Parametrizing Elements of SL(2,C): Exploring Different Approaches

[cristo]

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!

Thanks in advance!

 

[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.