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Reverie
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Let G be a 3-dimensional simply-connected Lie group. Then, G is either
1.)The unit quaternions(diffeomorphic as a manifold to S$^{3}$) with quaternionic multiplication as the group operation.
2.)The universal cover of PSL$\left( 2,\Bbb{R}\right) $
3.)The http://en.wikipedia.org/wiki/Semidirect_product#Outer_semidirect_products" of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$.
The last case is an infinite family. There are many possible functions $\phi $. My question is regarding these functions $\phi $.
The semi-direct product $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ can be written as a group operation on $\Bbb{R}^{3}$ as
$\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) $,
where $\alpha \left( z_{1}\right) $ , $\beta \left( z_{1}\right) $ , $\gamma \left( z_{1}\right) $ , and $\delta \left( z_{1}\right) $ are real-valued functions.
I would like to know what these real-valued functions are in terms of the constants occurring 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.
$\left[ X_{1},X_{2}\right] =\lambda X_{2}+\sigma X_{3}$.
$\left[ X_{1},X_{3}\right] =\theta X_{2}+\lambda X_{3}$.
$\left[ X_{2},X_{3}\right] =0$.
$\lambda $, $\theta $, and $\sigma $ are constants.
That is, I want to know the parameter dependent functions $\alpha \left( z_{1}\right) $ , $\beta \left( z_{1}\right) $ , $\gamma \left( z_{1}\right) $ , and $\delta \left( z_{1}\right) $.
1.)The unit quaternions(diffeomorphic as a manifold to S$^{3}$) with quaternionic multiplication as the group operation.
2.)The universal cover of PSL$\left( 2,\Bbb{R}\right) $
3.)The http://en.wikipedia.org/wiki/Semidirect_product#Outer_semidirect_products" of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$.
The last case is an infinite family. There are many possible functions $\phi $. My question is regarding these functions $\phi $.
The semi-direct product $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ can be written as a group operation on $\Bbb{R}^{3}$ as
$\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) $,
where $\alpha \left( z_{1}\right) $ , $\beta \left( z_{1}\right) $ , $\gamma \left( z_{1}\right) $ , and $\delta \left( z_{1}\right) $ are real-valued functions.
I would like to know what these real-valued functions are in terms of the constants occurring 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.
$\left[ X_{1},X_{2}\right] =\lambda X_{2}+\sigma X_{3}$.
$\left[ X_{1},X_{3}\right] =\theta X_{2}+\lambda X_{3}$.
$\left[ X_{2},X_{3}\right] =0$.
$\lambda $, $\theta $, and $\sigma $ are constants.
That is, I want to know the parameter dependent functions $\alpha \left( z_{1}\right) $ , $\beta \left( z_{1}\right) $ , $\gamma \left( z_{1}\right) $ , and $\delta \left( z_{1}\right) $.
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