Representations of Compact Groups

Abstract

If we leave the framework of pure algebra and decide to consider continuous homomorphisms, continuous representations, and so on, we discover that compact topological groups are in many respects similar to finite groups. In this chapter we examine the simplest examples of connected compact groups and their linear representations, and we generalize to compact linear groups the main theorems on matrix elements proven in the preceding chapter for finite groups. In the next chapter we will discuss the basic method of the theory of Lie groups and then use it, in particular, to describe all linear representations of the simplest compact groups. All this, however, can serve only as an introduction to the present day theory of linear representations of compact groups, the most important parts of which are

• 1) the classification of linear representations of compact connected Lie groups, and

• 2) the Peter-Weyl Theorem on the completeness of the set of matrix elements of an arbitrary compact group.