Lie Groups and Lie Algebras: Winter-2023 Practice Exercises 3 : Due Friday 1/27
The Lie algebra $\mathfrak{sl}(2,\mathbb{C})$
S0 4607S
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 \(\mathfrak{sl}(2,\mathbb{C})\).
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!
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.
What (other) Lie algebra do you think this one is the same as?
Bases for $SO(4)$
S0 4607S
Write down the standard basis for the Lie algebra \(\mathfrak{so}(4)\), that is, the infinitesimal rotations in four (Euclidean) dimensions. Note that rotations in orthogonal planes commute. Switch to new basis elements that are the sums and differences of rotations in orthogonal planes. Work out the commutators of these basis elements, showing that \(\mathfrak{so}(4)=\mathfrak{so}(3)\oplus\mathfrak{so}(3)\).
New basis for $\mathfrak{so}(3)$
S0 4607S
Recall the standard basis \(\{r_x,r_y,r_z\}\) for the Lie algebra \(\mathfrak{so}(3)\), satisfying \([r_x,r_y]=r_z\), etc.
What are the eigenvalues of \(r_z\)?
What are the eigenvectors of \(r_z\)?
What is the matrix representation of \(r_z\) in an eigenbasis, that is, in a basis consisting of eigenvectors? In other words, change the basis from the original \(x,y,z\) components to a new basis consisting of eigenvectors, and express \(r_z\) with respect to this basis.
If time permits, express \(r_x\) and \(r_y\) with respect to this basis.