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Originally, it was my contention that all Lie groups could be written as the semidirect product of a connected Lie group and a discrete Lie group. However, I no longer believe this is true.

The next best thing I could think of was to say that a Lie group is diffeomorphic to the (smooth) product manifold of a connected Lie group and a discrete one. This is proving to be difficult!

I'm a bit new to using general fiber bundles (I've only ever used vector bundles before!), but a friend of mine recommended me to use principal bundles. I've proven that ##\pi:G\to G/G_\mathrm{e}##, where ##G_\mathrm{e}## is the identity component of a Lie group ##G##, is a principal ##G_\mathrm{e}##-bundle. However, I really don't know what use this is to me.

Could someone explain how this is a step in the right direction?

Edit: Is there a particular way to get the LaTeX to render? It isn't working...

Edit2: In another post, display math works. What about inline...?

Edit3: Crisis averted.

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# Lie Group Decomposition

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