Inner Ideals of Real Simple Lie Algebras

A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise descriptions in the case of the exceptional Lie algebras.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Funding for open access charge: Universidad de Málaga / CBU

Geometries from structurable algebras and inner ideals

Se estudian ciertos tipos de geometrias en algebras estructurables via sus ideales internos abelianos.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Extremal elements in Lie algebras, buildings and structurable algebras

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

Convex subspaces of Lie incidence geometries

We classify the convex subspaces of all hexagonic Lie incidence geometries (among which all long root geometries of spherical Tits-buildings). We perform a similar classification for most other Lie incidence geometries of spherical Tits-buildings, in particular for all projective and polar Grassmannians, and for exceptional Grassmannians of diameter at most 3.Mathematics Subject Classifications: 51E24Keywords: Buildings, parapolar spaces, long root geometries, hexagonal Lie incidence geometrie