Example: and rotations

The Lie algebra of consists of real skew-symmetric matrices; exponentials are rotation matrices.

Example: and rotations

Let $SO(3)$ be the rotation group. Its Lie algebra is $\mathfrak{so}(3)$ , the real skew-symmetric $3\times 3$ matrices with bracket $[A,B]=AB-BA$ .

A concrete basis and bracket computation

Set

A_1=\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad A_2=\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix},\quad A_3=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}.

Direct multiplication gives the familiar relations

[A_1,A_2]=A_3,\qquad [A_2,A_3]=A_1,\qquad [A_3,A_1]=A_2.

Thus $\mathfrak{so}(3)$ is 3-dimensional and simple as a real Lie algebra (and closely related to $SU(2)$ via a covering).

Equivalently, identifying $(a_1,a_2,a_3)\in\mathbb R^3$ with $a_1A_1+a_2A_2+a_3A_3$ turns the Lie bracket into the cross product on $\mathbb R^3$ .

For $\theta\in\mathbb R$ , consider $\theta A_3$ . Since $A_3$ acts as an infinitesimal rotation in the $(x,y)$ -plane, the exponential map yields

\exp(\theta A_3)= \begin{pmatrix} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0&0&1 \end{pmatrix},

the rotation about the $z$ -axis by angle $\theta$ . Similar formulas hold for $\exp(\theta A_1)$ and $\exp(\theta A_2)$ .

$SO(3)$ is connected but not simply connected. Its universal cover is $SU(2)$ , with a 2-to-1 covering homomorphism (see covering Lie groups and $\mathrm{Spin}(3)$ ).