Parametrization manifold of SL(2,R)

In summary, the conversation discusses the decomposition of SL(2,R) into a product of a symmetric matrix and a rotation matrix. The symmetric matrix is parameterized by a hyperboloid, while the rotation matrix is parameterized by a point on a circle. The conversation also mentions the Iwasawa decomposition and the unimodular condition for both matrices. The main question is how to relate the symmetric matrix to the hyperboloid, which is clarified by a simple coordinate transformation.
  • #1
Wledig
69
1
I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a hyperboloid with equation given by ## z^2 - x^2 - y^2 = 1##, while the rotation matrix is parameterized by a point on a circle. I guess the rotation matrix being parameterized by a point on a circle kind of makes sense intuitively, but I can't see how the hyperboloid can possibly parameterize the symmetric matrix. What am I missing here? If anyone can demonstrate the parametrization for the rotation matrix in a rigorous way aswell, I'd be glad.
 
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  • #2
Wledig said:
I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a hyperboloid with equation given by ## z^2 - x^2 - y^2 = 1##, while the rotation matrix is parameterized by a point on a circle. I guess the rotation matrix being parameterized by a point on a circle kind of makes sense intuitively, but I can't see how the hyperboloid can possibly parameterize the symmetric matrix. What am I missing here? If anyone can demonstrate the parametrization for the rotation matrix in a rigorous way aswell, I'd be glad.
Can you show the decomposition? I know of the Iwasawa decomposition into a product diagonal times upper triangle times rotation.

The rotation part is easy: A rotation matrix is of the form ##K = \begin{bmatrix}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi\end{bmatrix}## which is parameterized by ##\varphi \in [0,2\pi )##.

The Iwasawa decomposition is ##G=ANK## with ##A=\begin{bmatrix}e^t&0\\ 0 & e^{-t}\end{bmatrix}\; , \;N=\begin{bmatrix}1&x\\0&1\end{bmatrix}\,.## Of course I could consider ##AN## but this isn't symmetric.
 
  • #3
Thanks for your reply. I think I should've mentioned this but the book also states that both matrices are unimodular. So we could have
## A=\begin{bmatrix}a&c\\ c & b\end{bmatrix}\ ## and ## K = \begin{bmatrix}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi\end{bmatrix} ## so that the product AK yields a matrix with determinant equal to that of the symmetric matrix which is 1 by the unimodular condition. The big question is how to relate the symmetric matrix to the hyperboloid though.
 
  • #5
I see, book would've done me a favor writing the equation in this second manner. Still, I guess I should've noticed that it was just a silly transformation. Thanks again for clarifying things.
 

Related to Parametrization manifold of SL(2,R)

1. What is the Parametrization Manifold of SL(2,R)?

The Parametrization Manifold of SL(2,R) is a mathematical concept used in the study of Lie groups. It refers to the set of all possible values that can be used to represent the group elements of SL(2,R), which is the special linear group of 2x2 real matrices with determinant equal to 1.

2. Why is the Parametrization Manifold of SL(2,R) important?

The Parametrization Manifold of SL(2,R) is important because it allows for a convenient and efficient way to describe the group elements of SL(2,R). This makes it easier to perform calculations and analyze the properties of this group.

3. How is the Parametrization Manifold of SL(2,R) represented?

The Parametrization Manifold of SL(2,R) is often represented as a set of parameters or coordinates, which correspond to the group elements of SL(2,R). These parameters can be real numbers, complex numbers, or other mathematical objects.

4. What is the relationship between the Parametrization Manifold of SL(2,R) and the Lie algebra of SL(2,R)?

The Parametrization Manifold of SL(2,R) and the Lie algebra of SL(2,R) are closely related. The Lie algebra is a vector space that describes the infinitesimal behavior of a Lie group, while the Parametrization Manifold describes the finite behavior. In other words, the Parametrization Manifold is a global representation of the Lie algebra.

5. How is the Parametrization Manifold of SL(2,R) used in physics?

The Parametrization Manifold of SL(2,R) has many applications in physics, particularly in the study of symmetries and conservation laws. It is used in the theory of special relativity, quantum mechanics, and other areas of physics to describe the behavior of physical systems with certain symmetries.

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