Generalized Derivations and Rota-Baxter Operators of n -ary Hom-Nambu Superalgebras
aa r X i v : . [ m a t h . QA ] M a r Generalized Derivations and Rota-Baxter Operators of n -ary Hom-Nambu Superalgebras Sami Mabrouk , Othmen Ncib , Sergei Silvestrov Faculty of Sciences, University of Gafsa, BP 2100, Gafsa, [email protected], [email protected] Division of Applied Mathematics, School of Education, Culture and Communication,M¨alardalen University, Box 883, 72123 V¨aster˚as, [email protected]
Abstract
The aim of this paper is to generalise the construction of n -ary Hom-Lie bracket bymeans of an ( n − n -Hom-Lie superalgebras. We study the notion of generalized derivation and Rota-Baxteroperators of n -ary Hom-Nambu and n -Hom-Lie superalgebras and their relation withgeneralized derivation and Rota-Baxter operators of Hom-Lie superalgebras. We alsointroduce the notion of 3-Hom-pre-Lie superalgebras which is the generalization of3-Hom-pre-Lie algebras. Introduction
Hom-Lie algebras and more general quasi-Hom-Lie algebras were introduced first by Hartwig,Larsson and Silvestrov in [48], where the general quasi-deformations and discretizations ofLie algebras of vector fields using more general σ -derivations (twisted derivations) and a gen-eral method for construction of deformations of Witt and Virasoro type algebras based ontwisted derivations have been developed. The general quasi-Lie algebras and the subclassesof quasi-Hom-Lie algebras and Hom-Lie algebras and their more general color Hom-algebra Mathematics subject classification : 17A30,17A36,17A40,17A42
Keywords : Hom-Lie superalgebras, n -ary Nambu superalgebras, n -ary Hom-Nambu superalgebras, n -Hom-Lie superalgebras, derivations, quasiderivations, Rota-Baxter operators, 3-Hom-pre-Lie algebras. G -Hom-associative algebras including Hom-associative al-gebras, Hom-Vinberg algebras (Hom-left symmetric algebras), Hom-pre-Lie algebras (Hom-right symmetric algebras) and some other Hom-algebra structures, were introduced andshown to be Hom-Lie admissible. Also, flexible Hom-algebras have been introduced andsome connections to some Hom-algebra generalizations of derivations and of adjoint mapshave been noticed, and the variety of n -dimensional Hom-Lie algebras have been consideredand some classes of low-dimensional Hom-Lie algebras have been described. Since the pi-oneering works [48, 59–63, 70, 77, 81], Hom-algebra structures have developed in a popularbroad area with increasing number of publications in various directions. In Hom-algebrastructures, defining algebra identities are twisted by linear maps. Hom-algebra structuresof a given type include their classical counterparts and open more possibilities for deforma-tions, Hom-algebra extensions of cohomological structures and representations, formal de-formations of Hom-associative and Hom-Lie algebras, Hom-Lie admissible Hom-coalgebras,Hom-coalgebras, Hom-Hopf algebras [9, 35, 60, 71–73, 80, 88, 89].The n -Lie algebras found their applications in many fields of Mathematics and Physics.Ternary Lie algebras appeared first in Nambu generalization of Hamiltonian mechanics [75]using ternary bracket generalization of Poisson algebras. The algebraic foundations of Nambumechanics and foundations of the theory of Nambu-Poisson manifolds have been developedin the works of Takhtajan and Daletskii in [40, 83, 84]. Filippov, in [42] introduced n -Liealgebras. Further properties, classification, and connections to other structures such as bial-gebras, Yang-Baxter equation and Manin triples for 3-Lie algebras of n -ary algebras werestudied in [19–28, 49]. Hom-algebra generalization of n -ary algebras, such as n -Hom-Liealgebras and other n -ary Hom algebras of Lie type and associative type, were introducedin [17], by twisting the defining identities using a set of linear maps. A way to generate ex-amples of such n -ary Hom-algebras from n -ary algebras of the same type has been described.Representations and cohomology of n -ary multiplicative Hom-Nambu-Lie algebras have beenconsidered in [10]. Further properties, construction methods, examples, cohomology and cen-tral extensions of n -ary Hom-algebras have been considered in [14–16, 55, 56, 89–92]. Thesegeneralizations include n -ary Hom-algebra structures generalizing the n -ary algebras of Lietype including n -ary Nambu algebras, 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 partiallyassociative algebras. 2he construction of ( n + 1)-Lie algebras induced by n -Lie algebras using combination ofbracket multiplication with a trace, motivated by the work of Awata et al. [18] on the quanti-zation of the Nambu brackets, was generalized using the brackets of general Hom-Lie algebraor n -Hom-Lie algebra and trace-like linear forms satisfying some conditions depending on thelinear maps defining the Hom-Lie or n -Hom-Lie algebras in [15, 16]. The structure of 3-Liealgebras induced by Lie algebras, classification of 3-Lie algebras and application to construc-tions of B.R.S. algebras have been considered in [4, 5, 7]. Interesting constructions of ternaryLie superalgebras in connection to superspace extension of Nambu-Hamilton equation is con-sidered in [8]. In [33], a method was demonstrated of how to construct n -ary multiplicationsfrom the binary multiplication of a Hom-Lie algebra and a ( n − n -Hom-Lie Algebras and( n + 1)-Hom-Lie Algebras Induced by n -Hom-Lie Algebras have been considered in [58].In [37], Leibniz n -algebras have been studied. The general cohomology theory for n -Liealgebras and Leibniz n -algebras was established in [79]. The structure and classification offinite-dimensional n -Lie algebras were considered in [66] and many other authors. For moredetails of the theory and applications of n -Lie algebras, see [41] and references therein.Derivations and generalized derivations of different algebraic structures are an importantsubject of study in algebra and diverse areas. They appear in many fields of Mathematics andPhysics. In particular, they appear in representation theory and cohomology theory amongother areas. They have various applications relating algebra to geometry and allow theconstruction of new algebraic structures. There are many generalizations of derivations. Forexample, Leibniz derivations [50] and δ -derivations of prime Lie and Malcev algebras [43–45].The properties and structure of generalized derivations algebras of a Lie algebra and theirsubalgebras and quasi-derivation algebras were systematically studied in [65], where it wasproved for example that the quasi-derivation algebra of a Lie algebra can be embedded intothe derivation algebra of a larger Lie algebra. Derivations and generalized derivations of n -ary algebras were considered in [76, 86] and it was demonstrated substantial differencesin structures and properties of derivations on Lie algebras and on n -ary Lie algebras for n >
2. Generalized derivations of Lie superalgebras have been considered in [87]. Generalizedderivations of Lie color algebras and n -ary (color) algebras have been studied in [38, 51–54].Generalized derivations of Lie triple systems have been considered in [39]. Generalizedderivations of various kinds can be viewed as a generalization of δ -derivation. Quasi-Hom-Lie and Hom-Lie structures for σ -derivations and ( σ, τ )-derivations have been consideredin [46, 48, 63, 77, 78]. Graded q -differential algebra and applications to semi-commutativeGalois Extensions and Reduced Quantum Plane and q -connection was studied in [2, 3, 6].Generalized N -complexes coming from twisted derivations where considered in [64].Generalizations of derivations in connection with extensions and enveloping algebras ofHom-Lie color algebras have been considered in [12, 13, 31]. Generalized derivations of mul-3iplicative n -ary Hom-Ω color algebras have been studied in [36]. Derivations, L -modules, L -comodules and Hom-Lie quasi-bialgebras have been considered in [29, 30]. In [57], con-structions of n -ary generalizations of BiHom-Lie algebras and BiHom-associative algebrashave been considered. Generalized Derivations of n -BiHom-Lie algebras have been studiedin [34]. Color Hom-algebra structures associated to Rota-Baxter operators have been con-sidered in context of Hom-dendriform color algebras in [32]. Rota-Baxter bisystems and co-variant bialgebras, Rota-Baxter cosystems, coquasitriangular mixed bialgebras, coassociativeYang-Baxter pairs, coassociative Yang-Baxter equation and generalizations of Rota-Baxtersystems and algebras, curved O -operator systems and their connections with (tri)dendriformsystems and pre-Lie algebras have been considered in [67–69]. Generalisations of derivationsare important for Hom-Gerstenhaber algebras, Hom-Lie algebroids and Hom-Lie-Rinehartalgebras and Hom-Poisson homology [74].This paper is organized as follows. In Section 1 we review basic concepts of Hom-Lie, n -ary Hom-Nambu superalgebras and n -Hom-Lie algebras. We also recall some examplesand classification of Hom-Lie superalgebras of dimension two. We recall the definition ofgeneralized derivations of n -Hom-Lie superalgebras and n -ary Hom-Nambu superalgebras.In Section 2 we provide a construction procedure of n -Hom-Lie superalgebras starting from abinary bracket of a Hom-Lie superalgebra and multilinear form satisfying certain conditions.To this end, we give the relation between generalized derivations of Hom-Lie superalgebraand generalized derivations n -Hom-Lie algebras. In Section 3, we provide a construction for n -ary Hom-Nambu algebra using Hom-Lie algebra. In Section 4 the notion of Rota-Baxteroperators of n -ary Hom-Nambu superalgebras are introduced and some results obtained.Finally, we give the definition of 3-Hom-pre-Lie superalgebras generalizing 3-Hom-pre-Liealgebras in graded case. n -ary Hom-Lie algebras and Hom-Lie superalgebras Throughout this paper, we will for simplicity of exposition assume that K is an algebraicallyclosed field of characteristic zero, even though for most of the general definitions and resultsin the paper this assumption is not essential.Let V = V ⊕ V be a finite-dimensional Z -graded linear space. Let H ( g ) = V ∪ V denotethe set of homogeneous elements of g . If v ∈ V is a homogenous element, then its degreewill be denoted by | v | , where | v | ∈ Z and Z = { , } . Let End ( V ) be the Z -graded linearspace of endomorphisms of a Z -graded linear space V = V ⊕ V . The composition of twoendomorphisms a ◦ b determines the structure of superalgebra in End ( V ), and the graded4inary commutator [ a, b ] = a ◦ b − ( − | a || b | b ◦ a induces the structure of Lie superalgebrasin End ( V ). Definition 1.1.
