1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Why are SOnR and SLnR Lie Groups?

  1. Oct 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove SOnR and SLnR are Lie groups, and determine their dimensions.

    SOnR = {nxn real hermitian matrices and determinant > 0}
    SLnR = {nxn real matrices with determinant 1}

    3. The attempt at a solution

    We can see that SLnR is level set at zero of the graph of a smooth function, namely the function f:Rm-1 -> R, x -> det[x] - 1, which is smooth because it is just a polynomial in the coefficients of elements of SLnR.
    We know that the graph of a smooth function is a smooth manifold, and by the implicit function theorem, so is a level set of a smooth function, and so is SLnR, in this case of dimension n^2 -1.

    The above is a very short version of what took the better part of several afternoons.

    I still have problems showing one thing:

    I want to make a similar map for SOnR, to show that this is too a smooth manifold. What is such map? I imagine it being slightly more complicated, since I looked up its dimension to be n(n-1)/2. I figured SOnR is also the set {nxn matrices with determinant +1 or -1}, and hence be the union of the previous level set, with som other graph, but this would never yiel a dimension of n(n-1)/2

    I feel quite lost, I hope someone can help me out.
    Last edited: Oct 7, 2008
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted