distinction between
neighborhood from which people are departing and the areas to which they are
arriving - this requires methods that can deal with directed graphs. All
developed codes can be found at https://github.com/Nikunj-Gupta/Spectral-
Clustering-Directed-Graphs.

## Data

Data was taken from the taxi trips for 2016, reported to the City of Chicago,
and available on Kaggle. (https://www.kaggle.com/datasets/chicago/chicago-
taxi-rides-2016?select=data“˙dictionary.csv) This data contained information
on pick-up community area and drop-off community area, the community areas
where taxi trips began and ended. A total of almost 1 300 000 observations
were included. We converted this into a graph considering the community areas
as nodes and mean travel time between nodes as the weighted edges. As such, we
established a directed graph consisting of 77 nodes representing the taxi
traffic between areas of Chicago.

## Methodology

### Clustering methods for undirected graphs

Several algorithms exist to cluster nodes of an undirected graph. We modified
our directed graph to an undirected graph by considering any trip between two
community areas to contribute to the edge weight, independent of the direction
of the trip. We then applied the following algorithms to establish cluster in
the graph:

#### Spectral clustering:

Spectral graph theory associates spectral properties of matrices to undirected
graphs. We define the graph Laplacian L, given by L = I - $D^{-1}$W, where D
is the diagonal matrix of degrees and W is the weighted adjacency matrix.
Matrix L is real and symmetric, hence diagonalizable, and has real eigenvalues
and eigenvectors, with zero as an eigenvalue (always) and the vector of all
ones as the corresponding eigenvector. The key incentive of spectral
clustering is to group the nodes of a graph into two or more clusters based on
relating them to the spectrum of the graph Laplacian. The spectral clustering
algorithm involves the computation of the k smallest eigenvalues and the use
of all k corresponding eigenvectors to cluster graph nodes into k clusters.

  1. 1.

Input: Adjacency matrix $W\in R^{n\times n}_{+}$ of a directed graph;

  2. 2.

Parameters: $k\in\\{2,3,..n\\}$

  3. 3.

Compute the graph Laplacian L;

  4. 4.

Find the k first eigenvectors and store them as the columns of a matrix

  5. 5.

Consider each row of the above matrix as a point in $R^{k}$ and cluster these
points using a k-means algorithm

  6. 6.

Output: estimation of cluster membership;

Algorithm 1 Spectral clustering algorithm

#### Leiden Algorithm:

The Leiden algorithm uses the Constant Potts Model defined as
$H=\sum[e_{c}-\gamma{\binom{n_{c}}{1}}]$, where $n_{c}$ is the number of nodes
in cluster $c$, $e_{c}$ is the number of edges in $c$, and $\gamma$ is a
tuning parameter. The Leiden algorithm works by iterating through the
following steps (Traag, Waltman, and vanEck 2019):

  1. 1.

Start with a singleton partition.

  2. 2.

Moves nodes from one cluster to another in order to find a partition.

  3. 3.

Refine the clusters, which may result in splitting the clusters into multiple
subcluster by merging well-connected nodes.

  4. 4.

Aggregate clusters to form a single node in the subsequent iteration of the
algorithm (now on a smaller graph).

Algorithm 2 Leiden Algorithm

The Leiden algorithm is implemented through the igraph package available for
python.

#### Walktrap:

The walktrap algorithm developed in the igraph package uses an algorithm
proposed in (Pons and Latapy 2005). This approach uses a partition of the
graph to define clusters. The intuition behind this algorithm is that a random
walk tends to get ”trapped” in the densely connected subregions of the graph
corresponding to a cluster. As such, Pons and Latapy create a notion of
distance, $r$, between nodes i and j in terms of the probability of going from
i to j through a random walk of length t. The walktrap algorithm then works as
follows:

  1. 1.

Choose two clusters based on the distance $r$ between them.

  2. 2.

Merge these two clusters to form a single cluster, and create a new partition
of the graph. In this algorithm, only adjacent clusters are merged.

  3. 3.

Update the distances between clusters.

Algorithm 3 Walktrap Algorithm

### Clustering methods for directed graphs

Directed graphs have their vertices connected by directional edges. Unlike
undirected graphs, the direction of the edges between two vertices convey
distinct information, in this case the direction of taxi travel. Hence the
weight matrix representing the edge is not symmetric. In our case, the value
in $W_{ij}$ represents the time taken to go from neighborhood $i$ to $j$.
Given any directed graph, we can symmetrize the graph to an undirected graph
by dropping the direction of each edge. We can then use any of the numerous
partitioning methods for undirected graphs. The problem with this approach is
that it loses key information.