A Hom-Lie superalgebra is a Z -graded linear space g = g ⊕ g over a field K equipped with an even bilinear map [ · , · ] : g × g → g , (meaning that [ g i , g j ] ⊂ g i + j , ∀ i, j ∈ Z )and an even linear map α : g → g (meaning that α ( g i ) ⊆ g i , ∀ i ∈ Z ).[ x, y ] = − ( − | x || y | [ y, x ] (super-skew-symmetry) (cid:9) x,y,z ( − | x || z | [ α ( x ) , [ y, z ]] = 0 (super-Hom-Jacobi identity)for all x, y, z ∈ H ( g ), where (cid:9) x,y,z denotes summation over the cyclic permutations of x, y, z . Definition 1.2.
A Hom-Lie superalgebra ( g , [ · , · ] , α ) is called multiplicative if α ([ x, y ]) =[ α ( x ) , α ( y )] for all x, y ∈ g .For any x ∈ g , define ad x ∈ End K ( g ) by ad x ( y ) = [ x, y ], for any y ∈ g . Then thesuper-Hom-Jacobi identity can be written asad [ x,y ] ( α ( z )) = ad α ( x ) ◦ ad y ( z ) − ( − | x || y | ad α ( y ) ◦ ad x ( z ) (1)for all x, y, z ∈ H ( g ). Remark . An ordinary Lie superalgebra is a Hom-Lie superalgebra when α = id . Example 1.4 ( [1]) . Let A be the complex superalgebra A = A ⊕ A where A = C [ t, t − ] is the Laurent polynomials in one variable and A = θ C [ t, t − ] , where θ is the Grassmanvariable ( θ = 0) . We assume that t and θ commute. The generators of A are of the form t n and θt n for n ∈ Z . For q ∈ C \{ , } and n ∈ N , we set { n } = − q n − 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 A q be a superspace with basis { L m , I m | m ∈ Z } of parity and { G m , T m | m ∈ Z } ofparity , where L m = − t m D, I m = − t m , G m = − θt m D, T m = − θt m and D is a q -derivationon A such that D ( t m ) = { m } t m , D ( θt m ) = { m + 1 } θt m . We define the bracket [ · , · ] q : A q × A q → A q , with respect the super-skew-symmetry for n, m ∈ Z by [ L m , L n ] q = ( { m } − { n } ) L m + n , (2)5 L m , I n ] q = −{ n } I m + n , (3)[ L m , G n ] q = ( { m } − { n + 1 } ) G m + n , (4)[ I m , G n ] q = { m } T m + n , (5)[ L m , T n ] q = −{ n + 1 } T m + n , (6)[ I m , I n ] q = [ I m , T n ] q = [ T m , G n ] q = [ T m , T n ] q = [ G m , G n ] q = 0 . (7) Let α q be an even linear map on A q defined on the generators by α q ( L n ) = (1 + q n ) L n , α q ( I n ) = (1 + q n ) I n ,α q ( T n ) = (1 + q n +1 ) G n , α q ( T n ) = (1 + q n +1 ) T n . The triple ( A q , [ · , · ] q , α q ) is a Hom-Lie superalgebra, called q -deformed Heisenberg-Virasorosuperalgebra of Hom-type. Example 1.5. In [11] , the authors construct an example of Hom-Lie superalgebra, which isnot a Lie superalgebra starting from the orthosymplectic Lie superalgebra. We consider inthe sequel the matrix realization of this Lie superalgebra.Let osp (1 ,
2) = V ⊕ V be the Lie superalgebra where V is generated by: H = − , X = , Y = , and V is generated by F = , G = −
10 0 0 . Those of the defining relations that have nonzero elements in the right-hand side are [ H, X ] = 2 X, [ H, Y ] = − Y, [ X, Y ] = H, [ Y, G ] = F, [ X, F ] = G, [ H, F ] = − F, [ H, G ] = G, [ G, F ] = H, [ G, G ] = − X, [ F, F ] = 2 Y. Let λ ∈ R ∗ , we consider the linear map α λ : osp (1 , → osp (1 , defined by: α λ ( X ) = λ X, α λ ( Y ) = 1 λ Y, α λ ( H ) = H, λ ( F ) = 1 λ F, α λ ( G ) = λG. We provide a family of Hom-Lie superalgebras osp (1 , λ = ( osp (1 , , [ · , · ] α λ , α λ ) , where theHom-Lie superalgebra bracket [ · , · ] α λ on the basis elements is given, for λ = 0 , by: [ H, X ] α λ = 2 λ X, [ H, Y ] α λ = − λ Y, [ X, Y ] α λ = H, [ Y, G ] α λ = 1 λ F, [ X, F ] α λ = λG, [ H, F ] α λ = − λ F, [ H, G ] α λ = λG, [ G, F ] α λ = H, [ G, G ] α λ = − λ X, [ F, F ] α λ = 2 λ Y. These Hom-Lie superalgebras are not Lie superalgebras for λ = 1 . Theorem 1.6 ( [85]) . Every -dimensional multiplicative Hom-Lie superalgebra ( g = g ⊕ g , [ · , · ] , α ) generated by { e , e } is isomorphic to one of the following nonisomorphic Hom-Lie superalgebras. Each algebra is denoted by g ki,j , where i is the dimension of g , j is thedimension of g , k is the number. g , : is an abelian Hom-Lie superalgebra. g , : is an abelian Hom-Lie superalgebra. g , : [ e , e ] = e , [ e , e ] = 0 and α = (cid:18) a (cid:19) , a ∈ K . g , : [ e , e ] = e , [ e , e ] = 0 and α = (cid:18) a
00 0 (cid:19) , a = 0 , . g , : [ e , e ] = 0 , [ e , e ] = e and α = (cid:18) a a (cid:19) , a = 0 . Now, we recall the definitions of n -ary Hom-Nambu superalgebras and n -Hom-Lie super-algebras, generalizing of n -ary Nambu superalgebras and n -Lie superalgebras (see [1]). Definition 1.7. An n -ary Hom-Nambu superalgebra ( N , [ · , . . . , · ] , e α ) is a triple consistingof a linear space N = N ⊕ N , an even n -linear map [ · , . . . , · ] : N n → N such that[ N k , . . . , N k n ] ⊂ N k + ··· + k n and a family e α = ( α i ) ≤ i ≤ n − of even linear maps α i : N → N ,satisfying ∀ ( x , . . . , x n − ) ∈ H ( N ) n − , ( y , . . . , y n ) ∈ H ( N ) n :7 α ( x ) , . . . ., α n − ( x n − ) , [ y , . . . ., y n ] (cid:3) = (8) n X i =1 ( − | X || Y | i − (cid:2) α ( y ) , . . . ., α i − ( y i − ) , [ x , . . . ., x n − , y i ] , α i ( y i +1 ) , . . . , α n − ( y n ) (cid:3) , where | X | = n − X k =1 | x k | and | Y | i − = i − X k =1 | y k | . The identity (8) is called super-Hom-Nambu identity .Let e α : N n − → N n − be even linear maps defined for all X = ( x , . . . , x n − ) ∈ N n − by e α ( X ) = ( α ( x ) , . . . , α n − ( x n − )) ∈ N n − . For all X = ( x , . . . , x n − ) ∈ N n − , the mapad X : N → N defined by ad X ( y ) = [ x , . . . , x n − , y ] , ∀ y ∈ N , (9)is called adjoint map. Then the super-Hom-Nambu identity (8) may be written in terms ofadjoint map asad e α ( X ) ([ y , . . . , y n ]) = n − X i =1 ( − | X || Y | i − [ α ( y ) , . . . , α i − ( y i − ) , ad X ( y i ) ,α i ( y i +1 ) . . . , α n − ( y n )] . Definition 1.8. An n -ary Hom-Nambu superalgebra ( N , [ · , . . . , · ] , e α ) is called n -Hom-Liesuperalgebra if the bracket [ · , . . . , · ] is super-skewsymmetric that is ∀ ≤ i ≤ n − x , . . . , x i , x i +1 , . . . , x n ] = − ( − | x i || x i +1 | [ x , . . . , x i +1 , x i , . . . , x n ] . (10)It is equivalent to ∀ ≤ i < j ≤ n :[ x , . . . , x i , . . . , x j , . . . , x n ] = − ( − | X | j − i +1 ( | x i | + | x j | )+ | x i || x j | [ x , . . . , x j , . . . , x i , . . . , x n ] (11)where x , . . . , x n ∈ H ( N ) and | X | ji = j X k = i | x k | . Remark . When the maps ( α i ) ≤ i ≤ n − are all identity maps, one recovers the classical n -ary Nambu superalgebras. 8et ( N , [ · , . . . , · ] , e α ) and ( N ′ , [ · , . . . , · ] ′ , e α ′ ) be two n -ary Hom-Nambu superalgebras where e α = ( α i ) ≤ i ≤ n − and e α ′ = ( α ′ i ) ≤ i ≤ n − . A linear map f : N → N ′ is an n -ary Hom-Nambusuperalgebras morphism if it satisfies f ([ x , . . . , x n ]) = [ f ( x ) , . . . , f ( x n )] ′ ,f ◦ α i = α ′ i ◦ f, ∀ i = 1 , . . . , n − . In the sequel we deal sometimes with a particular class of n -ary Hom-Nambu superalgebraswhich we call n -ary multiplicative Hom-Nambu superalgebras. Definition 1.10. A multiplicative n -ary Hom-Nambu superalgebra (resp. multiplicative n -Hom-Lie superalgebra ) is an n -ary Hom-Nambu superalgebra (resp. n -Hom-Lie superal-gebra) ( N , [ · , . . . , · ] , e α ) with e α = ( α i ) ≤ i ≤ n − where α = · · · = α n − = α and satisfying α ([ x , . . . , x n ]) = [ α ( x ) , . . . , α ( x n )] , ∀ x , . . . , x n ∈ N . (12)For simplicity, denote the n -ary multiplicative Hom-Nambu superalgebra as ( N , [ · , . . . , · ] , α )where α : N → N is an even linear map. Also by misuse of language an element X ∈ N n refers to X = ( x , . . . , x n ) and α ( X ) denotes ( α ( x ) , . . . , α ( x n )). Definition 1.11.
