Multiplicative n-Hom-Lie Color Algebras

Abstract

The purpose of this paper is to generalize some results on n-Lie algebras and n-Hom-Lie algebras to n-Hom-Lie color algebras. Then we introduce and give some constructions of n-Hom-Lie color algebras.

7.1 Introduction

The investigations of various q-deformations (quantum deformations) of Lie algebras began a period of rapid expansion in 1980s stimulated by introduction of quantum groups motivated by applications to the quantum Yang-Baxter equation, quantum inverse scattering methods and constructions of the quantum deformations of universal enveloping algebras of semi-simple Lie algebras. In [2, 22,23,24,25, 27,28,29, 35, 36, 50,51,52] various versions of q-deformed Lie algebras appeared in physical contexts such as string theory, vertex models in conformal field theory, quantum mechanics and quantum field theory, q-deformations of infinite-dimensional algebras, primarily the q-deformed Heisenberg algebras [34], q-deformed oscillator algebras and q-deformed Witt and q-deformed Virasoro algebras, and some interesting q-deformations of the Jacobi identity for Lie algebras in these q-deformed algebras were observed.

Hom-Lie algebras and more general quasi-Hom-Lie algebras were introduced first by Larsson, Hartwig and Silvestrov [33], where the general quasi-deformations and discretizations of Lie algebras of vector fields using more general $\sigma$-derivations (twisted derivations) and a general method for construction of deformations of Witt and Virasoro type algebras based on twisted derivations have been developed, initially motivated by the q-deformed Jacobi identities observed for the q-deformed algebras in physics, along with q-deformed versions of homological algebra and discrete modifications of differential calculi. The general abstract quasi-Lie algebras and the subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras as well as their general colored (graded) counterparts have been introduced [33, 45,46,47, 59]. Subsequently, various classes of Hom-Lie admissible algebras have been considered in [53]. In particular, in [53], 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 via commutator map as new product. In [53], moreover several other interesting classes of Hom-Lie admissible algebras generalising some classes of non-associative algebras, as well as examples of finite-dimensional Hom-Lie algebras have been described. Since these pioneering works [33, 43, 45,46,47,48, 53, 55, 58], Hom-algebra structures have developed in a popular broad area with increasing number of publications in various directions. In Hom-algebra structures, defining algebra identities are twisted by linear maps. Hom-algebras structures are very useful since Hom-algebra structures of a given type include their classical counterparts and open more possibilities for deformations, extensions of cohomological structures and representations (see for example [3, 4, 20, 45, 56, 63, 64] and references therein).

Ternary algebras and more generally n-ary Lie algebras first appeared in Nambu’s generalization of Hamiltonian mechanics, using a ternary generalization of Poisson algebras [54]. The mathematical algebraic foundations of Nambu mechanics have been developed by Takhtajan and Daletskii in [30, 60, 61]. Filippov, in [32] introduced n-Lie algebras. In [21], Leibnitz n-algebras have been studied. Properties and classification of n-ary algebras, including solvability and nilpotency, were studied in [10,11,12,13,14,15,16,17, 37]. The general cohomology theory for n-Lie algebras and Leibniz n-algebras was established in [57]. The structure and classification theory of finite-dimensional n-Lie algebras was considered in [49] and many other authors. For more details of the theory and applications of n-Lie algebras, see [31] and references therein.

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 [8], by twisting the identities defining them using a set of linear maps, together with the particular case where all these maps are equal and are algebra morphisms. A way to generate examples of such algebras from non Hom-algebras of the same type is introduced. Further properties, construction methods, examples, cohomology and central extensions of n-ary Hom-algebras have been considered in [5,6,7, 42, 43, 62, 65]. The construction of $(n+1)$-Lie algebras induced by n-Lie algebras using combination of bracket multiplication with a trace, motivated by the work of Awata et al. [9] on the quantization of the Nambu brackets, was generalized using the brackets of general Hom-Lie algebra or n-Hom-Lie and trace-like linear forms depending on the linear maps defining the Hom-Lie or n-Hom-Lie algebras [6, 7]. Generalized derivations of Lie color algebras and n-ary (color) algebras have been studied in [26, 38,39,40,41]. Derivations, L-modules, L-comodules and Hom-Lie quasi-bialgebras have been considered in [18, 19]. Super 3-Lie algebras induced by super Lie algebras in similar way have been considered in [1].

The purpose of this paper is to generalize some results on either n-Lie algebras or n-Hom-Lie algebras to the case of n-Hom-Lie color algebras. Then we introduce and give some constructions of n-Hom-Lie color algebras. Section 7.2 contains some necessary important basic notions and notations on graded spaces and algebras and n-ary algebras and used in other sections. Section 7.3 presents some useful methods for construction of n-Hom-Lie color algebras. In Sect. 7.4, Hom-modules over n-Hom-Lie color algebras are considered. Section 7.5 is devoted to generalized derivations of color Hom-algebras and their color Hom-subalgebras.

Throughout this paper, all graded linear spaces are assumed to be over a field $\mathbb {K}$ of characteristic different from 2.

7.2 Preliminaries

This section contains necessary important basic notions and notations on graded spaces and algebras and n-ary algebras used in other sections.

Definition 7.1

• (1) Let G be an abelian group. A linear space V is said to be a G-graded if, there exists a family $(V_a)_{a\in G}$ of linear subspaces of V such that

  $$V=\bigoplus _{a\in G} V_a.$$

• (2) An element $x\in V$ is said to be homogeneous of degree $a\in G$ if $x\in V_a$. We denote $\mathcal {H}(V)$ the set of all homogeneous elements in V.

• (3) Let $V=\oplus _{a\in G} V_a$ and $V'=\oplus _{a\in G} V'_a$ be two G-graded linear spaces. A linear mapping $f : V\rightarrow V'$ is said to be homogeneous of degree b if

  $$f(V_a)\subseteq V'_{a+b}, \quad \text { for all }\quad a\in G.$$

  If, f is homogeneous of degree zero i.e. $f(V_a)\subseteq V'_{a}$ holds for any $a\in G$, then f is said to be even.

Definition 7.2

• (1) An algebra $(A, \cdot )$ is said to be G-graded if its underlying linear space is G-graded i.e. $A=\bigoplus _{a\in G}A_a$, and if furthermore

  $$A_a\cdot A_b\subseteq A_{a+b}, \quad \text { for all } \quad a, b\in G.$$

• (2) A morphism $f : A\rightarrow A'$ of G-graded algebras A and $A'$ is by definition an algebra morphism from A to $A'$ which is, in addition an even mapping.

Definition 7.3

Let G be an abelian group. A map $\varepsilon :G\times G\rightarrow \mathbf{\mathbb {K}^*}$ is called a skew-symmetric bicharacter  on G if the following identities hold for all $a, b, c\in G$:

• (i) $\varepsilon (a, b)\varepsilon (b, a)=1$,

• (ii) $\varepsilon (a, b+c)=\varepsilon (a, b)\varepsilon (a, c)$,

• (iii) $\varepsilon (a+b, c)=\varepsilon (a, c)\varepsilon (b, c)$,

If x and y are two homogeneous elements of degree a and b respectively and $\varepsilon$ is a skew-symmetric bicharacter, then we shorten the notation by writing $\varepsilon (x, y)$ instead of $\varepsilon (a, b)$.

Example 7.1

Some standard examples of skew-symmetric bicharacters are:

• (1) $G=\mathbb {Z}_2,\quad \varepsilon (i, j)=(-1)^{ij}$, or more generally

  $$\begin{aligned} \displaystyle \begin{array}{c} G=\mathbb {Z}_2^n=\{(\alpha _1, \dots , \alpha _n)| \alpha _i\in \mathbb {Z}_2 \}, \\ \displaystyle \varepsilon ((\alpha _1, \dots , \alpha _n), (\beta _1, \dots , \beta _n)):= (-1)^{\alpha _1\beta _1+\dots +\alpha _n\beta _n}. \end{array} \end{aligned}$$

• (2) $G=\mathbb {Z}_2\times \mathbb {Z}_2,\quad \varepsilon ((i_1, i_2), (j_1, j_2)=(-1)^{i_1j_2-i_2j_1}$,

• (3) $G=\mathbb {Z}\times \mathbb {Z} ,\quad \varepsilon ((i_1, i_2), (j_1, j_2))=(-1)^{(i_1+i_2)(j_1+j_2)}$,

• (4) $G=\{-1, +1\} , \quad \varepsilon (i, j)=(-1)^{(i-1)(j-1)/{4}}$.

Definition 7.4

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

$$\begin{aligned} \begin{array}{l} [x_1, \dots , x_{n-1}, [y_1, \dots , y_n]] =\\ = \displaystyle \sum _{i=1}^n[y_1, \dots , y_{i-1}, [x_1, \dots , x_{n-1}, y_i], y_{i+1}, \dots y_{n}]. \end{array} \end{aligned}$$

(7.1)

Definition 7.5

An n-Hom-Lie color algebra is a graded linear space $L=\oplus L_a, a\in G$ with an n-linear map $[\cdot \dots , \cdot ]: L\times \dots \times L\rightarrow L$, a bicharacter $\varepsilon : G\times G\rightarrow \mathbf{K}^*$ and an even linear map $\alpha : L\rightarrow L$ such that

$$\begin{aligned}{}[x_1, \dots , x_i, x_{i+1}, \dots , x_n]= & {} -\varepsilon (x_i, x_{i+1})[x_1, \dots , x_{i+1}, x_i, \dots , x_n], \\&\quad \quad \quad \quad \quad \,\,\, i=1,2, \dots n-1. \nonumber \end{aligned}$$

(7.2)

$$\begin{aligned} \begin{array}{l} [\alpha (x_1), \dots , \alpha (x_{n-1}), [y_1, y_2, \dots , y_{n}]] = \\ =\displaystyle \sum _{i=1}^n\varepsilon (X, Y_i)[\alpha (y_1), \dots , \alpha (y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \alpha (y_{i+1}),\dots , \alpha (y_{n})] \end{array} \end{aligned}$$

(7.3)

where $x_i, y_j\in \mathcal {H}(L)$, $X=\sum _{i=1}^{n-1}x_i$, $Y_i=\sum _{j=1}^iy_{j-1}$ and $y_0=e$.

Remark 7.1

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

Remark 7.2

• (1) When $\alpha =id$, we get n-Lie color algebra.

• (2) When $G=\{e\}$ and $\alpha =id$, we get n-Lie algebra.

• (3) When $G=\{e\}$ and $\alpha \ne id$, we get n-Hom-Lie algebra.

Definition 7.6

A morphism $f : (L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )\rightarrow (L', [\cdot , \dots , \cdot ]', \varepsilon , \alpha ')$ of an n-Hom-Lie color algebras is an even linear map $f : L\rightarrow L$ such that $f\circ \alpha =\alpha \circ f$ and for any $x_i\in \mathcal {H}(L)$,

$$f([x_1, \dots , x_n])=[f(x_1), \dots , f(x_n)]'$$

Definition 7.7

(1) An n-Hom-Lie color algebra $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is said to be multiplicative if $\alpha$ is an endomorphism, i.e. a linear map on L which is also a homomorphism with respect to multiplication $[\cdot , \dots , \cdot ]$).

