The Exceptional Lie algebra G_2.

Abstract

The Killing-Cartan classification of finite-dimensional complex simple Lie algebras was one of the great milestones of 19th-century mathematics. According to it, there are four infinite families of classical simple Lie algebras (special linear, orthogonal, and symplectic) and five isolated "exceptional" examples, G_2, F_4, E_6, E_7 and E_8, of dimensions 14, 52, 78, 133 and 248 respectively. In this brief course, we would like to speak about the smallest of the exceptional algebras, G_2, as well as its relationship with another relevant nonassociative algebra, the octonion algebra, for which G_2 is the derivation algebra. We will use this example to illustrate the structure theory of simple Lie algebras over C, while giving some hints about the classification over the reals. Hopefully, we speak about the relevance of G_2 to Geometry or Physics.