Orbits in real $\Z_m$-graded semisimple Lie algebras

TL;DR

This paper introduces a method for classifying homogeneous nilpotent elements in real Z_m-graded semisimple Lie algebras, enabling orbit classification and applications to 4-vector and 4-form classifications in R^8.

Contribution

It provides a novel classification method for homogeneous nilpotent elements in real Z_m-graded semisimple Lie algebras and applies it to classify orbits in specific cases.

Findings

Classification of homogeneous nilpotent elements achieved

Set of orbits for homogeneous elements in real Z_2-graded Lie algebras described

Application to classify 4-vectors and 4-forms on R^8

Abstract

In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on $R^8$ can be given using this method.