(2) An n-Hom-Lie color algebra $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is said to be regular if $\alpha$ is an automorphism.

(3) An n-Hom-Lie color algebra $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is said to be involutive if $\alpha ^2=id$.

Example 7.2

Let $G=\mathbb {Z}_2,\quad \varepsilon (i, j)=(-1)^{ij}$, $L=L_0\oplus L_1=<e_2, e_4>\oplus <e_1, e_3>$,

$$[e_1, e_2, e_3]=e_2,\quad [e_1, e_2, e_4]=e_1,\quad [e_1, e_3, e_4]=[e_2, e_3, e_4]=0,$$

and $\alpha (e_1)=e_3, \quad \alpha (e_2)=e_4, \quad \alpha (e_3)=\alpha (e_4)=0.$ Then $(L, [\cdot , \cdot , \cdot ], \varepsilon , \alpha )$ is a 3-Hom-Lie color algebra.

Example 7.3

Let L be a graded linear space

$$L=L_{(0, 0)}\oplus L_{(0, 1)}\oplus L_{(1, 0)}\oplus L_{(1, 1)}$$

with $L_{(0, 0)}=<e_1, e_2>,\quad L_{(0, 1)}=<e_3>,\quad L_{(1, 0)}=<e_4>,\quad L_{(1, 1)}=<e_5>.$

The 4-ary even linear multiplication $[\cdot , \cdot , \cdot , \cdot ] : L\times L\times L\times L\rightarrow L$ defined for basis $\{e_i\}, i=1, \dots ,5$ by

$$\begin{aligned} \begin{array}{c} {[}e_2, e_3, e_4, e_5]=e_1, [e_1, e_3, e_4, e_5]=e_2, [e_1, e_2, e_4, e_5]=e_3, \\ {[}e_1, e_2, e_3, e_4]=0, [e_1, e_2, e_3, e_5]=0 \\ \end{array} \end{aligned}$$

makes L into the five dimensional 4-Lie color algebra.

Now define on $(L, [\cdot , \cdot , \cdot , \cdot ], \varepsilon )$ an even endomorphism $\alpha :L\rightarrow L$ by

$$ \alpha (e_1)=e_2,\quad \alpha (e_2)=e_1, \quad \alpha (e_i)=e_i, i=3, 4, 5. $$

Then $L_\alpha =(L, [\cdot , \cdot , \cdot , \cdot ]_\alpha , \varepsilon , \alpha )$ is a 4-Hom-Lie color algebra. Observe that $\alpha$ is involutive (bijective).

Definition 7.8

A graded subspace H of an n-Hom-Lie color algebra L is a color Hom-subalgebra of L if

• (i) $\alpha (H)\subseteq H$,

• (ii) $[H, H, \dots , H]\subseteq H$.

Definition 7.9

Let $L_1, L_2, \dots , L_n$ be Hom-subalgebras of an n-Hom-Lie color algebra L. Denote by $[L_1, L_2, \dots , L_n]$ the Hom-subalgebra of L generated by all elements $[x_1, x_2, \dots , x_n]$, where $x_i\in L_i, i=1, 2, \dots , n$.

• (i) The sequence $L_1, L_2, \dots , L_n, \dots$ defined by

  $$\begin{aligned} \begin{array}{c} L_0=L,\quad L_1=[L_0, L_0, \dots , L_0],\quad L_2=[L_1, L_1, \dots , L_1], \dots ,\\ L_n=[L_{n-1}, L_{n-1}, \dots , L_{n-1}], \dots \end{array} \end{aligned}$$

  is called the derived sequence. 

• (ii) The sequence $L^1, L^2, \dots , L^n, \dots$ defined by

  $$\begin{aligned} \begin{array}{c} \displaystyle L^0=L,\quad L^1=[L^0, L, \dots , L],\quad L^2=[L^1, L, \dots , L], \dots , \\ L^n=[L^{n-1}, L, \dots , L], \dots \end{array} \end{aligned}$$

  is called the descending central sequence. 

• (iii) The graded subspace Z(L) defined by

  $$\begin{aligned} Z(L)=\{x\in L| [x, L, L, \dots , L]=0\} \end{aligned}$$

  (7.4)

  is called the center of L.

Definition 7.10

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

Theorem 7.1

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra with surjective twisting map $\alpha : L\rightarrow L$. Then, $I_n, I^n$ and Z(L) are Hom-ideals of L.

Proof

We only prove, by induction, that $I_n$ is a Hom-ideal. For this, suppose, first, that $I_{n-1}$ is a Hom-subalgebra of L and show that $I_n$ is a Hom-subalgebra of L. For any $y\in \mathcal {H}(I_n)$, there exist $y_1, y_2, \dots , y_n\in \mathcal {H}(I_{n-1})$, such that

$$y=[y_1, y_2, \dots , y_n].$$

So, $\alpha (y)=\alpha ([y_1, y_2, \dots , y_n])=[\alpha (y_1), \alpha (y_2), \dots , \alpha (y_n)]$, which belong to $I_n$ because $I_{n-1}$ is a Hom-subalgebra. That is $\alpha (I_n)\subseteq I_n$.

For any $y_i\in \mathcal {H}(I_{n})$, there exist $y_i^1, y_i^2, \dots , y_i^n\in I_{n-1}, i=1, 2, \dots , n$ such that

$$[y_1, y_2, \dots , y_n]=[[y_1^1, y_1^2, \dots , y_1^n], [y_2^1, \dots , y_2^n], \dots , [y_n^1, \dots , y_n^n]].$$

$I_{n-1}$ being a Hom-subalgebra, by hypotheses, $[y_i^1, y_i^2, \dots , y_i^n]\in I_{n-1}$ for $1\le i\le n$, and so $[y_1, y_2, \dots , y_n]\in I_n$. Thus $I_n$ is a Hom-subalgebra.

Now, suppose that $I_{n-1}$ is a Hom-ideal. Let $x'_1, \dots , x'_{n-1}\in L, y\in I_n$, then there exist $x_1, \dots , x_{n-1}\in L$, $y_1, \dots , y_n\in I_{n-1}$ such that

$$\begin{aligned}{}[x'_1, \dots , x'_{n-1}, y]= & {} [\alpha (x_1), \dots , \alpha (x_{n-1}), [y_1, \dots , y_n]]\nonumber \\= & {} \sum _{i=1}^n\varepsilon (X, Y_i)[\alpha (y_1), \dots , \alpha (y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \alpha (y_{i+1}), \alpha (y_n)]\nonumber \end{aligned}$$

As $[x_1, \dots , x_{n-1}, y_i]\in I_{n-1}$, then $[x'_1, \dots , x'_{n-1}, y]\in I_n$. So, $I_n$ is a Hom-ideal of L. $\square$

7.3 Constructions of n-Hom-Lie Color Algebras

In this section we present some useful methods for construction of n-Hom-Lie color algebras.

Proposition 7.1

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\xi \in L_e$ such that $\alpha (\xi )=\xi$.

Then $(L, \{\cdot , \dots , \cdot \}, \varepsilon , \alpha )$ is an $(n-1)$-Hom-Lie color algebra with

$$\{x_1, \dots , x_{n-1}\}=[\xi , x_1, \dots , \dots , x_{n-1}].$$

Proof

With conditions in the statement,

$$\begin{aligned}&\{\alpha (x_1), \dots , \alpha (x_{n-2}), \{y_1, \dots , y_{n-1}\}\}=\nonumber \\&=[\xi ,\alpha ( x_1), \dots , \alpha (x_{n-2}), [\xi , y_1, \dots , y_{n-1}]]\nonumber \\&=[\alpha (\xi ),\alpha ( x_1), \dots , \alpha (x_{n-2}), [\xi , y_1, \dots , y_{n-1}]]\nonumber \\&=[[\xi , x_1, \dots , x_{n-2}, \xi ], \alpha (y_1), \dots , \alpha (y_{n-1})]\nonumber \\&\quad +\sum _{i=i}^{n-1}\varepsilon (X, Y_i)[\xi , \alpha (y_1), \dots , \alpha (y_{i-1}), [\xi , x_1, \dots , x_{n-2}, y_i], \alpha (y_{i+1}), \dots , \alpha (y_{n-1})]\nonumber \\&=\sum _{i=i}^{n-1}\varepsilon (X, Y_i)\{\alpha (y_1), \dots , \alpha (y_{i-1}), \{x_1, \dots , x_{n-2}, y_i\}, \alpha (y_{i+1}), \dots , \alpha (y_{n-1})\}\nonumber . \end{aligned}$$

which completes the proof. $\square$

Corollary 7.1

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\xi _i\in L_e$ such that $\alpha (\xi _i)=\xi _i, i=1,2, \dots , k$.

Then $L_k=(L, \{\cdot , \dots , \cdot \}_k, \varepsilon , \alpha )$ is an $(n-k)$-Hom-Lie color algebra with

$$\{x_1, \dots , x_{n-k}\}_k=[\xi _1, \dots , \xi _k, x_1\dots , \dots , x_{n-k}].$$

Corollary 7.2

Let $(L, [\cdot , \dots , \cdot ], \varepsilon )$ be an n-Lie color algebra and $\xi \in L_e$.

Then $(L, \{\cdot , \dots , \cdot \}, \varepsilon )$ is an $(n-1)$-Lie color algebra with

$$\{x_1, \dots , x_{n-1}\}=[\xi , x_1, \dots , \dots , x_{n-1}].$$

Theorem 7.2

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\beta$ be an even endomorphism of L. Then

$$L_\beta =(L, \{\cdot , \dots , \cdot \}=\beta [\cdot , \dots , \cdot ], \varepsilon , \beta \alpha )$$

is an n-Hom-Lie color algebra.

Moreover suppose that $(L', [\cdot , \dots , \cdot ]', \varepsilon , \alpha ')$ is another n-Hom-Lie color algebra and $\beta '$ be an even endomorphism of $L'$. If

$$f : (L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )\rightarrow (L', [\cdot , \dots , \cdot ]', \varepsilon , \alpha ')$$

is a morphism such that $f\beta =\beta 'f$, then $f : L_\beta \rightarrow L_{\beta '}$ is also a morphism.

