Lie Groups Part 2

Problem 4.70 Let $G$ be the group defined by $$ G=\left\{A \in \operatorname{GL}(2, \mathbb{R}): A^{t} A=r^{2} I, r>0, \operatorname{det} A>0\right\} . $$ (i) Find the explicit expression of the elements of $G$. (ii) Find its Lie algebra. (iii) Calculate the adjoint representation of $G$.

Problem 4.71 The algebra $\mathbb{H}$ of quaternions is an algebra of dimension 4 over the field $\mathbb{R}$ of real numbers. $\mathbb{H}$ has a basis formed by four elements $e_{0}, e_{1}, e_{2}, e_{3}$ satisfying $$ e_{0}^{2}=e_{0}, \quad e_{i}^{2}=-e_{0}, \quad e_{0} e_{i}=e_{i} e_{0}=e_{i}, \quad e_{i} e_{j}=-e_{j} e_{i}=e_{k}, $$ where $(i, j, k)$ is an even permutation of $(1,2,3)$. If $q=\sum_{i=0}^{3} a_{i} e_{i} \in \mathbb{H}$, the conjugate quaternion of $q$ is defined by $$ \bar{q}=a_{0} e_{0}-\left(a_{1} e_{1}+a_{2} e_{2}+a_{3} e_{3}\right) $$ and the real number $|q|=\sqrt{\sum_{i=0}^{3} a_{i}^{2}}$ is called the norm of $q$. Let $\mathbb{H}^{*}$ denote the multiplicative group of non-zero quaternions. (i) Prove that $\mathbb{H}^{*}$ is a Lie group. (ii) Consider the map $\rho$ that defines a correspondence from each $p \in \mathbb{H}^{*}$ into the $\mathbb{R}$-linear automorphism of $\mathbb{H}$ defined by $$ \rho(p): q \mapsto \rho(p) q=p q, \quad q \in \mathbb{H} $$ Which is the representative matrix of $\rho(p)$ with respect to the given basis of $\mathbb{H}$ ? Compute its determinant. (iii) Prove that $\rho$ is a representation of $\mathbb{H}^{*}$ on $\mathbb{H} \equiv \mathbb{R}^{4}$. (iv) Find the group of inner automorphisms Int $\mathfrak{g}$ of the Lie algebra of $\mathbb{H}^{*}$.

Problem 4.72 Prove that $S U(2) \cong S p(1)$ and apply it to prove that $S U(2)$ is a twofold covering of $\mathrm{SO}(3)$. Hint Apply that $\operatorname{Sp}(1)$ has centre $\mathbb{Z}_{2} \cong\{ \pm I\}$ and acts by conjugation on $\operatorname{Im} \mathbb{H} \cong \mathbb{R}^{3}$. One can find the relevant theory, for instance, in Ziller [15].