Some Constructions of Multiplicative $\varvec{n}$-ary hom–Nambu Algebras

Abstract

We show that given a hom–Lie algebra one can construct the n-ary hom–Lie bracket by means of an $(n-2)$-cochain of the given hom–Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov–Jacobi identity, thereby inducing the structure of n-hom–Lie algebra. We introduce the notion of a hom–Lie n-tuple system which is the generalization of a hom–Lie triple system. We construct hom–Lie n-tuple system using a hom–Lie algebra.

1 Introduction

The first instance of n-ary algebras in Physics appeared with a generalization of the Hamiltonian mechanics proposed in 1973 by Nambu [23]. More recent motivation comes from string theory and M-branes involving naturally an algebra with ternary operation called Bagger–Lambert algebra which gives impulse to a significant development. It was used in [7] as one of the main ingredients in the construction of a new type of supersymmetric gauge theory that is consistent with all the symmetries expected of a multiple M2-brane theory: 16 supersymmetries, conformal invariance, and SO(8) R-symmetry acting on the eight transverse scalars. On the other hand, in the study of supergravity solutions describing M2-branes ending on M5-branes, the Lie algebra appearing in the original Nahm equations has to be replaced with a generalization involving a ternary bracket in the lifted Nahm equations (see [8]).

In [6], generalizations of n-ary algebras of Lie type and associative type by twisting the identities using linear maps were introduced. The notions of representations, derivations, cohomology and deformations were studied in [3, 12, 15, 21, 24]. These generalizations include n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-Lie algebras (called also n-ary Nambu–Lie algebras) and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. In [4], a method was demonstrated how to construct ternary multiplications from the binary multiplication of a hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions; and it was shown that this method can be used to construct ternary hom–Nambu–Lie algebras from hom–Lie algebras. This construction was generalized to n-Lie algebras and n-hom–Nambu–Lie algebras in [5].

It is well known that algebras of derivations and generalized derivations are very important in the study of Lie algebras and its generalizations. The notion of $\delta$-derivation appeared in the paper of Filippov [14]. The results for $\delta$-derivations and generalized derivations were studied by many authors. For example, Zhang and Zhang [26] generalized the above results to the case of Lie superalgebras; Chen, Ma, Ni and Zhou considered the generalized derivations of color Lie algebras, hom–Lie superalgebras and Lie triple systems [10, 11]. Derivations and generalized derivations of n-ary algebras were considered in [17, 18] and other papers. In [9], the authors generalize these results in the color n-ary hom–Nambu case.

This paper is organized as follows. In Sect. 1, we review some basic concepts of hom–Lie algebras, n-ary hom–Nambu algebras and n-hom–Lie algebras. We also recall the definitions of derivations, $\alpha ^k$-derivations, $\alpha ^k$-quasiderivations and $\alpha ^k$-centroid. In Sect. 2, we provide a construction procedure of n-hom–Lie algebras starting from a binary bracket of a hom–Lie algebra and multilinear form satisfying certain conditions. To this end, we give the relation between $\alpha ^k$-derivations, (resp. $\alpha ^k$-quasiderivations and $\alpha ^k$-centroid) of hom–Lie algebras and $\alpha ^k$-derivations (resp. $\alpha ^k$-quasiderivations and $\alpha ^k$-centroid) of n-hom–Lie algebras. In Sect. 3, we introduce the notion of a hom–Lie n-tuple system which is the generalization of a Lie n-tuple system which is introduced in [13]. We construct a hom–Lie n-tuple system using a hom–Lie algebra. Finally, we give a relation between $\alpha ^k$-quasiderivations of a hom–Lie algebra and $(n + 1)$-ary $\alpha ^k$-derivations of associated hom–Lie n-tuple system.

2 hom–Lie Algebra and n-ary hom–Nambu Algebras

Throughout this paper, we will, for simplicity of exposition, assume that $\mathbb {K}$ is an algebraically closed field of characteristic zero, even though, for most of the general definitions and results in the paper, this assumption is not essential.

2.1 Definitions

The notion of a hom–Lie algebra was initially motivated by examples of deformed Lie algebras coming from twisted discretizations of vector fields (see [16, 19]). We will follow notation conventions in [22].

Definition 1.1

A hom–Lie algebra is a triple $({\mathfrak {g}}, [~,~],\alpha )$, where $[~ ,~]:{\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}$ is a bilinear map and $\alpha :{\mathfrak {g}}\rightarrow {\mathfrak {g}}$ a linear map satisfying

$$\begin{aligned}&{[}x,y] = - [y,x],\,\, \text {(skew-symmetry)} \\&\displaystyle \circlearrowleft _{x,y,z}[\alpha (x),[y,z]]=0,\,\, \text {(hom--Jacobi}~ \text {condition)} \end{aligned}$$

for all x, y, z from ${\mathfrak {g}}$, where $\displaystyle \circlearrowleft _{x,y,z}$ denotes summation over the cyclic permutations of x, y, z.

Definition 1.2

A hom–Lie algebra $(\mathfrak {g},[~,~],\alpha )$ is called multiplicative if $\alpha ([x,y])=[\alpha (x),\alpha (y)]$ for all $x,y\in \mathfrak {g}$.

We define a linear map $ad:\mathfrak {g}\rightarrow End(\mathfrak {g})$ by $\text {ad}_x(y)=[x,y]$. Thus, the hom–Jacobi identity is equivalent to

$$\begin{aligned} \text {ad}_{[x,y]}(\alpha (z))=\text {ad}_{\alpha (x)}\circ \text {ad}_y(z)- \text {ad}_{\alpha (y)}\circ \text {ad}_x(z),\quad {\text {for}}\ {\text {all}} \ x,y,z\in \mathfrak {g}. \end{aligned}$$

(1.1)

Remark 1.3

An ordinary Lie algebra is a hom–Lie algebra when $\alpha =id$.

Example 1.4

Let $\mathcal {A}$ be the complex algebra where $\mathcal {A}= \mathbb {C}[t, t^{-1}]$ is the ring of Laurent polynomials in one variable. The generators of $\mathcal {A}$ are of the form of $t^n$ for $n \in \mathbb {Z}$.

Let $q\in \mathbb {C}\backslash \{0, 1\}$ and $n\in \mathbb {N}$, we set $\{n\} = \frac{1-q^n}{1-q}$, a q-number. The q-numbers have the following properties: $\{n + 1\} = 1 + q\{n\} = \{n\} + q^n$ and $\{n + m\} = \{n\} + q^n\{m\}$.

Let $\mathfrak {A}_q$ be a space with basis $\{L_m,\ I_m| m\in \mathbb {Z}\}$ where $L_m=-t^mD,\ I_m=-t^m$ and D is a q-derivation on $\mathcal {A}$ such that

$$\begin{aligned} D(t^m)=\{m\}t^m. \end{aligned}$$

We define the bracket $[\ ,\ ]_q:\mathfrak {A}_q\times \mathfrak {A}_q\longrightarrow \mathfrak {A}_q$, with respect to the super-skew-symmetry for $n,m\in \mathbb {Z}$ by

$$\begin{aligned} {[}L_m,L_n]_q= & {} (\{m\}-\{n\})L_{m+n}, \end{aligned}$$

(1.2)

$$\begin{aligned} {[}L_m,I_n]_q= & {} -\{n\}I_{m+n}, \end{aligned}$$

(1.3)

$$\begin{aligned} {[}I_m,I_n]_q= & {} 0. \end{aligned}$$

(1.4)

Let $\alpha$ be an even linear map on $\mathfrak {A}_q$ defined on the generators by

$$\begin{aligned} \alpha _q(L_n)= & {} (1+q^n)L_n,\quad \alpha _q(I_n)=(1+q^n)I_n, \end{aligned}$$

The triple $(\mathfrak {A}_q, [\ ,\ ]_q, \alpha _q)$ is a hom–Lie algebra, called the q-deformed Heisenberg–Virasoro algebra of hom-type.