Proof

First part is proved as follows:

$$\begin{aligned}&\{\beta \alpha (x_1), \dots , \beta \alpha (x_{n-1}), \{y_1, \dots , y_n\}\}=\nonumber \\&=\beta ([\beta \alpha (x_1), \dots , \beta \alpha (x_{n-1}), \beta [y_1, \dots , y_n]])\nonumber \\&=\beta ^2\Big ([\alpha (x_1), \dots , \alpha (x_{n-1}), [y_1, y_2, \dots , y_{n}]]\Big )\nonumber \\&=\beta ^2\left( \sum _{i=1}^n\varepsilon (X, Y_i)[\alpha (y_1), \dots , \alpha (y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \alpha (y_{i+1}),\dots , \alpha (y_{n})]\right) \nonumber \\&=\sum _{i=1}^n\varepsilon (X, Y_i)\beta ^2\Big ([\alpha (y_1), \dots , \alpha (y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \alpha (y_{i+1}),\dots , \alpha (y_{n})]\Big )\nonumber \\&=\sum _{i=1}^n\varepsilon (X, Y_i)\beta [\beta \alpha (y_1), \dots , \beta \alpha (y_{i-1}), \beta [x_1, \dots , x_{n-1}, y_i], \beta \alpha (y_{i+1}),\dots , \beta \alpha (y_{n})]\nonumber \\&=\sum _{i=1}^n\varepsilon (X, Y_i)\{\beta \alpha (y_1), \dots , \beta \alpha (y_{i-1}), \{x_1, \dots , x_{n-1}, y_i\}, \beta \alpha (y_{i+1}),\dots , \beta \alpha (y_{n})\}\nonumber . \end{aligned}$$

Second part is proved as follows:

$$\begin{aligned} f(\{x_1, \dots , x_n\})= & {} f([x_1, \dots , x_n]_\beta ) =f\beta [x_1, \dots , x_n]=f[\beta (x_1), \dots , \beta (x_n)]\nonumber \\= & {} [f\beta (x_1), \dots , f\beta (x_n)]'=[\beta 'f(x_1), \dots , \beta 'f(x_n)]'\nonumber \\= & {} \beta '[f(x_1), \dots , f(x_n)]'=[f(x_1), \dots , f(x_n)]'_{\beta '}\nonumber \\= & {} \{ f(x_1), \dots , f(x_n)\}'\nonumber \end{aligned}$$

This completes the proof. $\square$

Taking $\beta =\alpha ^n$ leads to the following statement.

Corollary 7.3

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. Then, for any positive integer n,

$$(L, \alpha ^n[\cdot , \dots , \cdot ], \varepsilon , \alpha ^{n+1})$$

is also an n-Hom-Lie color algebra.

Taking $\beta =\alpha$ and $\alpha =id$ leads to the following statement.

Corollary 7.4

Let $(L, [\cdot , \dots , \cdot ], \varepsilon )$ be an n-Lie color algebra and $\alpha$ be an even endomorphism of L. Then

$$L_\alpha =(L, \{\cdot , \dots , \cdot \}=\alpha [\cdot , \dots , \cdot ], \varepsilon , \alpha )$$

is a multiplicative n-Hom-Lie color algebra.

Taking $\beta \in Aut(L), \beta =\alpha ^{-1}$ leads to the following statement.

Corollary 7.5

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a regular n-Hom-Lie color algebra. Then

$$L_{\alpha ^{-1}}=(L, \{\cdot , \dots , \cdot \}=\alpha ^{-1}[\cdot , \dots , \cdot ], \varepsilon )$$

is an n-Lie color algebra.

Taking $\beta =\alpha$ leads to the following statement.

Corollary 7.6

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an involutive n-Hom-Lie color algebra. Then

$$L_\beta =(L, \{\cdot , \dots , \cdot \}=\alpha [\cdot , \dots , \cdot ], \varepsilon )$$

is an n-Lie color algebra.

Theorem 7.3

Let $(A, \cdot )$ be a commutative associative algebra and $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra. The tensor product $A\otimes L=\sum _{g\in G}(A\otimes L)_g=\sum _{g\in G}A\otimes L_g$ with the bracket

$$ [a_1\otimes x_1, \dots , a_n\otimes x_n]'=a_1\dots a_n\otimes [x_1, \dots , x_n], $$

the even linear map

$$ \alpha '(a\otimes x):=a\otimes \alpha (x) $$

and the bicharacter

$$ \varepsilon (a+x, b+y)=\varepsilon (x, y), \forall a, b\in A, \forall x, y\in \mathcal {H}(L), $$

is an n-Hom-Lie color algebra.

Proof

It follows from a straightforward computation. $\square$

In the next definition, we introduce an element of the centroid (or semi-morphism) for n-Hom-Lie color algebra.

Definition 7.11

A semi-morphism  of an n-Hom-Lie color algebra $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is an even linear map $\beta : L\rightarrow L$ such that $\beta \alpha =\alpha \beta$ and

$$\beta [x_1, \dots , x_n]=[\beta (x_1), x_2, \dots , x_n].$$

Remark 7.3

Due to $\varepsilon$-skew-symmetry,

$$\beta [x_1, \dots , x_n]=[x_1, \dots \beta (x_i), \dots , x_n].$$

Theorem 7.4

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\beta : L\rightarrow L$ a semi-morphism of L. Define a new multiplication $\{\cdot , \dots , \cdot \}: L\times \dots \times L\rightarrow L$ by

$$\{x_1, \dots , x_n\}=[x_1, \dots , \beta (x_i)\dots , x_n]$$

Then $(L, \{\cdot , \dots , \cdot \}, \varepsilon , \alpha )$ is also an n-Hom-Lie color algebra.

Proof

The proof can be obtained as follows:

$$\begin{aligned}&\,\{\alpha (x_1), \dots , \alpha (x_{n-1}), \{y_1, \dots , y_n\}\}=\nonumber \\&=[\alpha (x_1), \dots , \beta \alpha (x_i), \dots , \alpha (x_{n-1}), [y_1, \dots , \beta (y_j), \dots , y_n]]\nonumber \\&=[\alpha (x_1), \dots , \alpha \beta (x_i), \dots , \alpha (x_{n-1}), [y_1, \dots , \beta (y_j), \dots , y_n]]\nonumber \\&=\sum _{k<j}\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha (y_{k-1}),[x_1, \dots , \beta (x_i), \dots , x_{n-1}, y_k], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\, \alpha (y_{k+1}), \dots , \alpha \beta (y_j), \dots \alpha (y_n)] \nonumber \\&+\varepsilon (X, Y_j)[\alpha (y_1), \dots , \alpha (y_{j-1}), [x_1, \dots , \beta (x_i), \dots , x_{n-1}, \beta (y_j)], \alpha (y_{j+1}), \dots , \alpha (y_n)]\nonumber \\&+\sum _{k>j}\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha \beta (y_{j}), \dots , \alpha (y_{k-1}) [x_1, \dots , \beta (x_i), \dots , x_{n-1}, y_k], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{k+1}), \dots , \alpha (y_n)] \nonumber \\&=\sum _{k<j}\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha (y_{k-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{k+1}), \dots , \beta \alpha (y_j), \dots \alpha (y_n)] \nonumber \\&+\varepsilon (X, Y_j)[\alpha (y_1), \dots , \alpha (y_{j-1}), \beta (\{x_1, \dots , x_i, \dots , x_{n-1}, y_j\}), \alpha (y_{j+1}), \dots , \alpha (y_n)]\nonumber \\&+\sum _{k>j}\varepsilon (X, Y_k)[\alpha (y_1), \dots , \beta \alpha (y_{j}), \dots , \alpha (y_{k-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{k+1}), \dots , \alpha (y_n)]\nonumber \\&=\sum _{k<j}\varepsilon (X, Y_k)\{\alpha (y_1), \dots , \alpha (y_{k-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\,\quad \alpha (y_{k+1}), \dots , \alpha (y_j), \dots \alpha (y_n)\}\nonumber \\&+\varepsilon (X, Y_j)\{\alpha (y_1), \dots , \alpha (y_{j-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_j\}, \alpha (y_{j+1}), \dots , \alpha (y_n)\}\nonumber \\&+\sum _{k>j}\varepsilon (X, Y_k)\{\alpha (y_1), \dots , \alpha (y_{j}), \dots , \alpha (y_{k-1}) \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{k+1}), \dots , \alpha (y_n)\}\nonumber . \end{aligned}$$

This completes the proof. $\square$

Corollary 7.7

Let $(L, [\cdot , \dots , \cdot ], \varepsilon )$ be an n-Lie color algebra and $\alpha : L\rightarrow L$ a semi-morphism of L. Then $(L, \{\cdot , \dots , \cdot \}, \varepsilon )$ is another n-Lie color algebra, with

$$\{x_1, \dots , x_n\}=[x_1, \dots , \alpha (x_i)\dots , x_n]$$

Definition 7.12

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra. An averaging operator of an n-Hom-Lie color algebra L is an even linear map $\beta : L\rightarrow L$ such that

• (1) $\beta \alpha =\alpha \beta$

• (2) $\beta [x_1, \dots , \beta (x_i), \dots , x_n]=[x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_n]$

Theorem 7.5

Let $(L, [\cdot , \dots , \cdot ], \varepsilon )$ be an n-Lie color algebra and $\alpha : L\rightarrow L$ an averaging operator of L. Define a new multiplication $\{\cdot , \dots , \cdot \}: L\times \dots \times L\rightarrow L$ by

$$\{x_1, \dots , x_n\}=[x_1, \dots , \alpha (x_i)\dots , x_n]$$

Then $L_\alpha =(L, \{\cdot , \dots , \cdot \}, \varepsilon , \alpha )$ is an n-Hom-Lie color algebra.

Proof

The proof can be obtained as follows:

$$\begin{aligned}&\{\alpha (x_1), \dots , \alpha (x_{n-1}), \{y_1, \dots , y_n\}\}=\nonumber \\&=[\alpha (x_1), \dots , \alpha ^2(x_i), \dots , \alpha (x_{n-1}), [y_1, \dots , \alpha (y_j), \dots , y_n]]\nonumber \\&=\sum _{k<j}\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha (y_{k-1}), [x_1, \dots , \alpha (x_i), \dots , x_{n-1}, y_k], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{k+1}), \dots , \alpha ^2(y_j), \dots \alpha (y_n)]\nonumber \\&+\varepsilon (X, Y_j)[\alpha (y_1), \dots , \alpha (y_{j-1}), [x_1, \dots , \alpha (x_i), \dots , x_{n-1}, \alpha (y_j)], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{j+1}), \dots , \alpha (y_n)]\nonumber \\&+\sum _{k>j}\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha ^2(y_{j}), \dots , \alpha (y_{k-1}) [x_1, \dots , \alpha (x_i), \dots , x_{n-1}, y_k], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \alpha (y_{k+1}), \dots , \alpha (y_n)]\nonumber \\&=\sum _{k<j}\varepsilon (X, Y_k)\{\alpha (y_1), \dots , \alpha (y_{k-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{k+1}), \dots , \alpha (y_j), \dots \alpha (y_n)\}\nonumber \\&+\varepsilon (X, Y_j)[\alpha (y_1), \dots , \alpha (y_{j-1}), \alpha (\{x_1, \dots , x_i, \dots , x_{n-1}, y_j\}), \alpha (y_{j+1}), \dots , \alpha (y_n)]\nonumber \\&\quad +\sum _{k>j}\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha (y_{j}), \dots , \alpha (y_{k-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \quad \alpha (y_{k+1}), \dots , \alpha (y_n)]\nonumber \\&=\sum _{k<j}\varepsilon (X, Y_k)\{\alpha (y_1), \dots , \alpha (y_{k-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\, \alpha (y_{k+1}), \dots , \alpha (y_j), \dots \alpha (y_n)\}\nonumber \\&+\varepsilon (X, Y_j)\{\alpha (y_1), \dots , \alpha (y_{j-1}), \{x_1, \dots , x_i, \dots , x_{n-1}, y_j\}, \alpha (y_{j+1}), \dots , \alpha (y_n)\}\nonumber \\&+\sum _{k>j}\varepsilon (X, Y_k)\{\alpha (y_1), \dots , \alpha (y_{j}), \dots , \alpha (y_{k-1}) \{x_1, \dots , x_i, \dots , x_{n-1}, y_k\}, \nonumber \\&\qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \alpha (y_{k+1}), \dots , \alpha (y_n)\}\nonumber . \end{aligned}$$

This ends the proof. $\square$

Theorem 7.6

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\beta : L\rightarrow L$ an averaging operator of L. Define a new multiplication $\{\cdot , \dots , \cdot \}: L\times \dots \times L\rightarrow L$ by

$$\{x_1, \dots , x_n\}=[x_1, \dots , \beta (x_i)\dots , x_n]$$

Then $(L, \{\cdot , \dots , \cdot \}, \varepsilon , \alpha )$ is also an n-Hom-Lie color algebra.

