Béchir Dali | Searcharxiv

arXiv2025

Splittable Lattices in the metabelian solvable Lie group $\mathbb{R}^n\rtimes\mathbb{R}^m$

The purpose of this note is describe and classify the splittable lattices in the completely solvable metabelian Lie group (semidirect product of abelian vector groups) $G:=\mathbb{R}^n\rtimes_η\mathbb{R}^m$, where $η$ is the continuous representation of the topological additive abelian group $\mathbb R^m$ in $\mathbb R^n$ given by $η(t_1,\dots, t_m)=\exp(\sum_{j=1}^{m}t_jΔ_j)$ with $(Δ_j)_{1\leq j\leq m}$ is a set of pairwise commuting diagonal matrices in $\mathbb R^{n\times n}$.

math.DG↗

arXiv2009

Deformation of Vect($\mathbb{R})$-Modules of Symbols

We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_δ^n=\bigoplus_{j=0}^n \mathcal{F}_{δ-j}$. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space $\mathrm{H}^2_{\rm diff}(\mathfrak{vect}(1),{\cal D}_{ν,μ})$ where ${\cal D}_{ν,μ}$ is the space of differential operators from $\mathcal{F}_ν$ to $\mathcal{F}_μ$. Necessary second-order integrability conditions of any infinitesimal deformations of $\mathcal{S}_δ^n$ are given. We describe completely the formal deformations for some spaces $\mathcal{S}_δ^n$ and we give concrete examples of non trivial deformations.

math.RT↗