On Properties and Classification of a Class of 4-Dimensional 3-Hom-Lie Algebras with a Nilpotent Twisting Map

: The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map α and eight structure constants as parameters. Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. Building upon that, all algebras of this class are classified up to Hom-algebra isomorphism. Necessary and sufficient conditions for multiplicativity of general ( n + 1 ) -dimensional n -Hom-Lie algebras, as well as for algebras in the considered class, are obtained in terms of the structure constants and the twisting map. Furthermore, for some algebras in this class, it is determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals.


Introduction
and discretizations of Lie algebras of vector fields using more general σ -derivations (twisted derivations) and a general method for construction of deformations of Witt and Virasoro type algebras based on twisted derivations have been developed, initially motivated by the q-deformed Jacobi identities observed for the q-deformed algebras in physics, q-deformed versions of homological algebra and discrete modifications of differential calculi [7, 34-37, 39-41, 53, 55, 70-72]. The general abstract quasi-Lie algebras and the subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras as well as their general colored (graded) counterparts have been introduced in [51,[64][65][66]85]. Subsequently, various classes of Hom-Lie admissible algebras have been considered in [74]. In particular, in [74], the Hom-associative algebras have been introduced and shown to be Hom-Lie admissible, that is leading to Hom-Lie algebras using commutator map as new product, and in this sense constituting a natural generalization of associative algebras, as Lie admissible algebras leading to Lie algebras via commutator map as new product. In [74], moreover, several other interesting classes of Hom-Lie admissible algebras generalizing some classes of non-associative algebras, as well as examples of finite-dimensional Hom-Lie algebras have been described. Hom-algebras structures are very useful since Homalgebra structures of a given type include their classical counterparts and open more possibilities for deformations, extensions of cohomological structures and representations. Since these pioneering works [51,[64][65][66][67]74], Hom-algebra structures have developed in a popular broad area with increasing number of publications in various directions (see for example [8, 29, 48, 63, 64, 68, 75-78, 80, 83, 84, 90, 91] and references therein).
Ternary Lie algebras appeared first in generalization of Hamiltonian mechanics by Nambu [79]. Besides Nambu mechanics, n-Lie algebras revealed to have many applications in physics. The mathematical algebraic foundations of Nambu mechanics have been developed by Takhtajan in [86]. Filippov, in [49] independently introduced and studied structure of n-Lie algebras and Kasymov [56] investigated their properties. Properties of n-ary algebras, including solvability and nilpotency, were studied in [32,56,88]. Kasymov [56] pointed out that n-ary multiplication allows for several different definitions of solvability and nilpotency in n-Lie algebras, and studied their properties. Further properties, classification, and connections of n-ary algebras to other structures such as bialgebras, Yang-Baxter equation and Manin triples for 3-Lie algebras were studied in [15-23, 25, 56]. The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras and application to constructions of B.R.S. algebras have been considered in [2][3][4]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation is considered in [5]. In [33], Leibniz nalgebras have been studied. The general cohomology theory for n-Lie algebras and Leibniz n-algebras was established in [42,82,87]. The structure and classification of finite-dimensional n-Lie algebras were considered in [21,49,69] and many other authors. For more details of the theory and applications of n-Lie algebras, see [45] and references therein.
Classifications of n-ary or Hom generalizations of Lie algebras have been considered, either in very special cases or in low dimensions. The classification of n-Lie algebras of dimension up to n + 1 over a field of characteristic p = 2 has been completed by Filippov [49] using the specific properties of (n + 1)-dimensional n-Lie algebras that make it possible to represent their bracket by a square matrix in a similar way as bilinear forms, the number of cases obtained depends on the properties of the base field, the list is ordered by ascending dimension of the derived ideal, and among them, one nilpotent algebra, and a class of simple algebras which are all isomorphic in the case of an algebraically closed field, the remaining algebras are k-solvable for some 2 ≤ k ≤ n depending on the algebra. These simple algebras are proven to be the only simple finite-dimensional n-Lie algebras in [69]. The classification of (n + 1)-dimensional n-Lie algebras over a field of characteristic 2 has been done by Bai, Wang, Xiao, and An [22] by finding and using a similar result in characteristic 2. Bai, Song and Zhang [21] classify the (n + 2)-dimensional n-Lie algebras over an algebraically closed field of characteristic 0 using the fact that an (n + 2)-dimensional n-Lie algebra has a subalgebra of codimension 1 if the dimension of its derived ideal is not 3, thus constructing most of the cases as extensions of the (n + 1)-dimensional n-Lie algebras listed by Filippov. In [31], Cantarini and Kac classified all simple linearly compact n-Lie superalgebras, which turned out to be n-Lie algebras, by finding a bijective correspondence between said algebras and a special class of transitive Z-graded Lie superalgebras, the list they obtained consists of four representatives, one of them is the (n + 1)-dimensional vector product n-Lie algebra, and the remaining three are infinite-dimensional n-Lie algebras.
