# Lie Groups

1. Jun 29, 2007

### Reverie

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 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.
$$\left[ X_{1},X_{2}\right] =\lambda X_{2}+\sigma X_{3}.$$
$$\left[ X_{1},X_{3}\right] =\theta X_{2}+\lambda X_{3}.$$
$$\left[ X_{2},X_{3}\right] =0.$$

$$\lambda$$, $$\theta$$, 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 $$\delta \left( z_{1}\right)$$.

Last edited by a moderator: Apr 22, 2017
2. Jun 29, 2007

### Reverie

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
3. Jun 29, 2007

### Reverie

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...

4. Jun 30, 2007

### Reverie

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.