Among the topological groups, compact groups are the most easy ones to handle. Since we would like to treat their representations next, it is quite natural that the appearance of topology leads to an additional requirement for an appropriate definition, namely a continuity requirement. We will describe this first and indicate the necessary changes for the general concepts from Sections 1.1 to 1.5 (to be used in the whole text later). In the next chapter we specialize to compact groups. We will find that their representation theory has a lot in common with that of the finite groups, in particular the fact that all irreducible representations are finite-dimensional and contained in the regular representation (the famous Theorem of Peter and Weyl). But there is an important difference: the number of equivalence classes of irreducible representations may be infinite.
KeywordsIrreducible Representation Topological Group Compact Group Unitary Representation Continuous Representation
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