Proof

It is similar to the one of Theorem 7.4. $\square$

Taking $\alpha =id$, yields the following statement.

Corollary 7.8

Let $(L, [\cdot , \dots , \cdot ], \varepsilon )$ be an n-Lie color algebra and $\alpha : L\rightarrow L$ an averaging operator of L. Then $(L, \{\cdot , \dots , \cdot \}, \varepsilon )$ is another n-Lie color algebra, with

$$\{x_1, \dots , x_n\}=[x_1, \dots , \alpha (x_i)\dots , x_n]$$

Taking $\alpha =\beta$, yields the following statement.

Corollary 7.9

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\alpha : L\rightarrow L$ an averaging operator. Define a new multiplication $\{\cdot , \dots , \cdot \}: L\times \dots \times L\rightarrow L$ by

$$\{x_1, \dots , x_n\}=[x_1, \dots , \alpha (x_i)\dots , x_n]$$

Then $(L, \{\cdot , \dots , \cdot \}, \varepsilon , \alpha )$ is also an n-Hom-Lie color algebra.

Theorem 7.7

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\beta : L\rightarrow L$ an averaging operator of L. Then the new bracket $\{\cdot , \dots , \cdot \}: L\times \dots \times L\rightarrow L$ makes L into an n-Hom-Lie color algebra with

$$\{x_1, \dots , x_n\}=[x_1, \dots , \beta (x_i)\dots , \beta (x_j), \dots , x_n].$$

Proof

The proof is obtained as follows:

$$\begin{aligned}&\{\alpha (x_1), \dots , \alpha (x_{n-1}), \{y_1, \dots , y_n\}\}=\nonumber \\&=[\alpha (x_1), \dots , \beta \alpha (x_i), \dots , \beta \alpha (x_j), \dots , \alpha (x_{n-1}), [y_1, \dots , \beta (y_k), \dots , \beta (y_l), \dots , y_n]]\nonumber \\&=\sum _{m<k}\varepsilon (X, Y_m)[\alpha (y_1), \dots , \alpha (y_{m-1}), [x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_{n-1}, y_m], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \,\,\,\, \alpha (y_{m+1}), \dots , \alpha \beta (y_k), \dots , \alpha \beta (y_l), \dots , \alpha (y_n)]\nonumber \\&\quad \ +\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha (y_{k-1}), [x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_{n-1}, \beta (y_k)],\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \,\, \alpha (y_{k+1}), \dots , \alpha \beta (y_l), \dots , \alpha (y_n)]\nonumber \\&\quad \ +\sum _{k<m<l}\varepsilon (X, Y_m)[\alpha (y_1), \dots , \alpha \beta (y_{k}), \dots , \alpha (y_{l-1}), \nonumber \\&\qquad \, [x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_{n-1}, y_m], \alpha (y_{m+1}), \dots , \alpha \beta (y_{l}), \dots , \alpha (y_n)]\nonumber \\&\quad \,\,\,\, +\varepsilon (X, Y_l)[\alpha (y_1), \dots , \alpha \beta (y_{k}),\dots , \alpha (y_{l-1}), \nonumber \\&\qquad \qquad \,\,\, [x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_{n-1}, \beta (y_l)],\alpha (y_{l+1}), \dots , \alpha (y_n)]\nonumber \\&\quad \,\,\,\, +\sum _{m>l}\varepsilon (X, Y_m)[\alpha (y_1), \dots , \alpha \beta (y_{k}), \dots , \alpha \beta (y_{l}), \dots , \alpha (y_{m-1}), \nonumber \\&\qquad \qquad \quad [x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_{n-1}, y_m],\alpha (y_{m+1}), \dots , \alpha (y_n)]\nonumber \\&=\sum _{m<k}\varepsilon (X, Y_m)\{\alpha (y_1), \dots , \alpha (y_{m-1}), \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_m\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \,\, \alpha (y_{m+1}),\dots , \alpha (y_k), \dots , \alpha (y_l), \dots , \alpha (y_n)\}\nonumber \\&\quad \ +\varepsilon (X, Y_k)[\alpha (y_1), \dots , \alpha (y_{k-1}), \beta ([x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_{n-1}, y_k]),\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\, \alpha (y_{k+1}), \dots , \beta \alpha (y_l), \dots , \alpha (y_n)]\nonumber \\&\quad \ +\sum _{k<m<l}\varepsilon (X, Y_m)\{\alpha (y_1), \dots , \alpha (y_{k}), \dots , \alpha (y_{m-1}), \nonumber \\&\qquad \qquad \,\, \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_m\},\alpha (y_{m+1}), \dots , \alpha (y_{l}), \dots , \alpha (y_n)]\nonumber \\&\quad \ +\varepsilon (X, Y_l)[\alpha (y_1), \dots , \beta \alpha (y_{k}),\dots , \alpha (y_{l-1}), \nonumber \\&\qquad \qquad \quad \,\,\, \beta ([x_1, \dots , \beta (x_i), \dots , \beta (x_j), \dots , x_{n-1}, y_l]), \alpha (y_{l+1}), \dots , \alpha (y_n)]\nonumber \\&\quad \,\,\, +\sum _{m>l}\varepsilon (X, Y_m)\{\alpha (y_1), \dots , \alpha (y_{k}), \dots , \alpha (y_{l}), \dots , \alpha (y_{m-1}), \nonumber \\&\qquad \qquad \qquad \,\,\,\, \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_m\},\alpha (y_{m+1}), \dots , \alpha (y_n)\}\nonumber \end{aligned}$$

$$\begin{aligned}&=\sum _{m<k}\varepsilon (X, Y_m)\{\alpha (y_1), \dots , \alpha (y_{m-1}), \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_m\}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \, \alpha (y_{m+1}), \dots ,\alpha (y_k), \dots , \alpha (y_l), \dots , \alpha (y_n)\}\nonumber \\&\qquad +\varepsilon (X, Y_k)\{\alpha (y_1), \dots , \alpha (y_{k-1}), \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_k\},\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\, \alpha (y_{k+1}), \dots , \alpha (y_l), \dots , \alpha (y_n)\}\nonumber \\&\qquad +\sum _{k<m<l}\varepsilon (X, Y_m)\{\alpha (y_1), \dots , \alpha (y_{k}), \dots , \alpha (y_{m-1}), \nonumber \\&\qquad \qquad \,\,\,\,\, \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_m\},\alpha (y_{m+1}), \dots , \alpha (y_{l}), \dots , \alpha (y_n)\}\nonumber \\&+\varepsilon (X, Y_l)\{\alpha (y_1), \dots , \alpha (y_{k}),\dots , \alpha (y_{l-1}), \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_l\}),\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \, \alpha (y_{l+1}), \dots , \alpha (y_n)\}\nonumber \\&+\sum _{m>l}\varepsilon (X, Y_m)\{\alpha (y_1), \dots , \alpha (y_{k}), \dots , \alpha (y_{l}), \dots , \alpha (y_{m-1}), \nonumber \\&\qquad \qquad \,\,\,\,\, \{x_1, \dots , x_i, \dots , x_j, \dots , x_{n-1}, y_m\},\alpha (y_{m+1}), \dots , \alpha (y_n)\}\nonumber . \end{aligned}$$

This finishes the proof. $\square$

7.4 Hom-Modules over n-Hom-Lie Color Algebras

In this section we consider Hom-modules over n-Hom-Lie color algebras.

Definition 7.13

Let G be an abelian group. A Hom-module is a pair $(M,\alpha _M)$ in which M is a G-graded linear space and $\alpha _M: M\longrightarrow M$ is an even linear map.

Definition 7.14

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $(M, \alpha _M)$ a Hom-module. The Hom-module $(M, \alpha _M)$ is called an n-Hom-Lie module over L if there are n polylinear maps:

$$\omega _i : L\otimes \dots L\otimes \underbrace{M}_{i}\otimes L\otimes \dots \otimes L\rightarrow M, \quad i=1, 2, \dots , n$$

such that, for any $x_i, y_i\in \mathcal {H}(L)$ and $m\in \mathcal {H}(M)$,

• (a) $\omega _i(x_1, \dots , x_{i-1}, m, x_{i+1}, \dots , x_n)$ is a $\varepsilon$-skew-symmetric by all x-type arguments.

• (b) $\omega _i(x_1, \dots , x_{i-1}, m, x_{i+1}, \dots , x_n)=-\varepsilon (m, x_{i+1})\omega _i(x_1, \dots , x_{i-1}, x_{i+1}, m\dots , x_n)$ for $i=1, 2, \dots , n-1$.

• (c) $\omega _n(\alpha (x_1), \dots , \alpha (x_{n-1}), \omega _n(y_1, \dots , y_{n-1}, m))=$

  $$\begin{aligned}&=\sum _{i=1}^{n-1}\varepsilon (X, Y_i)\omega _n(\alpha (y_1), \dots , \alpha (y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \alpha (y_{i+1}), \dots , \alpha _M(m))\nonumber \\&\qquad +\varepsilon (X, Y_n)\omega _n(\alpha (y_1), \dots , \alpha (y_{n-1}), \omega _n(x_1, \dots , x_{n-1}, m)),\nonumber \end{aligned}$$

  where $x_i, y_j\in \mathcal {H}(L)$, $X=\sum _{i=1}^{n-1}x_i$, $Y_i=\sum _{j=1}^iy_{j-1}, y_0=e$ and $m\in \mathcal {H}(M)$.

• (d) $\omega _{n-1}(\alpha (x_1), \dots , \alpha (x_{n-2}), \alpha _M(m), [y_1, \dots , y_{n}])=$

  $$\begin{aligned}&=\sum _{i=1}^{n}\varepsilon (X, Y_i)\omega _i(\alpha (y_1), \dots , \alpha (y_{i-1}), \omega _{n-1}(x_1, \dots , x_{n-2}, m, y_i), \nonumber \\&\qquad \qquad \qquad \qquad \quad \quad \quad \,\,\, \alpha (y_{i+1}), \dots , \alpha (y_n)]\nonumber , \end{aligned}$$

  where $x_i, y_j\in \mathcal {H}(L)$, $X=\sum _{i=1}^{n-2}x_i+m$, $Y_i=\sum _{j=1}^iy_{j-1}, y_0=e$ and $m\in \mathcal {H}(M)$.

