Lie Groups and Lie Algebras: Winter-2023
Practice Exercises 1 : Due Friday 1/13

1. Euler's Formula II S0 4603S
   1. Look up or derive the Taylor series expansions for \(\cos\theta\), \(\sin\theta\), and \(e^x\).
   2. Show that \(e^{i\theta}=\cos\theta+i\sin\theta\).
   3. Verify that \(e^{2i\theta}=(e^{i\theta})^2\) using trigonometric identities, that is, show that \[ \cos2\theta+i\sin2\theta = (\cos\theta+i\sin\theta)^2 \]
2. Orthogonal Groups S0 4603S 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.
   1. Show that \(SO(n)\) is a group.
   2. Show that \(\mathfrak{so}(n)\) is a vector space.
   3. Show that if \(A,B\in\mathfrak{so}(n)\) then \([A,B]=AB-BA\in\mathfrak{so}(n)\).
3. The Rotation Group $SO(2)$ S0 4603S Consider rotation matrices of the form \[ M(\theta) = \begin{pmatrix} \cos\theta&-\sin\theta\\\sin\theta&\cos\theta \end{pmatrix} \]
   1. Show that
      1. \(M(0)=1\)
      2. \(M(-\theta)=M(\theta)^{-1}\)
      3. \(M(\alpha+\beta)=M(\alpha)M(\beta)\)
   2. Compute \(A=\dot{M}=\frac{dM}{d\theta}\Big|_{\theta=0}\).
   3. Show that \(M(\theta)=\exp(A\theta)\).
4. Matrix Exponentation S0 4603S Exponentiate the following matrices by hand:
   1. \(P=\begin{pmatrix}i\alpha&0\\0&0\end{pmatrix}\)
   2. \(Q=\begin{pmatrix}0&i\beta\\0&0\end{pmatrix}\)
   (You may check your answers using computer algebra, but should attempt the calculation first without such assistance.)