A multiplicative n -ary Hom-Nambu superalgebra ( N , [ · , . . . , · ] , α ) is calledregular if α is bijective. n -ary Hom-Nambu superalgebras In this section we recall the definition of derivation, quasiderivation and generalized deriva-tion of multiplicative n -ary Hom-Nambu superalgebras.Let ( N , [ · , . . . , · ] , α ) be a multiplicative n -ary Hom-Nambu superalgebra. We denote by α k the k -times composition of α (i.e. α k = α ◦ · · · ◦ α | {z } k − times ). In particular α = Id and α = α . If( N , [ · , . . . , · ] , α ) is a regular Hom-Lie superalgebra. Definition 1.12.
For any k ≥
1, we call D ∈ End ( N ) an α k - derivation of the multiplicative n -ary Hom-Nambu superalgebra ( N , [ · , . . . , · ] , α ) if[ D , α ] = 0 i.e. D ◦ α = α ◦ D ; (13) D [ x , . . . , x n ] = n X i =1 ( − | D || X | i − [ α k ( x ) , . . . , α k ( x i − ) , D ( x i ) , α k ( x i +1 ) , . . . , α k ( x n )] . (14)We denote by Der α k ( N ) the set of α k -derivations of the multiplicative n -Hom-Lie superal-gebra N . 9or X = ( x , . . . , x n − ) ∈ N n − satisfying α ( X ) = X and k ≥
1, we define the mapad kX ∈ End ( N ) by ad kX ( y ) = [ x , . . . , x n − , α k ( y )] ∀ y ∈ N . (15)Then, we find the following result. Lemma 1.13.
The map ad kX is an α k +1 -derivation (called inner α k +1 -derivation), and | ad kX | = | X | . We denote by
Inn α k ( N ) the space generate by all the inner α k +1 -derivations. For any D ∈ Der α k ( N ) and D ′ ∈ Der α k ′ ( N ) we define their supercommutator [ D , D ′ ] as usual:[ D , D ′ ] = D ◦ D ′ − ( − | D || D ′ | D ′ ◦ D , (16)then [ D , D ′ ] ∈ Der α k + k ′ ( N ). Set Der ( N ) = M k ≥ Der α k ( N ) and Inn ( N ) = M k ≥ Inn α k ( N ),the pair ( Der ( N ) , [ · , · ]) is a Lie superalgebra. Definition 1.14.
Let ( N , [ · , . . . , · ] , α ) be a multiplicative n -ary Hom-Nambu superalgebra.An endomorphism D ∈ End ( N ) is said to be an α k − quasiderivation, if there exists anendomorphism D ′ ∈ End ( N ) such that n X i =1 ( − | D || X | i − [ α k ( x ) , . . . , D ( x i ) , . . . , α k ( x n )] = D ′ ([ x , . . . , x n ]) , for all x , . . . , x n ∈ N . We call D ′ the endomorphism associated to α k − quasiderivation D .We denote the set of α k -quasiderivations by QDer α k ( N ) and QDer ( N ) = M k ≥ QDer α k ( N ) . Definition 1.15.
An endomorphism D of a multiplicative n -ary Hom-Nambu superalgebra( N , [ · , . . . , · ] , α ) is called a generalized α k -derivation if there exist linear mappings D ′ , D ′′ , . . . , D ( n − , D ( n ) ∈ End ( N )such that D ( n ) ([ x , . . . , x n ]) = n X i =1 ( − | D ( i − || X | i − [ α k ( x ) , . . . , D ( i − ( x i ) , . . . , α k ( x n )] , (17)for all x , . . . , x n ∈ N . An ( n + 1)-tuple ( D , D ′ , D ′′ , . . . , D ( n − , D ( n ) ) is called an ( n + 1)-ary α k -derivation.We denote the set of generalized α k -derivations by GDer α k ( N ) and GDer ( N ) = M k ≥ GDer α k ( N ) . n -Hom-Lie superalgebras induced by Hom-Lie su-peralgebras In [47], the authors introduced a construction of a 3-Hom-Lie superalgebra from a Hom-Liesuperalgebra. It is called 3-Hom-Lie superalgebra induced by Hom-Lie superalgebra. In thissection we generalize this construction to the n -ary Hom-algebras by the approach in [4].Let ( g , [ · , · ] , α ) be a multiplicative Hom-Lie superalgebra and g ∗ be its dual superspace.Fix an even element of the dual space ϕ ∈ g ∗ . Define the triple product as follows ∀ x, y, z ∈ H ( g ) :[ x, y, z ] = ϕ ( x )[ y, z ] + ( − | x | ( | y | + | z | ) ϕ ( y )[ z, x ] + ( − | z | ( | x | + | y | ) ϕ ( z )[ x, y ] . (18)Obviously this triple product is super-skew-symmetric. It is straightforward to compute theleft-hand side and the right-hand side of the super-Hom-Nambu identity (8) if ϕ ◦ α = ϕ and ϕ ( x ) ϕ ([ y, z ]) + ( − | x | ( | y | + | z | ) ϕ ( y ) ϕ ([ z, x ]) + ( − | z | ( | x | + | y | ) ϕ ( z ) ϕ ([ x, y ]) = 0 . (19)Now we consider ϕ as an even K -valued cochain of degree one of the Chevalley-Eilenbergcomplex of a Hom-Lie superalgebra g . Let coboundary operator δ : ∧ k g ∗ → ∧ k g ∗ be definedby δf ( x , . . . , x k +1 ) = X i Let φ ∈ ∧ n − g ∗ be an even ( n − n -ary product asfollows [ x , . . . , x n ] φ = n X i The n -ary product [ · , . . . , · ] φ is super-skew-symmetric.Proof. Let x , . . . , x n ∈ H ( g ) and fix an integer 1 ≤ i ≤ n − 1. Then,[ x , . . . , x i , x i +1 , . . . , x n ] φ = X k Let ( g , [ · , · ] , α ) be a multiplicative Hom-Lie superalgebra, g ∗ its dual and φ be an even cochain of degree n − , i.e. φ ∈ ∧ n − g ∗ . The linear space g equipped with then-ary product (21) and the even linear map α is a multiplicative n-Hom-Lie superalgebra ifand only if φ ∧ δφ X = 0 , ∀ X ∈ ∧ n − H ( g ) , (22) φ ◦ ( α ⊗ Id ⊗ · · · ⊗ Id ) = φ. (23) Proof. Firstly, if ( x , . . . , x n ) ∈ ∧ n H ( g ), then[ α ( x ) , . . . , α ( x n )] φ = 13 n X i Let φ : g ∧ · · · ∧ g → K be an even super-skewsymmetric linear form of themultiplicative Hom-Lie superalgebra ( g , [ · , · ] , α ), then φ is called supertrace if: φ ◦ ([ · , · ] ∧ Id ∧ · · · ∧ Id ) = 0 and φ ◦ ( α ∧ Id ∧ · · · ∧ Id ) = φ. Corollary 2.6. Let φ : g ∧ n − → K be a supertrace of Hom-Lie superalgebra ( g , [ · , · ] , α ) , then g φ = ( g , [ ., . . . , . ] φ , α ) is a n -Hom-Lie superalgebra. Proposition 2.7. Let ( g , [ · , · ] , α ) be a Hom-Lie superalgebra, and let D ∈ Der ( g ) be an α k -derivation such that n − X i =1 ( − | D || X | i − φ ( x , . . . D ( x i ) , . . . , x n − ) = 0 . Then D is an α k -derivation of the n -Hom-Lie superalgebra ( g , [ · , . . . , · ] φ , α ) .Proof. Let X = ( x , . . . , x n ) ∈ ∧ n H ( g ), on the one hand we get D ([ x , . . . , x n ] φ ) = D (cid:16) X i 3, [ x , . . . , x n ] n = [[ x , . . . , x n − ] n − , α n − ( x n )]. Theorem 3.1. Let ( g , [ · , · ] , α ) be a multiplicative Hom-Lie superalgebra. Then g n = ( g , [ · , . . . , · ] n , α n − ) is a multiplicative n -ary Hom-Nambu superalgebra. To prove this theorem we need the following lemma. Lemma 3.2. Let ( g , [ · , · ] , α ) be a multiplicative Hom-Lie superalgebra, and ad the adjointmap defined by ad x ( y ) = ad x ( y ) = [ x, y ] . Then, we havead α n − ( x ) [ y , . . . , y n ] n = n X k =1 ( − | x || Y | k − [ α ( y ) , . . . , α ( y k − ) , ad x ( y k ) , α ( y k +1 ) , . . . , α ( y n )] n , where x ∈ H ( g ) , y ∈ H ( g ) and ( y , . . . , y n ) ∈ H ( g ) n . roof. For n = 2, using the super-Hom-Jacobi identity we havead α ( x ) [ y, z ] = [ α ( x ) , [ y, z ]] = [[ x, y ] , α ( z )] + ( − | x || y | [ α ( y ) , [ x, z ]]= [ad x ( y ) , α ( z )] + ( − | x || y | [ α ( y ) , ad x ( z )] . Assume that the property is true up to order n , that isad α n − ( X ) [ y , . . . , y n ] n = n X k =1 ( − | X || Y | k − [ α ( y ) , . . . , α ( y k − ) , ad X ( y k ) , α ( y k +1 ) , . . . , α ( y n )] n . Let x ∈ H ( g ) and ( y , . . . , y n +1 ) ∈ H ( g ) n +1 , we havead α n ( x ) [ y , . . . , y n +1 ] = ad α n ( x ) [[ y , . . . , y n ] n , α n − ( y n +1 )] = h ad α n − ( x ) [ y , . . . , y n ] n , α n ( y n +1 ) i + ( − | x || Y | h [ α ( y ) , . . . , α ( y n )] n , ad α n − ( x ) ( α n − ( y n +1 )) i = n X k =1 h [ α ( y ) , . . . , α ( y k − ) , ad x ( y k ) , α ( y k +1 ) , . . . , α ( y n )] n , α n ( y n +1 ) i + ( − | x || Y | k − h [ α ( y ) , . . . , α ( y n )] n , α n − (ad x ( y n +1 )) i = n X k =1 ( − | x || Y | k − h α ( y ) , . . . , α ( y k − ) , ad x ( y k ) , α ( y k +1 ) , . . . , α ( y n ) , α ( y n +1 ) i n +1 + h α ( y ) , . . . , α ( y n ) , ad x ( y n +1 ) i n +1 = n +1 X k =1 ( − | x || Y | k − h α ( y ) , . . . , α ( y k − ) , ad x ( y k ) , α ( y k +1 ) , . . . , α ( y n +1 ) i n +1 . The lemma is proved. Proof. ( Proof of Theorem 3.1 ) Let X = ( x , . . . , x n − ) ∈ H ( g ) n − and Y = ( y , . . . , y n ) ∈H ( g ) n . Using Lemma 3.2, we have h α n − ( x ) , . . . , α n − ( x n − ) , [ y , . . . , y n ] n i n = h [ α n − ( x ) , . . . , α n − ( x n − )] n − , [ α n − ( y ) , . . . , α n − ( y n )] n i =ad α n − [ x ,...,x n − ] n − ([ α n − ( y ) , . . . , α n − ( y n )] n )= n X k =1 ( − | X || Y | k − h α n − ( y ) , . . . , ad x ,...,x n − ] n − ( α n − ( y k )) , . . . , α n − ( y n ) i n n X k =1 ( − | X || Y | k − h α n − ( y ) , . . . , [[ x , . . . , x n − ] n − , α n − ( y k )] , . . . , α n − ( y n ) i n = n X k =1 ( − | X || Y | k − h α n − ( y ) , . . . , [ x , . . . , x n − , y k ] n , . . . , α n − ( y n ) i n . Example 3.3. Consider the -dimensional multiplicative Hom-Lie superalgebras g , and g , given in Theorem 1.6. We can construct a multiplicative n -ary Hom-Nambu superalgebrasstructures on g , and g , given respectively by: [ e , e , . . . , e ] n = ( − n − e and [ e , e , . . . , e ] n = − ( − a ) n − e . The other brackets are zero. Example 3.4 ( [85]) . Consider a -dimensional graded linear space L = L ⊕ L , where L is generated by e and L is generated by e , e . Define an even linear map α : L → L by α ( e ) = ae , α ( e ) = ae α ( e ) = e and an even super-skewsymmetric bilinear map [ · , · ] : L × L → L given by [ e , e ] = [ e , e ] = [ e , e ] = 0 , [ e , e ] = be , [ e , e ] = ce , where a, b, c are parameters and a = 0 . Then ( L, [ · , · ] , α ) is a multiplicative Hom-Lie super-algebra. Therefore, using Theorem 3.1, we can construct a multiplicative -ary Hom-Nambusuperalgebra ( L, [ · , · , · ] , α ) , where the ternary bracket [ · , · , · ] is given by [ e , e , e ] = bce and [ e , e , e ] = bce . We can also construct a multiplicative n -ary Hom-Nambu superalge-bra ( L, [ · , . . . , · ] n , α n − ) , where the n -ary bracket [ · , . . . , · ] n is given by: • If n = 4 p , then [ e , e , e , . . . , e ] = b p c p +1 e and [ e , e , e , . . . , e ] = b p +1 c p e . • If n = 4 p + 1 , [ e , e , e , . . . , e ] = b p +1 c p +1 e and [ e , e , e , . . . , e ] = b p +1 c p +1 e . • If n = 4 p + 2 , then [ e , e , e , . . . , e ] = b p +1 c p +2 e and [ e , e , e , . . . , e ] = b p +2 c p +1 e . • If n = 4 p + 3 , then [ e , e , e , . . . , e ] = b p +2 c p +2 e and [ e , e , e , . . . , e ] = b p +2 c p +2 e . Proposition 3.5. Let ( g , [ · , · ] , α ) be a multiplicative Hom-Lie superalgebra and D : g → g an α k -derivation of g for an integer k . Then D is an α k -derivation of g n .Proof. We use the mathematical induction. For n = 3, given x, y, z ∈ H ( g ), we have D ([ x, y, z ]) = D ([[ x, y ] , α ( z )]) 19[ D ([ x, y ]) , α k +1 ( z )] + ( − | D || [ x,y ] | [[ α k ( x ) , α k ( y )] , D ( α ( z ))]=[[ D ( x ) , α k ( y )] , α k +1 ( z )] + ( − | D || x | [[ α k ( x ) , D ( y )] , α k +1 ( z )]+ ( − | D | ( | x | + | y | ) [[ α k ( x ) , α k ( y )] , α ( D ( z ))]=[ D ( x ) , α k ( y ) , α k ( z )] + ( − | D || x | [ α k ( x ) , D ( y ) , α k ( z )] + ( − | D | ( | x | + | y | ) [ α k ( x ) , α k ( y ) , D ( z )] . Now, suppose that the property is true to order n − D ([ x , . . . , x n − ] n − ) = n X i =1 ( − | D || X | i − [ α k ( x ) , . . . , D ( x k ) , . . . , α k ( x n − )] n − . If ( x , . . . , x n ) ∈ g n , then D ([ x , . . . , x n ] n ) = D ( h [ x , . . . , x n − ] n − , α n − ( x n ) i )= h D ([ x , . . . , x n − ] n − ) , α n + k − ( x n ) i + ( − | D || [ x ,...,x n − ] n − | h [ α k ( x ) , . . . , α k ( x n − )] n − , D ( α n − ( x n )) i = h D ([ x , . . . , x n − ] n − ) , α n − ( α k ( x n )) i + ( − | D || X | n − h [ α k ( x ) , . . . , α k ( x n − )] n − , α n − ( D ( x n )) i = n − X i =1 ( − | D || X | i − h [ α k ( x ) , . . . , D ( x i ) , . . . , α k ( x n − )] n − , α n − ( α k ( x n )) i + ( − | D || X | n − [ α k ( x ) , . . . , α k ( x n − ) , D ( x n )] n = n − X i =1 ( − | D || X | i − [ α k ( x ) , . . . , D ( x i ) , . . . , α k ( x n − ) , α k ( x n )] n + ( − | D || X | n − [ α k ( x ) , . . . , α k ( x n − ) , D ( x n )] n = n X i =1 ( − | D || X | i − [ α k ( x ) , . . . , D ( x i ) , . . . , α k ( x n − ) , α k ( x n )] n , which completes the proof. Proposition 3.6. Let ( g , [ · , · ] , α ) be a multiplicative Hom-Lie superalgebra. For endomor-phisms D , D ′ , . . . , D ( n − of g such that D ( i ) is an α k -quasiderivation with associated endomor-phism D ( i +1) for ≤ i ≤ n − , the ( n + 1) -tuple ( D , D , D ′ , D ′′ , . . . , D ( n − ) is an ( n + 1) -ary α k -derivation of g n . roof. Let x , . . . , x n ∈ g , then D ( n − ([ x , . . . , x n ] n ) = D ( n − ([[ x , . . . , x n − ] n − , x n ])= [ D ( n − ([ x , . . . , x n − ] n − ) , α k ( x n )]+ ( − | D ( n − || X | n − [[ α k ( x ) , . . . , α k ( x n − )] n − , D ( n − ( x n )]= [[ D ( n − ([ x , . . . , x n − ]) , α k ( x n )] , α k ( x n )]+ ( − | D ( n − || X | n − hh [ α k ( x ) , . . . , α k ( x n − )] , D ( n − ( x n − ) i , α k ( x n ) i + ( − | D ( n − || X | n − h [ α k ( x ) , . . . , α k ( x n − )] n − , D ( n − ( x n ) i ...