Classifications of n-Lie algebras in higher dimensions have only been studied in particular cases. Metric n-Lie algebras, that is n-Lie algebras equipped with a nondegenerate compatible bilinear form, have been considered and classified, first in dimension n+2 by Ren, Chen and Liang [81] and dimension n+3 by Geng, Ren and Chen [50], and then in dimensions n + k for 2 ≤ k ≤ n + 1 by Bai, Wu and Chen [24]. The classification is based on the study of the Levi decomposition, the center and the isotropic ideals and properties around them. Another case that has been studied is the case of nilpotent n-Lie algebras, more specifically nilpotent n-Lie algebras of class 2. Eshrati, Saeedi and Darabi [46] classify (n + 3)-dimensional nilpotent n-Lie algebras and (n + 4)-dimensional nilpotent n-Lie algebras of class 2 using properties introduced in [43,47]. Similarly Hoseini, Saeedi and Darabi [52] classify (n + 5)-dimensional nilpotent n-Lie algebras of class 2. In [54], Jamshidi, Saeedi and Darabi classify (n + 6)-dimensional nilpotent n-Lie algebras of class 2 using the fact that such algebras factored by the span of a central element give (n + 5)dimensional nilpotent n-Lie algebras of class 2, which were classified before. There has been a study of the classification of 3-dimensional 3-Hom-Lie algebras with diagonal twisting maps by Ataguema, Makhlouf and Silvestrov in [13].
Hom-type generalization of n-ary algebras, such as n-Hom-Lie algebras and other n-ary Hom algebras of Lie type and associative type, were introduced in [13], by twisting the defining identities by a set of linear maps. The particular case, where all these maps are equal and are algebra morphisms has been considered and a way to generate examples of n-ary Hom-algebras from n-ary algebras of the same type have been described. Further properties, construction methods, examples, representations, cohomology and central extensions of n-ary Hom-algebras have been con-sidered in [9,11,12,58,89,92]. These generalizations include n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type 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 partially associative algebras. In [60], constructions of n-ary generalizations of BiHom-Lie algebras and BiHom-associative algebras have been considered. Generalized derivations of n-BiHom-Lie algebras have been studied in [28]. Generalized derivations of multiplicative n-ary Hom-Ω color algebras have been studied in [30]. Cohomology of Hom-Leibniz and n-ary Hom-Nambu-Lie superalgebras has been considered in [1] Generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu superalgebras have been considered in [73]. A construction of 3-Hom-Lie algebras based on σ -derivation and involution has been studied in [6]. Multiplicative n-Hom-Lie color algebras have been considered in [26].
In [14], Awata, Li, Minic and Yoneya introduced a construction of (n + 1)-Lie algebras induced by n-Lie algebras using combination of bracket multiplication with a trace in their work on quantization of the Nambu brackets. Further properties of this construction, including solvability and nilpotency, were studied in [10,17,57]. In [11,12], this construction was generalized using the brackets of general Hom-Lie algebra or n-Hom-Lie and trace-like linear forms satisfying conditions depending on the twisting linear maps defining the Hom-Lie or n-Hom-Lie algebras. In [27], a method was demonstrated of how to construct n-ary multiplications from the binary multiplication of a Hom-Lie algebra and a (n − 2)-linear function satisfying certain compatibility conditions. Solvability and nilpotency for n-Hom-Lie algebras and (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras have been considered in [59]. In [61], properties and classification of n-Hom-Lie algebras in dimension n + 1 were considered, and 4-dimensional 3-Hom-Lie algebras for various special cases of the twisting map have been computed in terms of structure constants as parameters and listed in classes in the way emphasizing the number of free parameters in each class.
