Quaternion representation and Haar measure of $SU(3)$ [closed]

Question

Do we have easily and practically useful quaternion representation for $SU(3)$ group element and for Haar measure?

Also, is $SU(2)$ really simplified in the quaternion base?

Answer 1

The Haar measure of $\text{SU}(3)$ is given by

$$ dV = \sin 2\beta \sin 2b \sin 2\theta \sin^2 \theta\; d\alpha\; d\beta\; d\gamma\; d\theta\; da\; db\; dc\; d\phi, $$ for the parameterization $$U(\alpha,\beta,\gamma,\theta,a,b,c,\phi) = e^{i\lambda_3 \alpha} e^{i\lambda_2 \beta}e^{i\lambda_3 \gamma} e^{i\lambda_5 \theta} e^{i\lambda_3 a} e^{i\lambda_2 b} e^{i\lambda_3 c} e^{i\lambda_8 \phi}, $$ in terms of the Gell-Mann matrices.

See The geometry of $\text{SU}(3)$. For the alternative representation in terms of quaternions, see Explicit closed-form parametrization of SU(3) and SU(4) in terms of complex quaternions and elementary functions.

---

For the record, I also give the corresponding (much simpler) result for $\text{SO}(3)$, $$dV=\sin 2\theta\;d\alpha \;d\beta\;d\theta,$$ in the quaternion parameterization \begin{align} O&=\begin{pmatrix} e^{i\alpha\sigma_2}&0\\ 0&1 \end{pmatrix}\begin{pmatrix} 1&0\\ 0&e^{i\theta\sigma_2} \end{pmatrix}\begin{pmatrix} e^{i\beta\sigma_2}&0\\ 0&1 \end{pmatrix},\;\;\sigma_2={{0\,-i}\choose{i\;\;\,0}}. \end{align} See arXiv:1405.3115