Reversibility and Real Adjoint Orbits of Linear Maps

TL;DR

This paper extends the classification of reversible elements and adjoint orbits in general linear groups over real, complex, and quaternionic fields, providing new proofs and a comprehensive classification of real adjoint orbits.

Contribution

It generalizes classical results on reversibility to quaternionic groups and classifies real adjoint orbits in various Lie algebras, offering new proofs and broader understanding.

Findings

Classification of reversible elements in quaternionic linear groups.

New proof of classical results over real and complex fields.

Complete classification of real adjoint orbits in Lie algebras.

Abstract

We extend classical results on the classification of reversible elements of the group $\mathrm{GL}(n, \mathbb{C})$ (and $\mathrm{GL}(n, \mathbb{R})$) to $\mathrm{GL}(n, \mathbb{H})$ using an infinitesimal version of the classical reversibility, namely adjoint reality in the Lie algebra set-up. We also provide a new proof of such a classification for the general linear groups over $\mathbb{R}$ and $\mathbb{C}$. Further, we classify the real adjoint orbits in the Lie algebra $\mathfrak{gl}(n, \mathbb{D})$ for $ \mathbb{D}=\mathbb{R}, \mathbb{C}$ or $\mathbb{H} $.