3
$\begingroup$

Working through homework and I run into this problem:

Suppose the Lie group $SO^{+}(2,2)$ is presented as the group of all transformations in its associated space. How do you determine whether a given 4x4 transformation matrix is an element of the group?

I'm slightly familiar with the $SO^{+}(1,3)$ Lie group, but have almost no experience with group theory as a whole. So far, I seem to remember the conditions require orthogonality between coordinates, preservation of the orientation (of subgroups?), and elements with a determinant of 1, but still feel like I'm missing something. I expect given a matrix representation of the form:

$\left( \begin{array}{ccc} a & b & 0 & 0\\ c & d & 0 & 0\\ 0 & 0 & e & f\\ 0 & 0 & g & h\end{array} \right) $

Do I need to ensure the determinants of the components

$\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) $ and $\left( \begin{array}{ccc} e & f \\ g & h \\ \end{array} \right) $

are related in a certain way to prevent orientation shifts (such as reflections), or do I only need to pay attention to the total determinant of a transformation?

$\endgroup$
2
+50
$\begingroup$

For $SO(2,2)$, You should check that it preserves the quadratic form $-x^2-y^2+z^2+w^2$, and that the matrix has determinant one. I guess the $+$ means that the transformation preserves both orientations, meaning that in you example both smaller determinants have to be equal to one.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.