Example 7.4

Any n-Hom-Lie color algebra $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is an n-Hom-Lie module over itself by taking $M=L$, $\alpha _M=\alpha$ and $\omega _i(\cdot ,\dots , \cdot )=[\cdot , \dots , \cdot ]$.

Theorem 7.8

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra, $(M, \alpha _M, \omega _i)$ an n-Hom-Lie color module and $\beta : L\rightarrow L$ be an endomorphism. Define

$$\tilde{\omega }_i=(\beta , \dots , \beta , \underbrace{id}_i, \beta , \dots , \beta ), i=1, 2, \dots , n.$$

Then $(M, \alpha _M, \tilde{\omega }_i)$ is an n-Hom-Lie color module.

Proof

The items (a) and (b) are obvious. So we only prove (c), item (d) being proved similarly.

$$\begin{aligned}&\tilde{\omega }_n(\alpha (x_1), \dots , \alpha (x_{n-1}), \tilde{\omega }_n(y_1, \dots , y_{n-1}, m))=\nonumber \\&=\omega _n(\beta \alpha (x_1), \dots , \beta \alpha (x_{n-1}), \omega _n(\beta (y_1), \dots , \beta (y_{n-1}), m))\nonumber \\&=\omega _n(\alpha \beta (x_1), \dots , \alpha \beta (x_{n-1}), \omega _n(\beta (y_1), \dots , \beta (y_{n-1}), m))\nonumber \\&=\sum _{i=1}^{n-1}\varepsilon (X, Y_i)\omega _n(\alpha \beta (y_1), \dots , \alpha \beta (y_{i-1}), [\beta (x_1), \dots , \beta (x_{n-1}), \beta (y_i)], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \,\, \alpha \beta (y_{i+1}), \dots , \alpha _M(m)) \nonumber \\&\qquad +\varepsilon (X, Y_n)\omega _n(\alpha \beta (y_1), \dots , \alpha \beta (y_{n-1}), \omega _n(\beta (x_1), \dots , \beta (x_{n-1}), m))\nonumber \\&=\sum _{i=1}^{n-1}\varepsilon (X, Y_i)\omega _n(\beta \alpha (y_1), \dots , \beta \alpha (y_{i-1}), \beta ([x_1, \dots , x_{n-1}, y_i]), \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\, \beta \alpha (y_{i+1}), \dots , \alpha _M(m))\nonumber \\&\qquad +\varepsilon (X, Y_n)\omega _n(\beta \alpha (y_1), \dots , \beta \alpha (y_{n-1}), \omega _n(\beta (x_1), \dots , \beta (x_{n-1}), m)),\nonumber \\&=\sum _{i=1}^{n-1}\varepsilon (X, Y_i)\omega _n(\beta \otimes \dots \otimes \beta \otimes id)(\alpha (y_1), \dots , \alpha (y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \qquad \alpha (y_{i+1}), \dots , \alpha _M(m))\nonumber \\&\qquad +\varepsilon (X, Y_n)\omega _n(\beta \otimes \dots \otimes \beta \otimes id)\Big (id\otimes \dots \otimes id\otimes \omega _n(\beta \otimes \dots \otimes \beta \otimes id)\Big ) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad \qquad (\alpha (y_1), \dots , \alpha (y_{n-1}), x_1, \dots , x_{n-1}, m)\nonumber \\&=\sum _{i=1}^{n-1}\varepsilon (X, Y_i)\tilde{\omega }_n(\alpha (y_1), \dots , \alpha (y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \alpha (y_{i+1}), \dots , \alpha _M(m))\nonumber \\&\qquad +\varepsilon (X, Y_n)\tilde{\omega }_n(\alpha (y_1), \dots , \alpha (y_{n-1}), \tilde{\omega }_n(x_1, \dots , x_{n-1}, m))\nonumber . \end{aligned}$$

This ends the proof. $\square$

Corollary 7.10

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be an n-Hom-Lie color algebra and $\beta : L\rightarrow L$ be an endomorphism. Then $(L, \{\cdot , \dots , \cdot \}_i, \alpha )$, with

$$\{\cdot , \dots , \cdot \}_i=[\beta , \dots , \beta ,\underbrace{id}_i, \beta , \dots , \beta ], i=1, 2, \dots , n,$$

is an n-Hom-Lie color module.

Corollary 7.11

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. Then, for nay $k\ge 1$, $(L, \{\cdot , \dots , \cdot \}_i^k, \alpha )$ is an n-Hom-Lie color module, with

$$\{\cdot , \dots , \cdot \}_i^{k}=[\alpha ^k, \dots , \alpha ^k,\underbrace{id}_i, \alpha ^k, \dots , \alpha ^k], i=1, 2, \dots , n.$$

We end this section by giving some results for trivial graduation i.e. $G=\{e\}$.

Proposition 7.2

Let $(M, \alpha _M, \omega _i)$ be a module over the n-Hom-Lie algebra $(L, [\cdot , \dots , \cdot ], \alpha _L)$. Consider the direct sum of linear spaces $A=L\oplus M$. Let us define on A the bracket

• $\{x_1, \dots , x_n\}=[x_1, \dots , x_n]$,

• $\{x_1, \dots , x_{i-1}, \dots , m, x_{i+1}, \dots , x_n\}=\omega _i(x_1, \dots , x_{i-1}, \dots , m, x_{i+1}, \dots , x_n)$,

• $\{x_1, \dots , x_{i}, \dots , x_{j}, \dots , x_n\}=0$, whenever $x_i, x_j\in M$.

Then $(A, \alpha _A=\alpha _L+\alpha _M)$ is an n-Hom-Lie algebra.

Proposition 7.3

Let $(M_1, \alpha _M^1, \omega _i^1)$ and $(M_2, \alpha _M^2, \omega _i^2)$ be two modules over the n-Hom-Lie algebra $(L, [\cdot , \dots , \cdot ], \alpha )$. Then $(M, \alpha _M, \omega _i)$ is an n-Hom-Lie module with

$$M=M_1\oplus M_2, \quad \alpha _M=\alpha _M^1\oplus \alpha _M^2\quad \text{ and }\quad \omega _i=\omega _i^1\oplus \omega _i^2.$$

7.5 Generalized Derivation of Color Hom-Algebras and Their Color Hom-Subalgebras

This section is devoted to generalized derivation of color Hom-algebras and their color Hom-subalgebras.

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

Definition 7.15

For any $k\ge 1$, we call $D\in End(L)$ an $\alpha ^k$-derivation of degree d of the multiplicative n-Hom-Lie color algebra $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ if

$$\begin{aligned}&\alpha \circ D=D\circ \alpha , \end{aligned}$$

(7.5)

$$\begin{aligned}&D([x_1, \dots , x_n])= \\&\sum _{i=1}^n\varepsilon (d, X_{i})[\alpha ^k(x_1), \dots , \alpha ^k(x_{i-1}), D(x_i), \alpha ^k(x_{i+1}), \dots , \alpha ^k(x_n)]. \nonumber \end{aligned}$$

(7.6)

Example 7.5

([18]) The Laplacian of any Hom-Lie quasi-bialgebra $(\mathcal {G}, \mu , \gamma , \phi , \alpha )$ is an $\alpha ^2$-derivation of degree 0 of $(\Lambda \mathcal {G}, [\cdot , \cdot ]^{\mu , \alpha })$, i.e.

$$\begin{aligned} L([X, Y ]^{\mu , \alpha } )= [L(X), \alpha ^2(Y)]^{\mu , \alpha } + [\alpha ^2(X), L(Y)]^{\mu , \alpha },\quad \forall X , Y \in \Lambda \mathcal {G}. \quad \end{aligned}$$

(7.7)

Example 7.6

Now, Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. For any homogeneous elements $x_1, \dots , x_{n-1}$ of L and any integer $k\ge 1$, one defines the adjoint action of $\Lambda L$ on L by

$$\begin{aligned}&ad^{[\cdot , \dots , \cdot ], \alpha ^{k}}_{x_1, \dots , x_{n-1}}([y_1, y_2, \dots , y_n]) := \\&\qquad \,\, \displaystyle \sum _{i=1}^k\varepsilon (X, Y_{i})[\alpha ^k(y_1), \dots \alpha ^k(y_{i-1}), [x_1, \dots , x_{n-1}, y_i], \alpha ^k(y_{i+1}), \dots , \alpha (y_n)], \end{aligned}$$

for any $y_1,\dots ,y_n\in \mathcal {H}(L)$. Then $ad^{[\cdot , \dots , \cdot ], \alpha ^{k}}_{x_1, \dots , x_{n-1}}$ is an $\alpha ^k$-derivation of L of degree X. We call $ad^{[\cdot , \dots , \cdot ], \alpha ^{k}}_{x_1, \dots , x_{n-1}}$ an inner $\alpha ^k$-derivation. Denote by $Inn(L)=\oplus _{k\ge -1}Inn_{\alpha ^k}(L)$ the space of all inner $\alpha ^k$-derivation.

The following proposition is proved by a straightforward computation.

Proposition 7.4

Let D be an $\alpha ^k$-derivation of an n-Hom-Lie color algebra L and $\beta : L\rightarrow L$ an even endomorphism of L such that $D\circ \beta =\beta \circ D$. Then, for any non-negative integer s, $\Delta _s=D\circ \beta ^s : L\rightarrow L$ is a $(\beta ^s\alpha ^k)$-derivation.

Corollary 7.12

If D is an $\alpha ^{k}$-derivation of an n-Hom-Lie color algebra L. Then $\Delta _s$ is an $\alpha ^{k+s}$-derivation of L.

We denote the set of $\alpha ^k$-derivations of the multiplicative n-Hom-Lie color algebras L by $Der_{\alpha ^k}(L)$. For any $D\in Der_{\alpha ^k}(L)$ and $D'\in Der_{\alpha ^k}(L)$, let us define their commutator $[D, D']$ as usual:

$$[D, D']=D\circ D'-\varepsilon (d, d')D'\circ D.$$

Lemma 7.1

For any $D\in Der_{\alpha ^k}(L)$ and $D'\in Der_{\alpha ^k}(L)$,

$$[D, D']\in Der_{\alpha ^{k+s}}(L).$$

Denote by $Der(L)=\oplus _{k\ge -1}Der_{\alpha ^k}(L)$.

Proposition 7.5

$(Der(L), [\cdot , \cdot ], \omega )$ is a Hom-Lie color algebra, with $\omega (D)\,{=}\,D\circ ~\alpha$.

