8.2 Lie Algebra

The Lie algebra $\mathrm{\mathfrak{so}}(3)$ is the set of antisymmetric $3\times3$ matrices, generated by the differential rotations about each axis:

$\displaystyle G_{1}={\scriptstyle \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & ... ...\left(\begin{array}{ccc} 0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)}$

(95)

The mapping $\mathrm{alg}:\mathbb{R}^{3}\rightarrow\mathrm{\mathfrak{so}}(3)$ sends 3-vectors to their skew matrix:

$\displaystyle \boldsymbol{\omega}$	$\displaystyle \equiv$	$\displaystyle \left(\begin{array}{c} a\\ b\\ c \end{array}\right)\in\mathbb{R}^{3}$	(96)
$\displaystyle \mathrm{alg}\left(\boldsymbol{\omega}\right)$	$\displaystyle =$	$\displaystyle \boldsymbol{\omega}_{\times}$	(97)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{ccc} 0 & -c & b\\ c & 0 & -a\\ -b & a & 0 \end{array}\right)$	(98)

Ethan Eade 2012-02-16