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Differential geometry of Lie groups

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In Weinberg's Classical Solutions of Quantum Field Theory, he states whilst introducing homotopy that groups, such as $SU(2)$, may be endowed with the structure of a smooth manifold after which they may be interpreted as Lie groups. My questions are:

  • If we formulate a quantum field theory on a manifold which is also a Lie group, does that quantum field theory inherit any special or useful properties?
  • Does a choice of metric exist for any Lie group?
  • Are there alternative interpretations of the significance of Killing vectors if they preserve a metric on a manifold which is also a Lie group?
This post imported from StackExchange Physics at 2014-08-07 15:38 (UCT), posted by SE-user user45389
asked Apr 27, 2014 in Theoretical Physics by user45389 (60 points) [ no revision ]
Comments on question (v1). Note that the moment you use the term "Lie group," you must be certain that the object being considered is a smooth manifold. So I'd like to suggest the rephrasing "groups, such as $\mathrm{SU}(2)$, may be endowed with the structure of a smooth manifold after which they may be interpreted as Lie groups." Also, do you have a compelling reason to believe that a "natural" choice of metric exists for an arbitrary Lie group? Perhaps "does there exist a natural choice of metric..." would be a better phrasing? Great question!

This post imported from StackExchange Physics at 2014-08-07 15:38 (UCT), posted by SE-user joshphysics
@joshphysics: Thank you for the suggested edit, I will incorporate it into the question. I will add the question of the existence of a choice of metric for an arbitrary Lie group, but I would also like to know how to select one if possible, given the appropriate Lie group.

This post imported from StackExchange Physics at 2014-08-07 15:38 (UCT), posted by SE-user user45389
This question is extremely similar to the following math.SE question. Coincidence? math.stackexchange.com/q/769080

This post imported from StackExchange Physics at 2014-08-07 15:38 (UCT), posted by SE-user joshphysics
@joshphysics: Thanks for the link, that answers part of my question.

This post imported from StackExchange Physics at 2014-08-07 15:38 (UCT), posted by SE-user user45389

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