Fine gradings on the Lie algebra

Abstract

As it is well known, there is an element in the family of classical Lie algebras which presents more symmetries than the others. We are speaking, of course, about the algebra which, together with the usual two-fold symmetry which is common to the whole family, presents an order three symmetry which makes of it an “exceptional” classical Lie algebra. In this work we describe the fine group gradings induced by outer automorphisms of the Lie algebra . The qualitative approach used here emphasizes some algebraic aspects, as triality, and it is devoted to gradings involving the outer automorphisms of order three. The computation of the remaining fine group gradings, the ones induced by inner automorphisms and outer ones of order two, follows the same pattern as the classical case with n ≥ 5 and so it has been previously considered in the literature.