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General element of SL(2,C)

  1. Jan 11, 2007 #1

    cristo

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    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!
     
  2. jcsd
  3. Jan 11, 2007 #2

    dextercioby

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    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.
     
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