Watching the Lie algebra \(\mathfrak{su}(3)\) at work

Just as in the post on \(\mathfrak{su}(2)\), we may attempt to visualize the effect of the \(\mathfrak{su}(3)\) rotations

\[
\begin{equation*}
t_x = \frac{i}{2}\,\begin{pmatrix}
0 & -1 & 0\\
-1 & 0 & 0\\
0 & 0 & 0 \end{pmatrix} \quad
t_y = \frac{i}{2}\,\begin{pmatrix}
0  & i & 0 \\
-i & 0 & 0 \\
0 & 0 & 0  \end{pmatrix} \quad
t_z = \frac{i}{2}\,\begin{pmatrix}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0 \end{pmatrix}\\
u_x = \frac{i}{2}\,\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & -1\\
0 & -1 & 0 \end{pmatrix}\quad
u_y = \frac{i}{2}\,\begin{pmatrix}
0 & 0 & 0 \\
0 & 0  & i \\
0 & -i & 0 \end{pmatrix}\quad
u_z = \frac{i}{2}\,\begin{pmatrix}
0 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1 \end{pmatrix}\\
v_x = \frac{i}{2}\,\begin{pmatrix}
0 & 0 & -1\\
0 & 0 & 0\\
-1 & 0 & 0 \end{pmatrix}\quad
v_y = \frac{i}{2}\,\begin{pmatrix}
0  & 0 & i \\
0 & 0 & 0 \\
-i & 0 & 0  \end{pmatrix}\quad
v_z = \frac{i}{2}\,\begin{pmatrix}
-1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1 \end{pmatrix}
\end{equation*}
\] As already noted \(v_z = t_z+u_z\), leaving only eight linear independent generators. The resulting animation is shown in Fig. 1.

Fig. 1: The Lie algebra su(3) at work

The red, green and blue symbols mark three unit squares [(0,0), (1,0), (1,i), (0,i)], one in each of the three complex planes which constitute the three-dimensional complex space. The animations show the motion of the three squares when rotated by \(T_x= \mathbb{1}+\alpha\cdot t_x\), \(T_y = \mathbb{1}+\alpha\cdot t_y\), etc.; here, \(\alpha\) denotes the (infinitesimal) rotation angle. Note that it requires an accumulated rotation of \(4\,\pi\) for the squares to return to their original position.

When we examine the right-most three (sub-)figures of Fig. 1 it actually can be seen that \(V_z\) corresponds to the combined effect of \(T_z\) and \(U_z\). E.g. the anti-clockwise rotation of the blue square generated by \(T_z\) and its clockwise rotation generated by \(U_z\) cancel out causing it to remain fixed when \(V_z\) is applied.