Example 1.5

We consider the matrix construction of the algebra ${\mathfrak {sl}}_2(\mathbb {R})$ generated by the following three vectors:

$$\begin{aligned}H=\left( \begin{array}{cc} 1 &{}\quad 0 \\ 0 &{}\quad -1 \\ \end{array} \right) ;\quad X=\left( \begin{array}{cc} 0 &{}\quad 1 \\ 0 &{}\quad 0 \\ \end{array} \right) ;\quad Y=\left( \begin{array}{cc} 0 &{}\quad 0 \\ 1 &{}\quad 0 \\ \end{array} \right) \end{aligned}$$

The defining relations are

$$\begin{aligned} {[}H,X]=2X;\quad [H,Y]=-2Y;\quad [X,Y]=H. \end{aligned}$$

Let $\lambda \in \mathbb {R}^*$ and consider the linear maps $\alpha _{\lambda }:{\mathfrak {sl}}_2(\mathbb {R})\rightarrow {\mathfrak {sl}}_2(\mathbb {R})$ defined by:

$$\begin{aligned} \alpha _{\lambda }(H)=H;\quad \alpha _{\lambda }(X)=\lambda ^2X;\quad \alpha _{\lambda }(Y)=\frac{1}{\lambda ^2}Y. \end{aligned}$$

Note that $\alpha _{\lambda }$ is a Lie algebra automorphism.

In [2], the authors have shown that $({\mathfrak {sl}}_2(\mathbb {R}))_\lambda =({\mathfrak {sl}}_2(\mathbb {R}),[\ ,\ ]_{\alpha _\lambda },\alpha _\lambda )$ is a family of multiplicative hom–Lie algebras where the hom–Lie bracket $[\ ,\ ]_{\alpha _\lambda }$ on the basis elements is given, for $\lambda \ne 0$, by

$$\begin{aligned} {[}H,X]_{\alpha _\lambda }=2\lambda ^2X;\quad [H,Y]_{\alpha _\lambda }=-\frac{2}{\lambda ^2}Y;\quad [X,Y]_{\alpha _\lambda }=H. \end{aligned}$$

Now, we recall the definitions of n-ary hom–Nambu algebras and n-ary hom–Nambu–Lie algebras, generalizing n-ary Nambu algebras and n-ary Nambu–Lie algebras (also called Filippov algebras), respectively, which were introduced by Ataguema et al. [6].

Definition 1.6

An n-ary hom–Nambu algebra $(\mathcal {N}, [\ ,\ldots , \ ], \widetilde{\alpha } )$ consists of a vector space $\mathcal {N}$, an n-linear map $[\ ,\ldots , \ ] : \mathcal {N}^{ n}\longrightarrow \mathcal {N}$ and a family $\widetilde{\alpha }=(\alpha _i)_{1\le i\le n-1}$ of linear maps $\alpha _i:\ \ \mathcal {N}\longrightarrow \mathcal {N}$, satisfying

$$\begin{aligned}&\big [\alpha _1(x_1),\ldots ,\alpha _{n-1}(x_{n-1}),[y_1,\ldots ,y_{n}]\big ] \nonumber \\&\quad = \sum _{i=1}^{n}\big [\alpha _1(y_1),\ldots ,\alpha _{i-1}(y_{i-1}),[x_1,\ldots ,x_{n-1},y_i] ,\alpha _i(y_{i+1}),\ldots ,\alpha _{n-1}(y_n)\big ],\nonumber \\ \end{aligned}$$

(1.5)

for all $(x_1,\ldots , x_{n-1})\in \mathcal {N}^{ n-1}$, $(y_1,\ldots , y_n)\in \mathcal {N}^{ n}.$

The identity (1.5) is called the hom–Nambu identity.

Let $X=(x_1,\ldots ,x_{n-1})\in \mathcal {N}^{n-1}$, $\widetilde{\alpha } (X)=(\alpha _1(x_1),\ldots ,\alpha _{n-1}(x_{n-1}))\in \mathcal {N}^{n-1}$ and $y\in \mathcal {N}$. We define an adjoint map $\text {ad}(X)$ as a linear map on $\mathcal {N}$, such that

$$\begin{aligned} \text {ad}_X(y)=[x_{1},\ldots ,x_{n-1},y]. \end{aligned}$$

(1.6)

Then, the hom–Nambu identity (1.5) may be written in terms of the adjoint map as

$$\begin{aligned}&\text {ad}_{\widetilde{\alpha } (X)}( [x_{n},\ldots ,x_{2n-1}])\\&\quad = \sum _{i=n}^{2n-1}{[\alpha _1(x_{n}),\ldots ,\alpha _{i-n}(x_{i-1}), \text {ad}_X(x_{i}), \alpha _{i-n+1}(x_{i+1}) \ldots ,\alpha _{n-1}(x_{2n-1})].} \end{aligned}$$

Definition 1.7

An n-ary hom–Nambu algebra is a triple $(\mathcal {N}, [\ ,\ldots , \ ], \widetilde{\alpha } )$ that is called n-hom–Lie algebra if the bracket $[\ ,\ldots , \ ]$ is skew-symmetric, i.e $[x_{\sigma (1)},\ldots ,x_{\sigma (n)}]=(-1)^{sign(\sigma )}[x_{1},\ldots ,x_{n}]$ for $\sigma \in S_n$.

Remark 1.8

When the maps $(\alpha _i)_{1\le i\le n-1}$ are all identity maps, one recovers the classical n-ary Nambu algebras. The hom–Nambu identity (1.5), for $n=2$, corresponds to the hom–Jacobi identity (see [22]), which reduces to the Jacobi identity when $\alpha _1=id$.

Let $(\mathcal {N},[\ ,\dots ,\ ],\widetilde{\alpha })$ and $(\mathcal {N}',[\cdot ,\dots ,\cdot ]',\widetilde{\alpha }')$ be two n-ary hom–Nambu algebras where $\widetilde{\alpha }=(\alpha _{i})_{i=1,\ldots ,n-1}$ and $\widetilde{\alpha }'=(\alpha '_{i})_{i=1,\ldots ,n-1}$. A linear map $f: \mathcal {N}\rightarrow \mathcal {N}'$ is an n-ary hom–Nambu algebra morphism if it satisfies

$$\begin{aligned} f ([x_{1},\ldots ,x_{2n-1}])= & {} [f (x_{1}),\ldots ,f (x_{2n-1})]'\\ f \circ \alpha _i= & {} \alpha '_i\circ f \quad \forall i=1,\ldots ,n-1. \end{aligned}$$

In the sequel, we deal sometimes with a particular class of n-ary hom–Nambu algebras which we call n-ary multiplicative hom–Nambu algebras.

Definition 1.9

A multiplicative n-ary hom–Nambu algebra (resp. multiplicative n-hom–Lie algebra) is an n-ary hom–Nambu algebra (resp. n-hom–Lie algebra) $(\mathcal {N}, [\ ,\ldots , \ ], \widetilde{\alpha })$ with $\widetilde{\alpha }=(\alpha _i)_{1\le i\le n-1}$ where $\alpha _1=\cdots =\alpha _{n-1}=\alpha$ and satisfying

$$\begin{aligned} \alpha ([x_1,\ldots ,x_n])=[\alpha (x_1),\ldots ,\alpha (x_n)],\quad \forall x_1,\ldots ,x_n\in \mathcal {N}. \end{aligned}$$

(1.7)

For simplicity, we will denote the n-ary multiplicative hom–Nambu algebra as $(\mathcal {N}, [\ ,\ldots , \ ], \alpha )$ where $\alpha :\mathcal {N}\rightarrow \mathcal {N}$ is a linear map. Also by misuse of language, an element $x\in \mathcal {N}^n$ refers to $x=(x_1,\ldots ,x_{n})$ and $\alpha (x)$ denotes $(\alpha (x_1),\dots ,\alpha (x_n))$.

2.2 Derivations, Quasiderivations and Centroids of Multiplicative n-hom–Lie Algebras

In this section, we recall the definition of derivation, generalized derivation, quasiderivation and centroids of multiplicative n-hom–Lie algebras.

Let $(\mathcal {N}, [\ ,\ldots , \ ], \alpha )$ be a multiplicative n-hom–Lie algebra. We denote by $\alpha ^k$ the k-times composition of $\alpha$ (i.e. $\alpha ^k=\alpha \circ \cdots \circ \alpha$ k-times). In particular, $\alpha ^{-1}=0$, $\alpha ^0=id$.

