quantum field theory - Differential geometry of Lie groups

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?