The exceptional groups

Abstract

AbstractIt is the aim of this chapter to describe the ten families of so-called ‘exceptional groups of Lie type’. There are three main ways to approach these groups. The first is via Lie algebras, as is wonderfully developed in Carter’s book 21. The second, more modern, approach is via algebraic groups (see for example Geck’s book ‘Introduction to algebraic geometry and algebraic groups’ [65]). The third is via ‘alternative’ algebras, as in ‘Octonions, Jordan algebras and exceptional groups’ by Springer and Veldkamp [158]. I shall adopt the ‘alternative’ approach, for a number of reasons: although it lacks the elegance and uniformity of the other approaches, it gains markedly when it comes to performing concrete calculations. We obtain not only the smallest representations in this way, but also construct the (generic) covering groups, whereas the Lie algebra approach only constructs the simple groups.KeywordsMaximal SubgroupParabolic SubgroupJordan AlgebraBorel SubgroupOuter AutomorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.