Definition 7.16

An endomorphism D of degree d of a multiplicative n-Hom-Lie color algebra $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is called a generalized $\alpha ^k$-derivation if there exist linear mappings $D', D'', \dots , D^{(n-1)}, D^{(n)}$ of degree d such that for any $x_1,\dots ,x_n\in \mathcal {H}(L)$:

$$\begin{aligned}&D\circ \alpha =\alpha \circ D\;\;\text{ and }\;\; D^{(i)}\circ \alpha =\alpha \circ D^{(i)},\end{aligned}$$

(7.8)

$$\begin{aligned}&\begin{array}{l} D^{(n)}([x_1, \dots , x_n])= \\ \displaystyle \sum _{i=1}^n\varepsilon (d, X_{i})[\alpha ^k(x_1), \dots , \alpha ^k(x_{i-1}), D^{(i-1)}(x_i), \alpha ^k(x_{i+1}), \dots , \alpha ^k(x_n)]. \end{array} \end{aligned}$$

(7.9)

An $(n{+}1)$-tuple $(D, D', D'', \dots , D^{(n-1)}, D^{(n)})$ is called an $(n{+}1)$-ary $\alpha ^k$-derivation.

The set of generalized $\alpha ^k$-derivation is denoted by $GDer_{\alpha ^k}$. Set

$$GDer(L)=\oplus _{k\ge -1}GDer_{\alpha ^k}(L).$$

Definition 7.17

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. A linear mapping $D\in End(L)$ is said to be an $\alpha ^k$-quasiderivation of degree d  if there exists a $D'\in End(L)$ of degree d such that

$$\begin{aligned}&D^{'}([x_1, \dots , x_n]) = \\&\sum _{i=1}^n\varepsilon (d, X_{i})[\alpha ^k(x_1), \dots , \alpha ^k(x_{i-1}), D(x_i), \alpha ^k(x_{i+1}), \dots , \alpha ^k(x_n)] \nonumber \end{aligned}$$

(7.10)

for all $x_1,\dots ,x_n\in \mathcal {H}(L)$

We call $D'$ the endomorphism associated to the $\alpha ^k$-quasiderivation D. The set of $\alpha ^k$-quasiderivations will be denoted QDer(L). Set $QDer(L)=\oplus _{k\ge -1}QDer_{\alpha ^k}(L)$.

Definition 7.18

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. The set $C_{\alpha ^k}(L)$ consisting of linear mapping D of degree d with the property

$$\begin{aligned}&D([x_1, \dots , x_n])= \\&\varepsilon (d, X_{i})[\alpha ^k(x_1), \dots , \alpha ^k(x_{i-1}), D(x_i), \alpha ^k(x_{i+1}), \dots , \alpha ^k(x_n)] \nonumber \end{aligned}$$

(7.11)

for all $x_1,\dots ,x_n\in \mathcal {H}(L)$, is called the $\alpha ^k$-centroid of L. 

We recover the definition of the centroid when $k=0$. 

Definition 7.19

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. The set $QC_{\alpha ^k}(L)$ consisting of linear mapping D of degree d with the property

$$\begin{aligned}&[D(x_1), \dots , x_n]= \\&\varepsilon (d, X_{i})[\alpha ^k(x_1), \dots , \alpha ^k(x_{i-1}), D(x_i), \alpha ^k(x_{i+1}), \dots , \alpha ^k(x_n)], \nonumber \end{aligned}$$

(7.12)

for all $x_1,\dots ,x_n\in \mathcal {H}(L)$, is called the $\alpha ^k$-quasicentroid of L. 

Definition 7.20

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. The set $ZDer_{\alpha ^k}(L)$ consisting of linear mappings D of degree d, such that for all $x_1,\dots ,x_n\in \mathcal {H}(L)$:

$$\begin{aligned}&D([x_1, \dots , x_n])= \\&\varepsilon (d, X_i)[\alpha ^k(x_1), \dots , \alpha ^k(x_{i-1}), D(x_i), \alpha ^k(x_{i+1}), \dots , \alpha ^k(x_n)]=0, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\, i=1, 2, \dots , n, , \nonumber \end{aligned}$$

(7.13)

is called the set of central $\alpha ^k$-derivations of L. 

It is easy to see that

$$ ZDer(L)\subseteq Der(L)\subseteq QDer(L)\subseteq GDer(L)\subseteq End(L). $$

Proposition 7.6

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra.

• (1) GDer(L), QDer(L), C(L) are color Hom-subalgebras of $(End (L), [\cdot , \cdot ], \omega )$:

  • (1a) $\omega (GDer(L))\subseteq GDer(L)$ and $[GDer(L), GDer(L)]\subseteq GDer(L)$.

  • (2b) $\omega (QDer(L))\subseteq QDer(L)$ and $[QDer(L), QDer(L)]\subseteq QDer(L)$.

  • (3c) $\omega (C(L))\subseteq C(L)$ and $[C(L), C(L)]\subseteq C(L)$.

• (2) ZDer(L) is a color Hom-ideal of Der(L): $\omega (ZDer(L))\subseteq ZDer(L)$ and $[ZDer(L), Der(L)]\subseteq ZDer(L)$.

Proof

(1a) Let us prove that if $D\in GDer(L)$, then $\omega (D)\in GDer(L)$. For any $x_1,\dots ,x_n\in \mathcal {H}(L)$,

$$\begin{aligned}&(\omega (D^{(n)}))([x_1, \dots , x_n])=(D^{(n)}\circ \alpha )([x_1, \dots , x_i, \dots , x_n]) \nonumber \\&=D^{(n)}([\alpha (x_1), \dots , \alpha (x_i), \dots , \alpha (x_n)])\nonumber \\&=\sum _{i=1}^n\varepsilon (d, X_{i})[\alpha ^{k+1}(x_1), \dots , \alpha ^{k+1}(x_{i-1}), D^{(i-1)}\alpha (x_i), \alpha ^{k+1}(x_{i+1}), \dots , \alpha ^{k+1}(x_n)]\nonumber \\&=\sum _{i=1}^n\varepsilon (d, X_{i})[\alpha ^{k+1}(x_1), \dots , \alpha ^{k+1}(x_{i-1}), ( D^{(i-1)}\circ \alpha )(x_i), \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \alpha ^{k+1}(x_{i+1}), \dots ,\alpha ^{k+1}(x_n)]\nonumber \\&=\sum _{i=1}^n\varepsilon (d, X_{i})[\alpha ^{k+1}(x_1), \dots , \alpha ^{k+1}(x_{i-1}), \omega (D^{(i-1)})(x_i), \alpha ^{k+1}(x_{i+1}), \dots , \alpha ^{k+1}(x_n)]\nonumber . \end{aligned}$$

This means that $\omega (D)$ is an $\alpha ^{k+1}$-derivation i.e. $\omega (D)\in GDer(L)$. Now let $D_1\in GDer_{\alpha ^{k}}(L)$ and $D_2\in GDer_{\alpha ^{s}}(L)$, we have

$$\begin{aligned}&(D_2^{(n)}D_1^{(n)})([x_1, \dots , x_n])=D_2^{(n)}(D_1^{(n)}([x_1, \dots , x_n]))=\nonumber \\&=\sum _{i=1}^n\varepsilon (d_1, X_{i})D_2^{(n)}([\alpha ^{k}(x_1), \dots , \alpha ^{k}(x_{i-1}), D_1^{(i-1)}(x_i), \alpha ^{k}(x_{i+1}), \dots , \alpha ^{k}(x_n)])\nonumber \\&=\sum _{i=1}^n\sum _{j<i}^n\varepsilon (d_1, X_{i})\varepsilon (d_2, X_{j}) ([\alpha ^{k+s}(x_1), \dots , D_2^{(j-1)}(x_j),\dots , \alpha ^{k+s}(x_{i-1}), D_1^{(i-1)}\alpha ^{s}(x_i), \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\, \alpha ^{k+s}(x_{i+1}), \dots , \alpha ^{k+s}(x_n)])\nonumber \\&+\sum _{i=1}^n\varepsilon (d_1+d_2, X_{i}) ([\alpha ^{k+s}(x_1), \dots , \alpha ^{k+s}(x_{i-1}), D_2^{(i-1)}D_1^{(i-1)}(x_i), \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\, \alpha ^{k+s}(x_{i+1}), \dots , \alpha ^{k}(x_n)])\nonumber \\&+\sum _{i=1}^n\sum _{j>i}^n\varepsilon (d_1, X_{i})\varepsilon (d_2, d_1+X_{j}) ([\alpha ^{k+s}(x_1), ,\dots , \alpha ^{k+s}(x_{i-1}), D_1^{(i-1)}\alpha ^{s}(x_i), \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \,\,\, \alpha ^{k+s}(x_{i+1}), \dots , D_2^{(j-1)}(x_j), \dots , \alpha ^{k+s}(x_n)])\nonumber . \end{aligned}$$

It follows that

$$\begin{aligned}&([D_1^{(n)}, D_2^{(n)}])([x_1, \dots , x_n])=(D_1^{(n)}D_2^{(n)}-\varepsilon (d_1, d_2)D_2^{(n)}(D_1^{(n)})([x_1, \dots , x_n]))=\nonumber \\&=\sum _{i=1}^n\varepsilon (d_1+d_2, X_{i}) ([\alpha ^{k+s}(x_1), \dots , \alpha ^{k+s}(x_{i-1}), \nonumber \\&\qquad \qquad (D_1^{(i-1)}D_2^{(i-1)}-\varepsilon (d_1, d_2)D_2^{(i-1)}D_1^{(i-1)}](x_i), \alpha ^{k+s}(x_{i+1}), \dots , \alpha ^{k+s}(x_n)])\nonumber \\&=\sum _{i=1}^n\varepsilon (d_1+d_2, X_{i}) ([\alpha ^{k+s}(x_1), \dots , \alpha ^{k+s}(x_{i-1}), \nonumber \\&\qquad \qquad [D_1^{(i-1)}, D_2^{(i-1)}](x_i), \alpha ^{k+s}(x_{i+1}), \dots , \alpha ^{k+s}(x_n)])\nonumber . \end{aligned}$$

Thus we obtain that $[D_1, D_2]\in GDer_{\alpha ^{k+s}}(L)$.

• (1b) That QDer(L) is a color Hom-subalgebra of $(End (L), [\cdot , \cdot ], \omega )$ is proved in the similar way.

