arXiv subjects

Fattoum Harrathi

Publications and source records attributed to Fattoum Harrathi.

5 recordsLinked to original sources

arXiv2025

Nijenhuis operators and twisted $\mathcal{O}$-operators on Nambu-Poisson algebras

A ternary Nambu-Poisson algebra (which we call a Nambu-Poisson algebra in the paper) is the underlying algebraic structure of Nambu-Poisson manifolds of order $3$ that appeared in the generalized Hamiltonian mechanics. First, we consider the 2nd cohomology group of a Nambu-Poisson algebra with coefficients in a given representation. Next, we discuss suitable linear deformations of a Nambu-Poisson algebra and show that any such trivial deformation yields a Nijenhuis operator on it. To understand the deformed Nambu-Poisson algebra obtained from a Nijenhuis operator, we introduce a new algebraic structure, which we name NS-Nambu-Poisson algebras. Finally, we consider $\mathcal{O}$-operators twisted by $2$-cocycles and find their close relationships with NS-Nambu-Poisson algebras.

math.RA↗

arXiv2025

Bialgebras, Manin triples of Malcev-Poisson algebras and post-Malcev-Poisson algebras

A Malcev-Poisson algebra is a Malcev algebra together with a commutative associative algebra structure related by a Leibniz rule. In this paper, we introduce the notion of Malcev-Poisson bialgebra as an analogue of a Malcev bialgebra and establish the equivalence between matched pairs, Manin triples and Malcev-Poisson bialgebras. Moreover, we introduce a new algebraic structure, called post-Malcev-Poisson algebras. Post-Malcev-Poisson algebras can be viewed as the underlying algebraic structures of weighted relative Rota-Baxter operators on Malcev-Poisson algebras.

math.RA↗

arXiv2024

Skew-supersymmetric solution of the super Malcev Yang-Baxter equation and Pre-Malcev superalgebras

The purpose of this paper is to introduce the notion of pre-Malcev superalgebras as the algebraic structure behind the super $\mathcal{O}$-operators on Malcev superalgebras. Moreover, the relations among Malcev superalgebras, pre-Malcev superalgebras and pre-alternative superalgebras are established. Then, we study the operator forms of the classical Yang-Baxter equation (CYBE) in Malcev superalgebras and give their relationship with super $\mathcal{O}$-operators. There are close relationships between the CYBE in Malcev superalgebras and pre-Malcev superalgebras which can be interpreted through the super $\mathcal{O}$-operators.

math.RA↗

arXiv2022

Kupershmidt operators on Hom-Malcev algebras and their deformation

The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras, called Hom-pre-Malcev algebras. We also introduce the notion of Kupershmidt operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev and Hom-pre-Malcev algebras using Kupershmidt operators. Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras to the Hom-alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alternative algebras. Finally, we establish a deformation theory of Kupershmidt operators on a Hom-Malcev algebra in consistence with the general principles of deformation theories and introduce the notion of Nijenhuis elements.

math.RA↗

arXiv2021

Hom-pre-Malcev and Hom-M-Dendriform algebras

The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras and M-dendriform algebras, called Hom-pre-Malcev algebras and Hom-M-dendriform algebras. We also introduce the notion of $\mathcal{O}$-operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev, Hom-pre-Malcev and Hom-M-dendriform algebras using $\mathcal{O}$-operators. Hom-pre-Malcev algebras and Hom-M-dendriform algebras generalize Hom-pre-Lie algebras and Hom-L-dendriform algebras respectively to the alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alternative algebras and Hom-alternative quadri-algebras respectively.

math.RA↗