Definition 1.10

For any $k\ge 1$, we call $D\in End(\mathcal {N})$ an $\alpha ^k$-derivation of the multiplicative n-hom–Lie algebra $(\mathcal {N}, [\ ,\ldots ,\ ], \alpha )$ if

$$\begin{aligned} {[}D,\alpha ]=0\ \ {\text {i.e.}}\ \ D\circ \alpha =\alpha \circ D, \end{aligned}$$

(1.8)

and

$$\begin{aligned} D[x_1,\ldots ,x_n]=\sum _{i=1}^n\Big [\alpha ^k(x_1),\ldots ,\alpha ^k(x_{i-1}),D(x_i),\alpha ^k(x_{i+1}),\ldots ,\alpha ^k(x_n)\Big ]. \end{aligned}$$

(1.9)

We denote by $Der_{\alpha ^k}(\mathcal {N})$ the set of $\alpha ^k$-derivations of the multiplicative n-hom–Lie algebra $\mathcal {N}$.

For $X=(x_1,\ldots ,x_{n-1})\in \mathcal {N}^{ n-1}$ satisfying $\alpha (X)=X$ and $k\ge 1$, we define the map $\text {ad}^k_X\in End(\mathcal {N})$ by

$$\begin{aligned} \text {ad}^k_X(y)=\Big [x_1,\ldots ,x_{n-1},\alpha ^k(y)\Big ]\quad \forall y\in \mathcal {N}. \end{aligned}$$

(1.10)

Lemma 1.11

The map $\text {ad}^k_X$ is an $\alpha ^{k+1}$-derivation that we call the inner $\alpha ^{k+1}$-derivation.

We denote by $Inn_{\alpha ^k}(\mathcal {N})$ the space generated by all the inner $\alpha ^{k+1}$-derivations. For any $D\in Der_{\alpha ^k}(\mathcal {N})$ and $D'\in Der_{\alpha ^k}(\mathcal {N})$, we define their commutator $[D,D']$ as usual:

$$\begin{aligned} {[}D,D']=D\circ D'-D'\circ D. \end{aligned}$$

(1.11)

Set $Der(\mathcal {N})=\displaystyle \bigoplus \nolimits _{k\ge -1}Der_{\alpha ^k}(\mathcal {N})$ and $Inn(\mathcal {N})=\displaystyle \bigoplus \nolimits _{k\ge -1}Inn_{\alpha ^k}(\mathcal {N})$.

Definition 1.12

An endomorphism D of a multiplicative n-ary hom–Nambu algebra $(\mathcal N, [~,\ldots ,~], \alpha )$ is called a generalized $\alpha ^k$-derivation if there exist linear mappings $D',D'', \ldots ,D^{(n-1)},D^{(n)} \in End(\mathcal {N})$ such that

$$\begin{aligned} D^{(n)}([x_1, \ldots , x_n])=\sum _{i=1}^n\Big [\alpha ^k(x_1), \ldots ,D^{(i-1)}(x_i), \ldots , \alpha ^k(x_n)\Big ], \end{aligned}$$

(1.12)

for all $x_1,\ldots , x_n\in \mathcal {N}$. An $(n + 1)$-tuple $(D,D',D'', \ldots ,D^{(n-1)},D^{(n)})$ is called an $(n + 1)$-ary $\alpha ^k$-derivation.

The set of generalized $\alpha ^k$-derivations is denoted by $GDer_{\alpha ^k}(\mathcal {N})$. Set $GDer(\mathcal {N})=\displaystyle \bigoplus \nolimits _{k\ge -1}GDer_{\alpha ^k}(\mathcal {N})$.

Definition 1.13

Let $(\mathcal N, [~,\ldots ,~], \alpha )$ be a multiplicative n-ary hom–Nambu algebra and $End(\mathcal N)$ be the endomorphism algebra of $\mathcal N$. An endomorphism $D\in End(\mathcal N)$ is said to be an $\alpha ^k$-quasiderivation, if there exists an endomorphism $D'\in End(\mathcal N)$ such that

$$\begin{aligned} \displaystyle \sum _{i=1}^n\Big [\alpha ^k(x_1),\dots ,D(x_i),\dots , \alpha ^k(x_n)\Big ]=D'([x_1,\dots , x_n]), \end{aligned}$$

for all $x_1,\dots ,x_n\in \mathcal N$. We call $D'$ the endomorphism associated with the $\alpha ^k$-quasiderivation D.

The set of $\alpha ^k$-quasiderivations will be denoted by $QDer_{\alpha ^k}(\mathcal N)$. Set $QDer(\mathcal N)=\displaystyle \bigoplus \nolimits _{k\ge -1}QDer_{\alpha ^k}(\mathcal N)$.

Definition 1.14

Let $(\mathcal N, [~,\ldots ,~], \alpha )$ be a multiplicative n-ary hom–Nambu algebra and $End(\mathcal N)$ be the endomorphism algebra of $\mathcal N$. Then the following subalgebra of $End(\mathcal N)$

$$\begin{aligned} Cent(\mathcal N) = \{\theta \in End(N) : \theta ([x_1,\dots , x_n])= [\theta (x_1),\dots , x_n], ~~\forall x_i\in \mathcal N\}\end{aligned}$$

is said to be the centroid of the n-ary hom–Nambu algebra. The definition is the same for the classical case of n-ary Nambu algebra. We may also consider the same definition for any n-ary hom–Nambu algebra.

Now, let $(\mathcal N, [~,\ldots ,~], \alpha )$ be a multiplicative n-ary hom–Nambu algebra.

Definition 1.15

An $\alpha ^k$-centroid of a multiplicative n-ary hom–Nambu algebra $(\mathcal N, [~,\ldots ,~], \alpha )$ is a subalgebra of $End(\mathcal N)$, denoted $Cent_{\alpha ^k}(\mathcal N)$, given by

$$\begin{aligned}&Cent_{\alpha ^k} (\mathcal N)\\&\quad =\Big \{\theta \in End(\mathcal N): \theta [x_1,\ldots , x_n]=\Big [\theta (x_1), \alpha ^k(x_2),\dots , \alpha ^k(x_n)\Big ], \forall x_i\in \mathcal N\Big \}. \end{aligned}$$

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

If $\mathcal N$ is a multiplicative n-hom–Lie algebra, then it is a simple fact that

$$\begin{aligned} \theta [x_1,\dots , x_n]=\Big [\alpha ^k(x_1),\dots , \theta (x_p),\dots , \alpha ^k(x_n)\Big ],\, \forall p \in \{1,\dots , n\}. \end{aligned}$$

3 n-hom–Lie Algebras Induced by hom–Lie Algebras

In [4], the authors introduced a construction of a 3-hom–Lie algebra from a hom–Lie algebra, and more generally of an $(n+1)$-hom–Lie algebra from an n-hom–Lie algebra. It is called the $(n + 1)$-hom–Lie algebra induced by n-hom–Lie algebra. In this context, Abramov gave a new approach of this construction (see [1]). Now, we generalize this approach in the Hom case.

Let $(\mathfrak {g},[~,~],\alpha )$ be a multiplicative hom–Lie algebra and $\mathfrak {g}^*$ be its dual space. Fix an element of the dual space $\varphi \in \mathfrak {g}^*$. Define the triple product as follows:

$$\begin{aligned} {[}x,y,z]=\varphi (x)[y,z]+\varphi (y)[z,x]+\varphi (z)[x,y],\quad \forall \ x,\ y,\ z\in \mathfrak {g}. \end{aligned}$$

(2.1)

Obviously, this triple product is skew-symmetric. Straightforward computation of the left hand side and the right hand side of the Filippov–Jacobi identity (1.5) if $\varphi \circ \alpha =\varphi$ yields

$$\begin{aligned} \varphi (x)\varphi ([y,z])+\varphi (y)\varphi ([z,x])+\varphi (z)\varphi ([x,y])=0. \end{aligned}$$

(2.2)

Now, we consider $\varphi$ as a $\mathbb {K}$-valued cochain of degree one of the Chevalley–Eilenberg complex of a Lie algebra ${\mathfrak {g}}$. Making use of the coboundary operator $\delta :\wedge ^{k}\mathfrak {g}^*\rightarrow \wedge ^{k+1}\mathfrak {g}^*$ defined by

$$\begin{aligned}&\delta f(u_1,\ldots ,u_{k+1})\nonumber \\&\quad =\sum _{i<j}(-1)^{i+j+1}f([u_i,u_j]_{{\mathfrak {g}}},\alpha (u_1)\ldots ,{\widehat{u_i}},\ldots ,{\widehat{u_j}},\ldots ,\alpha (u_{k+1})),\qquad \end{aligned}$$

(2.3)

for $f\in \wedge ^{k}\mathfrak {g}^*$ and for all $u_1,\ldots ,u_{k+1}\in \mathfrak {g}$, we obtain that $\delta \varphi (x, y) = \varphi ([x, y])$.

