A note on adjoint reality in simple complex Lie algebras

TL;DR

This paper explores the concept of adjoint reality in simple complex Lie algebras, establishing conditions for semisimple elements and proving that all elements in complex symplectic Lie algebras are adjoint real.

Contribution

It introduces the notion of adjoint reality for Lie algebra elements and characterizes it for semisimple elements in complex simple classical Lie algebras.

Findings

Every element in a complex symplectic Lie algebra is adjoint real.

Characterization of adjoint real and strongly adjoint real semisimple elements.

Extension of classical reality notions to the infinitesimal Lie algebra setting.

Abstract

Let $G$ be a Lie group with Lie algebra $\mathfrak g$. In the paper "Reality of unipotent elements in simple Lie groups, Bull. Sci. Math., 185, 2023, 103261" by K. Gongopadhyay and C. Maity, an infinitesimal version of the notion of classical reality, namely adjoint reality, has been introduced. An element $X \in \mathfrak g$ is adjoint real if $-X$ belongs to the adjoint orbit of $X$ in $\mathfrak g$. In this paper, we investigate the adjoint real and the strongly adjoint real semisimple elements in complex simple classical Lie algebras. We also prove that every element in a complex symplectic Lie algebra is adjoint real.