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

TL;DR

This paper classifies splittable lattices within a specific class of completely solvable, metabelian Lie groups formed as semidirect products of abelian groups, focusing on their algebraic and geometric structure.

Contribution

It provides a detailed description and classification of splittable lattices in the metabelian Lie group constructed from commuting diagonal matrices, expanding understanding of lattice structures in solvable Lie groups.

Findings

Classification of splittable lattices in the given Lie group.

Characterization of the lattices via the representation ta.

Identification of conditions for lattice splittability.

Abstract

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) $G:=\mathbb{R}^n\rtimes_\eta\mathbb{R}^m$, where $\eta$ is the continuous representation of the topological additive abelian group $\mathbb R^m$ in $\mathbb R^n$ given by $\eta(t_1,\dots, t_m)=\exp(\sum_{j=1}^{m}t_j\Delta_j)$ with $(\Delta_j)_{1\leq j\leq m}$ is a set of pairwise commuting diagonal matrices in $\mathbb R^{n\times n}$.