Here is a nice question(adsbygoogle = window.adsbygoogle || []).push({});

I know that exponentiating elements of a Lie-Algebra gives you back an element of the Lie-Group. These Lie-algebra-elements generate the Lie-Group transformations. Like the Galilei-group, these Lie-groups are used in theoretical fysics as the great START, I mean they contain the transformations under which all the fysical interactions (equations) have to be invariant. Like the Galilei-transformation for Newtonian Mechanics...

Now, my question concernes the background independence of formalisms like LQG, for example. On a certain manifold you can study it's structure by parallel transporting elements of the Lie Algebra, like vektors of a tangent vektor-space. The reason why we take just these vektors, is because the Lie-Algebra provides us with differentials and that's always nice in fysics.

Now we take a vektor and transport it around some loop on the manifold. Exponentiation the differential motion of this vektor when one step allong the loop is taken gives us back an element from the Lie-Group that represents the total movement the vektor made allong the transport. For example a rotation of 90 degrees when taken around the loop.

Is this vision correct, or is this science fiction. I think it must be ok, just wanting to check it.

AAAh, can someone give me a PRACTICAL example (i know the theoretical definition) of the use of a fiber. Can these trnaformations be used to go from a manifold to a tangent space ???

regards

marlon

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# Lie algebra

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