= [ D ( x ) , α k ( x ) , . . . , α k ( x n )] n + ( − | D || x | [ α k ( x ) , D ( x ) , . . . , α k ( x n )] n + ( − | D | ( | x | + | x | ) [ α k ( x ) , α k ( x ) , D ′ ( x ) , . . . , α k ( x n )] n + · · · + ( − | D ( n − || X | n − [ α k ( x ) , . . . , α k ( x n − ) , D ( n − ( x n )] n ) . Therefore ( n + 1)-tuple ( D , D , D ′ , D ′′ , . . . , D ( n − ) is a generalized α k -derivation of g n . n -ary Hom-Nambu superalgebras In this section, we introduce the notion of Rota-Baxter operators of Hom-Nambu super-algebras and 3-Hom-pre-Lie algebras. Then we introduce the notion of a 3-Hom-pre-Liesuperalgebra which is closely related to Rota-Baxter operators of weight 0. n -ary Hom-Nambu superalgebras Let ( A, · , α ) be a K -super-linear space with an even binary operation · and linear map α : A → A and let λ ∈ K . If an even linear map R : A → A satisfies, for all x, y ∈ A , Rα = αR, R ( x ) · R ( y ) = R ( R ( x ) · y + x · R ( y ) + λx · y ) , (25)then R is called a Rota-Baxter operator of weight λ on Hom-superalgebra ( A, · , α ).We generalize the concepts of a Rota-Baxter operator to n -ary Hom-Nambu superalgebras. Definition 4.1. Let λ ∈ K and an n -ary Hom-Nambu superalgebra ( N , [ · , . . . , · ] , α ). ARota-Baxter operator of weight λ on ( A, [ · , . . . , · ] , α ) is an even linear map R : N → N such21hat Rα = αR satisfying[ R ( x ) , . . . , R ( x n )] = R (cid:0) X ∅6 = I ⊆ [ n ] λ | I |− [ ˆ R ( x ) , . . . , ˆ R ( x i ) , . . . , ˆ R ( x n )] (cid:1) , (26)where ˆ R ( x i ) := ˆ R I ( x i ) := (cid:26) x i , i ∈ I,R ( x i ) , i I for all x , . . . , x n ∈ N . In particular, a Rota-Baxter operator of weight λ of ternary Hom-Nambu superalgebra ( N , [ · , . . . , · ] , α ) is an evenlinear map R : N → N commuting with α such that[ R ( x ) , R ( x ) , R ( x )] = R (cid:16) [ R ( x ) , R ( x ) , x ] + [ R ( x ) , x , R ( x )] + [ x , R ( x ) , R ( x )]+ λ [ R ( x ) , x , x ] + λ [ x , R ( x ) , x ] + λ [ x , x , R ( x )]+ λ [ x , x , x ] (cid:17) . Proposition 4.2. Let ( N , [ · , . . . , · ] , α ) be a n -ary Hom-Nambu superalgebra over a field K .An invertible even linear mapping R : N → N is a Rota-Baxter operator of weight on N if and only if R − is an even derivation on N .Proof. R is an even invertible Rota-Baxter operator of weight 0 on N if and only if ∀ x , . . . , x n ∈ A : [ R ( x ) , . . . , R ( x n )] = R (cid:16) n X i =1 [ R ( x ) , . . . , x i , . . . , R ( x n )] (cid:17) . For X k = R ( x k ) , k ∈ { , . . . , n } : [ X , . . . , X n ] = R (cid:16) n X i =1 [ X , . . . , R − ( X i ) , . . . , X n ] (cid:17) . Hence, R − ([ X , . . . , X n ]) = n X i =1 [ X , . . . , R − ( X i ) , . . . , X n ] . Thus R − is an even derivation on A . Proposition 4.3. Let R be a Rota-Baxter of weight of Hom-Lie superalgebra ( g , [ · , · ] , α ) and φ ∈ ∧ n − g ∗ an even ( n − -cochaine satisfying the conditions (22) and (23) . Then R isa Rota-Baxter operator of weight on the n -ary Hom-Nambu superalgebra ( g , [ · , . . . , · ] φ , α ) defined in (3) if and only if R satisfies ∀ x , . . . , x n ∈ H ( g ) : n X k A Rota-Baxter R of weight of Hom-Lie superalgebra ( g , [ · , · ] , α ) is aRota-Baxter operator on the associated n -ary Hom-Nambu algebra ( g , [ · , . . . , · ] n , α n − ) definedin (24) .Proof. It easy to show that α p R = Rα p for any integer p . We use the mathematical inductionon the integer n ≥ n = 3: Let x, y, z ∈ H ( g ), we have:[ R ( x ) , R ( y ) , R ( z )] = [[ R ( x ) , R ( y )] , α ( R ( z ))] == [ R ([ R ( x ) , y ]) , R ( α ( z ))] + [ R ([ x, R ( y )]) , R ( α ( z ))]= R ([ R ([ R ( x ) , y ]) , α ( z )]) + R ([[ R ( x ) , y ] , R ( α ( z ))])+ R ([ R ([ x, R ( y )]) , α ( z )]) + R ([[ x, R ( y )] , R ( α ( z ))])= R ([ R ([ R ( x ) , y ]) , α ( z )]) + R ([ R ([ x, R ( y )]) , α ( z )])+ R ([ R ( x ) , y, R ( z )] ) + R ([ x, R ( y ) , R ( α ( z ))] )= R ([ R ([ R ( x ) , y ]) + R ([ x, R ( y )]) , α ( z )])+ R ([ R ( x ) , y, R ( z )] ) + R ([ x, R ( y ) , R ( z )] )= R ([ R ( x ) , R ( y ) , z ] ) + R ([ R ( x ) , y, R ( z )] ) + R ([ x, R ( y ) , R ( z )] )= [ R ( x ) , R ( y ) , R ( z )] ii) Assume the property is true to order n > 3, that is: ∀ ( x , . . . , x n − ) ∈ H ( g ) ⊗ n − :[ R ( x ) , . . . , R ( x n − )] n − = R (cid:16) n − X i =1 [ R ( x ) , . . . , x i , . . . , R ( x n − )] n − (cid:17) . For ( x , . . . , x n ) ∈ H ( g ) ⊗ n ,[ R ( x ) , . . . , R ( x n )] n = [[ R ( x ) , . . . , R ( x n − )] n − , α n − ( R ( x n ))]24 n − X i =1 h R ([ R ( x ) , . . . , x i , . . . , R ( x n − )] n − ) , R ( α n − ( x n )) i = R (cid:16) n − X i =1 h [ R ( x ) , . . . , x i , . . . , R ( x n − )] n − , α n − ( R ( x n )) i(cid:17) + R (cid:16) n − X i =1 h R ([ R ( x ) , . . . , x i , . . . , R ( x n − )] n − ) , α n − ( x n ) i(cid:17) = R (cid:16) n − X i =1 [ R ( x ) , . . . , x i , . . . , R ( x n − ) , R ( x n )] n (cid:17) + R (cid:16)h n − X i =1 R ([ R ( x ) , . . . , x i , . . . , R ( x n − )] n − ) , α n − ( x n ) i(cid:17) = R (cid:16) n − X i =1 [ R ( x ) , . . . , x i , . . . , R ( x n − ) , R ( x n )] n (cid:17) + R (cid:0) [[ R ( x ) , . . . , R ( x n − )] n − , α n − ( x n )] (cid:17) = R (cid:16) n − X i =1 [ R ( x ) , . . . , x i , . . . , R ( x n − ) , R ( x n )] n (cid:17) + R (cid:0) [ R ( x ) , . . . , R ( x n − ) , x n ] n (cid:17) = R (cid:16) n X i =1 [ R ( x ) , . . . , x i , R ( x n )] n (cid:17) . The theorem is proved. -Hom-pre-Lie superalgebras In this subsection, we generalize the notion of a 3-Hom-pre-Lie algebra introduced in [19]to the super case, which is closely related to Rota Baxter operators. In particular, there isa construction of 3-Hom-pre-Lie superalgebras obtained from 3-Hom-Lie superalgebras. Definition 4.5. A triple ( A, {· , · , ·} , α ), consisting of a linear super-space A and two evenlinear maps {· , · , ·} : A ⊗ A ⊗ A → A and α : A → A , is called a 3 -Hom-pre-Lie superalgebra if the following identities hold: { x, y, z } = − ( − | x || y | { y, x, z } , (28) { α ( x ) , α ( x ) , { x , x , x }} = { [ x , x , x ] C , α ( x ) , α ( x ) } 25 ( − | x | ( | x | + | x | ) { α ( x ) , [ x , x , x ] C , α ( x ) } + ( − ( | x | + | x | )( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} , (29) { [ x , x , x ] C , α ( x ) , α ( x ) } = { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} , (30)where x, y, z, x i ∈ H ( A ) , ≤ i ≤ · , · , · ] C is called 3-supercommutator and defined by ∀ x, y, z ∈ H ( A ) :[ x, y, z ] C = { x, y, z } + ( − | x | ( | y | + | z | ) { y, z, x } + ( − | z | ( | x | + | y | ) { z, x, y } . (31) Proposition 4.6. Let ( A, {· , · , ·} , α ) be a -Hom-pre-Lie superalgebra. Then the induced -supercommutator in (31) and the linear map α define a -Hom-Lie superalgebra on A .Proof. By (28), the induced 3-supercommutator [ · , · , · ] C in (31) is super-skew-symmetric.For x , x , x , x , x ∈ H ( A ),[ α ( x ) , α ( x ) , [ x , x , x ] C ] C − [[ x , x , x ] C , α ( x ) , α ( x )] C − ( − | x | ( | x | + | x | ) [ α ( x ) , [ x , x , x ] C , α ( x )] C − ( − ( | x | + | x | )( | x | + | x | ) [ α ( x ) , α ( x ) , [ x , x , x ] C ] C = { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | + | x | + | x | ) { α ( x ) , [ x , x , x ] C , α ( x ) } + ( − ( | x | + | x | )( | x | + | x | + | x | ) { [ x , x , x ] C , α ( x ) , α ( x ) }− { [ x , x , x ] C , α ( x ) , α ( x ) }− ( − ( | x | + | x | + | x | )( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }}− ( − ( | x | + | x | )( | x | + | x | + | x | ) ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }}− ( − ( | x | + | x | )( | x | + | x | + | x | ) ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }}− ( − | x | ( | x | + | x | + | x | + | x | + | x | ) { α ( x ) , [ x , x , x ] C , α ( x ) }− ( − | x | ( | x | + | x | ) { α ( x ) , [ x , x , x ] C , α ( x ) }− ( − | x | ( | x | + | x | ) { [ x , x , x ] C , α ( x ) , α ( x ) }− ( − | x | ( | x | + | x | + | x | + | x | )( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }}− ( − | x | ( | x | + | x | + | x | + | x | ) ( − | x | ( | x | + | x | )+ | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} ( − | x | ( | x | + | x | + | x | + | x | ) ( − | x | ( | x | + | x | )+ | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }}− ( − | x | ( | x | + | x | )+ | x | ( | x | + | x | ) { α ( x ) , [ x , x , x ] C , α ( x ) }− ( − | x | ( | x | + | x | ) { [ x , x , x ] C , α ( x ) , α ( x ) }− ( − ( | x | + | x | )( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }}− ( − ( | x | + | x | )( | x | + | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }}− ( − | x | ( | x | + | x | ) ( − ( | x | + | x | )( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} = 0 . when applying identities (29) and (30). Thus the proof is completed. Definition 4.7. Let ( A, {· , · , ·} , α ) be a 3-Hom-pre-Lie superalgebra. The 3-Hom-Lie su-peralgebra ( A, [ · , · , · ] C , α ) is called the sub-adjacent -Hom-Lie superalgebra of ( A, {· , · , ·} , α )and ( A, {· , · , ·} , α ) is called a compatible -Hom-pre-Lie superalgebra of the 3-Hom-Lie super-algebra ( A, [ · , · , · ] C , α ).New identities of 3-pre-Hom-Lie superalgebras can be derived from Proposition 4.6. Corollary 4.8. Let ( A, {· , · , ·} , α ) be a -Hom-pre-Lie algebra. The following identities hold: { [ x , x , x ] C , α ( x ) , α ( x ) } − ( − | x || x | { [ x , x , x ] C , α ( x ) , α ( x ) } + ( − | x | ( | x | + | x | ) { [ x , x , x ] C , α ( x ) , α ( x ) }− ( − | x | ( | x | + | x | + | x | ) { [ x , x , x ] C , α ( x ) , α ( x ) } = 0 , { α ( x ) , α ( x ) , { x , x , x }} + ( − ( | x | + | x | )( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | + | x | )+ | x || x | { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} + ( − | x | ( | x | + | x | ) { α ( x ) , α ( x ) , { x , x , x }} = 0 , for x i ∈ H ( A ) , ≤ i ≤ . Proposition 4.9. Let ( A, [ · , · , · ] , α ) be a -Hom-Lie superalgebra and R : A → A is anoperator Rota-baxter of weight . Then there exists a -Hom-pre-Lie superalgebra structureon A given by { x, y, z } = [ R ( x ) , R ( y ) , z ] , ∀ x, y, z ∈ H ( A ) . (32)27 roof. Let x, y, z ∈ H ( A ). It is obvious that { x, y, z } = [ R ( x ) , R ( y ) , z ] = − ( − | x || y | [ R ( y ) , R ( x ) , z ] = − ( − | x || y | { y, x, z } . Furthermore, the following equation holds:[ x, y, z ] C = [ R ( x ) , R ( y ) , z ] + ( − | z | ( | x | + | y | ) [ R ( z ) , R ( x ) , y ] + ( − | x | ( | y | + | z | ) [ R ( y ) , R ( z ) , x ] . Since R is a Rota-Baxter operator, we have R ([ x, y, z ] C ) = [ R ( x ) , R ( y ) , R ( z )] . For x , x , x , x , x ∈ H ( A ), { α ( x ) , α ( x ) , { x , x , x }} =[ R ( α ( x )) , R ( α ( x )) , [ R ( x ) , R ( x ) , x ]]=[ α ( R ( x )) , α ( R ( x )) , [ R ( x ) , R ( x ) , x ]]; { [ x , x , x ] C , α ( x ) , α ( x ) } =[ R ([ x , x , x ] C ) , R ( α ( x )) , α ( x )]=[[ R ( x ) , R ( x ) , R ( x )] , α ( R ( x )) , α ( x )]; { α ( x ) , [ x , x , x ] C , x } =[ R ( α ( x )) , R ([ x , x , x ] C ) , α ( x )]=[ α ( R ( x )) , [ R ( x ) , R ( x ) , R ( x )] , α ( x )]; { α ( x ) , α ( x ) , { x , x , x }} =[ R ( α ( x )) , R ( α ( x )) , [ R ( x ) , R ( x ) , x ]]= α ( R ( x )) , α ( R ( x )) , [ R ( x ) , R ( x ) , x ]] . By Condition (8), (29) holds. On the other hand, we have { [ x , x , x ] C , α ( x ) , α ( x ) } =[ R ([ x , x , x ] C ) , R ( α ( x )) , α ( x )]=[[ R ( x ) , R ( x ) , R ( x )] , α ( R ( x )) , α ( x )]; { α ( x ) , α ( x ) , { x , x , x }} =[ R ( α ( x )) , R ( α ( x )) , [ R ( x ) , R ( x ) , x ]]=[ α ( R ( x )) , α ( R ( x )) , [ R ( x ) , R ( x ) , x ]; { α ( x ) , α ( x ) , { x , x , x }} =[ R ( α ( x )) , R ( α ( x )) , [ R ( x ) , R ( x ) , x ]]=[ α ( R ( x )) , α ( R ( x )) , [ R ( x ) , R ( x ) , x ]; { α ( x ) , α ( x ) , { x , x , x }} =[ R ( α ( x )) , R ( α ( x )) , [ R ( x ) , R ( x ) , x ]]=[ α ( R ( x )) , α ( R ( x )) , [ R ( x ) , R ( x ) , x ]] . By super-Nambu identity, (30) holds. This completes the proof. Corollary 4.10. With the above conditions, ( A, [ · , · , · ] C , α ) is a -Hom-Lie superalgebra asthe sub-adjacent -Hom-Lie superalgebra of the -Hom-pre-Lie superalgebra given in Propo-sition 4.9, and R is a -Hom-Lie superalgebra morphism from ( A, [ · , · , · ] C , α ) to ( A, [ · , · , · ] , α ) . urthermore, R ( A ) = { R ( x ) | x ∈ A } ⊂ A is a -Hom-Lie super-subalgebra of A and there isan induced -Hom-pre-Lie superalgebra structure ( {· , · , ·} R ( A ) , α ) on R ( A ) given by { R ( x ) , R ( y ) , R ( z ) } R ( A ) := R ( { x, y, z } ) , ∀ x, y, z ∈ H ( A ) . (33) Proposition 4.11. Let ( A, [ · , · , · ] , α ) be a -Hom-Lie superalgebra. Then there exists a com-patible -Hom-pre-Lie superalgebra if and only if there exists an invertible Rota-Baxter op-erator R on A .Proof. Let R be an invertible Rota-Baxter operator of A . Then there exists a 3-Hom-pre-Liesuperalgebra structure ( { x, y, z } , α ) on A defined by { x, y, z } = ad ( R ( x ) ,R ( y )) ( z ) , ∀ x, y, z ∈ H ( A ) . Moreover, there is an induced 3-Hom-pre-Lie superalgebra structure ( {· , · , ·} A , α ) on A = R ( A ) given by { x, y, z } A = R { R − ( x ) , R − ( y ) , R − ( z ) } = R ( ad ( x,y ) ( R − ( z )))for all x, y, z ∈ H ( A ). Since R is an operator Rota-Baxter on A , we have[ x, y, z ] = R (cid:16) [ x, y, R − ( z )] + [ x, R − ( y ) , z ] + [ R − ( x ) , y, z ] (cid:17) = R (cid:16) ad ( x,y ) ( R − ( z )) + ( − | z | ( | x | + | y | ) ad ( z,x ) ( R − ( y )) + ( − | x | ( | y | + | z | ) ad ( y,z ) ( R − ( x )) (cid:17) = { x, y, z } A + ( − | z | ( | x | + | y | ) { z, x, y } A + ( − | x | ( | y | + | z | ) { y, z, x } A Therefore ( A, {· , · , ·} A , α ) is a compatible 3-Hom-pre-Lie superalgebra of ( A, [ · , · , · ]). Acknowlegments Dr. Sami Mabrouk is grateful to the research environment in Mathematics and AppliedMathematics MAM, the Division of Applied Mathematics of the School of Education, Cultureand Communication at M¨alardalen University for hospitality and an excellent and inspiringenvironment for research and research education and cooperation in Mathematics during hisvisit in Autumn 2019. 