Homework Help: General element of SL(2,C)

1. Jan 11, 2007

cristo

Staff Emeritus
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

$$\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$$ 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!

2. Jan 11, 2007

dextercioby

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.