# Classification of balanced toral elements of exceptional Lie algebras

## TL;DR

This paper provides a complete classification of balanced toral elements in exceptional Lie algebras over algebraically closed fields, extending to automorphisms of order p in characteristic zero.

## Contribution

It offers the first comprehensive classification of balanced toral elements in exceptional Lie algebras over fields of positive characteristic and relates these to automorphisms in characteristic zero.

## Key findings

- Classified all G-conjugacy classes of balanced toral elements in exceptional Lie algebras for char p > 0.
- Derived classification of balanced inner automorphisms of order p in characteristic zero.
- Established a link between toral elements and automorphisms in different characteristics.

## Abstract

Let $\mathfrak{g}$ be a simple Lie algebra of exceptional type over an algebraically closed field $k$, and let $G$ be a simple linear algebraic group with Lie algebra $\mathfrak{g}$. For $\mathrm{char} \, k =p >0$, we present a complete classification of the $G$-conjugacy classes of balanced toral elements of $\mathfrak{g}$. As a result, we also obtain the classification of conjugacy classes of balanced inner torsion automorphisms of $\mathfrak{g}$ of order $p$ when $\mathrm{char} \, k =0$.