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Bouzid Mosbahi

Publications and source records attributed to Bouzid Mosbahi.

At least 19 recordsLinked to original sources
arXiv2026

Classification of Nijenhuis operators on three-dimensional multiplicative simple Bihom-Lie algebras

We provide a complete classification of Nijenhuis operators on three-dimensional multiplicative simple BiHom-Lie algebras over the complex field C. Based on the classification of such algebras by Saadaoui into three families L1, L2, and L3, we derive the complete system of equations arising from the equivariance conditions and the Nijenhuis identity. We provide explicit matrix representations of all solutions, including all degenerate parameter cases. We construct the BiHom-Fr\"olicher-Nijenhuis bracket on the cochain complex of a BiHom-Lie algebra, showing that Maurer-Cartan elements of this graded Lie algebra are precisely Nijenhuis operators. This enables us to define the cohomology of Nijenhuis operators and study formal deformations. We introduce BiHom-NS-Lie algebras as the algebraic structure underlying Nijenhuis operators. We provide explicit deformed BiHom-Lie algebra structures and prove the non-isomorphism of different families. We also study the moduli space of Nijenhuis operators and provide computational verification of all results.

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arXiv2025

Cohomology and deformation theory of Reynolds--Nijenhuis associative algebras

In this paper, we introduce and study Reynolds--Nijenhuis operators on associative algebras a novel hybrid structure that simultaneously satisfies the defining identities of both Reynolds and Nijenhuis operators. We investigate their connections with Rota-Baxter and modified Rota-Baxter operators. We develop a representation theory for Reynolds--Nijenhuis associative algebras and introduce a corresponding cohomology theory. Furthermore, we establish a one-parameter formal deformation theory for these algebras, examining the role of infinitesimals, rigidity, and equivalence in the context of deformations.

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arXiv2025

Cohomology and deformation theory of Averaging Leibniz algebras

In this paper, we introduce the concepts of representation and dual representation for averaging Leibniz algebras. We also develop a cohomology theory for these algebras. Additionally, we explore the infinitesimal and formal deformation theories of averaging Leibniz algebras, showing that the cohomology we define is closely connected to deformation cohomology.

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arXiv2025

Hochschild cohomology groups of 5-dimensional complex nilpotent associative algebras

This paper explores the structure of low-dimensional cohomology groups in the context of complex nilpotent associative algebras. Specifically, we study 5-dimensional complex nilpotent associative algebras satisfying $\mathcal{A}^4 = 0$ and $\mathcal{A}^3 \neq 0$. Using their isomorphism invariants, we compute and present the zeroth and first Hochschild cohomology groups, $H^0(\mathcal{A}, \mathcal{A})$ and $H^1(\mathcal{A}, \mathcal{A})$, in explicit matrix form. These results show how cohomology helps to identify and classify different associative algebras.

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arXiv2025

Compatible Pairs of Low-Dimensional Associative Algebras and Their Invariants

A compatible associative algebra is a vector space endowed with two associative multiplication operations that satisfy a natural compatibility condition. In this paper, we investigate and classify compatible pairs of associative algebras of complex dimension less than four. Alongside these classifications, we systematically compute and analyze various algebraic invariants associated with them, including derivations, centroids, automorphism groups, quasi-centroids, Rota-Baxter operators, Nijenhuis operators, averaging operators, Reynolds operators, quasi-derivations, and generalized derivations.

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arXiv2025

Rota-Type Operators on 2-Dimensional Pre-Lie Algebras

This paper studies Rota-Baxter, Reynolds, Nijenhuis, and Averaging operators on 2-dimensional pre-Lie algebras over $\mathbb{C}$. Using the classification of 2-dimensional pre-Lie algebras and computational tools like Mathematica or Maple, we describe these Rota-type operators in detail. Our results provide a deeper understanding of these operators and their roles in algebraic structures.

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arXiv2025

Computational Methods for Biderivations of 4-dimensional nilpotent complex leibniz algebras

This paper focuses on the biderivations of 4-dimensional nilpotent complex Leibniz algebras. Using the existing classification of these algebras, we develop algorithms to compute derivations, antiderivations, and biderivations as pairs of matrices with respect to a fixed basis. By utilizing computer algebra software such as Mathematica and Maple, we provide detailed descriptions and examples to illustrate these computations.

