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

Cristina Draper; Alberto Elduque; Mikhail Kochetov

doi:10.1016/j.laa.2026.02.019

paper

Special pure gradings on simple Lie algebras of types $E_{6}$ , $E_{7}$ , $E_{8}$

arXiv:2507.03762 · doi:10.1016/j.laa.2026.02.019

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.

27 pages

Special pure gradings on simple Lie algebras of types E6​, E7​, E8​

Abstract

Special pure gradings on simple Lie algebras of types $E_{6}$ , $E_{7}$ , $E_{8}$