Let L be a simple finite-dimensional Lie algebra over an algebraically closed field of characteristic distinct from 2 and from 3. Then L contains an extremal element, that is, an element x such that [x, [x, L]] is contained in the linear span of x in L. Suppose that L contains no sandwich, that is, no element x such that [x, [x, L]] = 0. Then, up to very few exceptions in characteristic 5, the Lie algebra L is generated by extremal elements and we can construct a building of irreducible and spherical type on the set of extremal elements of L. Therefore, by Tits’ classification of such buildings, L is determined by a known shadow space of a building. This gives a geometric alternative to the classical classification of finite-dimensional simple Lie algebras over the complex numbers and of classical finite-dimensional simple modular Lie algebras over algebraically closed fields of characteristic = 5. This paper surveys developments pertaining to this kind of approach to classical Lie algebras.
|Title of host publication||Buildings, finite geometries and groups |
|Subtitle of host publication||proceedings of a Satellite Conference, International Congress of Mathematicians, Hyderabad, India, 2010|
|Editors||N.S. Narasimha Sastry|
|Place of Publication||New York|
|Number of pages||21|
|Publication status||Published - 2012|
|Name||Springer Proceedings in Mathematics (PROM)|