Gradings on simple Jordan and Lie algebras

Abstract

In this paper we describe all group gradings by a finite Abelian group G of several types of simple Jordan and Lie algebras over an algebraically closed field F of characteristic zero.  2004 Published by Elsevier Inc.

Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms

“…The question whether each grading is also a group grading seemed to be positively answered in [24]. However, A. Elduque in [7] gave an example of a grading (on a 16-dimensional complex non-simple algebra), whose grading subspaces cannot be indexed by elements of any Abelian group neither semigroup while satisfying the commutation relations (1).…”

“…A number of works followed [8,9,10,11,12,13], using the theoretical results of that article for applications on concrete Lie algebras. In recent years, gradings were intensively studied not only on the classical finite-dimensional Lie algebras in [1,2], but also on the exceptional Lie algebras in [4,5,6].…”

Fine gradings on $\mathfrak{e}_6$

“…For algebraically closed fields of characteristic zero, the gradings on the classical Lie algebras have been studied in [6,9,16,18] and the gradings in some exceptional ones, namely, f 4 and g 2 , in [14,15,13,7]. Lately, some authors have already studied the case of prime characteristic, [5] in the classical case (with the exception of d 4 ) and [20] in f 4 and g 2 .…”

“…About the results we would like to mention the great amount of fine gradings on e 6 . If we take into account that in g 2 there are 2 fine gradings, and in f 4 there are 4 ones, we find that the number in this case is much bigger, proportionally speaking.…”

“…Besides, in this section about generalities, we recall the elementary p-groups of Int e C 6 and try to translate the result to a more general field. All this will be used in Section 3, after describing some examples of fine gradings on e 6 , to prove that they are all the possible cases of fine gradings produced by inner automorphisms. This only needs a technical result, the fact that every MAD-group contains a minimal elementary non-toral p-group (even more, of minimum rank).…”