Normal forms of nilpotent elements in semisimple Lie algebras

TL;DR

This paper generalizes the Jordan normal form for nilpotent elements in semisimple Lie algebras using cyclic elements theory, providing a unified normal form framework.

Contribution

It introduces a generalized normal form for nilpotent elements in semisimple Lie algebras, extending classical Jordan form concepts.

Findings

Derived a normal form for nilpotent elements in semisimple Lie algebras.

Unified the classification of nilpotent elements across different Lie algebras.

Extended the theory of cyclic elements to broader algebraic contexts.

Abstract

We find the normal form of nilpotent elements in semisimple Lie algebras that generalizes the Jordan normal form in $\mathfrak{sl}_N$, using the theory of cyclic elements.