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

  • Abdelkader Ben Hassine
  • Sami MabroukEmail author
  • Othmen Ncib
Open Access


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.


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

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
$$\begin{aligned} {[}L_m,L_n]_q= & {} (\{m\}-\{n\})L_{m+n}, \end{aligned}$$
$$\begin{aligned} {[}L_m,I_n]_q= & {} -\{n\}I_{m+n}, \end{aligned}$$
$$\begin{aligned} {[}I_m,I_n]_q= & {} 0. \end{aligned}$$
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
$$\begin{aligned}&\big [\alpha _1(x_1),\ldots ,\alpha _{n-1}(x_{n-1}),[y_1,\ldots ,y_{n}]\big ] \nonumber \\&\quad = \sum _{i=1}^{n}\big [\alpha _1(y_1),\ldots ,\alpha _{i-1}(y_{i-1}),[x_1,\ldots ,x_{n-1},y_i] ,\alpha _i(y_{i+1}),\ldots ,\alpha _{n-1}(y_n)\big ],\nonumber \\ \end{aligned}$$
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
$$\begin{aligned} \text {ad}_X(y)=[x_{1},\ldots ,x_{n-1},y]. \end{aligned}$$
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
$$\begin{aligned} \alpha ([x_1,\ldots ,x_n])=[\alpha (x_1),\ldots ,\alpha (x_n)],\quad \forall x_1,\ldots ,x_n\in \mathcal {N}. \end{aligned}$$
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
$$\begin{aligned} {[}D,\alpha ]=0\ \ {\text {i.e.}}\ \ D\circ \alpha =\alpha \circ D, \end{aligned}$$
$$\begin{aligned} D[x_1,\ldots ,x_n]=\sum _{i=1}^n\Big [\alpha ^k(x_1),\ldots ,\alpha ^k(x_{i-1}),D(x_i),\alpha ^k(x_{i+1}),\ldots ,\alpha ^k(x_n)\Big ]. \end{aligned}$$
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
$$\begin{aligned} \text {ad}^k_X(y)=\Big [x_1,\ldots ,x_{n-1},\alpha ^k(y)\Big ]\quad \forall y\in \mathcal {N}. \end{aligned}$$

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:
$$\begin{aligned} {[}D,D']=D\circ D'-D'\circ D. \end{aligned}$$
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
$$\begin{aligned} D^{(n)}([x_1, \ldots , x_n])=\sum _{i=1}^n\Big [\alpha ^k(x_1), \ldots ,D^{(i-1)}(x_i), \ldots , \alpha ^k(x_n)\Big ], \end{aligned}$$
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:
$$\begin{aligned} {[}x,y,z]=\varphi (x)[y,z]+\varphi (y)[z,x]+\varphi (z)[x,y],\quad \forall \ x,\ y,\ z\in \mathfrak {g}. \end{aligned}$$
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
$$\begin{aligned} \varphi (x)\varphi ([y,z])+\varphi (y)\varphi ([z,x])+\varphi (z)\varphi ([x,y])=0. \end{aligned}$$
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
$$\begin{aligned}&\delta f(u_1,\ldots ,u_{k+1})\nonumber \\&\quad =\sum _{i<j}(-1)^{i+j+1}f([u_i,u_j]_{{\mathfrak {g}}},\alpha (u_1)\ldots ,{\widehat{u_i}},\ldots ,{\widehat{u_j}},\ldots ,\alpha (u_{k+1})),\qquad \end{aligned}$$
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:
$$\begin{aligned} {[}x_1,\ldots ,x_n]_\phi =\sum _{i<j}^{n}(-1)^{i+j+1}\phi (x_1,\ldots ,{\hat{x_i}},\ldots ,{\hat{x_j}},\ldots ,x_n)[x_i,x_j], \end{aligned}$$
for all \(x_1,\ldots ,x_n\in \mathfrak {g}\).

Proposition 2.2

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


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

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
$$\begin{aligned}&\phi \wedge \delta \phi _X=0,\quad \forall X\in \wedge ^{n-3}\mathfrak {g}, \end{aligned}$$
$$\begin{aligned}&\phi \circ (\alpha \otimes Id\otimes \cdots \otimes Id)=\phi . \end{aligned}$$


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 \(({\mathfrak {g}},[~,\ldots ,~]_\phi ,\alpha )\).


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 \(({\mathfrak {g}},[~,\ldots ,~]_\phi ,\alpha )\).


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


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.


  2. 2.


  3. 3.


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.


  2. 2.


  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


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:
$$\begin{aligned}&{[}~,\ldots ,~]_n:{\mathfrak {g}}^{\otimes n} \longrightarrow {\mathfrak {g}}\nonumber \\&(x_1,\ldots ,x_n)\longmapsto \Big [x_1,\ldots ,x_n\Big ]_n = \Big [\big [[\dots [x_1,x_2],\alpha (x_3)],\alpha ^2(x_4)\big ]\ldots \alpha ^{n-3}(x_{n-1})],\alpha ^{n-2}(x_{n})\Big ].\nonumber \\ \end{aligned}$$
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 \((y_1,\ldots ,y_n)\in \mathfrak {g}^n\).


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 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 \(\mathfrak {g}_n\).


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


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



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


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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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