Gradings on the exceptional Lie algebras \(F_4\) and \(G_2\) revisited
The authors describe all gradings on central simple Lie algebras of type \(G_2\) over an arbitrary field of characteristic other than \(2\) and \(3\) and classify the gradings up to equivalence and up to isomorphism, assuming the field to be algebraically closed. They also give constructions of four fine gradings in the Albert algebra over an algebraically closed field of characteristic different from \(2\) (as pointed out by the authors, one of them is not present in characteristic \(3\)). Then, they prove that these gradings exhaust the list of fine gradings up to equivalence. Finally they classify all gradings on the simple Lie algebra of type \(F_4\) under the same assumptions on the ground field. Since the authors work over field of prime characteristic, the techniques used involve algebraic groups (in the sense of affine group schemes). The relation of properties related to gradings with the automorphism group schemes is explained in section 2 of the paper.
scientific article; zbMATH DE number 6078123