Explicit generators of homotopy groups of unitary groups SU(2) and SU(3)

Question

Hello everyone,

in [1] I found some explicit generators of homotopy groups of unitary groups, for example

\(\pi_3[SU_2]\):

\(\begin{bmatrix}z_1\\z_2\end{bmatrix}\rightarrow \begin{bmatrix}z_1 &-\bar{z_2}\\z_2 &z_1\end{bmatrix}\)

and

\(\pi_5[SU_3]\):
\(\begin{bmatrix}z_1\\z_2\\z_3\end{bmatrix}\rightarrow \begin{bmatrix}z_1+\bar{z}_3z_2 & -\bar{z}_3^2 & -\bar{z}_2+\bar{z}_3\bar{z}_1\\z^2_2 & z_1-z_2\bar{z}_3 & z_3+z_2\bar{z}_1 \\ -z_3+\bar{z}_1z_2 & -\bar{z}_2-\bar{z}_1\bar{z}_3 & \bar{z}^2_1\end{bmatrix}\).

I am in search of explicit 2x2 and 3x3 unitary matrices of this form for \(\pi_4[SU_2]\) and \(\pi_5[SU_2]\), as well as \(\pi_n[SU_{2/3}] \ (n>5)\), but haven't been able to find them (maybe they don't exist). If anyone knows of a paper containing them or an easy method of constructing them, I would greatly appreciate it.

Many thanks,

Pascal

  [1]: http://arxiv.org/abs/math/0301192v1

Answer 1

Well, assuming the paper is correct, its section 3 and especially Lemma 3.1 do give an explicit construction of a generator of \(\pi_4[SU(2)]\), if you just set n=2 in that section.