Let $X \subseteq \mathbb { R } ^ { n }$ and $p \in X$. By a tangent vector of $X$ at $p$, we mean $\gamma ^ { \prime } ( 0 )$, where $\gamma : ( - \epsilon , \epsilon ) \longrightarrow X$ is a differentiable function with $\gamma ( 0 ) = p$. ($\epsilon \in \mathbb { R } , \epsilon > 0$.) The tangent space of $X$ at $p$ is the $\mathbb { R }$-vector space of all the tangent vectors at $p$. Think of $\mathrm { GL } _ { n } ( \mathbb { C } )$ as a subspace of $\mathbb { R } ^ { 2 n ^ { 2 } }$, with the euclidean topology. Let $G : = \left\{ A \in \mathrm { GL } _ { 2 } ( \mathbb { C } ) \mid A ^ { * } A = A A ^ { * } = I _ { 2 } , \operatorname { det } A = 1 \right\}$. (A) (2 marks) Show that every tangent vector of $\mathrm { GL } _ { n } ( \mathbb { C } )$ at $I _ { n }$ is of the form $\gamma _ { A } ^ { \prime } ( 0 )$ where $A$ is a $n \times n$ complex matrix and $\gamma _ { A } : \mathbb { R } \longrightarrow \mathrm { GL } _ { n } ( \mathbb { C } )$ is the function $t \mapsto e ^ { t A }$. (B) (3 marks) Show that the tangent space of $G$ at $I _ { 2 }$ is $V : = \left\{ \left. \left[ \begin{array} { c c } i a & z \\ - \bar { z } & - i a \end{array} \right] \right\rvert \, a \in \mathbb { R } , z \in \mathbb { C } \right\}$. (C) (5 marks) Consider the homeomorphism $\Phi : G \longrightarrow \mathbb { S } ^ { 3 }$ (where $\mathbb { S } ^ { 3 }$ denotes the unit sphere in $\mathbb { R } ^ { 4 }$) given by $$\left[ \begin{array} { c c } \alpha & \beta \\ \bar { \beta } & \bar { \alpha } \end{array} \right] \mapsto ( \Re ( \alpha ) , \Im ( \alpha ) , \Re ( \beta ) , \Im ( \beta ) )$$ Define a 'multiplication' on $V$ by $[ A , B ] = \frac { A B - B A } { 2 }$. Determine the multiplication on the tangent space at $\Phi \left( I _ { 2 } \right)$ induced by the derivative $D \Phi$. (Hint: The map $( A , B ) \longrightarrow [ A , B ]$ is $\mathbb { R }$-bilinear.)
Let $X \subseteq \mathbb { R } ^ { n }$ and $p \in X$. By a tangent vector of $X$ at $p$, we mean $\gamma ^ { \prime } ( 0 )$, where $\gamma : ( - \epsilon , \epsilon ) \longrightarrow X$ is a differentiable function with $\gamma ( 0 ) = p$. ($\epsilon \in \mathbb { R } , \epsilon > 0$.) The tangent space of $X$ at $p$ is the $\mathbb { R }$-vector space of all the tangent vectors at $p$. Think of $\mathrm { GL } _ { n } ( \mathbb { C } )$ as a subspace of $\mathbb { R } ^ { 2 n ^ { 2 } }$, with the euclidean topology.
Let $G : = \left\{ A \in \mathrm { GL } _ { 2 } ( \mathbb { C } ) \mid A ^ { * } A = A A ^ { * } = I _ { 2 } , \operatorname { det } A = 1 \right\}$.\\
(A) (2 marks) Show that every tangent vector of $\mathrm { GL } _ { n } ( \mathbb { C } )$ at $I _ { n }$ is of the form $\gamma _ { A } ^ { \prime } ( 0 )$ where $A$ is a $n \times n$ complex matrix and $\gamma _ { A } : \mathbb { R } \longrightarrow \mathrm { GL } _ { n } ( \mathbb { C } )$ is the function $t \mapsto e ^ { t A }$.\\
(B) (3 marks) Show that the tangent space of $G$ at $I _ { 2 }$ is $V : = \left\{ \left. \left[ \begin{array} { c c } i a & z \\ - \bar { z } & - i a \end{array} \right] \right\rvert \, a \in \mathbb { R } , z \in \mathbb { C } \right\}$.\\
(C) (5 marks) Consider the homeomorphism $\Phi : G \longrightarrow \mathbb { S } ^ { 3 }$ (where $\mathbb { S } ^ { 3 }$ denotes the unit sphere in $\mathbb { R } ^ { 4 }$) given by
$$\left[ \begin{array} { c c } \alpha & \beta \\ \bar { \beta } & \bar { \alpha } \end{array} \right] \mapsto ( \Re ( \alpha ) , \Im ( \alpha ) , \Re ( \beta ) , \Im ( \beta ) )$$
Define a 'multiplication' on $V$ by $[ A , B ] = \frac { A B - B A } { 2 }$. Determine the multiplication on the tangent space at $\Phi \left( I _ { 2 } \right)$ induced by the derivative $D \Phi$. (Hint: The map $( A , B ) \longrightarrow [ A , B ]$ is $\mathbb { R }$-bilinear.)