Finally, we can define the wedge product of two cochains $\varphi$ and $\delta \varphi$, which is a cochain of degree three, by

$$\begin{aligned} \varphi \wedge \delta \varphi (x, y, z) =\varphi (x)\varphi ([y,z])+\varphi (y)\varphi ([z,x])+\varphi (z)\varphi ([x,y]). \end{aligned}$$

Hence, (2.2) is equivalent to $\varphi \wedge \delta \varphi =0$. Thus, if a 1-cochain $\varphi$ satisfies the equation (2.2), then the triple product (2.1) is the ternary Lie bracket and we will call this multiplicative 3-hom–Lie bracket the quantum Nambu bracket induced by a 1-cochain.

Definition 2.1

For $\phi \in \wedge ^{n-2}\mathfrak {g}^*$, we define the n-ary product as follows:

$$\begin{aligned} {[}x_1,\ldots ,x_n]_\phi =\sum _{i<j}^{n}(-1)^{i+j+1}\phi (x_1,\ldots ,{\hat{x_i}},\ldots ,{\hat{x_j}},\ldots ,x_n)[x_i,x_j], \end{aligned}$$

(2.4)

for all $x_1,\ldots ,x_n\in \mathfrak {g}$.

Proposition 2.2

The n-ary product $[~,\ldots ,~ ]_\phi$ is skew-symmetric.

Proof

Let $x_1,\ldots ,x_n\in {\mathfrak {g}}$ and, fixing two integers $i<j$, we have

$$\begin{aligned}&{[}x_1,\ldots ,x_i,\ldots ,x_j,\ldots ,x_n]_{\phi }\\&\quad =\displaystyle \sum _{k<l:k,l\ne i,j}(-1)^{k+l+1}\phi (x_1,\ldots ,x_i\ldots ,\widehat{x}_k,\ldots ,x_j,\ldots ,\widehat{x}_l,\ldots ,x_n)[x_l,x_k]\\&\qquad +\displaystyle \sum _{i<l\ne j} (-1)^{i+l+1}\phi (x_1,\ldots ,\widehat{x}_i\ldots ,x_j,\ldots ,\widehat{x}_l,\ldots ,x_n)[x_i,x_l]\\&\qquad +\displaystyle \sum _{l< i} (-1)^{i+l+1}\phi (x_1,\ldots ,\widehat{x}_l,\ldots ,\widehat{x}_i,\ldots ,x_j,\ldots ,x_n)[x_l,x_i]\\&\qquad +\displaystyle \sum _{j<l} (-1)^{j+l+1}\phi (x_1,\ldots ,x_i\ldots ,\widehat{x}_j,\ldots ,\widehat{x}_l,\ldots ,x_n)[x_j,x_l]\\&\qquad +\displaystyle \sum _{l<j, i\ne l} (-1)^{j+l+1}\phi (x_1,\ldots ,x_i,\ldots ,\widehat{x}_l,\ldots ,\widehat{x}_j,\ldots ,x_n)[x_l,x_j]\\&\qquad + (-1)^{i+j+1}\phi (x_1,\ldots ,\widehat{x}_i,\ldots ,\widehat{x}_j,\ldots ,x_n)[x_i,x_j]\\&\quad =-[x_1,\ldots ,x_j,\ldots ,x_i,\ldots ,x_n]_{\phi }. \end{aligned}$$

Given $X=(x_1,\ldots ,x_{n-3})\in \wedge ^{n-3}{\mathfrak {g}}$, $Y=(y_1,\ldots ,y_{n})\in \wedge ^{n}{\mathfrak {g}}$ and $z\in {\mathfrak {g}}$, we define the linear map $\phi _X$ by

$$\begin{aligned} \phi _X(z)=\phi (X,z), \end{aligned}$$

and

$$\begin{aligned} \phi \wedge \delta \phi _X(Y)= & {} \sum _{i<j}^{n}(-1)^{i+j}\phi (y_1,\ldots {\hat{y_i}}\ldots {\hat{y_j}}\ldots ,y_{n})\delta \phi _X(y_i,y_j)\\= & {} \sum _{i<j}^{n}(-1)^{i+j}\phi (y_1,\ldots {\hat{y_i}}\ldots {\hat{y_j}}\ldots ,y_{n})\phi _X([y_i,y_j]). \end{aligned}$$

Theorem 2.3

Let $({\mathfrak {g}},[~,~],\alpha )$ be a multiplicative hom–Lie algebra, ${\mathfrak {g}}^*$ be its dual and $\phi$ be a cochain of degree $n-2$, i.e. $\phi \in \wedge ^{n-2}{\mathfrak {g}}^*$. The vector space ${\mathfrak {g}}$ is equipped with the n-ary product (2.4) and the linear map $\alpha$ is a multiplicative n-hom–Lie algebra if and only if

$$\begin{aligned}&\phi \wedge \delta \phi _X=0,\quad \forall X\in \wedge ^{n-3}\mathfrak {g}, \end{aligned}$$

(2.5)

$$\begin{aligned}&\phi \circ (\alpha \otimes Id\otimes \cdots \otimes Id)=\phi . \end{aligned}$$

(2.6)

Proof

Firstly, let $(x_1,\ldots ,x_n)\in \wedge ^{ n}{\mathfrak {g}}$. We have

$$\begin{aligned}&{[}\alpha (x_1),\ldots ,\alpha (x_n)]_{\phi }\\&\quad =\sum _{i<j}^{n}(-1)^{i+j+1}\phi (\alpha (x_1), \ldots ,{\hat{\alpha (x_i)}},\ldots ,{\hat{\alpha (x_j)}},\ldots ,\alpha (x_n))[\alpha (x_i),\alpha (x_j)]\\&\quad =\sum _{i<j}^{n}(-1)^{i+j+1}\phi (x_1, \ldots ,{\hat{x_i}},\ldots ,{\hat{x_j}},\ldots ,x_n)\alpha ([x_i,x_j])\\&\quad =\alpha ([x_1,\ldots ,x_n]_{\phi }). \end{aligned}$$

Secondly, for $(x_1,\ldots ,x_{n-1})\in \wedge ^{ n-1}{\mathfrak {g}}$ and $(y_1,\ldots ,y_n)\in \wedge ^{ n}{\mathfrak {g}}$, we have

$$\begin{aligned}&{[}\alpha (x_1),\ldots ,\alpha (x_{n-1}),[y_1,\ldots ,y_n]_{\phi }]_{\phi }\\&\quad =\displaystyle \sum _{i<j}(-1)^{i+j+1} \phi (y_1,\ldots ,\widehat{y}_i,\ldots ,\widehat{y}_j,\ldots ,y_n) \\&\qquad \times [\alpha (x_1),\ldots ,\alpha (x_{n-1}),[y_i,y_j]]_{\phi }\\&\quad =\displaystyle \sum _{i<j}\displaystyle \sum _{k<l\le n-1}(-1)^{i+j+k+l} \phi (\alpha (x_1),\ldots ,{\widehat{\alpha (x_k)}},\ldots ,{\widehat{\alpha (x_l)}},\ldots ,[y_i,y_j])\\&\qquad \times \phi (y_1,\ldots ,\widehat{y}_i,\ldots ,\widehat{y}_j,\ldots ,y_n)[\alpha (x_k),\alpha (x_l)]\\&\qquad +\displaystyle \sum _{i<j}\displaystyle \sum _{k<n}(-1)^{i+j+k} \phi (\alpha (x_1),\ldots ,{\widehat{\alpha (x_k)}},\ldots ,\alpha (x_{(n-1)}),\ldots ,{\widehat{[y_i,y_j]}})\\&\qquad \times \phi (y_1,\ldots ,\widehat{y}_i,\ldots ,\widehat{y}_j,\ldots ,y_n)[\alpha (x_k),[y_i,y_j]]. \end{aligned}$$

