Journal of Algebra | 2021
Extremal elements in Lie algebras, buildings and structurable algebras
Abstract
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.