There are also a wide variety of clustering methods for directed graphs.
Generally speaking, algorithms can be partitioned into two categories: given a
directed graph, either we build a related undirected graph and apply known
clustering algorithms to it or we work directly on the directed graph and
formulate new methods from scratch. In this section we explain some of the
methods we implemented on the directed graph we obtained for the data on
Chicago taxi trips.

#### Simple Transformation:

In this case, a new undirected graph with symmetric adjacency matrix is
created from the original directed graph using a simple matrix transformation
(shown below). Then clustering methods for undirected graphs are applied to
this transformed weight matrix.

$W_{u}=W+W^{T}$

This transformation simply ignores the directionality of edges. Hence, it
should be avoided if the goal is to take the directionality of edges into
account.

#### Bibliometric Symmetrization:

Satuluri et al. proposed this type of transformation from non-symmetric to
symmetric weight matrix (Satuluri and Parthasarathy 2011). The transformation
is defined as follows:

$W_{u}=W^{T}W+WW^{T}$

The reason that it is able to find patterns is that $W^{T}W$ measures common
in-neighbours and $WW^{T}$ measures common out-neighbours for every pair of
nodes in the directed graph. As there does not seem to be any obvious reason
for leaving out either in-neighbours or out-neighbours, a sum of both matrices
is taken so as to account for both.

#### Spectral clustering using Chung’s directed Laplacian:

A straightforward generalization of the Laplacian matrix to the directed case
can lead to a non-symmetric matrix because the adjacency matrix of a directed
graph is not symmetric. This makes spectral analysis more difficult (complex
eigenvalues, defective matrices, etc). Fan Chung proposed a symmetrized
version of the Laplacian that applies to directed graphs (Chung 2005). This
method was further developed by David Gleich in (Gleich 2006) to cluster nodes
in directed graphs. Given a strongly connected directed, aperiodic graph G =
(V, E) and a weight matrix $W\in R^{n\times n}_{+}$, Chung’s Directed
Laplacian is given by

$L(G)=I-\frac{1}{2}(\pi^{1/2}P\pi^{-1/2}+\pi^{-1/2}P^{T}\pi^{1/2})$

where, P the transition matrix of the graph given by $P=D^{-1}W$ and D is the
out degree matrix $D=W\mathbf{1}$ (1 is the column vector of 1’s).

  1. 1.

Input: Adjacency matrix $W\in R^{n\times n}_{+}$ of a directed graph;

  2. 2.

Parameters: $k\in\\{2,3,..n\\}$

  3. 3.

Compute the graph’s Chung’s directed Laplacian L;

  4. 4.

Find the k first eigenvectors and store them as the columns of a matrix

  5. 5.

Consider each row of the above matrix as a point in $R^{k}$ and cluster these
points using a k-means algorithm

  6. 6.

Output: estimation of cluster membership;

Algorithm 4 Chung’s directed Laplacian (CDL) Spectral clustering

#### SVD spectral clustering:

Sussman et al propose an algorithm based on the singular value decomposition
of the adjacency matrix (Sussman et al. 2012). The algorithm includes the
latent feature dimension (d), which should be number of singular values of W
significantly greater than 0. The adjacency matrix W is expected to be well
approximated by a matrix of rank d. The main difference with classical
spectral clustering algorithms is that this algorithm uses the non-symmetric
adjacency matrix instead of a Laplacian matrix. An advantage of SVD spectral
clustering is that it is very general in the type of clusters it is able to
detect.

  1. 1.

Input: Adjacency matrix $W\in R^{n\times n}_{+}$ of a directed graph;

  2. 2.

Parameters: $d\in\\{1,2,3,..n\\}$, $k\in\\{2,3,..n\\}$

  3. 3.

Compute the singular value decomposition W = US$V^{T}$ where S has decreasing
main diagonal;

  4. 4.

Let $\tilde{U}$ and $\tilde{V}$ be the first d columns of U and V , let
$\tilde{S}$ contain the first d columns and first d rows of S;

  5. 5.

