| In the second half of the 20th century striking classifications of finite simple groups and modular Lie algebras have been obtained. The classification of all finite simple groups states that a finite simple group is either cyclic, alternating, a group of Lie type, or one of 26 sporadic examples. The classification of finite-dimensional modular simple Lie algebras says that a simple modular Lie algebra in characteristic at least $5$ over an algebraically closed field is either classical, of Cartan type, or Melikian. The groups of Lie type and the classical (modular) Lie algebra are strongly related with each other and both form a central part in the conclusions of these classification results. They can be connected by Tits' unifying geometric concept of buildings. Within the theory of finite simple groups the interaction of groups and geometries has been very fruitful. The geometric method in finite group theory has been one of the key ingredients in the theory of finite simple groups. This successful interaction is a model for the relations between Lie algebras and geometries that we wish to explore. An important step towards this approach to Lie algebras is the study of extremal elements initiated by Cohen. An element x in a Lie algebra L is called extremal if [x,[x,L]] equals the 1-space spanned by x. Cohen et al. have shown that a natural geometry on the 1-spaces of a Lie algebra containing extremal elements is related to a building. |
