Unitary homomorphism with

The 0(3) group is homomorphic with the SU(2) group, that of 2 x 2 unitary matrices with unit determinant [6]. It is well known that there is a two to one mapping of the elements of SU(2) onto those of 0(3). However, the group space of SU(2) is simply connected in the vacuum, and so it cannot support an Aharonov-Bohm effect or physical potentials. It has to be modified [26] to SU(2)/Z2 SO(3). 

Our purpose is to construct an analogous parameterization for the 50(4) group. In order to reach this goal we mention that there exists a similar homomorphism between the 50(2) group of 2 x 2 unitary matrices with complex elements with the unit determinant and the 50(3) group. The correspondence establishes as follows for a rotation 1Z in the three dimensional space one can choose a quaternion representation r = (rn.r). This quaternion is normalized and it defines a 2 x 2 matrix ... 

Proof. Suppose every linear transformation 7 V —> V that commutes with p is a scalar multiple of the identity. Suppose also that W is an invariant subspace for (G, V, p. must show that IV = V. By Proposition 3.5, because V is finite dimensional there is an orthogonal projection fliy V V whose image is W. Since p is unitary, we can apply Proposition 5.4 to show that the linear transformation flyy is a homomorphism of representations. So, by... 

Proof. By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2. ... 

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