arXiv subjects

Hamza El Ouali

Publications and source records attributed to Hamza El Ouali.

9 recordsLinked to original sources

arXiv2026

On flat pseudo-Euclidean solvable Malcev algebras

A pseudo-Euclidean Malcev algebra is a Malcev algebra equipped with a non-degenerate symmetric bilinear form. In this paper, we introduce a curvature operator for pseudo-Euclidean Malcev algebras, generalizing the notion of curvature for pseudo-Euclidean Lie algebras. We define flat pseudo-Euclidean Malcev algebras and show that every flat Euclidean solvable Malcev algebra which is also a Lie algebra remains flat in the classical Lie algebra curvature sense. Furthermore, we develop the flat double extension construction for flat pseudo-Euclidean Malcev algebras and prove that every flat Lorentzian Malcev algebra with a degenerate center can be obtained via the flat double extension of a flat Euclidean Malcev algebra. Moreover, we demonstrate that all flat Lorentzian nilpotent Malcev algebras arise from the flat double extension of a Euclidean abelian Lie algebras. Finally, we establish that any flat Lorentzian nilpotent Malcev algebra is necessarily a Lie algebra and is flat in the classical Lie algebra curvature sense.

math.RA↗

arXiv2026

Pre-symplectic left-symmetric algebras

A pre-symplectic left-symmetric algebra $\mathcal{A}$ is a left-symmetric algebra endowed with a nondegenerate skew-symmetric bilinear form $ω$ such that all left multiplication operators are symmetric with respect to $ω$. In this setting, the underlying subadjacent Lie algebra $(\mathcal{A}^{-},ω)$ forms a flat $T$-symplectic Lie algebra. This paper provides a systematic investigation into the structural properties of pre-symplectic left-symmetric algebras. In particular, we introduce a distinguished subclass termed \emph{Milnor pre-symplectic algebras}, and prove that any pre-symplectic left-symmetric algebra whose commutator ideal is nondegenerate necessarily belongs to this subclass. Next, we investigate the Levi-Civita product associated with symplectic Lie algebras. We show that this product always yields a right-symmetric algebra, and we prove that it forms a left-symmetric algebra if and only if it is associative. Furthermore, we provide a characterization of symplectic Lie algebras in terms of representations of left-symmetric algebras, and conclude by establishing a construction method for these structures known as the $T^*$-extension. Furthermore, we develop a double extension procedure for pre-symplectic left-symmetric algebras by means of commutative associative algebras. We show that every such algebra with a degenerate commutator ideal can be reconstructed via this extension process. More generally, we show that any pre-symplectic left-symmetric algebra is either a Milnor pre-symplectic algebra or can be obtained through a finite sequence of successive double extensions starting from a Milnor pre-symplectic algebra. As a concrete application of these structural results, we provide a complete classification of pre-symplectic left-symmetric algebras of dimension less than or equal to $4$.

math.RA↗

arXiv2026

Structure of Flat Quadratic Quasi-Frobenius Lie Superalgebras via Double Extensions

A flat quadratic quasi-Frobenius Lie superalgebra is a quadratic Lie superalgebra equipped with an additional symplectic structure that is flat with respect to the natural symplectic product. In this paper, we introduce the notion of a flat quadratic double extension of a flat quadratic quasi-Frobenius Lie superalgebra, in the cases where both the symplectic structure and the quadratic structure are either even or odd. We show that, over an algebraically closed field, any such Lie superalgebra can be constructed through a sequence of flat quadratic double extensions starting from the trivial algebra $\{0\}$. Moreover, when the quadratic and symplectic structures have different parity, we introduce the notion of a planar double extension, which constitutes the main novelty of this paper. In this case, we prove that such Lie superalgebras have total dimension $4n$. Finally, we classify flat quadratic quasi-Frobenius Lie superalgebras of dimension at most four and present explicit examples in dimensions six and eight.

math.RA↗

arXiv2026

On anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras

We introduce the notions of anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras, which arise as nearly associative Levi-Civita products associated with pseudo-Euclidean Lie and Jordan algebras, respectively. We establish a correspondence between these classes of algebras and the Levi-Civita products of their associated Lie and Jordan structures. For anti-left-invariant pseudo-Euclidean nearly associative algebras, we prove that they are nilpotent of index at most five and characterize them as Jacobi--Jordan-admissible nearly associative algebras. We further show that the associated pseudo-Euclidean Jacobi--Jordan algebras are cyclic. Motivated by the classical double extension of Medina and Revoy, we introduce a double extension procedure for this class of algebras and prove that every anti-left-invariant pseudo-Euclidean nearly associative algebra can be obtained from a trivial pseudo-Euclidean algebra by a finite sequence of such extensions. For left-invariant pseudo-Euclidean nearly associative algebras, we prove that the associated pseudo-Euclidean Lie algebras are two-step solvable and cyclic. We then develop block, planar, and linear double extensions and show that every left-invariant pseudo-Euclidean nearly associative algebra can be recursively constructed from a quadratic commutative associative algebra by means of block double extensions. Moreover, we prove that over the field of real numbers, every such algebra can be recursively constructed from a quadratic commutative associative algebra using planar double extensions. These recursive constructions provide a unified framework for describing and classifying pseudo-Euclidean nearly associative algebras in both the anti-left-invariant and left-invariant settings.

