Lie group representation

Proof. Suppose (SU(2), V, p) is a finite-dimensional unitary irreducible Lie group representation. Let / denote its character. Define the function... 

Exercise 6.13 Suppose (G, V, p) is a Lie group representation where G is a Lie group with a volume-one invariant integral and V is a complex scalar product space ( , ), Then there is a complex scalar product ( , p on V such that p is a unitary representation on V with respect to ( , p. (Hint define... 

So pi is a projective unitary representation of SO(3). In fact, pi is a bona fide projective Lie group representation, i.e,. it is a differentiable ftinction, as we will show in Proposition 10.5. However, pi does not descend from any linear unitary representation of St/(2) (Exercise 10.20). 

All the irreducible linear Lie group representations of 5 U (2) correspond to spin representations of particles, i.e., to irreducible projective representations. The definition is quite natural. 

Next we show that pn is a Lie group representation, i.e., that it is a differentiable function from 50(3) to PU IjP"). To this end, consider Figure 10.8. By Proposition 4.5 we know that is surjective. So given an arbitrary element A 6 5 0(3), there is an element g e 50(2) such that 4>(g) = A. By Proposition B.L we know that local diffeomorphism (Definition B.2). Hence there is a neighborhood A of g such that has a differentiable inverse. By... 

In this appendix we prove Proposition 10.6 from Section 10.4, which states that the irreducible projective unitary Lie group representations of SO(3) are in one-to-one correspondence with the irreducible (linear) unitary Lie group representations of St/ (2). The proof requires some techniques from topology and differential geometry. 

Proof, (of Proposition 10.6) First we suppose that (S(/(2), V, p) is a linear irreducible unitary Lie group representation. By Proposition 6.14 we know that p is isomorphic to the representation R for some n. By Proposition 10.5 we know that R can be pushed forward to an irreducible projective representation of SO(3). Hence p can be pushed forward to an irreducible projective Lie group representation of SO(3). 

Conversely, suppose that (SO(3), P( V), cr) is a finite-dimensional projective rmitary representation. We want to show that cr is the pushforward of the projectivization of a linear unitary representation p of SO (2). In other words, we must show that there is a Lie group representation p that makes the diagram in Figure B.2 commutative, and that this p is a Lie group representation. 

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