The n-Hom-Lie algebras are fundamentally different from the n-Lie algebras especially when the twisting maps are not invertible or not diagonalizable. When the twisting maps are not invertible, the Hom-Nambu-Filippov identity becomes less restrictive since when elements of the kernel of the twisting maps are used, several terms or even the whole identity might vanish. Isomorphisms of Hom-algebras are also different from isomorphisms of algebras since they need to intertwine not only the multiplications but also the twisting maps. All of this make the classification problem different, interesting, rich and not simply following from the case of n-Lie algebras. In this work, we consider n-Hom-Lie algebras with a nilpotent twisting map α, which means in particular that α is not invertible.
To our knowledge, the classification of 4-dimensional 3-Hom-Lie algebras up to Hom-algebras isomorphism has not been achieved previously in the literature. The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map α and eight structure constants as parameters, namely 4 3,N(2),6 given in [61]. All 3-dimensional 3-Hom-Lie algebras with diagonal twisting maps have been listed unclassified in [13]. The algebras considered in our article are 4-dimensional, and the twisting maps are of a different type, namely nilpotent. Nilpotent linear maps are neither invertible nor diagonalizable, which makes the object of our study fundamentally different from the case of n-Hom-Lie algebras with diagonal twisting maps in the sense that when the twisting maps are not invertible, the Hom-Nambu-Filippov identity becomes less restrictive since when elements of the kernel of the twisting maps are used in the identity, several terms or even the whole identity might vanish, and when the twisting maps are not diagonalizable, the change induced by introducing them in the identity is more significant. In this work, we achieved a complete classification up to isomorphism of Hom-algebras of the considered class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map, computed derived series and central descending series for all of the 3-Hom-Lie algebras of this class, studied solvability and nilpotency, characterized the multiplicative 3-Hom-Lie algebras among them and studied the ideal properties of the terms of derived series and central descending series of some chosen examples of the Hom-algebras from the classification. These results improve understanding of the rich structure of nary Hom-algebras and in particular the important class of n-Hom-Lie algebras. It is also a step towards the complete classification of 4-dimensional 3-Hom-Lie algebras and in general (n + 1)-dimensional n-Hom-Lie algebras. Moreover, our results contribute to in-depth study of the structure and important properties and sub-classes of n-Hom-Lie algebras.
In Section 2, definitions and properties of n-Hom-Lie algebras that are used in the study are recalled, and new results characterizing nilpotency as well as necessary and sufficient conditions for multiplicativity of general (n + 1)-dimensional n-Hom-Lie algebras and for algebras in the considered class are obtained in terms of the structure constants and the twisting map. In Section 4, Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. In Section 5, building up on the previous sections, all algebras of this class are classified up to Hom-algebra isomorphism. In Section 6, for some algebras in this class, it has been determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals or Hom-ideals.

Definitions and properties of n-Hom-Lie algebras
In this section, we present the basic definitions and properties of n-Hom-Lie algebras needed for our study. Throughout this article, it is assumed that all linear spaces are over a field K of characteristic 0, and for any subset S of a linear space, S denotes the linear span of S. The arity of all the considered algebras is assumed to be greater than or equal to 2. Hom-Lie algebras are a generalization of Lie algebras introduced in [51] while studying σ -derivations. The n-ary case was introduced in [13].
Definition 1 ( [51,74] Hom-Jacobi identity (cyclic form) In Hom-Lie algebras, by skew-symmetry, the Hom-Jacobi identity is equivalent to Hom-Jacobi identity (Hom-derivation form) (1) Hom-algebras satisfying just the Hom-algebra identity (1), without requiring the skew-symmetry identity, are called Hom-Leibniz algebras [65,74]. Thus, Hom-Lie algebras are skew-symmetric Hom-Leibniz algebras. There are many Hom-Leibniz algebras which are not skew-symmetric and thus not Hom-Lie algebras. When the twisting map is the identity map α = Id A on A, Hom-Leibniz algebras become (left) Leibniz algebras, and Hom-Lie algebras become Lie algebras. A Hom-Leibniz algebra is also a Leibniz algebra, or a Hom-Lie algebra is also a Lie algebra, if and only if the map Id A belongs to the set of all linear maps α for which the identity (1) holds. Whether the map Id A belongs to the set of all linear maps α for which the identity (1) holds or not depends on the underlying algebra. The Hom-algebra identity (1) is linear with respect to α in the linear space of all linear maps on the algebra, and hence, the set of all such α, for which the identity (1) holds, is a linear subspace of the linear space of all linear maps on the algebra. There are many Hom-Leibniz algebra which are not Leibniz algebras, or Hom-Lie algebras which are not Lie algebras.
