AIMS Lie Groups: Fall-2022 Practice Exercises 2 : Due 02 Tue 6 Dec
Vector Fields on $\mathbb{S}^3$
S0 4482S
Show that the vector fields
\begin{align*}
r_z &= \partial_\phi ,\\
r_x &=
-\sin\phi \,\partial_\theta - \cot\theta\cos\phi \,\partial_\phi
+ \csc\theta\cos\phi \,\partial_\psi ,\\
r_y &=
\cos\phi \,\partial_\theta - \cot\theta\sin\phi \,\partial_\phi
+ \csc\theta\sin\phi \,\partial_\psi
\end{align*}
satisfy the expected commutation relations, that is,
\begin{equation*}
[r_x,r_y] = -r_z ,
\qquad [r_y,r_z] = -r_x ,
\qquad [r_z,r_x] = -r_y .
\end{equation*}
Orthogonal Groups
S0 4482S
Let \(SO(n)=\{M\in\mathbb{R}^{n\times n}:M^TM=1\}\) and \(\mathfrak{so}(n)=\{A\in\mathbb{R}^{n\times n}:A^T+A=0\}\), where \(\mathbb{R}^{n\times n}\) denotes the set of real \(n\times n\) matrices.
Show that \(SO(n)\) is a group.
Show that \(\mathfrak{so}(n)\) is a vector space.
Show that if \(A,B\in\mathfrak{so}(n)\) then \([A,B]=AB-BA\in\mathfrak{so}(n)\).