Real adjoint orbits of special linear groups

TL;DR

This paper classifies the orbits of elements in the special linear Lie algebra that are symmetric under conjugation by group elements or involutions, focusing on complex and quaternionic cases.

Contribution

It provides a complete classification of Ad_G-real and strongly Ad_G-real orbits in sl(n, F) for complex and quaternionic fields, extending previous understanding.

Findings

Classified Ad_G-real orbits in sl(n, C) and sl(n, H).

Classified strongly Ad_G-real orbits in these Lie algebras.

Established criteria for symmetry under involutions in the orbits.

Abstract

Let $ G $ be a Lie group with Lie algebra $ \mathfrak{g} $. An element $ X \in \mathfrak{g} $ is called $\mathrm{Ad}_G$-real if $ -X=gXg^{-1} $ for some $ g \in G $. Moreover, if $ -X=gXg^{-1} $ holds for some involution $ g\in G $, then $ X $ is called strongly $\mathrm{Ad}_G$-real. We have classified the $\mathrm{Ad}_G$-real and the strongly $\mathrm{Ad}_G$-real orbits in the special linear Lie algebra $\mathfrak{sl}(n,\mathbb{F}) $ for $ \mathbb{F}=\mathbb{C}$ or $\mathbb{H} $.