29 eferences [1] Abdaoui, K., Mabrouk, S., Makhlouf, A., Cohomology of Hom-Leibniz and n -ary Hom-Nambu-Lie superalgebras, 24pp, arXiv:1406.3776[math.RT][2] Abramov, V., On a graded q -differential algebra, Journal of Nonlinear MathematicalPhysics, 13: sup 1, 1-8, (2006)[3] Abramov, V., Graded q -Differential Algebra Approach to q -Connection, In: Silvestrov,S., Paal, E., Abramov, V., Stolin, A. (Eds.), Generalized Lie Theory in Mathematics,Physics and Beyond, Springer-Verlag, Berlin, Heidelberg, Ch. 6, 71-79 (2009)[4] Abramov, V., Super 3-Lie algebras induced by super Lie algebras. Adv. Appl. CliffordAlgebr. 27, no. 1, 9-16 (2017)[5] Abramov, V., L¨att, P., Classification of Low Dimensional 3-Lie Superalgebras, In: Silve-strov S., Rancic M. (eds), Engineering Mathematics II, Springer Proceedings in Math-ematics and Statistics, Vol 179, Springer, Cham, 1-12 (2016)[6] Abramov, V., Raknuzzaman, Md., Semi-commutative Galois Extensions and ReducedQuantum Plane, In: Silvestrov S., Rancic M. (eds), Engineering Mathematics II,Springer Proceedings in Mathematics and Statistics, Vol 179, Springer, Cham, 13-31(2016)[7] Abramov, V., Weil Algebra, 3-Lie algebra and B.R.S. algebra, In: Silvestrov,S., Malyarenko, A., Rancic, M. (Eds.), Algebraic Structures and Applications,Springer Proceedings in Mathematics and Statistics, Vol 317, Ch 1 (2020).arXiv:1802.05576[math.RA][8] Abramov, V., L¨att, P., Ternary Lie superalgebras and Nambu-Hamilton equation insuperspace, In: Silvestrov, S., Malyarenko, A., Rancic, M. (Eds.), Algebraic Structuresand Applications, Springer Proceedings in Mathematics and Statistics, Vol. 317, Ch. 3(2020)[9] Ammar, F., Ejbehi, Z., Makhlouf, A., Cohomology and deformations of Hom-algebras,J. Lie Theory , no. 4, 813836 (2011)[10] Ammar, F., Mabrouk, S., Makhlouf, A., Representation and cohomology of n -ary mul-tiplicative Hom-Nambu-Lie algebras, J. Geom. Phys. (10) 18981913 (2011)[11] Ammar, F., Makhlouf, A., Hom-Lie algebras and Hom-Lie admissible superalgebras,Journal of Algebra, Vol. 324 (7), 1513-1528 (2010)3012] Armakan, A., Silvestrov, S., Farhangdoost, M., Enveloping algebras of color hom-Liealgebras, Turk. J. Math. , 316-339 (2019) doi:10.3906/mat-1808-96. arXiv:1709.06164[math.QA][13] Armakan, A., Silvestrov, S., Farhangdoost, M., Extensions of hom-Lie color al-gebras, To appear in Georgian Mathematical Journal, doi:10.1515/gmj-2019-2033.arXiv:1709.08620 [math.QA][14] Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S.: Structure and Cohomology of 3-Lie algebras induced by Lie algebras, In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A.(Eds.), Algebra, Geometry and Mathematical Physics, Springer Proceedings in Mathe-matics and Statistics, Vol 85, 123-144 (2014). arXiv:1312.7599 [math.RA][15] Arnlind, J., Makhlouf, A., Silvestrov, S.: Ternary Hom-Nambu-Lie algebras induced byHom-Lie algebras, J. Math. Phys. , 043515, 11 pp. (2010)[16] Arnlind, J., Makhlouf, A., Silvestrov, S.: Construction of n -Lie algebras and n -aryHom-Nambu-Lie algebras, J. Math. Phys. , 123502, 13 pp. (2011)[17] Ataguema, H., Makhlouf, A., Silvestrov, S., Generalization of n -ary Nambu algebrasand beyond. J. Math. Phys. 50, 083501 (2009)[18] Awata, H., Li, M., Minic, D., Yoneya, T., On the quantization of Nambu brackets, J.High Energy Phys. , Paper 13, 17 pp. (2001)[19] Bai, C., Guo, L., Sheng, Y., Bialgebras, the classical Yang-Baxter equation and Manintriples for 3-Lie algebras, (2016), arXiv preprint arXiv:1604.05996.[20] Bai, R., Bai, C., Wang, J., Realizations of 3-Lie algebras, J. Math. Phys. , 063505(2010)[21] Bai, R., Wu, Y., Li, J., Zhou, H., Constructing ( n + 1)-Lie algebras from n -Lie algebras,J. Phys. A , no. 47 (2012)[22] Bai, R., Song, G., Zhang, Y., On classification of n -Lie algebras, Front. Math. China ,581-606 (2011)[23] Bai, R., Wang, X., Xiao, W., An, H., The structure of low dimensional n -Lie algebrasover the field of characteristic 2, Linear Algebra Appl. (89), 1912-1920 (2008)[24] Bai, R., Chen, L., Meng, D., The Frattini subalgebra of n -Lie algebras, Acta Math.Sinica, English Series, 23 (5) 847-856 (2007)3125] Bai, R., Meng, D., The central extension of n -Lie algebras, Chinese Ann. Math. 27 (4)491-502 (2006)[26] Bai, R., Meng, D., The centroid of n -Lie algebras, Algebras Groups Geom. 25 (2) 29-38(2004)[27] Bai, R., Zhang, Z., Li, H., Shi, H., The inner derivation algebras of ( n + 1)-dimensional n -Lie algebras, Comm. Algebra, 28 (6) 2927-2934 (2000)[28] Bai, R., An, H., Li, Z., Centroid structures of n -Lie algebras, Linear Algebra and itsApplications 430 (2009) 229240[29] Bakayoko, I., Laplacian of Hom-Lie quasi-bialgebras, International Journal of Algebra, (15), 713-727 (2014)[30] Bakayoko, I., L -modules, L -comodules and Hom-Lie quasi-bialgebras, African DiasporaJournal of Mathematics, Vol 17 49-64 (2014)[31] Bakayoko, I., Silvestrov, S., Multiplicative n -Hom-Lie color algebras, In: Silve-strov, S., Malyarenko, A., Rancic, M. (Eds.), Algebraic Structures and Applica-tions, Springer Proceedings in Mathematics and Statistics, Vol 317, Ch. 7, 2020.arXiv:1912.10216[math.QA][32] Bakayoko, I., Silvestrov, S., Hom-left-symmetric color dialgebras, Hom-tridendriformcolor algebras and Yau’s twisting generalizations, 24pp, arXiv:1912.01441[math.RA][33] Ben Hassine, A., Mabrouk S., Ncib, O., Some Constructions of Multiplicative n -aryhom-Nambu Algebras, Adv. Appl. Clifford Algebras 29, 88 (2019).[34] Ben Abdeljelil, A., Elhamdadi, M., Kaygorodov, I., Makhlouf, A., Generalized Deriva-tions of n -BiHom-Lie algebras, In: Silvestrov, S., Malyarenko, A., Rancic, M. (Eds.),Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statis-tics, Vol 317, Ch. 4, 2020. arXiv:1901.09750[math.RA][35] Benayadi, S., Makhlouf, A., Hom-Lie algebras with symmetric invariant nondegeneratebilinear forms, J. Geom. Phys. , 3860 (2014)[36] Beites, P. D., Kaygorodov, I., Popov, Y., Generalized derivations of multiplicative n -aryHom-Ω color algebras, Bull. of the Malay. Math. Sci. Soc. 41 (2018),[37] Casas, J. M., Loday, J.-L., Pirashvili, T., Leibniz n -algebras, Forum Math. 14, 189-207(2002) 3238] Chen, L., Ma, Y., Ni, L., Generalized Derivations of Lie color algebras, Results Math., (3-4), 923-936 (2013)[39] L. Chen, Y. Ma, J. Zhou, Generalized Derivations of Lie triple systems, arXiv:1412.7804[40] Daletskii, Y. L., Takhtajan, L. A., Leibniz and Lie Algebra Structuresfor Nambu Algebra, Letters in Mathematical Physics , 127141 (1997).https://doi.org/10.1023/A:1007316732705[41] De Azc´arraga, J. A., Izquierdo, J. M., n -Ary algebras: a review with applications, J.Phys. A: Math. Theor. 43, 293001 (2010)[42] Filippov, V. T., n -Lie algebras, Sib. Math. J. , 879-891 (1985) (Transl. from Russian:Sib. Mat. Zh., 26 126-140 (1985))[43] Filippov, V. T., On δ -derivations of Lie algebras. Sib. Math. J. , 1218-1230 (1998).