On rational and real elements in a class of Lie groups

TL;DR

This paper develops a unified method to identify rational and real elements across various groups, including Lie and algebraic groups, and applies it to classify elements in specific semidirect product groups.

Contribution

It introduces a general framework for determining rational and real elements in diverse groups, unifying previous results and providing new classifications.

Findings

Classified all real and rational elements in SL(2,R) semidirect products.

Proved that rationality of an element in GL(n,R) extends to its action on vectors.

Unified earlier group-specific results into a broader theoretical framework.

Abstract

For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into a wider framework. As an application, we classify all real and rational elements in the semidirect product ${\rm SL}(2,\mathbb{R}) \ltimes \mathrm{Sym}^n(\mathbb{R}^2)$. Furthermore, for affine groups of the form ${\rm GL}(n,\mathbb{R}) \ltimes \mathbb{R}^n$, we show that if $x \in {\rm GL}(n,\mathbb{R})$ is rational, then $(x,v)$ is rational for every $v \in \mathbb{R}^n$.