Maximal finite abelian subgroups of E₈

Abstract

The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E 8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the simple Lie algebra of type E 8 with trivial neutral homogeneous component.

An overview of fine gradings on simple Lie algebras

“…The fine gradings by finite groups on the simple Lie algebra of type E 8 have been independently classified in [DE16]. This problem is equivalent to the computation of the conjugacy classes of the maximal abelian finite subgroups of the simple algebraic group E 8 .…”

Fine gradings and automorphism groups on associative algebras

“…Besides being a natural question, it is important to know the fine gradings of an algebra, because the classification up to equivalence of gradings that are not necessarily fine is a difficult problem, as it is illustrated in [16, example 2.41 and figure 2.2]. Therefore the classification of gradings up to equivalence is usually restricted to fine gradings, see for example [18,5,10,13,12,14,11,17]. Once we have classified all gradings on a certain algebra, a possible next step is to compute the automorphism groups of those gradings.…”

Gradings on the real form e6,−14

“…This monograph was published in 2013 and it contains almost completely the classification of the gradings on the complex simple finite-dimensional Lie algebras. Precisely, the classification of the gradings on type-E algebras is only conjectured in the book, and had to wait until [DrV16], [DrE17] (finite groups) and [Y16] to be completed. A review of the fine gradings on Lie algebras of type-E is [DrEl14], which tries to give a version as unified as possible of the gradings on the three complex Lie algebras of type E. Having finished the complex case, it is the moment to return to the real case, which is relevant for many applications.…”