Extremal elements in Lie algebras, buildings and structurable algebras

Abstract

An extremal element in a Lie algebra g over a field of characteristic not 2 is an element x∈g such that [x,[x,g]] is contained in the linear span of x. The linear span of an extremal element, called an extremal point, is an inner ideal of g, i.e. a subspace I satisfying [I,[I,g]]≤I. We show that in characteristic different from 2,3 the geometry with point set the set of extremal points and as lines the minimal inner ideals containing at least two extremal points is a Moufang spherical building, or in case there are no lines a Moufang set.This last result on the Moufang sets is obtained by connecting Lie algebras to structurable algebras, a class of non-associative algebras with involution generalizing Jordan algebras. It is shown that in characteristic different from 2,3 each finite-dimensional simple Lie algebra generated by extremal elements is either a symplectic Lie algebra or can be obtained by applying the Tits-Kantor-Koecher construction to a skew-dimension one structurable algebra. Various relations between the Lie algebra g and its extremal geometry on the one hand and the associated structurable algebra on the other hand are investigated.