@ARTICLE{À venir, 

TITLE={[FORTHCOMING] Réseaux dans les groupes nilpotents métabéliens de type ℝ<sup>n</sup> ⋊ ℝ<sup>m</sup>}, 
  
AUTHOR={Béchir Dali 
, Yasmin Ben Soula, }, 
	
JOURNAL={Avancées en Mathématiques Pures et Appliquées}, 
	
VOLUME={}, 

NUMBER={Articles à paraître<br>}, 
	
YEAR={2026}, 

URL={https://www.openscience.fr/Reseaux-dans-les-groupes-nilpotents-metabeliens-de-type-%E2%84%9Dn-%E2%8B%8A-%E2%84%9Dm}, 
	
DOI={À venir}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={The main objective of this paper is  to characterize a class of lattices, called splittable lattices, in the nilpotent semidirect product Lie group $$$G=\mathbb R^n\rtimes_\eta\mathbb R^m$$$. Here, $$$\eta$$$ denotes the representation of the abelian Lie group $$$\mathbb R^m$$$ in $$$GL_n(\mathbb R)$$$ defined by $$$\eta(t_{1},\dots,t_{m})=\exp{(\sum_{i=1}^mt_iN_i)}$$$ with $$$(N_i)_{1\leq i\leq m}$$$ is a set of pairwise commuting nilpotent matrices in $$$\mathbb Q^{n\times n}$$$. As an application we study the case of the nilpotent Lie group $$$F_{n,m}=\mathbb R^n\rtimes_\eta\mathbb R^m$$$ where $$$N_i=J^i$$$ with $$$J=\sum_{k=1}^{n-1}E_{k+1,k}$$$ where $$$E_{k+1,k}\in\mathbb R^{n\times n}$$$ denotes the matrix given by the Kronecker delta, namely,   $$$\big(E_{k+1,k}\big)_{i,j}=\delta_{i,k+1}\delta_{j,k}$$$.}}