In the previous chapter we dealt with real Lie algebras built on real linear vector spaces, as these are what arise when analysing Lie groups whose elements are labelled by certain numbers of real essential independent coordinates. The further study of ‘all possible Lie algebras and Lie groups’, their survey and classification, is quite intricate. We will only try to convey the flavour of the concepts and methods involved. Two crucial steps are: complexification of real Lie algebras, and the direct definition of complex Lie algebras; and the exploration of the properties of solvability, semisimplicity and simplicity. Our final goal is to arrive at the compact simple Lie algebras – CSLA – and their associated Lie groups. We will only aim to give a ‘physical account’ of arguments and strategies in this field. It will be largely descriptive with limited derivations.

A few words about the concept of compactness may be useful at this point. This is a topological property – a given topological space T may either be compact or noncompact. A topology on T is defined through the notion and properties of a class of subsets of T that are called ‘open sets’. Familiar examples are bounded connected open intervals on the real line; bounded simply connected open regions in the plane, etc. The space T is compact if every covering of it by a family of open sets contains a finite subset of these open sets which also cover T : every open cover contains a finite subcover. Otherwise it is noncompact. In an intuitive sense both the real line and the plane are noncompact. On the other hand, both the groups SO (3) and SU (2) are compact. An intuitive indication of this is that with respect to the invariant volume elements for them given in Eqs. (3.34, 3.87) they both have finite total volumes.

From a Real Lie Algebra to Its Complexification

Let L be an n -dimensional real Lie algebra. We first complexify it as a vector space to arrive at LL consisting of formal expressions of the form