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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 alpha 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.

We compare and examine the influence of Hom-associativity, involving a linear map twisting the associativity axiom, on fundamental aspects important in study of Hom-algebras and (σ,τ)-derivations satisfying a (σ,τ)-twisted Leibniz product rule in connection to Hom-algebra structures. As divisibility may be not transitive in general not necessarily associative algebras, we explore factorization properties of elements in Hom-associative algebras, specially related to zero divisors, and develop an α-deformed divisibility sequence, formulated in terms of linear operators. We explore effects of the twisting maps σ and τ on the whole space of twisted derivations, unfold some partial results on the structure of (σ,τ)-derivations on arbitrary algebras based on a pivot element related to σ and τ and examine how general an algebra can be while preserving certain well-known relations between (σ,τ)-derivations. Furthermore, new more general axioms of Hom-associativity, Hom-alternativity and Hom-flexibility modulo kernel of a derivation are introduced leading to new classes of Hom-algebras motivated by (σ,τ)-Leibniz rule over multiplicative maps σ and τ and study of twisted derivations in arbitrary algebras and their connections to Hom-algebra structures.

This study is concerned with induced ternary Hom-Nambu-Lie algebras from Hom-Lie algebras and their classification. The induced algebras are constructed from a class of Hom-Lie algebra with nilpotent linear map. The families of ternary Hom-Nambu-Lie algebras arising in this way of construction are classified for a given class of nilpotent linear maps. In addition, some results giving conditions on when morphisms of Hom-Lie algebras can still remain morphisms for the induced ternary Hom-Nambu-Lie algebras are given.

The aim of this work is to study properties of n-Hom-Lie algebras in dimension n+1 allowing to explicitly find them and differentiate them, to eventually classify them. Specifically, the n-Hom-Lie algebras in dimension n+1 for n= 4, 5, 6 and nilpotent α with 2-dimensional kernel are computed and some detailed properties of these algebras are obtained.

The aim of this work is to study properties of n-Hom-Lie algebras in dimension n+1 allowing to explicitly find them and differentiate them, to eventually classify them. Some specific properties of (n+1) -dimensional n-Hom-Lie algebra such as nilpotence, solvability, center, ideals, derived series and central descending series are studied, the Hom-Nambu-Filippov identity for various classes of twisting maps in dimension n+1 is considered, and systems of equations corresponding to each case are described. All 4-dimensional 3-Hom-Lie algebras with some of the classes of twisting maps are computed in terms of structure constants as parameters and listed in the way emphasising the number of free parameters in each class, and also some detailed properties of the Hom-algebras are obtained.

The aim of this work is to explore some properties of n-ary skewsymmetric Hom-algebras and n-Hom-Lie algebras related to their ideals, derived series and central descending series. We extend the notions of derived series and central descending series to n-ary skew-symmetric Hom-algebras and provide various general conditions for their members to be Hom-subalgebras, weak ideals or Hom-ideals in the algebra or relatively to each other. In particular we study the invariance under the twisting maps of the derived series and central descending series and their subalgebra and ideal properties for a class of 3-dimensional Hom-Lie algebras and some 4-dimensional 3-Hom-Lie algebras. We also introduce a type of generalized ideals in n-ary Hom-algebras and present a few basic properties.

In this paper we examine interactions between (σ, τ) -derivations via commutator and consider new n-ary structures based on twisted derivation operators. We show that the sums of linear spaces of (σ^k, τ^l) -derivations and also of some of their subspaces, consisting of twisted derivations with some commutation relations with σ and τ, form Lie algebras, and moreover with the semigroup or group graded commutator product, yielding graded Lie algebras when the sum of the subspaces is direct. Furthermore, we extend these constructions of such Lie subalgebras spanned by twisted derivations of algebras to twisted derivations of n-ary algebras. Finally, we consider n-ary products defined by generalized Jacobian determinants based on (σ,τ) -derivations, and construct n-Hom-Lie algebras associated to the generalized Jacobian determinants based on twisted derivations extending some results of Filippov to (σ,τ) -derivations. We also establish commutation relations conditions for twisting maps and twisted derivations such that the generalised Jacobian determinant products yield (σ,τ,n) -Hom-Lie algebras, a new type of n-ary Hom-algebras different from n-Hom-Lie algebras in that the positions of twisting maps σ and τ are not fixed to positions of variables in n-ary products terms of the sum of defining identity as they were in Hom-Nambu-Filippov identity of n-Hom-Lie algebras.

The aim of this paper is to introduce n-ary BiHom-algebras, generalizing BiHom-algebras. We introduce an alternative concept of BiHom-Lie algebra called BiHom-Lie-Leibniz algebra and study various type of n-ary BiHom-Lie algebras and BiHom-associative algebras. We show that n-ary BiHom-Lie-Leibniz algebra can be represented by BiHom-Lie-Leibniz algebra through fundamental objects. Moreover, we provide some key constructions and study n-ary BiHom-Lie algebras induced by (n-1)-ary BiHom-Lie algebra.

The purpose of this work is to generalize the concepts of k-solvability and k-nilpotency, initially defined for n-Lie algebras, to n-Hom-Lie algebras and to study their properties. We define k-derived series, k-central descending series and study their properties, we show that k-solvability is a radical property and we apply all of the above to the case of (n+1)-Hom-Lie algebras induced by n-Hom-Lie algebras.

This study is concerned with induced ternary Hom-Lie-Nambu Lie algebras from Hom-Lie algebras and their classification. The induced algebras are constructed from a class of Hom-Lie algebra with nilpotent linear map. The families of ternary Hom-Nambu-Lie arising in this way of construction are classified for a given  class of  nilpotent linear maps. In addition, some results giving conditions on when morphisms of Hom-Lie algebras can still remain morphisms for the induced ternary Hom-Nambu-Lie algebras are given.