Exercise 2, § 3

Denote by K one of the fields $\mathbf{R},\mathbf{C}$ or $\mathbf{H}$, and by $n$ an integer $\geq 1$. Give the space $K^n_s$ the usual hermitian form.

a) Show that $\mathbf{U}(n,K)$ is a compact real Lie group.

b) Show that the sphere of radius 1 in $K^n_s$ is a (real) Lie homogeneous space for $\mathbf{U}(n,K)$; the fixer of a point is isomorphic to $\mathbf{U}(n-1,K)$.

c) Deduce that the groups $\mathbf{U}(n,\mathbf{C})$ and $\mathbf{U}(n,\mathbf{H})$ are connected, and that $\mathbf{O}(n,\mathbf{R})$ has two connected components.

d) Show that the groups $\mathbf{S}\mathbf{O}(n,\mathbf{R})$ and $\mathbf{S}\mathbf{U}(n,\mathbf{C})$ are connected.

e) Show that the group $\mathbf{O}(n,\mathbf{R})$ (resp. $\mathbf{U}(n,\mathbf{C})$) is the semi-direct product of $\mathbf{Z}/2\mathbf{Z}$ (resp. $\mathbf{T}$) by $\mathbf{S}\mathbf{O}(n,\mathbf{R})$ (resp. $\mathbf{S}\mathbf{U}(n,\mathbf{C})$).