Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$

TL;DR

This paper classifies special pure group gradings on simple Lie algebras of types E6, E7, and E8, using quadratic forms and Weyl group actions to distinguish isomorphism classes.

Contribution

It provides a complete classification of special pure gradings on E6, E7, and E8 Lie algebras, identifying the exact number of classes and their invariants.

Findings

3 equivalence classes for E6

4 equivalence classes for E7

5 equivalence classes for E8

Abstract

A group grading on a semisimple Lie algebra over an algebraically closed field of characteristic zero is special if its identity component is zero; it is pure if at least one of its components, other than the identity component, contains a Cartan subalgebra. We classify special pure gradings on Lie algebras of types $E_6$, $E_7$, $E_8$ up to equivalence and up to isomorphism. To this end, we use quadratic forms over the field of two elements to show that there are exactly three equivalence classes for $E_6$, four for $E_7$, and five for $E_8$. The computation of the corresponding Weyl groups and their actions on the universal groups yields a set of invariants that allow us to distinguish the isomorphism classes.