Gradings on Algebras over Algebraically Closed Fields

Abstract

The classification, both up to isomorphism or up to equivalence, of the gradings on a finite dimensional nonassociative algebra A over an algebraically closed field F such that the group scheme of automorphisms Aut(A) is smooth is shown to be equivalent to the corresponding problem for A K = A⊗ F K for any algebraically closed field extension K.

Graded simple Lie algebras and graded simple representations

“…Our proof is based on a detailed analysis of properties of the character group of Q and is directed towards showing existence of a non-graded simple ideal in the case when the underlying ungraded Lie algebra is not simple. Thanks to the recent classification of all gradings on finite dimensional simple Lie algebras, see [5,7,18], we thus obtain a full classification of all finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristic 0.…”

“…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].…”

“…From Corollary 16, we actually obtain a full classification of all finitedimensional Q-graded simple Lie algebras over any algebraically closed field of characteristic 0 due to the recent classification of all gradings on finite dimensional simple Lie algebras, see [5,7,18]. For a similar classification over an algebraically closed field of characteristic p > 0 it remains only to determine all gradings on finite dimensional simple Lie algebras.…”

Gradings on the Lie algebra D4 revisited

“…In the past two decades, there has been much interest in gradings on simple Lie algebras by arbitrary groups -see our recent monograph [EK13] and references therein. In particular, the classification of fine gradings (up to equivalence) on all finite-dimensional simple Lie algebras over an algebraically closed field of characteristic 0 is essentially complete ([EK13, Chapters 3-6], [Eld14], [Yu14]). For a given group G, the classification of G-gradings (up to isomorphism) on classical simple Lie algebras over an algebraically closed field of characteristic different from 2 was done in [BK10] (see also [EK13, Chapter 3]), excluding type D 4 , which exhibits exceptional behavior due to the phenomenon of triality.…”