arXiv subjects

Othmane Dani

Publications and source records attributed to Othmane Dani.

3 recordsLinked to original sources

arXiv2026

A complete description of solvable symplectic Lie algebras

In this paper, we present a complete characterization of solvable symplectic Lie algebras via a symplectic double extension process. We demonstrate that any such algebra is either symplectically irreducible or can be constructed through a finite sequence of symplectic double extensions by a line or a plane, starting from symplectically irreducible Lie algebras. Furthermore, we show that if a symplectic Lie algebra has a nondegenerate derived ideal, then it is necessarily unimodular and, in particular, solvable. Finally, we present a novel algebraic proof of a classical structural theorem on symplectically irreducible symplectic Lie algebras and classify all Lie algebras of dimension up to $6$ that admit such structures.

math.DG↗

arXiv2025

Some invariant connections on symplectic reductive homogeneous spaces

A symplectic reductive homogeneous space is a pair $(G/H,\Omega)$, where $G/H$ is a reductive homogeneous $G$-space and $\Omega$ is a $G$-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric spaces, and flag manifolds. This paper focuses on the existence of a natural symplectic connection on $(G/H,\Omega)$. First, we introduce a family $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$ of $G$-invariant connection on $G/H$, and establish that $\nabla^{0,1}$ is flat if and only if $(G/H,\Omega)$ is locally a symplectic Lie group. Next, we show that among all $\{\nabla^{a,b}\}_{(a,b)\in\mathbb{R}^2}$, there exists a unique symplectic connection, denoted by $\nabla^\mathbf{s}$, corresponding to $a=b=\tfrac{1}{3}$, a fact that seems to have previously gone unnoticed. We then compute its curvature and Ricci curvature tensors. Finally, we demonstrate that the $\operatorname{SU}(3)$-invariant preferred symplectic connection of the Wallach flag manifold $\operatorname{SU}(3)/\mathbb{T}^2$ (from Cahen-Gutt-Rawnsley) coincides with the natural symplectic connection $\nabla^\mathbf{s}$, which is furthermore Ricci-parallel.

math.DG↗

arXiv2023

Special Affine Connections on Symmetric Spaces

Let $(G,H,σ)$ be a symmetric pair and $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ the canonical decomposition of the Lie algebra $\mathfrak{g}$ of $G$. We denote by ${\nabla}^0$ the canonical affine connection on the symmetric space $G/H$. A torsion-free $G$-invariant affine connection on $G/H$ is called special if it has the same curvature as ${\nabla}^0$. A special product on $\mathfrak{m}$ is a commutative, associative, and ${\operatorname{Ad}}(H)$-invariant product. We show a one-to-one correspondence between the set of special affine connections on $G/H$ and the set of special products on $\mathfrak{m}$. We introduce a subclass of symmetric pairs called strongly semi-simple for which we prove that the canonical affine connection on $G/H$ is the only special affine connection, and we give many examples. We study a subclass of commutative, associative algebra, allowing us to give examples of symmetric spaces with special affine connections. Finally, we compute the holonomy Lie algebra of special affine connections.

math.DG↗