VIII - Lie algebras

Summary

The end of this book will concern connections between Freudenthal and composition algebras on the one hand and Lie algebras and group schemes on the other. We begin with Lie algebras, the subject of this chapter. The classification of finite-dimensional simple Lie algebras over the complex numbers leads to the notion of root system, a language that will be used for the rest of the book. In that classification, one finds infinite families that are related to the unitary, orthogonal and symplectic involutions of n-by-n matrices. The five isolated cases are usually referred to as exceptional, and those cases are where we find the closest links with Albert and octonion algebras. Most of this chapter is devoted to the study of the algebra of derivations of a non-associative or para-quadratic algebra.