Reversibility of Affine Transformations

TL;DR

This paper classifies which elements in affine transformation groups over real, complex, or quaternionic fields are reversible or strongly reversible, providing a comprehensive understanding of their conjugacy properties.

Contribution

It offers a complete classification of reversible and strongly reversible elements in affine groups over various fields, extending previous work on conjugacy and symmetry.

Findings

Characterization of reversible elements in affine groups

Criteria for strong reversibility involving involutions

Extension of conjugacy classification to quaternionic fields

Abstract

An element $g$ in a group $G$ is called reversible if $g$ is conjugate to $g^{-1}$ in $ G $. An element $g$ in $G$ is strongly reversible if $ g $ is conjugate to $g^{-1}$ by an involution in $G$. The group of affine transformations of $\mathbb{D}^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $, where $\mathbb{D}:=\mathbb{R}, \mathbb{C}$ or $ \mathbb{H} $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $.