Generators and relations for compact Lie algebras

The main purpose of this paper is to provide a system of generators and relations for each of the nine types of compact simple Lie algebras.Indeed, we are able to give a presentation of each such algebra which depends only on the finite Cartan matrix (Aᵢⱼ) which is attached to the complexification of our com- pact algebra.One of the main results that lies behind our work is the Theorem of Serre which gives a presentation of the simple Lie algebras over the complex field attached to (Aᵢⱼ) .Although our main interest is with the compact Lie algebras, we work in the generality of Kac-Moody Lie algebras (see [1], [4], [7], [8]).In this setting we will be able to provide a generalization of the compact simple Lie algebras.We realize these algebras as certain forms of the Kac-Moody algebra.More specifically, if (Aᵢⱼ) is any indecomposable Cartan matrix which is non-Euclidean we let LC (resp.L̄C ) be the reduced (resp.standard) Kac-Moody Lie algebra over the complex field C , (see Section 1 for more details).We define a real form LC (resp.X C ) of LC (resp.L̄C ), and show that LC is the only simple homomorphic image of L̄C .We then give generators and relations for L̄C .The question of when L̄C=LC is equivalent to the question of when LC=L̄C , and is a major unsolved question about Kac-Moody algebras.However, thanks to Serre's Theorem, we know LC=L̄C , and hence LC=L̄C , when (Aᵢⱼ) is of finite type.This yields a presentation of LC in this case.The content of the paper is as follows.In Section 1 we recall the notation and a few facts about Kac-Moody algebras.In Section 2, the final section, we begin by making a study of the algebras LC and L̄C .We then go on to obtain a presentation of L̄C , and then use this in dealing with the compact simple Lie algebras.We think it is interesting that analogues of the compact Lie algebras exist in the Kac-Moody setting.Moreover, just as the compact algebras are