• (1c) Let $D_1\in C_{\alpha ^{k}}(L)$ and $D_2\in C_{\alpha ^{s}}(L)$. Then

  $$\begin{aligned}&\omega (D_1)([x_1, x_2, \dots , x_n])=\alpha D_1([x_1, x_2, \dots , x_n])\nonumber \\&=\varepsilon (d_1, X_i)\alpha ([\alpha ^k(x_1), \alpha ^k(x_2), \dots , D_1(x_i), \dots , \alpha ^k(x_n])\nonumber \\&=\varepsilon (d_1, X_i)[\alpha ^{k+1}(x_1), \alpha ^{k+1}(x_2), \dots , D_1(x_i), \dots , \alpha ^{k+1}(x_n])\nonumber . \end{aligned}$$

  Thus $\omega (D)\in C_{\alpha ^{k+1}}(L)$. Moreover,

  $$\begin{aligned}&[D_1, D_2]([x_1, \dots , x_n])=D_1D_2([x_1, \dots , x_n])- \varepsilon (d_1, d_2)D_2D_1([x_1, \dots , x_n])\nonumber \\&=\varepsilon (d_2, X_i)D_1[\alpha ^k(x_1), \alpha ^k(x_2), \dots , D_2(x_i), \dots , \alpha ^k(x_n)]\nonumber \\&-\varepsilon (d_1, d_2)\varepsilon (d_1, X_i)D_2[\alpha ^s(x_1), \alpha ^s(x_2), \dots , D_1(x_i), \dots , \alpha ^s(x_n)]\nonumber \\&=\varepsilon (d_1+d_2, X_i)[\alpha ^{k+s}(x_1), \alpha ^{k+s}(x_2), \dots ,D_1 D_2(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&-\varepsilon (d_1+d_2, X_i)[\alpha ^{k+s}(x_1), \alpha ^{k+s}(x_2), \varepsilon (d_1, d_2)\dots ,D_2 D_1(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (d_1+d_2, X_i)[\alpha ^{k+s}(x_1), \alpha ^{k+s}(x_2), \dots , [D_1 D_2](x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber . \end{aligned}$$

  So, $[D_1, D_2]\in C_{\alpha ^{k+s}}(L)$ and finally $[D_1, D_2]\in C(L)$.

• (2) By the same method as previously one can show that $\omega (D)\in ZDer_{\alpha ^{k+1}}(L)$ and $[D_1, D_2]\in ZDer_{\alpha ^{k+s}}(L)$, where $D_1\in ZDer_{\alpha ^{k}}(L)$ and $D_2\in Der_{\alpha ^{s}}(L)$. $\square$

Proposition 7.7

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra.

• (1) If $\varphi \in C(L)$ and $D\in Der(L)$, then $\varphi D$ is a derivation i.e.

  $$C(L)\cdot Der(L)\subseteq Der(L).$$

• (2) Any element of centroid is a quasiderivation i.e.

  $$C(L)\subseteq QDer(L).$$

Proof

(1) For any $x_1, \dots , x_n\in \mathcal {H}(L)$,

$$\begin{aligned} \varphi D([x_1, \dots , x_n])= & {} \sum _{i=1}^n\varepsilon (d, X_i)\varphi ([\alpha ^k(x_1), \dots , D(x_i), \dots , \alpha ^k(x_n)])\nonumber \\= & {} \sum _{i=1}^n\varepsilon (d, X_i)\varepsilon (\varphi , X_i)[\alpha ^{k+s}(x_1), \dots , \varphi D(x_i), \dots , \alpha ^{k+s}(x_n)])\nonumber \\= & {} \sum _{i=1}^n\varepsilon (d+\varphi , X_i)[\alpha ^{k+s}(x_1), \dots , \varphi D(x_i), \dots , \alpha ^{k+s}(x_n)])\nonumber . \end{aligned}$$

Thus $\varphi D$ is an $\alpha ^{k+s}$-derivation of degree $d+\varphi$.

(2) Let D be an $\alpha ^{k}$-centroid, then for any $x_1, \dots , x_n\in \mathcal {H}(L),$

$$\begin{aligned} D([x_1, \dots , x_n])=\varepsilon (d, X_i)[\alpha ^k(x_1), \dots D(x_i), \dots , \alpha ^k(x_n)], i=1, 2, \dots , n.\qquad \quad \end{aligned}$$

(7.14)

It follows that

$$\begin{aligned} \sum _{i=1}^n\varepsilon (d, X_i)[\alpha ^k(x_1), \dots D(x_i), \dots , \alpha ^k(x_n)]=nD([x_1, \dots , x_n]). \end{aligned}$$

(7.15)

It suffises to take $D'=nD$. $\square$

Lemma 7.2

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ be a multiplicative n-Hom-Lie color algebra. Then

• (1) The $\varepsilon$-commutator of two elements of quasicentroid is a quasiderivation i.e.

  $$[QC(L), QC(L)]\subseteq QDer(L).$$

• (2) $QDer(L)+QC(L)\subseteq GDer(L)$.

Proof

For any $x_1, x_2, \dots , x_n\in \mathcal {H}(L)$,

• (1) Let $D_1\in QC_{\alpha ^k}(L)$ and $D_2\in QC_{\alpha ^s}(L)$. We have, on the one hand

  $$\begin{aligned}&[D_1D_2(x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (D_1, D_2+X_i)[D_2(\alpha ^{k}(x_1)), \alpha ^{k+s}(x_2), \dots , D_1(\alpha ^{s}(x_i)), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (D_1, D_2+X_i)\varepsilon (D_2, X_i)[\alpha ^{k+s}(x_1), \dots , D_2D_1(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (D_1, D_2)\varepsilon (D_1+D_2, X_i)[\alpha ^{k+s}(x_1), \dots , D_2D_1(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber . \end{aligned}$$

  On the other hand,

  $$\begin{aligned}&[D_1D_2(x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_n)]=\nonumber \\&=\varepsilon (D_1, D_2+x_1)[D_2(\alpha ^{k}(x_1)), D_1(\alpha ^{s}(x_2)), \dots , \alpha ^{k+s}(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (D_1, D_2+x_1)\varepsilon (D_2, D_1+X_i) \nonumber \\&\qquad \qquad \quad \qquad \,\,\,\, [\alpha ^{k+s}(x_1), D_1(\alpha ^{s}(x_2)), \dots , D_2(\alpha ^{k}(x_i)), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (D_1, x_1)\varepsilon (D_2, X_i)\varepsilon (x_1, D_1) \nonumber \\&[D_1(\alpha ^{s}(x_1)), \alpha ^{k+s}(x_2), \dots , D_2(\alpha ^{k}(x_i)), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (D_2, X_i)\varepsilon (D_1, X_i)[\alpha ^{k+s}(x_1), \dots , D_1D_2(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=\varepsilon (D_1+D_2, X_i)[\alpha ^{k+s}(x_1), \dots , D_1D_2(x_i), \dots , \alpha ^{k+s}(x_n)],\nonumber \end{aligned}$$

  and so

  $$\begin{aligned}&\varepsilon (D_1+D_2, X_i)[\alpha ^{k+s}(x_1), \dots , [D_1, D_2](x_i), \dots , \alpha ^{k+s}(x_n)]=\nonumber \\&=\varepsilon (D_1+D_2, X_i)\Big ([\alpha ^{k+s}(x_1), \dots , D_1D_2(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&\quad -\varepsilon (D_1, D_2)[\alpha ^{k+s}(x_1), \dots , D_2D_1(x_i), \dots , \alpha ^{k+s}(x_n)]\Big )=0.\nonumber \end{aligned}$$

  It follows that

  $$\begin{aligned} \sum _{i=1}^n\varepsilon (D_1+D_2, X_i)[\alpha ^{k+s}(x_1), \dots , [D_1, D_2](x_i), \dots , \alpha ^{k+s}(x_n)]=0\nonumber . \end{aligned}$$

  Therefore $D'\equiv 0$, and $[D_1, D_2]\in QDer(L)$.

• (2) Let $D_1\in QDer_{\alpha ^k}(L)$ and $D_2\in QC_{\alpha ^k}(L)$ with $|D_1|=|D_2|$. Then there exists $D'_1\in End(L)$ such that

  $$\begin{aligned}&D'_1([x_1, \dots , x_n]) =\sum _{i=1}^n \varepsilon (D_1, X_i)[\alpha ^{k}(x_1), \dots , D_1(x_i), \dots , \alpha ^{k}(x_n)]\nonumber \\&=[D_1(x_1), \alpha ^{k}(x_2), \dots , \alpha ^{k}(x_n)]+\varepsilon (D_1, x_1)[\alpha ^{k}(x_1), D_1(x_2), \dots , \alpha ^{k}(x_n)]\nonumber \\&\qquad +\sum _{i=3}^n\varepsilon (D_1, X_i)[\alpha ^{k}(x_1), \dots , D_1(x_i), \dots , \alpha ^{k}(x_n)]\nonumber \\&=[(D_1+D_2)(x_1), \alpha ^{k}(x_2), \dots , \alpha ^{k}(x_n)]-[D_2(x_1), \alpha ^{k}(x_2), \dots , \alpha ^{k}(x_n)]\nonumber \\&\qquad +\varepsilon (D_1, x_1)[\alpha ^{k}(x_1), D_1(x_2), \dots , \alpha ^{k}(x_n)] \nonumber \\&\qquad + \sum _{i=3}^n\varepsilon (D_1, X_i)[\alpha ^{k}(x_1), \dots , D_1(x_i), \dots , \alpha ^{k}(x_n)]\nonumber \\&=[(D_1+D_2)(x_1), \alpha ^{k}(x_2), \dots , \alpha ^{k}(x_n)]-\varepsilon (D_2, x_1)[\alpha ^{k}(x_1), D_2(x_2), \dots , \alpha ^{k}(x_n)]\nonumber \\&\quad +\varepsilon (D_1, x_1)[\alpha ^{k}(x_1), D_1(x_2), \dots , \alpha ^{k}(x_n)] \nonumber \\&\qquad + \sum _{i=3}^n\varepsilon (D_1, X_i)[\alpha ^{k}(x_1), \dots , D_1(x_i), \dots , \alpha ^{k}(x_n)]\nonumber \\&=[(D_1+D_2)(x_1), \alpha ^{k}(x_2), \dots , \alpha ^{k}(x_n)] \nonumber \\&\qquad +\varepsilon (D_2, x_1)[\alpha ^{k}(x_1), (D_1-D_2)(x_2), \dots , \alpha ^{k}(x_n)] \nonumber \\&\qquad +\sum _{i=3}^n\varepsilon (D_1, X_i)[\alpha ^{k}(x_1), \dots , D_1(x_i), \dots , \alpha ^{k}(x_n)]\nonumber . \end{aligned}$$

The conclusion follows by taking

$$ D^{(n)}=D'_1,\quad D=D_1+D_2,\quad D'=D_1-D_2,\quad D^{(i)}=D_1,\quad 2\le i\le n-1.$$

This proved that $D_1+D_2\in GDe(L)$. $\square$

Proposition 7.8

If $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is a multiplicative n-Hom-Lie color algebra, then

$$QC(L)+[QC(L), QC(L)]$$

is a color Hom-subalgebra of GDer(L).

Proof

It follows from Lemma 7.2 by using the same arguments as in Proposition 2.4 in [41]. $\square$

Proposition 7.9

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is a multiplicative n-Hom-Lie color algebra such that $\alpha$ be a surjective mapping, then $[C(L), QC(L)]\subseteq Hom(L, Z(L))$. Moreover, if $Z(L)=\{0\}$, then $[C(L), QC(L)]=\{0\}$.

