Generators and relations for compact Lie algebras

Abstract

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 finiteCartan matrix (Aij) which is attached to the complexification of our compact 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 fieldattached to (Ai}). 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 (Ai}) is any indecomposable Cartan matrix which is non-Euclidean we let Xc (resp. Xc) 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 Xc (resp. Xc) of Xc (resp. Xc), and show that Xc is the only simple homomorphic image of Xc. We then give generators and relations for Ic. The question of when I'C=XC is equivalent to the question of when Xc―Xc, and is a major unsolved question about Kac-Moody algebras. However, thanks to Serre's Theorem, we know XC=XC, and hence XC=XC, when {Aij) is of finite type. This yields a presentation of Xc 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 Xc and XcWe then go on to obtain a presentation of Xc, and then use thisin 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