Gradings on Simple Lie Algebras

Classification of involutions on graded-division simple real algebras

“…In this section we review, following [9], the basic definitions and properties of gradings that will be used in the rest of the paper. Here we only deal with associative algebras.…”

“…The study of gradings on various algebras has recently become an active research field -see the monograph [9] and the references therein for an overview of this topic. One of the milestone results in that monograph (following [5,2,8]) is the classification of gradings on classical simple Lie algebras over algebraically closed fields of characteristic different from 2.…”

“…We consider only abelian grading groups here because of our intended applications: the support of a grading on a simple Lie algebra always generates an abelian subgroup of the grading group (see for example [9,Proposition 1.12]). Our main classification results are achieved in Sections 6, 7, 8 and 9.…”

Graded simple Lie algebras and graded simple representations

“…In the past two decades, there was a significant interest to the study of gradings on simple Lie algebras by arbitrary groups, see the recent monograph [7] and references therein. In particular, there is an essentially complete classification of fine gradings (up to equivalence) on all finite-dimensional simple Lie algebras over an algebraically closed field of characteristic 0, see [5,7,18]. Some properties of simple Z 2 -graded Lie algebras were studied in [19].…”

“…A classification of simple Lie algebras with a Z ngrading such that all homogeneous spaces are one-dimensional was obtained in [9]. For a given abelian group Q, a classification of Q-gradings (up to isomorphism) on classical simple Lie algebras over an algebraically closed field of characteristic different from 2 was obtained in [2,5], see also [7]. In [1], one finds some characterizations of graded-central-simple algebras with split centroid, see Correspondence Theorem 7.1.1 in [1].…”