The terms $[\alpha (x_k),[y_i,y_j]]$ are simplified by the hom–Jacobi condition in the second half of the Filippov identity. Now, we group together the other terms according to their coefficient $[\alpha (x_i),\alpha (x_j)]$. For example, if we fix (k, l), and if we collect all the terms containing the commutator $[\alpha (x_k),\alpha (x_l)]$, then we get the expression

$$\begin{aligned}&\left( \displaystyle \sum _{i<j}(-1)^{i+j+k+l} \phi (\alpha (x_1),\ldots ,{\widehat{\alpha (x_k)}},\ldots ,{\widehat{\alpha (x_l)}},\ldots ,[y_i,y_j])\right. \\&\quad \left. \phantom {\left( \displaystyle \sum _{i<j}(-1)^{i+j+k+l} \phi (\alpha (x_1),\ldots ,{\widehat{\alpha (x_k)}},\ldots ,{\widehat{\alpha (x_l)}},\ldots ,[y_i,y_j])\right. }\times \phi (y_1,\ldots ,\widehat{y}_i,\ldots ,\widehat{y}_j,\ldots ,y_n)\right) [\alpha (x_k),\alpha (x_l)]. \end{aligned}$$

Hence, the n-ary product (2.4) will satisfy the n-ary Filippov–Jacobi identity; if for any elements $X=(x_1,\ldots ,x_{n-3})\in \wedge ^{n-3}{\mathfrak {g}}$ and $Y=(y_1,\ldots ,y_n)\in \wedge ^n {\mathfrak {g}}$ we require

$$\begin{aligned} \left( \displaystyle \sum _{i<j}^n(-1)^{i+j} \phi (\alpha (x_1),\ldots ,\alpha (x_{n-3}),[y_i,y_j]) \phi (y_1,\ldots ,\widehat{y}_i,\ldots ,\widehat{y}_j,\ldots ,y_n)\right) =0. \end{aligned}$$

Definition 2.4

Let $\phi :{\mathfrak {g}}\otimes \cdots \otimes \mathfrak {g}\rightarrow \mathbb {K}$ be a skew-symmetric multilinear form of the multiplicative hom–Lie algebras $({\mathfrak {g}},[~,~],\alpha )$, then $\phi$ is called a trace if

$$\begin{aligned} \phi \circ (Id\otimes \ldots \otimes Id\otimes [~,~])=0~~~~~\text {and}~~ \phi \circ (\alpha \otimes Id\otimes \ldots \otimes Id)=\phi .~ \end{aligned}$$

Corollary 2.5

If $\phi :{\mathfrak {g}}^{\otimes n-2}\rightarrow \mathbb {K}$ is a trace of the hom–Lie algebra $({\mathfrak {g}},[~,~],\alpha )$, then ${\mathfrak {g}}_\phi =({\mathfrak {g}},[.,\ldots ,.]_\phi ,\alpha )$ is a n-hom–Lie algebra.

Proposition 2.6

Let $({\mathfrak {g}},[~,~],\alpha )$ be a hom–Lie algebra and $D \in Der(\mathfrak {g})$ be an $\alpha ^k$-derivation such that

$$\begin{aligned} \sum _{i=1}^{n-2}\phi (x_1,\ldots D(x_i),\ldots ,x_{n-2})=0. \end{aligned}$$

Then, D is an $\alpha ^k$-derivation of the n-hom–Lie algebra $({\mathfrak {g}},[~,\ldots ,~]_\phi ,\alpha )$.

Proof

Let $X=(x_1,\ldots ,x_n)\in \wedge ^n {\mathfrak {g}}$. On the one hand, we get

$$\begin{aligned}&D([x_1,\ldots ,x_n]_\phi )\\&\quad = D\left( \displaystyle \sum _{i<j}(-1)^{i+j+1} \phi (\alpha (x_1),\ldots ,\widehat{x}_i,\ldots ,\widehat{x}_j,\ldots ,\alpha (x_n))[\alpha (x_i),\alpha (x_j)]\right) \\&\quad =\displaystyle \sum _{i<j}(-1)^{i+j+1} \phi (\alpha (x_1),\ldots ,\widehat{x}_i,\ldots ,\widehat{x}_j,\ldots ,\alpha (x_n))D([\alpha (x_i),\alpha (x_j)])\\&\quad =\displaystyle \sum _{i<j}(-1)^{i+j+1} \phi (x_1,\ldots ,\widehat{x}_i,\ldots ,\widehat{x}_j,\ldots ,x_n)\Big [\alpha (D(x_i)),\alpha ^{k+1}(x_j)\Big ]\\&\qquad +\displaystyle \sum _{i<j}(-1)^{i+j+1} \phi (x_1,\ldots ,\widehat{x}_i,\ldots ,\widehat{x}_j,\ldots ,x_n)\Big [\alpha ^{k+1}(x_i),\alpha (D(x_j))\Big ], \end{aligned}$$

and, on the other hand, we have

$$\begin{aligned}&\displaystyle \sum _{l=1}^n\Big [\alpha ^k(x_1),\ldots ,\alpha ^k(x_{l-1}),D(x_l),\ldots ,\alpha ^k(x_{l+1}),\ldots ,\alpha ^k(x_n)\Big ]_\phi \\&\quad = \displaystyle \sum _{l=1}^n\displaystyle \sum _{i<j\;;\;i,j\ne l}(-1)^{i+j+1} \phi (\alpha ^k(x_1),\ldots ,{\widehat{\alpha ^k(x_i)}},\ldots ,\\&\qquad D(x_l),\ldots ,{\widehat{\alpha ^k(x_j)}},\ldots ,\alpha ^k(x_n))\Big [\alpha ^k(x_i),\alpha ^k(x_j)\Big ]\\&\qquad +\displaystyle \sum _{l=1}^n\displaystyle \sum _{i<l}(-1)^{i+l+1} \phi (\alpha ^k(x_1),\ldots ,{\widehat{\alpha ^k(x_i)}},\ldots ,\\&\qquad {\widehat{D(x_l)}},\ldots ,\alpha ^k(x_n))\Big [\alpha ^k(x_i),D(x_l)\Big ]\\&\qquad +\displaystyle \sum _{l=1}^n\displaystyle \sum _{l=i<j}(-1)^{j+l+1} \phi \Big (\alpha ^k(x_1),\ldots ,{\widehat{D(x_l)}},\ldots ,\\&\qquad {\widehat{\alpha ^k(x_j)}},\ldots ,\alpha ^k(x_n)\Big )\Big [D(x_l),\alpha ^k(x_j)\Big ]. \end{aligned}$$

If D is an $\alpha ^k$-derivation, then $D([x_1,\ldots ,x_n]_\phi )=\displaystyle \sum _{l=1}^n[\alpha ^k(x_1),\ldots ,\alpha ^k(x_{l-1}), D(x_l),\ldots ,\alpha ^k(x_{l+1}),\ldots ,\alpha ^k(x_n)]_\phi$, which gives

$$\begin{aligned}&\displaystyle \sum _{i<j\;;\;i,j\ne l}(-1)^{i+j+1}\left( \displaystyle \sum _{l=1}^n\displaystyle \phi \Big (\alpha ^k(x_1),\ldots ,{\widehat{\alpha ^k(x_i)}},\ldots ,\right. \\&\qquad \left. \phantom {\left( \displaystyle \sum _{l=1}^n\displaystyle \phi \Big (\alpha ^k(x_1),\ldots ,{\widehat{\alpha ^k(x_i)}},\ldots ,\right. } D(x_l),\ldots ,{\widehat{\alpha ^k(x_j)}},\ldots ,\alpha ^k(x_n)\Big )\right) \Big [\alpha ^k(x_i),\alpha ^k(x_j)\Big ]=0. \end{aligned}$$

Finally, if we fix (i, j), we have

$$\begin{aligned} \displaystyle \sum _{l=1}^{n-2}\displaystyle \phi \Big (\alpha ^k(x_1),\ldots ,D(x_l),\ldots ,\alpha ^k(x_{n-2})\Big )=0. \end{aligned}$$

Proposition 2.7

Let $({\mathfrak {g}},[~,~],\alpha )$ be a hom–Lie algebra and $D\in QDer({\mathfrak {g}})$ be an $\alpha ^k$-quasiderivation and $D':\mathfrak {g}\rightarrow \mathfrak {g}$ be the endomorphism associated with D such that

$$\begin{aligned} \sum _{i=1}^{n-2}\phi (x_1,\ldots D(x_i),\ldots ,x_{n-2})=0. \end{aligned}$$

Then, D is an $\alpha ^k$-quasiderivation of the n-hom–Lie algebra $({\mathfrak {g}},[~,\ldots ,~]_\phi ,\alpha )$ with the same associated endomorphism $D'$.

