Week 4

Session 1

Pre-class videos

Pre-class exercises

Exercise:

In an earlier video, we constructed a map R : S U ( 2 ) S O ( 3 ) . Check that R ( M 1 ) = R ( M 2 ) if and only if M 1 = ± M 2 . Given a representation S : S O ( 3 ) G L ( n , 𝐂 ) , we get a representation R S : S U ( 2 ) S O ( 3 ) . Show that a representation T : S U ( 2 ) S O ( 3 ) has this form if and only if T ( - M ) = T ( M ) for all M S U ( 2 ) .

Exercise:

Suppose we have a Lie algebra 𝔤 𝔤 𝔩 ( n , 𝐑 ) consisting of real matrices. Consider the subspace 𝔤 𝐂 𝔤 𝔩 ( n , 𝐂 ) consisting of matrices of the form M + i N with M and N in 𝔤 . Show that:

  • this is a Lie subalgebra, i.e. that it is preserved by Lie bracket

  • if f : 𝔤 𝔤 𝔩 ( m , 𝐂 ) is a real-linear Lie algebra homomorphism then f 𝐂 : 𝔤 𝐂 𝔤 𝔩 ( m , 𝐂 ) defined by f 𝐂 ( M + i N ) = f ( M ) + i f ( N ) for M and N in 𝔤 is also a Lie algebra homomorphism.

Session 2

Pre-class videos

Pre-class exercises

Exercise:

Check that ( ± i , 1 ) is an eigenvector of ( cos θ - sin θ sin θ cos θ ) with eigenvalue e ± i θ .

Exercise:

Show that if , is a Hermitian inner product on 𝐂 n and R : U ( 1 ) G L ( n , 𝐂 ) is a representation then u , v i n v = 0 2 π R ( e i θ ) u , R ( e i θ ) v d θ 2 π is also a Hermitian inner product on 𝐂 n .