arXiv subjects

Abdennour Kitouni

Publications and source records attributed to Abdennour Kitouni.

5 recordsLinked to original sources

arXiv2026

Hom-unitality and hom-associative structures

We study hom-associative structures on general possibly non-associative algebras focusing on one-sided and two-sided unital algebras. New characterizations and aspects of these structures, along with some important subclasses, are explored for nonassociative algebras. By exploiting the observation that the twisting linear map in the hom-associativity axiom of one-sided unital hom-associative algebras is a left or right multiplication operator by an element of the algebra (obtained by the action of the twisting map on a corresponding one-sided unity), a new characterization of the multiplicative twisting operators (or, in other words, the multiplicative hom-associative algebras) is established for one-sided unital algebras. This demonstrates a strong connection between multiplicativity in hom-associative structures and the idempotents of the algebra, thereby further enhancing our understanding of the structure and special nature of multiplicative hom-associative algebra structures as a special subclass of arbitrary hom-associative structures with arbitrary linear twisting maps. Furthermore, new insights into subspaces and a subalgebra of hom-unities, that is, elements that induce hom-associativity by multiplication, are obtained. Moreover, because non-unital hom-associative algebras need not be twisted by a multiplication operator, a unitalization process is employed to describe a subalgebra of two-sided hom-unities that induce hom-associative structures on such algebras. This is formulated in terms of the eigenspaces of multiplication operators within the algebra. Additionally, the obtained insights and general results about the structure and characterization of hom-algebra structures are applied to some important known general classes of non-associative algebras, such as commutative, possibly non-associative algebras, Cayley-Dickson algebras, Leibniz algebras, and hom-Leibniz algebras.

math.RA↗

arXiv2023

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 up on 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 has been determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals or Hom-ideals.

math.RA↗

arXiv2018

On $n$-ary Generalization of BiHom-Lie Algebras and BiHom-Associative 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 objets. Moreover, we provide some key constructions and study $n$-ary BiHom-Lie algebras induced by $(n-1)$-ary BiHom-Lie algebras.

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arXiv2015

On $(n+1)$-Hom-Lie Algebras Induced by $n$-Hom-Lie Algebras

The purpose of this paper is to study the relationships between an $n$-Hom-Lie algebra and its induced $(n+1)$-Hom-Lie algebra. We provide an overview of the theory and explore the structure properties such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology.

math.RA↗

arXiv2014

On Structure and Central Extensions of $(n+1)$-Lie Algebras Induced by $n$-Lie Algebras

The purpose of this paper is to investigate $(n+1)$-Lie algebras induced by $n$-Lie algebras and trace maps. We highlight a comparison of their structure properties (solvability, nilpotency) and the cohomology groups as well as central extensions. Moreover, we provide for dimensions $n$, $n+1$ and $n+2$, the classification of $n$-Lie algebras which are induced by $(n-1)$-Lie algebras.

math.RA↗