Definition 3 ( [29,74]). A Hom-Lie algebra (A, [·, ·], α) is said to be multiplicative if α is an algebra morphism, and it is said to be regular if α is an isomorphism.
An n-ary Hom-algebra is said to be skew-symmetric if its n-ary operation is skew-symmetric, that is satisfying for all x 1 , . . . , x n−1 , y 1 , . . . , y n ∈ A, The n-Hom-Lie algebras are an n-ary generalization of Hom-Lie algebras to nary algebras satisfying a generalisation of the Hom-algebra identity (1) involving n-ary product and n − 1 linear maps.
The following proposition, providing a way to construct an n-Hom-Lie algebra from an n-Lie algebra and an algebra morphism, was first introduced in the case of Lie algebras and then generalized to the n-ary case in [13]. A more general version of this theorem, given in [92], states that the category of n-Hom-Lie algebras is closed under twisting by weak morphisms.
The following particular case of Proposition 1 is obtained if α = Id A . The following definition is a specialization of the standard definition of a subalgebra in general algebraic structures to the case of n-Hom-Lie algebras and n-ary skew-symmetric Hom-algebras considered in this paper.
The following definitions are a direct extension of the corresponding definitions in [59] to arbitrary n-ary skew-symmetric Hom-algebras.
Definition 10. Let (A, [·, . . . , ·] , α 1 , . . . , α n−1 ) be an n-Hom-Lie algebra or more generally an n-ary skew-symmetric Hom-algebra, and let I be an ideal of A. For 2 ≤ k ≤ n and p ∈ N, we define the k-derived series of the ideal I by Definition 11. Let (A, [·, . . . , ·] , α 1 , . . . , α n−1 ) be an n-Hom-Lie algebra or more generally an n-ary skew-symmetric Hom-algebra, and let I be an ideal of A. For 2 ≤ k ≤ n, the ideal I is said to be k-solvable (resp. k-nilpotent) if there exists r ∈ N such that D r k (I) = {0} (resp. C r k (I) = {0}), and the smallest r ∈ N satisfying this condition is called the class of k-solvability (resp. the class of nilpotency) of I.
The following direct extension of the corresponding result in [59] to arbitrary n-ary skew-symmetric Hom-algebras is proved in the same way as in [59] since the proof does not involve the Hom-Nambu-Filippov identity. This lemma also implies that if two n-Hom-Lie algebras are isomorphic, they would also have isomorphic terms of the derived series and central descending series, which also means that if two algebras have a significant difference in the derived series or the central descending series, for example different dimensions of given corresponding terms, then these algebras cannot be isomorphic.

Proposition 4 ( [61]).
Let (e i ) 1≤i≤n+1 be a basis of a linear space A, let σ be a permutation of the set {1, . . . , n + 1} of n + 1 elements, and let B = (b i, j ) 1≤i, j≤n+1 be a matrix representing a skew-symmetric n-ary bracket in this basis, then the matrix representing the same bracket in the basis (e σ (i) ) 1≤i≤n+1 is given by the matrix

Remark 3. ( [61]
). For the whole algebra A, all the k-central descending series, for all 2 ≤ k ≤ n, are equal. Therefore all the notions of k-nilpotency, for all 2 ≤ k ≤ n, are equivalent, and we denote C p k (A) for any 2 ≤ k ≤ n by C p (A).
Definition 12. Let (A, [·, . . . , ·] , α 1 , . . . , α n−1 ) be an n-Hom-Lie algebra or more generally an n-ary skew-symmetric Hom-algebra. Define Z(A), the center of A, by Proof. (i) The first statement is a generalization of Lemma 3 to the case of n-ary skew-symmetric Hom-algebras, and is proved in the same way, since the original proof does not use the Hom-Nambu-Filippov identity.
The following direct extension of the corresponding result in [61] to arbitrary n-ary skew-symmetric Hom-algebras is proved in the same way as in [61] since the proof does not involve the Hom-Nambu-Filippov identity.  Proof. Let f : A → B be a surjective homomorphism, then for all y 1 , . . . , y n ∈ B there exists x 1 , . . . , If f is an isomorphism, then the converse can be proved by applying the same argument using f −1 instead of f .
