Fatima-Ezzahrae Abid
Fatima-Ezzahrae Abid has a total of 4 co-authors. Here are the top co-authors by number of shared publications:
- Mohamed Boucetta
Université Cadi-Ayyad Faculté des sciences et techniques BP 549 Marrakech Maroc
3 shared publications - Hamza El Ouali
Faculté des sciences et techniques, Université Cadi-Ayyad, Marrakech, Morocco
1 shared publication - Hicham Lebzioui
Univ Sultan Moulay Slimane, Ecole Super Technol Khenifra, BP 170, Khenifra, Morocco
1 shared publication - Saïd Benayadi
Laboratoire IECL, CNRS-UMR 7502, UFR MIM, Université de Lorraine, Metz Cedex 03, France
1 shared publication
Latest Publications
- Open Access
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.Open AccessCyclic 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. - Open Access
Symplectic Leibniz algebras as a non-commutative version of symplectic Lie algebras
We introduce symplectic left Leibniz algebras and symplectic right Leibniz algebras as generalizations of symplectic Lie algebras. These algebras possess a left symmetric product and are Lie-admissible. We describe completely symmetric Leibniz algebras that are symplectic as both left and right Leibniz algebras. Additionally, we show that symplectic left or right Leibniz algebras can be constructed from a symplectic Lie algebra and a vector space through a method that combines the double extension process and the $T^*$-extension. This approach allows us to generate a broad class of examples.
Open AccessSymplectic Leibniz algebras as a non-commutative version of symplectic Lie algebras
We introduce symplectic left Leibniz algebras and symplectic right Leibniz algebras as generalizations of symplectic Lie algebras. These algebras possess a left symmetric product and are Lie-admissible. We describe completely symmetric Leibniz algebras that are symplectic as both left and right Leibniz algebras. Additionally, we show that symplectic left or right Leibniz algebras can be constructed from a symplectic Lie algebra and a vector space through a method that combines the double extension process and the $T^*$-extension. This approach allows us to generate a broad class of examples.
- Open Access
Complete Description of Invariant, Associative Pseudo-Euclidean Metrics on Left Leibniz Algebras via Quadratic Lie Algebras
A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered $\mathrm{L}$-invariant (resp. $\mathrm{R}$-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if $\langle u\bullet v,w\rangle= \langle u,v\bullet w\rangle$ for all $u, v, w \in \mathfrak{g}$. These three notions coincide when $\mathfrak{g}$ is a Lie algebra and in this case $\mathfrak{g}$ endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of $\mathrm{L}$-invariant, $\mathrm{R}$-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a $\mathrm{L}$-invariant or $\mathrm{R}$-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from these complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an $\mathrm{L}$-invariant metric, up to dimension 4, and $\mathrm{R}$-invariant metric up to dimension 5.Open AccessComplete Description of Invariant, Associative Pseudo-Euclidean Metrics on Left Leibniz Algebras via Quadratic Lie Algebras
A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered $\mathrm{L}$-invariant (resp. $\mathrm{R}$-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if $\langle u\bullet v,w\rangle= \langle u,v\bullet w\rangle$ for all $u, v, w \in \mathfrak{g}$. These three notions coincide when $\mathfrak{g}$ is a Lie algebra and in this case $\mathfrak{g}$ endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of $\mathrm{L}$-invariant, $\mathrm{R}$-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a $\mathrm{L}$-invariant or $\mathrm{R}$-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from these complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an $\mathrm{L}$-invariant metric, up to dimension 4, and $\mathrm{R}$-invariant metric up to dimension 5.