• Support PF! Buy your school textbooks, materials and every day products Here!

General element of SL(2,C)

  • Thread starter cristo
  • Start date
cristo
Staff Emeritus
Science Advisor
8,056
72
I'm halfway through a question on a past differential geometry exam, and suddenly in comes a matrix g a member of SL(2,C) (where C denotes the complex numbers)

Now, I can't remember how to express a general element of this group: I know the matrix must be

[tex]\left(\begin{array}{cc}a&b\\c&d\end{array}\right)[/tex] such that ad-bc=1, but can this be expressed in a more precise way in the complex case (i.e. with fewer than four unknowns, maybe by utilising the complex conjugate)?

I've tried looking on the internet, but to no avail. I would really appreciate someone helping, since I could do with getting on with the question!

Thanks in advance!
 

Answers and Replies

dextercioby
Science Advisor
Homework Helper
Insights Author
12,965
536
What you have is the simplest possiblity of parametrizing an arbitrary element of Sl(2,C). To see that, answer the questions below:

1.How many parameter does SL(2,C) have ?
2.How many does the matrix assume?
3.How many does the ad-bc=1 condition fix?

As for other parametrizations of SL(2,C), search for the "polar decomposition theorem for SL(2,C)". Also for the "Cayley-Klein parametrization of SL(2,C)".
Daniel.
 

Related Threads for: General element of SL(2,C)

Replies
4
Views
384
Replies
11
Views
14K
Replies
4
Views
5K
Replies
10
Views
8K
Replies
8
Views
2K
Replies
7
Views
439
Replies
1
Views
2K
Replies
10
Views
1K
Top