Math 7320, Problems 1

1) Show that the group O(p, q) has 4-connected components if p, q > 0.

2) Let 𝔤 = 𝔰𝔩(2, ℝ) and G = SL(2, ℝ). Show the following:

a) The center of 𝔤 is zero.

b) [𝔤, 𝔤] = 𝔤.

c) The center of G is {±Id}.

d) G/ {±Id} ≃ SO_{o}(2, 1).

3) Let 𝔤 and 𝔥 be Lie algebras and φ : 𝔤 → 𝔥 a homomorphism. Then Ker(φ) is an ideal in 𝔤 and Im(φ) is a subalgebra of 𝔤. Furthermore the map

𝔤/Ker(φ) ∋ X + Ker(φ) ↦ φ(X) ∈ Im(φ)

is an isomorphism of Lie algebras.

4) Show that ℝ^{3} with the usual cross-product is a Lie algebra isomorphic to 𝔰𝔬(3).

5) Find the dimension of the following Lie algebras:

a) 𝔰𝔬(n)

b) 𝔰𝔩(n, ℝ)

c) 𝔰𝔬(p, q).

5) Let SU(2) = {A ∈ GL(2, ℂ) | A^{∗} = A}. Then SU(2) is a linear group.

a) Show that dim SU(2) = 3.

b) Show that the adjoint representation Ad defines a surjective homomorphism onto SO(3) with kernel {±Id}.

6) Show that the exponential map exp : 𝔤 → G is surjective for G = SO(n).