Proposition 2.8

Let $({\mathfrak {g}},[~,~],\alpha )$ be a hom–Lie algebra and $\theta :{\mathfrak {g}}\rightarrow {\mathfrak {g}}$ be an $\alpha ^k$-centroid such that

$$\begin{aligned} \phi (\theta (x_1),\ldots x_i,\ldots ,x_{n-2})\Big [\alpha ^k(x),y\Big ]=\phi (x_1,\ldots x_i,\ldots ,x_{n-2})[\theta (x),y]. \end{aligned}$$

Then, D is an $\alpha ^k$-centroid on the n-hom–Lie algebra $({\mathfrak {g}},[~,\ldots ,~]_\phi ,\alpha )$.

Proof

If $x_1,\ldots ,x_n\in \mathfrak {g}$, we have

$$\begin{aligned} \theta ([x_1,\ldots ,x_n]_\phi )= & {} \sum _{i<j}^{n}(-1)^{i+j+1}\phi (x_1,\ldots ,{\hat{x_i}},\ldots ,{\hat{x_j}},\ldots ,x_n)\theta ([x_i,x_j])\\= & {} \sum _{i<j}^{n}(-1)^{i+j+1}\phi (x_1,\ldots ,{\hat{x_i}},\ldots ,{\hat{x_j}},\ldots ,x_n)\Big [\theta (x_i),\alpha ^k(x_j)\Big ]. \end{aligned}$$

On the other hand, we have

$$\begin{aligned}&{\Big [}\theta (x_1),\alpha ^k(x_2),\ldots ,\alpha ^k(x_n)\Big ]_\phi \\&\quad =\sum _{i<j}^{n}(-1)^{i+j+1}\phi (\theta (x_1),\alpha ^k(x_2),\ldots ,{\hat{x_i}},\ldots ,{\hat{x_j}},\ldots ,\alpha ^k(x_n))\Big [\alpha ^k(x_i),\alpha ^k(x_j)\Big ]\\&\quad =\sum _{i<j}^{n}(-1)^{i+j+1}\phi (x_1,\ldots ,{\hat{x_i}},\ldots ,{\hat{x_j}},\ldots ,x_n)\Big [\theta (x_i),\alpha ^k(x_j)\Big ]\\&\quad =\theta ([x_1,\ldots ,x_n]_\phi ). \end{aligned}$$

4 hom–Lie n-Tuple Systems

4.1 hom–Lie Triple Systems

In this section, we start by recalling the definitions of Lie triple systems and hom–Lie triple systems.

Definition 3.1

[20]

A vector space T together with a trilinear map $(x, y, z)\rightarrow [x,y,z]$ is called a Lie triple system (LTS) if

• 1. $[x,x,z]=0,$

• 2. $[x,y,z]+[y,z,x]+[z,x,y]=0$,

• 3. $[u,v,[x,y,z]]=[[u,v,x],y,z]+[x,[u,v,y],z]+[x,y,[u,v,z]],$

for all $x,y,z,u,v\in T$.

Definition 3.2

[25] A hom–Lie triple system (hom-LTS for short) is denoted by $(T,[\cdot ,\cdot ,\cdot ], \alpha )$, which consists of a $\mathbb {K}$-vector space T, a trilinear product $[\cdot ,\cdot ,\cdot ]: T\times T\times T\rightarrow T$, and a linear map $\alpha :T\rightarrow T$, called the twisted map, such that $\alpha$ preserves the product and for all $x,y,z,u,v\in T$,

• 1. $[x,x,z]=0,$

• 2. $[x,y,z]+[y,z,x]+[z,x,y]=0$,

• 3. $[\alpha (u),\alpha (v),[x,y,z]]=[[u,v,x],\alpha (y),\alpha (z)]+[\alpha (x),[u,v,y],\alpha (z)]+[\alpha (x),\alpha (y),[u,v,z]]$.

Remark 3.3

When the twisted map $\alpha$ is equal to the identity map, a hom-LTS is an LTS. So LTS are special examples of hom-LTS.

Definition 3.4

A hom–Lie triple system $(T,[\cdot ,\cdot ,\cdot ], \alpha )$ is called multiplicative if $\alpha ([x,y,z])=[\alpha (x),\alpha (y),\alpha (z)]$, for all $x,y,z\in T$.

Theorem 3.5

[25]

Let $(\mathfrak {g},[\cdot ,\cdot ], \alpha )$ be a multiplicative hom–Lie algebra. Then

$$\begin{aligned} \mathfrak {g}_T=(\mathfrak {g},[\cdot ,\cdot ,\cdot ]=[\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes \alpha ), \alpha ^2),\end{aligned}$$

is a multiplicative hom–Lie triple system.

4.2 hom–Lie n-Tuple System

In this section, we introduce the definitions of Lie n-tuple systems and multiplicative hom–Lie n-tuple systems. We give the analogue of Theorem 3.5 in the hom–Lie n-tuple systems case.

Definition 3.6

A vector space $\mathcal {G}$ together with a n-linear map $(x_1,\ldots , x_n)\rightarrow [x_1,\ldots , x_n]$ is called a Lie n-tuple system if

• 1. $[x,x,y_1,\ldots ,y_{n-2}]=0,$ for all $x,y_1,\ldots ,y_{n-2}\in \mathcal {G}$.

• 2. $\displaystyle \circlearrowleft _{x_1,x_2,x_3}[x_1,\ldots ,x_{n}]=0,$ for all $x_1,\ldots ,x_{n}\in \mathcal {G}$.

• 3. $\big [x_1,\ldots ,x_{n-1},[y_1,\ldots ,y_{n}]\big ]= \displaystyle \sum _{i=1}^{n}\big [y_1,\ldots ,y_{i-1},[x_1,\ldots ,x_{n-1},y_i], y_{i+1},\ldots ,y_n\big ],$

for all $x_1,\ldots ,x_{n-1},y_1,\ldots ,y_{n}\in \mathcal {G}$.

Definition 3.7

A vector space $\mathcal {G}$ together with a n-linear map $(x_1,\ldots , x_n)\rightarrow [x_1,\ldots , x_n]$ and a family $\widetilde{\alpha }=(\alpha _i)_{1\le i\le n-1}$ of linear maps $\alpha _i:\ \ \mathcal {G}\longrightarrow \mathcal {G}$ is called a hom–Lie n-tuple system if

• 1. $[x,x,y_1,\ldots ,y_{n-2}]=0,$ for all $x,y_1,\ldots ,y_{n-2}\in \mathcal {G}$.

• 2. $\displaystyle \circlearrowleft _{x_1,x_2,x_3}[x_1,\ldots ,x_{n}]=0,$ for all $x_1,\ldots ,x_{n}\in \mathcal {G}$.

• 3. $\big [\alpha _1(x_1),\dots ,\alpha _{n-1}(x_{n-1}),[y_1,\dots ,y_{n}]\big ] =\displaystyle \sum _{i=1}^{n}\big [\alpha _1(y_1),\dots ,\alpha _{i-1}(y_{i-1}),[x_1,\dots ,x_{n-1},y_i] ,\alpha _i(y_{i+1}),\dots ,\alpha _{n-1}(y_n)\big ],$ for all $x_1,\ldots ,x_{n-1},y_1,\ldots ,y_{n}\in \mathcal {G}$.

Definition 3.8

A hom–Lie n-tuple system $(\mathcal {G},[~,\ldots , ~],\widetilde{\alpha })$ is called a multiplicative hom–Lie n-tuple system if $\alpha _1=\dots =\alpha _{n-1}=\alpha$ and $\alpha ([x_1,\ldots , x_n])=[\alpha (x_1),\ldots , \alpha (x_n)]$ for all $x_1,\ldots , x_n\in \mathcal G$.

Remark 3.9

When the twisted maps $\alpha _i$ are equal to the identity map, hom–Lie n-tuple systems are Lie n-tuple systems. So Lie n-tuple systems are special examples of hom–Lie n-tuple systems.

The following result gives a way to construct hom–Lie n-tuple systems starting from classical Lie n-tuple systems and algebra endomorphisms.

