Classification Of Finite-Dimensional Complex Semi-Simple Lie Algbras And Serre’s Theorem

We consider nite-dimensional complex semi-simple Lie algebras g. Any such Lie algebra hasa Cartan subalgebra h, and its adjoint representation on g yields a root space decomposition of g, whichin turn gives rise to a root system. These are in turn classi ed by the Dynkin diagrams. Conversely,for any root system, there is a corresponding semi-simple Lie algebra, and the complex semi-simpleLie algebras are therefore classi ed by the root system. Given a root system, Serre's theorem statesexplicitly how to reconstruct corresponding semi-simple Lie algebra from this root system