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arXiv2025

Computational Approaches to Derivations and Automorphism Groups of Associative Algebras

This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with computational tools like Mathematica and Maple, we offer detailed descriptions of the derivations and automorphism groups for these algebras. Our analysis of these groups helps to uncover important structural features and symmetries in low-dimensional associative algebras.

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arXiv2024

An Algorithmic Approach to Inner Derivations of Low-Dimensional Zinbiel Algebras

In this paper, we introduce the concept of inner derivations of low-dimensional Zinbiel algebras and investigate their properties. The primary objective of this study is to develop an algorithm to characterize the inner derivations of any n-dimensional Zinbiel algebra in matrix form. Additionally, we apply this algorithm to two, three and four-dimensional complex Zinbiel algebras, providing explicit descriptions of their inner derivations.

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arXiv2024

Compatible Associative Algebras and Some Invariants

A compatible associative algebra is a vector space equipped with two associative multiplication structures that interact in a certain natural way. This article presents the classification of these algebras with dimension less than four, as well as the classifications of their corresponding derivations, centroids, automorphisms, and quasi-centroids. We then characterize a selection of further invariants such as Rota-Baxter operators and second cohomology for some specific examples.

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arXiv2024

Central derivations of low-dimensional Zinbiel algebras

The study of central derivations in low-dimensional algebraic structures is a crucial area of research in mathematics, with applications in understanding the internal symmetries and deformations of these structures. In this article, we investigate the central derivations of complex Zinbiel algebras of dimension $\leq 4$. Key properties of the central derivation algebras are presented, including their structures and dimensions. The results are summarized in a tabular format, providing a clear classification of decomposable and indecomposable centroids based on these derivations. Specifically, we show that the centroid of two-dimensional Zinbiel algebras is indecomposable, while in three-dimensional Zinbiel algebras, centroids such as $\A_3^3$, $\A_4^3$, $\A_6^3$, and $\A_7^3$ are decomposable. For four-dimensional Zinbiel algebras, centroids including $\A_1^4$, $\A_3^4$, $\A_5^4$, $\A_9^4$, $\A_{10}^4$, $\A_{11}^4$, and $\A_{16}^4$ are decomposable. Furthermore, the dimensions of central derivation algebras vary across different dimensions: two-dimensional Zinbiel algebras have central derivation dimensions of one, while in three-dimensional and four-dimensional cases, these dimensions range from zero to nine.

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arXiv2024

Quasi-Centroids and Quasi-Derivations of low-dimensional Zinbiel algebras

In this paper, we introduce the concepts of quasi-centroid and quasi-derivation for Zinbiel algebras. Utilizing the classification results of Zinbiel algebras established previously, we describe the quasi-centroids and quasi-derivations of low-dimensional Zinbiel algebras. Additionally, we explore certain properties of quasi-centroids in the context of Zinbiel algebras and employ these properties to classify algebras with so-called small quasi-centroids. This description of quasi-derivations allows us to identify a significant subclass of Zinbiel algebras characterized as quasi-characteristically nilpotent.

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arXiv2024

Cohomology of BiHom-Associative Trialgebras

The paper concerns the cohomology of (multiplicative) BiHom-associative trialgebras. We first detail the correspondence between central extensions and second cohomology. This is followed by a general cohomology theory that unifies those of BiHom-associative algebras and associative trialgebras. Finally, we introduce one-parameter formal deformations and classify generalized $αβ$-derivations of 3-dimensional BiHom-associative trialgebras.

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arXiv2024

Generalized derivations of BiHom-supertrialgebras

In this note, our goal is to describe the concept of generalized derivations in the context of BiHom-supertrialgebras. We provide a comprehensive analysis of the properties and applications of these generalized derivations, including their relationship with other algebraic structures. We also explore various examples and applications of BiHom-supertrialgebras in different fields of mathematics and physics. Our findings contribute to a deeper understanding of the algebraic properties and applications of BiHom-supertrialgebras, and pave the way for further research in this area.

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