The study of SO (3) and SU (2) has shown how elements of a continuous group can be labelled by (a certain number of) real independent continuous coordinates or parameters; how the composition law can be expressed using these coordinates; how in a representation we encounter generators, commutation relations, structure constants; the representation of finite group elements by (products of) exponentials in the generators; and so on. Now we will try to understand all this in a more basic manner and in a general situation.

The work of this chapter and the next one will lead us to a vast generalisation of SO (3) and SU (2) resulting in the so-called classical families of continuous groups which are all, like SO (3) and SU (2), compact. (The concept of compactness will be briefly described in a heuristic manner in Chapter 5.) These are mathematical results from the late nineteenth and early twentieth centuries, associated with the names of Killing, Cartan and Weyl and are truly beautiful.

Local Coordinates, Group Composition, Inverses

Let a Lie group G be given. The dimension of G , also called its order, will hereafter be denoted by r rather than n . (In the development of the theory of compact simple Lie groups, the order is traditionally denoted by r ; and another important property called the rank, which we will come to in Chapter 5, by l . These are the notations used, for instance, in the classic 1951 Princeton lectures by Giulio Racah on Group Theory and Spectroscopy.) In some neighborhood N of e ∈ G , we use r essential real independent parameters to label group elements:

It is understood that a , b , · · · ∈ N ⊂ G. As a convention we always assume

As a ∈ G runs over N , α runs over some open set around the origin in r -dimensional Euclidean space. So in this region and inN , coordinates and group elements determine one another uniquely.