On real and rational elements in a class of Lie groups

Abstract

For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we provide a general method to determine real and rational 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 $\mathrm{SL}(2,\mathbb{R})⋉{\mathrm{Sym}}^n({\mathbb{R}}^2)$. Moreover, we study real elements in connected real solvable Lie groups. It is easy to observe that if $G$ is a simply connected solvable real Lie group, then the identity is the only real element of $G$. Conversely, if the identity is the only real element of a connected real Lie group $G$, then $G$ is simply connected, solvable, and exponential, which turns out to be of independent interest.

arXiv comment18 pages, comments are welcome