The Classification of Complex Simple Lie Algebras

Abstract

AbstractIn the first section of this lecture we introduce the Dynkin diagram associated to a semisimple Lie algebra g. This is an amazingly efficient way of conveying the structure of g: it is a simple diagram that not only determines g up to isomorphism in theory, but in practice exhibits many of the properties of g. The main use of Dynkin diagrams in this lecture, however, will be to provide a framework for the basic classification theorem, which says that with exactly five exceptions the Lie algebras discussed so far in these lectures are all the simple Lie algebras. To do this, in §21.2 we show how to list all diagrams that arise from semisimple Lie algebras. In §21.3 we show how to recover such a Lie algebra from the data of its diagram, completing the proof of the classification theorem. All three sections are completely elementary, though §21.3 gets a little complicated; it may be useful to read it in conjunction with §22.1, where the process described is carried out in detail for the exceptional algebra \( {\mathfrak{g}_2}\) . (Note that neither §21.3 or §22.1 is a prerequisite for §22.3, where another description of \({{\mathfrak{g}}_{2}} \) will be given.)