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?