9.2 Lie Algebra

$\displaystyle G_{1}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)},$	$\displaystyle G_{2}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)},$	$\displaystyle G_{3}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \hline 0 & 0 & 0 & 0 \end{array}\right)}$	(115)
$\displaystyle G_{4}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)},$	$\displaystyle G_{5}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)},$	$\displaystyle G_{6}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)}$	(116)

Thus the mapping $\mathrm{alg}:\mathbb{R}^{3}\rightarrow\mathrm{\mathfrak{se}}(3)$ :

$\displaystyle \mathbf{u},\boldsymbol{\omega}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(117)
$\displaystyle \mathrm{alg}\left(\begin{array}{c} \mathbf{u}\\ \boldsymbol{\omega} \end{array}\right)$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c\vert c} \boldsymbol{\omega}_{\times} & \mathbf{u}\\ \hline 0 & 0 \end{array}\right)$	(118)