Lie Groups and Lie Algebras: Winter-2023
HW 2 : Due Wednesday 1/25

1. Quaternionic rotations S0 4606S Identify imaginary quaternions \(v=xi+yj+zk\in\mathrm{Im}\mathbb{H}\) with \((x,y,z)\in\mathbb{R}^3\) and let \(q=\cos\frac\alpha2+u\sin\frac\alpha2\), with \(u\in\mathrm{Im}\mathbb{H}\) of unit norm, that is, \(u\bar{u}=1\). Consider the transformation \[ T_q(v) = v\longmapsto qvq^{-1} \]
   1. Show that the only direction fixed by \(T_q\) is \(u\), that is, show that \(T_q(w)=w\) for \(w\in\mathrm{Im}\mathbb{H}\) if and only if \(w\) is a (real) multiple of \(u\).
   2. If \(u\in\mathrm{Im}\mathbb{H}\) is perpendicular to \(v\), show that \(T_q(v)\) with \(\alpha=\frac\pi2\) is perpendicular to both \(u\) and \(v\).
   3. A reasonable conclusion (which you do not need to verify) is that \(T_q\) represents rotation through an angle \(\alpha\) about the \(u\) direction. (Every rigid rotation in \(\mathbb{R}^3\) can in fact be accomplished as a single rotation about some axis.) Using this interpretation, determine the effective angle of a rotation about the \(z\) axis by \(\beta\) followed by a rotation about the \(u\) axis by \(\alpha\). You do not need to determine the effective axis of rotation.
   4. For the special case where \(u=j\) (the \(y\) axis) and \(\alpha=\frac\pi2=\beta\), determine both the angle of rotation and its axis.
      You are encouraged to check your result by actually performing these rotations on a physical object!
2. Generators of $SL(2,\mathbb{R})$ S0 4606S An explicit basis for the Lie algebra \(\mathfrak{sl}(2,\mathbb{C}\)) is given in GELG.
   1. Find a subbasis that spans the Lie subalgebra \(\mathfrak{sl}(2,\mathbb{R})\) of \(\mathfrak{sl}(2,\mathbb{C})\).
   2. Show that the commutators of these elements are linear combinations of these elements, thus verifying that the vector space \(\mathfrak{sl}(2,\mathbb{R})\) is in fact a Lie algebra.
   3. How many boosts (hyperbolic rotations) and how many (ordinary) rotations are in \(\mathfrak{sl}(2,\mathbb{R})\)? Explain your reasoning!
   4. The Killing form \(B\) on a (real) matrix Lie algebra \(\mathfrak{g}\) is given for \(X,Y\in\mathfrak{g}\) by \[B(X,Y)=\mathrm{tr}(XY)\] Determine the Killing form on \(\mathfrak{sl}(2,\mathbb{R})\). That is, determine the matrix representation of the Killing form when acting on the basis you found above.