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

  • Abdelkader Ben Hassine
  • Sami MabroukEmail author
  • Othmen Ncib
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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.

Keywords

n-ary Nambu algebra hom–Lie triple system hom–Lie n-tuple system Derivation Quasiderivation 

Mathematics Subject Classification

17A30 17A36 17A40 17A42 

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 xyz from \({\mathfrak {g}}\), where \(\displaystyle \circlearrowleft _{x,y,z}\) denotes summation over the cyclic permutations of xyz.

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 (kl), 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 (ij), 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. 1.

    \([x,x,z]=0,\)

     
  2. 2.

    \([x,y,z]+[y,z,x]+[z,x,y]=0\),

     
  3. 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. 1.

    \([x,x,z]=0,\)

     
  2. 2.

    \([x,y,z]+[y,z,x]+[z,x,y]=0\),

     
  3. 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. 1.

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

     
  2. 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. 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. 1.

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

     
  2. 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. 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 \(Y=(y_1,\ldots ,y_n)\in \mathfrak {g}^n\).
  1. (i)

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

     
  2. (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 \(x_1,\ldots ,x_{n}\in \mathcal {G}\).

     
  3. (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 \(\mathfrak {g}_{n}\). \(\square \)

Notes

Acknowledgements

We would like to thank Abdenacer Makhlouf and Viktor Abramov for helpful discussions and for their interest in this work.

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Abdelkader Ben Hassine
    • 1
    • 2
  • Sami Mabrouk
    • 3
    Email author
  • Othmen Ncib
    • 3
  1. 1.Department of Mathematics, Faculty of Science and Arts at BelqarnUniversity of BishaBishaKingdom of Saudi Arabia
  2. 2.Faculty of SciencesUniversity of SfaxSfaxTunisia
  3. 3.Faculty of SciencesUniversity of GafsaGafsaTunisia

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