AIMS Lie Groups: Fall-2022
Practice Exercises 7 : Due 07 Tue 13 Dec

1. The Lie algebra $\mathfrak{sl}(2,\mathbb{C})$ S0 4488S Recall that a basis for \(\mathfrak{su}(2)\) is given by \(\{s_m\}\), where \(s_m=-i\sigma_m\) and the \(\sigma_m\) are the Pauli matrices \[ \sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix} \qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix} \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix} \] Consider instead the set of six matrices \(\{s_m,\sigma_m\}\), which is a basis for \(\mathfrak{sl}(2,\mathbb{C})\).
   1. Show that \(\mathfrak{sl}(2,\mathbb{C})\) is a Lie algebra, that is, show that the commutator of any two elements in \(\mathfrak{sl}(2,\mathbb{C})\) is again an element of \(\mathfrak{sl}(2,\mathbb{C})\)
      HINT: You can do this without explicitly working out any of the commutators!
   2. The Killing product on this Lie algebra turns out to be \[ B(X,Y)=\mathrm{Re}\bigl(\mathrm{tr}(XY)\bigr) \] Compute the Killing product of every pair of basis elements of \(\mathfrak{sl}(2,\mathbb{C})\).
      HINT: Again, you can avoid most (but not all) of the explicit computation by making a suitable argument.
   3. What (other) Lie algebra do you think this one is the same as?