Proposition 3.10

Let $(\mathcal {G},[~,\ldots , ~])$ be a Lie n-tuple system and $\alpha :\mathcal {G}\rightarrow \mathcal {G}$ be a linear map such that $\alpha ([x_1,\ldots ,x_n])=[\alpha (x_1),\ldots ,\alpha (x_n)]$. Then, $(\mathcal {G},[~,\ldots , ~]_\alpha ,\alpha )$ is a hom–Lie n-tuple system, where $[x_1,\ldots ,x_n]_\alpha =[\alpha (x_1),\ldots ,\alpha (x_n)]$, for all $x_1,\ldots ,x_n\in \mathcal {G}$.

Let $(\mathfrak {g},[~,~],\alpha )$ be a hom–Lie algebra. We define the following n-linear map:

$$\begin{aligned}&{[}~,\ldots ,~]_n:{\mathfrak {g}}^{\otimes n} \longrightarrow {\mathfrak {g}}\nonumber \\&(x_1,\ldots ,x_n)\longmapsto \Big [x_1,\ldots ,x_n\Big ]_n = \Big [\big [[\dots [x_1,x_2],\alpha (x_3)],\alpha ^2(x_4)\big ]\ldots \alpha ^{n-3}(x_{n-1})],\alpha ^{n-2}(x_{n})\Big ].\nonumber \\ \end{aligned}$$

(3.1)

For $n=2$, $[x_1,x_2]_2=[x_1,x_2]$ and for $n\ge 3$ we have $[x_1,\ldots ,x_n]_n=[[x_1,\ldots ,x_{n-1}]_{n-1},\alpha ^{n-2}(x_{n})]$.

Theorem 3.11

Let $(\mathfrak {g},[\ ,\ ], \alpha )$ be a multiplicative hom–Lie algebra. Then

$$\begin{aligned} \mathfrak {g}_n=(\mathfrak {g},[\ ,\ldots ,\ ]_n, \alpha ^{n-1}) \end{aligned}$$

is a multiplicative hom–Lie n-tuple system.

When $n=3$ we obtain the multiplicative hom–Lie triple system constructed in Theorem 3.5. To prove this theorem, we need the following lemma.

Lemma 3.12

Let $(\mathfrak {g},[\ ,\ ], \alpha )$ be a multiplicative hom–Lie algebra, and $\text {ad}^2$ the adjoint map defined by

$$\begin{aligned} \text {ad}_x^2(y)=\text {ad}_x(y)=[x,y]. \end{aligned}$$

Then, we have

$$\begin{aligned} \text {ad}_{\alpha ^{n-1}(x)}^2[y_1,\ldots ,y_n]_n=\displaystyle \sum _{k=1}^n \Big [\alpha (y_1),\ldots ,\alpha (y_{k-1}),\text {ad}_x^2(y_k),\alpha (y_{k+1}),\ldots ,\alpha (y_n)\Big ]_n, \end{aligned}$$

where $x\in \mathfrak {g}, y\in {\mathfrak {g}}$ and $(y_1,\ldots ,y_n)\in \mathfrak {g}^n$.

Proof

For $n=2$, using the hom–Jacobi identity we have

$$\begin{aligned} \text {ad}_{\alpha (x)}^2[y,z]= & {} [\alpha (x),[y,z]]=[[x,y],\alpha (z)]+[\alpha (y),[x,z]]\\= & {} \Big [\text {ad}_x^2(y),\alpha (z)\Big ]+\Big [\alpha (y),\text {ad}_x^2(z)\Big ]. \end{aligned}$$

Assume that the property is true up to order n, that is

$$\begin{aligned}&\text {ad}_{\alpha ^{n-1}(X)}^2[y_1,\ldots ,y_n]_n \\&\quad =\displaystyle \sum _{k=1}^n [\alpha (y_1),\ldots ,\alpha (y_{k-1}),\text {ad}_X^2(y_k),\alpha (y_{k+1}),\ldots ,\alpha (y_n)]_n. \end{aligned}$$

If $x\in \mathfrak {g}$ and $(y_1,\ldots ,y_{n+1})\in \mathfrak {g}^{n+1}$, we have

$$\begin{aligned}&\text {ad}^2_{\alpha ^n(x)}[y_1,\ldots ,y_{n+1}] \\&\quad =\text {ad}^2_{\alpha ^n(x)}[[y_1,\ldots ,y_n]_n,\alpha ^{n-1}(y_{n+1})]_2\\&\quad = \Big [\text {ad}^2_{\alpha ^{n-1}(x)}[y_1,\ldots ,y_n]_n,\alpha ^n(y_{n+1})\Big ]_2 \\&\qquad + \Big [[\alpha (y_1),\ldots ,\alpha (y_n)]_n,\text {ad}^2_{\alpha ^{n-1}(x)}(\alpha ^{n-1}(y_{n+1}))\Big ]_2\\&\quad = \displaystyle \sum _{k=1}^n\Big [[\alpha (y_1),\ldots ,\alpha (y_{k-1}),\text {ad}^2_x(y_k),\alpha (y_{k+1}),\ldots ,\alpha (y_n)]_n,\alpha ^n(y_{n+1})\Big ]\\&\qquad + \Big [[\alpha (y_1),\ldots ,\alpha (y_n)]_n,\alpha ^{n-1}(\text {ad}^2_x(y_{n+1}))\Big ]_2\\&\quad = \displaystyle \sum _{k=1}^n\Big [\alpha (y_1),\ldots ,\alpha (y_{k-1}),\text {ad}^2_x(y_k),\alpha (y_{k+1}),\ldots ,\alpha (y_n),\alpha (y_{n+1})\Big ]_{n+1}\\&\qquad + \Big [\alpha (y_1),\ldots ,\alpha (y_n),\text {ad}^2_x(y_{n+1})\Big ]_{n+1}\\&\quad = \displaystyle \sum _{k=1}^{n+1}\Big [\alpha (y_1),\ldots ,\alpha (y_{k-1}),\text {ad}^2_x(y_k),\alpha (y_{k+1}),\ldots ,\alpha (y_{n+1})\Big ]_{n+1}. \end{aligned}$$

The lemma is proved. $\square$

Proof

(Proof of Theorem 3.11) Let $X=(x_1,\ldots ,x_{n-1})\in \mathfrak {g}^{n-1}$ and $Y=(y_1,\ldots ,y_n)\in \mathfrak {g}^n$.

• (i) It is easy to see that $[x_1,x_1,x_2,\ldots ,x_{n-1}]_n=[[\ldots [[x_1,x_1]_2,\alpha (x_2)]_2, \alpha ^2(x_3)]_2,\ldots ]_2,\alpha ^{n-2}(x_{n-1})]_2=0$

• (ii) Using the hom–Jacobi condition, it is easy to prove $\displaystyle \circlearrowleft _{x_1,x_2,x_3}[x_1,\ldots ,x_{n}] =0,$ for all $x_1,\ldots ,x_{n}\in \mathcal {G}$.

• (iii) Using Lemma (3.12), we have

  $$\begin{aligned}&\Big [\alpha ^{n-1}(x_1),\ldots ,\alpha ^{n-1}(x_{n-1}),[y_1,\ldots ,y_n]_n\Big ]_n \\&\quad = \Big [[\alpha ^{n-1}(x_1),\ldots ,\alpha ^{n-1}(x_{n-1})]_{n-1},[\alpha ^{n-2}(y_1),\ldots ,\alpha ^{n-2}(y_n)]_n\Big ]_2\\&\quad = \text {ad}^2_{\alpha ^{n-1}[x_1,\ldots ,x_{n-1}]}([\alpha ^{n-2}(y_1),\ldots ,\alpha ^{n-2}(y_n)]_n)\\&\quad = \displaystyle \sum _{k=1}^n\Big [\alpha ^{n-1}(y_1),\ldots ,\text {ad}^2_{[x_1,\ldots ,x_{n-1}]}(\alpha ^{n-2}(y_k)),\ldots ,\alpha ^{n-1}(y_n)\Big ]_n\\&\quad = \displaystyle \sum _{k=1}^n\Big [\alpha ^{n-1}(y_1),\ldots ,[[x_1,\ldots ,x_{n-1}],\alpha ^{n-2}(y_k)]_2,\ldots ,\alpha ^{n-1}(y_n)\Big ]_n\\&\quad = \displaystyle \sum _{k=1}^n\Big [\alpha ^{n-1}(y_1),\ldots ,[x_1,\ldots ,x_{n-1},y_k]_n,\ldots ,\alpha ^{n-1}(y_n)\Big ]_n. \end{aligned}$$

Example 3.13

Using Example 1.5 and Theorem 3.11, for $\lambda \in \mathbb {R}^*$, we have the following.

