Week 2

Session 1

Exercise:

Prove that if $X$ and $Y$ are in $\mathfrak{o}(n)$ then $[X,Y]$ is also in $\mathfrak{o}(n)$ .

Exercise:

Show that $X=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ and $Y=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}$ are in $\mathfrak{sl}(2,\mathbf{C})$ .

Exercise:

Check that the Jacobi identity holds for the commutator bracket $[X,Y]:=XY-YX$ .

Exercise:

Why is the tangent vector to $\exp(sX)$ at $s=0$ equal to $X$ ?