Remark 4. Let us compare the polynomial equations obtained from the Nambu-Filippov identity and the Hom-Nambu-Filippov identity in dimension n + 1 with various types of twisting maps: Diagonalizable and invertible with eigenvalues λ i , 1 ≤ i ≤ n + 1: Diagonalizable with dim ker α = 1 with eigenvalues λ i , 1 ≤ i ≤ n + 1: Nilpotent with dim ker α = 1: These different cases are separate from each other, and the case of n-Lie algebras is the special case of (9) where all the λ i are equal. Notice that the higher the dimension of ker α the less equation we have and the less terms we have in each equation, that is, in these cases, the Hom-Nambu-Filippov identity is considerably less restrictive. Another difference from the case of n-Lie algebras is that the isomorphisms in Hom-algebras intertwine the multiplications and the twisting maps, which leads to different, more restrictive isomorphism conditions and, in general, more isomorphism classes.
skew-symmetric and α nilpotent. Let (e i ) 1≤i≤n+1 be a basis of A where α is in its Jordan form, and consider [·, . . . , ·] to be defined as in (8).
Proof. Suppose that dim ker α ≥ 2, then for all 1 ≤ i ≤ n + 1, Suppose now that dim ker α = 1, then we have α(e 1 ) = 0 and α(e i ) = e i−1 for 2 ≤ i ≤ n + 1. We get and B are the matrices representing the twisting map α and the bracket in any given basis.
Corollary 3. Let (A, [·, . . . , ·] , α) be an n-ary Hom-algebra with dim A = n + 1, [·, . . . , ·] skew-symmetric and α nilpotent. Let (e i ) 1≤i≤n+1 be a basis of A where α is in its Jordan form, and consider [·, . . . , ·] to be defined by its structure constants in this basis, that is, Remark 5. Note that when dim A = n + 1, it is sufficient to define the bracket by its structure constants as [e 1 , . . . , e i , . . . , Applying Lemma 5 to the class of 3-Hom-Lie algebras 4 3,N(2),6 , we get the following result describing all multiplicative 3-Hom-Lie algebras in the class 4 3,N(2),6 . A consequence of Lemma 1 is that the derived series and the central descending series of an n-Hom-Lie algebra are algebraic invariants. Here, we divide the considered class of 3-Hom-Lie algebras into five subclasses following their derived series and central descending series. Two 3-Hom-Lie algebras in two different subclasses will necessarily be non-isomorphic, and we use this as an intermediate step towards the full classification up to isomorphism of the algebras in this class.
In the case of n-Hom-Lie algebras, the terms of the derived series and the central descending series are in general not ideals as in the case of n-Lie algebras. In the most general case, they are weak subalgebras, and they can be subalgebras or ideals if the twisting maps are algebra morphisms or surjective algebra morphisms respectively, as it has been shown in [59]. For the case of 4 3,N(2),6,M , we have the following result.
If RankB = 1 or equivalently d(p, q) = 0, for all 1 ≤ p < q ≤ 4, then 4 3,N(2),6 is 2-solvable of class 2, and also dim Z(A ) = 1, and Proof. By Remark 2, we know that 4 3,N(2),6 is 3-solvable. The derived series of A are given by is less than 3 (the arity). We compute now the 2-derived series, and has the same dimension. We conclude in this case that A is not 2-solvable.  The following theorem gives the classification up to isomorphism of the class of 3-Hom-Lie algebras 4 3,N(2),6 . Note that isomorphisms are considered in the sense of Hom-algebras, that is they are required to intertwine not only the multiplications, but also the twisting maps.    Two such algebras, given by the structure constants (c ′ (i, j, k, p)) and (c ′′ (i, j, k, p)) respectively are isomorphic if and only if c ′ (1,2,4,1) c ′′ (1,2,4,1) is a square in K.

Any two different brackets of this form give non-isomorphic 3-Hom-Lie
algebras.
which is equivalent to p(1, 1) = 0 and p(3, 3) = 0. We denote by c ′ (i, j, k, p) the structure constants of the bracket after the transformation by P.
In this case w 2 and w 3 are linearly dependent. If
Since D 2 2 (A ) is not a Hom-subalgebra of A , it is not a Hom-ideal either. Let us study now whether D 2 2 (A ) is a weak ideal of A . We have