For $n=3,\;({\mathfrak {sl}}_2(\mathbb {R}),[~,~,~]_3,\alpha ^2_\lambda )$ is a hom–Lie triple system. The different brackets are as follows:

$$\begin{aligned}&{[}H,X,Y]_3=[[H,X]_{\alpha _{\lambda }},\alpha _{\lambda }(Y)]_{\alpha _{\lambda }}=2H; \,\,\, [H,X,H]_3=-4\lambda ^4X; \\&[H,Y,X]_3=4H.\\&{[}H,Y,H]_3=-\frac{4}{\lambda ^4}Y; \,\,\, [X,Y,Y]_3=-\frac{2}{\lambda ^4}Y; \,\,\, [X,Y,X]_3=2\lambda ^4X. \end{aligned}$$

Each of the other brackets is equal to zero.

For $n=4,\;({\mathfrak {sl}}_2(\mathbb {R}),[~,~,~,~]_4,\alpha ^3_\lambda )$ is a hom–Lie 4-uplet system. The different brackets are defined as follows:

$$\begin{aligned}&{[}H,X,H,H]_4=[[H,X,H]_3,\alpha ^2(H)]_{\alpha _{\lambda }}=-4\lambda ^4[X,H]_{\alpha _{\lambda }}=8\lambda ^6X; \\&[H,X,H,Y]_4=-4H;\\&{[}H,Y,H,H]_4=-\frac{8}{\lambda ^6}Y;\;\;\;[H,Y,H,X]_4=4H; \\&[H,X,Y,X]_4=4\lambda ^6X;\;\;\;[H,X,Y,Y]_4=-\frac{2}{\lambda ^6}Y;\\&{[}H,Y,X,X]_4=8\lambda ^6X;\;\;\;[H,Y,X,Y]_4=-\frac{8}{\lambda ^6}Y; \\&[X,Y,X,Y]_4=2H;\\&{[}X,Y,X,H]_4=-4\lambda ^6X;\;\;\;[X,Y,Y,X]_4=2H; \\&[X,Y,Y,H]_4=-\frac{4}{\lambda ^6}Y.\\ \end{aligned}$$

Each of the other brackets is equal to zero.

Proposition 3.14

Let $(\mathfrak {g},[\ ,\ ], \alpha )$ be a multiplicative hom–Lie algebra and $D:\mathfrak {g}\rightarrow \mathfrak {g}$ be an $\alpha ^k$-derivation of $\mathfrak {g}$ for an integer k. Then, D is an $\alpha ^k$-derivation of $\mathfrak {g}_n$.

Proof

By recurrence

Fix $n=3$. For $x,y,z\in \mathfrak {g}$, we have

$$\begin{aligned} D([x,y,z])= & {} D([[x,y],\alpha (z)])\\= & {} \Big [D([x,y]),\alpha ^{k+1}(z)\Big ]+\Big [\Big [\alpha ^k(x),\alpha ^k(y)\Big ],D(\alpha (z))\Big ]\\= & {} \Big [\Big [D(x),\alpha ^k(y)\Big ],\alpha ^{k+1}(z)\Big ]+ \Big [\Big [\alpha ^k(x),D(y)\Big ],\alpha ^{k+1}(z)\Big ]\\&+\Big [\Big [\alpha ^k(x),\alpha ^k(y)\Big ],\alpha (D(z))\Big ]\\= & {} \Big [D(x),\alpha ^k(y),\alpha ^k(z)\Big ]+\Big [\alpha ^k(x),D(y),\alpha ^k(z)\Big ] \\&+ \Big [\alpha ^k(x),\alpha ^k(y),D(z)\Big ]. \end{aligned}$$

Now, suppose that the property is true to order $n-1$, i.e:

$$\begin{aligned} D([x_1,\ldots ,x_{n-1}]_{n-1})=\displaystyle \sum _{i=1}^n\Big [\alpha ^k(x_1),\ldots ,D(x_k),\ldots ,\alpha ^k(x_{n-1})\Big ]_{n-1}. \end{aligned}$$

If $(x_1,\ldots ,x_n)\in \mathfrak {g}^n$, then

$$\begin{aligned} D([x_1,\ldots ,x_n]_n)= & {} D\Big (\Big [[x_1,\ldots ,x_{n-1}]_{n-1},\alpha ^{n-2}(x_n)\Big ]\Big )\\= & {} \Big [D([x_1,\ldots ,x_{n-1}]_{n-1}),\alpha ^{n+k-2}(x_n)\Big ]\\&+\Big [[\alpha ^k(x_1),\ldots ,\alpha ^k(x_{n-1})]_{n-1},D(\alpha ^{n-2}(x_n))\Big ]\\= & {} \Big [D([x_1,\ldots ,x_{n-1}]_{n-1}),\alpha ^{n-2}\Big (\alpha ^k(x_n)\Big )\Big ]\\&+\Big [\Big [\alpha ^k(x_1),\ldots ,\alpha ^k(x_{n-1})\Big ]_{n-1},\alpha ^{n-2}(D(x_n))\Big ]\\= & {} \displaystyle \sum _{i=1}^{n-1}\Big [\Big [\alpha ^k(x_1),\ldots ,D(x_i),\ldots ,\alpha ^k(x_{n-1})\Big ]_{n-1},\alpha ^{n-2}\Big (\alpha ^k(x_n)\Big )\Big ]\\&+ \Big [\alpha ^k(x_1),\ldots ,\alpha ^k(x_{n-1}),D(x_n)\Big ]_n\\= & {} \displaystyle \sum _{i=1}^{n-1}\Big [\alpha ^k(x_1),\ldots ,D(x_i),\ldots ,\alpha ^k(x_{n-1}),\alpha ^k(x_n)\Big ]_n\\&+ \Big [\alpha ^k(x_1),\ldots ,\alpha ^k(x_{n-1}),D(x_n)\Big ]_n\\= & {} \displaystyle \sum _{i=1}^n\Big [\alpha ^k(x_1),\ldots ,D(x_i),\ldots ,\alpha ^k(x_{n-1}),\alpha ^k(x_n)\Big ]_n. \end{aligned}$$

Proposition 3.15

Let $(\mathfrak {g},[\ ,\ ], \alpha )$ be a multiplicative hom–Lie algebra and $D,D',\ldots ,D^{(n-1)}$ be endomorphisms of $\mathfrak {g}$ such that $D^{(i)}$ is $\alpha ^k$-quasiderivation with associated endomorphism $D^{(i+1)}$ for $0\le i\le n-2$. Then, the $(n + 1)$-tuple $(D,D,D' ,D'', \ldots ,D^{(n-1)})$ is an $(n + 1)$-ary $\alpha ^k$-derivation of $\mathfrak {g}_{n}$.

Proof

If $x_1,\ldots ,x_n\in \mathfrak {g}$, then

$$\begin{aligned} D^{(n-1)}([x_1,\ldots ,x_n]_n)= & {} D^{(n-1)}([[x_1,\ldots ,x_{n-1}]_{n-1},\alpha ^{n-2}(x_n)])\\= & {} [D^{(n-2)}([x_1,\ldots ,x_{n-1}]_{n-1}),\alpha ^k(x_n)] \\&+\Big [\Big [\alpha ^k(x_1),\ldots ,\alpha ^k(x_{n-1})\Big ]_{n-1},D^{(n-2)}(\alpha ^{n-2}(x_n))\Big ]\\&\vdots \\= & {} \Big [D(x_1),\alpha ^k(x_2),\ldots ,\alpha ^k(x_n)\Big ]_n \\&+\Big [\alpha ^k(x_1),D(x_2),\ldots ,\alpha ^k(x_n)\Big ]_n\\&+\Big [\alpha ^k(x_1),\alpha ^k(x_2),D'(x_3),\ldots ,\alpha ^k(x_n)\Big ]_n \\&+\cdots +\Big [\alpha ^k(x_1),\ldots ,\alpha ^{k}(x_{n-1}),D^{(n-2)}(x_n)\Big ]_n. \end{aligned}$$

Therefore, the $(n + 1)$-tuple $(D,D,D',D'', \ldots ,D^{(n-1)})$ is an $(n + 1)$-ary $\alpha ^k$-derivation of $\mathfrak {g}_{n}$. $\square$