Student Seminar: Classical and Quantum Integrable Systems

is a representation of the group SU(2). Here
g =
( ) α β
γ δ
with α = ¯δ and γ = − ¯β is a group element of SU(2).
Exercise 8
Prove that
⎛
1 + c
2
2 1t 2 c 1 c 2 t 2 − c 3 t c 1 c 3 t 2 ⎞
+ c 2 t
α(φ) = −I + ⎝ c
1 + c 2 t 2 2 c 1 t 2 + c 3 t 1 + c 2 2t 2 c 2 c 3 t 2 − c 1 t ⎠
c 3 c 1 t 2 − c 2 t c 3 c 2 t 2 + c 1 t 1 + c 2 3t 2
is a one-parameter subgroup in SO(3), where tan φ 2 = ct, c2 = c 2 1 + c 2 2 + c 2 3 and
⃗c = (c 1 , c 2 , c 3 ) is a constant vector.
Exercise 9
Let
⎛
cos ϕ − sin ϕ
⎞
0
⎛
1 0
⎞
0
B ϕ = ⎝ sin ϕ cos ϕ 0 ⎠ , C θ = ⎝ 0 cos θ − sin θ ⎠ .
0 0 1
0 sin θ cos θ
Show that any matrix A ∈ SO(3) can be represented in the form
A = B ϕ C θ B ψ .
Write the one-parameter subgroup from the exercise 8 in the coordinates (ϕ, θ, ψ).
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