Hicham Lebzioui

• Université Moulay Ismail de Meknes

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...

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=Ω(e))a symplectic Lie algebra. The product • on g defined by 3ω(x•y,z)=ω([x,y],z)+ω([x,z],y) extends to a left invariant connection ∇ on G which is torsion free and symplectic (∇Ω=0). When ∇ has vanish...

Let $(G,\Omega)$ 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 $(\G,\omega=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$ defined by $3\omega\left(x\bullet y,z\right)=\omega\left([x,y],z\right)+\omega\left([x,z],y\right)$ extends to a left...

A flat quadratic Lie algebra (g,〈,〉,k) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈,〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a q...

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature $(2,n-2)$ must be degenerate and...

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature $(2,n-2)$ must be degenerate and...

A Lorentzian flat Lie group is a Lie group $G$ with a flat left invariant metric $\mu$ with signature $(1,n-1)=(-,+,\ldots,+)$. The Lie algebra $\mathfrak{g}=T_eG$ of $G$ endowed with $\mu(e)$ is called flat Lorentzian Lie algebra. It is known that the metric of a flat Lorentzian Lie group is geodesically complete if and only if its Lie algebra is...

We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate center. We show that the double...