# Parametrization of SL(3,R)

Are there any known common parametrizations of SL(3,R)? I know that it is easy to obtain a local parametrization by just exponentiating generators from the Lie algebra, but I do not know if they are global or not. How can I know if a parametrization obtained by exponentiating Lie algebra generators is a global parametrization?

As a second part of this question, SL(2,R) can be easily parametrized with three variables $t,\phi, \chi$ as $$\left[\begin{array}{cc}\cosh\chi\cos t+\sinh\chi\sin\phi&\sinh\chi\cos\phi+\cosh\chi\sin t\\\sinh\chi\cos\phi-\cosh\chi\sin t&\cosh\chi\cos t-\sinh\chi\sin\phi\end{array}\right]$$

Are there similar parametrizations for SL(3,R), that have reasonably simple expressions for the elements of the matrix? If I try to exponentiate the Lie algebra elements the resulting matrix elements are a mess.

• Perhaps the Cartan decomposition en.wikipedia.org/wiki/Cartan_decomposition would be along the lines you want? – Yemon Choi Feb 3 '17 at 13:47
• It's not clear how strictly you mean the word 'parametrization': Your 'parametrization' of $\mathrm{SL}(2,\mathbb{R})$ is degenerate along the plane $\chi=0$ in $\chi\phi t$-space, as this whole plane is mapped to the circle $\mathrm{SO}(2)\subset\mathrm{SL}(2,\mathbb{R})$. – Robert Bryant Feb 3 '17 at 15:50
• Indeed it's diffeomorphic to $\mathbf{R}^5\times SO(3)$ and you won't get a homeomorphic parameterization by $\mathbf{R}^8$. On the other hand surjective parametrizations by 8 parameters exist, and probably also injective parameterizations by 8 parameters of dense Zariski open subsets (of course I at least mean continuous, probably analytic, and polynomial at least for the surjective parameterization). – YCor Feb 3 '17 at 23:29
• How do you know it is diffeomorphic to R^5 x SO(3)? Do mean globally or locally? – user2133437 Jun 26 '17 at 9:30

## 1 Answer

By singular value decomposition, every matrix in $SL_3$ factors as $UDV$ with $U$ and $V$ in $SO(3)$ and $D$ of the form $\mathrm{diag}(x,y,1/(xy))$ (or $\mathrm{diag}(e^u,e^v,e^{-u-v})$ if you prefer). Any matrix in $SO(3)$ can be factored as $$\begin{bmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0&0&1 \\ \end{bmatrix} \begin{bmatrix} \cos \beta & 0 & \sin \beta \\ 0&1&0 \\ -\sin \beta & 0 &\cos \beta \\ \end{bmatrix} \begin{bmatrix} 1&0&0 \\ 0 & \cos \gamma & -\sin \gamma \\ 0 & \sin \gamma &\cos \gamma \\ \end{bmatrix}$$ (Euler angle factorization). Is that the sort of formula you want?

• Thanks! Just want to point out that you have a typo in your last matrix (a gamma that became a beta) – user2133437 May 11 '17 at 9:44
• @DavidSpeyer: It seems that the determinant of your last matrix is not 1 (it is $\cos^2\gamma +\sin^2\beta\$). – Mikhail Borovoi May 11 '17 at 13:36
• Looks like Ben McKay fixed it before I got there. Thanks to both of you! – David E Speyer May 11 '17 at 13:46