Let G be a 3-dimensional simply-connected Lie group. Then, G is either(adsbygoogle = window.adsbygoogle || []).push({});

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 http://en.wikipedia.org/wiki/Semidirect_product#Outer_semidirect_products" 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].

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# Lie Groups

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