Discovering the Functions of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ in 3D Lie Groups

In summary, the three possible forms for a 3-dimensional simply-connected Lie group are the unit quaternions, the universal cover of PSL(2,R), and the semi-direct product of R^2 and R. The last case has an infinite number of possible functions, which are real-valued and can be written in terms of constants occurring in the Lie-brackets. The functions can be found explicitly using infinitesimal generators and involve hyperbolic sines and cosines.
  • #1
Reverie
27
0
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) $[/tex]
3.)The http://en.wikipedia.org/wiki/Semidirect_product#Outer_semidirect_products" of [tex]$\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 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.
[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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  • #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.
 
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  • #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...
 
  • #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.
 

1. What is the purpose of studying the functions of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ in 3D Lie Groups?

The functions of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ in 3D Lie Groups have many applications in mathematics, physics, and engineering. They allow us to model and understand complex systems, such as fluid dynamics, elasticity, and quantum mechanics. Additionally, studying these functions helps us to develop new methods and techniques for solving problems in these fields.

2. What is the difference between $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ and a traditional 3D Lie Group?

While a traditional 3D Lie Group has three dimensions, $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ has four dimensions. This extra dimension allows for more complex and nuanced mathematical structures, making it a powerful tool for modeling and analyzing systems in three dimensions.

3. How can understanding the functions of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ in 3D Lie Groups benefit real-world applications?

By studying these functions, we can gain insights into the behavior of physical systems, which can then be applied to real-world problems. For example, understanding the functions of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ in fluid dynamics can lead to more efficient designs for aircraft wings or wind turbines.

4. What are some common techniques used to study the functions of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ in 3D Lie Groups?

Some common techniques include group theory, differential geometry, and representation theory. These tools allow us to analyze the symmetries and structures of these functions, making it easier to understand their properties and behavior.

5. Are there any limitations or challenges in studying the functions of $\Bbb{R}^{2}\times _{\phi }\Bbb{R}$ in 3D Lie Groups?

One potential challenge is the complexity of the mathematics involved. These functions often involve advanced mathematical concepts, such as non-commutative algebra and tensor calculus, which can be difficult to grasp. Additionally, the high dimensionality of these functions can make it challenging to visualize and analyze their behavior. However, with proper training and practice, these challenges can be overcome.

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