Proof

Let $D_1\in C_{\alpha ^k}(L)$, $D_2\in QC_{\alpha ^s}(L)$ and $x_1, \dots , x_n\in \mathcal {H}(L)$. Since $\alpha$ is surjective, for any $y'_i\in L$, there exists $y_i\in L$ such that $y'_i=\alpha ^{k+s}(y_i), i= 2, \dots , n$. Thus

$$\begin{aligned}&[[D_1, D_2](x_1), y'_2, \dots , y'_n]=\nonumber \\&=[[D_1, D_2](x_1), \alpha ^{k+s}(y_2), \dots , \alpha ^{k+s}(y_n)]\nonumber \\&=[D_1D_2(x_1), \alpha ^{k+s}(y_2), \dots , \alpha ^{k+s}(y_n)] \nonumber \\&\qquad -\varepsilon (d_1, d_2)[D_2D_1(x_1), \alpha ^{k+s}(y_2), \dots , \alpha ^{k+s}(y_n)]\nonumber \\&=D_1([D_2(x_1), \alpha ^{s}(y_2), \dots , \alpha ^{s}(y_n)]) \nonumber \\&\qquad -\varepsilon (d_1, d_2)\varepsilon (d_2,x_1+ d_1)[D_1\alpha ^{s}(x_1), D_2\alpha ^{k}(y_2), \dots , \alpha ^{k+s}(y_n)]\nonumber \\&=D_1([D_2(x_1), \alpha ^{k+s}(y_2), \dots , \alpha ^{k+s}(y_n)]) \nonumber \\&\qquad -\varepsilon (d_2, x_1)D_1[\alpha ^{s}(x_1), \alpha ^{s}D_2(y_2), \dots , \alpha ^{s}(y_n)]\nonumber \\&=D_1\Big ([D_2(x_1), \alpha ^{k+s}(y_2), \dots , \alpha ^{k+s}(y_n)]) \nonumber \\&\qquad -\varepsilon (d_2, x_1)[\alpha ^{s}(x_1), \alpha ^{s}D_2(y_2), \dots , \alpha ^{s}(y_n)]\Big )\nonumber \\&=D_1\Big ([D_2(x_1), \alpha ^{k+s}(y_2), \dots , \alpha ^{k+s}(y_n)]) -[D_2(x_1), \alpha ^{s}(y_2), \dots , \alpha ^{s}(y_n)]\Big )=0.\nonumber \end{aligned}$$

Hence, $[D_1, D_2](x_1)\in Z(L)$, and $[D_1, D_2]\in Hom(L, Z(L))$. Furthermore, if $Z(L)=\{0\}$, we know that $[C(L), QC(L)]=\{0\}$. $\square$

Proposition 7.10

Let $(L, [\cdot , \dots , \cdot ], \varepsilon , \alpha )$ is a multiplicative n-Hom-Lie color algebra with surjective twisting $\alpha$ and H be a graded subset of L. Then

• (i) $Z_L(H)$ is invariant under C(L).

• (ii) Every perfect color Hom-ideal of L is invariant under C(L).

Proof

• (i) For any $\varphi \in C(L)$ and $x\in Z_L(H)$, by (7.4), we have

  $$\begin{aligned} 0=\varphi ([x, H, L, \dots , L])=[\varphi (x), \alpha ^k(H), \alpha ^k(L), \dots , \alpha ^k(L)]=[\varphi (x), H, L, \dots , L].\nonumber \end{aligned}$$

  Therefore $\varphi (x)\in Z_L(H)$, which implies that $Z_L(H)$ is invariant under C(L).

• (ii) Let H be a perfect color Hom-ideal of L. Then $H^1=H$, and so for any $x\in H$ there exist $x_1^i, x_2^i, \dots , x_n^i\in H$ with $0<i<\infty$ such that $x{=}\sum _i[x_1^i, x_2^i, \dots , x_n^i]$. If $\varphi \in C(L)$, then

  $$\begin{aligned} \varphi (x)= & {} \varphi \left( \sum _i[x_1^i, x_2^i, \dots , x_n^i]\right) =\sum _i\varphi ([x_1^i, x_2^i, \dots , x_n^i]) \nonumber \\= & {} \sum _i[\varphi (x_1^i), \alpha ^k(x_2^i), \dots , \alpha ^k(x_n^i)])\in H. \end{aligned}$$

  This shows that H is invariant under C(L). $\square$

Proposition 7.11

If the characteristic of $\mathbb {K}$ is 0 or not a factor of $n-1$. Then

$$ZDer(L)=C(L)\cap Der(L).$$

Proof

If $\varphi \in C(L)\cap Der(L)$, then by (7.6) we have

$$\varphi ([x_1, \dots , x_n])=\sum _{i=1}^n\varepsilon (d, X_i)[\alpha ^k(x_1), \dots , \varphi (x_i), \dots , \alpha ^k(x_n)],$$

and by (), for $i=1, 2, \dots , n$,

$$\varepsilon (d, X_i)[\alpha ^k(x_1), \dots , \varphi (x_i), \dots , \alpha ^k(x_n)]=\varphi ([x_1, \dots , x_n]).$$

Thus

$$\varphi ([x_1, \dots , x_n])=n \varphi ([x_1, \dots , x_n])$$

The characteristic of $\mathbb {K}$ being 0 or not a factor of $n-1$, we have

$$0=\varphi ([x_1, \dots , x_n])=\varepsilon (d, X_i)[\alpha ^k(x_1), \dots , \varphi (x_i), \dots , \alpha ^k(x_n)], i=1, 2, \dots , n.$$

Which means that $\varphi \in ZDer(L)$.

Conversly, let $\varphi \in ZDer(L)$, Then

$$\varphi ([x_1, \dots , x_n])=\varepsilon (d, X_i)[\alpha ^k(x_1), \dots , \varphi (x_i), \dots , \alpha ^k(x_n)]=0, 1\le i\le n$$

and thus $\varphi \in C(L)\cap Der(L)$. Therefore $ZDer(L)=C(L)\cap Der(L)$. $\square$

Proposition 7.12

Let L be an n-Hom-Lie color algebra. For any $D\in Der(L)$ and $\varphi \in C(L)$

• (1) Der(L) is contained in the normalizer of C(L) in End(L) i.e.

  $$[Der(L), C(L)]\subseteq C(L).$$

• (2) QDer(L) is contained in the normalizer of QC(L) in End(L) i.e.

  $$[QDer(L), QC(L)]\subseteq QC(L).$$

Proof

(1) For any $D\in Der(L), \varphi \in C(L)$ and $x_1, x_2, \dots , x_n\in \mathcal {H}(L)$,

$$\begin{aligned}&D\varphi ([x_1, \dots , x_n])=D([\varphi (x_1), \alpha ^{k}(x_2), \dots , \alpha ^{k}(x_i), \dots , \alpha ^{k}(x_n)])\nonumber \\&=[D\varphi (x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&\quad +\sum _{i=2}^n\varepsilon (d, \varphi + X_i)[\alpha ^{s}\varphi (x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k}D(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&=[D\varphi (x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&\quad +\sum _{i=2}^n\varepsilon (d,\varphi + X_i)\varepsilon (\varphi , X_i)([\alpha ^{k+s}(x_1), \alpha ^{k+s}(x_2), \dots , \varphi D(x_i), \dots , \alpha ^{k+s}(x_n)])\nonumber \\= & {} [D\varphi (x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&\quad +\varepsilon (d,\varphi )\sum _{i=2}^n\varepsilon (d+\varphi , X_i)([\alpha ^{k+s}(x_1), \alpha ^{k+s}(x_2), \dots , \varphi D(x_i), \dots , \alpha ^{k+s}(x_n)])\nonumber \\= & {} [D\varphi (x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_i), \dots , \alpha ^{k+s}(x_n)]\nonumber \\&\quad +\varepsilon (d,\varphi )\Big (\varphi D[x_1, x_2, \dots , x_i, \dots , x_n]\nonumber \\&\quad -[\varphi D(x_1), \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_i), \dots , \alpha ^{k+s}(x_n)] \Big )\nonumber . \end{aligned}$$

Then we get

$$\begin{aligned}&(D\varphi -\varepsilon (d, \varphi )\varphi D)([x_1, \dots , x_n]) \nonumber \\&\qquad =[(D\varphi -\varepsilon (d, \varphi )\varphi D)(x_1), \dots , \alpha ^{k+s}(x_2), \dots , \alpha ^{k+s}(x_i), \dots , \alpha ^{k+s}(x_n)])\nonumber , \end{aligned}$$

that is $[D, \varphi ]=D\varphi -\varepsilon (d, \varphi )\varphi D\in C(L)$.

(2) It is proved by using a similar method. $\square$

Proposition 7.13

Let L be an n-Hom-Lie color algebra. For any $D\in Der(L)$ and $\varphi \in C(L)$

• (1) $D\varphi$ is contained in C(L) if and only if $\varphi D$ is a central derivation of L.

• (2) $D\varphi$ is a derivation of L if and only if $[D, \varphi ]$ is a central derivation of L.

Proof

(1) From Proposition 7.12, $D\varphi$ is an element of C(L) if and only if $\varphi D\in Der(L)\cap C(L)$. Thanks to Proposition 7.11, we get the result.

(2) The conclusion follows from (1), Propositions 7.11 and 7.12. $\square$

If A is a commutative associative algebra and L is an n-Hom-Lie color algebra, the n-Hom-Lie algebra $A\otimes L$ (Theorem 7.3) is called the tensor product n-Hom-Lie color algebra of A and L. For $f\in End(A)$ and $\varphi \in End(L)$ let $f\otimes \varphi : A\otimes L\rightarrow A\otimes L$ be given by $f\otimes \varphi (a\otimes x)=f(a)\otimes \varphi (x)$, for $a\in A, x\in L$. Then $f\otimes \varphi \in End(A\otimes L)$.

Recall that if A is a commutative associative algebra, the centroid C(A) of A is by definition

$$C(A)=\{f\in End(A) \mid f(ab)=f(a)b=af(b), \forall a, b\in A\}.$$

We now state the following

Proposition 7.14

By the above notation, we have

$$C(A)\otimes C(L)\subseteq C(A\otimes L).$$

Proof

For any $a_i\in A, x_i\in \mathcal {H}(L), 1\le i\le n$, and any $f\in C(A)$ and $\varphi \in C(L),$

$$\begin{aligned}&(f\otimes \varphi )[a_1\otimes x_1, \dots , a_n\otimes x_n]=(f\otimes \varphi )(a_1\dots a_n)\otimes [x_1, \dots , x_n]\nonumber \\&=f(a_1\dots a_n)\otimes \varphi [x_1, \dots , x_n]\nonumber \\&=\varepsilon (\varphi , X_i)a_1\dots f(a_i)\dots a_n\otimes [\alpha ^{k}(x_1), \dots , \varphi (x_i), \dots , \alpha ^{k}(x_n)]\nonumber \\&=\varepsilon (\varphi , X_i)[a_1\otimes \alpha ^{k}(x_1)\dots f(a_i)\otimes \varphi (x_i), \dots , a_n\otimes \alpha ^{k}(x_n)]\nonumber \\&=\varepsilon (\varphi , X_i)[\alpha '^{k}(a_1\otimes x_1)\dots (f\otimes \varphi )( a_i\otimes x_i), \dots , \alpha '^{k}(a_n\otimes x_n)]\nonumber . \end{aligned}$$

Therefore, $f\otimes \varphi \in C(A\otimes L)$. $\square$