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Lie Groups

  1. Jun 29, 2007 #1
    Let G be a 3-dimensional simply-connected Lie group. Then, G is either

    1.)The unit quaternions(diffeomorphic as a manifold to S[tex]$^{3}$[/tex]) with quaternionic multiplication as the group operation.
    2.)The universal cover of PSL[tex]$\left( 2,\Bbb{R}\right) $[/tex]
    3.)The semi-direct product of [tex]$\Bbb{R}^{2}\times _{\phi }\Bbb{R}$[/tex].

    The last case is an infinite family. There are many possible functions [tex]$\phi $[/tex]. My question is regarding these functions [tex]$\phi $[/tex].

    The semi-direct product [tex]$\Bbb{R}^{2}\times _{\phi }\Bbb{R}$[/tex] can be written as a group operation on [tex]$\Bbb{R}^{3}$[/tex] as

    [tex]$\left( x_{1},y_{1},z_{1}\right) \ast \left( x_{2},y_{2},z_{2}\right) =\left( x_{1}+\alpha \left( z_{1}\right) x_{2}+\beta \left( z_{1}\right) y_{2},y_{1}+\gamma \left( z_{1}\right) x_{2}+\delta \left( z_{1}\right) y_{2},z_{1}+z_{2}\right) $[/tex],
    where [tex]$\alpha \left( z_{1}\right) $[/tex] , [tex]$\beta \left( z_{1}\right) $[/tex] , [tex]$\gamma \left( z_{1}\right) $[/tex] , and [tex]$\delta \left( z_{1}\right) $[/tex] are real-valued functions.

    I would like to know what these real-valued functions are in terms of the constants occuring in the Lie-brackets. The Lie algebra at the identity of this infinite family of Lie groups(the semi-direct products) is isomorphic to a Lie algebra of the following form.
    [tex]$\left[ X_{1},X_{2}\right] =\lambda X_{2}+\sigma X_{3}$.[/tex]
    [tex]$\left[ X_{1},X_{3}\right] =\theta X_{2}+\lambda X_{3}$.[/tex]
    [tex]$\left[ X_{2},X_{3}\right] =0$.[/tex]

    [tex]$\lambda $[/tex], [tex]$\theta $[/tex], and [tex]$\sigma $[/tex] are constants.

    That is, I want to know the parameter dependent functions [tex]$\alpha \left( z_{1}\right) $[/tex] , [tex]$\beta \left( z_{1}\right) $[/tex] , [tex]$\gamma \left( z_{1}\right) $[/tex] , and [tex]$\delta \left( z_{1}\right) $[/tex].
     
    Last edited: Jun 29, 2007
  2. jcsd
  3. Jun 29, 2007 #2
    It appears someone fixed the LaTeX issue... Thanks...

    By the way, the functions alpha, beta, gamma, and delta comprise the automorphism phi. phi is a map from R to the automorphisms of R^2. For some values of lambda, theta, and sigma; I know these functions explicitly. If anyone believes that information would be useful to solve this problem, I could post it. However, I cannot seem to determine these functions explicity in the general case given arbitrary values of the constants lambda, theta, and sigma.
     
    Last edited: Jun 29, 2007
  4. Jun 29, 2007 #3
    Oh... it occurred to me that it may be possible to do this using infinitesimal generators... which basically means exponentiating a matrix... I'll post again after working it out...
     
  5. Jun 30, 2007 #4
    I solved the problem. The functions can be found explicitly... it is fairly straightforward, but the calculation is a bit tedious. The functions involve hyperbolic sines and hyperbolic cosines.
     
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