math.RA↗

arXiv2026

Pseudo-Euclidean Novikov Superalgebras: Structure and Properties

A pseudo-Euclidean Novikov superalgebra $A$ is a Novikov superalgebra endowed with a non-degenerate symmetric bilinear form $\langle,\rangle$ such that all left multiplication operators are $\langle,\rangle$-antisymmetric. In this case, the associated Lie superalgebra $(A^{-},$\langle,\rangle$)$ is a flat pseudo-Euclidean Lie superalgebra. In this paper, we investigate the structure of pseudo-Euclidean Novikov superalgebras. In particular, we introduce a distinguished subclass, called Milnor superalgebras, and prove that any pseudo-Euclidean Novikov superalgebra whose two-sided ideal is non-degenerate belongs to this class. We provide a method for constructing pseudo-Euclidean Novikov superalgebras. We also introduce a double extension procedure for pseudo-Euclidean Novikov superalgebras and show that every such superalgebra with a degenerate two-sided ideal can be obtained via this method. Furthermore, we establish that any pseudo-Euclidean Novikov superalgebra is either a Milnor superalgebra or can be obtained by a sequence of double extensions starting from a Milnor superalgebra. As an application, we provide a complete classification of pseudo-Euclidean Novikov superalgebras of total dimension at most four.

math.RA↗

arXiv2026

Flat quasi-Frobenius Lie superalgebras

A non-associative superalgebra is called pre-symplectic if it is equipped with a non-degenerate, anti-symmetric bilinear form. It is called quasi-Frobenius if, in addition, is a Lie superalgebra and the form is closed. We introduce the Levi-Civita product associated with pre-symplectic superalgebras and establish its existence and uniqueness. We then introduce the symplectic product associated with quasi-Frobenius Lie superalgebras. We prove that while such a product always exists, it is not unique. We therefore define a natural symplectic product that depends only on the Lie structure and the bilinear form. When the curvature of this product vanishes, the superalgebra is called a flat quasi-Frobenius Lie superalgebra. In this paper, we study flat quasi-Frobenius Lie superalgebras and introduce the notion of a flat double extension. We prove that the double extension process characterizes such superalgebras. More precisely, every flat orthosymplectic (resp. periplectic) quasi-Frobenius Lie superalgebra can be obtained by a sequence of flat double extensions starting from an abelian one (resp. the trivial one). Moreover, we show that every flat quasi-Frobenius Lie superalgebra is nilpotent with a degenerate center. We apply our results to obtain a complete classification of such superalgebras of total dimension at most five.

math.RA↗

arXiv2025

Flat pseudo-Euclidean Leibniz superalgebras

In this paper, we introduce pre-Lie and pre-Leibniz superalgebras, which generalize pre-Lie and pre-Leibniz algebras to the super setting. Additionally, we define a Levi-Civita product associated with a symmetric non-degenerate bilinear form on a non-associative superalgebra. This leads to the definition of flat pseudo-Euclidean left Leibniz superalgebras as those whose Levi-Civita product induces a pre-Leibniz structure. We study the structure of flat pseudo-Euclidean left Leibniz superalgebras and provide a characterization theorem. In the second part, we focus on quadratic Leibniz superalgebras and show that such a superalgebra is flat if and only if it is symmetric Leibniz and 2-step nilpotent. We further study the structure of quadratic 2-step nilpotent symmetric Leibniz superalgebras. Finally, we introduce the notion of double extension for flat pseudo-Euclidean (resp. Lie) left Leibniz superalgebras and prove that any flat pseudo-Euclidean non-Lie left Leibniz superalgebra can be obtained by a sequence of double extensions starting from a flat pseudo-Euclidean Lie superalgebra.

math.RA↗

arXiv2025

Cyclic Riemannian Lie groups: description and curvatures

A cyclic Riemannian Lie group is a Lie group $G$ equipped with a left-invariant Riemannian metric $h$ that satisfies $\oint_{X,Y,Z}h([X,Y],Z)=0$ for any left-invariant vector fields $X,Y,Z$. The initial concept and exploration of these Lie groups were presented in Monatsh. Math. \textbf{176} (2015), 219-239. This paper builds upon the results from the aforementioned study by providing a complete description of cyclic Riemannian Lie groups and an in-depth analysis of their various curvatures.

math.DG↗

arXiv2022

Flat symplectic Lie algebras

Let $(G,Ω)$ be a symplectic Lie group, i.e, a Lie group endowed with a left invariant symplectic form. If $\G$ is the Lie algebra of $G$ then we call $(\G,ω=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$ defined by $3ω\left(x\bullet y,z\right)=ω\left([x,y],z\right)+ω\left([x,z],y\right)$ extends to a left invariant connection $\na$ on $G$ which is torsion free and symplectic ($\na\Om=0)$. When $\na$ has vanishing curvature, we call $(G,Ω)$ a flat symplectic Lie group and $(\G,\om)$ a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to $\om$. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection $\na$ of a flat symplectic Lie group is always complete. We prove that the double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of double extension of flat symplectic Lie algebras starting from $\{0\}$. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension $\leq6$.

math.DG↗