18.755 ninth problems, due Monday, November 23, 2015
1. Find as many non-isomorphic four-dimensional compact connected Lie groups as you
can. You should prove that your examples are not isomorphic.
2. Suppose V is a real vector space of dimension n, and X ⊂ V is a discrete (additive)
subgroup. (“Discrete” means that for each ξ ∈ X there is an ǫ > 0 so that
Bǫ (ξ) ∩ X = {ξ} :
the ball of radius ǫ about ξ meets X only in ξ. Prove that there are m linearly independent
vectors (v1 , . . . , vm ) in V so that X is the group generated by the Vj . Deduce that the pair
(V, X) is isomorphic to (Rn , Zm ). Here the subgroup consists of vectors in which the first m
coordinates are integers, and the rest are zero. Therefore V /X is isomorphic to the product
of an m-dimensional torus and an n − m-dimensional vector space.
3. Recall that Sp(n) is the group of n × n quaternionic unitary matrices. For n ≥ 1,
prove that
Sp(n)/Sp(n − 1) ≃ S 4n−1 ,
a sphere of dimension 4n − 1. Deduce that Sp(n) is connected and simply connected.
Calculate the dimension of Sp(n). (Hint: you can use the fact that if H is a closed subgroup
of a Lie group G, then G/H is a manifold of dimension equal to dim G − dim H.)
4. Prove that U (n) is diffeomorphic to SU (n) × U (1). Prove that the Lie groups U (n)
and SU (n) × U (1) are not isomorphic.
5. Recall that Sp(1) is by definition the group of quaternions of norm 1; so the left
multiplication action of Sp(1) on H defines an inclusion
λ: Sp(1) → GLR (H),
the group of R-linear transformations of H. (The λ is meant to stand for “left.”)
(1) Prove that (with an appropriate choice of coordinates)
λ: Sp(1) → SO(4).
(2) Find a second inclusion
ρ: Sp(1) → GLR (H)
with the property that the images of ρ and λ commute.
(3) Prove that the universal cover of SO(4) is Sp(1) × Sp(1).