Representations Definition and Examples

So there are precisely two elements of SU (2) in the preimage of the identity in SO(3) under t , namely I and —1. From Exercise 4.9 we conclude that 4 is a two-to-one function.  

In Proposition 4.8 and in Section 6.3 we will use the Spectral Theorem to simplify calculations in SU (2), In Section 6.6 we will use the homomorphism T between S(7(2) and SO(3) to make some calculations about SO(3) that would be harder to make directly. 

In this section we define representations and give examples. We also define homomorphisms and isomorphisms of representations, as well as unitary representations and isomorphisms. 

If G and p are clear from context, one can call V the representation. This 

Representations are powerful mathematical tools because they allow us to use the concepts of linear algebra to study a group. For instance, we cannot talk about the eigenvalues of an element of an abstract group G, but given a representation (G, V, p) we can talk about the eigenvalues of /)(g) for any g e G. 

---