(Translated from Sibirskii Matematicheskii Zhurnal, Vol. 39, No. 6, pp. 14091422,NovemberDecember, 1998.)[44] Filippov, V. T., δ -Derivations of prime Lie algebras, Sib. Math. J. 1999;40:174-184.[45] Filippov, V. T., δ -derivations of prime alternative and Maltsev algebras, Algebra andLogic , 354358 (2000). (Translated from Algebra i Logika, Vol. 39, No. 5, pp. 618625,SeptemberOctober, 2000)[46] Elchinger, O., Lundeng˚ard, K., Makhlouf, A., Silvestrov, S., Brackets with ( τ, σ )-derivations and ( p, q )-deformations of Witt and Virasoro algebras. Forum Math., Vol.28, pp. 657-673 (2016).[47] Guan, B., Chen, L., Sun, B., 3-Ary Hom-Lie Superalgebras Induced By Hom-Lie Su-peralgebras, Adv. Appl. Clifford Algebras, 27(4), 3063-3082.[48] Hartwig, J. T., Larsson, D., Silvestrov, S. D., Deformations of Lie algebras using σ -derivations, J. Algebra , 314-361 (2006) (Preprint in Mathematical Sciences 2003:32,LUTFMA-5036-2003, Centre for Mathematical Sciences, Department of Mathematics,Lund Institute of Technology, 52 pp. (2003))[49] Kasymov, Sh. M., Theory of n -Lie algebras, Algebra and Logic, , 155-166 (1987)(Transl. from Russian: Algebra i Logika, Vol. 26, No. 3, pp. 277297, (1987))[50] Kaygorodov, I, Popov, Yu., Alternative algebras admitting derivations with invertiblevalues and invertible derivations, Izv. Math. δ -Derivations of n -ary algebras, Izvestiya: Mathematics, (5)1150-1162 (2012)[52] Kaygorodov, I., ( n + 1)-Ary derivations of simple n -ary algebras, Algebra and Logic, (5) 470-471 (2011)[53] Kaygorodov, I., ( n + 1)-Ary derivations of semisimple Filippov algebras, Math. Notes, (2) 208-216 (2014)[54] Kaygorodov, I., Popov, Yu., Generalized derivations of (color) n -ary algebras, Linearand Multilinear Algebra, (6) (2016)[55] Kitouni, A., Makhlouf, A., On structure and central extensions of ( n + 1)-Lie algebrasinduced by n -Lie algebras, arXiv:1405.5930 [math.RA] (2014)[56] Kitouni, A., Makhlouf, A., Silvestrov, S., On ( n + 1)-Hom-Lie algebras induced by n -Hom-Lie algebras, Georgian Math. J. no. 1, 75-95 (2016)[57] Kitouni, A., Makhlouf, A., Silvestrov, S., On n -ary generalization of BiHom-Lie alge-bras and BiHom-associative algebras, In: Silvestrov, S., Malyarenko, A., Rancic, M.(Eds.), Algebraic Structures and Applications, Springer Proceedings in Mathematicsand Statistics, Vol 317, Ch 5 (2020)[58] Kitouni, A., Makhlouf, A., Silvestrov, S., On Solvability and Nilpotency for n -Hom-LieAlgebras and ( n + 1)-Hom-Lie Algebras Induced by n -Hom-Lie Algebras, In: Silvestrov,S., Malyarenko, A., Rancic, M. (Eds.), Algebraic Structures and Applications, SpringerProceedings in Mathematics and Statistics, Vol 317, Ch 6 (2020)[59] Larsson, D., Sigurdsson, G., Silvestrov, S. D., Quasi-Lie deformations on the algebra F [ t ] / ( t N ), J. Gen. Lie Theory Appl. Vol. 2, No. 3, 201-205 (2008)[60] Larsson, D., Silvestrov, S. D., Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra ,1473-1478 (2005)[63] Larsson, D., Silvestrov, S. D., Quasi-deformations of sl ( F ) using twisted derivations,Comm. in Algebra , 4303-4318 (2007)[64] Larsson, D., Silvestrov, S. D., On Generalized N -Complexes Comming from TwistedDerivations, In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), GeneralizedLie Theory in Mathematics, Physics and Beyond, Springer-Verlag, Berlin, Heidelberg,Ch. 7, 81-88 (2009)[65] Leger, G., Luks, E., Generalized derivations of Lie algebras, J. Algebra , 165-203(2000)[66] Ling, W. X., On the structure of n -Lie algebras, PhD Thesis, University-GHS-Siegen,Siegen (1993)[67] Ma, T., Makhlouf, A., Silvestrov, S., Curved O -operator systems, 17pp (2017),arXiv:1710.05232 [math.RA][68] Ma, T., Makhlouf, A., Silvestrov, S., Rota-Baxter bisystems and covariant bialgebras,30 pp, (2017). arXiv:1710.05161[math.RA][69] Ma, T., Makhlouf, A., Silvestrov, S., Rota-Baxter Cosystems and CoquasitriangularMixed Bialgebras, J. Algebra Appl. Accepted 2019.[70] Makhlouf, A., Silvestrov, S. D., Hom-algebra structures, J. Gen. Lie Theory Appl. , No. 04, pp. 553-589 (2010). arXiv:0811.0400[math.RA][73] Makhlouf, A., Silvestrov, S. D., Notes on Formal deformations of Hom-Associative andHom-Lie algebras, Forum Math. (4), 715-739 (2010)3574] Mishra, S. K., Silvestrov, S., A Review on Hom-Gerstenhaber algebras and Hom-Liealgebroids, In: Silvestrov, S., Malyarenko, A., Rancic, M. (Eds.), Algebraic Structuresand Applications, Springer Proceedings in Mathematics and Statistics, Vol 317, Ch 11,2020.[75] Nambu, Y., Generalized Hamiltonian dynamics, Phys. Rev. D (8) 2405-2412 (1973)[76] Pojidaev, A, Saraiva, P., On derivations of the ternary Malcev algebra M8, Comm.Algebra. , 3593-3608 (2006)[77] Richard, L., Silvestrov, S. D., Quasi-Lie structure of σ -derivations of C [ t ± ], J. Algebra319, no. 3, 1285-1304 (2008)[78] Richard, L., Silvestrov, S., A Note on Quasi-Lie and Hom-Lie Structures of σ -Derivations of C [ z ± , · · · , z ± n ], In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A.(Eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, Springer-Verlag,Berlin, Heidelberg, Ch. 22, 257-262 (2009)[79] Rotkiewicz, M., Cohomology ring of n -Lie algebras, Extracta Math. , 219-232 (2005)[80] Sheng, Y., Representation of Hom-Lie algebras, Algebr. Represent. Theory , no. 6,10811098 (2012). arXiv:1005.0140 [math-ph][81] Sigurdsson, G., Silvestrov, S., Lie color and Hom-Lie algebras of Witt type and their cen-tral extensions, In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), GeneralizedLie Theory in Mathematics, Physics and Beyond, Springer-Verlag, Berlin, Heidelberg,Ch. 21, 247-255 (2009)[82] Sigurdsson, G., Silvestrov, S., Graded quasi-Lie algebras of Witt type, Czech. J. Phys.56: 1287-1291 (2006)[83] Takhtajan, L. A., On foundation of the generalized Nambu mechanics, Comm. Math.Phys., , no. 2, 295-315 (1994)[84] Takhtajan, L. A., Higher order analog of Chevalley-Eilenberg complex and deformationtheory of n -gebras, St. Petersburg Math. J. no. 2, 429-438 (1995)[85] Wang, C., Zhang, Q., Wei, Z., A classification of low dimensional multiplicative Hom-Liesuperalgebras, Open Mathematics, (1), 613-628 (2016).[86] Williams, M. P., Nilpotent n -Lie Algebras, Comm. in Algebra, 37:6, 1843-1849, (2009).3687] Zhang, R., Zhang, Y., Generalized derivations of Lie superalgebras, Comm. Algebra,38:10 (2010), 3737-3751[88] Yau, D., Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl. , no. 2,95-108 (2008)[89] Yau, D., Hom-algebras and homology, Journal of Lie Theory 19, No. 2, 409-421 (2009)[90] Yau, D., A Hom-associative analogue of n -ary Hom-Nambu algebras, arXiv:1005.2373[91] Yau, D., On n -ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys. ,506-522 (2012). arXiv:1004.2080[math.RA][92] Yau, D., On nn