Define $\tilde{Z}$ = [$\tilde{U}$ $\tilde{S}^{1/2}$ |$\tilde{V}$
$\tilde{S}^{1/2}$] by concatenation;

  6. 6.

Cluster the rows of $\tilde{Z}$ using a k-means algorithm with Euclidean
distance.

  7. 7.

Output: estimation of cluster membership;

Algorithm 5 SVD spectral clustering

#### Spectral clustering algorithm based on random walk:

This algorithm mainly extract information from eigenvectors and eigenvalues of
the transition matrix of a directed graph. The method focuses on finding the
second largest eigenvalue of transition matrix P in terms of magnitude and
then look at the components of the corresponding eigenvector to partition the
network into two clusters. Finding this second eigenvector seems to be
sufficient to partition the network in more than two clusters.

  1. 1.

Input: $W\in\\{0,1\\}^{n\times n}$;

  2. 2.

Parameters: $d\in\\{1,2,...,n\\},k\in\\{2,3,...,n\\}$

  3. 3.

Compute the transition matrix P = $D^{-1}$W

  4. 4.

Calculate the second eigenvector u of P and calculate vector v such that
$v_{i}=Re(u_{i})+Im(u_{i}),$ for $1\leq i\leq n;$

  5. 5.

Apply the Gaussian kernel function to the components of v and store the result
in a vector w;

$w_{i}=\Phi_{v}(v_{i})$

  6. 6.

Cluster the components of w into k clusters (for instance by using k-means
algorithm).

  7. 7.

Output: estimation of cluster membership;

Algorithm 6 Random Walk Spectral Clustering Algorithm

## Results and Conclusion

The results of various clustering algorithms on undirected graphs are shown in
Figure 1. Neither spectral clustering with an unnormalized Laplacian nor
clustering using the Leiden algorithm produced meaningful clusters, other than
identifying outlier nodes. This is likely because the nodes were highly
connected and the weights of the edges had a wide distribution. However
spectral clustering with a normalized Laplacian and clustering using the
Walktrap algorithm both found two clusters in the data. That two algorithms
identified similar clusters indicates that there is some robustness to these
results. For directed graphs (figure 2), we can see that, for all the
algorithms, one cluster can be identified that has nodes with a lot of
incoming edges, whereas other clusters have relatively decreasing densities.
All algorithms identified similar clusters indicating that there is some
robustness to these results.

(a) Unclustered Undirected Graph from taxi trip data

(b) Spectral Clustering (unnormalized Laplacian)

(c) Spectral Clustering using normalized Laplacian

(d) Clustered using Leiden algorithm

(e) Clustered using Walktrap algorithm

Figure 1: Spectral clustering algorithms for Undirected Graphs

(a) Unclustered Directed Graph from taxi trip data

(b) Clustered using Simple Transformation

(c) Clustered using Bibliometric Symmetrization

(d) Clustered using Chung’s Directed Laplacian

(e) Clustered using SVD Spectral Clustering

(f) Spectral clustering algorithm based on random walk

Figure 2: Spectral clustering algorithms for Directed Graphs

In subsequent work, we could stratify the data by weekday/weekend in order to
better understand the very different types of traffic flow that occur
throughout the week.

## References

  * Chung (2005) Chung, F. 2005.  Laplacians and the Cheeger inequality for directed graphs.  _Annals of Combinatorics_ , 9(1): 1–19.
  * Gleich (2006) Gleich, D. 2006.  Hierarchical directed spectral graph partitioning.  _Information Networks_ , 443.
  * Pons and Latapy (2005) Pons, P.; and Latapy, M. 2005.  Computing communities in large networks using random walks.  In _International symposium on computer and information sciences_ , 284–293. Springer.
  * Satuluri and Parthasarathy (2011) Satuluri, V.; and Parthasarathy, S. 2011.  Symmetrizations for clustering directed graphs.  In _Proceedings of the 14th International Conference on Extending Database Technology_ , 343–354.
  * Sussman et al. (2012) Sussman, D. L.; Tang, M.; Fishkind, D. E.; and Priebe, C. E. 2012.  A consistent adjacency spectral embedding for stochastic blockmodel graphs.  _Journal of the American Statistical Association_ , 107(499): 1119–1128.
  * Traag, Waltman, and vanEck (2019) Traag, V.; Waltman, L.; and vanEck, N. 2019.  From Louvain to Leiden: guaranteeing well-connected communities.  _Science_ , 9(5233).